# kelly-capital-growth --- ## Page 1 World Scientific Handbook in Financial Economic Series - Vol. 3 THEORY and PRACTICE - THE __________ _ KELLY CAPITAL GROWTH INVESTMENT CRITERION I , I • I ~ ~ 1 I I leonard ( Maclean Edward 0 Thorp . . . Wilham T Ziemba editors ,I» World Scientific --- ## Page 2 THEORY and PRACTICE _ THE ________ _ KELLY CAPITAL GROWTH INVESTMENT CRITERION --- ## Page 3 World Scientific Handbook in Financial Economic Series (ISSN: 2010-1732) Series Editor: William T. Ziemba Professor Emeritus, University of British Columbia, Canada Visiting Professor, Oxford University and University of Reading, UK Advisory Editors: Kenneth J. Arrow Stanford University, USA George C. Constantinides University of Chicago, USA Espen Eckbo Dartmouth College, USA Harry M. Markowitz University of California, USA Robert C. Merton Harvard University, USA Stewart C. Myers Massachusetts institute of Technology, USA Paul A. Samuelson Massachusetts institute of Technology, USA William F. Sharpe Stanford University, USA The Handbooks in Financial Economics (HIFE) are intended to be a definitive source for comprehensive and accessible information in the field of finance. Each individual volume in the series presents an accurate self-contained survey ofa sub-field of finance, suitable for use by finance, economics and financial engineering professors and lecturers, professional researchers, investments, pension fund and insurance portfolio mangers, risk managers, graduate students and as a teaching supplement. The HIFE series will broadly cover various areas of finance in a multi-handbook series. The HIFE series has its own web page that include detailed information such as the introductory chapter to each volume, an abstract of each chapter and biographies of editors. The series will be promoted by the publisher at major academic meetings and through other sources. There will be links with research articles in major journals. The goal is to have a broad group of outstanding volumes in various areas of financial economics. The evidence is that acceptance of all the books is strengthened over time and by the presence of other strong volumes. Sales, citations, royalties and recognition tend to grow over time faster than the number of volumes published. Published Vol. 1 Stochastic Optimization Models in Finance (2006 Edition) edited by William T Ziemba & Raymond G. Vickson Vol. 2 Efficiency of Racetrack Betting Markets (2008 Edition) edited by Donald B. Hausch, Victor S. Y. Lo & William T Ziemba Vol. 3 The Kelly Capital Growth Investment Criterion: Theory and Practice edited by Leonard C. MacLean, Edward 0. Thorp & William T Ziemba --- ## Page 4 World Scientific Handbook in Financial Economic Series - Vol. 3 THEORY and PRACTICE ___ THE ___________________ _ KELLY CAPITAL GROWTH INVESTMENT CRITERION Editors leonard C Maclean Dalhousie University, USA Edward 0 Thorp University of California, Irvine, USA William T Ziemba Mathematical Institute, Oxford University, UK and University of British Columbia, Canada '~World Scientific NEW JERSEY· LONDON· SINGAPORE· BEIJING· SHANGHAI · HONG KONG· TAIPEI· CHENNAI --- ## Page 5 Published by World Scientific PublisWng Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA offlce: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK offlce: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data The Kelly capital growth investment criterion : theory and practice I edited by Leonard C. MacLean, Edward O. Thorp, William T. Ziemba. p. em. -- (World Scientific handbook in financial economic series, 2010-1732 ; 3) Includes bibliograpWcal references. ISBN-13: 978-9814293495 ISBN-lO : 9814293490 ISBN-13: 978-9814293501 ISBN-lO : 9814293504 1. Investments--Mathematical models. 2. Portfolio management--Mathematical models. 1. MacLean, L. C. (Leonard C.) II. Thorp, Edward O. III. Ziemba, W. T. HG45 15.2.K45201O 332.63'2042--dc22 British Library Cataloguing-in-Publication Data 2010044902 A catalogue record for this book is available fro m the British Library. Copyright © 2011 by World Scientific PublisWng Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopyi ng of material in tWs volume, please pay a copyi ng fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 0 1923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore. --- ## Page 6 To my wife Gwenolyn, for her patience and encouragement, and Dalhousie University for its constant support over many years Leonard C. MacLean To Vivian, with whom I've shared "the long run" Edward O. Thorp To Sandra for companionship, help, patience, and understanding over a long time and to the memory of Kelly criterion pioneeers, John L. Kelly, Henry A. Latane, Leo Breiman, and Kelly critic Paul A. Samuelson William T. Ziemba --- ## Page 7 This page is intentionally left blank --- ## Page 8 Contents Preface List of Contributors Acknow ledgments Pictures Part I: The Early Ideas and Contributions 1. Introduction to the Early Ideas and Contributions 2. Exposition of a New Theory on the Measurement of Risk (translated by Louise Sommer) D. Bernoulli Econometrica, 22, 23-36 (1954) 3. A New Interpretation of Information Rate J. R. Kelly, Jr. Bell System Technical Journal, 35, 917- 926 (1956) 4. Criteria for Choice among Risky Ventures H. A. Latane Journal of Political Economy, 67, 144- 155 (1959) 5. Optimal Gambling Systems for Favorable Games L. Breiman Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, 1, 63- 68 (1961) VII xv XVll XXI xxv 3 11 25 35 47 --- ## Page 9 viii Contents 6. Optimal Gambling Systems for Favorable Games 61 E. O. Thorp Review of the International Statistical Institute, 37(3) , 273-293 (1969) 7. Portfolio Choice and the Kelly Criterion 81 E. O. Thorp Proceedings of the Business and Economics Section of the American Statistical Association, 215- 224 (1971) 8. Optimal Investment and Consumption Strategies under Risk 91 for a Class of Utility Functions N. H. Hakansson Econometrica, 38, 587- 607 (1970) 9. On Optimal Myopic Portfolio Policies, with and without 113 Serial Correlation of Yields N. H. Hakansson Journal of Business, 44, 324- 334 (1971) 10. Evidence on the "Growth-Optimum-Model" 125 R. Roll The Journal of Finance, 28(3), 551- 566 (1973) Part II: Classic Papers and Theories 11. Introduction to the Classic Papers and Theories 143 12. Competitive Optimality of Logarithmic Investment 147 R. M. Bell and T . M. Cover Mathematics of Operations Research, 5(2) , 161- 166 (1980) 13. A Bound on the Financial Value of Information 153 A. R. Barron and T. M. Cover IEEE Transactions of Information Theory, 34(5) , 1097- 1100 (1988) --- ## Page 10 14. 15. 16. 17. 18. 19. 20. Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment P. H. Algoet and T. M. Cover Annals of Probability, 16(2), 876- 898 (1988) Universal Portfolios T . M. Cover Mathematical Finance, 1(1), 1- 29 (1991) The Cost of Achieving the Best Portfolio in Hindsight E. Ordentlich and T. M. Cover Mathematics of Operations Research, 23(4), 960- 982 (1998) Optimal Strategies for Repeated Games M. Finkelstein and R. Whitley Advanced Applied Probability, 13, 415- 428 (1981) The Effect of Errors in Means, Variances and Co-Variances on Optimal Portfolio Choice V. K. Chopra and W. T. Ziemba Journal of Portfolio Management, 19, 6- 11 (1993) Time to Wealth Goals in Capital Accumulation L. C. MacLean, W . T . Ziemba, and Y. Li Quantitative Finance, 5(4), 343- 355 (2005) Survival and evolutionary Stability of Rule the Kelly 1. V. Evstigneev, T. Hens, and K. R. Schenk-Hoppe (2010) Contents IX 157 181 211 235 249 259 273 --- ## Page 11 x Contents 21. Application of the Kelly Criterion to Ornstein-Uhlenbeck Processes Y. Lv and B. K. Meister Lecture Notes of the Institute for Computer Sciences, 4, 1051- 1062 (2009) Part III: The Relationship of Kelly Optimization to Asset Allocation 22. Introduction to the Relationship of Kelly Optimization to Asset Allocation 23. Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time 24. 25. 26. 27. 28. S. Browne Mathematics of Operations Research, 22(2) , 468- 493 (1997) Growth versus Security in Dynamic Investment Analysis L. C. MacLean, W. T. Ziemba, and G. Blazenko Management Science, 38(11), 1562-1585 (1992) Capital Growth with Security L. C. MacLean, R. Sanegre, Y. Zhao, and W. T . Ziemba Journal of Economic Dynamics and Control, 28(4), 937-954 (2004) Risk-Constrained Dynamic Active Portfolio Management S. Browne Management Science, 46(9), 1188-1199 (2000) Fractional Kelly Strategies for Benchmark Asset Management M. Davis and S. Lleo (2010) A Benchmark Approach to Investing and Pricing E. Platen (2010) 285 301 307 331 355 373 385 409 --- ## Page 12 Contents Xl 29. Growing Wealth with Fixed-Mix Strategies M. A. H. Dempster, 1. V. Evstigneev, and K. R. Schenk-Hoppe (2010) P art IV : Critics and Assessing the Good and Bad P roperties of K elly 30. Introduction to the Good and Bad Properties of Kelly 31. Lifetime Portfolio Selection by Dynamic Stochastic Programming P. A. Samuelson Review of Economics and Statistics, 51, 239- 246 (1969) 32. Models of Optimal Capital Accumulation and Portfolio Selection and the Captial Growth Criterion W . T. Ziemba and R. G. Vickson (2010) 33. The "Fallacy" of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling P. A. Samuelson Proceedings National Academy of Science, 68(10), 2493- 2496 (1971) 34. Why We Should Not Make Mean Log of Wealth Big Though Years to Act Are Long P. A. Samuelson Journal of Banking and Finance, 3, 305- 307 (1979) 35. Investment for the Long Run: New Evidence for an Old Rule H. M. Markowitz Journal of Finance, 31(5), 1273- 1286 (1976) 36. Understanding the Kelly Criterion E. O. Thorp Wilmott, May and September (2008) 427 459 465 473 487 491 495 509 --- ## Page 13 Xli Contents 37. Concave Utilities Are Distinguished by Their 525 Optimal Strategies E. O. Thorp and R. Whitley Colloquia Mathematica Societatis Janos Bolyai, 813-830 (1972) 38. Medium Term Simulations of the Full Kelly and 543 Fractional Kelly Strategies Investment L. C. MacLean, E. O. Thorp, Y. Zhao, and W. T. Ziemba (2010) 39. Good and Bad Kelly Properties of the Kelly Criterion 563 L. C. MacLean, E. O. Thorp, and W. T. Ziemba (2010) Part V: Utility Foundations 40. Introduction to the Utility Foundations of Kelly 575 41. Capital Growth Theory 577 N. H. Hakansson and W. T. Ziemba In R. A. J arrow , V. Maksimovic, and W. T. Ziemba (Eds.) , Finance, Handbooks in OR fj MS, Volume 9, 65- 86. North Holland (1995) 42. A Preference Foundation for Log Mean-Variance Criteria in 599 Portfolio Choice Problems D. G. Luenberger Journal of Economic Dynamics and Control, 17, 88- 906 (1993) 43. Portfolio Choice with Endogenous Utility: A Large 619 Deviations Approach M. Stutzer Journal of Econometrics, 116, 365-386 (2003) --- ## Page 14 44. 45. 46. 47. 48. 49. 50. Contents On Growth-Optimality vs. Security against Underperformance M. Stutzer (2010) Part VI: Evidence of the Use of Kelly Type Strategies by the Great Investors and Others Introduction to the Evidence of the Use of Kelly Type Strategies by the Great Investors and Others Efficiency of the Market for Racetrack Betting D. B. Hausch, W. T. Ziemba, and M. E. Rubinstein Management Science, 27, 1435- 1452 (1981) Transactions Costs, Extent of Inefficiencies, Entries and Multiple Wagers in a Racetrack Betting Model D. B. Hausch and W. T. Ziemba Management Science, 31, 381- 394 (1985) The Dr. Z Betting System in England W. T. Ziemba and D. B. Hausch In D. B. Hausch, V. Lo, and W. T . Ziemba (Eds.), Efficiency of Racetrack Betting Markets, 567- 574. World Scientific (2008) A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds and Bills, with and without Small Stocks R. R. Grauer and N. H. Hakansson Journal of Business, 592, 287- 318 (1986) A Dynamic Portfolio of Investment Strategies: Applying Capital Growth with Drawdown Penalties J. M. Mulvey, M. Bilgili, and T. M. Vural (2010) Xlll 641 657 663 681 695 703 735 --- ## Page 15 XIV Contents 51. Intertemporal Surplus Management M. Rudolf and W. T. Ziemba Journal of Economic Dynamics and Control, 28, 975-990 (2004) 52. The Symmetric Downside-Risk Sharpe Ratio and the Evaluation of Great Investors and Speculators W. T. Ziemba Journal of Portfolio Management, 32(1), 108- 122 (2005) 53. Postscript: The Renaissance Medallion Fund R. E. S. Ziemba and W. T. Ziemba In Scenarios for Risk Management and Global Investment Strategies, 295- 298. Wiley (2007) 54. The Kelly Criterion in Blackjack Sports Betting and the Stock Market E. O. Thorp In S. A. Zenios and W. T. Ziemba (Eds.), Handbook of Asset and Liability Management, Volume 1, 387- 428. Elsevier (2006) Bibliography Author Index Subject Index 753 769 785 789 833 839 843 --- ## Page 16 xv Preface The modern development of the Kelly, or growth optimal, approach to allocating invest- ments began with J .L. Kelly's seminal 1956 paper, over two hundred years after Daniel Bernoulli's 1738 introduction to the notion of logarithmic utility. Kelly's paper was fol- lowed by Latane's (1959) intuitive economic analysis and theoretical advances by Breiman (1960, 1961). Breiman showed that the Kelly maximization of expected utility with a log- arithmic utility function also maximized the long run asymptotic growth of wealth while minimizing the expected time to reach arbitrarily large goals. Thorp (1962, 1966, 1969, 1971) pioneered the application of the Kelly criterion to actual gambling and investment. Ziemba and Vickson (1975) surveyed the literature presenting key papers, introductions and problems up to that time; for an update, see Ziemba and Vickson (2006). Algoet and Cover (1988) generalized the Breiman results to wider classes of assets and arbitrary ergodic market processes. MacLean, Ziemba and Blazenko (1992) show applications to a wide variety of sports and gambling events following Hausch, Ziemba and Rubinstein's (1981) application to racetrack betting. Thorp (1960) suggested the term Fort'une '8 Formula which later became the title of William Poundstone's 2005 book. This was in an abstract for a talk Thorp gave to the American Mathematical Society in January 1961, presenting his blackjack card counting discovery and his use of the Kelly approach to size bets in favorable situations. The term Kelly criterion appears to date from Thorp (1966) and is used in Thorp (1969) Over the years both theory and practice have developed prolifically. The theory has been extended to managing portfolios of investments, results have been obtained for a broad range of distributional assumptions, the simultaneous management of assets and liabilities has been elaborated upon, and the various properties, advantages and disadvantages have been clarified. We now have a fuller understanding of the tradeoff between risk and reward for fractional Kelly versus full Kelly and for Kelly subject to minimizing the underperformance of a benchmark or specified desired wealth path. The theory has also benefitted from the practical experience of gamblers, traders, hedge v --- ## Page 17 XVI Preface fund managers and investors, especially from some of the greatest investors. In this volume we present a selection of many of the most important papers from the now vast and growing literature on the subject. While we could not publish all the important papers, we feel that the main results appear here. The volume is organized into six sections that cover the early ideas and contributions, classic papers and theories, relations to asset allocation including optimization with with- drawals, fractional Kelly wagering and its relations to benchmarks, assessing the good and bad properties of Kelly wagering, utility foundations and the use of Kelly type strategies by various investors including the greatest investors. We thank our authors for their contributions, those who helped us with the editing and production especially Sandra Schwartz and our publisher, World Scientific, for their pro- duction and promotion of this volume. Special thanks go to Tom Cover for many helpful comments on earlier versions of the introductions and to Bryan Fitzgerald for valuable data. Leonard C. MacLean Edward O. Thorp William T. Ziemba March 2010 --- ## Page 18 List of Contributors Contributors Andrew R. Barron Department of Statistics, Yale University, Connecticut, USA Bernhard K. Meister Department of Physics, Renmin University of China, China Daniel Bernoulli University of Basel, Switzerland David G. Luenberger Stanford University, Stanford, USA Donald B. Hausch School of Business, University of Wisconsin, Wisconsin, USA Eckhard Platen School of Finance and Economics and Department of Mathematical Sciences, University of Technology, Sydney, Australia Edward O. Thorp Edward O. Thorp and Associates, Newport Beach, CA, USA Erik Ordentlich Hewlett Packard Labs, Palo Alto, California, USA George Blazenko School of Business Administration, Simon Fraser University, British Columbia, Canada xvii Chapters 13 21 2 42 46, 47, 48 28 6,7,36,37,38,39, 54 16 24 --- ## Page 19 XV111 List a/Contributors Contributors Harry M. Markowitz IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA Henry A. Latane Chapel Hill, University of North Carolina, USA Igor V. Evstigneev Economics Department, School of Social Sciences, University of Manchester, UK John L. Kelly Jr. Bell Labs, New Jersey, USA John M. Mulvey Princeton University, New Jersey, USA Klaus R. Schenk-Hoppe Leeds University Business School and School of Mathematics, University of Leeds, UK Leo Breiman University of California, Los Angeles, USA Leonard C. MacLean School of Business Administration, Dalhousie University, Halifax, Canada Mark Davis Department of Mathematics, Imperial College London, London, UK Mark Finkelstein University of California, Irvine, USA Markus Rudolf WHU-Otto Beisheim Graduate School of Management, Dresdner Bank Chair of Finance, Germany Mehmet Bilgili Alliance Bernstein, Equity Trading, New York, USA Chapters 35 4 20,29 3 50 20,29 5 19, 24, 25, 38, 39 27 17 51 50 --- ## Page 20 Contributors Michael Stutzer Burridge Center for Securities Analysis and Valuation, Leeds School of Business, University of Colorado, Boulder, USA Nils H. Hakansson Walter A. Haas School of Business, University of California, Berkeley, USA Paul A. Samuelson Department of Economics, Massachusetts Institute of Technology, Cambridge, USA Paul H. Algoet Boston University, Massachusetts, USA Rafael Sanegre University of British Columbia, Canada Rachel E. S. Ziemba Roubini Global Economics, London, UK Raymond G. Vickson Professor Emeritus, University of Waterloo, Canada Robert M. Bell Stanford University, Stanford, USA Robert R. Grauer Simon Fraser University, Canada Robert Whitley University of California, Irvine, USA Richard Roll Anderson School of Management, UCLA, Los Angeles, USA Sebastien Lleo Department of Mathematics, Imperial College London, London, UK List of Contributors xix Chapters 43, 44 8,9,41,49 31, 33, 34 14 25 53 32 12 49 17,37 10 27 --- ## Page 21 xx List a/Contributors Contributors Sid Browne Graduate School of Business, Columbia University, New York, USA Taha M. Vural Princeton University, New Jersey, USA Thomas M. Cover Department of Statistics and Electrical Engineering, Stanford University, Stanford, USA Thorsten Hens University of Zurich, Switzerland Vijay K. Chopra Frank Russell Company William T. Ziemba Professor Emeritus, University of British Columbia, Canada Visiting Professor, Oxford University and University of Reading, UK Yingdong Lv Department of Physics, Renmin University of China, China Yonggan Zhao School of Business, Dallhousie University, Halifax, Canada Yuming Li School of Business, California State University, Fullerton, USA Chapters 23,26 50 12, 13, 14, 15, 16 20 18 18, 19, 24, 25, 32, 38, 39, 41, 46, 47, 48, 51, 52 21 25,28 19 --- ## Page 22 XXI Acknowledgements We thank the following publishers and authors for permission to reproduce the articles listed below. Advanced Applied Probability Finkelstein, M. and R. Whitley (1981). Optimal strategies for repeated games. 13, 415- 428. Annals of Probability Algoet, P. H. and T. M. Cover (1988) . Asymptotic optimality and asymp- totic equipartition properties of log-optimum investment. 16(2), 876- 898. Bell System Technical Journal Kelly, Jr., J . R. (1956). A new interpretation of information rate. 35, 917- 926. Colloquia Mathematica Societatis Janos Bolyai Thorp, E. O. and R. Whitley (1972). Concave utilities are distinguished by their optimal strategies. 813- 830. Econometrica Bernoulli, D. (1954). Exposition of a new theory on the measurement of risk (translated by Louise Sommer). 22, 23- 36. Hakansson, N. H. (1970). Optimal investment and consumption strategies under risk for a class of utility functions. 38, 587- 607. Elsevier/ North Holland Hakansson, N. H. and W. T . Ziemba (1995). Capital growth theory. In R. A. Jarrow, V. Maksimovic, and W. T. Ziemba (Eds.), Finance, Hand- books in OR & MS, Vol. 9, 65- 86. Thorp, E. O. (2006). The Kelly criterion in blackjack sports betting and the stock market. In S. A. Zenios and W. T. Ziemba (Eds.), Handbook of Asset and Liability Management, Vol. 1, 387- 428. --- ## Page 23 xxii Acknowledgements Review of the International Statistical Institute Thorp, E. O. (1969). Optimal gambling systems for favorable games. 37(3), 273- 293. Springer Lv, Y. and B. K. Meister (2009). Application of the Kelly criterion to Ornstein-Uhlenbeck processes. Lecture Notes of the Institute for Computer Sciences, 4, 1051- 1062. The Journal of Finance Wiley Markowitz, H. M. (1976). Investment for the long run: New evidence for an old rule. 31(5), 1273- 1286. Roll, R. (1973). Evidence on the "growth-optimum" model. 28(3) ,551- 566. Thorp, E. O. (2008). Understanding the Kelly criterion. Wilmott, May and September. Ziemba, R. E. S. and W. T. Ziemba (2007). Postscript: The Renaissance Medallion Fund. In Scenarios for Risk Management and Global Invest- ment Strategies, 295-298. World Scientific Ziemba, W. T. and D. B. Hausch (2008). The Dr. Z betting system in England. In D. B. Hausch, V. Lo, and W. T. Ziemba (Eds.), Efficiency of Racetrack Betting Markets, 567- 574. We thank the following authors for permission to publish their new papers: M. Bilgili I. V. Evstigneev M. H. A. Davis M. A. H. Dempster T. Hens S. Lleo L. C. Maclean J. M. Mulvey E. Platen K. R. Schenk-Hoppe M. Stutzer E. O. Thorp T. M. Vural W. T. Ziemba --- ## Page 24 Acknowledgements XXIII IEEE Transactions of Information Theory Barron, A. R. and T. M. Cover (1988). A bound on the financial value of information. 34(5), 1097-1100. Journal of Banking and Finance Samuelson, P. A. (1979). Why we should not make mean log of wealth big though years to act are long. 3, 305- 307. Journal of Business Grauer, R. R. and N. H. Hakansson (1986). A half century of returns on levered and unlevered portfolios of stocks, bonds and bills, with and without small stocks. 592, 287- 318. Hakansson, N. H. (1971). On optimal myopic portfolio policies, with and without serial correlation of yields. 44, 324- 334. Journal of Econometrics Stutzer, M. (2003). Portfolio choice with endogenous utility: A large devi- ations approach. 116, 365- 386. Journal of Economic Dynamics and Control Luenberger, D. G. (1993). A preference foundation for log mean-variance criteria in portfolio choice problems. 17, 887- 906. MacLean, L. C. , R. Sanegre, Y. Zhao, and W. T. Ziemba (2004). Capital growth with security. 28(4) , 937- 954. Rudolf, M. and W. T. Ziemba (2004). Intertemporal surplus management. 28, 975- 990. Journal of Political Economy Latane, H. A. (1959). Criteria for choice among risky ventures. 67, 144- 155. Journal of Portfolio Management Chopra, V. K. and W. T . Ziemba (1993). The effect of errors in means, variances and co-variances on optimal portfolio choice. 19, 6- 11. Ziemba, W. T . (2005). The symmetric downside-risk Sharpe ratio and the evaluation of great investors and speculators. 32(1), 108- 122. Mathematical Finance Cover, T. M. (1991). Universal portfolios. 1(1), 1- 29. --- ## Page 25 XXIV Acknowledgements Mathematics of Operations Research Bell, R. M. and T. M. Cover (1980). Competitive optimality of logarithmic investment. 5(2), 161- 166. Browne, S. (1997). Survival and growth with a liability: Optimal portfolio strategies in continuous time. 22(2),468- 493. Ordentlich, E. and T. M. Cover (1998). The cost of achieving the best portfolio in hindsight. 23(4), 960- 982. Management Science Browne, S. (2000). Risk-constrained dynamic active portfolio management. 46(9), 1188- 1199. Hausch, D. B., W. T. Ziemba, and M. E. Rubinstein (1981). Efficiency of the market for racetrack betting. 27, 1435- 1452. Hausch, D. B. and W. T. Ziemba (1985). Transactions costs, extent of inefficiencies, entries and multiple wagers in a racetrack betting model. 31, 381- 394. MacLean, L., W. T. Ziemba, and G. Blazenko (1992). Growth versus secu- rity in dynamic investment analysis. 38(11), 1562- 1585. Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability Breiman, L. (1961). Optimal gambling systems for favorable games. 1, 63-68. Proceedings of the Business and Economics Section of the American Statistical Association Thorp, E. O. (1971). Portfolio choice and the Kelly criterion. 215- 224. Proceedings National Academy of Science Samuelson, P. A. (1971). The "fallacy" of maximizing the geometric mean in long sequences of investing or gambling. 68, 2493-2496. Quantitative Finance MacLean, L. C., W. T. Ziemba, and Y. Li (2005). Time to wealth goals in capital accumulation. 5(4) , 343-355. Review of Economics and Statistics Samuelson, P. A. (1969). Lifetime portfolio selection by dynamic stochastic programming. 51, 239- 246. --- ## Page 26 xxv Leonard MacLean at the 12th International Conference on Stochastic Programming, Dalhousie University, Halifax, Canada, August 19, 2010. Vivian and Edward Thorp, Newport Beach, California, 2004. --- ## Page 27 XXVI Professor William T. Ziemba while visiting as a Christianson Fellow at St. Catherines College, Oxford University in 2003. Beginning in 2003 and yearly thereafter he has given a half-day lecture on the Kelly criterion theory and practice to the Masters students in the Mathematical Finance program at the Mathematical Institute of Oxford University. --- ## Page 28 Part I The early ideas and contributions --- ## Page 29 This page is intentionally left blank --- ## Page 30 3 1 Introduction to the Early Ideas and Contributions We live in an age of instant contact through email, twitter, TV, and other modes of communication. Back in the 1700s, communication was slower and mail would take weeks or months between receipt and response. The first paper in this volume and arguably the first on log utility is Daniel Bernoulli's article, written in 1738 in Basel, Switzerland where he was professor of physics and philosophy. Bernoulli, at age 25, studied in Basel, went to St. Petersburg and then returned to Basel. He is a member of the famous family of Swiss mathematicians, who were known as the first to apply mathematical analysis to the movement of liquid bodies. His article on log utility and the St. Petersburg paradox reprinted here was translated by Dr. Louise Sommer of the American University with assistance from Karl Menger, mathemat- ics professor at the Illinois Institute of Technology, and William J. Baumol, eco- nomics professor at Princeton University. The article was published in Econometrica in 1954. A great paper often has only one new idea well developed with no major error. In his paper, Bernoulli develops two new ideas. The first idea is the development of declining marginal utility of wealth leading to logarithmic utility. His simple idea is that marginal utility should be proportional to current wealth. So upon integration, one has log utility. Bernoulli postulated monotone utility so that utility was increasing in wealth. Prior to this, it was assumed that decisions were made on an expected value or linear utility basis. The general idea of declining marginal utility or what we would later call "risk aversion" or "concavity" is crucial in modern decision theory. The second idea is his contribution to the St. Petersburg paradox. This prob- lem actually originates from Daniel Bernoulli's cousin, Nicolas Bernoulli, profes- sor at the University of Basel. In 1708, he submitted five important problems to professor Pierre Montmort, one of which was the St. Petersburg paradox. The idea is to determine the expected value and what you would pay for the following gamble: A fair coin with ~ probability of heads is repeatedly tossed until heads occurs, ending the game. The investor pays c dollars and receives in return 2k - 1 with probability 2- k for k = 1,2, ... should a head occur. Thus, after each succeeding loss, assuming a head does not appear, the bet is doubled to 2, 4, 8, ... etc. Clearly the expected value is ~ + ~ + ~ + ... or infinity with linear utility. --- ## Page 31 4 L. C. MacLean, E. 0. Thorp and W T. Ziemba Bell and Cover (1980) argue that the St. Petersburg gamble is attractive at any price c, but the investor wants less of it as c ---+ 00. The proportion of the investor's wealth invested in the St. Petersburg gamble is always positive but decreases with the cost c as c increases. The rest of the wealth is in cash. Bernoulli offers two solutions since he feels that this gamble is worth a lot less than infinity. In the first solution, he arbitrarily sets a limit to the utility of very large payoffs. Specifically, any amount over 10 million is assumed to be equal to 224 . Under that assumption, the expected value is = 12 + the original 1 = 13 If utility is ,jW, the expected value is 1 1 1 1 - Vi + - v'2 + - V4+ ... = -- ~ 2.9 2 4 8 2-v'2 When utility is log, as Bernoulli proposed, the expected value is I I I '2 log 1 + 4'log 2 + slog 4 + ... = log 2 = 0.69315 The use of a concave utility function does not eliminate the paradox. For exam- ple, the utility function U(x) = xl log(x + A), where A > 2 is a constant, is strictly concave, strictly increasing, and infinitely differentiable yet the expected value for the St. Petersburg gamble is +00. As Menger (1934) pointed out, the log, the square root and many others, but not all, concave utility functions eliminate the original St. Petersburg paradox but it does not solve one where the payoffs grow faster than 2n. So if log is the utility function, one creates a new paradox by having the payoffs increase at least as fast as log reduces them so one still has an infinite sum for the expected utility. With exponentially growing payoffs one has 1 1 '2log(e1 ) + 4'log(e2 ) + ... = 00 The super St. Petersburg paradox, in which even E log X = 00, is examined in Cover and Thomas (2006: 181, 182) where a satisfactory resolution is reached by looking at relative growth rates of wealth. Another solution to such paradoxes is to have bounded utility, for example, as Bernoulli suggested above 10 million. To solve the St. Petersburg paradox with exponentially growing payoffs, or any other growth rate, a second solution, in addition to that of bounding the utility function above, is simply to choose a utility function which, though unbounded, grows "sufficiently more slowly" than the inverse of the payoff function, e.g., like the log of the inverse function to the payoff function. The key is whether the valuation using a utility function is finite or not; if finite, the specific value does not matter since utilities are equivalent to within a positive linear transformation (V = aU + b, a > 0). So --- ## Page 32 Introduction to the Early Ideas and Contributions 5 for any utility giving a finite result, there is an equivalent one that will give you any specified finite value as a result. Only the behavior of U(x) as x ---+ 00 matters and strict monotonicity is necessary for a paradox. For example, U(x) = x, x ~ A will not produce a paradox, but the continuous concave utility function x A U(x) = - + - x > A 2 2 ' will have a paradox. Samuelson (1977) provides an extensive survey of the paradox (see also Menger (1967) and Aase (2001)). Kelly (1956) is given credit for the idea of using log utility in gambling and repeated investment problems, as such it is known as the Kelly criterion. Kelly's analyses use Bernoulli trials. Not only does he show that log is the utility function which maximizes the long run growth rate, but that this utility function is myopic in the sense that period by period maximization based only on current capital is optimal. Working at Bell Labs, Kelly was strongly influenced by information theorist Claude Shannon. Kelly defined the long run growth rate of the investor's fortune using WN G = lim log- N--7OO WO where Wo is the initial wealth and W N is the wealth after N trials in sequence. With Bernoulli trials, one wins = +1 with probability p and loses - 1 with probability q = 1 - p. The wealth with M wins and L = N - M loses is WN = (1 + f)M (1 - j)N-MWO where f is the fraction of wealth wagered on each of the N trials. Substituting this into G yields G = 1,EToo (~ 10g(1 + j) + (N ~ M) 10g(1 - j)) = plog(l + f) + qlog(l - f) = ElogW by the strong law of large numbers. Maximizing G is equivalent to maximizing the expected value of the log of each period's wealth. The optimal wager for this is 1* = p - q, p~q>O which is the expected gain per trial, or the edge. If there is no edge, the bet is zero. If the payoff is + B for a win and - 1 for a loss, then the edge is Bp - q, the odds are B , and 1* - Bp - q _ edge - -B-- - odds Latane (1959) introduced log utility as an investment criterion to the finance world independent of Kelly's work. Focusing, like Kelly, on simple intuitive versions of the expected log criteria, he suggested that it had superior long run properties. --- ## Page 33 6 L. C. MacLean, E. 0. Thorp and W T Ziemba Breiman (1961), following his earlier intuitive paper Breiman (1960), established the basic mathematical properties of the expected log criterion in a rigorous fash- ion. He proves three basic asymptotic results in a general discrete time setting with intertemporally independent assets. Suppose in each period, N, there are K investment opportunities with returns per unit invested XNI, ... ,XNK. Let A = (AI, ... , AK ) be the fraction of wealth invested in each asset. The wealth at the end of period N is Property 1. In each time period, two portfolio managers have the same family of investment opportunities with returns, X, and one uses a A * which maximizes E log W N whereas the other uses an essentially different strategy, A, so they differ infinitely often, that is Then . WN(A*) J~oo WN(A) --> 00 So the wealth exceeds that with any other strategy by more and more as the horizon becomes more distant. This generalizes the Kelly-Bernoulli trial setting to inter temporally independent and stationary returns. Property 2. The expected time to reach a preassigned goal A is, asymptotically least as A increases with a strategy maximizing E log W N. Property 3. Assuming a fixed opportunity set, there is a fixed fraction strategy that maximizes E log W N, which is independent of N. Thorp (1969) discusses the general theory of optimal betting over time on fa- vorable games or investments. Favorable games are those with a strategy such that Frob [ lim W N > 0] = 1 N--+oo where W N is the investor's capital after N trials. Thorp follows the footsteps of Kelly and Breiman, Bellman and Kalaba (1957), and Ferguson (1965), by discussing some favorable games such as blackjack, the side bet in Nevada-style baccarat (also called chemin de fer), roulette, the wheel of fortune, and the stock market. Once one has an edge, with positive expectation, Thorp outlines a general theory for optimal wagering on such games. For the stock market, Thorp discusses mispricings in --- ## Page 34 Introduction to the Early Ideas and Contributions 7 warrants and how to hedge them with advantage and to choose the optimal amount to invest using the Kelly approach. Markowitz arithmetic mean-variance (MV) efficiency applies to single period returns. For multiperiod returns we are interested in geometric or compound rates of return and the associated variance. Arithmetic MV-efficient portfolios are, in general, not geometric MV-efficient and conversely. The Kelly strategy is always geometric MV-efficient and has the highest growth rate of all such strategies. Any strategy which bets more is not geometric MV-efficient. Thorp (1971) focuses on the theory of logarithmic utility as applied to portfolio selection and contrasts it with Markowitz mean-variance portfolio theory. He shows that the Kelly strategy is not necessarily arithmetic mean-variance efficient. Hakansson (1970) presents arithmetic optimal strategies for the class of utility functions 00 L o:t-lu(Ct), 0<0:<1 t= l where Ct is the consumption in period t and either the relative -cu//(c) / u'(c) or the absolute Arrow-Pratt risk aversion u//(c)/u'(c) is a positive constant for all c ?': O. This includes positive power u(c) = c'Y, 0 < I < 1, negative power u(c) = -c'Y, I> 0, logarithmic u(c) = log c, and exponential u(c) = _e-'Yc , I> O. Using this model, Hakansson is able to determine, in closed form, optimal con- sumption, investment and borrowing strategies. The investor has an initial capital position, which could be negative, and a known deterministic non-capital income stream, and an arbitrary number of possible investments that can be held long or shorted whose probability distributions satisfy an arbitrage free condition called no easy money. It means that no combination of risky assets can with probability one out return the constant rate of interest, that no combination of short sale invest- ments exists where the probability is zero that a loss will exceed the lending rate of interest, and that no combination of short sale investments can guarantee against loss. It is assumed that there are no taxes or transaction costs, and that the joint distribution functions of the assets are known and stationary and have stochasti- cally constant returns to scale. Double the investment provides double returns. The stationarity assumption can be generalized and the same results obtained. The optimal investment strategies are independent of wealth, noncapital income, age and impatience to consume. Optimal consumption is linear and increasing in current wealth and in the present value of the noncapital income stream. The opti- mal asset mix depends only on the probability distribution of returns, the interest rate and the investor's one-period utility function of consumption. Necessary and sufficient conditions for capital growth are derived. With loga- rithmic utility of consumption, the investor will always invest the capital available after consumption is paid so as to maximize the expected growth rate of capital plus the present value of the noncapital income stream. --- ## Page 35 8 L. C. MacLean, E. 0. Thorp and W T Ziemba Hakansson's definition of capital growth is asymptotic lim Pr(Wt > WI) = 1 t-+oo When there is statistical independence period by period, this condition implies that but not the converse. A necessary and sufficient condition for capital growth is ar > 1 or that E[log W] > O. Here a is the time discount factor and r is the rate of interest. A sufficient but not necessary condition for growth is that there be a non-zero in- vestment in at least one of the risky investments since then Pr(Wt+l > Wt) > 0 for all t. A key property that aids multiperiod optimization calculations is myopia. Let u( w) be a utility function of final wealth. The u is myopic if induced interme- diate time utility functions of wealth are independent of the past and the future, in particular, they are independent of yields beyond the current period so they are positive linear transformations of u(w). Hakansson (1971) shows that log util- ity induces myopia for general asset return distributions. Hakansson also clarifies earlier results of Mossin (1968) and Leland (1968) for power utility functions; for these one needs independence of period by period returns to induce myopia. For u(w) = logw, for all time dependent markets one should, with myopia, maximize E{logWNIWl' W2 , ... , WN-I} , the conditional mean. Mossin and Leland showed that with zero interest rates, myopia obtains for linear risk tolerance utility u'(w)jul/(w) = a + bw When interest rates are non-zero, there is myopia if these future interest rates are known. The linear risk tolerance family includes exponential, log and the negative and positive power utility families. All of these results are under the assumption of stochastically constant returns to scale, perfect liquidity and divisibility of the assets, no transaction costs, withdrawals, capital additions, taxes, and short sales. Roll (1973) follows the finance literature starting with Latane (1959) rather than the Kelly (1956) literature. He investigates the relationship between the Kelly capital growth model, mean-variance and capital asset pricing (CAPM) analysis. Additional results on this topic are presented by Thorp (1969,1971), who shows that Kelly weightings are not necessarily mean-variance efficient and Markowitz (1976) , who argues that the log-optimal portfolio is a limiting mean-variance portfolio. MacLean, Ziemba and Blazenko (1992) argue that, from a growth versus security prospective, the Kelly portfolio is the most aggressive utility function that can be used and betting more leads to lower long run growth and less security. If estimation error is considered, the log utility is even riskier. Since, as Chopra and Ziemba (1993) show, with such low Arrow-Pratt absolute risk aversion (almost zero), errors in means are about 100 times errors in variances and co-variances in --- ## Page 36 Introduction to the Early Ideas and Contributions 9 certainty equivalent value. Hence, obtaining accurate mean estimates is crucial to successful investing. Roll (1973) observes the result proved by Breiman (1961) that growth-optimum "portfolios maximize the probability of exceeding a given level of wealth in a fixed time". Roll focuses on an empirical study of the implications for observed common stock returns of all investors selecting such a portfolio. Mean-variance type mod- els have dominated finance theory and practice largely because of the relationship with the capital asset pricing model and its theoretical justification of index funds which beat about 75% of managers with less cost and effort. On the other hand, growth-optimal Kelly portfolios have been used not by ordinary investors but those, like Warren Buffett, who attempt to have superior returns. This may lead to less diversification, larger positions and more monthly losses, see Ziemba (2005). For his test, Roll does not specify specific return distributions but rather relies on the fact that in each period the value of 1 + Rjt 1 + RFt + Li [Ai,t-dRit - R Ft )] expected by each individual is equal for all risky assets, where Rjt is the return from asset j in period t, RFt is the risk free return in period t and Ait is the investment is asset i in period t. Under typical, rather strict, financial economics assumptions to get to the aggregate level, it is assumed that: (a) all investors hold identical probability beliefs or (b) that a "representative" investor exists with the invested proportions X* s being equal to relative values of existing asset supplies. In either case, the denominator equals 1 + R m .t , a "market return" defined as a value-weighted average of all individual asset returns. It is only approximately observable because no comprehensive value-weighted asset indices exist. Z. = 1 + Rjt Jt - 1 + Rmt will have the same mean for all securities. Then the return on stock j relative to the market will have the same mean for all securities. Such assumptions fiy in the face of standard Kelly application which picks "the best stocks" , and we see clearly the difference in philosophy from academic financial economics and other investment approaches. Using 1962-1969 data and rather clever ways of looking at the two models: growth-optimium and CAPM, Roll concludes that there is: " ... a close correspondence was demonstrated between their quali- tative implications. For example, both models imply that an asset's expected return will equal the risk-free interest rate if the covari- ance between the asset's return and the average return on all assets, Cov(Rj , Rm), is zero. For most cases, the growth-optimum model also shares the Sharpe-Lintner implication that an asset's expected return will exceed the risk-free rate if and only if Cov( Rj , Rm) > O. --- ## Page 37 IO L. C. MacLean, E. 0. Thorp and W T Ziemba There are, however , some cases of highly-skewed probability dis- tributions where this implication does not follow for the growth- optimum model. A close empirical correspondence between the two models was demonstrated for common stock returns. The procedure: (1) esti- mated returns and risk premia implied by the two models from time series; (2) calculated cross-sectional relations between estimated returns and risks; and (3) compared the cross sectional relations to the theoretical predictions of the two models. They could not be distinguished on an empirical basis. In every period, estimated corresponding coefficients of the two models were nearly equal; and indeed, they deviated much further from their theoretically antici- pated levels than they deviated from each other." The investment time horizon is, as Roll points out, important here since, for short intervals, the models are essentially equivalent after quadratic approximations and for long horizons, the normality of returns kicks in. --- ## Page 38 Econometrica, 22, 23- 36 (1954) 2 EXPOSITION OF A NEW THEORY ON THE MEASUREMENT OF RISKl By DANIEL BERNOULLI 11 §1. EVER SINCE mathematicians first began to study the measurement of risk there has been general agreement on the following proposition: Expected values are computed by multiplying each possible gain by the number of ways in which it can occur, and then dividing the sum of these products by the total number of possible cases where, in this theory, the consideration of cases which are all of the same probability is insisted upon. If this rule be accepted, what remains to be done within the framework of this theory amounts to the enumeration of all alterna- tives, their breakdown into equi-probable cases and, finally, their insertion into corresponding classifications. §2. Proper examination of the numerous demonstrations of this proposition that have come forth indicates that they all rest upon one hypothesis: since there is no reason to assume that of two persons encountering identical risks,2 either 1 Translated from Latin into English by Dr. Louise Sommer, The American University, Washington, D. C., from "Specimen Theoriae Novae de Mensura Sortis," Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tomus V [Papers oj the Imperial Academy oj Sciences in Petersburg, Vol. V), 1738, pp. 175-192. Professor Karl Menger, Illinois Insti- tute of Technology has written footnotes 4, 9, 10, and 15. EDITOR'S NOTE: In view of the frequency with which Bernoulli's famous paper has been referred to in recent economic discussion, it has been thought appropriate to make it more generally available by publishing this English version. In her translation Professor Sommer has sought, in so far as possible, to retain the eighteenth century spirit of the original. The mathematical notation and much of the punctuation are reproduced without change. References to some of the recent literature concerned with Bernoulli's theory are given at the end of the article. TRANSLATOR'S NOTE : I highly appreciate the help of Karl Menger, Professor of Mathe- matics, Illinois Institute of Technology, a distinguished authority on the Bernoulli problem, who has read this translation and given me expert advice. I am also grateful to Mr. William J. Baumol, Professor of Economics, Princeton University, for his valuable assistance in interpreting Bernoulli's paper in the light of modern econometrics. I wish to thank also Mr. John H. Klingenfeld, Economist, U. S. Department of Labor, for his cooperation in the English rendition of this paper. The translation is based solely upon the original Latin text. BIOGRAPHICAL NOTE : Daniel Bernoulli, a member of the famous Swiss family of distin- . guished mathematicians, was born in Groningen, January 29,1700 and died in Basle, March 17, 1782. He studied mathematics and medical sciences at the University of Basle. In 1725 he accepted an invitation to the newly established academy in Petersburg, but returned to Basle in 1733 where he was appointed professor of physics and philosophy. Bernoulli was a. member of the academies of Paris, Berlin, and Petersburg and the Royal Academy in London. He was the first to apply mathematical analysis to the problem of the movement of liquid bodies. . (On Bernoulli see: HandwlJrterbuch der Naturwissenschajten, second edition, 1931, pp. 800-801; "Die Basler Mathematiker Daniel Bernoulli und Leonhard Euler. Hundert Jahre nach ihrem Tode gefeiert von der N aturforschenden Gesellschaft," Basle, 1884 (Annex to part VII of the proceedings of this Society); and Correspondance mathematique ... , edited by Paul Heinrich FUBB, 1843 containing letters written by Daniel Bernoulli to Leonhard Euler, Nicolaus Fuss, and C. Goldbach.) 2 i.e., risky propositions (gambles). [Translator) 23 --- ## Page 39 12 D. Bernoulli 24 DANIEL BERNOULLI should expect to have his desires more closely fulfilled, the risks anticipated by each must be deemed equal in value. No characteristic of the persons themselves ought to be taken into consideration; only those matters should be weighed carefully that pertain to the terms of the risk. The relevant finding might then be made by the highest judges established by public authority. But really there is here no need for judgment but of deliberation, i.e., rules would be set up whereby anyone could estimate his prospects from any risky undertaking in light of one's specific financial circumstances. §3. To make this clear it is perhaps advisable to consider the following exam- ple: Somehow a very poor fellow obtains a lottery ticket that will yield with equal probability either nothing or twenty thousand ducats. Will this man evaluate his chance of winning at ten thousand ducats? Would he not be ill- advised to sell this lottery ticket for nine thousand ducats? To me it seems that the answer is in the negative. On the other hand I am inclined to believe that a rich man would be ill-advised to refuse to buy the lottery ticket for nine thou- sand ducats. If I am not wrong then it seems clear that all men cannot use the same rule to evaluate the gamble. The rule established in §l must, therefore, be discarded. But anyone who considers the problem with perspicacity and in- terest will ascertain that the concept of value which we have used in this rule may be defined in a way which renders the entire procedure universally accept- able without reservation. To do this the determination of the value of an item must not be based on its price, but rather on the utility it yields. The price of the item is dependent only on the thing itself and is equal for everyone; the utility, however, is dependent on the particular circumstances of the person making the estimate. Thus there is no doubt that a gain of one thousand ducats is more significant to a pauper than to a rich man though both gain the same amount. §4. The discussion has now been developed to a point where anyone may proceed with the investigation by the mere paraphrasing of one and the same principle. However, since the hypothesis is entirely new, it may nevertheless require some elucidation. I have, therefore, decided to explain by example what I have explored. Meanwhile, let us use this as a fundamental rule: If the utility of each possible profit expectation is multiplied by the number of ways in which it can occur, and we t/J,en divide the sum of these products by the total number of pos8'tole cases, a mean utility3 [moral expectation] will be obtained, and the profit which corresponds to this utility will equal the value of the risk in question. §5. Thus it becomes evident that no valid measurement of the value of a risk can be obtained without consideration being given to its utility, that is to say, the utility of whatever gain accrues to the individual or, conversely, how much profit is required to yield a given utility. However it hardly seems plausible to make any precise generalizations since the utility of an item may change with circumstances. Thus, though a poor man generally obtains more utility than does a rich man from an equal gain, it is nevertheless conceivable, for example, 3 Free translation of Bernoulli's "emolumentum medium," literally: "mean utility." [Translator) --- ## Page 40 Exposition of a New Theory on the Measurement of Risk 13 THE MEASUREMENT OF RISK 25 that a rich prisoner who possesses two thousand ducats but needs two thousand ducats more to repurchase his freedom, will place a higher value on a gain of two thousand ducats than does another man who has less money than he. Though innumerable examples of this kind may be constructed, they repre- sent exceedingly rare exceptions. We shall, therefore, do better to consider what usually happens, and in order to perceive the problem more correctly we shall assume that there is an imperceptibly small growth in the individ- ual's wealth which proceeds continuously by infinitesimal increments. Now it is highly probable that any increase in wealth, no matter how insignificant, will always result in an increase in utility which is inversely proportionate to the quantity of goods already possessed. To explain this hypothesis it is necessary to define what is meant by the quantity of goods. By this expression I mean to con- note food, clothing, all things which add to the conveniences of life, and even to luxury-anything that can' contribute to the adequate satisfaction of any sort of want. There is then nobody who can be said to possess nothing at all in this sense unless he starves to death. For the great majority the most valuable portion of their possessions so defined will consist in their productive capacity, this term being taken to include even the beggar's talent: a man who is able to acquire ten ducats yel1rly by begging will scarcely be willing to accept a sum of fifty ducats on condition that he henceforth refrain from begging or otherwise trying to earn money. For he would have to live on this amount, and after he had spent it his existence must also come to an end. I doubt whether even those who do not possess a farthing and are burdened with financial obligations would be willing to free themselves of their debts or even to accept a still greater gift on such a condition. But if the beggar were to refuse such a contract unless immediately paid no less than one hundred ducats and the man pressed by credi- tors similarly demanded one thousand ducats, we might say that the former is possessed of wealth worth one hundred, and the latter of one thousand ducats, though in common parlance the former owns nothing and the latter less than nothing. §6. Having stated this definition, I return to the statement made in the pre- vious paragraph which maintained that, in the absence of the unusual, the utility resulting from any small increase in wealth will be inversely proportionate to the quantity of goods previously possessed. Considering the nature of man, it seems to me that the foregoing hypothesis is apt to be valid for many people to whom this sort of comparison' can be applied. Only a few do not spend their entire yearly incomes. But, if among these, one has a fortune worth a hundred thousand ducats and another a fortune worth the same number of semi-ducats and if the former receives from it a yearly income of five thousand ducats while the latter obtains the same number of semi-ducats it is quite clear that to the former a ducat has exactly the same significance as a semi-ducat to the latter, and that, therefore, the gain of one ducat will have to the former no higher value than the gain of a semi-ducat to the latter. Accordingly, if each makes a gain of one ducat the latter receives twice as much utility from it, having been enriched by two semi- ducats. This argument applies to many other cases which, therefore, need not --- ## Page 41 14 D. Bernoulli 26 DANIEL BERNOULLI be discussed separately. The proposition is all the more valid for the majority of men who possess no fortune apart from their working capacity which is their only source of livelihood. True, there are men to whom one ducat means more than many ducats do to others who are less rich but more generous than they. But since we shall now concern ourselves only with one individual (in different states of affluence) distinctions of this sort do not concern us. The man who is emotionally less affected by a gain will support a loss with greater patience. Since, however, in special cases things can conceivably occur otherwise, I shall first deal with the most general case and then develop our special hypothesis in order thereby to satisfy everyone. Q s N AI-----:,..--=;'----~~~----=~---R s §7. Therefore, let AB represent the quantity of goods initially possessed. Then after extending AB, a curve BGLS must be constructed, whose ordinates CG, DB, EL, FM,· etc., designate utilities corresponding to the abscissas BC, BD, BE, BF, etc., designating gains in wealth. Further, let m, n, p, q, etc., be the numbers which indicate the number of ways in which gains in wealth BC, BD, BE, BF [misprinted in the original as CF], etc., can occur. Then (in accord with §4) the moral expectation of the risky proposition referred to is given by: PO = m .CG + n.DH + p.EL + q.FM + .. . m+n+p+q+'" Now, if we erect AQ perpendicular to AR, and on it measure off AN = PO, the straight line NO - AB represents the gain which may properly be expected, or the value of the risky proposition in question. If we wish, further, to know how --- ## Page 42 Exposition of a New Theory on the Measurement of Risk 15 THE MEASUREMENT OF RISK 27 large a stake the individual should be willing to venture on this risky proposi- tion, our curve must be extended in the opposite direction in such a way that the abscissa Bp now represents a loss and the ordinate po represents the cor- responding decline in utility. Since in a fair game the disutility to be suffered by losing must be equal to the utility to be derived by winning, we must assume that An = AN, or po = PO. Thus Bp will indicate the stake more than which persons who consider their own pecuniary status should not venture. COROLLARY I §8. Until now scientists have usually rested their hypothesis on the assump- tion that all gains must be evaluated exclusively in terms of themselves, i.e., on the basis of their intrinsic qualities, and that these gains will always produce a utility directly proportionate to the gain. On this hypothesis the curve BS becomes a straight line. Now if we again have: PO = m.CG + n.DH + p.EL + q.FM + ... , m+n+p+q+··· and if, on both sides, the respective factors are introduced it follows that: BP = m.BC + n.BD + p.BE + q.BF + ... , m+n+p+q+··· which is in conformity with the usually accepted rule. COROLLARY II §9. If AB were infinitely great, even in proportion to BF, the greatest possible gain, the arc BM may be considered very like an infinitesimally small straight line. Again in this case the usual rule [for the evaluation of risky propositions] is applicable, and may continue to be considered approximately valid in games of insignificant moment. §1O. Having dealt with the problem in the most general way we turn now to the aforementioned particular hypothesis, which, indeed, deserves prior atten- tion to all others. First of all the nature of curve sBS must be investigated under the conditions postulated in §7. Since on our hypothesis we must consider in- finitesimally small gains, we shall take gains BC and BD to be nearly equal, so that their difference CD becomes infinitesimally small. If we draw Gr parallel to BR, then rH will represent the infinitesimally small gain in utility to a man whose fortune is AC and who obtains the small gain, CD. This utility, however, should be related not only to the tiny gain CD, to which it is, other things being equal, proportionate, but also to AC, the fortune previously owned to which it is inversely proportionate. We therefore set: AC = x, CD = dx, CG = y, rH = dy and AB = a; and if b designates some constant we obtain dy = bdx or y = x --- ## Page 43 16 D. Bernoulli 28 DANIEL BERNOULLI x b log~. The curve sBS is therefore a logarithmic curve, the subtangent4 of which is everywhere b and whose asymptote is Qq. §11. If we now compare this result with what has been said in paragraph 7 it will appear that: PO = b log AP/AB, CG = b log AC/AB, DH = b 10gAD/AB and so on; but since we have it follows that PO = m.CG + n.DH + p.EL + q.FM + ... m+n+p+q+,,· AP ( AC AD AE AF ) b log AB = mb log AB + nb log AB + pb log AB + qb log AB + ... : (m + n + p + q + ... ) and therefore AP = (AC"'.AD" .AEP .AFq • ••. )l/m+,,+p+q+ ... and if we subtract AB from this, the remaining magnitude, BP, will represent the value of the risky proposition in question. §12. Thus the preceding paragraph suggests the following rule: Any gain must be added to the fortune previously possessed, then this sum must be raised to the power given by the number of possible ways in which the gain may be obtained; these terms should then be multiplied together. Then of this product a root must be extracted the degree of which is given by the number of all possible cases, and finally the value of the initial possessions must be subtracted therefrom; what then remains indicates the value of the risky proposition in question. This principle is essential for the measurement of the value of risky propositions in various cases. I would elab- orate it into a complete theory as has been done with the traditional analysis, were it not that, despite its usefulness and originality, previous obligations do not permit me to undertake this'task. I shall therefore, at this time, mention only the more significant points among those which have at first glance occurred to me. , The tangent to the curve y = b log:: at the point (xo, 10g~) is the line y - b log ~ = a a a ~ (x - xo). This tangent intersects the Y-axis (x = 0) at the point with the ordinate Xo b log::'o - b. The point of contact of the tangent with the curve has the ordinate b log ~ . a a So also does the projection of this point on the Y-axis. The segment between the two points on the Y-axis that have been mentioned has the length b. That segment is the projection of the segment on the tangent between its intersection with the Y-axis and the point of contact. The length of this projection (which is b) is what Bernoulli here calls the "sub- tangent." Today, by the subtangent of the curve y - f(x) at the point (xo .!(xo» is meant the length of the segment on the X -axis (and not the Y-axis) between its intersection with the tangent and the projection of the point of contact. This length is f(xo) If' (xo). In the case of the logarithmic curve it equals Xo log~.-Karl Menger. a --- ## Page 44 Exposition of a New Theory on the Measurement of Risk 17 THE MEASUREMENT OF RISK 29 §13. First, it appears that in many games, even those that are absolutely fair, both of the players may expect to suffer a loss j indeed this is Nature's admoni- tion to avoid the dice altogether .... This follows from the concavity of curve sBS to BR. For in making the stake, Bp, equal to the expected gain, BP, it is clear that the disutility po which results from a loss will always exceed the ex- pected gain in utility, PO. Although this result will be quite clear to the mathe- matician, I shall nevertheless explain it by example, so that it will be clear to everyone. Let us assume that of two players, both possessing one hundred ducats, each puts up half this sum as a stake in a game that offers the same probabilities to both players. Under this assumption each will then have fifty ducats plus the expectation of winning yet one hundred ducats more. However, the sum of the values of these two items amounts, by the rule of §12, to only (501.1501)j or -vi 50.150, i.e., less than eighty-seven ducats, so that, though the game be played under perfectly equal conditions for both, either will suffer an expected loss of more than thirteen ducats. We must strongly emphasize this truth, although it be self evident: the imprudence of a gambler will be the greater the larger the part of his fortune which he exposes to a game of chance. For this purpose we shall modify the previous example by assuming that one of the gamblers, before putting up his fifty ducat stake possessed two hundred ducats. This gambler suffers an expected loss of 200 - -vi150.250, which is not much greater than six ducats. §14. Since, therefore, everyone who bets any part of his fortune, however small, on a mathematically fair game of chance acts irrationally, it may be of interest to inquire how great an advantage the gambler must enjoy over his opponent in order to avoid any expected loss. Let us again consider a game which is as simple as possible, defined by two equiprobable outcomes one of which is favorable and the other unfavorable. Let us take a to be the gain to be won in case of a favorable outcome, and x to be the stake which is lost in the unfavorable case. If the initial quantity of goods possessed is a we have AB = aj BP = a; PO = b log a + a (see §1O), and sinc~ (by §7) po = PO it follows by the nature a of a logarithmic curve that Bp = a+a. Since however Bp represents the stake a a x, we have x = a+a a magnitude which is always smaller than a, the expected a a gain. It also follows from this that a man who risks his entire fortune acts like a simpleton, however great may be the possible gain. Noone will have difficulty in being persuaded of this if he has carefully examined our definitions given above. Moreover, this result sheds light on a statement which is universally accepted in practice: it may be reasonable for some individuals to invest in a doubtful enterprise and yet be unreasonable for others to do so. §15. The procedure customarily employed by merchants in the insurance of commodities transported by sea seems to merit special attention. This may again be explained by an example. Suppose Caius,5 a Petersburg merchant, has pur- 5 Caius is a Roman name, used here in the sense of our "Mr. Jones." Caius is the older form; in the later Roman period it was spelled "Gaius." [Translator) --- ## Page 45 18 D. Bernoulli 30 DANIEL BERNOULLI chased. commodities in Amsterdam which he could sell for ten thousand rubles if he had them in Petersburg. He therefore orders them to be shipped there by sea, but is in doubt whether or not to insure them. He is well aware of the fact that at this time of year of one hundred ships which sail from Amsterdam to Petersburg, five are usually lost. However, there is no insurance available below the price of eight hundred rubles a cargo, an amount which he considers out- rageously high. The question is, therefore, how much wealth must Caius possess apart from the goods under consideration in order that it be sensible for him to abstain from insuring them? If x represents his fortune, then this together with the value of the expectation of the safe arrival of his goods is given by IV' (x + l0000)9bx5 = V (x + l0000)19x in case he abstains. With insurance he will have a certain fortune of x + 9200. Equating these two magnitudes we get: (x + l0000)19x = (x + 9200)20 or, approximately, x = 5043. If, therefore, Caius, apart from the expectation of receiving his commodities, possesses an amount greater than 5043 rubles he will be right in not buying insurance. If, on the contrary, his wealth is less than this amount he should insure his cargo. And if the question be asked "What minimum fortune should be possessed by the man who offers to provide this insurance in order for him to be rational in doing so?" We must answer thus: let y be his fortune, then V' (y + 800)19. (y - 9200) = y or approximately, y = 14243, a figure which is obtained from the foregoing without additional calculation. A man less wealthy than this would be foolish to provide the surety, but it makes sense for a wealthier man to do so. From this it is clear that the introduction of this sort of insurance has been so useful since it offers advantages to all persons concerned. Similarly, had Caius been able to obtain the insurance for six hundred rubles he would have been unwise to refuse it if he possessed. less than 20478 rubles, but he would have acted much too cautiously had he insured his commodities at this rate when his fortune was greater than this amount. On the other hand a man would act unadvisedly if he were to offer to sponsor this insurance for six hundred. rubles when he himself possesses less than 29878 rubles. However, he would be well advised. to do so if he possessed more than that amount. But no one, however rich, would be manag- ing his affairs properly if he individually undertook the insurance for less than five hundred rubles. §16. Another rule which may prove useful can be derived from our theory. This is the rule that it is advisable to divide goods which are exposed to some danger into several portions rather than to risk them all together. Again I shall explain this more precisely by an example. Sempronius owns goods at home worth a total of 4000 ducats and in addition possesses 8000 ducats worth of commodities in foreign countries from where they can only be transported by sea. However, our daily experience teaches us that of ten ships one perishes. Under these conditions I maintain that if Sempronius trusted all his 8000 ducats of goods to one ship his expectation of the commodities is worth 6751 ducats. That IS V 120000.40001 - 4000. --- ## Page 46 Exposition of a New Theory on the Measurement of Risk 19 THE MEASUREMENT OF RISK 31 If, however, he were to trust equal portions of these commodities to two ships the value of his expectation would be V1200()81.80oo18 .4000 - 4000, i.e., 7033 ducats. In this way the value of Sempronius' prospects of success will grow more favor- able the smaller the proportion comInitted to each ship. However, his expectation will never rise in value above 7200 ducats. This counsel will be equally service- able for those who invest their fortunes in foreign bills of exchange and other hazardous enterprises. §17. I am forced to oInit many novel remarks though these would clearly not be unserviceable. And, though a person who is fairly judicious by natural instinct Inight have realized and spontaneously applied much of what I have here ex- plained, hardly anyone believed it possible to define these problems with the precision we have employed in our examples. Since all our propositions harmonize perfectly with experience it would be wrong to neglect them as abstractions rest- ing upon precarious hypotheses. This is further confirmed by the following ex- ample which inspired these thoughts, and whose history is as follows: My most honorable cousin the celebrated Nicolas Bernoulli, Professor utriusque iuris8 at the University of Basle, once subInitted five problems to the highly distinguished7 mathematician Montmort.s These problems are reproduced in the work L'analyse sur les jeux de hazard de M. de Montmort, p. 402. The last of these problems runs as follows: Peter tosses a coin and continues to do so until it slwuld land "heads" when it comes to the ground. He agrees to give Paul one ducat if he gets "heads" on the very first throw, two ducats if he gets it on the second, four if on the third, eight if on the fourth, and so on, so that with each additional throw the number of ducats he must pay is doubled. Suppose we seek to determine the value of Paul's expectation. My aforementioned cousin discussed this problem in a letter to me asking for my opinion. Although the standard calculation shows9 that the value of Paul's expectation is infinitely great, it has, he said, to be admitted that any fairly reasonable man would sell his chance, with great pleasure, for twenty ducats. The accepted method of calculation does, indeed, value Paul's prospects at infinity though no one would be willing to purchase it at a moderately high price. e Faculties of law of continental European universities bestow up to the present time the title of a Doctor utriusque juris, which means Doctor of both systems of laws, the Roman and the canon law. [Translator] 7 Cl., i.e., Vir Clarissimus, a title of respect. [Translator] 8 Montmort, Pierre Remond, de (1678-1719). The work referred to here is the then famous "Essai d'analyse sur les jeux de hazard," Paris, 1708. Appended to the second edition, published in 1713, is Montmort's correspondence with Jean and Nicolas Bernoulli referring to the problems of chance and probabilities. [Translator]. i The probability of heads turning up on the 1st throw is 1/2. Since in this case Paul receives one ducat, this probability contributes 1/2·1 ... 1/2 ducats to his expectation. The probability of heads turning up on the 2nd throw is 1/4. Since in this case Paul receives 2 ducats, this possibility contributes 1/4·2 ... 1/2 to his expectation. Similarly, for every integer n, the possibility of heads turning up on the n-th throw contributes 1/2n ·2n- 1 ... 1/2 ducats to his expectation. Paul's total expectation is therefore 1/2 + 1/2 + ... + 1/2 + ... , and that is infinite.-Karl Menger. --- ## Page 47 20 D. Bernoulli 32 DANIEL BERNOULLI If, however, we apply our new rule to this problem we may see the solution and thus unravel the knot. The solution of the problem by our principles is as follows. §18. The number of cases to be considered here is infinite: in one half of the cases the game will end at the first throw, in one quarter of the cases it will conclude at the second, in an eighth part of the cases with the third, in a six- teenth part with the fourth, and so on.10 If we designate the number of cases through infinity by N it is clear that there are 75,N cases in which Paul gains one ducat, YiN cases in which he gains two ducats, ygN in which he gains four, YJ..6N in which he gains eight, and so on, ad infinitum. Let us represent Paul's fortune by aj the proposition in question will then be worth {I(a + 1)NI2.(a + 2)NI4.(a + 4)NI8.(a + 8)N116 ... - a = V(a + l).V'(a + 2).V(a + 4) . ~(a + 8) . .. - a. §19. From this formula which evaluates Paul's prospective gain it follows that this value will increase with the size of Paul's fortune and will never attain an infinite value unless Paul's wealth simultaneously becomes infinite. In addi- tion we obtain the following corollaries. If Paul owned nothing at all the value of his expectation would be which amounts to two ducats, precisely. If he owned ten ducats his opportunity would be worth approximately three ducatsj it would be worth approximately four if his wealth were one hundred, and six if he possessed one thousand. From this we can easily see what a tremendous fortune a man must own for it to make sense for him to purchase Paul's opportunity for twenty ducats. The amount which the buyer ought to pay for this proposition differs somewhat from the amount it would be worth to him were it already in his possession. Since, however, this difference is exceedingly small if a (paul's fortune) is great, 10 Since the number of cases is infinite, it is impossible to speak about one half of the cases, one quarter of the cases, etc., and the letter N in Bernoulli's argument is meaning- less. However, Paul's expectation on the basis of Bernoulli's hypothesis concerning evalua- tion can be found by the same method by which, in footnote 9, Paul's classical expectation was determined. If Paul's fortune is a ducats, then, according to Bernoulli, he attributes + 2,,-1 to a gain of 2,,-1 ducats the value b log a . If the probability of this gain is 1/2", his a + 2,,-1 expectation is b/2" log a • Paul's expectation resulting from the game is therefore a b a + 1 b a + 2 b a + 2,,-1 -log --+-log--+'" +-log + ... 2 a 4 a 2" a = b log [(a + 1)1/1(01 + 2)1'" .... (a + 2,,-1)1/1" •... ] - b log 0/. a+D What addition D to Paul's fortune has the same value for him? Clearly, b log -- must a equal the above sum. Therefore D = (a + 1)1/1(01 + 2)1/4 ..... (a + 2,,-1)1/2" .... - a . -Karl Menger. --- ## Page 48 Exposition of a New Theory on the Measurement of Risk 21 THE MEASUREMENT OF RISK 33 we can take them to be equal. If we designate the purchase price by x its value can be determined by means of the equation ...y(a + 1 - x).~(a + 2 - x).V"(a + 4 - x).~(a + 8 - x) .. , = a and if a is a large number this equation will be approximately satisfied by _2/-- _4;-;-"n _8/-- 18/-- X = V a + 1. V a + 2. V a + 4. V a + 8 . .. - a. After having read this paper to the Societyll I sent a copy to the aforementioned Mr. Nicolas Berrwulli, to obtain his opinion of my proposed solution to the diffic:ulty he had indicated. Ina letter to me written in 1732 he declared that he was in no way dissatisfied with my proposition on the evaluation of risky propositions when applied to the case of a man who is to evaluate his own prospects. However, he thinks that the case is different if a third person, somewhat in the position of a judge, is to evaluate the prospects of any participant in a game in accord with equity and jus- tice. I myself have disc:ussed this problem in §2. Then this distinguished scholar informed me that the celebrated mathematician, Cramer/2 had developed a theory on the same subject several years before I produced my paper. Indeed I have found his theory so similar to mine that it seems mirac:ulous that we independently reached such close agreement on this sort of subject. Therefore it seems worth quoting the words with which the celebrated Cramer himself first descn"bed his theory in his letter of 1728 to my cousin. His words are as followsl 3 "Perhaps I am mistaken, but I believe that I have solved the extraordinary" problem which you submitted to M. de Montmort, in your letter of September 9," 1713, (problem 5, page 402). For the sake of simplicity I shall assume that A" tosses a coin into the air and B commits himself to give A 1 ducat if, at the" first throw, the coin falls with its cross upward; 2 if it falls thus only at the" second throw, 4 if at the third throw, 8 if at the fourth throw, etc. The paradox" consists in the infinite sum which calculation yields as the equivalent which" A must pay to B. This seems absurd since no reasonable man would be willing" to pay 20 ducats as equivalent. You ask for an explanation of the discrepancy" between the mathematical calculation and the vulgar evaluation. I believe" that it results from the fact that, in their theory, mathematicians evaluate" money in proportion to its quantity while, in practice, people with common" sense evaluate money in proportion to the utility they can obtain from it. The" mathematical expectation is rendered infinite by the enormous amount which" I can win if the coin does not fall with its cross upward until rather late, perhaps" at the hundredth or thousandth throw. Now, as a matter of fact, if I reason" as a sensible man, this sum is worth no more to me, causes me no more pleasure" 11 Bernoulli's paper had been submitted to the Imperial Academy of Sciences in Peters- burg. [Translator] 12 Cramer, Gabriel, famous mathematician, born in Geneva, Switzerland (1704-1752). [Translator I U The following passage of the original text is in French. (Translator] --- ## Page 49 22 D. Bernoulli 34 DANIEL BERNOULLI "and influences me no more to accept the game than does a sum amounting "only to ten or twenty million ducats. Let us suppose, therefore, that any "amount above 10 millions, or (for the sake of simplicity) above 224 = 166777216 "ducats be deemed by him equal in value to 224 ducats or, better yet, that I "can never win more than that amount, no matter how long it takes before the "coin falls with its cross upward. In this case, my expectation is .%.1 + 74.2 + "~.4 . .. + *".224 + *u .224 + *27.224 + ... = ~ + .% + ~ + ... "(24 times) . . . + ~ + 74 + % + ... = 12 + 1 = 13. Thus, my moral ex- "pectation is reduced in value to 13 ducats and the equivalent to be paid for "it is similarly reduced-a result which seems much more reasonable than does "rendering it infinite." Thus farH the exposition is somewhat vague and subject to counter argument. If it, indeed, be true that the amount 22& appears to us to be no greater than ~., no attention whatsoever should be paid to the amount that may be won after the twenty-fourth throw, since just before making the twenty-fifth throw I am certain to end up with no less than 224 - 1,1& an amount that, according to this theory, may be considered equivalent to 224. Therefore it may be said oorrectly that my expectation is only worth twelve ducats, not thirteen. However, in view of the coincidence between the basic principle developed by the aforementioned author and my own, the fore- going is clearly not intended to be taken to invalidate that principle. I refer to the ptoposition that reasonable men should evaluate money in accord with the utility they derive therefrom. I state this to avoid leading anyone to judge that entire theory adversely. And this is exactly what Cl. C.10 Cramer states, expressing in the following manner precisely what we would ourselves conclude. He continues thus:17 "The equivalent can turn out to be smaller yet if we adopt some alternative "hypothesis on the moral value of wealth. For that which I have just assumed "is not entirely valid since, while it is true that 100 millions yield more satis- "faction than do 10 millions, they do not give ten times as much. If, for example, "we suppose the moral value of goods to be directly proportionate to the square "root of their mathematical quantities, e.g., that the satisfaction provided by ''40000000 is double that provided by 10000000, my psychic expectation "becomes 1 "However this magnitude is not the equivalent we seek, for this equivalent "need not be equal to my moral expectation but should rather be of such a "magnitude that the pain caused by its loss is equal to the moral expectation "of the pleasure I hope to derive from my gain. Therefore, the equivalent must, It From here on the text is again translated from Latin. (Translator] 15 This remark of Bernoulli's is obscure. Under the conditions of the game a gain of 2" - 1 ducats is impossible.-Karl Menger. 18 To be translated as "the distinguished Gabriel." (Translator) 11 Text continues in French. (Translator] --- ## Page 50 Exposition of a New Theory on the Measurement of Risk 23 THE MEASUREMENT OF RISK 35 on our hypothesis, amount to (2 _1 v'2Y = (6 _ ~ v'2) = 2.9 ... , which" is consequently less than 3, truly a trifling amount, but nevertheless, I believe," closer than is 13 to the vulgar evaluation." REFERENCES There exists only one other translation of Bernoulli's paper: Pringsheim, Alfred, Die Grundlage der modernen Wertlehre: Daniel Bernoulli, Versuch einer neuen Theorie der Wertbestimmung von Gluck8f(J,llen (Specimen Theoriae novae de Mensura Sortis). Aua dem Lateinischen iibersetzt und mit Erlituterungen versehen von Alfred Pringsheim. Leipzig, Duncker und Humblot, 1896, Sammlung itlterer und neuerer staats- wissenschaftlicher Schriften des In- und Auslandes hrsg. von L. Brentano und E. Leser, No.9. For an early discussion of the Bernoulli problem, reference is made to Malfatti, Gianfrancesco, "Esame critico di un problema di probabilita del Signor Daniele Bernoulli, e soluzione d'un altro problema analogo al Bernoulliano" in "Memorie di Mate- matica e Fisica della Societa italiana" Vol. I, Verona, 1782, pp. 768-824. For more on the "St. Petersburg Paradox," including material on later discussions, see Menger, Karl, "Das Unsicherheitsmoment in der Wertlehre. Betrachtungen im An- schluss an das sogenannte Petersburger Spiel," Zeit8chrift fur NationaliJkonomie, Vol. 5, 1934. This paper by Professor Menger, is the most extensive study on the literature of the prob- lem, and the problem itself. Recent interest in the Bernoulli hypothesis was aroused by its appearance in von Neumann, John, and Oskar Morgenstern, The Theory of Games and Economic Be- havior, second edition, Princeton: Princeton University Press, 1947, Ch. III and Appendix: "The Axiomatic Treatment of Utility." Many contemporary references and a discussion of the utility maximization hypothesis are to be found in Arrow, Kenneth J ., "Alternative Approaches to the Theory of Choice in Risk-Taking Situations," ECONOMETRICA, Vol. 19, October, 1951. More recent writings ~n the field include Alchian, A. A., "The Meaning of Utility Measurement," American Economic Review, Vol. XLIII, March, 1953. Friedman, M., and Savage, L. J., "The Expected Utility-Hypothesis and the Measura- bility of Utility," Journal of Political Economy, Vol. LX, December, 1952. Herstein, I. N., and John Milnor, "An Axiomatic Approach to Measurable Utility," ECONOMETRICA, Vol. 21, April, 1953. Marschak, J., "Why 'Should' Statisticians and Businessmen Maximize 'Moral Expecta- tion'?", Second Berkeley Symposium on Mathematical Statistics and Probability, 1953. Mosteller, Frederick, and Philip Nogee, "An Experimental Measurement of Utility," Journal of Political Economy, lix, 5, Oct., 1951. Samuelson, Paul A., "Probability, Utility, and the Independence Axiom," ECONO- METRICA, Vol. 20, Oct. 1952. Strotz, Robert H., "Cardinal Utility," Papers and Proceedings of the Sixty-Fifth Annual Meeting of the American Economic Association, American Economic Review, Vol. 43, May, 1953, and the comment by W. J. Baumol. For dissenting views, see : Allais, M., "Les Theories de la Psychologie du Risque de I'Ecole Americaine", Revue d'Economie Politique, Vol. 63,1953. -- "Le Comportement de l'Homme Rationnel devant Ie Risque : Critique des Postu- --- ## Page 51 24 36 DANIEL BERNOULLI lats et Axiomes de l'Ecole Americaine," ECONOMETRICA, Oct., 1953 and D. Bernoulli Edwards, Ward, "Probability-Preferences in Gambling," The American Journal of Psy- chology, Vol. 66, July, 1953. Textbooks dealing with Bernoulli: Anderson, Oskar, Einfilhrung in die mathematische Statistik, Wien: J. Springer, 1935. Davis, Harold, The Theory of Econometrics, Bloomington, Ind.: Principia Press, 1941. Loria, Gino, Storia delle Matematiche, dall'alba della civiltd al secolo XIX, Second re- vised edition, Milan: U. Hopli, 1950. --- ## Page 52 Bell System Technical Journal, 35, 917- 926 (1956) 25 3 ANew Interpretation of Information Rate reproduced with permission of AT&T By J. L. KELLY, JR. If the input symbols to a communication channel represent the outcomes of a chance event on which bets are available at odds consistent with their probabilities (i.e. , "fair" odds), a gambler can use the knowledge given him by the received symbols to cause his money to grow exponentially. The maximum exponential rate of growth of the gambler's capital is equal to the rate of transmission of information over the channel. This result is generalized to include the case of arbitrary odds. Thus we find a situation in which the transmission rate is significant even though no coding is contemplated. Previously this quantity was given significance . only by a theorem of Shannon's which asserted that, with suitable encoding, binary digits could be transmitted over the channel at this rate with an arbitrarily small probability of error. INTRODUCTION Shannon defines the rate of transmission over a noisy communication channel in terms of various probabilities.1 This definition is given significance by a theorem which asserts that binary digits may be encoded and transmitted over the channel at this rate with arbitrarily small probability of error. Many workers in the field of communication theory have felt a desire to attach significance to the rate of transmission in cases where no coding was contemplated. Some have even proceeded on the assumption that such a significance did, in fact, exist. For example, in systems where no coding was desirable or even possible (such as radar), detectors have been designed by the criterion of maximum transmission rate or, what is the same thing, minimum equivocation. Without further analysis such a procedure is unjustified. The problem then remains of attaching a value measure to a communication IC.E. Shannon, A Mathematical Theory of Communication, B.S.T.J., 27, pp. 379-423, 623-656, Oct., 1948. 917 --- ## Page 53 26 J L. Kelly, Jr. 918 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 system in which errors are being made at a non-negligible rate, Le., where optimum coding is not being used. In its most general formulation this problem seems to have but one solution. A cost function must be defined on pairs of symbols which tells how bad it is to receive a certain symbol when a specified signal is transmitted. Furthermore, this cost function must be such that its expected value has significance, i.e., a system must be preferable to another if its average cost is less. The utility theory of Von Neumann2 shows us one way to obtain such a cost function. Generally this cost function would depend on things external to the system and not on the probabilities which describe the system, so that its average value could not be identified with the rate as defined by Shannon. The cost function approach is, of course, not limited to studies of commu- nication systems, but can actually be used to analyze nearly any branch of human endeavor. The author believes that it is too general to shed any light on the specific problems of communication theory. The distinguishing feature of a communication system is that the ultimate receiver (thought of here as a person) is in a position to profit from any knowledge of the input symbols or even from a better estimate of their probabilities. A cost function, if it is sup- posed to apply to a communication system, must somehow reflect this feature. The point here is that an arbitrary combination of a statistical transducer (Le., a channel) and a cost function does not necessarily constitute a communication system. In fact (not knowing the exact definition of a communication system on which the above statements are tacitly based) the author would not know how to test such an arbitrary combination to see if it were a communication system. What can be done, however, is to take some real-life situation which seems to possess the essential features of a communication problem, and to analyze it without the introduction of an arbitrary cost function. The situation which will be chosen here is one in which a gambler uses knowledge of the received symbols of a communication channel in order to make profitable bets on the transmitted symbols. THE GAMBLER WITH A PRIVATE WIRE Let us consider a communication channel which is used to transmit the results of a chance situation before those results become common knowledge, so that a gambler may still place bets at the original odds. Consider first the case of a noiseless binary channel, which might be 2Von Neumann and Morgenstein, Theory of Games and Economic Behavior, Princeton Univ. Press, 2nd Edition, 1947. --- ## Page 54 A New Interpretation of Information Rate 27 A NEW INTERPRETATION OF INFORMATION RATE 919 used, for example, to transmit the results of a series of baseball games between two equally matched teams. The gambler could obtain even money bets even though he already knew the result of each game. The amount of money he could make would depend only on how much he chose to bet. How much would he bet? Probably all he had since he would win with certainty. In this case his capital would grow exponentially and after N bets he would have 2N times his original bankroll. This exponential growth of capital is not uncommon in economics. In fact, if the binary digits in the above channel were arriving at the rate of one per week, the sequence of bets would have the value of an investment paying 100 per cent interest per week compounded weekly. We will make use of a quantity G called the exponential rate of growth of the gambler's capital, where . 1 VN G == hm -log- N-+oo N Vo where VN is the gambler's capital after N bets, Vo is his starting capital, and the logarithm is to the base two. In the above example G = 1. Consider the case now of a noisy binary channel, where each transmitted symbol has probability, p, of error and q of correct transmission. Now the gambler could still bet his entire capital each time, and, in fact, this would maximize the expected value of his capital, (VN) , which in this case would be given by This would be little comfort, however, since when N was large he would probably be broke and, in fact, would be broke with probability one if he continued indefinitely. Let us, instead, assume that he bets a fraction, £, of his capital each time. Then where W and L are the number of wins and losses in the N bets. Then G = J~oo [~ 10g(1 + £) + t log(1 - £)] = qlog(1 + e) + plog(l - £) with probability one Let us maximize G with respect to £. The maximum value with respect to the Yi of a quantity of the form Z = L Xi log Yi, subject to the constraint L Yi = Y, is obtained by putting --- ## Page 55 28 J L. Kelly, Jr. 920 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 where X = L Xi' This may be shown directly from the convexity of the logarithm. Thus we put and (1 + £) = 2q (1 - f) = 2p Gmax = 1 + p logp + q logq =R which is the rate of transmission as defined by Shannon. One might still argue that the gambler should bet all his money (make e = 1) in order to maximize his expected win after N times. It is surely true that if the game were to be stopped after N bets the answer to this question would depend on the relative values (to the gambler) of being broke or possessing a fortune. If we compare the fates of two gamblers, however, playing a nonterminating game, the one which uses the value f found above will, with probability one, eventually get ahead and stay ahead of one using any other f. At any rate, we will assume that the gambler will always bet so as to maximize G. THE GENERAL CASE Let us now consider the case in which the channel has several input symbols, not necessarily equally likely, which represent the outcome of chance events. We will use the following notation: pes) the probability that the transmitted symbol is the s'th one. p{r/s) the conditional probability that the received symbol is the r'th on the hypothesis that the transmitted symbol is the s'th one. pes, r) the joint probability of the s'th transmitted and r'th received symbol. q(r) received symbol probability. q(s/r) conditional probability of transmitted symbol on hypothesis of received symbol. 0: 8 the odds paid on the occurrence of the s'th transmitted symbol, i.e., 0:8 is the number of dollars returned for a one-dollar bet (including that one dollar). a(s/r) the fraction of the gambler's capital that he decides to bet on the oc- currence of the s'th transmitted symbol after observing the r'th received symbol. --- ## Page 56 A New Interpretation of Information Rate 29 A NEW INTERPRETATION OF INFORMATION RATE 921 Only the case of independent transmitted symbols and noise will be consid- ered. We will consider first the case of "fair" odds, i.e., In any sort of parimutuel betting there is a tendency for the odds to be fair (ignoring the "track take"). To see this first note that if there is no "track take" since all the money collected is paid out to the winner. Next note that if for some s a bettor could insure a profit by making repeated bets on the sth outcome. The extra betting which would result would lower as. The same feed- back mechanism probably takes place in more complicated betting situations, such as stock market speculation. There is no loss in generality in assuming that La{s/r) = 1 s i.e., the gambler bets his total capital regardless of the received symbol. Since he can effectively hold back money by placing canceling bets. Now r,s where War is the number of times that the transmitted symbol is s and the received symbol is r. log ~n = L War logasa{s/r) o rs . 1 VN ~ G = hm -log - = L."p{s, r) logasa{s/r) N-;oo N Vo rs with probability one. Since 1 as = p{s) (I) --- ## Page 57 30 J L. Kelly, Jr. 922 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 here G = 2::p(s, r) log a(s/r) TS pes) = 2:: pes, r) loga(s/r) + H(X) TS where H(X) is the source rate as defined by Shannon. The first term is maxi- mized by putting ( / ) _ p( s, r) _ p( s, r) _ (/) as r - LkP(k,r) - q(r) -q 8 r Then Gmax = H(X) - H(X/Y), which is the rate of transmission defined by Shannon. WHEN THE ODDS ARE NOT FAIR Consider the case where there is no track take, i.e., but where as is not necessarily 1 pes) It is still permissible to set La a(s/r) = 1 since the gambler can effectively hold back any amount of money by betting it in proportion to the l/as . Equation (1) now can be written G = LP(s,r) Ioga(s/r) + LP(s) Iogas . TS 8 G is still maximized by placing a(s/r) = q(s/r) and G max = -H(X/Y) + LP(s)logas S = H(a) - H(X/Y) where s Several interesting facts emerge here (a) In this case G is maximized as before by putting a(s/r) = q(s/r). That is, the gambler ignores the posted odds in placing his bets! --- ## Page 58 A New Interpretation of Information Rate A NEW INTERPRETATION OF INFORMATION RATE (b) Since the minimum value of H(a) subject to obtains when L~=l s as 1 as = pes) and H(X) = H(a), any deviation from fair odds helps the gambler. 31 923 (c) Since the gambler's exponential gain would be H(a) - H(X) if he had no inside information, we can interpret R = H(X) - H(X/Y) as the increase of Gmax due to the communication channel. When there is no channel, i.e., H(X/Y) = H(X) , Gmax is minimized (at zero) by setting 1 as =- Ps This gives further meaning to the concept "fair odds." WHEN THERE IS A "TRACK TAKE" In the case there is a "track take" the situation is more complicated. It can no longer be assumed that L:s a(s/r) = 1. The gambler cannot make canceling bets since he loses a percentage to the track. Let br = 1 - L:s a(s/r), i.e., the fraction not bet when the received symbol is the rth one. Then the quantity to be maximized is G = LP(s,r)log[br +asa(s/r)), (2) rs subject to the constraints br + L a(s/r) = 1. 8 In maximizing (2) it is sufficient to maximize the terms involving a particular value of r and to do this separately for each value of r since both in (2) and in the associated constraints, terms involving different r's are independent. That is, we must maximize terms of the type Gr = q(r) L q(s/r) log[br + asa(s/r)] s subject to the constraint br + La(s/r) = 1 s --- ## Page 59 32 J L. Kelly, Jr. 924 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 Actually, each of these terms is the same form as that of the gambler's exponential gain where there is no channel G = LP(S) log[b + Qsa(s)]. (3) 8 We will maximize (3) and interpret the results either as a typical term in the general problem or as the total exponential gain in the case of no communication channel. Let us designate by A the set of indices, s, for which a(s) > 0, and by >.' the set for which a(s) = O. Now at the desired maximum fJG p(s)Qs 10 e = k for S£A = fJa(s) b + a{s)Qs g fJG L pes) loge = k = fJb b + a(s)Qs s fJG p(s)Q s I < k for S£A' = -b- oge = fJa(s) where k is a constant. The equations yield k = loge, b a(s) = pes) - - Q s b = l=2. 1-0" for S£A where p = L),P(S), a = L),(I/Qs), and the inequalities yield I-p p(s)Qs ;£ b = 1 _ a for S£A' We will see that the conditions 0'<1 p(s)Qs > 1-p for S£A I-a p{s)Qs ~ 1-p for S£A' 1-0' completely determine A. If we permute indices so that p(S)Qs ~ pes + 1)Qs+1 --- ## Page 60 A New Interpretation of Information Rate 33 A NEW INTERPRETATION OF INFORMATION RATE 925 then A must consist of all s ~ t where t is a positive integer or zero. Consider how the fraction varies with t, where t t l Pt = I>(s), 1 Ft decreases with t until p(t + l)at+1 < Ft or Ft and the fraction increases until 1 some bets might be made for which p(s)as < 1, i.e., the expected gain is negative. This violates the criterion of the classical gambler who never bets on such an event. CONCLUSION The gambler introduced here follows an essentially different criterion from the classical gambler. At every bet he maximizes the expected value of the logarithm of his capital. The reason has nothing to do with --- ## Page 61 34 J L. Kelly, Jr. 926 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1956 the value function which he attached to his money, but merely with the fact that it is the logarithm which is additive in repeated bets and to which the law of large numbers applies. Suppose the situation were different; for example, suppose the gambler'S wife allowed him to bet one dollar each week but not to reinvest his winnings. He should then maximize his expectation (expected value of capital) on each bet. He would bet all his available capital (one dollar) on the event yielding the highest expectation. With probability one he would get ahead of anyone dividing his money differently. It should be noted that we have only shown that our gambler's capital will surpass, with probability one, that of any gambler apportioning his money dif- ferently from ours but still in a fixed way for each received symbol, independent of time or past events. Theorems remain to be proved showing in what sense, if any, our strategy is superior to others involving a(sJr) which are not constant. Although the model adopted here is drawn from the real-life situation of gambling it is possible that it could apply to certain other economic situations. The essential requirements for the validity of the theory are the possibility of reinvestment of profits and the ability to control or vary the amount of money invested or bet in different categories. The "channel" of the theory might cor- respond to a real communication channel or simply to the totality of inside information available to the investor. Let us summarize briefly the results of this paper. If a gambler places bets on the input symbol to a communication channel and bets his money in the same proportion each time a particular symbol is received, his capital will grow (or shrink) exponentially. If the odds are consistent with the probabilities of occur- rence of the transmitted symbols (Le., equal to their reciprocals), the maximum value of this exponential rate of growth will be equal to the rate of transmission of information. If the odds are not fair, i.e., not consistent with the transmit- ted symbol probabilities but consistent with some other set of probabilities, the maximum exponential rate of growth will be larger than it would have been with no channel by an amount equal to the rate of transmission of information. In case there is a "track take" similar results are obtained, but the formulae involved are more complex and have less direct information theoretic interpre- tations. ACKNOWLEDGMENTS I am indebted to R. E. Graham and C. E. Shannon for their assistance in the preparation of this paper. --- ## Page 62 Journal o/Political Economy, 67, 144-155 (1959) 35 4 CRITERIA FOR CHOICE AMONG RISKY VENTURES HENRY ALLEN LATANE Chapel Hill, North Carolina THE SUB GOAL T HIS paper is concerned with the problem of how to make rational choices among strategies in situa- tions involving uncertainty. Such choices can be expressed through payout matrices stated in terms of some measure of value to be maximized. These matrices show the probabilities of all relevant future oc- currences and the payouts resulting from the combined effects of each possible strategy, on the one hand, and each relevant future occurrence, on the other.l All this information is needed to choose the proper strategy rationally. It would 1 Payout matrices are shown in Tables 1, 2, and 3. The payouts represent the possible final outcomes of choices among strategies. The matrices have single-valued payouts and probabilities. Many, if not all, decision problems can be reduced to such form. Consider first the probabilities. A subjective probability distribution of an imperfectly known underlying probability can be reduced to a subjec- tive probability of the event itself. For example, sup- pose a gambler believes that there is a 0.5 probabil- ity that a coin is biased so that it always comes up tails and a 0.5 probability that it is unbiased. He has a subjective probability of 0.25 for heads and 0.75 for tails, and these probabilities would be used in his payout matrix as long as his probability be- liefs remain unchanged. Consider next the payouts. In much discussion of decision theory the payouts are taken as given, with only the probabilities sub- ject to uncertainty. However, in real life the sizes of the payouts often are as subject to uncertainty as are the probabilities. But, even when the payouts are uncertain a matrix filled in with single values can be constructed. If we have probability distribu- tions of payouts for all specified occurrences (such as heads and tail. in the toss of a coin), a payout matrix can be constructed listing each possible payout and the subjective probability of its occurrence. These large matrices often can be reduced to simple two- valued distributions of payouts without much loss of information. be impossible for a decision-maker to choose rationally among strategies if he disregarded either the probability of the relevant future occurrences or any of the possible payouts. The problem of rational decision- making can be broken down into three steps: (1) deciding upon an objective and criteria for choosing among strategies; (2) filling out a payout matrix; and (3) choosing among available strategies on the basis of this matrix and the criteria. In real life the second step-deciding upon the size of the payout matrix, meas- ured by the number of columns represen t- ing relevant future occurrences and rows representing available strategies, and filling in the matrix with reasonable esti- mates of payouts and probabilities- is by far the most difficult part of the decision-maker's job. This paper has little to say about these problems. It deals largely with the first step: the prob- lem of setting up cri teria for choosing among strategies on the basis of a filled- in payout matrix. A hierarchy of goals and guides for reaching these goals is involved in ra- tional choices among strategies. This hierarchy consists of (1) a goal; (2) a subgoal; and (3) a criterion for choosing among strategies to reach the subgoal, that is, a measure that must be maxi- mized to attain the subgoal. The goal in rational decision-making is the maxi- mization of some measure of value. Each decision is made for the sake of the difference it will make in terms of this 144 --- ## Page 63 36 H A. Lalane CRITERIA FOR CHOICE AMONG RISKY VENTURES 145 objective. The decision-maker is con- fronted with a payout matrix expressed in terms of either a subjective utility measure such as utiles or an objective measure such as money or bushels of wheat. He wishes to choose the strategy that will give him the maximum payout. This is his goal. When some one strategy gives a higher payout than any other strategy in all relevant future occur- rences, the goal itself enables the de- cision-maker to choose among strategies. He merely chooses the dominant strategy. When there is no strategy superior to all the rest in all possible future occur- rences, the decision-maker needs some other guide for making decisions, since the goal itself does not enable him to make his choice. This guide is here called the "subgoal." The need for a subgoal exists because the outcome of specific strategies is subject to probabilistic un- certainty. In utility theory the payout matrix is expressed in terms of some measure of subjective utility, say, utiles. Choice of the strategy that will give the maximum payout in utiles is the goal, and choice of the strategy with the maxi- mum expected utility2 is taken as the subgoal. Given a completely filled-in matrix, this subgoal can surely be reached. Whether or not the goal is reached depends on future occurrences, but, in any event, the subgoal of maxi- mization of the expected value of the pay- outs expressed in utiles is logically re- lated to the goal of maximization of the forthcoming payout also expressed in utiles. In this paper a second subgoal is pro- posed for use when the choice is repe- titious and has cumulative effects and 2 The expected utility of a strategy is computed by mUltiplying all possible payouts expressed in u tiles by their respective probabilities and then slimming the products. when the goal is maximization of wealth at the end of a large number of choices. Under these conditions the choice of the strategy that has a greater probability (PI) of leading to as much or more wealth than any other significantly dif- ferent strategy at the end of a large number of choices also is a logical sub- goal. The P' subgoal is not as general as the maximum expected utility subgoal. For example, it would not apply to unique choices. When a man is faced with a once-in-a-lifetime choice of risking his whole fortune and his life on a venture that will produce great rewards if suc- cessful, it does not help him to know that he is almost certain to be ruined if he takes such a risk often enough. The P' subgoal is not logically related to the goal in this case. 3 Such a man, however, could set up a payout matrix expressed in utiles and decide which course of action maximized his expected utility. Here the maximum expected utility sub- goal is logically related to the goal even though the P' subgoal is not. The P' sub- goal is less general bu t would seem to be more operational than the expected util- ity subgoal because of the difficulty of constructing a payout matrix expressed in terms of utiles, especially if the de- cision involves a firm or group of people. The P' subgoal would seem to be par- ticularly applicable to many business 3 For certain utility functions and for certain re- peated gambles, no amount of repetition justifies the rule that the gamble which is almost sure to bring the greatest wealth is the preferable one. For ex- ample, the pI subgoal is not appropriate for a decision-maker for whom the possibility of great gain, however small and diminishing, is more im- portant than maximization of the probability of as much or more wealth than can be obtained by any other strategy in the long run. Such a decision· maker may adopt a course of action that is almost certain to result in less wealth in the long run. Whether or not his utility function is compatihle with the specified goal of maximum long-run wealth is not at issue here. --- ## Page 64 Criteria/or Choice among Risky Ventures 37 146 HENRY ALLEN LATANE decisions such as those involved in port- folio management. Wealth-holders have the option of holding their wealth in many different combinations of stocks, bonds, and cash. The allocation of wealth among these types of assets in- volves a series of choices extending over time. The fact that these choices are repetitive in nature with cumulative ef- fects may be used as the key factor in defining a goal, a subgoal, and a cri- terion for choosing among portfolios. The problem of choice among port- folios may be stated in terms of the pay- out matrix in Table 1. In this table Pi TABLE 1 PAYOUT MATRIX FOR VARYING PORTFOLIOS PORTFOLIO RELEVANT FUTURE O CCURRENCES 1, . .. ,j, .. . , k 1. . . . . . . . . . . . . . . . a iI, . .. , au, ... , al ~: 'to . • . • • • • . • • . • • • • . • . ail, ... , Qij, .• . , ai/~ t. Probability of occur- rence . . .. . .. . . . PI, ... , Pi> . .. , PI, represents the probability of the jth oc- currence, with 'J;Pi = 1, and a ii repre- sents the return from the ith portfolio with i = 1, .. . , t, if the jth occurrence takes place, withj = 1, ... , k. A return is the payout, including return of princi- pal, per dollar of portfolio value per in- ves tmen t period (here called "year"). Returns cannot be negative, so that ai ~ O. The portfolio manager is faced with such a payout matrix for n years and wants to choose in a rational manner one portfolio from all available port- folios in each of the n years. 4 The goal of portfolio management is taken to be to select a portfolio so as to maximize wealth at the end of a period of years, assuming reinvestment of all re- turns.5 Let Wi be the final value of $1.00 placed in portfolio i if returns are rein- vested n times. Then the goal of port- folio management is taken to be to select the optimum portfolio so that W:P1 ~ Wi, with i = 1, ... ,t. This goal cannot be used as a basis for choice among portfolios, since which portfolio will have the maximum Wn depends on future occurrences. The sub goal proposed here is the choice of the portfolio that has a greater probability (PI) of being as valuable or more valuable than any other signifi- cantly different portfolio at the end of n years, n being large. It is shown below 4 The idea of maximizing wealth at the end of a large number of separate decisions based on the same payout matriK may appear unrealistic, but portfolio managers are continually being faced with choices having cumulative effects and involving approxi- mately the same payouts and probabilities time after time. For example, year after year a portfolio man- ager may have probability beliefs such as; "I look for conditions in the next ten years to be very similar to those prevailing in 1926 through 1935. Bonds will yield about 4 per cent per annum during the whole period. Some day we are going to have a boom and a bust in the stock market, but I do not know which is going to come first." Choosing one portfolio to hold in each of the n years is not the same as choos- ing one portfolio at the beginning of the period to hold throughout the n years. For example, if the probability beliefs at the beginning of one year are such that the maximum pi allocation of the port- folio is 40 per cent in bonds and 60 per cent in stock and if these beliefs remain the same at the beginning of the next year, then the maximum pi allocation again will be 40 per cent in bonds and 60 per cent in stock at the beginning of the second year. If the rela- tive prices of stocks and bonds have changed be- tween the two dates, it will be necessary for the portfolio manager to make some sales and purchases in his portfolio to bring it into line with the desired proportions even if these proportions themselves have not changed. • Few wealth-holders reinvest all returns, so the problem of maximizing wealth assuming no with- drawals is somethat unrealistic. However, this re- striction can be modified. If withdrawals per unit of time are a fixed proportion of wealth (considered as interest, for example), they will not affect proper maximizing action. \Vhatever would maKimize wealth, assuming no withdrawals, 1I'0uld maximize wealth, assuming proportionate withdrawals. --- ## Page 65 38 H A. Lalane CRITERIA FOR CHOICE AMONG RISKY VENTURES 147 that the portfolio having a probability distribution of returns with the highest geometric mean, G, also has the greatest P'. The central fact of this paper is a simple one: If the value of an asset, say, portfolio i, priced initially at $1.00 is be- lieved to change after a year to, alterna- tively, ail or ai2, ... ,or aik, with re- spective probabilities PI, pz, . .. , Pk, and if the proceeds are reinvested n times, then the final value of the invest- ment, Wi, "converges in probability" to Gi = aft • aft' • • • afk". The prob- ability that the absolute difference be- tween W7 and G7 is smaller than any preassigned positive number will ap- proach 1 as n increases indefini tely. In other words, the final return from $1 .00 invested in portfolio i, assuming rein- vestment of all annual returns for n years will converge in probability on Gi, the nth power of the geometric mean of the probability distribution of annual re- turns from that portfolio. This relation- ship is intuitively obvious, since the ail return will "tend" to occur npl times, the aii return will tend to occur npi times, and so forth, if n is large. It can be proved rigorously by use of the law of large numbers applied to the logarithms of the annual returns. Let nj be the number of occurrences of the jth relevant future occurrence, with 27.nj = n, with j = 1, ... ,k, then nj n i: pj and njn log aij i: Pi log aij as n ~ aJ. But log Wi = 27.njn log aij, with j = 1, ... , k and log Gi = 27.pj log aij, so log Wi i~ log Gi and Wi I~ Gi as n ~ aJ.6 It follows from this that, if Gi > Gj , then the probability, pi, that Wi > WJ at the end of n years ap- proaches 1 as n increases indefinitely. The portfolio with the highest G is al- most certain to be more valuable than any other significantly different port- folio in the long run. For this reason G is accepted here as a rational criterion for choice among portfolios. SUB GOALS AND SUBJECTIVE UTIJ.ITY Rational choice among strategies is the ancient problem of the gambler who has the option to choose among bets. Classical writers on probability theory recommended that problems of this kind be solved by first computing the ex- pected winnings (possibly negative) for each available bet and then choosing the bet with the highest mathematical ex- pectation of winning. Since there was no reason to assume that, of two persons encountering identical risks, either should expect to have his desires more closely fulfilled, the classical writers thought that no characteristic of the risk-takers themselves ought to be taken into consideration; only those matters' should be weighed carefully that pertain to the terms of the risk. 7 In 1738 Daniel Bernoulli in four short paragraphs dem- onstrated that the use of the mathe- matical expectation of winnings did not always apply and proposed instead that gamblers should evaluate bets on the basis of the mathematical expectation of the utilities of winnings. ~ In terms of subgoals as defined in this study, Bernoulli showed that use of the 6 The asymptotic quality of G is used in informa- tion theory as developed by Dr. Claude Shannon and was applied to a gambling situation by John Kelly in "A New Interpretation of Information Rate," Bell System Technical Journal, August, 1956, pp. 917-26. See also R. Bellman and R. Kalaba, "Dynamic Programming and Statistical Communi- cation Theory," Proceedings of the NlLtional Academy of Science, XLIII (1957), 749-51. I had no knowl- edge of this work when I first proposed the pi sub- goal at a Cowles Foundation Seminar in February, 1956. 7 See Daniel Bernoulli, "Exposition of a New Theory on the Measurement of Risk," trans. Louise Sommer, Econometrica, XXII (January, 1954) , 23. "[bid., p. 24. --- ## Page 66 Criteria/or Choice among Risky Ventures 39 148 HENRY ALLEN LATANl!: expected-value subgoal did not always lead to choices that seemed rational to him and proposed instead the use of the expected-utility subgoal. He used the fol- lowing example: Somehow a very poor fellow obtains a lottery ticket that will yield with equal probability either nothing or twenty thousand ducats. Will this man evaluate his chances of winning at ten thousand ducats? Would he not be ill-advised to sell this lottery ticket for nine thousand ducats? To me it seems that the answer is in the negative. On the other hand I am inclined to believe that a rich man would be ill-advised to refuse to buy the lottery ticket for nine thou- sand ducats. If I am not wrong then it seems clear that all men cannot use the same rule to evaluate the gamble. 9 Bernoulli's example is somewhat aside from the daily business of living, but, when stripped of its gambling wrappings and expressed in terms of payouts and returns, it is seen to represent a major segment of economic decision-making. The hypothetical market price of the ticket, which has an equal probability of paying 20,000 ducats or nothing, is 9,000 ducats. Both the poor man and the rich man have the option either to hold the lottery ticket or to hold 9,000 ducats. Possible payouts range from 2.22 per ducat risked to 0. Payouts with ranges such as this-indeed, much greater ranges- are ordinary economic occur- rences. The magnitude of the choice faced by the rich man is well within the range of ordinary business decisions, and the "poor man" today is continually faced with implicit or explicit decisions as serious as that faced by Bernoulli's lottery-ticket owner. He must decide whether to move to a new job, buy a new home, sign a second mortgage. He is con- tinually offered the opportunity to un- dertake such risky ventures as purchas- ing his own truck, opening a restaurant, 9 Ibid. buying some uranium stock, some oil stock, or some investment shares. Some of these options may be highly ad- vantageous, and he must choose some one course of action in each case. The ef- fects of these choices are cumulative; that is, the decision-maker never comes back to exactly the same position he occupied before making his choice. The major difference between Bernoulli's problem and other choices among courses of action is that the ticket-holder's choice is dearly defined, while the other opportunities are usually ignored, or the choices are muddled. TABLE 2 PAYOUT MATRIX OF GAINS AND LOSSES FUTURE O CCUR.RENCE Ticket Ticket CRI' STRA.TEGY Wins Loses TElUON A a) Poor man: Hold ticket . . . 20 0 10 Sell ticket ... . 9 9 9 b) Rich man: Buy ticket. .. 11 - 9 1 Not buy ticket 0 0 0 Probability of oc- currence .. .. .. . . 0.5 0.5 Thus Bernoulli's example is represent- ative of a wide class of choices. The decision-maker is being faced continually with such choices, and the outcome of each decision affects his entire future. In the following discussion this example is stated in payout matrices constructed to illustrate choices based on <.a) classical mathematical expectation (the expected value subgoal) ; (b) Bernoulli's subjective utility (the expected-utility subgoal); and \c) the maximum chance (Pi) sub- goal. Table 2 shows the classical approach to choosing among risky ventures. The payout matrix, expressed in terms of thousands of ducats, shows the probabil- ity of the lottery ticket paying off or not and the net payout to the poor man and to the rich man for each of two courses of --- ## Page 67 40 H A. Lalane CRITERIA FOR CHOICE AMONG RISKY VENTURES 149 action. The classical writers would calcu- late the mathematical expectation, A, of the net payouts and choose that strategy which maximizes A. In this case they would recommend that the rich man buy the ticket for 9,000 ducats and that the poor man refuse to sell it at this price. The mathematical expectation (that is, the arithmetic mean) of the probabil- ity distribution of payouts is, indeed, a good criterion when there are large numbers of independent trials. Even decision-makers who make repeated choices with cumulative effects (for ex- ample, the operators of roulette wheels and insurance companies) are rightly interested in this average when each risk is small in relation to total wealth. There is little or no conflict under these condi- tions between the use of the arithmetic mean as a criterion and the use of the geometric mean of the probability dis- tribution of payouts per dollar of wealth as a criterion./O When a decision-maker can surely bet the same small amount on a large number of independent trials, he can maximize the expected value of his gain, and also the likelihood of having more gain than can be obtained by any other plan, by choosing that set of bets which gives him the greatest mathe- matically expected payout. For example, if Bernoulli's poor man had found 10,000 tickets involving 10,000 independent drawings, each with a payout equally likely to be 2 ducats or nothing, he clear- ly would be unwise to sell his block of tickets for 9,000 ducats. His winnings on 10,000 different trials would be almost 10 When a gambler who has the choice of betting or not betting bets all his wealth on the toss of a fair coin with a payout of $3.00 per $1.00 bet if heads occur and nothing per $1.00 bet if tails occur, he is maximizing the expected value of the payout but not G. When he can bet only 1 per cent of his wealth, however, he will maximize both A and G by betting. certainly very close to 10,000 ducats, the mathematical expectation of the value of the set of tickets, and the advice of the classical writers would be sound. Bernoulli used the lottery-ticket ex- ample to show that the expected values of the payouts are not good guides in mak- ing choices involving large risks. He pro- posed instead that the expected value of the utilities of the payouts be used as a criterion. He would fill in the payout matrix in Table 2 not with the money value of the gains and losses but with their utilities and then would use the mathematical expectation of these utili- ties as his criterion. Whether or not particular payout matrices expressed in terms of subjective utility are realistic is not a problem here. But Bernoulli's procedure is very much at issue. He defines the "mean utility" of a course of action as the mathematical expectation of the probability distribu- tion of the possible utilities from that course of action. He then states, with no discussion, that this mean utility can be used as a basis for valuing risks, that is, as a basis for choosing among courses of action. In other words, he explains why he expresses his profits (or losses) in terms of subjective utility, but he does not give any justification for maximizing the mathematical expectation of these utilities. Bernoulli's use of subjective utility has had wide recognition, and his use of mathematical expectation also has been widely adopted with little or no discussion.ll 11 Mathematical expectation now is used as a basis for defining utility. The present emphasis on the axiomatic approach to utility is largely derived from John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior (rev. cd.; Princeton, N.J. : Princeton University Press, 1935). They say : "We have practically defined numerical utility as being that thing for which the calculus of mathematical expectations is legitimate" (p. 28). --- ## Page 68 Criteria/or Choice among Risky Ventures 41 150 HENRY ALLEN LATANf: Bernoulli's problem can also be solved by the use of the maximum-chance (P') subgoal. Table 3 shows the payout matrix of returns (that is, payouts, in- cluding return of principal, per dollar of wealth) for the poor man, who is as- sumed to have a wealth of 1,000 ducats aside from his lottery ticket, and for the rich man, who is assumed to have a wealth of 100,000 ducats. The arithmetic mean, A, of the probability distribution of returns is higher for the poor man when he holds the ticket and for the rich man when he buys the ticket. The geo- TABLE 3 PAYOUT MATRIX OF RETURNS FUTURE OCCURRENCE Ticket Ticket CRITERWN S TRATEGY Wins Loses A G a) Poor man: Hold ticket. 2. 1 0 . 1 1.1 0.46 Sell ticket .. 1.0 1.0 1.0 1.0 b) Rich man: Buy ticket. 1.11 0.91 1.01 1.005 Not buy ticket .. 1.00 1.00 1.00 1.00 Probability of oc- currence .. .. . . 0 .5 0 .5 metric mean, G, of returns for the poor man is higher when the ticket is sold, however, and G for the rich man is higher when he buys the ticket. Over a long enough period of time many economic choices involving returns of the same order of magnitude repeat themselves. Bernoulli's poor man may never find another lottery ticket, but he probably will have many options among courses of action with as wide, or wider, a range of returns. It is assumed here that both the rich man and the poor man will have many opportunities to risk the same proportions of their respective fortunes on approximately the same terms and that both men prefer more wealth to less wealth, everything else being equal. If these assumptions are valid, the maximization of P', the prob- ability of having more wealth at the end of a long series of such choices than can be obtained by any other specified course of action, is a rational subgoal, and G is a rational criterion. The use of the maximum-chance subgoal results in courses of action for both the rich man and the poor man which seemed rational to Bernoulli. The decision-maker who is interested in maximizing his wealth at the end of a long series of choices should ask himself how he would come out in the long run if he made the same choice on the same terms over and over again. It is not necessary for him to ask himself what his individual subjective utility of winning is. This is not to say that other goals, rather than the goal of maximum wealth at the end of a long series of choices, are irrational. Indeed, the use of subgoals based on the goal of maximum wealth often may be irrational. For example, the man who desperately needs $10.00 to escape a jail sentence and who has only $1.00 may well be justified in taking a gamble to get his money, even though this gamble would not stand the maxi- mum-chance subgoal test. Even under these conditions, however, it would be useful for the man to know that he should not often act in such a manner, if he wants to build up his fortune so as to avoid similar predicaments in the future. In his paper Bernoulli uses the ex- pected utilities of the payouts as his criterion. He then reaches the conclusion that the utility resulting from any small increase in wealth usually is inversely proportional to the quantity of goods previously possessed.12 Under these con- ditions the utilities of the returns vary as their logarithms, and the geometric mean, G, of the probability distributions 12 This is generally crediten with being the first use of a utili ty function. --- ## Page 69 42 H A. Lalane CRITERIA FOR CHOICE AMONG RISKY VENTURES 151 of returns can be used as a criterion in- stead of expected utility. The arithmetic mean of the logarithms (utilities) of re- turns is maximized when G is maxi- mized. 13 Bernoulli gives a number of applica- tions of his formula to gambling and to insurance. In each instance he is able to give a specific answer. He says that everyone who bets any part of his fortune on a mathematically fair game of chance is acting irrationally, and he then de- termines what odds a gambler with a specified fortune must obtain to break even in the long run. Most of his prob- lems still are interesting in their own right, and many have a bearing on proper portfolio management. For instance, he demonstrates with numerical examples the advantages of diversification among equally risky ventures and between risky and safe assets. Bernoulli's approach to the valuation of risky ventures is not contradictory to the maximum-chance (P') approach. Not only do the two approaches lead to the same conclusion when they both can be applied but they tend to support each other. Wealth-holders may be divided into two groups. The first group contains those to whom each risk is a unique event either because they do not expect it to recur or because they keep its effects en- tirely separate from the results of other risks. For example, the man who each year sets aside a small sum to bet on the races during his vacation, with the in- tention of "living it up" if he wins and writing it off to experience if he loses, presumably is not actuated by long-run profit-maximizing motives. The effects of ,3 As pointed out to me by Professor L. J. Savage (in correspondence), not only is the maximization of G the rule for maximum expected utility in connec- tion with Bernoulli's function but (insofar as cerlain approximations are permissible) this same rule is ap- proximately valid for all utility functions. each risk are kept separate. AnaJysis based on maximum chance has nothing to offer this first class of wealth-holders. The choice between profit and safety or expected return and variance is a matter of subjective utility. Bernoulli's assump- tion that the satisfaction derived from a small gain tends to vary in inverse pro- portion to the initial wealth mayor may not be a shrewd guess. The second class of wealth-holders in- cludes those who expect to be faced re- peatedly with risks of the same general type and magnitude. This group in- cludes those making most business and portfolio decisions and hence is of great importance. It includes, specifically, all those who want to maximize the value of their portfolio at the end of n years, assuming reinvestment of all returns. Here there is a definite rule for choosing between risk and return, the P' subgoal, based on maximum-chance principles. This class may be subdivided further into (a) those who undertake only one risky venture at a time and (b) those who are able to diversify their risky ventures. Because so many economic phenomena, including yields on stocks, tend to fluctu- ate together over time, diversification among risky ventures cannot go as far toward eliminating risk as otherwise would be the case. Final choice among efficient portfolios for both groups, (a) and (b), is based on maximization of G, not because this maximizes subjective utility, but because it maximizes P'. Bernoulli states that the wealth- holder should ask himself whether the added satisfaction associated with the expected gain justifies undertaking the risky venture. He bases an exact rule of behavior on his assumption as to how the added satisfaction varies with the size of the potential gain or loss in relation to the size of the portfolio. The rule mayor --- ## Page 70 Criteria/or Choice among Risky Ventures 43 152 HENRY ALLEN LATANE may not be empirically useful, but it is grounded on rather shaky evidence about the exact shape of the utility function. According to maximum-chance analysis, the wealth-holder or portfolio manager should ask himself how he can maximize his chances of getting as good or better return than can be obtained with any other specified plan, assuming that he risks the same proportion of his portfolio on the same terms over and over again. It turns out that the formula which enables the portfolio manager to answer the maximum-chance question is the same as that developed by Bernoulli on grounds of subjective utility. In conclusion Bernoulli says: Though a person who is fairly judicious by natural instinct might have realized and spon- taneously applied much of what I have here explained, hardly anyone believed it possible to define these problems with the precision we have employed in our examples. Since all of our propositions harmonize perfectly with experi- ence it would be wrong to neglect them as abstractions resting upon precarious hy- potheses.14 Professor Stigler, in a review article,15 gives considerable space to Bernoulli's hypothesis about the slope of the wealth- holder's utility function, even though the major emphasis of the article is on utility not affected by probability. He mentions that Laplace and Marshall, among others, have accepted the law as a realistic guide. He also points out the similarity of Bernoulli's law to the Weber-Fechner psychological hypothesis that the just noticeable increment to any stimulus is proportional to the stimulus. Stigler says: "Bernoulli was right in seeking the explanationl6 in utility and u Op. cit., p. 31. i6 George J. Stigler, "The Development of Util- ity Theory ," Journal of Political &onomy, L VIn (1950), 373- 77. he was wrong only in making a special assumption with respect to the slope of the utility curve for which there was no evidence and which he submitted to no tests. "17 More recently Savage in a section on "Historical and Critical Comments on Utility" had this to say: Bernoulli went further than the law of diminishing marginal utility and suggested that the slope of utility as a function of wealth might, at least as a rule of thumb, be supposed, not only to decrease with, but to be inversely proportional to, the cash value of wealth. To this day, no other function has been suggested as a better prototype for Everyman's utility function .... Though it might be a reasonable approximation to a person's utility in a moder- ate range of wealth, it cannot be taken seriously over extreme ranges.18 INDIVIDUAL RISK PREFERENCE As indicated in the previous section, Bernoulli took the following steps to de- velop his utility function and to justify diversification among risky ventures and between risk assets and safe assets. (1) He showed-subject to the implicit as- sumption about subgoals previously dis- cussed~that the value of a risky venture to the individual wealth-holder is not the arithmetic mean of the probability dis- tribution of returns (the mathematical expectation of returns) but may be taken to be the arithmetic mean of the prob- ability distribution of the utilities of the returns. (2) He stated that, in the ab- sence of the unusual, the gain in utility ,. Bernoulli is explaining the reason for the limited value of the game involved in the St. Peters- burg paradox. This game is a type of risky venture with an infinitely large mathematically expected value but with an extremely small probability of winning. 11 Stigler, op. cit., p. 375. 13 Leonard J. Savage, The Foundations of Sta- tistics (New York: John Wiley & Sons, 1954), p. 94. --- ## Page 71 44 H A. Lalane CRITERIA FOR CHOICE AMONG RISKY VENTURES 153 resulting from any small increase in wealth may be assumed to be inversely proportional to the quantity of goods previously possessed. (3) He developed a formula for calculating the utility of a risk asset to the individual wealth-holder using as a criterion the utility function developed in step 2. According to Ber- noulli, the subjective utility of the wealth-holder's assets, including the risky venture, is measured by the geo- metric mean, G, of the probability dis- tribution of payouts from such assets. (4) Using this formula, he was able to calcu- late exactly the utility of the wealth- holder's assets, including the risky ven- ture, and to show that diversification among risky ventures increases the utility.19 Bernoulli's step 2 may be a reasonable assumption about utility,2° but it is sub- ject to so many qualifications and excep- tions (it does not explain gambling, for example) that it has not been accepted as a suitable basis for erecting the super- structure of steps 3 and 4. The valuation of risky ventures has been left to indi- vidual risk preference without any cri- terion for deciding what this preference is likely to be. For example, Makower and Marschak present a hypothetical table in which an asset's marginal con- tribution is determined by adding to- gether its contribution to "lucrativity" and safety measured in "lucrativity 19 Bernoulli, op. cit., pp. 24, 25, 28, 30. 20 Cf. Alfred Marshall, Principles of Economics (8th ed.; New York: Macmillan Co., 1950), p. 135. Marshall says: "In accordance with a suggestion made by Daniel Bernoulli, we may regard the satis- faction which a person derives from his income as commencing when he has enough to support life, and afterwards as increasing by equal amounts with every equal successive percentage that is added to his income; and vice versa for loss of income." See also Savage's comment quoted previously. units" determined by the safety prefer- ence rate for a single individual.~l These individual safety preference rates, in turn, are a matter of taste and must be accepted as given. Friedman and Savage build on Bernoulli's step 1 but modify step 2 by developing a doubly inflected curve comparing utility with income.22 Markowitz begins his analysis of port- folio selection by pointing out that "the portfolio with the maximum expected re- turn is not necessarily the one with the minimum variance. There is a rate at which the investor can gain expected re- turn by taking on variance, or reduce variance by giving up expected return.'m He assumes that the investor considers, or should consider, expected return a de- sirable thing and variance of return an undesirable thing, and he defines an ef- ficient portfolio as a portfolio with minimum variance for a given expected return or more and a maximum expected return for a given variance or less. He develops a method for selecting efficient portfolios from the set of all possible portfolios but does not give any basis for choice among the efficient portfolios ex- cept the individual's safety preference rate. THE NEED FOR AN OBJECTIVE CRITERION The difficulty of evaluating subjective risk preference and the need of an objec- tive criterion are well indicated in the 21 Helen Makower and Jacob Marschak, "Assets, Prices and Marketing Theory," Economica, V (1938), 261-88. Reprinted in American Economic Association, Readings in Price Theory (Chicago: Richard D. Irwin, Inc., 1952), pp. 301-2. 22 Milton Friedman and L. J. Savage, "The Util- ity Analysis of Choices Involving Risk," Journal of Political Economy, LVI (1948), 279-304. 2> Harry Markowitz, "Portfolio Selection," Jour- nal of Finance, VII (March, 1952), 79. --- ## Page 72 Criteria/or Choice among Risky Ventures 45 154 HENRY ALLEN LATANE following quotation from a recent journal article dealing with selection of an opti- mum combination of crops for a farmer: The introduction of risk into an economic model of a firm and consequently into a linear programming model of a firm has been accom- plished by describing risky outcomes as prob- ability distributions and choosing from among alternate possible distributions by the expected utility hypothesis. Two basic weaknesses have appeared in applying this method of incorporating risk. One difficulty arises in choosing a value for the constant a, which in this case is some sort of risk aversion indicator, and is, to some degree, governed by the personal characteristics of the entrepreneur. A large value for a indicates that the entrepreneur places a great weight on the variance as a deciding factor and is consequent- ly highly averse to risk, and vice versa. The estimation of such a constant to be used in a model is thus quite important; the wrong choice will invalidate any results obtained. The derivation of this constant is a delicate task beyond the scope of this paper.24 A major advantage of the criterion for choice among risky ventures developed in this paper is that it avoids the necessity for direct subjective determi- nation of such factors as Marschak's "lucrativity units" or Freund's "risk aversion indicator." As Roy remarks, "A man who seeks advice about his actions will not be grateful for the suggestion that he maximize expected utility."26 The criteria for choice between risk and safety in portfolio management can be illustrated by assuming that a gam- bler has the choice of holding his money in cash or of betting on a gambling de- vice which, with equal probability, will return R-s on loss occasions and R+s on gain occasions with an expected re- .. Rudolph J. Freund, "The Introduction of Risk into a Programming Model," Econometrica, XXIV (July, 1956),253-63. 2. A. D. Roy, "Safety First and the Holding of Assets," Econometrica XX (1952),433. turn of R per dollar played. The gam- bler's portfolio at any time consists of the proportion of his wealth held in cash plus the proportion bet on the gambling device. When the gambler bets none of his wealth, the expected return from his portfolio is 1, and the standard deviation of returns is O. As the proportion bet in- creases, both the expected portfolio re- turn and the standard deviation of re- turns increase. When he bets all his wealth, the expected portfolio return is R, and the expected standard deviation of returns is s. As long as R is greater than 1, and R - s is less than 1, all pos- sible combinations of the two assets in this range are efficient portfolios in that anyone of the' combinations gives the maximum possible expected return for some standard deviation or variance and the minimum standard deviation or vari- ance for some expected return. Neither Marschak nor Friedman and Savage nor Markowitz would be able to help the gambler in choosing among these efficient portfolios beyond telling him that he should gamble heavily if he has a high preference for risk and should be very conservati ve in his betting if he has a high risk-aversion factor. In this paper an attempt is made to give the gambler (and wealth-holders, in general) an ob- jective criterion for making this choice. The wealth-holder who adopts the maximum-chance (PI) subgoal can reach this subgoal by using the geometric mean, G, of the probability distribution of returns as his criterion and choose the strategy that has the probability distri- bution of returns with the highest G . Bernoulli also has shown that choice of that risky venture with the highest G is a rational choice (1) if maximization of the mathematical expectation of the utilities --- ## Page 73 46 H A. Lalane CRITERIA FOR CHOICE AMONG RISKY VENTURES 155 of the payouts is a rational subgoal and (2) if the utility of a small gain or loss varies inversely with the amount of wealth already possessed. Most economists recognize that the mathematical expectation and the vari- ance of the probability distribution of returns and the chance of ruin are im- portant to the wealth-holder-but they leave it to individual risk preference to balance one factor against the others. Since G depends on both the mathe- matical expectation and the variance of the probability distribution of returns, when G is maximized, there is no chance of ruin if the wealth-holder's probability beliefs are correct. Consequently, maxi- mization of G falls within the generally accepted range of rational behavior. This is not to say that G is the only rational criterion for choice among strategies; it is to say, however, that it is a useful criterion in dealing with a broad range of problems. --- ## Page 74 Proc. a/the 4th Berkeley Symp. on Math. Statistics and Probability, 1, 63--68 (1961) 5 OPTIMAL GAMBLING SYSTEMS FOR FAVORABLE GAMES L. BREIMAN UNIVERSITY OF CALIFORNIA, LOS ANGELES 1. Introduction 47 Assume that we are hardened and unscrupulous types with an infinitely wealthy friend. We induce him to match any bet we wish to make on the event that a coin biased in our favor will turn up heads. That is, at every toss we have probability p > 1/2 of doubling the amount of our bet. If we are clever, as well as unscrupulous, we soon begin to worry about how much of our available for- tune to bet at every toSS. Betting everything we have on heads on every to~s williead to almost certain bankruptcy. On the other hand, if we bet a small, but fixed, fraction (we assume throughout that money is infinitely divisible) of our available fortune at every toss, then the law of large numbers informs us that our fortune converges almost surely to plus infinity. What to do? More generally, let X be a random variable taking values in the set I = {I, ... , s} such that P{X = i} = p; and let there be a class e of subsets Ai of I, where e = {AI, ... , A r}, with Ui Ai = I, together with positive numbers (01, •.• , Or). We play this game by betting amounts f31, •.. , f3r on the events {X E Aj} and if the event {X = i} is realized, we receive back the amount L;EA; f310j where the sum is over allJ' such that i E A j • We may assume that our entire fortune is distributed at every play over the betting sets e, because the possibility of holding part of our fortune in reserve is realized by taking AI, say, such that Al = I, and 01 = 1. Let 8" be the fortune after n plays; we say that the game is favorable if there is a gambling strategy such that almost surely 8 ft -+ 00. We give in the next section a simple necessary and sufficient condition for a game to be favorable. How much to bet on the various alternatives in a sequence of independent repetitions of a favorable game depends, of course, on what our goal utility is. There are two criterions, among the many possibilities, that seem pre-eminently reasonable. One is the minimal time requirement, that is, we fix an amount x we wish to win and inquire after that gambling strategy which will minimize the expected number of trials needed to win or exceed x. The other is a magnitude condition; we fix at n the number of trials we are going to play and examine the size of our fortune after the n plays. This research was supported in part by the Office of Naval Research under Contract Nonr-222(53). 65 --- ## Page 75 48 L. Breiman 66 FOURTH BERKELEY SYMPOSIUM: BREIMAN In this work, we are especially interested in the asymptotic point of view. We show that in the long run, from either of the two above criterions, there is one strategy A * which is optimal. This strategy is found as that system of betting (essentially unique) which maximizes E(log Sn). The reason for this result is heuristically clear. Under reasonable betting systems Sn increases exponentially and maximizing E(log Sn) maximizes the rate of growth. In the second section we investigate the nature of A *. It is a conservative policy which consists in betting fixed fractions of the available fortune on the various A j • For example, in the coin-tossing game A * is: bet a fraction p - q of our fortune on heads at every game. It is also, in general, a policy of diversifica- tion involving the placing of bets on many of the Aj rather than the single one with the largest expected return. The minimal expected time property is covered in the tlt,ird section. We show} .~ by an examination of the excess in Wald's formula, that"the desired fortune X i becomes infinite, that the expected time under A * to amass x becomes less than · that under any other strategy. Section four is involved with the magnitude problem. The content here is that A * magnitudewise, does as well as any other strategy, and that if one picks a policy which in the long run does not become close to A *, then we are asymptot- ically infinitely worse off. Finally, in section five, we discuss the finite (nonasymptotic) case for the coin-tossing game. We have been unsuccessful in our efforts to find a strategy which minimizes the expected time for x fixed, but we state a conjecture which expresses a moderate faith in the simplicity of things. It is not difficult, however, to find a strategy which maximizes P{Sn ~ x} for fixed n, x and we state the results with only a scant indication of proof, and then launch into a comparison with the strategy A * for large n. The conclusion of these investigations is that the strategy A * seems by all reasonable standards to be asymptotically best, and that, in the finite case, it is suboptimal in the sense of providing a uniformly good approximation to the optimal results. Since completing this work we have been allowed to examine the most sig- nificant manuscript of L. Dubins and L. J. Savage [1], which will soon be pub- lished. Although gambling has been associated with probability since its birth, only quite recently has the question of gambling systems optimal with respect to some goal utility been investigated carefully. To the beautiful and deep results of Dubins and Savage, upon which work was commenced in 1956, must be given priority as the first to formulate systematically and solve the problems of optimal gambling strategies. We strongly recommend their work to every student of probability theory. Although our original impetus came from a different source, and although their manuscript is almost wholly concerned with unfavorable and fair games, there are a few small areas of overlap which I should like to point out and acknowledge priority. Dubins and Savage did, of course, formulate the concept --- ## Page 76 Optimal Gambling Systems for Favorable Games 49 OPTIMAL GAMBLING SYSTEMS 67 of a favorable game. For these games they considered the class of "fractionalizing strategies," which consist in betting a fixed fraction of one's fortune at every play, and noticed the interesting phenomenon that there was a critical fraction such that if one bets a fixed fraction less than this critical value, then S" ~ 00 a.s. and if one bets a fixed fraction greater than this critical value, then Sn ~ 0 a.s. In addition, our proposition 3 is an almost exact duplication of one of their theorems. In their work, also, will be found the solution to maximizing P {Sn ~ x} for an unfavorable game, and it is interesting to observe here the abrupt dis- continuity in strategies as the game changes from unfavorable to favorable. My original curiosity concerning favorable games dates from a paper of J. L. Kelly, Jr. [2J in which there is an intriguing interpretation of information theory from a gambling point of view. Finally, some of the last section, in prob- lem and solution, is closely related to the theory of dynamic programming as originated by R. Bellman [3]. 2. The nature of A * We introduce some notation. Let the outcome of the kth game be X k and Rn = (Xn, ... , Xl)' Take the initial fortune So to be unity, and Sn the fortune after n games. To specify a strategy A we specify for every n, the fractions [Xin+l), ... ,x~n+I)J = Xn+l,bf our available fortune after the nth game, Sn, that we will bet on alternative AI, ... , Ar in the (n + l)st game. Hence (2.1) Note that X may depend on R". Denote A = (Xl, X2, ••• ). Define the random '>1-1-1 variables V n by (2.2) V n = :E xJn) OJ, Xn = i, i1' ;EAj\ so that Sn+l = V n+ISn' Let W n = log V n, so we have (2.3) To define A *, consider the set of vectors};. = (AI, . .. , AI') with r nonnegative components such that Al + ... + Xr = 1 and define a function W(X) on this space 5 by (2.4) W(X) = :E pdog ( :E A;o;). i iEAj The function W(};.) achieves its maximum on 5 and we denote W = maX~E(f W(};.). PROPOSITION 1. Let };.(1), };.(2) be in 5 such that W = W(};.(L» = W(};.(2» , then for all i, we have LtEAjAJI) OJ = LtEAj xj2) OJ. -· PRC)OF. Let a, {3 be positive numbers SliC~ that a + (3 = 1. Then if X = a};.(l) + (3};.(2), we have W(X) ;;;; W. But by tKe;J~ncavity of log (2.5) W(};.) ~ aW(};.(l) + (3W(};.(2») with equality if and only if the conclusion of the proposition holds. --- ## Page 77 50 L. Breiman 68 FOURTH BERKELEY SYMPOSIUM: BREIMAN Now let >:* be such that W = W(>:*) and define A * as (>:*, >:*, ... ). Although >:* may not be unique, the random variables Wi, W~, . .. arising from ~ are by proposition 1 uniquely defined, and form a sequence of indepelldent,--identically distributed random variables. Questions of uniqueness and description of >:* are complicated in the general case. But some insight into the type of strategy we get using A * is afforded by PROPOSITION 2. Let the sets AI, • • . , Ar be disjoint, then no matter what the odds OJ are, >:* is given by Xj = P{X E A j }. The proof is a simple computation and is omitted. From now on we restrict attention to favorable games and give the following criterion. PROPOSITION 3. A game is favorable if and only if W > O. PROOF. We have n (2.6) log S~ = l: Wt I If W = EW: is positive, then the strong law of large numbers yields S~ -? 00 a.s. (Conversely, if there is a strategy A such that Sn -? 00 a.s. we use the result of i section 4, which says that for any strategy A, limn SnIS~ exists a.s. finite. Hence S~ -? 00 a.s. and therefore W ~ O. Suppose W = 0, then the law of the iterated logarithm comes to our rescue and provides a contradiction to S~ -? 00 • 3. The asymptotic time minimization problem For any strategy A and any number x > I, define the random variable T(x) by (3.1) T(x) = {smallest n such that Sn ~ x}, and T*(x) the corresponding random variable using the strategy A*. That is, T(x) is the number of plays needed under A to amass or exceed the fortune x. This section is concerned with the proof of the following theorem. THEOREM 1. If the random variables Wi, W;, ... are nonlattice,t then for any strategy (3.2) 1 ., lim [ET(x) - ET*(x)] = W l: (W - EWn ) x~~ 1 and there is a constant a, indepMdent of A and x such that (3.3) ET*(x) - ET(x) ~ a. Notice that the right side of (3.2) is always nonnegative and is zero only if A is equivalent to A * in the sense that for every n, we have W n = W~. The reason for the restriction that W: be nonlattice is f~_ .~pI>~.re.~t. But as this restriction is on log V: rather than on V~ itself, the common games with rational values of the odds OJ and probabilities Pi usually will be nonlattice. For instance, a little number-theoretic juggling proves that in the coin-tossing case the countable set of values of P for which W~ is lattice consists only of irrationals.'" --- ## Page 78 Optimal Gambling Systems for Favorable Games 51 OPTIMAL GAMBLING SYSTEMS 69 The proof of the above theorem is long and will be carried out in a sequence of propositions. The heart is an asymptotic estimate of the excess in Wald's identity [4]. PROPOSI'l'ION 4. Let Xl, X 2, • •• be a sequence of identically distributed, inde- pendent nonlattice random variables with 0 < EX I < 00. Let Y n = Xl + ... + X n' For any real numbers x, ~, with ~ > 0, let F,,,W = P{first Y" ~ x is < x + ~}. Then there is a continuous distribution GW such that for every value of ~, (3.4) lim F",W = GW. PROOF. The above statement is contained in known results concerning the renewal theorem. If Xl > 0 a.s. and has the distribution function F, it is known (see, for example, [5]) that limx-.", F",(~) = (1/ EXI ) fo~ [1 - F(t)] dt. If Xl is not positive, we use a device due to Blackwell [6]. Define the integer-valued random variables nl < n2 < ... by nl = {first n such that Xl + ... + Xn > O}, n2 = {first n such that X n1+1 + ... + Xn > O}, and so forth. Then the random variables Xi = Xl + ... + X m , X~ = X n1+1 + ... + X m , ••. are independent, identically distributed, positive, and EXi < 00 (see [6]). Letting y~ = Xi + ... + X~, note that P{first Y" ~ x is < x + n = P{first Y~ ~ x is < x + ~}, which completes the proof. We find it useful to transform this problem by defining for any strategy A, a random variable N(y), (3.5) N(y) = {smallest n such that W" + ... + WI ~ y} with N*(y) the analogous thing for A *. To prove (3.2) we need to prove 1 00 (3.6) lim [EN(y) - EN*(y)] = W L: (W - EWn), y_oo 1 and we use a result very close to Wald's identity. PROPOSITION 5. For any strategy A such that Sn -t 00 a.s. and any y 1 {N(Y) } 1 [N(Y) ] EN(y) = WE L: [W - E(Wk!Rk_ I)] + WE L: Wk • k=l k~l (3.7) PROOF. The above identity is derived in a very similar fashion to Doob's derivation ~ of Wald's identity. The difficult point is an integrability condition and we get around this by using, instead of the strategy A, a modification AJ which consists in using A for the first J plays and then switching to X*. The condition Sn -t 00 a.s. implies that none of the W k may take on the value - 00 and that N(y) is well defined. Let N J(Y) be the random variable analogous to N(y) under AJ and WY) to Wk. Define a sequence of random variabJes Zn by (3.8) This sequence is a martingale with EZn = O. By Wald's identity, ENJ(y) < 00 and it is seen that the conditions of the optional sampling theorem ([7], theorem 2.2-C3) are validated with the conclusion that EZNJ = O. Therefore --- ## Page 79 52 L. Breiman 70 FOURTH BERKELEY SYMPOSIUM: BREIMAN (3.9) WENJ = E [I: W] k-l The second term on the right satisfies (3.10) y ~ E [~ WYl] ~ Y + ex, k=l where ex = maXj (log OJ). Hence, if EN = 00, then limJ EN J = 00, so that E {I:fJ [W - E(WkIRk_ 1)]} = 00 and (3.7) is degenerately true. Now assume that EN < 00, and let J -? 00. The first term on the right in (3.9) converges to E{I:f [W - E(WkIRk- 1)]} monotonically. The random variables I:fJ WYl converge a.s. to I:f W k and are bounded below and above by y and y + ex so that the expectations conve~ge. It remains to show that limJ EN J = EN. Since (3.11) ENJ = ;;N~JI N dP + ;;N>JI NJ dP, we need to show that the extreme right term converges to zero. Let J n+J (3.12) UJ = I: W k , N(UJ) = {first n such that I: Wfl ~ Y - UJ} 1 J+l so that (3.13) ;;N>JI NJ dP = JP(N > J) + ;;N>J} N(UJ) dP. Since EN < 00, we have limJ JP{N > J} = O. We write the second term as E{E[N(UJ)IUJ]IN> J} P{N > J}. By Wald's identity, (3.14) On the other hand, since the most we can win at any play is ex, the inequality (3.15) N ~ y - UJ + J ex holds on the set {N > J}. Putting together the pieces, (3.16) r N(UJ) dP ~ ;, r (N - J) dP + ; peN > J). ) {N >J} ) {N>J} The right side converges to zero and the proposition is proven. If we subtract from (3.7) the analogous result for A* we get (3.17) EN(y) - EN*(y) 1 { N } 1 [N N*] = WE I: [W - E(WkIRk _ 1)] + WE I: W k - I: W; . k=l k=l k=! --- ## Page 80 Optimal Gambling Systems for Favorable Games 53 OPTIMAL GAMBLING SYSTEMS 71 This last result establishes inequality (3.3) of the theorem. As we let y ~ 00, then N(y) ~ 00 a.s. and we see that (3.18) lim E {t [W - E(WkIRk_I)]} = t (W - EWk ). y ..... '" k=l k=l By proposition 4, the distribution Ft of Lf" Wk - y converges, as y ~ 00, to some continuous distribution F* and we finish by proving that the distribution Fy of Lf W k - y also converges to F*. PROPOSITION 6. Let Y n, En be two sequences of random variables such that Y n ~ 00, Y n + En ~ 00 a.s. If Z is any random variable, if E = Supn ~l lEn\, and if we define (3.19) Hy(~) = P{first Y" ~ Z + y is < Z + y + ~}, (3.20) DvW = P {first Y n + En ~ Z + y is < Z + y + ~}, then for any u > 0, (3.21) F,,+u(~ - 2u) - P{E ~ u} ~ DuW ~ D.y(~) ~ H,,-'U(~ + 2u) + P{E ~ u}. PROOF. (3.22) D"W ~ P{first Y n + En ~ Z + y is < Z + y +~, E < u} + P{E ~ u} (3.23) ~ P{first Y n > Z + y - u is < Z+ y + ~ + u, E < u} + P{e ~ u} ~ Hu-u(~ + 2u) + P{E ~ u}. DvW ~ P{first Yn + En ~ Z + y i~ < Z + y +~, E < U} ~ P {first Y n ~ Z + y + u is < Z + y + ~ - U, E < u} ~ H11+1t(~ - 2u) - P{E ~ u} . PROPOSITION 7. Let XI, X 2, ••• be a sequence of independent identically dis- tributed nonlattice random variables, ° < EX I < 00, with Y n = X I + ... + X n· If Z is any random variable independent of Xl, X 2, ••• , G the limiting distribu- tion of proposition 4, and (3.24) F".zW = P{first Yn ~ Z + y is < Z + y +~}, then lim" F".zW = GW· PROOF. (3.25) F".zW = E[P{first Y n ~ Z + y is < Z + y + ~IZ}] = E[FII+zW], where F,l~) = P{first Yn ~ y is < y + n. But limy FlI+zW = GW a.s. which, together with the boundedness of F11+zW, establishes the result. We start putting things together with --- ## Page 81 54 L. Breiman 72 FOURTH BERKELEY SYMPOSIUM: BREIMAN PROPOSITION 8. Let:2:1 W k - :2:~ W: converge a.s. to an everywhere finite limit. If the W: are nonlattice, if Film is the distribution function for :2:f W .. - y, then limll Film = F*(~). PROOF. Fix m, let n n (3.26) ~m.n = :2: W k - :2: W;, Em = SUp IEm.nl, m m n and by assumption ~m ~ ° a.s. Now n (3.27) Film = P{first :2: Wk ~ Y is < y + ~} 1 = P{first (~ W; + Em.n) ~ Z", + y is < Zm + y + t}. If (3.28) n Hllm = P{first :2: W~ ~ Zm + y is < Zm + y + n, m then by proposition 6, for any u > 0, (3.29) HII+u(~ - 2u) - P{~m ~ u} ~ Film ~ HlI-'U(~ + 2u) + P{Em ~ tt}. Letting y ~ 00 and applying proposition 7, (3.30) F*(~ - 2u) - P{Em ~ u} ~ lim Film ~ lim Film ~ F*(~ + 2u) + P{Em ~ u}. V II Taking first m ~ 00 and then u ~ ° we get (3.31) lim Film = lim Film = F*m· -11- II To finish the proof, we invoke theorems 2 and 3 of section 4. The content we use is that if :2:1 W k - :2:~ W: does not converge a.s. to an everywhere finite limit, then :2:i [W - E(WkIRk _ 1)] = +00 on a set of positive probability. Therefore, if the conditions of propositions 5 and 8 are not validated, then by (3.17) both sides of (3.2) are infinite. Thus the theorem is proved . .,;._ .f ~ la. Asymptotic magnitude problem The main results of this section can be stated roughly as: asymptotically, S~ is as large as the Sn provided by any strategy A, and if A is not asymptotically close to 11.*, then S~ is infinitely larger than Sn. The results are valid whether or not the games are favorable. THEOREM 2. Let A be any strategy leading to the fortune Sn after n plays. Then limn Sn/S: exists a.s. and E(limn Sn/S~) ~ 1. For the statement of theorem 3 we need DEFINITION. A is a nonterminating strategy if there are no values of Xn stlch that :2:tEAI >-.In)Oj = 0, for any n. --- ## Page 82 Optimal Gambling Systems for Favorable Games 55 OPTIMAL GAMBLING SYSTEMS 73 THEOREM 3. If A is a nonterminating strategy, then almost surely . ~ L [W - E(Wk [Rk_ I )] = 00 ¢=> lim S" = 00. 1 n n (4.1) PROOFS. We present the theorems together as their proofs are similar and hinge on the martingale theorems. For every n (4.2) If we prove that E(VnjV~IRn_l) ~ 1 a.s., then SnjS~ IS a decreasing semi- martingale with limn Snj S~ existing a.s. and (4.3) E 1· Sn < E So - 1 Im S·= S·- · n n 0 By the definition of A*, for every E > 0, (4.4) E{log [(1 - E) V: + EVn] - log V:[Rn_l} ~ o. Manipulating gives (4.5) 1 [( E V n)1 ] 1 1 - E log 1 + -- -. Rn_ 1 ~ - log --. E 1 - E Vn E 1 - E By Fatou's lemma, as E --t 0 (4.6) E (~;IRn_l) = E [lim (; log 1 + 1 ~ E ~;)IRn-1 ] :$; lim.! log _1_ = 1. --E 1-E Theorem 3 resembles a martingale theorem given by Doob ([6], pp. 323-324), but integrability conditions get in our way and force some deviousness. Fix a number M > 0 and take A to be the event {W - E(W nIRn - 1) !?; M Lo.}. If p = min; pi, then E(W: - W nIRn- l ) !?; M implies P {W~ - W n !?; MIRn- 1} !?; p. By the conditional version of the Borel-Cantelli lemma ([7], p. 324), the set on which Li P{W~ - Wn !?; M[Rn_l} = 00 and the set {W~ - Wn !?; M i.o.} are a.s. the same. Therefore, a.s. on A, we have W~ - Wn !?; M i.o. and log (S~jSn) = L~ (W~ - Wk ) cannot converge. We conclude that both sides of (4.1) diverge a.s. on A. Starting with a strategy A, define an amended strategy AM by: if W- E(W n[Rn- l ) < M, use A on the nth play, otherwise use A * on the nth play. The random variables (4.7) form a martingale sequence with (4.8) Un - Un- I = W~ - W~Ml - [W - E(Wi.MlIRn_l)]. --- ## Page 83 56 L. Breiman 74 FOURTH BERKELEY SYMPOSIUM: BREIMAN For AM, we have E(W: - W~M)IRn_l) < 111, leading to the inequalities, (4.9) sup (W: - W~M)) :s;; lVI, U~ _ U,,-1 :s;; lVI. _ p u _ p On the other side, if (4.10) a = min log ( I: Xjoi) , /3 = max log 0h j iEAi j then Un - U n-1 ~ a - /3 - M. These bounds allow the use of a known martin- gale theorem ([7], pp. 319-320) to conclude that limn Un exists a.s. whenever one of lim U" < 00, lim Un > - 00 is satisfied. This implies the statement (4.11) However, on the complement of the set A the convergence or divergence of the above expressions involves the convergence or divergence of the corresponding quantities in (4.1) which proves the theorem. COROLLARY l. If for some strategy A, we have I:i [W - E(WkIRk _ 1) ] = 00 with probability l' > 0, then for every E > 0, there is a strategy A such that with probability at least l' - E, lim SnIS" = 0 and except for a set of probability at most E, lim SnlSn ~ l. PROOF. Let E be the set on which lim SnIS~ = 0, with P{E} = 1'. For any € > 0, for N sufficiently large, there is a set EN, measurable with respect to the field generated by RN such that P{EN DoE} < E, where Do denotes the symmetric set difference. Define A as follows: if n < N, use A, if Rn , with n ~ N, is such that the first N outcomes (Xl, ... ,XN) is not in EN, use A, otherwise use A*. On EN, we have I:i [W - E(WkIRk-l)] < 00, hence limSnlS: > 0 so that lim SnIS" = 0 on EN n E. Further, P{EN n E} ~ P{E} - E = r - E. On the complement of EN, we have Sn = Sn, leading to lim SnlSn ~ 1, except for a set with probability at most E. 5. Problems with finite goals in coin tossing In this section we consider first the problem: fix an integer n > 0, and two numbers y > x > 0, find a strategy which maximizes P{Sn ~ ylSo = x}. In this situation, then, only n plays of the game are allowed and we wish to maxi- mize the probability of exceeding a certain return. We will also be interested in what happens as n, y become large. By changing the unit of money, note that (5.1) sup P{S" ~ ylSo = x} = sup P {Sn ~ 11So = ~} where the supremum is over all strategies. Thus, the problem reduces to the unit interval, and we may evidently translate back to the general case if we find an optimum strategy in the reduced case. Define, for ~ ~ 0, n ~ 1, --- ## Page 84 Optimal Gambling Systems for Favorable Games (5.2) and (5.3) OPTIMAL GAMBLING SYSTEMS { SUP P{Sn ~ 11So = ~}, cf>,,(~) = 1, {a, cf>oW = 1 , ~ < 1, ~ ~ 1. ~ < 1, ~ ~ 1, In addition cf>"W satisfies (5.4) cf>nm = supE[P{Sn ~ 11S1, So} ISo = ~] ~ sup E[cf>n-l(SI) ISo = ~] ' ~ sup [Pcf>n-l(~ + z) + qcf>n-l(~ - z)]' O;>z ~~ To find cf>nW and an optimal strategy, we define functions ¢n(O by (5.5) ¢om = cf>om, ¢nm = sup [p¢n-l(~ + z) + q¢n-1(~ - Z») o;>z ;:>~ 57 75 having the property cf>nm ~ ¢nm, for all n, ~. If we can find a strategy A such that under A we have ¢n(O = P{Sn ~ 11So = n, then, evidently, A is optimum, and ¢n = cf>n. But, if for every n ~ 1, and ~ there is a ZnW, with ° ~ ZnW ~ ~, such that (5.6) then we assert that the optimum strategy is A defined as: if there are m plays left and we have fortune ~, bet the amount ZmW, Because, suppose that under A, for n = 0,1, ... ,m we have ¢nW = P{Sn ~ 11So = n, then (5.7) P{Sm+l ~ 11So = n = E[P{Sm+l ~ 11S1, So} ISo = ~] = E[t,bm(S1)ISo = ~] = ¢m+lW, Hence, we need only solve recursively the functional equation (5.5) and then look for solutions of (5.6) in order to find an optimal strategy. We will not go through the complicated but straightforward computation of ¢"W. It can be described by dividing the unit interval into 2" equal intervals h '" , 12n such that h = [k/2 n , (k + 1)/2"]. In tossing a coin with P{H} = p, rank the prob- abilities of the 2" outcomes of n tosses in descending order PI ~ P2 ~ .•. ~ P2-, that is, P l = p", P2- = q". Then, as shown in figure 1, (5.8) cf> .. (O = L Ph j 1/2, then lim" cf>n(O = 1, with ~ > 0; and in the limiting case ]J = 1/2, then limn cf>n(~) = ~, with ~ ~ 1, in agreement with the Dubins-Savage result [2]. . There are many different optimum strategies, and we describe the one which seems simplest. Divide the unit interval into n + 1 subintervals Ibn), '" , nn), such that the length of Ikn) is 2-n(~) where the (~) are binomial coefficients. On --- ## Page 85 58 L. Breiman 76 FOURTH BERKELEY SYMPOSIUM: BREIMAN o FIGURE 1 Graph of 4>nW for the case n = 3. each lin) as base, erect a 45°-45° isosceles triangle. Then the graph of Zn+lW is formed by the sides of these triangles, as shown in figure 2. Roughly, this strategy calls for a preliminary "jockeying for position," with the preferred posi- tions with m plays remaining being the midpoints of the intervals 11m). Notice that the endpoints of the intervals {lin)} form the midpoints of the intervals {11n- 1)}. So that if with n plays remaining we are at a midpoint of {lin)}, then at all remaining plays we will be at midpoints of the appropriate system of intervals. Very interestingly, this strategy is independent of the values of p so o I "4 I "2 FIGURE 2 Graph of Zn+lW for the case n = 3. 3 "4 --- ## Page 86 Optimal Gambling Systems for Favorable Games 59 OPTIMAL GAMBLING SYSTEMS 77 long as p > 1/2. The strategy A * in this case is: bet a fraction p - q of our fortune at every play. Let cf>:W = P{S: ~ 11So = n. In light of the above remark, the following result is not without gratification. THEOREM 4. limn sup~ [c:PnW - c:P:WJ = 0. PROOF. The proof is somewhat tedious, using the central limit theorem and tail estimates. However, some interesting properties of c:P"W will be discovered along the way. Let P{kll/2} be the probability of k or fewer tails in tossing a fair coin n times, P{klp} the probability of k or fewer tails in n tosses of a coin with P{H} = p. If ~ = P{kI1/2} + 2-", note that cjJ,,(~-) = P{klp}, Let u = Vpq, by the central limit theorem, if ~t.n = P{qn + tuV;ll/2} + 2-", then (5.9) uniformly in t. Thus, if we establish that (5.10) uniformly for t in any bounded interval, then by the monotonicity of c:PnW, c:P:W, the theorem will follow. By definition, (5.11) c:P:(~) = P{Wi + ... + W: ~ 0IWo = log n = P {Wt + ," . + W: ~ -log~}, where the WI are independent, and identically distributed with probabilities P{Wt = log 2p} = p and P{W: = log 2q} = q. Again using the central limit theorem, the problem reduces to showing that (5.12) lim log ~n.t + nEWt = t n Vnu(Wt) uniformly in any bounded interval. By a theorem on tail estimates [8J, if Xl, X 2, • •• are independent random variables with P{Xk = 1} = 1/2 and P{Xk = O} = 1/2, then (5.13) log P{Xl + ... + Xn ~ na} = n6(a) + /L(n, a) log n, where /L(n, a) is bounded for all n, with 1/2 + 0 ~ a ~ 1 - a, and 6(a) = -a log (2a) - (1 - a) log [2(1 - a)]. Now (5.14) log ~n.t = log [P{Xl + ... + Xn ~ np - tuVn) + 2-n] so that the appropriate a = p - tu/Vn with (5. 15) 6(a) = 6(p) - ~ log 2 + 0 (1). Vn p n Since 6(p) > -log 2, we may ignore the 2-" term and estimate --- ## Page 87 60 78 (5.16) But 8(p) (5.17) FOURTH BERKELEY SYMPOSIUM: BREIMAN log ~n.t = nO(p) - i 0 where Sn is the player's capital after n trials. PART 1. FAVORABLE GAMES 1. BLACKJACK Blackjack, or twenty-one, is a card game played throughout the world. The casinos in Nevada currently realize an annual net profit of roughly eighty million dollars from the game. TaKing a price/earnings ratio of 15 as typical for present day common stocks, the Nevada blackjack operation might be compared to a $ 1.2 billion corpo- ration. To begin the game a dealer randomly shuffles n decks of cards and players place their bets. (The value of n does not materially affect our discussion. It generally is 1, 2, or 4, and we shall use 1 throughout.) There are a maximum and a minimum allowed bet. The minimum insures a positive probability of eventual ruin for the player who continues to bet. The maximum protects the casino from large adverse fluctuations and in particular prevents the game from being beaten by a martingale (e.g. doubling up), especially one starting with a massive bet. In fact, without a maximum, a casino 1 The research for this paper was supported in part by the Air Force Office of Scientific Research through Grant AF-AFOSR 1113-66. The paper is intended in large part to be an exposition for the general mathematical reader with some probability background, rather than for the expert. --- ## Page 89 62 E. 0. Thorp 274 with finite resources can in general be ruined a.s. (almost surely) by a player with infinite resources. The player simply bets enough at each trial so that the casino is ruined if it loses that trial. A practical way for the player to have infinite resources would be for the casino to extend unlimited credit for the finite time it might be needed. The players' hands are dealt after they have placed their bets. Each player then uses skill in his choice of a strategy for improving his hand. Finally, the dealer plays out his hand according to a fixed strategy which does not allow skill, and bets are settled. In the case where play begins from one complete randomly shuffled deck, an approxi- mate best strategy (i.e. one giving greatest expected return) was first given in 1956 [I]. Though the rules of blackjack vary slightly, the player following [I] typicaUy has the tiny edge of + .10 %. (The pessimistic figure of - .62 % cited in [I] was erroneous and may have discouraged the authors from further analysis.) These mathematical results were in sharp contrast to the earlier and very different intuitive strategies generally recommended by card experts, and the associated player disadvantage of two or three per cent. We call the best strategy against a complete deck the basic strategy. Determined in 1965, it is almost identical with the strategy in [I] and it gives the player an edge of + 0.13 % [22]. If the game were always dealt from a complete shuffled deck, we would have repeated independent trials. But for compelling practical reasons, the deck is not generally reshuffled after each round of play. Thus as successive rounds are played from a given deck, we have sampling without replacement and dependent trials. It is necessary to show the players most or all of the cards used on a given rOllnd of play before they place their bets for the next round. They can then use this knowledge of which cards have been played both to sharpen their strategy, and to more precisely estimate their edge. (The strategies for various card counting procedures, and their expectations, were determined directly from probability theory with the aid of com- puters. The results were reverified by independent Monte Carlo calculations.) For a given card counting procedure and associated strategy, there is a probability distribution Fe describing the player's expectation on the next hand, provided c cards have been counted. As c increases, Fe spreads out. (This is a theorem. whose proof resembles that for the similar theorem in Baccarat, mentioned in [24], page 316). This spread in Fe can be exploited by placing large bets when the expectation is posi- tive and small bets when it is negative. Part II indicates how best to do this. If the basic strategy is always used, E(Fc) = + 0.13 %,just as from a complete deck. But if an improved strategy, based on the card count, is used, E(Fc) increases as c increases, approaching values of one to two per cent or more. Ties, in which no money is won or lost, may be discounted. They occur about one tenth of the time. Most, but not all, of the other outcomes result in the player either winning or losing an amount equal to his original bet. The conditional means E(Fe I Fe-d; k = 1,2, ... , c, of the successive Fe are non- decreasing. The Fe are dependent; in particular when a deck "goes good", it tends to stay good. Probabilistic summary To a good first approximation, Blackjack is a coin toss where the probability p 01 success is selected independently on each trial from a known distribution F (which is a suitably weighted average of the Fe) and announced before each trial. A more accurate model considers that the p's are dependent in short consecutive groups, corresponding to successive rounds of play from the same deck. Another --- ## Page 90 Optimal Gambling Systems f or Favorable Games 63 275 more accurate observation is that insurance, naturals, doubling down, and pair splitting. each win or lose an amount different from the amount initially bet. We do not consider this more accurate model in part II because the improvement in results is slight and the increase in complexity is considerable. 2. BACCARAT The terms Baccarat and Chemin de Fer are used, sometimes interchangeably, to refer to several closely related variants of what is essentially one card game. The game is currently popular in England and France, where it is sometimes played for un- limited stakes. It is also played in Nevada. The game-theoretic aspects of Baccarat have been discussed in [11,14]. The Nevada game is analyzed in [24] which includes results of extensive computer calculations. The studies of Baccarat show that the available bets generally offer an expectation on the order of -1 %. The use of mixed strategies, to the very limited extent that this is possible in some variants of the game, has but slight effect on the expectation. Despite the resemblances between Baccarat and Blackjack, the favorable situations detected by perfect card counting methods are not sufficient to make the game favorable. Thus Baccarat is not in general a favorable game. The game as played in Nevada sometimes permits certain side bets. The minimum on the side bets was observed to be $ 5 to $ 20 and the maximum was $ 200. The bets either won nine times the amount bet or lost the amount bet. The game was played with eight well shuffled decks dealt from a dealing box, or shoe. Using the card counting techniques described in [24], the side bets were favorable about 20 % of the time. When they were favorable, the expectations ranged as high as + 100 %. The expectation initially was about - 5 % and as the number c of cards seen increased, the distribution Fe of expectations spread out ([24], page 316) as in Blackjack. In practice the betting methods discussed in part II, in which the bet increased with the expectation, doubled initial capital in twenty hours. Unlike the Blackjack player, the Baccarat side bettor has no strategic decisions to make so E(Fe) does not vary as c changes. When the expectation of the side bet falls below a certain value, it is best to make a "waiting" bet on one of the main bets. There are either two or four side bets, similar and dependent. How to apportion funds on the side bets is complicated by the fact that there are several of them. These com- plexities are treated in [24]. Probabilistic summary When only one side bet is available, the pay-off for a one unit side bet is either + 9 or - 1. If p is the probability of success, we may suppose that p is selected indepen- dently from a known distribution F and announced before each trial. When several side bets are available, the situation is more complex. It illustrates the general setting of [3], page 65. As in Blackjack, a more accurate model considers that the p's are dependent in consecutive groups, corresponding to successive rounds of play from the same en- semble of (eight) decks. It also considers the effect of waiting bets. The situation here is more complex than in Blackjack. First, it is important to exploit any opportunities of making simultaneous bets on two or more favorable side bet situations. Second, the pay-off is never one to one. --- ## Page 91 64 E. 0. Thorp 276 3. ROULETTE Roulette has long been the prototype of unbeatable gambling games. It is normally regarded as a repeated independent trials process which generates at each trial precisely one from a set of random numbers. In Monte Carlo these numbers are 0, 1, 2, .. . , 36. Players may wager on particular conventional subsets of random num- bers (e.g. the first dozen, even, {27}, etc.), winning if the number which comes up is a number of the chosen subset. A player may wager on several subsets simultaneously, and each bet is settled without reference to the others. The expectation of each bet is negative (in Nevada generally - 5.26 %, except for one worse bet, and in Monte Carlo - 1.35 %.) Thus it has been long known that the classical laws of large numbers insure that the player will with probability one fall behind and stay behind, tending to lose in the long run at a rate close to the expectation of his bets. Despite this, Henri Poincare and Karl Pearson each examined roulette. Poincare ([20], pages 69-70, pages 76-77; [21], pages 201-203; [9], pages 61-62) supposes that the uncertainty in initial conditions (e.g. the angular position and velocity of the ball and of the rotor at a given time) leads to a continuous probability density / in the ball's final position. He shows by an argument involving continuity only that if/has sufficient spread, then the finitely many final ball positions are to very high approxi- mation equally likely. Karl Pearson statistically analyzed certain published roulette data and found very significant patterns. In particular Pearson says, "If Monte Carlo roulette had gone on since the beginning of geological time on this earth, we should not have expected such an occurrence as this fortnight's play to have occurred once on the supposition that the game is one of chance." And again, "To sum up, then: Monte Carlo roulette ... is ... the most prodigious miracle of the nineteenth century." I've been told that it was later learned that the roulette data was supplied for a newspaper by journalists hired to sit at the wheel and record outcomes. The journalists instead simply made up numbers and submitted them. It was their personal bias that Pearson detected as statistically significant. It brings to mind David Hume's essay 0./ Miracles: "No testimony is sufficient to establish a miracle, unless the testimony be of such a kind that its falsehood would be more miraculous than the fact that it endeavors to establish .... it is nothing strange .. . that men should lie in all ages." Poincare assumed a mechanically perfect roulette wheel. However, wheels some- times have considerable bias due to mechanical imperfections. Some observed in- stances and their exploitation are discussed in detail in [25]. In Blackjack and Baccarat, we used the following fundamental principle: The payoff random variables, hence the favorability of a game to an optimal player, depend on the information set used to determine the optimal strategy. For instance, if used cards are ignored in Blackjack, then we simply have Bernoulli trials with p = + 0.13 %. However, as more card counting information is employed, the distribution of p spreads out (has more structure), its expected value increases, and it can be more effectively exploited. The roulette system we now describe illustrates the use of an enlarged information set. Play at roulette begins when the croupier launches the ball on a circular track which inclines towards the center so the ball will fall into the center when it slows down sufficiently. The center contains a rotor with a circle of congruent numbered pockets rotating in the opposite direction to the ball. The ball eventually slows and falls from its track on the stator, spiralling into the moving rotor and eventually coming to rest --- ## Page 92 Optimal Gambling Systems f or Favorable Games 65 277 in a numbered pocket, the "winning number". Bets may generally be placed until the ball leaves its track. This is crucial for what follows. A collaborator and I tried to use the mechanical perfection of the wheel - the very perfection needed to eliminate the bias method - to gain positive expectation. Our basic idea was to determine an initial position and velocity for the ball and rotor. We then hoped to predict the final position of the ball in much the same way that a planet's later position around the sun is predicted from initial conditions, hence the nickname "the Newtonian method". The Newtonian method occurred to me in 1957, and by 1961 the work described here had been completed. Although the wheel of fortune device was mentioned in LIFE Magazine, March 27, 1964, pp. 80-91, we pointedly did not mention the roulette work there. However, we do so in [22], page 181-182. The Newtonian method is also mentioned in the significant book by R. A. Epstein, The Theory of Gambling and Statistical Logic, Academic Press, pp. 135-136, (1967). The stator has metal deflectors placed to scatter the ball when it spirals down and the pockets are separated by vertical dividers ("frets") which also introduce scattering. These scatterings were measured and found to be far from sufficient to frustrate the Newtonian approach. However, there were additional sources of randomness which did frustrate this approach. (We never satisfactorily identified these causes and can only speculate - perhaps the causes included minute imperfections in track or ball or high sensitivity of the coefficient of friction to dirt or atmospheric humidity.) We were led to a variation we called the quantum method. If a roulette wheel is tilted slightly the ball will not fall from a sector of the track on the "high" side. The effect is strong with a tilt of just 0.2°, which creates a forbidden zone of a quarter to a third of the wheel. The non-linear differential equation governing the ball's motion on the track is the equation for a pendulum which at first swings completely around its pivot, but is gradually slowed by air resistance. (It is illuminating to sketch the orbits of the equation, as indicated in [4], page 402, problem 3.) The experimental orbits of angle versus time could be plotted easily in the laboratory by taking a movie of the system in motion, along with a large electric clock whose hand swept out one revolution per second! The existence of a forbidden zone partially quantizes the angle at which the ball can exit, and hence quantizes the final angular position of the ball on the rotor. The physics involved suggests that the quantization is in fact very sharp: Suppose the ball is going to exit beyond the low point of the tilted wheel. Then it must have been moving faster than a ball exiting at the low point, so it reaches its destination sooner. But it has also gone farther, and the two effects tend to cancel. They in fact cancel very well. A similar argument shows that balls which exit before the low point have been slower, hence later, offsetting the fact they have not gone as far. Observation verifies the conclusions of this heuristic argument. The sharp quantization of ball final position, as a function of initial conditions, makes remarkably accurate prediction possible. Using algorithms, it was possible by eye judgements alone to estimate the ball's final position three or four revolutions before exit (perhaps five to seven seconds before exit, which was ample time in which to bet) well enough to have a + 15% expectation on each of the five most favored numbers. A cigarette pack sized tran- sistorized computer which we designed and built was able to predict up to eight revo- lutions in advance. The expectation in tests was + 44 %. One third of the Nevada roulette wheels which we observed had the desired tilt of at least 0.2°. The input to the computer consisted of four push-button hits: two when --- ## Page 93 66 E. 0. Thorp 278 the 0 of the rotor crossed a fiducial mark during successive revolutions and two when the ball crossed a fiducial mark on successive revolutions. The decay constants of ball and rotor, approximately constant over the class of wheels observed, had been deter- mined earlier by simple observations. The ultimate weakness of the system was that the house could foil it by forbidding bets after the ball had been launched. Probabilistic summary Roulette on a slightly tilted wheel is repeated independent trials. At each trial the player may wager on one or more subsets of the finitely many elementary outcomes. A wager on a subset wins if and only if it contains the elementary outcome that occurs. There are subsets with expectations of 44 %. Our procedure in practice was to bet on one of eight neighborhoods of five numbers. Thus the payoff for a bet of .2 units on each of five numbers was either -lor + 6.2. The expectation of + 44 % corresponds to a probability of success of .2. We remark that our knowledge of p increases with the sample size. 4. THE WHEEL OF FORTUNE The wheel of fortune, featured in many Nevada casinos, is a six foot vertical wheel with horizontal equally spaced pegs in its rim. As the wheel spins, a rubber flapper strikes successive pegs, slowing the wheel. There are generally 48 to 54 spaces between the pegs, numbered with Is, 2s, 5s, lOs, 20s, and two distinct 40s. A player betting a unit on one of these outcomes is paid that number of units if his outcome occurs. The wheel behaves to good approximation as though a constant increment of energy is lost each time a peg passes the flapper. Thus e, the total angle of rotation, is propor- tional to the energy E, which equals Iw 2, where I is the moment of inertia and w is the angular velocity of the wheel. In practice, a transistor timing device of match box size (a "spinoff" from the roulette technology) produced a faint click a chosen time after a push-button was hit. The button was hit when a specified 40 passed the flapper. The timer was set so the click was approximately when the second 40 reached the flapper. Ifit clicked after the second 40 reached the flapper, the wheel was "fast" and would go farther than average before stopping. If it clicked before the second 40 reached the flapper, the wheel was "slow". For a given timer setting, a table was constructed empirically, giving the approxi- mate final position of the wheel as a function of the number of spaces the second 40 was fast or slow when the click was heard. In practice one could determine with certainty which of the two 40s could not occur. Thus, one could always bet on the "right" 40. On a wheel observed in the Riviera Hotel there were 50 numbers, including 22 ones, 14 twos, 7 fives, 3 tens, 2 twenties and 2 forties. Betting on the "right" 40 would win on average 80 units in 50 trials and lose 48, for an expectation of 32/50 or 64 %. Probabilistic summary Ignoring obvious refinements, we have repeated independent trials with probability p = 1/25 of success at each trial, a payoff for a I unit bet of - 1 or + 40, and an expectation of + 64 %. --- ## Page 94 Optimal Gambling Systems for Favorable Games 67 279 5. THE STOCK MARKET The stock market is a natural economic object for mathematical analysis because vast quantities of precise historical data are available in numerical form. There have been many attempts to mathematically predict future price behavior, using as a basis various subsets of the available information. Most notable are the attempts to predict future prices from past price behavior. These attempts have caused the view to be widespread in academic circles that, to first order, common stock prices are a random walk and changes in common stock prices are log normally distributed with a certain mean and standard deviation [5). Practitioners hotly contest this view. Part of the dispute is caused by practitioners who are unwilling or unable to test their claims scientifically and part of it is due to the success of a few practitioners who use much more information than past price history alone. A recent study suggests strongly that "relative strength" in a price series is continued and, consequently, that past prices do have some value in predicting future prices [16]. Whether or not we can predict the future course of stock prices!, there are invest- ments in combinations of securities which can yield high expected return [23]. These investments involve convertible securities. A convertible security is one which, in some cases with the addition of money, is exchangeable (per share) for a certain num- ber of shares of another security. Convertible securities include convertible bonds, convertible preferreds, stock options, stock rights, and warrants. There are several billion dollars worth of convertibles listed on the New York and American Stock Exchanges. The analysis of other convertibles follows from the analysis of the common stock purchase warrant. We therefore restrict ourselves to these in our discussion, and shall refer to them simply as warrants. A warrant is the right or option to buy a certain number of shares of common for a certain price, until a certain expiration date (warrants which do not expire are called perpetual). The terms ordinarily read: A warrants plus E dollars buy C shares until D date. To avoid normalization problems, we suppose A = C = I. Then E is the "exercise price" of the warrant. The prices of warrant and common are related and it is this which allows successful investments. One observes: (I) The price Wof the warrant should increase as the price S of the stock increases. (2) If W + E < S, warrants can be bought and common sold short, simultaneously. The warrants are then converted to common which is delivered against the short position. Neglecting commissions, a profit of S - W - E per warrant results. The purchase of warrants tends to increase Wand the sale of common tends to decrease S, until W + E > S. Thus W > S - E normally holds. (3) The common has advantages over the warrant such as possible dividends, or voting rights, hence we also normally expect W < S. Thus for practical purposes points (S, W) representing (nearly) simultaneous prices of a common stock and its warrant are confined to the part of the positive quadrant between the lines W = Sand W = S - E. The prices Wand S at a future time are random variables but they are related. As E(S) increases we would expect, and past history verifies ([12, 23]), that E( W) tends to increase. In fact the points (S, W) tend to lie on certain curves which depend 1 The great mathematician Karl Friedrich Gauss was successful in the market but we have little knowledge of his methods. On a basic salary of 1000 thalers per year he left an estate in cash plus securities of 170,857 thalers ([7], page 237). --- ## Page 95 68 E. 0. Thorp 280 on several variables, most notably the time remaining until expiration of the warrant. Thus, although we may not know the price S of the common, or the price W of the warrant at a given future time, we do know that ( W, S) is near one of these curves. The family of curves qualitatively resembles the family W = (SZ + £Z)l/Z - E, where z = 1.3 + 5.3/T and T is the number of months remaining until expiration. A historical study of expiring warrants (from here on we limit ourselves for con- venience to warrants traded on the American Stock Exchange) suggests that during the last two years or so before expiration they tend to trade at prices which are much too high. For instance the average loss from buying each of a certain 11 listed war- rants 18 months before expiration and holding until 2 months before expiration was 46.0 %, an annual rate of 34.5 % ([23], page 37). Thus selling warrants short seems to yield high expectation. However, it also happens to result in occasional large losses which, by the criterion of Part II, are extremely undesirable despite the high overall expectation. We can sharply reduce this high variance and yet retain a high expecta- tion by using the so-called warrant hedge. The technique is to simultaneously sell short overpriced warrants and buy common in a fixed ratio (generally from one to three warrants will be shorted for each share of common bought). The position is held until just before expiration of the warrant (at which time the warrant sells at a "correct" price) and then it is liquidated. Here is the rationale. We are mixing two investments with positive annual expecta- tions of say 34.5 % for the warrants and 10 % for the common, resulting in an invest- ment whose overall expectation must therefore be somewhere between these figures. (We suggest 10% for the common because this approximates the observed mean rate of return from common stocks during this century due to price appreciation plus dividends.) Buying the common leads to a gain when the common rises and a loss when it falls whereas shorting warrants leads to a gain when the common falls and leads to a loss only if the common rises substantially. Thus the risks tend to cancel out. In fact, the hedge generally yields a profit upon expiration of the warrant, for a wide range of prices of the common. If we make assumptions about the probability distribution of the price of the com- mon at expiration of the warrant, we get more precise information about the random variable representing the payoff from the hedge. Let the probability measure P with support [0, 00) describe the distribution of the stock price Slat expiration. Then 00 E ( Sf) = J x" d P ( x ) . o Let So be the present price and let E be the exercise price. Assume that P( S I > So + t) > P(SI < So - t) for each t > 0, i.e. for any t, the chance of a price rise of at least t is no less than the chance of a price drop of at least t. This is a very weak assumption. Note that it does imply E( S I) > So· Just before expiration WI == 0 if SI < E and WI == SI - E if SI > E. Thus the final gain from shorting a warrant at Wo is Wo if S I < E and is Wo - Sf + E if S I > E. The gain from buying a share of common at So is, of course, Sf - So· Hence if we assume one share of common is purchased at .5E and one warrant is shorted at .2E, the final gain G I is S I - .3E if S I < E and. 7 E jf Sf > E. A standard measure-theoretic argument yields E( G I) > .2E. Using 100% margin, the percent profit is E( G I) / .7 E > 28 %. With 100 % margin on the warrants and 70 % margin on the common, it is at least .2E / .55E > 36 %. With 70 % margin on each, it is at least .2 / .49 > 40 %, an annual rate of more than 20 % jf the warrant expires in two years. --- ## Page 96 Optimal Gambling Systems for Favorable Games 69 281 It is interesting to calculate E( G J ) by assuming that S J is log normally distributed. Letting S J = S J I E, we thus assume that S J has the density function f ( x) = (x cr .J2"it) -1 exp [ - (In x - Jl)2 I 2 cr2 ], where Jl and cr are parameters de- pending on the stock. We note E (s J) = exp (Jl + cr2 /2). If t is the time in months remaining until expiration (which is when S J is realized), then we assume (J. = log So + mt and 0"2 = a2t, where So = So E is the present stock price and m and a are constants depending on the stock. Thus E(s J) = So exp [em + a2/2)t]. A mean increase of 10% per year is approximated by setting 12(m + a2/2) = .1. If we estimate a2 from past price changes we can solve for m. Letting w, = WIlE, where W, is the final warrant price, a calculation yields E( w,) = E(s,) N«(J./O" + 0") - N«(J./O"), where N is the normal distribution. (Compare the equivalent expression from pp. 464-466 of [5].) Now suppose that So = .5, that a has the realistic value of .Iso or .05, and that 12(m + a2/2) = .1, whence m = .085/12. Then for t = 24 we have cr = ).06 = .245 and Jl = log .5 + .17 = -.523. This yields E(w,) = .0015 and E(s,) = .61, whence E(G,) = .20 + .11 = .31. Thus the profit, with 70% margin on both warrant and common, is .31/.49 or 63.3% and the annual rate is 31.6 %. Note that the warrant is virtually worthless! Instead of selling one warrant short and buying one share of common, we can sell short w warrants and buy s shares of common. Neglecting commissions, which we do throughout for simplicity, the gain G at any point (S, W) is s(S - So) -we W - Wo). Thus the line G = 0, the zero profit line, is the line through (Wo, So) with positive slope slw = 11m. We call m the mix. Points below the zero profit line represent gain and points above it represent loss. If 1 < m < co, the zero profit line intersects the S axis at So - m Wo and it intersects the line W = S - E at S = [me Wo + E) - So] / (m - 1); W = (m Wo + E - So) I (m - 1). When the warrant expires the hedge position will yield a profit if S, is between the S values of the two intersections and it will yield a loss if S, is beyond the inter- sections. For instance, if So = .5E and Wo = .2E, the choice m = 2 insures a final profit if .IE < s, < 1.9E. Such safety is characteristic of the warrant hedge. The final gain G, is s(S,- So) + wWo if S, < E and it is sCSI - So) + w( Wo + E - S,) if S, > E. Thus as a function of S, it is an inverted "V" with apex above S, = E. With 100 % margin, the initial investment is s So + w Wo so the gain per unit invested is g, = G ,/(s So + w Wo). With margin of ot on the common and f3 on the warrants it is g, = G,/(ot s So + f3 w Wo). We have assumed so far that a hedge position is held unchanged until expiration, then closed out. This static or "desert island" strategy is not optimal. In practice intermediate decisions in the spirit of dynamic programming lead to considerably superior dynamic strategies. The methods, technical details, and probabilistic sum- mary are more complex so we defer the details for possible subsequent publication. Probabilistic summary The warrant hedge may offer high expectation with low risk. The gain per unit g,isg, = [(Sf + So) + m Wo] / (ot So + f3m Wo) when SJ < Eandg, = [(S,- So) + m( Wo + E - Sf)] / (ot So + f3m Wo) if SJ > E. The gain per unit depends only on the random variable S ,. This has an unknown distribution but it can be estimated. The other quantities are constants depending on circumstances. --- ## Page 97 70 E. 0. Thorp 282 PART II. A MATHEMATICAL THEORY FOR COMMITTING RESOURCES IN FAVORABLE GAMES 1. INTRODUCTION: COIN TOSSING Suppose we are confronted with an infinitely rich adversary who will match all bets we make on repeated independent trials of a biased coin (whose two outcomes are "heads" and "tails".) Assume that we have finite capital Xo, that we bet Bi on the outcome of the ith trial, where Xi is our capital after the ith trial, and that the proba- bility of heads is p, where -1 < p < I. (This is approximately the situation in Nevada blackjack, except that the game is played with a "mix" of biased coins.) Our problem is to decide how much to bet at each trial. A classic criterion is to choose Bi so that our expected gain E(Xi - Xi-I) is maximized at each trial, which is equivalent to maximizing E(Xn) for all n. Define Tj by Tj = I if the jth trial results in success and Tj = -1 if the jth trial n results in failure. Then X j = X j_1 + TjBj, j = 1,2, ... , and Xn = Xo + l: TjBj. j=1 We assume that Tj , Xj' and Bj are all random variables on a suitable sample space Q. If, for example, Bj is a function of Xo, XI' . . . , Xj _ 1 as it is in the common gambling systems, e.g. Martingale, Labouchere, etc. (note that Bk = I Xk - Xk - 1 I so we need not add the Bk , k = I, ... ,j - I), then we see by induction that Bj is a function of Xo, T I , T2 , •• • Hence the underlying sample space can be taken to be the space of all sequences of successes and failures, with the usual product measure. Suppose, more generally, that the player determined Bj by examining Xo, ... , Xj _ I' and then "consulting" a chance device, e.g., a near-by roulette wheel. Then the sample space consisting of an infinite product of spaces, each of them a joint outcome of the roulette wheel and the latest trial, might be suitable. Such possibilities are in- cluded if we simply assume T j , Xj and Bj are all random variables on some suitable sample space n. When Bj > Xj-I' the player is betting more than he has. He is asking for credit. This is common in gambling casinos, in the stock market (buying on margin), in real estate (mortgages) and is not unrealistic. When Bj < 0, the player is making a "negative" bet. To interpret this, we note that in our sequence of Bernoulli trials, or coin toss between two players, that what one wins, Xj - Xo, the other loses. To make a negative bet may be interpreted as "backing" the other side of the game, to taking the role of the "other" player. In particular, the payoff BjTj from trialj may be written as (-B) (-T). If Bj;£ 0, then -Bj > ° and may be interpreted as a nonnegative bet by a player who succeeds when -Tj = I, i.e., with probability q, and who fails with probability p. The -Tj are independent so we have Bernoulli trials with success probabilities q, i.e., the other side of the game. For simplicity we shall assume in what follows that ° < Bj ::;; X j - 1, but we may wish at a future time to remove one or both of these limitations. We also assume that Bj is independent of Tj , i.e., the amount bet on the jth out- come is independent of that outcome. Definition: A betting strategy is a family {B j } such that ° ::: B j ::;; Xj _ I ,j = I, 2, ... Theorem I: The betting strategies Bj = Xj _ 1 when p > t; Bj = 0, p < t; Bj arbi- trary when p = t; are precisely the ones which maximize E(X) for eachj. n n Proof: Since Xn = Xo + l: BJj, E(Xn) = Xo + l: E(BjTj ) = j=1 j=l --- ## Page 98 Optimal Gambling Systems for Favorable Games 71 283 n = Xo + r (p-q)E(Bj).Ifp-q = O, i.e.,p = t,thenE(Bj)doesnotaffectE(Xn ). j=l If p - q > 0, i.e. p > t, then E(Bj ) should be maximized, i.e., Bj = Xj' to maximize E(Xn). Similarly, if p - q < 0, i.e., p < t, then E(Bj ) should be minimized to maxi- mize the jth term, i.e., Bj = 0. Clearly, these maxima are not attained with other choices for Bj • This establishes the theorem. Remark. In the foregoing discussion, the Bernoulli trials and the Tj can be general- ized, yielding a more general theorem. (The Tj become "payoff functions" that are not necessarily identically distributed; roulette is the classic example.) The particular case of blackjack is covered, for instance, by replacing p and q throughout by Pj and qj for the respective probabilities that Tj = I or-l. To maximize our expected gain we must bet our total resources at each trial. Thus if we lose once we are ruined, and the probability of this is I - pn -+ I so maximizing expected gain is undesirable. 2. MINIMIZING THE PROBABILITY OF RUIN Suppose instead that we play to minimize the probability of eventual ruin, where ruin occurs after the jth outcome if Xj = o. If we impose no further restriction on Bj , then many strategies minimize the probability of ruin. For example, it suffices to choose Bj < X j _ d2. The discreteness of money makes it realisti·c to assume B j ;::::: C > 0, where C is a non-zero constant. We further restrict ourselves to the subclass of strategies where Bj equals C whenever ° < Xj - 1 < a, Bj = 0 if Xj - 1 < 0 or Xj - 1 > a, and C divides both a - z and z, where we have set z = Xo. This lets us use the gambler's ruin formulae ([8], page 3(4). Consider the gambler's ruin situation: Xo = z, Bj = I if 0 < Xj _ 1 < a, Bj = 0 if X j - 1 = 0 or Xj - 1 = a, a and z are integers. Let r be a positive number (necessarily rational) such that zr and ar are integers. Let R(r) be the ruin probability when z and a are replaced by zr and ar, respectively. This is equivalent to betting r -1 units when o < X j - 1 < a, in the original problem. We have R(r) = (6or - 6U ) / (oar - I), where 0 < p =l= 1- and 6 = q/p. Theorem 2: (a) If 1 > p > t, R(r) is a strictly decreasing function of r. (b) If o < p < 1. R(r) is a strictly increasing function of r. Proof: Follows from Lemma 3 below. Part (a) of the Theorem says that in a favorable game, the chance of ruin is decreased by decreasing stakes. Note that for p > t, i.e., e < 1, lim R (r) = 0, hence by r-+", making stakes sufficiently small, the chance of ruin can be made arbitrarily small. Lemma 3. Let a> z > 0, x> O. IfO < 6 < 1, then I(x) = (6 ZX - 6ax ) / (1- 6°X ) is strictly decreasing as x increases, x > 0. If 6 > 1, I (x) is strictly increasing as x increases, x > o. Proof: Elementary calculations which we omit. Theorem 2(a) shows that, at least in the limited subclass of strategies to which it applies, we minimize ruin by making a minimum bet on each trial. In fact, this holds for a broader class of strategies: Theorem 3' : If 1 > p > t, the strategy Bj = I if 1 < z < a-I, Bj = ° otherwise (timid play), uniquely minimizes the probability of ruin among the strategies where B j is an integer satisfying 1 < Bj < min (z, a-z) if I < z < a-I, Bj = 0 otherwise. Proof: We first show that if timid play is optimal, then it is uniquely so. Let qz be the probability of ruin, starting from z, under timid play. To establish uniqueness it suffices to show --- ## Page 99 72 E. 0. Thorp 284 qn < pqz+k + q qz-k' 2 < k < a-2, z-k > 0, z + k < a. (1) Using qz = (6°_6Z ) I (6°-1) and simplifying (1), we find that it is equivalent to show I(p) =p2k-l + q2k-l_ pk-l qk-t > 0, 1/2

0, t p > t and Bj = I Xj-t, where ° < 1< I is a constant. (This is sometimes called "fixed fraction" or "proportional" betting.) Let Sn and Fn be the number of successes and failures, respectively, in n trials. Then Observe that I = ° andl = 1 are uninteresting; we assume ° < I < 1. Note too that if 1< 1. there is no chance that Xn = 0, ever. Hence ruin, in the sense of the gambler's ruin problem, cannot occur. We reinterpret "ruin" to mean that for each s > 0, lim P [Xn > s] = 0, and we shall see that this can occur. Note too that we n are now assuming that capital is infinitely divisible. However, this assumption is not a serious problem in practical applications of the theory. Remark: The min-max criterion of game theory is an inappropriate criterion in Bernoulli trials. If Bj is a positive integer for all}, the maximum loss, i.e. ruin, is always possible and all strategies have the same maximum possible loss, hence all are equiva- lent. If capital is infinitely divisible, ruin is as we redefined it, and we restrict ourselves to fixed fractions, then for an infinite series of trials the min-max criterion (suitably probabilitistically modified) considers all I with 1 > I > Ie equivalent and all I with ° < 1< Ic equivalent. It chooses the latter class. For a fixed number n of trials, smaller I are preferred over larger f The criteria of minimizing ruin or of maximizing expectation likewise fail to make desirable distinctions between the fixed fraction strategies. --- ## Page 100 Optimal Gambling Systems for Favorable Games 73 285 The quantity log [Xn / XO]l/n = (Sn / n) log (1 + f) + (Fn / n) log (1 - f) mea- sures the rate of increase per trial. Since time is important it is plausible to in some sense maximize this. Kelly's choice [13] was to maximize E log [ Xn / Xo ]l/n = p log (1 + f) + q log (1 - f) == G (I), which we call the exponential rate of growth. The following theorems show the advantages of maximizing G (I). Theorem 4. If 1 > p > t, G (I) has a unique maximum at/* = p - q, ° <1* < 1, where G (1*) = p log p + q log q + log 2 > 0. There is a unique fraction fc > ° such that G (/c) = 0, and Ie satisfies f* < fc < 1. Further, we have G (I) > 0, ° < I < Ie; G (I) < 0, I> /C, with G (I) strictly increasing, from ° to G (1*), on [0,/*], and G (I) strictly decreasing, from G (1*) to - 00 on [/*, 1]. Theorem 5(a). If G (f) > 0, then lim Xn = 00 a.s., i.e., for each M, P.[ lim n n Xn> MJ = 1. (b) If G (f) < 0, then lim Xn = 0 a.s., i.e., for each c: > 0, P [lim Xn < c:] = 1. n n (c) If G (f) = 0, then lim Xn = OCJ a.s. and lim Xn = 0 a.s. n -n- Thus for ° < I < fc, the player's fortune will eventually permanently exceed any fixed bounds with probability one. For I = Ie it will almost surely oscillate wildly between 0 and + 00. If I> fe, ruin is almost sure. l/n Proof: (a) By the Borel strong law ([17], page 19),limlog[Xn/Xo] = G(f) > ° n with probability 1. Hence, a.s., for w € n, where n is the space of all sequences of Bernoulli trials, there exists N (w) such that for n > N (w), log [Xn/ Xop/n> G(I)/2> O. But then Xn > Xo en GU)j2 for n > N(w) so XnJ' 00. (b) The proof is similar to part (a). (c) We use the fact that, given any M, lim Sn > np + M + 1 and n . . / ~+M lIm Sn < np - M -1. Then If Sn > np + M, log [Xn / XO]l n > n n n-(np+M) M 1+/ M 1+/ log (1 + f) + log (l - f) = G (f) + - log -1 / = -log -1 -I' n n - n - whence Xn > Xo (~ = j) M. Since Sn > np + M infinitely often, a.s., then li~ Xn > Xo (~ ~ j) M a.s. Since the right side may be chosen arbitrarily large, lim Xn = 00 a.s. n The proof that lim Xn = 0 a.s. is similar. n Theorem 6: IfG(fd > G(f2), then lim X n(f1)/Xn(f2) = 00 a.s. n Proof: log [Xn (ld / Xo]l/n -log [Xn (12) / XO)l/n = log [Xn (f1) / Xn (f2)]1/n = ~ log G :j:) + :n log G=j:). Therefore, by the Borel strong law of large numbers, lim log [X n (f1) / X n (f2 )] -+ G (fd - G (f2) > 0 n with probability 1. Now proceed as in the proof of Theorem 5(a). In particular, we see that if one player uses/* and another, betting on the same favor- --- ## Page 101 74 E. 0. Thorp 286 able situation, uses any other fixed fraction strategy I, then lim Xn(f*) / Xn (f) = co n with probability I. This is one of the important justifications of the criterion "bet to maximize E log X n." Bellman and Kalaba ([2], pages 200-201) show thatf* not only maximizes E log Xn within the class of all fixed fraction betting strategies but in the class of "all" betting strategies. This also is a consequence of the following theorem, part of which was suggested by a conversation with J. Holladay. Consider a series of independent trials in which the return on one unit bet on the ith outcome is the random variable Qi. Then n n Xn = n (Xi/Xi_dandElogXn = L Elog(Xi/Xi_ I ). We have Xi = Xi-I + i= 1 i= 1 BiQi and Xi / Xi-I = 1 + (Bi / Xi-I) Qi. Thus each term is of the form E log (1 + FiQ;) where the random variable Fi depends only on the first i-I trials, Qi depends only on the ith trial, and hence Fi and Qi are independent. We are free to choose the Fi to maximize E log (1 + FiQ;), subject to the constraint 0 ~ Fi ~ 1. Theorem 7: If for each i there is anfi' 0 < fi < I, such that E log (1 + fiQi) is defined and positive, then for each i there is a number Ii* such that E log ( 1 + FiQi) attains its unique maximum for Fi = Ir a.s. To avoid trivialities we assume Qi * 0 a.s., each i. Proof: It follows that the domain of definition of E log (I + fiQi) is an interval [0, a;) or [0, ad, where ai = min (1, b;) and bi = sup U; :fiQi > ° a.s. } > O. Since the second derivative with respect to h of E log ( 1 + hQi) is - E ( Q? i (1 + j;Qi)2) , which is defined and negative, any maximum of E log (I + /;Q i) is unique. The func- tion is continuous on its domain so if it is defined at ai' there is a maximum. If it is not defined at ai' then lim E log (1 + IiQ;) = - oc> so again there is a maximum. I I to, By the independence of Fi and Qi' we can consider Fi(sl) and Qi(S2) as functions on a product measure space SI x S2. Then Elog(l+FiQi) = S S log(1+Fi(sl)Qi(s2» = EElog(l+Fi(sl)Qi) S, S2 < E log (1 + !;* Qi) with equality if and only if E log (1 + Fi (s I) Qi) = E log (1 + !;* Q;) a.s., which is equivalent to Ii * Qi = Fi (Sl) Qi a.s., and by the in- dependence this means either It = Fi a.s. or Qi = ° a.s. hence It = Fi a.s., and the theorem is established. We see in particular from the preceding theorem that for Bernoulli trials with suc- cess probability Pi on the ith trial and 1 > Pi > t, E log Xn is maximized by simply choosing on each trial the fraction Ii * = Pi - q i which maximizes E log ( 1 + h Qi ). 4. THE ADVANTAGES OF MAXIMIZING E LOG Xn The desirability of maximizing E log Xn was established in a fairly general setting by Breiman [3]. Consider repeated independent trials with finitely many outcomes I = {I, ... , S } for each trial. Let P(i) = Pi' i = 1, .. . , s, and suppose that { AI' .. . , A, } is a collection of (betting) subsets of I, that each i is in some Ak , and that payoff odds Ok correspond to the Ak • We bet amounts B I, . . . , B, on the respective Ak and if outcome i occurs, we receive L Bjo j where the sum is over {j : i e: A j }. We make the convention that A I = I and ° I = 1, which allows us to hold part of our fortune in reserve by simply betting it on AI. We have, in effect, a generalized roulette game. --- ## Page 102 Optimal Gambling Systems for Favorable Games 75 287 Roulette and the wheel of fortune, as described in Part I, are covered directly by Breiman's theory. The theory easily extends to independent trials with finitely many outcomes which are not identically distributed but which are a mix of finitely many distinct distribu- tions, each occurring on a given trial with specified probabilities. The theory so ex- tended applies to Blackjack and Nevada Baccarat, as described in Part 1. Breiman calls a game (i.e., a series of such trials) favorable if there is a gambling strategy such that Xn ~ 00 a.s. Thus the infinite divisibility of capital is tacitly assumed. However, this is not a serious limitation of the theory. If the probability is "negligible" that the player's capital will at some time be "small," then the theory based on the assumption that capital is infinitely divisible applies to good approximation when the player's capital is discrete. This problem is considered for Nevada Baccarat in [24], pages 319 and 321. Breiman establishes the following about strategies which maximize E log X n• 1. Allowing arbitrary strategies, there is a fixed fraction strategy B i = Ui, ... ,f,.) which maximizes E log X n• 2. If two players bet on the same game, one using a strategy A * which maximizes E log X n and the other using an "essentially different" strategy A, then lim n Xn (A*) / Xn (A) ~ 00 a.s. 3. The expected time to reach a fixed preassigned goal x is, asymptotically as x in- creases, least with a strategy which maximizes E log X n • Thus strategies which maximize E log Xn are (asymptotically) best by two reason- able criteria. 5. A STOCK MARKET EXAMPLE Though in practice there are only finitely many outcomes of a bet in the stock market, it is technically convenient to approximate the finite distributions by discrete countably infinite distributions or by continuous distributions. In fact it is generally difficult not to do this. The additional hypotheses and difficulties which occur are, from the practical point of view, artificial consequences of the technique. Hence the new theory must preserve the conclusions of the finite theory so again we apportion our resources to maximize E log X n• As a first example, consider the following stock market investment. It was the first to catch our interest, and was based on a tip from a company insider. Suppose a certain stock now sells at 20 and that the anticipated price of the stock in one year is uniformly distributed on the interval [15, 35]. We first computeJ* and G (J*), assuming the stock is purchased and fully paid for now, and sold in one year. The purchase and selling fees have been included in the price. Thus, the outcome of this gamble, per unit bet, is described by dF (s) = C (_ t. t) ( s) ds, where F is the asso- ciated probability distribution and Cis) is I for s in A and 0 for s not in A. The mean m of F is t > O. Also t . ~- S log2 e G(f)= S log2(1+fs)ds,G'(j) = S -1 f,ds,and lim G'(f) =-00. -t -t + S It 4 Therefore Theorem 8 below applies and there is a uniqueJ* such that 0 0, G' (f) > 0 for 0 <1< 3. Therefore 3 < 1* < 4; calculation yields 1* = 3.60-. Thus if the maximum fraction of current capital which can be bet is 1, we should bet all our capital. However, if margin buying is allowed, we should (consistent with our ability to cover later) be willing to bet as much as possible, up to a fraction 1* which is 3.6 times our current capital. The mathematical expectation for buying outright is 0.25 Vo if buying on margin is excluded and 0.90 Vo if unlimited buying on margin is permitted, and additional coverings can be made later as required, and we betf* = 3.60 of our current capital. Integration yields G(f) = [(log2 e) If] { (1 + 3114)[ In (1 + 3114) - 1 ] - (1-114)[ In (1 - fl 4) - 1 J} from which we find G (1) = 0.28 and G (3.60) = .59. Next, we compute 1* and G (/*), assuming that calls are purchased for 2 points per share. Thus the outcome of this gamble per unit bet is described by the probability distribution F with mass 5/20 at -1, and dF(s) = 2/20 if -1 < s < 6.5. 6.5 The mean m = 1.8125> O. Also G(I) = (5/20) log2 (1-1) + (2/20) J log2 -1 (1 + fs) ds and G'(I) = -(5/20»)Og2 e + (2/20) 6/ S110g2fe ds, from which it is 1- -1 + s clear that lim G' (I) = - 00 . Therefore, again by Theorem 8 below, there is a unique fil f* such that G' (1*) = 0 and 0 < J* < 1. It suffices to solve h (f) = 0 where h (f) = 20G' (I) I log2 e = ;~f+ y{ 7.5-].IOg.(17 ~'j)}. We findJ* = 0.57. The mathematical expec- tation of the call purchase process is 1.8125 f* Vo or about 1.03 Yo. Integration yields G (f) = 0- ) log2 (1 -1)+ (Jn2 e 1101)( 1 + 6.51) [In ( 1 + 6.51) - 1 ] - ( 1 - f) [ In ( 1 -I) - 1 ]. We find G ( 0.57 ) = 0.55. Thus we have the interesting result that the expectation from buying calls is higher than from buying on unlimited margin but that the growth coefficient is higher from buying on unlimited margin. Our criterion selects the latter investment. For buying on margin G ( 3 -) = .55 so our criterion selects buying on margin if the margin requirement is less than t - and buying calls, if possible, if the margin requirement exceeds t -. . In the preceding example we needed the following theorem to establish the unique- ness of/*. We define a = sup {t : F(- 00, t) = 0 } and note that if 1 + fa > 0 and <:tJ 00 1 the integral G (I) = J log2 (1 + fs) dF (s) is defined, then G' (I) = J sl Ogj:2 e dF (s). a u + s (See, e.g. [17], page 126). <:tJ 1 Theorem 8: The function G' (I) = J s og2 e dF (s) is monotone strictly de- a 1 + fs <:tJ creasing on [0, - 1/ a). If the mean m = J s dF (s) > 0, then the equation G' (I) = a j S log2 e dF (S) = 0 has exactly one solution f* in the interval (0, - 1/ a) iff lim a 1 + fs J7" ,.- I /a G' (f) < O. In this event, G (I) is monotonely strictly increasing for fin [f*, - 1/ a). Proof: If 0 l~f (0 f= s> a)soG'(ll) > G'(l2)' + IS + 2 S --- ## Page 104 Optimal Gambling Systems f or Favorable Games 77 289 From this and the right-hand continuity of G' (f) at 0, G' is monotone strictly de- creasing on [0, -1 / a). By hypothesis G' ( ° ) = m log2 e > 0. Therefore, from the continuity of G' (f) on [0, -1 / a ), G' (f) attains all values t on the interval lim G' (f) < t < G' (0) exactly once. Thus there is exactly one solution f* in f 7"-1/0 (O,-l/a)iff lim G'(f) < 0. f7"-I /o The description of G (f) is now evident. 6. WARRANT HEDGING We next apply the criterion of maximizing E log Xn to the warrant hedge described in Part 1. With the notation of Part I, and an assumed mix of 1, the gain X from a one unit bet is X = (sf - So + Wo ) / ( (X So + ~ Wo ), Sf < 1 , and X = (Wo + 1 - So ) / ( (X So + ~ Wo ), Sf> I . We wish to maximize the exponential rate of growth G (f), given by G (f) = E log ( 1 + f X). It can be shown that the situation is essentially the same as in Theorem 8 and that this depends on the a.s. boundedness of X; we have in fact a.s. sup X = ( Wo + 1 - so) / «x So + ~ wo) and a.s. inf X = - (so - wo) / «x So + ~ wo). Thus f* can be com- puted when the mix is 1, though the details are tedious. When the mix is greater than 1, more serious difficulties appear. The payoff function X has a.s. inf X = - co and a.s. sup X < co. This means that, no matter what fraction f > 0 of our unit capital is bet, there is positive probability of losing at least the entire unit. Thus any bet is rejected! Yet this is unrealistic. We now find out what is wrong. First, the assumption that arbitrarily large losses have positive probability of occur- rence is not realistic. (a) The broker will automatically act to liquidate the position before the equity is lost. (b) The strategies for investing in hedges automatically lead to liquidating the position after the common is substantially above exercise price. There is, then, a maximum imposed on X by practice but it is not easy in practice to specify this maximum. Further, this maximum will, in general, be a random variable (a.s. bounded, however) which is a function of the individual's investment strategy. It is not easy to determine the consequent probability distribution of sf' yet this is required to calculate E log ( 1 + f X). More generally, we might consider an individual's lifetime sequence of bets of various kinds. It is plausible to assume that Xn = ° only upon the death of the indivi- dual, for although the individual may have no cash equity at a given instant, he does have a cash "worth", based on his future income, serendipity, etc., and this should be included in X n• This is true even of a (Billie Sol Estes) bettor who loses more than he owns. The subtlety here, then, is that the accountant's figure for net assets (plus or minus) is not an accurate figure for Xn as Xn decreases below small positive amounts. One can also object to Xn at death being assigned the value 0, by arguing that the chance of death in a time interval always has a small positive probability, thus making E log Xn = - co always. Also, individuals when choosing between two alternatives each involving a low probability of death generally do not meticulously select the safer alternative (e.g., air travel versus train travel). Thus death should really be treated as an event with a large but finite negative value. Another common objection to E log Xn as a measure of "utility" is that, like all such measures which are not a.s. bounded, it allows the St. Petersburg paradox. --- ## Page 105 78 E. 0. Thorp 290 The foregoing objections to E log Xn only arise when we leave the case of finitely many outcomes. We say that these are artificial technical difficulties which can all be removed in the cases of practical importance. This may be tedious, as it is for the warrant hedge, so we defer such matters for a subsequent paper. 7. PORTFOLIO SELECTION USING E LOG X The Breiman results were obtained for repeated independent trials with finitely many outcomes and finitely many ways to apportion our capital (amongst finitely many betting sets). The results extend, as we have remarked, to independent trials which are a mix of finitely many differently distributed trials (i.e., finitely many out- (:omes and betting sets) provided that as n tends to infinity, the number of trials with each distribution also tends to infinity. There are significant real world situations, such as the selection and continuous revision of a portfolio of securities, to which this extended theory does not generally .apply. A difficulty which we have already discussed is that it may be technically con- venient to introduce continuously distributed and possibly unbounded payoffs, but now generalized to the apportionment of capital among a finite number of alter- natives, rather than just betting a fraction on one alternative. Another problem is that the sequence of betting situations may change so that no two are ever the same. Further difficulties arise when we consider the possible dependence of trials. Still other problems appear when we consider that in the real world the spectrum of situa- tions is changing continuously and that a potentially continuous portfolio revision is part of an optimal approach. (Actually, because of the transactions costs which occur in practice, portfolio revision is likely to occur in discrete steps.) The extent to which Breiman's conclusions for the finite case can be generalized in these directions will be considered subsequently. For now we simply remark that the possible generalizations promise to be adequate for the real world problems of port- folio selection. Assuming this to be the case, we shall see in the next section that economists and others now have for the first time an accurate guide for portfolio selection and revision. 8. THE KELLY CRITERION AND DEFICIENCIES IN THE MARKOWITZ THEORY OF PORTFOLIO SELECTION How to apportion funds among investments has endlessly puzzled economists and decision-makers. The literature was noted for its lack of instruction in such matters. When Markowitz' work on portfolio selection appeared, first in articles and later in the monograph [I8], it became the standard reference. Markowitz considers situations in which there are r alternative and, in general correlated, investments, with the gain per unit invested of XI ' .. . , X" respectively. (It is so much more dignified to call bets investments; we shall try to remember to do this in this section.) One of the investments is, of course, cash. The gain is given by X k = 0 a.s. To select a portfolio is to apportion our resources so that fj is placed in the ith investment. Markowitz' basic idea is that a portfolio is better if it has higher expecta- tion and at least as small a variance or if it has at least as great an expectation and has a lower variance. If two portfolios have the same expectation and variance, neither is preferable. As the h range over all possible admissible values, the set of portfolios is generated. Typically the assumptions on theh are 'i.fi = I, andfi ~ 0 for i = I, . .. , r . --- ## Page 106 Optimal Gambling Systems for Favorable Games 79 291 If a portfolio has the property that no other portfolio in the set is preferable, then it is called efficient. Markowitz says that the investor should always choose an efficient portfolio. Which efficient portfolio to choose depends on factors outside the theory. such as the investor's "needs". The Markowitz theory has the obvious deficiency that if Ei and cr;, i = 1, 2, are the expectation and variance of portfolios 1 and 2, then if E 1 < E2 and cri < cr~, the theory cannot choose between the portfolios. Yet there are obvious instances where "everyone" will choose the second portfolio over the first, such as when Fl ( x ) < F2 ( X ) for all x. Specifically, let Xl be distributed uniformly on [1, 3], let X2 be uni- formly distributed on [10, 100] and let X3 = 0 a.s. represent the possibility of holding some of our resources in cash. Suppose Xl' X2 , and X3 are independent. Then E 'E}; Xi = 2fl + 55f2 and cr2 'Efi Xi = 'Efi2cr/ = f~/3 + 675f~. All cash, or f3 = 1, is an efficient portfolio since this is the unique portfolio with zero expectation. The portfolio j~ = 1 also is efficient since this is the unique portfolio with greatest expecta- tion. There are, in fact, infinitely many efficient portfolios. (They lie on a curve in the 11'/2 plane connecting (0, 0) and (0, I).) The theory doesn't tell us which is best, yet 12 = I is clearly preferable to any alternative. In the case where there are the two alternatives Xl = 0 a.s. (cash) and X2 with E2 > 0 and (1'2 > 0, all portfolios are efficient and Markowitz' theory gives no infor- mation on which to choose. The Kelly criterion tells us to choose 12 to maximize E log ( 1 + 12X2) and we know further from the theory of the Kelly criterion why this choice is good. As we have seen, repeated trials of such an investment with 12 greater than the fraction Ie will lead to ruin a.s. Remark: This incompleteness of Markowitz' theory is understandable since he only uses probability information about first and second moments. We note though that the examples he gives, and the real world applications, generally assume that more detailed structure is known. Hence, it is reasonable that the criterion E log Xn , which does use higher moment information, can provide a sharper theory. Next consider those two-point probability distributions with masses m i located at x j, i = I, 2, and with mean and variance 1. These are indistinguishable by Marko- witz' criterion. A calculation shows, however, that for Xl defined by Xl = -J, X2 = 3/2, ml = 1/5, m2 = 4/5, the optimalfractionfi is 1 and G(ji) is -(1/5) log 3 + (4/5) log 2. For X 2 defined by Xl = -2, X2 = 4/3, m 1 = 1/10, m2 = 9/10, we havef: = i and Gun = -(1/10) log 4 + (9/10) log (3/2), which is smaller than Gun· Hence if X:. 1 is the fortune after n repeated independent trials of an investor who invests fi in X 1 at each trial and X n. 2 is the fortune after n trials of an investor who invests in any manner whatsoever in X 2 at each trial, we have lim X:. 1/ X n• 2 = 00 a.s. As a final example, suppose we are to apportion our resources between the fore- going Xl and X2 , which we now suppose to be independent, and cash, represented by X3 . We impose the constraints Ii > 0, i = 1, 2, 3; II + 12 + 13 = 1, and II + 2/2 < 1. The latter constraint prevents investments where our losses exceed our total resources. (The analysis and conclusion are essentially the same without this con- straint.) The admissible portfolios are represented by the closed triangular region of the positive quadrant bounded by the axes and the line II + 212 = 1. We have E'L/iXi = II + Ii and, because of the independence of Xl and X2 , (J2'EJ;Xi =!f + !~. The efficient portfolios are the points ofthe!1.J2 plane 011 the two closed line segments joining ( t, t) to (0, 0) and to (1, 0). The function E log( 1+ 11 Xl +.f2X2) == G(flJ2) is given by 50 G(flJ2) = --- ## Page 107 80 E. 0. Thorp 292 36 log (1 + 311/2 + 412/3) + 4 log ( 1 + 311/2 - 212) + 9 log ( 1 -11 + + 412 / 3) + log ( 1 - j~ - 212)' This function is undefined on the line joining (0,1/3) and (1, 0). It is defined and continuous elsewhere on the triangle of portfolios and as (/1'/2) tends to the segment from this triangle, G (/1'/2) --+ - 00. It follows (by the continuity) that G (11,j~) attains an absolute maximum in the region of the triangle where it is defined. We also know that any such maximum is positive. It follows that, if an efficient portfolio maximizes G (/1'/2), then it must be a portfolio from the interior of the segment joining (0, 0) and (1/3, 1/3). Hence the coordinates must simultaneously satisfy the equations 0 G (/1'/2) / 011 = 0 and 0 G (/1 '/2) / 012 = O. (We note that in repeated independent trials where the investor selects an efficient portfolio from the segment joining (1/3, 1/3) to (1,0), he will be ruined with probability one.) Setting 11 = 12 = t in the equations a G / 011 = 0 and a G / 012 = 0 and attempt- ing to solve simultaneously yields, upon elimination between the two equations of the last of the four fractions, the necessary condition -2796 + 3761 + Illt2 = O. Since this is negative at t = 0 and t = 1, there are no roots in the interval 0 < t < 1/3. Hence no efficient portfolio maximizes G (11'/2)' We conclude that if X:. 1 is the fortune after n trials of a player who bets to maxi- mize G (fl'/2) on each trial, and X n, 2 is the fortune of a player who chooses any efficient portfolio on each trial, then lim X:, 1/ X n, 2 = 00 a.s. Furthermore, the Kelly investor will reach a fixed goal x in less time, asymptotically as x --+ 00, than a Markowitz investor. The Kelly criterion should replace the Markowitz criterion as the guide to port- folio selection. REFERENCES [1] Baldwin, Cantey, Maisel, and McDermott (1956). The optimum strategy in blackjack, J. Amer. Statist. Assoc., 51, 429-439. (2] Bellman, R., Kalaba, R. (Sept. 1957). On the role of dynamic programming in statistical com- munication theory. IRE Trans. of the professional group on information theory, IT-3 no. 3, 197 -203. (3) Breiman, L. (1961). Optimal gambling systems for favorable games. Fourth Berkeley Symposium on probability and statistics, J, 65-7B. [4] Coddington, E. A., Levinson, N. (1955). Theory of Ordinary Differential Equations. New York, McGraw-Hill. [5) Cootner, P. H., editor (1964). The Random Character of Stock Market Prices. The M.LT. press. [6) Dubins, L., Savage, L. (1965). How to Gamble if You Must. New York, McGraw-HilI. (7) Dunnington, G. Waldo (1955). Carl Friedrich Gauss, Titan of Science. New York, Hafner. [B) Feller, W. (1957). An Introduction to Probability Theory and Its Applications, Vol. I, Revised. New York, Wiley. [9] Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. 11. New York, Wiley. (lO) Ferguson, T. S. (1965). Betting systems which minimize the probability of ruin. J. Soc. Indust. Appl. Math., 13, no. 3. [11] Foster, F. G. (1964). A computer technique for game-theoretic problems. 1. Chemin-de-fer analyzed. Comput. J., 1, 124--130. [12] Kassouf. Sheen T. (1965). A Theory and an Econometric Model for Common Stock Purchase Warrants. Ph. D. Thesis, Columbia University, New York. [13] Kelly, J. L. (1956). A new interpretation of information rate. Bell System Technical Journal, 35, 917--926. (14) Kemeny, J. G., Snell, J. L. (1957). Game theoretic solution of baccarat, Amer. Math. Monthly, 114, no. 7, 465-9. [15] Kendall, M. G., Murchland, J. D. Statistical aspects of the legality of gambling, J. Roy. Statist. Soc. Ser. A. To be published. --- ## Page 108 Procs. o/the Business and Economics Sect. o/the Amer. Stat. Assoc., 215- 224 (1971) 81 7 PORTFOLIO CHOICE AND THE KELLY CRITERION Edward O. Thorp, University of California at Irvine:O:: I. Introduction. The Kelly (or capital growt!.» criterion is to maximize the expected value E log X of the logarithm of tr.e wealth random variable X. Logaritlunic utility has been widely dis c ussed since Daniel Bernoulli introduced it about 1730 in connection with the Petersburg galne [3, 28]. However, it was not until certain mathclnatical results were proven in a limited setting by Kelly in 1956 and then in much more general setting by Breiman in 1960 and 1961 thai logarithmic utility was clearly distinguished ~. its properties from other utilities as a guide to portfolio selection. (Sec also [2, 4, 15], and the very significant paper of Hakansson [II). ) Suppose for each timp. period (n ;:.1, 2, ... ) there are k investment opportunities with re- sults per unit invested denoted by the family of random variables Xn,l'Xn,2"" ,Xn,k' Suppose also that these random variables have only fi- nitely many distinct values, that for distinct n the farrdlies are independent of each other, and that the joint probability distributions of distinct farnilies (as subscripted) are identical. Then Breiman's results imply that portfolio strategies 1\ which Inaximize E log Xn , where Xn is the wealth at the end of the n-th time period, have the fol1owing properties: Property 1. (Maximizing E log Xn asymp- totic:c.:tlly maxiInizcs the rate of asset growth. ) If, for each tim~ p~riod, two portfolio managers have the same falnily of investmt'nt opportunities or investn1.ent universes, and one uses a strategy 1\':: maximiz.ing Elog Xn whereas the other uses an 'lessentially different l1 (1. e., ElogXn(II.*)- E log Xn(/\)"~) strategy 1\, then lim XJI\*)/Xn(I\)" = almost surely (a. s.). Property 2. The expected time to reach a fixed preassigned goal x is, asymptotically as x increases I least with a strategy maxhni zing ElogX;,. The qualification '1essentially different" con- ceals subtleties which are not generally appreci- at~d. For instance [11], which is close in method to this article, and whose conclusions we heartily endorse, contains nUITlCroUS rnathematically incorrect statemp.nts and several incorrect con- clusions, mostly from overlooking the require- mp.nt 11essentially different. 11 We intend to present a detailed analysis elsewhere and only indie.,te the problem here: If Xj = Xj!Xj-l' then even though E log Xj>O for all j it need not be the ca.'>c that P(lim Xn =00) = 1. "In fact, we can have 215 (just .as in the case of Bernoulli trials and E log Xj = 0; sec (26)) P(lim sup Xn = m) = I and P(lim inf Xn = 0) = I (contrary to [11, p. 522, eq. (18)) and following assertions) . Similarly, when ElogxjO is given, N'I- is the strategy which maximizes ElogXn · and A is an "essentially different" strat- egy. Thus in any application it is important to have an idea of how rapidly· e -to O. Work needs to be done on this in order to reduce E log X to a guide that is useful (not merely valuable) for portfolio manage.rs. Some illustrative examples for n=6 appear in[ll]. Property 2 shows us that nUl.xiITIizing E log X also is appropriate [or individua~s who have a set goal (c. g., to becorn0. a millionaire). Appreciation of the compelling properties of the Kelly criterion may have been impeded by Reprinted from the 1971 Business and Economics Statistics Section Proceedings of the American Statistical Association --- ## Page 109 82 certain l1.1.iSlll'ldcrstandings about it that persist. in the literature of mathematical economics. The first misunderstanding involve~ failure to distinguish among kinds of utility theories . \Ve COlnpare and contrast three types of utility theories: (1) descriptive, where data on observed behavior is fitted mathem.atically. Many differ- ent utility functions might be needed', corres- ponding to widely varying circumstances, cultures, or behavior typesl ; (2) predictive, which "explains" observed data: fits for ob- sCl"vf'd data arc deduced from hypotheses, with the hope future data will also be found to fit. Many different utility functions may be needed. corresponding to the many sets of hypotheses that lllay be put forward; (3) prescriptive (also called normativeL which is a guide to behavior, i. e., a recipe f;r optimally achieving a stated goal. It is not necessarily either descriptive or predictive nor is it intended to be so, \Ve use logaritrunic utility in this last way, and nHlch of the n1isunderstanding of it comes frol11. those who think it is being proposed as a descriptive or a predictive theory. The E log X theory is a prescription for allocating resourCl~5 so as to (asymptotically) maximize the rate of growth of as scts. Another .iobjcetion" voiced by SOrnf! ecollO- n1:'sts to E log X and, in fact to all unbounded utility functions, is that it doesn't resolve the (g(.'ncralized\ P(~ter5burg paradox. The rebuttal is blunt and pragmatic: The generalized Peters- burg paradox dues not arise'in the rea.l world btcause anyone real world random variable is bourdcd (as is any finite collection). Thus in a.ny real application the paradox does not arise .. To insist that a utility function resolve the paradox is an artificial requirement, certainly pern1is sible, but obstructive and tangential to the goal of building a theory which is also a practical guide. 2. Samllelson's objections to logarithmic utility. Samllelson [21, pp. 245-6; 22, pp. 4-5] says that repeatedly authorities [S, 6,14, IS, 30] ..... have proposed a drastic simplification of the decision problen1 whenever T [the number of investmcnt periodsj2 is large. Rule: Act in each. period to maximize the geol11etric Jl1l.'an or the expected value of log xt ' The plausibility of such a procedure Cornp.s fr0111 the recognition of the following valid asyn1p- toLic result. Theorem: Acting to ma.ximize the geometric lnl~il.n at ~very step will if the period is "suffi- ci(!mly long, II "aln"lOst certainly"3 result in hj gilt' r tern"linal wealth and terminal utility than (rOlli \· corollary~ False corollary: If maximiz~ng -the geo- 216 E. 0. Thorp metric mean alnlost certainly leads to a bl."ttl " outcomc, then the expc.cted value utility of it::; outcomes exceeds that of any other rule, pro· vided T is sufficiently large. It Samuelson then gives count.cr(~xafnple5 t ( ) the corollary, Wt.~ heartily ag~'ec that the COl"l.!- lary is false. In fact we had already sho'wn tin ·· for one of the utilities Salnllelson uses, for \Vl' not.ed [26] that in the case o( Bernoulli u·i"l. with probability 1/2 0; otherwise VI and Uz are inequivalent.) In this connection we have proved: Theorem: Let U and V be utilities defined and differentiable on (0, m), with U'(x) and V'(.'I positive and. strictly decreasing as x increa.!W5, Then if U and V are inequivalent, there is a one period lnv·estment setting such that U and V have distinct optimal strategies. 4 All this is in the nature of an aside for Samuelson's correct criticism of the "false --- ## Page 110 Portfolio Choice and the Kelly Criterion corollaryH docs not apply to our use of logarith- mic utility. Our point of view is: if your goal i"s property 1 or property 2 , then a recipe for achieving either goal is to n"laximize E log X . These properties distinguish log from the pro- lixity of utility functions in the literature . Furthermore, we consider these goals appro- priate for n1.any (but not all) investors . Investors with other uti1it.ie~ ;;-with goals incompatible with logarithn1ic utility, will of course, find it inappropriate . Property I implies that if It' maximizes E log Xn{/\) and A' is "essentially different, " then Xn(Ir"') tends almost certainly to be better than Xn(A') as n""'. Samuelson [21. p. 246). apparently referring to this, says a!t~r refuting the !lIaIse corollary": "Moreover, as 1 sho'wed elsewhere [20. p. 4). the ordering principle of selecting between two actions in terms of which has the greater probability of producing a higher result does not even possess the property of be- ing transitive . . . . we could have wt,:*t; better than \Y)~;:', and Wi,l~:t better than w*, and also have. w::t better than wi,'*:';. II For SOlne entertaining e xamples, ·see the discussion of no.n-transit1.vc dice in [91 . (Consider the dice with equiprobable faces numbered as follows: X ~ -(3.3.3.3,3.3); y ~ (4.4,4.4.1.1); Z ~ (5.5.2,2.2..2) . Then P(Z>Y) ~ 5/9. P(y>X) ~ 2/3, P(X>Z) ~ 2/3.) What Samuelson does. not tell us is that the property of producing a high;;-result ahnost certainly, as in property 1, is transitive. If 'wc have w ;;n:,::= >w~q almost certainly, and w~;t~' >w::': alnlost ccrtainly, then we must have w*~"'.: >w);t almost certainly. One might· object [20. p . 6J that in a real in- vestrnent sequence the limit as n -+(0 is not reached . Instead the process stops at some finite N. Thus we do not have XJI(o') > Xn(A') almost certainly. Instead we have P(Xn(N'»Xn(A')) ~ 1- £N where ~"'O as N-t co , and "transitivity can be shown to fail. This is correct. But an approximate form of transitivity does hold: Let X, Y.Z be random var- iables with P(X>Y)~I-€I ' P(Y>Z)=l-'2. Then P(X>Z);, 1- (£1 + '"2'. To prove this, let A be the event X>Y, B be the event Y>Z. and· C be the event X>Z. Then P(A)+P(B) ~ P(AUB)+p(AnB) ~ l+p(AnB) . But AnBCC so PIC) ;o;p(AnB) ;0; P(A)+P(B) - I. i. e .• P(X>Z);o; I - ('I + '2.). Thus our approach is not aHectcd by the vart- ous Samuelson objections to the uses of logarith- n1ic utility. Mark~witz [16. pp. ix-x] says .... : inI955-56. I concluded ... that the investor' who is currently r e invC'sting everything for lithe long run" .should maxilnize the expected yalue of the logarithm of 217 83 wealth. 11 (This assertion s e ems to be regal"Clle s s of the investor's utility and so indicates b<:li e f in the Bfalse coronary ." ") Mossin [181 and Sannlclson [20] "II ... have each shown that this concluf>ion i::; not true for a wide range of [utility] functions ... The fascinating ~1ossin-Samuelson result, CO!Y\- bined with the straightforward arguments sup- porting the earlie r conclusions, sE'.emcd pat'adux- ical at first. 1 have since returned to the view of- Chapter 6 (concluding that: for large T. the Mossin-Samuelson Inan acts absurdly ... . I ' Markowitz says here, in effect, ,that alternate utility functions (to log) are absurd , This position is unsubstantiated and unreasonable. He continues " ... like a player who would pay an unlimited amount for the St. Petersburg game ... ... If you agre~ with us that the St. Peters- burg galYle is not realizable and may be ignored when fashioning utility theories for the real 'world, then his continuation I I •• , the tern1inal utility'func- tion must be bounded to avoid this absurdity; .. , ! ' does not follow . Finally, .J\1arkowitz says " .. . and the argl.: ~ ment in Chapter 6 applies when utility of tern1i- nal wealth is bounded." If he means by this that the "false corollaryll holds if we restrict our- selves to bounded utility functions, then he is mistaken. Mossin [18] already showed that the optimal strategies for logx and xY/y, yt 0, are fixed fraction fo r these and only these utilitie s. Thus any bounde d utility besides x'l/v. 1'<0. \'."ill have optimal strategies which are not fixed frac.- tion. hence not optimal for logx .. ""Sa";nuelson[22] gives counterexan1ples which include·. the bounded utilities xY/y. Y < O. Since Mossin assumes U" exists and our theorem only aSSUlnes that U' exists, it provides additional counterexao1ples. 3. An outline of the theory of logarithmic _utility as applied to portfolio selection. 'The sim?lest case is Bernoulli trials with probability p of success, Ol!2 and to bet nothing otherwise. To o1aximize E log Xn is equivalent to nlaxi- mizing Elog[X IX ]Im ".G(f). which we call the n 0 (exponential) rate of growth (pel" time period). It turns out that for p >-li2. G(f) has a unique posi- tive maxinlun1 at P:< and that there is a critical fraction fc ' Of • limX ~ 0 a.s . 11 . c n· (llruinl!~. Bernoulli trials exhibit many of the features --- ## Page 111 84 of the following more general case. Suppose we· have at each trial n = 1, 2, ... the k investlnent opportunities Xn,l,Xn,2' ... ,Xn,k and that the con- ditions of property 1, section 1 are satisfied. This means that the joint distributions of {X . ,X ., ... ,X . j"l are the same for all n J for \. n.l1 n,lZ n,lj each subset of indices l;;il0 for each set of indices ml ... ·.Il).. Thus the do- main of G(f1 ... · .~) is the intersection of all these open half-spaces with the k-dimensional 216 E. 0. Thorp simplex ~. Note that the dOlnain is convex and includes ~ll of 1.0 in some neighborhood of the origin. Note too that the domain of G is all of ~ if (and only if) R/ -I for all j. i.e . • if there is no probability of total loss on any investment. The domain of G includes the interior of \ if R;; -I. Both domains are particularly simple J . and most cases of interest are included. If fl' .... ~ arc chosen so that I+fr + .. ·+fr ;;0 for some in ... ·.m. I l.ml K k.,,\ I k then P(fIXn •1 + ... + ~Xn.k;; 0) = € > 0 for all nand ruin occurs with probability I . . Computational procedures for finding an op- timal fixed · fraction strategy (generally unicjue in our present setting) are based on the theory of concave (dually. convex) functions [29J and will be presented elsewhere . (As Hakansson [11, p . 552] has noted. " ... the computational aspects of the capital growth model arc [presently) much less advanced" than for the Markowitz model. J The theory lnay be extended to n10re general randoITl variables and to dependence between dif- ferent time periods . Most important, we may include the case where tht! investment universe changes with the time period, provided only that there be some mild regularity conditions on the X ..• such as that they be uniformly a . s. bounded 1,) and that they do not tend to 0 uniforml y as i" ~ . (See [IS). and the generalization of the Bernoulli trials case as applied to blackjack in [26).) The techniques rely heavily on those used to genera- lize the law of large numbers. Transactions costs, the use of margin, and the effect of tax'es can be incorporated into the theory. Bellman's dynamic programming method is used here. The general procedure for developing the theory into a practical tool imitates Markowitz [16]. Markowitz requires as inputs estimates of the expectations, standard deviations, and co- variances of the Xi,j' We require joint proba- bility distributions. This would seem to be a much more severe requirement, but in practice does not seem to be so [16. pp. 193-4. 198-201). Among the actual i;"puts which Markowitz chose were (I) past history [16. ex .• 8-20). (2) probability beliefs of analysts (pp. 26-33). and (3) models, most"notably regression models, to predict future performance from past data (p. 33 • . pp. 99-100). In each instance one can get enough additional information to estimate E log (XnlXn _1). There are, however, two great difficulties which all theories of portfolio selection have, in- cluding ours and that of Markowitz. First, there seems to be no established method for generally --- ## Page 112 Portfolio Choice and the Kelly Criterion predicting sC"curity prices which gives an edge of even a few pc 1" cent. The random walk is the best model for security prices today. (See [7, 10j.) The second difficulty is that for portfolios v ... ith 11lany securities the volume of inputs called for is prohibitive: for 100 securities, Markowitz requires 100 expectations and 4950 cQvar.iances;. and our theory requires somewhat more infor- mation . Although considerable attention has been given to finding condensed inputs that can be used instead, this aspect of portfolio theory still seems unsatisfactory. In the fifth section we will show how both the se difficulties were overcome in practice by an institutional investor. That investor, guided by the Kelly criterion, then outperformed for the year 1970 everyone of the approximately400 Mutual Funds listed by the S & P stock guide! But fir.t we relate our theory to that of Markowitz. 4. Relation to the Markowitz theory; solution to problems therein. The most widely used guide to portfolio selection today is probably the Mar- kowitz theory . The basic idea is that a portfolio PI is supedor to a portfolio Pz If the expecta- tion ("gain'l) is at least as great, i.e., E(P1) 1;E(PZ) and the standard deviation ("risk") is no greater, i.e., a{PI)~a(pZ)' with at least one in- equality. This partially orders the set (} of port- folios. A portfolio such that no portfolio is superior (i.e., a maximal portfolio in the partial ordering) is called efficient. The goal of the portfolio manager is to determine the sct of effi- cient portfolios, from which he then makes a choice ba sed on his needs. This is intuitively very appealing: It is based on standard quantities for the securities in the portfolio, namely expectation .. standard deviation, and covariance (needed to compute the variance of the portfolio from that of the component secur- ities). It also gives the portfolio manager ''choice.1I As Markowitz [16, Chapter 6) has pointed out, the optimal Kelly portfolio is approximately one of the Markowitz efficient portfolios under cer - tain circmTIstances. If E= E{P) and R= P-I is the return per unit of the portfolio P, let log P = log (ltR) = log ( (ltE)+ (R - E)). Expanding in Taylor's series about ItE gives log P= 10g(I+E) + (R-E)/{l +E) - (R_E)Z /2{l +E)Z +higher order terms. Taking expectations and neglecting higher order terms gives ElogP= 10g{l+E)- J(P)/2{l+E)Z. This leads to a simple pictorial relationship with the Markowitz theory. Consider the E-a plane, and plot (E(P), a(P)) for the efficient portfolios. The locus of efficient portfolios is a convex non-decreasing curve which includes its endpoints (Figure I). Then constant values of the growth. rate 219 85 FIGURE 1. GROWTH RATE G (RETURN RATE' R) IN THE E-a PLANE ASSUMING THE VALIDITY OF THE POWER SERIES APPROXIMATION. 1.00 .90 and A.eli-I .80 .7 .60 .W AO .50 .60 .70 ' R<49% R'6~% G=ElogP approximately satisfy G=log{l+E) _ aZ(p)/2{l+El. This family of curves is illus- trated in Figure I and the {efficientj portfolio which maximizes logarithmic utility is (approxi- mately) the one which lies on the greatest G curve. Because of the convexity of the curve of efficient portfolios and the concavity of the G curves, the (E, a) value where this occurs is unique . The · approximation to G breaks down badly in some significant practical settings, including that of the next section. But for portfolios with large numbers of "typical" securities, the a"pprax ... imation for G will generally -provide an efficient pot.'tiolio which approximately maxilnizes asset growth. This solves the portfolio manager's problem of which efficient portfolio to choose. Also, if he repeatedly chooses his portfolio in successive time periods by this criterion he wi ll tend to nlaximize the rate of growth of his as:i (' l s, i.c.,. nlaximize "performance. II We see also that in this instance the problem is reduced to that. of finding the efficient portfolios plus the easy step --- ## Page 113 86 of using Figure 1. Thus if the Markowitz theory can be applied in practice in this setting, so can our theory. We have already remarked on the ambiguity of the set of efficient portfolios. and how our theory resolves them .. To illustrate further that such ambiguity repre.sent~ a defect in the Marko- witz theory, let X1 be uniformly distributed over [1.3]. let Xz be uniformly distributed over [10. 100], let cor(~.Xz)= 1. and suppose these are the only securities. Then ~ and Xz are both efficient with 01 < 0z and El < ~ so Markowitz' theory does not choose between them. Yet "everyone ll would choose Xz over ~ because the worst outcome with Xz is far better than the best outcome from ~. (We presented this example in [Z6). Hakansson [11] presents further examples and an extended analysis. He formalizes the idea by introducing the notion of stochastic dominance: X stochastically dominates Y if P(X~Y) = 1 and P(X>Y) > O. It is easy to prove the Lemma: An E log X optimal portfolio is never stochastically dominated. Thus our portfolio theory does not have this defect. ) There are investment universes ()\" .. JXn~ such that a unique portfolio P maximizes ElogP yet P is not efficient in the sense of Markowitz . Hence choosing P in repeated independent trials will outperform any strategy limited to choosing efficient portfolios. In addition. the optimal Kelly strategy gives positive growth rate, yet sOlne of the Markowitz- efficient strategies ·give negative growth rate and ruin after repeated trials. We gave such an example in [Z6] and another appears in [ll]. See also [ll. pp. 553-4] for further discussion of defects in the Marko .. witz model. 5. The theory in action: re suIts for a real insti- tutional portfolio. The elements of a practical profitable theory of convertible hedging were pub- lished in [Z7]. Thorp and I0 years after .the hedge was initiated, is lognor mally distributed with density 221 DJIA Starting Gain Over Closing Chg. Even With DJIA DJIA+ ('/0)++ DJIA+ (%) 855 0. 0 855 0.0 800 - 6.3 889 + 10.3 758 -11.3 974 + 25.3 839 1.8 1034 + 22.8 891 + 4 .2 1196 + 35.7 r::-: -1 [ 2 2] t (x)= (xo,,2n) exp -(Iogx-IJ) 120 ,hence t 2 mean E(St) = 1xp(lJ+0 /2) and standard deviation O(St) = E(St) (eO -I J12. The functions 1.1 "U(t) and 0" oCt) depend on the stock and on the time t. If t is the tin"le in years until St is realized, it is plausible to assume U(t) = log S +mt and O(tr = a2t, o where So is the present stock price and n1. and a are constants depending on the stock. For a de- tailed discussion, see [I, 19]. Then E(S)=S exp[(m+a2/l)t] and a mean in- t 0 crease o'f 10%/yea~ is approximated by setting m+}/l = .1 . If we estimate} from past price changes we can solve for m . In the case of Kaufman and Broad it is plausible to take 0;'.45 whence }= ,}o, .20. This yields m,. O. We then find O(St) = .5250 • It is by no means established that the log- norm~l model is the app·ropriate one for stock price series [7, 10]. However, once we clarify certain. general principles by working through our example on the basis of the lognormal model, it can be shown that the results are substantially unchanged by choosing instead any distribution that roughly fit. observation! For a time of one year, a computation shows the return R(S) on the stocl< to be +10.5%, the r("!- turn R(W) on the warrant to be +34.8%, O(S) = .52. cr(W) = . 92 I and the correlation coefficient cor(S, W) = .99. The difference" in R(S) and R(W) shows that the·warrant is a nnlch. better buy than the comnlono Thus a hedge long warrants and short common has.a substantial positive expec- tation. The value cor(S, W)= .99 shows that a --- ## Page 115 88 E. 0. Thorp FIGURE l. PRICE HISTORY OF KAUFMAN AND BROAD COMMON, S, VERSUS THE WARI{ANTS, W. The points moved up and to the right until they reached the neighborhood of (38, 17). At this point a 3:2 hedge (15,000 warrants long, 11,200 common short) was instituted. As the points continued to move up and to the right during the next few n'lonths , the position was closed out in stages with the final liquidation at about (58, 36). 2 w 25 hedge corresponding to the best linear fit of W to S has a standard deviation of approximately (1- . 99)1/2= . 1 which suggests that o(P) for the optimal hedged portfolio is probably going to be close to .1. The high return and low risk (or the bedge will remain, it can be shown, under wide variations in the choice of m and a. To calculate the optimal mix of warrants long to common short we Inaximize G(fl'~) = E log (I +)S+~ W) . The detailed computational procedures are too lengthy and involved to be presented here . We plan to present them elsewhere. Our instit\ltional investor considered posi- tions already held, some of which might have to be closed out to release a s sets, and also other current candidates for inve stment. The decision was made to short COnllTIon and buy warrants ill the ratio of three shares to four. The initial Jnarket value of the long side was about 14% of assets and for the short side about 20% of assets The net profit, in te rms of the initial nl~rket value of the long side, was about 20 ~o in six months . This resulted fron1 a 'move in the 222 35 w·s 40 . E{Sll,; <8.63 WIS, ) i. 29.66 Tc)'111S of warrant : 1 \va r r anl + $21 . 67 .... I sha r e com nion stock un til 3-1-74 . Full p r 0 - tcc tion again s t diluti on . The con'rany has thf! right tl) r t"!ou ce the e xerc i se pl'ice fOl" lC'nl po- r a ry periods ; 750,OOO ... n i !Tt.ulb ; and 5, 940,000 conunon out. standing . COlllmon dividends Q .05 ex 10-26-70 . comlnon frail,} about 40 to a l nlost 60, 6. Cdntluding rema r ks , As remarked a bo .... e . we do not propose logarithnlic utility a s dc ::-. ~~ riptiv e oC actual invcstJne nt be havior, nor do we be li e ve ,anyone utility fun c tion could suffi ce . It wou ld be, o! interest, howe ve r, to have eln pirkc:d c-..tide nc:e showing areas of b e havio r which are cha.rac ter- ized adequately b y logarithm ic utility, Neitl: c rdo we intend Ibgarithmic utility to be prcdictivl!; again, it would be of inter e st to know what it does predict. We only propose the theory t.o be nonn ative or prescript'ivc, and only for those in. stitu tion s, groups, or individuals who s e ove rriding curren t obje ctive is n1aximization of the rate of as se t growth. Those with a diffe rcnt "prin1c eli r c ctive" may find that another \ltility function i$ a he tt e l' gUide. We have found ElogX to be ':alu"bl ~ a ' a qualitative guide and su ggest that thi s could. be its 'most inlportant \1 se , Once fa ni ilia rity Y\ l~ ' h its properties is gained, Inany invc~tI1"H~, n t. fl ,)" cisions can 'be guided by it without c Olnple x supporting ca lc:ulation s, What sort of econom,ic. b ehavi or can be E:.x ~ --- ## Page 116 Portfolio Choice and the Kelly Criterion pected from followers of E log X ? Insurance is' "explained,11 Le., even though it is a ne gative expectation investment {or the insured and we aS5unH: both insurer and insured have the same probability information, it is often optitnal for hin") (as well as for the insurance company) to in- sure [3]. It usually tUrns out that insurance against large losses is indicated and insurance against sIl1all losses is not. (Don It insure an old car fo,' collision, take $200 deductible, not $25, etc. ) We find that if all parties to a security trans- action are followers of E log X they will often find it mutually optimal to trade. This may be true whether the transactions be two party (no brokerage), or three party (brokerage), and whether or not the parties have the same proba- bility information about the security involved, or even about the entire investment universe. Maximizing logarithmic utility excludes port- folios which have positive probability of total loss of assets. Yet it can be argued that an impover- ished follower of E log X might in some instances risk "everyt.hing." This agrees with some ob- served behavior, but is not what we might at first e xpect in view of the prohibition against positive probability of tGtal loss . But consider each indi- vidual as a piece of capital equipment with an assignable monetary value. Then if he risks and loses all his cash assets, he hasn't really lost everything [3J. All of us behave as though death itself does not have infinite negative utility. Since the risk of death, although generally srnall, is ever pre- sent, a negative infinite utility for death would n1ake all expected utilities negative infinite and utility theory meaningless. In" the case of loga- rithlnic utility as applied to the extended case of the (monetized) individual plus all his resources, death should be assigned a fin"ite, though large and negative, utility. The value of this "death constant ll is an additional arbitrary assumption for the enlarged theory of logarithmic utility. In the case of investor s who behave accord- ing to E log X (or other utilities unbounded helow), it might be possible to discover their tacit , 'death constants. ,. Hakansson [l!, p. 551J observes that loga- rithnlic utility exhibits decreasing absolute risk aversion in agreement with deductions of Arrow and other s on the qualities of "reasonable ll utility functions . Hakansson says, IIWhat the relative risk aversion index [given by -xU'(x)!U'(x)]would look like [or a meaningful utility function is less cleaf . .. In. view of Arrow' s conclusion that ' . . . broadly speaking, the relative risk aversion must. hover around 1, being, if anything, some- what less for low wealths and somewhat higher fo1' high wealths ... I the optinial growth model scerns to be on safe ground. 1; As he notes, for U(x):;: logx, the relative risk aversion "is pre- ci~cly 1. However", in both the exte~sion to 223 89 valuing the individual as capital equipment, and the further extension to include the death constant, we are led to U(x)=log(x+c) where c is positive. But then the relative risk aversion index is x/(x+c) which behaves strikingly like Arrow's description. See also the discussion of U(x) ~ log(x+c) in [8, p . 103, p . 112J. Morgenstern [17J has forcefully obs cr\'~d that assets are random variables, not num bers , and that econolni~ theory generally does not in- corporate" this. To replace assets by their ex- pected utilit>' in valuing companies, portfolios, property and the like, allows for comparisons when asset values are given as random variables. We think logarithmic utility will often be c.ppr,?- priate for such valuation. I wish to thank Jame5 Bicksler for several stimulating and helpful convers~tions. FOOTNOTES '::Edward O. Thorp is prafes sor, Department of lvlathematics, University of California at Irvine. This .research was supported in part by the Air Force Office of Scientific Research under Grant AF-AFOSR 1870A. An expanded Version of this paper will be submitted for publication elsewhere. lInformation on descriptive utility is sparse; how rnany writer s on the subj~ct have even been able to dcterlnine for us their own personal utility? 2parenthetical explanation added since we have used n. 3"Almost certainly" and "almost surely" are synonymous. 4The proof of this theorem, and some further re- sults obtained with R. Whitley, will appeal' else- where. REFERENCES [1) Ayres, Herbert F., "Risl~ Aversion in the Warrant Markets, II S.M. Thesis, M. 1. T . , Industrial Management Review 5: 1 (1963), 45-53 . Reprinted in Cootner, pp. 479-505. [2J Bellman, R. and I(alaba, R . , "On the Role of Dynamic Programming in Statistical Comn1un- ication Theory, 11 IRE Transactions of the Pn~"­ fessional Group on Information Theory, IT- 3: 3 (1957), 197-203. [3] Bernoulli, Daniel, IIExposition of a New Theory on Hit:~ :t-..{easurcment of Risk, 11 Ec~~~­ metrica, XXlI (January 1954), 23-36, trans. ~Sommer. [4J Borch, Karl H., The Economics ofUncerla;n'J:, Princeton Univer sity Presf.i, 1968 . [5] Brein1an, Leo, "Investment Policies for Ex- panding Businesses Optimal in a Long R\lIl Sens~," Naval Research Logistics Quarter ly, 7:4 (1960), 647-651. --- ## Page 117 90 [6] Breiman, Leo, "Optimal Gambling Systems for Favorable Games, II Fourth Berkeley Sym- posium on Probability and Statistics, I, (1961), 65-78. [7] Cootne.r, P. H., ed., The Random Character of Stock Market Prices, The M.1. T. Press, 19M. [8] Freimer, MarshaH and Gordon, Myron S. , IIlnvestment Behavior . With Utili~y a Concave Function of V/ealth,lI in K. Borch and J. MOBsin, eds., Risk and Uncertainty, New York: St. Martin'. Press, 1968, 94-115. [9] Gardner, Martin, IIMathematicalGames: The Paradox of the Non-Transitive Dice and the Elusive Principle of Indifference,lt Scientific American (December 1970), 110. [lOJ Gr"nger, Clive and Morgenstern, Oskar, Pre- dictability of Stock Market Prices, Lexington, Massachusetts: D. C. Heath and Company, 1970. [11] Hakansson, Nils, "Capital Growth and the Mean- Variance Approach to Portfolio Selection, II Journal of Finance and Quantitative Analysis (January 1971), 517-557. [12J Kassouf, Sheen T., "A Theory and an Econo- nletric model for CCmlTIori stock purcha se warrants,1I Thesis. Columbia University, 1965; New York: Analytic Publishers Com- pany, 1965. A regression model statistical fit of nornlal price curves for warrants ... There are la'rge systen1atic errors in the model due to faulty (strongly biased) regreS- sion techniques. The average n1.ean square error in the fit is large. Thus, it is not safe to use the model in practice as a predictor of warrant prices. However, the model and the methodology are valuable as a first qualita- tive description of warrant behavior and as a guide to a more precise analysis. [13J Kassouf, Sheen T., "An econometric model {or option price with in1plications for inves- tors' expectations and audacity, II Econo- metrica, 37:4 (1969), 685-694. Based on the thesis . The \'ariance of residuals is given as .24"8, or a standard deviation of about .50 in y, the normalized warrant price, and a standard error of about .34 ,. The mid-range of y varies {ron1 0 to .5 and is never greater, thus the caveat about not using the model for practical predictions! [I4J Kelly, J. L., "A New Interpretation ofInfor- ITlation Rate, 'I Bell System Technical Journal, 35 (1956), 917- 926. [15] Latane, HenryA., F~Criteria{orChoiceAn1.ong Risky Venture 5, II Journal of Political Economy, 67 (1959), 144-155. [16] Markowitz, H., Portfolio Selection, New York: 224 E. 0. Thorp John Wiley and Sons, Inc., 1959. Se e also the preface to the second printing of the Yale University Press, 1970 reprint. [17] Morgenstern, Oskar, On the Accuracy of Economic Observations, 2nd cd. revised, Princeton Univer.sity Press, 1'963. (18] lo..fossin, Jan, "OptiInal Multipcriod Portfolio Policies, 1\ Journalof Business {April 1968l. [19] Osborne, M. F. M . , "Brownian hlotion in the StockMarket,l1 Operations Research, 7 (1959). 145-173. Reprinted in Cootner, pp. 100- 128. [20] San1.uclson, Paul A., "Risk and Uncertainty: A Fallacy of Large Numbers," Scientia, 6th Ser., 57th Y,·., (April- May l~ [21] Samuelson, PaulA., "Lifetime Portfolio Se- lection by Dynan1ic Stochastic Prograrruning, 11 The Review of Economics and Statistics (August 1969), 239-246. [22] Samuelson, Paul A . , "The 'Fallacy' of Maxi- mizing the Geometric Mean in Long Sequen- ces of Investing or Gambling, II Unpublished preliITlinar)' preprint, 1971. [23J Schrock, Nicholas W., "The Theory ·of Asset Choice: Simultaneous Holding of Short and Long Positions in the FutUl'CS Market," Journal of Political Economy, 79:2 (1971 l, 270-293. [24J Shelton, John P . , "The Relation of the Price of a Warrant to Its As sociated Con1n:on Stock, II Financial Analysts Journal, 23: 3 (1967), 143-151; and Financial Analysts Journal, 23:4 (1967), 88- 99. [25J Thorp, Edward, "A Winning Bet in Nevada B~ccarat,1I Journal of the American Statis- tical Association, 61, Part 1(1966), 313- 328. [26J Thorp, Edward, "Optimal Gambling Systems for Favorable Garnes," Review of the Interna- tional Statistical Institute, 37:3 (1969), 273-293. [27] Thorp, E. and Kassouf, S., Beat the Market, New York : Random House, 1967. [28] Todhunter, 1. , A History oHhe Mathematical Theory of Probability, lsted., Cambridge, 1865, as reprinted by Chelsea, New York, 1965. (See pp. 213 ff. for details on Daniel Bernoulli's use of logarithmic utility.) [29] Wagner, Harvey M., Principles of Opera- tions Research, With Application to Mana- gerial Decisions, New Jersey: Prentice .. Hall, 1969. [30J Williams, J. B., "Speculation and the Carry- over, II Quarterly .TournaI of Economics, 50, (May 1936), 436-455. --- ## Page 118 91 ECONOMETRICA VOLUME 38 September, ]970 NUMBER 5 8 OPTIMAL INVESTMENT AND CONSUMPTION STRATEGIES UNDER RISK FOR A CLASS OF UTILITY FUNCTIONS I By NILS H. HAKANSSON l This paper develops a sequential model of the individual's economic decision problem under risk. On the basis of this model, optimal consumption, investment, and borrowing- lending strategies are obtained in closed form for a class of utility functions. For a subset of this class the optimal consumption strategy satisfies the permanent income hypothesis precisely. The optimal investment strategies have the property that the optimal mix of risky investments is independent of wealth, noncapital income, age, and impatience to consume. Necessary and sufficient conditions for long-run capital growth are also given. I . INTRODUCTION AND SUMMARY TillS PAPER presents a normative model of the individual's economic decision problem under risk. On the basis of this model, optimal consumption, investment, and borrowing-lending strategies are obtained in closed form for a class of utility functions. The model itself may be viewed as a formalization of Irving Fisher's model ofthe individual under risk, as presented in The Theory of Interest [4] ; at the same time, it represents a generalization of Phelps' model of personal saving (10). The various components of the decision problem are developed and assembled into a formal model in Section 2. The objective of the individual is postulated to be the maximization of expected utility from consumption over time. His resources are assumed to consist of an initial capital position (which may be negative) and a noncapital income stream which is known with certainty. The individual faces both financial opportunities (borrowing and lending) and an arbitrary number of productive investment opportunities. The returns from the productive opportuni- ties are assumed to be random variables, whose probability distributions satisfy the "no-easy-money condition." The fundamental characteristic of the approach taken is that the portfolio composition decision, the financing decision, and the consumption decision are all analyzed simultaneously in one model. The vehicle of analysis is discrete-time dynamic programming. In Section 3, optimal strategies are derived for the class of utility functions Lf=, I aJ-1u(cj ), 0 < a < 1, where cj is the amount of consumption in periodj, such that either the relative risk aversion index, -cu"(c)/u'(c), or the absolute risk I This paper was presented at the winter meeting of the Econometric Society, San Francisco, California, December, 1966. 2 This article is based on my dissertation which was submitted to the Graduate School of Business Administration of the University of California, Los Angeles, in June, 1966. I am greatly indebted to Professors George W. Brown (committee chairman), Jacob Manchak, and Jacques Dreze for many valuable suggestions and comments and to Professors Jack Hirshleifer, Leo Breiman, James Jackson, and Fred Weston for constructive criticisms. I am also grateful to the Ford Foundation for financial support over a three-year period. 587 --- ## Page 119 92 N. H Hakansson 588 NILS H. HAKANSSON aversion index, -u"(c)/u'(c), isa positive constant for all c ~ 0, ie., u(c) = cY, o < y < 1, u(c) = -c- r, y > 0, u(c) = log c, and u(c) = -e- r., y > O. Section 4 is devoted to a discussion of the properties ofthe optimal consumption strategies, which turn out to be linear and increasing in wealth and in the present value of the noncapital income stream. [n three of the four models studied, the optimal consumption strategies precisely satisfy the properties specified by the consumption hypotheses of Modigliani and Brumberg [9] and of Friedman [5]. The effects of changes in impatience and in risk aversion on the optimal amount to consume are found to coincide with one's expectations. [n response to changes in the "favorableness" of the investment opportunities, however, the four models exhibit an exceptionally diverse pattern with respect to consumption behavior. The optimal investment strategies have the property that the optimal mix of risky (productive) investments in each model is independent of the individual's wealth, noncapital income stream, and impatience to consume. It is shown in Section 5 that the optimal q1ix depends in each case only on the probability dis- tributions of the returns, the interest rate, and the individual's one-period utility function of consumption. This section also discusses the properties of the optimal lending and borrowing strategies, which are linear in wealth. Three of the models always call for borrowing when the individual is poor while the fourth model always calls for lending when he is sufficiently rich. The effect of differing borrowing and lending rates is also examined. Necessary and sufficient conditions for capital growth are derived in Section 6. It is found that when the one-period utility function of consumption is logarithmic, the individual will always invest the capital available after the allotment to current consumption so as to maximize the expected growth rate of capital plus the present value of the noncapital income stream. Finally, Section 7 indicates how the preced- ing results are modified in the non stationary case and under a finite horizon. 2. THE MODEL In this section we shaH combine the building blocks discussed in the previous section into a formal model. The following notation and assumptions will be employed : Cj : amount of consumption in period j. where cJ ~ 0 (decision variable). U(c •• c1• c3••• .): the utility function. defined over all possible consumption programs (e •• C1• e3 • . . . ). The class of functions to be considered is that of the form (1) U(el.c1. c1 . .. ·) = u(c l ) + aU(c2.cJ.C4• · .. ) '" = r aJ-'u(c j ). 0 < a < I. j- I It is assumed that u(c) is monotone increasing. twice differentiable. and strictly concave for c ~ O. The objective in each case is to maximize E[U(c •• c, •. . . )]. i.e .• the expected utility derived from con- sumption oyer time. J x j : amount of capital (debt) on hand at decision pointj(the beginning ofthejth period)(state variable). y : income received from noncapital sources at the end of each period. where 0 '" y < 00 . J While we make use ofthe e.xpec!ed utility theorem. we assume that the von Neumann-Morgenstern postulates (12] have been modIfied to such a way as to permit unbounded utility functions. --- ## Page 120 Optimal Investment and Consumption Strategies under Risk/ or a Class o/Utility Functions 93 OPTIMAL INVESTMENT 589 M; the number of available investment opportunities. S : the subset of investment opportunities which it is possible to sell short. z/,: amount invested in opportunity i, i =< I, . . . , M, at the beginning of the jth period (decision variable). r - I ; rate of interest, where r > I. p/ ; transformation of each unit of capital invested in opportunity i in any period j (random variable); that is, if we invest an amount 8 in iat the beginningofa period. we will obtain 11,8 at the end of that period (stochastically constant returns to scale, no transaction costs or tues). The joint distribution functions ofthe P"~ i ... I •. .. • M. are assumed to be known and independent with respect to time j. The (P,) have the following properties: (2) PI = r, (3) 0 ~ p, < ex> (i = 2 •. .. • M). (4) Pr { I: (fJ, - r)Oj < o} > 0, j: Z for all finite 8, such that 9, ~ 0 for all j, S and OJ '" 0 for at least one i. fj.xJ) : expected utility obtainable from consumption over all future time. evaluated at decision pointj. when capital at that point is xJ and an optimal strategy is followed with respect to consumption and investment. Y: present value at any decision point of the noncapital income stream capitalized at the rate of interest, i.e .• Y = yllr - I). ii == (Vl" .'. , II.,) ; a vector of real numbers. h(ii) == E[ ~t (fJ, - rJII; + r)J. k : maximum of h{ii) subject to (27) and (28) (see (26». ii· : vector ii which gives mallimum k of h(ii) (see (26)). IW v· : ~ v;. i* 1 c·(x) ; an optimal consumption strategy. z~(x): an optimal lending strategy. z;(x) : an optimal investment strategy for opportunity i. j = 2 •.. . • M. Sj == xJ + Y. The limitations of utility functions of the form (1) are well known and need not be elaborated here. Condition (4) will be referred to as the "no-easy-money condition." In essence. this condition states (i) that no combination of productive investment opportunities exists which provides. with probability 1, a return at least as high as the (borrowing) rate of interest; (ii) that no combination of short sales exists in which the probability is zero that a loss will exceed the (lending) rate of interest; (iii) that no combination of productive investments made from the proceeds of any short sale can guarantee against loss. For these reasons, (4) may be viewed as a condition that the prices of'the various assets in the market must satisfy in equilibrium. Consumption and investment decisions are assumed to be made at the beginning of each period. The amount allocated to consumption is assumed to be spent immediately or, if spent gradually over the period, to be set aside in a nonearning account. We also assume that any debt incurred by the individual must at all times be fully secured, i.e., that the individual must be solvent at each decision point. In view of the "no-easy-money condition" (4), this implies that his (net) debt cannot exceed the present value, on the basis of the (borrowing) rate of interest, of his noncapital income stream at the end of any period. --- ## Page 121 94 N H Hakansson 590 NILS H. HAKANSSON We shall now identify the relation which determines the amount of capital (debt) on hand at each decision point in terms of the amount on hand at the previous decision point. This leads to the difference equation (5) where (6) M X}+, = rZ'j + L PIZij + Y 1= 2 M L Zlj = x} - c) 1= , (}=1.2 •. . . ) U=I.2 •.. . ). The first term of (5) represents the payment of the debt or the proceeds from savings. the second term the proceeds from productive investments, and the third term the noncapital income received. Combining (5) and (6) we obtain (7) M X)+, = L (P, - r)zij + rex) - cj} + Y (j= 1,2, ... ). i=1 This is the difference equation. then, which governs the process we are about to study. The definition of fJ.x j) may formally be written (8) flx}) == max E[U(c,. Cj+ " C)+1. ·· ·)]IXj . From (I) we obtain. by the principle of optimality.4 for allj, (9) fiXj) = max E[u(cj ) + oc{max E[U(cj + " Cj+ 1'" ·nIXj+, nlxj' since we have assumed the {PI} to be independently distributed with respect to time j. By (8), (9) reduces to (l0) fJ.x,) = max {u(c}) + rxE[Jj+ ,(x)+ d]}, allj. Since by our assumptions we are faced with exactly the same problem at decision pointj + 1 as when we are at decision pointj. the time subscript may be dropped. Using (7), (10) then becomes (11) f(x) = ~~j~ {U(C) + (XE[ft~2 (Pi - r)zi + rex - c) + Y) J} subject to (12) C ~ 0, (13) ZI ~ 0, ijS. and (14) Pr { f (Pi - r)zl + rex - c) + Y ~ - Y} = 1 1-1 at each decision point. Expression (14), of course, represents the solvency constraint. • The principle of optimality states that an optimal strategy has the property that whatever the initial state and the initial decision. the remaining decisions must constitute an optimal strategy with regard to the state resulting (rom the first decision (2. p. 83]. --- ## Page 122 Optimal Investment and Consumption Strategies under Risk/ or a Class o/Utility Functions 95 OPTIMAL INVESTMENT 591 For comparison, the model studied by Phelps [10] is given by the functional equation (15) I(x) = max {u(c) + (XE(f(fJ(x - c) + y)]} . 0"''',,, In this model, all capital not currently consumed obeys the transformation p, which is identicalJy and independently distributed in each period. Since the amount invested, x - c, is determined ODce c is known, (15) has only one decision variable (c).J Since x represents capital, I(x) is clearly the utility of money at any decision poiDt j. Instead of being assumed, as is generally the case, the utility function of money has in this model been induced from inputs which are more basic than the preferences for money itself. As (11) shows, I(x) depends on the individual's preferences with respect to consumption, his noncapital income stream, the interest rate, and the available investment opportunities and their rIskiness. 3. THE MAIN THEOREMS We shall now give the solution to (11) for the class of one-period utility functions 1 (16) u(c)=-cY , 0<'1<1 (Modell); (17) (18) (19) y 1 u(c) = -c1 , y u(c) = log c, u(c) = _e- YC , '/<0 '/>0 (Model II); (ModeJ III); (Model IV). 'Phelps gives the solution to (15) for the utility functions u(c) = c',O < }' < I, u(c) = - c-', }' < 0, and for u(c) = log c when y = O. Unfortunately. this solution is incorrect in the general case, i.e., whenever y> 0 and the distribution of fJ is nondegenerate. For example, when u(c) = -c - >, the solution is asserted to be, letting P ;:; E[P- >], [ (,,/J)-II(,+II J'+I[ Y J-Y (I Sa) fIx) = - (a./J)-III>+I1- 1 x + /1-11, _ 1 ' (lSb) c(x) = [1 - (17./1)11(" 1I{ x + p-I;' _ IJ, whenever a./J < 1. But for this to be a solution, it would be necessary that one be able to write (lSc) E[(P(x - c) + y)-') = E[P-'1(~ - c -+ F.[p!.)"II,r' which is clearly impossible unless the distribution of P is degenerate or y = 0 or both. The right side of (1 Sc) may, of course, be regarded as a first-order approximation of the left side when the variance of fJ is small, bUllhis negates the presence ofuncerlainty. In fact,the preceding solution holds even under certainty only when "P ~ 1 and x ~ [("p)-I/('+ I) - I]YI

0 and the distribution of fJ is nondegenerate. It is ironic, therefore, that when one generalizes Phelps' problem by introducing the possibility of choice among risky investment opportunities and the opportunity to borrow and lend (see (11)), an analytic solution does exist (as will be shown). It is the second of these generalizations which guarantees the solution in closed form. --- ## Page 123 96 N H Hakansson 592 NILS H. HAKANSSON Pratt [11] notes that (16H18) are the only monotone increasing and strictly concave utility functions for which the relative risk aversion index u"(c)c q*(c):: --- u'(c) (20) is a positive constant and that (19) is the only monotone increasing and strictly concave utility function for which the absolute risk aversion index (21) u"(c) q(c) E - u'(c) is a positive constant.6 THEOREM 1 : Let u(c),«, y, r, {PI}' and Y be defined as in Section 2. Then, whenever u(c) is one of the functions (16H18) and Icy < 1/« in Model I, a solution to (11) subject to (12H14) existsfor x ~ - Yand is given by (22) f(x) = Au(x + Y) + C, (23) c*(x) = B(x + Y). (24) z~(x) = (1 - B}(l - v*)(x + Y) - Y, (25) z;(x) = (l - B)v;(x + Y) where the constants v; (v· :: L~2 v;) and k are given by (26) k E E[ ut~2' (PI - r)v; + r)] = max E[U( ~ (PI - r)v, + r)]. {v,} 1_ 2 subject to (27) VI ~ 0, i ; S. and (28) pr{ ~ (PI - r)vl + r ~ o} = I, 1-2 and the constants A, B, and C are given by (i) in the case of Models I-II, A = (I - (< O. 1 The author gratefully acknowledges a debt to Professor George W. Brown for several valuable suggestions concerning the proof of the closure and the boundedness of D. --- ## Page 125 98 N. H Hakansson 594 NILS H. HAKANSSON By the "no-easy-money condition" (4), b(v') ~ -r for v' E D, b(vo) = 0, and b(v) < o for all v :I: vO. Applying the "no-easy-money condition" with respect to the point vA and using the inequality Pr {.f {fJ1 - r)-tvi < ).b} > 0, , .. 2 we obtain that -tb(v') = b(Av'). But when -tb(v') < - r, or A > - r/b(v'), the point vA cannot lie in D since -t > - r/b(v') implies that Pr { f {fJ, - r)-tv; + r ~ o} < 1. 1-2 Thus, AO := - r/b(jj') is the greatest lower bound on -t such that jjA t D. Since Aob(v') = -r, jjAO ED and is in fact the point farthest from VO lying on the line through jjO and jj' and belonging to D. We shall only sketch the remainder of the proof establishing the closure and bounded ness of D. Let v :I: VO be the limit of a sequence of points V(II' e D. Since each point in the sequence belongs to D, b(V(II') ~ - r for all n. It can now be shown, by utilizing the fact that I;t!. 2 (PI - r)vl is continuous at any v :I: vO, uniformly with respect to the P,'s on any bounded set, that limll ... co b(V'"') ~ b(v), which implies that iJ E D. Consequently, D must be closed. The bounded ness of D is established as follows. Let S R be the set of points jj such that liJl = R > O. SR is then clearly both closed and bounded. If D' := D (', SR is empty, the boundedness of D follows immediately. Let us therefore assume that D' is nonempty ; in this case D' is also bounded and closed since D is closed and S R is bounded and closed. If v is a limit point of the sequence < v,n,) such that v,n' e D', we must have that jj E D' since D' is closed. But b(v) < 0 by the "no-easy-money condition" (4), since jj :I: jjO by assumption. Therefore, since we already have that limn .... co b(jj'JI') ~ b(v), 0 cannot be a limit point to the sequence ),u(w.) + (I - A.)u(wl ), 0 < ), < 1. Consequently, (34) implies E[u(..J.w. + (1 - ..J.)w2 )] > ..1.E[u(w.)] + (1 - ..1.)E[U(W2»), ii::#: ii~ E D, 0<), < 1, which, by (32) and (33), in turn implies that h is strictly concave on D. Since our problem has now been shown to be one of maximizing a strictly concave function over a nonempty, closed, bounded, convex set, it follows directly that the function h has a maximum and that the v; are finite and unique. A number of corollaries obtain from this lemma which we shall also require in the proof of Theorem 1. COROLLARY 1: Let u(c), {PI}, and r be defined as in the Lemma. Moreover, let u(c) be such that it has no lower bound. Then the v; which maximize (31) subject to (27) and (28) are such that Pr { . f (PI - r)v; + r > o} = 1. 1=2 The proof is immediate from the observation that h -+ - ex::: as the greatest lower bound on b such that Pr I:~ 2 (jJ1 - r)Vi + r < b} > 0 approaches 0 from above. COROLLARY 2 : Let u(c), {Pi}, and r be defined as in the Lemma. Then the maximum of the function (31) subject to the constraints (27) and (28) is greater than or equal to u(r). PROOF: When Vi = 0 for all i, ~hich is always feasible, we obtain by (31) that h = u(r). COROLLARY 3: Let u(c), {PI}, and r be defined as in the Lemma. Moreover, let u(c) be such that u(c) ~ b. Then the vectors fj which satisfy (27) and (28) are such that h(ii) ;;;;; E[ ut~2 (jJ1 - r)vl + r) ] < b. --- ## Page 127 100 N. H Hakansson 596 NILS H. HAKANSSON The proof is immediate from the observation that u(c) is monotone increasing and that r, {PI}' and the feasible VI are bounded. We are now ready to prove the theorem. The method of proof will be to verify that (22H25) is the (only) solution to (11).8 PROOF OF THEOREM 1 FOR MODELS I-II: Denote the right side of(11) by T(x) upon inserting (22) for f(x). This gives, for all decision pointsj, (35) T(x) = max{!cY + cr(1 - (crky)J/(\-YIY-JE[!( f (PI - r)zl <.(0" Y y 1= 2 + r(x - c) + Y + Y)]} subject to (12) c ~ 0, (13) Z/ ~ 0, itlS, and (14) prt~2 0 and (39) Pr { f (PI - 7)z/l(s - c) + 7 ~ o} = 1, 1 .. 2 where s == x + Y. Under feasibility with respect to (14), we then obtain (40) T(x) = [max{~sy, T(x)}, 0 < y < 1, max {-oo, T(x)}, )' < 0, • A proof based on the method o( successive approximations may be (ound in (7). --- ## Page 128 Optimal Investment and Consumption Strategies under Risk/or a Class o/Utility Functions 101 OPTIMAL INVESTMENT 597 where (41) T(x) = sup{!eY + a(l - (aky)l/(1-Y'y.-l(s - c)y t.{f,) Y X E[~t~}PI - r)zj/(s - e) + r rJ} subject to (12), (38), (39), and (42) zd(s - e) ~ 0, i tI S, since (42) is equivalent to (13) in view of (38). But by (31) the expectation factor in (41) may be written (43) h(Z2/(S - e), ... , ZM/(S - e» and (26), the Lemma, and Corollary2 give (44) ky ~ r Y > 0 (Model I), (45) ky ~ r Y < 1 (Model II), while (26), the Lemma, and Corollary 3 give (46) ky > 0 (Model II). Thus, af/ah > 0 always in Model II and in Model I whenever 1 (47) ky < -a under feasibility. When ky > 1/a. in Modell, T(x) does not exist; when ky = l/a, (41) and (40) give T(x) = (l/y)sY :# f(x). Consequently, it remains to consider the case when aT/ah > O. Since the maximum of (43) subject to (42) and (39) is k by (26) and the Lemma, we obtain by the Lemma that the strategy Zi • --=Vj S - C (i = 2, ... , M) or (48) (i = 2, .. . ,M) is optimal and unique for every e which satisfies (12) and (38) when (38) holds. It is clearly also optimal when (36) and (37) hold. Consequently, (40) reduces to (49) T(x) = max {!eY + a.k(l - (aky)I/(1-Y'y- I(S - eY}. O~t~' y Since u(e) is strictly concave and u'(O) ;: co in Models I and II, T(x) is strictly concave and differentiable with an "interior" unique solution e*(x) whenever (50) { > 0 (Model I), a.k(l - (aky)I/(l-Y'y- 1 < 0 (Model II), --- ## Page 129 102 598 NILS H. HAKANSSON and s ~ o. In this case, setting dT/de = 0 and solving for e, we get, cy- I - aky(l - (aky)I/('-YI)Y- I(S - cy-l = 0 or (51) c·(x) = (1 - (aky)I/(I- YI)(x + f). N. H Hakansson In Model I, (50) is satisfied whenever (44) and (47) hold; as noted earlier, no solution exists in Modell for those cases in which ky ~ l/a. In Model II, (50) is always satisfied as seen from (45) and (46). Inserting (51) in (48) we obtain (i = 2, . .. • M) and (24) follows from (6) upon insertion of e*(x) and the z;(x). T(x) now becomes, upon insertion of c·(x) in (49), T(x) =!(1 - (aky)I}('-Y»)1sY + ak(l - (ocky)l/(l-Yly-lsY(aky)y/(I-YI y = !(1 _ (CXky)l/(l-Yly-lsY Y = f(x) and the solution clearly exists for s ~ 0 or (52) Xj ~ - f. Since (52) is an induced constraint with respect to period j - 1, it remains to be verified that (52) is either redundant or not effective in period j - 1. Because (52) is already present in period j - 1 through (14). the induced constraint (52) is redundant, which completes the proof. PROOF OF THEOREM I FOR MODEL III : Denote the righ t side of (II) T(x) upon inserting (22) for f(x). This gives, for all decision points j, T(x) = max {lOge + _CX_E[IOg (f (Pi - r)Zi <,(." 1 - 0: i = 1 + rex - c) + Y + f) ] + K} where subject to (12), (13), and (14). By the reasoning for Models I and II, we obtain (53) T(x) = max { - 00, rex)} --- ## Page 130 Optimal Investment and Consumption Strategies under Risk/or a Class o/Utility Functions 103 OPTIMAL INVESTMENT 599 where (54) T(x) = sup {lOg e + _a_log (s - e) <.{z,} 1 - a + I : aE[IOg t~2 (pj - r)zJ(s - e) + r)]} + K subject to (12), (38) s - e > 0, (42) and zJ(s - e) ~ 0 (39) Pr t ~2 (PI - r)zJ(s - e) + r ~ o} = 1. By (31), the next to last term in (54) can be written (55) a. -1 --h(Z2/{S - c), . .. , ZM/(S - e)) -a (i = 2, ... , M), where aT /oh > O. Since the maximum of (55) subject to (42) and (39) is (ak/l - a) by (26) and the Lemma, we obtain from the Lemma that (48) z;(x) = v;(x + Y - c) (i = 2, ... , M) is optimal and unique for every c which satisfies (12) and (38). Thus, (53) reduces, in analogy with Models I and II, to (56) T(x) = max loge + --Iog(s - c) + -- + K { a. ak} 0'<" I-a. I-a. where T(x) always exists since 0 < ex < 1; furthermore, T(x) is strictly concave and differentiable. Setting aT lac = 0 we obtain (57) c*(x) = (1 - a.)(x + Y) , z;(x) = a.v;(x + Y) (i = 2, ... , M), and (24), all unique. Inserting (57) into (56) gives a. a T(x) = log (1 - a.) + log s + 1 _ a log a. + 1 _ a. log s a.k a a 2 log a. 1X2k + 1 - IX + 1 - a. log (l - IX) + (1 _ a)2 + (1 _ a)2 = /(x). $ince/(x)exists for Xj ~ - Y, which as an induced constraint with respect to period j - 1 is made redundant by (14) for that period, the proof is complete. --- ## Page 131 104 600 NILS H. HAKANSSON When y = 0, the solution to (11) reduces to f(x) = Au(x) + C, c*(x) = Bx, z;(x) = (1 - B)(1 - v*)x, z;(x) = (l - B)v;x But then, letting s == x + Y, f(s) = Au(s) + C, c*(s) = Bs. Z·l(S) = (l - B)(1 - v*)s, z;(s) = (1 - B)v;s N. H Hakansson (i=2, . . . ,M). (i = 2, ... , M). As a result, except for z;(x + f). the solution to the original problem is not altered when the individual, instead of receiving the noncapital income stream in install- ments, is given its present value Y in advance. Thus, instead of letting x be the state variable when there is a noncapital income, one could let x + Ybe the state variable (pretending there is no income), as long as Y is deducted from z~(x + Y). Note that it is sufficient, though not necessary, for a solution not to exist in Model I that r Y ~ 1/a. (Corollary 2). THEOREM 2 : Let Ct, {P,}, r, y, and Y be defined as in Section 2. Moreover, let u(c) = _e- Yt for c ~ 0 where y > O. Then a solution to (11) subject to (12H14) exists/or x ~ - Y + [r/(y(r - 1 )2)] log (- akr) and is given by (58) (59) (60) (61) f(x) = __ r_(_a.kr)l/C,-l).-IYC'-ll/'Jex+ YJ , r - 1 r - 1 1 c*(x) = --(x + Y) - ( ) log (-akr), r y r - 1 .() x y log (-a.kr) - rv· ZlX =---+~"":"---'--- r r y(r - 1) , where the constants k and v; (v· == t~ 2 v;) are given by (i=2, . . .• M), (62) k == E[ _e-r.~2C1l j -')U1) = max E[ _e-r.~2C.8I-')tlj] subject CO (27) (VI) provided that (63) log ( - a.kr) + b(v·) ~ 0 --- ## Page 132 Optimal Investment and Consumption Strategies under Risk/or a Class o/Utility Functions 105 OPTIMAL INVESTMENT 601 where b(v*) is the greatest lower bound on b such that prL~2 (Pi - r)v: < b} > 0 and v· == (v;, . .. , v;'). Moreover, the solution is unique. Since the conditions under which Theorem 2 holds are quite restrictive. the reader is referred to [7] for the proof. Condition (63) insures that the individual's capital position x is nondecreasing over time with probability 1 ; it must hold for a solution to exist in closed form. The condition ar ~ 1 is a necessary, but not sufficient, condition for (63) to be satisfied. 9 4. PROPERTIES OF THE OPTIMAL CONSUMPTION STRATEGIES In each of the four models we note that the optimal consumption function c·(x) . is linear increasing in capital x and in noncapital income y. Whenever y > 0, posi- tive consumption is called for even when the individual's net worth is negative, as long as it is greater than - Yin Models I-III and greater than - Y + [r/(y(r - 1)2)) log ( - akr) in Model IV. Only at these end points would the individual consume nothing. Since x + Y may be viewed as permanent (normal) income and consumption is proportional (0 < B < 1) to x + Y in Models I-lII, we see that the optimal con- sumption functions in these models satisfy the permanent (normal) income hypotheses precisely [9, S, 3]. In each model, c*(x) is decreasing in a. Thus, the greater the individual's im- patience 1 - a is, the greater his present consumption would be. This, of course, is what we would expect. By (20) and (21), the relative and absolute risk aversion indices of Models I-IV are as follows: q*(c) = I - y q·(c) = I q(c) = y (Models I-II), (Model III), (Model IV). " For example, when u(c) = - e - .000 It, a = .99, y = S 10,000, r = 1.06, M = 2, and {J 2 assumes each of the values .96 and 1.17 with probability .5, a solution exists for :c ;;. $ - 22,986. For selected capital positions. the optimal amounts to consum~. lend. and invest in this case are as follows: x CO(x) z~ (x) z;(x) $ - 22.986 o 50.000 100.000 SOO.OOO 1.000.000 o $ 1.301 4,131 6.961 29.601 57,901 $-102,488 - 80.803 - »,633 13.537 390.897 862.597 $79.502 79.502 79.502 79.502 79.502 79.502 The maximum loss in each period from risky investmenl is $3,180. --- ## Page 133 106 N. H Hakansson 602 NILS H. HAKANSSON In Models I-II, we obtain (64) oB = _ (Ilk )1/(1- YI{~d(kY)/d(l - y)] _ log. (llkY)} 0(1 - y) Y ky(l - y) (l - y)2 where d{ky)/d(l - y) is negative whenever b(ii*) ~ J - r; otherwise the sign is ambiguous. Since ky > 0 and Ilky < 1, the sign of (64) is ambiguous in both cases; i.e., a change in relative risk aversion may either decrease or increase present consumption. In Model IV, on the other hand, c·(x) is increasing in y; i.e., a more risk averse individual consumes more, ceterus paribus. From (26) and (62) we observe that k is a natural measure of the "favorableness" of the investment opportunities. This is because k is a maximum determined by (the one-period utility function and) the distribution function (F); moreover, F is reflected in the solution only through k, andf(x) is increasing in k. Let us examine the effect of k on the marginal propensities to consume out of capital and non- capital income. Equation (29) gives oB = ~(llky)Y/(l- Yl{ < 0 ok y - 1 > 0 (Model I), (Model II). Thus, we find that the propensity to consume is decreasing in k in the case of Modell. This phenomenon can at least in part be attributed to the fact that the utility function is bounded from below but not from above; the loss from postpone- ment of current consumption is small compared to the gain from the much higher rate of consumption thereby made possible later. In Model II, on the other hand, where the utility function has an upper bound but no lower bound, the optimal amount of present consumption is increasing in k, which seems more plausible from an intuitive standpoint. In Model Ill, we observe from (30) the curious phenomenon that the optimal consumption strategy is independent of the investment opportunities in every respect. While the marginal propensity to consume is independent of k in Model IV also, the level of consumption in this case is an increasing function of k as is apparent from (59). We recall that the utility function in Model III is unbounded while that in Model IV is bounded both (rom below and from above. Thus, the class of utility functions we have examined implies an exceptionally rich pattern of consumption behavior with respect to the "favorableness" of ~he investment opportunities. 5. PROPERTIES Of THE OPTIMAL INVESTMENT AND BORROWING-LENDING STRATEGIES The properties exhibited by the optimal investment strategies are in a sense the most interesting. Turning first to Model IV, we note that the portfolio of productive investments is constant, both in mix and amount, at all levels of wealth. The optimal portfolio is also independent of the noncapital income stream and the level of impatience 1 - IX possessed by the individual, as shown by (61) and (62). Similarly, we find in Models 1-111 that, since for all i, m > 1, z;(x)/z:(x) = v;/v~ (which is a constant), the mix of risky investments is independent of wealth, --- ## Page 134 Optimal Investment and Consumption Strategies under Riskfor a Class of Utility Functions 107 OPTIMAL INVESTMENT 603 noncapital income, and impatience to spend. In addition, the size of the total investment commitment in each period is proportional to x + Y. We also note that when y = 0, the ratio that the risky portfolio l:~ 2 z;(x) bears to the total portfolio l:~ I z;(x) is independent of wealth in each model. In summary, then, we have the surprising result that the optimal mix of risky (productive) investments in each of Models I-IV is independent of the individual's wealth, noncapital income stream, and rate of impatience to consume; the optimal mix depends in each case only on the probability distributions of the returns, the interest rate, and the individual's one-period utility function of consumption. In each case, we find that lending is linear in wealth. Turning firstto Models I-III, we find that borrowing always takes place at the lower end of the wealth scale; (24) evaluated at x = - Y gives - Y < 0 as the optimal amount to lend. From (24) we also find that z~(x) is increasing in x if and only if 1 - v· > 0 since 1 - B is always positive. As a result, the models always call for borrowing at least when the individual is poor; whenever 1 - v· > 0, they also always call for lending when he is sufficiently rich. In Model IV, we observe that lending is always increasing in x. Thus, when an individual in this model becomes sufficiently wealthy, he will always become a lender. At the other extreme, when x is at the lower boundary point of the solution set, he will generally be a borrower, though not r)ecessarily, since z;(x) evaluated at x = - Y + [r/(y(r - W)J log (-a.kr) gives _ y + r log (-a.kr) _ rv· y(r - 1)2 y(r - 1) which may be either negative or positive. We shall now consider the case when the lending rate differs from the borrowing rateas is usually the case in the real world. Let'B - 1 and,c. - I denote the borrow- ing and lending rates, respectively, where rB > rc.' Unfortunately, the sign of dv*/dr is not readily determinable. However, since lex) is increasing in k, the analysis is straight-forward.' 0 10 When 'B > 'L' the "no-easy-money condition" requires that the joint distribution function of fll.· · .. PM satisfies (4a) pr{ E (P, - ,,)8; < o} > 0 '-2 for all finite numbers 8, ~ 0 such that 8; > 0 for at least one;; (4b) Pr {L(P, - 'L)8; < o} > 0 i4S (or all finite numbers 8; ,.; 0 such that /J, < 0 for at least one i; and (4c) Pr{ E {J,8, - L P.O. < o} > ~ '-1 hS- for all finite numbers 9" 9. ~ 0 and all S· S; S such that 101 L II, = L 8 •• i>: 1 ... oS- I,S' and 8, > 0 for at least one i. When r, = r L • 4{a)-4(c) reduce to (4). --- ## Page 135 108 N. H Hakansson 604 NILS H. HAKANSSON Consider first Models I-III when noncapital income y = O. In that case, it is apparent from (24) that when the individual is not in the trapping state (i.e., x > - Y), he either always borrows, always lends. or does neither, depending on whether 1 - v· is negative, positive, or zero. Let kt denote the maximum of (31) when the lending rate is used and the constraint M (65) L {Ii ~ 1 i~l is added to constraints (27) and (28). Since the set of vectors r which satisfy (65) is convex and includes ii = (0, .. . ,0), the Lemma still holds when (65) is added to the constraint set. Analogously,let kBdenote the maximum of(31) under the borrowing rate 'B subject to (27), (28). and M (66) L v, ~ 1. 1=2 Again. the Lemma holds since the set of ii satisfying (66) is convex and any jj such that I::!. 2 Vi = I. Vi ~ O. for example. satisfies all constraints. Setting k == max (kB• kd. Theorem I holds as before when y = O. When y > 0 in Models I-III and in the case of Model IV . no "simple" solution appears to exist when 's > ',.' 6. THE BEHAVIOR OF CAPITAL We shall now examine the behavior of capital implied by the optimal investment and consumption strategies of the different models. According to one school. capital growth is said to exist whenever (67) E(xj + tJ > Xj (j = 1.2 . . . . ), that is. capital growth is defined as expected growth (10). We shall reject this measure since under this definition. asj ~ ~. Xi may approach a value less than x 1 with a probability which tends to I. We shall instead define growth as asymp- totic growth : that is, capital growth is said to exist if (68) lim Pr ·(x) > xtl = I. ) •• ao When the> sign is replaced by the ~ sign, we shall say that we have capital non- decline. If there is statistical independence with respect to j. (67) is implied by (68) but the converse does not hold. as noted. Model IV will be considered first. From (63) it follows that nondecline of capital is always implied (in fact, the solution to the problem is contingent upon the condition that capital does not decrease. as pointed out earlier). It is readily seen that a sufficient. but not necessary, condition for growth is that there be a nonzero investment in at least one of the risky investment opportunities since in that case Pr {x j + I > x)} > O,j = 1,2, .. . , by (63). A necessary and sufficient condition for asymptotic capital growth is a.T > I, which is readily verified by reference to (62). (63), and the foregoing statement. --- ## Page 136 Optimal Investment and Consumption Strategies under Risk/ or a Class o/Utility Functions 109 OPTIMAL INVFSTMENT 605 Let us now turn to Models I-II[ and lei. as before. Sj = Xj + Y. From (7). (23). and (25) we now obtain (69) Sj. I = 5)1 - B)[ f (Pi - r)u; + r] 1= 2 (j=1,2 . .. . ) where W is a random variable. By (28). W ~ O. Attaching the subscript n to W for the purpose of period identification. we note that since j- I (70) Sj = SI n w". "=1 (70) verifies that Sj ~ 0 for allj whenever sl ~ 0 (Models I-III). Moreover. since Pr { W > O} = 1 in Models II and III by Corollary I. it follows that (71) 5 j > 0 whenever 51 > Oforallfinitej (Models II-III). From (70) we also observe that Sj = 0 whenever s. = 0 for allj > k. Consequently. x = - Y is a trapping state which. once entered. cannot be left. In this state~ the optimal strategies in each case call for zero consumption. no productive invest- ments. the borrowing of Y, and the payment of noncapital income y as interest on the debt. In Models II and III, it follows from (71) that the trapping state will never be reached in a finite number of time periods if initial capital is greater than - Y. Equation (70) may be written The random variable r~:: log Wn is by the Central Limit Theorem asymptotically normally distributed; its mean is (j - I)E[log W). By the law of large numbers, j- I L log Wn "= I -+ E[log W] as j -+ 00. j-I Thus. since Sj > S I if and only if Xj > x I' it is necessary and sufficient for capital growth to exist that E[log W] > O. I t is clear that jJ given by II = t'£(l"1 WI may be interpreted as the mean growth rate of capital. By (69), we obtain E[log W] = log (I - B) + E[IOg {J2 ({1.- r)t'; + r} J --- ## Page 137 110 N. H Hakansson 606 NILS H. HAKANSSON For Model III, this becomes, by (30) and (26), E[log W] = log at: + max E[IOg { ~ (Pi - r)vi + r}] !v,1 i= 2 subject to (27) and (28). Thus, a person whose one-period utility function of consumption is logarithmic will always invest the capital available after the allot- ment to current consumption so as to maximize the mean growth rate of capital plus the present value of the noncapital income stream. 7 . GENERALIZATIONS We shall now generalize the preceding model to the nonstationary case. We then obtain, by the same approach as in the stationary case, for all j, (72) jj(x) = max {U(C) + at:jE[jj+l( r (Pi) - r)zij + rj.Xj - e) + YJ}]} 0 and bi are constants), j = 1, .. . , J - 1, that is, that the induced short-run utility functions at decision points 1, . . . , J - 1 are completely myopic, if and only if (1) fJ(x) is either logarithmic or a power function, or (2) 'I = '2 = . . . = rJ_I = 1 andfJ(x) is one of fJ(x) = - e-"S ; (5) (6) fJ(x) = log (x + p.) ; 1 fJ(x) = A-I (Ax + p.)I-I/X >. ~ 0, >. ~ 1, (7) where J.L ~ 0 and X are constants. Note that X and J.L cannot both be negative. While the first conclusion is beyond dispute, the second is incorrect, as are the conclusions concerning partial myopia in general, except in a severely restricted sense. We shall first demonstrate the assertion in the preceding case and then show that it also holds when borrowing and short sales are not ruled out. III. AN EXAMPLE Assume that 'I = '2 = . . . = 'J-I = 1 and that there is only one risky oppor- tunity (Le., M; = 2) in each period. Moreover, assume that the proceeds fJ2j of a Richard Bellman, Dynamic Programming (Princeton, N.J.: Princeton University Press, 1957). --- ## Page 142 On Optimal Myopic Portfolio Policies, with and without Serial Correlation of Yields 115 326 THE JOURNAL OF BUSINESS this opportunity are and that (3 = j 0 with probability 1/2 2J 13 with probability 1/2 all j (8) !J(XJ) = (xJ + d)I/2 d> O. (9) (9) clearly belongs to the class (7), (4) and (3) now give h(xj) = max E{fi+I[({32j - l)z2j + Xj]} OS'ljS"j j = 1, ... , J - 1, (10) where! J(xJ) is given by (9). It is easily verified that Thus, where ( does not exist Zi.J_l(XJ_l) = XJ_l 1/2(xJ_l + d) XJ_I < 0 o ~ XJ_l < d. XJ_I ~ d jl/2(3XJ_l + d)I/2 + 1/2d1/2 1 aJ_l(xJ_1 + d)1I2 o ~ XJ_l < d XJ_I ~ d GJ_l = 1/2[(1/2)112 + 21/2) . (11) (12) (13) We now observe that !J_I(X) is a positive linear transformation of hex) only for x ~ d; for x < d, !J_I(X) < aJ_I!J(x). Proceeding with the solution to (10), we obtain o ~ Xj < bj Xi ~ bj j = 1, . .. , J - 1, (14) where ai is a positive constant, gi(Xj) < aj(xj + d)1I2 for 0 ~ Xj < bi> and bj = d + 2bi+1 (bJ = 0), j = 1, . .. , J - 1 , (15) o 5 Xj < bj Xj ~ bj j = 1, .. . , J - 1. (16) It is easily determined that hj(xj) is highly irregular except for j = J - 1. When J = 11 and d = 1,000, we obtain from (15) that bl = 1.023 million. Thus, when the horizon is ten periods distant, the optimal amount to invest in oppor- tunity 2 is, in this example, proportional to Xj + d only if initial wealth XI exceeds $1 million by a substantial margin. Furthermore, while!J(x) is a positive linear trans- formation of !Ax) for x ~ bi> j = 1, ... , J - 1, it is not for x < bi> that is, for XI < 1.023 million, X2 < 511,000, etc., in the above example. Since the constant b i > 0 depends on the distribution functions F h . . . , F J-I, the short-run utility functions induced by the terminal utility function (9) are clearly not myopic. In other words, to make an optimal decision at decision point}, not only F j but F j+l, ... , FJ - I must be known. IV. BORROWING AND SOLVENCY The nonmyopic nature of the induced utility functions!l(xl), !2(X2), . . . ,jJ-I(XJ-I) in the preceding example is clearly attributable to the constraint j = 1, ... , J - 1, (17) which precludes borrowing and short sales. We shall now relax this constraint. --- ## Page 143 116 N. H Hakansson ON OPTIMAL MYOPIC PORTFOLIO POLICIES Case T.-(17) will first be replaced with 327 o ~ ~j ~ mXj m > 1 j = 1, . . . , J - 1, (18) that is, l00/m is assumed to be the percentage margin requirement. The solution to (10) for J - 1 now becomes d d o ~ %J-l < 2m _ 1 (19) d XJ_I ~ 2 1 ' . m- and the total solution is represented by (14)-(16) with bj > 0, j = I, ... , J - 1. Consequently, the optimal portfolio policy is nonmyopic in the case of constraint (18) also. Case I I.-Let us now introduce an absolute borrowing limit of m, that is, sub- stitute o ~ Z2j ~ Xj + m m > 0 j = 1, ... , J - 1 (20) for (17). When m < d/2 the solution to (9) is again given by (14)-(16) with bj > 0, j = 1, ... , J - 1. However, when m ~ L = d/2, the solution becomes h(Xj) = akcj + d)1/2 j -= 1, ... , J - 1 ; zi/(x/) = 1/2(x/ - d) zi;(xj) = 1/2(xj + d) j = 1, ... , J - 1 j (21) j = 1, . .. , J - 1; (22) that is, the optimal investment policy would seem to be completely myopic on the basis of our assumptions. But L clearly depends on Ff.fh . . • • FJ- I • Thus to know whether m ~ L, knowledge of future returns is necessary. Consequently, the optimal investment policy is not myopic in Case II either. Let us now consider the realism of assumptions (18) and (20). With respect to (20), we observe from (21) that borrowing takes place, considering decision point J - 1, only when XJ_I < d. By (3), (8), (21), and (22), we obtain . _ ! 1/2(xJ_1 - d) with probability 1/2 XJ - ~ 2xJ_1 + d with probability 1/2 . Thus, the terminal wealth position has a 1/2 chance of being negative if and only if %J-I < d, that is, when borrowing takes place. If the first event ({i2,J-l =- 0) takes place and the investor declares bankruptcy at time J, the lender will stand to lose the entire loan of /1I2(%J_1 - d) I. The point here is that it would be unreasonable for anyone to lend money to his investor when his optimal strategy cal1s for it; that is, m should be zero in (20)- which converts (20) to (17). In fact, (18) and (20) may be said to be inconsistent --- ## Page 144 On Optimal Myopic Portfolio Policies, with and without Serial Correlation of Yields 117 328 THE JOURNAL OF BUSINESS with the portfolio model itself. This is because a wealthier investor (one whose wealth exceeds d) with the same preferences and probability beliefs as a poorer one is, by (21), a possible lender to the poorer one whose wealth is less than d. But the model assumes that lending is safe" that is, that aU loans are repaid with probability 1 while, as we have seen, the poorer investor may not be able to repay. Case lll.-A borrowing arrangement that is consistent with the assumed riskless- ness of lending is one which permits borrowing to the extent that ability to repay, that is, solvency, is guaranteed. Thus, a reasonable constraint on borrowing and short sales, with considerable intuitive appeal as well, is given by Pr {Xi+l ~ O} = 1 j = 1, ... , J - 1. (23) When (23) is substituted for (17), the solution to (to) is the same as when (17) is used; that is, it is given by (14), (15), and (16). Thus, myopia is not optimal in this case either. V. THE GENERAL CASE It is readily verified that the conclusions of Sections III and IV are not changed if the number of risky investment opportunities is arbitrary. Moreover, the con- clusions hold for all of the functions (5), (6), and (7) whenever p. > 0, both with no borrowing and in each of Cases I-III. Finally, when ,. j ¢ 1, j = 1, . . . , J - 1, partial myopia, as defined by Mossin, is not optimal either in any of the preceding cases. It should be noted that a solution need not exist in Case III unless the "no- easy-money condition" holds.4 A generalization of this condition (for the case when yields are serially correlated) is given in Section VI. In the most general version of Case III, the set Z/(x;) in (4) is given by those z/ which satisfy Zij ~ 0 i~ Sj (24) and (23). When p. = ° in (6) and (7), [(5) is of no interest when p. 5 0), complete myopia is optimal in both Cases I and III but not in Case II, as is easily shown. The Mossin- Leland conclusions concerning complete myopia when 1'1 = 1'2 = ... = l' J-I = 1 and partial myopia do not apply in (6) and (7) when p. < ° either, except in Case III, as we shall demonstrate below. In doing so, we shall also show that, when p. ~ 0, (5), (6), and (7) imply that the optimal investment policies at decision points 1, ... , J - 1 are never myopic in the presence of explicit borrowing limits of any kind, with one exception. When a solution to the portfolio problem at decision point J - 1 exists in the presence of constraints (24) only, the optimal lending strategy Z1.J_I(XJ_I) has the form !1.J-l(XJ-I) = (1 - MJ-1)xJ-l - AJ-Ip. in the case of (6) (A = 1) and (7) and the form ZI,J-l(XJ-I) = XJ-I - BJ_ , in the case of (5), where A J- 1 and BJ- 1 are constants, generally positive,6 which depend on FJ _ I .' Let us consider (6) and (7) when A,p. > O. Since AI-I, and hence, -Z),J-I(XJ-I), • Nils Hakansson, "Optimal Investment and Consumption Strategies under Risk for a Class of Utility Functions," &onomel,ica 38 (September 1970): 587-607. 'Nonpositive A J_ I and B J_ I imply that total short sales exceed or equal total long investments, • Nils Hakansson, "Risk Disposition and the Separation Property in Portfolio Selection," Journal (>f Financial and Quantitative Analysis 4 (December 1969): 401-16. --- ## Page 145 118 N. H. Hakansson ON OPTIMAL MYOPIC PORTFOLIO POLICIES 329 may be arbitrarily large, any finite borrowing limit has a chance to be binding. Consequently, for /J_I(X) to be a positive linear transformation of /J(x) in the pres- ence of a borrowing limit, it must be a positive linear transformation of/J(x) whether the borrowing limit is binding or not. From Section IV, it is apparent that, when M j = 2 and (17) holds, a necessary and sufficient condition for / J-I (x) to be a posi- tive linear transformation of hex) is that Z:.J-l(XJ-l) has the form Z;'.J_I(XJ_I) = a2.J-I(~J-1 + J.I.), where a!.J-1 is a nonnegative constant. When there is more than one risky asset and (24) holds, this condition generalizes to Z1.J_l(XJ_l) = ai.J_l(AXJ_l + J.I.) i = 2, ... , MJ-l, (25) where the ai.J-l are constants, nonnegative only for i ~ SJ_I.7 It is now clear that the optimal investment strategy 21-1 will have the form (25) if and only if (1) the borrowing limit is not binding or (2) the borrowing limit has the form -ZI.J_l ( = MtIZi.J_l - XJ_I) ~ (ACJ_I - l)XJ_l + CJ_IJ.I. ._2 XCJ _ 1 > 1, X, J.I. > O. (26) The latter assertion follows from (2) and the fact that this form of the borrowing limit does give the solution (25) for any F J-l, as is easily verified; moreover, only (26) is capable of giving a solution of form (25) when the borrowing limit is binding. Since knowledge of whether any given borrowing limit is binding or not requires knowledge of F J - I , it follows that/J_I(X) is myopic in the presence of a borrowing limit only if this limit has the form (26). By induction, fl(x), . . . ,/J_I(X) are then myopic in the case of (6) and (7) for 'A,J.L > 0 if and only if the borrowing limit in period j is given by (ACj - l)x; + C;J.I. }l.C; > 1, A, J.I. > 0, j = 1, ... , J - 1. (27) When Jl < 0 or 'A < 0 in (6) and (7), any borrowing limit would, to be consistent with myopia, again have to have the form (27). But when Jl < 0, we must have ). > 0 and vice versa so that (27) cannot be nonnegative for all Xi > 0 for which borrowing may be desired, a basic requirement of any "true" borrowing limit. The situation in the case of function (5) is analogous. As a result, fl(xI), ... ,fJ-I(xJ-I) can never be myopic for (5), (6), and (7) when 'A < 0 or Jl < 0 in the presence of a borrowing limit. Turning now to the solvency constraint (23), we obtain whenever a solution exists for (6) and (7) that the greatest lower bound on b such that Pr r XJ < b} > 0, for any decision at decision point J - 1 which satisfies (24), is KJ_I('AxJ_I + Jl) + r~l . J-I(XJ-I)' where K J_I is a. constant which depends on F J-I. Since f;( - JlI'A) = co for ). > 0, we obtain, letting XJ 5 KJ_1('AxJ_I + Jl) + r~1.J-I(XJ_I)' that XxJ + J.I. > 0, which implies, since A and J.I. cannot both be negative, XJ > 0 when Il ~ O. Thus the solvency constraint (23) is not binding when Il ~ 0 but may be when J.I. > O. Consequently, the induced utility functions !I(X), .. . ,!J_I(X) are myopic for the class (6) and (7) when J.I. < 0 in the presence of (24) and the solvency constraint (23). 7 Ibid. --- ## Page 146 On Optimal Myopic Portfolio Policies, with and without Serial Correlation of Yields 119 330 THE JOURNAL OF BUSINESS In sum then, when I-' ~ 0 and interest rates are zero, the induced utility of wealth functions/l(xl), ... ,/J-I(XJ-I) are myopic in the presence of borrowing constraints if and only if the borrowing limits are of the form (27) and /J(x) is of the form (6) (7) with X,,", > 0; in the presence of· the solvency constraint,/I(xl), ... ,h-l(xJ-I) are myopic only if /1 (x) has the form (6) or (7) with I-' < 0 (and hence). > 0). It should ~so be noted that, when I-' < 0, hex) is undefined for x < /1-'/)./ and f;(Xi) , j = 1, . . . , J - I is undefined for small Xii a significant drawback. In addi- tion, the relative risk aversion index -x/,/(x)/fi(x) is decreasing for these functions, whereas Arrow,s for example, suggests that plausible utility functions of money exhibit increasing relative risk aversion. VI. SERIALLY CORRELATED YIELDS We shall now consider the sequential investment problem when yields are serially correlated. In contrast to Mossin's assertion,' we shall find that the optimal invest- ment policy is myopic in this case only for a small subset of the terminal utility functions which induce myopic short-run utility functions when returns are serially independent. For simplicity, we assume that yields and the interest rate obey a Markov process. A distinction between risk due to general market forces, called . the economy, and risk due to individual assets and periods is made. As a result, the assumptions and notation of Section II are modified as follows: Xi: amount of investment capital at decision point j; N;: number of states of the economy at decision point j; M im: number of investment opporturiities available at decision point j, given that the economy is at state m at that time; Sim: the subset of investment opportunities which it is possible to sell short at decision pointj, given that the economy is in state m at that time; ri., - 1: interest rate in periodj, given that the economy is in state m at decision pointj (Tim> 1); l3iimn: proceeds at the end of period j, given that the economy is in state n at that time, per unit of investment in opportunity i, i = 2, . . . , M ;"', at decision point j, given that the economy was in state m at that time; Pi"" : probability that the economy makes a transition from state m to state n in period j N'+ I ( Pi,"n ~ 0, f Pim" = 1) ; n_1 Slim: amount lent in period j, given that the economy is in state m at decision point j (nega- tive Zlim indicate borrowing) (decision variable); Zii .. : amount invested in opportunity i, i = 2, . .. , M j"" at decision point j, given that the economy is in state m at that point; !; ... (Xi): utility of money at decision pointj, given that the economy is in state m at that time; zii": an optimal lending policy for state m at decision pointj; Z~im: an optimal investment policy for state m at decision point j, i = 2, . . . , M im. _ liim • Vij",~-, Xi i = 1, ... , M im • • Kenneth Arrow, Aspects of the Theory of Risk-bearing (Helsinki: Yrjo Jahnssonin Siiiitio, 1965). I Mossin, p. 222. --- ## Page 147 120 N. H Hakansson ON OPTIMAL MYOPIC PORTFOLIO POLICIES 331 Clearly, Vi,,,. represents the proportion of capital Xj invested in opportunity i at decision point j, given that the economy is in state m at that point; thus M j ". VI' .. = 1 - L Vij .... i_t It will be assumed that the joint distribution functions F j",~ are independent with respect to j. In addition, we postulate that the {Pij",~ I satisfy the following condi- tions: all j, m, and n Pr {O ~ {jijm~ < ex>} = 1 , i = 2, ... , M,,,,, (28) M· Pr j f (ftii"'~ - r",.)6i < ot > 0 1 ._t 5 (29) for all j, all m, some n for which Pi"''' > 0, and all finite 9; such that 9; ~ 0 for all i ~ S,,,, and 9i ~ 0 for at least one i. (29) is a modification of the "no-easy-money- condition" for the case when the lending rate equals the interest rate. 10 This condi- tion states that no combination of risky investment opportunities exists in any period which provides, with probability 1, a return at least as high as the (borrowing) rate of interest; no combination of short sales is available for which the probability is zero that a loss will exceed the (lending) rate of interest; and no combination of risky investments made from the proceeds of any combination of short sales can guarantee against loss. (29) may be viewed as a condition which the prices of all assets must satisfy in equilibrium. (3) is now replaced by the conditional difference equations M"M xi+ll mn = L (ft;j",,. - 'j",)Zijm + 'j",Xj j = 1, ... ,J - 1, all m, n, (30) i_2 and (4) becomes, for j = 1, .. . , J - 1 and all m, N i+1 j;",(Xj) = max L Pj",,.E[ji+l.n(X,+ll mn») , 'im ".1 (31) where /J",(XJ) is given for all m, subject to and Zijm ~ 0 Pr {xj+11 mn ~ O} = 1 VII. OPTIMAL MYOPIC POLICIES i~ Sj"., (32) n = 1, ... , N j+1 • (33) On the basis of the finite yield a.ssumption (28) and the "no-easy-money-condi- tion" (29), we obtain the following: Theorem.-Let rj"" Fi "" and Pi",. be defined as in Section VI and let u(x) be a monotone increasing and strictly concave function for all x ~ O. Then the functions (34) 10 Hakansson, "Optimal Investment . . . " (see n. 4 above). --- ## Page 148 On Optimal Myopic Portfolio Policies, with and without Serial Correlation of Yields 332 subject to and THE JOURNAL OF BUSINESS 121 M. Pr j t-(f3ii",,, - rim)Viim + rim;::: ot ... 1 n ... 1, ... ,Ni+1 (36) 1 ,-2 ~ have maxima for all j = 1, ... , J - 1 and all m. Moreover, the maximizing vec- tors, v; m, are finite and unique. The proof may be found in Hakansson (1968).11 Let us now assume that /I",(X) ... X1I2 all m ; (37) and let kim denote the maximum of (34) subject to (35) and (36) when u(x) = x1/2 ; that is, N;+1 l [M;.. ]1/2f kim == L pim"E :E ({3ii",,, - ri"')V!i'" + rim j ... 1, ... , J - I all m. (38) n_l .-2' By the theorem, we know that kjm exists. Let us now determine!J_I.m(%I_I)' From (31) we obtain for all m subject to and NJ /J_I .... (XI_I) .... max LPI-l ..... E[ (XI I mn)1I2) , J J-l. '" ".1 %,.J-I.m ;::: 0 (39) i~ SI_I.". (40) M J - 1 ... Pr j ~ ({3i,J-I ...... - r J-I.",)%,.J-l,., + r 1-I.",XI-I ;::: ot ... 1 n = 1, ... , N J. (41) 1 .-2 · ~ By (29) and (41),fI-.(:tJ-I) does not exist for %1-1 < O. For XI-I ~ 0, (39) may be written, since (32) and (33) are equivalent to (35) and (36) when XI-I> 0, HJ (I_I.",(XI_I) = XJ~21 max LPI-I.",,, iJ-1,M "-1 j [MJ - 1... ]1/2t E I ~ ({3U-I.m" - r I_I ... )VU_I, ... + r I-I.... ~, subject to and V',I_I. ... ;::: 0 (42) (43) M J - 1, .. Pr j :E (f3,.J-I."." - rl_I, .. )V,.J_I ... + rl_I, .. ~ ot ... 1 n ... 1, ... ,NJ • (44) 1 ,-2 5 By the theorem, we now obtain that /I-I .... (XI-I) exists for all m and %J-I ~ 0 and is given by, using (38), all m. (45) 11 Nils Hakansson, "Optimal Entrepreneurial Decisions in a Completely Stocha.stic Environment," Managemenl Sciem;e: Theory 17 (March 1971): 427-49. --- ## Page 149 122 N. H Hakansson ON OPTIMAL MYOPIC PORTFOLIO POLICIES 333 By (31), the expression to be maximized at decision point J - 2, given that the economy is in state m at that time, becomes M1_1 ... ~ PJ-2.mnkJ-I "E(xJ_II mn)1/2). (46) n_1 Since the constants kJ-1.n will in general be different for different n, (46) and, there- fore, the optimal portfolio i'_2.m. depend on the yields in period J - 1. Thus the optimal investment policy is not myopic at decision point J - 2; neither is it myopic at decision points 1, . . . , J - 3, which is easily shown by induction. The existence of positive constants ai, . .. ,aJ_1 and of constants bll , . . . , bJ_t.N,_. such that Ji,..(X) = a;jJ".(x) + bjm all m, j = 1, ... , J - 1 (47) are clearly both necessary and sufficient for myopia to be optimal in this model. As noted, (45) violates (47) for j = J - 1 whenever N J-I > 1. However, when N j = 1, j = 1, . . . ,J , (48) (47) is satisfied; but (48) also implies that yields are statistically independent in the various periods. This confirms Mossin's result that the optimal investment policy is myopic when returns are stochastically independent over time and the terminal utility function is X1/2• Let us now assume that!Jm(x) has the form fJ",(x) = log x all m. (49) Letting H J - 1• m denote the maximum of (34) subject to (35) and (36) when u(x) = log x, that is, (50) we obtain from (31)-(33), solving recursively, hm(x) = log x + hjm all m j = 1, ... , J - 1, (51) (where bJ_t.m = H J - I •m, all m), which is consistent with (47). As a result, the in- duced utility functions !u(x), . . . ,/tN1(X), . .. ,/J-1I(X), ... ,!J-I.NJ_JX) are myopic when the terminal utility function is logarithmic, both when yields are serially correlated and when they are not. Just as in the case of (37), which is a special case of (7), the optimal investment policy is not myopic for any function (7) (whether p. = 0 or ~ot), nor for any func- tion (5), when yields 3.l'e serially dependent, as is easily verified. Since interest rates are assumed to be positive and state-dependent, myopia is not optimal for log (x + p.), p. ¢ 0 either, or any other nonlogarithmic function, under serial depen- dence. Thus, when yields are serially correlated, only the logarithmic utility func- tion of terminal wealth induces short-run utility of wealth functions which are --- ## Page 150 On Optimal Myopic Portfolio Policies, with and without Serial Correlation of Yields 123 334 THE JOURNAL OF BUSINESS myopic when yields in the various periods are nonindependent, contrary to Mossin's assertion .12 VIII. CONCLUDING REMARKS In view of the difficulty of estimating future yields and their apparent serial cor- relation, the myopic property of the logarithmic utility functions is, of course, highly significant. However, this function also has other attractive properties. 13 Per- haps the most important of these is the property that maximization of the expected logarithm of end-of-period capital subject to (32) and (33) in each period also maximizes the expected growth rate of capital, whether returns are serially corre- lated14 or not.l~ As Mossin points out, the portfolio decision is in general not independent of the consumption decision. A realistic model of the investor's decision problem must, therefore, include consumption as a decision variable and a preference function for evaluating consumption programs. Consumption-investment models of this type have been developed by Hakansson, both when investment yields are serially corre- lated16 and when they are not. 17 II Mossin, p. 222. II Some of the properties are reviewed in Nils Hakansson and Tien-Ching Liu, "Optimal Growth Port- folios When Yields Are Serially Correlated," Review of Economics and Statistics 52 (November 1970): 385-94. " Ibid. 11 Henry Latane, "Criteria for Choice among Risky Ventures," Journal of Political Economy 67 (April 1959): 144-55; and Leo Breiman, "Optimal Gambling Systems for Favorable Games," Fourth Berkeley Symposium on Probability and Mathematical Statistics (Berkeley: University of California Press, 1961). 16 Hakansson, "Optimal Entrepreneurial Decisions . . . " (see n. 11 above) . 17 Hakansson, "Optimal Investment .. . " (n. 4 above); and "Optimal Investment and Consumption Strategies under Risk, an Uncertain Lifetime, and Insurance," International &onomic Review 10 (October 1969): 443-66. --- ## Page 151 This page is intentionally left blank --- ## Page 152 125 The Journal of FINANCE VOL XXVIII JUNE 1973 No.3 10 EVIDENCE ON THE "GROWTH-OPTIMUM" MODEL RICHARD ROLL * 1. INTRODUCTION A PORTFOLIO OWNER may hope to maximize the long run growth rate of his real wealth. In 1959, Latane suggested maximum growth as an operational cri- terion for portfolio selection, contending that its (possible) su boptimality on theoretical grounds was practically unimportant and emphasizing Roy's [ 1952] warning that "A man who seeks advice about his actions will not be grateful for the suggestion that he maximize expected utility." In the past ten years, however, little work on growth-maximization of port- folio value has appeared in the academic literature of finance or economics. The neglect was due to competing norms for asset selection, particularly to norms based on two-parameter, two-period portfolio models deriving from the work of Markowitz [1959], Tobin [1958], Sharpe [1964], and Lintner [1965]. These were developed into full theories of capital market equilibrium and the empirical evidence collected in their support seemed at least sufficient to justify continued research along the lines of relaxing assumptions and performing more tests on observed portfolio behavior. Recently, Hakansson [1971) and Hakansson and Liu [1970] again brought the growth maximization criterion to our attention. Hakansson presented a persuasive theoretical argument that" ... the mean-variance model [a special case of the aforementioned two-parameter model, was] severely compromised by the capital growth model in several significant respects. III Most readers will find the following a significant respect: Given temporally independent returns, a number of mean-variance efficient portfolios can be shown to bring complete ruin after an infinite sequence of re-investments. It is true, of course, that such sequences may indeed be optimal from an expected * Graduate School of Industrial Administration, Carnegie-Mellon University. The comments of Eugene Fama, Haim Levy, Robert Litzenberger and Myron Scholes are gratefully acknowledged. Remaining errors are due to the author alone. This project was supported by the Ford Foundation which does not necessarily agree with the results and opinions. 1. Hakansson [1971, p. 517J. 551 --- ## Page 153 126 R.Roll 552 The Journal oj Finance utility viewpoint, despite ultimate ruin. No mathematician can prove the contrary, especially when it is recognized that a typical investor's horizon will probably fall short of infinity and that he will likely consume a fraction of his assets each period. However, one of our basic concerns should not be with formal proof, but with practical and intuitively credible models of investor behavior. In this regard, "growth-optimum" portfolios possess appealing fea- tures even during a finite time span. For example, such portfolios maximize the probability of exceeding a given level of wealth within a fixed time.2 Hakansson also pointed out other features of the growth-optimum model that may be more or less appealing. It implies a logarithmic (in wealth) utility function, which displays decreasing absolute risk aversion, and it implies optimal decision rules that are myopic. As a further embellishment, Hakansson and Liu derived a "separation theorem" that holds the optimal sequence of investments to be independent of the sequence of wealth levels even when returns are stochastically dependent across time. (Myopia also holds with tem- poraldependence). Although these are strong challenges to the practical superiority of two- parameter portfolio models, we should not abandon the latter too hurriedly. They have been successfully used in many empirical contexts and their com- petitors should be required to weather empirical examination; so the purpose of this paper is to report on some empirical tests of growth-optimum theory using common stock returns. Briefly, the growth-optimum model receives mixed support. In some tests it performs extremely well while the results of other tests are puzzling. In com- parison to the mean-variance model it also performs well but the test results are clouded by the close operational similarity of the two models. II. A TEST STATISTIC FOR THE GROWTH-OPTIMUM MODEL The quantitative derivation of the growth-optimum rule will employ the following convenient NOTATION: nJ-number of shares purchased of security j initially p'ot-price per share of security j in period t. N Vt = L n,p'ot-Value of a portfolio of N distinct securities in period t. J=1 X'ot == n,PJotlf n,PJ.t-fraction of resources invested in security j in t. RJot = {[ (Pjot + D,.t) /PJ.t _ d -l}-rate of return to security j from t - 1 to t. D,. t-per share dividend or coupon paid to security j between t - 1 and t. E-Mathematical expectation. 2. Breiman [1961, section 5). Hakansson seems to have erred slightly when he stl,tes that "Breiman has shown that if the objective is to achieve a certain level of capital as soon as possible, then the optimal-growth portfolio .. . minimizes the expected time to reach the given level," Hakansson [1971, p. 540]. Breiman conjectured that this was true but was unable to state a proof for a fixed level of wealth. As wealth grows indefinitely large, however, the limited expected minimum time is in fact achieved by the "growth-optimum" portfolio. --- ## Page 154 Evidence on the "Growth-Optimum " Model 127 Growth Optimum Model 553 Since the rule for maximizing the expected growth of portfolio value is myopic, one only needs to optimize between successive periods. Thus, the growth-optimum rule is maximize E[loge(Vt/Vt_d] == E {1oge[l:jXI.t_l (I + Ri.t )]} (1) X t _ 1 subject to l:iXi.t-l == 1. Negative values of X represent short sales which, by assumption, can be accomplished without penalty. Transaction costs are ne- glected. Although the problem is easily solved without further qualifications, for historical comparison purposes it is worth assuming that the Nth asset, denoted F, is risk-free and that it receives the residual portion from the amounts in- vested in all risky assets; i.e., ~-l ~-l Vt/Vt_ 1 = L XJ.t - 1 (1 + RJ.t ) + (1 + RF,t) (1 - L xJ.t-J. First-order conditions from problem (1) show that the investor's growth- optimum portfolio will be determined by proportions X* such that E N == 0; J == 1, ... , N - 1. (2) [ ~J,t- RF,t ]. 1 + RF" + t. X" ,H (i!", - RF,,) . As illustrated by Hakansson [1971] and Breiman [1960], these optimal invest- ment fractions generally will imply a diversified portfolio but their exact values cannot be determined without specifying the joint probability distribution of the Rt's. To obtain a testable proposition from (2), however, it will not be necessary to specify that distribution. We can rely instead on the fact that in a given period, the value of N , I + R.".t + l:rXI.t-t (Ri.t - R F .t ) 1 I expected by each individual is equal for every risky security. Advancing to an aggregate level will require either of two traditional assumptions: (a) that all investors hold identical probability beliefs or (b) that a "representative" in- vestor holds the expectations of equation (2) with the invested proportions (X*'s) being equal to relative values of existing asset supplies, In either case, the denominator of (2) is equal to 1 + Rm.I> a "market return" defined as a value-weighted average of all individual asset returns, The approximately ob- servable8 variable 1 + RJ,t z"ta! 1 + R m.t (3 ) 3. It is only approximately observable because no comprehensive value-weighted asset indexes exist. --- ## Page 155 128 R. Roll 554 The Journal of Finance will have the same mean for all securities. It will also be true, of course, that some unexplained variation may occur about these expectations so that the quantities Zj.t will not be exactly equal in every period. However, an appro- priate test of the identity of expectations only requires that corresponding sample means be statistically equal; that is, the temporally averaged means T - """ 1 + Rj,t / Zj=L..J T t=l 1 + Rm•t (4) must be insignificantly different across securities. Testing the equality of N sample means is an analysis of variance problem. Its application to New York and American Stock Exchange listed securities will be reported in the next section. III. A SIMPLE TEST OF THE GROWTH-OPTIMUM MODEL'S BASIC VALIDITY Data used in this section are rates of return obtained from the Wells Fargo rate-oC-return tape prepared by M. Scholes. The tape contains daily price changes for all common stocks listed on the New York and American exchanges from June 2, 1962 through July 11, 1969. It is a condensation and thus a tractable version of the ISL Quarterly Historical Stock Price Tapes. The Standard & Poor's Composite Price Index (The" SaO") is used to obtain the "market return." In the following test, returns were taken over weekly intervals. Thus, the statistic ZJ,t = (1 + Rj,t)/( 1 + Rm,t) was calculated for stock j at the end of week t; where 1 + Rj.t = (Pj,t + DJ,t)/PJ.t-l and 1 + R m•t was similarly calculated using the S&P Composite Index. The null hypothesis requires the expected return ratios to be equal, E(ZJ.t) = E(ZI.t), Cor all i, j, and t. Usually, this would be tested by one-way - 1 analysis of variance on the temporally-averaged means, Zj == T ~t Zj,t but simple ·analysis of variance procedures requires that all the ZJ.t'S be uncorre- lated cross-sectionally and have equal variances under the null hypothesis. These assumptions are obviously too strong and are not required by the (null) growth-optimum hypothesis anyway. Furthermore, we have the evidence of many previous studies to confirm the existence of positive covariation between stock returns and returns on a market index. Some researchers (notably King, [1966]) have calculated directly a substantial co-movement among stock returns. Although the co variation between two individual stocks returns, say Rj and Rk , may be reduced in the return ratios (Z's) as the result of division by 1 + R m , it would be too audacious to assert a complete ~limination:...,Further­ more, there is no a priori reason to suppose that Var (ZJ) = V ar (Zk ), as is required by a simple one-way of' variance. Fortunately, the Hotelling T2 statistic4 is available for precisely those cases 4. See Morrison [1967, pp. 117-1241 or Graybill, [1961, pp. 205-2061. --- ## Page 156 Evidence on the "Growth-Optimum" Model 129 Growth Optimum Model 555 where independence among observations and equal variances do not hold. At the expense of considerable computer time, which is patiently borne by Car- negie-Mellon's undergraduates and the Nation's taxpayers, and is thus free to its current beneficiaries, this statistic was calculated for groups of 31 stocks selected alphabetically according to the procedure described below. Hotelling's T2 is computed as a quadratic form in the sample vector of differences between adjacent return ratios and the sample covariance matrix of those differences.s It is distributed as an F distribution with k - 1 and N - k + 1 degrees of freedom where N is the sample size (weeks) and k is the number of stocks in a group.n Since stocks are not always traded over coinci- dental calendar periods, and since coincidental observations are required in order to calculate sample covariances, the sample size N of each group was reduced to the number of weeks when all 31 stocks had been traded and re- corded. At most, this was the number of weeks for the stock that had the minimum in its group and it was generally somewhat less. In fact, realizing in advance that some groups might be reduced to a very low number of coinci- dental observations, I decided to discard a stock if keeping it in the group meant reducing the second number of degrees of freedom, N - k + 1, below 30. When a stock was discarded, the next alphabetical one on the' tape was added to the group. Of course this meant that a stock was then missing from the subsequent group and another had to be taken from the group following that and similarly to the end of the tape. Finally, 68 groups of 31 stocks re- mained for analysis and this number comprises all the stocks on the tape with sufficient coincidental observations for the test procedure. These 68 F statistics are depicted in Figure 1 and tabulated in Table 1. The distribution of Figure 1 is stochastically below the expected null distribu- tion. For example, using a Chi-square goodness-of-fit test with 17 classes to compare the F sampling distribution with the null distribution, the test statistic is about 70, which is far above the .005 level of significant difference between the two distributions. It should be emphasized, however, that only high F values reject the null growth-optimum hypothesis. Thus, growth-optimum theory is strongly, even too strongly, supported by this test of its basic validity.' 5. As further cxplanation, recall that Zj.t == (1 + Rj.t)/(l + Rm.t) is the ratio of return on stock j to the market return. For a given group of 31 stocks, the differences in return ratios are calculated as Yj,t == Zj,t - Zj+1.I for j = I, . . , ,30. The means of Yj,t and covariances of yj,t and Yu are then calculated ovcr time. Hotelling's statistic is based on the sample quadratic form 9'S-I}' where 5' is the vector of sample mean differences and 5 is the sample covariance matrix of differences. Differences were calculated between adjacent alphabetic pairs but any other random scheme for selecting pairs would have been equally acceptable. Cf. Morrison [1967, pp. 135-1381. 6. This is the rationa1e for using a group size of 31 stocks: since the number of degrees of freedom for the test is k-l, where k is the number of stocks, a group size of 31 is both large and makes tabular comparison easy. If the group size had been chosen larger, the second degrees of freedom parameter, N-k+l, (where N is the number of available time points), would be reduced to a low number. Thus, I thought k=31 would balance the two d.f. parameters and still leave them quite large. 7. Because the Hotelling test supports the null hypothesis too strongly, I decided to check several potential causes, One obvious possibility is the thick-tailed distributions of stock returns that hnc been pointed out by many researchers, (Cf. Fama [19651, Blume r 1970 /), The appropriate way to check this problem is to use a non-parametric analysis of variance, Friedman's multi-sample test, (Bradley, r1968, p, 1271) . This was done for exactly the same sample of stocks grouped in --- ## Page 157 130 556 Frequency 10 8 6 4 2 Computed F Statistics The Journal oj Finance FIGURE 1 F DIstributIon expected under null hypothesis R. Roll Distribution of F Statistics from Hotelling's T2 Test of Equality Among Mean Return Ratios IV. RELATIONS BETWEEN THE GROWTH-OPTIMUM AND THE SHARPE-LINTNER MODELS A. Theory When one compares the first order conditions for the growth-optimum model 1 + RJ 1 E ;( = (1 + Rr). E R ; j = 1, .. . N - 1 I+Km 1+ m (5) with those for the Sharpe [1964]-Lintner [1965] model E(1 + RJ) (1 + Rr) E(l + Rm} = ~J + (l - ~J) E( 1 + Rm) j = 1, ... ,N - 1 (6) (where ~j = Cov (R" ~m) jVar (Rul)), some correspondence appears but it is rather difficult to evaluate fully just by inspection. For example, when the Sharpe-Lintner risk coefficient, ~j, is equal to zero, equation (6) becomes E(1 + Rj ) 1 + Rr E(l + Rm} E(1 + Rm} But when ~j = 0, COy (R), RnJ = 0, and the growth optimum condition ex- pressed in equation (5) becomes the same way. The results were identical to those obtained by using HotelUng's T2, the growth- optimum model was too strongly supported. A second possible misspecification is a deficiency in the market price index. The Standard & Poor's Composite Price Index, used in the preceding tests, is heavily-weighted in favor of a few stocks. Also, since it is essentially a "buy-and-hold" portfolio, weights of individual stocks change over time as relative prices change. Evans [1968] and Cheng and Deets [1971 J have provided empirical evidence that such an index performs Quite differently from 'a "fixed-investment propor- tion" or "rebalanced" index. Therefore, a rebalanced index composed of stocks on the tape was constructed and used in reporting the tests already done. Again, no difference was detected. Thirdly, the possibility that an unrepresentative episode biased the entire sample period of seven years of weekly observations was checked by repeating the tests for annual sub-periods. There was no perceptible difference among the sub-periods or between the overall period and any sub-period. In every case the growth-optimum model was strongly supported. Further details of all the tests in this footnote are available in an earlier working paper. (It was edited to conserve Journal space.) --- ## Page 158 Evidence on the "Growth-Optimum " Model 131 Growth Optimum Model 557 TABLE 1 CALCULATED VALUES OF HOTELLlNC'S T2 STATISTIC FROM COMMON STOCK MEAN RETURN RATIOS, 1962·1969 Sample size Sample size Group (weeks) T2 Group (weeks) 1'2 1 113 36.0 36 98 30.8 2 81 20.2 37 76 35.0 3 61 53.3 38 101 29.2 4 61 45.4 39 79 54.1 5 143 28.4 40 69 56.3 6 76 35.3 41 107 32.3 7 132 55.5 42 61 6S.9 8 61 14.4 43 61 72.0 9 148 29.7 44 78 32.0 10 131 24.1 45 64 35.9 11 90 22.6 46 97 46.3 12 68 28.7 47 66 56.9 13 lOS 41.3 48 62 57.1 14 76 27.5 49 91 24.9 15 109 41.5 50 60 50.1 16 273 29.4 51 61 37.6 17 III 33.5 52 64 40.3 IS 110 35.0 53 63 38.3 19 175 36.5 54 65 25.6 20 75 42.0 55 60 35.1 21 95 27.9 56 67 63.S 22 76 51.3 57 80 4S.2 23 112 22.4 58 60 107.0 24 74 64.5 S9 96 24.9 25 11 2 23.4 60 79 39.0 26 127 31.8 61 73 57.8 27 158 38.6 62 82 56.2 28 78 23.0 63 72 28.5 29 123 39.6 64 66 65.7 30 112 40.2 65 61 47.3 31 104 49.7 66 65 53.5 32 61 32.3 67 75 42.3 33 115 35.2 68 63 52.9 34 62 65.6 Note: The F statistic is calculated as 35 70 30.6 N -30 T2. (N - 1) 30 E(l + RJ) E ( 1 +Rm ) = (I + Rr) E ( 1; Rm ). Since the terms containing market returns cancel in both of the displayed equations just above, the growth optimum model provides a market diversifica- tion result which is well-known from Sharpe-Lintner theory; namely, a security whose portfolio risk is zero will sell at an expected return equal to the riskless rate. --- ## Page 159 132 R.Roll 558 The Journal 0/ Finance Stock B Stock A 1 + Rm o FIG1.1RE 2 Stocks Satisfying the Growth-Optimum Model with Different Degrees of Market Response It is not true that the growth-optimum model implies a constant ~ coefficient although it may seem to after a first glance at equation (6) ,8 A simple example is sufficient to show the contrary, Figure 2 illustrates the discrete probability 8, Because (6) is and (5) is EO + Rj ) E(l + Rm) 1 + RF [ ----+~j 1 E(l + Rm) ( 1 + Rj ) ( 1 + RF ) E ;( =E _ , 1 + J(.m 1 + Rill a reader not cautious about quotients and reciprocals of random variables mi~ht think that the latter model implies ~j = zero, independent of j. --- ## Page 160 Evidence on the "Growth-Optimum " Model 133 Growth Optimum Model 559 distributions of stocks A and B and of the market. In this example, only four equally likely returns are possible for the market and each stock can return corresponding amounts to those four. The growth-optimum model requires that ( 1 + RA ) _ (1 + RJl ) E "" _E "" 1 + Rm 1 + Rm which in geometric terms implies that the expected slopes of rays from the origin through each possible point in the 1 + RJ, 1 + Rno plane is equal for both stocks. This is clearly satisfied by the radially symmetric solid lines which pass through the points of possible occurrence in Figure 2 . Nevertheless, the slopes of regression lines of RJ on Rm, indicated by dashes, are quite different. Stock A has low portfolio risk because its ~ is low, while stock B has much greater response to the market and thus higher risk. Both securities' returns are perfectly functionally related to the market return but it is already apparent that correlation per se will have the same relatively unimportant role in a growth-optimum as it has in the Sharpe-Lintner framework (i.e., the expected slopes of rays can be equal no matter what correlation occurs between a stock's return and the market's). A close correspondence between the two models can be made more ap- parent by rearranging a few terms. The result (derived in the footnote9 ), is E(RJ - RF ) = [ COY ( Rj, 1.., ) / COY ( Rm, 1",) ] E(Rm - RF ); • 1 + Rm 1 + Rno (7) which is very similar indeed to the Sharpe-Lintner equilibrium equation. In fact, since Cov (Rm' 1,.,,) is negative, one is tempted to suggest that 1 + Rno a second implication of Sharpe-Lintner theory is also satisfied by the growth- optimum model: namely, that security j's expected return will exceed the riskless return if and only if Cov (RJ' Rm) is positive. One would need to prove, 9. To obtain (7) , note that equation (5) is equivalent to CovlRj , I/ (l + Rill) 1 = (I + R~.)K - EO + Rj)K where K ;: Ef I/(l + Rm)l. ( Sa) Multiplying both sides of (Sa) by Xj' the proportion of wealth invested in security j, (or the fraction of aggregate economic wealth represented by security j), Covfl:XJRj , I/O + Rill) J = -K l: fXjEdtJ - R .. ) J J J and substituting for the definition of Rill' i.e., for Rm ;: RF + l:XJ(RJ - RF), CovlRm - RF(l -l: XJ)' 1/ (1 + Rm) 1 = -K E(Rm - R I._). j Since RF(l-l:x) is a constant, K = -Cov[Rm, 1/ (1 + Rill) ]/ E(Rm - Rt,) . Substituting this for K in (Sa) provides equation (7). --- ## Page 161 134 R. Roll 560 The Journal of Finance however, that Cov (Rj , Rm) > 0 implies Cov (Rj , __ 1~_) < 0 and this 1 + Rm is definitely not true in general. Counterexamples can be displayed for highly- skewed distributions.1o B. Testing Sharpe-Lintner vs. Growth-Optimum The two models will provide approximately equivalent empirical implications if certain restrictions are placed on the ranges of individual rates of return. To verify this, one only needs to note the following facts: (a) quadratic utility functions and homogenous anticipations will lead to the Sharpe-Lintner equi- librium result of equation (6) and (b) the logarithmic utility function implied by portfolio growth maximization can be approximated by a quadratic as log (1 + RT ) .... RT - 1/2 RT2 provided that the portfolio's total return is restricted to less than 100 per cent per periodY It is indeed trivial to show that the two models are identical when this approximation to the logarithm is made. Thus, given the truth of one theory, we should not be surprised to find that an empirical test of the other supports it very well, especially when the observed rates of return used in test- ing fall predominantly near zero. Of course, this leads us to ask whether the strong empirical support for the growth-optimum model reported in the pre- ceding section is really damnation for Sharpe-Lintner or just an accidental stroke of choosing a short time interval (one week) which guaranteed that returns were never observed far from zero. An obvious way to test this is suggested by the logarithmic approximation. If growth-optimum theory appears to satisfy the data only because the loga- rithm approximates a quadratic when returns are near zero, one should choose a longer time interval for empirical testing so that many more large and small returns are observed.l2 This was done for both four week and twenty-six week periods with the same common stock data as used previously and the conclu- sions were identical to those for weekly periods already reported. A more refined and direct test to discriminate between the two models can be based on equilibrium conditions of the two competing theories, E(RJ - RF ) = {cov [RJ' 1..., JI 1 + Rm COY [ Rm, 1 R J} E(Rm - Rd = Yj (8) 1 + m 10. However, for at least one special asymmetric case, (lognormal distributions) the growth- optimum model does agree completely with the Sharpe-Lintner result that a security's expected return will be a linear function of systematic risk i i.e., of /iJ' I am indebted to Robert Litzenberger for demonstrating this point. 11. And more than minus 100 per cent per period. Without short sales, this last restriction is presumably satisfied for common stocks by the elristence of limited liability. Samuelson [1970) has derived a broader "fundamental approximation theorem" which shows the close match of mean- variance to any correct portfolio theory when the joint distribution of returns has a small dispersion. 12. This completely ignores the crucial question of investor horizon period that may have a sig- nificant effect on the form of the Sharpe-Lintner market model. Cf. Jensen [1969, pp. 186-191 J. --- ## Page 162 Evidence on the "Growth-Optimum " Model 135 Growth Optimum Model 561 which is the growth-optimum condition (7), and E(Rj - R F ) = {COy [Rj, RmllVar (Rm)} E(~m - R F ) == bj (9) which is the familiar Sharpe-Lintner condition. A suggested test procedure obtains the best estimates of all components of (8) and (9) from time series and then performs cross-sectional regression with those estimates. A series of comparative tests of these two equations was conducted with the common stock returns mentioned previously. In each case, time series were used to calculate Rj , the mean return, and ?j or ~J' the estimated risk measures im- plied by the growth-optimum or the Sharpe-Lintner model, respectively.13 Then, cross-sectional computations were performed for the regression models Rj = 30 + a;~J (10) and (II) Estimated coefficients from (10) and (11) should be compared to their theo- retical counterparts: depending on which theory is correct, ~ or bo should equal RF, the average risk-free interest rate, and ~I or 1)1 should equal unity. In the first test, all calculations were carried out with individual security returns during the same time periodY For example, 1192 separate values of R, ~, and t were obtained from weekly data covering the annual sub-period July, 1962 through June, 1963. Cross-sectional regressions using these 1192 estimates gave fto = 20.1 and 1)11 = 20.3 per cent. The value of RF , as measured by the weekly average interest rate on short-term government debt obligations during the year, was only 3.27 per cent; so neither model satisfied its theoreti- cal prediction very well in this particular annual sub-period.15 OVer all the seven years of available data, the estimated values of ~1 and bl were very sig- nificantly positive and they were scattered around unity in nice accord with their expected level. fto and 1)0 were also reasonably close to RF, at least on average. As a distinguishing test of the two competing theories, however, these results failed miserably; for the estimates were very highly correlated between the models. The two competitive intercepts were practically identical in every 13. To be precise, and 8J == rCSv(RJ,RIlI)/Vir(Rm)J(R", - RF ) where' indicates the sample analog of a population parameter, calculated from weekly observations over a specified period, and - indicates sample mean from the same period. 14. To save computation expense, only New York Exchange listed stocks with at least 30 weekly quotations during a year were included in the sample. The risk-free rate was measured by a weekly "average of short-term government debt obligations" taken from Standard & Poor's Trade Statistics. The market indexes used were: The S&P Composite Index and a rebalanced index constructed by weighting all NYSE Stock Returns equally each week. The results were very similar but only tbe results ' for the rebalanced index are quoted in the text. IS. To save space, only one annual sub-period is reported here but all the results are available from the author upon request. --- ## Page 163 136 R.Roll 562 The Journal oj Finance period as were the two competitive slope coefficients. They deviated much more from theoretical predictions than from each other. In order to sharpen the discriminatory resolution of the data, a different procedure1G was necessary. The first problem to be alleviated was an extremely low explanatory power which was indicated by low R2's (on the order of .05), for the cross-sectional models with individual stock returns. A technique to remedy this is to form portfolios of stocks and conduct cross-sectional tests on portfolios rather than on individual securities. A difficulty arises, however, be- cause randomly-selected portfolios would have very similar values of the risk measures t:t and l To create a cross-sectional spread in risk measures, portfolios ~ust be selected on the basis of risk by grouping stocks with the lowest 1"s or 6's in one portfolio, stocks with higher ?'s or fI's in the next portfolio, and so on. This procedure for forming portfolios makes another econometric problem obvious: In the cross-sectional regression, the explanatory variables contain errors which will tend to bias slope coefficient estimates toward zeroY To alleviate this problem somewhat, stocks were placed in portfolios based on their risk measures calculated from weekly data of a given year and then the mean portfolio return and risk measures for the portfolio as a whole were calculated from weekly data in the subsequent year. Cross-sectional models (10) and (11) were then estimated using these latter estimates and the results are given in Table 2 and Figure 3. To recapitulate, the procedure which resulted in the output of Table 2 and Figure 3 was as follows: l. For each stock (j) in year t, the total risk premia ?J and 8J were calcu- lated from the time series of year t using the rebalanced market index (see note 7). 2. Stocks were ranked from smallest to largest ?J and from smallest to largest ~J' 3. Twenty portfolios were selected by assigning the lowest five per cent of ranked stocks to one portfolio, the next lowest five per cent to a second portfolio, etc. Thus, two sets of 20 portfolios each were formed; one set based on y rankings and. one set on b rankings. 4. Each portfolio's mean return, R-p, was calculated from time series in year t + 1. For each portfolio that had been formed on the basis of y rankings, the growth-optimum risk premium ~II was calculated from year t + 1 data. Similarly, the Sharpe-Lintner premium 8p was calculated from year t + 1 data for each portfolio that had been formed by ?) rankings. 5. For each of six years (1963-64, 1964-65, ... 1968-69), these calculated portfolio mean returns and risk measures are plotted in Figure 3 and regressions across portfolios are reported in Table 2. In Figure 3, the scatters of mean portfolio returns versus the two compet- ing risk measures are displayed for six different years. The solid lines mark 16. Blume [1970] I Miller and Scholes {i9721, Black, Jensen, and SchQles [19721. and Farna and MacBeth [1972], have originated and developed the procedures used here in their work with the empirical validity of the two· parameter portfolio model. 17. This is true, of course, for regressions using individual stocks as well as for those usinl1 portfolios. --- ## Page 164 TABLE 2 RELATIONS BETWEEN AVERAGE RETURNS AND RISK COEFFICIENTS OF 20 PORTFOLIOS SELECTED ON THE BASIS OF INDIVIDUAL RISK COEFFICIENTS CALCULATED ONE PERIOD EARLIER (Rebalanced Index) Growth Optimum Model Sharpe-Lintner Model Period RFe 3.0 to Size of (July through % per % per % per tId Portfolio" June) a annum Rm annum aId R2 annum R2 (No. of Stocks) 1963-1964 3.71 14.4 15.8 - .0817 .0077 15.S -.0781 .0065 59 (6.29) (-.374) ~6.05) (.343) 1964-1965 3.81 12.6 12.4 .0195 .0005 12.5 .00604 .0001 58 (6.53) (.0918) (6.90) (.0297) 1965-1966 4.34 21.0 -9.91 1.85 .940 -9.6S 1.83 .951 62 ( -5.14) (I6.S) ( -5.62) (IS.6) 1966-1967 4.61 29.4 -2.72 1.27 .896 -3.02 1.28 .895 63 ( -1.04) (}2.5) ( -1.14) (12.4) 1967-1968 5.1S 29.2 21.7 .261 .247 22.2 .241 .172 62 (S.20) (2.43) (7.23 ) (1.93) 1968-1969 5.36 -2AS 13.3 1.97 .500 13.2 1.96 .537 59 (3.55) ( 4.24) (3.84) (4.57) a The last date in 1969 was July 11. In other years the first date was the first Thuxsday in July, 196X, and the last date was the last Thursday in June, 196(X + 1). b t-ratios are in parentheses . ., This is equal to the total numher of stocks that have at least 30 available prices in periods t - 1 and t, N, divided by 20. The remainder from N/20, J = MOD(N,20), was distributed such that one extra stock was included in each of the fixst ' J portfolios. d Means of sample values al = .881; t1 = .873; RF = 4.50; ao = 8.43; to = 8.50. e Mean of weekly observations of short-term government debt obligations, Standard and Poor's Tratk Statistics. C) ~ <;) ~ .... ~ a ~ .... ... ;! ;: ;! ~ <;) ~ '" ..... "" 0- c... ~ ~ ;: " '" <::> ;: ;;.:. '" Q c; :; ;;.:. ~ §'. :::: ;: " ~ ~ W -.J --- ## Page 165 138 564 " " '" 1: z · "'. c" ~o . ... . , • ..... c- · . .. .. wb.oo '.00 I . 00 RISK . (S-U. 1963-6ij g i • z ~g ., . a ..... aa lI!- ~ • '. • • • 0 .. '11. 00 '.00 1' . 00 "15K, (e-Ol, 1963-64 g / ,; , z "'" =>0 .... ...... c .. • .. ~;:I.~oo:--"""'~::-"""'~"".-:oo-:- "15K. 1966-67 " D ~ / .. '. . . • g""",,-....,,.,...,-:---.,.,::--::,- ~ . oo 2 .00 ~0.00 "ISK. (C-Ol. 1966-67 The Journal oj Finance ,. " " .. . ... . . z "'0 .. ~" ... . w" c- . • " " ~.oo RISK. g :2 z • "'" "\, =>0 ..... w" ",- 0 " "i.. 00 "15K. o D :: z c" => .. ... . .... .. ",'" '.00 (S-LJ • • • • . • '.00 ((;-01. IS . 00 196~-65 12.00 1964-65 .. g ";:+""'-+--:0:-::-:----.-- ' . 00 00.00 "IS • 1967-68 FIGURE 3 " " " " z c =>" ..." ~o • ; le.oo RISK. g g z '" =>0 "'0 ~o • " 0 " .. le.oo " " " .. RISK. R. Roll 11.00 26.00 (S-LI • 1965-66 16. 00 11. 00 (C-O) • 1965-66 z '" ~: .. ~ ~. g ~+-L-_~ ___ _ '_15.00 - 10.00 -5.00 RISK. (S-L). 1968-69 g ~+-------:--~- 1. 13 . 00 ·a.oo ·S.GO RISK, IC-OI. 1968-69 Mean Portfolio Returns and Estimated Portfolio Risk Premia, New York Exchange Stocks, 1962-1969, Rebalanced Index * (S-L) Denotes Sharpe-Lintner Risk Premia and (G-O) denotes Growth-Optimum Premia. the theoretical predictions, an intercept of RF and a slope of unity.ls Table 2 contains results from cross-sectional models (10) and (11) applied to these data. On average across the six years, both the growth-optimum and the Sharpe- Lintner model seem to have excessive intercepts, (~I' bo > RF), and deficient slopes, (il' bl < 1). The averages are given in footnote d of Table 2. These deviations from the anticipated can no doubt be attributed, at least in part, 18. Since the axes are scaled differently in each plot, the lines do not appear to have slopes of unity upon first examination. Note that the plotted lines are the theoretical predictions and are not the regression lines reported in Table 2. --- ## Page 166 Evidence on the "Growth-Optimum" Model 139 Growth Optimum Model 565 to errors in the measurement of risk premia. In the first two years and in year 5 (1967 -68) the wide scatter suggests that either the mean portfolio returns or the risk premia were measured inaccurately. In the two years of best fit, 1965-66 and 1966-67, portfolios with low risk premia return less than anticipated while portfolios with high premia return more. This is unlikely to be just a sampling phenomena if the standard errors of al and bl are credible. For example, the growth-optimum slope for 1965-66 is al = 1.85 which is nearly eight standard errors above unity.19 Perhaps the most striking characteristic of the two models is their very close relation over time. The slope coefficients and intercepts given in Table 2 vary widely across periods but are extremely close, between the two models, in each period. This is obvious from even a quick look at Figure 3.20 Based on these results, one can only conclude that the two models are empirically identical. A qualification is in order, of course; assets whose returns are much more highly skewed (e.g., warrants), may permit a finer discriminatory test. V. SUMMARY AND CONCLUSIONS If investors wish to maximize the probability of achieving a given level of wealth within a fixed time, they should choose the "growth-optimum" port- folio; that is, the portfolio with highest expected rate of increase in value. This paper has examined the implications for observed common stock returns of all investors selecting such a portfolio. Given some widely-used (and useful) aggregation assumptions, the growth- optimum model implies that the expected return ratio E[(l + RJ)/(1 + Rm)] is equal for all securities.21 This implied equality of expected return ratios was utilized in analysis of variance tests with New York and American Ex- change listed stocks from 1962-1969 in order to ascertain the basic validity of the growth-optimum model. The model was well-supported by the data. The growth-optimum model was compared algebraically to Sharpe-Lintner theory, which is probably the most widely-used portfolio result in empirical work. A close correspondence was demonstrated between their qualitative im- plications. For example, both models imply that an asset's expected return will equal the risk-free interest rate if the covariance between the asset's return and the average return on all assets, Cov(~J' Rm), is zero. For most cases, the growth-optimum model also shares the Sharpe-Lintner implication that an asset's expected return will exceed the risk-free rate if and only if Cov (RJ' Rm) > O. There are, however, some cases of highly-skewed probability distributions where this implication does not follow for the growth-optimum model. A close empirical correspondence between the two models was demonstrated for common stock returns. The procedure (1) estimated returns and risk premia 19. For a more detailed discussion of this point, see Friend and Blume [1970], 20. The greatest difference between '3.1 and £1 is .02 which is only about 2.3 per cent of their average value. Between 10 'So, the greatest difference is about six per cent of their average value . .'\ close connection between the two models was previously implied by the work of Young and Trent [1969], They showed that the geometric mean of portfolio returns was closely approximated by functions of the arithmetic m~an and variance of returns. These functions were developed as approximations to the geometric mean. For accuracy, they require a minimal amount of skewness and are, therefore, analogous to the truncated (after two terms), Taylor series expansions of log. (l+R) . 21. RJ is the rate of return on security j and Rm is the rate of return on a portfolio of all assets. --- ## Page 167 140 R.Roll 566 The Journal of Finance implied by the two models from time series; (2) calculated cross-sectional relations between estimated returns and risks; and (3) compared the cross- sectional relations to the theoretical predictions of the two models. They could not be distinguished on an empirical basis. In every period, estimated corre- sponding coefficients of the two models were nearly equal; and indeed, they deviated much further from their theoretically anticipated levels than they deviated from each other. REFERENCES Fischer Black, Michael C. Jensen, and Myron Scholes. "The Capital Asset Pricing Model: Some Empirical Tests," in Jensen, ed., [1972]. Marshall E. Blume. "Portfolio Theory : A Step Towards Its Practical Application," Journal 0/ Business, 43, (April, 1970), 152-173. James V. Bradley. Distribfltion-/ree Statistical Tests, (Englewood Cliffs, :-;;.J.: Prentice-Hall), 1968. Leo Breiman. "Investment Policies for Expanding Businesses Optimal in a Lonl!:-Run Sense," Saval Research Logistics Quarterly, 7, (December, 1960), 647-651. L. Breiman. "Optimal Gambling Systems for Favorable Games," Proceedings 0/ the Fourth Berkeley Symposium on Mathematical Statistics and Probability, I (Berkeley. University of California Press), 1961, 65-78. Pao L. Cheng and M. King Deets. "Portfolio Returns and the Random Walk Theory," Journal 0/ Finance, 26 (March, 1971), 11-30. John L. Evans. "The Random Walk Hypothesis, Portfolio Analysis and the Buy-and-Hold Crite- rion," Journal 0/ Financial and Quantitative Analysis, 3 (September, 1968), 327-342 . Eugene F. Fama and James MacBeth. "Risk, Return and Equilibrium: Empirical Tests," (Workin!( paper, University of Chicago, Graduate School of Business, February, 1972). Eugene F. Fama. "The Behavior of Stock Market Prices," Journal 0/ Bllsiness, 38 (January, 1965), 34-105. Irwin Friend and Marshall Blume. "Measurement of Portfolio Performance Under Uncertainty," American Economic Review, 60, (September, 1970), 561-575. Franklin A. Graybill. An Introduction to Linear Statistical Models, Vol. I, (:-;;ew York: McGraw- HilI,1961). Nils H. Hakansson. "Capital Growth and the Mean-Variance Approach to Portfolio Selection," Journal 0/ Financial and Quantitative Analysis, VI (January, 1971), 517-557. Nils H. Hakansson and Tien-Ching Liu. "Optimal Growth Portfolios When Yields are Serially Correlated," Review 0/ Economics and Statistics, 52 (November, 1970), .185-394. Michael C. Jensen, ed., Studies in The Theory 0/ Capital Markets, (:-':ew York, Praeger Publishing Co., 1972). Michael C. Jensen. "Risk, The Pricing of Capital Assets, and the Evaluation of Investment Port- folios," Journal 0/ Business, LXII (April, 1969), 167-247. Benjamin F. King. "Market and Industry Factors in Stock Price Behavior," Journal 0/ Bllsiness, 39 (January, 1966 supp.), 131}-190. . Henry Allen Latane. "Criteria for Choice Among Risky Ventures," Journal 0/ Political Economy, 67 (April, 1959), 144-155 . John Lintner. "The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets," Review 0/ Economics and Statistics, 47 (February, 1965), 13-37. Harry M. Markowitz. Port/olio Selection : Efficient Diversification 0/ investments, (New York: John Wiley & Sons, Inc.), 1959. Merton H. Miller and Myron Scholes. "Rates of Return in Relation to Risk: A Re-examination of Some Recent Findings," in Jensen, ed., (1972). Donald F. Morrison. Multivariate Statistical Methods, (New York: McGraw-Hili, 1967) . A. D. Roy. "Safety First and the Holding of Assets," Econometrica, 20 (July, 1952), 431-449. P. A. Samuelson. "The Fundamental Approximation Theorem of Portfoli(l Analysis in Terms of Means, Variances and Higher Moments," Review 0/ Economic Stlldies, 37 (October, 1970) , 537-542. William F. Sharpe. "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk," Journal 0/ Finance, 19 Q, £2:,1-1 tii = I. The payoff to player I for policy til vs. ti2 is P (til! ;;.ti~!}. The optimal policy is shown to be ti· = U!?·, where U is a random variable uniformly distributed on [0, 2), and !?. maximizes £ In !?/! over!? ;.Q, 2:, bi = I. Curiously, this competitively optimal investment policy!?' is the same policy that achieves the max.imum possible growth rate of capital in repeated independent investments (Breiman (1961) and Kelly (1956». Thus the immediate goal of outperforming another investor is perfectly compatible with maximizing the asymptotic rate of return. 147 I. Introduction. An investor is faced with a collection of stocks (XI' X 2, ••. , Xm) drawn according to some known joint distribution function F. We shall assume that stock values Xi are nonnegative. A portfolio is a vector Q = (b l , ... , bnr)', bi ;;;. O,2:,bi = I, with the interpretation that bi is the proportion of capital a\loc&ted to stock i. The capital return S from investment portfolio Q is m S = J?I X = 2: biXi • (I) ;=1 How should Q be chosen? A currently accepted procedure is the efficient portfolio selection approach of Markowitz (1952, 1959). A portfolio Q is said to be efficienl if (EQ',K, Var QIK) is undominatt:d. Criticisms of this approach are many. Only the first two moments are used in the analysis; there is no optimality of this procedure with respect to other obvious investment goals, and no choice procedure among the efficient portfolios is provided. (See Thorp (1971) and Samuelson (1969) for further comments.) Also, such a portfolio is not necessarily admissible [Hakansson (197 L p. 529), Thorp (1971 , p. 20)] in the sense that it may be stochastically dominated by some other mixture S. Another criterion for selecting Q, that of maximizing E In S, has been put forth by Kelly (1956) and Breiman (1961), and persuasively advocated by Thorp (1969, 197 L 1973). (Also see Latane (1959) and Williams (1936).) This portfolio is admissible, since it maximizes the expectation of a monotonic function of S. The resulting portfolio investment policy Iz* has been demonstrated by Breiman to have the following properties: Pl. In repeated independent sequential investment, 12* maximizes lim inf(l / n)ln Sn' Thus the asymptotic "interest rate" is maximized. "Received January 19, 1979; revised September I, 1979. AMS 1970 subject classification. Primary 90040. Secondary 90015. IAOR 1973 subject classification. Main: Investment. Cross reference: Games. ORj MS Index 1978 subject classification. Primary 202 Finance, portfolio. Secondary 234 Games, noncoop- erative. Key words. Competitive portfolio policy, two-person zero-sum games, logarithmic investment, maximum rate or" return. t This work was partially supported by National Science Foundation Grant ENG 76-03684 and JSEP NOOOI4-C-0601. 161 0364-765X j 80j 0502 j O 161$01.25 Copy right cr:J 1980. The Institute or Management Sciences --- ## Page 175 148 R. M Bell and T. M Cover P2. The time required to achieve a certain capital -1 i~ nlll1imized hy f!." (in a sense that can be made precise). in the limit as A ~ 'X. Yet Q* is not accepted in current economic practice. Perhaps one reason is that maximizing E In S suggests that the investor has a logarithmic utility for money. However, the criticism of the choice of utility functions ignores the fact that maximiz- ing E In S is a consequence of the goals represented by properties P I and P2, and has nothing to do with utility theory. (See Thorp (1971).) A thorough discllssion of investment strategies and their relation to utility theory is developed in Arrow (1971). We are not interested in utility theory in this paper insofar as utility theory deals with the consistency of subjective preferences. We wish instead to emphasize the objective aspects of portfolio selection, i.e., properties of optimal portfolios that have some objective appeal. In particular we wish to add another goal to the list PI , P2. namely that of outperforming another investor (or even of outperforming oneself with respect to what one could have done). If we can show that all three goals are uniquely achieved by a given policy, we are on our way to making an objective case for utility-independent optimality of the stated portfolio. An objection to PI and P2 and, by extension, to lz*, held by Samuelson (1967, 1969) and others, is that not all investors are interested in long term goals. Samuelson (1969, p. 245) writes, "Our analysis enables us to dispel a fallacy that has been borrowed into portfolio theory from information theory of the Shannon type. Associated with independent discoveries by J.B. Williams (1936), John Kelly (1956), and H.A. Latani: (1959) is the notion that if one is investing for many periods. the proper behavior is to maximize the geometric mean of return rather than the arithmetic mean. I believe this to be incorrect (except in the Bernoulli logarithmic case where it happens to be correct for reasons quite distinct from the Williams-Kelly-Latane reasoning) .... It is a mistake to think that. just because a w** decision ends up with almost-certain probability to be better than a 11'* decision. this implies that \1** must yield a better expected value of utility." Another possible objection is that lz* may be too consen'ative, since it optimizes a concave (risk averse) function of the return S. One interpretation of "too conserva- tive" could be that 1:.* will be outperformed (i.e .. I:.'.J{ >I:. *r~) with high probability by a more ambitious policy 1:.. Alternatively. too conservati\'e might mean that with substantial probability Q* will be outperformed by a large factor (i.e., lz'K > cQ*'K, for some constant c> I) by a more risky policy Q. Thus a reasonable goal for an individual investor or a mutual fund would be good short term competitive perfor- mance. With the above objections in mind, we are led to the analysis of one-stage investments. Consider the two-person zero-sum game in which two investors seek portfolio policies that are competitively best in the sense that at least half the time one achieves more capital than one's opponent. Surprisingly. the game theoretic optimal strategy will be shown to be VQ* where U is an independent uniform [0,2] random variable, and 1:.* is the same log optimal policy as before. Furthermore, among non-randomized strategies, Q* is shown to be competitively best in the sense that it will not be beaten by very much very often. Thus the alleged conservatism of Q* must be established on other grounds: and the short term value of VQ* is established competitively. In the next section. we shall argue for the naturalness or the random variable V in the competitive investment game. Theorem l. establishing L'I}* as the solution of the game, will be proved in *3. 2. A game-theoretic digressioll. Before proceeulng, we establish the necessity for randomization in the competitive investment game. Suppose 2 players each have I unit of capital. Their competitive positions are equal. --- ## Page 176 Competitive Optimality a/Logarithmic Investment 149 However. let us now suppose that player 2 has available to him any fair gamble (EX = 1, X ;;. 0). By selecting the distribution of the gamble X judiciously, he can beat player I with probability I - £. Simply let P(X = I/(l - £» = I - £, P(X = 0) = (. Then P(X > I) = I - (. Therefore, player I must protect himself by randomizing his capital. This is a purely game theoretic maneuver and has nothing to do with maximizing investment return. We now solve the following two-person zero-sum game. Let players I and 2 choose d.f.'s F and G, respectively, fx dF = fy dG = I, F(O-) = G(O- ) = O. Assume X - F and Y - G are independently drawn. The freedom of choice we allow in the choice of F and G makes physical sense, since any capital distribution F(x), J x dF(x) = I, F(O-) = 0, is achievable from initial capital I by a sequential gambling scheme on fair coin tosses (Cover (1974». The payoff to player 1 is P (X ;;. Y) = J G dF. LEMMA. The value of this game is t, and the unique optimal strategies are P(/) = G*(t) = { t/2, 1, PROOf. For F* and for any G, 0.;; t .;; 2, I> 2. P(Y>X)= JPdG= !ooomin{t/2,I}dG(t) .;; t LootdG(f) = t· Thus F* achieves } against any G. (2) (3) (4) Uniqueness of the optimal distribution P(x) is proved by assuming P( Y > X) .;; 1 for (i) Y uniform [0, 2J, (ii) Ya two point distribution at ° and a point c E [I , 2], and (iii) Y a two point distribution at c E [0, I] and 2. Then (i)~ P(2) = I; (ii)~ F*(c) .;;c/2, l';;c.;;2; and (iii)~F*(c)';;c/2, O 0, a.e., £2:7'=1 Bi = 1. Note that the random investment policy !1 can be achieved by first exchanging the I unit initial capital for a fair random return W drawn according to the distribution of 2:7'-1 Bi . Observe that W > 0, £W = I. Then W is distributed across the stocks according to the conditional joint distribution of (BI' B 2, ••• , Bm) given 2:7'= I Bi = W. The latter distribution can be performed on paper. Happily, the allowed conditional randomization is not necessary in the game theoretic optimal policy in Theorem I below. Let the investment vector J.: = (XI' X 2, . .. , Xm)' be a r.v. with known distribution function F~). We assume that J.: > 0, a.e. To eliminate degeneracy, we also assume --- ## Page 177 150 R. M Bell and T M Cover that --00 0, bt = 0, i = l, 2, ... , m, where A is chosen so that 2:b;* = I. But we see A = l, since A = 2:btA = 2:bt EXj(2:b/ X;) = E(2:bt X, )/(2:b/ Xj ) = I. (6) (7) We now investigate the payoff of fl* = UQ* against any other investment policy fl E ~l": P { B' X ;;'Jl..*1 X} = P {Jl..I X;;. U 12*1 X} = P { U ~ (B' X)/(b*' X)} ~ 1/2E(( Jl..I X)/(12*' X») m = 1/22: EBiE(Xj(2:btXi)) ~ 1/2'2,EB;A 1=1 = Aj2E'L,E; = A/2 = 1/2. (8) Thus fl· = UIz* achieves the value of the game against any fl, and the proof is complete. The above strategy fl· can be implemented by first ex.changing the I unit initial capital for the fair gamble U, uniformly distributed over [0, 2J, then distributing U on the investments according to the solution 12* max.imizing E In Q'K This result can be generalized to show that fl* = UQ* will not be beaten by very much very often: 0 COROLLARY I. P {!l.1){ ;;. eUQ*'K} ~! c, for all Ii E ~.j) . e > 0. PROOF. P {eU ~ fl'KI 1z*'K} ,,; (1 e)E(fl'KI f2*IK), and the proof proceeds as in Theorem I. Dropping the randomization U increases this probability by at most a factor of 2: COROLLARY 2. P{ Jl..' K ;;. e 12*' X) ~ II c, jiJr all !l. E ','i), C > 0, (9) PROOF. By Markov's inequality and (8). P(Jl..' X;;;, C12*1 X) ";(llc)E(Jl..f XI 12*fK) ~ lie. --- ## Page 178 Competitive Optimality of Logarithmic Investment 151 RnlARK I. This is the best that can be attained by any nonrandomized strategy, as can be seen from the: d ISCUSSIOn at the beginning of ~2. RDI ARK 2. Corollary 2 bears a strong resemblance to Markov's lemma. i.e., Y > 0, EY = fL= PO > q.t.) ';; l i e. ":Ie> O. This suggests that t!.*?{ acts like the fixed amount of capital ,u in \1 arko\'s lemma and that (z*'l ca n be changed from a competitive standpoint only by fair randomization. Inequality (9) is true despite the fact that E!tK may be greater than E(z*'K 4. Example: The St. Petersburg paradox. In the St. Petersburg paradox. a gambler pays an entry fee c. He receives in return a random amount of capital X . where P(X = 2') = 2 - I . i = I. 2 . . .. . 00. Note that EX = 00. Suppose that a gambler has total initial capital So. He is allowed to receive I unit of St. Petersburg investment for each c units that he pays as an entry fee. Let him invest the amount bSo' 0 < b .;; 1. and retain (I - b)So in cash. Thus his return S is given by S = So«(I - b) + (b / c)X). In the framework of the previous sections, the investment vector is K = (XI' X 2Y = (I. X / c)'. Let b* E [0, I] maximize Eln S. We calculate dE In S - I + X / c --- = E ------:-,.-- db (I - b) + (b / c)X ""(I)i (2i/e-l) = i~1 ."2 2'b / c + (I - b) ( 10) Letting b = I, we see that dE In S/ db = I - (e/3), which IS > 0 for c .;; 3. Thus b* = I, for 0 .;; c .;; 3. For c > 3, the solution b* to (10) tends monotonically to zero as the entry fee c ~ 00 . Finally it can be seen that b* and max E In S are always strictly positive. Investing a proportion of capital b* guarantees that (I) The investor is acting in accordance with an investment policy maximizing lim inf(l / n)ln Sn' regardless of whether or not the other investment opportunities are of the St. Petersburg form ; and (2) the investor investing Vb* is competitively optimal in the St. Petersburg game. Moreover, we see that all entry fees c are "fair." However. the proportion b* of total capital invested varies as a function of c. Also. b* is independent of the total initial capital So. Finally, if the investment fee is low enough, i.e., 0 .;; c .;; 3. then b* = I and all of the capital is invested. This results in Sn - SoC 4/ cr in the sense that (1 / n)log2 Sn ~ 2 - log2 c, for 0 .;; c .;; 3. 5. Conclusions. It should now be clear that the investment policy /.2.* achieving max E In tlK has good short run as well as good long run properties. In addition, /.2.* is admissihle in the sense that no other policy /.2. stochastically dominates /.2.*. We wish to comment on the use of V/.2.* (as opposed to (z* alone) in practice. We have seen in §2 that the use of V is a purely game theoretic protection against competition and has nothing to do with increasing a player's capital. Thus we feel that lz* alone is sufficient to achieve all reasonable competitive investment goals, and we do not choose to advocate the additional randomiza tion U. Finally, it is tantalizing that I}* arises as the solution to such dissimilar problems as maximizing lim inf( I / n)ln Sn and maximizing P (!!\r >!ISK). The underlying reason for this coincidence will be investigated. References 111 Arrow, K. (1971). Essays in the Theory of Risk-Bearing. Markham Publishing Co., Chicago. [21 Bicksler, J. and Thorp. E. (1973). The Capital Growth Model: An Empirical Investigation. 1. of Financial and Quantitative Analysis. VIII 273 ··287. --- ## Page 179 152 R. M Bell and T. M Cover 166 ROBERT M. BELL AND THOMAS M. COVER (3) Breiman, L. (1961). Optimal Gambling Systems for Favorable Garnes. Fourth Berkeley Symposium. 1 65- 78. (4) Cover, T. (1974). Universal Gambling Schemes and the Complex.ity Measures of Kolmogorov and Chaitin. Stanford Statistics Dept. Tech. Report No. 12. (5) FeUer, W. (\950). An Introduction to Probability Theory and Applications. Vol. I, second edition, 235- 237. (6) Hakansson, N. (1971). Capital Growth and the Mean-Variance Approach to Portfolio Selection. J. of Financial and Quantitative Analysis. VI 517- 557. (7) -- and Liu, 1'. (1970). Optimal Growth Portfolios When Yields Are Serially Correlated. ReI', Econom. Statist. 385-394. (8) KeUy, 1. (1956). A New Interpretation of Information Rate. Bell System Tech. J. 917- 926. (9J Kuhn, H. and Tucker, A. (1951). Nonlinear Programming. Proc., Second Berkeley Symposium on Math. Stat. and Prob., University of California Press, Berkeley, Calif. 481-492. [IOJ Latane, H. (1959). Criteria for Choice Among Risk Ventures. J. of Political Economy. 67 144--155. (II) Markowitz, H. (\952). Portfolio Selection. The J. of Finance VII 77-91. (12) --. (1959). Portfolio Selection. Wiley and Sons, Inc., New York. (\3) Samuelson, P. (1967). General Proof that Diversification Pays. J. of Financial and Quantitalive A nalysis II 1- 13. (14) - -. (1969). Lifetime Portfolio Selection by Dynamic Stochastic Programming. Rev. Econom. Statisl. 239-246. [I5J Thorp, E. (1971). Portfolio Choice and the Kelly Criterion. Business and Economics Statistics Proceedings, American Statistical Association. 215-224. (16) --. (1969). Optimal Gambling Systems for Favorable Games. Rev. Internat. Statist. 37273-293. (17) Williams, 1. (1936). Speculation and the Carryover. Quarterly J. of Economics. SO 436-455. BELL: DEPARTMENT OF STATISTICS, STANFORD UNIVERSITY, STANFORD, CALIFORNIA 94305 COVER: DEPARTMENTS OF STATISTICS AND ELECTRICAL ENGINEERING, STANFORD UNIVERSITY, SEQUOIA HALL, ROOM 130, STANFORD, CALIFORNIA 94305 --- ## Page 180 153 13 IEEE TIlANSAC110NS ON INFORMATION THEORY, VOL 34, NO. S, ·sEPTEMBER 1988 1097 A Bound on the Financlal Value of Information ANDREW R. BARRON, MEM!lEIl,.IEEE, AND THOMAS M. COVER, FEllOW, IEEE Alntrod -II will be shown duol e8ch. bil of information at mosl dooblos Hoe resulting wO!llth in the general stock _ setup. This inforillation bound no the growth 0' wealth is IICIuaIly attained • .,. 0 for hl - O. (3) These are the Kubn-Tucker conditions characterizing b'(F) (see Bell and Cover 131, Cover 141, and Finkelstein and Whitley 15D. II current wealth is reaI10cated according to b' in repeated independent investments against stock vectors X" X" .. ·. inde- pendent identieaIly distributed (i.i.d.) according to F( r), then the wealth S: at time n is given by . S.: =- n b*'Xj • 1-1 (4) Manuscript received January 8, 1988; revised February 15, 1988. lbis work was supported in pari by the National Science Foundation under a research fellowship and under Contract NCR-85-20136 Al and in pari by the Office of Naval Research under Contract NOOO']4·86-K-06. This work was partially presented at the 6th Annual Symposium on Informahoo Theory. Tashkent, USSR. September 1984. A. R. Barron is with the Department of Statistics, University of Illinois. 725 SoUIt,. Wright Street. Champaign, IL 61282. T. M. Cover is with the Departments of FJectrical Engineering and Statis- tics. Stanford University, Durand, Room 121, Stanford. CA 94305. IEEE Log Numbe the value of the random variable X (if X is discrete), and since H( XIY) bits are required to describe X given knowledge of .y, the decrement in the expected description length of X is given by H(X)':'" H(XIY) =I(X; Y). In summary; the mutual information I( X; Y) is 1) the decrease in the entropy of X when Y is made available, 2) the number of bits by which the expected description length of X is reduced by knowledge of Y, 3) The rate in bits at which Y can communicate with X by appropriate choice of Y, 4) the error exponent for the hypothesis test (X, Y) independent versus (X, Y) dependent. IV. PORTFOUOS BASED ON INCORRECT DISTRIBUTIONS Suppose that it is believed that X - G(x) when in fact X- F( x). Thus the incorrect portfolio b*( G) is used instead of b*( F). The doubling rate associated with portfolio b and distri- bution F can be written W(b,F) = ! log IfxdF(x) (14) with resulting growth of wealth (15) The decrement in exponent from using b*( G) is IlW(F,G) -W(b*( F), F)- W(b*(G), F). (16) The following theorem is. central io our results. Theorem / : O$IlW(F,G) $D(FIIG). (17) Proof: The first inequality 0 $ Il follows by the optimality of b*( F) for the distribution F .. The second inequality Il $ D is shown to be a consequence of the optimality of b*(G) for the distribution G. Let F and G have densities f and g wilh respect to some dontinating measure. The result Il $ D is trivially true if D(FIIG) - co, so it is henceforth assumed Ihat D is finite (whence F O} has prObability one with respect to F. Then Ilw(i,G) - ~(IOg ~) dF -j log ~-- dF ( S* g f) A 52 f g -f log ~ - dF + D( FIIG) ( S* g) A 52 f S* $Iog~ ~ dG+D(FIIG) $ D( FIIG) (19) where . the first inequality follows from the concavity of the logarilbm and the second from the Kuhn-TuCker conditions for the optimality of b'( G) for the distribution G. We can improve Theorem 1 by normalizing X. Let f denote the distribution of X/EX,. We note !hat E(logIl,X/t>;X) de· pends on the ~stribution F(JC) only through Ihe distribution of X/E;:, X, - F. Corollary: IlW( F, G) $ D( fIlG). Remark: Another relationship between W and D is shown by Milri [13]. The doubling rate W- W(b*(F), F) is equaltQ Ihe ntinimum of D(FIIG) over all distributions G for which EcX, $1, for i--I,2,···,m. V. THE INFORMATION BoUND FOR SIDE INFORMATION We now ask how Il and 1 are related for the stoCk market. We have and b*'(Y) X Il - £ log --;;;;x- f(X,y) 1= Elog f(X)f(Y) (20) (21) where (X, Y) - F(x, y). The first involves wealth and depends on the values X takes on. The second involves information and depends on X and Yonly through the density f(JC, y). The foUowiitg theorem establishes that the increment Il in the dou- bling rate resulting from side information Y is less !han or equal to the mutual infOrmation I . Theorem 2: O$Il$T(X;Y) . (22) Proof" For anyy, Ie! Px1y be Ihe conditional distribution for X given that Y - y and let Px be the marginal distribution for X .. Also Jet b" - b*(PXly )' Apply Theorem 1, wilh PXI , and Px in place of F and G, respectively, to obtain . [ b."XI ]. 0$£ log b*'X Y-y $D(px1yIlPx ). (23) Averaging with respecl to the distribution of Y yields OSIl$~ (2~ St =b"(F)X 52 = b"( G)X Remark.: An alternative proof of tltis Iheorem, based on money (18) ratio tests and Stein's lemma, appears in [7]. be the wealth factors corresponding to Ihe optimal portfolios with respect to F andG. From the Kuhn-Tucker conditions the wealth factor 52 is strictly positive with probability one with respect to G (and with :respect to F since F« G). It foUows (again since F"" G) Ihat the set A - (x: 52 > 0, f(x) > 0, VI. SEQUENTIAL PORTFOUO EsTIMATION Here we show !hat Ii good scquence of estimates of the true market distribution leads to asymptotically optimal growlh rate of wealth. First we generalize Theorem 1 to handle the sequential setting. --- ## Page 182 A Bound on the Financial Value of Information 155 lEEE TRANSACTIONS 'ON INFORMATION THEORY, VOl. 34,.NO. 5, SEPTEMBER 1988 1099 Let XI.X,.· : " X" be a ~uence of random stock vectors with joint · probability distribution p.. The .. log'optimal ~uential strategy uses the portfolio b,' - b·(PX•I.\ .• , ....... _\). which maxi- mizes the .conditional expected value of log b' Xi given that XI = %1,' • " X;_l == Xj -1 ' Suppose that instead of IJ* , we use-portfolios 6, = b·(QX.I., ...... H) which are optimal for an incorrect distrib\1' tion Q" for the ~uence XI'" ., X •. Let PXIIX1 •• ••• XI _ 1 and QX1IX1 ..... XI _ 1 be the regular conditional distributions associated with p. and Q., respectively. We com· pare the resulting wealth . S.- n "ix, (25) '-I with the wealth " S: = n b,*TX,. (26) ;-1 Theorem 3: S* 0.:5 Elog i . .:5 D(p·IIQ·). (27) Proo!" Application of Theorem 1 shows that [( b~'X.)1 ] Os;E log ~x,' XI,"',X'_I S; D( Px,lx,-"IQx,IX'-')' (28) Averaging with respect to the distribution of X, - I = (XI" . " X, _ I) and then summing for i -1,2 •. . '. n yields • ( b~'X . ) Os;E E log i;' .-1 ,X, . S; E ED(Px.1x,- IIIQx.lx,-I) j-l =D(P"IIQ·) by the chain rule. completing the proof. (29) Suppose XI ' X,. . .. are independent with unknown density pIx). Clearly. the optimal portfolio b* does not depend on the time i or on the past. However. it p( x) is unknown. a series of estimators of the distribution P,(·) corresponding to density estimators p,(x) based on the pas! X'-I ¥lay be used to oblain asymplotically optimal portfolios b, = b'( P,). It is often the case (see Barron Ill), (12) that there exists a ~u!nce of estimators p. converging to P in the sense thai ED(PIIP.) - 0, alleast in the Cesaro sense, ie., 1-- " lim - E ED( PilI;) =0. (30) n-oo n 1-1 In this case Theorem 3 applies with Qx IX'-I given by the estimator I; (-) to yield • 1 S' limE-log-;"=O. (31) n S. It follows that the aClual wealth S. is close 10 the log·optimal wealth S: as shown in the fol\owing theorem. Theorem 4: Let XI' X" . " be i.i.d . ..; P. Let p. be a sequence of estimalors of the true distribution P such thai (32) and let where ", = b'( 1;). Let . S: - n b*'( P)X, 1:-1 be the optimal wealth seq~ce. Then ~,,_ S,,* 2" 0(1) , where 0(1) - 0 in probability. (33) (34) (35) (36) A Consequently, if S: has an exponential growth rale W', then S. has the same asymptotic exponent. Proof: To see that S.IS: = 2" 0(1) in probability. first ob· serve that by Markov's inequality p{ S. > 2"} ,; 2-.'E S. < r.' SrI· S,,"- (37) where the inequality E( S. IS: ) s; 1 follows from the Kuhn- Tucker conditions . for the optimality of b' (see Bell and Cover [8]). . ' . On the other hand, using the notation y+ - max {O, y}. y- = max{O.--y}. p{ f > 2.'} -p{logf > n,} 1 ( S*)+ s;-E log~ n( S • 1[1 S: 1] s;- -Elog--x-+- (n S" n. (38) where the first ineqUality follows from Markov's inequalily and the second from E(logS:IS.r = Elogmax{ S.IS: ,J} S; Elog(l+ S./S:) s;log(I+ E(S.IS:» S; log2=1 (39) by the concavity of the logarithm and the Kuh,.n-Tucker condi- tions. Combining (31), (37) and (38), we have S. IS: = 2"°(1) in probability, as claimed. VII. ExAMPLES We ftrst give an example due to Kelly [6) in whicb ,1 - I . Here the slock markel is a horse race, which, in .the selup of (I). ·consists of a probability mass function P{ X = O,e;} = p,. i = 1,2, .. . , m, where ~; is a unit vector with a 1 in the i th place and O's elsewhere, 0.' equals th~ win odds (0. for I), and p, is the probability that the ith horse -;vins the race. Then W(X) - max Elogb'X • ;;:I; m~ .f Pi logb;O; " i-1 where H(X)=-r:,,-,p,logp,. Also b" - p, i.e., the optimal --- ## Page 183 156 A. R. Barron and T M Cover 1100 IEEE TRANSACTIONS ON INFOllMAnON THEORY, VOL 34. NO. 5, SEPTEMBER 1988 portfolio is to bet in proportion to the win probabilities, regard- less of the odds. For side ,information Y, where (X, Y) has a given distribution, a similar ealcuI~tioil yields W(XIY)-Ep,logq-H(XIY) (41) and b,. - P(X-O,",ly), ;-1,2,·· ·,m. Here the optimal portfolio is to bet in proportion to the conditional probabilities, given Y. Subtracting (40) Crom (41), we have I>-W(XIY) - W(X) = H(X)- H(XIY) = I(X; Y). (42) Consequently, the information bound on I> is tight Of course, it sometimes happens that the information Yabout the market is useless for 'investment purposes, The next example has 1>- 0, I-I. Let X - (1,1/2) with probability 1/2, and X-(l,3/4) with probability 1/2. Let Y-X. An investment in the first stock always returns the investment, but an investment in the second stock may cut the investment capital to either 1/2 or 3/4 depending on the outcome X. It would be foolish to invest in the second stock, since the first. stock dominates its perfor- mance. Thus b'-b'(y) -(1,0) for all y, and I> =0. On the other band, since tbe outcomes of X are equally likely, and y .... x, we see leX; Y) = leX; X) = H(X) - H(XIX) - H(X) -1 bit. (43) Thus a bit of information is available, but d - 0 and the growth rate is not iIDproved. VIII. CONCLUSION We offer one final interpretation. Recall that H(X)- H(XIY) = l( X; Y) is the decrement in the expected description length of X due to the side information Y. Hence the inequality d ;S; I has the interpretation that the increment in the doubling rate of the market X is less than the decrement in the description rate of X. IX. ACKNOWLI!DGMENT We would like 10 thank R. O. Duda for speculations thai led to the statement of Theorem 2. REFERENCES (lJ l. Bmman. "Optimalaamblioa systems ror favorable 'games," in Proc. 4th 8erkdq Symp .• vol. 1. pp. 65:.... 78, 1961. (21 H. Chemotr, "Large-sample theory: Pa.-arnetric case," Ann, Math. SIal., pp. 1-22,1956. [31 R. Bdl and T . Cover, "Competitive optimality or logarithmic invest- meat," Math. Opera/ions Ro .• vol. 5, no. 2, pp., 161-166, 1980. (4) T Cover, .. An algorithm (or maximi2.ing expected log investment return," IEEE Trans. Inform. Theory, vol. IT-30, no. 2. pp. 369- 373, 1984. ' {51 M. Finkelstein and R. Whitley, " Optimal strategies (or repeated games," Ad", . . Appl. Prob., vol. 13, pp. 415- 428, 1981. (6) J. Kelly; "N~ interpretation of information rate," Bell Syst. Tech. J ., vol. 35;pp. 917- 926, 1956. . (7) T. Cover, "A bound on the monetary value of informati9n," Stanford Statist. Dept. Tech. Re.p, ,52, 1984. (8) R. Bell and T. Cover,,"Game theoretic optimal pon(olios," Management Science, vol. 34, no. 6, pp. 7.24-733, 1988: . (9) P. Aigaet . ,and T.' Cover. "Asymptotic optimality and ·asymptotic equipartition properties of log-optimum investment," Ann. Prob .• vol. :16. no. 2: pp. 876-898. 1988. [10J T. Cover and D. GJuss, "Empirical Bayes stock mark.et portfolios," Adu. ApR I. Math .• vol. 7, pp. 170-181. 1986_ (11) A. Barron, " Are Bayes rules CODSistent in infonn.ation?" in Open Prob- lems in Communicali~ and Computation, T. Cover and B. Gopiaath, Eds. New Yort: Springer-Verlag. 1987, pp. 85- 91. [121 A. Barron, "The exponential convergence of posterior probabilities with implications for Bayes estimators of density (unctions,~ ~bmiued to Ann. Slatist., 1987. (13) T. Mori, "{-divergence geometry of disln"butioos and stochastic gains," in Proc. 3rd Pannonian Symp. 01t MGt" . SIOI., 1. MO&YOf"6di. I. Vi.ocu, W. Wertz, Ed ... V;,.grid, Hunpry-, 1982. pp. 231-238. --- ## Page 184 The Annals of Probability 1988, Vol. 16, No, 2, 876-898 14 ASYMPTOTIC OPTIMALITY AND ASYMPTOTIC EQUIPARTITION PROPERTIES OF LOG-OPTIMUM INVESTMENT By PAUL H. ALGOET1 AND THOMAS M. COVER2 Boston University and Stanford University We ask how an investor (with knowledge of the past) should distribute his funds over various investment opportunities to maximize the growth rate of his compounded capital. Breiman (1961) answered this question when the stock returns for successive periods are independent, identically distributed random vectors. We prove that maximizing conditionally expected log return given currently available information at each stage is asymptotically opti- mum, with no restrictions on the distribution of the market process. If the market is stationary ergodic, then the maximum capital growth rate is shown to be a constant almost surely equal to the maximum expected log return given the infinite past. Indeed, log-optimum investment policies that at time n look at the n-past are sandwiched in asymptotic growth rate between policies that look at only the k-past and those that look at the infinite past, and'the sandwich closes as k -> ex;. 157 1. Introduction. Suppose an investor starts with an initial fortune So = 1. At the beginning of each period t (where t takes on discrete values 0,1, ... ), the current capital St is distributed over investment opportunities j = 1, ... , m according 'to some portfolio bt = (b!h s j s m' a vector of nonnegative weights summing to 1. Let Xf ~ ° denote the return per monetary unit allocated to stock j during period t, and X t = (X!)\ S j S m the vector of returns. The yield per unit invested according to portfolio bt is the weighted average of the return ratios of the individual stocks, i.e., the inner product (1) (bt , Xl) = L bfX{. \ ,;,i,;,m Given that St units are invested at the beginning of period t, the total amount collected at the end of the period when the random outcome XI is revealed is St+ 1 = StC bt> Xl)' This capital is redistributed at the beginning of the next round, and the compounded capital after n investment periods is (2) Sn = I1 (bt> Xt)· O,;, t < n Received September 1985; revised August 1987. 1 Partially supported by Joint Services Electronics Program Grant DAAG 29-84-K-0047 and National Science Foundation Grant ECS-82-11568. 2Partially supported by National Science Foundation Grant ECS-82-11568 and DARPA contract N00039-84-C-0211. AMS 1980 subject classifications. Primary 90A09, 94Al5, 28020; secondary 60Fl5, 60G40, 49A50. Key words and phrases. Portfolio theory, gambling, expected log return, log-optimum portfolio, capital growth rate, ergodic stock market, asymptotic optimality principle, asymptotic equipartition property (AEP), Shannon-McMillan-Breiman theorem. 876 --- ## Page 185 158 PH Algoet and T. M Cover LOG-OPTIMUM INVESTMENT 877 Portfolio bt must be chosen on the basis of ffe, a a-field that embodies what is known at the beginning of period t. It obviously makes a difference whether decisions may depend on the history of an aggregate quantity like the Dow-Jones average, on detailed records of the past, or perhaps on inside information or help of a clairvoyant oracle. Our default assumption is that ffe = a(Xo," " Xt- I) is the information contained in the past outcomes. We wish to distinguish an optimum strategy {bt}o ,; t < 00 among all nonanticipating strategies {bt}o ,; t < 00 such that bt is ffe-measurable for all t ~ O. We are dealing with a sequential version of the portfolio selection problem that has received much attention in the literature (not to speak of financial practice). Economic theory promotes the maximization of subjective expected utility as a guiding principle toward its solution, and this is certainly appropriate if the investor's preferences are sufficiently well elucidated so that they can be captured in a well-defined utility function. But subjective utilities are difficult to assess and many investors may prefer a less elusive and more objective criterion if there is some rationale for its use. The mean-variance analysis of Markowitz (1952, 1959) trades off expected return with risk as quantified by the standard deviation of the return. This approach is mathematically and computationally tractable, but it lacks generality [ef. Samuelson (1967,1970)] and it fails to single out an optimum among the portfolios located on the efficient frontier. However, its economic foundation becomes more solid when cast in the form of the capital asset pricing model [ef. Sharpe (1985)]. Breiman (1960,1961) considered a market with m stocks and independent, identically distributed discrete-valued return vectors X t = (X!)I ,; j ,; m' and proved asymptotic optimality of the portfolio b* that attains the maximum expected log return w* = sUPbE{log(b, X)}. Thorp (1971) exhibited certain optimality properties of the log return as a normative utility function, and Bell and Cover (1980, 1986) proved that log-optimum investment is also competitively optimum, from a game-theoretic point of view. Although some authors [e.g., Samuelson (1967,1971)] have suggested that the log return should be considered just one among many possible utility functions, we hope to convince the reader of its more fundamental character. We consider arbitrarily distributed outcomes {Xt } and prove that maximizing the conditional expected log return given currently available information at each stage is optimum in the long run. A nonanticipating portfolio bt = b*(Xo,'''' Xt- I) is called log-optimum (for period t) if it attains the maximum conditional expected log return (3) we* = E{log(bt , Xt)lffe} = sup E{log(b, Xt)IXt- I, · .. , Xo}· b=b(Xo,"" X, - I) Such bt also attains the maximum (unconditional) expected log return (4) We* = E{we*} = E{log(bt, XI)} = sup E{log(b, Xt)}. b=b(Xo,· ··, X, - I) A log-optimum portfolio bt always exists, and is unique if the conditional distribution of X t given ffe has full support not confined to a hyperplane in gem. In any case, the return (bt, Xt) is always uniquely defined, even if bt is not. --- ## Page 186 Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment 159 878 P. H. ALGOET AND T. M. COVER The results of Breiman (1960, 1961) and Finkelstein and Whitley (1981) for independent, identically distributed {Xt } are enhanced by the following theorem, which proves that bt is optimum to first order in growth exponent. THEOREM (Asymptotic optimality principle). Let Sn* = flo < t < n< bt, Xt) and Sn = flo < t < n( bt, Xt), respectively, denote the capital growth over n periods of investment according to the log-optimum strategy {bt}o :5 t < 00 and a competing strategy {bt}o < t < 00' Then {S,.ISn*' $,".}o < n < 00 is a nonnegative supermartingale - - converging almost surely to a random variable Y with E{Y} ::; 1, and (5) limsupn- 110g(S,.ISn*) ::; 0 a.s. n Thus Sn < exp(ne)Sn* eventually for large n and arbitrary e> 0, which means that no strategy can infinitely often exceed the log-optimum strategy by an amount that grows exponentially fast. The asymptotic optimality principle will be deduced from the Kuhn-Tucker conditions for log-optimality using Markov's inequality and the Borel-Cantelli lemma. Now suppose {Xt } _ 00 < t < 00 is a two-sided sequence of return vectors, and . be* = b*( X _I' ... , X _ t) is a log-optimum portfolio for period 0 based on the t-past ~ = a( X - 1> •• • , X _ t). Portfolio be* attains the maximum conditional expected log return for period 0 given ~, (6) wt = E{log(be*, Xo)~} = sup E{log(b, Xo)IX_l>"" X - t}. b= b(X - 1>"" X _ ,) The maximum expected log return for period 0 given ~ is given by (7) We* = E{wt} = E{log(be*, Xo)} = sup E{log(b, Xo)}. b= b(X_ I>'''' x _,) The supremum is taken over a larger set of portfolios as t increases, so that ~ * is monotonically increasing and {wt, ~}o < t < 00 is a submartingale [strictly speaking only if all ~* are finite]. - The information fields ~ = a(X_ 1, ... , X - t) increase to a limiting a-field .#'00 = a(X_ 1, X_ 2 , · •• ). Any accumulation point of {be*} is a log-optimum port- folio for period 0 based on .#'00' and bt = b*( X _I" .. , X _ t) almost surely converges to b:' = b*(X_l' X_ 2 , ... ) if the log-optimum portfolio for period 0 given.#'oo is unique. Furthermore, ~ * increases to the maximum expected log return given the infinite past, (8) ~* /' We: = E{log(b:', Xo)} = sup E{log( b, Xo)}. b=b(X_ 1 , X _ 2 ,· .. ) We may use the expanded notation ~ * = W*( X tIXt- 1, ... , Xo) and ~ * = W*(XOIX_ 1, ••• , X-t). Setting E{logSn*} = W*(Xo, ... , X n- 1) yields the chain rule (9) W*(Xo,· .. , Xn- 1) = L W*(XtIXt_1,· .. , Xo)· 0.$ t 0 a.s. The rate We: lS highest possible. The AEP is an immediate consequence of the ergodic theorem if {Xt } is finite order Markov. A sandwich argument and the asymptotic optimality principle will reduce the proof of the general case to applications of the ergodic theorem. In the first half of the paper we discuss log-optimum investment for a single period. The Kuhn-Tucker conditions for log-optimality of a portfolio b* are recalled in Section 2, and in Section 3 we examine log-optimum portfolio selections and the maximum expected log return as functions of the distribution P of the random outcome X = (Xi)l<;i <; m on f!ll';'. To simplify the analysis we use a divide-and-conquer approach. Namely, we consider the decomposition X = ({:J, X)U, where {:J = ({:Jih <; i <; m is a fixed reference portfolio and U = X/({:J, X) is the scaled outcome in the simplex iJ// = {u = (uJh <; i <; m E f!ll';': ({:J, u) = I}. The return (b, X) factors as ({:J, X)(b, U), and the maximum --- ## Page 188 Asymptotic Optimality and Asymptotic £quipartition Properties of Log-Optimum Investment 161 880 P. H. ALGOET AND T. M. COVER expected log return w*(P) = sUPbEp{log(b, X)} decomposes as the sum of a reference level r(P) = Ep{log(,B, X)} that is affine in P and an extra term w*(Q) = sUPbEQ{log(b, U)} that depends on P only through the marginal distribution Q of U. The term w*(Q) is nonnegative, bounded and continuous in Q when the space of probability measures on I¥f is equipped with the weak topology, whereas the irregular term r(P) = Ep{log(,B, X)} is irrelevant for portfolio selection. We need the decomposition w*(P) = r(P) + w*(Q) to show that the maxi- mum conditional expected log return w/ is always attained by an ~-measurable portfolio bt. Furthermore, the nonnegativity and lower semicontinuity of w*(Q) are essential in Section 4 when we argue that the maximum expected log return given the t-past converges to the maximum expected log return given the infinite past (i.e., ~ * /' We: at t -+ 00). The asymptotic optimality principle is proved in Section 5, for an arbitrarily distributed sequence of return vectors. In Section 6 we argue that Sn* has a well-defined growth rate if {Xt } is stationary ergodic, and in Section 7 we examine whether the same is true if the market is stationary, or stationary in an asymptotic sense. Although the ergodic theorem is generally valid for asymptoti- cally mean stationary processes (whose definition is recalled in Section 7), the AEP will hold for an asymptotically mean stationary market only if the investor can recover from transient losses before reaching the asymptotic regime. Finally, in Section 8 we specialize the investment game to gambling on the next outcome of a random process. 2. The Kuhn-Tucker conditions for log-optimality. When managing funds during a given investment period, an investor may diversify his risk by building a portfolio that includes several assets. The allocation of one unit of capital over elementary investment opportunities j = 1, ... , m is conveniently described by a vector of weights b = (bi)l';; i,;; m' The weights must be nonnega- tive (since no borrowing is allowed) and sum to 1. Thus a portfolio is a vector b in the unit simplex (13) Let X j ~ 0 denote the return per monetary unit invested in stock j, and let X = (Xjh,;;j,;;m denote the vector of returns. Capital ~nvested according to portfolio b will grow by the factor (b, X) = Ll,;;i,;;mbJXJ, that is, the weighted average of the per-unit returns of the individual stocks. Portfolio b must be selected at the beginning of the investment period, before the actual value of the random outcome X is revealed. However, the distribution of X on ~':' is assumed to be known. Let the expected log return of a portfolio b be denoted by (14) w( b) = E {log( b, X) } . We set w( b) = - 00 if the expectation is not well defined in the usual sense. --- ## Page 189 162 PH Algoet and T M Cover LOG-OPTIMUM INVESTMENT 881 DEFINITION. A portfolio b* is called log-optimum if no competing portfolio b can improve the expected log return relative to b*, i.e., if (15) { ( (b,X))} E log (b*, X) :s; 0, for all bE!!t. Every log-optimum portfolio b* attains the maximum expected log return (16) w* = supE{log(b, X)}. bE~ Conversely, if w* is finite, then every portfolio b* attaining w* = sUPbW( b) is log-optimum. However, condition (15) may single out a unique log-optimum portfolio b* even if w( b) is infinite for all b E !!to We recall the Kuhn-Tucker conditions for log-optimality derived in Bell and Cover (1980). Let the expected score vector be defined for each portfolio b as (17) o:(b) = E{Xj (b , X)}. THEOREM 1. Let 0:* = 0:( b*) denote the expected score vector for portfolio b*. Then b* is log-optimum iff the Kuhn-Tucker conditions o:*j :s; 1 hold for all 1 :s; j :s; m, or equivalently, iff (18) { (b,X)} ( b, 0:*) = E (b*, X) :s; 1, for all bE !!to PROOF. For bE !!t and 0 < X = 1 - A < 1 let bx = Xb* + Ab. Then (bx, X) _ (b, X) (b, X) (b* , X) =A+A(b*,X) =1+AZ, whereZ= (b*,X)-1. Using a Taylor series expansion we obtain, for any a > 0, AZ~log(1 + AZ) ~ 10g(1 + A(Z 1\ a)) = A(Z 1\ a) - ~8A2(Z 1\ a)2 (for some 0 < 8 < 1) ~ A(Z 1\ a) - ~A2a2. Choosing a = a( A) so that a( A) --+ 00 and Aa( A) --+ 0 as A "» 0, we see that A - lE{log(l + AZ)} --+ E{ Z} as A "» O. But E{ Z} = (b, 0:*) - 1, so the right derivative at A = 0 of w(bx) = E{log(bx' X)} is given by (19) ~w( bx)1 = lim E{log(l + AZ)} = E{Z} = (b, 0:*) - 1. dA X=o+ >-'>00 A The Kuhn-Tucker conditions assert that b* is log-optimum iff the directional derivative of the expected log return is nonpositive when moving from b* to any competing portfolio b (in particular, when moving from b* to any extreme point of !!t). The infinitesimal conditions dw(bx)j dAlx=o+:S; 0 are necessary for log- optimality of b* , and they are also sufficient because w( b) is concave in b. 0 --- ## Page 190 Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment 163 882 P. H. ALGOET AND T. M. COVER The set B* of log-optimum portfolios is never empty [cf. Cover (1984)]. In fact, let!£' denote the linear hull of the support of the distribution of X, that is, the smallest linear subspace of !!Jm such that X E!£' with probability 1. Then w( b) is strictly concave when restricted to !£' and constant along fibers per- pendicular to !£'. It follows that B* is a polyhedral set (the intersection of !JI with a fiber orthogonal to !£'), and the log-optimum portfolio b* is unique if X has full support (!£' = !!J m). The return (b*, X) and log return log( b*, X) are unambiguously defined, independent of the choice of log-optimum portfolio b* in B*. 3. Continuity and attainability of the maximum expected log return. We make explicit how various quantities depend on the distribution P of X on !!Jr;'. Let w(b, P) = Ep{log(b, X)} denote the expected log return of portfolio b, w*(P) = sUPbw(b, P) the maximum expected log return and B *(P) the set of log-optimum portfolios. It is clear that w*( P) is convex in P, since w*( P) is the supremum of functions E p{log( b, X)} that are affine in P. The direction of the return vector X embodies everything an investor needs to know in order to maximize the expected log return. To justify this claim, we choose a fixed reference portfolio {3 = ({3i)l 5, i 5, m with {3i > 0 for all j, and we define the scaled return vector (20) U = u(X), where u(x) = x/ ({3, x). Thus U is obtained by projecting the return vector X on the simplex (21) If X = 0, then we set U = u(O) = U o for some arbitrary U o E %'. The distribution Q of U = u(X) on %' is obtained by integrating out the distribution P of X along rays through the origin. All mass accumulated along a ray is collected at the point where the ray crosses the simplex %', except that mass found at X = 0 is transferred to u(O) = Uo' Thus Q is the image measure of P through u: !!Jr;' ~ %', and for any Borel subset A ~ %' we have (22) Q{U E A} = P{u(X) E A} = P(u - 1(A)). Since X = ({3, X)u(X), the expected log return may be decomposed as the sum Ep{log(b, X)} = Ep{log({3, X)} + Ep{log(b, u(X))}, or equivalently, (23) w( b, P) = r(P) + w( b, Q). Here r(P) = w({3, P) denotes the expected log return of the reference portfolio {3. We interpret r(P) as a reference level for the expected log return, since it is an inherent property of the market over which the investor has no control. Whereas r(P) = Ep{log({3, X)} is affine in P, it is also a very irregular function of P, possibly infinite or ill defined. Since our choice of b cannot affect its value, we shall subtract r( P) from the expected log return w( b, P). The remaining quantity w( b, Q) = EQ{log( b, U)} depends on P only through the marginal distribution Q of the scaled outcome U, and represents the relative improvement in expected log return that results when portfolio b is chosen instead of {3. The --- ## Page 191 164 P H Algael and T. M Cover LOG-OPTIMUM INVESTMENT 883 maximum expected log return can be expressed as the sum (24) w*(P) = r(P) + w*(Q), where w*(Q) = supEQ{log(b, U)}. bE f$ Maximizing w( b, P) or w( b, Q) are equivalent operations, so that B*( P) = B*(Q). Notice that w*(Q) = w*(P) - reP) ~ 0, with equality iff the reference portfolio {3 is log-optimum. It is an interesting fact that the maximum expected log return w*(Q) is always attained by some portfolio choice. However, we need a stronger result, namely, the existence of log-optimum portfolios b*(Q) that depend measurably on Q. To prove the existence of a measurable selection of log-optimum portfolios, we make use of topological properties, including compactness of !2 and upper semicontinuity of the expected log return web, Q) = EQ{log(b, U)} in (b, Q). The space 2 of probability measures on the compact metric space i¥f is compact and metrizable when equipped with the weak topology [that is the weakest topology on 2 such that Q ~ EQ{f(U)} is continuous in Q E 2 for all bounded continuous functions f: i¥f ~ .?l). Its Borel a-field is the smallest a-field on 2 such that A ~ Q(A) is measurable in Q for all Borel subsets A ~ i¥f. THEOREM 2. The maximum expect log return w*(Q) = sUPbE YtEQ{log(b, U)} is convex, bounded [between 0 and max/-log f31)] and uniformly continuous when the space 2 of probability measures on i¥f is equipped with the weak topology. The set of log-optimum portfolios B *( Q) is a nonempty compact convex subset of !2 for every distribution Q on i¥f, and a log-optimum portfolio b*(Q) E B*(Q) can be selected for each Q E 2 so that b*(Q) is measurable in Q. PROOF. Clearly w*(Q) is convex in Q for the same reason that w*(P) is convex in P . We argue that w*(Q) is bounded below and lower semicontinuous on 2, because ({3, u) is bounded below on i¥f and (b, u) is concave in bE !2 and lower semicontinuous in u E i¥f. We also prove that w( b, Q) is bounded above and upper semicontinuous, using compactness of !2 and boundedness above and upper semicontinuity of (b, u) on !2 X i¥f. Boundedness and uniform continuity of w*(Q) and existence of a measurable selection of log-optimum portfolios b*(Q) will follow automatically. First, we argue that w*(Q) is nonnegative and lower semicontinuous on 2. For 0 ~ A ~ 1 and bE !2, let X = 1 - A, b>-. = X{3 + Ab, !2>-. = {b>-.: bE !2} and (25) wx(Q) = sup EQ{log(b,U)} = supEQ{log(b>-.,U)} . bE f$~ bE f$ Observe that wx(Q) is monotonically increasing in A, since the supremum is taken over a larger set !2 >-. as A increases. Furthermore, !2 1 = !2, !2 0 = {{3} and ({3, u) = 1 for all u E i¥f, so that w*( Q) = wt( Q) ~ wx( Q) ~ wo*( Q) = EQ{log({3, U)} = o. If A < 1, then loge b>-., u) is bounded below (by log X) and lower semicontinuous in u, so that w(b>-., Q) = EQ{log(b>-., U)} and wx(Q) = SUPbE YtW(b>-., Q) are lower semicontinuous in Q. On the other hand, the inequality (b>-., u) ~ A(b, u) --- ## Page 192 Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment 165 884 P. H. ALGOET AND T. M. COVER implies that w>.* ( Q) ::; w* ( Q) ::; w: ( Q) - log A, and hence w>.*(Q).l' w*(Q) as A.l' 1. Since w*(Q) is the supremum of lower semicontinuous functions w:(Q), it follows that w*(Q) is lower semicontinuous as well. The expected log return w( b, Q) = EQ{log( b, U)} is bounded above and upper semicontinuous on 86 X 2, since (b, u) is bounded and upper semicontinuous on 86 X CiJ/. Since 86 is compact, it follows [ef. Bertsekas and Shreve (1978), Proposi- tion 7.33] that w*( Q) = SUPb E &Bw( b, Q) is bounded above and upper semicon- tinuous on 2 . Furthermore, log-optimum portfolios b*(Q) E B*(Q) can be selected in a measurable fashion for all Q E 2 by the measurable selection theorem of Kuratowski and Ryll-Nardzewski (1961). The upper bound w*(Q) ::; max/-log,8i) holds since (b,u)::; LiUi ::; maxi(l/,8i) for all bE86 if u satisfies (,8, u) = 1. 0 It is impossible to select a portfolio b*(Q) E B*(Q) for all distributions Q on CiJ/ so that b*(Q) is continuous in Q. However, if Qn -+ Qoo and b: E B*(Qn) for all n, then any accumulation point b:' of the sequence {b:} is a point in B*(Qoo)' Furthermore, (b:, U) -+ (b:', U) almost surely under Qoo' These con- tinuity properties of the multivalued correspondence Q ~ B*(Q) follow from the following. THEOREM 3. The set Gr(B*) = {(Q, b*): b* E B*(Q)} is closed in 2 X 86. Consequently, any selection of log-optimum portfolios Q ~ b*(Q) E B*(Q) is continuous at any Q E 2 such that B*(Q) = {b*(Q)} is a singleton set. PROOF. Since 86 is compact, the theorem will follow from the following claim: If Qn -+ Qoo in 2, b: -+ b:' in 86 and b: E B*(Qn) for all n, then b:, E B*(Qoo)' To prove the claim we consider the sequence of maximum expected log returns w*(Qn) = w(b:, Qn). It is clear that w*(Qn) -+ w*(Qoo) since Qn -+ Qoo in 2 and w*(Q) is continuous in Q. On the other hand (see the proof of Theorem 2), w( b, Q) is upper semicontinuous on 86 X 2 and hence limsupw(b:,Qn)::; w(limb:, limQn) = w(b:' , Qoo)' n n n The claim b:' E B*(Qoo) and Theorem 3 follow, since w(b:',Qoo ) ~ limsupw(b:,Qn) = limw*(Qn) = w*(Qoo) = supw(b,Qoo)' 0 n n b The maximum expected log return w*( P) is neither bounded nor continuous (for the weak topology) as P ranges over the space of probability measures on !Je';:. But if the support of P is constrained to a closed subset f of !Je';:, then w*(P) is lower semicontinuous and bounded below iff f is bounded away from 0, upper semicontinuous and bounded above iff f is bounded, and bounded and --- ## Page 193 166 P H Algael and T. M Cover LOG-OPTIMUM INVESTMENT 885 uniformly continuous iff f is bounded away from o. In particular, if f = 01/ (i.e., if X is distributed on the simplex 01/), then P = Q and w*(P) = w*(Q) is bounded and continuous. Some of the conclusions of Theorems 2 and 3 continue to hold if the investor may distribute his funds over a countable set or even a separable metrizable space d of investment opportunities. Indeed, suppose every realization of the return X is a nonnegative lower semi continuous function x( a) on d. (This is no restriction if d is a countable set with the discrete topology.) The average return (b, x) = L.-x( a )b( cia) is then well defined for every portfolio b [i.e., for every normalized measure b( cia) on the Borel a-field of d]. Further assume the existence of a reference portfolio {1 such that ({1, x) > 0 is strictly positive for any return function x( a) that is not identically o. [Such {1 exists if d is locally compact, and, in particular, if d is countable.] If P and Q denote the distribu- tion of X and U = X / ({1, X), then the maximum expected log return w*(P) admits the decomposition r(P) + w*(Q), and w*(Q) is nonnegative and lower semicontinuous by the argument presented in the proof of Theorem 2. If, moreover, d is compact and the return functions x( a) are continuous and bounded by a fixed constant, then w*(Q) is bounded and continuous and a measurable selection of log-optimum portfolios b*(Q) exists by Theorem 2, and Gr( B*) is closed by Theorem 3. 4. Martingale properties. It will be shown that the maximum expected log return given increasing information fields tends to the maximum expected log return given the limiti~g a-field. We assume that the random return vector X( w) E !1f';' is defined on a perfect probability space (Q, $", P), so that X admits a regular conditional probability distribution given any sub-a-field of $". See Jifina (1954) for a proof of this fact, and Ramachandran (1979) for a complete discussion of perfect measures. THEOREM 4. Suppose the random vector X is defined on a perfect probability space (Q, $", P), and {~}o < t < 00 is an increasing sequence of sub-a-fields of $" with limiting a-field ~oo ~ ff. (a) If Pt is a regular conditional probability distribution of X given ~, then (26) Pt -+ Poo weaklya.s. (b) If b*( ·) is a measurable selector of log-optimum portfolios, then be* = b*( Pt) is an ~-measurable portfolio attaining the maximum conditional ex- pected log return given ~ . Moreover, (be*, X) -+ (b~, X) a.s., and hence (27) log(be* , X) -+ log(b~, X) a.s. If the log-optimum portfolio given ~oo is unique [B*(Poo) = {b;;}], then be* -+ b;; a.s. as weU. (c) If w*(·) denotes the maximum expected log return function , then the maximum conditional expected log return given ~ is given by (28) w/ = w*( Pt) = sup E{log( b, X)~} = E{log(be*, X)I~} a.s. bEff, --- ## Page 194 Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment 167 886 P. H. ALGOET AND T. M. COVER Furthermore, {w/, ~}o" t" 00 is a submartingale and (29) w/ - W'; a.s. (and in V it We: < 00). (d) The maximum expected log return given ~ is given by (30) ~ * = E {w/} = sup E {log( b, X)} = E (log( be* , X) } . bEff, Furthermore, (31) ~*)" We:, ast- 00. PROOF. Levy's martingale convergence theorem for conditional expectations of a bounded continuous (or nonnegative measurable) function t(x) states that This proves (a), and assertion (b) follows in view of Theorem 3. Notice that be* = b*(Pe) and w/ = w*(Pt) are ~-measurable, since Pt is measurable on (rl, ~) and both b*(·) and w*(·) are measurable functions. If 0:::; s :::; t:::; 00, then ~ ~~, so that every ~-measurable portfolio (includ- ing bs*) is also ~-measurable. It follows that E{log(bs*, X)I~} :::; w/ = sup E{log( b, X)I~}. bEff, Taking ~-conditional expectations proves that ws* = E{log(bs*' X)I~} :::; E{wt*I~}' and hence {w/, ~}o < t 0 for all 1 :::; j :::; m), and we recall the decomposition w*( P) = r( P) + w*( Q) of the maximum expected log return into a reference level r(P) = Ep{log({3, X)} and a relative improvement w*(Q) that only depends on the distribution Q of the scaled return vector U = u(X) = X/({3, X). Let Qt designate a regular conditional probability distribution of U = u(X) given ~, for 0:::; t :::; 00. Then Qt - Qoo weakly almost surely and {w*(Qt), ~}o"t - 00 or SUPkE{W*(Qk)} < 00 . 5. The asymptotic optimality principle. We now prove the asymptotic optimality principle for sequential log-optimum investment. The market is described by a sequence of return vectors {Xt}o ,;; t <00 defined on a perfect probability space (12, .%, P), and capital invested according to a portfolio bt at the beginning of period t will grow by a factor (bt> Xt) when the random outcome X t is revealed at the end of that period. If the initial fortune is normalized to So = 1, then the compounded capital Sn after n periods is given by (35) Sn = n (bt , Xt)· O,;; t < n The objective is to select portfolios bt so as to maximize the capital growth rate lim inf n n - 1 log Sn. Portfolio bt must be selected on the basis of an information field ~ that embodies what is known at the beginning of period t. In other words, bt must be ~-measurable (bt E ~, for short). Let Fe denote a regular conditional probability distribution of X t given ~, and let b*(·) be a measurable selector of log-optimum portfolios. Then bt = --- ## Page 196 Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment 169 888 P. H. ALGOET AND T. M. COVER b*(Pe) is an ~-measurable portfolio attaining the maximum conditional ex- pected log return (36) w/ = w*(Pt) = E{log( bt, Xt)I~} = sup E{log( b, Xt)I~}. bEff, The expectation of the log return log( bt, Xt) and of its conditional expectation wt* are both equal to the maximum expected log return for period t given ~, (37) ~* = E{w/} = E{log(bt, Xt)} = sup E{log(b, Xt)}. bEff, We argue that {bno Xt)· Os;t 0, P{S,,/Sn* ;;:: rn} .:s; rn- lE{S,,/Sn*} .:s; r;l . If rn increases sufficiently fast so that Lnr;l < 00, then L:P{S,,/Sn* ;;:: rn} .:s; L:r;l < 00 , n n and hence S,,/Sn* < rn eventually for large n, by the Borel-Cantelli lemma. In particular, choosing rn = exp(ne) with e > 0 proves that p{ n - llog(S,,/Sn*) ;;:: €infinitely often} = O. Since e> 0 was arbitrary we may conclude that limsuPnn- 1log(S,,/Sn*) .:s; 0 a.s. [This fact can be proved also by observing that Sn/Sn* converges to a random variable Y with E{Y} .:s; 1 and hence 0 .:s; Y < 00 a.s. Indeed, Sn/Sn* .:s; (1 + Y) for large n and hence limsuPnn-1log(S,,/Sn*) .:s; limnn - 1log(1 + Y) = 0 a.s.] 0 Theorem 5 asserts that any alternative is dominated in the long run by the log-optimum strategy. Indeed, E{S,,/Sn*} .:s; 1 for all n, and the Borel-Cantelli lemma implies that S,,/Sn* < rn eventually for any sequence {rn} such that Lnrn-1 < 00 (e.g., rn = n1+ E or rn = eM). The maximal inequality for nonnegative supermartingales [cf. Neveu (1972), Proposition II-2-7, page 23] asserts that (40) p{ supS,,/Sn* ;;:: A} .:s; I/A. n Thus with probability at least 1 - I/ A, a competing investor will never outper- form Sn* by a factor greater than A. The random variable sUPnSn/Sn* is finite almost surely, although its expectation is generally infinite. A game-theoretic sense in which Sn* dominates Sn for games with payoff E{ Ut)/( bt, Ut) is completely determined by the history of the scaled outcomes Ut = u( Xt). 6. The asymptotic equiparitition property. Breiman (1960, 1961) consid- ered a market with outcomes {Xt } that are independent and identically distrib- uted according to an atomic measure, and he argued that repeated choice of the log-optimum portfolio b* is optimum according to various criteria. In particular, the capital Sn* = no < t < n( b*, Xt) will grow exponentially fast almost surely with limiting rate equal to the maximum expected log return w * = sUPbE{log(b, X)}, by the strong law of large numbers, (41) n-llogSn* = n - 1 L: log(b*, Xt) ~ w* = E{log(b*, X)} a.s. O:s; t'''' X _t) equals ~* = W*(XtIXt_I , . .. , Xo) by stationarity. If b:' is a log-optimum portfolio for period 0 based on the limiting a-field ffoo = a(X _I, X_ 2 , ••• ), then ~* = ~* increases monotonically to We: = E{log(b:', Xo)}' This limiting expectation is equal to the maximum expected log return given the infinite past, and is denoted by We: = W*(XoIX _p X- 2, .. . ) . It may be noted that W*(XOIX _I, ... , X - h) is the maximum expected log return given the infinite past under the stationary kth-order Markov process having the same (k + 1 )st-order marginal distribution as {X t}. Let Sn* = IIo Y - t,···, X _I' Y- I) are monotonically increasing, and n-llogSn* ~ We: almost surely where We: = W*(XOIX _I' Y- I, X _2 , Y- 2 , • •• ) is the maximum expected log return given the infinite past. The proof is identical to that of Theorem 6, except that bt = b*(Qt) and bt = b*(Qt) are now defined by applying a measurable selector of log-optimum portfolios b*(·) to regular conditional probability distributions Qt and Qt of Vt = u(Xt) given ~ and of Vo = u(Xo) given ~. The true log return log( bt, Xt) will generally differ from the conditional expected log return wt = E{log(bt, Xt)I~}. If conditional expected log returns were always exactly realized then the capital growth over n periods would be not Sn* but rather (53) Sn* = TI exp[E{log( bt, Xt)I~}]. O~t < n If Sn = no < t < nexp[E{log( bt, Xt)I~}] denotes the corresponding quantity un- der the co~peting strategy {btl, then Sn ~ Sn* for all n, and hence (54) lim supn -Ilog( S";Sn*) ~ 0 a.s. n This may be called an asymptotic optimality principle for the hypothetical growth rate Sn*. If the market is stationary ergodic, then an asymptotic equipar- tition property for Sn* can be proved as well, under certain integrability condi- tions. Let L log L designate the class of random variables g( w) such that E{lglloglgl} < 00. THEOREM 7. If the market is stationary ergodic and E{log(,B, Xo)l~oo} belongs to L log L, then (55) n - Ilog Sn* ~ We: a.s. and in V . PROOF. Breiman's (1957/1960) extension of the ergodic theorem asserts that n-lLO ~ t < nge supE{logp(Xo,"" X n - 1 )} a.s. n See Barron (1985) and Orey (1985) for a proof of this generalized Shannon- McMillan-Breiman theorem using Breiman's extension of the ergodic theorem and Algoet and Cover (1988) for a sandwich proof. Acknowledgments. We wish to acknowledge supporting discussions with John Gill and David Larson. REFERENCES ALFSEN, E. M. (1971). Compact Convex Sets and Boundary Integrals. Springer, New York. ALGOET, P. H. and COVER, T. M. (1988). A sandwich proof of the Shannon-McMillan-Breiman theorem. Ann. Probab. 16 899-909. BARRON, A. R. (1985). The strong ergodic theorem for densities: Generalized Shannon-McMillan- Breiman theorem. Ann. Probab. 13 1292-1303. BELL, R. and COVER, T. M. (1980). Competitive optimality of logarithmic investment. Math. Oper. Res. 5 161-166. BELL, R. and COVER, T. M. (1986). Game-theoretic optimal portfolios. Preprint. BERTSEKAS, D. P. and SHREVE, S. E. (1978). Stochastic Optimal Control, the Discrete Time Case. Academic, New York. BREI MAN, L. (1957/1960), The individual ergodic theorem of information theory. Ann. Math. Statist. 28809-811. Correction 31 809-810. BREIMAN, L. (1960). Investment policies for expanding businesses optimal in a long run sense. Naval Res. Logist. Quart. 7 647-651. --- ## Page 206 Asymptotic Optimality and Asymptotic £quipartition Properties of Log-Optimum Investment 179 898 P. H. ALGOET AND T. M. COVER BREIMAN, L. (1961). Optimal gambling systems for favorable games. Proc. Fourth Berkeley Symp. Math. Skltist. Probab.l 65-78. Univ. California Press. CHUNG, K. L. (1974). A Course in Probability Theory, 2nd ed. Academic, New York. COVER, T. M. (1984). An algorithm for maximizing expected log investment return. IEEE Trans. Inform. Theory IT-30 369-373. FINKELSTEIN, M. and WHITLEY, R. (1981). Optimal strategies for repeated games. Adv. in Appl. Probab. 13 415-428. GRAY, R. M. and KIEFFER, J. C. (1980). Asymptotically mean stationary measures. Ann. Probab.8 962-973. Jd~INA, M. (1954). Conditional probabilities on a-algebras with countable basis. Czechoslavak Math. J. 4 372-380. (English translation in A mer. Math. Soc. Transl. Ser. 2 2 (1962) 79-86.) KELLY, J . L., JR. (1956). A new interpretation of information rate. Bell System Tech. J . 35917-926. KURATOWSKI, K. and RYLL-NARDZEWSKI, C. (1961). A general theorem on selectors. Bull. A cad. Polon. Sci. 13 397-403. LoEVE, M. (1978). Probability Theory 2, 4th ed. Springer, New York. MAITRA, A. (1977). Integral representations of integral measures. Trans. Amer. Math. Soc. 229 209-225. MARKOWITZ, H. M. (1952). Portfolio selection. J. Finance 777-91. MARKOWITZ, H. M. (1959). Portfolio Selection. Wiley, New York. NEVEU, J. (1972). Martingales a Temps Discret. Masson, Paris. OREY, S. (1985). On the Shannon-Perez-Moy theorem. Contemp. Math. 41 319-327. RAMACHANDRAN, D. (1979). Perfect Measures I-Basic Theory and II-Special Topics. lSI Lec- ture Notes 5,7. Macmillan, New York. SAMUELSON, P. (1967). General proof that diversification pays. J. Financial and Quantitative Anal. 2 1-13. SAMUEI.SON, P. (1970). The fundamental approximation theorem of portfolio analysis in terms of means, variances and higher moments. Rev. Econom. Stud. 37 537-542. SAMUELSON, P. (1971). The "fallacy" of maximizing the geometric mean in long sequences of investing or gambling. Proc. Nat. Acad. Sci. U.S.A. 68 2493-2496. SHARPE, W. F. (1985). Investments, 3rd ed. McGraw-Hili, New York. THORP, E. O. (1971). Portfolio choice and the Kelly criterion. In Stochastic Optimization Models in Finance (W. T. Ziemba and R. G. Vickson, eds.) 599-619. Academic, New York. COLLEGE OF ENGINEERING BOSTON UNIVERSITY 110 CUMMINGTON STREET BOSTON, MASSACHUSETTS 02215 DEPARTMENTS OF STATISTICS AND ELECTRICAL ENGINEERING STANFORD UNIVERSITY STANFORD, CALIFORNIA 94305 --- ## Page 207 This page is intentionally left blank --- ## Page 208 Mathematical Finance, Vol. I, No.1 (January 1991), 1-29 15 UNIVERSAL PORTFOLIOS THOMAS M. COVER I Departments of Statistics and Electrical Engineering, Stanford University, Stanford, CA We exhibit an algorithm for portfolio selection that asymptotically outperforms the best stock in the market. Let Xj == (xi), xi2' ... ,Xjm)' denote the performance of the stock market on day i, where xi) is the factor by which the jth stock increases on day i. Let bj == (bjl , bj2, ... , bjm )', bi) ~ 0, Ejbi) == I, denote the proportion bi) of wealth invested in the jth stock on day i. Then Sn == II?= I b)xj is the factor by which wealth is increased in n trading days. Consider as a goal the wealth S: == maxb II?= I b'xj that can be achieved by the best constant rebalanced portfolio chosen after the stock outcomes are revealed. It can be shown that S: exceeds the best stock, the Dow Jones average, and the value line index at time n. In fact, S: usually exceeds these quantities by an exponential factor. Let xI' x2' . . . , be an arbitrary sequence of market vectors. It will be shown that the .Q0nanticipating sequence of portfolios bk ~ j bII t=-II b'xj db/j II ;k=-II b'x; db yields wealth Sn == IIZ=I b~Xk such that (I/n)ln(S:/Sn) -+ 0, for every bounded sequence XI' x2' .. . , and, under mild conditions, achieves " S:(m - 1)!(211'/n)(m-I)J2 Sn - IJnl l / 2 where I n is an (m - I) x (m - I) sensitivity matrix. Thus this portfolio strategy has the same exponential rate of growth as the apparently unachievable S: . KEYWORDS: portfolio selection, robust trading strategies, performance weighting, rebalancing I. INTRODUCTION 181 We consider a sequential portfolio selection procedure for investing in the stock market with the goal of performing as well as if we knew the empirical distribution of future market performance. Throughout the paper we are unwilling to make any statistical assumption about the behavior of the market. In particular, we allow for the possibility of market crashes such as those occurring in 1929 and 1987. We seek a robust procedure with respect to the arbitrary market sequences that occur in the real world. We first investigate what a natural goal might be for the growth of wealth for arbitrary market sequences. For example, a natural goal might be to outperform the best buy-and-hold strategy, thus beating an investor who is given a look at a newspaper n days in the future. We propose a more ambitious goal. To motivate this goal let us consider all constant rebalanced portfolio strategies. Let x = (XI> X2, ••• , xm)\ ~ 0 denote a stock market vector for one investment period, where Xi is the price relative for I I wish to thank Hal Stern for his invaluable contributions in obtaining and analyzing the stock market data. I also wish to thank the referee for helpful comments. This work was partially supported by NSF Grant NCR 89-14538. Manuscript received January 1990; final revision received July 1990. --- ## Page 209 182 T M Cover 2 THOMAS M. COVER the ith stock - i.e., the ratio of closing to opening price for stock i. A portfolio b = (b l , b2, . . . ,bm)t, bi ;;:: 0, E bi = 1, is the proportion of the current wealth invested in each of the m stocks. Thus S = btx = E bixi, where b and x are considered to be column vectors, is the factor by which wealth increases in one investment period using portfolio b. Consider an arbitrary (nonrandom) sequence of stock vectors xl> X2, ... , Xn E R~. Here xij is the price relative of stock j on day i. A constant rebalanced portfolio strategy b achieves wealth n 0.1) Sn(b) = II btXi' i=1 where the initial wealth So (b) = 1 is normalized to 1. Let (1.2) denote the maximum wealth achievable on the given stock sequence maximized over all constant rebalanced portfolios. Our goal is to achieve S:. We will be able to show that there is a "universal" portfolio strategy hh where hk is based only on the past xl> X2, . . . , Xk-l> that will perform asymptotically as well as the best constant rebalanced portfolio based on foreknowledge of the sequence of price relatives. At first it may seem surprising that the portfolio hk should depend on the past, because the future has no relationship to the past. Indeed the stock sequence is arbitrary, and a malicious nature can structure future Xk'S to take advantage of past beliefs as expressed in the portfolio hk • Nonetheless the resulting wealth can be made to track S:. The proposed universal adaptive portfolio strategy is the performance weighted strategy specified by (1.3) hi = (~,~, . .. , ~), m m m where (1.4) k Sdb) = II btXi' i=1 and the integration is over the set of (m - 1 )-dimensional portfolios B = [bERm: bi ;;:: O"~ bi = I). 1=1 0.5) The wealth Sn resulting from the universal portfolio is given by n 0 .6) SA II Abt n = kXk· k=1 Thus the initial universal portfolio hi is uniform over the stocks, and the port- --- ## Page 210 Universal Portfolios 183 UNIVERSAL PORTFOLIOS 3 folio bk at time k is the performance weighted average of all portfolios bE B. An approximate computation will be given in Section 8, and a generalization of this algorithm will be given in Section 9. We will show that (1.7) (lln)ln Sn - (lIn)ln S; --+ 0, for arbitrary bounded stock sequences Xl> X2, . .•• Thus Sn and S; have the same exponent to first order. A more refined analysis for two stocks shows A {h * (1.8) Sn - ~;iJ,; Sn' in a sense that will be made precise. It is difficult to summarize the behavior of Sn relative to S; because of the arbitrariness of the sequence and the fact that we cannot assume a limiting distribution. For example, even the limit of (lIn)ln S; cannot be assumed to exist. The goal of uniformly achieving S;(XI, X2, .. . , xn ), as specified in (1.7), was partially achieved by Cover and Gluss (1986) for discrete-valued stock markets by using the theory of compound sequential Bayes decision rules developed in Robbins (1951), Hannan and Robbins (1955), and the game-theoretic approachability-excludability theory of Blackwell (1956a, b). Work on natural investment goals can be found in Samuelson (1967) and Arrow (1974). The vast theory of undominated portfolios in the mean-variance plane is exemplified in Markowitz (1952) and Sharpe (1963), wbile the theory of rebalanced portfolios for known underlying distributions is developed in Kelly (1956), Mossin (1968), Thorp (1971), Markowitz (1976), Hakansson (1979), Bell and Cover (1980, 1988), Cover and King (1978), Cover (1984), Barron and Cover (1988), and Algoet and Cover (1988). A spirited defense of utility theory and the incompatibility of utility theory with the asymptotic growth rate approach is made in Samuelson (1967, 1969, 1979) and Merton and Samuelson (1974). We see the present work as a departure from the above model-based investment theories, whether they be based on utility theory or growth rate optimality. Here the goal S; = maxbII;= I btx; depends solely on the data and does not depend upon underlying statistical assumptions. Moreover, Theorem 5.1, for example, provides a finite sample lower bound for the performance Sn of the universal portfolio with respect to S;. Therefore the case for success rests almost entirely on the acceptance of S; as a natural investment goal. The performance of the universal portfolio is exhibited in Section 8, where numerous examples are given of Sn (b), S;, and Sn for various pairs of stocks. In general, volatile uncorrelated stocks lead to great gains of S; and Sn over the best buy-and-hold strategy. However, ponderous stocks like IBM and Coca-Cola show only modest improvements. --- ## Page 211 184 T M Cover 4 THOMAS M. COVER 2. ELEMENTARY PROPERTIES We wish to show that the wealth t generated by the universal portfolio strategy bk exceeds the value line index and that Sn is invariant under permutations of the stock sequence xl> X2, ••• ,Xn• We will use the notation (2.1) (2.2) W{b, F) = fin btx dF{x), W*{F) = max W{b, F), b and we will denote by Fn the empirical distribution associated with XI' X2, ••• , Xm where Fn places mass lin at each Xi. In particular, we note that n (2.3) Sri' = max Sn{b) = max n btXi = enW*(Fn ). b b i=1 For purposes of comparison, we pay special attention to buy-and-hold strategies b = ej = (O, 0, ... ,0,1,0, ... ,0), where ej is the jth basis vector. Note that n n (2.4) Sn{ej) = n ejxk = n Xkj k=1 k=1 is the factor by which thejth stock increases in n investment periods. Thus Sn (ej) is the result of the buy-and-hold strategy associated with the jth stock. We now note some properties of the target wealth S:: PROPOSITION 2.1 (Target Exceeds Best Stock). (2.5) S: ~. max Sn{ej). J=I.2 •... • m Proof. S: is a maximization of Sn{b) over the simplex, while the right-hand side is a maximization over the vertices of the simplex. 0 PROPOSITION 2.2 (Target Exceeds Value Line). (2.6) S: ~ C~ Sn(ej) ) 11m. Proof. Each Sn (ej) is =:; S: . o The next proposition shows that the target exceeds the DJIA. PROPOSITION 2.3 (Target Exceeds Arithmetic Mean). If OIj ~ 0, E OIj = 1, then (2.7) m S: ~ ~ OIjSn{e). j=1 --- ## Page 212 Universal Portfolios 185 UNIVERSAL PORTFOLIOS 5 Proof. (2.8) j = 1,2, ... , m. o Thus S: exceeds the arithmetic mean, the geometric mean, and the maximum of the component stocks. Finally, it follows by inspection that S: does not depend on the order in which Xl' X2, ... , Xn occur. PROPOSITION 2.4 S:(Xi> X2, •. • ,Xn ) is invariant under permutations of the sequence Xl, X2, • .. , Xn • Now recall the proposed portfolio algorithm in (1.3) with the resulting wealth (2.9) It will be useful to recharacterize Sn in the following way. LEMMA 2.5. (2.10) where n (2.11 ) Sn(b) = II blXi· i=l Thus the wealth Sn resulting from the universal portfolio is the average of Sn(b) over the simplex. Proof. Note from (1.3) and (1.4) that Thus the product in (2.9) telescopes into (2.14) o We observe two properties of the wealth Sn achieved by the universal portfolio. --- ## Page 213 186 T M Cover 6 THOMAS M. COVER PROPOSITION 2.5 (Universal Portfolio Exceeds Value Line Index). (2.15) Sn ~ (IT Sn(eJ»11m. J=I Proof. Let Fn be the empirical cumulative distribution function induced by XI> X2, ... , xn. By two applications of Jensen's inequality and writing (2.16) we have (2.17) Sn = EbSn(b) = Eb exp(nW(b, Fn) I ~ exp(nEbW(b, Fn) 1= exp(nEbJ lnbtx dFn(X)j =exp(nEbJlnC~ bje)x) dFn(X») ~exp[nEbj~ bjJln(e)X) dFn(X») =exp[n(~ ~Jlne)XdFn(X»)) = C~ Sn(ej») 11m. o Thus the wealth induced by the proposed portfolio dominates the value line index for any stock sequence XI> X2, ... , xI),! for all n. Next, we observe that although bk depends on the order of the sequence xI> X2, ... , Xn, the resulting wealth Sn = n blexk does not. PROPOSITION 2.6. Sn is invariant under permutations of the sequence XI, X2, ... ,xn· Proof. Since the integrand in (2.18) Sn = IT bleXk =J Sn(b) dbj·J db =J IT btX;dbjJ db k= I B B B ;= I B is invariant under permutations, so is Sn . 0 This observation guarantees that the crash of 1929 will have no worse conse- quences for wealth Sn than if the bad days of that time had been sprinkled out among the good. 3. THE REASON THE PORTFOLIO WORKS The main idea of the portfolio algorithm is quite simple. The idea is to give an amount db/IB db to each portfolio manager indexed by rebalancing strategy b, let him or her make Sn(b) = enW(b, Fn) db at exponential rate W(b, Fn), and --- ## Page 214 Universal Portfolios 187 UNIVERSAL PORTFOLIOS 7 pool the wealth at the end. Of course, all dividing and repooling is done "on paper" at time k, resulting in bk• Since the average of exponentials has, under suitable smoothness conditions, the same asymptotic exponential growth rate as the maximum, one achieves almost as much as the wealth S: achieved by the best constant rebalanced portfolio. The trap to be avoided is to put a mass distribution on the market distributions F( x). It seems that this cannot be done in a satisfactory way. 4. PRELIMINARIES We now introduce definitions and conditions that will allow characterization of the behavior of SnIS:. Let Fn (x) denote the empirical probability mass function putting mass lin on each of the points XI> x2, ... , xn E R~ . Let the portfolio b* == b*(Fn) achieve the maximum of Sn(b) == rr7=I blXi. Equivalently, since Sn(b)==enW(b,Fn), the portfolio b*(Fn) achieves the maximum of W(b,Fn). Thus, (4.1) DEFINITION. We shall say all stocks are active (at time n) if (b*(Fn» i > 0, i > 1,2, . .. ,m, for some b* achieving W*(Fn ). All stocks are strictly active if inequality is strict for all i and all b* achieving W*(Fn). DEFINITION. We shall say XI, X2, •• • , Xn E Rm are of full rank if XI , X2, ..• , Xn spans Rm. The condition of full rank is usually true for observed stock market sequences if n is somewhat larger than m, but the condition that all stocks be active often fails when certain stocks are dominated. The next definition measures the curvature of Sn(b) about its maximum and accounts for the second-order behavior of Sn with respect to S:. DEFINITION. The sensitivity matrix function J(b) of a market with respect to distribution F(x), X E R~, is the (m - 1) x (m - 1) matrix defined by (4.2) J .(b) == J (Xi - Xm) (Xi - Xm) dF(x) IJ (b1x)2 ' 1 ~ i, j ~ m - I. The sensitivity matrix j* is J(b*), where b* == b*(F) maximizes W(b, F) . We note that (4.3) 2 * * b* ~m-I b* F * (J W( (b l , b2 , • •• , m-(o I - ""i=1 d, ) Jij == - (Jb. (Jb . I J LEMMA 4.1. j* is nonnegative definite. It is positive definite if all stocks are strictly active. --- ## Page 215 188 T M Cover 8 THOMAS M. COVER 5. ANALYSIS FOR TWO ASSETS We now wish to show that SnIS: - ,J27rlnJm where I n is the curvature or volatility index. We show in detail that ,J27rlnJn is an asymptotic lower bound on Snl S:, and we develop explicit lower bounds on Snl S: for all n and any market sequence XI' • •• , Xn • We develop an upper bound by invoking strong conditions on the market sequence. Section 6 outlines the proof for m assets. We investigate the behavior of Sn for m = 2 stocks. Consider the arbitrary stock vector sequence (5.l) i = 1,2, . .. . We now proceed to recast this two-variable problem in terms of a single variable. Since the portfolio choice requires the specification of one parameter, we write (5.2) b= (b, I-b), O::s; b ::s; 1, and rewrite Sn (b) as n (5.3) Sn(b) = IT (bXil + (l - b)xd, i=1 Let (5.4) S: = max Sn(b), Osbsl and let b~ denote the value of b achieving this maximum. Section 8 contains examples. The universal portfolio (5.5) is defined by (5.6) and achieves wealth (5.7) Let (5.8) (5.9) (5.10) bk = J~ bSdb) db I J~ Sdb) db n Sn = IT (biXil + (l - bi)Xi2)' i=1 1 n = - ~ In(bxiI + (l - b)xi2) n i=1 --- ## Page 216 Universal Portfolios 189 UNIVERSAL PORTFOLIOS 9 where Fn (x) is the empirical cdf of I Xi 17= I' By Lemma 4.1, the wealth Sn achieved by the universal portfolio bk is given by (5.1l) In order to characterize the behavior of Sn we define the following functions of the sequence XI> X2, ••• , Xn. Define the relative range Tn of the sequence XI,X2,'" 'Xn to be (5.12) Tn = 21/3(m~xIXijl - 1), mmlxijl where the minimum and maximum are taken over i = 1,2, ... , n; j = 1,2. Let (5.13) I n = ~ t (Xii - xd 2 n i=1 (b~xiI + (1- b~)Xi2)2' where b~ maximizes Wn(b). Let (5.14) Thus Tn corresponds to the relative range of the price relatives and I n denotes the curvature of In Sn(b) at the maximum. THEOREM 5.1. Let xI> X2, ••• , be an arbitrary sequence of stock vectors in R~, and let an = min I b~, 1 - b~, 3Jnh~ I. Then for any 0 < e < 1, and for any n, (5.15) REMARKS. This theorem says roughly that SnIS:;::: .j27rlnJn. So the universal wealth is within a factor of CI -!n of the (presumably) exponentially large S:. It will turn out that every additional stock in the universal portfolio costs an additional factor of 1/ -!n. But these factors become negligible to first order in exponent. It is important to mention that this theorem is a bound for each n. The bound holds for any stock sequence with bound an and volatility I n. Proof We wish to bound Sn = f~ enWn(b) db. We expand Wn(b) about the maximizing portfolio b~, noting that Wn(b) has different local properties for each n and indeed a different maximizing b~. We have (5.16) (b - b~)2 Wn(b) = Wn(b~) + (b - b~) W~(b~) + 2 W~'(b~) (b - bn*)3 + 3! w;'(f>n), --- ## Page 217 190 10 THOMAS M. COVER where bn lies between b and b~ . We now examine the terms. (i) The first term is (5.17) where S: is the target wealth at time n. (ii) The second term is (5.18) by the optimality of b~. (iii) The third term is = 0, if 0 < b~ < 1, (5.19) Wn"(bn*) = -J (XI - X2)2 dF () J* (b~IX)2 n x = - n ' T M Cover Thus, W; (b~) ~ 0, with strict inequality if 0 < b~ < 1 and Xii * Xi2 for some time i. This term provides the constant in the second-order behavior of Sn. (iv) The fourth term is III - r (XI - X2)3 (5 .20) Wn (bn) = 2j (bxl + (1 _ b)X2)3 dFn(x). We have the bound (5.21) (5.22) Thus (5.23) for 0 ~ b ~ 1, where < 3 - Tn> for all bn E [0, 1]. (5.24) J (XI - X2)2 I n = (b~xl + (1 _ b~)X2)2 dFn(x) . We now make the change of variable (5 .25) u = [ri (b - b~), where the new range of integration is (5.26) - [ri b~ ~ u ~ [ri (1 - b~). --- ## Page 218 Universal Portfolios 191 UNIVERSAL PORTFOLIOS II Then, noting enW! = S~, we have (5.27) Sn = r~ Sn(b) db (5.28) S; r,r,;(I-b~) ( 1 2 1 3 3) ~ - exp - -u I n - - lu ITn dUo .[n -,r,;b~ 2 6.[n We wish to approximate this by the normal integral. To do so let 0 < e ::s 1 and note that (5.29) for (5.30) u::s 3r,.[n JnIT~. Let cfl denote the cdf of the standard normal (5.31) cfl (x) = _1_ rx e -u212 du, J2; J-c» and let (5.32) Thus an is a measure of the degree to which Sn(b) has a maximum of reasonable curvature within the unit interval. Then from (5.28), for any 0 < 8 ::s I, .[n S lJii(I-b~) (I 1 ) (5.33) ~ ~ exp - -u 2Jn - - lul3T~ du Sn -Jiib~ 2 6fo (5.34) ~ rJiian £ exp[ - -21 u2 JnO + 8») du J - Jiian' (5.35) (5.36) = We use the inequality (5.37) 1 (X2) ( 1 ) 1 (X2) -- exp - - 1 - 2 < cfl ( -x) < -- exp - - , J27rx 2 2 X J27rx 2 2 for x > 0, to obtain the bound Hence, --- ## Page 219 192 T M Cover 12 THOMAS M. COVER for any ° < f, S 1, for all n, and all XIo X2, • •• , which proves the theorem. 0 The explicit bounds in Theorem 5.1 may be useful in practice, but a cleaner summary of performance is given in the following weaker theorem. THEOREM 5.2. Let XI, X2, '" be a sequence oj stock vectors in R~ and suppose ° s b! s 1 - 0, 7 n S 7 < ex:>, and In ;:::: J > 0, jor a subsequence oj times nto n2, .... Then (5.40) along this subsequence. Prooj. The conditions of the theorem, together with Theorem 5.1 imply (5.41) SnIS: 11 2fT" J21flnJn ;:::: ~~ - f,nJ21fnJmin(0, 3J1T3} , where 7 is the bound ratio, and where we are free to choose f,n E [0, 1] at each n. Noting that I n s 7 2 < ex:> and letting f,n = n -1 / 4 proves the theorem. 0 We have just shown that SnIS: is as good as .J21flnJn. We now show that it is no better. For this we consider a subsequence of times such that Wn(b) is approximately equal to some function W(b) , and we argue that upper bounds on f~ enW(b) db suffice to limit the performance of the wealth Sn. Toward that end, let us consider functions W such that (5.42) (i) W(b) is strictly concave on [0, 1]. (ii) W"'(b) is bounded on [0, 1]. (iii) W(b) achieves its maximum at b* E (0, 1). We plan to pick out a subsequence of times such that Wn(b) = (lin) ' E7=lln btx; approaches W(b). We can expect such limit points from Arzelil's theorem on the compactness of equicontinuous functions on compact sets. Let b! maximize Wn(b). Let (n;l be a subsequence of times such that for (i) Wn(b) s W(b) , Os b s 1. (ii) W~(b~) -+ W"(b*). (5.43) Recall the notation I n = - W~/(b~). The following theorem establishes the tightness of the lower bound in Theorem 5.2. THEOREM 5.3. For any x I, X2, ..• E R~ and jor any subsequence oj times nlo n2, .. . such that Wn(b) satisjies conditions (5.43) jor some W(b) satisjying (5.42), we have --- ## Page 220 Universal Portfolios 193 UNIVERSAL PORTFOLIOS 13 (5.44) along the subsequence. Proof The lower bound follows from Theorem 5.2. From Laplace's method of integration we have (5.45) r I eng(u) du _ eng(u') 211' Jon I g II (u*) I if g is three times differentiable with bounded third derivative, strictly concave, and the u * maximizing g ( . ) is in the open interval (0, 1). Consequently, (5.46) and the theorem is proved. o 6. MAIN THEOREM Here we prove the result for m assets under the assumption that all stocks are active and of full rank and b~(Fn) -+ b* Eint(B) . We discuss removing the conditions in Section 9. For example, lack of full rank reduces the dimension from m to m' , as does the existence of inactive stocks. Finally, b;(Fn ) need not have a limit, in which case we can describe the behavior of Sn for convergent subsequences of b~(Fn), as ~ell as develop explicit bounds for all n. From Lemma 2.1, we have where Sn = IB Sn(b) db lIB db, k Sk(b) = II btXi ' i =1 bk + 1 = I bSdb) db I J Sdb) db, n Sn = II b~Xk ' k=1 A summary of the performance ofbk is given by the following theorem. THEOREM 6.1. Suppose XI> X2, ... E [a, c] m, 0 < a :5 c < 00, and at a subsequence oj times nl' n2, .. . , Wn(b) /' W(b) Jor bE B, J: -+ J*, b~ -+ b*, --- ## Page 221 194 T M Cover 14 THOMAS M. COVER where W (b) is strictly concave, the third partial derivatives of Ware bounded on B, and W(b) achieves its maximum at b* in the interior of B. Then Sn _ ( fh)m-I (m - 1)! (6.1) S; ~-;- 11*1112 in the sense that the ratio of the right- and left-hand sides converges to 1 along the subsequence. Proof. (Outline) We define (6.2) C= (CI>C2, ••• ,Cm_I):Ci~0, ~Ci~ 1) and n (6.3) Sn(e) = II b1(e)xj, eEC, j=1 where (6.4) ( m-I ) b(e) = Cl> C2,' •• , cm-I> 1 - .~ Cj • 1=1 Note that (6.5) Vol(C) = J de = 1 . c (m-1)! We shall prove only the lower bound associated with (6.1). From Lemma 2.1, the universal portfolio algorithm yields (6.6) where b is uniformly distributed over the simplex B. Since a uniform distribution over B induces a uniform distribution over C, we have (6.7) We now expand Sn(e) in a Taylor series about e* = (br, .. . , b!_I), where b* maximizes W(b, Fn). We drop the dependence of b* on n for notational convenience. By assumption, bi > 0, for all i. We have (6.8) Sn(e) = enWn(C), where (6.9) 1 n Wn(e) =;; j~ In b1xj = J In b1x dFn(x) g, EFn Inb1X --- ## Page 222 Universal Portfolios 195 UNIVERSAL PORTFOLIOS 15 and (6.10) Expanding Wn(e), we have 2(X; - Xm) (Xi - Xm) (Xk - Xm) . S3(c) where C = Ae· + (1 - A)e, for some 0 SA S 1, where A may depend on e, and (6.12) m-I (m-I) See) = ;~ c;X; + 1 - ;~ C; X m • Here n (6.13) Sn(e) = II b'(e)x;, ; = 1 ( m-I ) bee) = c., C2, ••• , 1 - .L: Ci , 1=1 ( m-I (m-I)) Wee) = J In i~ C;X; + 1 - ~ c; Xk dFn(x), (6.14) The condition that all stocks be strictly active implies by Lemma 4.1 that IJ,; I > 0, where 1'1 denotes determinant. We treat the terms one by one. (i) By definition of b*, (6.15) W(e*) = W(b*, Fn) = W*(Fn). (ii) The second term is 0 because b* is in the interior of B, Wn(b) is differenti- able, and b* maximizes Wn• Thus, --- ## Page 223 196 16 THOMAS M. COVER (6.16) = 0, i = 1, 2, ... , m - 1. (iii) The third term is a positive definite quadratic form, where J: = J*(b*(Fn». (iv) For the fourth term, we examine (6.18) E (Xi - Xm) (Xj - Xm) (Xk - Xm) Fn S3 (c) We note (6.19) since Xi ~ a for all i. Also since Xi - Xm s; 2e, we have (6.20) _ 8e 3 s; E (Xi - Xm) (Xj - Xm) (Xk - Xm) S; 8e 3 • a3 S3(C) a 3 T M Cover We now make the change of variable u = Jir(c - c*), where we note the new range of integration u E U = Jir (C - c *). Thus (6.21) ( n * t * * n~J Sn(c) = exp nWn(c*) -"2 (c - c ) I n (c - c ) +"3 LJ 3 = exp( n W: - ~ utFnu + 3Jn L:3) , where (6.22) Note that (6.23) Observing (6.24) --- ## Page 224 Universal Portfolios 197 UNIVERSAL PORTFOLIOS 17 yields (6.25) (6.26) Sn = (m - I)! J Sn(e) de ceC ~ (m - I)! S,i J (_~ IJ* _ m312 IIU1I 38C3) (_1 )m-I d ueuexp 2 u n U r: 3 r: U, 3...jn a ...jn which can now be bounded using the techniques in the two-stock proof. The upper bound follows from Laplace's method of integration, as in Theorem 5.3, from which the theorem follows. 0 7. STOCHASTIC MARKETS Another way to see the naturalness of the goal S,i = enW(b*(Fn), Fn) is to consider random investment opportunities. Let X" X2, ••. be independent identically distributed (ij.d.) random vectors drawn according to F(x), xeRm , where F is some known distribution function. Let Sn (b) = rr7= I blXj denote the wealth at time n resulting from an initial wealth So = 1 and a reinvestment of assets according to portfolio b at each investment opportunity. Then (7.1) Sn(b) = g blXj = exp(~ InblX) = exp (n (E In btX + op (1» I = exp (n ( W( b, F) + op (1 » I by the strong law of large numbers, where the random variable op( 1) --+ 0, a.e. We observe from the above that, to first order in the exponent, the growth rate of wealth Sn (b) is determined by the expected log wealth (7.2) W(b, F) = J In b1x dF(x) for portfolio b and stock distribution F(x). It follows for X" X2, ••. , Li.d. - F that b*(F) achieves an exponential growth rate of wealth with exponent W* (F). Moreover Breiman (1961) establishes for LLd. stock vectors for any nonanticipating time-varying portfolio strategy with associated wealth sequence Sn that (7.3 ) lim ~ In Sn :s W* (FI, a.e. n Finally, it follows from Breiman (1961), Finkelstein and Whitley (1981), Barron and Cover (1988), and Algoet and Cover (1988), in increasing levels of generality --- ## Page 225 198 T M Cover 18 THOMAS M. COVER on the stochastic process, that limn--+ n- 1 In Sn/S~ ::s 0, a.e., for every sequential portfolio. Thus b*(F) is asymptotically optimal in this sense, and W*(F) is the highest possible exponent for the growth rate of wealth. Thus S~ is asympto- tically optimal. We omit the proof of the following. THEOREM 7.1. Let Xi be U.d. - F(x). Let b*(F) be unique and lie in the interior of B. Then the universal portfolio bk yields a wealth sequence t satisfying (7.4) I A - In Sn -+ W*(F), a.e. n Thus, in the special case where the stocks are independent and identically distributed according to some unknown distribution F, the universal portfolio essentially learns F in the sense that the associated growth rate of wealth is equal to that achievable when F is known. 8. EXAMPLES We now test the portfolio algorithm on real data. Consider, for example, Iroquois Brands Ltd. and Kin Ark Corp., two stocks chosen for their volatility listed on the New York Stock Exchange. During the 22-year period ending in 1985, Iroquois Brands Ltd. increased in price (adjusted in the usual manner for dividends) by a factor of 8.9151, while Kin Ark increased in price by a factor of 4.1276, as shown in Figure 8.1. Prior knowledge (in 1963) of this information would have enabled an investor to buy and hold the best stock (Iroquois) and earn a 791070 profit. However, a closer look at the time series reveals some cause for regret. Table 8.1 lists the performance of the constant rebalanced portfolios b = (b, I - b). The graph of Sn(b) is given in Figure 8.2. For example, reinvesting current wealth in the proportions b = (0.8,0.2) at the start of each trading day would have resulted in an increase by a factor of 37.5. In fact, the best rebalanced portfolio for this 22-year period is b* = (0.55,0.45), yielding a factor Sri = 73.619. Here S~ is the target wealth (with respect to the coarse quantization of B = [0, I] we have chosen). The universal portfolio bk achieves a factor of Sn = 38.6727. While Sn is short of the target, as it must be, Sn dominates the 8.9 and 4.1 factors of the constituent stocks. The daily performance of both stocks, the universal portfolio, and the target wealth are exhibited in Figure 8.3. The portfolio choice bk as a function of time k is given in Figure 8.4. To be explicit in the above analysis, we have quantized all integrals, resulting in the replacements of (8.l) S~ = max Sn(b) by S~ = max Sn(i120) b ;=0. 1 . ...• 20 and --- ## Page 226 Universal Portfolios UNIVERSAL PORTFOLIOS "iroqu" and "kinar" 16 ,-------~------~----~~----_.------_.------_, 14 12 l 10 8 6 4 "kin " 2 O L-------L-----~~ ____ ~ ______ ~ ______ ~ ______ ~ o 1000 2000 3000 Days 4000 5000 FIGURE 8.1. Performance of Iroquois brands and Kin Ark. 20 Year Return vs. mix of "iroqu" and "kinar" 6000 80r---~----.----.-----r----.---~----.----.-----r---' 70 60 50 40 ~ __________ ~R~e~~fr~o~m~U~n~iv~ e~rs~ al~P~o~n~fo~l~io~ ______ -\ ________ ~ 30 20 10 O L---~----~--~----~----L---~----~--~----~--~ o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Fraction of "iroqu" in Ponfolio FIGURE 8.2. Performance of rebalanced portfolio. 199 19 --- ## Page 227 200 20 (8.2) THOMAS M. COVER TABLE 8.1 Iroquois Brands Ltd versus Kin Ark Corp b 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 Target wealth: S~ = 73.619 Best rebalanced portfolio: b~ = 0.55 Best constituent stock: 8.915 Universal weahh: ~n = 38.6727 8.9151 13.7712 20.2276 28.2560 37.5429 47.4513 57.0581 65.2793 71.0652 73.6190 72.5766 68.0915 60.7981 51.6645 41.7831 32.1593 23.5559 16.4196 10.8910 6.8737 4.1276 T M Cover bk+ 1 = f~ bSk(b) db /tSk(b) db by bk+ 1 = i~ ;OSk(:O) /i~ Sk(:O)' The resulting wealth factor (8.3) is calculated using (8.4) Telescoping still takes place under this quantization and it can be verified that Sn in (8.3) can be expressed in the equivalent form (8.5) Thus Sn is the arithmetic average of the wealths associated with the constant rebalanced portfolios. --- ## Page 228 Universal Portfolios 201 UNIVERSAL PORTFOLIOS 21 Universal Portfolios with "iroqu" and "kinar" 45 ,-------~-------.--------.-------~-------,------~ 40 35 30 25 20 15 10 5 1000 2000 3000 4000 5000 6000 Days FIGURE 8.3. Performance of universal portfolio. Mix of "iroqu" and "kinar" in Universal Portfolio 0.6 0.55 0.5 '" go 0.45 ':= ..... 0 c:: .S: 0.4 ti '" ... ~ 0.35 0.3 0.25 0 1000 2000 3000 4000 5000 6000 Days FIGURE 8.4. The portfolio bk . --- ## Page 229 .~ .~ .. g '~ il: "cammeR and "kiD.,.- M ,r-----~--------------------~----~------, 70 60 50 40 30 20 10 ~ I rl' j I ~il"°'f" I 'I' : l ' : "AI I j 'i , Ii~ Y I ~"" I:. I" I I ;V~ j,i , l ,,' , (O"" ..... rv..."ri Ll ~ ~ . .,. -..,___ • ...oar o -~~~ I o 1000 2000 3000 4000 5000 6000 pays Mix of "comme" aDd "kinu" it!. Uaiversal Portfolio 0,7 0.65 0,6 0,55 0,5 0.45 0,' 0.35 0,3 0 1000 2000 3000 4000 5000 6000 Days Universal Portfolio. with "comme" and "lcinu" 140rl ------~--------------~--------------------, 120 100 M 60 j "comme" 40 20 I ~~--....r- , i • o ~"Jcin ... o 1000 2000 3000 4000 5000 6000 Days 20 Yeu Retum vs_ mix of "comme" and "kinu· 160,' ~--,--::.:...:.=.::::::~~~~~~~- 140 120 100 80 ~ Renarn froyUnivenal Portfolio \ 60 40 20 °OL-~0~ , I~~O~,2---0~,~3--~O~.4~~0,~5---0~,6~~0~ ,7---0~,8~~0~ ,9--~ Fraction of "cam me" in Pomolio FIGURE 8.5. Commercial Metals and Kin Ark; Performance of Universal Portfolio; Universal Portfolio; Performance of Rebalanced Portfolio. N N -l l: o 3:: > en 3:: () o <: tTl ;:.::I tv o tv ~ ~ 6' '" (\) ': --- ## Page 230 Universal Portfolios 203 UNIVERSAL PORTFOLIOS 23 Finally, note the calculation of the portfolio bn+ 1 = (bn+l> I - bn+ l) in this example. Merely compute the inner product of the band Sn(b) columns in Table 8.1 and divide by the sum of the Sn(b) column to obtain bn + l . Note in particular that the universal portfolio bn + I is not equal to the log optimal portfolio b*(Fn) = (0.55,0.45) with respect to the empirical distribution of the past. A similar analysis can be performed on Commercial Metals and Kin Ark over the same period. Here Commercial Metals increased by the factor 52.0203 and Kin Ark by the factor 4.1276 (Figure 8.5). It seems that an investor would not want any part of Kin Ark with an alternative like Commercial Metals available. Not so. The optimal constant rebalanced portfolio is b* = (0.65,0.35), and the universal portfolio achieves Sn = 78.4742, outperforming each stock. See Table 8.2. Next we put Commercial Metals (52.0203) up against Mei Corp (22.9160). Here Sri = 102.95 and Sn = 72.6289, as shown in Figure 8.6 and Table 8.3. However, IBM and Coca-Cola show a lockstep performance, and, indeed, Sn barely out- performs them, as shown in Figure 8.7. A final example crudely models buying on 50070 margin. Suppose we have four investment choices each day: Commercial Metals, Kin Ark, and these same two stocks on 50% margin. Margin loans are settled daily at a 6% annual interest rate. The stock vector on the ith day is (8.6) TABLE 8.2 Commercial Metals versus Kin Ark b 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 Target wealth: Sri = 144.0035 Best rebalanced portfolio: b~ = 0.65 Best constituent stock: 52.0203 Universal wealth: ~n = 78.4742 52.0203 68.2890 85.9255 103.6415 119.8472 132.8752 141.2588 144.0035 140.7803 131.9910 118.6854 102.3564 84.6655 67.1703 51.1127 37.3042 26.1131 17.5315 11 .2883 6.9704 4.1276 --- ## Page 231 N I~ "commc" and "meico" Univcnal Portfolios with "commc" and "meico" +;. 80 . 90 70 f ~I 80 60f in J 70 I r . ~ "¢ommc" 60 50 f i ,,, " ! . i 50 .of "A I ' il '~l\ y '10 30f ~ ' . I iJ ' W 30 20f l ' . '~"m.i"'" 20 10 , 'f--.""- .r! ) ~~' ~1V~..t "'V 10 oL , ,~"v.,\J ... '._./''''''' ~ OL .JP?~ ~ --l -: ~ :r: 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000 0 Days Days 3!:: )-en 3!:: Mix of "commc" aDd "mcico" in Universal Portfolio 20 Year Rerum VI. mix of"comme ~ aod "meico" 0.75 . 110 ~ () oH /\ ~ 0 <: 100 tTl 90 ::0 ;) VJ ;l> r Mix of "ibm" and "coke" in Uoiversal Portfolio 20 Year Rerum vs. mix of" ibm ~ and "coke" 't! 0.52 15.5 0 I p~flI\IIA :>;) 0.51 \~ -l "r1 15 0 0.5 r 0.49 14.5 (3 VJ ~ 0.48 Rerum from U Diverul Portfolio 14 ~ 0.41 II· .~ 0.46 13.5 "' 0.45 ~\ 13 0.44 12.5 0.43 I 0.42 12 0 1000 2000 3000 4000 .5000 6000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.8 0.9 Days Fraction of ~ibm~ in Portfolio FIGU RE 8.7. IBM and Coca-Cola. N IV 0 VI VI --- ## Page 233 206 26 (8.7) THOMAS M. COVER TABLE 8.3 Commercial Metals versus Mei Corp b 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 Target wealth: Sh = 102.9589 Best rebalanced portfolio: bh = 0.60 Best constituent stock: 52.0203 Universal wealth: S"n = 72.6289 r = 0.000233, 52.0203 61.0165 70.0625 78.7602 86.6815 93.4026 98.5414 101.7927 102.9589 101 .9691 98.8869 93.9033 87.3172 79.5057 70.8890 61.8932 52.9162 44.3012 36.3178 29.1538 22.9160 T M Cover where Xi and Yi are the respective price relatives for Commercial Metals and Kin Ark on day i. Plunging on margin into Commercial Metals yields a factor 19.73, plunging into Kin Ark a factor of 0.0 (to four significant digits). Good as these stocks are, they cannot survive the down factors induced by the leverage. But a random sample of the simplex of portfolios listed in Table 8.4 reveals Sn = 98.4240, while the optimal rebalanced portfolio b* = (0.2, 0.5, 0.1, 0.2) results in a factor S: = 262.4021. Clearly 98.4 beats the factor of 78 achieved when margin is unavailable. Both factors exceed the performance 52.02 of the best stock. We observe that Sn = 98.4 exceeds the factor Sn = 78.47 obtained for these stocks when margin is unavailable. This is borne out by the fact that b* is positive in each component, calling for a small amount of leverage in the a posteriori optimal rebalanced portfolio. 9. THE GENERAL UNIVERSAL PORTFOLIO If the best rebalanced portfolio b: lies in the interior of a boundary k-face, then only k stocks are active in the best rebalanced portfolio. Thus we expect to obtain the previous bounds on SnIS: with m replaced by k. This is achieved if we start --- ## Page 234 Universal Portfolios UNIVERSAL PORTFOLIOS TABLE 8.4 Two Stocks with Margin Commercial metals Commercial metals on margin Kin Ark Kin Ark on margin r = 0.OOO233/day = 6~o/year S: = 262.4021 Best constituent stock 52.0203 = n 1= I Xi 19.7335 = nl=1 (2xi -I-r), 4.1276 = nl=IYi 0.0000 = nl=1 (2Yi-1-r) b~ = (0.2, 0.5, 0.1, 0.2) 52.0203 Wealth achieved by universal portfolio §n = 98.4240 b (0.8, 0.2, 0.0, 0.0) (0.8, 0.1, 0.0, 0.1) (0.6, 0.1, 0.1, 0.2) (0.6, 0.0, 0.4, 0.0) (0.5, 0.0, 0.2, 0.3) (0.4, 0.0, 0.4, 0.2) (0.3, 0.5, 0.1, 0.1) (0.3, 0.4, 0.1, 0.2) (0.3, 0.2, 0.2, 0.3) (0.3, 0.1, 0.2, 0.4) (0.3, 0.0, 0.1, 0.6) (0.2, 0.7, 0.0, 0.1) (0.2, 0.2, 0.3, 0.3) (0.1, 0.8, 0.1, 0.0) (0.1, 0.5, 0.2, 0.2) (0.1, 0.4, 0.2, 0.3) (0.1, 0.3, 0.1, 0.5) (0.1, 0.2, 0.4, 0.3) (0.1, 0.1, 0.2, 0.6) (0.0, 0.5, 0.4, 0.1) (0.0, 0.4, 0.2, 0.4) (0.2, 0.5, 0.1, 0.2) Sn(b) 57.0535 148.9951 207.1143 140.7803 60.8358 47.6074 212.8928 261.0452 89.0330 19.4840 0.7700 121.0142 45.2562 67.5882 233.6328 112.6695 12.7702 19.4840 0.2354 225.2524 31.8076 262.4021 207 27 with some mass on each face. To accomplish this, we let /J-s be the measure corresponding to the uniform distribution on B (S) = (b E Rm : E bi = 1, bi = 0, j E SC), where S S; (1, 2, ... , m). Thus /J-s puts unit mass on the I S I-dimensional face of the portfolio simplex. Let /J- be the mixture of these measures given by (9.1) where the sum is over all S '* 0, S S; (1, 2, ... , m). The generalized universal portfolio now becomes --- ## Page 235 208 T. M Cover 28 THOMAS M. COVER (9.2) with n (9.3 ) Sn(b) = II blXj, So(b) = 1. j=1 To state the results we define J~k)(Fn} to be the k x k sensitivity matrix with respect to the active stocks S, I S I = k, where S is the smallest set of stocks such that all optimal rebalanced portfolios b*(F} are in the interior of 8(S). Then (9.4) ;;.- (;m-_l~! e7rr-II2/IJ~k)(Fn)'112 will be the asymptotic behavior of SnIS:. 10. CONCLUDING REMARKS We now try to be sensible and ask how the universal portfolio works in practice. Of course, the examples are encouraging, as the universal portfolio outperforms the constituent stocks. However, we have ignored trading costs. In practice we would not trade daily, but only when the current empirical holdings were far enough from the recommended bk • (A rule of thumb might be to trade only if the increase in W is greater than the logarithm of the normalized transaction costs.) We are really interested in whether Sn will "take off," leaving the stocks behind. We first discuss the target wealth S:. The best rebalanced portfolio b*(Fn} based on prior knowledge of the stock sequence XI> X2, ••• ,xn yields wealth S: = enWh . Now S: grows exponentially fast to infinity under mild conditions. For example, if one of the constituent stocks is a risk-free asset with interest rate r > 0, then W: ~ In(l + r) > 0, for all n, and S: ~ (1 + rr --+ 00 . Since the universal port- folio yields (10.1) it follows that Sn will tend to infinity, and t will have the same exponent as S:, differing only in terms of order (In n}/n. What state of affairs do we expect in the real world? Certainly we expect the stock sequence to be of full dimension m for n slightly greater than m. However, we do not expect all stocks to be active. But we do expect that two or more stocks will be active. This is important because it guarantees that the target growth rate W: will be strictly greater than the growth rate of the constituent stocks. Conse- quently, we believe that the universal portfolio will achieve i = 1,2, ... ,m, exponentially fast, where Sn(ej) is the wealth relative of the ith stock at time n. --- ## Page 236 Universal Portfolios 209 UNIVERSAL PORTFOLIOS 29 However, n may need to be quite large before this exponential dominance manifests itself. In particular, we need n large enough that the difference in exponents between S: and the stocks overcomes the O((ln n)/n) penalties incurred by universality. We conclude that Sn will leave the constituent stocks exponentially behind if there are at least two strictly active stocks in the best rebalanced portfolio. REFERENCES ALGOET, P., AND T. M. COVER (1988): "Asymptotic Optimality and Asymptotic Equipartition Properties of Log·Optimum Investment," Ann. Probab., 16,876-898. ARROW, K. (1974): Essays in the Theory of Risk Bearing. Amsterdam North-Holland; New York: American Elsevier. BARRON, A., AND T. M. COVER (1988): "A Bound on the Financial Value of Information," IEEE Trans. Inform. Theory, 34, 1097-1100. BELL, R., AND T. M. COVER (1980): "Competitive Optimality of Logarithmic Investment," Math. Oper. Res., 5,161-166. BELL, R., AND T. M. COVER (1988): "Game-Theoretic Optimal Portfolios," Management Sci., 34, 724-733. BLACKWELL, D. (l956a): "Controlled Random Walks," in Proc. Int. Congress Math. III, Amsterdam: North-Holland, 336-338. BLACKWELL, D. (l956b): "An Analog of the Minimax Theorem for Vector Payoffs," Pacific J. Math. 6, 1-8. BREI MAN, L. (1961): "Optimal Gambling Systems for Favorable Games." in Fourth Berkeley Symp. on Mathematical Statistics and Probability, 1,65-78. COVER, T. M. (1984): "An Algorithm for Maximizing Expected Log Investment Return," IEEE Trans. Inform. Theory, IT-30, 369-373. COVER, T. M., AND D. GLUSS (1986): "Empirical Bayes Stock Market Portfolios," Adv. Appl. Math., 7, 170-181. (A summary also appears in Proceedings of Conference Honoring Herbert Robbins, Springer-Verlag, 1986.) COVER, T. M., AND R. KING (1978): "A Convergent Gambling Estimate of the Entropy of English," IEEE Trans. Inform. Theory, 24, 413-421. FINKELSTEIN, M., AND R. WHITLEY (1981): "Optimal Strategies for Repeated Games," Adv. Appl. Probab., 13, 415-428. HAKANSSON, N. (1979): "A Characterization of Optimal Multiperiod Portfolio Policies," in Portfolio Theory, 25 Years After: Essays in Honor of Harry Markowitz, ed. E. Elton and M. Gruber. TIMS Studies in the Management Sciences, 11 , Amsterdam; North-Holland, 169-177. HANNAN, J. F., AND H. ROBBINS (1955): "Asymptotic Solutions of the Compound Decision Problem," Ann. Math. Stat., 16,37-51. KELLY, J. L. (1956): "A New Interpretation of Information Rate," Bell Systems Tech. J., 917-926. MARKOWITZ, H. (1952): "Portfolio Selection," J. Finance, 8, 77-91. MARKOWITZ, H. (1976): "Investment for the Long Run: New Evidence for an Old Rule," J. Finance, 31, 1273-1286. MERTON, R. c., AND P . A. SAMUELSON (1974): "Fallacy of the Log-Normal Approximation to Optimal Portfolio Decision-Making over Many Periods," J. Financial Econ., I, 67-94. MOSSIN, J . (1968): "Optimal Multiperiod Portfolio Policies," J. Business, 41, 215-229. ROBBINS, H. (1951): "Asymptotically Subminimax Solutions of Compound Statistical Decision Problems," Proc. Second Berkeley Symp. on Mathematical Statistics and Probability, Berkeley: University of California Press. SAMUELSON, P. A. (1967): "General Proof that Diversification Pays," J. Financial Quant. Anal. II, 1-13. SAMUELSON, P. A. (1969): "Lifetime Portfolio Selection by Dynamic Stochastic Programming," Rev. Econ. Statist., 239-246. SAMUELSON, P. A. (1979): "Why We Should Not Make Mean Log of Wealth Big Though Years to Act Are Long," J. Banking Finance, 3, 305-307. SHARPE, W. F. (1963); "A Simplified Model for Portfolio Analysis," Management Sci., 9, 277-293. THORP, E. (1971): "Portfolio Choice and the Kelly Criterion," Business Econ. Statist. Proc. A mer. Stat. Assoc., 215-224. --- ## Page 237 MATHEMATICS OF OPERATIONS RESEARCH Vol. 13. No. 4, November 1998 Primed ill U.S.A. 16 THE COST OF ACHIEVING THE BEST PORTFOLIO IN HINDSIGHT ERIK ORDENTLICH AN D THOMAS M. COVER For a market with m assets consider the minimum, over all possible sequences of asset prices through time 11, of the ratio of the fin al wealth of a nonanticipating investment strategy to the wealth obtained by the best constant rebalanced portfolio computed in hindsight for that price sequence. We show that the maximum value of this ratio over all nonanticipating investment strategies is V" = [L 2 ~ "",,,, I ,, ..... ,,.,,I"' ( I1!/( I1 ,!·· '11", '))]-' , where H (· ) is the Shannon entropy, and we specify a strategy achieving it. The optimal ratio V" is shown to decrease only polynomially in 11 , indicating that the rate of return of the optimal strategy converges uniformly to that of the best constant rebalanced portfolio determined with full hindsight. We also relate this result to the pricing of a new derivative security which might be called the hindsight allocation option. 211 1. Introduction. Hindsight is not available when it is most useful. This is true in investing where hindsight into market performance makes obvious how one should have invested all along. In this paper we investigate the extent to which a nonantici- pating investment strategy can achieve the performance of the best strategy determined in hindsight. Obviously. with hindsight, the best investment strategy is to shift one's wealth daily into the asset with the largest percentage increase in price. Unfortunately, it is hopeless to match the performance of this strategy in any meaningful way, and therefore we must restrict the class of investment strategies over which the hindsight optimization is per- formed. Here we focus on the class of investment strategies called the constant rebalanced portfolios. A constant rebalanced portfolio rebalances the allocation of wealth among the available assets to the same proportions each day. Using all wealth to buy and hold a single asset is a special case. Therefore the best constant rebalanced portfolio, at the very least, outperforms the best asset. In practice, one would expect the wealth achieved by the best constant rebalanced portfolio computed in hindsight to grow exponentially with a rate determined by asset price drift and volatility. Even if the prices of individual assets are going nowhere in the long run, short-term fluctuations in conjunction with constant rebalancing may lead to substantial profits. Furthermore, the best constant rebalanced portfolio will in all likelihood exponentially outperform any fixed constant rebalanced portfolio which includes buying and holding the best asset in hindsight. The intuition that the best constant rebalanced portfolio is a good performance target is motivated by the well-known fact that if market returns are independent and identically distributed from one day Ito the next, the expected utility, for a wide range of utility functions including the log utility, is maximized by a constant rebalanced portfolio strat- egy. Additionally, " turnpike" theory (see Huberman and Ross 1983, Cox and Huang 1992, and references therein) finds an even broader class of utility functions for which, by virtue of their behavior at large wealths, constant rebalancing becomes optimal as the investment horizoi111 tends to infinity. In all these settings, the optimal constant rebalanced Recoived October 3, 1996; revised September 2, 1997; November 27, 1997. AMS 1991 subject classification. Primary: 90A09. ORIMS Illdex :.:ubject classificatioll. Primary: Finance/Portfolio/ Investment. Key words. portfolio selection, asset all ucation, derivati ve security, optimal investment. 960 03M -765X/98/2304/0960/$05 .00 Copyright © 1998, Institute for Oper O. Cover ( 1991 ) defines a family of f.t-weighted universal portfolios and uses Laplace' s method of integration to show, for a bounded ratio of maximum to minimum daily asset returns, that S"IS; ;?: c"ln(", - 1112 for m stocks, where c" is the determinant of a certain sensitivity matrix measuring the empirical volatility of the price sequence. Merhav and Feder ( 1993) establish polynomial bounds on S,./ S,t- under the same constraints. The first individual sequence (worst-case ) analysis of the universal portfolio of Cover (1991) is given in Cover and Ordentlich (1996) , where it is shown that a Dirichletd) weighted universal portfolio achieves a worst case performance of S"IS,t- ;?: c/n( m- 11/2. This analysis is also extended to investment with side information, with similar results. lamshidian ( 1992) applies the universal portfolio of Cover ( 1991 ) (with f.1 uniform) to a geometric Wiener market, establishing the asymptotic behavior of 5 (t)1 S*( t) , and sholll- ing (lit) log 5(t)/S*(/) -> 0, for such markets. The paper is organized as follows. Section 2 establishes notation and some basic defi- nitions. The individual-sequence performance and game-theoretic analysis are established in ~3 . Section 4 contains the hindsight allocation option pricing analysis. 2. Notation and definitions. We represent the behavior of a market of m assets for n trading periods by a sequence of nonnegative, nonzero (at least one nonzero component) price-relative vectors XI, .. . , X" E IR'~. We refer to x" = XI, ... , x" as the market sequence. The jth component of the ith vector denotes the ratio of closing to opening price of thejth asset for the ith trading period. Thus an investment in assetj on day i increases by a factor of xu' Investment in the market is specified by a portfolio vector b = (b l , • • • , b",)' with nonnegative entries summing to one. That is, b E "B, where A portfolio vector b denotes the fraction of wealth invested in each of the !1~ assets. ' .n investment according to portfolio bi on day i multiplies wealth by a factor of --- ## Page 240 214 E. Ordentlich and T M Cover COST OF ACHIEVING THE BEST PORTFOLIO IN HINDSIGHT b;xi I hijxij. j=1 963 A sequence of n investments according to portfolio choices b l , .. . , b" changes wealth by a factor of A constant rebalanced portfolio investment strategy uses the same portfolio b for each trading day. Assuming normalized initial wealth So = 1, the final wealth will be i=1 For a sequence of price-relatives x" it is possible to compute the best constant rebalanced portfolio b * as b * = arg max S" ( x" , b) , b Et! which achieves a wealth factor of S;(x") = max S,,(x ll , b). b E ·9 The best constant rebalanced portfolio b * depends on knowledge of market performance for time 1,2, . .. , n; it is not a nonanticipating investment strategy. This brings up the definition of a nonanticipating investment strategy. DEFINITION 1. A nonanticipating investment strategy is a sequence of maps bi : 1R,;,(i- I) -> 73, i = 1,2, . . . where is the portfolio used on day i given past market outcomes x i- I = XI, .. . , Xi-I . 3. Worst-case analysis. We now present the main result, a theorem characterizing the extent to which the best constant rebalanced portfolio computed in hindsight can be tracked in the worst case. Our analysis is best expressed in terms of a contest between an investor, who announces a nonanticipating investment strategy bi ( . ), and nature, who, with full knowledge of the investor's strategy, selects a market sequence x ll = XI, X2, .. . , x" to minimize the ratio of wealths SIl(X")/ S,~ (XIl) , where SIl(XIl ) is the investor's wealth against sequence x il and is given by 11 S,,(x") = n b; (Xi- I )Xi . i=] --- ## Page 241 The Cost of Achieving the Best Portfolio in Hindsight 215 964 E. ORDENTLICH AND T . M. COVER Thus, nature attempts to induce poor performance on the part of the investor relative to the best constant rebalanced portfolio b * computed with complete knowledge of x". The investor, wishing to protect himself from this worst case, selects that nonanticipating investment strategy bi (. ) which maximizes the worst-case ratio of wealths. THEOREM I (Max-min ratio). For m assets and all n , . S,,(x") max mm --- = V"' b x" S,r(x") where (2) V" = [ L ( n )2-"H(",I" ... 11 1+ . '+ fl lI1=n n], ... , nm and H(PI , ' " ,p",) = - L Pj logpj j~ 1 is the Shannon entropy function. REMARK. For m = 2, the value V" is simply and it is sh~wn in ,.§3.2 that 2/[,;+\ :2: V" :2: 1/(2,);+1) for all n. Thus V" behaves essentIally like I r.jn . For m > 2, V" - c( I 1m )",-1. REMARK. It is noted in §3.3 that in the sense that For m = 2 this reduces to V" - hl7r( I 1m) . REMARK . The max-min optimal strategy for m = 2 will be specified in Equations (8) - ( 13). These equations are followed by an alternative definition of the optimal strat- egy in terms of extremal strategies. We note that the negative logarithm of the max-min ratio of wealths given by Equation (2) also corresponds to the solution of a min-max pointwise redundancy problem in universal data compression theory. The pointwise redundancy problem was studied and --- ## Page 242 216 E. Ordentlich and T. M Cover COST OF ACHIEVING THE BEST PORTFOLIO IN HINDSIGHT 965 solved by Shtarkov ( 1987) and earlier works referenced therein. A principal result of the present work, which is developed in greater information theoretic detail in Cover and Ordentlich (1996) and Ordentlich ( 1996), is that worst sequence market performance is bounded by worst sequence data compression. The strategy achieving the maximum in Theorem 1, as developed in the proof below, depends on the horizon n. We note, however, that Cover and Ordentlich ( 1996) exhibits an infinite horizon investment strategy, the Dirichlet-weighted universal portfolio, denoted by bDC), which for m = 2 assets achieves a wealth ratio S~(xn)/s,~(xn) satisfying (4) . S~(x") 1 mm--- :2: - V . S*( ") ~ II x" n X 'i2K At time i , the Dirichlet-weighted universal portfolio investment strategy uses the portfolio (5) where i SJb, Xi) = Si(b) = n b1xj , and So(b, xo) = 1. j ~ 1 The measure tJ on the portfolio simplex 'Bis the Dirichlet( 1/2, ... , 1/2) prior with density III-I L hj :5 I , hj :2: 0, j = 1, . . . , m - 1, j~ 1 where r( .) denotes the Gamma function. The running wealth factor achieved by the Dirichlet-weighted universal portfolio through each time n is S~(X " ) = f SII(b, x")dtJ(b) . '8 Thus the max-min ratio can be achieved to within a factor of -& for all n by a single infinite horizon strategy. The bound (4) generalizes to m > 2 so that for each m, the worst-case wealth achieved by the Dirichlet-weighted universal portfolio is within a con- stant factor (independent of n) of V". The significance of Theorem 1 can be appreciated by considering some naive choices for the optimum investor strategy b. Suppose, for m = 2 assets, that b corresponds to investing half of the initial wealth in a buy-and-hold of asset I and the other half in a buy-and-hold of asset 2. In this case the first two portfolio choices are --- ## Page 243 The Cost of Achieving the Best Portfolio in Hindsight 217 966 E. ORDENTLICH AND T . M. COVER (6) b = - - and (1 1)' I 2'2 Since we are allowing nature to select arbitrary price-relative vector sequences, nature could set XI = (0,2)' and X2 = (2, 0) ', in which case the investor using the split buy- and-hold strategy (6) goes broke after two days. On the other hand, for this two-day sequence, the best constant rebalanced portfolio is b * = (1/2, 1/2)' and yields a wealth factor S i (XI, X2) of I. Suppose the investor instead opts to rebalance his wealth daily to the initial ( 1/2, I 12) proportions. Here bi is the constant rebalanced portfolio b = (1/2, 112)'. If nature then chooses the sequence of price-relati ve vectors x" = (2, 0)', (2, 0) ', .. . , (2, 0) ' the investor earns a wealth factor S,,(x") = I while the best constant rebalanced portfolio b* = ( I, 0)' earns S;r (x") = 2" . The ratio S"I S,f of these two wealths decreases exponentially in n while the max-min ratio V" decreases only polynomia).!.yJ.-n particular, the wealth achieved by the max-min optimal strategy is at least 2"/(2vn + I) for this sequence. These two investment strategies are particularly naive. A more sophisticated scheme might start off with bl = (1/2, 1/2)' and then use the best constant rebalanced portfolio for the observed past. This scheme, however, is also flawed, since if nature chooses XI = ( I, 0)' , the investor would use b2 = ( I, 0)' the following day and then would go broke if nature set X2 = (0, I )'. One might think of fixing this scheme by using a time varying mixture of the ( I /2, 1/2) portfolio and the best constant rebalanced portfolio for the past. However, this class of strategies also fails to achieve V". We now proceed with the proof of Theorem I. The following lemma is used. In the sequel we adopt the conventions that a/O = IXJ if a > 0, and that 010 = 0. LEMMA I. If 01 I , ... ,OI,,:::=: 0, {31, " " {3,,:::=: 0, then (7) PROOF OF LEMMA I. Let . OIj J = arg min ~ . j (3j The lemma is trivially true if O:J = 0 since the right side of (7) is zero. So assume OIJ > O. Then, if (3J = 0 the lemma is true since both the left and right sides of (7) are infinity. Therefore assume OIj > ° and (3j > 0. Then because which implies (X) > - - (3) --- ## Page 244 218 E. Ordentlich and T M Cover COST OF ACHIEVING THE BEST PORTFOLIO IN HINDSIGHT 967 for allj. 0 PROOF OF THEOREM 1. For ease of exposition we prove the theorem for m = 2. The generalization of the argument to m > 2 is straightforward. Thus, for the case of m = 2 we must show that . S,,(x") max mm --- = V"' Ii x" S;(X/) where We prove that . S,,(x") max mm--- ~ V"' b x" S;(x n ) by explicitly specifying the max-min optimal strategy h. We define the strategy by keeping track of the indices of the terms in the product n i'~ I h: Xi. For sequences j" E {I, 2 }" let nl(j") and n2(j") denote, respectively, the number of l's and the number of2's inj" . That is, ifj" = (jJ, ... , jll)' (8) n,(j") = L l(ji = r), ;=) where 1(·) is the indicator function. Let (9) Then, since Lj" E{ I.2 }" w(j") = 1, w(j") is a probability measure on the set of sequences j" E {l, 2}". For 1< n, let (10) be the marginal probability mass of jl' . .. ,k This marginal probability may also be denoted by w (j'- I , j,) . Finally, define the nonanticipating investment strategy hi = b/l , b12r (11) and --- ## Page 245 The Cost of Achieving the Best Portfolio in Hindsight 219 968 E. ORDENTLI C H AND T. M. COV E R (12) with (13 ) Gil = w(1) and GI 2 = w(2). An alternative characterization of the max-min optimal strategy, which turns out to be equivalent to the above, is as follows. Break the initial wealth into 2" piles, one corre- sponding to each sequence)n, where the fraction of initial wealth assigned to pile)" is precisely w(jn) as given in (9). Now invest all the wealth in pile)" in asset)1 on day I. From then on, for each day i, shift the entirety of the running wealth for this pile into asset I . Do this in parallel for each of the 2" piles)n. We refer to the strategy used to manage pile)" as the extremal strategy corresponding to the sequence)". The wealth factor achieved by the investor using ( II ) and ( 12) is (14 ) where n Sn(x") = n b;x, ' ~ I = n L /E { 1,2 }1 w(J') II ~~ , X Ui L HE {I O}/-I W(j·'- I) II';; ', x ·. 1= I j.- I 'l, L w(j") n Xiii p iE {1.2 }" i= 1 k (k)k( n _ k)"-k = V" L - -- X(k) k~ O n n X(k) ~ pl:n] (/')=k i= ! and ( l4) follows from a telescoping of the product. It is apparent from Equation (14) that the extremal strategy formulation of the max- min optimal strategy is equivalent to the portfolio formulation (8) - ( l3) . The extremal strategies simply " pick off" the product of the price relatives corresponding to the se- quence of assets with indices)" . Equation ( 14) represents the sum of the wealths obtained by the extremal strategies operating in paralle\. Note that for 0 :5 k :5 n, (15) max b k ( I - b),,- k. Os /):::s: 1 Also note that S,r (x n ) can be rewritten as --- ## Page 246 220 E. Ordentlich and T M Cover COST OF ACHIEV ING THE BEST PORTFOLIO IN HINDSIGHT 969 n S,T(X") = IT b* 'x; j= 1 (16) IT b* X i, lJi i" E {1.2 }" i=) = I. b*k(1 - b*),,-kX(k), k ~ O where b* = (b*, I - b*), achieves the maximum in (16). Therefore, for any market sequence x", Lemma 1 and the above imply that S,,(x") ( k)k( k)"-k Vn ~k~O ~ ~ X(k) - --- S,T(X") 2:k~ O b*k( 1 - b*),,-kX(k) (17) ~ V" min --,-------,- b*k( 1 - b*),,-k O:-=:;ksn ( 18) where ( 17) follows from a combination of Lemma I and the cancellation of the sums of products of xu;, and ( 18) follows from (15). Since the above holds for all sequences x", we have shown that ( 19) . S,,(x") max mm S*( ") ~ V". b Xii 11 X To show equality in (19) we consider the following possibilities for x" . For each j" E {I, 2}" define x"(j") = XI(jI), "" xn (j,,) , as (20) Let { (1, 0)' x; (j;) = (0, I)' if }; = I, if j; = 2. 'K= {x"(j") : j" E {I, 2}"} be the set of such extremal sequences x". An important property shared by all nonanticipating investment strategies b( ') on the sequences (20) is that (21) I. S,,(x") = 1. Also note that, for x" (j") E 'K, the best constant rebalanced portfolio is easily verified to be --- ## Page 247 The Cost of Achieving the Best Portfolio in Hindsight 970 SO that Therefore E. ORDENTLICH A ND T . M. COV ER ( n (J''') )"'U") (n (J''') )"2U") S;(X"(j"» = _I _ _2 __ n n w(/' ) v" 1 L S;(XIl) = -. x " E "J( Vil Since the minimum is less than any average, we obtain equality in ( 19) from (22) . S,,(x") ~ ( S;(x n ) ) S,,(x") mm ---:s £.. --- x" E~ S,f(x") X"EW 2: x"E* S;(X") S;(x") (23) (24) (25 ) which holds for any b. Thus . Sn(X" ) max mm --- :s V". b x" S,f(x") Combining this with (19) completes the proof of the theorem. 0 221 Complexity. It appears from (II) and (12) that computing the max-min optimal port- folio requires keeping track of the products n;:: xijj for each sequence / -1. This quickly becomes prohibitively complex, since the number of such sequences is exponentially increasing in I. Fortunately, a simplification can be made. This follows from the observation that W(j" ) defined in (9) depends on/, only through its type (nl(j"), n2(j"», the number of I's and 2's. This implies that W(j,,- I,jll)' for fixedj,,, is a function of /,- 1 only through (nl(j ,,- I), n2(/,- I» . The same applies to W(j"- I) = 2:j " W(j"- I,j") , Thus, by induction W(j'- I,j,) and W(j'- I), for alii, are constant on /- 1 with the same type. Using this fact, the numerator and denominator of ( 11) and ( 12) can be evaluated by grouping the products n;:: xij; according to the type of / - 1. More specifically, the nu- merator of ( 11 ), for example, can be written as --- ## Page 248 222 E. Ordentlich and T M Cover COST OF AC HIEVIN G THE BEST PORTFOLIO IN HI NDSIGHT 971 I- I I- I I- I We/- I, I) TIXiji = I Wf- ,(k , I) TIXiji ;=1 k=O I-I = I wf- , (k, 1 )XI _ I (k) , k=O where wf- , (k, 1) equals We/- I, 1) when nl (/-1) = k and The denominator can be rewritten in a similar way. It is now clear that only the quantities X I_ I (k) need be computed and stored instead of the exponentially many products n~: : XUi . The complexity of this is linear in I, since there are only I such quantities. The simple recursions suffice to update the X I_ I (k) . The above generalizes in the obvious way to m > 2 assets resulting in a computational complexity growing like /111- 1. Therefore, the max-min optimal portfolio is, in fact, com- putationally feasible for moderate m. 3.1. Game-theoretic analysis. A full game-theoretic result can also be proved. Spe- cifically, we imagine the same contest as above, except that mixed strategies are allowed. The payoff function is A(b, x") = S,,(x") . S:;'(x") As before, the investor and nature respectively try to maximize and minimize the payoff. Let $" denote the game when played with this payoff function. A mixed strategy for the investor is a probability distribution 'P( b) on the space of nonanticipating investment strategies, b = (b" b2(XI) , ... , b,,( X,,- I» . Similarly, nature's mixed strategies are probability distributions on the space of price-relative sequences and will be denoted by !l(x"). The following theorem can then be proved. THEOREM 2. The value of the game 9" is max min EA(b , x") = min max EA(b, x") = V,,, 1\ 11 ) 'y( XIl ) .!;I(x") "\ ))) where V" is given by (2) . Further, the investor's optimum strategy 'J»" is the pure strategy specified by (8) - (13 ) . PROOF. We prove this for m = 2, the generalization being obvious. The pure strategy 'P* is precisely the max-min optimal strategy (8) - ( 13) achieving the maximum in The- --- ## Page 249 The Cost of Achieving the Best Portfolio in Hindsight 223 972 E. ORDENTLlCH AND T. M . COVER orem 1. Nature's optimum mixed strategy.2* (for m = 2) consists of choosing sequences from 'K = {X"(j") : j" E {1, 2}" } according to the probability distribution W(j") given by (9). The proof of (26) min max EA(b, x") :s VII' 2(x ll ) tlb) follows from Equations (22) through (25). The theorem follows from (26) and (] 9). 0 The full game-theoretic analysis brings out a nice symmetry between the optimal in- vestment strategy and nature's optimal strategy. The optimal investment strategy 'P* is a pure strategy constructed from the distribution W(j") on binary strings given by (9). Nature's optimal strategy, on the other hand, is to choose 0-1 price-relative vectors at random according to this same probability distribution. This analysis generalizes to games with payoff _ ( S,,(x") ) A,(b , x") = 1> S,t(x") , for which the following holds. THEOREM 3. For concave nondecreasing 1>, the game fJ,'{1» with payoff A",(b, x") has a value V ($11 (1) » given by V ($11 ( 1> » = 1>( V,,), where VII is given by (2) and the optimal strategies are the same as those for fJ" . 3.2. Bounds on VII' We prove the following lemma for m = 2. LEMMA 2. For all n, 1 2 --- :s VII :s --- . 2[,;+1 [,;+I PROOF. We first prove the lower bound. In Cover and Ordentlich ( 1996), a sequential portfolio selection strategy called the Dirichlet( 112, . . . , 1/2) weighted universal port- folio was shown to achieve a wealth S~(x") satisfying . S~(X") 1 mll1--- 2: . x" S;;'(x") 2[,;+1 Therefore --- ## Page 250 224 E. Ordentlich and T M Cover COST OF ACHIEVING THE BEST PORTFOLIO IN HINDSIGHT 973 . S~(x") 2': mlll--- x" S;i'(X") > --- -2[,;+1' proving the lower bound on VII' We now establish the upper bound. Write I/VII as VII (27) r(n + I) II k'(n - k)"- k = nil k~) r(k + 1 )r(n - k + 1) , where r(x) = .C (, - I e - / dt is the Gamma function. If x is an integer, then rex + 1) = x!. In Marshall and Olkin (1979), it is shown that (XI, X2) t-7 (x1'x'?) /(r(xl + 1 )r(X2 + I» is Schur convex. This implies that under the constraint XI + X2 = n, it is minimized by setting XI = X2 = nl2. Therefore, each term in the summation (27) can be bounded from below by 11/2 11/2 n n e(n - k)" - k 2 2 :> r(k+ I)r(n-k+ I) - r(~+ I)r(~+ 1) to obtain 11/2 n/2 n n (n + I)r(n + I) 2'T 2 (~ + I) The identity (see Rudin ( 1976)) (28) 2" (n + I) (n ) r(n + I) = r;;. r - 2- r "2 + 1 , can now be applied to obtain --- ## Page 251 The Cost of Achieving the Best Portfolio in Hindsight 225 974 E. ORDENTLICH AND T. M. COVER (29) The log convexity of nX) (see Rudin 1976) now implies that (30) = r(~ +!) mIl 2 2 V~' where we have used the identity n x + 1) = xn x). Combining (29) and (30) we obtain = V2(n 7r+ 1) thereby proving that 2 Vil :S )n + 1 for all n . D This bound can be generalized to m > 2 with the help of r(!!.2..!.) III m rc n + 1) = mil IT ( i ) , ~J r- m an extension of (28) to general m. 3.3. Asymptotics of Vil • The following lemma characterizes the asymptotic behavior of Vil for m stocks. LEMMA 3. For all m , Vil satisfies (31 ) v ~ r( ;) (~) (III- J)12 Il I v 7r n in the sense that --- ## Page 252 226 E. Ordentlich and T M Cover COST OF ACHIEVING THE BEST PORTFOLIO IN HINDSIGHT 975 The quantity VII arises in a variety of settings including the max-min data compression problem (see Shtarkov 1987), the distribution of the longest common subsequence be- tween two random sequences (Karlin 1996), and bounds on the probability of undetected errors by linear codes (see Kl0ve 1995, Massey 1978, and Szpankowski 1995). Lemma 3 is proved in Shtarkov, Tjalkens, and Willems (1995) and an asymptotic expansion of VII to arbitrary order is given in Szpankowski (1995, 1996). A direct proof of Lemma 3 based on a Riemann sum approximation is given in Ordentlich ( 1996). In addition, Shtarkov ( 1987) obtains the bound V ~ ['" (m) ~ (!!.) (i-11/2 ]-1 II i~ i rU!2) 2 implying one half of the asymptotic behavior in Equation (31 ). 4. The hindsight allocation option. The results of the previous section motivate the analysis of the hindsight allocation option, a derivative security which pays S,T(X/), the result of investing one dollar according to the best constant rebalanced portfolio computed in hindsight for the observed market behavior XII . Let - I HII =-. VII Certainly the price of the hindsight allocation option should be no higher than fill . This follows because fill dollars invested in the nonanticipating s\Iategy described in the proof of Theorem 1 is guaranteed to result in wealth at time n no less than S;r(xll ) for all market sequences x". If the price of the hindsight allocation option were more than fill, selling the option and investing only fill of the proceeds in the above strategy would be an arbitrage. Note that this argument assumes the existence of a riskless asset for investing the surplus. Therefore, fill is an upper bound on the price of the hindsight allocation option valid for any market model (with a risk free asset). Furthermore, while the return of the best constant rebalanced portfolio is expected to grow exponentially with n, the upper bound on the price of the hindsight allocation option fill behaves like r,;. This polynomial factor is exponentially negligible relative to S~' . Is fill a reasonable price for the hindsight allocation option? Probably not; the price should be lower. Pricing the option at fill may be appropriate if no assumptions about market behavior can be made. This is the case in §3, where no restrictions are placed on nature's choice for the market behavior. Returns on assets can be arbitrarily high or low, even zero. Actual markets, however, are typically less volatile. We gain more insight into this issue by using established derivative security pricing theory to determine the no- arbitrage price of the hindsight allocation option for two much studied models of market behavior, the binomial lattice and continuous time geometric Brownian motion models. 4.1. Binomial lattice price. We consider a risky stock and a riskless bond. Accord- ingly, the price-relatives Xi are assumed to take on one of two values --- ## Page 253 The Cost of Achieving the Best Portfolio in Hindsight 227 976 E. ORDENTLICH AND T. M. COVER Xi E {(l + u, 1 + r)', (1 + d, 1 + r)'} with r :2: 0, U > r > d. The first component of Xi reflects the change in the price of the stock as measured by the ratio of closing to opening price. The second component indicates that the riskless bond compounds at an interest rate of r for each investment period. The parameters of the model are thus u, d, and r. If the stock price changes by a factor of 1 + u it has gone " (u )p"; if it changes by a factor of 1 + d it has gone" (d )own." We will find that the no-arbitrage price H" of the hindsight allocation option for this model is closely related to fi", the upper bound obtained in the previous sections. It will be apparent that for certain choices of d, u, and r, the upper bound fill is essentially attained. For a sequence of n price-relatives x" = XI, .. . , X"' the wealth acquired by a constant rebalanced portfolio b = (b, 1 ~ b)' can be written as where k is the number of vectors Xi for which X i i = 1 + u . Since log Sn(b) is concave in b, the best constant rebalanced portfolio b * = (b *, 1 ~ b * )' is easily determined using calculus. For 0 < k < n, define b * as the solution to It is given by d log S,,(b) = O. db b* = (\ + r) (_k_ ~ n ~ k) . n r ~ d u ~ r For k = 0, set b * = 0, and for k = n, set b* = 1. Then b* is given by b * = max (0, min ( 1, b * ) ). We then obtain the wealth achieved by the best constant rebalanced portfolio as s,t(X") = [1 + r + b*(u ~ r)]k[l + r + b*(d ~ r)],,-k { (l + r)" if b* = 0, (1 + U)k(l + d),,- k if b* = \, [1 + r + h*(u ~ r)] k[l + r + h*(d ~ r)],,-k if 0 < b* < 1. If 0 < b* < 1, the wealth achieved can be written more explicitly as S,teXn)=(l+r + (l+r)[ k ~ n ~ k ](u~r»)k ner ~ d) neu ~ r) . 1 + r + e 1 + r) ~ (d ~ r) ( [ k n k ] )"-k n(r~d) neu~r) which simplifies to --- ## Page 254 228 E. Ordentlich and T M Cover COST OF ACHIEVING TH E BEST PORTFOLIO IN HINDSIGHT 977 ( k)k(n - k)"-k(U - d)k(U - d)"-k S;;'(x") = (1 + r) " - -- -- -- n 11 r-d u - r It is well known that for this model the no-arbitrage price P" of any derivative security with payoff S" at time 11 is given by 1 EQ S P" = (I + r)" ( ,,) , where the expectation is taken with respect to Q, the so called equivalent martingale measure on asset price changes. The unique equivalent martingale measure for this market is a Bernoulli distribution on the sequence of "up" and " down" moves of the asset price with the probability of an " up" equal to p" = (r - d)/ (u - d) and the " down" probability equal to Pd = I - (r - d)/(u - d) = (u - r)/(u - d). We note that for the case of 0 < h* < 1 ( k)k( k)"-k S,~ (x") = (I + r)" -;; 11 ~ p,-/ p ;;(I1- k) ~ S:':(k). Therefore, H = EQ(S:':) " (1 + r)" - S* k k ,,-k + I + r " k ,,-k I (n) I (n) -(I+r)"k()<~'< 1 ,,() k PuP" (I+r)" u~~ ()( ) k PuPd which simplifies to H" = I (11) ('5.) k (~)"-k + I (1I)p ~p;;_ k U)< /'''< I k n 11 k:/'*~ O k ( 11) ( (I + u ))k ( (I + d) ),,-k + !I~~ I k Pu ~ Pd ~ The range of k such that 0 < h* < I is k (U+I) Pu< - < p" -- . 11 r+1 Thus --- ## Page 255 The Cost of Achieving the Best Portfolio in Hindsight 229 978 E. ORDENTLICH AND T. M. COVE R H = L - - - + L pkp"-k (n) (k)k( n - k)"-k (n) 11 Pr/ < kln (n - 1 )/n , in which case the value of the hindsight allocation option is at least H" - 2. In summary, the no-arbitrage price H" of the hindsight allocation is given by H = L - -- + L pk p,,-k (n) (k)k( n - k)"-k (n) /I l'u< klll < p,lu+ l )/(r+i) k n 11 klfl'S PII k Ii d (n) ( (1 + u ) )k ( (1 + d) )"-k + kl" "'I'''(I~I )/(r+ l) k p" 1+r p" 1+r ' where the first summation comprises the bulk of the price for reasonable parameter values. The terms appearing in this sum are identical to those in the expression (3) for V" = 1/ H". The number of such terms appearing in the sum depends on the parameter values. A Reimann sum approximation argument shows that for fixed parameter values H" ~ ern, where the constant c depends only on the parameters. 4.2. Geometric Brownian motion price. In this section, we give the price of the hindsight allocation option for the classical continuous time Black-Scholes market model with one stock and one bond. The stock price X, follows a geometric Brownian motion and evolves according to the stochastic differential equation dX, = /1X,dt + aX,dB" where J1 and (J are constant, and B is a standard Brownian motion. Note that here X, denotes a price, not a price-relative. The bond price /3, obeys --- ## Page 256 230 E. Ordentlich and T M Cover COST OF ACHIEVING TH E BEST PORTFOLIO IN HINDSIGHT 979 d/3, = /3,rdt where r is constant and therefore /3, = e rt/3o· Let S,(b) be the wealth obtained by investing one dollar at t = 0 in the constant rebalanced portfolio b = (b , 1 - b)', where b is the proportion of wealth invested in the stock. Then S,(b) satisfies the stochastic differential equation (32) dSJb) = b dX, + (1 _ b) d/3, S,(b) X, /3, ' which can be solved to give (33) S,(b) = exp ( - b 2;2t + b( log ;~ + O";t) + (l - b)rt) . That this solves (32) can be verified directly using Ito's lemma (see Duffie 1996, Karatzas and Shreve 1991). Notice that, for fixed 0" 2 and r, the wealth S,(b) depends on the stock price path only through the final price X,. The best constant rebalanced portfolio in hindsight at time T is obtained by maximizing the exponent of (33) for t = T under the constraint that 0 :5 b :5 I. This results in (34) b*- (0 · ( ~ (LlT)IOg(XTIXo)-r)) T - max ,mm I , + 0 • 2 0"- The wealth achieved by the best constant rebalanced portfolio is then obtained by eval- uating (33) at b = b?f resulting in { erT S* - S (b*) - 1,,2T12)b?+rT T- T T - e XT Xo if b* =0 T if o :5 b ~ :5 if b~ 2 I. From the martingale approach to options pricing, the no-arbitrage price at t = 0 of the hindsight allocation option with duration T is given by (35) SrCb~) Ho:r = /3oEQ -/3-- T where Q is the equivalent martingale measure or the unique (in this case) probability measure under which XJ /3, is a martingale, and assuming that ST( bj.) is integrable under Q, which it is. It is well known (see Duffie 1996) that under the equivalent martingale measure Q the stock price X, obeys dX, = rX,dt + O"X,dB,. --- ## Page 257 The Cost of Achieving the Best Portfolio in Hindsight 231 980 E. ORDENTLlCH AND T. M. COVER This and Ito's lemma imply that under Q, the expression log(XTIXo) appearing in the exponent of (33) is normally distributed with mean (r - (112 )a 2 ) T and variance a 2T. Therefore, the random variable is standard normal. It can be rewritten as (36) so that, by equation (34), ~a2T(bi) = max(O, min(~a2T, Y». Equation (36) can be solved for XTIXo resulting in XT YWT +(r- u '!2)T -=e . Xo The expectation (35) is then easily evaluated as (37) The first expectation is clearly equal to !, since Y is standard normal. The middle expec- tation is Finally, the third expectation is 1 f ro - (1 /2)(V-WT )' d = - e . Y & ~ 1 2 --- ## Page 258 232 E. Ordentlich and T M Cover COST OF ACHIEVING THE BEST PORTFOLIO IN HINDSIGHT 9S1 Thus (37) reduces to a surprisingly simple form. The no-arbitrage price HOJ of the hind- sight allocation option is The price is affinely increasing in the volatility (J and increases like the square root of the duration T. The dependence on duration matches the r;; growth of the discrete-time upper bound fi" and the binomial lattice price H". If the hindsight allocation option payoff is redefined to be S J (x") - e rT (the excess return of the best constant rebalanced portfol io beyond the return of the bond) then the price is simply ) (J lTI (27r) . This can be thought of as a premium for volatility. 5. Conclusion. The worst sequence approach to the problem of achieving the best portfolio in hindsight leads to a favorable result: the max-min optimal portfolio strategy for m assets loses only « m - I )/2) (log n)1 n in the rate of return in the worst case. This yields an asymptotically negligible difference in growth rate as the number of investment periods n grows to infinity. In practice we would expect even better performance, since real markets are less volatile than the max-min market identified here. This intuition is partially validated by the hindsight allocation pricing analysis for the binomial and geo- metric Wiener market models which indicates that the cost of achieving the best portfolio in hindsight depends monotonically on market volatility. Acknowledgment. This work was supported by NSF grant NCR-9205663, JSEP con- tract DAAH04-94-G-005S, and ARPA contract JFBI-94-2IS-2. Portions of this paper were presented at CIFER 96, COLT 96, IMS 96. References Blackwell, D. ( 1956a). Controlled random walks. In Proceedings td Il1fernational Congress '!l Mathematics, vol. III , pages 336- 338, Amsterdam, North Holland. --- ( 1956b). An analog of the minimax theorem for vector payoffs. Pacific J . Mathe. 6. Cover, T. M. (1991). Universal portfolios. Math . Finance 1( I) 1- 29. - - - , D. Gluss ( 1986). Empirical Bayes stock market portfolios. Adv. Appl. Math. 7 170- 181. ---, E. Ordentlich ( 1996). Universal portfolios with side information. IEEE Trans. Itl/O. TheOlY 42( 2). 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Inequalities: Theory of Majorization and Its Applications, volume 143 of Mathematics in Science and Engineering . Academic Press, London. Massey, J. (1978). Coding techniques for digital networks. In Proceedings Intenwtional Conference on Infor- mation Theory Systems , Berlin. Germany. Merhav, N .. M. Feder ( 1993). Universal schemes for sequential decision from individual data sequences. IEEE Trans. Inli) . Theory 39(4) 1280- 1292. Ordentlich, E. ( 1996). Universal investment and universal data compression. Ph.D. thesis, Stanford University, Stanford. California. --- ## Page 259 The Cost of Achieving the Best Portfolio in Hindsight 233 982 E. ORDENTLICH AND T. M. COVER ---, T. M. Cover ( 1996). On-line portfolio selection. In Proceedings of Ninth Conference on Computational Learning Theory , Desenzano del Garda, Italy. Rudin, W. ( 1976). Principles of Mathematical Analysis. McGraw-Hill, third edilion. Shtarkov, Yu. M. (1987). Universal sequential coding of single messages. Problems of Information Transmission 23(3),3-17. - --, T. Tjalkens, F. M. Willems ( 1995) . Multi-alphabet universal coding of memoryless sources. Problems of Information Transmission 31 114- 127. Szpankowski, W. (1995). On asymptotics of certain sums arising in coding theory. IEEE Trans. Info. Theory 41(6). --- (1996). Some new sums arising in coding theory. Preprint. E. Ordentlich: Hewlett-Packard Laboratories, 1501 Page Mill Road 3U-4, Palo Alto, California 94304-1126 T. M. Cover: Departments of Statistics and Electrical Engineering, Stanford University, Stanford, California 94305 --- ## Page 260 This page is intentionally left blank --- ## Page 261 This page is intentionally left blank --- ## Page 262 235 17 Adv. Appl. Prob.13, 415-428 (1981) Printed in N. Ireland 0001-8678/81/020415-14$01.65 © Applied Probability Trust 1981 OPTIMAL STRATEGIES FOR REPEATED GAMES MARK FINKELSTEIN* AND ROBERT WHITLEY,· University of California, Iroine We extend the optimal strategy results of Kelly and Breiman and extend the class of random variables to which they apply from discrete to arbitrary random variables with expectations. Let Fn be the fortune obtained at the nth time period by using any given strategy and let F! be the fortune obtained by using the Kelly-Breiman strategy. We show ('Theorem 1(i» that Fn/F! is a supermartingale with E(Fn/f!):a 1 and, consequently, E(lim FJF!):a 1. This establishes one sense in which the Kelly-Breiman strategy is optimal. How- ever, this criterion for 'optimality' is blunted by our result ('Theorem 1(ii» that E(Fn/F!) = 1 for many strategies differing from the Kelly-Breiman strategy. This ambiguity is resolved, to some extent, by our result (Theorem 2) that F!/Fn is a submartingale with E(F!/Fn)S::1 and E(limF!lFn)ii:I ; and E(F!/Fn) = 1 if and only if at each time period i, 1;'ii i ~ n, the strategies leading to Fn and F! are 'the same'. KEU.Y CRITERION; OPTIMAL STRATEGY; FAVORABLE GAME; OPTIMAL GAMBLING SYSTEM; PORTFOUO SELECI'rON; CAPITAL GROWTH MODEL 1. Introduction Suppose a gambler is given the opportunity to bet a fixed fraction y of his (infinitely divisible) capital on successive flips of a biased coin: on each flip, with probability p >! he wins an amount equal to his bet and with probability q = 1- P he loses his bet. What is a good choice for y and why is it good? This question is subtle because the obvious answer has an obvious flaw. The obvious answer is for the gambler to choose y = 1 to maximize the expected value of his fortune. The obvious flaw is that he is then broke in n or fewer trials with probability 1- p", which tends to 1 as n tends to 1Xl. A germinal answer was given by Kelly [10]: a gambler should choose y = p - q so as to maximize the expected value of the log of his fortune. He shows that a gambler who chooses y = p - q will 'with probability 1 eventually get ahead and stay ahead of one using any other value of y' ([10], p. 920). In an important paper Breiman [4] generalizes and considers strategies other Received 23 April 1980; revision received 12 August 1980. • Postal address: Department of Mathematics, University of California, Irvine, CA 92717, U.S.A. 415 --- ## Page 263 236 M Finkelstein and R. Whitley 416 MARK FINKELSTEIN AND ROBERT WHITLEY than fixed-fraction strategies and generalizes the random variable as follows: Let X be a random variable taking values in {I, 2, ... ,s} = I, ~ be a class {AI> A 2, ••• , A,.} of subsets of I whose union is I, and 0b 02, ••• , 0, be positive numbers (odds). If for one round of betting a gambler bets fractional amounts f3I> f32, ... ,f3, of his capital on the events {X E AI}' ... , {X E A,.}, then when X = i he gets a payoff of L f3jOj summed over those j with i in A j • In this setting Breiman discusses several 'optimal' properties of the fixed-fraction strategy which chooses f31> f32, . . . ,f3, so as to maximize the expected value of the log of the fortune and then bets these fractions on each trial, leading to the fortune F: at the conclusion of the nth trial. He shows that if Fn is a fortune resulting from the use of any strategy, then lim F.JF~ almost surely exists and E(lim F.JF~)~ 1. In what follows we shall be concerned solely with magnitude results, like this asymptotic magnitude result of Breiman's, but the reader should be aware that under additional hypotheses Breiman also shows that T(x), the time required to have a fortune exceeding x, has an expectation which is asymptotically minimized by the above fixed-fraction strategy. The problem of how to apportion capital between various random variables is exactly the problem of portfolio selection, and so it is correct to suppose that these results on optimal allocation of capital are of considerable interest to economists, as Kelly recognized ([10], p. 926). He also prophetically realized that economists, familar with logarithmic utility, could easily misunderstand his result and think, incorrectly, that the choice of maximizing the expected value of the log of the fortune depended upon using logarithmic utility for money. For discussion see [15], p. 216 and [17]. An interesting concise discussion of the 'capital growth model of Kelly [10], Breiman [4], and Latane [11]' from an economic point of view can be found in [3]. A brief discussion of Kelly's proof will motivate his criterion and allow us to make an important conceptual distinction between his results and Breiman's. Suppose a gambler bets the fixed fraction 'Y of his capital at each toss of the p-coin. Kelly considers the exponential growth rate G = lim log [(F.JFo) lin]. If our gambler has W wins and L losses in the first n trials, Fn = (1 + 'Y)w(1- 'Y)LFO' so G = lim (: log (1 +'Y)+;IOg (1- 'Y») = p log (1 + 'Y)+q log (1- 'Y), by the law of large numbers. The growth rate G is maximized by 'Y = P - q, and if he uses another 'Y his G will be less and therefore eventually so will his fortune. A complication enters when we consider, as Kelly did not, strategies which are not fixed-fraction strategies. In that case we can have different --- ## Page 264 Optimal Strategies for Repeated Games 237 Optimal strategies for repeated games 417 strategies with the same G, e.g., use 'Y = 1 for the first 1000 trials and then use 'Y = P - q. This complication is intrinsic in the use of G and it has consequences which are quite serious for any application. For example, two strategies which at trial n give fortunes, respectively, of 1 and exp (.In), both have G = I! It is obviously unsatisfactory to regard these two strategies with the same G as 'the same', but it is done because using G makes it easy to extend the Kelly results to more general situations which involve more general random variables; using G the argument is a simple one employing either the law of large numbers or techniques which 'rely heavily on those used to generalize the law of large numbers' ([15], p. 218). Breiman understood the problems created by using G and so he considered F,JF!, not (F,JF!)l'''. This is mathematically more difficult, but the results are more useful. 2. Definitions and lemmas We shall consider situations with the property that at each time period a gambler can lose no more than the amount he invests, e.g., buying stock or betting on Las Vegas table games. Since there is a real limit to a gambler's liability, based on his total fortune, a broad interpretation of the phrase 'the amount he invests' will allow the inclusion of such situations as selling stock short or entering commodity futures contracts. We suppose that there are a finite number of situations 1,2, . .. ,N on which a gambler can bet various fractions of his (infinitely divisible) capital. The random variables Xl> X 2 , ••• , XN represent, respectively, the outcome of a unit bet on situations 1,2, ... , N. Because the loss can be no more than the investment, Xk ~ -1 for 1 ~ k ~ N. (Breiman considers the amount returned to the gambler after he has given up his bet in order to play, a real example of this sequence of events being betting on the horses. Here the amount the gambler gets back is ~O, which corresponds to the amount he wins being ~ -1). We further suppose, with no loss of applicability, that in all of what follows each X k has an expectation, i.e., that E(IXk D is finite. These will be the only restrictions on the random variables, and so we are considering a substantially larger class than those discrete random variables Breiman considers. We also suppose that the gambler can repeatedly reinvest and change the proportion of the capital bet on the situations. The outcome at time j corres- ponds to the random variables x:y),~), ... , XW. For each k, 1 ~ k ~ N, the results of repeated betting of one unit on the kth situation is a sequence X~/), Xl2), ... , x~m) , ... of independent random variables, each having the same distribution as Xk • In contrast to this independence, it is quite important for applications that Xl' X 2 , ••• , XN be allowed to be dependent. --- ## Page 265 238 M Finkelstein and R. Whitley 418 MARK FINKElSTEIN AND ROBERT WHITI..EY A strategy for the game will be a sequence 'Y(1\ • •• , 'Y(m>, ... of vectors, 'Y(m) = (y~m), 'Y~m), ... ,'Y~» giving the fractional amount 'Y~m) of the capital which at the mth bet is bet on the kth situation. Thus 'Y~m) ~ 0, 1 ~ k ~ N, and L~-l 'Y~m) ~ 1. We allow the possibility that 'Y(m) can depend, as a Borel- measurable function, on the past outcomes XV), ... , xW, X\2), ... ,X' ... ,x\m-1), ... ,X 'Y2, ... , 'YN) which bets the same amount 'Yk on situation k for all m. The result of using 'Y = 'Y(j), 1 ~ j ~ m, for m bets is Fm = Fo &-1 (1 + 'Y . X{J». We shall be particularly interested in 'the' fixed-fraction strategy 'Y* = ('Yt, 'Y~, ... , 'Y~) which maxuruzes E(log (Fm». In Lemma 3 we shall show that 'Y* exists, and Lemma 1 shows in what sense it is unique. The strategy 'Y* maximizes E(log (Fm» if and only if it maximizes the function (3) ( 'Y) = ( 'YI> ... , 'YN) = E(Iog (1 + L 'YkXk» = E(Iog (1 +"1 . X» over the domain (4) D = {('Yl' "', 'YN): 'Yk~O, 1 ~k ~N, L 'Yk ~ 1}. Lemma 1. (i) The function of (3) is concave. (ii) If (aa+(1-a){3)=a(a)+(I-a)({3) for 0(a) and ({3) finite, a . X = {3 . X almost surely. In particular, if a = (at> a2, ... ,aN) and {3 = ({31' {32, ... , {3N) both maximize over its domain D, then L akXk = L {3kXk a.s. (iii) At 'Y = ('YI>' .. ,'YN) in D with L 'Yk < 1, the partial derivative d/d'Yi exists and equals E(K;/(1 + 'Y . X», 1 ~ i ~N. Proof. (i) The function f(x)=log(1+x) is strictly concave on (-1,00), and so for X=(Xl"" ,XN) a value of X,a and {3 in D, and 0(aa+(1-a)/3)= Jf(aa ' X+(1-a)/3' X) dP ~ J (af(a . X) +(1- a)f(/3 . X) dP = a(a)+(1- a)(/3). (ii) Since is concave the set where it attains its max is convex. If (a) = <1>(/3) is a max, then for 0< a < 1, <1>(00 + (1- a)/3) = a(a)+(1-a)(/3), or (6) J (f(a· aX+(1-a)/3 ' X)-af(a ' X)-(1-a)f(/3' X» dP=O. From (5) and (6), (7) f(aa · X+(1-a)/3' X}=af(a' X)+(1-a)f(/3' X) a.s. Because f is strictly concave, aa • X +(1- a)/3 . X = a . X = /3 • X at all values of X where f is finite. Both sides of (7) are -00 only at values of X where a . X = /3 • X = -1. In any case, a . X = /3 • X almost surely. (iii) Choose E > 0 so that L 'Yk < 1- E. Then 1111 + 'Y . XI ;:a 11 E. The difference quotient for iJ/iJ'Yi is J log ( 1 + L 'YkXk + ('Yi + A 'YI)X; ) -log (1 + L 'YkXk) k .. 1 dP A~ . (8) Let x = (Xl' ... ,XN) be a value of X and consider the function g('Yi) = log (1 + Lk .. i 'YkXk + 'YiXj) . By the mean value theorem, I g(yj + A'YJ - g(yj) 1= IXj I A 'Yi 11 + L 'YkXn + ~Xj I ' o < ~i < A'Yi' So the integrand in (8) is dominated by the L 1 function IX; liE for A'Yi small. Result (iii) follows from the Lebesgue dominated convergence theorem. Here is a simple example which conceptually illustrates a practical use of the Kelly-Breiman criterion: maximize E(log Fm). Example 1. Define two random variables Xl and X2 by flipping a fair coin: if heads, then Xl = 100 and X 2 = -10, if tails, then Xl = -1 and X 2 = 1. The payoff from Xl is far superior to the payoff from X 2, but because Xl and X 2 are (completely) correlated and have payoffs with opposite signs, the criterion --- ## Page 267 240 M. Finkelstein and R. Whitley 420 MARK FINKELSTEIN AND ROBERT WHITLEY will mix both in order to smooth out the rate of capital growth. A simple calculation shows that <1>( 'Y) = E(log (1 + 'YIXl + 'Y2X2» is maximized over D on the face 'Yl + 'Y2 = 1 at 'Yf == 0.54 and 'Y~ == 0.46. (Lemma 3 will discuss the basic problem created by maxima occurring at non-interior points of D.) The extent to which the criterion will sacrifice expectation is surprising: E(O.54X1 + 0.46X2) ==24.7 vs. E(X1) =49.5. A fascinating example of the use of this criterion, in which the underlying idea is the same as this example, is in hedging a warrant against its stock as described in [15], pp. 220-222. Example 2. For A> 0, let X have density Ae-"(x+1) for x ~ -1, and 0 for x < -1; an exponential shifted to allow losses. We shall show that there is a unique 'Y*, 0 ~ 'Y* < 1, which maximizes {y) = E(log (1 + yX», 0 ~ y ~ 1; y* = o iff A ~ 1. By Lemma 1, Further, '(y) = - rex> _x_ Ae-"(x+1) dx. 1-1 1+yx <1>"( y) = rex> -x2 Ae-Mx+1) dx < 0 1-1 (1 + yX)2 , so is strictly concave on [0, 1). Since <1>'(0) = E(X) = A -} -1, the strict concavity of shows that y* = 0 iff <1>'(0) ~ O. It remains to show that for 0 < A < 1 there is a unique maximizing point y* with O'(y) = (Aly2)ea[g(a)-E1(a)], where a = A«1/y) -1), g(a) = e-a/(a + A), and the exponential integral E1(a) = J:e-'/tdt. Since El(O) =00 and g(0)=1/A, '{Y) is strictly convex, there is thus a unique point y*,0<'Y*<1, at which is maximized. For future reference we note that it is not obvious that is continuous at 1; part of the computation involves an integration by parts and a change of variable to obtain <1>( y) = log (1- 'Y) + ea E 1( a), a as above. The expansion El(a)=e-a(-loga-yo+o(a» ([1], p. 229), where yo=0.577··· is Euler's constant, shows that li.my-+l <1>( 'Y) = -log (A) - 'Yo. In Example 2 'Y* = 0 iff E(X) ~ 0, i.e., a gambler bets on X only if it has positive expectation. This is a special case of a more general result. Breiman [4], p. 65, calls a game favorable if there is a strategy such that the associated fortune Fn tends almost surely to 00 with n, and he shows that this condition is equivalent to ('Y*) being positive ([4], Proposition 3, p. 68). Lemma 2 establishes the equivalence with the intuitive Condition (iv). --- ## Page 268 Optimal Strategies f or Repeated Games Optimal strategies for repeated games Lemma 2. The following are equivalent: (i) There is a strategy with the associated fortune (ii) F! -+ 00 a.s. (iii) cf>( 1'*) > O. Fn -+ 00 a.s. as n-+ oo• (iv) E(X;) > 0 for at least one i, 1 ~ i ~ N. Proof· 241 421 (i) implies (ii). In Theorem 1 we show that F,JF~ tends almost surely to a finite limit. (ii) implies (iii). If cf>( 1'*) = 0, then F! = Fo for all n. (iii) implies (iv). If 1'* . X = 0, then cf>( 1'*) = O. So 'Y~ > 0 for some i with X; not identically O. Fix all variables in

('Y!, 'Y!, ' .. , 'Y~-l> 1';, 'Y~+l"'" 'Y~), a concave function which has a positive max at 'Yf. If E(Xi ) ~ 0, then '11"(0) ~ 0, and 'I' has a local max at 0 and so has a global max there because it is concave. Hence E(X;) > O. (iv) implies (i). Define 'I' by setting all the variables but 1'; equal to 0 in cf>: '1'(0, 0, ' ",0, 'Yi, 0,"',0). As E(X;»O, '1"(0»0 and so '1'(1';»0 for 'Yi close to O. Thus cf>( 1'*) > O. Since log (F!/Fo) = g log (1 + 1'* . X(j) and E((log F!/Fo)/n) = cf>( 1'*) > 0, the strong law of large numbers shows that F!/Fo -+ 00 almost surely. Example 3. Let Xl and X2 be the coordinates of a point distributed uniformly on [-1, b]x[ -1, b]. Then cf>('Yh 1'2) = E(log (1 +'YlXl +'Y2X2» has a maximum at 1'* = O,!) if b ~ log (16)-1. If (1'1' 1'2) is a point where cf> attains its max, then so is (1'2' 1'1) by symmetry. Since cf> is concave, G>(cf>('Yh'Y2)+cf>('Y2,'Yl»~cf>G('Yl+'Y2),!('Yl+'Y2»' and we may look for the maximum of cf> along the diagonal (1', 1'), 0:;; l' :;;!. Then d 1 rb rb Xl +X2 d'Y cf>( 1',1') = (b + 1)2 tl tl (1 + 'Y(X I + X2» dxl dx2· The second derivative is <0, and cf>' is decreasing. A direct but tedious integration and calculation shows that . d ((log 16)) 11m - cf>('Y, 1') =2 1- --- . ..,-!- d'Y (b + 1) Hence cf>('Y, 1'), which can be shown to be continuous on [0, n increases up to its max at G,!) as long as b $;log (16)-1 = 1.77 · . . . It is interesting to compare this situation with betting on only one variable, say Xl' Then cf>l( 1'1) = E(log (1 + 'YIXl» is continuous on [0, 1]. Continuity on [0, 1) follows from Lemma 1 or inspection. Because of the singularity at --- ## Page 269 242 M Finkelstein and R. Whitley 422 MARK. FINKELSTEIN AND ROBERT WHm..EY Yt . Xl = -1 in the integrand, continuity at 1 is not by inspection; one must compute cf>l (1) = log (1 + b) - 1 and show it is equal to the limit. (The situation in two variables, which we dismissed with a word, is not easier.) The function cf>1 is differentiable on [0, 1) by inspection or Lemma 1. It is only differentiable at 1 in an extended sense; we can show that <1>'1(1) = -00 and that lillly_l cf>'kf) = -00. The function <1>1 has a unique maximum at a point yf, 0< yf< 1. From Lemma 2, yf = 0 iff E(X) = (b -1)/2 ~ O. The existence of yt < 1 follows from 't (0) = (b -1)/2 and cf>'l (1) = -00, the uniqueness from strict concavity. The surprising fact is that for one variable Xl a gambler does not bet all his fortune no matter what b is, but he does bet all his fortune on two independent copies of Xl for b large enough. The reader who has carried out the calculations of Examples 2 and 3 knows that, because of the possible singularity on L Yk = 1, it is not clear that attains a maximum, and the differentiability of on the boundary L Yk = 1 is even less clear. Think of a continuous strictly increasing concave function 1 on [0,1] and redefine it at 1 so that its value there is less than 1(0). If this redefined function were E(log (1 + yX», then X would be a most interesting game with no Kelly-Breiman optimal strategy: with unit fortune, if a gambler bet an amount less than 1 he could always do better by betting slightly more, but betting all would be worst. One result of Lemma 3 is that there is an optimal y* so no game can have the property discussed in the paragraph above. Another result of Lemma 3 is that is continuous, when finite. This is important because when we compute y*, a numerical calculation which will generally give y* to a certain number of decimals, we want to know that using this approximation to the exact y* will give close to optimal perfonnance. The other result is a substitute for differentiation when y* has I y~ = 1, which allows us to derive the basic ineqUalities (9) and (10). Note that if all the random variables Xl> X 2 , ••• , XN are discrete, with a finite number of values, as they are in [4], then we can differentiate cf> at y*: for then if y •. X equals -1 it does so with positive probability and <1>( y*) = -00 < <1>(0) = 0, contrary to <1>( y*) a maximum; thus is actually defined on a neighborhood of y. (which may extend outside D) and is differentiable as in Lemma 1. The problems which Lemma 3 resolves are those which arise from more general random variables. Lemma 3. (i) The function is continuous where finite, and attains a maximum at a point y* in D. --- ## Page 270 Optimal Strategies for Repeated Games 243 Optimal Slrategies for repeated games 423 (ij) Let K = {k : yt 1= O}. For k in K, E(X,,/(1 + 1'* . X) is finite and non- negative. If k belongs to K, then for all m (9) E(XJ(1 + 1'* . X) ~E(x..J(1 + 1'* . X) . If both k and m belong to K, (10) E(Xk/(1 + 1'* . X) = E(X"J(1 + 1'* . X). Proof. Suppose that y(n) converges to 1'(0) with 4>(1'(0» finite. Denote the positive and negative parts of log by log+ and log- : log+(x) = log x if x ~ 1, log+ (x) = 0 if x ~ 1 and log-ex) = -log (x) if x;i 1, log-ex) = 0 if x ~ 1. Since log+ (1 + l' . X) ~ 11' . XI ;i max IXk I, 4> is infinite only if it is -00. By the Lebes- gue dominated convergence theorem lim J log+ (1 + y(n) . X) dP = J log+ (1 + 1'(0) . X) dP, by Fatou's lemma lim inf J log- (1 + y(n) . X) dP ~ J log- (1 + 1'(0) . X) dP, and putting these two facts together, lim sup 4>( y(n» ~ 4>(1'(0» . Thus 4> is upper semicontinuous and therefore attains its maximum on the compact set {y in D: 4>(y)~O}. For 0< a < 1, 4>(ay+(I- a)y(O» ~ aq,{y)+(1- a)4>(y(O», and so lim infa-o 4>(ay + (1- a)y(O» ~ 4>( 1'(0» if 4>(1') is finite, i.e., 4> is continuous along lines directed towards 1'(0) from points where 4> is finite. Suppose that yO) and 1'(2) are in D with 4>(1'(2» finite, and let a E [0, 1). Now 1 +ay(1)· X +(1-a)y(2). X ~ 1 +a L yLl)(-l)+(1- a)y(2). X ~(1- a)+ (1- a)y(2) . X = (1- a)(1 + 1'(2) . X). Since log- (1 + z) is a decreasing function of z, log- (1 + ay(1) · X + (1- a)y(2). X)~log-(1-a)(1 +1'(2). X) which equals -log (1-a)- log (1 + 1'(2) . X) when (1- a)(1 + 1'(2) . X) ~ 1, and equals 0 otherwise. Hence JIog- (1 + ay(l)· X +(1- a)y(2). X) dP~-log (1-a)+J log- (1 + 1'(2). X) dP < 00, and thus 4>(ay(1) + (1- a )y(2» is finite for a 1= 1. Let a point l' be a given at which 4> is finite. For 1;i k ~ N, let elk) be the vector in D whose kth coordinate is 1, e~k) = a/let e(O) = 0, and set y(k) = ae(k)+(1- a)y, a in (0,1), for O~ k ~N. Note that D is the convex hull co(e(O), e(1), . .. , e(N» . As we have seen above, cf>( y(k» is finite and so cf> is continuous on the line joining y(k) to y. Given e > 0, by choosing a small enough we have 14>( y(k» - cf>{y)\ ~ e for 0 ~ k ~ N. For any vector v in the convex hull U = co (1'(0), 1'(1), .. . , y(N», v = L a,. y(k), ale ~ 0, L a,. = 1, we have cf>(V)~L ak4>(y(k»~cf>(y)-e. The convex hull U is easily seen to have an interior (relative to D). Since 4> is upper semicontinuous, V = {a : cf>(a) < cf>(y) + e} is open. Therefore for v in the neighborhood VOn V, Icf>( v) - cf>( y)\;i e and cf> is continuous at y. --- ## Page 271 244 M Finkelstein and R. Whitley 424 MARK FINKEl.STEIN AND ROBERT WHITLEY For y in D with LYi<1, and 0~a~1, define qr(a) = 4>(ay+(1-a)y*). We have seen that qr is continuous on [0,1]. Given 0 < e < 1, for a in [e, 1], 1 +(ay+(1-a)y*) . X~E(1-L Yi»O. As in Lemma 1, \(Y-Y*)'X/(1+(ay+(1-a)y*)'x)\ is bounded by the Ll function \(y- y*) . X/E(1-L Yi)\ and we may use the Lebesgue dominated convergence theorem to justify differentiating under the integral to obtain qr'(a) = f (y-y*). X dP 1 +(ay +(1- a)y*) . X . Since e was arbitrary, this holds on (0, 1J. If L y: < 1, then Lemma 1 establishes the fact that the expectations in (10) are O. As in Lemma 2, if m is not in K, then fixing all the variables in 4> but Ym and considering that concave function with a max at 0 shows that a4>/ aYm (Y*) ~ o and (9) follows. Now suppose that L Y: = 1 and let x = (Xl> • •• , XN) be some value of the random variable X in which not all the x" = -1 for k in K. Note that the event x" = -1 for all k in K has probability 0 since the integrand in the integral defining 4> is -00 there and 4>( y*) is finite. For 0 ~ a < 1 and Y in DO, define f(a) = log (1 + (ay + (1- a)y*) . x). The function f is finite because x was so chosen, and is differentiable with (y - y*) . x f'(a)=-~~~-- 1 +(ay +(1-a)y*)· x . Because f"~0, f'(a) increases to (y-y*) . x/(1 + y*. x) as a decreases to O. For a+O, (y-y*). X/(l+(ay+(l-a)y*)' X) in Ll we may apply the B. Levi theorem to obtain . f ( y - y*) . X lIm ¥(a) = 1 * X dP. aJ.O +y . By tl)e mean value theorem, (qr(a)-qr(O)/a = 'I"(b), 0< b < a. Then, because qr' is increasing, limaJ.o qr'(a) = limaJ.o «'I'(a) -'I'(O»/a). Since 4>( -y*) is a maximum the right-hand side is ~O and we obtain the basic (11) f(y-y*),X < 1 * X dP=O. +y . For 0 < e < 1 and k in K, the choice Yi = y1 for j =f k and Yk = y:(1- e) in (11) gives - S X,J(1 + y* . X) dP ~ 0, and S Xk/(1 + y* . X) dP is non-negative and therefore finite. For kinK and any m, and 0 < e < y:, the choice y/ = yr for j neither k nor --- ## Page 272 Optimal Strategies f or Repeated Games 245 Optimal strategies for repeated ga~s 425 m, Yn = y~ - e, Ym = Y! + e in (11) gives (9). Finally, (10) follows from (9) by symmetry. 3. Theorems If Fn is a gambler's fortune at the nth time period, obtained by using some strategy y(1), y(2), ... , y(n), ... , and F! is the fortune obtained by using the fixed fraction strategy y*, y*, ... , y*, . • . , then Breiman concludes ([4], Theorem 2, p. 72) that lim F,J F! almost surely exists and E(lim F,J F!) ~ 1. We extend these results in Theorem 1 below. Two comments are in order. First, the advantage in using y*, as indicated by the fact that E(lim F,JF!);:a 1, does not require passage to the limit. This was noted by Durham, for the case of two branching processes, in the proof of Theorem 1 of [6], p. 571. In fact, given that the limiting result is true and given a finite strategy y(1), ... , y(m), extend it by setting yO) = y* for j > m; then F,J F~ = F ml F! for n ~ m and E(FmIF!);:a 1. This should reassure the careful investor who won- ders whether a strategy good in the long run may not be inferior in any practical number of trials-disregard of this point leads to an overevaluation of games of the type which produces the St. Petersburg paradox. Second, by our analysis of q" we are able to show in Theorem 1 that the presence of the expectation in E(lim Fn/F~);:a 1 raises serious problems in any superficial attempt to use this as an indication of the superiority of y*. Theorem 1. Let Fn be the fortune obtained by using a strategy y = y(l), ... ,y(n) for n repeated investment periods, and let F! be the fortune obtained by using the fixed fraction strategy y*. Then (i) F,JF! is a supermartingale with B(F,JF!)~ 1. Consequently, lim F,JF! exists almost surely as a finite number and E(lim F,JF!) ~ 1. (ii) Suppose that y bets only on those X" with y: > 0, i.e., that yp) = 0 if I ¢ K, 1 ;:a I ;:a N and all j. If L y: = 1, then funher suppose that L y~) = 1 for all j. Then Fn/F! is a martingale with E(Fn/F!) = 1. Proof. Let m be given and let f€m be the sigma-algebra generated by X1t), l;:ak~N, l~j~m. Then fEm+l\ )= (l+ y(m+l).x(m+l).Fm \ ) E\F* f€m E 1 * . X I ) F! E 1 + y* . X f€m . Because y(m+l) is a strategy depending on the past values of the X~), it too is --- ## Page 273 246 M Finkelstein and R. Whitley 426 MARK FINKELSI'EIN AND ROBERT WHITLEY C€m-measurable. That, together with the independence of x(m+1) and the previous XO), shows this equals Fm E( 1 ) ~ (m+l) ( Xkm +1) ) F! 1 + y* . X + k'=l Yk E 1 + Y* . X . Since x(m+l) has the same distribution as X, this can be written (12) ~ [ E(l + y1* . X) + k~l y~m+l)E(l +~: . X)] = A, say. By Lemma 3, E(Xk/(1 + y* . X) equals a constant c for k in K and is ~c for all k; this constant c = 0 if L yt < 1. Hence A ~=~ [E(1+y1* . x)+ L yt· c]= ;;EG:~:: ~]= ;;. We have shown that F,JF! is a positive supermartingale, with E(F,JF!)~ E(Fn-1/F!-1) ~ ... ~ E(Fo/Fo) = 1, and so by the supermartingale convergence theorem it converges almost surely to a finite limit. Using Fatou's lemma, E(Iim F,JF~ ~ lim inf E(F,JF!) ~ 1. An examination of (12) shows that under the conditions of (ii), E(Fm+l/F!+l I C€m) = Fm/F,!, and (ii) follows. The requirement in Theorem l(ii) that if L y! = 1 then we must have L y~) = 1, for all j, in order to be sure to get a martingale, is made clear by a one-variable example, Let X=2. Then cP(y) = log (1+2y) is maximal at y*= 1. The gambler will do worse betting any amount less than 1, even though he still bets on the same random variable as y. does, the key observation being cf/(l) > O. In general, the 'partial derivatives' E(X,J(l + y* . X), k in K, may be positive if L yt = 1, whereas they are all 0 if L yt < 1. The surprising result of Theorem 1 is the broad conditions in (li) under which E(F,JF!) = 1. To see what the surprise is, we shall superficially interpret Theorem 1(i): since 'on the average', and 'for large n', F,JF!~ 1, the gambler 'does better' with F! than with Fn. But then Theorem l(ii) tells us that if the gambler simply bets on the same variables as y. does, but in any proportions at all, and if y* bets all so does he, then E(FnlF!) = 1. So with the same intuitive interpretation as above, 'on the average' the gambler does the same with Fn as with F!, so it really does not matter which strategy he uses! But we know that it does matter. For example, in a repeated biased-coin toss, if he plays a fixed fraction strategy betting an amount y f y. = p - q, then almost surely F,J F! -+ O. Yet we have E(F,JF!) = 1. In general it will not help to look at lim F,JF~ . For example, if on the first flip of the coin he bets all his fortune, and from then on he bets p - q, F,J F! > 1 with probability p. --- ## Page 274 Optimal Strategies for Repeated Games 247 Optimal strategies for repeated games 427 Theorem 2 will help our understanding of this situation by showing that F! is the only denominator with E(F,./F!) ~ 1 for all F,,; in fact E(F!/Fn) > 1 if Fn does not come from a strategy equivalent to using ,.* repeatedly. The suspi- cious reader will note that this characterization of the sense in which ,.* is optimal contains an expectation. Anyone attempting to state intuitively the result of Theorem 2(i) in the form 'F! is better than Fn because, on the average, F!/Fn!: 1', should also be willing to apply the same interpretation to Theorem 1(ii) and conclude that often, 'F,./F! = 1 on the average and so Fn and F! are often the same after all'. Theorem 2. (i) F!/Fn is a submartingale with E(F!/Fn) ~ 1. Urn F!/Fn almost surely exists as an extended real number and E(lim F!/Fn) iE;; 1. (ii) E(F!/Fn ) = 1 iff ,.0), ,.(2), •.• ,,.(n) are all equivalent to ,.*, i.e., iff y(j) . X'= y* . X almost surely for almost all values of X 1 unless ,.*. X = ,.m+1(w)· X almost surely, in which case C= 1. Thus (15) with equality holding jff,.*· X = ,.~::.,>+1) . X almost surely for almost all values w in the range of (X, ... , X< .... ». If E(F!+dFm +1) = 1, then 1 = E(E(F!+I/Fm+l I ~m» iE;; E(F!/Fm) iE;; ... iE;;E(Fo/Fo) = 1. By (15), equality holds in (15) and therefore ,.* . X = ,.(m+1)(w) . X for almost all w in the range of (X<1), ... ,X<"'». Continue for m, m -1, ... , I, to obtain (ii). --- ## Page 275 248 M Finkelstein and R. Whitley 428 MARK FINKEI..STEIN AND ROBERT WHITl...EY Let Y=limF~/F". By Theorem 1, P(Y=O)=O and E(1/¥)~l. The func- tion g(x) = l/x is convex in (0,00) and Jensen's inequality applies, to obtain 1 ~ E(1/¥) ~ l/E(¥) which completes the proof. References [1] ABRAMOWITZ, M. AND S'TEGUN, A. (1965) Handbook of Mathematical Functions. National Bureau of Standards, Washington, D.C. [2] AuCAMP, D. (1975) Comment on "the random nature of stock market prices" authored by Barrett and Wright. Operat. Res. 13, 587-591. [3] BICKSI.J:R, J. AND THORP, E. (1973) The capital growth model: an empirical investigation. 1. Finance and Quant. Anal. vm, 273-287. [4] BREIMAN, L. (1961) Optimal gambling systems for favorable games. Proc. 4th Berkeley Symp. Math. Statist Prob. 1, 65-78. [5] BREIMAN, L. (1968) Probability. Addison-Wesley, New York. [6] DURHAM, S. (1975) An optimal branching migration process. 1. Appl. Prob. 1l, 569-573. [7] FERGUSON, T. (1965) Betting systems which minimize the probability of ruin. 1. SIAM 13, 795-818. (8) HEWITT, E. AND STROMBERG, K. (1965) Real and Abstract Analysis. Springer-Verlag, New York. (9) HAKANSSON, N. (1971) Capital growth and the mean-variance approach to portfolio selection. 1. Finance and Quant. Anal. VI, 517-557. [10] KmLv, J. (1956) A new interpretation of information rate. Bell System Tech. 1. 35, 917-926. [11] LATANE, H. (1959) Criteria for choice among risky ventures. 1. Political Beon. 67, 144-155. [12) LATANE, H., TlJm.E, D., AND JAMES, C. (1975) Security Analysis and Portfolio Manage- ment. Wiley, New York. [13] OLVER, F. (1974) Introduction to Asymptotics and Special Functions. Academic Press, New York. [14) ROBERTS, A. AND VARBERG, D. (1973) Convex Functions. Academic Press, New York. [15) THORP, E. (1971) Portfolio choice and the Kelly criterion. Proc. Amer. Statist. Assoc, Business, Beon. and Stat. Section, 215-224. [16] THORP, E. (1969) Optimal gambling systems for favorable games. Rev. Internal. Statisl. Inst. 37, 273-293. [17] THORP, E. AND WHm..EV, R. (1972) Concave utilities are distinguished by their optimal strategies. Coil. Math. Soc. Janos Bolya, European Meeting of Statisticians, Budapest (Hungary), 813-830. --- ## Page 276 Journal of Portfolio Management, 19, 6-11 (1993) 249 18 The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice* Vijay K. Chopra and William T. Ziemba Good mean forecasts are critical to the mean-variance framework There is considerable literature on the strengths and limitations of mean- variance analysis. The basic theory and extensions of MV analysis are dis- cussed in Markowitz [1987] and Ziemba & Vickson [1975]. Bawa, Brown & Klein [1979] and Michaud [1989] review some of its problems. MV optimization is very sensitive to errors in the estimates of the inputs. Chopra [1993] shows that small changes in the input parameters can result in large changes in composition of the optimal portfolio. Best & Grauer [1991] present some empirical and theoretical results on the sensitivity of optimal portfolios to changes in means. This article examines the relative impact of estimation errors in means, variances, and covariances. Kallberg & Ziemba [1984] examine the question of mis-specification in normally distributed portfolio selection problems. They discuss three areas of misspecification: the investor's utility function, the mean vector, and the covariance matrix of the return distribution. They find that utility functions with similar levels of Arrow-Pratt absolute risk aversion result in similar optimal portfolios irrespective of the functional form of the utilityl; Thus, mis-specification of the utility function is not a major concern because several different utility functions (quadratic, nega- tive exponential, logarithmic, power) result in similar portfolio allocations for similar levels of risk aversion. Misspecification of the parameters of the return distribution, however, does make a significant difference. Specifically, errors in means are at least ten times as important as errors in variances and covariances. We show that it is important to distinguish between errors in variances and covariances. The relative impact of errors in means, variances, and co- variances also depends on the investor's risk tolerance. For a risk tolerance of -Reprinted, with permission, from Journal 0/ Portfolio Management, 1993. Copyright 1993 Institutional Investor Journals. IFor an investor with utility function U and wealth W, the Arrow- Pratt absolute risk aversion is ARA == -U"(W)jU'(W). Friend and Blume 11975] show that investor behavior is consistent with decreasing ARA; that is, as investors' wealth increases, their aversion to a given risk decreases. 53 --- ## Page 277 250 V K. Chopra and W T Ziemba 54 Chopra and Ziemba 50, errors in means are about eleven times as important as errors in variances, a result similar to that of Kallberg & Ziemba.2 Errors in variances are about twice as important as errors in covariances. At higher risk tolerances, errors in means are even more important relative to errors in variances and covariances. At lower risk tolerances, the relative impact of errors in means, variances, and covariances is closer. Even though errors in means are more important than those in variances and covariances, the difference in importance diminishes with a decline in risk tolerance. These results have an implication for allocation of resources according to the MV framework. The primary emphasis should be on obtaining superior estimates of means, followed by good estimates of variances. Estimates of covariances are the least important in terms of their influence on the optimal portfolio. Theory For a utility function U and gross returns r - i (or return relatives) for assets i = 1,2, .. . , N, an investor's optimal portfolio is the solution to: N maximize Z(x) = E[U(Wo ~]ri)xi)] i=l N such that Xi> 0, L = 1, i=1 where Z(x) is the investor's expected utility of wealth, Wo is the investor's initial wealth, the returns ri have a distribution F(r), and Xi are the portfolio weights that sum to one. Assuming a negative exponential utility function U(W) = - exp( -aW) and a joint normal distribution of returns, the expected utility maximization problem is equivalent to the MV-optimization problem: N 1 N N maximize Z(x) = L E[ri]Xj - t L L xixjE[Uij] i=1 i=lj=1 N such that Xi > 0, LXi = 1, i=l 2The risk tolerance reflects the investor's desired trade-off between extra return and extra risk (variance). It is the inverse slope of the investor's indifference curve in mean- variance space. The greater the risk tolerance, the more risk an investor is willing to take for a little extra return. Under fairly general input assumptions, a risk tolerance of 50 describes the typical portfolio allocations of large US pensions funds and other institutional investors. Risk tolerances of 25 and 75 characterize extremely conservative and aggressive investors, respectively. --- ## Page 278 The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice 251 The Effect of Errors 55 EXHIBIT 1: List of Ten Randomly Chosen DJIA Securities 1. Aluminum Co. of America 2. American Express Co. 3. Boeing Co. 4. Chevron Co. 5. Coca Cola Co. 6. E.!. Du Pont De Nemours & Co. 7. Minnesota Mining and Manufacturing Co. 8. Procter & Gamble Co. 9. Sears, Roebuck & Co. 10. United Technologies Co. where E[TiJ is the expected return for asset i, t is the risk tolerance of the investor, and E[OijJ is the covariance between the returns on assets i and j.3 A natural question arises: How much worse off is the investor if the distri- bution of returns is estimated with an error? This is an important considera- tion because the future distribution of returns is unknown. Investors rely on limited data to estimate the parameters of the distribution, and estimation errors are unavoidable. Our investigation assumes that the distribution of returns is stationary over the sample period. If it is time-varying or non- stationary, the estimated parameters will be erroneous. To measure how close one portfolio is to another, we compare the cash equivalent (CE) values of the two portfolios. The cash equivalent of a risky portfolio is the certain amount of cash that provides the same utility as the risky portfolio, that is, U(CE) = Z(x) or CE = U- 1[Z(x)] where, as defined before, Z(x) is the expected utility ofthe risky portfolio.4 The cash equivalent is an appropriate measure because it takes into account the investor's risk tolerance and the inherent uncertainty in returns, and it is independent of utility units. For a risk-free portfolio, the cash equivalent is equal to the certain return. Given a set of asset parameters and the investors risk tolerance, a MY- optimal portfolio has the largest CE value of any portfolio of those assets. The 3 Although the exponential utility function is convenient for deriving the MV problem with normally distributed returns, the MV framework is consistent with expected utility maximization for any concave utility function, assuming normality. 4For negative exponential utility, Freund [19561 shows that the expected utility of port- folio x is Z(x) = 1 - exp( -aE[x] + (a2/2)Var[x]), where E[X] and Var[xl are the expected return and variance of the portfolio. The cash equivalent is eEl: = (lla) log(l - Z(x)). If returns are assumed to have a multivariate normal distribution, this is also the cash equivalent of an MV-optimal portfolio. See Dexter, Yu & Ziemba [1980] for more details. --- ## Page 279 252 V K. Chopra and W T. Ziemba 56 Chopra and Ziemba percentage cash equivalent loss (CEL) from holding an arbitrary portfolio, x instead of an optimal portfolio 0 is CEL = CEo - CEx CEo where CEo and CEx are the cash equivalents of portfolio 0 and portfolio x respectively. Data and Methodology The data consist of monthly observations from January 1980 through Decem- ber 1989 on ten randomly selected Dow Jones Industrial Average (DJIA) se- curities. We use the Center for Research in Security Prices (CRSP) database, having deleted one security (Allied-Signal, Inc.) because of lack of data prior to 1985. Each of the remaining twenty-nine securities had an equal probabil- ity of being chosen. The securities are listed in Exhibit 1. MV optimization requires as inputs forecasts for: mean returns, variances, and covariances. We computed historical means (Ti), variances (aii), and covariances (aij), and assumed that these are the 'true' values of these pa- rameters. Thus, we assumed that E[ri] = Ti, E[aii] = aii, and E[aij] = aij' A base optimal portfolio allocation is computed on the basis of these parameters for a risk tolerance of 50 (equivalent to the parameter a = 0.04). Our results are independent of the source of the inputs. Whether we use historical inputs or those based on a complete forecasting scheme, the results continue to hold as long as the inputs have errors. Exhibit 2 gives the input parameters and the optimal base portfolio re- sulting from these inputs. To examine the influence of errors in parameter estimates, we change the true parameters slightly and compute the resulting optimal portfolio. This portfolio will be suboptimal for the investor because it is not based on the true input parameters. Next we compute the cash equivalent values of the base portfolio and the new optimal portfolio. The percentage cash equivalent loss from holding the suboptimal portfolio instead of the true optimal portfolio measures the impact of errors in input parameters on investor utility. To evaluate the impact of errors in means, we replaced the assumed true mean Ti for asset i by the approximation Ti(l + kzi) where Zi has a standard normal distribution. The parameter k is varied from 0.05 through 0.20 in steps of 0.05 to examine the impact of errors of different sizes. Larger values of k represent larger errors in the estimates. The variances and covariances are left unchanged in this case to isolate the influence of errors in means. The percentage cash equivalent loss from holding a portfolio that is optimal for approximate means Ti (1 + kzi ) but is suboptimal for the true means r, is --- ## Page 280 The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice 253 The Effect of Errors 57 EXHIBIT 2: Inputs to the Optimization and the Resulting Optimal Portfolio for a Risk Tolerance of 50 (January 198G-December 1989) Alcoa Amex Boeing Chev. Coke Du Pont MMM P&G Sears UTech Means ('Yo per month) 1.5617 1.9477 1.907 1.5801 2.1643 1.6010 1.4892 1.6248 1.4075 1.1537 Std. Dev. ('Yo per month) 8.8308 8.4585 10.040 8.6215 5.988 6.8767 5.8162 5.6385 8.0047 8.212 Correlations Alcoa 1.0000 Amex 0.3660 1.0000 Boeing 0.3457 0.5379 1.0000 Chev. 0.1606 0.2165 0.2218 1.0000 Coke 0.2279 0.4986 0.4283 0.0569 1.0000 Du Pont 0.5133 0.5823 0.4051 0.3609 0.3619 1.0000 MMM 0.5203 0.5569 0.4492 0.2325 0.4811 0.6167 1.0000 P&G 0.2176 0.4760 0.3867 0.2289 0.5952 0.4996 0.6037 1.0000 Sears 0.3267 0.6517 0.4883 0.1726 0.4378 0.58Il 0.5671 0.5012 1.0000 UTech 0.5101 0.5853 0.6569 0.3814 0.4368 0.5644 0.6032 0.4772 0.6039 1.0000 Optimal Port. Weights 0 .0350 0.0082 0.0 0.1626 0.7940 0.0 0.0 0.0 0.00 0.00 then computed. This procedure is repeated with a new set of Z values for a total of 100 iterations for each value of k. To investigate the impact of errors in variances each variance forecast O'ii was replaced by O'ii(l + kZj ). To isolate the influence of variance errors, the means and covariances are left unchanged. Finally, the influence of errors in covariances is examined by replacing each covariance O'ij (i # j) by O'ij + kZij where Zij has a standard normal distribution, while retaining the original means and variances. The procedure is repeated 100 times for each value of k, each time with a new set of Z values, and the cash equivalent loss computed. The entire procedure is repeated for risk tolerances of 25 and 75 to examine how the results vary with investors' risk tolerance. Results Exhibit 3 shows the mean, minimum, and maximum cash equivalent loss over the 100 iterations for a risk tolerance of 50. Exhibit 4 plots the average eEL --- ## Page 281 254 V K. Chopra and W T. Ziemba 58 Chopra and Ziemba EXHIBIT 3: Cash Equivalent Loss (CEL) for Errors of Different Sizes k (size of Parameter Mean Min. Max. error) with Error CEL CEL CEL 0.05 Means 0.66 0.01 5.05 0.05 Variances 0.05 0.00 0.34 0.05 Covariances 0.02 0.00 0.25 0.10 Means 2.45 0.01 15.61 0.05 Variances 0.22 0.00 1.39 0.10 Covariances 0.11 0.00 0.66 0.15 Means 5.12 0.15 24.35 0.15 Variances 0.55 0.00 3.35 0.15 Covariances 0.27 0.00 1.11 0.20 Means 10.16 0.17 3609 0.20 Variances 0.90 0.01 4.16 0.20 Covariances 0.47 0.00 1.94 as a function of k. The CEL for errors in means is approximately eleven times that for errors in variances and over twenty times that for errors in covariances. Thus, it is important to distinguish between errors in variances and errors in covariances.:; For example, for k = 0.10, the CEL is 2.45 for errors in means, 0.22 for errors in variances, and 0.11 for errors in covariances. Our results on the relative importance of errors in means and variances are similar to those of Kallberg & Ziemba [1984J. They find that errors in means are approximately ten times as important as errors in variances and covariances considered together (they do not distinguish between variances and covariances). Our results show that for a risk tolerance of 50 the importance of errors in covariances is only half as much as previously believed. Furthermore, the relative importance of errors in means, variances, and covariances depends upon the investor's risk tolerance. Exhibit 5 shows the average ratio (averaged over errors of different sizes, k) of the CELs for errors in means, variances, and covariances. An investor with a high risk tolerance focuses on raising the expected return of the portfolio 5The result for covariances also applies to correlation coefficients, as the correlations differ from the covariances only by a scale factor equal to the product of two standard deviations. --- ## Page 282 The Effect a/Errors in Means, Variances, and Covariances on Optimal Portfolio Choice 255 The Effect of Errors 59 % Cash Equivalant lo .. 11,---------------------------~----_, Means 10 9 8 7 6 5 4 3 2 ___ .... __ .... . V.ri.~ce • ... ~_.:_-::::: .... _ ................ Covlnances O~~--~-*~~~~~~----~----~ o 0.05 0.10 0.15 0.20 Magnituda of error (kl EXHIBIT 4 Mean percentage cash equivalent loss due to errors in inputs EXHIBIT 5 A verage Ratio of CELs for Errors in Means, Variances, and Covariances Risk Errors in Means Errors in Means Errors in Variances Tolerance versus Variances versus Covariances 25 3.22 5.38 50 1.98 22.50 75 21.42 56.84 versus Covariances 1.67 2.05 2.68 and discounts the variance more relative to the expected return. To this investor, errors in expected returns are considerably more important than errors in variances and covariances. For an investor with a risk tolerance of 75, the average CEL for errors in means is over twenty-one times that for errors in variances and over fifty-six times that for errors in covariances. Minimizing the variance of the portfolio is more important to an investor with a low risk tolerance than raising the expected return. To this investor, errors in means are somewhat less important than errors in variances and covariances. For an investor with a risk tolerance of 25, the average CEL for errors in expected returns is about three times that for errors in variances and about five times that for errors in covariances. Most large institutional investors have a risk tolerance in the 40 to 60 range. Over that range, there is considerable difference in the relative importance of errors in means, variances, and covariances. Irrespective of the level of risk tolerance, errors in means are the most important, followed by errors --- ## Page 283 256 V K. Chopra and W T. Ziemba 60 Chopra and Ziemba in variances. Errors in covariances are the least important in terms of their influence on portfolio optimality. Implications and Conclusions Investors have limited resources available to spend on obtaining estimates of necessarily unknowable future parameters of risk and reward. This analysis indicates that the bulk of these resources should be spent on obtaining the best estimates of expected returns of the asset classes under consideration. Sometimes, investors using the MV framework to allocate wealth among individual stocks set all the expected returns to zero (or a non-zero constant). This can lead to a better portfolio allocation because it is often very difficult to obtain good forecasts for expected returns. Using forecasts that do not accurately reflect the relative expected returns of different securities can sub- stantially degrade MV performance. In some cases it may be preferable to set all forecasts equal. 6 The opti- mization then focuses on minimizing portfolio variance and does not suffer from the error-in-means problem. In such cases it is important to have good estimates of variances and covariances for the securities, as MV optimizes only with respect to these characteristics. Of course, if investors truly believe that they have superior estimates of the means, they should use them. In this case it may be acceptable to use historical values for variances and covariances. For investors with moderate to high risk tolerance, the cash equivalent loss for errors in means is an order of magnitude greater than that for errors in variances or covariances. As variances and covariances do not much influence the optimal MV allocation (relative to the means), investors with moderate- to-high risk tolerance need not expend considerable resources to obtain better estimates of these parameters. References Bawa, Vijay S., Stephen J. Brown and Roger W. Klein (1979). 'Estimation Risk and Optimal Portfolio Choice.' Studies in Bayesian Econometrics, Bell Laboratories Series. North Holland. 6This approach is in the spirit of Stein estimation and is discussed in Chopra, Hensel, and 'Thrner [1993]. As a practical matter, it should be used for assets that belong to the same asset class. e.g., equity indexes of different countries or stocks within a country. It would be inappropriate to apply it to financial instruments with very different characteristics; for example, stocks and T-bills. --- ## Page 284 The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice 257 The Effect of Errors 61 Best, Michael J. and Robert R. Grauer (1991). 'On the Sensitivity of Means- Variance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results.' Review of Financial Studies 4, No.2, 315-342. Chopra, Vijay K. (1991). 'Mean-Variance Revisited: Near-Optimal Portfolios and Sensitivity to Input Variations.' Russell Research Commentary. Chopra, Vijay K., Chris R. Hensel and Andrew L. Thrner (1993). 'Massaging Mean-Variance Inputs: Returns from Alternative Global Investment Strategies in the 1980s.' Management Science, (July): 845-855. Dexter, Albert S., Johnny N.W. Yu and William T. Ziemba (1980). 'Portfolio Selection in a Lognormal Market when the Investor has a Power Utility Func- tion: Computational Results.' In Proceedings of the International Conference on Stochastic Programming, M.A.H. Dempster (ed.), Academic Press, 507-523. Freund, Robert A. (1956). 'The Introduction of Risk into a Programming Model.' Econometrica 24253-263. Freund, L. and M. Blume (1975). 'The Demand for Risky Assets.' The American Economic Review, December, 900-922. Kallberg, Jarl G. and William T. Ziemba (1984). 'Mis-specification in Portfolio Selection Problems'. In Risk and Capital, G. Bamberg and K. Spremann (eds.), Lecture Notes in Econometrics and Mathematical Systems. Springer-Verlag. Klein, Roger W, and Vijay S. Bawa (1976). 'The Effect of Estimation Risk on Optimal Portfolio Choice. J. of Financial Economics 3 (June), 215-231. Markowitz, Harry M. (1987) . Mean- Variance Analysis in Portfolio Choice and Capital Markets. Basil Blackwell. Michaud. Richard O. (1989). 'The Markowitz Optimization Enigma: is 'Opti- mized' Optimal?' Financial Analysts Journal 45 (January-February), 31-42. Ziemba, William T. and Raymond G. Vickson, eds. (1975) Stochastic Optimization Models in Finance. Academic Press. --- ## Page 285 This page is intentionally left blank --- ## Page 286 259 QU(llItilatilie Fillance, Vol. 5, No. 4, August 2005, 343- 355 11 Routledge ~ TayIor& Fraoci' Groop 19 Time to wealth goals in capital accumulation LEONARD c. MACLEAN*t , WILLIAM T. ZIEMBAt§ and YUMING LI ~ tSchool of Business Administration, Dalhousie University, Halifax, NS, Canada B3H IZ5 t Sauder School of Business, University of British Columbia, Vancouver, BC, Canada V6T IZ2 §Department of Finance, Sloan School of Management, 50 Memorial Drive E52-410, Massachusetts Institute of Technology, Cambridge, MA 02142-1 347, USA I. Introduction -,rSchool of Business, California State University, Fullerton, CA 92834, USA (Received 26 January 2004; infina/form /5 Apri/ 2005) This paper considers the problem of investment of capital in risky assets in a dynamic capital market in continuolls time. The model controls risk, and in particular the risk associated with errors in the estimation of asset returns. The framework for investment risk is a geometric Brownian motion model for asset prices. with random rates of return. The information filtration process and the capital allocation decisions are considered separately. The fi ltration is based on a Bayesian model for asset prices, and an (empirical) Bayes estimator for current price dynamics is developed from the price history. Given the conditional price dynamics, investors allocate wealth to achieve their fi nancial goals efficiently over time. The price updat- ing and wealth reallocations occur when control limits on the wealth process are attained. A Bayesian fractional Kelly strategy is optimal at each rebalancing, assuming that the risky assets are jointly lognormal distributed. The strategy minimizes the expected time to the upper wealth limit while maintaining a high probability of reaching that goal before falling to a lower wealth limit. The fractional Kelly strategy is a blend of the log-optimal portfolio and cash and is eq uivalently represented by a negative power utility function, under the multi- va riate lognormal distribution assumption. By rebalancing when control limits are reached, the wealth goals approach provides greater control over downside risk and upside growth. The wealth goals approach with random rebalancing times is compared to the expected utility approach with fixed rebalancing ti mes in an asset allocation problem involving stocks, bonds, and cash. Keywords: Capital accumulation; Wealth goals; Investment of capital In capital accumulation under uncertainty, a decision- maker must determine how much capital to invest in riskless and risky investment opportunities at each point in time. The investment strategy yields a stream of capital over time, with investment decisions made so that the distribution of wealth has desirable properties. An investment strategy which has generated considerable interest is the growth optimal or Kelly strategy, where the expected logarithm of wealth is maximized (Kelly 1956). The wealth distribution of this strategy has many attrac- tive characteristics (see e.g. Hakansson 1970, 1971, Markowitz 1976, Hakansson and Ziemba 1995). As Breiman (1 960, 1961 ), and Algoet and Cover (1 988) have shown, the Kelly strategy maximizes the long run expected rate of growth of capital and minimizes the expected time to reach a fixed level of wealth for suffi- ciently large goals under mild conditions. Researchers such as Thorp (1975), Hausch et al. (1981), Grauer and Hakansson (1986, 1987), and Mulvey and Vladimirow (1992) have used the optimal growth strategy to compute optimal portfolio weights in multi-asset and worldwide asset allocation problems. The literature considers the stream of capital foll owing from an investment policy from either a wealth or a time perspective. In most situations the expected utility of accumulated capital at a fixed point in time is analysed. There is particular interest in the logarithm of accumu- lated capital, and the corresponding rate of capital growth. Alternatively, the growth rate can be viewed as the time it takes for accumulated capital to reach wealth milestones. *Correspondi ng author. Email: Imac1ean@mgmt.dal.ca The distribution of accumulated capital to a fixed point in time and the distribution of the first passage Qual/fi/afire Fillallce ISSN 1469- 7688 print/ISSN 1469- 7696 online © 2005 Taylor & Francis http://www.tandf.co.uk/journals DO lo 10.1080/1 4697680500149552 --- ## Page 287 260 L. C. MacLean, W T. Ziemba and Y Li 344 L. C. MacLean et al. time to a fixed level of accumulated capital are variables controlled by the investment decisions. The Kelly strategy maximizes the expected logarithm of accumulated capital or the expected growth rate. However, the strategy is very aggressive. As Hausch and Ziemba (1985) and Clark and Ziemba (1 987) have demonstrated, the optimal portfolio weights in the risky assets given by this strategy tend to be so large for favourable investments that the chances of losing a substantial portion of wealth are very high, par- ticularly if the probability estimates are in error. In the time domain, the chances are high that the first passage to subsistence wealth occurs before achieving the established wealth goals. When the investor is more risk averse, then this can be reflected in the utility function choice. If the utility function has an Arrow-Pratt risk aversion parameter, e.g. the constant relative risk aversion (CRRA) utility, then the value of this parameter captures risk tolerance; see Kallberg and Ziemba (1 983). Another approach is to define a measure of risk which depends on the investment decision. A standard measure of risk is volatility as defined by the variance of wealth at a point in time or the variance of the passage time to a wealth target. Mean-variance analysis of wealth has been widely used to determine investment strategies; see Markowitz (1952, 1987). In the time domain the mean- variance approach yields different strategies. However, the logarithm of wealth and the first passage time have consistent mean-variance properties; see Burkhardt (1998). An alternative to variance is to use a downside risk measure (Breitmeyer et al. 1999). MacLean et al. (1992) considered, as risk measures, quantiles for wealth, log- wealth and first passage time in identifying investment strategies which achieve capital growth with a required level of security. Security is defined as controlling down- side risk. Growth is traded for security with fractional Kelly strategies. In discrete time models with general return distributions this strategy is generally suboptimal, but it has attractive wealth/time distribution properties. See MacLean and Ziemba (1999) for extensions of this research. The emphasis in that trade-off work is the prop- erties of the wealth process for a given fraction. For example, the probability of reaching upper wealth U before lower wealth L is calculated for various strategies. In this paper, the reverse problem of findin g a strategy which achieves a specified probability (risk) is studied. The most common downside risk measure is Value at Risk (VaR) (Jorion 1997). Va R has been studied exten- sively (Artzner et al. 1999, Gaivoronski and Pflug 2005). Basak and Shapiro (2001 ) consider VaR in a model with C RRA utility. Although VaR is an industry standard it has weaknesses in controlling risk. The most serious shortcoming is the insensitivity to very large losses which have small probability- the essence of risk. Measures based on lower partial moments (incomplete means) such as Cva R (Rockafeller and Uryasev 2000) and convex risk measures based on target violations (Carino and Ziemba 1998, Rockafeller and Ziemba 2000) attempt to deal with this problem. Similar to mean-variance, a variety of mean-risk problems have been studied. Bi-criteria problems such as maximizing expected logarithm of wealth subject to a VaR constraint are consistent with stochastic dominance, the traditional concept for ordering wealth distributions. (Ogryczak and Ruszczynski 2002). There is, however, another issue complicating the measurement of risk and return, which is the estimation of parameters defining the returns distribution. The value of the risk/return measures in practice are estimates of the true values, and the error of the estimates of measures is very sensitive to errors in the estimation of the returns distribution. Furthermore, the sensitivity to estimation errors of expected value measures (mean, CVaR) is much greater than quantile measures (median, VaR). In this paper parameter estimation and risk control are considered in a model where the filtration and control processes are separate. A dynamic stochastic model for asset prices is presented in section 2. The model is a gen- eralization to the multi-asset case of the random coeffi- cients model of Browne and Whitt (1996). The inclusion of many assets facilitates the estimation of model coeffi- cients since the correlation between assets is related to the intrinsic model structure. The existence of latent factors generating price movements is implied by the equations for price dynamics. Alternatively, price parameters could be related to observable state variables, as is the case in the model of Xia (200 I). (See also Brennan 1998) Given the estimated price dynamics, an investment decision is made to control the path of future wealth. In section 3, an approach to control is developed based upon wealth levels as stopping rules. In practice, an investment portfolio cannot be continuously rebalanced and a realistic approach is to reconsider the investment decision at regular discrete time intervals (Rogers 2000). At each rebalance time, with additional data and a change in wealth, price parameters are re-estimated and a new investment strategy is developed. The process is illustrated in figure I. The most significant aspect of the rebalancing is the accuracy of the estimated returns distribution. If forecasts are accurate, then wealth will accumulate as expected. If prices are not as forecast over the next hold interval then unacceptable wealth levels may result. To protect against that outcome Ivea/th limils can be placed on the wealth process. If the wealth trajectory is within the limits, the wealth process is under control. However, the asset pricing model and the invest- ment decisions are reconsidered when a control limit is reached. So rebalancing occurs at a random time rather than at a given point in time. Intervention occurs when the wealth trajectory is not proceeding as expected and it can be concluded that the investment decision does not fit the true securities price process. Whether rebalancing occurs at fixed or random times, at the rebalancing the investor: 1. generates new estimates for returns distributions; 2. establishes next lVea/lh limits for random time rebal- ancing, or the next hold time for the fi xed rebalance interval; --- ## Page 288 Time to Wealth Goals in Capital Accumulation 261 Time 10 Iveallh goals in capilal accumulation 345 Portfolio Decision Parameter Estimates D Figure 1. Dynamic investment process. 3. determines a nell' investment slralegy based on the updated returns estimates, and control limits or hold time. At a rebala nce point, the decision on investment is based on accumulated wealth at a planning ho rizon. A number of bi-criteria decision models have been men- tioned, but the model considered is maximization of expected utility at the horizon, with a negative power utility. It is significant that there exist wealth goals and an associated bi-criteria decision model which yields an identical investment strategy. Although demonstrated for the special case of negative power utility, this result is general, so that wealth goals reflect a wide range of preferences. The wealth goals decision is put in the context of capital growth with security. So a strategy is determined which minimizes the expected time to the upper limit (grolVlh) while maintaining a high probability of achieving the upper limit before falling to the lower limit (security). The optimal strategy for the power utility and growth-security models is shown to be fracl ional Kelly-a blend of the risk free (cash) strategy and the optimal growth strategy. Although the negative power utility and wealth goals models are linked by the strategy, the approaches to risk are different. The expected utility model considers risk at the ho rizon, whereas the wealth goals model considers risk along the trajectory. In section 4, the wealth goals and power utility approaches are compared for the funda- mental problem of investing in stocks, bonds, and cash over time. It is shown that there is an advantage from wealth goals, since rebalancing occurs at times indicated by the wealth trajecto ry, i.e. the wealth conditio n. This result is comparable to the superiority of condition based intervention in repairable systems (Aven a nd Jensen 1999). Appendix A contains proofs for proposi- tions in the text. 2. Dynamic estimation of asset price distributions In the dynamic investment process shown in fi gure I, the returns o n risky assets are re-estimated at the time of rebalancing a portfolio. The updated returns distribu- tions are inputs to the investment decision models. In this section a linear pricing model is proposed where the esti- mation problem is separated from the control problem. The model is Bayesian, so that the updating of parameter estimates follows from Bayes theorem, and fits naturally into the dynamic investment process. 2.1. Price model Suppose there are 111 risky assets, with P'(I) equal to the trading price of asset i at time I. i = I, . ,m, and a risk- less asset with rate of return I' at time I. The risky asset prices are assumed to have a joint log-normal distribu- tion. Letting Y,(I) = CnP;(t), i = 0, .. . ,111, the price dynamics are defined by the stochastic differential equations dYo(I)= rd t, (I) dY;(I) = "',(I)dl +8; dU;, i= I, ... ,m, where d U;, i = I, ... , 171, are independent standard Brownian motions. They represent the variation in price specific to each security. The other component of price variation is generated by the instantaneous mean rate of return ",,{I), i = I, ... , m. Assume that ",;(1) is a random variable with distribution defined by ",;(t) = /1.; + y;Z ,(I), i = 1, ... ,171. (2) The Z;, i = I, . . . ,117, are correlated Ga ussian variables, with Pu being the correlation between Z; and Z j- So the rates of return are correlated, with the covariance between ",;(1) and "'J{I) given by Y;YjPij' The price process defined by ( I) and (2) is a variation on the Merton model (1992) with the conditio n that the rates Clj, i = I , . , 111, are stochastic and the volatilities OJ, i = I, . , 117, are not stochastic. The intention is to empha- size uncertainty in the mean since errors in estimating the mean have by far the greatest impact on portfolio deci- sions (Chopra and Ziemba 1993). The asset prices in the model are correlated, with the correlatio n generated by the relationship between the random rates of return o n assets. Presumably those rates of return are related through some cOl11l11on factors. The factors may be lalem, as is assumed in this paper, o r they may be observable state variables as presented by Xia (2001), who considers the single risky asset case. If the model for prices is correct then the error in forecasting prices arises from errors in parameter estimates. Error from incorrect modeling may be con- founded with estimation error, so attention to the correct model in an applicatio n may be significant. There are --- ## Page 289 262 L. C. MacLean, W T. Ziemba and Y Li 346 L. C. MacLean et al. generalizations of (I) and (2) which can be incorporated into the approach developed in this section. For example, the rates of return could be defined by stochastic differential equations: dai(l) = fLi d l + Yi dq" where dqi are correlated Brownian motions, i = I, ... , m. Then a,(l) = ai(O) + fLil + Yi0Z,(I), and the rates are dynamic. The assumption in (2) simplifies the presentation, so that formulation is followed, but reference to the dynamic linear model will be made where appropriate. With regard to the stationary process (2), Ihe model assumplion applies lmlif Ihe nexl rebalance poinl. That is, the distribution for the random rates is the same between rebalance points, but the distribution at the next point may change. The model in (2) can be considered as an approximation between rebalance points. When used with control limits, where rebalancing occurs when the actual prices depart from model forecasts, this is a useful approach. From (I) and (2) the relevant distributions for asset prices follow. Let Y(l) = (YI (I), ... , Y",(l))' , fL = (fLJ, .. , fL",)', a = (a l, ' . , a",)', 6. = diag(8;' .. , 8~, ) , and r = (Yij), where Yij = YiYA; (Prior) a ~ N(fL, r), (3) (Conditional) (Y(1)la, 6.) ~ N(al, 16.), (4) (Marginal) Y(I) ~ N(fL(I), E(I)), (5) where fL(l) = IfL and E(I) = 12r + 16. = ret) + 6.(1). The Bayesian pricing model defined by (3}-(5) has an alternative structural model representation. The prior covariance has a decomposition r = AA', where A is an m x I matrix with I=" m. Consider the independent lalel1l jaclors F ' = (FI , . .. , FI) , F ' ~ N(O, I), and the errors / (1) = (£1 (I), ... , "",(1)), e'(I) ~ N (O, 6.(1)). Let A (I) = I A. Then the log prices at time I are Y(I) = fLU) + A(I)F + set). (6) From (6), Y(I) ~ N(fL(!), E(/)), where E(I) = A(1)A'(I)+ 6.(1). The relationship between prices is driven by under- lying common factors which are unobserved. This manifests itself in rates of return, a = fL + AF, which are random. The factor model in (6) has been the subject of numer- ous asset pricing studies, with the factors representing traded portfolio's in some cases (Campbell el al. 1997). Here, the factor model is used to define estimates for parameters in the inter-temporal pricing model of (I) and (2). 2.2. Bayes estimation At the time an investor is rebalancing a portfolio, infor- mation is available on past realized securities prices. Consider the data availa ble at time I, (Y(I), ° =" s =" I}, and the corresponding filtrati on F,v = a( Y(I), ° =" s =" I), the a -algebra generated by the process Y up to time I. Conditioned on the data, the distribution for the rate of return can be determined from Bayes theorem. That is, a(l) = (a(I)IFt') ex N(&(I), ret)), (7) where &(1) = fL(l) + (I - 6.(I)E- I (1))(1'(1) - fL(I)), with 1'(1) = ~ Y(I), I and - I I r(1) = (2 (I - 6.(I)E- (1))6.(1). The Bayes eSlimale for the rate of return a(l) is the conditional expectation &(1) = Ea(I). This estimate is the minimum mean squared error forecast for the rate given the data. In the context of rebalancing, this is the planning value for a until the next decision time 1+ f (when the stopping boundaries are reached or the fixed rebalance time occurs). Then, since no informa- tion is added in the hold interval, E(a(I)!F,v) = E(E(a(I)IF,~, )IF,v) , and the best estimate is &(1) through- out the interval. The Bayes estimate a(l) has an appealing form. It is inherently dynamic, so that estimates can be updated as more information on prices is obtained. In particular, at the next rebalance time 1+ f , the posterior &(1) becomes the next prior and Bayes theorem is used to update with the new information-Y(I + f). This approach is described in MacLean el al. (2004). There is another approach which emphasizes the current dynamics. The pricing model parameters (fL, r, 6.) are interpreted as values for the most recent rebalance cycle, and only data from that time interval is used in estimating the parameters. With control limits, the recent data contains information on a change in dynamics, and a change in prior parameters. The technique for implying the values for the prior parameters from data is empirical Bayes estin1ation. To develop the estimation consider a rebalance interval (0, I) and prices at the discrele times kIln, k = 0, ... , n, with corresponding log prices Y(k) := Y[(kl n)I]. The change in log prices between times kil n and [(k + 1)l n]1 is e(k) = Y(k) - Y(k - I), k = I, ... , n. From the model equations e(k)=':'a + fi6. I/2Z, k=I , ... ,n, (8) n '1-;; where Z ~ N(O, I). The increment vectors defined by (8) are independent and Gaussian for each k . The covariance matrix for each vector is 12 I E,,(I) = ,. r + - 6. = r ,,(I) + 6.,,(1), n- 11 and I " 1'(1) = - L e(k) n k= 1 Consider observed log prices on risky securities at the discrete time points: {Yik, i= 1, ... ,111, k=O, ... , n}. With eik = Yik - Yi k- J, the data on increments (first- order differences i~ log prices) are {eik> i = I, ... , m, k = I, ... , n}. Let the covariance matrix for the observed increments be Su" the estimate of E,,(I). --- ## Page 290 Time to Wealth Goals in Capital Accumulation 263 Time to \Vealth goals in capital accumulation 347 From the random effects model (8) the theoretical covariance is factored as l:,,(t) = r,,(t) + "',,(t). [f the rank (r,,(t» = I < m, then the sample covariance matrix can be factored to produce estimates of r,,(t) and "',,(t). In terms of the structural model for prices in (6), the assumption rank (r,,(t» = I < m is equivalent to the assumption Ihe number of latent faclors I is strictly less Ihan the number of risky assets m. Assuming 1< m, a maximum likelihood factor analysis of S'" with I factors will yield a loading matrix L", and a specific error matrix D,,,, where D", is diagonal. Since the number of factors is known, (L"" D",) is an efficient esti- mator of (A,,(t), "',,(I», with A,,(I) = (l/n)A (Lawley and Maxwell 1971). Then G", = L", L;" is an efficient estimator of r ,,(t). The prior mean J.i for the rates of return on securities in a rebalance interval needs to be estimated. If it is assumed that the rates have a common prior ~l1eal1, i.e. J.i; = J.i, i = I, ... , m, then the overall mean yet) = (1 / 1) L:=I Y;(t) is an estimate of J.i, where Y;(t) = (1 / t)Y;(t). The assumption of a common value is consis- tent with a long run rate of return on secu rities. (Alternatively, securities in the same class could have a common mean, with differences between classes of securities.) The discussion has identified estimates for all model parameters within a rebalance interval. The conditional distribution for securities prices can be estimated, and in particular the conditional mean rate of return for the next time interval can be forecast. Since the forecast is the Bayes formula with the prior parameters estimated from data it is an empirical Bayes estimate. Proposition 1: Let Yb k = 0, ... , n, be observations on Ihe veclor of log prices of risky assels 01 regular inlervals of widlil I/n, IVilh the Slalistics ~omputed from first-order increments: S"" L"" D", and Y,. Theil /:,. = (n/t )D,,,, A = (n/I)L,,, are estimates of Ihe model parameters '" and A, respeclively. With /:,.(1) = I/:", A(I) = tA and S; = A(t)A'(t) + /:,.(t), an eslimate for Ihe conditional mean rate of return at time I is &,,(1) = yet) + (I - /:,.(t)S;)(Y(t) - yet)) (9) Furthermore, the limit as II --> 00 of &,,(1) is &(1). The empirical Bayes estimate in (9) is a natural form u- lation given the proposed pricing model. If the model is correct then &,,(1) has smaller mean squared error than well known estimates for the rate of return, such as the maximum likelihood estimate and the James-Stein esti- mate. [f the model is correct but the number of latent factors is unknown, then there is additional estimation error from estimating that number. However, results from similar models indicate such error is small; see Macl ean and Weldon (1996). If a dynamic model for rates, a;(1) = a;(O) + J.i;! + Y;0Z;(I), i = I, ... , ! , is the correct formulation then the approach described above can be modified to obtain empirical Bayes estimates for parameters. The modification involves calculating second- order differences to obtain a random effects model with iid second-order increments. With this estimation procedure, an essential component of the dynamic investment process is in place. The next component, the planning model, will have the forecasts from the estimated model as inputs. 3. Portfolio planning models Assuming that at a rebalance point the data has been filtered to obtain estimates for the parameters, the decision on how much capital to allocate to the various assets is now considered. The objective is to control the path of accumulated capital generated by the decision and the unfolding asset prices. The decision will be developed from (he estimated price parameters whereas capital will accumulate from the (rue process. Of course, if the esti- mates used in computing a strategy are substantially in error, then the trajectory of wealth will not proceed as anticipated, but the trajectory may still be under control. At the rebalancing time t, there are estimates for the conditional rate of return art), and the volatility /:"(1), based on the methods of section 2. The forecast dynamics for the price process, conditional on the estimates for parameters, are d Y(I) = &(t)dt + /:,.1 /2(I)dZ, (10) where the innovations process dZ is standard Brownian motion. An investment strategy is a vector process !(xo(t), X(t», t ~ 01 = ! Xo(I), (X I(t), ... , x",(I» , t ~ 01, (II) where L;~o x;(t) = I for any I , with xo(t) the investment in the risk-free asset. The proportions of wealth invested in the m risky assets are unconstrained since xo(t) can always be chosen, with borrowing or lending, to satisfy the budget constraint. The forecast change in wealth from an investment deci- sion X(t) is determined by the conditional price process (10). Suppose the investor at time t has wealth W(I) and let ¢;(t) = &;(t) + (1 / 2)81(1), i = I , ... ,m. Then the fore- cast for the instantaneous change in wealth is d Wet) = [~ X;(t)(¢;(I) - r) + r] W(t)dl + Wet) t x;(t)8;(t)dZ;, ([2) i=l where dZ; is standard Brownian motion. There is a grolvth condilion which will be required of any feasible investment strategy: With a fixed mix strategy over the hold interval satisfy- ing the growth condition, the forecast wealth process W( r), r ~ I, defin ed by (12) follows geometric Brownian motion such that W(r) ~ O. --- ## Page 291 264 L. C. MacLean, W T Ziemba and Y Li 348 L. C. MacLean et al. To determine an investment decision which controls wealth, two approaches are considered. First, a model which considers preferences for wealth at the planning horizon is defined. The standard approach is to maximize expected utility at a planning horizon. Because of uncer- tainty in the parameters for the asset pricing model, parameters are re-estimated and decisions revised at regular intervals in time up to the horizon. In the second approach, the preferences at the horizon are used to develop wealth goals (limits). Then a wealth goals model is defined, which has an equivalent investment strategy. The contrast from the approaches comes from rebalancing times. The wealth goals approach involves a random hold interval, with stopping/rebalancing occur- ring when specified upper or lower wealth goals are reached. The matching of preferences in the two approaches does not mean that the same fixed mix strategy is used for both models. The rebalance times are different, and therefore there are different parameter estimates used in the strategy calculation. The link between the planning models is stochastic dominance. Wealth WI dominates wealth W 2 iff Eu(WI) is greater than or equal to Eu( W 2), with strict inequality for at least one utility function u E V. Orders of stochastic dominance are determined by classes of utilities. If Vk = {ul( _l)i-IuU) 0:: O,} = I, ... , kl, so that VI :::) V2 :::) ... , then VI is the class of monotone utilities, V2 the class of concave monotone utilities, . . . , and Voocontains the CRRA utilities. Since the CRRA utilities are a limiting subset, they are a reasonable reference class. There are alternative formulations of the various orders of stochastic dominance. In particular, first order is equivalent to dominance in the cumulative distribution of wealth. The wealth goals model will be defined on the distribution of wealth, and therefore is based on first-order dominance principles. However, it is a relaxa- tion of first-order dominance in that it focuses on parts of the distribution in a bi-criteria problem. 3.1. Expected utility strategy with fixed rehalallce times Consider an investor whose objective is to maximize the expected utility of wealth at the end of a finite planning horizon T. To obtain an explicit solution we assume a constant relative risk aversion (CRRA) utility. The utility of wealth w is At the current decision time, the parameters in the pricing model are estimated and an investment strategy is deter- mined, conditional on those estimates. Consider the conditional CRRA wealth problem. Definition 1: The conditional CRRA wealth problem at rebalance time I, with current wealth w" horizon T and estimated parameters &(1), 3.(1), is to find the strategy X(I) E X, which maximizes £[W~I)~l where the forecast dynamics of wealth W(t) are given by (12). Proposition 2: The oplimal slralegy lor Ihe condilional CRRA weallh problem is X*(I) = _ 1_3. - I (t)(¢(t) - re). (13) 1-,8 As ,8 --+ 0 in definition I the utility is log and the efficient strategy is the benchmark optimal growth solution: X(t) = Do - I (¢(t) - re). The fractionI= 1/ (1 -,8) captures the aversion to risk (Pratt 1964). When 0 < ,8 < 1, the positive power utility case, the optimal portfolio invests more than the benchmark, with the over-investment financed through borrowing (levered strategies). Although the strategy X~ (t) , 0 < ,8 < I, is optimal, it is problematic from the perspective of the distribution of terminal wealth W(I). Let Fp be the distribution function for terminal wealth following strategy X#(I), with Fo the terminal wealth distribution following the benchmark strategy. Distributions can be compared with the first- order stochastic dominance relation: Fp, dominates Fp, iff F~ ,(IV) ::S Fp, (IV) for IV E R with strict inequality for some IV. Proposition 3: The terminal weallh dislribution Folor Ihe benchmark portfolio first -order stochaslically dominales Ihe dislribulion Fplor 0 < ,8 < I. This result provides a rationale for a restriction ,8 ::s 0 to the negative power and log utility functions, and the resulting class of fractional Kelly strategies pX(t), where p ::s I. First-order stochastic dominance is defined on the class of all monotone utilities, of which the CRRA utility is a small subset. Imposing the first-order dominance con- dition on the fraction from a CRRA utility model is a restriction, but other work, where a risk constraint such as VaR is included in the expected CRRA utility model, has shown that levered strategies are excluded (Basak and Shapiro 2001). The solution in (13) is independent of current wealth IV, and the planning horizon T. The decision is sensitive to estimation and model- ing error, and at regular intervals of width s model parameters are re-estimated using the empirical Bayes methodology. The actual investment decision at times I, I + s, . would change with the new estimates. 3.2. Wealth goals stl'ategy ami ralldom I'ehalallce times The uncertainty in returns on investment, whether from natural variability, variability in underlying factors, or errors in estimation, is the motivation for strategies that control risk. In the conditional CRRA utility problem the risk associated with the forecast wealth at the horizon is controlled with the risk aversion parameter ,8, with the --- ## Page 292 Time to Wealth Goals in Capital Accumulation 265 Time 10 weallh goals in capilal accumulalion 349 estimation risk controlled by regularly updating estimates for parameters. Updates are appropriate when there is evidence that current estimates are substantially in error. This concept of updating based on the gap between expected and actual processes suggests another way to control the wealth process. That is, to set control limits (or wealth goals), as is the practice in process control. The wealth goals are action levels so that the investment portfolio is rebalanced if either of the goals is reached. With wealth w, at time I, the lower and upper wealth goals .!:!::( and WI are set, where !!:, < W t < \tit. If the investment strategy X(/) is chosen, then the strategy is followed until either ~, or IV, is reached. At that point the portfolio is rebalanced. So new estimates are generated, new goals are set and a new strategy is chosen. Between rebalancing points a fixed mix strategy is followed. Information on prices is obtained between rebalance points, but it is only used when the accumulated information on actual prices signals that the forecasts for price movements are substantially in error, that is, when a control limit is reached. The choice of wealth goals is an important component of the control process. Wealth goals can be an expression of preferences as naturally as a utility function. Our approach is to first consider the form of an investment strategy for given weallh goals, assuming the wealth goals are feasible, i.e. admit an investment strategy. Then a methodology for specifying goals based on expected utility is developed. The risk and return characteristics of the wealth pro- cess have traditionally guided the choice of strategy X(/). Within the wealth goals context, therefore: (i) the chance of hitting the lower wealth goal should be small. That is, the downside risk should be controlled; (ii) the upper wealth goal, the preferred target, should be reached as quickly as possible. That is, the growlh rale of capital should be maximized. To formalize downside risk in the wealth goals approach, consider T,.,(X(I), w, 1&(1),3.(/» = firsl passage lime to weallh w starting with lvealth w, at time I andfollolVing investment slralegy X ( I) , condilioned on Ihe eSlimales[or paramelers at time t. Let the set of wealth goals be denoted by B( = {btlb/ = U:!:p WI), 0 <'!:!::t < WI < WI }' For b, E B" downside risk for a strategy X(/) E X, is measured by mh,(X(/» = Pr[ T!!C/( X(I), IV,IO:(I), 3.(1») .:': T;;;, ( X(I), W, I&(I), 3.(/»)]. (14) The risk is the probability the first passage time to ~, is smaller than the first passage time to IV,. Risk measures are associated with acceptance sets of capital accumulation paths (Artzner el 01. 1999). Let Q(X(I» be the set of all paths of WeT), T ::: I, and let A(X(I» be the set of acceptable paths. If A' (X(I» is the complement, then the downside risk for X(I) is 9l(X(I» = a(A' (X(t)) . That is, the risk for a strategy is the probability measure of the unacceptable set of paths. Desirable properties of risk measures have been con- sidered in the literature. The following basic axioms for a dmvl1side risk measure are proposed in Breitmeyer el 01. (1999). A: (non-negativity) m(X(/» ::: o. B: (normalization) If AC(X(I)) = b, link explored. Consider, then, the condi- tional f3-class of goals, with W, = w~ , y = l (21) The set ~(f3) ,; B, defines wealth goals with the same Arrow-Pratt relative risk aversion given the current wealth I V~ and risk probability l. Proposition 5 estab- lishes that such goals actually exist. Proposition 5: At the rebalance time t consider currel1l weallh IV~. risk probabililY yO and risk aversion --- ## Page 294 Time to Wealth Goals in Capital Accumulation 267 Time 10 Ivealth goals in capital accumulation 351 parameter fJ. Assume that the benchmark portfolio sati4ies the growlh conciilion, i.e. o/,(X(t)) I --- - > - . o}(X(t)) 2 Then B?(fJ) is non-empty Jar fJ < O. Consider the following matching heuristic for setting wealth goals at any rebalancing time T: I. TV; = E(WEU(r + s)) is the expected wealth at time r+s with the CRRA strategy I - X"(r) = (I _ fJ) X(r). 2. 1£; = max{l£,lb, E B~(fJ)}. Setting goals at b; = (l!';, TV;) maintains the investment fracti on at p, = 1/ (1 - fJ) for any rebalancing time T, and links the goals to the holding time s. 4. Comparisons The accumulated capital from the wealth goals with ran- dom rebalancing times, and expected utility with fixed rebalancing times is now analysed for the fundamental problem of allocating investment capital to stocks, bonds and cash over time. The matching heuristic approach is used to insure the problems are comparable, with the main difference being rebalancing time. Daily prices for stocks and bonds are simulated using the dynamic pricing model, discretized to days. If P;(l) is the price for asset i on day t, where i = 0 (cash), I (stock), and 2 (bonds), then consider the incre- ments e;(l)= lnP;{I + 1) - lnP;(i), i= O, 1,2. From the model, e;(t) = a; + 8;2;{t), where 2;(t) are independent Gaussian variables. For cash returns, ao = In(1 + r). It is assumed that (ai , a2) are Gaussian variables dependent on a single latent state variable. So a ; = f.L + A;F, i = 1, 2, with F standard Gaussian. There is a simple structure to model parameters in this single factor case. If the covar- iance for risky asset log prices is ~ , The model implies ~ = /1/1' + t., where /I = (A j, A21' and t. = diag(81, 8~ ). The factor solution is /I = (..,rfJOI1 , ..,rfJ022)', t. = diag«1 - Ipl)uij, (\ - I pl)u~2) ' If the true covariance is known, then model parameters are specified and daily prices can be simulated. Conversely, if a sample covariance matrix is determined from a sequence of prices then the parameter values can be estimated using these equations. The estimates are empirical Bayes, based on the methods in section 2. For the simulation of asset prices, data on yearly asset returns corresponding to the S&P500, Solomon Brothers bond index, and U.S. T-bills for 1980- 1990 were used to generate seed parameters. Statistics on the associated Mean Variance Covariance Table I. Daily rates of return. Stocks Bonds 0.00050 0.00031 0.00062 0.00035 0.00014 Cash 0.00019 o 07 O'----~50:---1~O.,. 0 ---,1"'5:-0 ---::2.;-00:-----=25::0,----:-:300 Figure 2. Daily prices for stocks and bonds. daily returns are in table I. The corresponding statistics for log-rates were used as seeds. Daily prices were generated for one year (260 trading days) and this was repeated 200 times. The one year horizon is appropriate for the simulation since it provides sufficient time for differentiation of capital accumula- tion from the PC and EU strategies. The contrasting approaches to selecting an investment strategy were applied to each trajectory of prices. Initial parameter esti- mates were taken from average daily returns for the prior year, 1979. For expected utility, rebalancing (revising estimates of price model parameters; calculating new strategy) was done every 30 days. For process control, rebalancing occurs when control limits are reached. The estimation is based on the covariance matrix. To increase stability the covariance analysed at the jth rebalancing at time I is As time increases the updating is weighted to the observed covariance S. The procedure for setting limits (wealth goals) is described in section 4, where the process control fraction is maintained at p, = 1/ (\ - fJ). The expected utility and wealth goals (process control) methods are shown for the trajectory of prices in figure 2. This trajectory has a volatile return on stock and is a useful example. The expected utility strategies and portfolio values for the setting I - fJ = 1.1 are given in table 2. The rebalanc- ing occurred at 30 day intervals. The strategy emphasizes stocks and is quite stable. The return on the portfolio at the end of the year was 24%. --- ## Page 295 268 L. C. MacLean, W T. Ziemba and Y Li 352 L. C. M aeLean et al. Time I 31 61 91 121 151 181 211 241 Time I 64 65 66 67 74 88 107 108 115 11 8 120 123 130 Table 2. Expected utility performance. Stocks 0.4670 0.7439 0.8688 0.8963 0.9701 0.9432 0.9864 1.0125 0.9820 Strategy Bonds 0. 1465 - 0.1 319 - 0.1 770 - 0.2292 - 04437 - 04726 - 04 900 - 04181 - 0.3989 Port value I 0.8943 0.9930 1.1 287 1.4089 1.3114 1.2164 1.1 508 1.21 78 Table 3. Wealth goals performance. Strategy Wealth limits Stocks Bonds Port value Upper Lower 04670 0.1465 I 1.0112 0.5765 0.5805 0.1140 1.0148 1.028 1 0.5309 0.6346 0.1010 1.0379 1.0526 0.52 12 0.6656 0.09 \0 1.0562 1.0717 0.5 196 0.6744 0.0864 1.0787 1.0946 0.530 1 0.6804 0.0898 1.1065 1.1229 0.5427 0.7106 0.067 1 1.1433 1.1 615 0.5385 0.7566 - 0.0011 1.1 644 1.1 844 0.523 1 0.7889 - 0.0509 1.1910 1.21 27 0.5 177 0.7995 - 0.0936 1.2375 1.2603 0.535 1 0.8060 - 0.1302 1.277 1 1.3006 0.55 14 0.8089 - 0.1 571 1.3387 1.3634 0.5779 0.8242 - 0.1957 1.3640 1.3898 0.5798 0.7864 - 0.2 152 1.4043 1.4290 0.6234 The process control methodology was run on the same trajectory, with settings I - fJ = 1.1 , I - y = 0.03. Those results are presented in table 3. There is initially more frequent rebalancing with wealth goals, because of the volatility. When prices settle down, the fi xed mix strategy does not change. The return at the end of the year was 42%. It is noteworthy that the return was monotone and the advantage of the PC strategy increases with time. The comparison of the expected utility and process control approaches is the expected wealth ratio at the end of the year. The wealth ratio for each trajectory is the accumulated wealth with the process control strategy divided by the accumulated wealth with the expected utility strategy. Table 4 presents mean wealth ratios for values for the C RRA utility parameter fJ, and the process control risk probability 1- y. The levered strategies (O< fJ < I) are included, although those strategies are not growth- security efficient. The wealth goals (process control) approach has a sub- stantial advantage over expected utility, demonstrating the effect of accounting for current wealth (at the control limit) in determining a strategy. Since the PC strategy is determined (matched) from the EU strategy, the expected wealth advantage of being less risk averse (decreasing I - fJ) is embedded in the PC strategy, but the PC advan- tage over E U grows since the volatility is controlled by Table 4. Expected wealth ratios. I - y I - /l 0.0 1 0.02 0.03 0.04 0.05 1.1 5 1.291 1.290 1.289 1.265 1.264 1.10 1.322 1.321 1.320 1.298 1.293 1.05 1.365 1.364 1.359 1.340 1.337 1.00 14 10 1408 1.403 1.384 1.38 1 0.95 1.460 1.459 1.457 1.443 1.433 0.90 1.523 1.522 1.520 1.507 1.491 0.85 1.608 1.608 1.607 1.602 1.584 the upper and lower limits. As the security requirement is relaxed (y decreasing) the expected wealth advantage for PC decreases. This results from the specific construction of the strategy, where the lower control limit drops with decreased y and given upper control limit. The cases where (I - fJ) < I , the levered strategies, are high risk and are stochastically dominated as established in Proposition 4. 5. Conclusion This paper considers the risk to accumulated capital in a multi-asset investment problem where the distribution of asset prices has random parameters. The basis for invest- ment strategies is either process control, where wealth is controlled by upper and lower goals, or expected utility of terminal wealth at the horizon. In the process control approach, the portfolio is re-balanced when either wealth goal is reached. At a re-balancing time the history of asset prices is used to update the conditional price distribution. An empirical Bayes estimator of distribution parameters is defined based on the observed correlation structure of asset prices. With the revised price distribution, new upper and lower wealth goals or the next re-balance time are selected and a new optimal strategy is derived. The pro- cess control strategy is growth- securi ty efficient in the sense that it has maximum growth among strategies with a specified level of risk. Assuming a lognormal dis- tribution for asset returns, the growth-security strategy is a blend of the risk-free portfolio and the optimal growth portfolio. The wealth goals approach and the expected utility approach with constant relative risk aversion are com- pared. An equivalence between the wealth goals and the risk aversion parameter is developed. Although the alter- native models have the same portfolio solution with the appropriate choices of parameter values, the wealth goals approach has the advantage of a direct control of down- side risk and a re-balancing of the portfolio based on significa nt change in the price process and change in wealth. The benefit of intervening when the condition of wealth deviates from expectations rather than at regular intervals in time is clear. Re-balancing at specified points in time when wealth is proceeding as expected is analogous to tampering in process control, and typically introduces extra risk. --- ## Page 296 Time to Wealth Goals in Capital Accumulation 269 Time to weallh goals in capital accumulation 353 Acknowledgements This work was presented at the Bachelier World Congress 2002, and seminars at The University of British Columbia and Dalhousie University. The authors wish to thank Yonggan Zhao for assistance with the computations. This research was supported by grants from the Natural Sciences and Engineering Research Council of Canada and the National Center of Competence in Research FlNRISK, Swiss National Foundation. Then The first-order conditions _ d_ E ( W#(T- t)) _ 0 Appendix A dX;(t) f3 - This appendix contains proofs for the propositions in the reduce to paper. Proposition 2: The optimal Slrategy for the CRRA wealth problem is given by X'(t) =_I _ t,- I(t)(¢(t) - re). ( 13) 1 -f3 Proof: The analysis is similar to theorem 6.2 in Janecek (1998). For investment strategy X(t) the fo recast dynamics of wealth are [ '" ] dW(I) = 8 x;(t)(¢;(t) - r) + r W(t)d t '" + Wet) L x;(t)8;(t)dZ ;. i= ] With X(r) = X(I) and ¢;(r) = ¢;(t), r :O: I, based on info r- mation to time I, integration from t to T gives the forecast wealth at the horizon W(T- t) {[ '" I"' ] = w, exp 8 x;(t)(¢;(I) - r) + r -"28 x1(t)iJ(t) x (T - t) + (T - 1)1/2 ~X;(t)8;(I)Z; } , where Z; ~ N(O, I). The expected utility of wealth is Consider the expression (¢;(t) - r) - (I - f3)x7(t)8;(1) = 0, i= I , ... ,1, and X*(/) is as stated. D Proposition 3: The lerminal weallh dislribution Fo for Ihe benchmark portfolio .firsl-order stochastically dominates the dislribution Fp for 0 < f3 < I. Proof: Let 1/I(X) and O"(X) be the instantaneous mean rate of return and standard deviation, respectively, of the optimal growth (benchmark) portfolio. Let f = 1/( 1 - f3) and define 1/I(XP) = 1/I(f) =f(1/I(X) - r) + r),0"2(f) =/0"2(X) and Under the assumptions of the model, terminal wealth W(I + T ) from the strategy xP is log-nol"lnally distributed with F ( ) _ N[ln{(IV/IV,) - [ I (optimality of benchmark), the statement in the proposition follows. D Proposition 4: Suppose at rebalance time I, the investor has weallh IV" fo recast dynamics of asset prices given the history {Y" 0 S s s tl, and the return on Ihe risk-Fee assel is r. If COl1lrol limits (weallh goals) are b, E B, and Ihe risk level is I - y, Ihell Ihe growlh- security efficient strategy at lime t is Xi,,(t) = (Xib,(/), ... , X;',b,(t))' wilh Xi,,(t) = prClV" b" y)t, - I (t)(¢(t) - re), ( 19) where e is a veclor of ones. Proof: Let!.!:, = IV,/k and w, = k"wi' where k > I and c > O. For simplicity the time variable I is deleted. From (16) and (1 7) the growth- security efficiency problem is { I I - k -O(X) } m~x .p(X) I _ I« I+I)8(,\") :0: y, X feasible . --- ## Page 297 270 L. C. MacLean, W T. Ziemba and Y Li 354 L. C. MacLean et al. Letting y = k - O(¥) , the constraint requires yy,+1 - y+ (I - y) :0:: O. Consider the minimum positive root y* of the equation yy,+1 - y + (I - y) = 0, where y' ~ I since y* = I is a root. Then yy,+1 - Y + (I - y) :0:: 0 for y ~ y' and the constraint is satisfied if k - 9(¥) ~ y' or II(X) :0:: -C~~~) = q* :0:: 0, where q* depends upon ~" w" and y. But 1I(X) :o:: q' if 21/t(X) - ( I - q*)(J2(X) :0:: O. Hence, the efficiency problem is m~xj21/t(x) - (J2(X)121/t(x) -(I + q')(J2(X):o:: 0, X feasiblel. If X* is a solution to this problem then it also solves, for optimal multiplier A *, the Lagrangian problem m~x L(X, 1.*), where The first-order conditions 'VxL(X*, A' ) = 0 imply that the optimal X* satisfies the linear system 2(1 + A')(4)i - 1') - 2( 1 + 1.' (1 + q*))[xioT] = 0, i= 1, ... ,1. The solution to this system is X* = p,(IV" ~,, WI' y)x t, - I (4)(t) - re) where 1 +1.' p ,(IV"~,, WI' y) = I + 1.*(1 + q' )' D k -IJ(¥) = y'. Let i = t, - 1(4) - re) and X* = pi. Then the equation becomes [( I - IOgY' )(J2(X)]/ - 2[{L(X) - r]p - 21' = O. logk The statement in the corollary follows. D Proposition 5: At the rebalance lime I consider curren I weallh w?, risk probability l and risk aversion param- eler fJ. Assume Ihal Ihe benchmark portfolio salisfies the growlh condition, i.e. 1/t,(i(t)) I ----- >-. (Jl(X(I)) 2 It follOlvs thai lfi(fJ) is non-emply for fJ < 0 Proof: From corollary l, p,(IV" b" y)= 1/(I-fJ) implies I _ (J2(X) " - 2r(l - fJ)2 + 2H(X)(J2(X)(l - fJ) K > O. Since hllV , WI) = log IV, - log w, , -, log IV, - lo g~, - logy' any (~" WI) E B?(fJ) must satisfy the equation h(~" WI) = K. From the assumption fJ < O and 1/t/(J2> 1/2 it follows that 0 < K < I. The solution y* of the equation yyC+I _ y + (I - y) = 0 has the property y' t I as wt t w( , for fixed ~~ < WI- SO there exists (~~, w7) such that h(~~ , w;) > K. Also for fixed w;, h~ " 1i0) t 0 as ~l t w,. That is, there is a .!£: with h(]f:, w;) = K. 0 Corollory 1: For IVealth goals b, E B" the /raetion of References Iveallh invesled in risky assets 01 Ihe rebalancing time t is min{l , PI} for p,(IV" hI' y) = h, . H,(i(/)) + - 2 2rh, [hI . H,(X(t))] + - 2- - ' (J,(X(I)) where H,(i(1)) = [1/ttCi(~) - 1'] , (Jl(X(t)) h, = [log IV, - log ~, ] , [log 11', -Iog~, -logy'] (20) and y* is the minimum positive 1'001 of yy,+1 - y + (I - y) = o for Proof: [Iogw, - log 11',] c, = [log w, - log~ ,l' C l ea rly , p, ~ I, with 1 +1.' p, = 1 +1.' 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Rogers, L.C.G., The relaxed investor and parameter uncertainty. Finance and Slochastics, 2000, 5, \31- 154. Thorp, E.O., Portfolio choice and the Kelly criterion. In Stochastic Optimization Models in Finance, edited by W.T. Ziemba and R.G. Vickson, pp. 599- 619, 1975 (Academic Press: New York). Xia, Y., l earning about predictability: the effects of parameter uncertainty on dynamic asset allocation. JOllrnal oj Finance, 200 I, 56, 205- 246. --- ## Page 299 This page is intentionally left blank --- ## Page 300 20 Survival and Evolutionary Stability of the Kelly Rule* Igor V. Evstigneev Economics Department, School of Social Sciences, University of Manchester, United Kingdom igor. evstigneev@manchester.ac.uk Thorsten Hens Swiss Banking Institute, University of Zurich, Switzerland thens@isb. uzh. ch Klaus Reiner Schenk-Hoppe Leeds University Business School and School of Mathematics, University of Leeds, United Kingdom K.R.Schenk-Hoppe@leeds.ac.uk Abstract This chapter gives an overview of current research in evolutionary finance. We mainly focus on the survival and stability properties of investment strategies associated with the Kelly rule. Our approach to the study of the wealth dynamics of investment strategies is inspired by Darwinian ideas on selection and mutation. The goal of this research is to develop an evolutionary framework for practical investment advice. 1 Introduction 273 The principal objective of our evolutionary approach to the study of financial market dynamics is the development and analysis of models that constitute a plausible alternative to the conventional general equilibrium framework. Our aim is to provide a framework that is suitable for delivering practical investment advice. The equilibrium concept most commonly used in financial economics is due to Radner (1972). This equilibrium notion involves the plans and price expectations *Financial support by the Finance Market Fund, Norway (project Stability of Financial Markets: An Evolutionary Approach) and the National Center of Competence in Research "Financial Val- uation and Risk Management" , Switzerland (project Behavioural and Evolutionary Finance) is gratefully acknowledged. This chapter was written during KRSH's visit to the Department of Finance and Management Science at the Norwegian School of Economics and Business Admin- istration in August 2009. We are grateful to Terje Lensberg, Edward O. Thorp and William T. Ziemba for their helpful comments. --- ## Page 301 274 I V. Evstigneev, T Hens and K. R. Schenk-Hoppe of agents as well as market prices. A well-known drawback of that framework is the necessity of agents to have perfect foresight (rational expectations) in order to establish an equilibrium (see the discussion in Laffont (1989) and Dubey et al. (1987)) market participants have to agree on the future prices for each of the possible future realizations of the states of the world (without knowing which particular state will be realized). Our evolutionary model differs radically from that approach: only historical observations and the current state of the world influence the agents' behavior; no agreement about the future market structure is required and no coordinated actions by the agents are assumed. From a practical finance perspective, the important dis- tinction between our approach and the conventional general equilibrium paradigm lies in the data required to formulate the model. The evolutionary model does not use agents' characteristics that are unobservable (such as individual utilities or subjective beliefs). We describe aims of investors in terms of properties (sur- vival, evolutionary stability, etc.) holding almost surely, rather than in terms of the maximization of expected utilities. We consider this robust modeling approach as the basis for developing a new generation of dynamic equilibrium models that could be used for practical quantitative recommendations applicable in the financial industry. The general approach underlying this direction of work is to apply evolutionary dynamics - mutation and selection - to the analysis of the long-run performance of investment strategies. A stock market is understood as a heterogeneous population of frequently interacting portfolio rules in competition for market capital. The ultimate goal is to build a Darwinian theory of portfolio selection. Evolutionary ideas have a long history in the social sciences going back to Malthus, who played an inspirational role for Darwin (for a review of the subject see, e.g., Hodgeson (1993). A more recent stage of development of these ideas began in the 1950s with the publications of Alchian (1950) and others. An important role in this line of work has been played by the interdisciplinary research conducted in the 1980s and 1990s under the auspices of the Santa Fe Institute in New Mexico, USA - see, e.g., Arthur et al. (1997) , Farmer and Lo (1999) , LeBaron et al. (1999) , Blume and Easley (1992) , and Blume and Durlauf (2005). Our model also revives the literature on stochastic dynamic games going back to Shapley (1953). The framework also shares some conceptual features with the games of survival pioneered by Milnor and Shapley (1957) and the dynamic market game in Shubik and Whitt (1973). The main difference is that we use the notion of survival strategies rather than that of Nash equilibrium. In a finance context, the focus on survival is an advantage (over those approaches invoking expectations) because it is a property holding almost surely and not requiring discounted or undiscounted utility. This survey is based on the research carried out by Amir et al. (2005, 2008, 2009), Evstigneev et al. (2002, 2006, 2008) and Hens and Schenk-Hoppe (2005). --- ## Page 302 Survival and Evolutionary Stability of the Kelly Rule 275 A general survey of evolutionary finance is provided by Evstigneev et al. (2009). The issue of survival of traders in Radner's setting has been studied by Blume and Easley (1992); see also the surveys Blume and Easley (2008, 2009). Section 2 explains in details the model, Section 3 discusses its dynamics and defines the concepts of survival and evolutionary stability, Section 4 surveys the results obtained in the literature, and Section 5 concludes. 2 Model Consider a market in which K ~ 2 assets are traded. Each asset k = 1, 2, ... , K pays dividends at dates t = 1,2, .... The non-negative payoff Dt,k(st) ~ 0 depends on the history st = (Sl' ... , St) of states of the world up to date t. The states are random factors modeled in terms of an exogenous stochastic process Sl, S2, ... , where St is a random element of a measurable space St. We assume functions to be measurable throughout the following. At each date (and in each random situation), at least one asset is assumed to pay a strictly positive dividend: K L Dt,k(st) > 0 for all t, st. (1) k=l The total net supply of asset k is equal to the random amount Vt,k(st) > 0 (a constant VO,k > 0 at the initial time t = 0). The vector of market prices is Pt = (Pt,l, ···,Pt,K) E RIf., where Pt,k is the price per unit. We assume vt,k(st)/vt_l ,k(st-l) ~ '"Y > 0 for t ~ 0 and all st. This condition is satisfied, e.g., if the asset supply is constant or increasing; the supply however cannot decrease at an ever-increasing rate. There are N ~ 2 investors, or traders, acting in the market. A portfolio of investor i at date t = 0, 1, ... is specified by a vector e~ = (e~ , l' ... , e~ , K) E RIf. where e~ k is the number of units of asset k held in the portfolio. The value of investor i'~ portfolio at date tis (Pt, e~) = 'Lf=l pt,keLk' The state of the market at each date t is characterized by the set (Pt, et, ... , ef) consisting of the price vector and all traders' portfolios. Investors i = 1,2, ... , N enter the market at date t = 0 with endowments wb > 0 that form their initial budgets. At date t ~ 1, investor i's budget is (Dt(st) + Pt, eLl), where Dt(st) = (Dt,l(St), ... , Dt,K(St)). This budget consists of two components linked to the portfolio eLl: the dividends (Dt(st), eLl) received and the market value (Pt, eLl) expressed in terms of the current prices Pt. A fraction ex of the budget is invested in assets. We will assume that the invest- ment rate ex E (0,1) is a fixed number, the same for all the traders. The remaining amount of wealth is withdrawn from the market through a tax rate 1 - ex (or, if you dislike taxes, through a consumption rate 1 - ex). The assumption of a common 1 - ex for all the investors is quite natural when interpreted as a tax rate. As a consumption rate it might seem a restrictive condition, however, it is indispensable --- ## Page 303 276 1. V. Evstigneev, T Hens and K. R, Schenk-Hoppe since we focus on the analysis of the comparative performance of trading strategies in the long run, Without this assumption, an analysis of this kind does not make sense: a seemingly worse long-run performance of an investment strategy might be simply due to a higher consumption rate, For each t ;::: 0, every trader i = 1,2, "" N selects a vector of portfolio weights A~ = (A~ 1, "" A~ K) according to which he/she plans to distribute the available bud- get bet';een ass~ts, Vectors A~ belong to the unit simplex /j. K = {( aI, "" a K) ;::: 0 : a1 + '" + aK = I}, In game-theoretic terms, the vectors A~ represent the investors' actions, The portfolio weights at each date t ;::: 0 are selected by the N investors si- multaneously and independently- the investors participate in a simultaneous-move N -person dynamic game, The investors' actions can depend (for t ;::: 1) on the history st of the states of the world and the history of the game (pt-1, ()t-1, At - l ), where pt-1 = (Po, ""pt-d is the history of asset price vectors and ()t-1 = (()o, ()l, "" ()t-d, ()I = (()t, "" ()r) At- 1 = (AO, AI, "" At-d, Al = (Ai, "" Ar) are the sets of vectors describing the portfolios and the portfolio weights of all the traders at all the dates up to t - 1. The history of the game contains information about the market history - the sequence (Po, ()o), "" (Pt-l, ()t-d ofthe states ofthe market - and about the actions At of all the investors i = 1, "" N at all the dates l = 0, "" t - 1. A vector Ab E /j.K and a sequence of measurable functions with values in /j. K Ai( t t-1 ()t-1 \i-1) t 1 2 t S,P , , /\ , = , "" form an investment strategy Ai of trader i, specifying a portfolio rule according to which trader i selects portfolio weights at each date t ;::: 0, This is a general game-theoretic definition of a strategy, assuming full information about the history of the game, including the players' previous actions, and the knowledge of all the past and present states of the world, Among general investment strategies, we will distinguish those for which A~ depends only on st and not on the market history, We will call such strategies basic, Note that basic strategies are in general not simple (constant), To complete the description of the market, it remains to define a dynamic equi- librium for the assets, Suppose that at date 0, each investor i has selected some portfolio weights Xo = (Ab l' "" Xo K) E /j. K, Then the amount invested in asset k by trader i is exAb,k Wo and 'the tot~l amount invested in asset k is ex L~l Ab,k wo' It is assumed that the market is always in equilibrium (asset supply is equal to asset demand), which makes it possible to determine the equilibrium price PO,k of each asset k from the equation N PO,k VO,k = ex L Xb,k wb (2) i=l --- ## Page 304 Survival and Evolutionary Stability a/the Kelly Rule 277 The left-hand side is the total market value PO,k VO,k of the supply of asset k while the right-hand side represents the total wealth invested in asset k by all the investors. Equilibrium ensures that both sides are equal, i.e., (2) holds. The portfolio weights >"0 chosen by the traders at date 0 determine their portfolios eo at date 0 by . o>"hkwO eo k = ' , PO,k (3) i.e., the current market value PO,keO,k of the kth position of investor i's portfolio is equal to the fraction >"O,k of the investor's investment budget owb. Consider now the situation at any date t ~ 1. Suppose all investors have chosen their portfolio weights >..~ = (>..~ I, ... , >..~ K)' Then the equilibrium of asset supply , , and demand determines the market-clearing prices of assets k = 1, ... , K through the relation N Pt,k vt,k = a L >"~ , k(Dt(st) + Pt , e~_ l ) i=l (4) The investment budgets o(Dt(st) + Pt, e~_ l ) of the traders i = 1,2, ... , N are dis- tributed between assets in the proportions >..tk' so that the kth position of the trader i's portfolio e~ is (5) The price vector Pt is determined implicitly as the solution to the system of equations (4). The existence and uniqueness of a non-negative vector Pt solving (4) (for any st and any feasible eL l and >"D is proved in Amir et al. (2009, Proposition 1). For a given strategy profile (AI , ... , AN ) of investors and their endowments W6, ... , wr/, a path of the market game is generated by setting (6) (7) for all t = 1, 2, ... , and i = 1, ... , N; and by defining Pt and e~ recursively according to equations (2)- (5). The random dynamical system (Arnold 1998) described above defines step by step the vectors of portfolio weights >"~(st) , the equilibrium prices Pt(st) and the investors' portfolios e ~ (s t ) as measurable vector functions of st for each moment of time t ~ 0 (for t = 0 these vectors are constant). Thus, we obtain a random path of the game (Pt(st); ei(st) , ... , ef (st); >..i(st) , ... , >..t' (st)) (8) as a vector stochastic process in Rf. x Rf. N X Rf. N . The above description is correct only if the price of each asset is strictly positive. Strategy profiles which guarantee this property at each period will be called admis- sible. Admissibility is satisfied, e.g., if the strategy profile contains an investor who --- ## Page 305 278 1. V. Evstigneev, T Hens and K. R. Schenk-Hoppe uses a strategy with strictly positive portfolio weights. We will deal only with such strategy profiles from now on and, therefore, have well-definedness of the random dynamical system. Then (using induction) the equilibrium path all the portfolios e~ = (e~, l ' ... , e~,K) are non-zero and the wealth (9) is strictly positive. Equation (5) implies L~l e~,k = vt,k, i.e., market clearing for every asset k and each date t 2': 1 (analogously for t = 0). For every equilibrium state of the market one therefore has that markets clear, Pt > 0 and e~ =I- 0 for all i. The description of a financial market is game-theoretic: the market dynamics is formulated as a simultaneous-move N -person dynamic game. Less general versions of the model can be motivated by invoking Marshall's (1949) principle of temporary equilibrium (Evstigneev et al., 2008) or evolutionary dynamics (Evstigneev et al., 2006). 3 Dynamics and Stability This section provides details on the dynamics of the model and defines notions of survival and evolutionary stability. An explicit formulation of the dynamics for the investors' wealth as well as their relative wealth is given. 3.1 Dynamics and the role of basic strategies Let (AI , ... , AN) be an admissible strategy profile of the investors. Consider the path (8) of the random dynamical system generated by this strategy profile and the given initial budgets. Let w~ > 0 denote the investor i's wealth at date t 2': O. For t 2': 1, w~ = wHst) is given by formula (9) while each investor i's endowment, wh, is a constant. When studying the dynamics of this wealth process for a fixed strategy profile, it is sufficient to consider the class of basic strategies, i.e., those strategies for which A~ depends only on the history of states until time t, st, and not on the market history. This assertion holds because any sequence of vectors Wt = (wi , ... , wf) of wealth generated by some strategy profile (A!, ... , AN) can be generated by a strategy profile (Ai (st), ... , Af (st)) consisting of basic portfolio rules. The corresponding vector functions A~(st) can be defined recursively by (6) and (7), using (2)- (5). This observation is very useful when studying the dynamic properties (such as survival or evolutionary stability) for a basic strategy - as we will do here. However this 'reduction' of the model to basic strategies has its limitations when dealing with the characteristics of a general (non-basic) strategy which competes in markets with different opponents. Such a strategy will in general take on different basic strategies, dependent on the pool of competitors - it is therefore not basic. --- ## Page 306 Survival and Evolutionary Stability of the Kelly Rule 279 The procedure described above gives the system of equations (10) i = 1, ... , N. This dynamic on the space {w E RN I w 2:: 0 and w i=- O} is well-defined under the assumptions imposed in Section 2 (e.g., Amir et al. (2009) or Section 4.1 in Evstigneev et al. (2008) which applies with minor modifications). Our analysis will focus on the long-run behavior of the relative wealth or the market shares r~ = wU L~l wi of the traders. Both the stability and survival of portfolio rules will be studied in this framework. An explicit random dynamical system can be derived for the vector rt as follows. Assume that the asset supply changes over time at the same rate --y > 0: (11) where Vk > 0 (k = 1,2, ... , K) are the initial amounts ofthe assets. In the case of real dividend-paying assets - involving long-term investments in the real economy (e.g., real estate, transportation, media, infrastructure, etc.) - the above assumption means that the economic system under consideration is on a balanced growth path. Define the relative dividends of the assets k = 1, ... , K by R t = (Rt,l, ... , Rt,K) where R - R (t) _ Dt,k(st)Vk t,k - t,k S - K (12) L Dt,m(st)Vm m=l for t 2:: 1. Further, assume that (t < --y and define p = (th, and Pt = pt-l(l - p) One finds the dynamic for the vector of market shares: (13) with i = 1, ... , N. This equation can be written in explicit form (e.g., Evstigneev et al., 2009): (14) where Ai = (Ail' ... , Ai K) E R NxK is the matrix of portfolio weights and 8 t E R NxK is the m~trix of portfolios given by 8~ , k = A~ ,krU (At ,k, rt). Setting p = 0 in (14) one obtains a model with short-lived assets akin to parimutuel betting markets, see e.g. Evstigneev et al. (2009). --- ## Page 307 280 1. V. Evstigneev, T Hens and K. R. Schenk-Hoppe 3.2 Survival and extinction of portfolio rules We study the dynamics of wealth shares from an evolutionary perspective. Our focus is on the questions of "survival and extinction" of portfolio rules. A portfolio rule Ai = (A~ (st)) (or the investor i using it) survives with probability one if inft;=o:o r~ > 0 almost surely. This means that for almost all realizations of the process of states of the world Sl, S2, ... , the market share of the first investor is bounded away from zero by a strictly positive random constant. We say that Ai becomes extinct with probability one if limt---7oo r~ = 0 almost surely. Survival and extinction can be defined for general non-basic investment strategies without any changes. An investment strategy A is called a survival strategy if the investor using it survives with probability one regardless of what other strategies are present in the market. In terms of the wealth process wL i = 1,2, ... , N, no investor's wealth can grow asymptotically faster than the wealth of investors who use survival strategies. In this sense, all survival strategies are competitive. A portfolio rule A = (At (st)) is called globally evolutionarily stable if the following condition holds. Suppose, in a group of investors i = 1,2, ... , J (1 ::; J < N), all use the portfolio rule A, while all the others, i = J + 1, ... , N use portfolio rules ),i distinct from A. Then those investors who belong to the former group (i = 1, ... , J) survive with probability one, whereas those who belong to the latter (i = J + 1, ... , N) become extinct with probability one (cf. Evstigneev et al. , 2008)). An analogous concept of local evolutionary stability can be defined. In that definition of stability, the initial market share rg+1 + ... + rf/ of the group of investors who use strategies ),i distinct from A is supposed to be small enough (cf. Evstigneev et al., 2006). 4 Results The findings on the long-run outcome of the dynamics of the above game fall in two categories, survival and evolutionary stability. The study of the first is carried out in the general model while the latter requires placing restrictions on the set of admissible strategies. This is simply caused by the extreme generality of the asset market game introduced above. For instance, if one strategy is constant while some other strategy always imitates the decision made in the previous period, both end up with the same portfolio weights. However, it can be shown that mimicking strategies are not survival strategies. The central role in our analysis is played by a (generalized) Kelly rule. Define the investment strategy A * with the vectors of portfolio weights A; (st) by 00 A;,k = Et L PZRt+Z,k Z=1 (15) where EtC) = E('lst) is the conditional expectation given st (unconditional expec- tation EC) if t = 0). --- ## Page 308 Survival and Evolutionary Stability a/the Kelly Rule 281 The portfolio rule specified by (15) prescribes to distribute wealth across assets in accordance with the proportions of the expected flow of their discounted future relative dividends. The discount rate Pt+l/ Pt = P is equal to the investment rate a divided by the growth rate ,. The portfolio weights of the strategy A * generally depend on time t and the history of states of the world st, but do not depend on the history of the game (pt-l, gt-l, A t-l) - the strategy is basic. The strategy A * is a generalization of the Kelly portfolio rule of 'betting your beliefs' playing an important role in capital growth theory - see Kelly (1956), Breiman (1961) , Thorp (1971), Algoet and Cover (1988), and Hakansson and Ziemba (1995). 4.1 Survival of the Kelly rule This section collects the results on the survival of the Kelly rule in different speci- fications of the model. Survival of the Kelly rule is ensured for the case of general investment strategies in both stock and betting markets. Amir et al. (2009, Theorem 1) shows that the Kelly rule (15) is a survival strategy. The only additional assumption needed for this result is that for all k and t, EtRt+l ,k(St+d > 15 almost surely for some constant 15 > O. This results extends the findings in Amir et al. (2008) to the stock market case (long-lived dividend- paying assets, P > 0). In Amir et al. (2008), betting markets (p = 0) are studied in which the assets are short-lived and are reissued at each date. The Kelly rule in this setting is obtained from (15) by using the discount rates PI = 1 and pz = 0 for l > 1. One has (16) for k = 1, ... , K. Under the assumption ElnEtRt+l,k(St+I) > -00 (which ensures strict positivity of all A;,k)' Amir et al. (2008, Theorem 1) asserts that A;,k defined in (16) is a survival strategy. The existence of a non-basic survival strategy is an open problem. In both cases, the Kelly rule is asymptotic unique among all basic survival strategies. One has that 2::: 0 IIA; - Atl1 2 < 00 almost surely for any basic sur- vival strategy A = (At) (see Amir et al. (2008, Theorem 2) and Amir et al. (2009, Theorem 2)). 4.2 Evolutionary stability of the Kelly rule This section reports on properties of the Kelly rule that are stronger than 'merely' ensuring survival. These features however only surface when restricting the set of admissible strategies (as well as the type of randomness driving the dividend process). The main results are that evolutionary stability with iid asset payments holds when the set of admissible strategies is restricted to constant strategies; and that local evolutionary stability hold is strategies and states are Markov. These --- ## Page 309 282 1. V. Evstigneev, T Hens and K. R. Schenk-Hoppe theoretical findings are complemented by simulation studies of markets in which the pool of investment strategies changes over time through a process of mutation. Assume that there are finitely many states of the world S E S, states Sl, S2 , ... are independent identically distributed (iid) with P{ St = s} > 0 for each S E Sand that the relative dividends Rt,k(st) = Rk(St) depend only on the current state St and do not explicitly depend on t. In addition to (1), we assume that ERk(St) > 0 for k = 1, ... , K. Under these conditions, (15) takes the special form >-';,k = L ptERk(St) = ERk(St) (17) which means that the strategy A * is formed by the sequence of constant vectors (ER1 (St), ... , ERK(St)) (independent of t and st). Note that in this special case, the formula for A * does not involve the investment rate p. In this case, the 'beliefs' at each date t are concerned simply with the expected relative dividends (which do not depend on t). We call an investment strategy A simple, or fixed-mix, if all the portfolio weights are constant. The investment strategy (17) is simple and completely mixed (Le., all components are strictly positive). Suppose there are no redundant assets, i.e., the functions Rds), ... , RK(S) are linearly independent. Evstigneev et al. (2008) , Theorem 1, shows that the Kelly rule defined in (17) is globally evolutionarily stable in any market in which all the portfolio rules are simple. This result demonstrates that the Kelly rule A * has good properties beside ensuring survival: the group of investors following this rule outperform all other simple strategies and dominate the market. These investors eventually gather the total market wealth, while those who use simple strategies distinct from A * become extinct. The related result for short-lived assets (p = 0) is proved in Evstigneev et al. (2002) and Amir et al. (2005). The latter deals with a more general setting in which the state follows a Markov process and the relative asset payoffs at time t + 1 depend on St and St+l. Results on the local evolutionary stability of the Kelly rule are obtained in Hens and Schenk-Hoppe (2005) (dealing with short-lived assets) and Evstigneev et al. (2006) (considering long-lived assets and basic Markovian strategies). The Kelly rule is the only portfolio rule that is locally evolutionary stable - and, thus, the only candidate for a global evolutionary stable strategy. The advantage of consider- ing the local dynamics is that stability can be characterized by growth rates (of the linearized dynamics) that do have explicit representations. It is therefore straight- forward to verify whether particular portfolio rules co-exist or whether selection can occur through extinction of some portfolio rules. In a strict sense, all of the above papers deal with the issue of selection rather than the process of mutation that creates novel behavior. The question whether the Kelly rule emerges in a market where traders adapt their strategies is investigated in Lensberg and Schenk-Hoppe (2007). They combine the evolutionary model with traders whose decisions are made by genetic programs. In that setting, the strategies and the dynamics of wealth shares co-evolve in a process of market interaction and tournament selection of strategies (in which poor strategies are replaced by --- ## Page 310 Survival and Evolutionary Stability of the Kelly Rule 283 novel rules). Their numerical analysis shows that the Kelly rule evolves in the long-run but prices converge much faster than the individual strategies to those prescribed by the Kelly rule. Successful traders, interestingly, invest according to a fractional Kelly 'rule rather than its pure form. This approach drastically reduces the volatility of investment returns, which helps to avoid deletion by the tournament selection process. The optimality properties of the fractional Kelly rule for practical investment are discussed in MacLean et al. (1992). 5 Conclusion This survey presents in detail an evolutionary model of financial markets. The model is described as a simultaneous-move N -person dynamic game. Restricting the space of strategies to those depending only on the history of states (rather than the entire information consisting of the market history and revealed strategies of competing traders), one obtains the earlier, behavioral models evolutionary finance. The main findings of the survival and evolutionary stability properties of investment strategies are discussed. It turns out that the Kelly rule (appropriately defined to fit the framework under consideration) ensures the survival of the traders following this portfolio rule. The Kelly rule brings the additional benefit of (global) evolutionary stability. References Alchian, A. (1950). Uncertainty, evolution and economic theory. Journal of Political Economy, 58, 211- 221. Algoet, P. H. and T. M. Cover (1988). Asymptotic optimality and asymptotic equipartition properties of log-optimum investment. Annals of Probability, 16, 876- 898. Amir, R , I. V. Evstigneev, T. Hens and K. R Schenk-Hoppe (2005). Market selection and survival of investment strategies. Journal of MathematicaL Economics, 41, 105- 122. Amir, R , I. V. Evstigneev, T. Hens and L. Xu (2009) . Evolutionary finance and dynamic games. Working Paper No. 581. NCCR "Financial Valuation and Risk Management" , Switzerland, August 2009. Amir, R, I. V. Evstigneev and K. R Schenk-Hoppe (2008). Asset market games of sur- vival. Working Paper No. 505. NCCR Financial Valuation and Risk Management, Switzerland, Revised version, June 2010. Arnold, L. (1998) . Random Dynamical Systems. US: Springer. Arthur, W. B., J. H. Holland, B. LeBaron, R G. Palmer and P. Taylor (1997). Asset pricing under endogenous expectations in an artificial stock market. In The Economy as an Evolving Complex System II (eds. Arthur, W. B., Durlauf, S. and Lane, D.) , pp. 15- 44. MA: Addison Wesley. Blume, L. and S. Durlauf, (eds.) (2005). The Economy as an EvoLving CompLex System III. New York: Oxford University Press. Blume, L. and D. Easley (1992). Evolution and market behavior. Journal of Economic Theory, 58, 9- 40. Blume, L. and D. Easley (2008). Market competition and selection. In The New Palgrave Dictionary of Economics, Vol. 5 (eds. Blume, L. and Durlauf, S. N.) , pp. 296- 300. UK: Macmillan. --- ## Page 311 284 1. V. Evstigneev, T Hens and K. R. Schenk-Hoppe Blume, L. and D. Easley (2009). Market selection and asset pricing. In Handbook of Finan- cial Markets: Dynamics and EvoLution (eds. Hens, T. and Schenk-Hoppe, K. R), Chapter 7, pp. 403-437. Amsterdam: North-Holland. Breiman, L. (1961) . Optimal gambling systems for favorable games. In Proceedings of the Fourth BerkeLey Symposium on Mathematical Statistics and Probability, Vol. 1 (ed. Neyman, J.) , pp. 65-78. US: University of California Press. Dubey, P., J . Geanakoplos and M. Shubik (1987). The revelation of information in strategic market games: A critique of rational expectations equilibrium. JournaL of Mathe- matical Economics, 16, 105-137. Evstigneev, I. V., T. Hens and K. R Schenk-Hoppe (2002). Market selection of financial trading strategies: Global stability. MathematicaL Finance, 12, 329- 339. Evstigneev, I. V., T. Hens and K. R Schenk-Hoppe (2006). Evolutionary stable stock markets. Economic Theory, 27, 449- 468. Evstigneev, I. V., T. Hens and K. R Schenk-Hoppe (2008). Globally evolutionarily stable portfolio rules. Journal of Economic Theory, 140, 197-228. Evstigneev, I. V., T. Hens and K. R Schenk-Hoppe (2009). Evolutionary Finance. In Handbook of Financial Markets: Dynamics and EvoLution (eds. Hens, T. and Schenk- Hoppe, K. R), Chapter 9, pp. 507-566. Amsterdam: North-Holland. Farmer, J. D. and A. W. Lo (1999). Frontiers of finance: Evolution and efficient markets. Proceedings of the NationaL Academy of Sciences, 96, 9991- 9992. Hakansson, N. H. and W . T. Ziemba (1995). Capital growth theory. In Handbooks in Operations Research and Management Science, Vol. 9, Finance (eds. Jarrow, R. A. , Maksimovic, V. and Ziemba, W. T.) , Chapter 3, pp. 65- 86. Amsterdam: North- Holland. Hens, T. and K. R Schenk-Hoppe (2005). Evolutionary stability of portfolio rules in in- complete markets. Journal of Mathematical Economics, 41, 43- 66. Hodgeson, G. M. (1993) . Economics and Evolution: Bringing Life Back into Economics. US: Blackwell. Kelly, J. (1956). A new interpretation of information rate. Belt System TechnicaL JournaL, 35, 917-926. Laffont, J.-J. (1989) . The Economics of Uncertainty and Information. MA: MIT Press. LeBaron, B., W. B. Arthur and R Palmer (1999). Time series properties of an artificial stock market. Journal of Economic Dynamics and Control, 23, 1487-1516. Lensberg, T. and K. R Schenk-Hoppe (2007). On the evolution of investment strategies and the Kelly rule - A Darwinian approach. Review of Finance, 11, 25- 50. MacLean, L. C. , W. T. Ziemba and G. Blazenko (1992). Growth versus security in dynamic investment analysis. Management Science, 38, 1562- 1585. Marshall, A. (1949). Principles of Economics (8th edition). UK: Macmillan. Milnor, J . and L. S. Shapley (1957). On games of survival. In Contributions to the Theory of Games III (eds. Dresher, M., Tucker, A. W. and Wolfe, P.) Annals of Mathematics Studies 39, pp. 15- 45. US: Princeton University Press. Radner, R (1972). Existence of equilibrium of plans, prices, and price expectations in a sequence of markets. Econometrica, 40, 289- 303. Shapley, L. S. (1953). Stochastic games. Proceedings of the National Academy of Sciences of the USA, 39, 1095-1100. Shubik, M. and W. Whitt (1973). Fiat money in an economy with one durable good and no credit. In: Topics in Differential Games (ed. Blaquiere, A.), pp. 401- 448. Amsterdam: North-Holland. Thorp, E. O. (1971). Portfolio choice and the Kelly criterion. In Business and Economics Statistics Section, Proceedings of the American StatisticaL Association, pp. 215- 224. --- ## Page 312 Lecture Notes of the Institute for Computer Science, 4, 1051- 1062 (2009) 21 Application of the Kelly Criterion to Ornstein- Uhlen beck Processes Yingdong Lv & Bernhard K. Meister Department of Physics, Renmin University of China Email:lyd08250@163.com & b_meister@ruc.edu . cn Abstract. In this paper, we study the Kelly criterion in the continuous time framework building on the work of E.O. Thorp and others. The existence of an optimal strategy is proven in a general setting and the corresponding optimal wealth process is found. A simple formula is provided for calculating the optimal portfolio for a set of price processes satisfYing some simple conditions. Properties of the optimal investment strategy for assets governed by multiple Ornstein-Uhlenbeck processes are studied. The paper ends with a short discussion of the implications of these ideas for financial markets. Keywords: utility function; optimal investment strategy; self-fmancing; complete market; risk-neutral measure; Brownian motion; Ornstein-Uhlenbeck. 1 Introduction 285 The Kelly Criterion [1], [2] was initially introduced in 1956 to find the optimal betting amount in games with fixed known odds, and was later extended to the field of financial investments by E. O. Thorp and others. The strategy maximizes the entropy and with probability one outperforms any other strategy asymptotically [3]. This approach was recently further developed by Kargin [4], who applied the criterion to a mean-reverting asset process under liquidity and credit constraints. The Kelly Criterion tells us that the optimal betting fraction is given by p-q, if a gambler is faced with a bet, where the probability to double the money is p and to lose the initial stake is q (p>q). The optimal betting fraction maximizes the expected log wealth. The question, why investors should choose to maximize the log wealth, has a simple answer: according to Breiman's theorem [3], it gives the asymptotically optimal pay-out and dominates any other strategy. In this paper, we start by extending the original idea to the general continuous time framework with n correlated assets. Our task is to find the optimal self-financing trading strategy. We will prove that if the market is complete, this optimal self- financing trading strategy always exists. A limited number of applications are discussed in the context ofOrnstein-Uhlenbeck processes. The paper is organized as follows. In section 2 we review the standard assumptions and prove the optimization theorem. The theorem covers both the existence of the optimal trading strategy and the explicit form of the associated optimal investment fraction. In section 3 we apply the theory to a market of n correlated assets given by --- ## Page 313 286 Y Lv and B. K. Meister Application o/the Kelly Criterion to Ornstein-Uhlenbeck Processes Omstein-Uhlenbeck mean-reverting processes. The optimal investment strategy is calculated for some representative examples. In the last section we put the results into the financial context and describe some open problems. 2 General Theory In the first section, we will cover the assumptions and the theoretical framework. In 2.2 we will provide the main result, which is contained in Theorem 1. 2.1 Basic Assumptions and other Preliminaries We will use the standard notation and conventions of financial mathematics. In section 2 the basic assumption is that the market is complete and frictionless [5]. Let us further assume that we consider all the processes in the finite time interval ° to the terminal time T. There exists a probability space (.0, PT , Yr), on which all of the random variables are constructed, where .0 is the sample space, Yr is a (J -algebra which denotes the information accumulated up to time T and PT is the spot probability measure [5]. The filtration Sf ,t E [0, TJ ' represents the information accumulated up to time t. The sub-probability space (.0, PI'..r,) is introduced at time t, where Pt is the restriction ofPT on the filtration:if . We assume there are n+ 1 investable assets in the market including the wealth process B" representing a saving account with value 1 at the initial time 0. We assume B, follows (1) where r, is the short term rate at time t. The other n assets in the market are denoted by Si (t), 1 E [0, T] ,i = 1,2, ... , n, and we T T define a n-dimensional vector byS, ={SI (/),S2(/), ... ,Sn (I)) ,where' 'represents the transposition of a matrix. Let us define the relative assets price process by S, = S,B,-l . Let rPo (t) denotes the number of units of B, an investor holds at time t and rPi (I) , t E [0, TJ ' i = 1,2, ... , n , denotes the number of units of the /h asset an investor holds at time I. In addition, the n-dimensional vector rP, is defined as rP, = {rPl (I ),rPz (t ), ... ,rPn (t) r . v, (lJI) is the total value of the portfolio lJI, = (rPo (I ),rP, ). So we have (2) where rP, . S, is the inner product of two vectors. 2 --- ## Page 314 Application of the Kelly Criterion to Ornstein-Uhlenbeck Processes 287 Application of the Kelly Criterion to Ornstein-Uhlenbeck Processes Definition 1 A self-financing trading strategy lj/, = (cPo (t) ,cP, ) is a strategy that satisfies: dV, (lj/) = cPo (t ) dB, + cP, . dS" \it E [OJ] . (3) We assume Vo ( lj/ ) = J • Definition 2 A self-financing trading strategy lj/, = (cPo (t), cP,) is said to be admissible if and only if (4) U (x) , x ;:: 0, is defined to be a concave function representing the utility of wealth. Here concaveness means Further it is assumed that U (x) has a first order derivative for \ix E (0, +00). The first order derivative at x = 0 can be either finite or infinite, and the first order derivative of U (x) , x;:: 0 , is a strictly decreasing function of x with lim U' (x) = 0 . x-->+oo IfU' (0) = +00 , let I (x) , x;:: 0, be the inverse function ofU' (x) with 1(0) = +00 and I (+00) = 0 . For U' (0) = b > 0 , we denote by Ib (x), X E [0, b]' the inverse function ofU' (x) ,with 1(0) = +00 . In this case, we detine 1 (x) as (6) Let us denote by ~ the class of all of the admissible self-financing trading strategies. We say a self-financing trading strategy lj/' E ~ is the optimal trading strategy, if and only if Our task is to find an optimallj/' E §J , which satisfies eq.7. 2.2 The Optimal Strategy To find the optimal strategy, we will first need to introduce the following lemma .. Lemma J The function / (x), X E [0, +00) satisfies the following inequality: U(I (y))- yI(y);:: U (c)- yc, \iy,c E [0,+00) . (8) 3 --- ## Page 315 288 Y. Lv and B. K. Meister Application of the Kelly Criterion to Ornstein-Uhlenbeck Processes Proof: If I (y) = c , then eq.8 is obviously satisfied. If I (y) > c , then the average growth rate of U (x) from c to I (y) should be larger than the first order derivative of U (x) at I (y) , which is given by U' (I (y)) , since the first order derivative of U (x) is a strictly decreasing function of x. The average growth rate of U (x) from c to I (y) is This yields the following inequality U(I(y))-U(c) I(y)-c U(I(y))-U(c) > '{ ( ))= ( ) _U 1 y Y 1 y -c =:} U (/(Y))- yI(y) 2 U(c)- yc An almost identical argument can be applied in the case I (y) < c . o - & Let us define PT as the martingale of the market and Z{ = -d-. Then (Z{,.5D is &( a Pr martingale [5]. Define ,,; = y{Z{-1 B{-I , V;. = 1(,,;), where we assume y{ is a deterministic function of t and is defined in such a way that v,' = V;* B{-l is a Pr martingale. So y{ solves the equation E[B-1I ( Z-I B-1)] = 1 ( y" , (9) The deterministic property ofy{ seems contrived, but is necessary for the proof of Proposition 1. As we shall see in section 3, in the case of the log utility function the deterministic function y, indeed exists and is a constant. Proposition 1 v; = 1(,,; ) satisfies the inequality given by eq. 7. Proof: Let VT (If!) , V fj/ E f!» , be the wealth process corresponding to a special trading strategy fj/ , then Epr [ U (V;~ ) ] - Epr [ U (VT ( fj/ ) ) ] = EpT [( U ( 1 ( ,,; ) ) - ,,;1 ( ,,; ) ) - ( U (VT (fj/ ) ) - ,,;VT ( fj/ ) ) ] + Epr [ ,,; (V; - Vr ( IfI ) ) ] (10) 4 --- ## Page 316 Application of the Kelly Criterion to Ornstein-Uhlenbeck Processes 289 Application of the Kelly Criterion to Ornstein-Uhlenbeck Processes According to lemma 1, the first term of the right hand side of eq.1 ° is positive. The second term is equal to zero, The last equality of the above equation is deduced from the fact that both v,' and V, (1.jI ) are martingales under the martingale measure. Combining eq.lO and eq.11, we directly get o Proposition 1 only state the fact that! (1J:) satisfies eq.7. It doesn't necessarily mean that! ('l,') is the optimal wealth process. We will prove in the following theorem that J ( 1J: ) is in fact the optimal wealth process. Theorem 1 Given a concave utility function U (x), there exists an optimal self financing trading strategy 1.jI', such that for each time t E [0, T], the wealth process ~ ( 1.jI* ) of this strategy satisfies And the optimal wealth process is given by: ~ ( 1.jI') = J ( 1J:) , t E [0, T] . Proof: Define V; to be Br- ' J (1J:). For t= T, we have V; = BrV' = J (1J;) , which represents a general contingent claim in the market. Since the market is complete, the contingent claim ~~ is attainable. This means there exists a self-financing trading strategy 1.jI' such that V;. = Vr (1.jI') Br- ' , where ~ (1.jI') is the wealth process of this self- financing trading strategy. So the relative wealth processv,(I.jI·)=~(I.jI·)Br-'is a martingale under the martingale measure i\ . V; is also a martingale under the martingale measure Pr , and we have Eq.12 shows that 1.jI* is a selt:'financing trading strategy and replicates the optimal wealth process V; = B,-' J (1J;) . From the combination of v,', satisfYing eq.7 for any 5 --- ## Page 317 290 Y Lv and B. K. Meister Application of the Kelly Criterion to Ornstein-Uhlenbeck Processes time t before T, and eq.12, we can see that V, (If") also satisfies eq.7. This proves that the strategy If" is both self-financing and optimal. 0 It follows from Proposition 1 and Theorem 1 that the existence of a self-financing trading strategy If" E §" , where the total wealth at a fixed time T is consistent with eq. 7, implies that the wealth process V, ( If' +) satisfies Therefore, an optimal trading strategy for a fixed time T will be optimal for any time before T. It follows further that an optimal trading strategy is only based on the information up to time t. The optimal trading strategy If': is an adapted process with respect to the filtration { Sf, t E [0, -t 0 is some nonnegative real number andCTi,j are constants. W, = (W;I,W;l, ... ,w;nf is a standard n-dimensional Brownian motion. Define a to be the vector (ai' al , ... , an f (' T ' is the transposed of a matrix), b to be the { b . = b, i = j n Xn matrix of the form b = I,J 0' , and (J to be the matrix (J .. = CT . , bi,j = , i #- j I,J I,J for 1:'0: i,j :'0: n .The matrix (J has a non-zero determinant. Then the dynamic equation of x, = (Xl (t) 'Xl (t),,,,,xn (t))' can be expressed as dx, = [a- bx, ]dt + (JdW, (13) LetS, = (Sl (t),S2 (t),,,,,Sn (t))' ,S, = B,-IS, . According to Ito's lemma, the dynamic ofS, is 6 --- ## Page 318 Application of the Kelly Criterion to Ornstein-Uhlenbeck Processes Application of the Kelly Criterion to Ornstein-Uhlenbeck Processes dSj (I) = S; (/),lI; (t)dt+ Sj (t) IO"j,jdW/ ,i = 1,2, ... , n j ~1 where,ll, (t) = a; - b; log (S; (t)) + ~11(fi 1/2 and (fi = (0";,1, 0";,2' ""O"i,n f . Define :7 =exp fo" ,dW" -~ II/o" 1/2 du , (T T) 7 0 0 291 (14) where 9/1 = ( tH u ), cq u ), ... , en ( u ) f is a n-dimensional adapted stochastic process and 119" 1/ = Jel2 (u) + ... + e; (u) is the Euclidean vector norm, and Wt a Girsanov t transformed Brownian motion, i.e. W t = Wt - f9"du. o Then under i\, St it follows Define cj (t) = ,lIj (I) - r and in vector form ct = (cl (t) , c2 (t) , ,." cn (t) t . If9t solves then under 1\ it follows that (15) (16) - . -_ {sfj (t ) = S; (t ) , i = j - where the matrix Y, is defined as. Y, - - ( ) = .' . St is a martingale y"j to, 1* J under 1\ . To apply theorem 1, we need first to prove the completeness of the market price processes under consideration. The next lemma tells us that indeed the market is complete. The proof is given in an earlier presentation [7]. Lemma 2 The mean-reverting market given above is complete. In case U (x) = log(x) , we find J (x) = 11 x. Using eq.9, we can show y, = 1. Then the optimal discounted wealth process is Now we are in a position to derive a general result for Omstein-Uhlenbeck processes. 7 --- ## Page 319 292 Y Lv and B. K. Meister Application of the Kelly Criterion to Ornstein-Uhlenbeck Processes Theorem 2 The optima! trading strategy f//; = (< 00 . (3) The mean first passage time to reach the set [U, (0), that is, how long on average does it take the investor to reach a specific level of wealth. The measures of security are: (1) The probability that the investor will have a specific accumulated wealth at time t, that is what are the chances of reaching a given target in a fixed amount of time. --- ## Page 329 302 L. C. MacLean, E. 0. Thorp and W. T. Ziemba (2) The probability that the investor's wealth is above a specified path. (3) The probability of reaching a goal U which is higher than the initial wealth Yo before falling to a wealth level L below Yo. MacLean et al. study growth-security efficiency analgously to mean-variance efficiency as well as the weaker conditions of growth-security monotonicity. This latter tradeoff is called effective. They show that fractional Kelly strategies trace out effective growth-security tradeoffs. Later in MacLean, Ziemba, and Li (2005), it is shown that these fractionally Kelly strategies are efficient, if the assets are log-normally distributed, so they maximize growth for any given security level. The analysis leads to simple two dimensional graphs that compare growth rates with the probability of achieving a higher goal before falling to a lower level. Applications are made to blackjack, horse racing , lotto games, and futures trading. This range of applications runs the full range of asset weights from very small to very large percentages of the investor's wealth. MacLean, Sanegre, Zhao, and Ziemba (MSZZ) (2004) provide an approach to calculate the optimal fractional Kelly weights at discrete intervals of time. The idea is to maximize the expected log subject to being above a specified wealth path with high probability. This is a value at risk type of constraint and leads to a model where a modified Kelly strategy can be computed at the discrete intervals of the planning horizon. This strategy is not necessarily fractional Kelly. In general, as risk rises, the expected return will fall so there is an effective tradeoff. With geomet- ric Brownian motion in continuous time, fractional Kelly is optimal for the value at risk (VaR) and conditional value at risk (CVaR) where returns in the VaR tails are lineraly penalized constraints. The algorithm proposed uses a disjunctive form for the probabilistic constraints which identifies an outer problem of choosing an optimal set of scenarios and an inner (conditional) problem of finding the optimal decisions for a given discrete scenario set. The multiperiod inner problem is com- posed of a sequence of conditional one period problems. The theory is illustrated for the dynamic allocation of wealth in stocks, bonds and cash. There are two ways to model the application, maximize the expected log of final wealth subject to (1) wealth being above the specified wealth path with high probability; and (2) a secured annual drawdown with a drawdown amount and a security level both exogenously specified. The VaR model only controls loss at the horizon but not at intermediate decision points. The more stringent risk control constraint in the drawdown model considers the loss in each period. At low levels of risk control, the full Kelly strategy is optimal. As the risk control requirements are raised, the strategy becomes more conservative, especially when close to the planning horizon. Assets are assumed to be log-normally distributed with geometric random walk but the computational procedure is general and applies to general price distributions. --- ## Page 330 Introduction to the Relationship of Kelly Optimization to Asset Allocation 303 In an extension, MacLean, Zhao, and Ziemba (2009) add one more feature to the MSZZ model, namely that when the predetermined exogenous path is violated, there is a convex penalty for there shortfall violations (see Mulvey, Bilgili, and Vural (2010) in part VI for an application along these lines). The next three papers discuss the impact of benchmarks. Browne (2000) con- siders a dynamic portfolio management problem where the objective concerns the tradeoff of performance goals and the risk of a shortfall below a benchmark target. Return is measured by the expected time to reach investment goals relative to the benchmark. This return is maximized subject to the risk constraint in terms of the probability of shortfall relative to the benchmark. The setting is Merton's (1971) continuous time framework. Davis and 11eo (2009) extend the definition of fractional Kelly strategies to the situation in which the investor's objective is to outperform a benchmark. These benchmarked fractional Kelly strategies are efficient portfolios even when the asset returns are not log-normally distributed. They find the benchmarked fractional Kelly strategies for various types of benchmarks such as the S&P500 and the Sa- lomon Smith Barney World Government Bond Index. They also determine the re- lations between the investor's risk aversion and the given benchmarks. The setting is a stochastic control model in continuous time. They develop, following Merton, two and three mutual fund theorems. This means that the optimal weights over n assets can be found by optimally weighting the two or three mutual funds which are linear combinations of the original n assets, where n > 3, see Rudolf and Ziemba (2004) for a four mutual fund theory with liabilities in part VI; see also Davis and 11eo (2008a,b). In addition to single index benchmarks, there can be composite benchmarks plus alpha or composite weighted multiple index benchmarks with or without an alpha. The Davis and 11eo two fund theorem for the case without and with benchmarks is that the assets can be split between the full Kelly portfolio and a correction portfolio C with the weights dependent on the investor's risk sensitivity. C is an intertemporal hedging portfolio, which can be the risk free asset under certain assumptions. They develop a three mutual fund theorem for the case of benchmarks with risky and risk free assets. Formulas are developed for the optimal equity portfolio weightings in the various mutual funds. Any portfolio can be expressed as a linear combination of a mutual fund, an index fund and a long-short hedge fund with risky and risk free asset allocations. Again, the weights are dependent on the risk sensitivity. The risk sensitivity varies from zero (the full Kelly expected log case) to infinity where the investor replicates the benchmark with an index fund. With moderate risk sensitivity there will be a blend of all three asset funds. Platen (2010) discusses the benchmark approach for dynamic investing and pric- ing. A numeraire portfolio exists which is strictly positive that makes all bench- marked nonnegative portfolios downward trending or trendless. The benchmark --- ## Page 331 304 L. C. MacLean, E. 0. Thorp and W T. Ziemba portfolio is the Kelly E log maximizing portfolio which cannot be outperformed by another long only portfolio. The benchmark portfolio can be used for pricing and computing conditional expectations. The final paper in this part deals with fixed-mix and volatility induced wealth growth. Dempster, Evstigneev, and Schenk-Hoppe (2009) discuss the literature con- cerning the growth of wealth over time through volatility using fixed-mix strategies. These self-financing strategies rebalance the portfolio at each decision point to keep constant the proportions of wealth invested in various assets. Through volatility, the portfolio will gain wealth even with assets that do not have strictly positive means. This is a generalization of the buy-low sell-high investment strategy. But they show that there is much more to the story than this implies. The analysis does not need utility assumptions and compares the fixed-mix rebalancing with buy and hold strategies. The key assumption is that the period by period returns of the var- ious assets are stationary. Dempster, Evstigneev, and Schenk-Hoppe discuss some counter-intuitive examples such as growth with assets having zero means. They discuss the effects of transaction costs on their results and determine conditions for growth to occur with sufficiently small transaction costs. They also show that the conjecture, the higher the volatility, the higher the induced growth rate is not true in general. The key idea is to rebalance to an arbitrary fixed mix portfolio. --- ## Page 332 This page is intentionally left blank --- ## Page 333 This page is intentionally left blank --- ## Page 334 MATHEMATICS OF OPERATIONS RESEARCH 23 Vol 22, No 2, May 1997 Pmucdlll USA SURVIVAL AND GROWTH WITH A LlABILITY: OPTIMAL PORTFOLIO STRATEGIES IN CONTINUOUS TIME SID BROWNE We siudy the optimal behavior of an investor who is forced to withdraw funds continuously at a fixed rate per unit time (e.g., to pay for a liability, to consume, or to pay dividends). The investor is allowed to invest In any or all of a given number of risky stocks, whosc prices follow geometnc Brownian motion, as well as in a nskless asset which has a constant rate of return. The fact that the withdrawal is continuously enforced, regardless of the wealth level, ensures that there IS a region where there is a positive probability of ruin. In the complemen- tary region ruin can be avoided with certainty. Call the former region the danger-zone and the latter region the safe-region. We first consider the problem of maximizing the probability that the safe-region is reached before bankruptcy, which we caU the survival problem. While we show, among other results, that an opltmal policy does not eXIst for this problem. we are able to construct explicit f-optimal policies, for any E > O. In the safe-region, where ultimate survival is assured, we turn our attention to grm'1h. Among other results, we find the optimlli growth poitcy for the Investor, i.e., the pohcy which reaches another {higher valued} goal as quickly as possible. Other varIants of both the survival problem as well as the growth problem are also discussed. Our results for the latter are Intimately related to the theory of Constant Proportions PortfolIo Insurance. 307 1. Introduction. The problem considered here is to solve for the optimal invest- ment decision of an investor who must withdraw funds (e.g., to pay for some liability or to consume) continuously at a given rate per unit time. Income can be obtained only from investment in any of n + 1 assets: II risky stocks, and a bond with a deterministic constant return. The objectives considered here relate solely to what can be termed "goal problems," in that we assume the investor is interested in reaching some given values of wealth (called goals) with as high a probability as possible and/or as quickly as possible. The fact that the investor must continuously withdraw funds at a fixed rate introduces a new difficulty that was not present in the previous studies of objectives related to reaching goals quickly (cf. Heath et al. 1987). Specifically the forced withdrawals ensure that at certain levels of wealth, there is a positive probability of going bankrupt, and thus the investor is forced to invest in the risky stocks to avoid ruin. In this paper we address the two basic problems faced by such an investor: how to survive, and how to grow. The survival problem turns out to be somewhat tricky, in that we prove that no fully optimal policy exists. Nevertheless, we are able to construct e-optimal policies, for any e > O. The growth problem is answered com- pletely, once the survival aspect is clarified. While this model is directly applicable to the workings of certain economic enterprises, such as a pension fund manager with fixed expenses that must be paid continuously (regardless of the level of wealth in the fund), our results are also related to investment strategies that are referred to as Constant Proportion Portfolio Received January 11, 1995; reVised October 3,1995; July 20,1996. AMS 1991 subject claSSIfication. Primary: 90A09, 60HlO; Secondary: 93E20, 60G40, 60J60. OR/MS Index 1978 subject classificallon. Primary: Finance/Portfoho; Secondary: Optimal controljStochas- tic. Key words. Stochastic control, portfolio theory, diffuSions, martingales, bankruptcy, optimal gambling, Hamilton-Jacobi-Bellman equatIOns, portfolio insurance. 468 0364-765X/97/2202/0468/$05.00 CopYflght S 1997, lnc;htutc for Opera lion!!. Rt'-.c.tTch and the Managcmt:nl SClencc'!' --- ## Page 335 308 S. Browne OPTIMAL PORTFOLIO STRATEGIES 469 Insurance (eppI). In fact a related model was used as the economic justification of epPl in Black and Perold (1992), where both the theory and application of such strategies is described. In Black and Perold (1992), optimal strategies were obtained for the objective of maximizing utility of consumption, for a very specific utility function, subject to a minimum consumption constraint. However, the analysis and policies of Black and Perold (1992) are relevant only when initial wealth is in a particular region (specifically, when initial wealth is above the "floor"), wherein for that policy, there is no possibility of bankruptcy. Black and Perold (1992) did not address the fact that for the model described there, ruin, or bankruptcy, is a very real possibility when initial wealth is in the complementary region (below the "floor"). Here we focus on the objectives of survival and growth, which are intrinsic objective· criteria that are independent of any specific individual utility function. As such, our results for both aspects of the problem will therefore complement the results of Black and Perold (1992) (as well as the more recent related work of Dybvig 1995). Firstly, the survival problem has not been addressed before for this model (although see Majumdar and Radner 1991 and Roy 1995), and secondly, the optimal growth policies we obtain provides another objective justification for the use of the epPI policies prescribed in Black and Perold (1992), since for this problem we get similar policies as those obtained there. The remainder of the paper is organized as follows: In the next section, we will describe the model in greater detail, and prove a general theorem in stochastic control from which all our subsequent results will follow. To facilitate the exposition, we will at first consider the case where there is only one risky stock and where the withdrawal rate is constant per unit time. It turns out that the state space (for wealth) can be divided into two regions, which we will call the "danger-zone" and the "safe-region." In the latter region. the investor need never face the possibility of ruin, and so we can concentrate purely on the growth aspects of the investor. (The aforementioned studies of Black and Perold 1992 and Dybvig 1995 considered only this region in their analyses of the maximization of utility from consumption problem.) In the former region, ruin, or bankruptcy, is a possibility (hence the term "danger- zone") and therefore we first concentrate on passing from the danger-zone into the safe-region. This is the survival problem and it is completely analyzed in §3. In particular, two problems are considered, maximizing the probability of reaching the "safe-region" before going bankrupt, and minimizing the discounted penalty that must be paid upon reaching bankruptcy. It is the former problem that does not admit an optimal policy, although we are able to explicitly construct an e-optimal policy. The latter problem does admit an optimal policy, which we find explicitly. The structure of both policies are quite similar, in that they both essentially invest a (different) proportional amount of the distance to the safe-region. In §4 we consider the growth problem in the safe-region. We define growth as reaching a given (high) level of wealth as quickly as possible. Two related problems are then solved completely. First, we find the policy that minimizes the expected time to the (good) goal, and then we find the policy that maximizes the expected discounted reward of getting to the goal. Our resulting optimal growth policies turn out to be quite similar to the epPI policies obtained for a different problem by Black and Perold (1992), in that they invest a (different) proportional amount of the distance from the danger-zone. Extensions to the multiple asset case as well as the case of a wealth-dependent withdrawal rate are discussed in §§5 and 6. All the control problems considered in this paper are special cases of a particular general control problem that is solved in Theorem 2.1 in §2 below. For this problem we use the Hamiiton-lacobi-Bellman (H1B) equations of stochastic control (see, e.g., Fleming and Rishel 1975, or Krylov 1980) to obtain a candidate optimal policy in --- ## Page 336 Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time 309 470 s. BROWNE terms of a candidate value function and this value function is then in turn given as the solution to a particular nonlinear Dirichlet problem. These candidate values are then verified and rigorously proved to be optimal by the martingale optimality principle (see §V.lS in Rogers and Williams 1987, or §2 in Davis and Norman 1990). The resulting nonlinear differential equations are then solved in turn for each of the problems considered below, yielding the optimal solutions in explicit form. 2. The model and continuous-time stochastic control. Without loss of generality, we assume that there is only one risky stock available for investment (e.g., a mutual fund), whose price at time t will be denoted by Pl. {Extension to the multidimen- sional case (for a complete market) is quite straightforward, and since the excess notation required adds little to the understanding, we will simply outline how to obtain the results for the multidimensional case in a later section.) As is quite standard (see, e.g., Merton 1971, 1990, Davis and Norman 1990, Black and Perold 1992, Grossman and Zhou 1993, Pliska 1986), we will assume that the price process of the risky stock follows a geometric Brownian motion, i.e., P, satisfies the stochastic differential equation ( 1) where f.L and 0" are positive constants and {I¥,: t :?: O} is a standard Brownian motion defined on the complete probability space (0,7, P), where {9;} is the P-augmenta- tion of the natural filtration 9;w := O"{J¥.; ° ::S: s ::S: f}. (Thus the instantaneous return on the risky stock, dPtl PI' is a linear Brownian motion.) The other investment opportunity is a bond, whose price at time t is denoted by Bt • We will assume that (2) where r > O. To avoid triviality, we assume f.L > r. We assume, for now, that the investor must withdraw funds continuously at a constant rate, say e > 0 per unit time, regardless of the level of wealth. (This would be applicable for example if the investor faces a constant liability to which $e must be paid continuously.) In a later section we generalize this to a case where the withdrawals are wealth-dependent. Let It denote the total amount of money invested in the risky stock at time t under an investment policy f. An investment policy f is admissible if {f" t :?: O} is a measurable, {9;}-adapted process for which /[f/ dt < co, a.s., for every T < co. Let :9' denote the set of admissible policies. For each admissible control process f E :9', let {XI, t :?: O} denote the associated wealth process, i.e., X! is the wealth of the investor at time f, if he follows policy f. Since any amount not invested in the risky stock is held in the bond, this process then evolves as (3) dPt (f ) dBt dX[ = It p + X, - I, B - edt , , = [rX! + f,( f.L - r) - e] dt + ftO" d~ upon substituting from (1) and (2). --- ## Page 337 310 S. Browne OPTIMAL PORTFOLIO STRATEGIES 471 Thus, for Markov control processes f , and functions \ft(t , x) E ~,,j. 2, the generator of the wealth process is We will put no constraints on the control f, (other than admissibility). In particu- lar, we will allow /, < 0, as weJI as /, > X(. In the first instance, the company is selling the stock short, while in the second instance it is borrowing money to invest long in the stock. (While we do allow shortselling, it turns out that none of our optimal policies will ever in fact do this.) What will differentiate our model and results from previous work are the objectives considered and the fact that here the withdrawal rate c is constant, and not a decision variable. The usual portfolio and asset allocation problems considered in the financial economics literature deal with an investor whose wealth also evolves according to a stochastic differential equation as in (3), where instead of being constant, c is now a control variable as well, i.e., the consumption function c, = c(xf). For a specific utility function u(·), the investor's objective is then to maximize the expected utility of consumption and terminal wealth over some finite horizon, i.e., for T > 0, and some "bequest function" '1'(.), the investor wishes to solve (5) for some discount factor A ~ O. Alternatively, the investor may wish to solve the discounted infinite horizon problem (6) In both of these cases, since the process (e,l is usually assumed to be completely controllable, it is clear that for certain utility functions at least, ruin need never occur, since we may simply stop consuming at some level. Alternatively, as is the case when the utility function is of the form u(c) = C 1- R /0 - R) for some R < 1, or u(c) = In(c), the resulting optimal policy takes both investment I, and consumption c, to be proportional to wealth, i.e., f, = 1T'l{t)XI' C, = 1T'2(t)X" which in turns makes the optimal wealth process into a geometric Brownian motion, and thus the origin becomes an inaccessible barrier. Classical accounts of such (and more sophisticated) problems are discussed in Merton (1971, 1990) and Davis and Norman (1990) among others. Optimal investment decisions with constraints on consumption have also been considered in the literature previously. Most relevant to our model is the literature on constant proportion portfolio insurance (CPPI), ' as introduced in Black and Perold (1992), where the resulting policy is to invest a constant proportion of the excess of wealth over a given constant floor. (As its name suggests, portfolio insurance can be loosely considered any trading and investment strategy that ensures that the value of a portfolio never decrease below some limit. Alternative approaches to portfolio insurance using options and other techniques are described in e.g., Luskin 1988.) Black and Perold (1992) introduced this policy as the solution to the discounted infinite horizon problem of (6) subject to the constraint that c, ~ cmlO ' where cm," is --- ## Page 338 Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time 311 472 S. BROWNE some given constant. The specific utility function considered there was { C I - R/(l - R) U(C) = K - K c I 2 for c ~ c* , for c .:-;:; c*, where c* is a given constant, R ~ 1 and K], K2 are constants chosen to ensure u(') continuous throughout. While others (e.g., Dybvig 1995) have raised some technical questions about the analysis in Black and Perold (1992), more relevant to our point of view is the fact that this model (and the resulting optimal policy) allows for the possibility of ruin, or bankruptcy, if wealth is initially below the given floor. This possibility was never addressed in Black and Perold (1992). In this paper we do not concentrate on the usual utility maximization problems of (5) and (6). Rather, here we are concerned with the objective problems of survival and growth. In particular, we first study the problem of how the investor (whose wealth evolves according to (3» should invest to maximize the probability that the investor survives forever (which turns out to be related to maximizing the probability of achieving a given fixed fortune before going bankrupt), as well as the problem of how the investor should invest so as to minimize the time until a given level of wealth has been achieved. The former problem is called the survival problem, and is discussed in §3. The latter is called the growth problem and is the content of §4. Related problems have been studied in general under the label of "goal problems" in the works of Pestien and Sudderth (1985, 1988), Heath et al. (1987) and Orey et al. (1987). The survival problem for some specific related models were studied in Browne (1995) and Majumdar and Radner (991). The former treated an "incomplete market" model, where the withdrawals are not fixed but rather follow a stochastic process, and the latter treated a model with forced constant consumption but without the possibility of investing in a risk free asset. Recently, in order to provide a consumption based economic justification for the interesting portfolio strategies introduced in Grossman and Zhou (1993) (where the optimal policy invests a constant proportion of wealth over a stochastic floor), Dybvig (1995) considered the consumption-investment problem of (6) with the constraint that consumption never decrease, i.e., that C1 > cs ' for all t ~ s, with Co > O. Thus consump- tion is forced in his model as well. He considered utility functions of the form u(c) = c) - R /0 - R) and u(e) = In(e). However he only considered the problem in the feasible region, where initial wealth Xu , satisfies Xu > co/r, and so for which ruin need not occur. Dybvig (1995) did not consider the case when Xo > cu/r, and hence where ruin is possibility, and so our results on this problem in §3 will complement his analysis as well. Since in this paper our objectives deals solely with the achievement of particular goals associated with wealth, it is clear that if there is a constraint on consumption as in the models of Black and Perold (1992) and Dybvig (1995), we should always set consumption at the minimum level, which in both cases is a constant (cmm in Black and Perold 1992 and Co in Dybvig 1995). This is consistent with the model we analyze here, where we will (at least at first) take consumption as a fixed constant c per unit time. This implies that at least for some values of wealth, the origin is accessible, and thus ruin is in fact a possibility. In the next section we consider the problem of how to invest in order to survive. However, before we study that problem, we need a preliminary result from control theory that will provide the basis of all our future results. 2.1. Optimal control. The problems of survival and growth considered in this paper are all special cases of (Dirichlet-type) optimal control problems of the --- ## Page 339 312 S. Browne OPTIMAL PORTFOLIO STRATEGIES 473 following form: For each admissible control process {f" t ~ a}, let T! := inf{t > 0: Xf = z} denote the first hitting time to the point z of the associated wealth process {X!} of (3), under policy f. For given numbers (t. u) with I < Xo < u, let Tf := min{T!, Tj} denote the first escape time from the interval (I, u). For a given nonnegative continuous function A(x) ~ 0, a given real bounded continuous function g(x), and a function hex) given for x = I, x = u, let vf(x) be defined by with vex) = sup J1 f(x) and fv*(x) = argsup vf(x). ~ . ~ , We note at the outset that we are only interested in controls (and initial values x) for which vf(x) < 00. As a matter of notation, we note first that here, and throughout the remainder of the paper, the parameter 'Y will be defined by (8) THEOREM 2.1. Suppose that w(x): (I, u) ~ (- 00, (0) is a ~2 Junction that is the concave increasing (i.e., Wx > 0 and Wxx < 0) solution to the nonlinear Dirichlet problem (9) W2(X) (1X-C)WX to be solved for a function II is SUPfE ~{.wfv + g - All} = 0, subject to the Dirichlet boundary conditions v(1) = h(l) and v(u) = heu) (cf. Theorem 1.4.5 of Krylov 1980). Since vex) is independent of --- ## Page 340 Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time 313 474 S. BROWNE time, the generator of (4) shows that this is equivalent to (12) sup {U( p- - r) + rx - c)vx + if2(T2Vxx + g - >.v} = O. fE~' Assuming now that (12) admits a classical solution with /Ix > 0 and IIxx < 0 (see, e.g., Fleming and Saner 1993), we may then use standard calculus to optimize with respect to f in (12) to obtain the maximizer f.* = -« p- - r)/(T2)/lx/llxx (compare with (11)). When this r:(x) is then substituted back into (12) and the resulting equation is simplified, we obtain the nonlinear Dirichlet problem of (9) (with II = w). It remains only to verify that the policy g is indeed optimal. The aforementioned theorem in Krylav (1980) does not apply here, since in particular the degeneracy condition (Krylov 1980, page 23) is not met. We will use instead the martingale optimality principle, which entails finding an appropriate functional which is a uniformly integrable martingale under the (candidate) optimal policy, but a super- martingale under any other admissible policy, with respect to the filtration .7, (see Rogers and Williams 1987, Davis and Norman 1990). To that end, let A/(s, t) := J,'>'(X[) du, and define the process (13) M(t, Xl) := e - A1(O·l)w(X!) + j'e -,\ i(O")g(Xf) ds, for 0 S t S Tt, o where w is the concave increasing solution to (9). Optimality of g of (11) is then a direct consequence of the following lemma. LEMMA 2.2. For allY admissible policy f, and M(t, . ) as defined in (13), we have (14) with equality holding if and only iff = g, where f .* is the policy given in (11). Moreover, under policy fv* ' the process {M(t A TI, X,*" Tf )} is a unifonnly integrable martingale. PROOF. Applying Ito's formula to M(t, X() of (13) using (3) shows that for Os sst S Tt where Q(z; y) denotes the quadratic (in z) defined by Q(z;y) ,= z 2[ ~ 0, and hence the drift can be negative at certain wealth levels. This feature differentiates this model from those usually studied in the investment literature (e.g., Merton 1971, 1990, Pliska 1986, Davis and Norman 1990), with Majumdar and Radner (991) being a notable exception. (For results on an "incomplete market" model where the variance, as well as the drift, is also not completely controllable, see Browne 1995.) Specifically, let a denote the bankruptcy level or point, with corresponding "bankruptcy time"(or ruin time) 7/, where 0 s a < XO' One survival objective is then to choose an investment policy which minimizes the probability of ruin, i.e., one which minimizes P(7j < x), or equivalently, maximizes P(7! = 00) (see, e.g., Majumdar and Radner 1991, Browne 1995, Roy 1995). Clearly this objective is meaningless for xl ~ e/r. To see this directly, consider the case where the wealth level is x> e/r. We may then choose a policy which puts all wealth into the bond, and then under this policy the probability of bankruptcy is O. Specifically, if we take f = 0 for x > c /r, (3) shows that the wealth will then follow the deterministic differential equation dX, = (rX, - c)dt, Xo = x> e/r, which ex- hibits exponential growth and for which P(7x _ . = x) = 1, for all € > O. Thus the survival problem is interesting and relevant only in the region a < x < c/r, which we will call the "danger-zone." This is of course due to the fact that e /r = c J; e-n dt is the amount that is needed to be invested in the perpetual bond to payoff the forced withdrawals forever. Since the investor need never face the possibility of ruin for x > e Ir, we will call the region (c Ir, 00) the "safe-region." Our objective in this section therefore is to determine a strategy that maximizes the probability of hitting the safe-region or "safe pOint," c/r, prior 10 the "bankruptcy point," a, when initial wealth is in the danger-zone, i.e., a < x < c/r. As noted above, we will show that an optimal policy for this problem does 110t exist, necessitating the construction of an €-optimal strategy. A somewhat related survival problem with constant withdrawals was studied in Majumdar and Radner (1991) in a different setting, although without a risk-free investment, and hence without a safe-region. Moreover, their results are not applica- ble to our case since here inf! v2(J, x, t} = 0, which violates the conditions of the model in Majumdar and Radner (1991). As we shall see, it is in fact precisely this fact that negates the existence of an optimal policy for our problem. A related survival model which allows for investment in a risk-free asset, but where the "withdrawals" are assumed to follow another (possibly dependent). Brownian motion with drift, was treated in Browne (1995). Since the Brownian motion is unbounded, there was no safe-region in Browne (1995) either. A discrete-time model with constant withdrawals that does allow for a risk-free investment was treated in Roy (1995), but with no borrowing allowed and bounded support for the return on the risky asset. --- ## Page 343 316 S. Browne OPTIMAL PORTFOLIO STRATEGIES 477 To show explicitly why no policy obtains optimality for the model treated here, and how we may construct to-optimal strategies, we will first consider the following problem: for any point b in the danger-zone, i.e., with a < x < b < clr, we will find the optimal policy to maximize the probability of hitting b before a. For b strictly less than clr, an optimal policy does exist for this problem, and we will identify it in the following theorem. To that end let V( x: a, b) = SUpPx(7! > Tn, and let N = arg sup PAT! > 7{). ~? ~? THEOREM 3.1. The optimal policy is to invest, at each wealth level a < x < b, the state dependent amount (20) f;(x) = ~2~r(7:-x). The optimal value junction is (21) (c - ra r / r + 1 - (c _ 7X)y/ r +l Vex : a, b) = ( )y/ r+! ( b)y/r +" fora:s x:s b, c-ra - c-r where 'Y is defined by (8). REMARK 3.1. Note that the policy of (20) invests less as the wealth gets closer to the goal b. In fact, it invests a constant proportion of the distance to the "safe point" clr, regardless of the value of the goal b, and the bankruptcy point a. It is interesting to observe that while here this constant proportion is independent of the underlying diffusion parameter (T 2, this does not hold when there are multiple risky stocks in which to invest in (see §5 beloW). The constant proportion is greater (less) than 1 as ~/r < ( » 3. Thus it is interesting to observe that as the wealth gets closer to the bankruptcy point, a, the optimal policy does not " panic" and start investing an enormous amount, rather the optimal policy stays calm and invests at most f:(a) = 2(c - ra)/( ~ - d. The investor does get increasingly more cautious as his wealth gets closer to the goal b. (This behavior should be compared with the "timid" vs. "bold" play in the discrete-time problems considered in the classic book of Dubins and Savage 1965. See also Majumdar and Radner 1991 and Roy 1995.) Observe further that the investor is borrowing money to invest in the stock only when x < 2cl(jl + r) but not when 2c/(~ + r) x and in the latter case f:(x) < x. (The fact that 2c/( jl + r) < clr follows from the assumption that ~ > r.) PROOF. While we could prove Theorem 3.1 from a more general theorem in Pestien and Sudderth (1985) (see also Pestien and Sudderth 1988) which we will discuss later (see Remark 3.4 beloW), recognize that this is simply a special case of the control problem solved in Theorem 2.1 for I = a, U = b with A = 0, g = 0 and h(b) = 1, h(a) = O. As such the nonlinear Dirichlet problem of (9) for the optimal value function V becomes in this case (22) v2 (IX - c) Vx - ,.,; = 0, for a < x < b xx subject to the (Dirichlet) boundary conditions V(a) = 0, V(b) = 1. --- ## Page 344 Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time 317 478 S. BROWNE The general solution to the second-order nonlinear ordinary differential equation of (22) is KI - K 2(c - IX)"y/ r+ I, where Kp K2 are arbitrary constants which will be determined from the boundary conditions. The boundary condition V(a) = 0 deter- mines that K1 = Kic - ra)y/ r+l, and the boundary condition V(b) = 1 then deter- mines K 2, which then leads directly to the function Vex) given in (21). It is clear that this function V is in %,Z and does in fact satisfy V. > 0 and v.. < 0, and moreover satisfies conditions (0, (ii) and Gii) of Theorem 2.1 on the interval (a, b). (Condition (ij) is trivially met since V is bounded on (a, b).) As such V is indeed the optimal value function and the associated optimal control function ft of (20) is then obtained by substituting the function V of (21) for w in (11). 0 Note that under policy ft, the wealth process, say X,*, satisfies the stochastic differential equation (23) dX,* = (c - rX,*) dt + JL2~ r(c - rX,*) d~, for t ~ T*, where T* = min{Ta*, Tt}, and Tz* = inf{t > 0; X,* = z}. This is obtained by placing the control (20) into the evolutionary equation (3). Equation (23) defines a linear stochastic differential equation, i.e., X* is a time-homogeneous diffusion on (a, b) with drift function JL*(x) = c -IX, and diffusion coefficient function O'z*(x) = «20'1(JL - r»(c -IX»Z == (2/'YXc -IX)2. As such its scale function is defined by (24) S*(x) = {exp{ - t 2JLt(u/ dU} dy:= -( 'Y + r) -I(C - IX»)'/r+l, for a :::;; x ~ b 0' * (u where 'Y = i« JL - r) 1 0' )2. For this process therefore, * * S*(x) - S*(a) _ (c - ra)y/r+l - (c - IXr/r+ 1 P.(T. >Tb)= S*(b)-S*(a) = (c_ray>'/r+l_(c_rb)'Y/r+l' which of course agrees with (21). Thus the process (S*(X,*)} is a diffusion in natural scale, and is therefore a (uniformly integrable) martingale with respect to the filtration g; (as is the optimal value function), i.e., E(S*(X,*)Ig;) = S*(X.*) for o ~ s :::;; t ~ T*, where TOo := min{Ta*' Ttl. Note further that the scale function S*(x) of (24), is increasing in x (although negative) for 0:::;; x < clr. 3.1.1. Inaccessibility of the safe-region under f~ and E-optimal strategies. While we have found a policy that maximizes the probability of reaching any b < c Ir before any a < b, it is important to realize that if we extend b to clr, then this policy will never achieve the safe point c Ir with positive probability in finite time. We can of course extend the function displayed in (21) to the point c Ir to get S*(x)-S*(a) =1_(C-IX)y/r+1 < clr (i.e., for x> elr, we have ~(x) > 0 and ~/x) < 0), and thus Theorem 2.1 is not valid for any u > clr. REMARK 3.3. This difficulty disappears if r = 0, since if there is no risk-free investment, the investor always faces a positive probability of ruin and the only way to survive is to always invest in the risky stock. To see this, note that letting a = 0 and taking limits as r ! 0 (so that the "safe point" goes to infinity, i.e., when r = 0, there is always a positive probability of bankruptcy) shows that the value function, V(x: 0, e Ir), then goes to an exponential, i.e., as r J. 0, (27) Vex: O,clr) -+ 1 - exp { - 2:2eX} and for this case the (unconstrained) optimal control to minimize the probability of ruin is to always invest the fixed constant 2 e I 11-. (This model then becomes a --- ## Page 346 Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time 319 480 S. BROWNE degenerate special case of Browne 1995.) In this case the optimal wealth process follows a linear Brownian motion with drift c and diffusion coefficient 2cU" I J.L, for which the probability of ruin is the exponential (27). Ferguson (1965) conjectured that an ordinary investor (in discrete-time and space) can asymptotically minimize the probability of ruin by maximizing the exponential utility of terminal wealth, for some risk aversion parameter. It is interesting to observe that for this model the conjecture turns out to be true. To verify this, one would have to solve the finite-horizon utility maximization problem for the utility function u(x) = 0 - 1/ exp{ - 2exl J.L}, with arbi- trary 1] > 0 and o. Since this problem is then essentially a special case of the (Cauchy) problem considered in §3 of Browne (1995), we refer the reader there for further details. If we impose the constraint that the investor is not allowed to borrow, then it can be shown (Browne 1995, Theorem 3) that the optimal control in this case is f* = max{x, 2cl J.L}, whereby the investor must invest all his wealth in the risky stock when wealth is below the critical level 2c / J.L. In this case the value function is no longer concave below 2c I J.L. Such extremal behavior (or "bold" play, ala Dubins and Savage 1965) and nonconcavity of the value function below a threshold is also a feature of the optimal policies in the related survival models studied in Majumdar and Radner (1991) and Roy (1995), where borrowing is not allowed. REMARK 3.4. This inaccessibility and the resulting nonexistence of an optimal policy can be best understood in the context of the more general "goal" problem: Consider a controlled diffusion {Y/l on the interval (a, b) satisfying dY/ = m(f,y) dt + v(f,y) dW;, with the objective of determining a control to maximize the probability of hitting b before a. Let 'It(x) denote the optimal value function for this problem, i.e., 'It(x) = SUPt E go Px(Tj > TD with optimal control t/I(x) = arg SUPt E .If Px(Tj > 'rD· This prob- lem was first studied by Pestien and Sudderth (1985, 1988), who showed-using a different formulation-that (28) { m(f,x)} t/I( x) = argsj v2(f, x) , and indeed our f~(x) of (20) can be obtained from maximizing m/v2 for m and v2 in (19). However as noted in Pestien and Sudderth (1985, 1988), this is the case only when infx v~ (f, x) > 0, where t/I = m* Iv 2 *. These results can be obtained from somewhat simpler methods (albeit with some lesser generality) then those used in Pestien and Sudderth (1985, 1988) as follows: 'It(x) must satisfy the HJB equation (29) == sup {[m(f,x) 'It +.!.'It ] ·v2(f X)} = 0 2( f ) x 2 xx , , f v ,x subject to the Dirichlet boundary conditions 'I'(a) = 0, 'I'(b) = 1. If 'I' is a classical solution to the HJB equation (29), then we must have 'l'x > 0 and 'lJfxx < O. Therefore as long as v2(f, x) > 0, it is clear from (29) that the maximum of (29) occurs at the maximum of m/v2, which by (28) is denoted by t/I. If we now let p(x) = sUPr{m(f, x)/v2(J, x)}, i.e., p(x) = m(t/I(x), x)/v2(",(x), x), then the solu- tion to (29) subject to the Dirichlet conditions is simply 'I'(x) = faXs(z) dzlfabs(z) dz, --- ## Page 347 320 S. Browne OPTIMAL PORTFOLIO STRATEGIES 481 where s(z) = exp{ - 2 J 'p(y) dy}, with which our value function (21) of course agrees, for b < clr. However, it is also clear from (29) that for v2(f, x) = 0, the HJB equation need not hold, and therefore, no policy is in general optimal when this is the case, which is precisely what is happening here for b = clr (see also Example 4.1 in Pestien and Sudderth 1988). For more details on the general problem from a different perspective, we refer the reader to the fundamental papers of Pestien and Sudderth (1985, 1988). We now return to the problem of determining a 'good' strategy for crossing the clr barrier. An E-optimal strategy. As we have just seen, the inaccessibility of clr is due to the fact that N dictates an investment policy that causes the drift and variance of the resulting wealth process to go to zero as the clr barrier is approached from below. A practical way around this difficulty is to modify N as follows: Let i8* denote the (suboptimal) policy which agrees with It below the point elr - 8, and then above it invests K in the risky stock until the el r barrier is crossed, i.e., 18*(X) = {;(x) for x ~ elr - 0, for x> elr - o. Now V(xo, a, elr) as given in (25) is an upper bound on the probability of escaping the interval (a, elr) into the safe-region starting from an initial wealth level Xo < elr (see Krylov 1980, page 5). Without loss of generality, we may take a = 0 here. Thus for any € > 0, and initial wealth Xo < elr, the best we can do is find a policy which gives (30) ( rx ) ,,(/ r+ 1 V(Xo: 0, elr) - e = 1 - 1 - -f - e as its value. Therefore for any given e, and initial wealth Xo < elr, we need to find 0= 8(xo, €) and K = K(XO' e) which will achieve the value (30). To keep the drift and diffusion parameters continuous, we must take K = (2r I( I-L - r»o(xo, e), and so is(x) = (2r I( I-L - r »max{e Ir - x, O), which then gives a corresponding wealth pro- cess X 8 which has the (continuous) drift function I-Lo(x) and diffusion function u/(x) given by J-Lo(x) = max{e - rx, 2ro + rx - C}, and u/(x) = max{2(e - rx)2 11', 2r20 2 I 1'}' The scale density for this new process, defined by { f y 21-L8 ( Z ) } SIl(Y) = exp - u/(z) dz can then be written as for y ~ elr - 0, for y ~ elr - 0, where cp denotes the standard normal p.d.f. --- ## Page 348 Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time 321 482 S. BROWNE The probability of reaching the safe-region from initial wealth Xo < c/r under this policy is therefore . fJos8(y)dy _I Vs(xo·O,c/r) = je / r ( )d == V(xu: O,c/r)(1 + oy/ r+IH(y,r,c») o 58 Y Y where V is as in (25) and H is given by H( y, r, c) = (1 + y/r)(r/c)y/ r+1 ey/(2rl./27Tr/y [cJ>(2'h/r) - <1>( fy/r)], where denotes the standard normal c.dJ. Setting v.., = V - €, and then solving for 8 therefore gives (31) Therefore, for the particular o(xo, €) given in (31), the policy fa* is within e of optimality. Since we chose a = 0 here purely for notational convenience, we summa- rize this in the following theorem for the case with an arbitrary bankruptcy point a, with 0 ~ a < xo' THEOREM 3.2. The policy fa*, given by (32) ( N(X) fs*(x) = 2r r ( +r) p,_r[f./(H(y,r,c)[V(xo:a,clr) -e])] / y fora 0, then the amount due upon hitting this point is therefore Me - AT!, and we would like to find a policy that minimizes the expected value of this penalty. Clearly, this policy is equivalent to the policy that minimizes E/e - ),~J). To that end, let F(x) = infro E/e - AT!), and let t: denote the associated optimal policy, i.e., t: = arg inf] E r; EA (e -AT!). For reasons that will become clear --- ## Page 349 322 S. Browne OPTIMAL PORTfOLIO STRATEGIES 483 soon, define the constants 1'/ + and D by (33) (34) D = D( >.) = (y + >. - r)2 + 4ry, where 'Y is defined by (8). The optimal policy and optimal value function for this problem is then given in the following theorem. THEOREM 3.3. The optimal control is (35) and the optimal value fimction is (36) ( c-rx)'Tf ~ F(x) = -- c - ra ' fora :s: x s clr. REMARK 3.5. Note that TJ+> 1, and that F(a) = 1, F(clr) = 0, and F(x) is monotonically decreasing on the interval (0, clr), as is the optima} policy r;, which once again invests a constant proportion of the distance to the goal. Observe too that we are therefore once again faced with the problem that under this policy, the safe-point clr is inaccessible. However, in this case it is indeed the unique optimal policy. The condition F(a) = 1 is by construction, but the fact that F(cl r) = 0 is determined by the optimality equation itself, i.e., optimality determines that the clr barrier is inaccessible. The intuition behind this is that this policy-although it never allows the fortune to cross the clr-barrier-does indeed minimize the expected discounted time until ruin. The best one can do in this case is to get trapped in an asymptote approaching clr, which this policy tries to do. Any additional investment near the c Ir-barrier (such as in the E-optimal strategy of the previous problem) allows a greater possibility of hitting a, thus increasing the value of Ex(e - hu ) . REMARK 3.6. As a consistency check, note too that when we substitute the control N of (35) back into the evolutionary equation (3), we obtain a wealth process, say X/" that satisfies the stochastic differential equation (37) dX/' = [ (Tj+~ l)r - 1 ](c - rXn dt + u( ~+-=-rl)r (c - rX/) dW;, for t < T\ where TA = min{'T:, 'TC~TJ, where 'T/ = inf{t > 0: X/ = z}. For this process, it is well known that the Laplace transform of T} evaluated at the point A, say L(x: A) = E,(e- h :'), is the unique solution of the Dirichlet problem with L(a: A) = 1 and L(clr: A) = O. It can be easily checked that we do in fact have F(x) == L(x: A). It should be noted that 'To" is a defective random variable, with Ex('T:) = 00, as can be seen from the fact that ( c - rx )y/ r+l L(x: 0) = c _ ra == 1 - V(x: a, clr), --- ## Page 350 Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time 323 484 S. BROWNE where VeX: ',' ) is the function defined by (21). This of course is due to the fact that c Ir essentially acts as an absorbing barrier, and it can be hit with positive probability (albeit in infinite time). Specifically, as A l 0, it is clear that Tf +( A) ---) Tf +(0) == 'Y Ir + 1, and thus F(x) converges (uniformly in x) to the probability that the bankruptcy point a is hit before the safe point clr, which implies that the control t:(x) converges (uniformly in x) to the control f~(x) of (20), i.e. as A lO: F(x) ---) 1 - Vex: a,cl r) 2 and t: ( x) ---) p, _ r (c - rx) == g ( x ) . Note also that for A > 0, we have f~ > f~ , which is of course consistent with the fact that a "bolder" strategy maximizes the probability of survival, while a " timid"strategy maximizes expected playing time (for subfair games). PROOF. Theorem 2.1 is again relevant, however since Theorem 2.1 deals with the maximization pro_blem, recognize that F = -sup/{ _E\(e ~ AT!)}. We can now apply Theorem 2.1 to F:= -F with A(x) = A, g = 0, h(a) = -1. Reverting back to F, we then see that the nonlinear Dirichlet problem of (9) for F becomes then: (38) F2 (rx-c)F,-'Y/ -AF=O, fora O. The nonlinear second-order ordinary differential equation in (38) admits the two solutions C(c - rx)TJ+, and K(c - rx)TJ~, where C and K are constants to be deter- mined from the boundary condition, and where Tf+' Tf ~ are the roots to the quadratic equation Q(Tf) = 0, where (39) To determine which (if any) of these two solutions are appropriate we need to examine these roots in greater detail. The discriminant of (39) is the constant D of (34) which is clearly positive, and thus the two roots are real and, for A > 0, distinct. In particular Since Tf + Tf ~ = A I r > 0, both roots are of the same sign, and since Tf + > 0, they are both positive. The boundary condition F(a) = 1 determines the constants C, K as C = (c - ra)~TJ + and K = (c - ra)~TJ - and so clearly C> 0 and K> 0, and there- fore, Fx < 0 for both solutions. However it is easy to check the roots in (40) to see that Tf + > 1, while Tf ~ < 1, and so Fxx. > 0 only for the root Tf +. Thus we find that the (unique) solution to the (38) that satisfies F(a) = 1 and Fx < 0 Fxx > 0 is given by the function F(x) defined in (36). Moreover, it is a simple matter to check that conditions (j), (ij) and (iii) of Theorem 2.1 are indeed met for F (F is bounded on (a, b», and so we may conclude that F is optimal. The associated optimal control function, t: of (35), is then obtained by placing F (or F = - F) into (11). 0 REMARK 3.7. An alternative proof of Theorem 3.4 can be constructed by modify- ing the arguments in Orey et al. (1987), who treat the converse problem of maximiz- ing discounted time to a goal, to deal with the minimization problem treated here. --- ## Page 351 324 S. Browne OPTIMAL PORTFOLIO STRATEGIES 485 The evolutionary equation (3) would have to be reparameterized by taking /1 = 1T1 • (elr - Xl), for admissible control processes 1T, and then applying the results of Orey et a!. (1987) to the further transformed process Y,7T = In[(c - rX;r')/(c - rb)). Note that since the wealth process, say X/" under the policy ft, satisfies the stochastic differential equation (37), we can apply Ito's formula to the function FO given by (36) to show, after simplification, that (41) dF(X/,) = F(Xn[ (1-,+)2r ;+ ~+fr + Y) dt - {iY T/:;~ I da'l forO < 1< TA. The quadratic of (39) then shows that (T/+)2 r - T/+(r + Y) = A(T/+- 0, and thus substituting this into the r.h.s. of (40, and then solving the resulting (linear) stochastic differential equation gives which shows that the value function FO operating on the process X/, is a geometric Brownian motion on the interval (0, 1), for a .:::; X/, S; c Ir. Unfortunately, as noted above, this policy, while optimal for the stated problem, will never cross the c Ir bam·er into the safe-region, and thus the investor should utilize a policy similar to the e-optimal policy described earlier to get into the safe-region. Since this can be achieved at relatively little cost, we will assume for the sequel that the investor does in fact invest in a way that will allow a positive probability of getting into the safe-region. When (if) the safe-region is achieved, the investor no longer faces the problem of bankruptcy, and should then be concerned with other optimality criteria. We consider two such criteria in the next section. 4. Optimal growth policies, in the safe-region. Suppose now that we have survived, i.e., we have achieved a level x > clr. As noted earlier it is clear that in this region there need never be a possibility of ruin, and therefore the investor who has achieved this safe-region will be interested in criteria other than survival. In particu- lar, we assume here that in this region the investor is interested in growth, by which we mean achieving a high level of wealth as quickly as possible. Suppose therefore that there is now some target goal, which we will denote again by b with b > x, which the investor wants to get to (e.g., to payout dividends) as quickly as possible. In this section we consider two related aspects of this problem. First we consider the problem of minimizing the expected time to the goaL, and then we consider the related problem of maximizing the expected discounted reward of achieving the goal. In both cases, the optimal strategies are interesting generalizations of the Kelly criterion that has been studied in discrete-time in Kelly (1956), Breiman (1961) and Thorp (1969), and in continuous-time in Pestien and Sudderth (1985) and Heath et a!. (1987). (See also Theorem 6.5 in Merton 1990, where it is called the growth-optimum strategy. For Bayesian versions of both the discrete and continuous-time Kelly criterion, see Browne and Whitt 1996.) Such policies dictate investing a constant multiple of the wealth in the risky stock. Here our policies invest a constant multiple of the excess wealth over the boundary cl r, in the risky stock. This will make the clr boundary inaccessible from above, ensuring that the investor will stay in the safe-region forever, almost surely. --- ## Page 352 Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time 325 486 S. BROWNE 4.1. Minimizing the time to a goal. To formalize this, let Xo = x, but now with elr < x < b. For Tt:= int{t > 0: xl = b}, let C(x) = infE\(Tt), and let fJ=arginfEATt). ~ . ~ . THEOREM 4.1. For the problem of minimizing the expected lime to the goal b, lhe optimal policy is to invest (42) l1--r( C) fJ(x) = --;;'2 x - r ' forclr < x < b. The optimal value function is (43) 1 (rb-c) C(x) = --In -- r+1' rx-c' forelr < x::; b. REMARK 4.1. Note that the proportion ( 11- - r) I u 2 in (42) is the same proportion as in the ordinary continuous-time Kelly criterion (or optimal growth policy) (see Heath et al. 1987, Merton 1990, Browne and Whitt 1996). However in our policy fJ, this proportion operates only on the excess wealth over the boundary elr. Under this policy therefore, the lower boundary, c Ir is inaccessible. It is quite interesting to obseIVe that this policy is independent of the goal b. This is quite remarkable, since while it was to be expected a priori that the optimal policy should look something like (42) near the point clr, which ensures that clr is inaccessible from above, it is not clear why one should expect such behavior to continue throughout even when the wealth is far away from elr. Nevertheless, it appears that the best one can do is to simply put c Ir into the safe asset, and leave it there forever, continuously compound- ing at rate r . This is the endowment which will finance the withdrawal at the constant rate c forever. (Recall, e Ir = c /; e - r1 d/.) Once this is done, the optimal policy then plays the best ordinary optimal growth game with the remainder of the wealth, x - e Ir. This policy is quite similar to the policy prescribed in Proposition 11 of Black and Perold (1992) as a form of CPPJ (see also Dybvig 1995). Thus, we have shown that CPPI has another optimality property associated with it, namely that of optimal growth. PROOF. Since here we are minimizing expected time, we could apply Theorem 2.1 to q(x) = sUPJ{ -E/Tl)}, with g(x) = -1, A = 0, h(b) = 0. Recognizing that C = -G, it is then seen that in terms of G, Theorem 2.1 now requires Cx < 0 and Gxx > 0, and that the nonlinear Dirichlet problem of (9) specializes to (44) G2 (rx - c)q" - 1'C x + 1 = 0, for clr 0 for all clr < x < b. The control function fJ(x) of (42) is obtained by substituting C of (43) for /I in (11). However, note that while it is easy to see that conditions (i) and (iii) of Theorem 2.1 are satisfied by C, it is also clear that G is unbounded on (elr, b), since G(x) -> :0 as x J., elr. Thus it is doubtful that condition (ii) of --- ## Page 353 326 S. Browne OPTIMAL PORTFOLIO STRATEGIES 487 Theorem 2.1 holds for this case. Nevertheless, we will show that Theorem 4.1 holds and j:J is indeed the optimal policy, however the final proof of this awaits the development in §4.2, and we will complete the proof there after Lemma 4.3. 0 Note that when we substitute the control j:J of (42) back into the evolutionary equation (3), we obtain an (optimal) wealth process, say X b, that satisfies where 7t := inf{t > 0: X rb = b}, which is again a linear stochastic differential equa- tion. (It is clear from this that c Ir is in fact an inaccessible lower boundary for X b.) The solution to (45) is X/ = (XS - 7 )exp{(r + y)t + y'2Yw;} + 7' for 0 s: t < 76, from which it follows that ( 46) for 0 s: t < 7; , i.e., under the (optimal) policy j:J, the process {G(Xn - G(Xo)}, follows a simple Brownian motion on (0,00) with a drift coefficient equal to -1. (From this it is easy to recover the value function (43) from (46) by evaluating the expected value of (46) at t = 7; using the fact that G(b) = 0, which then gives Ex(rt) = G(x).) REMARK 4.2. The "minimal time to a goal" problem for the case c = 0 was first solved in the fundamental paper of Heath et al. (1987) without direct recourse to HJB methods (see also Schill 1993). The result in that case is simply (42) with c = 0 (see §4 in Heath et al. 1987). Merton (1990, Theorem 6.5), also obtained this policy via another, rather complicated, argument. Since the proof given here holds too for the case c = 0, our results also provide an alternative and complementary proof for that case to the ones in Heath et al. (1987) and Merton (990). In fact, it is possible to apply the results of Heath et a1. (1987) to construct a different proof of Theorem 4.1. First one would need to reparameterize the wealth equation (3) by taking fr = TTr ' (X! - clr), and then applying results of Heath et al. (1987) to the further transformed process 1';" = In[(rXr" - c)/(rb - c)]. However the results in Heath et al. (1987) are specific to the case where the controls must lie on a given constant set that is independent of the current wealth, while the approach here, based on the HJB methods of Theorem 2.1 could be modified to allow for a state dependent opportunity set. 4.2. Maximizing expected discounted reward of achieving the goal. Suppose now that instead of minimizing the expected time to the goal b, we are instead interested in maximizing E/e- AT1), for cl r < x s: b. To that end Jet U(x) = supEAe- ATb ), and let fJ(x) = argsupEAe - AT6 ). ~s ~s As we show in the following theorem, the optimal policy for this problem also invests a (different) constant proportion of the excess wealth above the c/r barrier, and is hence another version of the CPPI strategy as in Black and Perold (1992). --- ## Page 354 Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time 327 488 s. BROWNE THEOREM 4.2. The optimal control is ( 47) p,-r ( C) f~(x) = u 2(1 _ T/ - ) x - r' forc/r 0, g = 0 and h(b) = 1. Thus the nonlinear Dirichlet problem of (9) for this case specializes to (49) V 2 (IX - e)Vx - y_X - AU = 0, for e/r < x < b, Uu subject to the boundary condition U(b) = 1. Since we require Vx > 0 and Vxx < 0, it is clear the solution of interest here involves the smaller root, T/ -, to the quadratic Q( T/) = 0 (see (39» , since T/- < 1. The control function f(J(x) of (47) is then obtained by substituting V of (48) for }J into (11). Finally, it is easy to check that V of (48) satisfies conditions (i), (ii) and (iii) of Theorem 2.1, and we may therefore conclude that ft is indeed optimal. 0 1t is interesting to observe that when we place the control ft back into tpe evolutionary equation (3), we find that the resulting optimal wealth process, say X/" satisfies the stochastic differential equation (50) dX-A = (2Y ] ( -A) I2Y (-A) I (1 _ T/ )r + 1 rXI - c dt + (1 _ T/ )r rXt - c d~, for t < 'Tt where 'Tt = inf{t > 0: X/, = b}. An application now of Ito's formula to the function UO of (48) using (50) (and (39» gives which shows that the value function U(·) operatina on the process X/, is a geometric Brownian motion on the interval (0, 1), for c Ir < X/, < b. REMARK 4.4. Orey, Pestien and Sudderth (1987), using different methods, studied some general goal problems with a similar objective as that considered here, and as a particular example study a version of our problem with r = c = 0 (Orey Pestien and Sudderth 1987, page 1258). An alternative proof of Theorem 4.3 can therefore be --- ## Page 355 328 S. Browne OPTIMAL PORTFOLIO STRA TEG IES 489 constructed by using the results of Orey et aJ. (l987) using the transformation and reparameterization described above in Remark 4.2. We may now use the results of Theorem 4.2 to complete the proof of Theorem 4.1. However, we first need the following lemma, which is of independent interest since it is applicable to more general processes than those considered here (for related results, see SchiH 1993, §4). LEMMA 4.3. Suppose that for every A > 0, we have ( I - liT' ) V ( x; A) = i?f Ex - ~ ,with optima} control/" ( x; A), with v(x; A) < 00, limA) {) p(x; A) = v(x; 0) < 00, and limA> {) f*(x; A) = [(x; 0). Then (51) (1 -}OTf) (1 -AT') Jim infE, - ~ = infE. lim - ~ == infEX< 7'f), . .qo f f AtO f with infl Ex(Ti) = v(x;O) and with optimal controlf(x;O). PROOF. It is the first equality in (51) that needs to be established since the second is just an identity. To proceed, it is obvious that A -Ill - e- h () ::;; 1'f for all .A ~ 0, and hence £ .. 0 - '[1 - e- Mf ]) ::;; Ex(T f ), as well as infi E t(A- 1(1 - e- AT']) ::;; inf! £/7'i). Since the r.h.s. of this inequality is independent of the parameter A, it follows that we may take limits on A to get (52) For notational convenience now, Jet f1l* denote the policy f*(.; A), and for any policy I, let 1'[f) = Tf. Note that under this notation, we may write v(x; A) = E/A -1[1 - e - ~ T(in]). To go the other way now, suppose that there is an admissible policy, say f, such that T[N) ~a S T[/J as A ~ O. Then it is clear that An application of Fatou's lemma then shows that (53) The inequalities (52) and (53) yield (51). 0 COMPLETION OF PROOF OF THEOREM 4.1. ObselVe first that 77-(A) ~ 0 as ,q 0, and that therefore fi; --7 fJ as A ~ 0, where fiJ and ft are given by (47) and (42). --- ## Page 356 Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time 329 490 S. BROWNE Note further that for any c Ir < x < b, we have I· 1 - U( x) = G( ) 1m A x , ,qo where U and_ G are given by (48) and (43). Finally, since 1/ - ( A) ~ ° as A lO, we have X/,, ". ~"' . X,\ "; as At 0, where X,b and i"/ are defined by (4S) and (SO), from which it is clear that 7l ~a s 7" h as A lO. Therefore, Lemma 4.3 may be applied directly to Theorem 4.2 to deduce Theorem 4.1. 0 5. The multiple asset case. As promised earlier, here we show how all of our previous results extend in a very straightforward way to the case with multiple risky stocks. The model here is that of a complete market (as in, e.g., Karatzas and Shreve 1988) where there are n risky assets generated by n independent Brownian motions. The prices of these stocks evolve as (54) i=l, ... ,n, while the riskless asset, B" still evolves as dB, = rB, dt. The wealth of the investor therefore evolves as (S5) dXr = [rxr - c + ,~f,( J.i-, - r)] + ,~ Jt f,u,} dHl;(J), where now f, denotes the total amount of money invested in the ith stock. If we introduce now the matrix a = (0' \ ' and the (column) vectors f.l. = (J.i-l' .•. , J.i-,,)T, f = (fl'" . , fn)T, and then set A = aaT, we may write the generator of the (one-dimensional) wealth process, for functions \)I(x) E W 2 as (S6) where 1 denotes a vector of 1 'so The assumption of completeness implies that A - I exists, and thus all our results will go through exactly as before. In particular, if an optimal value function for a specific problem is denoted by vex), the optimal control vector is f~(x) where (57) The differential equations «22), (38), (44) and (49» , and hence the value functions «21), (36), (43) and (48», all remain the same except for the fact that now the scalar 'Y is evaluated as (58) It is interesting to note that for the problem of maximizing the probability of reaching b before a when b is in the danger-zone, considered in §3.1, the optimal policy now does depend on the variances and covariances of the risky assets, since --- ## Page 357 330 S. Browne OPTIMAL PORTFOLIO STRATEGI ES 491 instead of (20), in the multiple asset case we now get (59) The €-optimal policy of §3 needs to be modified, but the extension is straightforward and we leave the details for the reader. For reference, we note further that if we define the vector K by K := A - I (fA. - r1), then the optimal controls (35), (42) and (47) of §§3.2, 4.1 and 4.2 become, respectively (60) ft( x) = K( 7] + - 1) - 1 (f - x) , ft ( x) = K( 1 - 7] - ) - I ( X - f ). f~(x) = K(X - f), 6. Linear withdrawal rate. In this section we show how all of our previous results and analysis for the case of forced withdrawals at the constant rate c > 0 can be generalized to the case where there is a wealth-dependent withdrawal rate, c(x) where c(x) = c + Ox. Here we will only consider the case where 0 S; 6 < r. For notational ease, we will consider again only the case with one risky stock. The generalization to the multiple stock case as in the previous section is very straightforward, and so we leave the details for the reader. For this case the evolutionary equation (3) becomes (61) dP dB . dJ(I = [,_' + (Xl - [,)_' - (c + eX') dt t 'P, " B, I = [Cr - e)x( + It( JL - r) - c] dt + f,udW, . If we now define r:= r - e > 0, then for Markov control processes f, and 'I' E %,2 the generator of the wealth process is (62) The parameter r is simply the adjusted (risk-free) compounding rate. Essentially, nothing really changes except for the fact that the danger-zone is now the region x < elf, and the safe-region is its complement. The differential equations (22), (38), (44) and (49) all remain the same except for the fact that we must replace rx - c with rx - e. The parameter 'Y in all those equations, as well as here, is still defined as in (8), i.e., 'Y = ~« JL - r) I u )2, where r is the standard interest rate. Thus, the previous analysis will go through with relatively little change, and so we will only point out the essential differences. In particular, the structure of the policies remain the same, in that the optimal survival policies of §3 invest a fixed proportion of the distance to the (new) safe-region barrier, elf, while the optimal growth policies of §4 invest a fixed proportion of the excess of wealth over the barrier. 6.1. Sunival problems in the danger-zone. The analysis of §§3.1 and 3.2 can be repeated almost verbatim. What changes is that now for a < x < b < eli, the value --- ## Page 358 Survival and Growth with a Liability: Optimal Portfolio Strategies in Continuous Time 331 492 S. BROWNE function veX; a, b) of (21) becomes instead (c - Fa)Y!f+1 - (c - i'x)Y!f+l V( x: a , b) = . , for a s x s b. (c - Fa)"l r+l - (c - Fb)""!f+1 (63) Since (11) still holds, the resulting optimal control becomes, instead of (20), (64) For the disc0!1nted problem of §3.2 (~s well as for the discounted problem of §4.2), the quadratic Q(.) oj (39) changes to Q('Yj) = 'Yj2;: - 'Yj(y + A + F) + A, and thus the two (real) roots to Q('Yj) = 0, denoted by 1,+ and 1,-, become, instead of (40), (65) It is easy to check that once again, we have 0 .s 1,- < 1 < 1,+, and so the optimal value function (36) and the optimal control function (35) become, respectively (66) ( c i'x)T,+ F(x)= c-fa ' 6.2. Growth policies in the safe-region. Once again, the analysis is almost identical to that in §§4.1 and 4.2. The value and the optimal control functions for the minimal expected time to the goal problem, (43) and (42) are replaced respectively by (67) 1 (Fb-C) G(x) = -_ -In -_- r+y rx-c' p,-r( C) f6 ( x) = -;;'2 x - j , for c If < x < b. Note that the optimal (Kelly) proportion of the excess wealth invested in the stock, (p, - r) I u 2, is unchanged in this case. Similarly for the discounted problem considered in §4.2, the value function (48) and optimal policy (47) become (68) ( fx c )T,- Vex) = -- , Fb - c 1'* ( ) P, - r ( c ) }U X = 2( __ ) x - -= . u 1 - 'Yj r The case where e> r, and hence with f < 0, introduces new difficulties that will be discussed elsewhere. Acknowledgments. The author is most grateful to Professors Ioannis Karatzas of Columbia University and Bill Sudderth of the University of Minnesota for very helpful discussions. In particular, the proof of Lemma 4.3 is due to Ioannis Karatzas. He is also thankful to two anonymous referees for very helpful comments and suggestions and for bringing the references Schiil (1993), Dutta (1994) and Roy (1995) to his attention. References Black, F., A. F. Perold (1992). Theory of constant proportion portfolio insurance. Jour. Econ. Dyn. and elllri. 16 403-426. Brelman, L. (]961). Optimal gambling systems for favorable games FO/lrrh Berkeley Symp. Malh. Slal. and Prob. 1 65-78. --- ## Page 359 332 S. Browne OPTIMAL PORTFOLIO STRATEGIES 493 Browne, S. (995). Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin. Math. Oper. Res. 20 937-958. __ , W. Whitt (1996). Portfolio choice and the Bayesian Kelly criterion. Adv. Appl. Prob. 28 1145-1176. Davis, M. H. A, A R. Norman (1990). Portfolio selection with transactions costs, Math. Oper. Res. 15 676-713. Dubins, L. E., L. J. Savage (965). How to Gamble If You Must: Inequalities for Stochastic Processes, McGraw-Hili, New York. Dutta, P. (1994). Bankruptcy and expected utility maximization. Jour. ECOfl. Dyfl. and Cntrl. 18 539-560. Dybvig, P. H. (1995). Dusenberry's ratcheting of consumption: Optimal dynamic consumption and invest- ment given intolerance for any decline in standard of living. Review of EconomIc StudIes 62287-313. Ferguson, T. (1965). Betting systems which mimmize the probability of ruin. J. SlAM 13 795- 818. Fleming, W. H., R. W. Rishel (975). Detenninistic and Stochastic OptImal Control, Springer-Verlag, New York. __ , H. M. Soner (I993). Controlled Markov Processes and Viscoslly SolutIOns, Springer-Verlag, New York. Grossman, S. J., Z. Zhou (J993). Optimal investment strategies for controlling drawdowns. Math. Fin. 3 241-276. Heath, D., S. Orey, V. Pestien, W. Sudderth (1987). Minimizing or maximizing the expected time to reach zero. SlAM J. Contr. and Opt. 25 195- 205. Karatzas, L, S. Shreve (1988). Brownian Motion and StochastiC Calculus, Springer-Verlag, New York. Karlin, S., H. M. Taylor (1981). A Second Course on Stochastic Processes, Academic, New York. Kelly, 1. (1956). A new interpretation of information rate. BeJi Sys. Tech. 1. 35 917-926. Krylov, N. V. (1980). Controlled DiffusIOn Processes, Springer-Verlag, New York. Luskin, D. L. (Editor) (1988). PortfolIO Insurance: A Guide to DynamIC Hedging, Wiley, New York. Majumdar, M., R. Radner 099l). Linear models of economic survival under production uncertamty. Econ. Theol)' I 13-30. Merton, R. (1971). Optimum consumption and portfolio rules in a continuous time model. J. Ecoll. Theol)' 3373-413. __ ([990). ContinuoUl Time Finance, Blackwell, Massachusetts. Orey, S., V. Pestien, W. Sudderth (1987). Reaching zero rapidly. SlAM J. COni. and Opt. 2S 1253-1265. Pestien, V. C., W. Sudderth (1985). Continuous-time red and black: How to control a diffusion to a goal. Malh. Oper. Res. 10599-611. __ , __ (1988). Continuous-time casino problems. Math. Oper. Res. 13 364-376. Pliska, S. R. (1986). A stochastic calculus model of continuous trading: Optimal portfolios. Math. Oper. Res. 11 371-382. Rogers, L C. G., D. Williams (]987). Diffusions, Markov Processes, and Mamngales, Vol. 2, Wiley, New York. Roy, S. (1995). Theory of dynamic portfolio choice for survival under uncertainty. Math. Soc. Sci. 30 171-194. Schiil, M. (J993). On hitting times for Jump-diffusion processes with pasl dependent local charactenstics. Stoch. Proc. Appl. 47 131- 142. Thorp, E. o. (1969). Optimal gambling systems for favorable games. Rev. In/. Stat. Inst. 37 273-292. S. Browne: Graduate School of Business, Columbia University, 402 Uris Hall, New York, NY 10027; e-mail: sb30@columbia.edu --- ## Page 360 MANAGEMENT SCIENCE Vol. 38. No. I I. November 1992 Primed in U.S.A. 24 GROWTH VERSUS SECURITY IN DYNAMIC INVESTMENT ANALYSIS * L. C. MACLEAN, W. T. ZIEMBA AND G. BLAZENKO School 0/ Business Administration, Dalhousie University, Halifax , Nova Scotia, Canada B3H lZ5 Faculty o/Commerce, University 0/ British Columbia, Vancouver, British Columbia, Canada V6T lZ2 School of Business Administration, Simon Fraser University, Burnaby, British Columbia, Canada V5A IS6 This paper concerns the problem of optimal dynamic choice in discrete time for an investor. In each period the investor is faced with one or more risky investments. The maximization of the expected logarithm of the period by period wealth, referred to as the Kelly criterion, is a very desirable investment procedure. It has many attractive properties, such as maximizing the asymp- totic rate of growth of the investor's fortune. On the other hand, instead of focusing on maximal growth, one can develop strategies based on maximum security. For example, one can minimize the ruin probability subject to making a positive return or compute a confidence level of increasing the investor's initial fortune to a given final wealth goal. This paper is concerned with methods to combine these two approaches. We derive computational formulas for a variety of growth and security measures. Utilizing fractional Kelly strategies, we can develop a complete tradeoff of growth versus security. The theory is applicable to favorable investment situations such as blackjack, horseracing, lotto games, index and commodity futures and options trading. The results provide insight into how one should properly invest in these situations. (CAPITAL ACCUMULATION; FRACTIONAL KELLY STRATEGIES; EFFECTIVE GROWTH-SECURITY TRADEOFF; BLACKJACK; HORSERACING; LOTTO GAMES; TURN OF THE YEAR EFFECT) 331 This paper develops an approach to the analysis of risky investment problems for practical use by individuals. The idea is to provide simple-to-understand two-dimensional graphs that provide essential information for intelligent investment choice. Although the situation studied is multiperiod, the approach is couched in a growth versus security fashion akin to the static Markowitz mean-variance portfolio selection tradeoff. The approach is a marriage of the capital growth literature of Kelly (1956), Breiman (1961 ), Thorp (1966, 1975), Hakansson (1971, 1979), Algoet and Cover (1988) and others, which emphasizes maximal growth, with the maximal security literature of Ferguson (1965), Epstein (1977), and Feller (1962). In §I, we formulate a general investment model and desr:ribe three growth measures and three security measures. The measures all relate to single-valued aggregates of the investment results over mUltiple periods such as the mean first passage time to a particular wealth level and the probability of doubling one's wealth before halving it. In §2, we model the investment process as a random walk. Then, using familiar procedures from probability theory, we can easily generate com- putable quantities which are usually close approximations to the measures of interest. To generate tradeoffs of growth versus security one can utilize fractional Kelly strategies as discussed in §3. A simple but powerful result is that a complete trade-off of growth versus security for the most interesting growth and security measures is implementable simply by choosing various fractional Kelly strategies. Theoretical justification for the fractional Kelly strategies in a multi period context can be made using a continuous time approach and log normality assumptions (see Li et al. 1990 and Wu and Ziemba 1990). 1562 0025-1909/92/3811/1562$01.25 Copyright © 1992, The Institute of Management Sciences --- ## Page 361 332 L. C. MacLean, W T. Ziemba and G. Blazenko GROWTH VERSUS SECURITY IN DYNAMIC INVESTMENT ANALYSIS 1563 Application of the theory to four favorable investment situations is made in §4. In each case the basic game or investment situation is unfavorable to the typical or average player. However, systems have been developed that beat the game in the sense that they have positive expected value. The question then remains how large should the wagers be and how confident is one that particular goals will be achieved. The various games, blackjack, horseracing, lotto games and the turn of the year effect differ markedly in their character. The size of the wagers vary from over half of one's fortune to less than one millionth of the fortune. The graphs that are outputted for each of these applications show how to trade off risk and return and provide crucial insight into how one should invest intelligently in these situations. 1. The Basic Investment Problem An investor has initial wealth Yo E IR and is facing n risky investments in periods 1, 2, ... , t, .... The return on investments follows a stochastic process defined on the prob- ability space (Q, B, P) with corresponding product spaces (QI, B I, PI) . Given the realization history W 1- I E Q t-I , the investor's capital at the beginning of period t is Y I_I ( W 1- I ). The investment in period t in each opportunity i, i = I, ... , n , is Xjl(w t-I ), Xj,(W I- I ) :$: Y I_I (w 1- I ). The in vestment decisions in terms of proportions are X'I ( W 1- I ) = Pjl( WI- I )YI_I (WI- I) for i = I, .. . , n, andpI( WI- I) = (POI( WI- I), ... ,Pnt( WI- I », where L7~oPjl(WI) = I andpoI(wl- I)YI_I(wl- l) is the investment at time t in riskless cash-like instruments. In this general form the investment strategy PI( WI- I) depends on time and the history QI- I of the investment process. The net return per unit of capital invested in i, i = I, . .. , n, given the outcome WI E Q, in period t is given by K'I ( wJ , i = I, ... , n, t = I, .. . . The return on the risk-free asset is given by KOl = O. It is assumed that K jl( Wj) is defined by the multiplicative model Kjl(wl) = O'j(t)Ej(wl), where 0',(1) > 0 is the average time path and E,( Wi) is an independent error term. Special cases result when O'j(t) = O'j or O'j(t) = a: for t = I, . ... The error terms in this model are not autocorrelated. A more general approach would be to consider the conditional return at time t given the history WI- I, denoted by K jl( WI I WI- I). A Bayesian analysis with such a model is theoretically tractable, but it is not as suited to computations as the multiplicative model. An investment environment is said to be favorable if EKj ( w) > 0 for some i. We are only concerned with favorable environments. The total return on investment is KI( Wi ) = I + K jl ( WI), and the investment decisions are P = (Ph P2( Wi), ... , PI(W I- I ), •.. ). Then the accumulated wealth at the end of period t is wIEQI, t = I , ... . (I) For the stochastic capital accumulation process {YI(p)} ~ I we are interested in char- acterizing the accumulation paths as the investment decisions vary. Consider the following measures of growth and security. 1.1. Measures of Growth G 1. J./.I(YO , p) = EYI(p). This is the mean accumulation of wealth at the end of time t. That is, how much do we expect to have, on average, after t periods. G2. (f>t(yo, p) = E In (YI(p)I /I). This is the mean exponential growth rate over t periods. It is a measure of how fast the investor is accumulating wealth. We also consider the long-run growth rate cJ>(p) = liml_ co cJ>1(YO , p) . --- ## Page 362 Growth versus Security in Dynamic Investment Analysis 333 1564 L. C. MACLEAN, W. T. ZIEMBA AND G. BLAZENKO G3. 7/(Yo , p) = ET { Y(p );?U) . This is the mean first passage time T to reach the set [V, OC! ). That is, how long on average does it take the investor to reach a specific level of wealth. For example, how long must the investor wait before he is a millionaire. 1.2. Measures of Security Sl. "It(Yo, p) = Pr[Yt(p) ~ btl· This is the probability that the investor will have a specific accumulated wealth bt E ~ at time I . That is, what are his chances of reaching a given target in a fixed amount of time. For example, what chance does the investor have of accumulating $200,000 one hundred days from now? S2. a(yo, p) = Pr [YI(p) ~ bl , 1 = I, ... l. This is the probability that the investor's wealth is above a specified path. Our concern for an investor is that his wealth will fall back too much in any period. He may want protection against losing more than, say, 10% of current wealth at any point in time. S3. i3(yo, p) = Pr[ T { Y(p );,oU) < T { y( p);,o Ld. This is the probability of reaching a goal V which is higher than his initial wealth Yo, before falling to a wealth level L which is less than Yo . For example, if V = 2yo and L = Yo/2, this is the probability of doubling before halving. The rationale for the measures selected is to profile the growth and security dimensions of the accumulation process. In each case there is a natural pairing of the measures: (ILl, "II) for accumulated wealth at a point in time; (c/J, a) for the behavior of growth paths; and (7/, 13) for first passage to terminal or stopping states. The choice of profile (pair of measures) is largely a function of personal preference and problem context. It is expected that the information contained in each of the profiles would be useful in selecting an investment strategy. The usual criterion for evaluating a decision rule is tc/J/(YO , p) , the expected log of accumulated wealth. When Ki/(w/) = ai(t)Ei(w/), i = I, ... , n, 1 = I, . .. , then the log optimal strategy is proportional, Pil( wt- I ) = Pil. Furthermore, if Pi , i = I, . . . , n, solves the one-period problem Pi ~ 0, i = I· .. n} , , thenpil = inf {a i l (t)Pi , I} for i = I, . .. , n, 1 = I, . . .. In the case where ai(t) = ai, i = I, . . . , n, the log optimal strategy is a fixed fraction (independent of time) , and is referred to as the Kelly (1956) strategy. Considering the other growth measures we find that the Kelly strategy (i) maximizes the long run exponential growth rate c/J(Yo, p) , and (ii) minimizes the expected time to reach large goals 7/(Yo , p) (Breiman 1961, Algoet and Cover 1988). The main properties of the Kelly strategy are summarized in Table 1. The performance of the Kelly strategy on the security measures is not so favorable. As with any fixed fraction rule, the Kelly strategy never risks ruin, but in general it entails a considerable risk oflosing a substantial portion of wealth. In terms of security an optimal policy is often to keep virtually all wealth in the riskless asset (Eithier 1987). We will let Po = (I , 0, . .. , 0) be the extreme security strategy, where all assets are held in cash. Our purpose is to monitor the growth and security measures as the investment decisions vary with the goal of balancing the measures to obtain a strategy which performs well on both the growth and security dimensions. For a particular (growth, security) com- bination, the complete set of values is given by the graph (see Figure 1) Vi = {( Gi(p) , Si(P» I p is a feasible investment strategy } . --- ## Page 363 334 Good/Bad G G G B G B G B L. C. MacLean, W. T. Ziemba and G. Blazenko GROWTH VERSUS SECURITY IN DYNAMIC INVESTMENT ANALYSIS 1565 TABLE I-PART I Main Properties for the Kelly Criterion Property Maximizing E log X asymptotically maximizes the rate of asset growth. The expected time to reach a preassigned goal is asymptotically as X increases least with a strategy maximizing E log XN . Maximizing median log X. False Property: If maximizing E log X N almost certainly leads to a better outcome then the expected utility of its outcome exceeds that of any other rule provided n is sufficiently large. Counterexample: u(x) = X, 1 < p < I, Bernoulli trials,] = I maximizes EU(x) butf* = 2p - I < I maximizes E log XN • The E log X bettor never risks ruin. If the E log X bettor wins then loses or loses then wins he is behind. The order of wi n and loss is immaterial for one, two, . . . sets of trials. The absolute amount het is monotone in wealth. The bets are extremely large when the wager is favorable and the risk is very low. Reference Breiman (1961), Algoet and Cover (1988) Breiman (1961), Algoet and Cover (1988) Ethier (1987) Thorp (1975) Hakansson and Miller (1975) (I + ')')(1 - ')')Xo = (I - ')")Xo <0 (aE log X)/8Wo > 0 Roughly the optimal wager is proportional to the edge divided by the odds. Hence for low risk situations and corresponding low odds the wager can be extremely large. For one such example, see Ziemha and Hausch (1985, pp. 159-160). There in a $3 million race the optimal fractional wager on a 3-5 shot was 64%. The most obvious points in V; to identify are efficient points given by (G;(p,,), S;(p,,» E V; with G;(p,,) = max { G;(p) I S;(p) ~ lX, P feasible}. Alternatively the efficient points are ( G;(pf3), S;(pf3» with S;(PIi) = max {S;(p) I G;(p) ~ fl, p feasible} . Given the required level of security (or growth) the efficient points are undominated. The set of efficient points constitutes the efficient frontier. Two key efficient points are the unconstrained optimal growth and optimal security points, respectively (G;(p), S;(p» with G;(p) = max {G;(p) I p feasible} and (G,(po), S;(Po» with S;(Po) = max {S;(p) I p feasible}. The optimal growth strategy is the Kelly strategy p and the optimal security strategy is the cash strategy Po. These are both fixed fraction strategies. Other strategies yielding points on the efficient frontier would not typically be fixed fraction, but rather would be dynamic strategies, depending on current wealth and/ or time. Gottlieb ( 1985) has derived such a strategy for the criteria of max- imizing mean first passage time to a goal (G3 ), subject to a high probability of reaching the goal (S3)' --- ## Page 364 Growth versus Security in Dynamic Investment Analysis 335 1566 Good/Bad B B B G G G G L. C. MACLEAN, W. T. ZIEMBA AND G. BLAZENKO TABLE I-PART 2 Main Properliesjor the Kelly Criterion Property One overbets when the problem data is uncertain. The total amount wagered swamps the winnings- that is, there is much "churning." The unweighted average rate of return converges to half the arithmetic rate of return. The E log X bettor is never behind any other bettor on average in I, 2, ... trials. The E log X bettor has an optimal myopic policy. He does not have to consider prior to subsequent investment opportunities. The chance that an E log X wagerer will be ahead of any other wagerer after the first play is at least 50%. Simulation studies show that the E log X bettor's fortune pulls way ahead of other strategies' wealth for reasonably-sized sequences of investments. Reference Betting more than the optimal Kelly wager is dominated in a growth-security sense. Hence if the problem data provides probabilities, edges and odds that may be in error, then the suggested wager will be too large. This property is discussed in and largely motivates this paper. Ethier and Tavare (1983) and Griffin (1985) show that the Expected Gain/ E Bet is arbitrarily small and converges to zero in a Bernoulli game where one wins the expected fraction p of games. Related to property 5 this indicates that you do not seem to win as much as you expect; see Ethier and Tavare (1983) and Griffin (1985). Finkelstein and Whitley (1981) This is a crucially important result for practical use. Hakansson (1971 b) proved that the myopic policy obtains for dependent investments with the log utility function. For independent investments and power utility a myopic policy is optimal. Bell and Cover (1980) Ziemba and Hausch (1985) Rather than trying to move from optimal growth to optimal security along the efficient frontier, we will consider the path generated by convex combinations of the optimal growth and optimal security strategies, namely p(A) = Aft + (l - A)PO , Clearly p( A) is a fixed fraction strategy and we will call it fractional Kelly since it blends part of the Kelly with the cash strategy. The fractional Kelly strategy is n-dimen- sional, specifying investments in all opportunities, and the key parameter is A, which we can consider as a tradeoffindex. The path traced by the fractional Kelly strategies is u9 = {(G;(p(A)), S;(p(A)))IO ~ A ~ I}. The fractional Kelly path is illustrated in Figure I. This path has attractive properties. First, it is easily computable, as shown in the next section. When A varies from 0 to I we continuously trade growth for security. Although this tradeoff is not always efficient, it is effective in that we see a monotone increase in security as we decrease growth. However, the rate of change in growth or security with respect to A is not constant and thus the Kelly path is curvilinear. In fact, MacLean and Ziemba ( 1990) established that the Kelly path is above the straight line (secant) joining the optimal growth and optimal security points. These properties are discussed in §3. --- ## Page 365 336 L. C. MacLean, W T. Ziemba and G. Blazenko GROWTH VERSUS SECURITY IN DYNAMIC INVESTMENT ANALYSIS 1567 Si Efficient frontier Ui Fractional Kelly path (effective) Gi FIGURE 1. Graph of (Growth, Security) Profile Showing Efficient Frontier and Effective Path. 2. Computation of Measures If we select a strategy which specifies our investment in the various opportunities and we stick with that strategy, then each of the measures described in §2 is easily computable. We will now develop the computational formulae. Since most of our discussion focuses on fractional Kelly strategies, we work with them. However, more general fixed strategies could be used. Consider the fractional Kelly strategy p(}..) and use the transformation Z/(p(}..) , Wi) = In YI(p(}..), Wi). From (I) , ZI(P(}..), Wi) = Zo +\~ In C~ R;(Ws)P;(}..») = Z'_I(P(}..), w H ) + J(p(}..) , WI), where J(p(}..) , w) = In ( L ;'=0 R;( w )p;(}..» is a stationary jump process and Zo = In Yo. Hence, Z/> t = I, . .. , is a random walk and we can evaluate the various measures of interest using this process. An alternative approach to evaluating these measures is pre- sented in Ethier ( 1987) where the discrete time (and maybe discrete state) process YI(p(}..» is approximated by geometric Brownian motion. The mean wealth accumulation measure G I is iL/(Yo, p(}..» = EYI(p(}..» = YOED C~ R;(Ws)P;(}..)) = YOC~ R;p;(}..) r (2) where R; = ER;(w), i = 0, ... , n. For the mean growth rate measure G2 (Myo , p(}..» ~ E In (Y,(p(}..»I /I ) = l/tEZ,(p(}..» = E In L R;(w)p;(}..) + - Zo = EJ(p(}..), w) + - zo, ( n ) I I ;~ o t t (3) which is an easy calculation to make. For measure S I assume the process Z,(p(}..), Wi) = Zo + L J(p(}..), ws) s= I --- ## Page 366 Growth versus Security in Dynamic Investment Analysis 337 1568 L. C. MACLEAN, W. T. ZIEMBA AND G. BLAZENKO has an approximate normal distribution. Using this approximation yields >. _ F(bt - E(Z,(P(>'»») 'yAyo, p( » - 1 - u(Z,(p(>'») , (4) where F is the cumulative for the standard normal, bt = In b" and E(Z,(p(A»), u(Z,(p(>'») are the mean and standard deviation respectively. Measure S2 is given by a(yo, p(A» = Pr[Z/(p(>.»:2 bt, t = I, ... ] 00 = II Pr[Z,(p(>.»:2 btIZs(p(>'»:2 b:, s = 1, . .. , /- I] ' ~ I 00 = II a,(zo, p(A». ' ~ I Then with h,( z,1 Zs :2 b: , s = I, ... , t - I) the conditional density a,=J .h,(z,I Zs :2b:,s= 1, ... ,t-l)dz/ z/~ bl where 7r(x) is the density for J(p(>.», - J {J h'- I(Z,-xI Zs:2M,s=I, ... ,t-2) d}d - 7r(x) x Z,. z,?::b; XSZ,- b;_l (Xt- 2 (5) (6) (7) Equation (7) provides a sequential procedure to compute measure S2, requiring only the distribution from the previous stage and the jump probabilities. For the other measures assume that the random variable J(p( >'» has finite support given by the interval [Jm(P(>'», JM(p(>.»]. Consider measure S3 !1(yo,p(>'» = Pr[T{y(p(~» " U ) < T { Y(p(~» " L) Iyo]. Transforming to an additive process with z = In y and letting !1(z, p(>.» = Pr[ T{Z(p(X» " lnU ) < T{Z(p(~» " lnL) Iz] yields !1(zo,p(>'» = E{!1(zo + J(p(>,»,p(A»}. (8) The expectation in (8) is over the random variable J(p(>'». We must solve (8) for!1 subject to the boundary conditions !1(z, p(>.» = 0 !1( z, p( >.» = 1 if if z::o;; In L z :2 In U. (8a) (8b) A solution to (8) is of the form !1( z, p( >.» = L: 1~ 1 AkOL where Ak are chosen to satisfy (8a, 8b) and the roots 01 , ••• , Os satisfy the equation EOJ(p(~» = 1. It is easy to show that there are two positive roots 01 = 1 and O2 = 0, where O2 < 1 or O2 > 1 depending on whether EJ(p(A» > 0 or EJ(p(>.» < 0 respectively. Similar to the methods discussed in Feller (1962, pp. 330-334), we take the minimum Jm(p(>.» < 0 and maximum --- ## Page 367 338 L. C. MacLean, W T. Ziemba and G. Blazenko GROWTH VERSUS SECURITY IN DYNAMIC INVESTMENT ANALYSIS 1569 TABLE 2 Computational Formulas for Growth Measures GI-G3 and Security Measures SI-S3 GI G2 q,(p) = E In (~ Ri(W)Pi) G3 - Z oz. (U*(P) u*(P») 20 'I( 0, p) = 2EJ(p) OU.(p) + OU'(p) - EJ(p) SI _ (b~ - Zo - tEJ(P») ")'(20, p) = I - F Vtu(J(p» 00 S2 <>(20, p) = IT <>/(Zo, p), (=) S3 JM(p("A» > 0 jumps and find bounds on the solution by solving a simple system with the extreme boundary conditions satisfied as equalities. For the upper bound, !J(ln L + Jm(p("A» , p("A» = 0 and !J(ln U, p("A» = 1. For the lower bound, !J(ln L, p("A» = 0 and Let L * = In L, L *(p("A» = In L + Jm(p("A» , U* = In U, and U*(p("A» = In U + JM(p("A». The bounds for measure S3 are then 0=0 - 01.' 0=0 _ OL'(P( A» OU'(P(A» _ OL* ~ (3(yo, p("A» ~ Ou· _ OL*(P(A» . For measure G3, 1)(Yo,p("A» = ET I Y(p(A» " UIJ'o } = ET IZ(P(A» ,dnUlzo}' we use the recursive equation l1(Zo, p(>..» = E {1)(zo + J(p(>..», p(>"»} + I subject to the boundary condition 1)(Z, p(>..» = 0 if Z~ In U. (9) ( 10) ( lOa) One solution to (10) is 1)*(z, p("A» = - z/(EJ(p("A»). Any other solution can be written as 1)( z, p( >..» = 1)*( z, p("A» + ~(z, p("A» where ~(z , p( >..» satisfies the system ~(z, p(>..» = E{ ~(z + J(p(>..», p("A»}. ( lOb) Solutions of ( lOb) have the form ~(z, p("A» = L AkO%. Solutions of ( 10) are then l1(Z, p) = L AkOZ - z/(EJ(p(>"») where the Ak are chosen to satisfy the boundary conditions (lOa). With U*(p(>"» --- ## Page 368 Growth versus Security in Dynamic Investment Analysis 339 1570 L. C. MACLEAN, W. T. ZIEMBA AND G. BLAZENKO = In U - Jm(p( X» we can solve the extreme boundary conditions as equalities, namely 1]( U*(p(X» , p(X» = 0 and 1]( U*(p(X», p(X» = 0 yields the following bounds on G3: [ U*(P(X»]( 0=0 ) Zo EJ(p(X» OU.( p ( I-.» - EJ(p(X» ~ 1](Yo, p(X» [ u*(P(X»]( OZO) Zo ~ EJ(p(A» O U *(p(I-.» - EJ(p(X»' (II) The bounds in (9) and (II) are quite tight when the jumps are small relative to the initial wealth Zo. Table 2 summarizes the formulas we can use to compute the six growth and security measures. For G3 and S3 we have bounds and the formulas given provide midpoint estimates of these bounds. A similar table appears in Ethier ( 1987) for the geometric Brownian motion model. 3. Effective Growth-security Tradeoff The fact that the various growth and security measures can be easily evaluated for fractional Kelly strategies provides an opportunity to interactively investigate the per- formance of strategies for various measures and scenarios. The ability to evaluate various performance measures and determine a preferred strategy will depend on an effective tradeoffbetween complementary measures of growth and security. A tradeoff is effective if the loss of performance in one dimension (say growth) is compensated by a gain in the other dimension (security). In this section we establish that the tradeoff using fractional Kelly strategies is effective for each of the profile combinations. In calculating the rates of change for the various measures we will see that the tradeoff between growth and security is not constant for all X, 0 ~ X ~ I. (i) Rate Profile: (4), a). LEMMA 3.1. Suppose p( X) is a Factional Kelly strategy and p is the Kelly strategy. Then a4>(yo , p(X» > 0 ax ' PROOF. Consider the optimal growth strategy given by p = (Po , .. . , Pn), and the optimal security strategy Po = ( I, 0, ... , 0). Then p(X) = (XPo + (I - A) , .. . , Xp n) = (Po(X) , . .. , Pn(X» and d d dXPo(X) <0, dXP;(X» 0, i = I, ... , n. We have the optimal growth problem max 4>(p(X» = max E In C~ R i/(w)P ,(X») . Then and - 24>(p(X»=-E 11 , ~ I 1/W <0. a2 [ I - 2.: ~1 R ( )]2 apo 2.:i~O R;,(W)Pi(X) --- ## Page 369 340 L. C. MacLean, W T. Ziemba and G. Blazenko GROWTH VERSUS SECURITY IN DYNAMIC INVESTMENT ANALYSIS 1571 So 4>(p(A)) is concave in Po and with Po such that a4>(p(A))/apo = 0, we have Therefore a -a 4>(p(A)) < 0, Po d dpo ( a ) dA 4>(p(A)) = dA apo 4>(p(A)) > 0 Po < Po. and 4>(p(A)) is increasing in A. D LEMMA 3.2. Let p( A) be a/ractional Kelly strategy and In a < 0 be a given[allback rate. Then a aA a(a, p(A)) $ O. PROOF. We have for the security measure a(a, p(A)) = Prob [Y,(p(A)) ~ yoa', t = I, ... ] = II Prob [Y,(p(A)) ~ a'l Y,(p(A)) ~ a', s = 1, ... , t - I] I ~ I = II a,(a, p(A)). ,,,,1 Along any path I'YI' ... 'YI_ lwithy,~ a ',s= 1, ... , t-1. a,(a, p(A)) = Prob [In (L R"(W)P;(A)) > In a - t ~ I :~ z,], where z, = In y, and In a - (t - I ) - 1 L .~: II z, < O. With b = In a - (t - I) - I L z, and R,P(A) = L R,,(w)p,(A) we have a,(A) == a,(a, p(A)) = Prob [In (R,P(A)) ~ b] = Prob [In (AR,P + (I - A)R,po) ~ b] = Prob [A In (R,P) + (I - A) In (R,po) + b. ~ b] [ b ~ b.] = Prob In (R,P) ~ - A- = I - .r~ (x(A)), where X(A) = (b - b.)/A and F, is the distribution function for the optimal growth process In (R,po) . Then dX(A)/dA > 0 and Fnondecreasing imply da,(A)/dA $ O. Hence da(a, p(A))/ dA s O. D Thus along the fractional Kelly path growth is increasing and security is decreasing and an eff"cclive tradeoff is possible. (ii) Wealth Profile: (/-L" 1'1). LEMMA 3.3. J[p(A) is a Factional Kelly strategy then d dA /-L,(p(A)) > 0, O s A s l. --- ## Page 370 Growth versus Security in Dynamic Investment Analysis 341 1572 L. C. MACLEAN, W. T. ZIEMBA AND G. BLAZENKO PROOF. We have = E[YO ,~ a~O (~ R;(W, )Pj (A)) ~I C~ Rj(Ws)Pj(A))] = -tYoE[K(w' )Y'_ 1 (W'- I)] = -tyo(EK)(EYH ) < O. Then d dPo(A) a dA J.L,(p(A)) = ---;n:- apo J.L,(p(A)) > O. o LEMMA 3.4. Suppose p( A) is ajractional Kelly strategy and consider the growth rate (\' = 1. Then for O ~ A < 1. PROOF. We have 'Y,(Yu, peA)) = Prob [Y,(p(A)) ~ yoa'] = Prob [.'*1 In R,p(A) ~ tin a] = Prob [ :z.: In (ARJj + (I - A)RsPo) ~ 0] = Prob [ A :z.: In (R,ij) + :z.: Ils ~ 0] where Ils > 0 [ - L: Ils ] = Prob :z.: In (RJj) ~ -A- = I - F'(x,( A)) , where X,(A) = (-L: Ils)/A and F' is the distribution function for L:'\ ~ I In (Rjj). Then (d/dA)x(A) > 0 and F ' nondecreasing gives the desired result. 0 Hence for the wealth profile an effective tradeoff is possible along the fractional Kelly path. (iii) SlOpping Rule Profile : ( 11 - 1, (3). Our analysis of the stopping rule profile focuses on the computational formulas ~(p(A)) and (3(p(A)). Since we have upper and lower bounds for the true values, the accuracy of the approximation is easy to determine in a particular application. Our choice of strategy is based on the profile graph and that requires ~ and {3. The formulas for ~ and (3 are based on the random walk Z,(p( A)) with jump process J(p( A)). The process J(p( A)) isjlexible at the fractional Kelly strategy p( A) if the con- dition L '*( p( A)) (J L *( p ( ~)) E1' (p( A) )(JJ(P( ~ )) U'*( p( A)) (J U*( p( ~)) L*(p(A))(J L*( " (~ ) ) < EJ(p(A))(JJ( P(A )) < U*(p(A))(J U*(p(A) ) holds, where (J < I is a positive root of the equation E(J(p(A))J( P(A)) = I , and prime denotes the derivative W.r.t. A. LEMMA 3.5. Suppose p( A) is ajractional Kelly strategy and consider the wealth process Y,(p(A)) , t = I , ... and the absorbing states U = kyo , L = k - 1yo, k > 1.IjJ(p(A)) is jlexible at p( A) then .for 0 ~ A ~ 1 --- ## Page 371 342 L. C. MacLean, W. T. Ziemba and G. Blazenko GROWTH VERSUS SECURITY IN DYNAMIC INVESTMENT ANALYSIS 1573 (i) a _ ax 1)(Yo,p(X» <0, (ii) a - ax {3(yo, p(X» < O. PROOF. (i) For the mean first passage time consider the lower bound 2 X _ U*(p(X» 1) (p( » - EJ(p(X»OU.(P(A» , where without loss of generality assume Yo = 1 (zo = 0). Then where and N2(p(X» = [EJ(p(X»Ou,(P(X»J;n(P(X» - U*(p(X»EJ'(p(X»Ou,(p(X» - U*(p(X»EJ(p(X»OU.(f,(A»J;Il(p(X» In 0 - U* (p( X) )EJ(p( X»)O U.(J,(A» U* (p( X) )0' / 0], where the prime denotes the first derivative W.r.t. Po. Since EJ'(p(X» < 0 and EJ(p(X» > 0, J;I1(P(X» > 0, 0' < 0, we get all terms in N2(p(X» positive and (a/apo)1) 2(p(X» > O. In the same way for the upper bound 1)1(p(X» we get (a;apo)1)I(p(X» > O. With (d/ dX)po( X) < 0 the average of the bounds yields ( 1 ). (ii) For the first passage probability we utilize the upper bound 1 - OI'*(P(A» {31(p(X» = Ou* _ OI.*(P(A»· Then the numerator of(a;apo){3l(p(X» is N(p(X» = (Ou* - 1) ~ OI.*(p(A» - (1 - OL*(p(A») ~ OU*. apo apo We have EO(p(X»J(p( A» = 1 and (a;apo)EO(p(X»J(p(X» = 0 from which Then and a - O(p(X» = -O(p(X» In O(p(X))[EJ'(p(X))OJ(P(A))/ EJ(p(X»OJ(p(A»] apo = (-O(p(X» In O(p(X»)y;(p(X». ~ OI-*{J'(X» = OI.*(P(A»( L *(p( X» ~ 0 + In 0 ~ L *(p( X») apo O(p( X» apo apo = - L*(p(X»OL*(P(A»y;(p(X» In 0 + L'*(p(X»OI.*(P(A» In 0, --- ## Page 372 Growth versus Security in Dynamic Investment Analysis 1574 L. C. MACLEAN, W. T. ZIEMBA AND G. BLAZENKO Substituting into N(p( 'J...)) with a little algebra yields N(p('J...)) = -(O U" - I)L *(p('J...))OL"(P(A))l/;(p('J...)) In 0 + (J - OL"(p(A»)U*O U"l/;(p('J...) In 0 - (O U" - J)L'*(p('J...»)OL"(P( A» In O. With U = kyo and L = k - 1yo it is easy to show that U*O U" L(p('J...))OL"(P( A)) I - OU' < I - OL'(p(A» and therefore A = (J - eL"(p(A»)U*OU" > -(OU" - I )L*(p('J...))OL'(p(A)) = B. We have, then, substituting B for A N(p('J...)) > 2( 1- OU") In O(L*(p('J...))OL"(p( A))l/;(p('J...)) + L'*(p('J...))OL*(P(A))) > 0 since J flexible at p( 'J...) implies So L *(p('J...))OL*(p( A))l/;(p('J...)) + L'*(p('J...))OL"(p(A)) < O. a - {31 > O. apo 343 With the same result for the lower bound and with (d/d'J...)p('J...) < 0 we get (ii). 0 The proof of the above lemma depends on the flexibility condition but not the re- quirement that the strategy be fractional Kelly. The significance of the fractional Kelly strategies is that the flexibility condition is always satisfied if it is satisfied for the Kelly strategy. For the stopping rule profile (~ - I , if) we have along the fractional Kelly path growth increasing and security decreasing. The inverse of expected stopping time is a measure of growth since faster growth would imply getting to the target U sooner. In summary, for each of the profile types we have the same type of behavior. The complementary measures, growth and security, move in opposite directions as we change the blend between the Kelly and cash strategies. This enables a continuous tradeoff between growth and security. TRADEOFF THEOREM. Suppose p( 'J...) is a fractional Kelly strategy, Then for each profile there is an effective tradeoff between growth and security by varying 'J... , the trade- qfJ index, Additional theoretical results concerning these tradeoffs are presented in MacLean and Ziemba ( 1990), Friedman ( 1982) seems to have first considered fractional Kelly strategies in the context of blackjack, His results are along the lines discussed in §4, I below. Other references on fractional Kelly strategies are Gottlieb ( 1984); Ethier ( 1987); Li, MacLean and Ziemba (1990) ; and Wu and Ziemba (1990). The tradeoff theorem in this section establishes the use of growth and security measures in determining an investment strategy. There are many possible tradeoffs depending upon the choice of measures and the parameters specified in these measures. Rather than further considering the preferred measures and tradeoff in theory, we will apply the concepts to some real world examples and explore the choice of investment decisions in practice. --- ## Page 373 344 L. C. MacLean, W. T. Ziemba and G. Blazenko GROWTH VERSUS SECURITY IN DYNAMIC INVESTMENT ANALYSIS 1575 4. Applications The methods described in previous sections are now applied to a variety of well-known problems in gambling and investment. Rather than concentrating on the results in the tradeoff theorem, we will explore a variety of growth and security measures to see how interactively a satisfactory decision is achieved from both perspectives. In each case we emphasize a pair of measures (growth, security). There are a few points we should make about our approach to these applications. Satisfaction with a particular investment strategy will depend on many factors. The measures of performance we have described are helpful, but they are functions of inputs such as initial wealth, return distribution, and performance standards. We must vary those inputs in appropriate ways to see a decision rule under different scenarios. In cases where inputs are soft we may need many diverse scenarios. Throughout the theory and examples we use fixed strategies. This is not to imply that the particular fixed strategy is used forever, but rather a statement that ifall inputs remain constant we can predict the results from the fixed strategy. Of course, the inputs ,are dynamic and we would regularly (continuously) update the strategy and measures of performance based on new input data. This process is facilitated by easily computable measures. 4.1. Blackjack: (¢, (3) The game of blackjack seems to have evolved from several related card games in the 19th century. It became popular in World War I and has since reached enormous pop- ularity. It is played by millions of people in casinos around the world. Billions of dollars are lost each year by people playing the game. A relatively small number of professionals and advanced amateurs, using various methods such as card counting, are able to beat the game. The object is to reach, or be close to, twenty-one with two or more cards. Scores above twenty-one are said to bust or lose. Cards two to ten are worth their face value, Jacks, Queens, and Kings are worth ten points and Aces are worth one or eleven at the player's choice. The game is called blackjack because an Ace and a ten-valued card was paid three for two and an additional bonus accrued if the two cards were the Ace of Spades and the Jack of Spades or Clubs. While this extra bonus has been dropped by current casinos, the name has stuck. Dealers normally playa fixed strategy of hit until Probability 1.0 0.8 0.6 0.4 0.2 R.lativ. growth Optimal K.lly wag.r 0.0 -\------,,...------..-----,-------, 0.0 0.01 0.02 0.03 Fraction of W.alth Wag.r.d FIGURE 2. Probability of Doubling and Quadrupling before Halving and Relative Growth Rates Versus Fraction of Wealth Wagered for Blackjack (2% Advantage, p = 0.5\ and q = 0.49) . --- ## Page 374 Growth versus Security in Dynamic Investment Analysis 1576 Range for Blackjack Teams Overkill --+ Too Risky L. C. MACLEAN, W. T. ZIEMBA AND G. BLAZENKO TABLE 3 Growth Rates Versus Probability of Doubling before Halving for Blackjack /3(0, p) = P[Doubling "- before Halving] 0.1 0.999 0.2 0.998 0.3 t 0.98 t 0.4 SAFER 0.94 LESS GROWTH 0.5 0.89 0.6 RISKIER 0.83 MORE GROWTH 0.7 ~ 0.78 ~ 0.8 0.74 0.9 0.70 1.0 KELLY 0.67 1.5 0.56 2.0 0.50 345 0; p* = 0 if EK ~ O. (This optimal strategy may be interpreted as the edge divided by the odds (I-I in this case).) In general, for win or lose for two outcome situations where the size of the wager does not influence the odds, the same intuitive formula holds. Hence with a 2% edge betting on a 10-1 shot, the optimal wager is 0.2% of one's fortune.) The growth rate of the investor's fortune is rJ>(p) = 7fln(1 +p)+(I - 7f)ln(l - p) and this is shown in Figure 2. It is nearly symmetrical around p* = 0.02. Security measure S3 is also displayed in Figure 2, in terms of the probability of doubling or quadrupling before halving. The bounds, from (5), are fairly sharp. Since the growth rate and the security are both decreasing for p > p*, it follows that it is never advisable to wager more than p*. However, one may wish to trade off lower growth for more security using a fractional Kelly strategy. Table 3 illustrates the relationship between the fraction A and growth and security. For example, a drop from p = 0.02 to 0.01 for a 0.5 fractional Kelly strategy, drops the growth rate by 25%, but increases the chance of doubling before halving from 67% to 89%. --- ## Page 375 346 L. C. MacLean, W. T. Ziemba and G. Blazenko GROWTH VERSUS SECURITY IN DYNAMIC INVESTMENT ANALYSIS 1577 In the blackjack example we see how the additional information provided by the security measure (3(yo, p) was valuable in reaching a final investment decision. In a flexible or adaptive decision environment, competing criteria would be balanced to achieve a satisfactory path of accumulated wealth. With this approach professional blackjack teams typically use a fractional Kelly wagering strategy with the fraction 'A = 0.2 to 0.8. See Gottlieb ( 1985) for discussion including the use of adaptive strategies. 4.2. Horseracing: (JIt, 'Yt) Suppose we have n horses entered in a race. Only the first three finishers have positive return to the bettor. For the remaining positions, you lose your bet. Then n = {( 1, 2, 3), ... , (i, j, k), ... , (n - 2, n - 1, n) } is the set of all outcomes with probability 1fijk . We wager the fractions Pi I, Pn, Pi 3 of our fortune Yo on horse i to win, place, or show, respectively. One collects on a win bet only when the horse is first, on a place bet when the horse is first or second, and on a show bet when the horse is first, second, or third. The order of finish does not matter for place and show bets. The bettors, wagering on a particular horse, share the net pool in proportion to the amount wagered, once the original amount of the bets are refunded and the winning horses share the resulting profits. Let P be the n X 3 matrix of wager fractions, where L:;'~ ) L:]~ ) Pij:$ 1. The return function for a particular (i, j, k) outcome is K((i,j, k) , p) = (QW - Wi) ~[) + (QP - ;i - Pj) (~i2 + ~j2) + - +-+- ( QS - Si - Sj - Sk)(Pi3 Pj3 Pk3) 3 Si Sj Sk - (~i PI) + I~j PI2 + Il.k P13) , where Q = I - the track take, Wi, Pj and Sk are the total amounts bet to win, place, and show on the various horses, respectively, and W = L: Wi, P = L: p;, S = L: Sk . As an example we will consider a simplified scheme where a wager is made on a single horse per race and there are five races occurring simultaneously across tracks (Hausch and Ziemba 1990). The required information on the races is given below. Each of the horses has an expected return on a $1 bet of $1.14, so this is a very favorable situation. Probability of Collecting Race Horse on Wager Odds I A 0.570 I-I 2 B 0.380 2-1 3 C 0.285 3-1 4 D 0.228 4-1 5 E 0.190 5- 1 The Kelly strategy is P* = (p~, p~, pi, pr, p!, p~) = (0.8529,0.0140,0.0210,0.0141,0.0700,0.0280) . So 1.4% of your fortune is bet on horse A in race 1. Using the growth measure JIt(p) = mean accumulation of wealth, and the security measure 'Yt(p) = probability that accumulated wealth will exceed b[, the Kelly strategy was compared to half Kelly and the fixed fraction strategies which bet 0.0 I, 0.05 and 0.10 respectively, distributed across --- ## Page 376 Growth versus Security in Dynamic Investment Analysis 1578 b, 500 1,000 5,000 10,000 100,000 I-',(p)* L. C. MACLEAN, W . . T. ZIEMBA AND G. BLAZENKO TABLE 4 -y,( Yo, p) = Prob Y,(p) Will Exceed b, (t = 700, Yo = 1000) Strategy - p Proportional Kelly 0.5 Kelly 1% 5% 0.940 0.992 1.000 0.915 0.892 0.961 0.980 0.856 0.713 0.691 0.061 0.631 0.603 0.445 0.003 0.512 0.231 0.028 0.000 0.148 $17,497 $8,670 $2,394 $10,464 * The median is used here since Y,(p) is approximately log normal (skewed) (Ethier 1987). 347 10% 0.518 0.517 0.358 0.304 0.141 $1 ,176 horses in the proportions (0.1 , 0.3, 0.3, 0.2, 0.1). These fixed fraction strategies are not fractional Kelly. The results for J.Lr(P) and "(r(p) are given in Table 4. The Kelly strategy with a 6% betting fraction and the 5% fixed fraction are similar, both having favorable growth and security. The smaller betting fractions provide slightly more security but have much less growth. Since the 10% strategy is beyond the Kelly fraction it has less growth and less security. An interesting point is that the total fraction bet is more significant than the distribution of the bets across the horses. Implicit in the above example is the ability to identify races where there is a substantial edge in the bettor's favor. There has been considerable research into that question. Hausch et al. ( 1981) demonstrated the existence of anomalies in the place and show market. At thoroughbred racetracks about 2-4 profitable wagers with an edge of 10% or more exist on an average day. The profitable wagers occur mainly because: ( I ) the public has a distaste for the high probability-low payoff wagers that occur on short priced horses to place and show, and (2) the public's inability to properly evaluate the worth of place and show wagers because of their complexity-for example, in a ten-horse race there are 72 possible show finishes, each with a different payoff and chance of occurrence. In Hausch et al. ( 1981 ) and more fully in Hausch and Ziemba ( 1985) equations are de- veloped that approximate the expected return and optimal Kelly wagers based on minimal amounts of data to make the edge operational in the limited time available at the track. Ziemba and Hausch ( 1987) implement and discuss these ideas and explore various applications, extensions, simulations and results. A laymen's article discussing this appears in the May 1989 issue of OMNI. A survey of the academic literature on racetrack and other forms of sports betting is in Hausch and Ziemba (1992) . 4.3. Lollo Games: (1/, (3) Lotteries have been played since before the birth of Christ. Organizations and govern- ments have long realized the enormous profits that can be made, based on the greed and hopes of the players. Lotteries tend to go through various periods of government control and when excessive abuses occur, they have often been shut down or outlawcd, only to resurface later. Since 1964, there has been an unprecedented growth in lottery games in the United States and Canada. Current yearly sales are over thirty billion dollars, while net profits to various governmental bodies are well over ten billion dollars. Since the prizes are very large, the public cares little that the expected return per dollar wagered is only 40-50 cents. Typically, half the money goes to prizes, a sixth to expenses, and a third to profits. With such a low payback, it is exceedingly difficult to win at these games --- ## Page 377 348 L. C. MacLean, W. T. Ziemba and G. Blazenko GROWTH VERSUS SECURITY IN DYNAMIC INVESTMENT ANALYSIS l579 Prizes Jackpot Bonus, 5/6+ 5/6 4/6 3/6 Edge Optimal Kelly Bet Probability of Winning 1/13,983,8 16 1/2,330,636 1/55,492 1/1,032 1/57 Optimal Number of Tickets Purchased per Draw with $10 M Bankroll TABLE 5 Lollo Games Prize $6M $0.8 M $5,000 $150 $10 Case A Contribution to Expected Value 42.9 34.3 9.0 14.5 ~ 118.1 18.1 % 0.000,000, I I II Prize $10 M $1.2 M $10,000 $250 $10 Case B Contribution to Expected Value 71.5 51.5 18.0 24.2 -B.1 182.7 82.7% 0.000,000,65 65 and the chances of winning any prize at all, let alone one of the big prizes, is very small. Is it possible to beat such a game with a scientific system? The only hope of winning is to wager on unpopular numbers in pari-mutuel lotto games. A lotto game is based on a small selection of numbers chosen by the players. Typically, games have forty to forty- nine numbers and you must choose five or six numbers. Management then draws the winning numbers and prizes typically accrue to those with three, four or more correctly matching numbers. By pari-mutuel, it is meant that the net pools for the various prizes are shared by those with winning tickets. Hence, if a small number of people, with so- called unpopular numbers, win a prize, it will be larger than if more people with popular numbers are sharing. Ziemba et al. ( 1986) investigated the Canadian 6/49, which played across the country, and other regional games. They found that numbers ending in nines and zeros and high numbers tended to be unpopular. Collections of unpopular numbers have a slight edge, a dollar wagered was worth more than a dollar on average, and this edge became fairly substantial when there was a large carryover. With a large carryover, that is, a jackpot pool that is steadily growing because the jackpot has not been won lately, the edge can be as high as 100% or more. Despite this promise, investors may still lose because of the very reasons that Chernoff found, essentially reverberation to the mean and gamblers ruin. Still, with such substantial edges, it is interesting to see how an investor might do playing such numbers over a long period of time. Two realistic cases are developed in Table 5. Case A corresponds to the situation when the investor wagers only when there is a medium-sized carryover and the numbers that are drawn are quite unpopular. Case B corresponds to the situation where there is a huge carryover and the numbers drawn are the most unpopular. About one draw in every five to ten is similar to Case A and one draw in every twenty or more corresponds to Case B. Also, we have been generous in the suggested prizes. In short, we are giving the un- popular number system an at least fair chance to see ifit has any hope of being a winning system in the long run. These cases correspond to the Canadian situation of paying the lotto winnings in cash up front tax free. The US situation of paying the lotto winnings over twenty years and taxing these winnings yields prizes with expected returns about one-third of those in Canada. First we observe that the optimal Kelly wagers are miniscule. This is not surprising since for Case A, 77.2 cents of the $1.18 expected value and for Case B, $1.23 of the --- ## Page 378 Growth versus Security in Dynamic Investment Analysis 1580 L. C. MACLEAN, W. T. ZIEMBA AND G. BLAZENKO Probability .9 .8 .7 .6 .5 .4 .3 .2 .1 .0 O.OE+OO Optimal K!llly wager 9.7E-08 1.9E-07 2.9E-07 Fraction of wealth wagered Double Quadruple Tenfold 349 FIGURE 3. Probability of Doubling, Quadrupling and Tenfolding before Halving for Lotto 6/49, Case A. $1.83 expected value, respectively, is composed of less than a one in a million chance of winning the jackpot or the bonus prize. One needs a bankroll of $1 million to justify even one $1 ticket for Case A and over $150,000 for Case B. If one had a bankroll of $10 million, the optimal Kelly wagers are just II and 65 $1 tickets respectively. Figures 3 and 4 show the chance that the investor will double, quadruple, or tenfold his fortune before it is halved, using Kelly and fractional Kelly strategies for Cases A and B respectively. For the Kelly strategies, these chances are 0.4 to 0.6 for Case A and 0.55 to 0.80 for Case B. With fractional Kelly strategies in the range of 0.00000004 and 0.00000025 or less, the chance oftenfolding one's initial fortune before halving it is 95% or more with Cases A and B respectively. This is encouraging, but it takes an average 294 and 55 billion Probability 1.0 Growth rate 0.8 0.6 0.4 0.2 0.0 "-------,-------,----0"----,-----,-------, O.OE+OO 2.0E-07 4.0E-07 Optimal Kelly Wager 6.0E-07 8.0E-07 I.OE-06 I.2E-06 Fraction of Wealth Wagered FIGURE 4. Probability of Doubling, Quadrupling and Tenfolding before Halving for Lotto 6/49, Case B. --- ## Page 379 350 L. C. MacLean, W T. Ziemba and G. Blazenko GROWTH VERSUS SECURITY IN DYNAMIC INVESTMENT ANALYSIS 1581 o 1/2 2 6 8 10 Initial Wealth ($ million) FIGURE 5. Probability of Reaching the Goal of$IO Million before Falling to $! Million with Various Initial Wealth Levels for Kelly, ! Kelly and ~ Kelly Wagering Strategies for Case A. years, respectively, to achieve this goal. These calculations assume that there are 100 draws per year. Figure 5 shows the probability of reaching $10 million before falling to $! million for various initial wealth levels in the range H-$IO million for Cases A and B with full Kelly, half Kelly and quarter Kelly wagering strategies. The results are encouraging to the millionaire lotto player especially with the smaller wagers. For example, starting with $1 million there is over a 95% chance of achieving this goal with the quarter Kelly strategy for Cases A and B. Again, however, the time needed to do this is very long being 914 million years for Case A and 482 million years for Case B. More generally we have that Case A with full Kelly will take 22 million years, half Kelly 384 million years and quarter Kelly 915 million years. Case B with full Kelly will take 2.5 million years, half Kelly Probability 1.00 0.8 0.6 0.4 0.2 0.00 o 1/2 2 3.08 6 8 10 Initial Wealth ($ million) FIGURE 6. Probability of Reaching $10 million before Falling to $25,000 with Various Initial Wealth Levels for Kelly and! Kelly Wagering Strategies for Case B. --- ## Page 380 Growth versus Security in Dynamic Investment Analysis 351 1582 L. C. MACLEAN, W. T. ZIEMBA AND G. BLAZENKO 19.3 million years and quarter Kelly 482 million years. It will take a lot less time to merely double one's fortune rather than tenfolding it, but it is still millions of years. For Case A it will take 4.6, 2.6 and 82.3 million years for full, half, and quarter Kelly, respectively. For Case B it will take 0.792, 2.6 and 12.7 million years for full, half and quarter Kelly. Finally, we investigate the situation for the nonmillionaire wishing to become one. First, our aspiring gambler must pool his funds with some colleagues to get enough bankroll to proceed since at least $150,000 is needed for Case Band $1 million for Case A. Such a tactic is legal in Canada and in fact highly encouraged by the lottery corporations who supply legal forms for such an arrangement. For Case A, our player needs a pool of $1 million even if the group wagers only $1 per draw. Hence the situation is well modeled by Figure 5. Our aspiring millionaire "puts up" $100,000 along with nine other friends for the $1 million bankroll and when they reach $10 million each share is worth $1 million. The pool must play full Kelly and has a chance of success of nearly 50% before disbanding if they lose half their stake. Each participant does not need to put up the whole $100,000 at the start. Indeed, the cash outflow is easy to bankroll, namely 10 cents per week per participant. However, to have a 50% chance of reaching the $1 million goal each participant (and his heirs) must have $50,000 at risk. On average it will take 22 million years to achieve the goal. The situation is improved for Case B players. First, the bankroll needed is about $154,000 since 65 tickets are purchased per draw for a $10 million wealth level. Suppose our aspiring nouveau riche is satisfied with $500,000 and is willing to put all but $25,000 I 2 or $12,500 of the $154,000 at risk. With one partner he can play half Kelly strategy and buy one ticket per Case B type draw. Figure 6 indicates that the probability of success is about 0.95. On average with initial wealth of $308,000 and full Kelly it will take ~ million years to achieve this goal. With half Kelly it will take 2.7 million years and with quarter Kelly it will take 300 million years. The conclusion then seems to be: millionaires who play lotto games can enhance their dynasties' long-run wealth provided their wagers are sufficiently small and made only when carryovers are reasonably large; and it is not possible for non-already-rich people, except in pooled syndicates, to use the unpopular numbers in a scientific way to beat the lotto and have confidence of becoming rich; moreover these aspiring millionaires are most likely going to be residing in a cemetery when their distant heir finally reaches the goal. 4.4. Playing the Turn q(the Year Ejject with Index FlIlures: ( t = 1, ... , T}. 1. The value at risk measure for horizon T and wealth value w is qJ)(H, w) = Pr[W(T) ~ w] = 1 - HT(W) , --- ## Page 389 360 L. C. MacLean, R. Sanegre, Y. Zhao and W. T. Ziemba 942 L. C. MacLean el al. 110l//'I1ul of Ecol1omic Dynamics & Control 28 (2004) 937- 954 2. The drawdown measure for decay fraction b E [0, 1] is (P2(H,h) = Pr[W(t + 1) ~ bW(t), t = 1, . .. , T]. 3. The incomplete mean measure for horizon T and specified percentile IX is where Pr[W(T) ~ wa ] = IX. Each of these measures satisfies the basic axioms for downside risk. 4>1(H, w,, ) and 2(H,b(w)) ~ rj)I(H, w). So rj)2 and r/)3 are more stringent risk measures than r/)I' The purpose in defining a risk measure is to develop an investment strategy which achieves capital growth while controlling for risk. Subsequent discussion of risk will concentrate on the VaR measure 4>1 . The approach followed is easily adapted to other risk measures. Consider the general growth with security problem sup{E[G(X)] J(/),(H(X), w) ~ 1 - IX}, X (6) where H(X) = (H,(X), . .. ,HT(X)) is the distribution over wealth generated by invest- ment strategy X = {X(!), t = 1, ... , T}, w is a prespecified wealth floor and 1 - IX is a confidence level. It is assumed that the measurability conditions previously discussed for the return process and the investment process are imposed. Let w*= w/ Wo . Problem (6) can be written more explicitly in terms of rate of returns as (CCP): s~p { t, E In(R(t) T X(t)) Ipr [t,ln(R(t) T X(t)) ~ In w* 1 ~ 1 - IX} . (7) The rate of return process R = {R( t), t = I, ... , T} and the investment process X = {XU),t = 1, .... T} are defined on (Q,B, {BI} ,P). For a given trajectory OJ E Q let the associated return path be R( (j)) = {R( (0, t), t = 1, .. . , T} and the investment path be X(w) = {X(w,t),t = I, .. . , T}. The risk measure refers to acceptable and unac- ceptable sets of wealth trajectories. Consider sets of measure 1 - IX in the probability space, given by BCI. = {A JP(A) ~ I - IX} . There are associated sets of wealth trajectories for A and its complement A. An equivalent fonnulation to (7) based on trajectories --- ## Page 390 Capital Growth with Security 361 L.c. lvlacLean el al./Journal of Economic Dynall1ics & COlltrol28 (2004) 937-954 943 is (DP) I~~I~, [s~p {t E In(R(t) T X(t» 1t.>n(R«(V, I) T X(w,t) ~ lnw*,w EA }]. (8) The disjunctive fOI111Ulation in (8) defines a sequence of stochastic convex dynamic programming problems, Rockafeller and Wets (1978) referred to as inner problems, with a constraint for each wE A. Another reformulation of (7), found by introducing a weighing variable 8(w), wE Q, with 0 ~ 8(w) ~ 1, is (NCP): ~x~h {tEln(R(t) T X(t) 18(W) [t In(R(t)T X(t» ~ lnw* ] ~ 0, E[8(w)] ~ 1 - C(}. (9) At the optimal solution to (9) the weighing variable is an indicator function. That is, if (X*, 8 *) is optimal then there is a set A such that 8 *(w) = 1 for wE A, 8 *(w) = 0 for wE A.. Hence, for the optimal investment strategy: (CCP) ¢:} (DP) ¢:} (NCP). The growth with security problems (7)-(9) provide a framework for individual port- folio choice, where risk is controlled at a specific level. In multidimensional financial markets it is common to identify the underlying sources of systematic risk, and to define a small set of generating portfolios or mutual funds. These funds serve an inter- mediary role for fund managers to create products which satisfy investor preferences. It is important that the growth with security problem works within this intermediation theory. To understand the generating portfolios (mutual funds) recall for the price process it is assumed there exist p < m independent latent factors U T = (Ul, ... , Up), Ui ~ N (0, 1), i = 1, .. . ,p, so that the log-prices are represented as yet) = pt + tAU + Vi~, (10) where pT = C!11 , ... ,Pm), }, = Uij) for loadings )"j, i = l, ... ,m and j = l , ... ,p, and the covariance Cou( 0 = d iay( bt, ... , (5;,). Assume that E[ ¢] = 0, and U and ¢ are independent. yCt) has the distribution N(pl,L(t», where L(t) = tL1 + t2r and r=AAT . The factor model (l 0) and the associated conditions are referred to as the structural model assumptions. In this structure imposed on the price process, the factors U T = ( U1 , ••• , Up) are the standardized log-prices for the mutual funds. Theorem 1 (Financial intermediation). Suppose there exist m risky securities and a risk-ji-ee asset satisfying the market and structural model assumptions. Then there exists a risk-ji-ee and p < m risky mutual funds, so that growth with security investors as defined by (7) are indifferent between choosing portfolios from among the original securities and choosing portfolios ji-om the mutual funds. --- ## Page 391 362 L. C. MacLean, R. Sanegre, Y. Zhao and W. T. Ziemba 944 L. C. MacLean el al. I Journal of Economic Dynamics & Control 28 (2004) 937- 954 Proof. Consider the return in period t, R(t) T X(t) = 2:;:0 R;(t)Xi(t). From the model for securities prices Ri(f) = exp(Pi+ Ai u + ()iBi), where Ai is the ith row of the m x p loading matrix A in (10). Let Ao = (aij) , where aij = ),t/(o}- b7), and Define q(t) = A . X(t). From (10), 2:;=1 Ali = o-T- ()7 and m = L:Xi(t) = 1. i=O With A-, the generalized inverse of A, let X(t)=A-q(t). Hence, the return in period t is R(t) T X(t) = (R(t) T A- )q(t) =M(t) T q(t) = 2:)=0 Mj(t)qj(t), with MJCt) the return on risky mutual fund j, j = 1, ... , P and Mo(t) = 1 + r(t). Since the returns in each period are the same for the mutual funds and the original securities, the statement in the theorem holds. 0 The investment process X = {X(t), t= 1, ... , T} is defined on the space of trajectories (Q,B, {Bt} ,P), where X(t) is Bt predictable. An important property of the investment process is path independence (Cox and Leland, 2000), so that it depends only on the reinvested return on the securities, not on the whole price history. This property is satisfied by the growth with security investor. Theorem 2 (Path independence). The optimal growth with security investment strat- egy X* is path independent, i.e. X(t)* depends on the level of wealth at time t, Wt, but not on the particular path to achieve that wealth. Proof. Consider the growth with security problem in the disjunctive form If fA is the indicator function for the set A and ),( OJ) ? 0 is a multiplier so that ), E L(Q,B,P), the space of Lebesgue integrable functions on Q, then a Lagrangian --- ## Page 392 Capital Growth with Security 363 L. c. MacLean el al. /Jollrnal o( Economic Dynamics & Control 28 (20(M) 937- 954 945 for (DP) is 2 (X,)" A) = £ [t,ln(RCU),t)TX(U),{» + 1;1),«(1) (tln(R(W,I)TX({J),t) - lnw')] ' ( II ) If a solution exists for (DP) then there exist elements CA*, ).* ) so that the solution to (DP) is given by sUPx 2 (X, ;,*, A* ). The Lagrangian is l' 2 (X,),*, A* ) = 2:£[(1 + h · ),* )ln(R(t)TX(t))]. 1=1 Consider 116 = £[1 + 14' ;'*] and 17; = £1'- ,(1 + 1;/*;: ), where £1' - 1 is thc cxpectation over the data process from time t + 1 to time T. Then, II; = 11;- 1 . (I;, where I~; depends only on the additional data in period t. Since In 11( w') = L ~.= I m.v( (V, ), where m.,(w \. ) = £ '- sm(u/) - £ t- s+lm(co'), then l' l' sup 2: £[(1 + lA- ) In(R(t) T XCt»] = sup 2: £[1/;_ 1 . fJ; InCR(t) T X(t»)] x 1=1 X ,= 1 l' = sup 2:E[fJ,' · E,_ 1[11;_lln(R(t)TX(t))]], x , - I where £ 1- 1 is the expectation with respect to data for periods 1 to t - 1. With il;_ 1 = £ , ... 11/;_ 1' ~ 11;_lln(£H [R(t)T ~;= : XU)]) from Jensen's inequality and B,_I measurability of 117_ 1 and X(t). But Ri(t )=exp(Zi(t»), where the increments Zi(t) are intertemporally independent. With X(t) = £ 1- 1 [(11; -1/ /1;_1 )X(t)], the problem becomes l' sup 2: r;7- 1£[fJ; In(R(t) T X(t) )], x I - I where X(t) does not depend on the path 0/ - 1, t = 1, . . . , T . 0 The path independence property is within the context of planning for T periods with a given distribution over returns. As described in Section 6, at the start of the next T period plan the history of prices is used to update the distribution over returns. --- ## Page 393 364 L. C. MacLean, R. Sanegre, Y. Zhao and W T. Ziemba 946 L. C MacLean ef al. /Journal of Ecol1omic Dynamics & Control 28 (2004) 937- 954 So decisions depend on trajectories through the estimation (filtration) process, which is separated tl·om the optimization. 4. Scenario selection The growth with security problem in its disjunctive fOlm (8) or equivalent La- grangian form (11) involves complicated multivariate integration. Some form of dis- crete approximation to distributions for securities prices is required to proceed with computation. Consider the distributions F, on the returns Ra( t), t = 1, ... , T. Assume that in rytm a grid is constructured with rectangles so that the probability mass from F, in each rectangle is equal. Sample a point from each rectangle, generating the empirical distribution F;. Let the space of trajectories corresponding to the empirical distribution F;, t= I, ... ,T, be Q (n ), and the corresponding probability space be (Q(n),B(n),p(n». So Q(n) = {WI, ... , wr} and P(Wi) = l /f. With this lattice structure, a discrete approximation to the growth with security problem is (DPn ) : A ("~~~(n) {s~p { Et,ln(Rn(t) T Xn(t» It In(Rn(w, t) T X"(w, t) ~ In, w',w EA(n)} }. (12) The inner problem and the set selection problem are now discrete. For a given A and the corresponding A(II) the convergence of the solution of the discrete inner problem to the solution of the continuous inner problem has been established and bounds on the error for given n developed (Pflug, 1999). The identification of the optimal set A(n), where p(n)(A(n» ~ 1 - IX, is required. Assume that the specified target w' at the horizon T is such that problem (12) has a solution for A(n) = Q(n). The process for identifying the optimal set of scenarios is backward elimination (Culioli. 1996). If k is the largest integer such that k/f ~ IX, then up to k constraints corresponding to scenarios in [2(11 ) get eliminated and the remaining constraints yield the optimal A(n). The elimination is one constraint at a time starting from [2(n) and working backward. In describing the elimination procedure the notation A(n)(ijlij _ I , . .. , id refers to a set of scenarios in the family of sets A)")(ij- I, ... , i l ) defined by the j - 1 elimination steps, where scenarios Wil' ... , Wi;_1 have been eliminated sequentially. The basis for the algorithm is the inner problem P(A"'), s~p {t, E In(R'(t) T X'(t)) It.ln(R''(W'') T X'(w, I)) ;" In w"w E A'" }, The algorithm proceeds as follows: --- ## Page 394 Capital Growth with Security 365 L. C. MacLean el a/./Journa/ of Economic Dynamics & Con/rol28 (2004) 937- 954 947 [2] (Solution step) For each A(II)(ii!-) E Ajll)(-), solve the problem p(A(n)(ijl· )), denot- ing the optimal (strategy, value) by (X(A(n)(iil' », G*(A(n)(ij l' »)' and the set of scenarios corresponding to the active constraints by Ci+iCA(n)(ii l· ». If j = k or Cf+1 (A(II)Ui!-» = 0, then designate the problem P(A(I1)U;!- » as k-reduced and go to step [4]. [3] (Reduction step) For each A(II)(ii l·)EAjl/\.), generate a family of reduced sets, indexed by ij+ l, Aj~1 (ij, . . . , i l ) = {A (II )(ij+ t!. ) IA(II )(£;+11· ) = A(n)(ii!- )/ {OJ}, W E Cj + I (A(I1)(i;!- »)}. Return to step [2] with j updated to j + 1. [4] From the set of all k-reduccd problems select the set A(II) with the maximum optimal value G*(A(II». The optimal strategy for the growth with security problem (DPI/) is X * (A(I1». The reduction step in the elimination algorithm is founded on the following simple result which establishes the existence of candidate scenarios for elimination. Theorem 3 (Scenario elimination). Assume there exists an "optimar' set A(I1) fiJI' the problem (DPn ) and consider the j step reduced problem P(A(II)(ij l·» where j active constraillts have been sequentially eliminated. That is, the correspondinq scenarios are in Q(n)/A(I1 ). If cj~I(A(I1)(ijl ' » 1 0 are the scenarios correspolldinq to active constraints in the solution of the reduced prohlem, then Cf+ I(A(n)(iil '» n [Q(n)/A(I1)] 10. Proof. It suffices to consider the first elimination. In the solution to the problem with all constraints, p(Q(n», there arc sets of scenarios CI(Q(n» and Q(n )/CI(Q(fI» corre- sponding to active and inactive constraints, respectively, where C1(Q(n» 1 0. Since the inactive constraints exceed the goal In w* CI(Q(n» S;;; A(Il) and CI(Q(n» n [Q(n)/A(I/)] 10. The procedure for identifying A(I1) has a tree structure for reduced problems. If, for example, the number of active constraints in each reduced problem is r ~ 2, then the k eliminations require solving I:~= I rJ problems. This could be unmanageable. A simple heuristic is to solve one reduced problem at each stage corresponding to eliminating the active constraint in the previous stage with the best result (highest expected log wealth). With this heuristic method at most rk problems are solved. The heuristic is equivalent to the exact method if there is at most one active constraint at each stage. 5. Application to the fundamental problem of asset allocation over time The computation of growth with security strategies is now illustrated with the de- tennination of optimal fractions over time in cash, bonds and stocks. Table 1 has information on the yearly asset returns of the S&P500, the Salomon Brothers Bond index and U.S. T-bills for 1980-1990 with data from Data Resources, Inc. Without --- ## Page 395 366 L. C. MacLean, R. Sanegre, Y Zhao and W T. Ziemba 948 L. C MacLean el at./ Journal of Economic Dynamics & Controt 28 (2004) 937- 954 Table Yearly rate of return on assets relative to cash (%) Parameter Mean: It Standard deviation: rr Correlation: fI Table 2 Rates of return scenarios Stocks 108.75 12.36 0.32 Bonds 103.75 5.97 Cash 100 o Scenarios Stocks Bonds Cash Probability 95.00 101.50 100 0.25 2 106.50 110.00 100 0.25 3 108.50 96.50 100 0.25 4 125.00 107.00 100 0.25 loss of generality cash returns are set to one in each period and the mean returns for other assets are adjusted for this shift. The standard deviation for cash is small and is set to 0 for convenience. A simple grid was constructed from the assumed lognormal distribution for stocks and bonds by partitioning 9{2 at the centroid along the principal axes. A sample point was selected from each quadrant with the goal of approximating the parameter values. The sample points are shown in Table 2. The planning horizon is T = 3, so that there are 64 scenarios each with probability 1/64. Problems are solved with the VaR constraint and then for comparison, with the stronger drawdown constraint. (i) VaR control with w* = a. Consider the problem n~x {Et,ln (R(t)TX(t) ipr [tln(R(t)TX(t» ~ 3lna] ~ 1 - rx }. With initial wealth W(O) = I, the value at risk is a3. Table 3 presents the optimal investment decisions and optimal growth rate for several values of a, the secured average annual growth rate and 1 - rx, the security level. The heuristic was used to determine A, the set of scenarios for the security constraint. Since only a single constraint was active at each stage the solution is optimal. The mean return structure for stocks is quite favorable in this example, as is typical over long horizons (see Keim and Ziemba, 2000), and the Kelly strategy, not surprisingly, is to invest all capital in stock most of the time. It is only when security requirements are high that some capital is in bonds. As the requirements increase the fraction invested in the more secure bonds increases. The three-period investment decisions become more --- ## Page 396 Capital Growth with Security 367 L. C. MacLean ('I al. / jOl/mal (If Economic Dynamics & Con/rol 28 (2004) 937- 954 949 Table 3 Growth wi th secured rate Sccurcd Secured Period Optimal growth level growth rate (/ I-:y, rate (%) 2 3 Stocks Bonds Cash Stocks Bonds Cash Stoch Bonds Cash 0.95 0 0 0 0 0 0 0 23.7 0.85 0 0 0 0 0 0 23.7 0.9 0 0 0 0 0 0 23.7 0.95 0 0 I 0 0 0 0 23.7 0.99 0 0 0.492 0.508 0 0.492 0.508 0 19.6 0.97 0 0 0 0 0 0 0 23.7 0.85 0 0 0 0 0 0 23.7 0.9 0 0 0 0 0 0 23.7 0.95 0 0 1 0 0 0 0 23.7 0.99 0 0 0.333 0.667 0 0.333 0.667 0 1 R.2 0.99 0 0 0 0 0 0 0 23.7 0.85 0 0 0 0 0 0 23.7 0.9 0 0 0 0 0 0 23.7 0.95 0 0 0.867 0.133 0 0.867 0.133 0 19.4 0.99 0.456 0.544 0 0.27 0.73 0 0.27 0.73 0 12.7 0.995 0 0 0 0 0 0 0 23.7 0.85 0 0 0.996 0.004 0 0.996 0.004 0 23.7 0.9 0 0 0.996 0.004 0 0.996 0.004 0 23.7 0.95 0 0 0.511 0.489 0 0.442 0.558 0 19.4 0.99 0.27 0.73 0 0.219 0.59 0.191 0.218 0.59 0.192 12.7 0.999 0 0 0 0 0 1 0 0 23.7 0.85 0 0 0.956 0.044 0 0.956 0.044 0 23.4 0.9 0 0 0.956 0.044 0 0.956 0.044 0 23.4 0.95 0 0 0.38 1 0.619 0 0.51 0.49 0 19.1 0.99 0.27 0.73 0 0.008 0.02 0.972 0.008 0.02 0.972 5.27 conservative as the horizon approaches. Although this example is simplified the patterns observed illustrate the effect of security constraints on decisions and growth. Oi) Secured annual drawdown: b. The VaR condition only controls loss at the horizon. At intermediate times the investor could experience substantial loss, and in practice be unable to continue. The more stringent risk control constraint, referred to as drawdown, considers the loss 111 each period. Consider, then, the problem mr { E t In(R(t) T x(t))1 P,[ln(R(t) 'X(t)) " In b, t ~ 1,2,3] " I _. " } --- ## Page 397 368 L. C. MacLean, R. Sanegre, Y. Zhao and W T. Ziemba 950 L.c. MacLean el a1. / Journal of Economic Dynamics & Control 28 (2004) 937-954 Table 4 Growth with seclired maximum drawdown Draw- Secured Period Optimal down level growth b 1-:< rate (%) 2 3 Stocks Bonds Cash Stocks Bonds Cash Stocks Bonds Cash 0.96 0 0 0 0 0 0 0 23.7 50 0 0 0 0 0.846 0.154 0 23. 1 75 0 0 0.846 0.154 0 0.846 0.154 0 22.5 100 0.846 0. 154 0 0.846 0.154 0 0.846 0.154 0 21.9 0.97 0 0 0 0 0 0 0 23.7 50 0 0 0 0 0.692 0.308 0 22.5 75 0 0 0.692 0.308 0 0.692 0.308 0 21.3 100 0.692 0.308 0 0.692 0.308 0 0.692 0.308 0 20.1 0.98 0 0 0 0 0 0 0 23.7 50 0 0 I 0 0 0.538 0.462 0 21.2 75 0 0 0.538 0.462 0 0.538 0.462 0 18.6 100 0.538 0.462 0 0.538 0.462 0 0.538 0.462 0 16.1 0.99 0 0 0 0 0 0 0 23.7 50 0 0 1 0 0 0.385 0.615 0 21.2 75 1 0 0 0.385 0.615 0 0.385 0.6 15 0 18.6 100 0.385 0.615 0 0.385 0.615 0 0.385 0.615 0 16.1 0.999 0 0 0 0 0 0 0 23 .7 50 0 0 0 0 0.105 0.284 0.611 17.7 75 0 0 0.105 0.284 0.611 0.105 0.284 0.611 11.8 100 0.105 0.284 0.611 0.105 0.284 OJi I I 0.105 0.284 0.611 5.84 The simple form of the constraint follows from the arithmetic random walk In W(I), where Pr[W(t+l)~bW(t), t=0, 1 ,2]=Pr[lnW(t+l)- lnW(t) ~lnb, t=0,1 , 2] = Pr[ln(R(t)TX(t)) ~ lnb, t = 1,2,3]. In Table 4 the optimal investment decisions and growth rate for several values of b, the drawdown and 1 - rx, the security level are presented. The heuristic is used in determining scenarios in the solution. The security levels are different in the table since constraints are active at different probability levels in this discretized problem. As with the VaR constraint, investment in the more secure bonds and cash increases as the drawdown rate and/or the security level increases. Also the strategy is more conservative as the horizon approaches. For similar requirements (compare a = 0.97, J-rx=0.85 and b=0.97, l-rx=0.75), the drawdown condition is more stringent, with the Kelly strategy (all stock) optimal for VaR constraint, but the drawdown constraint --- ## Page 398 Capital Growth with Security 369 L.C MacLean el al./Journal of Economic Dynamics & Control 28 (2004) 937- 954 951 requires substantial investment in bonds in the second and third periods. In general, consideration of drawdown requires a heavier investment in secure assets and at an earlier time point. It is not a feature of this aggregate example, but both the VaR and drawdown constraints are insensitive to large losses, which occur with small probabil- ity. Control of that effect would require the lower patiial mean violations condition or a model with a convex risk measure that penalizes more and more as larger constraint violations occur, see e.g. Carino and Ziemba (1998). These results can be compared with those of Grauer and Hakansson (1997) who do calculations with the standard capital growth-Kelly model and Brennan and Schwartz (1998) who use a Merton, con- tinuous time model with the instantaneous mean returns dependent upon fundamental factors. These models have hair trigger type behavior that is very sensitive to small changes in mean values (see Chopra and Ziemba, 1993). 6. Portfolio rebalancing The approach to investment planning with downside risk control is to develop an optimal strategy over a T period planning horizon using projections of the multivariate returns distributions on securities. Although securities prices are dynamic, the changes are generated from a pricing model with seed parameters (}1, 1, LI). It is anticipated that the planning horizon is short, and the values of the seed pa- rameters will be reconsidered at the end of the horizon. An important feature of the proposed pricing model is the ability to revise the seed parameters using data collected during the planning period. Consider the observations {Y(1), ... ,Y(T)}. Since (Y(T)ln) '" N(nT,TLI) and n '" N(}1, r), from Bayes Theorem where and (n(T)IY(T» ~ N(nT,lT), - I - nT = It + (l - LlrLr )(Y r - }1), I ( A " I A 1T = T2 1- LJrTi )LJT , - 1 Yr=rY(T), Llr = TLl (13 ) (14 ) Furthennore, from the increments Zi( t) = f i ( t + I) - f i ( t), t = 1, .. . , T - 1, the covariance matrix Sr can be computed. With the number of factors (mutual funds) set at p < In, perfonn a factor analysis on ST obtaining a loading matrix Lr and the specific variance matrix Dr = diay(DT, ... ,D~,) , Let S; = LrLJ + Dr. Then S; is an estimate of L1', and Dr is an estimate of LI T . If there is a common mean, Jl T = ( p, ... , Jl) --- ## Page 399 370 L. C. MacLean, R. Sanegre, Y Zhao and W T. Ziemba 952 L.c. MacLean el al./Journal of Economic D)'Iwl!1ics & COillrol28 (2004) 937- 954 and YT = ~ I:7~ ] Y;(T), then YT is an estimate of II. All parameters in (13) and (14) are now estimated and the rate of return distribution for the next T period planning cycle can be specified. The revision of the rate of return distributions produces a rebalanced portfolio in the next planning cycle in light of new infonnation on securities prices. The fonnula for nT in (14) displays reversion to the grand mean. Considering the impact of errors in estimating mean values (Chopra and Ziemba, 1993), the improved estimates lead to more reliable investment decisions. An alternative to blending the mean estimate with a grand mean would be to blend the mean with the prior Bayes estimate. That is, for successive planning pe- riods {I, ... , T], T] + I, .. . , T] + T2 } the revised estimate for the mean rate of return is nT, = nT, + (I - LlT,L;,1 )(Y T, -nT,). This approach provides smoothed estimates where the full history of prices is considered with the past being weighted in the manner of exponential smoothing. 7. Conclusion This paper considers the problem of investment in risky securities with the objective of achieving maximal capital growth while controlling for downside risk. Working in discrete time, a geometric random walk model for asset prices was developed. The model has two important features. The increments in the random walk have a Bayes framework, so that the asset prices depend on hyper parameters. In addition the corre- lation in the asset price distributions was related to a structural model and thereby the hyper parameters were identified from data. A variety of risk measures were defined and corresponding capital growth with security problems were presented. The emphasis was on the Value at Risk, but control of period-by-period drawdown was also considered in the application. An algorithm for the computation of growth with security strategies was presented for the problems using discrete approximations to the distributions on asset returns. The computational procedure is general and applies to any price distributions, although it was presented in the context of the geometric random walk and log nonnal asset prices. The methods were applied to an example where investment capital is allocated to stocks, bonds and cash over time. At low levels of risk control the capital growth or Kelly strategy is optimal. As the risk control requirements are tightened the strategy becomes more conservative, particularly close to the planning horizon. The solved problems are discrete in time and state, and in contrast to the continuous time lognonnal model (MacLean et aI., 2000) a fractional Kelly strategy is generally not optimal. Acknowledgements This research was partially supported by The Natural Sciences and Engineering Re- search Council of Canada. It was presented at the 16th International Symposium of --- ## Page 400 Capital Growth with Security 371 L. C. MacLean el al. / JOllrnal of Economic Dynamics & Conlrol 28 (2004) 937- 954 953 Mathematical Programming, Lausanne, Switzerland, August 1997, the 21st Meeting of the EURO Working Group on Financial Modelling, Venice, Italy, October 1997, the 9th International Conference on Stochastic Programming, Berlin, August 2001 and the Bachelier Finance Society, Crete, June 2002. References Algoet, P., Cover, T., 1988. Asymptotic optimality and asymptotic equipartition properties of log-optimum investments. Annals of Probability 16, 876-898. Artzner, P., Delbaen, F, Eber, J., Heath, D., 1999. Coherent measures of risk. Mathematical Finance 9, 203-228. Aucamp, D., 1993. On the extensive number of plays to achieve superior performance with the geometric mean strategy. Management Science 39, 1163-1172. Basak, S., Shaprio, A., 2001. Value-at-risk based risk management: optimal policies and asset prices. Review of Financial Studies 14, 371-405. Breiman, L., 1961. 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On dynamic investment strategies. Journal of Economic Dynamics and Control 24, 1859-1880. Culioli, J.c., 1996. Problemes de minimax partial. Rapport Interne CAS, Ecole des Mines de Paris. Ethier, S.N., Tavare, 1983. The proportional bettor's return on investment. Journal of Applied Probability 20, 563-573. Finkelstein, M., Whitley, R., 1981. Optimal strategies for repeated games. Advanced Applied Probability 13, 415- 428. Grauer, R.R., Hakansson, N.H., 1997. On naive approaches to timing the market: the empirical probability assessment approach with an inflation adapter. In: Ziemba, W.T., Mulvey, J.M. (Eds.), Worldwide Asset and Liability Modelling. Cambridge University Press, Cambridge, pp. 149-181. Grossman, S.J., Zhou, Z., 1993. Optimal investment strategies for controlling drawdowns. Mathematical Finance 3, 241 - 276. Hakansson, N.H., 1972. On optimal myopic portfolio policies with and without serial correlation. Journal of Business 44, 324-334. Hakansson, N.H., Ziemba, W.T., 1995. Capital growth theory. In: Jarrow, R.A., Maksimovic, V., Ziemba, W.T. (Eds.), Finance Handbook. North-Holland, Amsterdam, pp. 123-144. Helmbold, D., Schapire, R., Singer, Y., Warmuth, M., 1996. On-line portfolio selection using multiplicative updates. In: Machine Learning. Proceedings of the 13th International Conference, pp. 243- 251. Jorion, P., 1997. Value-at-Risk: The New Benchmark for Controlling Market Risk. Irwin, Chicago. Keim, D., Ziemba, W.T. (Eds.), 2000. Security Market Imperfections in Worldwide Equity Markets. Cambridge University Press, Cambridge. Kelly, J., 1956. A new interpretation of information rate. Bell System Technology Journal 35, 917-926. MacLean, L., Ziemba, W.T., 1991. Growth-security profiles in capital accumulation under risk. Annal of Operations Research 31 , 50 I-51 o. --- ## Page 401 372 L. C. MacLean, R. Sanegre, Y Zhao and W T. Ziemba 954 L.C MacLI'Ili1 el al.ljournal of Econumic Dyna/l1ics & COlllrol28 (2004) 937-954 MacLean, L., Ziemba, W.T., 1999. Growth versus security tradeoffs in dynamic investment analysis. Annals of Operations Research 85, 193-227. MacLean, L., Ziemba, W.T., Blazenko, G., 1992. Growth versus security in dynamic investment analysis. Management Science 38, 1562-1585. MacLean, L., Ziemba, W.T., Li, Y., 2000. Time to wealth goals in capital accumulation and the optimal trade-off of growth versus security. Working Paper, School of Business Administration, Dalhousie University. Markowitz, H., 1959. Portfolio Selection. Yale University Press, New Haven, CT. Markowitz, H., 1976. Investment tor the long run: New evidence tor an old rule. Journal of Finance 31, 1273-1286. Merton, R.C., 1969. Lifetime porttolio selection under uncertainty: the continuous time case. Review of Economics and Statistics 51, 247-259. Merton, R.C., 1992. Continuous-Time Finance, 2nd Edition. Blackwell Publishers, Cambridge, MA. Pflug, G., 1999. Stochastic programs and statistical data. Annals of Operations Research 85, 59-79. Rockafeller, T., Wets, R.1.B., 1978. The optimal recourse problem in discrete time: L' -multipliers for inequality constraints. SIAM Journal of Control and Optimization 16, 16-36. Rotando, L.M., Thorp, E.O., 1992. The Kelly criterion and the stock market. The American Mathematical Monthly December, 992-\ 032. Rubinstein, M., 1977. The strong case for log as the premier model for financial modeling. In: Levy, H., Samet, M. (Eds.), Financial Decisions Under Uncertainty. Academic Press, New York. Rubinstein, M., 1991. Continuously rebalanced investment strategies. Journal of Portfolio Management 17, 78-81. Thorp, E.O., 1998. The Kelly criterion in blackjack, sports betting, and the stock market. Mimeo. Ziemba, W.T., Hausch, D., 1986. Betting at the Racetrack. Dr. Z Investments Inc., Los Angeles, CA. --- ## Page 402 Management Science,46(9), 1188- 1199 (2000) 373 26 Risk-Constrained Dynamic Active Portfolio Management Sid Browne Goldman, Sachs and Company, Firmwide Risk Management, 10 Hanover Square, New York, New York 10005, and Graduate School of Business, Columbia University, New York, New York 10027 sid.browne@gs.com • sb30@columbia.edu A ctive portfolio management is concerned with objectives related to the outperformance of the return of a target benchmark portfolio. In this paper, we consider a dynamic ac- tive portfolio management problem where the objective is related to the tradeoff between the achievement of performance goals and the risk of a shortfall. Specifically, we consider an ob- jective that relates the probability of achieving a given performance objective to the time it takes to achieve the objective. This allows a new direct quantitative analysis of the risk/return tradeoff, with risk defined directly in terms of probability of shortfall relative to the bench- mark, and return defined in terms of the expected time to reach investment goals relative to the benchmark. The resulting optimal policy is a state-dependent policy that provides new insights. As a special case, our analysis includes the case where the investor wants to mini- mize the expected time until a given performance goal is reached subject to a constraint on the shortfall probability. (Portfolio Theory; Benchmarking; Active Portfolio Management; Stochastic Control) 1. Introduction In this paper we analyze an optimal dynamic port- folio and asset allocation policy for an investor who is concerned about the performance of a portfolio relative to the performance of a given benchmark. We take as our setting the standard continuous-time framework pioneered by Merton (1971) and others. Portfolio problems where the objective is to exceed the performance of a selected target benchmark is sometimes referred to as active portfolio management, whereas passive portfolio management just tries to track a benchmark, see, for example, Sharpe et al. (1995). Many professional and institutional investors in fact follow this benchmarking procedure: For example, many mutual funds take the Standard and Poors (S&P) 500 Index as a benchmark; commodity funds seek to beat the Goldman Sachs Commodity Index; bond funds try to beat the Lehman Brothers Bond Index, etc. Moreover, benchmarking is not specific to profes- sional investors, as many ordinary investors implicitly MANAGEMENT ScIENCE ill 2000 INFORMS Vol. 46, No.9, September 2000 pp. 1188-1199 follow a benchmarking procedure, for example, by trying to beat inflation, exchange rates, or other in- dices. In other applications, such as pension funds, the benchmark might be a liability. See Litterman and Winkelman (1996) for more detail on these and other benchmarks. For a treatment of active portfolio man- agement in a static setting, see Grinold and Kahn (1995). This paper extends the earlier analysis in Browne (1999a) of active portfolio management problems with objectives related to the achievement of relative performance goals and shortfalls (see Browne 2000 for related stochastic differential games). Specifically, Browne (1999a) considered a general problem in an incomplete market where the benchmark was only partially correlated with the active investor's invest- ment opportunities and with investment objectives related to the achievement of investment goals and shortfalls relative to the benchmark. The specific ob- jectives that were explicitly solved for there include: 0025-1909/00/4609/1188$05.00 1526-5501 electronic ISSN --- ## Page 403 374 S. Browne BROWNE Risk-Constrained Dynamic Active Portfolio Management maximizing the probability that the investor's wealth achieves a certain performance goal relative to the benchmark before falling below it to a predetermined shortfall; minimizing the expected time to reach the performance goal; and maximizing the expected re- ward obtained upon reaching the goal, as well as minimizing the expected penalty paid upon falling to the shortfall level. The corresponding optimal policies obtained there are all constant proportion, or constant mix, portfolio allocation strategies, whereby the port- folio is continuously rebalanced so as to always keep a constant proportion of wealth in the various asset classes, regardless of the level of wealth. (Observe that if the proportion associated with an asset class is positive, then this rebalancing requires selling an asset when its price rises relative to the other prices, and conversely, buying the asset when its price drops relative to the others.) It is well-known that such policies have a variety of optimality properties asso- ciated with them for the ordinary portfolio problem (see, e.g., Merton 1990 or Browne 1998 for surveys) and are widely used in asset allocation practice (see Perold and Sharpe 1988 and Black and Perold 1992). Nevertheless, some investors object to using constant proportion strategies in that they dictate the same strategy for every wealth level, while their individual intuition would suggest otherwise. In this paper we address some of these issues for the complete market case where the investor is allowed to invest in all the individual components of the benchmark. From an an- alytic point of view, the problem addressed here is solvable only in the complete market case, which is a somewhat more restrictive setting than that of Browne (1999a). However, it is in fact the complete market case that is of most interest to active portfolio practition- ers. Our analysis allows us to extend the domain of goal/shortfall-related objectives with known explicit solutions to a case that allows for a very interest- ing and intuitive state-dependent optimal policy. (A continuous-time active portfolio management prob- lem with a finite-horizon probability-maximizing ob- jective in a complete market setting was studied in Browne (1999b, 1999c). The optimal portfolio policy in that case turns out to be intimately related to hedging strategies for certain options, and as such is both time and state dependent.) MANAGEMENT ScIENCE/Vol. 46, No.9, September 2000 An outline of the remainder of the paper, and a summary of our main results are as follows: In the next section, we provide a description of the model and the problems studied. For the objectives con- sidered here, the relevant state variable is the ratio of the investor's wealth to the benchmark. We then state for reference a general theorem in stochastic control for our model, which contains the specific goal-related objectives considered in the sequel as a special case. The upshot of this theorem is that it shows how the optimal value function and associated optimal control function for a general control problem can be obtained as the solution to a particular nonlin- ear Dirichlet problem. The theorem is a special case of the more general result in Browne (1999a), and so it is stated without proof. Because the specific goal-related problems considered in the sequel are special cases, we need only identify and then solve the appropriate nonlinear Dirichlet problem. In §3 we apply the theorem to show that the ordi- nary optimal-growth portfolio policy for the case of an investor without a benchmark is once again optimal in our extended model, in that regardless of the underly- ing benchmark, the ordinary optimal-growth strategy will minimize the expected time until that benchmark is outperformed by any given percentage. While this result is not new, it is quite important for the sequel. In particular, it highlights the rather disturbing prop- erty of this optimal-growth policy in that it yields no insight for the portfolio manager as to how the bench- mark affects the investment decision, because for this objective the benchmark is in fact irrelevant. Moreover, as we show below, not only is the policy independent of the benchmark, but the probability that the active manager using the optimal-growth policy reaches an investment goal before falling to a shortfall level rela- tive to the benchmark is also independent of the bench- mark as well as any other parameters of the assets. Given these disturbing results, we move on in §4 to consider a fractional objective that relates the time to beat the benchmark to the probability of a shortfall rel- ative to the benchmark. For this objective, the optimal strategy is no longer a constant proportion, but rather a state-dependent amount that modulates the amount invested in the risky assets in inverse proportion to 1189 --- ## Page 404 Risk-Constrained Dynamic Active Portfolio Management 375 BROWNE Risk-Constrained Dynamic Active Port/olio Management wealth. As a special case, our results are then applied to the objective of minimizing the expected time to reach the goal subject to a constraint on the shortfall probability. This objective is motivated by the interest- ing gambling model of Gottlieb (1985). 2. The Model The model under consideration here consists of k + I underlying processes: k (correlated) risky assets or stocks 5(1), ... , S(k ) and a riskless asset B called a bond. The investor may invest in the risky stocks and the bond, whose price processes will be denoted, respec- tively, by {S~i: I ~ 0} 7~ , and {B" I ~ O}. The probabilistic setting is as follows: We are given a filtered probability space (fI,.?, {Si'i }, P), supporting k independent standard Brownian mo- tions, (W(I), ... ,W(k) , where.?, is the P-augmentation of the natural filtration .?,W = cr{W~I),WF), ... ,W~k ); o ::; s ::; I} (see e.g., Duffie 1996 for a brief review of the relevant terminology). It is assumed that these k Brownian motions generate the prices of the k risky stocks. Specifically, following Merton (1971) and many others, we will assume that the risky stock prices are correlated geometric Brown- ian motions, i.e., Siil satisfies the stochastic differential equation dS(i) S(i)d ~ S(i)dW(j ) f' k (1) t = j1i t t+waij tt' Of/ = ], . .. " j= i where {Ili: i = I, .. . ,k} and {cri( i,j = 1, .. . ,k} are con- stants. The price of the risk-free asset is assumed to evolve according to dB, = r B,dt, (2) where r ~ O. We assume that ).li >r for all i = I, .. ,k. An investment policy is a (column) vector control process t = {i, : t ~ O} in Rk with individual compo- nents t:i), i = 1, ... , k, where nil is the fraction (we use I for fraction) or proportion of the investor's wealth invested in the risky stock i at time I, for i = 1, ... ,k, with the remainder invested in the risk-free bond. It is assumed that {t" I ~ O} is a suitable, admissible .?,- adapted control process, i.e., II is a nonanticipative function that satisfies foT NI,dt < 00 a.s. for every T < 00. We place no other restrictions on I, for example, we 1190 allow E;~ d~i) ~ I, whereby the investor is leveraged and has borrowed to purchase the stocks, as well as I~ J) < 0, whereby the investor is selling stock i short. Let X; denote the wealth of the investor at time I under policy I, with Xo = x. Since any amount not invested in the risky stock is held in the bond, this process then evolves as dXI = X/(tt 1dS:il)+ Xl (I _ ttl) dB, I t {= 1 t S ~ ' ) t {= 1 t Bt = X{ (r + f, liil().li - r»)dt + X /~ ~ /(i l crdW (j l I LL t IJ t i=l j= l (3) upon substituting from (1) and (2). This is the wealth equation first studied by Merton (1971). If we introduce now the matrix cr = (cr )ij and the column vectors ).l = ().ll, ... ,).lkl', 1 = (1 , .. ,1)" and W, = (W,(lI, ... ,W,lk)" we can rewrite the wealth pro- cess of (3) as dX{ = X,[(r + I;().l - r1))dt + l;crdW,j. (4) For the sequel, we will also need the matrix E = crcr'. It is assumed for the sequel that the square matrix cr is of full rank, hence cr- I (and E- 1) exists. 2.1. The Benchmark Portfolio As described above, our interest lies in determining investment strategies that are optimal relative to the performance of a benchmark. The benchmark we work with here is the wealth associated with another portfo- lio strategy n = (n( 1), ... , n(k»)' where n(i) denotes the fraction of the benchmark wealth invested in the i-th stock. Accordingly, the benchmark portfolio evolves similarly to (4), as dX~ = X~[(r + n'(Il - r1»dt + n'crdW,). (5) For example, if n(i) = 0 for each i = 1, ... , k, then the benchmark is simply "cash," which is the relevant benchmark in a variety of situations (see Litterman and Winkelman 1996). Alternatively, if n(i) = 1, with n(j) = 0 for all j oF i, then the benchmark is just the i-th stock or asset. We note that while the problems studied in this paper can in fact be treated with more MANAGEMENT ScIENCE/Vol. 46, No. 9, September 2000 --- ## Page 405 376 S. Browne BROWNE Risk-Cor/strained Dyflflmic Active Portfolio Management general benchmarks (and in more general settings), here we only consider the constant coefficients case for analytical and economic simplicity. 2.2. Optimal Growth Let IT' be the constant vector defined by IT' = E- '(Il - rl). (6) The vector IT' plays a fundamental role in the theory of finance (see Merton 1990, Ch. 6) and will also play a fundamental role in the sequel. Following Merton (1990), we refer to the vector IT' as the optimal-growth portfolio strategy. The reason for this is the policy fl = IT', for all t, has many optimality properties asso- ciated with it in an ordinary portfolio setting (where there is no benchmark) that are relevant for growth- related objectives. In particular, for an investor whose wealth evolves according to (4), and who is not con- cerned with performance relative to any benchmark, (i) IT' maximizes the expected logarithm of terminal wealth, for any fixed terminal time T, hence (ii) IT' max- imizes the (actual and expected) rate at which wealth compounds. More interesting, perhaps, and certainly more relevant to our concerns here is the property (iii): IT' minimizes the expected time until any given level of wealth is achieved (needless to say, so long as that level is greater than the initial wealth). Merton (1990, Ch. 6) contains a comprehensive review of these properties (see also Browne 1998 for further optimality proper- ties). Given these results, it is not surprising that the policy IT' has extended optimality properties in our benchmark-based model as well, as we show below. Most relevant to our concerns is the fact that indeed IT' is the policy that minimizes the expected time until the benchmark portfolio strategy is beaten by any given percentage (see Corollary 1 below). The fact that this holds for any benchmark, however, severely limits the applicability of this result in that it does not provide any insight for the active portfolio manager as to the role the benchmark plays in the investment decision, and as such might not be a reasonable objective for an active portfolio manager. We note also that the ratio of the wealth process of any (admissible) portfolio strategy to the wealth pro- cess determined by the optimal growth strategy is a supermartingale. This fact has important consequences MANAGEMENT ScIENCE/Vol. 46, No. 9, September 2000 for pricing contingent claims, as described, for exam- ple, in Merton (1990, Ch. 6). 2.3. Active Portfolio Management There are of course many possible objectives related to outperforming a benchmark. Here, as in Browne (1999a), we consider objectives related solely to the achievement of relative performance goals and short- falls or drawdowns. Specifically, for numbers I, u with Ixg 0, and that per- formance shortfall level 1 occurs if xl = IX~ for some I> O. The active portfolio management problems con- sidered for an incomplete market (i.e., where there are more sources of risk than there are traded securities) in Browne (1999a) are: (i) maximizing the probability that performance goal u is reached before shortfall I occurs; (ii) minimizing the expected time until the performance goal u is reached; (iii) maximizing the expected time until shortfall 1 is reached; (iv) maximiz- ing the expected discounted reward obtained upon achieving goal u; and (v) minimizing the expected discounted penalty paid upon falling to shortfall level l. Among other scenarios, these objectives are relevant to institutional money managers, whose performance is typically judged by the return on their managed portfolio relative to the return of a benchmark. Browne (1999a) showed that the optimal strategy for each of these objectives was in fact a constant proportions asset allocation strategy. While constant proportion strate- gies are optimal in a variety of other settings as well, many professional investors object to them on the grounds that they do not take into account the wealth level of the investor. In this paper we consider an extended fractional ob- jective that relates the probability in Objective (j) to the expected time in Objective (ii). It turns out that for this objective the optimal strategy is no longer constant, but rather state dependent. In particular, the policy is hyperbolic in the state variable, where the relevant state variable is the ratio of the wealth process to the bench- mark. We use standard techniques of stochastic control theory (e.g., Krylov 1980) to establish our results since this ratio is a controlled diffusion. 1191 --- ## Page 406 Risk-Constrained Dynamic Active Portfolio Management 377 BROWNE Risk-Constrained Dynamic Active Portfolio Management In particular, because xl is a controlled geometric Brownian motion, and X,rr is another geometric Brown- ian motion, it follows directly that the ratio process, Z/, where Z{ = X{ IX,", is also a controlled geometric Brownian motion. Specifically, a direct application to Ito's formula gives: PROPOSITION 1. For xl, X,' defined by (4) and (5), let Z{ be defined by Z{ = xl/x,". Then, using the definition of the vector rr' as given in (6), we have dZ{ =Z{U, - rr)' ~ (rr' - rr)dt + Z{U, - rr)' crdW,. (7) Alternatively, in integral form we have Z{ = Zo exp{[ Us - rr)' ~ (rr> - Ws + rr») ds + l(fs -rr)' ~(fs -rr ) dWs }. (8) Next, we provide a general theorem in stochastic optimal control for the process {Z{, t ~ O} of (7) that covers the specific problems treated here as special cases. 2.4. Optimal Control The active portfolio management problems considered in this paper are special cases of optimal control prob- lems of the following (Dirichlet-type) form: For the ratio process {Z{, t ~ O} given by (7), let ,£= inf{t > O:Z{ = x} (9) denote the first hitting time to the point x under a spe- cific policy f = {f" t ~ O}. For given numbers I, u, with l 0 and w" < 0) solution to the nonlinear Dirichlet problem w~(z) -y-- + g(z) = O, for l rr(i), and negative if rr' (i) < rr( i). That is, the active manager will invest more heavily in asset i than the benchmark if the benchmark is underinvested in asset i relative to the vector rr' , and vice versa. The extent to which this occurs depends on the specifics of the value function w(z). R EMARK 3. As noted earlier, Theorem 1 is a special case of a more general result in Browne 1999a, and as such we do not provide a formal proof. However, to provide some insight into the result, observe that the Hamilton-Jacobi-Bellman (HJB) optimality equation of dynamic programming for maximizing vf(z) of (10) M ANAGEMENT ScIENCE/Vo!. 46, No.9, September 2000 over control policies f, to be solved for an optimal value function V, is sup{(f- rr)' E(rr' -rr) zVz+(f- rr)' E(f - rr) z2 vzz+g} =0, f (17) subject to the Dirichlet boundary conditions 1'(1) = h(l) and v(u) = h(u) (see, e.g., Krylov 1980, Theorem 1.4.5, or Fleming and Soner 1993, §IV.5). Assuming now that (17) admits a classical solution with v, > ° and v" < 0, we may then use standard cal- culus to optimize with respect to f in (17) to obtain the optimal control function f:(x) of (16), with v = w. When (16) is then substituted back into (17) and simplified, we obtain the nonlinear Dirichlet problem of (13) (with 1' = 10). To complete the proof now, one only needs to verify that the solution to the HJB equation is indeed the optimal value function, and hence that the policy f: is indeed optimal. A verification argument based on the martingale optimality principle given in Browne (1999a) covers the case at hand, provided that Con- ditions (i), (ii), and (iii) hold. A stochastic differential game-theoretic version of the verification argument and martingale principle appears in Browne (2000). 3. Minimizing the Expected Time to Beat the Benchmark 3.1. Optimality of rr' We can use Theorem 1 to show that, as claimed ear- lier, the ordinary optimal growth portfolio policy, rr' of (6), is indeed also optimal for minimizing the ex- pected time to beat the benchmark by any predeter- mined amount, regardless of the underlying benchmark strategy rr. Indeed, we state this formally in the follow- ing corollary to Theorem 1. COROLLARY I. Let C'(z)= infrE,(T[ ) with optimizer /'(z)= arg inffE, (T[)., Then for 'any rr # rr', and y as defined in (12), we have C'(Z)=~ln(~), withf*(z) = rr', forallz <::, u. (18) PROOF. Observe first that while Theorem 1 is stated in terms of a maximization problem, it obviously contains the minimization case, as we can apply 1193 --- ## Page 408 Risk-Constrained Dynamic Active Portfolio Management 379 BROWNE Risk-Constrained Dynamic Active Portfolio Management Theorem 1 to G(z) = sUPf{ -E,(Tt )}, and then recog- nize that G' = - G. As such, Theorem 1 applied with g(z) = I and h(u) = O shows that G' must solve the ordinary differential equation G~(z) -y Gzz(z) + 1= 0, (19) together with the boundary condition G' (u) = 0. Moreover, G' must be convex decreasing (since it is the solution to a minimization problem). It is easy to sub- stitute the claimed values from (18) into (19) to verify that in fact that is the case. Furthermore, we have G; /zG; = - I, and as such (16) of Theorem 1 shows that the optimal control for this case reduces to rr' . It remains to verify whether the Conditions (i) (ii) and (iii) of Theorem 1 hold: It is clear that (i) and (iii) hold. Condition (ii) is seen to hold for this case since we have dG'(z )/dz = - I/ (zy), and as such Re- quirement (15) reduces here to J; y-'ds < 00, which holds trivially. 0 REMARK 4. We have just shown that the ordinary optimal growth policy, rr', minimizes the expected time to a goal in the presence of a benchmark. To see that this same policy maximizes logarithmic util- ity of the ratio for the investor, simply observe that sUPf{E[ln(Zf )]} = sUPf{E[ln(Xf)]} - E[ln(XTJJ. 3.2. Properties of the Growth-Optimal Ratio When rr' is substituted back into (7), we obtain the following stochastic differential equation for the ratio of the growth-optimal wealth to the benchmark dZ,(rr',rr) = Z,(rr', rr)[ydt + (rr' - rr)'crdWtJ, (20) which implies that under rr' , this ratio process is the geometric Brownian motion given by Z,(rr', rr)=Zoexp{yt + (rr' - rr)'crW,}. (21) While we did not consider a lower shortfall barrier, I, in the development above proving optimality of the standard growth-optimal portfolio policy, it is of im- portance in many applications to consider one, since many investors indeed are interested in avoiding sub- stantial shortfalls. In the following proposition we give two fundamental results for the wealth process for an investor following the optimal-growth strategy rr': (i) 1194 the probability that the investment goal u is reached before a shortfall of size I occurs; and (ii) the expected time to escape the interval (/, u) (which is not the same as the expected time to the goal). PROPOSITION 2. For the process Z,(rr' , rr ) of (21), let T denote the first escape time from the interval (/, u), and lei O(z: I, u) denote the probability of "suc- cessful"' escape, i.e., T= inf{I: Z,(rr',rr) \t (/,u)}, and 8(z : I, u) = P,(Zt( rr', rr) = u). Then 8(z) is given by u (z -I) 8(z :l,u)=:z ~ . (22) Also, Ihe expected time of first escape from the interval is given by Ez(T(rr' , rr» = y-I [8(z:l,u)ln('T) - In(Y) ] . (23) Thus, we see that while rr' is the policy under which any given investment goal will be reached in mini- mal expected time, and hence in a sense maximizes expected return, it does so with a risk of a shortfall of size I occurring with probability I - 8(z: I, u), where 8(z : I, u ) is the probability given in (22). REMARK 5. It is important to note that the probability 8(z: I, u) is independent of any of the underlying parameters associated with the underlying model. In particular, this probability is independent of the benchmark policy rr. (For example, the probability of the ratio doubling (u = 2z) before being halved (/ = 0.5z) is always p This limits the usefulness of the optimal-growth pol- icy in the active portfolio management setting, since it provides no guide to the manager in how to choose a benchmark. Of course, the expected time to escape the interval (/, ll), given by (23), does depend on the underlying parameters, but only through the para- meter y. REMARK 6. Observe that for I = 0, we obtain 8(z : 0, u) = 1, and the expected escape time from the inter- val Ez( T( rr', rr» reduces, as it should, to the opti- mal expected first passage time to the upper barrier E,(Tu(rr',rr)). Of course, for 1> 0, the expected hitting time of the optimal ratio process to the lower shortfall level, Ez(T/(rr',rr», is infinite due to the fact that the drift in (20) is positive. MANAGEMENT ScIENCE/ Va!. 46, No.9, September 2000 --- ## Page 409 380 S. Browne BROWNE Risk-Constrained Dynamic Active Portfolio Management REMARK 7. The probability in (22) and the expected hitting time in (23) can be established directly via a variety of different ways. Most directly we have the following lemma for geometric Brownian motion: LEMMA I. Let X, denote a geometric Brownian motion that satisfies dX, = mX,dt + v'2s X,dW" with Xo = x (24) and let T,= inf{t > O:X,=z}. For OS::a S:: x S:: b, define now 1 = min{ 1" 1b}, and H(x) = Ex(1). Then for m i s we have X V - aV m K(x)= bv _ aV' where v = 1 - 5' while for m = s we have K(x) = In(x/a) , In(b/a) H(x) = 2~ ([(lnb)2 - (lna)21~:i~~:; (25) (26) (27) (28) (29) - [(lnx)2 - (lna)2]) . (30) PROOF. Recognize first that {X,} is a diffusion pro- cess with drift function ~(x) = mx and diffusion func- tion a2(x) = 2sx. Therefore, if follows from elementary results about one-dimensional diffusions that K and H are the unique solutions to the respective (Dirichlet) problems: mxKx +sx2Kxx=0: K(a)=O, K(b) = l (31) mxHx + sx2Hxx + 1 = 0: H(a) = 0, H(b) = ° (32) (see e.g., Karlin and Taylor 1981, pp. 192-193). The gen- eral solution to the second order differential equation in the left-hand side of (31), for mis, is K(x) = C,x" + C2, and the boundary conditions determine the MANAGEMENT Sc'ENCE/Vol. 46, No.9, September 2000 constants C" C2, as in (27). For m = s, the general solu- tion is C, In x + C2, and the boundary conditions give (29). The general solution to the left-hand side of (32) is C, + C2xv - (m - s)- 'Inx for m i s, and from the boundary conditions we determine C, and C" giving 1(1) bVV -- --- [-( -a )Inx m - s bv - aV + (XV _ aV) Inb + (bV - xV) lna], which is equivalent to (28) above when we simplify using the definition of K(x). For m = s, the general so- lution of (32) is c, - (im)(lnxf + C2 lnx, which gives (30) after applying the boundary conditions. 0 Using this lemma for our purposes, we first note that (20) is distributionally equivalent to a diffusion that evolves according to the stochastic differential equa- tion dX, = X, (2ydt + /2Y dW,), where W is an independent one-dimensional Brown- ian motion. As such, identify m = 2y, s = y, and hence v = - I, and then using b = u and a = I, (22) and (23) follow upon substitution. 0 4. Risk-Related Objectives 4.1. Objective Function and Optimal Policy The results of the previous section indicate that the optimal growth strategy, 1t*, regardless of its many optimality properties in the ordinary portfolio setting, may not be appropriate for an active portfolio man- ager who cares about downside risk as well as upside growth relative to a benchmark. As such, in this section we treat an objective that is perhaps more appropriate in that it allows the active portfolio manager to incorporate the shortfall prob- ability directly in the risk/return tradeoff relative to expected growth. In particular, we consider a linear tradeoff between the shortfall probability and the ex- pected time to get to the surplus level. More specifi- cally, we now treat the following objective: For given 1195 --- ## Page 410 Risk-Constrained Dynamic Active Portfolio Management 381 BROWNE Risk-Constrained Dynamic Active Portfolio Management nonnegative constants a and p, let V'(z)= sup{ap,(Z~f = u) - PEz(,iJ}. (33) f As we will show, the optimal dynamic portfolio strat- egy for this objective is no longer a constant asset al- location strategy. Rather, as we will establish below, the optimal strategy hyperbolically modulates the frac- tions invested in the risky assets by the level of the ratio process. THEOREM 2. Let V'(z) denote the optimal value function in (33), and let fit(z) denote the associated optimal control function. Then, V' (z ) is given by , ( z + b)/ (u + b) V (z)=a In T+b In I+ b ' forl<:z<:u, where the scalar b = b( a, p: u, I, y) is given by b ue-Y. /P - I I - e- Y'/P . (34) (35) The optimal portfolio policy, fv(z), is given by fv(Z)=1t-(1t' - 1t)(I+n = 1t' + (1t' - 1t)~, (36) where b is given by (35), It' is the ordinary optimal growth vector given earlier in (6), It' = ~ - l(l' - rl), and 1t is the benchmark strategy. REMARK 8. Observe that the portfolio strategy fit(z) is a state-dependent policy that is inversely modulated by the level of the ratio process Z. The final repre- sentation in (36) shows that the policy is composed of two parts: First it just uses the optimal-growth pol- icy 1t', and then multiplies the difference between the optimal-growth policy 1t' and the tracking portfolio 1t by the correction term biz. The sign of the correc- tion factor is determined by the sign of b. Some direct manipulations on (35) shows that the sign of b is the sign of ~ln(T) - ~. Comparison now with (18) reveals that we can write this quantity as C'U) - a!p, where C'(l) is the minimal 1196 possible expected time to get from the shortfall level 1 to the surplus goal u. Thus, b is positive (negative) if the ratio alP is less (more) than this minimal expected time. Observe further that if b > 0, then the active manager invests more heavily in the i-th stock than does It' so long as the benchmark is underinvested in that stock relative to the optimal-growth policy, i.e., so long as It'(i)>1t(i). Finally, note that (35) shows that we must always have b~ -I. PROOF. Theorem 1 applies directly to this case with g(z)= -p, with h(u)=a and hU) = O. As such, we re- quire that V' be the concave increasing solution to the nonlinear Dirichlet problem: v2 -y-.£ _ p = O for l(2: b, I, u ) denote the probability of successful escape MANAGEMENT ScIENCE !Vol. 46, No.9, September 2000 from the interval, i.e., (z : b, I, u) = P,(Z;. = u). Then (z:b, I, u)= (z - I)(u + b) (z + b)(u -I) The expected time of the first escape is given by (42) Ez(t') = ~ [(Z: b, I, U)ln(~::) -InC :n]. (43) REMARK 9. Observe that the probability in (42) is now dependent on the benchmark policy It through the parameter b. Note further that as intuition would suggest, the probability in (42) is larger than the as- sociated probability for the optimal-growth strategy, e obtained earlier in (22), if b < 0, and smaller if b > O. PROOF. The results above can be established di- rectly from the earlier Lemma 1 applied to the process Z; + b, because as (41) exhibits, Z; + b is a geometric Brownian motion, with initial state Zo + b. In fact, (41) implies that Z; + b is distributionaUy equivalent to a multiple of the optimal-growth ratio described in the last section, i.e., Z; + b,1" (I + bIZo)Z,(rr', It). As such, we have (z:b, I, u) '" P,(Z;. = u) = P,(Z;. +b= u + b) = Pz( (I + DZr(It" rr) = u+ b) (44) where in the latter t denotes the first escape time of the process (I +blz)Z,(rr', rr) from the interval (I + b, u +b). As such, the results of the previous section allows us to evaluate this latter probability in terms of the function Se) defined earlier in (22). Specifically, the argument in (44) shows that ( I+ b U+ b) <1>(2: b, I, u) = B z: 1 + biz' 1 + biz ", B(z + b:l+b,u + b), (45) and indeed (42) is obtained when we substitute appro- priately into (22). The optimal expected hitting time, Ez( t ') of (43) is derived directly from the fact that under the optimal policy fv, the value function is V' (z) = ClP,(Z;. = u) - I3Ez(t'), 1197 --- ## Page 412 Risk-Constrained Dynamic Active Portfolio Management 383 BROWNE Risk-Constrained Dynamic Active Portfolio Management and therefore E, (t' )= i[CtPz(Z;. = u) - V'(z)]. (46) Substituting now for P,(Z;. = u ) from (42) and for V' (z) from (34), and using I / ~ = In[(u + bl/(l + b)]/(CtY) gives (43). 0 4.3. Risk-Constrained Minimal Time The results of the previous section can now be applied directly to the active portfolio management case where the shortfall probability is prespecified. Specifically, suppose that the shortfall probability is prespecified to the active manager to be no more than 1 - p, where p is a given number between 0 and 1, i.e., the active manager is told that he must have PZ(Z~ f = /) ::; 1 - p, or equivalently, that he must have P, (Z~, = u) ;::: p. The risk-constrained active portfolio management problem is now to minimize the expected time to beat the bench- mark subject to a constraint on the shortfall proba- bility, specifically, to find the strategy {fr, t ;::: O} that minimizes E(tf ) subject to P(Z~, = u) ;::: p, where p is a given number in (0, 1 ). This is now related to the gam- bling problem first solved in Gottlieb (1985). Follow- ing Gottlieb, we observe first that the dual of the risk- constrained active portfolio management problem is to maximize the probability that P(Z~, = u) subject to a constraint on E(tf ). Moreover, observe that should a solution exist, then the constraint will be met at equal- ity, and so we would have P(Z~f = u) = p. Let us write the solution to the dual problem, should it exist, as W(z ) = sup[Pz(Zf, = u) - ~Ezt f ], f T (47) where ~ is now the value of a Lagrangian multiplier. The control problem in (47) is a special case of the problem treated above with ~ = I, and as such, from (34) we know that the solution is given by ( z + b)0 (U + b) W(z) = w(z) = ln --_ In --_ , 1+ b I+ b (48) where b is the value of b in (35) evaluated at C( = I, i.e., b= b(l, ~). The value of ~, the Lagrangian mul- tiplier, will be determined from the risk constraint P(Z~f= U) =P' 1198 We also know from (36) that the associated risk- constrained optimal portfolio strategy is given by f' , • b (z) = lt +(It - It)-. z (49) Observe now that we may invert the identity for b in (35) in this case to write the unknown ~ in terms of the unknown ii as (50) Because we require P > 0, this implies that we require b;::: -I (we always need b;::: - l, to ensure that the prob- ability (z: .,.) does not exceed 1, i.e., to keep ::; I). To determine the value of ii, we can use the risk constraint evaluated at the initial time 0, i.e., set (Zo;b) = p. Using (42) with b = ii, this gives us (Zo - l)(u + b) (Zo+ ii)(u - l) = p, which in turn now can be solved for b, giving - - pZo(u - l) - u(Zo-l) b= b(p,Zo,u,I) = Z 1 (I) (51) 0 - -PH - REMARK 9. Observe that - Zo- l ~i~~, (Zo: b) =--u=T and as such, the risk-constrained problem is feasible only for an initial probability level p that satisfies Zo - I p> --u=T' Observe too that for p = I, b reduces to b = - I, which makes the lower barrier 1 unattainable as in many "portfolio insurance" models (see Browne 1997). The insurance level ii in (51) is positive for values of p sat- isfying ZO-l ~ (Zo -I) == 9(Zo), Zo u - I MANAGEMENT ScIENCE/Va!. 46, No.9, September 2000 --- ## Page 413 384 S. Browne BROWNE Risk-Constrained Dynamic Active Portfolio Management where 8(Zo) is the initial probability that the opti- mal growth wealth/ratio hits u before I, as given in (22). Thus, as intuition suggests, to have a higher "success" probability than the optimal-growth strat- egy, the active portfolio manager must take less risk and invest less (since b < 0) than the ordinary optimal- growth investor. The optimal expected hitting time, E,( 1*) can be obtained directly as E,(t*)= ! [z - ~ (u + b) In(U + ~) _In(Z + ~)] . Y z+b u-/ I+b I+b (52) 5. Conclusions We have studied a goal-related objective for the prob- lem of outperforming a benchmark. The objective relates the time to outperform the benchmark to the shortfall probability. The resulting optimal policy is state-dependent in an intuitive way not captured by previous studies where the optimal policy was of a constant proportions type. Moreover, this opti- mal policy directly addresses and alleviates one of the undesirable features, in the benchmark oUtper- formance problem, of the ordinary optimal-growth policy, whose shortfall probability was shown to be independent of the benchmark and any other model- specific parameter. As a special case of this objective, we studied the problem of minimizing the expected first passage time subject to a constrained probability of successful escape. References Black, F., A. F. Perold. 1992. Theory of constant proportion portfolio insurance. J. Econom. Dynam. and Control 16 403- 426. Browne, S. 1995. Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin. Moth. Oper. Res. 20 937-958. --.1997. Survival and growth with a fixed liability: Optimal portfolios in continuous time. Math. Oper. Res. 22468-493. --. 1998. The return on investment from proportional portfolio strategies. Adv. App/. Probab. 30(1) 216-238. --. 1999a. Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark. Finance and Stochastics 3 275-294. --. 1999b. Reaching goals by a deadline: Digital options and continuous-time active portfolio management. Adv. Appl. Probab. 31551-577. --. 1999c. The risk and reward of minimizing shortfall probability. J. Portfolio Management 25(4) 76-85. --.2000 Stochastic differential portfolio games. J. Appl. Probab. In press, 37(1). Duffie, D. 1996. Dynamic Asset Pricing Theory, 2nd ed. Princeton University Press, Princeton, NJ. Fleming, W. H., H. M. Soner. 1993. Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, New York. Gottlieb, G. 1985. An optimal betting strategy for repeated games. J. App/. Probab. 22 787-795. Grinold, R c., R. N. Kahn. 1995. Active Portfolio Management. Irwin, Probus, IL. Heath, D., S. Orey, V. Pestien, W. Sudderth. 1987. Minimizing or maximizing the expected time to reach zero. SIAM J. Control and Optim. 25(1) 195-205. Karlin, S., H. M. Taylor. 1981. A Second Course i" Stochastic Processes. Academic, New York. Krylov, N. V. 1980. Controlled Diffusion Processes. Springer-Verlag, New York. Litterman, R, K. Winkelmann. 1996. Managing market exposure. J. Portfolio Management 22(4) 32-48. Merton, R 1971. Optimum consumption and portfolio rules in a continuous time model. J. Econom. Theory 3373-413. --. 1990. Continuous Time Finance. Blackwell, MA. Perold, A. F., W. F. Sharpe. 1988. Dynamic strategies for asset allocation. Financial Ana/. J. 44(1) 16-27. Pestien, V. c., W. D. Sudderth. 1985. Continuous-time red and black: How to control a diffusion to a goal. Math. Oper. Res. 10(4) 599-611. Sharpe, W. F., G. F. Alexander, J. V. Bailey. 1995. Investments, 5th ed. Prentice Hall, NJ. Accepted by Paul Glasserman; received April 1, 1999. This paper was with the authors 5 months for 1 revision. MANAGEMENT ScIENCE/Va!. 46, No.9, September 2000 1199 --- ## Page 414 27 Fractional Kelly Strategies for Benchmarked Asset Management Mark Davis Department of Mathematics, Imperial College London, London SW'l 2AZ, England mark. davis@imperial. ac. uk Sebastien Lleo Department of Mathematics, Imperial College London, London SW'l 2AZ, England sebastien.lleo@imperial.ac.uk Abstract In this paper, we extend the definition of fractional Kelly strategies to the case where the investor's objective is to outperform an investment benchmark. These benchmarked fractional Kelly strategies are efficient portfolios even when asset returns are not lognormally distributed. We deduce the benchmarked fractional Kelly strategies for various types of benchmarks and explore the interconnection between an investor's risk-aversion and the appropriateness of their investment benchmarks. 1 Introduction 385 Classically, a fractional Kelly strategy with fraction f consists in investing a pro- portion f of one's wealth in the Kelly criterion, or log utility, optimal portfolio and a proportion 1 - f in the risk-free asset. Fractional Kelly strategies play an impor- tant role in active investment management in the asset only case, that is when the investor's objective is to maximize the terminal value of his/her wealth, without any benchmark to track or liability to pay (see for example Thorp (2006) and Ziemba (2003) for discussion and additional references). In this chapter, we analyze the role of fractional Kelly strategies in the asset allocation of a benchmarked investor, that is an investor whose objective is to outperform a given investment benchmarks, such as the S&P 500 or the Salomon Smith Barney World Government Bond Index. The argument developed in this chapter builds on and applies the stochastic control-based model proposed by Davis and Lleo (2008a). Their methodology is founded on the theory of risk-sensitive stochastic control, rather than on the classical stochastic control theory-based approach proposed by Merton (e.g., Merton (1969, --- ## Page 415 386 M Davis and S. Lleo 1971, 1992)). In the asset only case, this choice has the benefit of being consistent with both the Merton model and with the mean-variance analysis while allowing for the explicit inclusion of underlying valuation factors and while admitting a simpler analytical solution than the Merton model. An added advantage of this model is the relative ease with which the asset only problem as formalized by Bielecki and Pliska (1999) can be extended to include a benchmark, as in Davis and Lleo (2008a), or a liability, as in Davis and Lleo (2008b). In in these paper, it is seen that fractional Kelly strategies as defined above are not necessarily optimal, with the notable exception of the Merton model where they arise naturally as a consequence of the assumption that asset prices are lognormally distributed. As a result, to be able to interpret the solution to our benchmarked asset allocation problem in terms of fractional Kelly strategies and therefore guarantee their optimality, we will find it necessary to expand slightly their definition. In Section 2, we introduce risk-sensitive asset management from the perspective of an asset only investor. The interpretation of the resulting optimal asset allo- cation formula is then formalized in Section 3 both as a Mutual Fund Theorem and in terms of a redefinition of fractional Kelly strategies. Section 4 presents a comparison of the classical Merton model with the risk-sensitive asset management approach from the perspective of fractional Kelly strategies. Then, in Section 5, we present the analytical solution to the risk-sensitive benchmarked asset allocation problem, before interpreting this solution in a Mutual Fund theorem and in terms of benchmarked fractional Kelly strategies in Section 6. Finally, Section 7 contains a number of case studies showing applications of these ideas to specific types of benchmarks and to Kelly criterion investors. 2 Risk Sensitive Asset Management 2.1 Risk sensitive control Risk-sensitive control is most simply defined as a generalization of classical stochas- tic control in which the degree of risk aversion or risk tolerance of the optimizing agent is explicitly parameterized in the objective criterion and influences directly the outcome of the optimization. Risk sensitive control was introduced by Jacobson (1973) and has been developed by many authors, notably Whittle (1990) and Ben- soussan and van Schuppen (1985) before being applied to finance by Lefebvre and Montulet (1994) and to asset management by Bielecki and Pliska (1999). While in classical stochastic control the objective of the decision maker is to maximize E[F], the expected value of some performance criterion F, in risk-sensitive control the decision maker's objective is to select a control policy h(t) maximizing the criterion J(t x h' e) '= - ~ lnE [e-!lF(t,x ,h)] , " . e (1) --- ## Page 416 Fractional Kelly Strategies for Benchmarked Asset Management 387 where • t and x are the time and the state variable; • F is a reward function; • the risk sensitivity e E (-1,0) U (0,00) represents the decision maker's degree of risk aversion. A formal Taylor expansion of J around e = 0 evidences the vital role played by the risk sensitivity: e J(x, t, h; e) = E [F(t, x, h)] - "2 Var [F(t, x, h)] + O(e2 ) (2) • e ---+ 0 corresponds to the "risk-null" case and to classical stochastic control; • when e < 0, we have the "risk-seeking" case that is a maximization of the expectation of a convex decreasing function of F(t,x, h); • finally, e > 0 is the "risk-averse" case that is a minimization of the expec- tation of a convex increasing function of F(t, x, h). To summarize, risk-sensitive control differs from traditional stochastic control in that it explicitly models the risk-aversion of the decision maker as an integral part of the control framework, rather than importing it in the problem via an externally defined utility function. 2.2 Risk sensitive asset management Bielecki and Pliska (1999) pioneered the application of risk-sensitive control to asset management. They proposed to take the logarithm of the investor's wealth V as the reward function, i.e. F(t, x, h) = In V(t, x, h) (3) The natural interpretation of this choice is that the investor's objective is to max- imize the risk-sensitive (log) return of the his/her portfolio. With this choice of reward function, the control criterion is J(t x h· e) .= - ~ In E [e- 1I 1n V (t ,x,h)] , " . e (4) and interpret the expectation E [e-1I 1nV(t ,x,h)] = E [v(t,x, h)-II] =: UII(vt) (5) as the expected utility of time t wealth under the power utility (HARA) function. The investor's objective is to maximize the utility of terminal wealth. The Taylor expansion becomes e J(t, x; e) = E [In V(t, x, h)] - "2 Var [In V(t, x, h)] + O(e2 ) (6) Ignoring higher order terms, we recover the mean-variance optimization criterion and the log utility or Kelly criterion portfolio in the limit as e ---+ o. --- ## Page 417 388 M Davis and S. Lleo 2.3 The risk sensitive asset management model 2.3.1 Asset and factor dynamics Embedding the investor's risk-sensitivity in the control criterion provides more lee- way in the specification of the asset market than would be obtained in the classical stochastic control of the Merton approach. Bielecki and Pliska (1999), in particular, propose a factor model in which the prices of the m risky assets follow a SDE of the form dS(t) n+m Si'(t) = (a + AX(t))idt + {; (JikdWk(t) Si(O) =Si, i = l, ... ,m (7) where W(t) is a N := n + m-dimensional Brownian motion and the market pa- rameters a, A, I; := [(Jij] , i = 1, ... , m , j = 1, ... , N are matrices of appropriate dimensions. To avoid redundancy, we assume I;I;' > O. To these m risky securities, we add a money market asset with dynamics dSo(t) , So(t) = (ao + AoX(t)) dt, So(O) = So (8) Finally, the asset prices drift depends on n valuation factors modelled as affine stochastic processes with constant diffusion dX(t) = (b + BX(t))dt + AdW(t), X(O) = x (9) where X(t) is the JRn-valued factor process with components Xj(t) and the pa- rameters b, B, A := [Aij] , i = 1, ... , n, j = 1, ... , N are matrices of appropri- ate dimensions. These valuation factors must be specified, but they could include macroeconomic, micro economic or abstract statistical variables. Under these conditions, the logarithm of the investor's wealth is given by the SDE In V(t) = In v + lot (ao + A~X(s)) + h(s)' (a + AX(s)) ds lit it - - h(s)'I;I;'h(s)ds + h(s)'I;dW(s) 2 0 0 (10) where V(O) = v, h is the m-dimensional vector of portfolio weights and we used the notation a := a - aol, A := A - lA~ and 1 is a n-element column vector with all entries set to 1. The equation for V solely depends on the valuation factors (the state process) and is independent of the asset prices. This implies that the effective dimension of the risk-sensitive control problem will be n , the number of factors, rather than m, the number of risky assets. The limited impact of the number of assets is significant since for practical applications we would typically use only a few factors (possibly 3 to 5) to parametrize a large cohort of assets and asset classes (possibly several dozens). The risk-sensitive asset management model is therefore particularly efficient from a computational perspective. --- ## Page 418 Fractional Kelly Strategies for Benchmarked Asset Management 389 2.3.2 The associated linear exponential-oj-quadratic Gaussian control problem The next step in the analysis is due to Kuroda and Nagai (2002) who ingeniously observed that under an appropriately chosen change of probability measure (via the Girsanov theorem) , the risk-sensitive criterion can be expressed as I(v,X; h ; t , T) = ln V- ~lnE~ [exp{e l Tg(Xs, h(s);e)ds}] (11) where the expectation E ~ [.J is taken with respect to a newly-defined measure lP'~ depending on the investment strategy h, the functional g is 1 ' g(x, h; e) = "2 (e + 1) h'2:,2:,'h - ao - A~ x - h'(a + Ax ) (12) and the factor dynamics under the new measure lP'~ is dXs = (b + BXs - eA2:,'h(s)) ds + AdW! (13) (see Davis and Lleo (2008a) for details). In this formulation, the problem is a standard Linear Exponential-of-Quadratic Gaussian (LEQG) control problem which can be solved exactly, up to the resolution of a system of Riccati equations. But before solving this control problem and deriv- ing the optimal asset allocation, we will first develop some intuition by considering the simple case in which the security and factor risk are uncorrelated, i.e., when A2:,' = O. 2.3.3 Special case: Uncorrelated assets and Jactors When A2:,' = 0, security risk and factor risk are uncorrelated and the evolution of X t under the measure lP'~ given in equation (13) simplifies to dXs = (b + BXs) ds + AdW! (14) The evolution of the state is therefore independent of the control variable h and, as a result, the control problem can be solved through a pointwise maximisation of the auxiliary criterion function I ( v, x; h; t, T ). In this case, the optimal control h * is simply the maximizer of the function g(x; h; t, T) given by h* = _1_(2:,2:,')-1 (a + AX) e+l which represents a position of 1I!1 in the Kelly criterion portfolio. (15) Let (t, x) be the value function corresponding to the auxiliary criterion function I(v, x; h; t, T) . Substituting the value of h* in the equation for g yields (t, x) = sup I(v,x; h;t,T) hEA(T) = -~ InE~ [exp {e IT- t g(x, h*(s); t, T ; e)ds } v-II] (16) The PDE for can now be obtained directly via an exponential transformation and an application of Feynman-Kac. --- ## Page 419 390 M Davis and S. Lleo 2.3.4 The general case In the general case, the value function iP for the auxiliary criterion function I(v,x;h;t,T), defined as iP(t, x) = sup I(v,x;h;t,T) ACT) (17) satisfies the Hamilton-Jacobi-Bellman Partial Differential Equation (HJB PDE) oiP ~ + sup L~iP(X(t)) = 0 (18) vt h EiR'" where L~iP( t, x) = (b + Ex - BA'L,' h( s))' DiP + ~tr (AA' D2iP) - ~(DiP)' AA' DiP - g(x, h; B) (19) where DiP is the gradient vector defined as DiP := ( 881), ... , 881>., ... , 881»' and Xl X t Xn D2iP is the Hessian matrix defined as D2iP := [ 8~:!j] , i, j = 1, ... ,n. The value function iP satisfies the terminal condition iP(T, x) = ln v. Solving the optimization problem gives the optimal investment policy h * (t) h*(t) = B ~ 1 ('L,'L,,)-l [a + AX(t) - B'L,A' DiP(t, X(t))] (20) Moreover, the solution of the PDE is of the form iP(t, x) = x'Q(t)x + x'q(t) + k(t) (21) where Q(t) solves a n-dimensional matrix Riccati equation and q(t) solves a n- dimensional linear ordinary differential equation depending on Q (see Kuroda and Nagai (2002) for details). 3 Fractional Kelly Strategies in the Risk-Sensitive Asset Management Model 3.1 A mutual fund theorem Theorem 1 (Mutual Fund Theorem (Davis and Lleo, 2008a)). Any port- folio can be expressed as a linear combination of investments into two "mutual funds" with respective risky asset allocations: hK(t) = ('L,'L,,)-l (a + AX(t)) hC(t) = -('L,'L,')-1'L,A' (q(t) + Q(t)X(t)) (22) and respective allocation to the money market account given by h{!(t) = 1-1'('L,'L,,)-1 (a+AX(t)) h~(t) = 1 + l'('L,'L,,)-l'L,A' (q(t) + Q(t)X(t)) (23) Moreover, if an investor has a risk sensitivity B, then the respective weights of each mutual fund in the investor's portfolio equal e!l and e!l' respectively. --- ## Page 420 Fractional Kelly Strategies for Benchmarked Asset Management 391 The main implication of this theorem is that the allocation between the two funds is a sole function of the investor's risk sensitivity e. As e -+ 0, the investor's wealth gets invested in the Kelly criterion portfolio (portfolio K). On the other hand, as e -+ 00, the investor's wealth gets invested in the correction portfolio C. The investment strategy of this portfolio can be interpreted as a large position in the short-term rate and a set of positions trading on the comovement of assets and valuation factors. In financial economics, portfolio C is referred to as the 'intertemporal hedging term'. When we assume that there are no underlying valuation factors, the risky secu- rities follow geometric Brownian motions with drift vector J.L and the money market account becomes the risk-free asset (i.e., ao = rand Ao = 0). In this case, ~A' = 0 and we can then easily see that fund C is fully invested in the risk-free asset. As a result, we recover Merton's Mutual Fund Theorem for m risky assets and a risk-free asset (e.g., Theorem 15.1, p. 489 in Merton (1992)). 3.2 Fractional Kelly strategies in the risk-sensitive asset management model Fractional Kelly strategies arise naturally in the Merton investment model, since Merton's Mutual Fund Theorems guarantees that the optimal investment strategy can be split in an allocation to the risk-free asset and an allocation to a mutual fund investing in the Kelly criterion portfolio. However, the optimality offractional Kelly strategies is the exception rather than the rule: in the Merton model, fractional Kelly strategies are optimal as a result of the assumption that asset prices are lognormally distributed. In the factor-based risk-sensitive asset management model, we cannot expect either that the 'classical' definition of fractional Kelly strategies to yield optimal or near optimal asset allocations as soon as e -=f=. O. To address this difficulty, Davis and Lleo (2008a) proposed a generalization of the concept of fractional Kelly strategy based on the findings expressed in the Mutual Fund Theorem 1. Rather than regarding the fractional Kelly strategy as a split between the Kelly portfolio and the short-term rate, Davis and Lleo propose to define it as a split between the Kelly portfolio and the portfolio C, as defined in the Mutual Fund Theorem 1. In this case, the Kelly fraction, which represents the proportion of wealth invested in the Kelly portfolio, is inversely proportional to the investor's risk sensitivity and is equal to Ii!l' This redefinition of fractional Kelly strategies has two important consequences: • The fractional Kelly portfolios are always optimal portfolios; • In the lognormal case (i.e. when n = 0), the generalized definition of frac- tional Kelly strategies reverts to the 'classical' definition. This can be ver- ified from the fact that in the lognormal case, the Mutual Fund Theorem 1 simplifies into Merton's Mutual Fund Theorem for m risky assets and a risk-free asset. --- ## Page 421 392 M Davis and S. Lleo 4 Fractional Kelly Strategies, Risk-Sensitive Asset Management and the Merton Model with Power Utility 4.1 Objective In this section, we go further in our analysis by drawing a comparison between risk- sensitive asset management and the Merton model from the perspective of fractional Kelly strategies. For comparison purpose, we consider: • a fractional Kelly strategy with fraction f; • the Merton model maximizing utility of terminal wealth, with the utility function chosen to be the homogeneous power utility or hyperbolic absolute risk aversion (HARA) function; and • the diffusion risk-sensitive asset management model without any underlying valuation factors. In order to perform a comparison between the risk-sensitive asset allocation model and the original Merton model, we assume that the market is comprised of m risky assets and that the investor's objective is to find an optimal m-dimensional portfolio allocation vector h(t), where hi (t) represents the proportion of the portfolio invested in risky security i. We also assume that there are no valuation factors and hence n = O. We will show that the treatment of fractional Kelly strategies is comparable in both cases up to a sign difference in the risk-aversion/risk-sensitive coefficient. 4.2 A brief review of the Merton model In the Merton model, the objective of an investor is to maximize the utility of terminal wealth represented by the criterion: where I(t, s, h; e) := E [U(V(t, s, h))] (24) • V(t, x, h) is the investor's wealth at time t in response to a securities market with price vector s and an investment policy h; • U is the homogeneous power utility or hyperbolic absolute risk aversion (HARA) function, defined as z'Y U(z) =- I where I E (-00,0) U (0,1) is the risk-aversion coefficient. (25) The dynamics of the prices of the m risky assets follows a geometric Brownian motion of the form dSi(t) m Si(t) = Midt + {; O"ikdWk(t) , Si(O) = Si, i = 1, ... , m (26) --- ## Page 422 Fractional Kelly Strategies for Benchmarked Asset Management 393 where W(t) is a m-dimensional Brownian motion. The price of the money market asset satisfies d50(t) _ d 50(t) - r t, 50(0) = So In this setting, the optimal asset allocation is given by h*(t) = _l_(I:I:/)-lP 1- '1 (27) (28) with P := fL - r l where 1 is the m-element unit vector and I: is the diffusion matrix, defined as I: = [(Tij]. 4.3 Observations and conclusions From a fractional Kelly perspective, the two models are strikingly similar. Taking the case of an investor allocating a fraction f of his/her wealth to the Kelly portfolio, we see that • in the Merton model, such strategy would be optimal for an investor with a level of risk aversion 'I equal to 1 '1= 1 - -f (29) • in the risk-sensitive model, such strategy would be optimal for an investor with a level of risk-sensitivity 8 equal to 1 8 = f - 1 (30) Looking solely at the Kelly component of the optimal investment policy, this would imply that (31) This similarity is in not surprising. Indeed, when we restrict the risk-sensitive approach to a 0 factor model, we get the same optimal asset allocation as in the Merton model with homogeneous power utility, but for the fact that 'I has been replaced by - 8, i.e. (32) This observation is confirmed by both the range of 8 and 'I and by the functional form of the utility function associated with the risk-sensitive approach. These findings are summarized in Table 1. Key Points: • The situation between the Merton model with homogeneous power utility and the risk-sensitive asset management model is parallel, and we can use the guideline correspondence 8 = -'I = 1/ f - 1 to link them with the fractional Kelly approach; --- ## Page 423 394 M Davis and S. Lleo Table 1 Comparison of the Merton Model and of the Risk-Sensitive Asset Management Model from a Kelly Perspective Merton model with Risk-Sensitive Asset Management homogeneous power utility Risk-sensitive parameter / Risk eE(-I,O)U(O,oo) "'( E (-00, 0) U (0,1) aversion coefficient Range of the risk-sensitive pa- (0,00) (-00, 0) rameter/ risk aversion coeffi- cient for risk-averse investors Form of Utility Function U(z) = z-8 z'Y U(z)=- "'( Optimal asset allocation h*(t) _ I _(~~')_l (P,(X(t)) + ~A' D1» e + I _ I _(~~')_l P, 1 -"'( Value of the risk aversion I/f - 1 I - I / f coefficient/ risk-sensitive parameter corresponding to a Kelly fraction f Range of Kelly fractions for (0, 1) (0, 1) risk-averse investors • The risk-sensitive approach is therefore consistent with the Merton model with homogeneous power utility; • In the case when there are no factors (i.e., n = 0) , the optimal asset alloca- tion obtained in the risk-sensitive approach reverts to that associated with the Merton model; • In the general case (n > 0), the definition of fractional Kelly strategies in the risk-sensitive asset management model can be extended as proposed above while remaining consistent with the 'classical' definition of fractional Kelly strategies as a split bewteen the Kelly portfolio and the risk-free asset. 5 Adding an Investment Benchmark: Risk Sensitive Benchmarked Asset Management Model So far, we have introduced risk sensitive asset management in the classical asset only context of an investor attempting to maximize the terminal utility of his/ her wealth. We will now examine the related benchmarked asset management problem in which the investor selects an asset allocation to outperform a given investment benchmark. In the benchmarked case, Davis and Lleo (2008a) propose that the reward func- tion F(t, x; h) be defined as the (log) excess return of the investor's portfolio over --- ## Page 424 Fractional Kelly Strategies for Benchmarked Asset Management the return of the benchmark, i.e. F( h) '= I V(t, x, h) t,x, . n L(t,x,h) where L is the level of the benchmark. Furthermore, the dynamics of the benchmark is modelled by the SDE: dL(t) L(t) = (c + G' X(t))dt +~' dW(t), L(O) = l 395 (33) (34) where G is a scalar constant, G is a n-element column vector, and ~ is aN-element column vector. This formulation is wide enough to encompass a multitude of situations such as: • the single benchmark case, where the benchmark is, e.g. an equity index such as the S&P500 or the FTSE 100. • the single benchmark plus alpha, where, for example, a hedge fund has for benchmark a target based on a short-term interest rate plus alpha. • the composite benchmark case, e.g., a benchmark constituted of 5% cash, 35% Citigroup World Government Bond Index, 25% S&P 500 and 35% MSCI EAFE. • the composite benchmark plus alpha that is a combination of the previous two cases. By Ito's lemma, the log of the excess return in response to a strategy h is F(t, x; h) = In y + lot dIn V(s) - lot dlnL(s) = lny+ lot (ao+A~X(s)+h( s)'(a+AX(s)))ds lit it - - h(s)'l:,l:,'h(s)ds + h(s)'l:,dW(s) 2 0 0 it lit - (c + G' X(s))ds + - ~'~ds o 2 0 - lot~, dW(s) v F(O,x;h) = fo:= lny (35) Following an appropriate change of measure, the criterion function can be expressed as J(fo , x; h; t, T) = In fo - ~ In E~ [exp {e loT-t g(Xs, h(s); e)ds } 1 (36) --- ## Page 425 396 M Davis and S. Lieo where 1 ' g(x, h; B) = 2" (B + 1) h'2:,2:,'h - ao - A~x - h'(a + Ax) 1 - Bh'2:,~ + (c + e'x) + 2" (B -1) ~'~ (37) Once again, the control problem simplifies into a LEQG problem. The value function q, for the auxiliary criterion function I(fo , x; h; t, T). Then q, is defined as q,(t, x) = sup I(fo ,x;h;t,T) (38) A(T) and it satisfies the HJB PDE where aq, h -a + sup L t q, = 0 t hEIR= L?q, = (b+ Bx - BA(2:,'h - ~))' Dq, + ~tr (AA'D2q,) -~(Dq,)'AA'Dq, - g(x,h;B) Solving the optimization problem gives the optimal investment policy h * (t) h * = B ~ 1 (2:,2:,') -1 (a + Ax - B2:,A'Dq, + B2:,~ ) The solution of the PDE is still of the form q,(t, x) = x'Q(t)x + x'q(t) + k(t) (39) (40) ( 41) ( 42) where Q(t) solves a n-dimensional matrix Riccati equation and q(t) solves a n- dimensional linear ordinary differential equation. 6 What About the Kelly Criterion? 6.1 A mutual fund theorem In the benchmarked case, we will follow the same logic as in the asset only and rewrite the optimal asset allocation as an allocation between two funds. We will then use this result to define benchmarked optimal fractional Kelly strategies, that is fractional Kelly strategies, which are also optimal investment policies in the bench- marked asset management model. All we will need to do after that is to verify that this new definition is consistent with the definition we gave earlier in the asset only case, and thus, with the 'classical' definition of fractional Kelly strategies in the limit as our model converges to the Merton model. The following benchmarked mutual fund theorem is due to Davis and LIeo (2008a): --- ## Page 426 Fractional Kelly Strategies for Benchmarked Asset Management 397 Theorem 2 (Benchmarked Mutual Fund Theorem). Given a time t and a state vector X (t), any portfolio can be expressed as a linear combination of invest- ments into two "mutual funds" with respective risky asset allocations hK (t) = (~~/)-l (a + AX(t») hC(t) = (~~/)-l [~c; - ~A' (q(t) + Q(t)X(t»] ( 43) and respective allocation to the money market account given by hff (t) = 1 - l/(~~/) -l (a + AX(t») h~(t) = 1- l/(~~/) -l [~c; - ~A' (q(t) + Q(t)X(t»] (44) Moreover, if an investor has a risk sensitivity e, then the respective weights of each mutual fund in the investor's portfolio equal e!l and e!l' respectively. There are two main differences between Theorems 1 and 2: the definition of portfolio C and the role played by the risk sensitivity e. In the asset only case of Theorem 1, portfolio C is comprised of the money market asset and of a strategy trading the comovement of assets and valuation factors. In the benchmark case of Theorem 2, portfolio C still includes an allocation to the money market asset and the asset-factor co-movement strategy, but it also contains an allocation to a strategy designed to replicate the risk profile of the benchmark. Indeed, the term U := (~~/)-l~c; represents an unbiased estimator of a linear relationship between asset risks and benchmark risk c; = ~/u. In financial economics, the interpretation of u is as the vector of asset systematic exposure, or "betas", computed with respect to the benchmark. Hence, when e is low, the investor will take more active risk by investing larger amounts into the log-utility or Kelly portfolio. On the other hand, when e is high, the investor will divert most of his/her wealth to the correction fund, which is dominated by the term (~~/)-l~c; and designed to track the index. In short, while an investor with e = 0 is a Kelly criterion investor, an investor with extremely high e is a passive investor. This first observation leads us to reconsider the role and definition of the risk sensitive parameter e. Indeed, while in the asset only case, e represents the sen- sitivity of an investor to total risk, in the benchmark case, e corresponds to the investor's sensitivity to active risk. To some extent, in the benchmark case the investor already takes the benchmark risk as granted. The main unknown is there- fore how much additional risk the investor is willing to take in order to outperform the benchmark. This amount of risk is directly quantified by the risk sensitive parameter e. 6.2 Fractional Kelly strategies revisited In the benchmark case, as in the asset only case, we can expand the classical def- inition of fractional Kelly strategies by defining them as a split between the Kelly --- ## Page 427 398 M Davis and S. Lleo portfolio K and the benchmark-tracking portfolio C , as per the Mutual Fund The- orem 2 and which is includes a benchmark replicating strategy. We can quickly check that the extended definition in the benchmarked case is consistent with the definition given earlier in the asset only case by setting the benchmark to 0, which de facto implies the absence of a benchmark. We see then that the definitions of fund C given in Theorems 2 and 1 are identical: the two theorems now coincide. As expected, the asset only problem can be viewed as a special case of the benchmarked allocation. Thus, if we set n = ° so that no valuation factor is considered and set the benchmark to 0, the risk-sensitive benchmarked asset management model is the Merton model and the definition of optimal benchmarked fractional Kelly strategies reverts to the classical definition of fractional Kelly strategies. 7 Case Studies We will now apply the model to study some specific benchmark structures and assess the appropriateness of benchmarks for Kelly criterion investors. 7.1 Benchmark as a portfolio of traded assets First , we revisit two examples considered by Davis and Lleo (2008a) in which the benchmark is a portfolio of traded assets respectively with or without the money market asset. 7.1.1 Benchmark as a portfolio of traded assets and the money market asset Developing the ideas from the previous paragraph, we consider a benchmark whose dynamics is given by dLt A - = (ao + A~X(t)) + v'(t)(& + AX(t))dt + v'(t)~dWt Lt = [( 1 - v'l) ao + v' a + (( 1 - v'l) A~ + v' A ) X ( t ) 1 dt + v'(t)~dWt where v is an m-element allocation vector satisfying the budget equation l'v = 1 - h~ (45) (46) and hf; is the allocation left in the money market account. The process L(t) repre- sents the (log) return of a constant proportion portfolio with risky allocation vector v and the remainder (i.e., 1 - l'v) invested in the money market account. Corollary 3. (Fund Separation Theorem with a Constant Proportion Benchmark (II)). Given a time t and a state vector X(t) , any portfolio can be expressed as --- ## Page 428 Fractional Kelly Strategies for Benchmarked Asset Management 399 a linear combination of investments into a "mutual fund", an index fund and a "long-short hedge fund" with respective risky asset allocations hK(t) = (~~')-1 (a+AX(t)) hI(t) = v hH (t) = _(~~')-l~A' (q(t) + Q(t)X(t)) and respective allocation to the money market account given by h{f(t) = 1-1'(~~')-1 (a + AX(t)) h6(t) = 1 - l'v hf! (t) = 0 ( 47) ( 48) Moreover, if an investor has a risk sensitivity 8, then the respective weights of each fund in the investor's portfolio equal 11!1' 11!1 and 11!1' respectively. We can interpret this Corollary in terms of optimal fractional Kelly strategies in a similar way as we did for Corollary 4. The second component of the strategy, the correction fund C, can again be split into: 1. a portfolio J with risky asset allocation given by v which is designed to strictly replicate the risk exposure of the index; 2. a "long-short hedge fund" H with risky asset allocation given by h H (t) and whose sole purpose is to trade the comovement of assets and factors. Here again, we could show that H is a zero net weight strategy, i.e. ( 49) 7.1.2 Benchmark as a portfolio of traded assets only We assume that the benchmark follows a constant proportion strategy invested in a combination of traded assets. Its dynamics is given by the equation dL _t = v'(a + AX(t))dt + v'~dWt L t where v is a m-element allocation vector satisfying the budget equation l'v = 1 In this setting, Corollary 3 can be expressed as: (50) (51) Corollary 4. (Fund Separation Theorem with a Constant Proportion Benchmark (J)). Given a time t and a state vector X(t), any portfolio can be expressed as --- ## Page 429 400 M Davis and S. Lleo a linear combination of investments into a "mutual fund", an index fund and a "long-short hedge fund" with respective risky asset allocations hK (t) = (~~I)-l (a + AX(t)) hI(t)=v hH(t) = _(~~I)- l~A' (q(t) + Q(t)X(t)) and respective allocation to the money market account given by hfJ (t) = 1 - 11(~~') -l (a + AX(t)) h6(t) = 0 h{f (t) = 11(~~') -l~A' (q(t) + Q(t)X(t)) (52) (53) Moreover, if an investor has a risk sensitivity (), then the respective weights of each mutual fund in the investor's portfolio equal e!l' e!l and e!l' respectively. From the perspective of optimal fractional Kelly strategies, the implication of this corollary is that the second component of the strategy, the correction fund C, can now be explicitly split into two sub portfolios: 1. a portfolio I with asset allocation given by v which is designed to strictly replicate the risk exposure of the index. Observe that the linear estimator u := (~~I) - l~<; has vanished and been replaced by the actual asset allocation v: since the bench- mark is now comprised of traded assets it can be replicated by a direct investment in the appropriate asset allocation and does not require any further estimation; 2. a "long-short hedge fund" H with risky asset allocation given by _ (~~I) - l~A' (q(t) + Q(t)X(t)) (54) and whose sole purpose is to trade the comovement of assets and factors. The "long-short hedge fund" H is particularly interesting because it has zero net weight in the sense that the sum of all the long positions in the portfolio is exactly matched by the sum of short positions. As a result, fund H satisfies the budget equation (55) This "long-short hedge fund" can therefore be viewed as a macro-oriented overlay strategy within the asset allocation. 7.2 Benchmark with alpha target We now consider two new examples involving a benchmark based on an interest rate plus some measure of risk-adjusted excess return known as alpha. --- ## Page 430 Fractional Kelly Strategies for Benchmarked Asset Management 401 7.2.1 Money market rate plus alpha benchmark Money market rate plus some predetermined alpha has been adopted as a bench- mark by a large number of hedge funds. What are the implications of this choice in terms of the optimal fractional Kelly strategies? The dynamics of such a money market rate plus alpha benchmark can be ex- pressed as dLt Lt = (ao + AoX(t))dt + adt (56) where a represents the instantaneous level of risk-adjusted excess return, or alpha, required of the fund manager. The optimal asset allocation in this case is (57) which is simply the asset only asset allocation as given in (20). As a result, the con- clusion of the Mutual Fund Theorem 1 hold and optimal fractional Kelly strategies can be defined accordingly. So, what has happened to the benchmark? In our risk-sensitive framework, the benchmark is tracked through its risk profile rather than its return profile. Since a money market plus alpha benchmark does not have any direct risk, there cannot be any comovement between the assets and the benchmark. Thus, the benchmark risk profile cannot be replicated, and as a result, the asset allocation is established without any regard to the benchmark. The money market plus alpha benchmark will have an impact on the value function q,. Indeed, in the benchmarked case, the auxiliary criterion function J(fo , Xi hi t, T) reflects excess return over the benchmark, rather than total return as in the asset only case. The value function for the benchmarked problem is therefore consistent with excess return rather than total return, resulting in different value functions for the benchmark problem and for the asset only case. The main conclusion from this case study is that from the perspective of a ratio- nal risk-sensitive investor with no investment constraints, money market plus alpha benchmarks results in the same investment strategy as not having any benchmark at all. To some extent, in a risk-sensitive setting, money market plus alpha is a benchmark-free benchmark. In particular, and somehow counter intuitively, the choice a higher value for alpha does not result in a the design of a riskier asset allocation. In fact, the value of alpha has no impact at all! 7.2.2 Bill rate or bond yield plus alpha benchmark The situation would however be slightly different if the benchmark was for example 3 month Treasury bill rate plus alpha or a 5 year Treasury note yield plus alpha. To generalize slightly, the benchmark dynamics could be expressed as an extension --- ## Page 431 402 M Davis and S. Lleo of the model considered above in the "Benchmark as a Portfolio of Traded Assets Only" case (in Section 7.1.2) 1: dL _ t = v'(a + AX(t))dt + adt + v'~dWt L t where v is a m-element allocation vector satisfying the budget equation l'v = 1 (58) (59) As expected, the level of alpha does not influence the optimal asset allocation and the conclusions stemming from Corollary 4 are still valid in this case. 7.2.3 Solving the alpha puzzle Why is the alpha not producing any impact on the asset allocation? The reason is that the risk-sensitive benchmarked asset management model does not penalize for a non-achievement of the benchmark return. As we have seen above, risk sensitivity implies that the risk of the asset portfolio relative to the benchmark matters, but not the expected return. Since an alpha target is in essence a "pure return" target, it does not change the behaviour of the risk sensitive investor. Should the investor be penalized for a non-achievement of the benchmark return? Not in our opinion. For an investor who decides of their own asset allocation, the benchmark is a measure of acceptable risk rather than an minimum return objective. Indeed, the portfolio optimization process will always select , for a given risk sensitivity, the asset allocation which maximizes the relative return of the asset portfolio with respect to its benchmark. The issue of an alpha target is irrelevant in this problem. The situation might however be different if the investor hires a manager to perform the asset allocation. The alpha imperative may in this case be understood as a control imposed by the investor to force the asset manager to select an optimal (or near optimal) asset allocation. The investor should still not be penalized for a non achievement of the benchmark, but it is conceivable that the investor may want to penalize the manager for non achievement of the benchmark return. Here, the penalization would occur at the level of the fee earned by the manager and not at the level of the investor's terminal wealth. This would result in a completely different control problem in which we would need to take the perspective of the manager whose objective is to allocate the investor's assets in order to maximize the management fee received and subject to a non-achievement penalty. IThis slight generalization is required in the event the benchmark is based on a bond. Indeed , in the risk-sensitive model, zero coupon bonds are the representatives of the fixed income asset class in the investment universe. Any coupon bond must therefore be "recreated" as a linear combination of zero coupon bonds. In the event, the benchmark is a Treasury bill, then we have the degenerate case in which v is a vector with the element corresponding to the Treasury bill set to 1 and all other elements equal to 0 --- ## Page 432 Fractional Kelly Strategies for Benchmarked Asset Management 403 7.3 Alpha-omega targets An alternative to the pure alpha target is an alpha-omega target, where omega represents the variability of alpha (see Grinold and Kahn (1999) for a view of the role of alpha and omega in active management). This approach implictly recognises that alpha generation is not only uncertain but also risky. The introduction of the omega terms changes only slightly the asset allocation problem. The dynamics of a benchmark with alpha - omega targets can be modelled as df(~~) = [(c + C' X(t))dt +~' dW(t)] + [adt + w'dW(t) ] =(c+a+C'X(t))dt+(~+ w) 'dW(t) , L(O) = l (60) where the scalar constant c, the n-element column vector C, and ~ is aN-element column vector are related to the benchmark itself and the scalar a and vector w refer to the excess return demanded by the investor. The last element of ~ is set to o while the first n + m element of the vector ware set to O. This condition ensures that the variability of alpha is uncorrelated with the factors and assets, as active asset management theory suggests. 7.3.1 The optimal investment policy Extending the reasoning in Davis and Lleo (2008a), we would find that the optimal investment policy h* is h* = e! 1 (~~')-l (a + Ax - e~A' Dif> + e~(~ + w)) The solution of the PDE is still of the form if>(t, x) = x'Q(t)x + x' q(t) + k(t) (61) (62) where Q(t) solves a n-dimensional matrix Riccati equation and q(t) solves a n- dimensional linear ordinary differential equation. Specifically, Q(t) - Q(t)KoQ(t) + K~ Q(t) + Q(t)Kl + -e 1 A'(~~')-l A = 0 (63) +1 for t E [0, T ], with terminal condition Q(T) = 0 and with K = B - _e_A~'(~~')-l A 1 e + 1 q(t) + (K~ - Q(t)Ko) q(t) + Q(t)b + eQ'(t)A(~ + w) + Ao - C + e! 1 (2A' - eQ'(t)A~') (~~')-l (a + e~(~ + w)) = 0 (64) (65) (66) --- ## Page 433 404 M Davis and S. Lleo with terminal condition q(T) = O. k(s) = fo + iT l(t)dt (67) for 0 <:::: s <:::: T and where 1 e l(t) = 2tr (AA'Q(t)) - 2q'(t)AA'q(t) + b'q(t) + _1_0,1(2:2:/)-10, + -1-e2q'(t)A2:/(2:2:/)-12:A'q(t) e+1 e+1 __ e_q'(t)A2:/(2:2:/)-1O, _ 2e2 q'(t)A2:/(2:2:/)-12:')' e+1 e+1 1 e + e(~ + W)I A'q(t) - - (e - 1)(~ + w)'(~ + w) + -e -O,I(2:2:/)-12:(~ + w) 2 +1 + -e 1 e2')'/2:(2:2:/)-12:(~ + w) + ao - (c + a) (68) +1 7.3.2 Mutual fund theorem We can now restate the optimal asset allocation in terms of the following Mutual Fund Theorem: Theorem 5 (Alpha-Omega Benchmarked Mutual Fund Theorem). Given a time t and a state vector X(t), any portfolio can be expressed as a lin- ear combination of investments into two "mutual funds" with respective risky asset allocations hK (t) = (2:2:/)-1 (a + AX(t)) hC(t) = (2:2:/)-1 [2:(<; + w) - 2:A' (q(t) + Q(t)X(t))] and respective allocation to the money market account given by h{! (t) = 1 - 1/(2:2:/)- 1 (a + AX(t)) h~(t) = 1- 1/(2:2:/)-1 [2:(<; + w) - 2:A' (q(t) + Q(t)X(t))] (69) (70) Moreover, if an investor has a risk sensitivity e, then the respective weights of each mutual fund in the investor's portfolio equal e!1 and e!1' respectively. Proof. This Corollary can be proved in a similar fashion to Theorem 3 in Davis and Lleo (2008a). 0 7.3.3 Kelly strategies In terms of Kelly strategies, this Mutual Fund Theorem represents a split between a fraction e!1 invested in the Kelly portfolio K, and a fraction e!1 invested in the correction fund C. As in Theorem 2, portfolio C includes an allocation to the money market asset and the asset-factor co-movement strategy and an allocation to a strategy designed to replicate the risk profile of the benchmark. In addition, --- ## Page 434 Fractional Kelly Strategies for Benchmarked Asset Management 405 the allocation to portfolio C features a new term related to the variability of alpha, defined as it := (~~/)-l~w and which represents an unbiased estimator of a linear relationship between asset risks and alpha risk w = ~I u. When e is high, the investor will divert most of the wealth to the correction fund in order to replicate the risk profile of the benchmark and the appropriate level of active risk given by w. On the other hand, when e is low, the investor will take more active risk by investing larger amounts into the Kelly portfolio and will not attempt to stay within the bounds of active risk imposed by w. 7.4 Benchmarks and Kelly investors: No benchmark for buffett! Our last application of the idea of benchmarked fractional Kelly strategies takes the form of an anecdote concerning the appropriateness of benchmarks for Kelly investors. A few years ago, a controversy shook the North American investment industry: what would be a proper investment benchmark to assess the performance of legendary investor Warren Buffettt's Berkshire Hathaway? Although this ques- tion may appear, at first glance, trivial, it is in fact the reflection of a wider concern shared by asset managers and investors alike on what constitutes an appropriate investment benchmark. This question has received an increasing deal of attention in the past two decades and it is considered so important in the investment man- agement industry that the Research Foundation of the CFA Institute (then called AIMR) , a leading professional association in the investment management indus- try, has recently devoted a monograph to the question (see Siegel (2003) for more details) . Although we do not intend on entering the benchmark design and specification debate2 , the results we derived reveal a new dimension to the problem: the impact of the investment benchmark on a fund manager's investment strategy depends to a significant extent on the risk-aversion of the manager or investor. Indeed, we have shown that in the risk-sensitive benchmarked asset management model, the importance of the benchmark in the investment decision increases as the risk aversion of the investor increases. In fact , the risk-sensitivity e is a direct indication of the amount of active risk that an investor is willing to take. Thus, the exercise consisting in setting an investment benchmark becomes increasingly irrelevant as risk-aversion gets nearer to 0: Kelly criterion investors invest in the log-utility portfolio and have no regards for a benchmark. Before setting a benchmark, it is imperative to know the investor's risk prefer- ences as this will influences his/her investment style. A very rough correspondence would be to identify extremely high risk aversion investors to passive portfolio man- agers whose mandate is to track an index, low risk aversion investors to purely active managers, whose mandate is to generate the best possible risk-adjusted return in 2We have, after all, conveniently assumed throughout that an appropriate care had been taken beforehand to select the benchmark. --- ## Page 435 406 M Davis and S. Lleo a specific market, and medium risk-aversion investors to so called "core plus" or "index plus" managers who track closely and index while trying to generate some incremental extra return or "alpha". The only time these three broad categories of investors will coincide is when their benchmark is the Kelly portfolio. Going back to our original question, Warren Buffet focuses on long-term growth maximization rather than the avoidance of short-term losses (see Ziemba (2005) for a discussion of Berkshire Hathaway's risk-adjusted performance and Thorp (2006) for a Kelly criterion perspective). In this sense, Mr. Buffet behaves similarly to a Kelly criterion bettor. The RSBAM model then demonstrates that benchmarks are irrelevant to Mr. Buffet's investment strategy. As a concluding note, since at least 1995, the Berkshire Hathaway annual report features a table presenting a proxy for the performance of the Berkshire Hathaway share against the return of the S&P500 going back to 1965. Warren Buffet, however, does not recognize the S&P500 as his benchmark. He made it clear that this comparative table had been included in the annual reports at the request of investors but that he personally sees little value in it. 8 Conclusion Historically, Kelly and fractional Kelly strategies have played an important part in asset-only investment management. But the essential role of fractional Kelly strategies does not stop at this level: it also extends to benchmarks and even to asset and libaility management (see Davis and Lleo (2008b)). In benchmarked asset allocation problems, fractional Kelly strategies highlight the fundamental split between a purely active growth maximizing strategy (the Kelly criterion portfolio) and pure benchmark replication. Moreover, the Kelly fraction, which represent the fraction of one's wealth invested in the Kelly portfolio, is a sole function of the investor's degree of risk-sensitivity. This has profound implications. A Kelly investor, with 0 risk-sensitivity, will invest solely in the purely active portfolio and as a result, benchmarks are irrelevant to Kelly investors. On the other end of the spectrum, investors with extremely large risk-sensitivity will tend to opt for full replication of the benchmark: they are passive investors. Somewhere in the middle, we find investors with an average risk-sensitivity, who adopt core-plus strategies: a basic replication of the benchmark with some departure in order to generate excess returns. The only time these three broad categories of investors will coincide is when their benchmark is the Kelly portfolio. References A. Bensoussan and J. H. van Schuppen (1985). Optimal control of partially observable stochastic systems with an exponential-of-integral performance index. SIAM Journal on Control and Optimization, 23(4), 599- 613. --- ## Page 436 Fractional Kelly Strategies for Benchmarked Asset Management 407 Bielecki, T . R. and S. R. Pliska (1999) . Risk-sensitive dynamic asset management. Applied Mathematics and Optimization, 39, 337- 360. Davis, M. H. A. and S. Lleo (2008a) . Risk-sensitive benchmarked asset management. Quan- titative Finance, 8(4), 415- 426. Davis, M. H. A. and S. Lleo (2008b). A risk sensitive asset and liability management model. Working Paper. Grinold, R. and R. Kahn (1999) . Active Portfolio Management: A Quantative Approach for Producing Superior Returns and Selecting Superior Money Managers. New York: McGraw-Hill. Jacobson, D. H. (1973). Optimal stochastic linear systems with exponential criteria and their relation to deterministic differential games. IEEE Transactions on Automatic Control, 18(2), 114- 13l. Kuroda, K. and H. Nagai (2002) . Risk-sensitive portfolio optimization on infinite time horizon. Stochastics and Stochastics Reports, 73, 309- 33l. Lefebvre, M. and P. Montulet (1994) . Risk-sensitive optimal investment policy. Interna- tional Journal of Systems Science, 22, 183- 192. Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous time case. Review of Economics and Statitsics, 51, 247- 257. Merton, R. C. (1971). Optimal consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3, 373- 4113. Merton, R. C. (1992). Continuous-Time Finance. US: Blackwell Publishers. Siegel, L. (2003). Benchmarks and Investment Management. The Research Foundation of AIMR. Thorp, E. (2006). The Kelly criterion in blackjack, sports betting and the stock mar- ket. In S. A. Zenios and W. T. Ziemba, editors, Handbook of Asset and Liability Management, Volume 1, Chapter 9, pp. 385-428. Amersdam: North Holland. P. Whittle (1990). Risk Sensitive Optimal Control. New York: John Wiley & Sons. Ziemba, W. T. (2003). The Stochastic Programming Approach to Asset, Liability, and Wealth Management. Research Foundation Publication. CFA Institute. Ziemba, W. T. (2005). The symmetric downside-risk sharpe ratio. Journal of Portfolio Management, 32(1), 108- 122. --- ## Page 437 This page is intentionally left blank --- ## Page 438 28 A Benchmark Approach to Investing and Pricing Eckhard Platen University of Technology Sydney, School of Finance €3 Economics and Department of Mathematical Sciences, PO Box 123, Broadway, NSW, 2007, Australia Abstract This paper introduces a general market modeling framework, the benchmark approach, which assumes the existence of the numeraire portfolio. This is the strictly positive portfolio that when used as benchmark makes all benchmarked non-negative portfolios supermartingales, that is intuitively speaking downward trending or trendless. It can be shown to equal the Kelly portfolio, which maxi- mizes expected logarithmic utility. In several ways, the Kelly or numernire portfo- lio is the "best" performing portfolio and cannot be outperformed systematically by any other non-negative portfolio. Its use in pricing as numeraire leads di- rectly to the real world pricing formula, which employs the real world probability when calculating conditional expectations. In a large regular financial market , the Kelly portfolio is shown to be approximated by well-diversified portfolios. JEL Classification: G 10, G 13 1991 Mathematics Subject Classification: primary 90A12; secondary 60G30, 62P20. Keywords and phrases: Kelly portfolio, real world pricing, numeraire portfolio, strong arbitrage, diversification. 1 Introduction 409 The classical asset pricing theories, as developed in Debreu (1959), Sharpe (1964), Lintner (1965), Merton (1973a,b), Ross (1976), Harrison and Kreps (1979), Con- stantinides (1992), and Cochrane (2001) represent forms of relative pricing, since the existence of an equivalent risk neutral probability measure is typically requested. However, relative pricing ignores in the long run the presence of the equity premium. This paper presents an extension of classical risk neutral pricing in a general model- ing framework, the benchmark approach, where a form of absolute pricing naturally emerges, see Platen (2002) Platen (2002, 2006) and Platen and Heath (2006). The benchmark represents here the "best" performing, strictly positive, tradable port- folio, which turns out to be the K elly portfolio, see Kelly (1956). Most results we --- ## Page 439 410 E. Platen present can be stated in a model independent manner. The benchmark approach only requires the existence of the benchmark, playing the role of the numeraire portfolio, originally discovered in Long (1990). The benchmark approach with the Kelly portfolio as benchmark is directly applicable in portfolio optimization and covers a wider modeling world than classical theories allow. We will see that non-negative portfolios, when denominated in units of the benchmark are supermartingales. This supermartingale property has been already observed for particular settings, for instance, in Long (1990) , Bajeux-Besnainou and Portait (1997), Becherer (2001) , Platen (2002), Biihlmann and Platen (2003), Platen and Heath (2006), Karatzas and Kardaras (2007), and Kardaras and Platen (2008b). For the purpose of pricing, the inverse of the benchmark plays the role of the stochastic discount factor in the language of Cochrane (2001). Within this paper, there will be no major additional assumption made beyond the request on the existence of a numeraire portfolio. A series of fundamental results follows from this assumption by a few basic arguments. Some of these describe "best" performance properties of the Kelly portfolio which underline its fundamental importance in investing. In Platen and Heath (2006), the benchmark approach has been described for jump-diffusion markets. Probably the most striking feature of the rich benchmark framework is the possible co-existence of several self-financing portfolios that may perfectly replicate one and the same payoff. The presence of different replicating portfolios is not consistent with the classical Law of One Price. The proposed Law of the Minimal Price identifies for a given contingent claim the corresponding minimal replicating portfolio process, which characterizes also in an incomplete market the economically correct price process. The Law of the Minimal Price yields directly the real world pricing formula, where the expectation is taken with respect to the real world probability measure and the Kelly portfolio appears as the numeraire. No change of probability measure is performed under real world pricing, which extends the classical risk neutral approach. Consequently, the request on the existence of an equivalent risk neutral probability measure is here avoided and a much wider modeling world is available. The real world pricing formula generalizes the risk neutral pricing formula as well as the actuarial pricing formula. These formulae represent the central pricing rules in their respective streams of literature and nothing else is here requested than the existence of the Kelly portfolio. The fact that the Law of One Price does not hold does not create strong arbitrage opportunities in the sense of this paper. We will see that there is no economic reason to exclude any weaker form of arbitrage as under the classical no-arbitrage approach. A Diversification Theorem will be derived along the lines of Platen (2005). It states that any diversified portfolio in a regular financial market approximates the Kelly portfolio. This makes the benchmark approach rather practical and allows one to interpret a well-diversified portfolio as proxy for the Kelly portfolio. For --- ## Page 440 A Benchmark Approach to Investing and Pricing 411 further results on the benchmark approach the reader is referred to Platen and Heath (2006). The remainder of the paper is organized as follows: It introduces the numeraire portfolio in Section 2. Section 3 derives various manifestations of "best" perfor- mance for the Kelly portfolio. The Law of the Minimal Price is proposed in Sec- tion 4. In Section 5 we discuss the concept of real world pricing. Section 6 introduces a strong form of arbitrage. Finally, Section 7 provides a version of the Diversification Theorem. 2 Benchmark Approach Along the lines of Platen (2002) (Platen, 2002, 2006) and Platen and Heath (2006), we consider a general financial market in continuous time with d risky, non-negative, primary securities, d E {I, 2, . . . }. These securities could be, for instance, shares, currencies or other traded securities. Denote by st the value of the corresponding /h primary security account, j E {O, 1, ... , d}, at time t ~ O. This non-negative account holds units of the jth primary security together with all dividends or interest payments reinvested. The Oth primary security account S~ denotes the value of the locally riskless savings account at time t ~ o. The dynamics of the primary security accounts need not be specified when formulating the main statements. The market participants can form self-financing portfolios with primary security accounts as constituents. A portfolio value sf at time t is described by the number of of units held in the jth primary security account Sf for all j E {O, 1, ... , d} , t ~ o. For simplicity, assume that the units of the primary security accounts are perfectly divisible, and that for all t E [0,(0) the values o~, of, ... , Of, for any given strategy 0= {Ot = (o~,of, ... ,of)T, t ~ O}, depend only on information available at time t. The portfolio value at this time is given by the sum d sf = Lot st j=O We consider only self-financing portfolios where changes in their value are only due to changes in the values of the primary security accounts. We neglect any market frictions or liquidity effects. By V: denote the set of all strictly positive, finite, self-financing portfolios with initial capital x > O. The benchmark approach employs a very special strictly positive portfolio as benchmark, which we denote by S 8* E V:. Later it will become apparent that it is the Kelly portfolio, see Kelly (1956), which is in several ways the "best" performing strictly positive tradable portfolio. On the other hand, it will turn out that it is also the natural numeraire , see Long (1990), for pricing any type of claim when employing the real world probability for calculating the respective conditional expectation in the resulting pricing formula. --- ## Page 441 412 E. Platen Let Et(X) denote the conditional expectation under the real world probability measure P given the information available at time t ;:::: O. Definition 2.1. For given x> 0, a strictly positive, finite, self-financing portfolio SfH E V: is called a numemire portfolio if all non-negative portfolios S6, when denominated in units of S6*, are supermartingales. The notion of a numemire portfolio was originally introduced by Long (1990) in a rather special setting. Later it was generalized in Bajeux-Besnainou and Portait (1997) and Becherer (2001). These authors worked under classical no-arbitrage assumptions that are consistent with the existence of an equivalent risk neutral probability measure, see Delbaen and Schachermayer (199S). More recently, Platen (2002), Biihlmann and Platen (2003), Platen and Heath (2006), and Platen (2006) emphasized that in a more general setting one still obtains a viable financial market model, as long as a numemire portfolio exists. Also Fernholz and Karatzas (2005), Karatzas and Kardaras (2007) , and Kardaras and Platen (200Sb) consider financial market models beyond the classical no-arbitrage framework. To provide a basis, let us formulate the only major assumption of the paper: Assumption 2.2. For given x > 0, there exists a numemire portfolio S6* E V:. This assumption is satisfied for a wide range of financial market models used in practice. For instance, in Platen and Heath (2006) it has been verified for jump- diffusion markets. Karatzas and Kardaras (2007) and Kardaras and Platen (200Sb) confirm the validity of Assumption 2.2 for a wide range of semi-martingale markets. Now, under the benchmark approach we choose the numemire portfolio as benchmark. The benchmarked value Sf of a portfolio S6 is given by the ratio ' 6 Sf S - - t - S6* t for all t;:::: O. Definition 2.1 leads by Assumption 2.2 directly to the following conclusion: Corollary 2.3. The benchmarked value Sf of any non-negative portfolio S6 sat- isfies the supermartingale property S6 > E (S6) t - t s (2.1) for all 0 ::; t ::; s < 00 . Consequently, the currently observed benchmarked value of a non-negative port- folio is always greater than or equal to its expected future benchmarked value. This means intuitively, if there were any trend in a benchmarked non-negative portfolio, then this trend could only point downward. The supermartingale property (2.1) is the fundamental property of a financial market. For instance, it yields easily the uniqueness of the benchmark by the --- ## Page 442 A Benchmark Approach to Investing and Pricing 413 following argument: Consider two strictly positive portfolios that are supposed to be numeraire portfolios. According to Corollary 2.3, the first portfolio, when expressed in units of the second one, must satisfy the supermartingale property (2.1). By the same argument, the second portfolio, when expressed in units of the first one, must also satisfy the supermartingale property. Consequently, by Jensen's inequality the portfolios have to be identical, and the value process S5* E Vi of a numeraire portfolio is unique. Note that the stated uniqueness does not imply that the number of units invested has to be unique, which is due to potential redundancies in primary security accounts. 3 Best Performance of the Benchmark In this section we list several manifestations of the fact that our benchmark S5* is the "best" performing, strictly positive, tradable portfolio and equals the Kelly portfolio: 3.1 Numeraire portfolio First, the definition of the numeraire portfolio S5* itself, given by Definition 2.1, expresses the fact that this portfolio performs "best" in the sense that the expected returns of benchmarked non-negative portfolios never become strictly positive. 3.2 Kelly portfolio As a second manifestation of "best" performance, we derive the growth optimality of the benchmark, which identifies it very generally as the Kelly portfolio, a fact that has been documented for special models in the literature, for instance, in Long (1990). The expected growth gf,h of a strictly positive portfolio S5 over the time period (t, t + h] for t, h > ° is given by the conditional expectation gf,h = E t (In (A~, h)) of the logarithm of the portfolio ratio A5 _ Sf+h t,h - S5 t To identify the strictly positive portfolio that maximizes the expected growth, let us perturb at time t :;:. 0, the investment in a given strictly positive portfolio S~ E Vi, x > 0, by some small fraction E E (o,~) of some non-negative portfolio S5. For analyzing the changes in the expected growth of the perturbed portfolio S5 g define the derivative of expected growth in the direction of S5 as the right hand limit 8g~,\ I 1· 1 ( 5g ~) -- - lm- 9 -g 8E 0'=0 - 0'-->0 E t,h t ,h (3.1) --- ## Page 443 414 E. Platen for t, h ~ O. Obviously, if the portfolio that maximizes expected growth coincides in (3.1) with the portfolio SQ., then the resulting derivative of expected growth will never be greater than zero for all non-negative portfolios Sli. This leads to the following definition of growth optimality: Definition 3.1. A strictly positive portfolio S Q. is called growth optimal if the corresponding derivative of expected growth is less than or equal to zero for all non-negative portfolios SIi , that is, gt ,h < 0 o lie I oc E=O - for all t, h ~ O. Note that this definition is different to the classical characterization of the Kelly portfolio or growth optimal portfolio. In the literature, it is typically based on the maximization of expected logarithmic utility from terminal wealth, as used in Kelly (1956) and later also employed in a stream of literature, including Latane (1959) , Breiman (1960) , Hakansson (1971) , Merton (1973a), Roll (1973) and Markowitz (1976), among many others. It is clear from Definition 3.1 and the standard log- utility definition of the Kelly portfolio, see for instance Kelly (1956) and Thorp (1972), that the above growth optimal portfolio equals the Kelly portfolio. The following result provides a convenient method for the identification of the benchmark or numeraire portfolio in a given investment universe by searching for its Kelly portfolio. Theorem 3.2. The numeraire portfolio is growth optimal. Proof: For c E (0, ~) , two consecutive times t and t + h with h > 0, and a non- negative portfolio SIi , with Sf > 0, consider the perturbed portfolio Slie under the choice sf = Sr in (3.1), yielding a portfolio ratio A~~h = c A~, h - (1 - c) A~,h > O. One then obtains by the well-known inequality In(x) <:::; x -I for x ~ 0, the relations and Ali AIi* t ,h - t ,h Alie t ,h Since A~eh > 0 one obtains from (3.3) for A~ h - A~*h ~ 0 the inequality , , , clie > 0 t ,h - (3.2) (3.3) (3.4) --- ## Page 444 A Benchmark Approach to Investing and Pricing and for Ai,h - Ath < 0, because of c E (0, ~) and Ai,h ::::: 0, the relation A,h 1 1 COg > - ~ = - > - -- > -2 t h - AO A O - 1 - , t gh 1 - c + c ~ - c , A1,*h Summarizing (3.2)- (3.5) yields the upper and lower bounds AO -2 < COg < ~-1 - t ,h - AO* t,h where by Definition 2.1 one has 415 (3.5) (3.6) (3.7) By using (3.6) and (3.7) it follows by the Dominated Convergence Theorem, see Shiryaev (1984), that This proves by Definitions 2.1 and 3.1 that the numeraire portfolio SO* is growth optimal. D 3.3 Long term growth To formulate a third manifestation of "best" performance for the benchmark, define the long term growth rate gO of a strictly positive portfolio So E Vi as the upper limit l = lim sup ~ In (S~) t--->oo t So (3.8) The long term growth rate (3.8) is defined pathwise almost surely and does not involve any expectation. By exploiting the supermartingale property (2.1), the following fascinating property of the Kelly portfolio will be shown very generally below. Theorem 3.3. The numeraire portfolio SO* E Vi achieves the maximum long term growth rate. This means, when compared with any other strictly positive port- folio SO E Vi, one has (3.9) --- ## Page 445 416 E. Platen Proof: Similar as in Karatzas and Shreve (1998) consider a strictly positive port- folio S6 E Vi, x > 0, with the same initial capital as the numeraire portfolio, that is, S8 = S8* = x > 0. By Corollary 2.3, we can use the following maximal inequality, derived in Doob (1953) , where for any k E {I, 2, ... } and E E (0, 1) one has exp{Ek}P ( sup sf> eXP{Ek}) S; Eo (SZ) S; sg = 1 k$t E k) S; ~ exp{ -E k} < 00 By the Borel-Cantelli lemma, see Shiryaev (1984), there exists a random variable kc such that for all k 2> kc and t 2> k it holds that In (Sf) S; E k S; E t Therefore, it follows for all k > kc the estimate sup ~ In (Sf) S; E t?k t which implies that 1 (S6) 1 (S6*) lim sup - In -% S; lim sup - In -i; + E t--->oo t So t--->oo t So (3.10) Since the inequality (3.10) holds for all E E (0, 1) one obtains with (3.8) the relation (3.9). 0 According to Theorem 3.3, the trajectory of the Kelly portfolio outperforms in the long run those of all other strictly positive portfolios that start with the same initial capital. This property is independent of any model choice and, therefore, very robust. An investor, who is aiming in the long run for the highest possible wealth, has to invest her or his total tradable wealth into the Kelly portfolio. 3.4 Systematic outperformance Over short and medium time horizons, almost any strictly positive portfolio can gen- erate larger returns than those exhibited by the Kelly portfolio. However, the fourth manifestation of "best" performance will show that such short term out performance cannot be achieved systematically. To formulate a corresponding statement prop- erly, we will employ the following definition: Definition 3.4. A non-negative portfolio S6 systematically outperforms a strictly positive portfolio S8 if (i) both portfolios start with the same initial capital Sfa = sfa; --- ## Page 446 A Benchmark Approach to Investing and Pricing 417 (ii) at a later time t the portfolio value sg is at least equal to sf, that is P(sg 2:: Sf) = 1, and (iii) the probability for sg being strictly greater than sf is strictly positive so that p (sg > Sf) > O. Systematic outperformance of one portfolio by another one is possible under the benchmark approach, see Platen and Heath (2006). The above notion of systematic out performance was introduced in Platen (2004) and was motivated by the super- martingale property (2.1). It relates in some sense to the notion ofrelative arbitrage, later studied in Fernholz and Karatzas (2005), and also to the notion of a maxi- mal element earlier employed in Delbaen and Schachermayer (1998). The following result presents a fourth manifestation of "best" performance of the benchmark: Theorem 3.5. The numemire portfolio cannot be systematically outperformed by any non-negative portfolio. Proof: Consider a non-negative portfolio So with benchmarked value sg = 1 at a given time t 2:: 0, where s~ 2:: 1 almost surely at some later time s E [t, (0). Then it follows by the supermartingale property given in Corollary 2.3 that Since one has s~ 2:: 1 almost surely and Et(S~) :s; 1, it can only follow that S~ = l. This means that one has at time s the equality S~ = S~* almost surely. Therefore, according to Definition 3.4 the portfolio So does not systematically outperform the numemire portfolio. D As a consequence of Theorem 3.5, one can conclude that in the given very general modeling setting, no active fund manager can systematically outperform the benchmark. On the other hand, if the market portfolio is not the numemire portfolio, which is most likely the case, and a fund approximates well the Kelly portfolio, then this fund will not be outperformed systematically by the market portfolio or any other significantly different portfolio. In the long run its path will by property (3.8) beat the path of the market portfolio almost surely. There are further manifestations of "best" performance of the Kelly portfolio. For instance, in Kardaras and Platen (2008a), it is shown that the Kelly portfolio minimizes some expected market time to reach a given wealth level. Results in this direction can be found, for instance, in Browne (1998). 4 The Law of the Minimal Price The fundamental supermartingale property (2.1) ensures that the maximum ex- pected return of a benchmarked non-negative portfolio can at most equal zero. In --- ## Page 447 418 E. Platen the case when it equals zero for a given benchmarked price process for all time in- stances, then the current benchmarked value of the price is always the best forecast of its future benchmarked values. In this case, one has equality in relation (2.1) and we call such a price process fair. A benchmarked fair price process forms a martingale. In general, not all primary security accounts and portfolios need to be fair under the benchmark approach. This is the key property that differentiates this approach from the classical approaches. For instance, there may exist benchmarked portfolios that form local martingales but are not true martingales, as is exten- sively discussed in Platen and Heath (2006). Furthermore, note that non-negative fair price processes are somehow minimal, as we shall see below. It may be puzzling to some readers that discounting with another discount fac- tor than the savings account ought to lead to "fair" prices or could be particularly meaningful. Below we will give a valid reason why the Kelly portfolio is the "uni- versal currency" that should be used for discounting when pricing under the real world probability. It stems from the fact that the Kelly portfolio represents the best performing portfolio and, thus, the natural numeraire for valuation when the real world probability measure for calculating expectations. The following Law of the Minimal Price substitutes the widely postulated clas- sical Law of One Price, which no longer holds in our general setting: Theorem 4.1. Law of the Minimal Price. If a fair portfolio process replicates a given non-negative payoff at a given maturity date, then this portfolio represents the minimal replicating portfolio among all non-negative portfolios that replicate this payoff· Proof: A stochastic process [:;8, which satisfies relation (2.1) is a supermartingale, see Shiryaev (1984). When equality holds in (2.1) then the process is fair and its benchmarked value process is a martingale. Within a family of non-negative super- martingales with the same value at a given future payoff date, it is the martingale among these supermartingales which attains almost surely the minimal possible value at all times before the maturity date. This fundamental fact about the opti- mality of martingales in a family of supermartingales proves directly Theorem 4.l. D The existence of unfair price processes under the benchmark approach creates new realistic effects that can be modeled and are not captured by any classical no-arbitrage framework. For a given replicable payoff, it follows by Theorem 4.1 that the corresponding fair replicating portfolio describes the least expensive hedge portfolio. This is also the economically correct price process in a competitive mar- ket. We emphasize that the Law of the Minimal Price generates a unique price system for contracts and derivatives under the benchmark approach which only re- lies on the existence of a numeraire portfolio. As shown in Platen (2009), pricing purely based on hedging or classical no-arbitrage arguments, as employed under the --- ## Page 448 A Benchmark Approach to investing and Pricing 419 classical Arbitrage Pricing Theory, see Ross (1976), can lead in our general setting to significantly more expensive prices than provided by fair price processes. 5 Real World Pricing Define a contingent claim HT as a non-negative payoff delivered at maturity T E (0,00), which is expressed in units of the domestic currency and has finite expectation Eo (:;) < 00. (5.1) By the Law of the Minimal Price one can identify the corresponding fair price process via the following real world pricing formula: Corollary 5.1. If for a contingent claim HT, T E (0,00), there exists a fair portfolio S8}; that replicates this claim at maturity T such that HT = Sg.II , then its minimal replicating price at time t E [0, T] is given by the real world pricing formula S8 H = S8* E (HT) t t t S8* T (5.2) Relation (5.2) is called the real world pricing formula because it involves the conditional expectation E t with respect to the real world probability measure. This formula can be interpreted in the sense of Cochrane (2001) as a pricing formula that uses the stochastic discount factor (S[*)-1. Also the closely related use of the corresponding state price density, pricing kernel and deflator are consistent with the real world pricing formula (5.2) under appropriate assumptions. However, all the classical pricing approaches exclude classical arbitrage, which is equivalent to the existence of an equivalent risk neutral probability measure, see Delbaen and Schachermayer (1998). In this manner, they postulate the Law of One Price, see for instance Ingersoll (1987), Long (1990), Constantinides (1992), Duffie (2001), and Cochrane (2001) for classical no-arbitrage settings. The benchmark approach with its real world pricing concept does not require a risk neutral probability measure to exist or equivalent classical no-arbitrage constraints and goes beyond the Law of One Price. The real world pricing formula (5.2) only requires the existence of the numemire portfolio and the finiteness of the expectation in (5.1). No measure transformation needs to be applied. An important special case of the real world pricing formula (5.2) arises when HT is independent of S~*. In this case one obtains the actuarial pricing formula (5.3) with the fair zero coupon bond price (5.4) --- ## Page 449 420 E. Platen that pays one monetary unit at maturity T. The fair zero coupon bond price P(t, T) provides the discount factor in (5.3) in a way as it has been used intuitively by actuaries for obtaining the, so called, net present value of HT. The real world pricing formula provides a rigorous and general derivation of the actuarial pricing formula that has been in use for centuries. Let us now derive risk neutral pricing as a special case of real world pricing. For this purpose we rewrite for t = 0 the real world pricing formula (5.2) in the form sgll = Eo (AT ~~ HT) (5.5) " 0 while employing the benchmarked normalized savings account AT = ~~" By the o supermartingale property (2.1) ofthe normalized benchmarked savings account pro- cess A = {At = fa, t 2:> O} we have 1 = Ao 2:> Eo(AT), Together with equation (5.5) o this yields the inequality Eo (AT ¥ HT) SIiH < T o - Eo(AT) (5.6) If the benchmarked savings account is not a martingale, then equality does not hold in relation (5.6). To ensure equality in (5.6), one needs to impose the strong assumption that the savings account is fair, that is, intuitively its benchmarked value has no trend and is a martingale. In this particular case, the expression on the right hand side of (5.6) can be interpreted by Bayes' formula as the conditional expectation of the discounted contingent claim under the, in this case existing, equivalent risk neutral probability measure Q with Radon-Nikodym derivative AT = ~~. Only in this case when A is a martingale the relation (5.6) yields, in general, for a contingent claim HT the classical risk neutral pricing formula Sli II - EO (S8 H ) 0 - 0 SO T T see, for instance, Harrison and Kreps (1979) or Karatzas and Shreve (1998). Here E~ denotes the conditional expectation at time t = 0 under the equivalent risk neutral probability measure Q. By inequality (5.6) it follows that the fair derivative price is never more expensive than the price obtained under some formal application of the standard risk neutral pricing rule. Finally, it shall be noted for not perfectly hedgable contingent claims that utility indifference pricing, in the sense of Davis (1997), leads in the case of incomplete jump-diffusion markets again to the real world pricing formula (5.2), see Platen and Heath (2006). Under real world pricing, the hedgable part of the claim is replicated via the minimal possible hedge portfolio and the benchmarked unhedgable part remains untouched. In some sense, the real world pricing formula provides the least squares projection of a given unhedgable benchmarked contingent claim into the set of current benchmarked prices, The benchmarked hedge error has zero mean and --- ## Page 450 A Benchmark Approach to Investing and Pricing 421 its variance is minimized. This generalizes also the notion of local-risk minimization for pricing in incomplete markets, as advocated in Follmer and Schweizer (1991), Hofmann et al. (1992), and Schweizer (1995). In practice, when benchmarked hedge errors can be diversified in the large book of a well diversified bank or insurance company, then the total benchmarked hedge error vanishes by the Law of Large Numbers asymptotically from the bank's trading book and the market becomes asymptotically complete from this perspective. This shows that in practice real world pricing makes perfect sense. Any systematically more expensive pricing would make an institution less competitive. On the other hand, any lower prices would make it unsustainable. 6 Strong Arbitrage Since the benchmark approach goes significantly beyond the classical no-arbitrage world, it is of importance to clarify the potential existence of arbitrage opportu- nities. In the literature, there exist many different mathematical definitions of arbitrage, and one has to ensure that the given modeling framework is economi- cally viable. Obviously, arbitrage opportunities can only be exploited by market participants. These have to use their portfolios of total tradable wealth when trying to exploit potential arbitrage opportunities. Due to the established legal concept of limited liability, only non-negative total tradable wealth processes have to be considered when studying the exploitation of potential arbitrage. Therefore, any realistic arbitrage concept has to focus on non-negative, self-financing portfolios. An obvious, strong form of arbitrage arises when a market participant can gen- erate strictly positive wealth from zero initial capital. This leads to the following definition of strong arbitrage, which was introduced in Platen (2002) motivated by the supermartingale property (2.1): Definition 6.1. A non-negative portfolio S6 is a strong arbitrage if it starts with zero initial capital, that is S8 = 0, and generates strictly positive wealth with strictly positive probability at a later time t E (0,00), that is, P(st > 0) > O. The exclusion of the above form of arbitrage has been independently argued for on purely economic grounds by Loewenstein and Willard (2000). Important is that if only strong arbitrage is excluded, then weaker forms of arbitrage may still exist. However, this does not harm the economic viability of the market model. For instance, there may exist, so-called, free snacks and cheap thrills, in the sense of Loewenstein and Willard (2000), which usually represent situations of system- atic outperformance in the sense of Definition 3.4. Also free lunches with vanishing risk, as excluded in Delbaen and Schachermayer (1998), can exist without creating complications from an economic point of view. Furthermore, by looking at any of the weaker forms of arbitrage one realizes that these cannot be exploited in prac- --- ## Page 451 422 E. Platen tice without providing adequate collateral, see Platen and Heath (2006) . This re- quest, however, makes the corresponding theoretical notions of weaker forms of arbi- trage questionable from their practical relevance. By exploiting the supermartingale property (2.1) , the following result can be established: Theorem 6.2. arbitrage. There does not exist any non-negative portfolio that is a strong Proof: For a non-negative portfolio S6, which starts with zero initial capital, it follows by the supermartingale property given in Corollary 2.3 that 6 '6 ('6) '6 ° = So = x So ::::: x Eo St = x E(St) ::::: ° for t ::::: 0, where E(·) denotes expectation under the real world probability. By the nonnegativity of sf and the strict positivity of si*, the event sf > ° can only have zero probability, that is P (Sf> 0) = ° This leads to the conclusion that sf equals zero for all t ::::: 0, which proves by Definition 6.1 the Theorem 6.2. D Theorem 6.2 states that strong arbitrage is automatically excluded in the given general benchmark framework. Therefore, different to the classical no-arbitrage ap- proaches, pricing by excluding strong arbitrage does not make any sense. Instead, one should use real world pricing which is economically and also theoretically mean- ingful, as we have seen. 7 Diversification To conclude the paper, we consider the practical problem of identifying or approx- imating the Kelly portfolio of a given market. We will indicate how to construct proxies for the Kelly portfolio that can be used for portfolio optimization and val- uation under the benchmark approach. For simplicity, let us consider a continuous financial market with its benchmarked lh primary security account value Sf at time t satisfying the driftless stochastic differential equation (SDE) j dSj - sj '" (yj ,k dW k t - t~t t (7.1) k=l for t E [0,00), with sg > 0, see Platen and Heath (2006). The more general setting of jump diffusion markets has been considered in Platen (2005) and forthcoming work will generalize the results presented below, avoiding any particular assumptions. In (7.1) Wk = {Wtk, t E [O,oo)} denotes an independent standard Wiener process, k E {I, 2, . . . }. A benchmarked portfolio S6d with fractions 7rLt invested at time --- ## Page 452 A Benchmark Approach to Investing and Pricing 423 t in the jth primary security account, where j E {I, 2, . .. , d} with d E {I, 2, ... }, satisfies the SDE (7.2) for t E [0,00), with Sgd > O. It is clear that the benchmarked Kelly portfolio 5; equals the constant one and its diffusion coefficients vanish. This leads us to define a sequence (SOd )dE {1 ,2, ... } of benchmarked approximate K elly portfolios as a sequence of strictly positive portfolios such that for all c > 0 we have (7.3) for t E [0,00). Furthermore, a sequence (SOd)dE{1 ,2, ... } of benchmarked portfolios is called a sequence of benchmar ked diversified portfolios if some constants K 1 , K 2 E (0, 00 ) and K 3 E {I, 2, ... } exist independently of d, such that for d E {K3, K 3 + I , ... } each fraction 7rL,t is bounded in the form (7.4) almost surely for all j E {I , 2, ... , d} and t E [0,00). If most benchmarked primary security accounts would have the same driving Wiener process, then it may become difficult to form a benchmarked portfolio with vanishing volatility. To avoid such a situation we assume that the given continuous financial market is regular, which means that for all t E [0,00) and k E {I , 2, ... } there exists a finite adapted stochastic process C = {Ct , t E [O,oo)} such that (~Hk l) , <0 C, (7.5) almost surely. This allows us to prove, similar as in Platen (2005), and Platen and Heath (2006), the following Diversification Theorem: Theorem 7.1. In a regular continuous financial market each sequence of diver- sified portfolios is a sequence of approximate K elly portfolios. Proof: For a sequence of diversified portfolios it follows from (7.4) and (7.5) that --- ## Page 453 424 E. Platen for each d E {l, 2, . . . } one has ~ (t, nj" ai k )' <; t, (~H"I Hkl) , (~H 'k l) , < (K2)2 Ct - d2K1 . Consequently, we have for all c > 0 that }~~ P (t (t 7rL,t CTi,k) 2 ;:::: c) = 0, k=l J=l which proves (7.3). D 8 Conclusion A general financial modeling and pricing framework, the benchmark approach, has been presented, which only assumes the existence of the numeraire portfolio. It turns out that this portfolio coincides with the Kelly portfolio, which is in several ways the "best" performing strictly positive portfolio. It can be used as benchmark in the traditional sense of portfolio optimization but also as numeraire in deriva- tive pricing. Under the benchmark approach, the classical Law of One Price does generally not hold. It has been replaced by the Law of the Minimal Price, accord- ing to which the minimal replicating price process for a given contingent claim is trendless when expressed in units of the Kelly portfolio. By exploiting this fact, the real world pricing concept emerges with the Kelly portfolio as numeraire and the real world probability measure as pricing measure. Real world pricing turns out to be the natural pricing concept for nonhedgable contingent claims in an incomplete market. Certain weak forms of arbitrage may exist under the benchmark approach that are excluded under the classical no-arbitrage approach. However, this does not harm the economic viability of the resulting general modeling framework. It has been shown that diversified portfolios approximate asymptotically in a regular market the Kelly portfolio, as the number of constituents increases. References Bajeux-Besnainou, I. and R. Portait (1997). T he numeraire portfolio: A new perspective on financial theory. The European Journal of Finance, 3, 291- 309. --- ## Page 454 A Benchmark Approach to Investing and Pricing 425 Becherer, D. (2001). The numeraire portfolio for unbounded semimartingales. Finance and Stochastics, 5, 327- 34l. Breiman, L. (1960) . Investment policies for expanding business optimal in a long run sense. Naval Research Logistics Quarterly, 7 (4), 647 - 65l. 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In Proceedings of the 1971 Business and Economics Section of the American Statistical Association, Volume 21, pp.5- 224. --- ## Page 456 29 Growing Wealth with Fixed-Mix Strategies* Michael A. H. Dempster Centre for Financial Research, University of Cambridge, United Kingdom mahd2@cam.ac. uk Igor V. Evstigneev Economics Department, School of Social Sciences, University of Manchester, United Kingdom igor. evstigneev@manchester.ac.uk Klaus Reiner Schenk-Hoppe Leeds University Business School and School of Mathematics, University of Leeds, United Kingdom K.R.Schenk-Hoppe@leeds.ac.uk Abstract This chapter surveys theoretical research on the long-term performance of fixed- mix investment strategies. These self-financing strategies rebalance the portfolio over time so as to keep constant the proportions of wealth invested in various assets. The main result is that wealth can be grown from volatility. Our findings demonstrate the benefits of active portfolio management and the potential of financial engineering. 1 Introduction 427 Investment advice is usually based on some optimality principle such as the maxi- mization of expected utility or the growth rate. The present chapter takes a broader view by studying generic features of an investment style based on constant propor- tions, or fixed-mix, strategies. These self-financing strategies aim to maintain fixed proportions between the value of portfolio positions by trading in the market at 'Financial support by the Finance Market Fund, Norway (Stability of Financial Markets: An Evolutionary Approach) and the National Center of Competence in Research Financial Valua- tion and Risk Management, Switzerland (Behavioural and Evolutionary Finance) is gratefully acknowledged. This chapter was written during KRSH's visit to the Department of Finance and Management Science at the Norwegian School of Economics and Business Administration in June 2009. We are grateful to editors Edward O. Thorp and William T. Ziemba for their many helpful comments. --- ## Page 457 428 M A. H. Dempster, 1. V. Evstigneev and K. R. Schenk-Hoppe specific points in time. It is an active portfolio management that rebalances posi- tions by selling assets, whose portfolio value exceed the given benchmark to finance the purchase of those assets with a too low weight in the portfolio. The significance of constant proportions strategies for investment science was established in Kelly (1956)'s work on information theory and its application to betting markets. A detailed account is provided in Part I of this volume. Kelly's research was inspired by Claude Shannon's lectures on investment problems in which the founder of the mathematical theory of information outlined his pioneering ideas in the field of investment science (though he never published in it); see Cover (1998) for the history of this approach. The optimality property of the Kelly rule, which maximizes the growth rate of capital in the case of independent and identically distributed returns on investment, motivated the research on the log-optimum investment principle, Breiman (1961), Algoet and Cover (1988), MacLean, Ziemba and Blazenko (1992), Hakansson and Ziemba (1995), Browne and Whitt (1996), Cover (1991), Li (1998), and others. Con- stant proportions strategies have been studied in many different frameworks, e.g., Browne (1998), Luenberger (1998), Aurell et al. (2000) and Aurell and Muratore- Ginanneschi (2000), Fernholz (2002), Fernholz and Karatzas (2005), Fernholz et al. (2005), Evstigneev, Hens and Schenk-Hoppe (2009a, 2009b), Kuhn and Luenberger (forthcoming). This approach has proved quite successful in practical finance as witnessed by a considerable body of empirical literature, see Thorp (1971), Perold and Sharpe (1988), Ziemba and Mulvey (1998), Mulvey (2001, 2009), Dries et al. (2002), Dempster et al. (2003), Mulvey et al. (2007), and others as well as Part VI in this volume. In contrast to this literature on investment, we will ignore questions of opti- mality of trading strategies and will not make use of expected utility or related concepts. Our goal is to analyze the performance of arbitrary fixed-mix (constant proportions) strategies in markets with different characteristics. We are only in- terested in 'generic' properties of (not necessarily optimal) fixed-mix strategies and their performance relative to buy-and-hold strategies. l This survey is based on the authors' research, Evstigneev and Schenk-Hoppe (2002) and Dempster, Evstigneev, and Schenk-Hoppe (2003, 2007, 2008) . In these papers, we study the wealth dynamics of investors employing fixed-mix strategies. Very general (and mostly counter-intuitive) results on fixed-mix strategies are ob- tained under the assumption that markets will exhibit some degree of stationarity either of (relative) prices or returns. This assumption of stationarity of asset returns is widely accepted in financial theory (and, thus, the basis for practical investment advice) allowing, as it does, expected exponential price growth and mean reversion, volatility clustering and very general intertemporal dependence, such as long mem- ory effects, of returns. Stationarity of returns for instance is a salient feature of the Cox-Ross-Rubinstein binomial asset pricing model. lOf course it is well-known that optimally-chosen rebalancing strategies perform at least as well as any buy-and-hold portfolio, see Part II in this volume. --- ## Page 458 Growing Wealth with Fixed-Mix Strategies 429 The remainder of this chapter is organized as follows. Section 1.1 invites the readers to test their intuition while Section 1.2 provides a very simple illustrative example. The general theory of constant proportions strategies in stationary mar- kets is covered in Section 2. A generalization to fixed-mix strategies as well as an application to currency market models is given in Section 3. The most intrigu- ing case of stock markets with stationary returns, covered in Section 4, delivers counter-intuitive results that even experienced researchers find puzzling. Section 5 discusses several prominent explanations for volatility-induced growth and provides an interpretation that agrees with our findings. Section 6 concludes. 1.1 A test of your intuition There is barely any 'better' result than one that is (completely) counter-intuitive. But before one is tempted to make such a claim on one's own findings, it seems appropriate to test the audience's intuition. In the following, we ask a few questions to which the reader has to promptly guess the answer. Many scientists have been asked these questions casually in private, at seminars, and at conferences. The following insights were first reported in Dempster, Evstigneev, and Schenk-Hoppe (2007). Non-technically minded readers can skip to the next section which provides the informal discussion of an intriguing example. Stationary markets: puzzles and misconceptions. Consider a market with K as- sets whose price process Pt = (pi , ... ,pf) is ergodic and stationary. (It is assumed that prices are log-integrable.) The assumption of stationarity of asset prices, per- haps after some detrending, seems plausible when modeling currency markets where 'prices' are determined by exchange rates of all the currencies with respect to some selected reference currency. Question 1. Suppose vectors of asset prices Pt = (pi,· .. ,pf) fluctuate randomly, forming a stationary stochastic process (assume even that the vectors Pt are iid (in- dependent identically distributed)). Consider a fixed-mix self-financing investment strategy prescribing rebalancing one's portfolio at each of the dates t = 1,2, ... so as to keep equal investment proportions of wealth in all the assets. What is the tendency of the portfolio value in the long run, as t -> oo? Will the value: (a) decrease; (b) increase; or (c) fluctuate randomly, converging (in some sense) to a stationary process? The audience of our respondents was quite broad and professional, but practi- cally nobody succeeded in guessing the correct answer, which is (b). Among those with a firm view, nearly all selected (c). There were also a couple of respondents who decided to bet on (a). Common intuition suggests that if the market is sta- tionary, then the portfolio value for a constant proportions strategy must converge in one sense or another to a stationary process. The usual intuitive argument in support of this conjecture appeals to the self-financing property. The self-financing --- ## Page 459 430 M A. H. Dempster, 1. V. Evstigneev and K. R. Schenk-Hoppe constraint seems to exclude possibilities of unbounded growth. This argument is also substantiated by the fact that in the deterministic case both the prices and the portfolio value are constant. This way of reasoning makes the answer (c) to the above question more plausible a priori than the others. It might seem surprising that the wrong guess (c) has been put forward even by those who have known about examples of volatility pumping for a long time. The reason for this might lie in the non-traditional character of the setting where not only the asset returns but the prices themselves are stationary. Moreover, the phenomenon of volatility-induced growth is more paradoxical in the case of stationary prices, where growth emerges "from nothing". In the conventional setting of stationary returns, volatility serves as the cause of an acceleration of growth, rather than its emergence from prices with zero growth rates. A potentially promising attempt to understand the correct answer to Question 1 might be to refer to the concept of arbitrage. Getting something from nothing as a result of an arbitrage opportunity seems to be similar to the emergence of growth in a stationary setting where there are no obvious sources for growth. As long as we deal with an infinite time horizon, we would have to consider some kind of asymptotic arbitrage. All known concepts of this kind2 however are much weaker than what we would need in the present context. According to our results, growth is exponentially fast, unbounded wealth is achieved with probability one, and the effect of growth is demonstrated for specific (constant proportions) strategies. None of these properties can be directly deduced from asymptotic arbitrage. Thus, there are no convincing arguments showing that volatility-induced growth in stationary markets can be derived from, or explained by, asymptotic arbitrage over an infinite time horizon. But what can be said about relations between station- arity and arbitrage over finite time intervals? As is known, there are no arbitrage opportunities (over a finite time horizon) if and only if there exists an equivalent martingale measure. A stationary process can be viewed as an 'antipodal concept' to the notion of a martingale. This might lead to the conjecture that in a stationary market arbitrage is a typical situation. Is this true or not? Formally, the question can be stated as follows. Question 2. Suppose vectors of asset prices Pt = (pi , ... ,pf) form a stationary stochastic process (assume even that the vectors Pt are iid) Furthermore, suppose the first asset k = 1 is riskless with constant price pi = 1. The market is frictionless and there are no portfolio constraints (in particular, short selling is allowed). Does this market have arbitrage opportunities over a finite time horizon? An arbitrage opportunity over a fixed time horizon is understood in the conven- tional sense: the existence of a trading strategy that does not require any investment 2E.g. Ross (1976) , Huberman (1982) , Kabanov and Kramkov (1994), and Klein and Schacher mayer (1996). --- ## Page 460 Growing Wealth with Fixed-Mix Strategies 431 Table 1 Net return of the two assets in t he market Asset 1 Asset 2 Heads +50% -55% Tails - 40% + 100% at the initial time, is self-financing, does not incur a loss at the terminal time, and makes a gain with strictly positive probability. Again, the answer to this question is practically never guessed immediately. The correct answer depends on whether the distribution of the price vector of the risky assets is continuous or discrete. For example, if (PT, ... ,pf) takes on a finite number of values, then an arbitrage opportunity exists. But if its distribution is continuous, there are no arbitrage opportunities. For details see Evstigneev and Kapoor (2006). If your intuition led you astray, the remainder of this chapter will help to offer insights into these counter-intuitive results. Even if your intuition led you to the correct answers, you might be interested in the precise formulation of the results and the ideas behind their proofs. 1.2 An illustrative example The potential of constant proportions investment strategies for financial growth can be demonstrated by means of the following example which seems simple enough to be discussed at high-school level. A related example is presented in Dempster, Evstigneev, and Schenk-Hoppe (2008) (see also Luenberger, 1998, Chapter 15). A more demanding illustration, though with a sound empirical background, is given in Ziemba (2008). Two investment opportunities available at every point in time: 0, 1, 2 .... The realized returns of the assets between two points in time are determined by flipping a fair coin (only one!). Think of the outcome (heads/tails) as a economy-wide event that effects the two investments in different ways. The net returns are defined in Table l. Investors who buy and hold the first asset will see their wealth decline (expo- nentially fast) over time. The grow rate is negative: 91 := 0.5 In(l.5) + 0.5 In(0.6) :::::0 - 0.05268 < 0 Investment in only the second asset does not yield any better performance because the growth rate is negative as well and equal to 91: 92 := 0.5 In(0.45) + 0.5 In(2.0) = 91 < 0 Now consider an investor with a constant proportions strategy. Suppose the investor divides his wealth 50 : 50 between the two assets. After each change in the value of the two portfolio positions, the investor rebalances his portfolio by trading assets to restore the 50: 50 split of wealth. The growth rate of his wealth is 0.5 In(0.5 . l.5 + 0.5· 0.45) + 0.5 In(0.5 . 0.6 + 0.5· 2.0) :::::0 +0.1185 > 0 --- ## Page 461 432 M A. H. Dempster, 1. V. Evstigneev and K. R. Schenk-Hoppe 60 50 40 30 20 10 -10 -20 -30 -40 0 250 500 750 1000 Figure 1 Wealth dynamics of the two passive investors (both decreasing) and the active investor with 50: 50 constant proportions strategy (increasing). Initial wealth is 1.0. Logarithmic scale on y-axis. Both passive investors (holding only one asset) will see their wealth halve in less than 14 periods on average. In contrast, the investor following a constant proportions strategies with proportions 50: 50 will, on average, double his wealth more often than every 6th period. A typical realization of the wealth dynamics is depicted in Figure 1. The active investor's choice of the proportions is arbitrary. Indeed, qany con- stant proportions investment strategy holding both assets will generate growth in ex- cess of either buy-and-hold strategy. The 50: 50 proportions create positive growth while strategies with proportions close to 100: 0 (or 0 : 100) entail growth at negative rate but one that is higher than gl = g2. Examples, examples, examples. The above example is just one of many. Indeed most descriptions (and analyses) of the phenomenon of generating financial growth from volatility are restricted to examples involving specialized models. Since the deduction of general principles from examples is an unreliable route if a thorough mathematical analysis is feasible, it might come as no surprise that several terms have been coined to describe this phenomenon and many different explanations for its origins have been put forward. In the remainder of this chapter, we will present mathematical results on the growth-volatility nexus in general (discrete- time) models and attempt to de-mystify its origins and explanations. 1.3 Notation All models discussed in this chapter share the basic setup. The definitions and concepts are listed and discussed here. An investor observes prices and takes actions --- ## Page 462 Growing Wealth with Fixed-Mix Strategies 433 only at particular points in time. These time periods are denoted t = 0, 1, 2, .... The price dynamics are driven by random factors. These factors are described by a stochastic process St with t = 0, ±1, ±2, .... The process takes values in a measurable space S. The realization of the random parameter St corresponds to the state of the world at time t. Denote by P the probability measure induced by the stochastic process St, t = 0, ±1, ±2, . .. , on the space of its paths. There are K :;:. 2 assets traded in the market. Asset prices Pt = (pi, .. . ,pf ) > 0, at the time periods t = 0, 1, 2, ... are described as a sequence of strictly positive random vectors with values in the K -dimensional linear space RK. We will assume that the price vector Pt depends on the history of the process St up to time t: It is supposed, without further mentioning, that all functions of st will be measurable. A market is called stationary if the process St is stationary and the price vectors Pt do not explicitly depend on t, i.e., Pt = p(st ). We will further assume that the process St is ergodic and that E llnpk(st)1 < 00 for all k = 1, ... , K. A stochastic process 6 , 6, . .. is stationary if, for any m = 0, 1, 2, . .. and any measurable function ¢(Xo, Xl , . .. , xm ) , the distribution of the random vari- able ¢t := ¢ (~t, ~t+ l , ... '~t+m ) (t = 0, 1, ... ) does not depend on t. Accord- ing to this definition, all probabilistic characteristics of the process ~t are time- invariant. If ~t is stationary, then for any measurable function ¢ for which E I¢ (~t , ~t+l' ... '~t+m) 1 < 00 , the averages ¢ l + ... + ¢t t (1) converge almost surely (a.s.) as t --+ 00 by Birkhofi"'s ergodic theorem - see, e.g. , Billingsley (1965). If the limit of all averages of the form (1) is non-random (equal to a constant a.s.) , then the process ~t is called ergodic. In this case, the above limit is equal a.s. to the expectation E ¢t, which does not depend on t by virtue of stationarity of ~t. The exponential growth rate of a stochastic process 6 ,6 , ... with ~t > ° is defined by 1 lim - ln~t t ....,oo t (2) provided the limit exists. Suppose the process ~ is stationary and ergodic with E In ~m < 00, then its growth rate zero because r lln ~t --+ ° a.s. by Proposition 4.1.3 in Arnold (1998). One also has 1 In 6 + ... + In ~t tln(6 · ... ·~t) = t --+ Eln~m=const.(a. s .) (3) This property is frequently used below. --- ## Page 463 434 M A. H. Dempster, 1. V. Evstigneev and K. R. Schenk-Hoppe At each time period t, the investor chooses a portfolio ht (st) = (hi( st), ... , hr (st)) with a non-negative number of units of asset k, h~(st) 2: O. Short selling of assets is ruled out in our model by this assumption of non-negativity. A sequence ht(st) , t = 0,1,2, ... , of portfolios is called a trading strategy. The value of a portfolio ht at time t is Ptht = LkP~h~. A trading strategy H = (ho , h1 , ... ) with initial wealth wo > 0 is self-financing if Po ho = Wo and (4) The inequalities in (4) are supposed to hold almost surely with respect to the probability measure P. The market value of the portfolio after trade does not exceed the vale of the yesterday's portfolio at current prices. This budget constraint restricts the investor's choice. A balanced trading strategy H is of the form (5) with a scalar-valued function 'YO > 0 and a vector function h(.) 2: O. If t = 0, we assume that ho(sO) = h(sO). It is assumed that In 'Y(st) and In Ih(st) 1 are integrable with respect to the measure P , i.e., El ln 'Y(st)1 and El lnlh(st)11 are finite. Define Ih l = Lk Ihkl for a vector h = (hk). These strategies are called balanced because they are of balanced growth: (5) implies that all proportions between the amounts of different assets in the portfolio h~ (st) h~ (st) hj (st) hk(st) , j-=!=k (6) form stationary stochastic processes. The random growth rate of the amount of each asset k = 1, . . . , K , in the portfolio hk(st) hk(st) h~_: (st-l) = 'Y(st) hk(st-l) is a stationary process. A balanced strategy is self-financing (4) if and only if (7) Stationarity implies that if (7) holds for some t, it holds for all t. Suppose the vector of asset prices is stationary. Then every non-negative vector function h(st) with Elln Ih(st) 11 < 00 defines a self-financing balanced strategy (5) by p(st) h(st-l) 'Y( st):= p( st) h( st) since Ellnpk(st)1 < 00, and relations (4) and (7) hold as equalities. (8) --- ## Page 464 Growing Wealth with Fixed-Mix Strategies 435 Balanced trading strategies have the property that the growth rate of wealth of any investor employing it is completely determined by the expected value of "(. Proposition 1 in Evstigneev and Schenk-Hoppe (2002) states that for any balanced trading strategy (5) lim ~ In(p( st) ht( st)) = lim ~ In I ht (st) I = E In "(( so) (a.s.) (9) t->oo t t->oo t This result shows strict positivity of E In "(( sO) == E In "(( st) implies exponential growth of wealth, i.e., p(st) ht(st) ---> 00 a.s. exponentially fast. 2 Asset Markets with Stationary Prices We first discuss the simplest model in which one can analyze generic properties of fixed-mix strategies: a market in which prices are stationary processes. It turns out that any constant proportions strategy produces growth, though in this stationary market the growth rate of each asset price is zero and, therefore, buy-and-hold strategies do not yield positive growth. This results might seem, at the first glance, counter-intuitive. This section is based on Evstigneev and Schenk-Hoppe (2002). All notations are introduced in Section 1.3. 2.1 The model A constant proportions strategy in a market with K assets is characterized by a vector A = (Al' ... , AK) in the set ~ = {(Al"'" AK) E RK : Ak > 0, t Ak = I} k=l (10) To avoid pathological cases, we assume strict positivity of all components. Propor- tional investment rules of this kind are sometimes termed completely mixed. The trading strategy ht is a constant proportions strategy if (11) for all t = 1, 2, ... and k = 1, ... ,K. The investor rebalances the portfolio in every period in time by investing the constant share Ak of the wealth Pt ht- 1 into the kth asset. The investor's wealth at the beginning of period t is determined by evaluating the portfolio ht - 1 (bought in the previous period) at the current price system p~. We assume that all the coordinates of ht are non-negative: short sales are ruled out. Constant proportions strategies are self-financing because (11) implies Pt ht = Pt ht- 1 Given a vector A, the strategy is uniquely defined by its initial portfolio ho (sO). Recursive application of (11) determines ht(st) for every t = 1, 2, .... --- ## Page 465 436 M A. H. Dempster, 1. V. Evstigneev and K. R. Schenk-Hoppe In the study of the growth rate of wealth, the initial portfolio does not matter, see Evstigneev and Schenk-Hoppe (2002, pp. 568). Indeed if there are two constant proportions strategies ht and ht both generated by the same A E ~ but with different (non-zero) initial portfolios, then there exist~ , C > 0 such that ~h~ ::::; h~ ::::; ch~ for all k and all t ~ 1. Obviously, lim C 1 ln Ihtl > 0 a.s. if and only if lim C 1 ln Ihtl > 0 a.s. (and exponential growth is at the same speed). 2.2 The growth rate Constant proportions strategies generate balanced portfolios. From the representa- tion (5), the growth rate of the investors wealth is obtained as the expected value E In 1'( sO). The notion of balanced portfolios goes back, though in a somewhat dif- ferent form, to Radner (1971)'s study of stochastic generalizations of the von Neu- mann economic growth model. Arnold, Evstigneev, and Gundlach (1999) study this concept of balanced paths in a general setting. The application to financial problems is recent. The central result on the growth rate of constant proportions strategies in mar- kets with stationary asset prices is obtained by showing that they can be written as balanced strategies and that E In 1'( sO) ~ O. To have a strictly positive growth rate, the price process p( st) must be non-degenerate. We impose the following assumption: (A) With strictly positive probability, the random variable pk(st)jpk(st-l ) is not constant with respect to k = 1,2, . . . ,K , i.e., there exist m and n (that might depend on st) for which (12) All results in this Handbook section will be proved under this assumption, though it will be convenient to repeat this condition in different disguises- each best suited for the particular application. Under assumption (A), one has the following result, see Evstigneev and Schenk- Hoppe (2002, Theorem 1). Theorem 1. Fix any A = (AI"'" AK) E ~ , and let Wo be a strictly positive number. Then there exists a vector function ho(sO) ~ 0 such that the constant proportions strategy ht generated by A and ho is a balanced strategy with initial wealth wo, and we have . 1 . 1 hm -lnptht = hm -In Ihtl > 0 (a.s.) t->oo t t->oo t (13) The construction of the vector function ho(sO) ~ 0 is as follows. Suppose the proportions A E ~ and initial wealth Wo > 0 are given. To show that ht is a balanced --- ## Page 466 Growing Wealth with Fixed-Mix Strategies 437 trading strategy one needs to define a vector function he) :::: 0 and a scalar-valued function /,(.) > 0 such that ht coincides with the strategy defined in (5) (and that ht(sO ) = h(sO)) . Define - t ( Al Wo AK Wo ) h(s ) = pl(st) " ' " pK (st) (14) and t _ p(st) h(st-l ) [_ K pk(st) 1 /,(s)- - LAk k( t-l) Wo k= l P s (15) This is a balanced trading strategy which coincides with ht recursively defined by (11). Strictly positive growth (E In /,( st) > 0) holds because Jensen's inequality (ap- plied to the probability measure Ak on the set {I, ... , K} ) implies that K pk(st ) K pk(st) In L Ak pk (st-l ) :::: L Ak In pk(st- l) k=l k=l (16) with strict inequality on a set of positive probability by assumption (12), which ensures t K pk(st) _ E ln /,(s) > LAkE ln k(st-l) - 0 k=l P Strict positivity of E In /,( st) means that wealth tends to infinity at an expo- nential rate. If the non-degeneracy condition (A) is not satisfied, the market is essentially deterministic and all prices are constant. In this case, all strategies give zero growth. Indeed, (12) is a very weak requirement that is satisfied in virtually every market. The result presented in Theorem 1 holds under (sufficiently) small proportional transaction costs, see Evstigneev and Schenk-Hoppe (2002, Theorem 2). 2.3 Interpretation Our analysis shows that constant proportions strategies provide growth in a market in which buy-and-hold strategies do not deliver any. The intuition behind this result is that constant proportions strategies 'exploit ' the persistent fluctuation of prices. When keeping a fixed fraction of wealth invested in each asset , a change in prices leads the investor to sell those assets that are expensive relative to the other assets and to purchase relatively cheap assets. The stationarity of prices implies that this portfolio rule yields a strictly positive expected rate of growth, despite the fact that each asset price has growth rate zero. The notion of an asset being 'cheap' resp. 'expensive' makes sense when prices are stationary. If the current price is below the expected value, then the tomorrow's price has a higher-than-average chance of being larger than today's price. This asset is cheap today. In other words, if --- ## Page 467 438 M A. H. Dempster, 1. V. Evstigneev and K. R. Schenk-Hoppe prices are stationary, there is reversion to the median (which is also true for price ratios). Rebalancing strategies, on average, buy low and sell high. In the model with stationary returns (rather than prices), accepting this interpretation would be falling victim to the gambler's fallacy, see Section 4. The result highlights the benefit of financial engineering. The implementation of a constant proportions strategy requires to invest actively in the available assets by rebalancing at discrete time periods - the growth is financially engineered by making use of Jensen's inequality. 3 Fixed-Mix Strategies in Stationary Markets The model with constant proportions strategies can be generalized to accommodate the transfer of fractions between different portfolio positions. An application to currency markets is provided. In this model, the exchange rates fluctuate randomly in time as stationary stochastic processes. This aspect of our analysis is inspired, in particular, by recent work of Kabanov (1999) and Kabanov and Stricker (2001). This section follows Dempster, Evstigneev, and Schenk-Hoppe (2003). Again, the main result holds under small proportional transaction costs. 3.1 The model A fixed-mix strategy is determined by a (non-random) matrix Akj, k, j = 1, . . . ,K, such that K Akj > 0 , L Akj = 1 (17) k=l In each time period, this strategy prescribes the transfer of a fixed share Akj > 0 of the jth position of the portfolio to the kth position (k, j E {I, ... , K}). The dynamics of the wealth invested in the portfolio positions are given by K kk_ ,\", jj Pt ht - ~ AkjPt ht-1 (18) j=l For any matrix Akj satisfying (17)), a strategy H is called a fixed-mix strategy associated with the matrix A = (Akj), if (18) holds for all k, t and st. Any fixed-mix strategy is self-financing: Ptht = Ptht-1. As in the previous section, we are interested in the asymptotic behavior of the portfolio ht (and its market value) of a trader employing a fixed-mix investment rule. In the deterministic case, the analysis of the long-term dynamics is simple. The price Pt = P = (pI, ... ,pk) > 0 is a constant vector. The corresponding process ht defined by (18) is deterministic and will always converge to a steady state. This claim follows from results on positive matrices, Kemeny and Snell (1960). This result, combined with the assumption of stationarity, might seem to rule out --- ## Page 468 Growing Wealth with Fixed-Mix Strategies 439 unbounded growth and could lead to the (wrong) conjecture of the convergence of ht to a stationary distribution in the stochastic case. The case studied in Section 2 corresponds to the situation in which Akj does not depend on j: Akj = Ak with Ak > 0 , and Al + ... + AK = 1 because (18) reduces to (11). 3.2 Currency markets (19) The foreign exchange market is arguably the real-world example closest to a market in which prices are stationary. We show to formulate the wealth dynamics of a currency trader with a fixed-mix strategy in the above model. Currencies k = 1,2, ... , K are traded in a frictionless market. The exchange k" " k" rates 7r/ = 7rkl(st) > 0 fluctuate randomly in time. The real number 7r/ denotes the amount of currency k which can be purchased by selling one unit of currency j at time t. We assume absence of arbitrage at each point in time t, i.e.z the exchange rates must satisfy ~kj _ ~km ~mj Ht - Ht Ht (20) for all k, m and j. Assume the trader follows any fixed-mix strategy (17) by dividing the holdings hLl 2:: 0 of currency j purchased at time t-l according to the proportions Akj > 0, k = 1, ... , K at time t. The amount AkjhLl is exchanged into currency k. After execution of all these transactions, the amount of currency k obtained at time t is equal to (21) The dynamics (21) can be written in the form (18). Take currency 1 as a numeraire and define Pk _ ~lk t - Ht k"k "" k"" The relation (20) implies 7r/ = l/7ri and 7ril = 1. Therefore, 7r/ = pUp~. Multiplying (21) by 7rik and using these relations, yields the formulation (18) of the wealth dynamics. 3.3 The growth rate The analysis of the long-term growth of an investor following a fixed-mix strategy is conceptually identical to that of constant proportions strategies. First, one shows that there is an initial portfolio such that the fixed-mix strategy leads to a bal- anced portfolio. Then, one proves that the growth rate is independent of the initial --- ## Page 469 440 M A. H. Dempster, I. V Evstigneev and K. R. Schenk-Hoppe portfolio. Finally, one shows that this growth rate is strictly positive under a non- degeneracy condition. The mathematical tools required in the proof however are very different. The case of fixed-mix strategies requires considerably more advanced methods. As in Section 2 we assume that the market is stationary and the state of the world St is ergodic. The price process Pt = p(st) satisfies Ellnpk(st)l < 00 for all k = 1, ... ,K. The growth rate of each asset price is zero because C 1 lnpt --+ 0 a.s. and, therefore, buy-and-hold strategies do not yield positive growth. We assume non-degeneracy of the price process p(st): (B) The vector 15(st) = (151 (st), ... ,15K (st)) of normalized prices _ t pj(st) jY(s):= Lmpm(st) j E {l,,,.,K} is not constant a.s. with respect to st. This assumption says that there is no constant vector c for which 15(st) = c almost surely. By virtue of stationarity of (St), condition (B) holds for all t if is satisfied for some t. If St is ergodic, the condition (B) is equivalent to the following requirement (Dempster, Evstigneev, and Schenk-Hoppe, 2003, pp. 271): With positive probability, the ratios pk(st)jpk(st-1) are not constant with respect to k. This is condition (A). The main result on the performance of an investor employing any fixed-mix strategy is as follows, Dempster, Evstigneev, and Schenk-Hoppe (2003, Theorem 1). Theorem 2. Let ht(st), t ;:::: 0, be a fixed-mix strategy associated with the matrix A = (Akj) satisfying (17) . For each k E {1,2 , . . . ,K}, the limit 1 lim -lnh~ t ..... oo t (22) exists and is strictly positive almost surely. Furthermore, this limit does not depend on k, and . 1 k . 1 hm -In ht = hm -lnptht > 0 (a.s.) t ..... oo t t ..... oo t (23) The result ensures that the wealth of a fixed-mix investor tends to infinity at an exponential rate. All portfolio positions and the investor's wealth grow at the same positive exponential rate. This finding generalizes the results in Section 2 to the general case of fixed-mix investment strategies. The proof of Theorem 2 rests on the observation that one can define a balanced fixed-mix strategy for any given matrix A satisfying (17). Recall that a trading strategy H is balanced if ht(st) = " (Sl) . . . ,,(st) h(st) (a.s.) for t ;:::: l. For t = 0, we assume ho(sO) = h(sO). The function ,,(.) > 0 is scalar-valued and h(·) > 0 is a vector function such that Elln ,,(st) I < 00 and Ih(st) I = l. The norm Ihl of a vector h = (hk) is defined as Lk lhkl. The assumption lh(st) l = 1, which was not imposed in (5), will be satisfied automatically in the present case. --- ## Page 470 Growing Wealth with Fixed-Mix Strategies 441 For balanced strategies, all the proportions between the amounts of different assets in the portfolio and the random growth rate of the amount of each asset in the portfolio are stationary stochastic processes. To show existence of a balanced fixed-mix strategy associated with the matrix A = (Akj), one needs to show there are appropriate functions /'(.) and hC). Indeed, Theorem 2 in Dempster, Evstigneev, and Schenk-Hoppe (2003) ensures that for each matrix A satisfying (17), there is a unique balanced fixed-mix strategy with I h( st) I = l. The construction can be sketched as follows. Denote by At = A(st) = (Akj(st)) the positive random K x K matrix defined by t pj(st) Akj(S ) = Akj pk(st) (24) We have Elln Akj (st) I < 00. With this definition, the fixed-mix strategy H can be represented as (25) Stationarity of (St) implies that functions /'(.) and xC) (satisfying Elln/,(st)1 < 00 and Ix( st) I = 1) generate a balanced A-strategy if and only if /,(st)x(st) = A(st)X(st-1) (a.s.) (26) The existence of a solution to this equation follows from a stochastic version of the Perron- Frobenius theorem, Theorem A.l in Dempster, Evstigneev, and Schenk- Hoppe (2003). The problem (26) cannot be solved by applying the conventional Perron- Frobenius theorem because the vector x(st) on the left-hand side does not coincide with the vector X(st-1) on the right-hand side as the function x(st) is obtained by 'time-shifting' x( st-1) . The independence of the growth rate from the initial portfolio is proved by couching the one-step forward portfolio between multiples of the portfolio X(Sl), analogous to the outline in Section 2. All have the same growth rate. Finally, the strict positivity of the fixed-mix strategies' growth rate is asserted by proving that Eln /,(st) > O. This can be seen as follows. Denote by h(·),x(·)) the balanced strategy corresponding to the fixed mix strategy A. The relation (26) implies K pj (st) . 'V(st)xk (st) = ~ Ak· --Xl (st-1) k E {I K} I ~ lpk(st) , ... , 1=1 (27) The Perron- Frobenius theorem yields existence of a vector r = (r1, . . . ,rK) > 0 with K rk = LAkjrj k E {I, . . . ,K} j=l Put f3kj = r;;lAkjrj. One finds that t K pJ(st) pJ(st-1)xj(st-1)rk /,(s ) = L f3kj pj(st-1) pk(st)xk(st)r k E {I, ... , K} (28) 1=1 1 --- ## Page 471 442 M A. H. Dempster, 1. V. Evstigneev and K. R. Schenk-Hoppe From this representation of l'(st) one can conclude (using Jensen's inequality and several other arguments, see Dempster, Evstigneev and Schenk-Hoppe, 2003, Sec- tion 3) that Elnl'(st) > 0 under assumption (B). 3.4 Price processes with trend The concept of a stationary market, where asset prices Pt fluctuate as stationary stochastic processes, is an idealization. The assumption that only the relative pro- portions pi / p~ are stationary seems more realistic. Following Dempster, Evstigneev, and Schenk-Hoppe (2003), let us assume that the prices are of the form Pt = ~tfh where Pt = p(st) is a process satisfying the assumptions we previously imposed on Pt , and ~t = ~t(st) > 0 is any sequence of strictly positive random variables. The factors ~t represent the dynamics of a price index. The normalized prices Pt are free of this trend. The above analysis can directly be applied to this more general price process. The portfolio value Ptht satisfies 111 -lnptht = -ln~t + -lnptht t t t The growth rate of Ptht is determined by that of ~t and Ptht. Under assumption (B), the process Ptht grows exponentially fast almost surely. Consequently, if the price index ~t grows at an exogenous exponential rate r, then the investor's wealth Ptht will grow almost surely at a rate r' strictly greater than r. 4 Stock Markets with Stationary Returns The observation that rebalancing a portfolio by following any constant proportions or, more generally, any fixed-mix strategy leads to a strictly higher growth rate of wealth than any buy-and-hold portfolio has been confirmed in markets with stationary prices. After seeing the results and following the intuition that a reversion to the mean is a powerful source of capital growth, one might wonder about the case in which returns (rather than prices) are stationary processes. Indeed, as explained in Section 1.1, there is no general agreement that the previous results hold for stationary returns. This is quite sensible because the any notion of cheap (or expensive) assets based on the argument that reversion to the long run trend renders an asset cheap after a series of low returns would mean to fall victim to the gambler's fallacy. This lack of a simple intuition for the wealth dynamics of constant proportions strategies in markets where returns (not prices) are stationary demands a thorough analysis of this case. How can one rest without having gained an understanding of this problem? The presentation follows Dempster, Evstigneev, --- ## Page 472 Growing Wealth with Fixed-Mix Strategies 443 and Schenk-Hoppe (2007), which also contains a proof of the main result under small (proportional) transaction costs. 4.1 The model Denote the (gross) return on asset k between time t - 1 and t by k Rk.- ~ k 2 K t·- k = 1, , . . . , ,t2:1 Pt-l (29) Let Rt := (R}, ... ,Rn. We impose the assumption: (R) The vector stochastic process Rt , t = 1, 2, . . . , is stationary and ergodic. The expected values El ln R~ I, k = 1, 2, . . . , K , are finite. The typical example of a stationary ergodic process is a sequence of iid (inde- pendent identically distributed) random variables. To avoid misunderstandings, we emphasize that Brownian motion and a random walk are not stationary. The asset price at time t and the initial price are related as P~ = p~Rt .... . R~, where the random sequence R~ is stationary by (R ). This assumption on the structure of the price process is a fundamental hypothesis in finance. Moreover, it is quite often assumed that the random variables R~, t = 1, 2, ... are independent, i.e., the price process p~ forms a geometric random walk. This postulate, which is much stronger than the hypothesis of stationarity of R~ , lies at the heart of the classical theory of asset pricing (Black, Scholes, Merton), see e.g. Luenberger (1998). Birkhoff's ergodic theorem implies lIt lim -lnp~= lim - "'lnR~ = ElnR~ (a.s.) t-->oo t t-->oo t ~ n=l (30) for each k = 1, 2, . . . , K. This means that the price of each asset k has almost surely a well-defined and finite (asymptotic, exponential) growth rate, which turns out to be equal a.s. to the expectation Pk := E In R~, the drift of this asset's price. The drift can be positive, zero or negative. It does not depend on t in view of the stationarity of Rt . Consider an investor following a constant proportions strategy with proportions (10). Fix any vector ,\ E ~. Given an initial portfolio ho > 0, the (self-financing) trading strategy H is defined recursively by h~ = '\kPt ht-dp~ k = 1, 2, ... , K , t 2: 1 (31) This definition is equivalent to (11). Our aim is to study the asymptotic behavior of the portfolio value Vi = Ptht as t --> 00, i.e. the limit limt-->oo rlln(Vi) describing the (exponential) growth rate of the strategy. --- ## Page 473 444 M A. H. Dempster, 1. V. Evstigneev and K. R. Schenk-Hoppe 4.2 The growth rate We impose the following non-degeneracy condition: (C) With strictly positive probability, pk(st) pk (st-l) pf(st) i- p~~ (st-l) for some 1 :s; k, m :s; K and t ~ 1 This condition is a very mild assumption on the existence of volatility of the price process. Condition (C) does not hold if and only if the relative prices of the assets are constant in time (a.s.), i.e. , if with probability one, the ratio p~ /pr;:' of the prices of any two assets k and m does not depend on t. This condition is also equivalent to (B), see Dempster, Evstigneev, and Schenk-Hoppe (2003, pp. 271). The condition (C) on asset prices has an equivalent formulation for asset returns: (D) For some t ~ 1 (and hence, by virtue of stationarity, for each t ~ 1), the probability P{R~ i- Rr;:' for some 1 :S; k,m:S; K} is strictly positive. Equivalence can bee seen as follows. Note that p~ /pr;:' i- ptl/Pr'-l if and only if p~ /ptl i- pr;:' /Pr'-l' i.e. R~ i- Rr;:'. Denote by 6t the random variable that is equal to 1 if the event {R~ i- Rr;:' for some 1 :s; k, m :s; K} occurs and 0 otherwise. Condition (C) means that P{maXt;:'l 6t = I} > 0, while (D) states that, for some t (and hence for each t), P{ 6t = I} > O. The latter property is equivalent to the former because {maXt;:'l 6t = I} = U~l {6t = I}. We can now state the main result on the growth of wealth of investors following constant proportions strategies. Theorem 3. Fix any A E 6.. (i) The growth rate of the constant proportions strategy is almost surely equal to a constant which is strictly greater than Lk AkPk, where Pk is the drift of the price of asset k. (ii) Suppose all the assets have the same drift (therefore, almost surely the same asymptotic growth rate), i.e., ElnR~ = P fo r each k = 1, ... ,K with some real number p. Then the growth rate of the constant proportions strategy is almost surely strictly greater than the growth rate of each individual asset. In Theorem 3, assertion (ii) immediately follows from (i). The result (ii) shows that any completely mixed constant proportions strategy grows at a rate strictly greater than p, the growth rate of each particular asset. The growth of the investor's wealth is only driven by the volatility of the price process. This result seems to contradict conventional finance theory which usually regards the volatility of asset prices as an impediment to financial growth. In the present context, volatility serves as an endogenous source of its acceleration. Theorem 3 (ii) asserts the validity of --- ## Page 474 Growing Wealth with Fixed-Mix Strategies 445 the conclusion drawn in the illustrative example discussed in Section 1.2. Indeed, the constant proportions strategy (.5, .5) (as well as any other completely mixed constant proportions strategy) yields a higher growth rate than the price of each asset. The first part of Theorem 3 places a floor under the constant proportions strat- egy's growth rate. When asset prices grow at different rates, there is no general result that constant proportions strategy grow faster than any (completely diversi- fied) buy-and-hold strategy. The latter grows at rate maxk Pk which is not strictly dominated by L k AkPk · The growth-optimal constant proportions strategy however will always grow at least as fast as any buy-and-hold in the model. The proof of the above result is surprisingly simple and can be presented with success to an audience with knowledge of only elementary probability and calculus: Fix any vector A E .6... The random wealth dynamics of the corresponding constant proportions strategy (31) are given by KKk K k Vi = Ptht = LP~htl = L ~t pt1ht l = L ~t AkPt-lht-l k=l k=l Pt-l k=l Pt- l = [t,R~AklVi- l = (Rt A)Vi-l (32) for each t ::::: 1. This implies (33) for all t ::::: 1. The ergodic theorem ensures l I t lim -In Vi = lim - '" In(Rn A) = E In(Rt A) (a.s.) t-->oo t t--> oo t ~ n=l (34) It remains to show that E ln(Rt A) > Lf:=l AkPk under the condition (D). In- deed, Jensen's inequality and (D ) ensure the relation K K InLR~ A k > LA k(lnR~) k=l k=l with strictly positive probability, while the non-strict inequality holds always. Consequently, K K E In(Rt A) > L AkE(ln R~) = L AkPk (35) k=l k=l This proves Theorem 3. --- ## Page 475 446 M A. H. Dempster, 1. V. Evstigneev and K. R. Schenk-Hoppe 4.3 Interpretation The rigorous proof of the presence of volatility-induced growth in markets where asset returns are stationary leaves open the question on the intuition behind this result. It seems any explanation one can give is nothing but a repetition of the mathe- matical reasoning behind this result: If Ri, ... ,Rf are the random returns of the K assets, then the asymptotic growth rates of these assets are E In R~, while the asymptotic growth rate of a constant proportions strategy is Eln(Lk AkR~), which is strictly greater than Lk AkEln(R~) by Jensen's inequality because the logarith- mic function is strictly concave. The proof in Section 2 confirms that Jensen's inequality is the central tool used in the proof. Any explanation not based on this fact would be flawed. It is well-known that a linear combination of assets can produce non-linearity in a portfolio's characteristics. Indeed, this feature drives mean-variance portfolio choice, cf. Luenberger (1998). In the present model, the rebalancing of the portfolio so as to maintain constant proportions causes a non-linear effect in the portfolio's growth rate with the feature that Eln(Lk AkR~) - Lk AkEln(R~) > 0 for any A = (AI, ... , AK) E ~ provided assumption (C) holds. Other attempts to appeal to intuition are discusses (and rejected) in the next section. An interesting problem is the question under which (general) assumptions Eln(Lk AkR~) > 0 even if maxk Pk :::; 0 (or, less restrictive, mink Pk :::; 0). The example in Section 1.2 possesses this property. In that example, PI = P2 < 0, but E In(AIRi + A2R;) > 0 for the constant proportions strategy A = (.5, .5). But if A is close to (1,0) or (0,1) the growth rate of wealth becomes negative as well. A partial answer to this 'black swan' question is provided in Section 5. We discuss a case in which the addition of a slower growing asset always enhances the growth of a con- stant proportions strategy and raises the growth rate over that of any buy-and-hold strategy. 5 Myths and Misconceptions The phenomenon of volatility-induced growth is simply counter-intuitive as con- firmed by our test in Section 1.1. Providing a common-sense explanation for the mathematical results can therefore not be a simple task. In this section, we discuss some of the more prominent suggestions for intuition put forward in the literature. 5.1 Volatility pumping The term 'volatility pumping' appears to have been first used by Luenberger (1998). His suggestion is that constant proportions strategies force the investor to 'buy low --- ## Page 476 Growing Wealth with Fixed-Mix Strategies 447 and sell high'-the common sense dictum of stock market trading. Those assets whose prices have risen from the last rebalance date will be overweighted in the portfolio, and their holdings must be reduced to meet the required proportions and to be replaced in part by assets whose prices have fallen and whose holdings must therefore be increased. This behavior leads to growth if asset returns exhibit some stationarity properties. In Section 2.3, we argued that this explanation is correct for stationary prices. In the present case, accepting this explanation would mean to fall victim to the gambler's fallacy. This explanation is not valid because when the price follows a geometric random walk, the set of its values is generally unbounded and for every value there is a smaller as well as a larger value. Suppose returns are iid, then 'high' and 'low' do not have any meaning. Reversion to the long run trend (as postulated by the ergodic theorem) is not an explanation either: arguing that the longer the run of black numbers, the higher the odds of red numbers at the next spin of the roulette wheel is the gambler's fallacy. When returns are determined by the flip of a coin, an asset's upside and downside potential does not change over time. Such an asset is not cheap or expensive at any point in time. 5.2 The importance of constancy A more substantial lacuna in the above reasoning is that it does not reflect the assumption of constancy of investment proportions. This leads to the question: what will happen if the portfolio is rebalanced so as to sell all those assets that gain value and buy only those ones which lose it? (This should lead to an even higher growth rate - provided volatility could be 'pumped'.) Assume, for example, that there are two assets, the price PE of the first (riskless) is always 1, and the price P; of the second (risky) follows a geometric random walk, so that the gross return on it can be either 2 or 1/ 2 with equal probabilities. Suppose the investor sells the second asset and invests all wealth in the first if the price PZ goes up and performs the converse operation, betting all wealth on the risky asset, if P; goes down. Then the sequence At = (Al,t, A2,t) of the vectors of investment proportions will be iid with values (0, 1) and (1 ,0) taken on with equal probabilities. Furthermore, At- l will be independent of Rt . By virtue of (34), the growth rate of the portfolio value for this strategy is equal to Eln(RtAt-d = [In(O· 1 + 1 . 2)+ In(O· 1 + 1 . ~) + In(l . 1 + O· 2)+ In(l . 1 + O· ~)l/ 4 = 0, which is the same as the growth rate of each of the two assets k = 1, 2. But this growth rate is strictly lower than that of any completely mixed constant proportions strategy. 5.3 Energy-interpretation of volatility Our results highlight the importance of volatility because fixed-mix strategies do not produce growth beyond buy-and-hold portfolios in any of the above models when prices or returns are constant. It is tempting to see volatility as a source of energy that can be tapped to generate growth. --- ## Page 477 448 M A. H. Dempster, 1. V. Evstigneev and K. R. Schenk-Hoppe Indeed, Luenberger (1998) presents an intriguing continuous-time example which supports this view. (A closely related observation is made in Fernholz and Shay (1982).) There are K assets whose prices follow independent (but identical) geo- metric Brownian motions. The constant proportions strategy with equal weights on all assets has a higher growth rate than any asset price. The remarkable feature however is that increasing the number of assets K leads to a higher growth rate and - at the same time- to a reduction in the volatility of the return. It would seem as if fixed-mix strategies turn the classical return-volatility tradeoff on its head. The framework for this example is the well-known continuous-time model de- veloped by Merton and others, in which the price processes Sf, t :::: 0, of two assets k = 1, 2 are supposed to be solutions to the stochastic differential equa- tions dSf / Sf = J.Lkdt + CYkdWtk, where the Wtk are independent (standard) Wiener processes and S8 = 1. As is well-known, these equations admit explicit solutions Sf = exp[J.Lkt - (cyV2)t+CYkWn Given some B E (0,1), the value vt of the constant proportions portfolio prescribing investing the proportions Band 1- B of wealth into assets k = 1, 2 is the solution to the equation dvt/vt = [BJ.Ll + (1 - B)J.L2]dt + Bcy1dWl + (1 - B)CY2dW? Equivalently, vt can be represented as the solution to the equation dvt /vt = pdt + iTdWt, where p := BJ.Ll + (1- B)J.L2, iT2 := (Bcyd2 + [(1 - B)CY2j2 and Wt is a standard Wiener process. Thus, vt = exp[pt - (iT2/2)t + iTWt], and so the growth rate and the volatility of the portfolio value process Vt are given by p - (iT2/2) and iT. In particular, if J.Ll = J.L2 = J.L and CYl = CY2 = CY, then the growth rate and the volatility of vt are equal to (36) while for each individual asset , the growth rate and the volatility are J.L - (cy2/2) and CY, respectively. Thus, in this example, the use of a constant proportions strategy prescribing investing in a mixture of two assets leads (due to diversification) to an increase of the growth rate and to a simultaneous decrease of the volatility. When looking at the expressions in (36) , the temptation arises even to say that the volatility reduction is the cause of volatility-induced growth. Indeed, the growth rate J.L - (iT2/2) is greater than the growth rate J.L - (cy2/2) because iT < CY. This suggests speculation along the following lines. Volatility is something like energy. When constructing a mixed portfolio, it converts into growth and therefore decreases. The greater the volatility reduction, the higher the growth acceleration. 5.4 A counter-example Do the observations made in the previous section have any grounds in the general case, or do they have a justification only in the above example? To formalize this question and answer it, let us return to the discrete-time framework. Suppose there --- ## Page 478 Growing Wealth with Fixed-Mix Strategies 449 are two assets with iid vectors of returns R t = (R}, R;). Let (~, 'f}) := (Rt, Rr) and assume, to avoid technicalities, that the random vector (~, 'f}) takes on a finite number of values and is strictly positive. The value vt of the portfolio generated by a fixed-mix strategy with proportions x and 1 - x (0 < x < 1) is computed according to the formula t vt = V1 II[xR~ + (1- x)R;], t ~ 2 n=2 see (33). The growth rate of this process and its volatility are given, respectively, by the expectation E In (x and the standard deviationv'Var In (x of the random variable In (x, where (x := x~ + (1- x)'f}. We know from the above analysis that the growth rate increases when mixing assets with the same growth rate. What can be said about volatility? Specifically, we consider the following question. Question 3. (a) Suppose Varln~ = Var In 'f}. Is it true that Varln[x~ + (1- x)'f}] ::; Varln~ when x E (0, 1)? (b) More generally, is it true that Varln [x~ + (1 - x) 'f}] ::; max(Varln~, Var In 'f}) for x E (0,1)? Query (b) asks whether the logarithmic variance is a quasi-convex functional. Questions ( a) and (b) can also be stated for volatility defined as the square root of logarithmic variance. They will have the same answers because the square root is a strictly monotone function. Positive answers to these questions would substantiate the above conjecture of volatility reduction - negative, refute it. In general (without additional assumptions on ~ and 'f}) , the above questions 3(a) and 3(b) have negative answers. To show this, consider two iid random variables U and V with values 1 and a > 0 realized with equal probabilities. Consider the function f(y) := Varln [yU + (1- y)V], y E [0, 1] (37) By evaluating the first and the second derivatives of this function at y = 1/ 2, one can show the following. There exist some numbers 0 < a_ < 1 and a+ > 1 such that the function f(y) attains its minimum at the point y = 1/ 2 when a belongs to the closed interval [a_, a+] and it has a local maximum (!) at y = 1/ 2 when a does not belong to this interval. The numbers a_ and a+ are given by a± = 2e4 - 1 ± J(2e4 - 1)2 - 1 where a_ ~ 0.0046 and a+ ~ 216.388. If a E [a_, a+], the function f(y) is convex, but if a ~ [a_,a+], its graph has the shape illustrated in Figure 2. Fix any a for which the graph of f(y) looks like the one depicted in Figure 2. Consider any number Yo < 1/ 2 which is greater than the smallest local minimum of f(y) and define ~ := YoU + (1 - Yo)V and 'f} := Yo V + (1 - Yo)U. (U and V may be interpreted as "factors" on which the returns ~ and 'f} on the two assets depend.) Then Var In[ (~ + 'f}) / 2] > Var In ~ = Var In 'f}, which yields a negative answer both --- ## Page 479 450 M A. H. Dempster, I. V. Evstigneev and K. R. Schenk-Hoppe 16 15.5 ,1--------------------1 15~------------------_+ 14.5 tl-------=======-------I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Figure 2 Graph of the function j(y) in equation (37) for a = 104 . to (a) and (b). In this example, ~ and 7] are dependent. It would be of interest to investigate questions (a) and (b) for general independent random variables ~ and 7]. It can be shown that the answer to (b) is positive if one of the variables ~ and 7] is constant. But even in this case the function Var In[x~ + (1- x)7]] is not necessarily convex: it may have an inflection point in (0,1), which can be easily shown by examples involving two-valued random variables. Thus, it can happen that a fixed-mix portfolio may have a greater volatility than each of the assets from which it has been constructed. Consequently, the above conjecture and the 'energy interpretation' of volatility are generally not valid. It is interesting, however, to find additional conditions under which assertions regarding volatility reduction hold true. In this connection, we can assert the following fact, see Theorem 4 in Dempster, Evstigneev, and Schenk-Hoppe (2007). Theorem 4. Let U and V be independent random variables bounded above and below by strictly positive constants. If U is not constant, then one has that Varln[yU + (1- y)V] < VarlnU for all y E (0,1) sufficiently close to 1. Volatility can be regarded as a quantitative measure of instability of the portfolio value. The above result shows that small independent noise can reduce volatility. This result is akin to a number of known facts about noise-induced stability, e.g., Abbott (2001) and Mielke (2000). An analysis of links between the topic of the present work and results about stability under random noise might constitute an interesting theme for further research. 5.5 Growth under transaction costs In this section, we consider an example (a binomial model) in which quantitative estimates for the size of the transaction costs needed for the validity of the result --- ## Page 480 Growing Wealth with Fixed-Mix Strategies 451 on volatility-induced growth can be provided. Suppose that there are two assets k = 1,2: one riskless and one risky. The price of the former is constant and equal to l. The price of the latter follows a geometric random walk. It can either jump up by u > 1 or down by u- 1 with equal probabilities. Thus, both security prices have growth rate zero. Suppose the investor pursues the constant proportions strategy prescribing to keep 50% of wealth in each of the securities. There are no transaction costs for buying and selling the riskless asset, but there is a transaction cost rate for buying and selling the risky asset of c: E [0, 1). Assume the investor's portfolio at time t - 1 contains v units of cash; then the value of the risky position of the portfolio must be also equal to v. At time t, the riskless position of the portfolio will remain the same, and the value of the risky position will become either uv or u-1v with equal probability. In the former case, the investor rebalances his/her portfolio by selling an amount of the risky asset worth A so that v + (1 - c:)A = vu - A (38) By selling an amount of the risky asset of value A in the current prices, the investor receives (1 - c:)A, and this sum of cash is added to the riskless position of the portfolio. After rebalancing, the values of both portfolio positions must be equal, which is expressed in (38). From (38) we obtain A = v( u-l)(2-c:) -1. The positions of the new (rebalanced) portfolio, measured in terms of their current values, are equal to v + (1- c:)A = v[l+ (1 - c:)(2 - c:)-I(U - 1)]. In the latter case (when the value of the risky position becomes u- 1v), the investor buys some amount of the risky asset worth B, for which the amount of cash (1 + c:)B is needed, so that v - (1 + c:)B = u-1v + B From this, we find -B = v(u-1 - 1)(2 + c:)-I, and so v - (1 + c:)B = v[l + (1 + c:)(2 + c:)-1(u-1 - 1)]. Thus, the portfolio value at each time t is equal to its value at time t - 1 multiplied by the random variable ~ such that P{~ = g'} = P{~ = g"} = 1/2, where g' := 1 + (1 +c: )(2+c:)-1( u-1-1) and g" := 1 + (1-c:)(2 -c:)-1( u-l). Consequently, the asymptotic growth rate of the portfolio value, Eln~ = (1/2)(lng' + lng"), is equal to (1/2)ln¢(c:, u), where [ U-1 - 1] [ u - 1] ¢(c:,u):= 1+(1+c:) 2+c: 1+(1-C:)2_c: We have Eln~ > 0, i.e., the phenomenon of volatility induced growth takes place, if ¢(c:, u) > l. For c: E [0,1), this inequality turns out to be equivalent to the following very simple relation: 0::; c: < (u -l)(u + 1)-1. Thus, given u > 1, the asymptotic growth rate of the fixed-mix strategy under consideration is greater than zero if the transaction cost rate c: is less than c:*(u) := (u - l)(u + 1)-1. We call c:* (u) the threshold transaction cost rate. Volatility-induced growth takes place- in the present example, where the portfolio is rebalanced in everyone period- when o ::; c: < c:* ( u). --- ## Page 481 452 M A. H. Dempster, 1. V. Evstigneev and K. R. Schenk-Hoppe The volatility IJ of the risky asset under consideration (the standard deviation of its logarithmic return) is equal to In u. In the above considerations, we assumed that IJ-or equivalently, u-is fixed, and we examined ¢(s, u) as a function of s. Let us now examine ¢(s, u) as a function of u when the transaction cost rate s is fixed and strictly positive. For the derivative of ¢(s, u) with respect to u, we have "" ( ) _ 1 + s [1 -s -2] '!-' su - --- --- -u u' 4 - S2 1 + s If u = 1, then ¢~(s, 1) < O. Thus, if the volatility of the risky asset is small, the performance of the constant proportions strategy at hand is worse than the performance of each individual asset. This fact refutes the possible conjecture that 'the higher the volatility, the higher the induced growth rate '. Further, the derivative ¢~(s,u) is negative when u E [O,u*(s)), where u*(s) := (1-S)-1/2(1 +S)1/2. For u = u*(s), the asymptotic growth rate of the constant proportions strategy at hand attains its minimum value. For those values of u which are greater than u*(s), the growth rate increases, and it tends to infinity as u ---+ 00. Thus, although the assertion 'the greater the volatility, the greater the induced growth rate' is not valid always, it is valid (in the present example) under the additional assumption that the volatility is large enough. 6 Conclusion This chapter surveys the authors' recent theoretical work on the phenomenon of volatility-induced growth. We study the performance of fixed-mix strategies in markets where asset prices or returns are stationary ergodic. We have established the surprising result that these strategies generate portfolio growth rates in excess of the individual asset growth rates, provided some volatility is present. As a consequence, even if the growth rates of the individual securities all have mean zero, the value of a fixed-mix portfolio tends to infinity with probability one. By contrast with the twenty five years in which the effects of 'volatility pumping' have been investigated in the literature by example, our results are quite general. They are obtained under assumptions that accommodate virtually all the empirical market return properties discussed in the literature. We have in this paper also dispelled the notion that the demonstrated acceleration of portfolio growth is simply a matter of 'buying low and selling high' or stems from some 'volatility- energy' link. The example of Section 5.2 shows that our result depends critically on rebal- ancing to an arbitrary fixed mix of portfolio proportions. Any such mix defines the relative magnitudes of individual asset returns realized from volatility effects. This observation and our analysis of links between growth, arbitrage, and noise-induced stability suggest that financial growth driven by volatility is a subtle and delicate phenomenon. A major obstacle in practical applications of our results is the fact that in- vestment is not over an infinite period. This point has been forcefully argued by --- ## Page 482 Growing Wealth with Fixed-Mix Strategies 453 Samuelson (1979) , but its validity is not as clear cut as one may be led to believe, see MacLean, Zhao, and Ziemba (2006), Ziemba (2008) and the discussion in Part IV in this volume. Other issues are the size of transaction costs in real markets. Though our findings do hold under sufficiently small proportional transaction costs, this issue can only be resolved empirically. Another intriguing question is how much stationarity is present in real markets. All of these questions are left for future research. References Abbott, D. (2001). Overview: Unsolved problems of noise and fluctuations. Chaos 11, 526- 538. Algoet, P. H. and T. M. Cover (1988). Asymptotic optimality and asymptotic equipartition properties of log-optimum investment. Annals of Probability, 16, 876- 898. Arnold, L. (1998). Random Dynamical Systems. New York: Springer. Arnold, L., I. V. Evstigneev and V. M. Gundlach (1999). Convex-valued random dynam- ical systems: A variational principle for equilibrium states. Random Operators and Stochastic Equations, 7, 23- 28. Aurell, E., R Baviera, O. Hammarlid, M. Serva and A. Vulpiani (2000). A general method- ology to price and hedge derivatives in incomplete markets. International Journal of Theoretical and Applied Finance, 3, 1- 24. A urell , E. and P. Muratore-Ginanneschi (2000) . Financial friction and multiplicative Markov market games. International Journal of Theoretical and Applied Finance, 3, 501- 510. Billingsley, P. (1965). Ergodic Theory and Information. New York: John Wiley. Breiman, L. (1961). Optimal gambling systems for favorable games. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1 (ed. Neyman, J.) , pp. 65-78. LA: University of California Press. Browne, S. (1998). The return on investment from proportional portfolio strategies. Advances in Applied Probability, 30, 216- 238. Browne, S. and W. Whitt (1996). Portfolio choice and the Bayesian Kelly criterion. Advances in Applied Probability, 28, 1145- 1176. Cover, T. M. (1991). Universal portfolios. MathematicaL Finance, 1, 1- 29. Cover, T. M. (1998). Shannon and investment. IEEE Information Theory Society Newslet- ter, Summer 1998, Special Golden Jubilee Issue, pp. 10- 11. Dempster, M. A. H., I. V. Evstigneev and K. R Schenk-Hoppe (2003). Exponential growth of fixed-mix strategies in stationary asset markets. Finance (3 Stochastics, 7, 263- 276. Dempster, M. A. H., 1. V. Evstigneev and K. R Schenk-Hoppe (2007). Volatility-induced financial growth. Quantitative Finance, 7, 151- 160. Dempster, M. A. H., 1. V. Evstigneev and K. R Schenk-Hoppe (2008). Financial Markets: The joy of volatility. Quantitative Finance, 8, 1- 3. Dempster, M. A. H., M. Germano, E. A. Medova and M. Villaverde (2003). Global asset liability management. British Actuarial Journal, 9, 137- 195. Dries, D., A. Ilhan, J. Mulvey, K. D. Simsek and R Sircar (2002). Trend-following hedge funds and multi-period asset allocation. Quantitative Finance, 2, 354- 361. Evstigneev, LV., T. Hens and K. R Schenk-Hoppe (2009a). Evolutionary Finance. In Handbook of Financial Markets: Dynamics and Evolution (eds. Hens, T. and Schenk- Hoppe, K. R), Chapter 9, pp. 507-566. Amsterdam: North-Holland. --- ## Page 483 454 M A. H. Dempster, 1. V. Evstigneev and K. R. Schenk-Hoppe Evstigneev, 1. V., T . Hens and K. R . Schenk-Hoppe (2009b) . Evolutionary stability of investment rules. Evstigneev, 1. V. and D. Kapoor (2006). Arbitrage in stationary markets. Discussion Paper 0608, School of Economic Studies, University of Manchester. Evstigneev, 1. V. and K. R. Schenk-Hoppe (2002). From rags to riches: On constant proportions investment strategies. International Journal of Theoretical and Applied Finance, 5, 563-573. Fernholz, R. and B. Shay (1982). Stochastic portfolio theory and stock market equilibrium. Journal of Finance, 37, 615- 624. Fernholz, E. R. (2002) . Stochastic Portfolio Theory. New York: Springer. Fernholz, E. R. and 1. Karatzas (2005). Relative arbitrage in volatility-stabilized markets. Annals of Finance, 1, 149-177. Fernholz, E . R. , 1. Karatzas and C. Kardaras (2005). Diversity and arbitrage in equity markets. Finance CJ Stochastics, 9, 1- 27. Hakansson, N. H. and W. T. Ziemba (1995) . Capital Growth Theory. In Handbooks in Operations Research and Management Science, Vol. 9, Finance (eds. Jarrow, R. A. , Maksimovic, V. and Ziemba, W. T.) , Chapter 3, pp. 65- 86. Amsterdam: North-Holland. Huberman, G. (1982). A simple approach to Arbitrage Pricing Theory. Journal of Eco- nomic Theory, 28, 183- 19l. Kabanov, Yu. M. (1999). Hedging and liquidation under transaction costs in currency markets. Finance CJ Stochastics, 3, 237- 248. Kabanov, Yu. M. and D. A. Kramkov (1994) . Large financial markets: Asymptotic arbi- trage and contiguity. Theory of Probability and its Applications, 39, 222- 228. Kabanov, Yu. M. and Ch. Stricker (2001). The Harrison- Pliska arbitrage pricing theorem under transaction costs. Journal of Mathematical Economics, 35, 185- 196. Kelly, J. (1956). A new interpretation of information rate. Bell System Technical Journal, 35, 917- 926. Kemeny, J. G. and J. L. Snell (1960). Finite Markov Chains. New York: Van Nostrand. Klein, 1. and W. Schachermayer (1996). Asymptotic arbitrage in non-complete large finan- cial markets. Theory of Probability and its Applications, 41, 927- 934. Kuhn, D. and D. G. Luenberger (2010). Analysis of the rebalancing frequency in log- optimal portfolio selection. Quantitative Finance. (Forth coming). Li, y. (1998). Growth-security investment strategy for long and short runs. Management Science, 39, 915- 924. Luenberger, D. G. (1998). Investment Science. UK: Oxford University Press. MacLean, L. C., Y. Zhao and W. T . Ziemba (2006). Dynamic portfolio selection with process control. Journal of Banking and Finance, 30, 317- 339. MacLean, L. C., W. T. Ziemba and G. Blazenko (1992) . Growth versus security in dynamic investment analysis. Management Science, 38, 1562- 1585. Mielke, A. (2000). Noise induced stability in fluctuating bistable potentials. Physical Re- view Letters, 84, 818- 82l. Mulvey, J. M. (2001). Multi-period stochastic optimization models for long-term in- vestors. In M. Avellaneda (Ed.) , Quantitative Analysis in Financial Markets, Vol. 3, pp. 66- 85. Singapore: World Scientific. Mulvey, J. M. (2009). Applying Kelly and related strategies during turbulent markets. Mulvey, J. M., C. Ural and Z. Zhang (2007) . Improving performance for long-term in- vestors: Wide diversification, leverage, and overlay strategies. Quantitative Finance, 7, 175- 187. Perold, A. F. and W. F. Sharpe (1988). Dynamic strategies for asset allocation. Financial Analysts Journal, 44, 16-27. --- ## Page 484 Growing Wealth with Fixed-Mix Strategies 455 Radner, R. (1971). Balanced stochastic growth at the maximum rate. In C. Bruckman and W . Weber (Eds.), Contributions to the von Neumann Growth Model (Zeitschrijt fur Nationalokonomie, Suppl. 1), pp. 39- 62. New York: Springer. Ross, S. A. (1976). The arbitrage theory of asset pricing. JournaL of Economic Theory, 13, 341- 360. Samuelson, P. A. (1979). Why we should not make mean log of wealth big though years to act are long. Journal of Banking and Finance, 3, 307- 309. Thorp, E. O. (1971) . Portfolio choice and the Kelly criterion. In Business and Economics Statistics Section, Proceedings of the American StatisticaL Association, pp. 215- 224. Reprinted in: Stochastic Optimization Models in Finance (eds. Ziemba, W. T. and Vickson, R. C .) , pp. 599-619. US: Academic Press (1975, reprinted in 2006). Ziemba, W. T. (2008). The Kelly Criterion. The Actuary, September issue, Features: Investment. Ziemba, W. T. and J. M. Mulvey (eds.) (1998). Worldwide Asset and Liability Modeling. UK: Cambridge University Press. --- ## Page 485 Part IV Critics and assessing the good and bad properties of Kelly --- ## Page 486 This page is intentionally left blank --- ## Page 487 This page is intentionally left blank --- ## Page 488 Growing Wealth with Fixed-Mix Strategies 455 30 Introduction to the Good and Bad Properties of Kelly Multiperiod lifetime investment-savings optimization dates at least to Ramsey (1928). Phelps (1962) extended the model to include uncertainty while maximiz- ing expected utility of lifetime consumption by choosing between consumption and investment in a single risky asset using an additive utility function. He obtained explicit solutions for a constant member of the isolastic utility class. Samuelson (1969) and Merton (1969) in companion articles develop, following Ramsey (1928) and Phelps (1962), in both discrete-time and continuous time, lifetime portfolio selection models where the objective function is the discounted sum of concave functions of period by period consumption. Samuelson solves the case when there are interior maxima, and shows that for isoelastic period by period utility functions u' (C) = Co -1, 0 < 1, the optimal portfolio decisions are independent of current wealth at each stage and independent of all consumption-savings decisions with a stationary optimal policy to invest a fixed proportion of current wealth in each pe- riod. Ziemba and Vickson (2010) review this literature and point to some queries regarding the validity of the interior maxima as discussed in problems in Ziemba and Vickson (1975, 2006). For log utility, 0 -+ 0, Samuelson (1969) showed that the optimal decision splits into two independent parts with the Ramsey savings problem independent of the lifetime portfolio selection problem. Log utility frequently splits the problem in this way, see Rudolf and Ziemba (2004) reprinted in part VI of this book for a similar splitting in a multi period pension model. Samuelson (1971) focuses on the third point of his 1979 paper that follows. He states the following result which has broad agreement. Theorem: If one acts to maximize the geometric mean at each step, and if the period is "sufficiently long", "almost certainly" higher terminal wealth and terminal utility [for a log utility investor]l will result than from any other [essentially different] decision rule. Samuelson points out the following: False Corollary: If maximizing the geometric mean almost certainly leads to a better outcome, then the expected utility of its outcomes exceeds that of any other rule, provided T is sufficiently large. The editors of this book agree with Samuelson that the E log maximization is for the special utility function u( w) = log w and not for other utility functions. 1 [] indicate phrase added by the editors --- ## Page 489 460 L. C. MacLean, E. 0. Thorp and W. T. Ziemba Samuelson presents one example that illustrates this. Samuelson's criticisms are with the pure theory and not in conflict with any of the conclusions of this book, much of which is summarized in the papers in this part of the book. But he ends with the observation that his critical remarks " .. . do not deny that this criterion, arbitrary as it is, still avoids some of the even greater arbitrariness of conventional mean- variance analysis. Its essential defect is that it attempts to re- place the pair of "asymptotically sufficient parameters" E log X i, Variance(logXi ) by the first of these alone, thereby gratuitously ruling out arbitrary 'Y in the family u(x) = x '" in favor of "I u(x) = log x". Luenberger (1993), reprinted in part V, provides a careful analysis of these two parameters. The paper on Kelly simulations in part IV by MacLean et al. also considers both parameters in an analysis of proportional investment strategies. Samuelson' paper (1979) is written in a style, with one syllable words, that is hard to read, but when you cut through the language he makes a few points: (1) Those who follow the rule maximization of mean log of wealth will with higher and higher probability have more wealth in the long run than those that use an [essentially different] strategy. This is agreed and is one of the basic results due to Breiman and others. (2) Some of those who have favorable asset returns period by period and maximize the expected log of wealth can lose a lot and in fact almost all of their wealth. This is agreed and examples of this are in the simulation paper by MacLean, Thorp, Zhao and Ziemba (MTZZ) (2010) which appears in this part of the book. While the Kelly expected log criterion faced with favorable positive expectation bets will provide very high returns much of the time, in a small percent of the time there will be huge losses. In one example in Ziemba and Hausch (1986), which is redone in MTZZ with a 14% advantage on each play you can lose 98% of your initial wealth. Of course, with fairly high probability, you can achieve a 10- or 100- fold increase in your initial wealth. This is, of course, the pure theory and in actual applications by hedge funds and other investors, financial engineering methods may be used to limit losses should they occur. (3) Latane (1959, 1978) seems to have claimed (although even that is debatable) that the expected log maximizing strategy is in some sense better than a strat- egy based on some other utility function, this is not discussed or claimed by those involved with this book. The expected log maximizing strategy is about long run wealth maximizing most of the time. The only utility functions used are log in general for full Kelly and negative power in the special case with log normally distributed assets. The papers in this part of the book discuss these and other points further. --- ## Page 490 Introduction to the Good and Bad Properties of Kelly 461 Markowitz (1976) argues that when one traces out the set of mean-variance (MV) efficient portfolios, one passes through a portfolio which gives approximately the maximum E log X. Thorp (1969) gives an example of an asset distribution such that the Kelly log optimal portfolio is not on the efficient frontier, so the approximation is not exact. Markowitz argues that this Kelly portfolio should be considered the upper limit for conservative choice among E,V efficient portfolios, since portfolios with higher (arithmetic) mean give greater short-run variability with less return in the long run. A real investor might prefer a lower mean and variance giving up return in the long run for stability in the short run. Markowitz (1952) conjectured and Young and Trent (1969) proved that Elog(Xi ) ~ logE - ~ {;2} for a wide class of ex post distributions of portfolio returns, where X = gross return, E=EX and V=VarX. Thorp (2008) provides an introduction to the Kelly criterion and discusses a number of interesting topics about it. His paper and the following ones on simula- tions of Kelly returns and good and bad properties show what Kelly strategies do and do not do. Thorp begins with how he coined the term "Fortune's Formula" in 1960 in his blackjack application, where he developed favorable card counting strate- gies that have had a huge impact across the world. He discusses various examples and paradoxes and the issue of bet concentration, that is, limited diversification. Experts with good asset situations can plunge heavily into a few assets as Warren Buffett has done. Thorp's treatment and Ziemba's paper on great investors in part VI of this book provide clues that Buffett acts like a Kelly bettor. Buffett himself argues that amateurs are likely better off diversifying into index funds, after all, they beat about 75% of the professional managers in net returns with essentially no work. Thorp cautions the reader against common errors in the application of Kelly betting such as the failure to consider all investments, risk tolerance misjudgments, forgetting how aggressive Kelly betting is in the short run, overbetting because of data errors, forgetting about the impact of bad scenario black swans and forgetting that most of the superior Kelly betting properties are for the very long run. He presents some results on fractional Kelly wagering that supplement the papers in part III of this book. He discusses the subtle concept of "essentially different" that confounds even some top experts. He discusses some subtleties such as strategies that can beat Kelly strategies. An important topic is the use of Kelly strategies by great investors trying to multiply their fortunes, akin to the Kelly property that it will get you to a wealth goal faster than any other essentially different strategy for asymptotically high goals. Thorp shows the relationship between frac- tional Kelly strategies and Markowitz mean-variance strategies with the full Kelly being the limiting portfolio on this surface supplementing the Markowitz paper discussed above. --- ## Page 491 462 L. C. MacLean, E. O. Thorp and W T Ziemba Thorp also discusses and deals with Paul Samuelson's criticisms of Kelly invest- ment strategies and his discussion supplements the three Samuelson papers and the last papers in this part. The editors of this book agree that even when you make a long sequence of very good bets, you can, in fact, lose and the loss can be a lot. But most of the time you win a lot. Thorp mentions two papers he co-authored in the 1970s which show that different utility functions indeed have different opti- mal solutions. That, as fully discussed in the Thorp and Whitley's (1972) paper reprinted here, deals with the false notion that Kelly proponents think that log optimal strategies are optimal for other concave utility functions, which notion is neither claimed nor true. Thorp's paper concludes with a discussion of Proebsting's paradox which shows you can make a series of favorable bets and be asymptotically ruined. But the paradox needs a series of correlated bets made at different points in time to show this result. So it does not contradict the property that betting full Kelly or any fixed fraction less than full Kelly leads to exponential growth assuming the bets are independent (as in Breiman's paper in part I or have weak dependence as in Cover's papers in part II or Thorp's paper in part VI). One approach to successfully model black swans is to use a scenario optimization stochastic programming model where you include the possibility of an extreme event, specifying its consequences but not actually indicating the nature of the event (see Geyer and Ziemba (2008) for the application to the Siemens Austria Pension Fund). Correlations change as the scenario sets move from normal conditions to volatile to crash which include the black swans (see also Ziemba (2003) for additional applications of this approach). This is, of course, fully applicable to Kelly betting and will lower the wager. Thorp and Whitley (1972) show, for interesting concave utilities, that different utility functions, namely those which differ by more than a positive linear transfor- mation, have distinctly different sets of optimal strategies. Moreover, for the case of cash and a two valued risky investment, the optimal strategies are unique. This means that if two utility functions have the same set of optimal strategies for each such investment, then these two utility functions are equivalent, that is, they are positive linear transformations of one another. Hence, given any concave utility dif- ferent from log there are investment settings in which the two utilities have different optimal strategies. Thorp and Whitley show that utility functions that are close to one another may have very different optimal solutions. Kallberg and Ziemba (1983) show that two utility functions with the same Rubinstein risk aversion,2 under the assumption of normality, will have the same sets of optimal solutions. Thorp and Whitley show that for two continuous non-decreasing utility functions that are not equivalent, there is a one-period two security investment setting such that these two utility functions have distinct optimal strategies if either (a) the two utility 2The Rubinstein risk aversion - ~~::g::1 wo is a constant. --- ## Page 492 Introduction to the Good and Bad Properties of Kelly 463 functions are either concave or convex, or (b) they have a second derivative that exists except possibly for a set of isolated points. Short term Kelly and even fractional Kelly strategies are very risky since the Arrow-Pratt risk aversion index is very low. MacLean, Thorp, Zhao and Ziemba (2010) present three simple investment situations to simulate the behavior of full Kelly and fractional Kelly over medium time horizons. These simulations extend and redo over longer horizons with more scenarios earlier studies by Bicksler and Thorp (1973) and Ziemba and Hausch (1986). The results show that: (1) The great superiority of full Kelly and close to full Kelly strategies over medium term horizons (40 to 700 periods) with very large gains a large fraction of the time. (2) The short term performance of Kelly and high fractional Kelly strategies is very risky. (3) There is a well defined and consistent tradeoff of growth versus security as a function of the bet size determined by the various strategies. (4) No matter how favorable the investment opportunities are or how long the finite horizon is, a sequence of bad scenarios can lead to poor final wealth outcomes with a loss of most of the investor's initial capital. The calculations show the final wealth distributions including possible huge gains and huge losses and show the fraction of the time these are likely to occur. Mean- variance tradeoffs are shown versus the Kelly fraction. The mean is highest for full Kelly and decreases for lower and higher Kelly fractions. The variance increases with the Kelly fraction confirming that full Kelly is very volatile. The minimum and maximum wealth levels are shown versus the Kelly fraction. The wealth ac- cumulated from the full Kelly strategy does not stochastically dominate fractional Kelly strategies. To lower risk, one must lower the Kelly fraction. The results showed that for reasonable horizons with three different experiments, that the full Kelly and fractional Kelly strategies are not merely long run approaches. Proper use over short and medium time horizons can yield good wealth goals while protecting against drawdowns most of the time. The Kelly criterion has many good long-term properties and some favorable short-term properties. MacLean, Thorp, and Ziemba (2010) discuss the good and bad properties of the Kelly criterion. The main advantage is the long run growth rate maximization. That is an asymptotic mathematical result. Also the medium term simulation results show that most of the time the Kelly bettor has a lot higher final wealth than that from other strategies. But for a small percentage of the time, despite many periods of play with very favorable investments, there can be huge losses. Besides this, the main disadvantage of the Kelly criterion is that it is very risky short term and the near zero Arrow-Pratt risk aversion index typically leads to wild swings in the wealth path. But the Kelly strategy has favorable short run competitive optimality compared to other strategies. Moreover, the extreme --- ## Page 493 464 L. C. MacLean, E. O. Thorp and W T Ziemba sensitivity of E log to errors in the mean signals that mean estimates must be carefully and conservatively estimated to avoid overbetting which can lead to large portfolio losses. Fractional Kelly strategies such as half Kelly, moderate this but they can still have large losses. There is a tradeoff of lower growth for more security as the Kelly fraction is reduced and more is invested in riskless cash. Also it may take a long time for the Kelly bettor to dominate an essentially different strategy with high probability. No one disagrees with the Samuelson objection that E log maximizing policies are optimal for only log utility functions. Thorp and Whitley's (1972, 1974) result shows that this "objection" is simply a universal fact, true for all utilities. None the less, the long run properties indicate very high final wealth most of the time for Kelly and fractional Kelly bettors. --- ## Page 494 Review of Economics and Statistics, 51,239- 246 (1969) 465 31 LIFETIME PORTFOLIO SELECTION BY DYNAMIC STOCHASTIC PROGRAMMING Paul A. Samuelson * Introduction MOST .analyses of portfolio selection, whether they are of the Markowitz· Tobin mean-variance or of more general type, maximize over one period.1 I shaD here formu· late and solve a many-period generalization, corresponding to lifetime planning of consump- tion and investment decisions. For simplicity of exposition I shall confine my explicit dis- cussion to special and easy cases that suffice to illustrate the general principles involved. As an example of topics that can be investi- gated within the framework of the present model, consider the question of a "business- man risk" kind of investment. In the literature ' of finance, one often reads; "Security A should be avoided by widows as too risky, but is highly suitable as a businessman's risk." What is in- volved in this distinction? Many things. First, the «businessman" is more affluent than the widow; and being further removed from the threat of falling below some sub·- sistence level, he has a high propensity to embrace variance for the sake of better yield. Second, he can look forward to a high salary in the future; and with so high a present dis- counted value of wealth, it is only prudent for him to put more into common stocks compared to his present tangible wealth, borrowing if necessary for the purpose, or accomplishing the same thing by selecting volatile stocks that widows shun. ,. Aid from the National Science Foundation", gratefully acknowledged. Robert C. Merton has provided me with much stimulus; and In a companion paper in this issue of the RE\II£W he is tackling the much harder problem of optimal control in tbe presence of continuous-time sto- chastic variation. lowe thanks also to Stanley Fi!cher. 'See for example Harry Markowitz [51; James TobIn [14], Paul A. S"muelson [10]; Paul A. Samuelson and Robert C. M~ton [1.1]. See, however, James Tobin [151, for a pioneermg treatment of the multi.period portfolio problem; and Jan Massin [7] which overlaps with the pr=nt analysis in showing how to solve tbe basic dynamic stochastic program recursively by worldng backward from the end in tbe Bdlman fashion, and which prov,", the theorem tbat portfolio proportions wilt be invariant only if the marginal utility function is iso-elastic. Third, being still in the prime of life, the businessman can "recoup" any present losses in the future. The widow or retired man near- ing life's end has no such "second or nIh chance." Fourth (and apparently related to the last point), since the businessman will be investing for so many periods, "the law of averages will even out for him," and he can afford to act almost as if he were not subject to diminishing marginal utility. What are we to make of these arguments? It wiD be realized that the first could be purely a one-period argument. Arrow Pratt and h 2 ' , ot ers have shown that any investor who faces a range of wealth in which the elasticity of his marginal utility schedule is grea.t will have high fisk tolerance; and most writers seem to believe that the elasticity is at its highest for rich - but not ultra-rich!- people. Since the present model has no new insight to offer in connection with statical risk tolerance, I shall ignore the first point here and confine almost all my attention to utility functions with the same relative risk aversion at all levels of wealth. Is it then still true that lifetime considerations justify the concept of a businessman's risk in his prime of life? Point two above does justify leveraged in- vestment financed by borrowing against future earnings. But it does not rea.lIy involve any increase in relative risk-taking once we have related what is at risk to the proper larger base. (Admittedly, if market imperfections make loans difficult or costly, recourse to volatile, "leveraged" securities may be a rational pro- cedure.) The fourth point can easily involve the in- numerable fallacies connected with the "law of large numbers." I have commented elsewhere a on the mistaken notion that multiplying the same kind of risk leads to cancellation rather 'See K. Arrow [ll i J. Pratt [9] i P. A. Samuelson and R. C. Merton [L31. 'P. A. Samuelson [111. (H9 ] --- ## Page 495 466 P A. Samuelson 240 THE REVIEW OF ECONOMICS AND STATISTICS than augmentation of risk. I.e., insuring many ships adds to risk (but only as Vn) j hence, only by insuring more ships and by also subdividing those risks among more people is risk on each brought down (in ratio l/Vn). However, before writing this paper, I had thought that points three and four could be refmmulated so as to give a valid demonstra- tion of businessman's risk, my thought being that investing for each period is akin to agree- ing to take a l/ntb. interest in insuring n inde- pendent ships. The present lifetime model reveals that in- vesting for many periods does not itself in- troduce extra tolerance for riskiness at early, or any, stages of life. Basic Assumptions The familiar Ramsey model may be used as a point of departure. Let an individual maxi- mize 11' e-pt U[C(t) ]dt (1) subject to initial wealth Wo that can always be invested for an exogeneously-given certain . rate of yield Y; or subject to the constraint C{t) = rW(t) - W{t) (2) If there is no bequest at death, terminal wealth is zero. This leads to the standard calculus-of-varia- tions problem J = Max jTe_pj U[rW - W]dt (3) {W(t)} 0 This can be easily related' to a discrete- time formulation Max::S~_ o (l+p)-tU[Ctl (4) subject to WH1 ) Ct=W.--- (S 1+ r or, Max :s~~ o (l+p)-t U [ Wt - W1+t+l ] {W,} r (6) • See P. A. Samuelson [12], p. 273 for an exposition of discrete-time analogu.. to calculus-of-variations models. Note: here I assume tJu.t consumption, C t, takes plate at the beginning rath~r than at the end of the petlod. This change alters slightly the appearance of the equilibrium conditio,,", but not their substance. for prescribed (Wo, W'I'+l). Differentiating partially with respect to each WI in turn, we derive recursion conditions for a regular inte- rior maximum (1+ .. ) U'[ Wt - 1 - Wt ] 1+1' l+r = U' [ Wt _Wt +1 ] 1+1' (7) If U is concave, solving these second-order difference equations with boundary conditions (Wo, W 'l'+l) will suffice to give us an optimal lifetime consumption-investment program. Since there has thus far been one asset, and that a safe one, the time has come to introduce a stochastically-risky alternative asset and to face up to a portfolio problem. Let us postulate the existence, alongside of the safe asset that makes $1 invested in it at time t return to you at tbe end of the period $1 (1 + ,), a risk asset that makes $1 invested in, at time t, return to you after one period $lZ" where Z, is a random variable subject to the probability distribution Prob {Z, ;;a z} = P(2) . Z ~ 0 (8) Hence, Z'+l - 1 is the percentage "yield" of each outcome. The most general probability distribution is admissible: i.e., a probability density over continuous g'S, or finite positive probabilities at discrete values of z. Also I shall usually assume independence between yields at different times so that P(zo, 2 1, • • • , Zt, ... ,2.'1') = P(Zt)P(Zl) ... P(ZT)' For simplicity, the reader might care to deal with the easy case Prob {Z = .\.} = 1/ 2 = Prob{Z= >..-l}, .\.>1 (9) In order that risk averters with concave utility should not shun this risk asset when maximiz- ing the expected value of their portfolio, A must be large enough so that the expected value of the risk asset exceeds that of the safe asset, Le., 1 1 _,\. + _.\ -1 > 1 + 1', or 2 2 A > 1 + r + y2r + r~ . Thus, for A = 1.4, the risk asset has a mean yield of 0.057, which is greater than a safe asset's certain yield of , = .04. At each instant of time, what will be the optimal fraction, W" that you should put in --- ## Page 496 Lifetime Portfolio Selection by Dynamic Stochastic Programming 467 LIFETIME PORTFOLIO SELECTION 241 the risky asset, with 1 - w, going into the safe asset? Once these optimal portfolio fractions are known, the constraint of (5) must be written C - [W W.+ 1 ] ,- ,- [(l-w,)(I+r) + w,Zd . (10) ~ ow we use (to) instead of (4), and recogni:z- mg the stochastic nature of our problem specify that we maximize the expected valu~ of total utility over time. This gives us the stochastic generalizations of (4) and (5) or (6) Mill T {C"w,}E ~ (1 +p}-'U[Ct] (11) ._0 subject to C, = [Wt _ Wt+l ] (!+r)(1-w,) +w,Zt Wo given, W!'+I prescribed. If there is no bequeathing of wealth at death, presumably W 2'+1 = O. Alternatively, we could replace a prescribed W!' +, by a final bequest functi~n added to (11), of the form B(W2'+l), and with W 2'+1 a free decision variable to be chosen so as to maximi:ze (11) + B(WN ,). For the most part, I shall consider C2' = W T and W2'+l = O .. In (11), E stands for the "expected value of," so that, for example, E Zt = l"'ztdP(fl,) . In our simple case of C 9), 1 1 EZt = '2A+ '2,X-I. Equation (11) is our basic stochastic program- ming problem that needs to be solved simul- taneously for optimal saving-consumption and portfolio-selection decisions over time. Before proceeding to solve this problem, ref- erence may be made to similar problems that seem to have been dealt with explicitly in the economics literature. First, there is the valu- able paper by Phelps on the Ramsey problem in which capital's yield is a prescribed random variable. This corresponds, in my notation, to the (w,} strategy being frozen at some frac- tional level, there being no portfolio selection problem. (My analysis could be amplified to consider Phelps' 5 wage income, and even in the stochastic form that he cites Martin Beck- mann as having analyzed.) More recently Levhari and Srinivasan [4] have also treated the Phelps problem for T = 00 by means of the Bellman functional equations of dynamic programming, :and have indicated a proof that concavity of U is sufficient for a maximum. Then, there is Professor Mirrlees' important work on the Ramsey problem with Harrod- neutral technological change as a random vari- able.6 Our problems become equivalent if I ~eplace W. - W,+I[(l+I')(I-w.) + wtZ.J-1 m (10) by A,j(W,IA t ) - nW, - (W'+l - WI) let technical change be governed by the prob- ability distrihution Prob {A, ;:;; A,_,Z} = P(Z); rein.terpret my W, to be Mirrlees' per capita capital, K,IL" where L t is growing at the nat- ural rate of growth 11.; and posit that AdCW I At) is a homogeneous first degree, concave, ne~­ classical production function in terms of cap- ital 'and efficiency-units of labor. It should be remarked that I am confirming myself here to regular interior maxima and . . ' not gomg mto the Kuhn-Tucker inequalities that easily handle boundary maxima. Solution of the Problem The meaning of our basic problem T J~(Wo) = Max E ~ (1+p)-t U[C.l (11) {C"w.} ,-0 . subject to C, = W t - Wt+ 1[(1-w.) (1+1') + w,Z.] _1 is not easy to grasp. I act now at t == 0 to select Co and wo, knowing Wo but not yet knowing how Zo will turn out. I must act now, knowing that one period later, knowledge of Zo's outcome will be known and that W, will then be known. Depending upon knowledge of WI, a new decision will be made for C, and WI. Now I can only guess what that decision will be. As so often is the case in dynamic program- ming, it helps to begin at the end of the plan- ning period. This brings us to the well-known • E. S. Phelp' [8]. • J. A. Mirrlees [6]. I h~ve canverted his treatment into a discrete·Ume version. Robert Metton's companion paper throw. light on Mlrrlees' Brownian-motion model for A. •• --- ## Page 497 468 P. A. Samuelson 242 THE REVIEW OF ECONOMICS AND STATISTICS one-period portfolio problem. In our terms, this becomes J1(W2'_d '""' Max U[C2'-d { C2'-1,W2'_1} + E(l+p) - ~V[(W'l'_l- C:r-l) {(1- W'l'_1}(1+1') + WT-1ZT-I}-"'}. (12) Here the expected value operator E operates only on the random variable of the next period since current consumption C'l'-1 is known once we have made our decision. Writing the second term as EF(Z'l') , this becomes EF(Z",) :=: jF(Z'I')dP(Z",IZ"'_l'Z'l'_'" .. , ZQ) o = jF(ZT)dP(ZT), by our indepen:dence postulate. In the general case, at a later stage of decision making, say t = T-l, knowledge will be avail- able of the outcomes of earlier random vari- ables, Z'_Zl • . . ; since these might be relevant to the distribution of subsequent random vari- ables, conditional probabilities of the form P(Z"'_l IZ",_" . .. ) are thus involved. How- ever, in cases like the present one, where in- dependence of distributions is posited, condi- tional probabilities can be dispensed within favor of simple distributions. Note that in (12) we have substituted for C'I' its value as given by the constraint in (11) or (10). To determine this optimum (C'l'_l, W T _ 1 ), we differentiate with respect to each separately, to get 0= V' (CT-d - (1+1') -1 EV' [C'I') (I-WT_tl(1+I') + 2V1'_lZ",_d (12') 0= EV' [CTJ(WT- 1 -CI'_l) (Z'I'_1-1-r) = jlf' [(W'I'-I -C2'-I) o {(l-wT_l(l+r) - W'l'- lZT-dJ (W2'-1-CT- 1 ) (ZT_l-l-r)dP(Z:r_l) (12") Solving these simultaneously, we get our optimal decisions (C""'_l' W""_l) as functions of initial wealth W 2'-1 alone. Note that if somehow C"'l'-l were known, (12") would by itself be the familiar one-period portfolio op- timality condition, and could trivially be re- written to handle any number of alternative assets. Substituting (C*T_lt w*T-d into the elCpres- sion to be maximized gives us 11(WT - t ) ex- plicitly. From the equations in (12), we can, by standard calculus methods, relate the de- rivatives of U to those of I, namely, by the envelope relation J{(W"'_l) = u' [CT-d . (13) Now that we know Jt[WT- rJ, it is easy to determine optimal behavior one period earlier, namely by 12(W'l'-z} = Max U[C,,_z] (C"'_2,WT_'} + E(1+p)-IJ1 [(WT _ Z-C"_2) {(2 -WT_Z) (1+r) + W'l'_ZZ 'l'_2}]. (14) Differentiating (14) just as we did (11) gives the fallowing equations like those of (12) 0-- U' [C"-2] - (l+p)-'EJ,' [WT _ 2 ] {(1-w'l'_2) (1+r) + W'I'-2ZT_Z} (IS') 0= EIr' [WT-d (W",_z - C" _2)(Z"_2 - 1-1') = [jl' [(W:r-2 -C'l'- Z){(1-WT_2) (I+r) + WT_ZZ'I'_2}] (WT- 2 -C:r-2) (Z" _ 2 - 1-,.) dP(Z'l'_2). (15") These equations, which could by (13) be re- lated to U'[CT _ I ], can be solved simultaneous- ly to determine optimal (C*"- 2, W*'I'_2) and 12(WT _ 2 ). Continuing recursively in this way for 1'- 3, T-4, ... ,2, 1,0, we finally have our problem solved. The general recursive optimality equa- tions can be written as { 0 = V'[Co] - (1+p)-lEI',,_l[WO) {(l-wo)(l+r) + woZo) 0= EI'T_l[Wr](WO - Co)(Zo - I-r) o = U'[C'I'-lJ - (l+p) -, EJ'T_t(W,] {(l-w,-d (1+') + Wt_ ,Z,_r} (l6') 0= EJ''I'_. [W,_l -C._I) (Zt_l - l-r), (l = 1, ... ,T-1). (16") In (16')' of course, the proper substitutions must be made and the E operators must be over the proper probability distributions: Solv- ing (16") at any stage will give the optimal decision rules for consumption-saving and for portfolio selection, in the form c*. = ![W.; ZI_1, ... , zoJ = !",_t[Wt } if the Z's are independently distributed --- ## Page 498 Lifetime Portfolio Selection by Dynamic Stochastic Programming 469 LIFETIME PORTFOLIO SELECTION 243 W"'t = g[Wt ; Zt_L, ... , ZoJ = g",-,[Wtl if the Z'g are independently distributed. Our problem is now solved for every case but the important case of infinite-time horizon. For well-behaved cases, one can simply let T -+ 00 in the above formulas. Or, as often happens, the infinite case may be the easiest of all to solve, since for it C*t = feW,) , w*. = g(W,), independently of time and both these unknown functions can be deduced as solutions to the foHowing functional equations: 0= V' [f(W)J - (l+p)-l Ii'«W - j(W» «1+1') - g(W)(Z - 1 - r)}) [(1+1') - g(W)(Z - 1 - ,»)dP(Z) (17') 0= fi/'[(w - f(W)} o {1 + I' - g(W) (Z - 1 - r)}] [Z - 1 - r).:i p«-) (17") Equation (17'), by itself with g(W) pretended to be known, would be equivalent to equation (13) of Levha ... i and Srinivasan (4, p. £]. [n deriving (17')-( 17"), [ have utilized the enve- lope relation of my (13), which is equivalent to Levhari and Srinivasan's equation (12) [4, p.5J. Bernoulli and Isoelastic Cases To apply our results, let us consider the in- teresting Bernoulli case where U = log C. This does not have the bounded utility that Arrow [1} and many writers have convinced them- selves is desirable for an axiom system. Since I do not believe that Karl Menger paradoxes of the generalized St. Petersburg type hold any terrors for the economist, I have no particular interest in boundedness of utility and consider log C to be interesting and admissible. For this case, we have, from (12) , l l (W) = Max 10gC {C,w} +E(L+p)-1Iog [(W - C) {(1-w) (1+1') + wZ} ¥ = MaxlogC + (1+p)-ltog [W-C] {C} + Max jiog [(l-w)(I+r) o {w} + 14'ZJdP(Z)~ (IS) Hence, equations (12) and (16')-( 16") split into two independent parts and the Ramsey- Phelps saving problem becomes quite indepen- . dent of the lifetime portfolio selection problem. Now we have 0= (1/C) - (l+p)-l (W - C)-lor C"'_l = (1+p) (2+ p) - lW!'_1 (19') 0= l'(z-l-r)[(I - W) (1+r) + 14'Z) - 1 dP(Z) or 14''''_1 = w· independently of W 1'-1. (19") These independence results, of the C7'_l and W7'-I decisions and of the dependence of W7'-l on W"'_l, hold for an U functions with iso- elastic marginal utility. I.e., (16') and (16") become decomposable conditions for all V (C) = 1/y C'Y, y < 1 (20) as well as for U(C) = log C, corresponding by L 'Hopi tal's rule to 'Y = o. To see this, write (12) or (18) as C'Y (W-C)'Y 1, (W) = Max '---- + (1+p)-l __ - (C,w} y y j'"[(l-W)(l+r) + wZ}'YdP(Z) C'Y (W-C) 'Y =Max-+(l+p)_l X {C} y y Max!1(I-W)(1+r) w 0 + 14'Zp dp(Z). (21) Hence, (I 2") or ( 15") or (16") becomes .£~[ (1-14') (1+1') + 14'Z)'Y-l (Z-r-l) dP (Z) = 0, (22") which defines optimal w* and gives Max l'"[(t-w)U+r) +wZ]'YdP(Zl {w} 0 = i "'W- W*) (1+1') + w·Zj'Y dP(Z) = [l + r*) 'Y, for short. Here, r* is the subjective or util-prob mean return of the portfolio, where diminishing mar- ginal utility has been taken into account.7 To get optimal consumption-saving, differentiate (21) to get the new form of (12') , (15'), or ( 16') f See Samuelson and Merton for the utll-prab concept [131 . --- ## Page 499 470 P. A. Samuelson 244 THE REVIEW OF ECONOMICS AND STATISTICS 0= C-r- t - (l+p)-1 (1+"*)"1' (W_C)-r-1. (22') Solving, we have the consumption decision rule C*P_l = ~WI'_l 1 + al where al = [(l+,,*)-r/(1+p)j1h-1. Hence, by substitution, we find I t (Wp _ 1 ) = b1W'1',,_tly where hI = a,'1'(I+a'z)-')' (23) (24) (25) + (l+p)-1 (1+r*)'1' (l+at)-'Y. (26) Thus, I 1(·) is of the same elasticity form as U ( .) was. Evaluating indeterminate forms for i' = 0, we find I, to be of log form if U was. Now, by mathematical induction, it is easy to show that this isoelastic property must also hold for J2 (W"_2), Ja(W,,_a)," " since, whenever it holds for J,,(W'l'_ .. ) it is deducible that it holds for J,,+l(WT_,,_I)' Hence, at every stage, solving the general equations (16') and (16"), they decompose into two parts in the case of isoelastic utility. Hence, Theorem: For isoelastic marginal utility functions, U'(C} = C'1'-I, i' < 1, the optimal portfolio decision is independent of wealth at each stage and in~ dependent of all consumption-saving decisions, ieading to a constant w*, the solution to 0= ir. (l-w) (1+,) +WZl"-l(Z- L-,,)dP(Z). o Then optimal consumption decisions at each stage are, for a. no-bequest model, of the form C"'T_' = CtWI'_1 where one can deduce the recursion relations at Cl = --- I+Wl' a, = (l+p)/(L+,,*),>,jVl -'1' (1+,*)'>' = 1'" (I - w*)(l+r) + w* Z],>, dP (Z) alc' _l c. = ~"::""";--"- I +alcl-l al+'t _ I ' 1 = l+i ' In the limiting case, as 'Y -+ ° and we have Bernoulli's logarithmic function, al = (l+p), independent of 1'*, and all saving propensities depend on subjective time preference p only, being independent of technOlogical investment opportunities (except to the degree that Wt will itself definitely depend on those opportu- nities) . We can interpret 1 +1'* as kind of a "risk- corrected" mean yield; and behavior of a long- lived man depends criticaLLy on whether ( 1 +,*) 'Y .2:. (1 + p ), corresponding to al ..:::.. 1. < > (i) For O+r*)'>' = (l+p), one plans always to wnsumt at a uniform rate, dividingcun~nt W <_, evenly by remaining life, l/(l+i) . If young enough, one saves on the average; in the familiar "hump saving" fashion, one dissaves later as the end comes sufficiently close into sight. (ii) For (1+.*)'1' > (l+p), a, < I, and investment opportunities are, so to speak, so wnpting compared to psychological time preference that one consumes nothing at tbe beginning of a long-long life, i.e., rigor. ously Lim c. = 0, ~ < 1 i-+ CD and again hump saving must take place. For O+r*)'1' > (1+p), the perpetWJJ lifetime problem, with T = "", is divergent and ill-defined, i.e., /.(W) -> co as i -> 00, For 'Y";; a and p > 0, this case cannot arise. (iii) For (1+r*)'1 < (l+p), a, > I, consumption at very early ages drops only to a limiting positive fraction (ratber than zero), namely Lim ~. '" I - I/ d., < I, a, > L i~ Of.) Now whether there will be, on the average, initial hump saving depends upon the size of r* - c, r* - 1 - >0. (l+p)l/l - '1' This ends the Theorem. Although many of the results depend upon the no-bequest as- sumption, W,,+l. = 0, as Merton's companion paper shows (p. 247, this Review) we can easily generalize to the cases where a bequest func· tion B'l'(WT + t ) is added to ~?o (l+p)-iU(C t ). If B'l' is itself of isoelastic form, B'I' = bT(W N 1)'1' h, the algebra is little changed. Also, the same comparative statics put forward in Merton's continuous-time case will be applicable here, e.g" the Bernoulli 'Y = 0 case is a watershed between ca.ses where thrift is enhanced by risk- iness rather than reduced; etc. --- ## Page 500 Lifetime Portfolio Selection by Dynamic Stochastic Programming 471 LIFETIME PORTFOLIO SELECTION 245 Since proof of the theorem is straightfor- ward, I skip all details except to indicate how the recursion relations for c, and h, are derived, namely from the identities b'+lW"!y = J'+l(W) = Max (C'V/y C + b,(l+~*)'V(l+p)-I(W-C)'V!y} = (c'V'+1 + b,(1+r*)1' (l+p) -l(l-C'+ l )1'} W'V/y and the optimality condition 0 = 0 - 1 - b,(I+r*)'V(1+p)-1(W-C)'Y- 1 = (£,+IWP- 1 - 6,(1+,..*)1'(1 +p)-1 (1-£i+l)'V-1W'V- 1 , which defines C'+1 in terms of bi • What if we relax the assumption of isoelastic marginal utility functions? Then WT ~ J be- comes a function of WT - J- 1 (and, of course, of r, P, and a functional of the probability dis- tribution P). Now the Phelps-Ramsey optimal stochastic saving decisions do interact with the optimal portfolio decisions, and these have to be arrived at by simultaneous solution of the nondecomposable equations (16') and (16"). What if we have more than one alternative asset to safe cash? Then merely interpret Z, as a (column) vector of returns (Z2"za" ... ) on the respective risky assets; also interpret w, as a (row) vector (W2t,W3" .. . ), interpret P(Z) as vector notation for Prob {Z2, ~ Z2, Zs, ~ Z3, ... } = P(Z2, Z3, ... ) = P(Z) , interpret all integrals of the formJG(Z)dP(Z) as multiple integrals fG(Z2,Z3, . . . )dP(Z2,ZS, ... ). Then (16/1) becomes a vector-set of equations, one for each component of the vector Z" and these can be solved simultaneously for the unknown w, vector. If there are many consumption items, we can handle the general problem by giving a similar vector interpretation to C,. Thus, the most general portfolio lifetime problem is handled by our equations or obvious extensions thereof. Conclusion We have now come full circle. Our model denies the validity of the concept of business- man's risk j for isoelastic marginal utilities, in your prime of life you have the same relative risk-tolerance as toward the end of life! The "cbance to recoup" and tendency for the law of large numhers to operate in the case of re- peated investments is not relevant. (Note: if the elasticity of marginal utility, - UI (W) / WU"(W), rises empirically with wealth, and if the capital market is imperfect as far 'as lending and harrowing against future earnings is con- cerned, then it seems to me to be likely that a doctor of age 35-50 might rationally have his highest consumption then, and certain- ly show greatest risk tolerance then - in other words be open to a "businessman's risk." But not in the frictionless isoelastic model!) As usual, one expects w* and risk tolerance to be higher with algebraically large y . One expects C, to be higher late in life when rand r'" is high relative to p . As in a one-period model, one expects any increase in "riskiness" of Z" for the same mean, to decrease w*. One expects a similar increase in riskiness to lower or raise consumption depending upon whether marginal utility is greater or less than unity in its elasticity.s Our analysis enables us to dispel a fallacy that has been borrowed into portfolio theory from information theory of the Shannon type. Associated with independent discoveries by J. B. Williams [16], John Kelly [2 J, and H. A. Latane [3] is the notion that if one is invest- ing for many periods, the proper behavior is to maximize the geometric mean of return rather than the arithmetic mean. I believe this to be incorrect (except in the Bernoulli logarithmic case where it happens 9 to be correct for reasons • See Merton's cited companion paper in this issue, for explicit discussion of the comparative statical shifts of (16) 's C*t and W*t functions as the parameter-> (P, ""'1 Y, r*t and P(Z) or P(2" ... ) or B(Wr) functions chang~. The s~me results hold in the discrete·and·continuous-time models. ·S"'- Latane [3, p. lSlJ for explicit recognition of this point. I find somewhat mystifying his footnote there which <~y<, "As pointed out to me by Profes5or L. J. Savage (in correspondence), not only is the maximization of G Ithe geometric mean) the rule for ma"irnulll •• peeled utility in connection with Bemoulli's function but (in so far .. cer- tain appro: m*, then after a large number of stages we will have m**T > > m"'T, and the properly nor- malized probabilities will cluster around a higher value.) There is nothing wrong with the logical deduction from premise to theorem. But the implicit premise is faulty to begin with, as I have shown elsewhere in another connection [Samuelson, 10, p. 3]. It is a mistake to think that, just because a w** decision ends up with almost-certain probability to be better than a w* decision, this implies that w** must yield a better expected value of utility. Our analysis for marginal utility with elasticity differing from that of Bernoulli provides an effective counter example, if indeed a counter example is needed to refute a gratuitous assertion. Moreover, as I showed elsewhere, the ordering principle of selecting between two actions in terms of which has the greater probability of producing a higher result does not even possess the property of being transitive.10 By that principle, we could have w*** better than w*"', and w** better than w*, and also have w* better than w***. .. See Sao:nu~Json [Ill. REFERENCES [1] Arrow, K. ]., "Aspects of the Theory of Risk· Bearing" (Helsinld, Finland: Yrjo Jahnssonin Slilitio, 1965). [2] Kelly, J., "A New Interpretation of Information Rate," BeU System Technical Joumal (Aug. 1956),917-926. [3] Latane, H. A., "Criteria for Choice Among Risky Ventures," Journal oj Politi.Gn.l Economy 67 (Apr. 1959), 144-155. [4] Levh2.ri, D. and T. N. Srinivasan, "Optimal Sav- ings Under Uncertainty," Institute for Mathe- matical Studies in the Sodal Sciences, Technical Report No.8, Stanford University, Dec. 1967. [5] Markowitz, H., Portfolio Sekction: Effid.~nt Dive1'sificatwn 0/ investment (New York: John Wiley & Sons, 1959). [6] Mirrlees, J. A., "Optimum Accumulation Under Uncertainty," Dec. 1965, unpublished. [7] Mossin, J., "Optimal Multiperiod Portfolio Pol- icies," JouJ'tuIl of Busin.!ss 41, 2 (Apr. 19611), 215-229. [8] Phelps, E. S., "The Accumulation of Risky Cap· ital: A Sequential Utility Analysis," Economet- T~a 30, 4 (1962), 729-743. [9] Pratt, J., "Risk Aversion in the Small and in the Large," Econonutrica 32 (Jan. 1964). [10] Samuelson, P. A., "General Proof that Diversifica- tion Pays," J oUfntil 0/ Pinanciat tind Quantitative Analysis II (Mar. 1967), 1- 13. [11] --, "Risk and Un(;ertainty : A Fallacy of Large Numbers," Sc.~ntUl, 6th. Series, 57th. year (April-May, 1963). [12] --, "A Turnpike Refutation of the Golden Rule in a Welfare Maximizing Many-Year Plan," Essay XIV Essays on the Theo,y of Optimal EGonomi, Growth, Karl Shell (ed.) (Cambridge, Mass.: MIT Press, 1967). [13] --, and R. C. Merton, "A Complete Model of Warrant Pridng that Maximizes Utility," In- dustrUll Management Rf!vkw (in pr~). [14J Tobin, J., "Liquidity Preference as Behavior Towards Risk," Review of Econom.i& Studies, XXV, 67, Feb. 1958, 65-86. [15J --, "The Theol'Y of Portfolio Selection," The TMo,y of [merest Rates, F. H. Hahn and F. P. R. Brechling (eds.) (London: Macmillan, 1965). [16J Williams, J. B., "Speculation and the Carryover," Qut.u'terly Journal of Economics SO (May 1936), 436-455 . --- ## Page 502 473 32 Models of Optimal Capital Accumulation and Portfolio Selection and the Capital Growth Criterion* William T. Ziemba Professor Emeritus, University of British Columbia, Vancouver, BC Visiting Professor, Mathematical Institute, Oxford University, UK ICMA Centre, University of Reading, UK University of Bergamo, Italy Raymond G. Vickson Professor Emeritus, University of Waterloo Abstract T his edited and updated excerpt from the Ziemba and Vickson (1975, 2006) volume discusses various aspects of the history of optimal capital accumulation and the capital growth criterion, and supplements the introduction to Part IV of this volume. 1 Optimal Capital Accumulation Phelps (1962) extended the Ramsey (1928) model of lifetime saving to include un- certainty. In Phelps' problem, the choice in each period was between consumption and investment in a single risky asset, with the objective being the maximization of expected utility of life time consumption. Phelps assumed that the utility func- tion was additive in each period's utility for consumption, and he obtained explicit solutions when each utility function was the same and a member of the isoelastic marginal utility family. The papers reprinted in Ziemba and Vickson (1975, 2006) (ZV) and in this volume generalize and extend the work of Phelps in several direc- tions: Portfolio choice is included in Samuelson (1969) and Hakansson (1970, 1971a) papers, and more general utility functions are treated in Neave (1971b), which is in ZV. These papers also constitute generalizations of the multiperiod consumption- investment papers of Part IV.l Neave (1971 b) treats the multiperiod consumption-investment problem for a preference structure given by a sum of utility functions of consumption in different periods, including a utility for terminal bequests. Future utilities are discounted by • Reprinted, edited and updated from Ziemba and Vickson (1975, 2006). lSee Zabel (1973) for an interesting extension of the Phelps-Samuelson-Hakansson model to include proportional transaction costs. --- ## Page 503 474 W T Ziemba and R. G. Vickson an "impatience" factor, compounded in time. The portfolio problem is neglected, with the only decisions at any time being the amount of wealth to be consumed, the non-negative remainder being invested in a single risky asset. In each period, terminal wealth consists of a certain, fixed income plus gross return on investment, and this terminal wealth becomes initial wealth for the following period, from which consumption and investment decisions are again to be made, and so forth. Neave is concerned with conditions under which risk-aversion properties are preserved in the induced utility functions for wealth. He demonstrates the important result that the property of non-increasing Arrow-Pratt absolute risk aversion is preserved under rather general conditions. For inter temporally independent risky returns, if the single-period utilities for consumption and the utility for terminal bequests exhibit non-increasing absolute risk aversion, then all the induced utility functions for wealth also have this property. Mathematically, this result arises from the fact that convex combinations of functions having decreasing absolute risk aversion also have this property (see Pratt, 1964, reprinted in ZV). Thus, the decreasing absolute risk-aversion property is preserved under the expected-value operation, and is further preserved under maximization. The paper specifically includes the possibility of boundary as well as interior solutions in the presentation. Neave (1971b) also considers the behavior of the Arrow-Pratt relative risk- aversion index of induced utility for wealth. He presents sufficient conditions under which the property of non-decreasing relative risk aversion is preserved. Since this property is not, in general, preserved under convex combinations, results can be ob- tained only under special conditions. Unfortunately, these conditions are not easily interpreted, and involve the specific values of the optimal decision variables as well as the properties of the utility functions. In general, however, non-decreasing rel- ative risk aversion obtains for sufficiently large values of wealth. In ZV's Exercise V-ME-6, the reader is asked to attempt to widen Neave's classes of utility functions that generate desirable risk-aversion properties for the induced utility functions. Neave also relates properties of the relative risk-aversion measures to wealth elas- ticity of demand for risky investment. Arrow (1971) performed a similar study in the context of choice between a risk-free and a risky asset, with no consumption. In this case, the choice is between consumption and risky investment, with no risk-free asset. Neave shows that if the utilities for consumption and terminal wealth in any period exhibit constant relative risk aversion equal to unity, then the wealth elastic- ity for risky investment in that period is also equal to unity. He further shows that the wealth elasticity of risky investment exceeds unity if and only if the relative risk-aversion index of expected utility for wealth is less than or equal to that of utility for consumption in the relevant period. Samuelson (1969) studies the optimal consumption-investment problem for an investor whose utility for consumption over time is a discounted sum of single-period utilities, with the latter being constant over time and exhibiting constant relative risk aversion (power-law functions or logarithmic functions). Samuelson assumes --- ## Page 504 Models of Optimal Capital Accumulation 475 that the investor possesses in period t an initial wealth W t which can be consumed or invested between two assets. One of the assets is safe, with constant known rate of return r per period, while the other asset is risky, with known probability distribution of return Zt in period t. The Zt are assumed to be intertemporally independent and identically distributed. Samuelson thus generalizes Phelps' model to include portfolio choice as well as consumption. Terminal wealth Wt + 1 in period t consists of gross return on investment, and becomes initial wealth governing the consumption-investment decisions in period t + 1. The problem is to determine the fraction of wealth to be consumed and the optimal mix between riskless and risky investment in each period (given an initial wealth wo), so as to maximize expected utility of lifetime consumption. The usual backward induction procedure of dynamic programming yields a sequence of induced single-period utility functions Ut for terminal wealth in period t, with optimal consumption and investment in period t being determined by maximizing expected utility of consumption plus terminal wealth. Assuming interior maxima in each period, a recursive set of first-order conditions is derived for the optimal consumption-investment program. The explicit form of the optimal solution is derived for the special case of utility functions having constant relative risk aversion. Samuelson shows that the optimal portfolio decision is independent of time, of wealth, and of the consumption decision at each stage. This optimal portfolio is identical with the optimal portfolio result- ing from the investment of one dollar in the two assets so as to maximize expected utility of gross return, using the instantaneous utility for consumption as the "utility for wealth" function in the optimization procedure. Furthermore, in each period, the optimal amount to consume is a known fraction of initial wealth in the period, with the optimal fraction being dependent only on the subjective discount factor for consumption over time, the return on the risk-free asset, the probability distri- bution of risky return, and the relative risk-aversion index. For logarithmic utilities, the optimal consumption fraction further simplifies to a function of the subjective discount factor alone, and is independent of the asset returns. Samuelson utilizes his solutions to analyze optimal consumption and investment behavior as a function of time of life. As observed above, the optimal mix of riskless versus risky investment is (for constant relative risk aversion) completely independent of time of life. Samuelson relates the optimal consumption over time to the asset returns and the subjective discount factor, and discusses conditions under which the consumer-investor will save during early life and dissave in old age. Throughout the paper, Samuelson assumes the validity of interior maxima in the optimization problems. The validity of such an assumption in the multiperiod portfolio problem was seriously questioned in Hakansson (1970, 1971a), and in ZV- Exercises including IV-CR-3, 5, and 6. When consumption decisions are included, as they are in the present case, the question of boundary solutions is presumably even more important than it was in the pure investment case. The reader may wish to --- ## Page 505 476 W T. Ziemba and R. G. Vickson ponder whether these considerations can materially affect Samuelson's conclusions. ZV-Exercise V-CR-16 shows that a stationary policy is not optimal in a multiperiod mean-variance model even under strong independence assumptions. Hakansson (1970) studies a generalization of the problem treated by Samuelson (1969). Hakansson's assumptions regarding utility for consumption are identical to Samuelson's, except for an assumed infinite lifetime. The investment possibilities facing Hakansson's investor are, however, more general. The individual is assumed to possess a (possibly negative) initial capital position plus a non-capital steady income stream which is known with certainty. In each period, borrowing and lending at a known, time-independent rate of interest, and investment in a number of risky assets with known joint distribution of returns, are possible. The risky asset returns are assumed to be independent identically distributed (iid) over time. It is further assumed that a subset of the risky assets may be sold short. In each period, initial wealth to be consumed and invested consists of the certain income plus gross return on the previous period's investment. The objective is maximization of the expected utility of consumption over an infinite lifetime. Hakansson introduces two important and apparently realistic constraints on the risky returns and the investor's decision variables. The capital market is assumed to impose a so-called no-easy-money restriction on the risky asset returns. This restriction ensures that no mix of risky assets exists, which provides with proba- bility 1 a return exceeding the risk-free rate, and that no mix of long and short sales exists, which guarantees against loss in excess of the riskfree lending rate. The consumption-investment decision variables are assumed to satisfy the so-called solvency constraint. This requires that the investor always remain solvent with probability 1, i.e., that in any period t, the investor's initial capital position plus the capitalized value of the future income stream be non-negative with certainty. Hakansson uses the familiar Bellman backward induction procedure of dynamic programming to set up a functional equation for the optimal return function (i.e., the maximum expected utility of present and future consumption, given any value of initial wealth). Since the risky returns are iid and the lifetime infinite, the func- tional equation obtained is stationary over time: The optimal decision variables in period t depend only on initial wealth (in period t), on the single-period asset returns, and on the relative risk-aversion index of consumption. Hakansson's treat- ment of the infinite horizon dynamic programming problem is intuitively plausible, but not wholly satisfactory from a rigorous standpoint. (The results are correct, however.) First, he assumes that the optimal return function exists and satisfies the functional equation; in a totally rigorous development, these facts need proof. Next, he verifies the form of the optimal solution by direct substitution, that is, by showing that the assumed form of solution actually solves the functional equation. Here the question of uniqueness arises, and is not treated satisfactorily in the de- velopment. One way of treating the problem rigorously would be to deal first with the finite horizon problem (as Samuelson does) and subsequently go to the infinite --- ## Page 506 Models of Optimal Capital Accumulation 477 horizon limit. At each stage in the finite horizon backward induction procedure, one would have a functional equation similar to Hakansson's (but rigorously jus- tified). As shown in the paper, the decision variable feasibility region defined by the solvency constraint is compact and convex, and (as in Hakansson's proof of his lemma, and its corollaries) the functional equation has a finite, unique solution. By considering the limiting behavior of the optimal solution and optimal return func- tion, the given infinite horizon results could be justified. For the case of a power-law utility function unbounded from above, Hakansson needs a restriction on the utility function parameters, the optimal risky asset returns, and the subjective discount factor, in order to obtain a solution. In a limiting argument as outlined above, such a restriction would be required in order to obtain a finite return function in the limit of infinite horizon. For constant relative risk-aversion utilities, the form of the optimal solution is (i) a fixed fraction of initial wealth plus capitalized future income is consumed in each period; and (ii) a fixed proportion of initial wealth plus capitalized future income is invested in each of the risky assets, with the optimal asset proportions being independent of time, initial wealth, capitalized income stream, and impatience to consume. The remainder of initial wealth is invested in the risk-free asset. For logarithmic utilities, the optimal consumption fraction is completely independent of all asset returns, and is a function only of the impatience factor. Hakansson also considers the interesting case of constant absolute risk aversion (exponential utilities) , and states without proof the form of the optimal policy under special restrictions. The conditions required for the solution and the form of the optimal policy are more difficult to interpret than the corresponding results for the constant relative risk-aversion case. The reasons for this different behavior in the two classes of utilities may be understood as follows. For a power-law (or logarithmic) utility, expected utility of terminal wealth decomposes into a product (or sum) of utility of initial wealth and expected utility of rate of return on one dollar of investment (since terminal wealth equals initial wealth times rate of return). Such a factorization does not obtain for exponential utilities. See ZV Exercise IV-CR-3 for a clarification of this point in a simple example. A number of conclusions regarding consumption and investment behavior are derived as corollaries to the optimal solution. As stated previously, the optimal level of consumption is a fixed fraction of initial wealth plus capitalized future in- come. Furthermore, the optimal amount of consumption increases with increasing impatience to consume, as expected. The optimal solution is such that the indi- vidual borrows when poor and generally lends when rich. Additional behavioral implications of the solutions are discussed in the paper. In the Samuelson-Hakansson additive utility models of this section, the individ- ual's consumption versus saving and investment behavior may vary as a function of time of life, but attitudes toward risk do not. In these models, the same risky portfolios are chosen in youth and in old age. In ZV-Exercise V-ME-21, a multiplica- --- ## Page 507 478 W T. Ziemba and R. G. Vickson tive utility model, based on Pye (1972) , is developed. In this model, risk aversion in portfolio selection increases or decreases with age according as risk tolerance is greater or less than that of the logarithmic utility function. Logarithmic utility is a member of both the additive and multiplicative families. In the multiplicative fam- ily, risk aversion in portfolio selection depends on impatience to consume as well as on age. The multiplicative family thus allows greater richness in portfolio selection behavior over time. On the other hand, consumption behavior over time is simpler in the multiplicative model: Given the impatience factor and the length of remain- ing life, the propensity to consume is identical for all members of the multiplicative family, and is independent of present and future investment opportunities. Further- more, in the infinite horizon limit, portfolio selection behavior changes over time, but consumption behavior does not: A constant fraction of wealth is consumed in every period. To summarize, these papers show that an optimal lifetime consumption- investment program is simple to calculate for additive utilities belonging to the constant relative risk-aversion family. The optimal portfolio remains unchanged throughout life for asset returns which are iid over time. Consumption in any period is always proportional to wealth (including capitalized future income), the proportionality factor being dependent in general on impatience to consume, on the capital market, and possibly also on time of life. The optimal portfolio and the opti- mal consumption fraction are easily determined by solving a static, one-period, pure portfolio problem using utility for consumption as the appropriate utility for wealth function. Effects due to age-dependent risk aversion can be incorporated through a multiplicative form of utility function. In this case, the optimal consumption program may be calculated (in the absence of exogenous income, at least) without solving any "optimization" problem and without reference to the capital market. Optimal portfolio choice depends on time, however, and must be continually recal- culated in successive periods. In both the additive and multiplicative cases, a useful form of myopia obtains: In each period, past or future realized random outcomes need not be known in choosing the optimal portfolio, and past outcomes affect present consumption decisions only by fixing the available wealth. Some of the conclusions would remain unchanged under a slight weakening of hypotheses. In particular, the assumption of identically distributed asset returns over time may be dropped (while retaining independence). For additive utilities having constant relative risk aversion, the optimal amount of consumption will still be proportional to wealth. The relative risk aversion of induced utility for wealth will remain constant over time. The optimal asset proportions at any time t will be independent of wealth, consumption, and past or future asset returns, and will de- pend only on the nature of the capital market at time t. Calculation of the optimal consumption fraction at time t will, however, generally require the solution of all "future" consumption and portfolio problems, and will thus be much more complex than in the iid case. Hakansson (1971b) studied a significant generalization of his --- ## Page 508 Models of Optimal Capital Accumulation 479 model where the individual's impatience is variable, his lifetime is stochastic, and the capital assets (both risk free and risky) change in time. The individual con- sumes, borrows (or lends), invests in risky assets, and purchases life insurance. For additive utilities having constant relative risk aversion, solutions are obtained for the optimal lifetime consumption, investment, and insurance-buying program. For logarithmic utilities, Miller (1974ab) analyzed the optimal consumption strategies over an infinite lifetime with a stochastic (possible nonstationary) income stream and a single, safe asset. Rentz (1971, 1972, 1973) has analyzed optimal consump- tion and portfolio policies with life insurance, for stochastic lifetimes and changing family size. Additional extensions are in Hakansson (1969), Long (1972), and Neave (1971a). 2 The Capital Growth Criterion The papers by Breiman and Thorp are concerned with Kelly's capital growth cri- terion for long-term portfolio growth. The criterion states that in each period one should allocate funds to investments so that the expected logarithm of wealth is maximized. Hence, the investor behaves in a myopic fashion using the stationary logarithmic utility function and the current distribution of wealth. Kelly (1956) supposed that the maximization of the exponential rate of growth of wealth was a very desirable investment criterion. Mathematically the criterion is the limit as time t goes to infinity of the logarithm of period t's wealth relative to initial wealth divided by t. If one considers Bernoulli investments in which a fixed fraction of each period's present wealth is invested in a specific favorable double-or-nothing gamble, then the criterion is easily shown to be equivalent to the maximization of the expected logarithm of wealth. ZV Exercise V-CR-18 illustrates the calculations involved and related elementary properties of Kelly's criterion. The desirability of Kelly's criterion was further enhanced when Breiman [Breiman (1961) and his 1960 article reprinted here] showed that the expected log strategy produced a sequence of decisions that had two additional desirable proper- ties. First, if in each period two investors have the same investment opportunities and one uses the Kelly criterion while the other uses an essentially different strategy (i.e., the limiting difference in expected logs is infinite), then in the limit the former investor will have infinite times as much wealth as the latter investor with proba- bility 1. Second, the expected time to reach a pre-assigned goal is asymptotically (as the goal increases) least with the expected log strategy. Latane (1959) provided an earlier intuitive justification of the first property. Breiman (1961) established these results for discrete intertemporally indepen- dent identically distributed random variables. In this case, as in the Bernoulli case, a stationary policy is optimal. In ZV Exercise V-ME-22, the reader is asked to develop these and related results for the Bernoulli case. The main technical tool involved in the proof is the Borel strong law of large numbers, which states that --- ## Page 509 480 W T Ziemba and R. G. Vickson with probability 1 the limiting ratio of the number of successes to trials equals the Bernoulli probability of a success. Breiman's paper reprinted here proves the first result for positive bounded random variables. Results from the theory of martin- gales form the basis of proof of his conclusions. ZV Exercise V-CR-19 provides the reader with some elementary background on martingales. The reader is asked in ZV Exercise V-CR-20 to consider some questions concerning Breiman's analysis. The proof of the results in Breiman's paper involves difficult and advanced math- ematical tools. ZV Exercise V-ME-23 shows how similar results may be proved in a simple fashion using the Chebychev inequality. The crucial assumption is that the variance of the relative one-period gain be finite in every investment period. Breiman's assumption that the random returns are finite and bounded away from zero is sufficient but not necessary for the satisfaction of this assumption. ZV Ex- ercise V-CR-22 presents an extension of Breiman's model to allow for more general constraint sets and value functions. The development extends his Theorem 1 and indicates that no strategy has higher expected return than the expected log strat- egy in any period. The model applies to common investment circumstances that include borrowing, transactions and brokerage costs, taxes, possibilities. of short sales, and so on. In ZV Exercise V-ME-24, the reader is invited to consider the re- lationship between properties 1 and 2, to attempt to verify the validity of property 2 for Breiman's model reprinted here, and to consider the discrete asset allocation case. Thorp's paper provides a lucid expository treatment of the Kelly criterion and Breiman's results. He also discusses some relationships between the max expected log approach and Markowitz's mean-variance approach. In addition, he points out some of the misconceptions concerning the Kelly criterion, the most notable being the fact that decisions that maximize the expected log of wealth do not necessarily maximize expected utility of terminal wealth for arbitrarily large time horizons. The basic fallacy is that points that maximize expected log of wealth do not generally maximize the expected utility of wealth if an investor has nonlogarithmic utility function; see ZV Exercise V-ME-25 for one such example and Thorp and Whitley (1973) for a general analysis. See Markowitz (1976) for a refutation, in a limited sense, of the fallacy if the investor's utility function is bounded. For some enlighten- ing discussion of this and other fallacies in dynamic stochastic investment analysis see Merton and Samuelson (1974) and ZV Exercise V-ME-26. Miller (1974b) shows how one can avoid the fallacy altogether by utilizing what is called the utility of an infinite capital sequence criterion. Under this criterion, the investor's utility is assumed to depend only on the wealth levels in one or more periods infinitely dis- tant from the present; that is, capital is accumulated for its own sake, namely its prestige. In this formulation, the time and expectation limit operators are reversed (from the conventional formulation) and hence the improper limit exchange that yields the fallacy does not need to be made. One disadvantage of this formulation is that the admissible utility functions generally are variants of the unconventional --- ## Page 510 Models of Optimal Capital Accumulation 481 form: limit infimum of the utility of period t's wealth. The reader is invited in ZV Exercise V-CR-21 to consider some questions concerning Thorp's paper. In ZV Ex- ercise V-ME-27, the reader is asked to determine whether good decisions obtained from other utility functions have a property "similar" to the expected log strategy when they produce infinitely more expected utility. For additional discussion and results concerning the Kelly criterion the reader may consult Aucamp (1971), Breiman (1961), Dubins and Savage (1965), Goldman (1974) , Hakansson (1971a,b,d) , Hakansson and Miller (1972), Jen (1971, 1972), Latane (1959, 1972), Markowitz (1976), Roll (1972), Samuelson (1971) , Merton and Samuelson (1974) , Thorp (1969), Young and Trent (1969), and Ziemba (1972b). For an elementary presentation of the theory of martingales the reader may consult Doob (1971). More advanced material may be found in the work of Breiman (1968), Burrill (1972), and Chow et al. (1971). The highly technical Merton paper discusses the optimal consumption technical investment problem in continuous time. Because of its heavy reliance on stochastic differential equations and stochastic optimal control theory, the paper may be quite intimidating upon first reading. To make the paper intelligible to readers unfamiliar with these mathematical concepts, we deviate from our previous policy of keeping formalism out of the introduction: Appendix A contains a brief, intuitive introduc- tion to stochastic differential equations and stochastic control theory. Although not rigorous, the arguments are, hopefully, sufficiently plausible as to enable the reader to understand Merton's paper with relative ease. Merton assumes the investor's utility for lifetime consumption to be an integral of utilities for instantaneous consumption over time. This is a natural, continuous-time version of the additive utility assumption in discrete time. The general formalism of the optimal control problem is outlined first for utilities that vary arbitrarily in time and is specialized in later sections to the pure "impatience" case, as in the Samuelson and Hakansson discrete time models. Most of Merton's explicit solutions pertain to utility functions of the hyperbolic absolute risk aversion (HARA) class, that is, to the class where the reciprocal of the Arrow-Pratt absolute risk-aversion index is linear in wealth. This class contains as special cases utilities having constant absolute or relative risk aversion. The individual must decide at each point in time how to allocate his existing wealth to between consumption and investment in a number of financial assets. The risky asset returns are governed by a known Markov process. The objective is maximization of expected utility of lifetime consumption, including terminal bequests. In the spirit of dynamic programming, there exists at each time t a derived util- ity function for wealth w, namely, the maximum expected utility of future lifetime consumption given that wealth is w at time t. The instantaneous consumption- investment problem requires the maximization of utility of consumption and ter- minal wealth at time t, or rather, at an "infinitesimal" time later than t. For risky asset returns governed by a stationary log-normal process (see Appendix A), --- ## Page 511 482 W. T. Ziemba and R. G. Vickson terminal wealth at the end of the "infinitesimal" time span is almost deterministic. The expected utility maximization thus involves only means and variances of risky returns, as in the Samuelson paper in Part III, Chapter 1. The portfolio selection aspect of the continuous-time consumption-investment problem is thus solved ex- actly using mean-variance analysis. As in the Lintner and Ziemba papers in Part III, this implies that the optimal portfolio possesses the mutual fund separation property: All investors will choose a linear combination of two composite assets, which are, moreover, independent of individual preferences. If there exists a risk- free asset, it may be chosen as one of the mutual funds, and all portfolios reduce to a mix of this risk-free asset and a single composite risky asset which is the same for all investors. The returns on the mutual fund are also governed by a stationary log-normal process. Note that such a result is only true in continuous time. In discrete time, a non-negative linear combination of log-normally distributed random returns is not log-normally distributed. Through the use of a continuous- time formulation, the consumption-investment problem is thus reduced to a two- asset model with consumption. Using this reduction, Merton obtains explicit solu- tions for the optimal consumption-investment program for utility functions of the HARA class. As in the papers of Part V, the optimal amount of consumption and the optimal amount of investment are linear functions of wealth. Furthermore, when a steady noncapital income stream is included, this remains true with "wealth" reinterpreted as present wealth plus future income (capitalized at the risk-free rate of return). Merton discusses the effects on optimal consumption and investment due to the possibility of large but rare random events. Specifically, among such rare but significant events which he discusses are: (1) a bond which is otherwise risk free, may suddenly become worthless, or (2) the investor dies. The modeling of such processes in continuous time is achieved through the use of Poisson differential equations (see Appendix A). Merton derives the optimal consumption-investment policy for choice between a log-normally distributed common stock and a "suddenly worthless" bond, for utilities having constant relative risk aversion. As before, the optimal amount of consumption is linear in wealth. The optimal investment in the common stock is an increasing function of the probability of bond default, as expected. Merton also discusses the effect of uncertain lifetime, using a stationary Poisson process to model the arrival of the consumer's death. He shows that the optimal consumption-investment problem in this case (with no bequests) is identical to that of an infinite-lifetime model, with the consumer's utility discounted by a subjective rate of time preference equal to the reciprocal of life expectancy. There are a number of other topics covered in the paper, which we mention only briefly. These pertain to the important and usually neglected question of nonstationary random asset returns. Merton presents solutions for the optimal consumption-investment problem in three such nonstationary models. In the first model, he assumes there exists an asymptotic, deterministic price curve pet) , toward --- ## Page 512 Models of Optimal Capital Accumulation 483 which the stochastic asset prices tend (in expected value) in the distant future. In the second model, he assumes that the risky return is given by a log-normal distri- bution whose mean is itself a random process of a special form. Finally, in the third model, he assumes that the risky return is given by a stationary log-normal pro- cess whose parameters are unknown, but must be estimated from past behavior of the random variable. Explicit solutions for these three models are obtained in a two-asset setting, for utility functions having constant absolute risk aversion. ZV Exercise V-ME-28 considers a deterministic continuous-time consumption- investment model. If utility is intertemporally additive, then it is possible to develop explicit optimality criteria that yield an algorithm for constructing an optimal policy. 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Working paper, Queen's University, Kingston, Ontario, Canada. Neave, E. H. (1971b). Optimal consumption-investment decisions and discrete-time dy- namic programming. Journal of Economic Theory, 3, 40-53. Phelps, E . S. (1962) . The accumulation of risky capital: A sequential utility analysis. Econometrica, 30, 729- 743. Pratt, J. W. (1964) . Risk aversion in the small and the large. Econometrica, 32, 122-136. Pye, G. (1971) . Minimax policies for selling an asset and dollar averaging. Management Science, 17, 379- 393. Ramsey, F . P. (1928) . A mathematical theory of saving. Economic Journal, 38, 543-559. Rentz, W . F . (1971). Optimal consumption and portfolio policies. Unpublished disserta- tion, University of Rochester, Rochester, New York. Rentz, W. F . (1972). Optimal multi-period consumption and portfolio policies with life insurance. Working Paper No. 72-36, Graduate School of Business University of Texas, Austin. Rentz, W . F . (1973) . The family's optimal multi period consumption, term insurance and portfolio policies under risk. Working Paper No. 73-20. Graduate School of Business, University of Texas, Austin. Roll, R. (1972). Evidence on the 'growth-optimum' model. Journal of Finance, 28, 551- 566. Samuelson, P. A. (1969). Lifetime portfolio selection by dynamic stochastic programming. Review of Economics and Statistics, 51 , 239-246. Samuelson, P. A. (1971). The fallacy of maximizing the geometric mean in long sequences of investing or gambling. Proceedings National Academy of Science, 68, 2493- 2496. Thorp, E. O. (1969). Optimal gambling systems for favorable games. Review of the Inter- national Statistical Institute, 37(3), 273- 293. --- ## Page 514 Models of Optimal Capital Accumulation 485 Thorp, E. O. and R. Whitley (1973). Concave utilities are distinguished by their opti- mal strategies. Working Paper, Mathematics Department, University of California, Irvine. Young, W. E. and R . H. Trent (1969). Geometric mean approximations of individual security and portfolio performance. JFQA , 4, 179- 199. Zabel, E. (1973). Consumer choice, portfolio decisions and transaction costs. Econometrica, 44, 321- 335. Ziemba, W. T. (1972). Note on optimal growth portfolios when yields are serially correlated. JFQA , 7, 1995- 2000. Ziemba, W. T. and R. G. Vicks on (Eds.) (1975). Stochastic Optimization Models in Finance. NY: Academic Press. Ziemba, W. T. and R. G. Vicks on (Eds.) (2006). Stochastic Optimization Models in Finance, (2 ed.) . Singapore: World Scientific. --- ## Page 515 This page is intentionally left blank --- ## Page 516 487 33 Pro<. Nat.. Acad. Sci. USA Vol. 68, No. 10, pp. 2493-2496, Ocoober 1971 The "Fallacy" of Maximizing the Geometric Mean in Long Sequences of Investing or Gamhling (maximum geometric mean strategy/uniform stratqies/aeym.ptotieally auflicient param~terR) PAUL A. SAMUELSON Department of Economics, M888achuaetta Institute o( TethnoloQ', Cambridge, Ma&!. 02139 ABSTRACT Because the outcomes of repeated invest- ments or gambles involve products of variables, authori- ties have repeatedly been tempted to the belief that, in • long sequence, maximization of the expected value of terminal utility can be achieved or well ... approximated by a strategy of maximizing at each stage the geometric mean of outcome (or its equivalent, the expected value of the logarithm of principal plus return). The Jaw of large numbers or of the central limit theorem as applied to the logs can validate the conclusion that a maxiJnum-geo- metric-mean strategy does indeed. make it ''virtually certain" that, in a "long" sequence, ODe will end with a hia;her tenninal wealth and utility. However, this does not imply the false coroUary that the geometric-mean etratqy is optimal Cor aoy finite nutnber oC periodst however long, or that it becomea aaymptotically a good approximation. A. a trivial counter-example, it 1& shown that lor utility proportional to s'" /"1, whenever "I F 0, tbe geometric strategy iB sUboptimal Cor all T and never a good approd .. mation. For utility bounded above, &II when "I < 0, the same conclusion holds. If utility fa bounded above and finite at zero wealth, no uniform strategy ea.n be optimal, even though it can be that the best unilorm strategy will be that of the maIimulu geometric meaD. However, asymptotically the same level of utility can be reached by an infinity of nearby uniform Itrategiee. The true opti. mum in the bounded ease involvee nonuniform strategiest usually being more risky than the geometrio--lIlean maxi ... mizer's strategy at low wealth. and lees risky at high wealths. The Dovel criterion of muimiziDg the expected average colIlpound return, which uy:rn.ptotiealJy lead. to maximizing of geometric mean, is shown to be arbitrary. BACKGROUND Suppose one begins with initial wealth, X" and after a series of decisions one is left with terminal wealth, X T, suhj""t to a conditional probability distribution prob (XT ~ xix, = AI = Pr(X,A) (I) Then an "expected utility" maximizer, by definition, will choose his decisions to max E/u(Xr ) I = max f-== u(X)Pr(dX,A), (2) where E stands for the "expected value" and where u(x) is a specified utility function that is unique except for arbitrary scale and origin parameters, band 0, in a + bu(x), b > O. In a portfolio, or gambling situation, at each period one can make investments or bets proportional to wealth at the begin- ning of that period, X., so that the outcome of wealth for the next period is the random variable 2493 X.+, = X.(w,y, + .. + w.y.). tWI = I (3) 1 where the w's are the proportions decided upon for investment in the difierent securities (or gambling games). The returns per dollar invested in each alternative, respectively, are subject to known probability distributions. prob (y,::; y" ... ,Y.::; y./ = F(y" .. . ,y.), (4) where for simplicity the vector outcomes at any t are assumed each to be independent of outcomes at any other time periods, and to remain the same distribution over all time periods. Usually Y, is restricted to being nonnegative to avoid bank- ruptcy. A complete decision involves selecting over the inter- val of time t = 1,2, . .. ,T, all the vectors [WI(t»), which will be nonnegative if short-selling and bankruptcy are ruled out. In particular, if the same strategy is followed at all times, so that wit) a W10 the variables become Xl = XoXlr X, = XIX! = Xl)ElX~ (5) X, = X,-JXt = XoX,X2 . Xt and all x, are independently distributed according to the same distribution, which we may write as prob (x. ~ x/ = nix). (6) Of course, flex) will depend on the W strategy chosen, and is short for flex; W'" . . ,w.) in this particular case, and on the assumed F(y" . .. ,Y.) function. It can be ellSily shown that prob (X. ;:;; xiX, = Z/ = p.(X,Z) p.(X,Z) = p.(X/Z,I) (7) = P .(X/ Z) for short and P.(x) ""n(x) P,(x) = f-== P,(x/ s)dP,(s) (8) --- ## Page 517 488 2494 Applied Mathematical Sciences: Samuelson EXACT SOLUTIONS For general u(.) functions, a different decision-vector [wl(I)] is called for at each intermediate time period, I = 1,2, ... ,T - 1. This is tedious, but both inevitable and feasible. It is well known that for one, and only one, family of utility functions, namely ,,(x;,,) ~ x'h ,,7'0 (9) log x 1'~O it is optimal to use the same repeated strategy. For this case the common optimal strategy is that of a T = 1 period prob- lem, namely Xo' max Ex'/-y = Xo'max {m (x'/1')dP, (x;w'" .,w.) 'lD1 to; Jo (10) The log x case is included in this formulation, since as l' - 0, the indeterminate form is easily evaluated. PROPOSED CRITERION Often in probability problems, as the number of variables becomes large, T - 00. In this case, certain asymptotic simplifications become feasible. Repeatedly, authorities (1-3) have proposed a drastic simplification of the decision problem whenever T is large. Rule. Act in each period to maximize the geometric mean or the expected value of log x,. The plausibility of such a procedure comes from recognition of the following valid asymptotic result. TMorem. If one acts to maximize the geometric mean at every step, if the period is IIsufficiently long," Hahnost cer~ tainly" higher terminal wealth and terminal utility will result than from any other decision rule. To prove this obvious truth, one need only apply the central- limit theorem, or even the weaker law of large numbers, to the sum of independent variables T log X T = log Xo + L: 10gI, (11) I We may note the following fact about P1'(x; max g.m.) = Q1"(x) and P1'(x; other rule) = QT(X). for T > M(x) (12) The crucial point is that M is a function of x that is un- bounded in x. From this indisputable fact, it is tempting to believe in the truth of the following false corollary: False Corollary. If maximizing the geometric mean almost certainly lead. to a better outcome, then the expected utility of its outcomes exceed. that of any other rule, provided T is sufficiently large. The temptation to error is compounded by the considera- tion that both distributions approach asymptotically log normal distributions. 1 ~~g1'·(x) ~ VZ;~'T'/' P A. Samuelson Proc. Nal. Acad. Sci. USA 68 (1971) Since 1" > I' by hypothesis, it follows at once that, for T large enough LT"(x) can be made smaller than L1'(x) for any x. From this it is thought, apparently, that one can val.idly deduce that J.m U(X)dQT' > i m u(x)dQ1'(x). FALSITY OF COROLLARY A single example can show that the needed corollary is not generally valid. Suppose" = 1, and one acts, in the fashion recommended by Pascal, to maximize expected money wealth itself. Let the gambler-investor face a choice between invest- ing completely in safe cash, Y 1, or completely in a "security" that yields for each dollar invested, $2.70 with probability 1/2 or only $0.30 with probability 1/ 2. To maximize the geometric mean, one must stick only to cash, since [(2.7)(.3) ]'" = .9 < 1. But, Pascal will always put all his wealtb into the risky gamble. Isn't be a fool? If he wins and loses an equal number of bets, and in the long run that will be his median position , he ends up with (2.7)'(.3)1' ~ (.9)' _ 0 as T _ 00 (14) UAlmost certainly," he will be "virtually ruined" in a. "long enough" sequence of pLay. No, Pascal is not a fool according to his criterion. In those rare long sequences (and remember all sequences are finite, albeit very large) when he does experience relatively many wins, he ma.kes more than enough to compensate him, accord- ing to the max EX T criterion, for the more frequent times when he is ruined. Actually, f T } E{XT } ~ EI.XoI,Ix. = X.IT E{x,}, for independent variates (15) I = X.{EX.}T =XOJ.5T > X.F for aU T But, you may say, Pascal is foolish to court ruin just for a large money gain. A dollar he wins is surely worth less in utility than tbe dollar he loses. Very well, as with eighteenth- century writers on tbe St. Petersburg Paradox, let us assume a concave utility-function, u(x), with u'(x) < O. Let us now test the false corollary for u(x) = x'/-y, 1 > l' ;c O. Let the decision according to the geometric mean rule lead to E{logx,'}>Elogx, (16) But let the alternative decision produce E{x,'h} > E{x,*'h} (17) Then, as before, except for the changed value of tbe l' ex- ponent (18) 1 T~m Q1'(x) ~ ";'h"T'I> f lO" {I } -m exp - 2 [8 - "T]'/~ 'T ds = L1'(x) (13) --- ## Page 518 The "Fallacy" of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling 489 Pro<. Nat. Acad. Sci. USA 68 (1971) since T E{XT'hl = Xo'IIE{x.'hl I = Xo'[E{x,'h}]T (\9) > X'[E{x,*'hl]T = EXT*'h This strong inequality holds for all T, however large. Thus, the false corollary is seen to be invalid in general, even for concave utilities. BOUNDED UTILITIES Most geometric-mean maximizers are convinced by this reasoning (5). But not all. Thus, Markowitz, in the new 1971 preface to the reissue of his classic work on portfolio analysis (4), says that "bounded ness" of the utility function will aave the geometric-mean rule. For 'Y < 0, U - x'/'Y will be bounded from above, and a run of favorable gains will not bring the decision maker a. utility gain of more than 8. finite amount. But, as the case 'Y = -\ shows, that cannot save the false corollary. For in all cases when 'Y < 0; the false rule leads to over-riskiness, just as for 'Y > 0 it leads to under-riskines. Those few times, and they will happen for all T, however large, with a positive (albeit diminishing probability), whenever the false rule brings you closer to zero terminal wealth (or ruin), the unboundedness of u as x - 0 and u - - 00 puts a pro- hibitive penalty against the false rule. What about the case where utility is bounded above and is finite at X T = 0 or ruin? It is easy to show that 11b uniform decision rule can be optimal for such a case, where - OJ < u(O) ~ u(X) ~ M < OJ (20) But suppose we choose among all SUboptimal uniform strategies that have the property of "limited liability," so that each x, is confined to the range of nonnegative numbers. Then the following theorem, which seems to contain the germ of truth the geometric mean maximizers are groping fOT, is valid. Theorem. For u(x) bounded above and below in the range of nonnegative numbers, the uniform rule of maximizing the geometric mean E{log X T·}, will asymptotically outperform any other uniform strategy's result, E{ u(X T ) L in the follow- ing sense E{u(X T ·)} > E{u(X T»), T > T(X.) (21) The limited worth of this theorem is weakened a bit further by the consideration that an infinite number of blends of a suboptimal strategy with the geometric mean rule, of the form vx· + (I - v)x, v sufficiently small and positive, will do negligibly worse as T - "'. Thus for all Xo, lim E {u(X T·)} = u(O) or M = sup u(X), (22) T_w x depending upon whether E(log x,·) ~ 0 or > 0 For a range of v's, the same utility level will be approached as T- (I), OPTIMAL NONUNIFORM STRATEGY To illustrate that no uniform strategy can be optimal when utility is bounded, consider the well-known case of u .... _e-&S'. Maximizing the Geometric Mean 2495 Suppose X I can be invested at each stage into Z I dollars of a risky Y,such thatE/Y,} > 1, or into X, - Z. dollars of safe cash. Under "limited liability," if the lowest y, could get were r, Z. could not exceed X J(I - r). First, disregard limited liability and permit negative X T. Then a well-known result of Pfanaangl (6) shows that at every stage, it is optimal to set Z. = Z·, where m~x f-~~ - exp[ - b(X -Z) - ~Y ,z]dF,(Y,) = - [exp - b(X - Z·)] f-~~ exp - [bY,Z*jdF,(Y,) (23) Then optimally X T = X. - TZ· + (Y,(I) + Y,'" + ... + Y,(T»Z· (24) As T -+ (I), X T approaches a normal distribution, not a log- normal distribution as in the case of uniform strategies. How- ever, the utiUty level itself will approach a log-normal dis- tribution. T U .... exp - b[X. - TZ·] II (exp - bY,wZ·) I IimE/UT } =O=M T-~ (25) For this option of cash or a risky asset with positive return, it will necessarily be the case that max E{log[w + (I - w)Y,J) > 1 ID IimE{UT*} =O=M T_~ Nonetheless, at every wealth level, above or below a critical number, the optimal policy will generally differ from that of maximum geometric mean. When we reintroduce limited liability in the problem and never permit an investor to take a position that could leave him bankrupt with negative wealth, the optimal strategy at each X. will be to put min (X .,Z*) into the risky asset and an exact solution becomes more tedious. But clearly the optimal strategy is nonuniform and will outperform any and all uni- form strategies, including that of the geometric-mean maxi- mizer. MAXIMIZING EXPECI'ED AVERAGE COMPOUND GROWTH Hakansson (7) has presented an analysis with a bearing on geometric-mean maximization by the long-run investor. D ... fining the rate of return in any period as x, and the average compound rate of return 8S (XIX! . . • XTI/T), one can propose 85 a criterion of portfolio selection maximizing the expected value of this magnitude. After the conventional scaling-factor T is introduced, this gives max TX,E(x,x, ... XT)'/T = X.{ max E[xFT/ (I/T) J}T (26) VI This problem we have already met in (19) for'Y = liT. For finite T, this new criterion leads to slightly more risk-tak- ing than does geometric-mean or expected-logarithm m8xim~ ing: for T ~ 1, it leads to Pascal's maximizing of expected money gain; for T - 2, it leads t{) the eighteenth century square-root utility function proposed by Cramer to resolve the St. Petersburg Paradox. --- ## Page 519 490 2496 Applied Mathematical Sciences: Samuelson However, as T - IX) and"Y = li T - 0, lim ""/1' = log '" .,...0 (27) and we asymptotically approach the geometric-mean maxi- mizing. Thus, one wedded psychologically to a utility function _",-1 will find the new criterion leads to rash investing. Example: modify the numerical example of (14) above so that 2.7 and .3 are replaced by 2.4 and .6. Because the geo- metric mean of these numbers equals 1.2 > 1, none in cash is better for such an investor than is all in cash. But, since the harmonic means of these numbers equals .96 < 1, our hypothesized investor would prefer to satisfy his own pyscho- logical tastes and choose to invest all in cash rather than none in cash-no matter how great T is and in the full recog- nition that he is violating the new criterion. The few times that following that criterion leads him to comparative losses are important enough in his eyes to scare him off from use of that criterion. Indeed, if commissions were literally zero, then no matter how short were T in years, the number of transaction periods would become indefinitely large: Hence, with l' = 0, the novel criterion would lead to geometric-mean maximization, not just asymptotically for long-lived investors, but for any T. To be sure, as one shortens the time period between transactions, my assertion of independence of probabilities between periods might become unrealistic. This opens a Pandora's Box of difficulties. Fortuitously, the utility function log x is the one case that is least complicated to handle when probabilities are intertemporally dependent. This makes log x an attractive candidate for Santa Claus examples in textbooks, but will not endear it to anyone whose psychological tastes deviate signifi- cantly from log x. (For what it is worth, I may mention that I do not faU into that category, but that does not affect the logic of the problem.) These remarks critical of the criterion of maximum expected average compound growth do not deny that this criterion, P A. Samuelson Pro<. Nat. Acad. Sci. USA 68 (/971) arbitrary as it is, still avoids some of the even greater arbitrari- ness of conventional mean-variance analysis. Its essential defect is that it attempts to replace the pair of "asymptotically sufficient parameters" [E{log xd, Variance{log xd I by the first of these alone, thereby gratuitously ruling out arbitrary l' in the family u(x) = x' h in favor of .. (x) = log x. This diagnosis can be substantiated by the valuable discussion in the cited Hakansson paper of the efficiency properties of the pair [E {average-compound-return 1, Variance { averagEHlom- pound-return }I, which are asymptotically surrogates for the above sufficient parameters. Financial aid from the National Science Foundation and edioorial ... istance from Mrs. Jillian Pappas are gratefully acknowledged. I have benefited from conversations with H. M. Markowitz, H. A. Latan6, and L. J. Savage, but cannot claim that they would hold my views. N. H. Hakansson bas explicitly warned that the purpose of his paper was not to favor maximiz- ing the expected 8verage-compound-return criterion. 1. Williams, J. B., I'Speculation and Carryover." Quarterly J&urnal of Ecqnqmica, 50, 436-455 (1936). 2. Kelley, J. L., Jr., flA New Interpretation of Information Rate," BeU System Technical Journal, 917-926 (1956). 3. Latane, H. A" "Criteria. for Choice Among Risky Ventures," Joomal of Political Eronomy, 67, 144-155 (1956); Kelley, J. L., Jr., snd L. Breiman, "Investment Policies for Expand- ing Business Optimal in 8. Long-Run SenBe," Naval Research Logiat1'co Quarterly, 7 (4), 647~1 (1960); Breiman, L., "Optimal Gambling Systems for Favorable Games," ed. J. Neyman, ProaedifllJ8 of /.he Fuurth Berkeley Sympo';um on Mathematical Si Ws(N). This Odd Rule is odd since it can put you in this Fix: You may well pick A(N) from A(N) and B(N), and pick B(N) from B(N) and C(N), and yet still pick C(N) from A(N) and C(N). --- ## Page 522 Why We Should not Make Mean Log of Wealth Big Though Years to Act Are Long 493 P.A. Samuelson, Big mean log of wealth 307 References Latane, H.A., 1959, Criteria for choice among risky ventures, Journal of Political Economy 67, 144-155. Latane, H.A., 1978, The geometric-mean principle revisited: A reply, Journal of Banking and Finance 2, 395-398. Ophir, T., 1978, The geometric-mean principle revisited, Journal of Banking and Finance 2, 103- 107. Ophir, T., 1979, The geometric-mean principle revisited: A reply to a reply, Journal of Banking and Finance, this issue. Samuelson, P.A., 1963. Risk and uncertainty: A fallacy of large numbers, Scientia 1-6. Reproduced in: 1966, Collected scientific papers of Paul A. Samuelson - I (MIT Press, Cambridge, MA) 153-158. Samuelson, P.A., 1969, Lifetime portfolio selection, Review of Economics and Statistics 51, 239- 246. Reproduced in: 1972, Collected scientific papers of Paul A. Samuelson - II (MIT Press, Cambridge, MA) 883 890 (cf. p. 889). Thorp, E., 1971, Portfolio choice and the Kelley criterion. Reproduced in: W.T. Ziemba and R.G. Vickson, eds., 1975, Stochastic optimization models in finance (Academic Press, New York) 599-619. This Fix can come for all N, as large as we choose to make N. It can do so though the truth of Thorp (1971, p. 603) holds, plim W .. (N) = Jlt .. > WB=plim WB(N), WB> Wc=plim WdW), N-oo N-~ N-ao rules out plim WdN»plim W .. (N). N-oo N-oo That is so since W .. > WB and WB > We means W .. > We. But I'd not said: 'When you act to make mean of log of wealth large, you could get in the Fix'. To see why you can't, we need only note that mean of log of W .. (N)=LA(N»LB(N)=mean of log of WB(N), and mean of log of WB(N)=LB(N»LdN)=mean of log of WdN), rules out Le(N»L .. (N), and it does so for all N, as when N is as small as one. Some goals are strange, but need not be as bad as the odd rules some seek to base them on. --- ## Page 523 This page is intentionally left blank --- ## Page 524 495 The Journal of FINANCE VOL. XXXI DECEMBER 1976 No.5 35 INVESTMENT FOR THE LONG RUN: NEW EVIDENCE FOR AN OLD RULE HARRY M. MARKOWITZ* I. BACKGROUND "INVESTMENT FOR THE LONG RUN," as defined by Kelly [7], Latane [8] [9], Marko- witz [10], and Breiman [1] [2], is concerned with a hypothetical investor who neither consumes nor deposits new cash into his portfolio, but reinvests his portfolio each period to achieve maximum growth of wealth over the indefinitely long run. (The hypothetical investor is assumed to be not subject to taxes, commissions, illiquidi- ties and indivisibilities.) In the long run, thus defined, a penny invested at 6.01% is better-eventually becomes and stays greater-than a million dollars invested at 6%. When returns are random, the consensus of the aforementioned authors is that the investor for the long run should invest each period so as to maximize the expected value of the logarithm of (l + single period return). The early arguments for this "maximum-expected-Iog" (MEL) rule are most easily illustrated if we assume independent draws from the same probability distribution each period. Starting with a wealth of Wo' after T periods the player's wealth is T W T = Woo II (I +'/) (I) I - I where '1 is the return on the portfolio in period t. Thus T log( W rI Wo) = L log( 1+ '1) (2) I - I If log(l +,) has a finite mean and variance, the weak law of large numbers assues us that for any t: > 0 (3) • IBM Thomas J. Watspn Research Center, Yorktown Heights, New York. 1273 --- ## Page 525 496 H M Markowitz 1274 The Journal of Finance and the strong law' assures us that lim Tl ·log(WdWo)=Elog(1 +r) T-+oo (4) with probability 1.0. Thus if Elog(l + r) for portfolio A exceeds that for portfolio B, then the weak law assures us that, for sufficiently large T, portfolio A has a probability as close to unity as you please of doing better than B when time = T; and the strong law assures us that with probability one. (5) Some authors have argued that the strategy which is optimal for the player for the long run is also a good rule for some or all real investors. My own interest in the subject stems from a different source. In tracing out the set of mean, variance (E, V) efficient portfolios one passes through a portfolio which gives approximately maximum Elog(l + r).2 I argued that this "Kelly-Latant!" point should be consid- ered the upper limit for conservative choice among E, V efficient portfolios, since portfolios with higher (arithmetic) mean give greater short-run variability with less return in the long run. A real investor might, however, perfer a smaller mean and variance, giving up return in the long run for stability in the short run. Samuelson [14] and [IS] objected to MEL as the solution to the problem posed in [I], [2], [7], [8], [9], [10]. Samuelson's objection may be illustrated as follows: suppose again that the same probability distributions of returns are available in each of T periods, t = 1,2, ... , T. (Samuelson has also treated the case in which t is continuous; but his objections are asserted as well for the original discussion of discrete time. The latter, discrete time, analysis is the subject of the present paper.) Assume that the utility associated with a play of a game is U=W,;-Ia a#O (6) where W T is final wealth. Samuelson shows that, in order to maximize expected utility for the game as a whole, the same portfolio should be chosen each period. This always chosen portfolio is the one which maximizes single period EU=E(1 +rt la. (7) Furthermore, if E14 is the expected -return provided by this strategy for a T 1. In most cases the early literature on investment for the long run used the weak law of large numbers. The results in Breiman [l], however, specialize to a strong law of large numbers in the particular case of unchanging probability distributions. See also the Doob [4] reference cited by Breiman. 2. Markowitz [lO] Chapters 6 and 13 conjectures, and Young and Trent [16] confirm that Elog(l + r)~log(l + E) -!. (V /(1 + E)2) for a wide class of actual ex post distributions of annual portfolio returns. --- ## Page 526 Investment for the Long Run: New Evidence f or an Old Rule 497 Investment for the Long Run: New Evidence for an Old Rule 1275 period game, and EU~ is that provided by MEL, usually we will have EU~/EU~~oo as T~oo (8) Thus, despite (3), (4) and (5), MEL does not appear to be asymptotically optimal for this apparently reasonable class of games. Von Newman and Morgenstern [17] have directly and indirectly persuaded many, including Samuelson and myself, that, subject to certain caveats, the expected utility maxim is the correct criterion for rational choice among risky alternatives. Thus if it were true that the laws of large numbers implied the general superiority of MEL, but utility analysis contradicted this conclusion, I am among those who would accept the conclusions of utility analysis as the final authority. But not every model involving "expected utility" is a valid formalization of the subject purported to be analyzed. In particular I will argue that, on closer examination, utility analysis supports rather than contradicts MEL as a quite general solution to the problem of investment for the long run. II. THE SEQUENCE OF GAMES It is important to note that (8) is a result concerning a sequence of games. For fixed T, say T= 100, EU = EWfool a is the expected utility (associated with a particular strategy) of a game involving precisely 100 periods. For T= 101, EWfOlI a is the expected utility of a game lasting precisely 101 periods; and so on for T = 102, 103, .... That (8) is a statement about a sequence of games may be seen either from the statement of the problem or from the method of solution. In Samuelson's formula- tion W T is final wealth-wealth at the end of the game. If we let T vary (as in "T ~ 00") we are talking about games of varying length. Viewed differently, imagine computing the solution by dynamic programming starting from the last period and working in the direction of earlier periods. (Here we may ignore the fact that essentially the same solution reemerges in each step of the present dynamic program. Our problem here is not how to compute a solution economically, but what problem is being solved). If we allow our dynamic pro- gramming computer to run backwards in time for 100 periods, we arrive at the optimum first move, and the expected utility for the game as a whole given any initial Wo, for a game that is to last 100 moves. If we allow the computer to continue for an additional 100 periods we arrive at the optimum first move, and the expected utility for the game as a whole given any initial Wo' for a game that is to last for 200 moves; and so on for T=201,202, .... In particular, equation (8) is not a proposition about a single game that lasts forever. This particular point will be seen most clearly later in the paper when we formalize the utility analysis of unending games. To explore the asymptotic optimality of MEL, we will need some notation concerning sequences of games in general. Let TI < T2 < T3 • •• be a sequence of strictly increasing positive integers. In this paper3 we will denote by GI , G2, G3 •• • a 3. A somewhat different, but equivalent, notation was used in [11]. --- ## Page 527 498 H M Markowitz 1276 The J ouma! of Finance sequence of games, where the ith game lasts Ti moves. (In case the reader feels uncomfortable with the notion of a sequence of games, as did at least one of our colleges who read [11], perhaps the following remarks may help. The notion of a sequence of games is similar to the notion of a sequence.of numbers, or a sequence of functions, or a sequence of probability distributions. In each .case there is a first object (i.e., a first number or function or distribution or game) which we may denote as G1; a second object (number, function, distribution, game) which we may denote by G2 ; etc.). In general we will not necessarily assume that the same opportunities are available in each of the T; periods of the game Gi • We will always assume that-as part of the rules that govern Gi-the game Gi is to last exactly Ti periods, and that the investor is to reinvest his entire wealth (without commissions, etc.) in each of the T; periods. Beyond this, specific assumptions are made in specific analyses. In addition to a sequence of games, we shall speak of a sequence of strategies sp S2' S3"" where Si is a strategy (i.e., a complete rule of action) which is valid for (may be followed in) the game Gi• By convention, we treat the utility function as part of the specification of the rules of the game. The rules of Gi and the strategy Si together imply an expected utility to playing that game in that manner. III. ALTERNATE SEQUENCE-OF-GAMES FORMALIZATIONS Let g equal the rate of return achieved during a play of the game Gi; i.e., writing T for Ti : (9) or (10) In the Samuelson sequence of games, here denoted by Gp G2, G3, ••• , the utility function of each game Gi was assumed to be (11 ) We can imagine another sequence of games-call them HI' H 2' H 3' • .. -which have the same number of moves and the same opportunities per move as Gp G2, G3, ••• , respectively, but have a different utility function. Specifically im- agine that the utility associated with a play of each game Hi is U= V(g). (lla) for some increasing function of g. For a fixed game of length T= Ti' we can always find a function V( g) which gives the same rankings of strategies as does some specific j(WT ). For example, for fixed T (11) associates the same U to each possible playas does U= V(g)= W~·(l +gtT/a. (lIb) --- ## Page 528 Investment for the Long Run: New Evidence for an Old Rule 499 Investment for the Long Run: New Evidence for an Old Rule 1277 Thus for a given T it is of no consequence whether we assume that utility is a function of final wealth W T or of rate of return g. On the other hand, the assumption that some utility function V( g) remains constant in a sequence of games, as in HI' H 2, H 3,.. . has quite different con- sequences than the assumption that some utility function f( W T) remains constant as in Gp G2, •• • • Markowitz [11] shows that if V(g) is continuous then as T~oo (12a) where EV~ is the expected utility provided by the MEL strategy for the game Hi' and EV~ is the expected utility provided by the optimum strategy (if such an optimum exists4); and if V(g) is discontinuous then (12b) where {j is the largest jump in V at a point of discontinuity, and €~O as T ~oo. (12a) and (12b) do not require the assumption that the same probability distribu- tions are available each period. It is sufficient to assume that the return r is bounded by two extremes [. and r: -l W2"")' Since Goo has no "last period", we cannot speak of "final wealth". We can, however, assume that the player has preferences among alternate wealth-sequences: e.g., he may prefer the sequence of passbook entries provided by a savings account which compounds his money at 6%, starting with Wo' to one that compounds it at 3%. Given any two sequences: WQ=(Wo, W~, W~, ... ) 6. The fact that some numbers have two decimal expansions, like 0.4999 .. . versus 0.5000 ... , may be resolved in any manner without effect on the analysis; since such numbers occur with zero probability. --- ## Page 531 502 H M Markowitz 1280 The Journal of Finance and we may assume that the player either prefers W a to W b, or W b to W a or is indifferent. Further, we may assume that given a choice between any two probabil- ity distributions among sequences of wealth versus he either prefers probability distribution A to B, or B to A, or is indifferent between the two probability distributions. We shall not only assume that the player has such preferences, but also that he maximizes expected utility. In other words, we assume that he attaches a (finite) number to each sequence of wealths, and chooses among alternate strategies so as to maximize E U. The only additional assumption we make about the utility function U( .. . ), is this: If the sequence W a = (Wo' W~, W~, ... ) eventually pulls even with, and then stays even with or ahead of the sequence then W a is at least as good as W b ; i.e., if there exists a T such that for t ~ T (16) then U( W~ ~ U( W b ). This ~sumption expresses the basic notion that, in the sense that we have used the terms throughout this controversy, if player A eventually gets and stays ahead of player B (or at least stays even with him) then player A has done at least as well as player B "in the long run". At first it may seem appropriate to make a stronger assumption that if W~ eventually pulls ahead of W~, and stays ahead, then the sequence W" is preferable to the sequence W b• In other words, if there is a T such that W~>W~ for t> T (16a) then --- ## Page 532 Investment for the Long Run: New Evidence for an Old Rule 503 Investment for the Long Run: New Evidence for an Old Rule 1281 As shown in the footnote7, this stronger assumption is too strong in that no utility function U(Wo' WI> W2, • • . ) can have this property. Utility functions can however have the weaker requirement in (16). The analysis of unending games is particularly easy if we assume that the same opportunities are available at each move, and that we only consider strategies which select the same probability distribution of returns each period. We shall make these assumptions at this point. Later we will summarize more general results derived in the appendix to this paper. Without further loss of generality we will confine our discussion to just two strategies, namely, MEL and any other strategy, and will consider when the expected utility supplied by MEL is at least as great as that supplied by the other strategy. Letting W~ and W? be the wealth at t for a particular play of the game using MEL or the other strategy, respectively, U( Wo' wt, Wi, ... ) ~ U( Wo, W~, w~, ... ) (17) is implied if there is a T such that for t~ T. (18) 7. If U orders all sequences W-(Wo, WI' W2, . .. ) in a manner consistent with (16a), then in particular it orders sequences of the form { Wo-given WI - any positive number W,-O +a)'W,_1 for t .. 2; a .. -\. Since this family of sequences depends only on WI and a, we may here write Then (16a) requires For any a let Then (N .3) implies as well as if either a A >a B or and W1>wf· Ulow(a)-GLB V(WI,a) Uhi(a)-LUB V(WI,a). for every a (N.\) (N.2) (N.3) (N.4) (N.5a) (N.Sb) But (N.Sb) implies that we can have Ulow(a) < Uhi(a) for at most a countable number of values of a, since at most a countable number of values of a can have Uhi(a)- Ulow(a) > liN for N-l,2,3, ... . But ihis contradicts (N .Sa). --- ## Page 533 504 H M Markowitz 1282 The Journal of Finance Equation (2) implies that we have W~ ~ W? if and only if I I I I - ~ log(l +1) > - ~ log(l +r?). 1 i - I 1 i= I (19) Thus (18) will hold in any play of the game in which there exists a T such that 1 I 1 I - ~ log(l +1) > - ~ log(l +r?) Ii_I I i _I for all I> T. (20) Or, if we let (21 ) (20) may be written as for I> T. (22) Under the present simplified assumptions (23) by definition of MEL. But for random variables YI,h, ... with identical distribu- tions and with (finite) expected value p., we have 1 I lim - ~ (y;-p.)=O 1-+00 1 i-I except for a set of probability measure zero. In other words 1 I lim - ~ y;=p. 1 ..... 00 1 i-I (24) (25) except for a set of sequences which have (in total) zero probability of occurrence (c.f. the strong law of large numbers in [4] or [5]). But (23) and (25) imply (as a simple corollary of the definition of the limit of sequence) that there exists T such that (26) except on a set of probability zero; hence (17) holds except on a set of measure zero. Since (27) is not affected by arbitrarily changing the value of U on a set of measure zero, we --- ## Page 534 Investment f or the Long Run: New Evidence f or an Old Rule 505 Investment for the Long Run: New Evidence for an Old Rule 1283 have EU( WO' wf, Wt,·.·);;;. EU( Wo, W~, W~, .. . ). (28) Thus, given our simplifying assumption of an unchanging probability distribu- tion of returns for a given strategy, the superiority of MEL follows quite generally. The case in which opportunities change from period to period and, whether or not opportunities change, strategies may select different distributions at different times, is treated in the appendix. It is shown there that if a certain continuity condition holds, then MEL is optimal quite generally. If this continuity condition does not hold, however, then there can exist games for which MEL is not optimal. In this respect the results for the unending game are similar to those for the sequence of games with constant V(g). In the latter case we found that MEL was asymptotically optimal for the sequence of games if V( g) was continuous, but could fail to be so if V(g) was discontinuous. In the case of the unending game, the theorem is not concerned with asymptotic optimality in a sequence of games, but optimality for a single game. Given a particular continuity condition, MEL is the optimum strategy. V. CONCLUSIONS The analysis of investment for the long run in terms of the weak law of large numbers, Breiman's strong law analysis, and the utility analysis of unending games presented here each imply the superiority of MEL under broad assumptions for the hypothetical investor of [I], [2], [7], [8], [9], [10]. The acceptance or rejection of a similar conclusion for the sequence-of-games formalization depends on the defini- tion of asymptotic optimality. For example, if constant V(g) rather than constant U( W T) is assumed, as this writer believed plausible on a priori grounds, then the conclusion of the asymptotic analysis is approximately the same (even in terms of where MEL fails) as those of the unending game. I conclude, therefore, that a portfolio analyst should not be faulted for warning an investor against choosing E, V efficient portfolios with higher E and V but smaller Elog(l + R), perhaps not even presenting that part of the E, V curve which lies above the point with approximate maximum Elog(l + R), on the grounds that such higher E, V combinations have greater variability in the short run and less "return in the long run". ApPENDIX Using the notation of footnote 7, we will show that if U1ow(a) = Uhi(a) for all a then MEL is an optimum strategy quite generally; whereas, if Uhi(a» U10w(a) for some ao' then a game can be constructed in which MEL is not optimum. U1ow(a) = Uhi(a) for all a is the "continuity condition" referred to in the text. For v= L or 0, indicating the MEL strategy or some other strategy, respectively, we define (29) --- ## Page 535 506 H M Markowitz 1284 The Journal of Finance where (30) is the expected value of Y~ given the events prior to time t. From this follows E {u~1 Lj', ... , u~: d =0. (31 ) The u~ (for a given v) are thus what Doob [4] refers to as "orthogonal" random variables, and Feller [5] calls "completely fair" random variables. Therefore, writing var for variance, 00 var(u~) ~ 2;:::00 n-I n (32) implies converges to 0 almost always. (33) (In particular, (32) holds if the var(u~) are bounded.) In addition to now assuming condition (32) we will also assume that the game is such that lim.!. ± L/ n-+oo n i - I exists almost always. (34) This is the case, for example, when the same distributions are available each time, whether or not "the other" strategy uses a constant distribution. Since L/ ~ L? always, we have 1 n 1 n 1 n lim - ~ L/ = lim sup - ~ L/;;' lim sup - ~ L? n n i- I n n i-Inn i - I always. Thus when (32) holds we have 1 n 1 n lim- ~ yf'>limsup- ~ Y~ almost always. n i - I n i - I In general, a = limsup.!. ~ YY n implies (since there al)"ays exists another series yr ,Yi, ... such that 1 n a=lim- ~ Yi n i - I and . for all n; (35) (36) (37) --- ## Page 536 Investment/or the Long Run: New Evidence/or an Old Rule 507 Investment for the Long Run: New Evidence for an Old Rule 1285 hence If we now add to the assumptions expressed in equations (32) and (34), the assumption that Uhi( a) = Ulow( a) for all a, we get directly from (36) and (37) that EUL;;'EUo. Conversely, the following is an example in which Uhi(ao) > Ulow(aJ and which MEL is not optimum: let Wo= 1 and suppose that for some fixed positive a we have U(1, 0.5, 0.5(1 + a), 0.5(1 + a)2, ... ) equals Up , (1 + a), (1 +a)2, (1 + a)3, . .. ) < U( 1,1.5,1.5(1 + a), 1.5(1 + a)2, .. . ). With such a U-function it would be better to take a 50-50 chance of WI = 0.5 or 1.5 followed by W t =(1+a)Wt _ p t>2, rather than have W t =(1+a).Wt _ 1 with certainty for t> 1, .... While the above shows that MEL can fail to be optimal when Uhi(a) > Ulow(a) for some a, recall that we can have Uhi(a) > Ulow(a) for at most a countable number of values of a. Thus MEL is optimal in a game in which 1 n a= lim - ~ LL n n ~ I i-I has a continuous distribution, or in which a has a discrete or mixed distribution but in which none of the points of discontinuity of the cumulative probability distribution of a have Uhi(a) > Ulow(a). REFERENCES 1. Leo Breiman. "Investment Policies for Expanding Businesses Optimal in a Long Run Sense," Naval Research Logistics Quarterly, 7:4, 1960, pp. 647~51. 2. --. "Optimal Gambling Systems for Favorable Games," Fourth Berkeley Symposium on Probability anfi Statistics, I, 1961, pp. 65-78. 3. Daniel Bernoulli. "Exposition of a New Theory on the Measurement of Risk," Econometrica, XXII, January 1954, pp. 23~3 . Translated by Louise Sommer-original 1738. 4. J. L. Doob. Stochastic Processes, John Wiley and Sons, New York, 1953. 5. William Feller. An Introduction to Probabality Theory and Its Applications, Volume II, John Wiley and Sons, New York, 1966. 6. M. B. Goldman. "A Negative Report on the 'Near Optimality' of the Max-Expected-Log Policy As Applied to Bounded Utilities for Long Lived Programs." Journal of Financial Economics, Vol. I, No.1, May 1974. 7. J. L. Kelly, Jr. "A New Interpretation of Information Rate," Bell System Technical Journal, pp. 917-926, 1956. 8. H. A. Latane. "Rational Decision Making in Portfolio Management," Ph.D. dissertation, Univer- sity of North Carolina, 1957. 9. --. "Criteria for Choice Among Risky Ventures," Journal of Political Economy, April 1959. 10. H. M. Markowitz. Portfolio Selection: EffiCient Diversification of Investments, John Wiley and Sons, New York, 1959; Yale University Press, 1972. 11. --. "Investment for the Long Run," Rodney L. White Center for Financial Research Working Paper no. 20-72 (University of Pennsylvania) 1972. --- ## Page 537 508 H M Markowitz 1286 The Journal of Finance 12. R. C. Merton and P. A. Samuelson. "Fallacy of the Log-Normal Approximation to Optimal Portfolio Decision-Making Over Many Periods," Journal of Financial Economics, Volume I No. I, May 1974. 13. Karl Menger. "Das Unsicherheitsmoment in der Wertlehre. Betrachtungen im Anschluss an das sogenannte Petersburger Spiel," Zeitschrift fur Nationalokonomie, Vol. 5, 1934. Translated in Essays in Mathematical Economics in Honor of Oskar Morgenstern, M. Shubik ed., Princeton University Press, 1967. 14. P. A. Samuelson. "Risk and Uncertainty: A Fallacy of Large Numbers," Scientia, 6th Series, 57th year, April-May 1963. IS. ---. "Lifetime Portfolio Selection by Dynamic Stochastic Programming," Review of Economics and Statistics, August 1969. 16. W. E. Yong and R. M. Trent. "Geometric Mean Approximation of Individual Security and Portfolio Performance," Journal of Financial and Quantitative AnalySiS, June 1969. 17. John von Neuman and Oskar Morgenstern. Theory of Games and Economic Behavior, Princeton University Press, 1944. John Wiley and Sons, 1967. --- ## Page 538 509 36 Understanding the Kelly Criterion* Edward O. Thorp In January 1961, I spoke at the annual meeting of the American Mathemati- cal Society on "Fortune's Formula: The Game of Blackjack". This announced the discovery of favorable card counting systems for blackjack. My 1962 book Beat the Dealer explained the detailed theory and practice. The 'optimal' way to bet in favorable situations was an important feature. In Beat the Dealer, I called this, nat- urallyenough, "The Kelly gambling system", since I learned about it from the 1956 paper by John L. Kelly (Claude Shannon, who refereed the Kelly paper, brought it to my attention in November of 1960). I have continued to use it successfully in gambling and in investing. Since 1966, I've called it "the Kelly Criterion". The rising tide of theory about and practical use of the Kelly Criterion by several lead- ing money managers received further impetus from William Poundstone's readable book about the Kelly Criterion, Fortune's Formula. (As this title came from that of my 1961 talk, I was asked to approve the use of the title) . At a value investor's conference held in Los Angeles in May, 2007, my son reported that 'everyone' said they were using the Kelly Criterion. The Kelly Criterion is simple: bet or invest so as to maximize (after each bet) the expected growth rate of capital, which is equivalent to maximizing the expected value of the logarithm of wealth; but the details can be mathematically subtle. Since they're not covered in Poundstone (2005), you may wish to refer to my ar- ticle, Thorp (2006), and other papers in this volume. Also some services such as Morningstar and Motley Fool have recommended it. These sources use the rule: "optimal Kelly bet equals edge/odds" that applies only to the very special case of a two-valued payoff. Hedge fund manager, Mohnish Pabrai (2007), gives examples of the use of the Kelly Criterion for investment situations (Pabrai won the bidding for the 2008 lunch with Warren Buffett, paying over $600,000). Consider his investment in Stewart Enterprises (Pabrai, 2007: 108-115), his analysis gave what he believed to be a list of worst case scenarios and payoffs over the next 24 months which I summarize in Table 1. The expected growth rate of capital g(f) if we bet a fraction f of our net *Reprinted revised from two columns from the series A Mathematician on Wall Street in Wilmott Magazine, May and September 2008. Edited by Bill Ziemba. --- ## Page 539 510 worth is Table 1 Stewart enterprises, payoff within 24 months. P robability PI = 8.80 P2 = 0.19 P3 = 0.01 Sum = 1.00. 3 Return RI > 100% R2 > 0% R 3 = - 100% g(1) = LPi In(l + Rd) i= l E. O. Thorp (1) where In means the logarithm to the base e. When we use Table 1 to insert the Pi values, replacing the Ri by their lower bounds gives the conservative estimate g(1) = O.SO In(l + i) + 0.011n(1 - 1) . (2) Setting g'(1) = 0 and solving gives the optimal Kelly fraction 1* = 0.975 noted by Pabrai. Not having heard of the Kelly Criterion in 2000, Pabrai only bet 10% of his fund on Stewart. Would he have bet more, or less, if he had then known about Kelly's Criterion? Would I have? Not necessarily. Here are some of the many reasons why: (1) Opportunity costs. A simplistic example illustrates the idea. Suppose Pabrai's portfolio already had one investment which was statistically independent of Stewart and with the same payoff probabilities. Then, by symmetry, an optimal strategy is to invest in both equally. Call the optimal Kelly fraction for each 1*, then 21* < 1 since 21* = 1 has a positive probability of total loss, which Kelly always avoids, so 1* < 0.50. The same reasoning for n such investments gives 1* < l / n . Hence, we need to know the other investments currently in the portfolio, any candidates for new investments, and their (joint) properties, in order to find the Kelly optimal fraction for each new investment, along with possible revisions for existing investments. Formally, we solve the nonlinear programming problem: maximize the expected logarithm of final wealth subject to the various constraints on the asset weights (see the papers in Section 6 of this volume for examples). Pabrai's discussion (e.g. pp. 7S- S1) of Buffett's concentrated bets gives consid- erable evidence that Buffet thinks like a Kelly investor, citing Buffett bets of 25% to 40% of his net worth on single situations. Since 1* < 1 is necessary to avoid total loss, Buffett must be betting more than 0.25 to 0.40 of i 8 in these cases. The opportunity cost principle suggests it must be higher, perhaps much higher. Here's what Buffett himself says, as reported in http://undergroundvalue.blogspot.com/ 200S/ 02/ notes-from-buffett-meeting-215200S..23.html, notes from a Q & A session with business students: --- ## Page 540 Understanding the Kelly Criterion Emory: With the popularity of "Fortune's Formula" and the Kelly Cri- terion, there seems to be a lot of debate in the value community regarding diversification vs. concentration. I know where you side in that discussion, but was curious if you could tell us more about your process for position sizing or averaging down. Buffett: I have 2 views on diversification. If you are a professional and have confidence, then I would advocate lots of concentra- tion. For everyone else, if it's not your game, participate in total diversification. So this means that professionals use Kelly and am- ateurs better off with index funds following the capital asset pricing model. If it's your game, diversification doesn't make sense. It 's crazy to put money in your 20th choice rather than your 1st choice. If you have LeBron James on your team, don't take him out of the game just to make room for some else. Charlie and I operated mostly with 5 positions. If I were running 50, 100, 200 million, I would have 80% in 5 positions, with 25% for the largest. In 1964, I found a position I was willing to go heavier into, up to 40%. I told investors they could pull their money out. None did. The position was American Express after the Salad Oil Scandal. In 1951 I put the bulk of my net worth into GEICO. With the spread between the on-the-run versus off-the-run 30 year Treasury bonds, I would have been willing to put 75% of my portfolio into it. There were various times I would have gone up to 75%, even in the past few years. If it 's your game and you really know your business, you can load up. 511 This supports the assertion in Rachel and Bill Ziemba's 2007 book, that Buffett thinks like a Kelly investor when choosing the size of an investment. They discuss Kelly and investment scenarios at length. Computing 1* without considering the available alternative investments is one of the most common oversights I've seen in the use of the Kelly Criterion. It is a dangerous error because it generally overestimates f*. (2) Risk tolerance. As discussed at length in Thorp (2006) , "full Kelly" is too risky for the tastes of many, perhaps most, investors and using instead an f = c1*, with fraction c where 0 < c < 1 or "fractional Kelly" is much more to their liking. Full Kelly is characterized by drawdowns which are too large for the comfort of many investors. l ISeveral papers in Section 3 in this volume, as do the following two papers in this section, discuss fractional Kelly strategies. --- ## Page 541 512 E. O. Thorp (3) The "true" scenario is worse than the supposedly conservative lower bound estimate. Then we are inadvertently betting more than 1* and, as dis- cussed in Thorp (2006), we get more risk and less return, a strongly suboptimal result. Betting i = cd*, 0 < c < 1 gives some protection against this (see the graphs in Section 3, (MacLean, Ziemba and Blazenko (1992)). (4) Black swans. As fellow Wilmott columnist Nassim Nicholas (Taleb 2007) has pointed out so eloquently in his bestseller The Black Swan, humans tend not to appreciate the effect of relatively infrequent unexpected high impact events. Failing to allow for these "black swans", scenarios often don't adequately consider the probabilities of large losses. These large loss probabilities may substantially reduce 1* . One approach to successfully model such black swans is to use a scenario optimization stochastic programming model. 2 For Kelly bets that simply means that you include such extreme scenarios and their consequences in the nonlinear programming optimization to compute the optimal asset weights. The 1* will be reduced by these negative events. (5) The "long run". The Kelly Criterion's superior properties are asymptotic, appearing with increasing probability as time increases. For instance: As time t tends to infinity the Kelly bettor's fortune will, with probability tend- ing to 1, permanently surpass that of any bettor following an "essentially different" strategy. The notion of "essentially different" has confounded some well known quants so I'll take time here to explore some of its subtleties. Consider for simplicity repeated tosses of a favorable coin, the outcome of the nth trial is Xn where P(Xn = 1) = p> 1/2 and P(Xn = - 1) is 1 - P = q > O. The {Xn} are independent identically distributed random variables. The Kelly fraction is 1* = p - q = E(Xn) > O. The Kelly strategy is to bet a fraction in = 1* at each trial n = 1,2, .... Now consider a strategy which bets 9n, n = 1,2, ... at each trial with 9n i= 1* for some n ::; N and 9n = 1* thereafter. The {9n} strategy differs from Kelly on at least one of the first N trials but copies it thereafter, but it does not differ infinitely often. There is a positive probability that {9n} is ahead of Kelly at time N, hence ahead for all n 2: N. For example consider the sequence of the first N outcomes such that Xn = 1 if 9n > 1* and Xn = - 1 if 9n ::; 1*. Then for this specific sequence, which has probability 2: qN, {9n} gains more than Kelly for each n ::; N where 9n i= 1*, hence exceeds Kelly for all n 2: N. What if instead in this coin tossing example we require that 9n i= 1* for infinitely many n? This question arose indirectly about 15 years ago in the newsletter Black- jack Forum when a well known anti Kellyite, John Leib, challenged a well known blackjack expert with (approximately) this proposition bet: Leib would produce a strategy which differed from Kelly at every trial but would (with probability as 2There you assume the possibility of an event, specifying its consequences but not what it is. See Geyer and Ziemba (2008) for the application to the Siemens Austria Pension Fund. Correlations change as the scenario sets move from normal conditions to volatile to crash which include the black swans. See also Ziemba (2003) for addit ional applications of this approach. --- ## Page 542 Understanding the Kelly Criterion 513 close to 1 as you wish), after a finite number of trials, get ahead of Kelly and stay ahead forever. When I read the challenge I immediately saw how Leib could win the bet. Leib's Paradox: Assuming capital is infinitely divisible,footnoteThe infinite di- visibility of capital is a minor assumption and can be dealt with as needed in examples where there is a minimum monetary unit by choosing a sufficiently large starting capital. then given E > 0 there is an N > 0 and a sequence {In} with In i: f* for all n, such that P(V; < Vn for all n 2' N) > 1 - varepsilon where Vn = I1~=1 (1 + IiXi) and V; = I1~=1 (1 + f* Xi) . Furthermore, there is a b > 1 such that P(Vn/V; 2' b,n 2' N) > I-varepsilon and P(Vn-V; ---+ (0) > I-varepsilon. That is, for some N there is a non Kelly sequence that beats Kelly "infinitely badly" with probability 1 - E for all n 2' N. Proof. The proof has two parts. First we want to establish the assertion for n = N. Second we show that once we have an {In, n <:::: N} that is ahead of Kelly at n = N, we can construct {f n i: f*, n > N} to stay ahead. To see the second part, suppose VM > VN. Then VN 2' a + bVN for some a > 0, b > 1. For instance, VN >'N2' C > 0 since there are only a finite number of sequences of outcomes in the first N trials, hence, only a finite number with VN > VN. So: VN 2' c + VN 2' c/2 + [(c/2) + VNl = c/2 + [dMax VN + VNl 2' c/2 + (d + I)VN where d Max VN = c/2 defines d > 0 and Max VN is over all sequences of the first N trials such that VN > VN. Setting c/2 = a > 0 and d + 1 = b > 1 suffices. Once we have VN 2' a + bVN we can, for bookkeeping purposes, partition our capital into two parts: a and bVN. For n > N we bet In = f* from bVN and an additional amount a/2n from the a part, for a total which is generally unequal to f* of our capital. If by chance for some n the total equals f* of our total capital we simply revise a/2n to a/3n for that n. The portion bVN will become bVN for n > Nand the portion a will never be exhausted so we have Vn > bV; for all n > N. Hence, since P(VN ---+ (0) = 1, we have P(Vn/VN 2' b) = 1 from which it follows that P(Vn - VN ---+ (0) = 1. To prove the first part, we show how to get ahead of Kelly with probability 1- E within a finite number of trials. The idea is to begin by betting less than Kelly by a very small amount. If the first outcome is a loss, then we have more than Kelly and use the strategy from the proof of the second part to stay ahead. If the first outcome is a win, we're behind Kelly and now underbet on the second trial by enough so that a loss on the second trial will put us ahead of Kelly. We continue this strategy until either there is a loss and we are ahead of Kelly or until even betting 0 is not enough to surpass Kelly after a loss. Given any N, if our initial under bet is small enough, we can continue this strategy for up to N trials. The probability of the strategy failing is pN, 1/2 < p < 1 Hence, given E > 0, we can --- ## Page 543 514 E. O. Thorp choose N such that pN < E and the strategy therefore succeeds on or before trial N with probability 1 - pN > 1 - E. More precisely: suppose the first n trials are wins and we have bet a fraction 1* - ai with ai > 0, i = 1, ... ,n, on the ith trial. Then: Vn (1 + 1* - ad' .. (1 + 1* - an) VN (1 + f*) " . (1 + f*) (1 - 1 ~1f* ) ". (1 - 1 ~nf* ) > (1 - ad'" (1- an) > 1 - (a1 +". + an) where the last inequality is proven easily by induction. Letting al + ... + an = a, so Vn/V; > 1 - a, what betting fraction 1* - b will put us ahead of Kelly if the next trial is a loss? A sufficient condition is b>_a_ -l -a provided b <::: 1* and 0 < a < 1. If a <::: 1/2 then b = 2a suffices. Proceeding recursively, we have these conditions on the ai: choose al > O. Then an+! = 2(al +-. ·+an), n = 1, 2, ... provided all the an <::: 1/2. Letting f(x) = alx+a2x2+ .. we get the equation f(x) - alX = 2xf(x)(1 + x + x2 + ".) = 2xf(x)/(1- x) whose solution is f(x) = al {x + 2 L~=2 3n- 2xn} from which an = 2a13n-2 if n :::;, 2. Then given E > 0 and an N such that pN < E it suffices to choose al so that aN = 2a13N-2 <::: min(f*, 1/2). 0 Although Leib did not have the mathematical background to give such a proof he understood the idea and indicated this sort of procedure. So far we've seen that all sequences which differ from Kelly for only a finite number of trials, and some sequences which differ infinitely often (even always), are not essentially different. How can we tell, then, if a betting sequence is essentially different than Kelly? Going to a more general setting than coin tossing, assume now for simplicity that the payoff random variables Xi are independent and bounded below but not necessarily identically distributed. At this point we come to an important distinction. In financial applications, one commonly assumes that the fi are constants that are dependent only on the current period payoff random variable (or variables). Such "myopic strategies" might arise for instance, by selecting a utility function and maximizing expected utility to determine the amount to bet. However, for gambling systems, the amount depend on previous outcomes, i.e., fn = fn(Xl ,X2, ... ,Xn-d, just as it does in the Leib --- ## Page 544 Understanding the Kelly Criterion 515 example. As Professor Stewart Ethier pointed out, our discussion of "essentially different" is for the constant fi case. For a more general case, including the Leib example and many of the classical gambling systems, I recommend Ethier's 2010 book on the mathematics of gambling. We assume R(Xi) > 0 for all Xi from which it follows that ft > 0 for all i. As before, Vn = Il ~=1 (1 + fiXi) and V; = Il ~=1 (1 + It Xi) from which In Vn = L: ~=1 In(1 + fiXi) and In V; = L: ~=1 In(1 + ft Xi). Note from the definition f* that E In(l + ft Xi) 2> E In(l + fiXi) , where E denotes the expected value, with equality if and only if ft = fi. Hence: n n i=1 i=1 where ai 2> 0 and ai = 0 if and only if It = J;. This series of non-negative terms either increases to infinity or to a positive limit M. We say {fi} is essentially different from Ut} if and only if L:~=1 ai tends to infinity as n increases. Otherwise, {fi} is not essentially different from {ft}. The basic idea here can be applied to more general settings. (6) Given a large fixed goal, e.g., to multiply your capital by 100, or 1000, the expected time for the Kelly investor to get there tends to be least. Is a wealth multiple of 100 or 1000 realistic? Indeed. In the 511/2 years from 1956 to mid 2007, Warren Buffett has increased his wealth to about $5 x 1010. If he had $2.5 x 104 in 1956, that's a multiple of 2 x 106 . We know he had about $2.5 x 107 in 1969 so his multiple over these 38 years is about 2 x 103 . Even my own efforts, as a late starter on a much smaller scale, have multiplied capital by more than 2 x 104 over the 41 years from 1967 to early 2007. I know many investors and hedge fund managers who have achieved such multiples. One of the best is Jim Simons, who recently retired from running the Renaissance Medallion Fund. His record to 2005 is analyzed in Section 6 of this book. The caveat here is that an investor or bettor many not choose to make, or be able to make, enough Kelly bets for the probability to be "high enough" for these asymptotic properties to prevail, i.e., he doesn't have enough opportunities to make it into this "long run". Below I explore investors for which Kelly or fractional Kelly may be a more or less appropriate approach. An important consideration will be the investor's expected future wealth multiple. Using Kelly Optimization at PIMCO During a recent interview in the Wall Street Journal (March 22- 23,2008), Bill Gross and I discussed turbulence in the markets, hedge funds, and risk management. Bill considered the question of risk management after he read Beat the Dealer in 1966. That summer he was off to Las Vegas to beat blackjack. Just as I did some years earlier, he sized his bets in proportion to his advantage, following the Kelly Criterion --- ## Page 545 516 E. O. Thorp as described in Beat the Dealer, and ran his $200 bankroll up to $10,000 over the summer. Bill has gone from managing risk for his tiny bankroll to managing risk for Pacific Investment Management Company's (PIMCO) investment pool of almost $1 trillion. 3 He still applies lessons he learned from the Kelly Criterion. As Bill said: "Here at PIMCO it doesn't matter how much you have, whether it's $200 or $1 trillion. . .. Professional blackjack is being played in this trading room from the standpoint of risk management and that's a big part of our success" . The Kelly Criterion applies to multi period investing and we can get some in- sights by comparing it with Markowitzs standard portfolio theory for single period investing. Compound Growth and Mean-Variance Optimality Nobel Prize winner Harry Markowitz introduced the idea of mean-variance optimal portfolios. This class is defined by the property that, among the set of admissible portfolios, no other portfolio has both higher mean return and lower variance. The set of such portfolios as you vary return or variance is known as the efficient frontier. The concept is a cornerstone of modern portfolio theory, and the mean and variance refer to one period arithmetic returns. 4 In contrast, the Kelly Criterion is used to maximize the long term compound rate of growth, a multiperiod problem. It seems natural, then to ask the question: is there an analog to the Markowitz efficient frontier for multiperiod growth rates, i.e., are there portfolios such that no other portfolio has both a higher expected growth rate and a lower variance in the growth rate? We'll call the set of such portfolios the compound growth mean-variance efficient frontier. Let's explore this in the simple setting of repeated independent identically dis- tributed returns per unit invested, where the payoff random variables are {Xi: i = 1, ... ,n} with E(Xi ) > 0 so the "game" is favorable, and where the non-negative fractions bet at each trial, specified in advance, are {fi : i = 1, ... , n}. To keep the math simpler, we also assume that the Xi have a finite number of distinct values. Af- ter n trials the compound, growth rate per period is G( {fi}) = ~ 2::~ 110g(1 + fiXi) and the expected growth rate g( {gd) = E[G( {fd)] = ~ 2:: ~=1 E 10g(1 + fiXi) = ~ 2:: ~= 1 Elog(l+ fiX) :s; Elog(l+ IX). The last step follows from the (strict) con- cavity of the log function, where as X has the common distribution of the Xi, we define I = ~ 2::[:1 fi and we have equality if and only if fi = I for all i. Therefore, if some fi differ from I, we have g( {fi}) < g( {f}). This tells us that betting the same fixed fraction always produces a higher expected growth rate than betting a varying fraction with the same average value. Note that whatever I turns out to be, it can always be written as I = cf*, a fraction c of the Kelly fraction. 3PIMCO is widely regarded as the top bond trading operation in the world. 4 A comprehensive survey of mean-variance theory is in Markowitz and Van Dijk (2006). --- ## Page 546 Understanding the Kelly Criterion Now consider the variance of G( {fd). If is a random variable with: P(X =a)=p, P(x=- I) =q and a>O then Var[ln(1 + fX)] = pq [In Cl~ aj) r (Compare Thorp, 2006, Section 3.1). 517 Note: the change of variable f = bh, b > 0, shows the results apply to any two valued random variable. We chose b = 1 for convenience.) A calculation shows that the second derivative with respect to f is strictly positive for 0 < f < 1 so Var[ln(1 + fX)] is strictly convex in f. It follows that: 1 n 1 n Var[G( {fi})] = - L Var[log(l+ fiXi)] = - L Var[log(l+ fiX)] ~ Var[log(l+ jX)] n n i=l i=l with equality if and only if fi = j for all i. Since every admissible strategy is there- fore "dominated" by a fractional Kelly strategy, it follows that the mean-variance efficient frontier for compound growth is a subset of the fractional Kelly strategies. If we now examine the set of fractional Kelly strategies {J} = {cf*}, we see that for 0 ::; c ::; 1, both the mean and the variance increase as c increases but for c ~ 1, the mean decreases and the variance increases as c increases. Consequently {f*} dominates the strategies for which c > 1 and they are not part of the efficient fron- tier. No fractional Kelly strategy is dominated for 0 ::; c ::; 1. We have established in this limited setting: Theorem. For repeated independent trials of a two valued random variable, the mean-variance efficient frontier for compound growth over a finite number of trials consists precisely of the fractional Kelly strategies {cf* : 0 ::; c ::; I}. So, given any admissible strategy, there is a fractional Kelly strategy with 0 ::; c ::; 1, which has a growth rate that is no lower and a variance of the growth rate that is no higher. The fractional Kelly strategies in this instance are preferable in this sense to all the other admissible strategies, regardless of any utility function upon which they may be based. This deals with yet another objection to the fractional Kelly strategies, namely that there is a wide spread in the distribution of wealth levels as the number of periods increases. In fact, this eventually enormous dispersion is simply the magnifying effect of compound growth on small differences in growth rate and we have shown in the theorem that in the two outcome setting this dispersion is minimized by the fractional Kelly strategies. Note that in this simple setting, a one-period utility function will choose a constant h = cf which will either be a fractional Kelly with c ::; 1 in the efficient frontier or will be too risky, with c> 1, and not be in the efficient frontier. As a second example, suppose we have a lognormal diffusion process with instan- taneous drift rate m and variance s2 where as before the admissible strategies are --- ## Page 547 518 E. O. Thorp to specify a set of fixed fractions {Id for each of n unit time periods, i = 1, . . . , n. Then, for a given I and unit time period Var G(f) = s2I2 as noted in (Thorp, 2006, eq. (7.3)). Over n periods Var G( {Ii}) = S2 2::~=1 Ii2 ;:::: S2 2::~=1 P with equality if and only if Ii = J for all i. This follows from the strict convexity of the function h(x) = x2 . So the theorem also is true in this setting. I don't currently know how generally the convexity of Var[ln(l + I X)] is true but whenever it is, and we also have Var[ln(l + IX)] increasing in I , then the compound growth mean variance efficient frontier is once again the set of fractional Kelly strategies with 0 ~ c ~ 1. In email correspondence, Stewart Ethier subsequently showed that Var[ln(l + I X)] need not be convex. Example (Ethier): Let X assume values -1, 0 and 100 with probabilities 0.5, 0.49 and 0.01, respectively. Then, on approximately the interval [0.019, 0.180] the second derivative of the variance is negative, hence the variance is strictly concave on that interval. The first derivative of Var[ln(l+ IX)] equals 2 Cov(ln(l+ IX), X/(l+ IX)), which is al- ways nonnegative because the two functions of X in the covariance are increasing in X. Thus Var[ln(l + I X)] is always increasing in I. The second derivative of Var[ln(l + I X)] equals 2 Var(X/(l + I X))- 2 Cov(ln(l+ I X)-l, X 2 /(1+ I X)2). However, the covariance term sometimes exceeds the variance term. Samuelson's Criticisms The best known "opponent" ofthe Kelly Criterion is Nobel Prize winning economist, Paul Samuelson, who has written numerous polemics, both published and private, over the last 40 years. William Poundstone's book Fortune's Formula gives an extensive account with references. The gist of it seems to be: (1) Some authors once made the error of claiming, or seeming to claim, that acting to maximize the expected growth rate (i.e., logarithmic utility) would ap- proximately maximize the expected value of any other continuous concave utility (the "false corollary"). Response: Samuelson's point was correct but, to others as well as me, obvious the first time I saw the false claim. However, the fact that some writers made mistakes has no bearing on an objective evaluation of the merits of the criterion. So this is of no further relevance. (2) In private correspondence to numerous people Samuelson has offered exam- ples and calculations in which he demonstrates, with a two valued X ("stock") and three utilities, H(W) = - l /W, K(W) = log W, and T(W) = W 1/ 2 , that if any one who values his wealth with one of these utilities uses one of the other utilities to choose how much to invest then he will suffer a loss as measured with his own utility in each period and the sum of these losses will tend to infinity as the number of periods increases. --- ## Page 548 Understanding the Kelly Criterion 519 Response: Samuelson's computations are simply instances of the following gen- eral fact proven 30 years earlier by Thorp and Whitley (1972, 1974).5 Theorem 1. Let U and V be utilities defined and differentiable on (0,00) with U'(x) and V'(x) positive and strictly decreasing as x increases. Then if U and V are inequivalent, there is a one period investment setting such that U and V have distinct sets of optimal strategies. Furthermore, the investment setting may be chosen to consist only of cash and a two-valued random investment, in which case the optimal strategies are unique. Corollary 2. If the utilities U and V have the same (sets of) optimal strategies fo r each finite sequence of investment settings, then U and V are equivalent. Two utilities U1 and U2 are equivalent if and only if there are constants a and b such that U2 (x) = aU1(x) + b(a > 0), otherwise U - 1 and U2 are inequivalent. Thus, no utility in the class described in the theorem either dominates or is dominated by any other member of the class. Samuelson offers us utilities without any indication as to how we ought to choose among them, except perhaps for this hint. He says that he and an apparent majority of the investment community believe that maximizing U (x) = - 1/ x explains the data better than maximizing U(x) = logx. How is it related to fractional Kelly? Does this matter? Here are two examples showing that this utility can choose cf* for any 0 < c < 1, c i=- 1/2, depending on the setting: For a favorable coin toss and U(x) = - 1/x, we have f* = p,/(yIP + ,fij)2 which increases from p,/2 or half Kelly to p, or full Kelly as p increases from 1/2 to 1, giving us the set 1/2 < c1. On the other hand, if P (X = A) = P (X = - 1) = 1/2 describe the returns and A> 1 so P, > 0, p, = (A-l)/2 and the Kelly f* = p,/A. For U(x) = - 1/x we find U maximized for f = {-2A± (4A2+A(A- l )2)1/2}/A(A- l ), which is asymptotic to A -1/2 as A increases, compared to the Kelly f*, which is asymptotic to 1/2 as A increases, giving us the set 0 < c < 1/2. In the continuous case, the relation between c, g(f) and a(G(f)) is simple and the tradeoff between growth and spread in growth rate as we adjust between 0 and 1 is easy to compute and it's easy to visualize the correspondence between fractional Kelly and the compound growth mean-variance efficient frontier. This is not the case for these two examples so the fact that U (x) = -1/ x can choose any c, 0 < c < 1, c i=- 1/2 doesn't necessarily make it undesirable. 6 I suggest that 5The first Thorp and Whitley paper is reprinted in this book in Section 4 where three of Samuel- son's papers are reprinted and discussed in the introduction to that part of this book. 6 MacLean, Ziemba and Li (2005) reprinted in Section 4 of this book, show that for lognormally distributed assets, a fractional Kelly strategy is uniquely related to the coefficient a < 0 in t he negative power utility function oow" via the formula c = 1/(1 - a) so 1/2 Kelly is -l/w. However, when assets are lognormal this is only an approximation and, as shown here, it can be a poor approximation. --- ## Page 549 520 E. 0. Thorp a useful way to look at the problem for any specific example involving n period compound growth is to map the admissible portfolios into the (O"(G{fi}), g({fd)) plane, analogous to the Markowitz one period mapping into the (standard deviation, return) plane. Then examine the efficient frontier and decide what tradeoff of growth versus variability of growth you like. Professor Tom Cover points out that there is no need to invoke utilities. Adopting this point of view, we're simply interested in portfolios on the compound growth efficient frontier whether or not any of them happen to be generated by utilities. The Samuelson's preoccupation with utilities becomes irrelevant. The Kelly or maximum growth portfolio, which as it happens can be computed using the utility U (x) = log x, has the distinction of being at the extreme high end of the efficient frontier. For another perspective on Samuelson's objections, consider the three concepts: normative, descriptive and prescriptive. A normative utility or other recipe tells us what portfolio we "ought" to choose, such as "bet according to log utility to maximize your own good". Samuelson has indicated that he wants to stop people from being deceived by such a pitch. I completely agree with him on this point. My view is instead prescriptive: how to achieve an objective. If you know future payoff for certain and want to maximize your long term growth rate then Kelly does it. If, as is usually the case, you only have estimates of future payoffs and want to come close to maximizing your long term growth rate, then to avoid damage from inadvertently betting more than Kelly you need to back off from your estimate of full Kelly and consider a fractional Kelly strategy. In any case, you may not like the large drawdowns that occur with Kelly fractions over 1/2 and may be well advised to choose lower values. The long term growth investor can construct the compound growth efficient frontier and choose his most desirable geometric growth Markowitz type combinations. Samuelson also says that U (x) = - 1/ x seems roughly consistent with the data. That is descriptive, i.e., an assertion about what people actually do. We don't argue with that claim - it's something to be determined by experimental economists and its correctness or lack thereof has no bearing on the prescriptive recipe for growth maximizing. I met the economist Oscar Morgenstern (1902- 1977), coauthor with John von Neumann of the great book, The Theory of Games and Economic Behavior, at his company, Mathematica; in Princeton, New Jersey, in November of 1967 and, when I outlined these views on normative, prescriptive and descriptive, he liked them so much that he asked if he could incorporate them into an article he was writing at the time. He also gave me an autographed copy of his book, On the Accuracy of Economic Observations, which has an honored place in my library today and which remains timely. (For instance, think about how the government has made successive revisions in the method of calculating inflation so as to produce lower numbers, thereby gaining political and budgetary benefits). --- ## Page 550 Understanding the Kelly Criterion 521 Proebsting's Paradox Next, we look at a curious paradox. Recall that one property of the Kelly Cri- terion is that if capital is infinitely divisible, arbitrarily small bets are allowed, and the bettor can choose to bet only on favorable situations, then the Kelly bet- tor can never be ruined absolutely (capital equals zero) or asymptotically (capital tends to zero with positive probability). Here's an example that seems to flatly contradict this property. The Kelly bettor can make a series of favorable bets yet be (asymptotically) ruined! Here's the email discussion through which I learned of this. From: Todd Proebsting Subject: FW: incremental Kelly Criterion Dear Dr. Thorp, I have tried to digest much of your writings on applying the Kelly Crite- rion to gambling but I have found a simple question that is unaddressed. I hope you find it interesting: Suppose that you believe an event will occur with 50% probability and somebody offers you 2:1 odds. Kelly would tell you to bet 25% of your capital. Similarly, if you were offered 5:1 odds, Kelly would tell you to bet 40%. Now, suppose that these events occur in sequence. You are offered 2:1 odds, and you place a 25% bet. Then another party offers you 5:1 odds. I assume you should place an additional bet, but for what amount? If you have any guidance or references on this question, I would appre- ciate it. Thank you. From: Ed Thorp To: Todd Proebsting Subject: Fw: incremental Kelly Criterion Interesting. After the first bet the situation is: A win gives a wealth relative of 1 + 0.25 * 2 A loss gives a wealth relative of 1 - 0.25 Now bet an additional fraction 1 at 5:1 odds and we have: A win gives a wealth relative of 1 + 0.25 * 2 + 51 A loss gives a wealth relative of 1 - 0.25 - 1 The exponential rate of growth g(f) = 0.5*ln(1.5+5f) +0.5*ln(0.75- f) Solving g'(f) = 0 yields 1 = 0.225 which was a bit of a surprise until I thought about it for a while and looked at other related situations. From: Todd Proebsting To: Ed T horp Subject: RE: incremental Kelly Criterion Thank you very much for the reply. --- ## Page 551 522 I, too, came to this result, but I thought it must be wrong since this tells me to bet a total of 0.475 (0.25 + 0.225) at odds that are on average worse than 5:1, and yet at 5:1, Kelly would say to bet only 0.400. Do you have an intuitive explanation for this paradox? From: Ed Thorp To: Todd Proebsting Subject: Re: incremental Kelly Criterion I don't know if this helps, but consider the example: A fair coin will be tossed (Pr Heads = Pr Tails = 0.5). You place a bet which gives a wealth relative of 1 + u if you win and 1 - d if you lose (u and d are both nonnegative). (No assumption about whether you should have made the bet.) Then you are offered odds of 5:1 on any additional bet you care to make. Now the wealth relatives are, each with Pr 0.5, 1 + u + 5f and 1 - d - f. The Kelly fraction is f = (4 - u - 5d)/10. It seems strange that increasing either u or d reduces f. To see why it happens, look at the In(l + x) function. This odd behavior follows from its concave shape. From: Todd Proebsting To: Ed Thorp Subject: RE: incremental Kelly Criterion Yes, this helps. Thank you. It is interesting to note that Kelly is often thought to avoid ruin. For instance, no matter how high the offered odds, Kelly would never have you bet more than 0.5 of bankroll on a fair coin with one single bet. Things change, however, when given these string bets. If I keep offering you better and better odds and you keep applying Kelly, then I can get you to bet an amount arbitrarily close to your bankroll. Thus, string bets can seduce people to risking ruin using Kelly. (Granted at the risk of potentially giant losses by the seductress.) From: Ed Thorp To: Todd Proebsting Subject: Re: incremental Kelly Criterion Thanks. I hadn't noticed this feature of Kelly (not having looked at string bets). To check your point with an example I chose consecutive odds to one of An : 1 where An = 2n, n = 1,2, ... and showed by induction that the amount bet at each n was fn = 3(n-l) /4n (where /\ is exponentiation and is done before division or multiplication) and that sum{fn : n = 1, 2, ... } = 1. A feature (virtue?) of fractional Kelly strategies, with the multiplier less than 1, e.g. f = c* f(kelly) , 0 < c < 1, is that it (presumably) avoids this. E. O. Thorp --- ## Page 552 Understanding the Kelly Criterion 523 In contrast to Proebsting's example, the property that betting Kelly or any fixed fraction thereof less than one leads to exponential growth is typically derived by assuming a series of independent bets or, more generally, with limitations on the de- gree of dependence between successive bets. For example, in blackjack there is weak dependence between the outcomes of successive deals from the same unreshuffied pack of cards but zero dependence between different packs of cards, or equivalently between different shuffiings of the same pack. Thus the paradox is a surprise but doesn't contradict the Kelly optimal growth property. R eferences Ethier, S. (2010) . The Doctrine of Chances. Berlin: Springer-Verlag. Geyer, A. and W. T . Ziemba (2008) . The innovest Austrian pension fund financial planning model InnoALM. Operations Research, 56(4), 797- 810. MacLean, L. C., W . T. Ziemba and G. Blazenko (1992). Growth versus security in dynamic investment analysis. Management Science, 38, 1562- 1585. Markowitz, H. M. and E. van Dijk (2006). Risk return analysis, in S. A. Zenios and W. T. Ziemba (eds.) , Handbook of Asset and Liability Management, Vol. I: Theory and Methodology. Amsterdam: North Holland, 139- 197. Pabrai, M. (2007). The Dhandho Investor. New York: Wiley. Poundstone, W. (2005). Fortune 's Formula. US: Hill and Wang. Taleb, N. N. (2007). The Black Swan: The Impact of the Highly Improbable. US: Barnes and Noble. Thorp, E. O. (2006). The Kelly Criterion in blackjack, sports betting and the stock market, in S. A. Zenios and W. T. Ziemba (eds.) , Handbook of Asset and Liability Manage- ment, Vol. I: Theory and Methodology. Amsterdam: North Holland, 385- 428. Thorp, E . O. and R. Whitley (1972). Concave utilities are distinguished by their optimal strategies. Colloquia Mathematica Societatis Janos Bolyai, 9. Thorp, E. O. and R. Whitley (1974). Progress in statistics, in Proceedings of the European Meeting of Statisticians, Budapest. North Holland, pp. 813- 830. Ziemba, R. E. S. and W. T. Ziemba (2007). Scenarios for Risk Management and Global Investment Strategies. New York: Wiley. Ziemba, W. T. (2003). The Stochastic Programming Approach to Asset Liability Manage- ment. AIMR. Ziemba, W . T . and R. G. Vickson, eds. (2006) . Stochastic Optimization Models in Finance, 2nd Edition. Singapore: World Scientific. --- ## Page 553 This page is intentionally left blank --- ## Page 554 525 COLLOQUIA MATHEMATICA SOCIETATIS JA.NOS BOLVAI 9 . EUROPEAN MEETING OF STATISTICIANS, BUDAPEST (HUNGARY) . '972. 37 CONCAVE UTILITIES ARE DISTINGUISHED BY THEIR OPTIMAL STRATEGIES E . THORP -- R. WHITLEY 1. INTRODUCTION Mossin [5], Thorp [7], and Samuelson [6] showed for spe- cific pairs of utility functions that different utilities can lead to different optimal strategies. In particular the optimal investment strategy for the utility logx is not necessarily the optimal strategy for the utility 1 x'Y 'Y These examples suggest the following generalization, of obvious im- portance to general utility theory. Consider a T stage investment process. At each stage allocate re- sources among the available investments. Each chosen sequence A of al- locations ("strategy") yields a corresponding terminal probability distribu- tion F1 of assets at the completion of stage T. For each utility func- tion U(.), consider those strategies A *( U) which maximize the expect- ed value f U(x)dF1 (x) of terminal utility. Assume sufficient hypotheses - 813 - --- ## Page 555 526 E. Thorp and R. Whitley on U and the set of F1 so that the integral is defined and that further- more the maximizing strategy A *( U) exists. Then is it true in general that A*(U1 ) is not A*(U2 ) for "distinct" utilities UI and U2? As we now show, the answer is yes: the Mossin - Thorp - Samuelson results for specific utility pairs generalizes to the principal class of interest in modern utility theory. 2. THE MAIN THEOREM We prove this for the class of "interesting" concave utilities. We be- gin with more special hypotheses. Theorem 1. Let U and V be utilities defined and differentiable on (0,00) with U'(x) and V'(x) positive and strictly decreasing as x increases. Then if U and V are inequivalent, there is a one period in- vestment setting such that U and V have distinct sets of optimal strat- egies. Furthermore, the investment setting may be chosen to consist only of cash and a two-valued random investment, in which case the optimal strategies are unique. Corollary 2. If the utilities U and V have the same (sets of) opti- mal strategies for each finite sequence of investment settings, then V and V are equivalent. Two utilities UI and U2 are equivalent if and only if there are constants a and b such that UzCx) = aUI (x) + b (a> 0), otherwise VI and U2 are inequivalent. Let X. (I ~ i ~ k) be the (random) outcome per unit invested in I the ith "security". We call (Xl"'" Xk ) the investment setting. We as- sume X. is independent of the amount invested. Let the initial capital be I Zo and let the final capital be ZI' A strategy is an allocation W = = (WI' . .. , wk ) where Wj is the fraction of Zo allocated to security i. We assume w. ~ ° for all i, that Z w. = I, and that wealth is infinite- I j I ly divisible. Thus the w. may assume any real values consistent with the I - 814 - --- ## Page 556 Concave Utilities are Distinguished by Their Optimal Strategies 527 constraints and with the requirement that ~ wiX, is in the domain of the utility function U. I Given a particular U satisfying the hypotheses of the theorem, sup- pose E U(Zl (W)) is maximized by some strategy W*. Then W* is an optimal (or best) strategy for U relative to the given investment setting. Proof of Theorem. Suppose that U and V have the same optimal strategies for everyone period investment setting consisting of cash and a two-valued random investment. It will be shown that U and V are equiv- alent, which will establish the logical contrapositive to the theorem and hence the theorem itself. In the proof of theorems we shall assume for technical simplicity that the initial capital Zo = 1. When theorems have been established for this case, consideration of the transformation Uo(s) = U(Zos) = UU) gives the theorems for arbitrary Zo > O. We shall therefore state the gen- eral results without further comment after proving the Zo = 1 case. Let the only investment (besides cash) be X where P(X = 1 - b) = = q = 1 - p and P(X = 1 + a) = p, where a> 0 and 0 < p, b < 1. The choice 0 < b < I, rather than simply b = I, has been made because for b = 1 arid w = I, the expression U(O) would arise and 0 is not necessarily in the domain of U (e.g., U(x) = log x). The available strat- egies are to allocate the fraction w of recources to X and 1 - w to cash, with 0 ~ w ~ 1. At the end of the period, we have (2.1) EU(ZI (w») = pU(l + aw) + qU(l - bw) = ftw) . To find the maximum, consider f'(w) = apU'(l + aw) - bqU'(l- bw). Since U'(t) strictly decreases as t increases, we have f'(w) decreasing strictly as w increases. Thus there is a unique maximum. If f'(w*) == 0 for some w* with 0 ~ w* ~ I, then the maximum is at this unique w* . If i'(w) > 0 for all w with 0 ~ w ~ 1, then the unique maximum is at w = 1. If instead f'(w) < 0 for 0 ~ w ~ 1, then the unique maxi- mum is at w = O. - 815 -- --- ## Page 557 528 E. Thorp and R. Whitley If f '( ) - 0 h u ~ 1 + aW) - !!!I SOd w - we ave U'(l _ bw) - ap' uppose a> an 1 , 1 . U' (I + 2~) 2 < b < 1 are given and we wish f (2b) = O. Lettmg A = I' U'("2) we can solve A = Pfp for p, with 0 < p < 1. Thus for each a> 0 1 there is a choice of p, hence an X, such that w* = 2b is optimal for U. Now suppose that U and V have the same optimal strategies for all such investment settings. Then w* = 2Ib for V also and we have U' (l + 2~) V' ( 1 + ~) 1 1 --~l- = 1 for all a > O. Letting V' ( -2) = o:U' ( -2 ) U'("2) V'( 2) we find V'(t) = o:U'(t) (t> 1) whence V(t) = o:U(t) + f3 (t> I). When t < 1, we proceed similarly. Choose X so that P(X = 2) = P and P(X = I - b) = q, where 0 < b < 1. Then EU(ZI (w)) = pU(1 + w) + qU(l - bw) = f(w) f'(w) = pU'(l + w) - bqU'(l - bw) and the maximum is unique and located as before. If f '() 0 h "\ U'( 1 - awl ~ d' b w = we ave 1\ = U'( I + w) = aq an glven w = , 0< b < 1, we can choose p with 0 < p < I such that A = L. Then aq as before we find V'(l - ab) = -yU'(l - ab) and since a and b can be any numbers such that 0 < a, b < 1, then V'(t) = -yU'(t) (0 < t < 1) V'(l + b) where -y = U'(l + b)' But -y was shown to be 0:. Thus V(t) = o:U(t) + 6 (0 < t < 1). Also V(l) = o:U(l) + E. Hence V(t) - o:U(t) = f3 if t> 1, 6 if t < 1 and E if t = 1. But V(t)- - o:U(t) is continuous so f3 = 6 = € so V(t) = o:U(t) +~. Thus U and - 816 - --- ## Page 558 Concave Utilities are Distinguished by Their Optimal Strategies 529 V are equivalent under the assumption that they have the same optimal strategies for all one period investment settings containing only (cash and) a two-valued random investment. The logical contrapositive assertion is the Theorem. This completes the proof. The Corollary follows a fortiori. Note that a single investment setting of the type in the proof will not in general distinguish inequivalent utility functions. For instance, if E(X) ~ ~ 0 then w = 0 is the unique optimal strategy for all the utilities of Theorem I (more generally, for all strictly concave utilities, as defined be- low) so such X distinguish between none of these utilities. It may be of interest to characterize each investment setting by the pairs of utility func- tions it distinguishes between or "separates", and to similarly characterize collections of investment settings. For a security X, let m(X) and M(X) be the greatest and least numbers, respectively, such that P(m(X) ~ X ~ M(X)) = 1. Then for a collection C of investment settings whose securities are {Xa: a E A }, where A is some index set, let m A = inf {m(X a): a E A} and M A = = sup {M(Xa ): a E A}. Evidently, if U(t) = V(t) for mA ~ t ~ M A , the collection C will not separate U and V. Thus a collection with m A = = 0 and M A = 00 will be needed in general to prove the conclusion of Theorem 1. Next we generalize Theorem I to concave non-decreasing utilities de- fined on (0, 00). We do not make the common assumption that first or even second derivatives exist. A function ! is concave on an interval I if for each pair of points xl*- X 2 in I and each number s with O sflx 1) + (1 - s)flx 2) always, then ! is strictly concave. (We use "concave" to mean "concave from below".) The more general definition includes such computationally and empir- ically natural functions as the "polygonal" utilities. In these, the utility is a sequence of linear segments. The vertices are such that the function lies on or below each segment extended, and the ordinates of the vertices in- crease as the abscissas increase. - 817 - --- ## Page 559 530 E. Thorp and R. Whitley First, recall some facts from the elementary theory of concave func- tions. (Most texts give results for convex functions. But I is concave ex- actly when -I is convex so the theories of concave and convex functions are equivalent.) A concave function is either continuous in the interior of its domain or non-measurable. An increasing function is always measurable so our utilities are continuous. A continuous concave function I defined on an open interval has a left derivative f~ and a right derivative I~ defined everywhere. (If the left endpoint a is includ~d in the interval of definition, then f~ (a) is not defined and f~ (a) mayor may not be de- fined. Similarly, if the right endpoint b is included in the interval of de- finition, then I~ (b) is not defined and I~ (b) mayor may not be de- fined.) Furthermore, I~ (t) ~ I~ (t) for all t except the endpoints in the domain of I and whenever t 1 < t 2 then I~ (t 1 ) ~ f~ (t 2) and I: (tl) ~ I: (t2)· There are at most countably many points where I~ (t) > > I~ (t); otherwise f~ (t) = I~ (t) = f'(t) and I is differentiable. Proofs of these assertions and further theorems on concave functions are given for instance in H a r d y, Lit tIe woo d, Pol y a [3]. Theorem 3. Let U and V be concave utilities defined on (0,00), one of which is strictly increasing on (0, 1 + e) for some e> 0. If U and V are inequivalent then there is a one period investment setting such that the sets of optimal strategies lor U and for V are distinct. The in- vestment setting may be chosen to consist only of cash and a two-valued random investment. If U and V are each strictly concave on the same one of the sets (0, Zo] or [Zo' (0), then the optimal strategies are unique and U and V therefore have distinct optimal strategies. Proof. We proceed as in the proof of Theorem 1 until we obtain equiation (2.1). Note that f is concave and that if U is strictly concave on either (0, I] or [I, (0) then I is strictly concave. Now f(w) is a continuous function defined on the closed bounded set {w: 0 ~ w ~ I} hence f has an absolute maximum. Let w* be a pOint where f attains its max- imum. It follows from the continuity of f that the set of all such w* is closed. - 818 - --- ## Page 560 Concave Utilities are Distinguished by Their Optimal Strategies 531 From the concavity of f, the set of points w* where f attains its maximum is also convex, hence it is a closed interval in [0, 1]. If f is strictly concave the maximum is unique. For any w* with ° < w* < l, f is a maximum if and only if f~ (w*) ~ ° ~ f~ (w*). A maximum occurs at w* = ° if and only if f~ (0) :s:;;; 0. A maximum occurs at w* = I if and only if f~ (I) ~ 0. If the maxima occur on an interval [a, b] with o:s:;;; a < b:S:;;; 1, then f~ (a) ~ ° and f~ (a) = 0, f~ (b) = ° and f~{b):S:;;; 0, and ['(w*) ex- ists and is zero for a < w* < b. Equation (2.1) yields f~ (w) = apU~ (1 + aw) - bqU~ (l - bw) ~ (2.2) ~ apU~ (l + aw) - bqU~ (l - bw) = f~ (w) . Since U~ (t) and U~ (t) are non-increasing as t increases, it follows from equation (2.2) that f~ (w) and f~ (w) are non-increasing as w in- creases. Let c be such that 0 < c < band U/(l - c) and V'(l - c) are defined. This is possible because U' and V' are both defined except at countably many points hence there are uncountably many points in (0, 1) where both U' and V' exist. With a and b already given, choose w = %. Consider now the case where U~ (1 + a:) > 0. Then we may choose p with 0 < p < 1 in equation (2.2) so that f~ (%) = 0. This means f~ ( %) :s:;;; 0 and since w = % is not an endpoint of [0, 1 J this means f attains its maximum at %, thus % is optimal for U in the given investment setting. I e . t' Since U and V have the same optima strategies, w = b IS op 1- mal for V hence V attains its maximum there so for w = %, g~ (w) = apV~ (1 + aw) - bqV~ (l - bw) ~ ° and apV~ (1 + aw)- - 819- --- ## Page 561 532 E. Thorp and R. Whitley - bq V~ (l - bw) = g~ (w) ~ O. Note that g~ (%) ~ 0 and the fact V~ (1 - e) > 0 implies that V~ ( 1 + at) > O. We may show similarly that if V~ ( 1 + ~) > 0 then U~ ( 1 + ~) > O. Since a is chosen in- dependently of band e this means that for each t> 1, U~ (t) > 0 if and only if V~ (t) > O. But this is readily shown to be equivalent to the statement that (t: U(t) = sup U(t)} = (t: V(t) = sup V(t)}, i.e. that if either U or V become horizontal for t ~ e > 1 then they both be- come horizontal for t ~ e> 1. For t> e, we have of course U'(t) = = V'(t) = O. For t < e, the argument continues as follows. From f' (w) = 0, apU~ (1 + ~) = bqU'(l - e), noting that U' ( 1 + ae) - b!!!J.. ' U~ (l - e) = U'(l - e). Thus U'(l _ e) = ap' From g_ (w) ~ 0, it . . V~ (1 + 1f)!!!J.. . V'(l - e) follows slIllilarly that V'( 1 _ e) ~ ap' Lettmg a = U'( 1 _ e) yields V~ ( 1 + ~) ~ aU~ ( 1 + at). Since the choices of band e were in- dependent of that a, the result holds for all a> 0, therefore V~ (t) ~ ~ aU~ (t) for all t > 1. A similar argument shows that V: (t) ~ aU: (t) for all t> 1. Thus, except for at most countably many points, V'(t) = aU'(t) for t> I. Now U and V are readily shown to be absolutely continuous on any closed subinterval of (I, 00), as a consequence of the fact they are con- tinuous, concave, and non-decreasing, thus V - aU is absolutely contin- uous. The absolute continuity of U - aV and the fact that (V - aU)' = = 0 almost everywhere implies that V - aU = (3, a constant (Goffman [2], p. 242, Prop. 12). A similar argument shows that Vet) = aU(t) + "Y for t < 1. The role of 2 in the proof of Theorem I is played by any number e such that 1 < e < e and U'(e) and V'(e) are both defined. One then shows as in the proof of Theorem I that V(i1'= aU(t) + {3 for 0 < t < 00. We - 820- --- ## Page 562 Concave Utilities are Distinguished by Their Optimal Strategies 533 have established the contrapositive assertion as in the proof of Theorem I. This completes the proof. The hypothesis that either U or V (hence both, from the proof) is strictly increasing for a positive distance to the right of 1 is required. If instead U and V are merely concave and non-decreasing, the conclu- sion of Theorem 3 need not hold. For instance, let U(t) = V(t) = 0 if t ~ d , where 0 < d ~ 1. Let U(t) and J]t)each be extended to (0, d) so that they are continuous, concave, and strictly increasing on (0, d). Then all such utilities have the same optimal strategies, yet many pairs are inequivalent. To obtain an inequivalent pair, let UU) = t - d if 0 < t < d and let V(t) = - (t - d)2. If for some constants a and 13, V(t) = exUU) + 13 then V'(t) = exU'(t). But V'(t) = - 2(t - d) ~ ex = exU'(t). To see that all such utilities U have the same optimal strategies, note that W = (wI' ... , wk ) is optimal for the investment setting (Xl"" ... , Xk ) if and only if P (~ wiXi ~ d) = I, in which case E U(ZI (W» = I = O. If instead P Cf WjXi < d) > 0 then for some e> 0, P (t: WjXj , :s;;; d - e) = 0> O. Then EU(ZI (W»:s;;; oU(d - e) < 0 so W is not optimal. 3. OTHER SEPARATING FAMILIES We next establish the conclusion of Theorem 1 using investment set- tings with n points in their range. We detennine the effect of varying the payoffs (xl"' " xn) and their probabilities (PI"'" Pn ) separately. One surprising conclusion (part (b» can be stated in tenns of an example. Suppose X consists of betting on a wheel of fortune divided into red, white and blue sectors, with payoffs of ~,1, and ~ respectively. Then if U and V are inequivalent on [i, ~] the areas of the sectors may be chosen so U and V have distinct optimal strategies. But if the wheel is divided into just red and blue sectors, with payoffs of ~ and 1, then - 821 - --- ## Page 563 534 E. Thorp and R. Whitley there are two inequivalent utilities on [~, ~] :which have the same opti- mal strategies for every choice of areas for the two sectors. Theorem 4. Suppose U and V are increasing strictly concave util- ities on (0,00). Let X be a random variable with outcomes ° ~ x 1 ~ ~ X2 ~ ... ~ xn with xl < 1 and xn > 1. Suppose P(X = xi) = Pi> 0, n 2' p. = l. i= 1 I (a) Let n and the Pi be given. If U and V have the same opti- mal strategies for each X (t.e. Xl" " 'Xn vary), then U and V are equivalent. (b) Let n and the xi be given. Suppose U' and V' exist and are continuous at 1. If U and V have the same optimal strategies for each X (t.e. PI"" 'Pn vary) and at least three xi are unequal to 1, then U and V are equivalent on [ZOxl,ZOxnJ. If exactly two of the x/ s are unequal to one, there are utilities U and V which are not equiv- alent on [ZOX l' ZOxn], but which have the same optimal strategy for each X. Proof. Assume Zo = 1. Let R = X-I and Yj = Xj - 1. Then in- vesting w in X gives an expected return (with respect to U) of n E(U(wR + 1»)::= Z p.U(wr. + O. Each function U(wr. + 1) is differen- i= 1 I I I tiab1e except at a countable set Cj of points, so except for w in the countable set C1 U ... U Cn the expectation E( U(wR + 1)) is differen- n tiable at w with dE(UC;R + 1) = L; p.,.U'(wr. + 1). Similarly each w i= Itt I function V(wr. + 1) is differentiable except at a countable set. Thus, ex- t cept at a countable set D of points in [0,00] both E(U(wR + 1)) and E( V(wR + I» are differentiable functions of w. They are also strictly concave functions of w. For part (a) let PI"'" Pn be given and choose Wo in (0, 1) - D. Consider the vectors ~ = (U/(wO'l + I), ... , U'(worn + 1)) and {3 = = (V'(wOrl + 1), ... , V'(worn + 1». Suppose that the non-zero vector - 822- --- ## Page 564 Concave Utilities are Distinguished by Their Optimal Strategies 535 'Y = (C I' C2 ' . .. , cn ) is perpendicular to a, i.e., the inner product (a, 'Y) = O. Choose C. ,. = I I Pi { f itt I ci I} with € = max 1 ~ i ~ n . p/ Since each component of a is positive, some ct > 0 and some cj < 0, hence some x . > 1 and some x. < 1. Also ,. + 1 = x. > 0 Then I J I I ' dE(U(wR + 1» I ~, . dw Iw:wo = i~ P;'jU (wO't + 1) = 0 and E(U(wR + 1) has a maximum at wOo By hypothesis E(V(wR + 1» has a maximum at w 0 and, since it is differentiable there, dEC V(;R + 1,» I = w w=wo n = .Z P;'i V'(wO'j + 1) = 0 Le., ({3, 'Y) = O. Hence the set of vectors per- I == I pendicular to a is also perpendicular to {3 which implies that {3 = aa. Since the components of a and (3 are non-negative, a ~ O. Equating components (3.1) U'(wor. + 1) = aV'(wor. + 1) I . I where a is a non-negative function of '1, . . . ,'n ~nd wOo Since U and V are strictly concave there is a point to not in D, Wo < to < 1, with V'(to) > 0 and U'(to) > O. Choose '1 so that wOrt + 1 = to' choose '2 > 0 with t = wor 2 + 1 not in D , and choose '3 < ... < rn so they are not in D. Then U'(to) = a('l"'" rn , wo)V'(to) and , , U'(to) . U (w2'2 + 1) = aC'I' ... ,'n' wo) V (wO'2 + 1). Thus a = V'Uo) > 0 IS constant. So V'(t) = aU'(t) for any t> 1 not in D. Since V and U are absolutely continuous on any closed subinterval, Vet) = aU(t) + b for all t> 1. A similar argument shows that V(t) = cU(t) + d for t < 1 V '(to) with C = YI'(t ) = a. The equivalence of U and V now follows (as in o the proof of Theorem 1) from their continuity. For part (b) suppose that the Xi are given, with 0 < xl < X 2 < ... . . . < Xp ~ 1 ~ xp + 1 < .. . < x n . We proceed as before, but now consider, for 0 < Wo < 1 and U, V differentiable at WO'j + 1, 1 ~ j ~ n, the - 823 - --- ## Page 565 536 E. Thorp and R. Whitley vectors a = ('1 U'(wO'1 + 1), ... ,'n U'(WO'n + 1» and 'if = = ('1 V'(WO'l + 1), .. ·"n V'(WO'n + 1». Since a has both positive and negative components there is a vector (dl , d2 , ••• , dn) perpendicular to ...., d. ~ a with each dt > 1. Choose Pt = n I thus Pi> 0 and £..J P; = 1, Zd I ;= 1 j and define X by P(X = xI) = PI. Thus = dE(U(wR + 1)) I dw w=wo . dE(U(wR+ 1))1 @,(dl,···,dn)) By hypotheSls dw w=wo = v = 0 so t dl ('P, (dt ,· •. , dn») = o. Suppose that :y = (e1 , e2 , • •• , en) is perpendicu- el + dod; lar to a. Let do > max I el I and choose PI = " Note that n P, > 0 and Z P, = 1. Thus 1= 1 ,~ e, + dod, n dE(U(wR + 1)) I = ~ P [, U'(w , + 1)] dw w = w 0 1= 1 I I 0 I n and letting D = Z (el + dod,.) gives 1= 1 DI i e.,.U'(wo' · + 1) + DI do Z d,·'IU'(wO'j + 1) = 1= 1 I I I j 1 ....,...., 1 ...., = D (a, "Y) + D do(a,(d1,·· · ,dn ))= o. - 824 - --- ## Page 566 Concave Utilities are Distinguished by Their Optimal Strategies 537 Hence, dE(V(;R + 1» I = O. Th· . Id (R -) d '" W w=wo IS Yle s ,.,I'Y + 0(/3, (dl , ... . . . , dn» = (ii, -:y) = O. Thus (3/2) U'(wO'i + 1) = a(pl' Pz' ... , Pn' wo) V'(wo'; + 1) (1 ~ i ~ n) . For W in (0, I) with U, V differentiable at W't + I (1 ~ i ~ n) we have (3.2) with Wo = w. Consider the quotient U'(w,; -Fl) (3.3) h( w) = a(p (, . . .. , P n' w) = V' (w,. + I) (l <; i ~ n) . , First look at the case where at least three x;'s are unequal to one. Sup- pose that x I < I < xn _ I < xn; the proof where two or more points fall to the left of I is similar. Let 'P; O. (Al- ternately, the [c, 00) result implies the (c, (0) result: if V(x) = V(x) on every interval [c+€,oo) (O<€ O}. For the class {3 of bounded - 827 - --- ## Page 569 540 E. Thorp and R. Whitley utilities, I.e., Me U) == sup VU) < 00, m( U) == inf VU) > - 00, we suggest h h V . "" V-M t at eac [ ] eqUIvalence class be represented by V = M + 1. -m Note that M( iJ) = 1 and m( iJ) = O. Then the "closeness" of V and V, i.e., of [UJ and [V], is defined to be sup [V(t) - V(t)l and written either d( U, V) or d([ UJ, [V]) or d( V, V). We now show that U and V can be "close" yet the optimal strat- egies for U and V need not be. For n~ 2, let Vn and V n be de- fined as follows: V (t) = ~ - 1 if 0 < t ~ 1 + 1. and 1 if t > 1 + 1. . n n+l n n' ~ 2n - I. 1 Y n (t) = n + 1 t - 1 If 0 <. t <. 1 + n ' t : ~ 13 if 1 + ~ < t <. 2, and 1 if t> 2 . Then d( Un' Vn) =~. Now choose an investment setting consisting only of cash and the security X, fihere P(X = 1 - E) = q, P(X = 1 + a) = 1 ~ = p, 11 < a < 1, and 0 < E, p, q < 1. Assume Zo = 1. A calculation (2n-l)(n-l) shows that if ap > q€ n + 1 ,then the unique optimal strategy for V is w* = _1 and for V the unique optimal strategy is w* = 1. n an n Thus for any f> > 0 we can construct sequences Un and Vn such tha t d( V , V ) ~ 0 as n ~ 00 and I w * (V ) - wit (V ) I ~ I - f>, where n n n n w* (iJ) means an optimal strategy for V. Even though a small "error" in the utility function can lead to a large change in optimal strategy, it can only lead to a small change in conse- quences, in the following sense. (We use the abbreviation V(W) for EU(Zo ~ wiXi ). Thus for each W, U(W) is a number and I U(Zo Z w.X.) is a random variable.) i I I - 828 - --- ## Page 570 Concave Utilities are Distinguished by Their Optimal Strategies 541 Lemma. If d( U, V) is" small," then U(W*( V») == U(W*( V)) and V(W*( U)) == V(W*( V)), ie., if U and V are" close," an optimal strate- gy for one is "nearly optima!' for the other. Proof. Let d( U, V) ~ € so V( t) + € ~ U(t). Then for any alloca- tion W, V(Zo ZZ" wtXi ) + € ~ U(Zo ~ wiXi ) and E (V(Zo ~ wiX,) + z z + €) = E (V(Zo ~ wjX,)) + € ~ EU(Zo ~WiXi), or V(W) + € ~ U(W). """ -., I "J -., Interchanging U and V in the argument yields U(W) + € ~ V(W) so I U(W) - V(W) I ~ €. The choices for W of W*( V) and W*( V) yield the conclusion of the lemma. The lemma and the example show us what may happen if we replace a U by a nearby V which may have more desirable properties, such as differentiability (of various orders), strictly increasing, etc.: The optimal strategies may change drastically but the maximum utility over all strate- gies changes only slightly. Note added in proof: The authors have since extended the central re- sult of the paper, Theorem 3, as follows. Theorem. Let U and V be continuous non-decreasing functions defined on an arbitrary interval I of the real line. Then if U and V are inequivalent, there is a one-period two security investment setting such that U and V have distinct optimal strategies if either (a) U and V are in the class of all functions which are either concave or convex, or (b) U and V are in the class of all functions with a second derivative which exists and is continuous, except perhaps for a set of isolated points. Thus the Theorem includes the utility functions generally encount- ered. - 829- --- ## Page 571 542 E. Thorp and R. Whitley REFERENCES [1] M. F r i e d man - L. J. S a v ag e, The Utility Analysis of Choices Involving Risk, Journal of Political Economy, 56 (1948), 279-304. [2] C. Go ffm an, Real Functions, Holt, Rinehart and Winston Inc., New York, 1953. [3] G. Hardy - J . Littlewood .~ G. Polya, Inequalities, Cambridge University Press, 1959. [4] H. Markowitz, The Utility of Wealth, Journal of Political Econ- omy, (1952), 151-158. [5] J. M 0 s si n, Optimal Multiperiod Portfolio Policies, Journal of Bus- iness, (April 1968) . . [6] P. A. Sam u e 1 son, The 'Fallacy' of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling, unpublished pre- liminary preprint, 1971. [7] E. O. Tho r p, Optimal Gambling Systems for Favorable Games, Review of the International Statistical Institute, 37 (1969), 273-293. - 830- --- ## Page 572 543 38 Medium Term Simulations of The Full Kelly and Fractional Kelly Investment Strategies Leonard C. MacLean; Edward O. Thorp! Yonggan Zhaotand William T. Ziemba§ Abstract Using three simple investment situations, we simulate the behavior of the Kelly and fractional Kelly proportional betting strategies over medium term horizons using a large number of sce- narios. We extend the work of Bicksler and Thorp (1973) and Ziemba and Hausch (1986) to more scenarios and decision periods. The results show: (1) the great superiority of full Kelly and close to full Kelly strategies over longer horizons with very large gains a large fraction of the time; (2) that the short term performance of Kelly and high fractional Kelly strategies is very risky; (3) that there is a consistent tradeoff of growth versus security as a function of the bet size determined by the various strategies; and (4) that no matter how favorable the investment opportunities are or how long the finite horizon is, a sequence of bad results can lead to poor final wealth outcomes, with a loss of most of the investor's initial capital. 1 Introduction The Kelly optimal capital growth investment strategy has many long term positive theoretical properties (MacLean, Thorp and Ziemba 2009). It has been dubbed" fortunes formula" by Thorp (see Poundstone, 2005). However, properties that hold in the long run may be countered by negative short to medium term behavior because of the low risk aversion of log utility. In this paper, three well known experiments are revisited. The objectives are: (i) to compare the Bicksler- Thorp (1973) and Ziemba - Hausch (1986) experiments in the same setting; and (ii) to study them using an expanded range of scenarios and investment strategies. The class of investment strategies generated by varying the fraction of investment capital allocated to the Kelly portfolio are applied to simulated returns from the experimental models, and the distribution of accumulated capital is described. The conclusions from the expanded experiments are compared to the original results. "Herbert Lamb Chair, School of Business Administration, Dalhousie University, Halifax, Canada B3H 3J5 l.c.maclean@dal.ca IE.O. Thorp and Associates, Newport Beach, CA, Professor Emeritus, University of Califirnia , Irvine, CA ICanada Research Chair, School of Business Administration, Dalhousie University, Halifax, Canada B3H 3J5 yonggan.zhao@dal.ca §Alumni Professor of Financial Modeling and Stochastic Optimization (Emeritus), University of British Columbia, Vanvouver, Canada, Visiting Professor, Mathematical Institute, Oxford University, UK, ICMA Centre, University of Reading, UK, and University of Bergamo, Italy wtzimi@mac.com --- ## Page 573 544 L. C. MacLean, E. O. Thorp, Y. Zhao and W. T. Ziemba 2 Fractional Kelly Strategies: The Ziemba and Hausch (1986) example We begin with an investment situation with five possible independent investments where one wagers $1 and either loses it with probability 1 - P or wins $ (0 + 1) with probability p, where 0 is the odds. The five wagers with odds of 0 = 1, 2,3,4 and 5 to one all have expected value of 1.14. The optimal Kelly wagers are the expected value edge of 14% over the odds. So the wagers run from 14%, down to 2.8% of initial and current wealth at each decision point. Table 1 describes these investments. The value 1.14 was chosen as it is the recommended cutoff for profitable place and show racing bets using the system described in Ziemba and Hausch (1986). I Win Probability I Odds I Prob of Selection in Simulation I Kelly Bets 0.570 1-1 0.1 0.140 0.380 2-1 0.3 0.070 0.285 3-1 0.3 0.047 0.228 4-1 0.2 0.035 0.190 5-1 0.1 0.028 Table 1: The Investment Opportunities Ziemba-Hausch (1986) used 700 decision points and 1000 scenarios and compared full with half Kelly strategies. We use the same 700 decision points and 2000 scenarios and calculate more attributes of the various strategies. We use full, 3/ 4, 1/ 2, 1/ 4, and 1/ 8 Kelly strategies and compute the maximum, mean, minimum, standard deviation, skewness, excess kurtosis and the number out of the 2000 scenarios that the final wealth starting from an initial wealth of $1000 is more than $50, $100, $500 (lose less than half), $1000 (breakeven), $10,000 (more than lO-fold), $100,000 (more than 100-fold), and $1 million (more than a thousand-fold). Table 2 shows these results and illustrates the conclusions stated in the abstract. The final wealth levels are much higher on average, the higher the Kelly fraction. With 1/ 8 Kelly, the average final wealth is $2072, starting with $1000. Its $4339 with 1/ 4 Kelly, $19,005 with half Kelly, $70,991 with 3/ 4 Kelly and $524,195 with full Kelly. So as you approach full Kelly, the typical final wealth escalates dramatically. This is shown also in the maximum wealth levels which for full Kelly is $318,854,673 versus $6330 for 1/ 8 Kelly. 2 --- ## Page 574 Medium Term Simulations of the Full Kelly and Fractional Kelly Investment Strategies 545 Kelly Fraction Statistic loOk 0.75k 0.50k 0.25k 0.125k Max 318854673 4370619 1117424 27067 6330 Mean 524195 70991 19005 4339 2072 Min 4 56 111 513 587 St. Dev. 8033178 242313 41289 2951 650 Skewness 35 11 13 2 1 Kurtosis 1299 155 278 9 2 > 5 x 10 1981 2000 20Do 2000 2000 1O ~ 1965 1996 2000 2000 2000 > 5 x 1O~ 1854 1936 1985 2000 2000 > 1O~ 1752 1855 1930 1957 1978 > 104 1175 1185 912 104 0 > 10" 479 284 50 0 0 >10 III 17 1 0 0 Table 2: Final Wealth Statistics by Kelly Fraction: Ziemba-Hausch (1986) Model Figure 1 shows the wealth paths of these maximum final wealth levels. Most of the gain is in the last 100 of the 700 decision points. Even with these maximum graphs, there is much volatility in the final wealth with the amount of volatility generally higher with higher Kelly fractions. Indeed with 3/ 4 Kelly, there were losses from about decision point 610 to 670. X 10" Full Kelly :[ /. ] a 100 200 300 400 500 600 700 800 x 10' 314 Kelly 5[ =:;;;?~-. _____ .I a 0 100 200 300 400 500 600 700 800 X 106 1/2 Kelly I ~ -I 0 100 200 300 400 500 600 700 800 x 10' 1/4 Kelly :[ ~ J a a 100 200 300 400 500 600 700 800 1/8 Kelly 10000l ....,...-- ~~ 5000 o . - . a 100 200 300 400 500 600 700 800 Maximum Wealth Path Figure 1: Highest Final Wealth Trajectory: Ziemba-Hausch (1986) Model Looking at t he chance of losses (final wealth is less than the initial $1000) in all cases, even with --- ## Page 575 546 L. C. MacLean, E. 0. Thorp, Y. Zhao and W T. Ziemba 1/8 Kelly with 1.1% and 1/4 Kelly with 2.15%, there are losses even with 700 independent bets each with an edge of 14%. For full Kelly, it is fully 12.4% losses, and it is 7.25% with 3/ 4 Kelly and 3.5% with half Kelly. These are just the percent of losses. But the size of the losses can be large as shown in the >50, > 100, and > 500 and columns of Table 2. The minimum final wealth levels were 587 for 1/ 8 and 513 for 1/ 4 Kelly so you never lose more than half your initial wealth with these lower risk betting strategies. But with 1/ 2, 3/ 4 and full Kelly, the minimums were 111, 56, and only $4. Figure 2 shows these minimum wealth paths. With full Kelly, and by inference 1/ 8, 1/ 4, 1/ 2, and 3/ 4 Kelly, the investor can actually never go fully bankrupt because of the proportional nature of Kelly betting. 300 Full Kelly 400 3/4 Kelly 500 600 ~~~: o 1 00 200 300 400 500 600 112 Kelly '~~: o 100 200 300 400 500 600 1/4 Kelly 700 800 I 700 800 700 800 ::~,-----------,j a 100 200 300 400 500 600 700 800 1/8 Kelly ;:::~~;;:;:::::: j 500 --- ______ ~ ____ ~~ ____ _L __ ~~~_=~~~ ____ ~ a 1 00 200 300 400 500 600 700 800 Minimum Wealth Paths Figure 2: Lowest Final Wealth Trajectory: Ziemba-Hausch (1986) Model If capital is infinitely divisible and there is no leveraging than the Kelly bettor cannot go bankrupt since one never bets everything (unless the probability of losing anything at all is zero and the probability of winning is positive). If capital is discrete, then presumably Kelly bets are rounded down to avoid overbetting, in which case, at least one unit is never bet. Hence, the worst case with Kelly is to be reduced to one unit, at which point betting stops. Since fractional Kelly bets less, the result follows for all such strategies. For levered wagers, that is, betting more than one's wealth with borrowed money, the investor can lose more than their initial wealth and become bankrupt. --- ## Page 576 Medium Term Simulations of the Full Kelly and Fractional Kelly Investment Strategies 547 3 Proportional Investment Strategies: Alternative Experi- ments The growth and risk characteristics for proportional investment strategies such as the Kelly depend upon the returns on risky investments. In this section we consider some alternative investment experiments where the distributions on returns are quite different. The mean return is similar: 14% for Ziemba-Hausch, 12.5%for Bicksler-Thorp I, and 10.2% for Bicksler-Thorp II. However, the variation around the mean is not similar and this produces much different Kelly strategies and corresponding wealth trajectories for scenarios. 3.1 The Ziemba and Hausch (1986) Model The first experiment is a repeat of the Ziemba - Hausch model in Section 2. A simulation was per- formed of 3000 scenarios over T = 40 decision points with the five types of independent investments for various investment strategies. The Kelly fractions and the proportion of wealth invested are reported in Table 3. Here, 1.0k is full Kelly, the strategy which maximizes the expected logarithm of wealth. Values below 1.0 are fractional Kelly and coincide in this setting with the decision from using a negative power utility function. Values above 1.0 coincide with those from some positive power utility function. This is overbetting according to MacLean, Ziemba and Blazenko (1992), because long run growth rate falls and security (measured by the chance of reaching a specific positive goal before falling to a negative growth level) also falls. I Kelly Fraction: f I Opportunity i.75k i.5k i.25k i.Ok O.75k O.50k O.25k A 0.245 0.210 0.175 0.140 0.105 0.070 0.035 B 0.1225 0.105 0.0875 0.070 0.0525 0.035 0.0175 C 0.08225 0.0705 0.05875 0.047 0.03525 0.0235 0.01175 D 0.06125 0.0525 0.04375 0.035 0.02625 0.0175 0.00875 E 0.049 0.042 0.035 0.028 0.021 0.014 0.007 Table 3: The Investment Proportions (>.) and Kelly Fractions The initial wealth for investment was 1000. Table 4 reports statistics on the final wealth for T = 40 with the various strategies. --- ## Page 577 548 L. C. MacLean, E. O. Thorp, Y. Zhao and W. T. Ziemba I Fraction Statistic l.75k 1.5k 1.25k 1.0k 0.75k 0.50k 0.25k Max 50364.73 25093.12 21730.90 8256.97 6632.08 3044.34 1854.53 Mean 1738.11 1625.63 1527.20 1386.80 1279.32 1172.74 1085.07 Min 42.77 80.79 83.55 193.07 281.25 456.29 664.31 St. Dev. 2360.73 1851.10 1296.72 849.73 587.16 359.94 160.76 Skewness 6.42 4.72 3.49 1.94 1.61 1.12 0.49 Kurtosis 85.30 38.22 27.94 6.66 5.17 2.17 0.47 > 5 x 10 2998 3000 3000 3000 3000 3000 3000 10' 2980 2995 2998 3000 3000 3000 3000 > 5 x 10' 2338 2454 2634 2815 2939 2994 3000 > lOS 1556 11606 1762 1836 1899 1938 2055 > 10' 43 24 4 0 0 0 0 > 10" 0 0 0 0 0 0 0 > IOU 0 0 0 0 0 0 0 Table 4: Wealth Statistics by Kelly Fraction: Ziemba-Hausch Model (1986) Since the Kelly bets are small, the proportion of current wealth invested is not high for any of the fractions. The upside and down side are not dramatic in this example, although there is a substantial gap between the maximum and minimum wealth with the highest fraction. Figure 3 shows the trajectories which have the highest and lowest final wealth for a selection of fractions. The log-wealth is displayed to show the rate of growth at each decision point. The lowest trajectories are almost a reflection of the highest ones. --- ## Page 578 Medium Term Simulations of the Full Kelly and Fractional Kelly Investment Strategies Ln(W.alth) 11 Variable 10 -- 1.75K -- 1.5 K 9 ---- 1.0K -- O.5K _ •• O.25K 8 7 6 5 4 3 0.0 Ln(W.alth) 11 0.2 Variable 10 -- 1.75K -- 1.5K 9 ---- 1.0K -- O.5K -·· O.25K 8 7 6 5 4 0.4 0.6 0.8 1.0 ewup. (a) Inverse Cumulative 3 "r----.---,,---,----,----,----,----.--~ -4 -3 -2 -1 o 2 3 4 NScore (b) Normal Plot Figure 3: Trajectories with Final Wealth Extremes: Ziemba-Hausch Model (1986) 549 The skewness and kurtosis indicate that final wealth is not normally distributed. This is expected since the geometric growth process suggests a log-normal wealth. Figure 4 displays the simulated log-wealth for selected fractions at the horizon T = 40. The normal probability plot will be linear if terminal wealth is distributed log-normally. The slope of the plot captures the shape of the log- wealth distribution. In this case the final wealth distribution is close to log-normal. As the Kelly fraction increases the slope increases, showing the longer right tail but also the increase in downside risk in the wealth distribution. --- ## Page 579 550 Ln(Wealth) 11 10 9 8 Variable: -- 1.75K - - 1.5K 1.0 K -- - 0,5K 0,25 K L. C. MacLean, E. 0. Thorp, Y. Zhao and W T Ziemba ! .. j ~~/' 7r-----~~~~~-=~~~~~ , ~ · ,~-~--=:=--~ ··~ ·~~ ' --~~~--.---- . -~- . --~ 6 5 4 3~.-___ .----. ___ -r ___ .-___ ~ 0.0 0.2 Ln(Wealth) 11 10 9 8 7 6 5 4 Variable: -- 1.75K 1.5 K I.... 1.0 K 0.5 K 0,25 K ,.' 0.4 0.6 (a) Inverse Cumulative 0.8 1.0 CumPr 3~ __ r-_-r--.---'--.--~'---r--~ -4 -3 -2 -1 o 1 2 4 NScore (b) Normal Plot Figure 4: Final Ln(Wealth) Distributions byFraction: Ziemba-Hausch Model (1986) On the inverse cumulative distribution plot, the initial wealth In(lOOO) = 6.91 is indicated to show the chance of losses. The inverse cumulative distribution of log-wealth is the basis of comparisons of accumulated wealth at the horizon. In particular, if the plots intersect then first order stochastic dominance by a wealth distribution does not exist (Hanoch and Levy, 1969). The mean and standard deviation of log-wealth are considered in Figure 5, where the trade-off as the Kelly fraction varies can be understood. Observe that the mean log-wealth peaks at the full Kelly strategy whereas the standard deviation is monotone increasing. Fractional strategies greater than full Kelly are inefficient in log-wealth, since the growth rate decreases and the the standard deviation of logo. wealth increases. --- ## Page 580 Medium Term Simulations of the Full Kelly and Fractional Kelly Investment Strategies 551 Ln(Wealth) 0,6 O,g 1.2 1,5 1,8 Mean StDev 7,08 O,g 7,07 / --------- / // -----" 0,8 / 7,06 / \ /1 0,7 7,05 ,/ 7,04 0,6 ,//' 7,03 / \ 0,5 // I /' 7,02 / 0.4 7.01 // 0,3 7,00 0,6 O,g 1.2 1.5 1,8 fraction Figure 5: Mean-Std Tradeoff: Ziemba-Hausch Model (1986) The results in Table 4 and Figures 3 - 5 support the following conclusions for Experiment 1. 1. The statistics describing end of horizon (T = 40) wealth are all monotone in the fraction of wealth invested in the Kelly portfolio. Specifically the maximum terminal wealth and the mean terminal wealth increase in the Kelly fraction. In contrast the minimum wealth decreases as the fraction increases and the standard deviation grows as the fraction increases, There is a trade-off between wealth growth and risk. The cumulative distribution in Figure 4 supports the theory for fractional strategies, as there is no dominance, and the distribution plots all intersect. 2. The maximum and minimum final wealth trajectories clearly show the wealth growth - risk trade-off of the strategies. The worst scenario is the same for all Kelly fractions so that the wealth decay is greater with higher fractions. The best scenario differs for the low fraction strategies, but the growth path is almost monotone in the fraction. The mean-standard deviation trade-off demonstrates the inefficiency of levered strategies (greater than full Kelly). 3.2 Bicksler - Thorp (1973) Case I - Uniform Returns There is one risky asset R having mean return of +12.5% , with the return uniformly distributed between 0.75 and 1.50 for each dollar invested. Assume we can lend or borrow capital at a risk free rate r = 0.0. Let>. = the proportion of capital invested in the risky asset, where>. ranges from 0.4 to 2.4 . So>. = 2.4 means $1.4 is borrowed for each $1 of current wealth. The Kelly optimal growth investment in the risky asset for r = 0.0 is x = 2.8655. The Kelly fractions for the different values of>. are shown in Table 3. (The formula relating>. and f for this expiriment is in the Appendix.) In their simulation, Bicksler and Thorp use 10 and 20 yearly decision periods, and 50 simulated scenarios. We use 40 yearly decision periods, with 3000 scenarios. 9 --- ## Page 581 552 L. C. MacLean, E. O. Thorp, Y. Zhao and W. T. Ziemba Table 5: The Investment Proportions and Kelly Fractions for Bicksler-Thorp (1973) Case I The numerical results from the simulation with T = 40 are in Table 6 and Figures 7 - 9. Although the Kelly investment is levered, the fractions in this case are less than 1. Fraction Statistic 0.14k 0.28k 0.42k 0.56k 0.70k 0.84k Max 34435.74 743361.14 11155417.33 124068469.50 1070576212.0 7399787898 Mean 7045.27 45675.75 275262.93 1538429.88 7877534.72 36387516.18 Min 728.45 425.57 197.43 70.97 18.91 3.46 St. Dev. 4016.18 60890.61 674415.54 6047844.60 44547205.57 272356844.8 Skewness 1.90 4.57 7.78 10.80 13.39 15.63 Kurtosis 6.00 31.54 83.19 150.51 223.70 301.38 > 5 x 10 3000 3000 3000 3000 2999 2998 10" 3000 3000 3000 2999 2999 2998 > 5 x 10" 3000 2999 2999 2997 2991 2976 > 103 2998 2997 2995 2991 2980 2965 > 104 529 2524 2808 2851 2847 2803 > 105 0 293 1414 2025 2243 2290 > 10" 0 0 161 696 1165 1407 Table 6: Final Wealth Statistics by Kelly Fraction for Bicksler-Thorp Case I In this experiment the Kelly proportion is high, based on the attractiveness of the investment in stock. The largest fraction (0.838k) shows strong returns, although in the worst scenario most of the wealth is lost. The trajectories for the highest and lowest terminal wealth scenarios are displayed in Figures 6. The highest rate of growth is for the highest fraction, and correspondingly it has the largest wealth fallback. 10 --- ## Page 582 Medium Term Simulations a/the Full Kelly and Fractional Kelly Investment Strategies Figure 6: Ln{Wealth) 25 20 15 10 5 0 V.:.riable -+- O.28k - II- 0,42k -,+ - O.5Gk -.I. - a.7ok ~ - 0.841<. Ln(w.alth) 8 7 6 5 4 3 Variable -+- 0.28k 2 - II- O.42k - +, ' O.5Gk -.I. - O.lOk ~ - 0.841<. 0 0 10 20 (a) Maximum 10 20 (b) Minimum Trajectories with Final Wealth Extremes: 30 40 Time 30 40 Tinl.@ Bicksler-Thorp (1973) Case I 553 The distribution of terminal wealth in Figure 7 illustrates the growth of the f = O.838k strategy. It intersects the normal probability plot for other strategies very early and increases its advantage. The linearity of the plots for all strategies is evidence of the log-normality of final wealth. The inverse cumulative distribution plot indicates that the chance of losses is small - the horizontal line indicates log of initial wealth. 11 --- ## Page 583 554 L. C. MacLean, E. O. Thorp, Y. Zhao and W. T. Ziemba Ln(\Vealth) 25 ,-------------------------------------------~ 20 15 10 5 Variable -- 0.28k - - O.42k ---- O.5&k -- 0.70k _ •• 0.841< o~,_----_,,------.------,-----_,------_.~ 0.0 0.2 Ln(Wealth) 25 20 15 10 5 Variable -- 0.28k - - O.42k ---- O.5&k -- 0.70k _ •• 0.841< 0.4 0.6 (a) Inverse Cumulative 0.8 1.0 Cw"Pr 0"r----.-----,----,-----.----,----.-----.----~ -4 -3 -2 -1 o 1 2 3 4 Nsc:ore (b) Figure 7: Final Ln(Wealth) Distributions: Bicksler-Thorp (1973) Case I As further evidence of the superiority of the f = O.838k strategy consider the mean and standard deviation of log-wealth in Figure 8. The growth rate (mean In(Wealth)) continues to increase since the fractional strategies are less then full Kelly. 12 --- ## Page 584 Medium Term Simulations of the Full Kelly and Fractional Kelly Investment Strategies 555 Ln(Wealth) 0.2 0.4 0.6 0.8 14}-________ ~M~e=a~n~ ________ _43.0r-----------s~ID~ev~--------~ ,// 13 / / /1' :: /' 10 "r-------y----.------,-----' 1.0 '--__________ ----' 2.5 12 11 0.2 0.4 0.6 0.8 Fraction Figure 8: Mean-Std Trade-off: Bicksler-Thorp (1973) Case I From the results of this experiment we can make the following statements. 1. The statistics describing end of horizon (T = 40) wealth are again monotone in the fraction of wealth invested in the Kelly portfolio. Specifically the maximum terminal wealth and the mean terminal wealth increase in the Kelly fraction. In contrast the minimum wealth decreases as the fraction increases and the standard deviation grows as the fraction increases. The growth and decay are much more pronounced than was the case in experiment 1. The minimum still remains above 0 since the fraction of Kelly is less than 1. There is a trade- off between wealth growth and risk, but the advantage of leveraged investment is clear. As illustrated with the cumulative distributions in Figure 7, the log-normality holds and the upside growth is more pronounced than the downside loss. Of course, the fractions are less than 1 so improved growth is expected. 2. The maximum and minimum final wealth trajectories clearly show the wealth growth - risk of various strategies. The mean-standard deviation trade-off favors the largest fraction, even though it is highly levered. 3.3 Bicksler - Thorp (1973) Case II - Equity Market Returns In the third experiment there are two assets: US equities and US T-bills. According to Siegel (2002), during 1926-2001 US equities returned of 10.2% with a yearly standard deviation of 20.3%, and the mean return was 3.9% for short term government T-bills with zero standard deviation. We assume the choice is between these two assets in each period. The Kelly strategy is to invest a proportion of wealth x = 1.5288 in equities and sell short the T-bill at 1 - x = - 0.5228 of current wealth. With the short selling and levered strategies, there is a chance of substantial losses. For the simulations, the proportion: A of wealth invested in equities and the corresponding Kelly fraction f are provided in Table 7. (The formula relating A and f for this expiriment is in the Appendix.) 13 --- ## Page 585 556 L. C. MacLean, E. O. Thorp, Y. Zhao and W T. Ziemba In their simulation, Bicksler and Thorp used 10 and 20 yearly decision periods, and 50 simulated scenarios. We use 40 yearly decision periods, with 3000 scenarios. I A I 0.4 I 0.8 I 1.2 1.6 2.0 2.4 I f I 0.26 I 0.52 I 0.78 1.05 1.31 1.57 Table 7: Kelly Fractions for Bicksler-Thorp (1973) Case II The results from the simulations with experiment 3 are contained in Table 8 and Figures 9, 10, and 11. This experiment is based on actual market returns. The striking aspects of the statistics in Table 8 are the sizable gains and losses. For the the most aggressive strategy (1.57k), it is possible to lose 10,000 times the initial wealth. This assumes that the shortselling is permissable through to the horizon. Table 8: Final Wealth Statistics by Kelly Fraction for Bicksler-Thorp (1973) Case II I I I Fraction I I I Statistic 0.26k 0.52k 0.78k 1.05k 1.31k 1.57k Max 65842.09 673058.45 5283234.28 33314627.67 174061071.4 769753090 Mean 12110.34 30937.03 76573.69 182645.07 416382.80 895952.14 Min 2367.92 701.28 -4969.78 -133456.35 -6862762.81 -102513723.8 St. Dev. 6147.30 35980.17 174683.09 815091.13 3634459.82 15004915.61 Skewness 1.54 4.88 13.01 25.92 38.22 45.45 Kurtosis 4.90 51.85 305.66 950.96 1755.18 2303.38 > 5 x 10 3000 3000 2998 2970· 2713 2184 10" 3000 3000 2998 2955 2671 2129 > 5 x 10" 3000 3000 2986 2866 2520 1960 > 1O~ 3000 2996 2954 2779 2409 1875 > 10' 1698 2276 2273 2112 1794 1375 > 105 0 132 575 838 877 751 > 10° 0 0 9 116 216 270 The highest and lowest final wealth trajectories are presented in Figures 9. In the worst case, the trajectory is terminated to indicate the timing of vanishing wealth. There is quick bankruptcy for the aggressive strategies. 14 --- ## Page 586 Medium Term Simulations of the Full Kelly and Fractional Kelly Investment Strategies 557 Ln(Weolth) 20.0 17.5 15.0 12.5 10.0 7.5 Vari.able -+- 0.57k - .- O.78k -'. '- l.osk ~ - 1.31k - ... . I.S7k 5.0 L-~------r------r-------.----~-.J o 10 20 (a) Maximum Ln(\Veolth) 14 12 10 8 6 4 2 0 0 10 20 (b) Minimum 30 Vari.able -+- 0.57k - .- O.78k - + - 1.0Sk ~ - 1.31k - ... · 1.57k 30 40 Tune 40 Time Figure 9: Trajectories with Final Wealth Extremes: Bicksler·Thorp (1973) Case II The strong downside is further illustrated in the distribution of final wealth plot in Figure 10. The normal probability plots are almost linear on the upside (log-normality) , but the downside is much more extreme than log-normal for all strategies except for 0.52k. Even the full Kelly is risky in this case. The inverse cumulative distribution shows a high probability of large losses with the most aggressive strategies. In constructing these plots the negative growth was incorporated with the formula growth = [signWT]ln(IWTI). 15 --- ## Page 587 558 Ln(Wealth) 20 10 o -10 L. C. MacLean, E. O. Thorp, Y Zhao and W T. Ziemba .~ I/.ariable --O.S7k -- a.78k ---- l.osk --1.Jlk _ •• 1.S7k -20~,-______ .-____ -. ______ ~ ______ ~ ______ ~~ 0.0 0.2 Ln(Wealth) 20 10 o -10 0.4 0.6 (a) Inverse Cumulative 0.8 1.0 CumPr Variable -- 0.57k - - 0.78k ---- 1.0Sk -·l .Jlk _ •• 1.S7k -20~ __ ~ ____ ~ __ ~ __ ~ ____ ~ __ ~ __ ~ __ ~ -4 -3 -2 -1 o 2 3 4 NScore (b) Normal Plot Figure 10: Final Ln(Wealth) Distributions: Bicksler-Thorp (1973) Case II The mean-standard deviation trade-off in Figure 11 provides more evidence to the riskyness of the high proportion strategies. When the fraction exceeds the full Kelly, the drop-off in growth rate is sharp, and that is matched by a sharp increase in standard deviation. 16 --- ## Page 588 Medium Term Simulations of the Full Kelly and Fractional Kelly Investment Strategies Ln(Wealth) 0,50 0,75 1,00 1,25 1.50 10 9 8 7 6 Mean Stdev ------------------\ 8 / \ 7 // \\ 6 5 / \ 4 / / 3 I ,/ 2 / ' \ ---.,.;~/ -.--- \ o~ ____________________ ~ 0,50 0,75 1.00 1.25 1.50 Fraction Figure 11: Mean-Std Tradeoff: Bicksler-Thorp (1973) Case II The results in experiment 3 lead to the following conclusions. 559 1. The statistics describing the end of the horizon (T = 40) wealth are again monotone in the fraction of wealth invested in the Kelly portfolio. Specifically (i) the maximum terminal wealth and the mean terminal wealth increase in the Kelly fraction; and (ii) the minimum wealth decreases as the fraction increases and the standard deviation grows as the fraction increases. The growth and decay are pronounced and it is possible to have large losses. The fraction of the Kelly optimal growth strategy exceeds 1 in the most levered strategies and this is very risky. There is a trade-off between return and risk, but the mean for the levered strategies is growing far less than the standard deviation. The disadvantage of leveraged investment is clearly illustrated with the cumulative distribution in Figure 10. The log-normality of final wealth does not hold for the levered strategies. 2. The maximum and minimum final wealth trajectories clearly show the return - risk of levered strategies. The worst and best scenarios are the not same for all Kelly fractions. The worst scenario for the most levered strategy shows the rapid decline in wealth. The mean-standard deviation trade-off confirms the riskyness/ folly of the aggressive strategies. 4 Discussion The Kelly optimal capital growth investment strategy is an attractive approach to wealth creation. In addition to maximizing the rate of growth of capital, it avoids bankruptcy and overwhelms any essentially different investment strategy in the long run (MacLean, Thorp and Ziemba, 2009). However, automatic use of the Kelly strategy in any investment situation is risky. It requires some adaptation to the investment environment: rates of return, volatilities, correlation of alternative assets, estimation error, risk aversion preferences, and planning horizon. The experiments in this paper represent some of the diversity in the investment environment. By considering the Kelly 17 --- ## Page 589 560 L. C. MacLean, E. 0. Thorp, Y. Zhao and W. T. Ziemba and its variants we get a concrete look at the plusses and minusses of the capital growth model. The main points from the Bicksler and Thorp (1973) and Ziemba and Hausch (1986) studies are confirmed. • The wealth accumulated from the full Kelly strategy does not stochastically dominate frac- tional Kelly wealth. The downside is often much more favorable with a fraction less than one. • There is a tradeoff of risk and return with the fraction invested in the Kelly portfolio. In cases of large uncertainty, either from intrinsic volatility or estimation error, security is gained by reducing the Kelly investment fraction. • The full Kelly strategy can be highly levered. While the use of borrowing can be effective in generating large returns on investment, increased leveraging beyond the full Kelly is not warranted. The returns from over-levered investment are offset by a growing probability of bankruptcy. • The Kelly strategy is not merely a long term approach. Proper use in the short and medium run can achieve wealth goals while protecting against drawdowns. References [1] Bicksler, J.L. and E.O. Thorp (1973). The capital growth model: an empirical investigation. Journal of Financial and Quantitative Analysis 8\ 2, 273~287. [2] Hanoch, G. and Levy, H. (1969). The Efficiency Analysis of Choices Involving Risk. The Review of Economic Studies 36, 335-346. [3] MacLean, L.C., Thorp, E.O., and Ziemba, W.T. (2010). Good and bad properties of the Kelly criterion. in The Kelly Capital Growth Investment Criterion: Theory and Practice. Scientific Press, Singapore. [4] MacLean, L.C., Ziemba, W.T. and Blazenko, G. (1992). Growth versus Security in Dynamic Investment Analysis. Management Science 38, 1562-85. [5] Merton, Robert C. (1990). Continuous Time Finance. Malden, MA Blackwell Publishers Inc .. [6] Poundstone, W. (2005). Fortunes Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street. Farrar Straus & Giroux, New York, NY. Paperback version (2005) from Hill and Wang, New York. [7] Siegel, J.J. (2002) . Stocks for the long run. Wiley. [8] Ziemba, W.T. and D.B. Hausch (1986). Betting at the Racetrack. Dr. Z. Investments Inc., San Luis Obispo, CA. 18 --- ## Page 590 Medium Term Simulations of the Full Kelly and Fractional Kelly Investment Strategies 561 5 Appendix The proportional investment strategies in the experiments of Bicksler and Thorp (1973) have frac- tional Kelly equivalents. The Kelly investment proportion for the experiments are deveolped in this appendix. 5.1 KELLY STRATEGY WITH UNIFORM RETURNS Consider the problem Max x {E(ln(l + r + x(R - rn , where R is uniform on [a, bJ and r =the risk free rate. We have the first order condition l b R - r 1 ----,-------,- x -b -dR = 0, a l+r+x(R-r) -a which reduces to (b ) ( )1 ( l+r+X(b-r)) [l+r+X(b-r)]! b - a X - a = 1 + r n {==? = e l + r • l+r+x(a-r) l+r+x(a-r) In the case considered in Experiment II, a = -0.25, b = 0.5, r = O. The equation becomes 1 [~];; = eO. 75 with a solution x = 2.8655. So the Kelly strategy is to invest 286.55% of 1-O.25x ' wealth in the risky asset. 5.2 Kelly Strategy with Normal Returns Consider the problem Max x {E(ln(l + r + x(R - rn, where R is Gaussian with mean J.LR and standard deviation (JR, and r =the risk free rate. The solution is given by Merton (1990) as The values in Experiment III are J.LR x = 1.5288. J.LR - r X=---. (JR 0.102 , (JR = 0.203, r 19 0.039, so the Kelly strategy is --- ## Page 591 This page is intentionally left blank --- ## Page 592 39 Good and Bad Properties of the Kelly Criterion* Leonard C. MacLean School of Business, Dalhousie University, Halifax, NS Edward O. Thorp E. O. Thorp and Associates, Newport Beach, CA Professor Emeritus, University of California, Irvine William T. Ziemba Professor Emeritus, University of British Columbia, Vancouver, BC Visiting Professor, Mathematical Institute, Oxford University, UK ICMA Centre, University of Reading, UK University of Bergamo, Italy Abstract We summarize what we regard as the good and bad properties of the Kelly cri- terion and its variants. Additional properties are discussed as observations. 563 The main advantage of the Kelly criterion, which maximizes the expected value of the logarithm of wealth period by period, is that it maximizes the limiting expo- nential growth rate of wealth. The main disadvantage of the Kelly criterion is that its suggested wagers may be very large. Hence, the Kelly criterion can be very risky in the short term. In the one asset two valued payoff case, the optimal Kelly wager is the edge (expected return) divided by the odds. Chopra and Ziemba (1993), reprinted in Part II of this volume, following earlier studies by Kallberg and Ziemba (1981, 1984) showed for any asset allocation problem that the mean is much more impor- tant than the variances and co-variances. Errors in means versus errors in variances were about 20: 2 : 1 in importance as measured by the cash equivalent value of final wealth. Table 1 and Figure 1 show this and illustrate that the relative importance depends on the degree of risk aversion. The lower is the Arrow-Pratt risk aver- sion, RA = -u" (w) / u' (w), the higher are the relative errors from incorrect means. Chopra (1993) further shows that portfolio turnover is larger for errors in means 'Special thanks go to Tom Cover and John Mulvey for helpful comments on an earlier draft of this paper. 1 --- ## Page 593 564 L. C. MacLean, E. 0. Thorp and W T Ziemba Table 1 Average Ratio of Certainty Equivalent Loss for Errors in Means, Vari- ances and Covariances. [Source: Chopra and Ziemba (1993)] Errors in Means Errors in Means Errors in Variances Risk Tolerance* vs Covariances vs Variances vs Covariances 25 5.38 3.22 1.67 50 22.50 10.98 2.05 75 56.84 21.42 2.68 1 1 1 20 10 2 Error Mean Error Var Error Covar 20 2 1 *Risk tolerance=RT(w) = ~C ) where RA(W) = _ u';CCw» zRA w U W % Cash Equivalent4lss 11~----------------------------~~----, 10 Means 9 8 7 6 5 4 3 2 o -' o 0.05 0.10 0.15 Magnitude of eor (k) _ Varianes ,-- Covarial1lS 0.20 Figure 1 Mean Percentage Cash Equivalent Loss Due to Errors in Inputs than for variances and for co-variances but the degree of difference in the size of the errors is much less than the performance as shown in Figure 2. Since log has RA(W) = l/w, which is close to zero, the Kelly bets may be ex- ceedingly large and risky for favorable bets. In MacLean et aZ. (2009), we present simulations of medium term Kelly, fractional Kelly, and proportional betting strate- gies. The results show that, with favorable investment opportunities, Kelly bettors attain large final wealth most of the time. However, because a long sequence of bad scenario outcomes is possible, any strategy can lose substantially even if there are many independent investment opportunities and the chance of losing at each investment decision point is small. The Kelly and fractional Kelly rules, like all other rules, are never a sure way of winning for a finite sequence. In Part VI of this volume, we describe the use of the Kelly criterion in many applications and by many great investors. Two of them, Keynes and Buffett, were long term investors whose wealth paths were quite rocky but with good long term 2 --- ## Page 594 Good and Bad Properties of the Kelly Criterion Average Turnover (% per month) 50 r----------------------------------------------=~~ 40 Means 30 20 Variances ...... ...... Covariances 10 I ... ~~=====~ 10~ 5 10 15 20 25 30 35 40 45 50 Change(%) 565 Figure 2 Average turnover for different percentage changes in means, variances and covariances. [Source: Based on data from Chopra (1993)] outcomes. Our analyses suggest that Buffett seems to act similar to a fully Kelly bettor (subject to the constraint of no borrowing) and Keynes like a 80% Kelly bettor with a negative power utility function _W-0 .25 , see Ziemba (2003) and the wealth graphs reprinted in Part VI from Ziemba (2005). Graphs such as Figure 3 show that growth is traded off for security with the use of fractional Kelly strategies and negative power utility functions. Log maximizes the long run growth rate. Utility functions such as positive power that bet more than Kelly have more risk and lower growth. One of the properties shown below that is illustrated in the graph is that for processes, which are well approximated by continuous time, the growth rate becomes zero plus the risk free rate when one bets exactly twice the Kelly wager. Hence, it never pays to bet more than the Kelly strategy because then risk in- creases (lower security) and growth decreases, so Kelly dominates all these strategies in geometric risk-return or mean-variance space. See Ziemba (2009) in this volume. As you exceed the Kelly bets more and more, risk increases and long term growth falls, eventually becoming more and more negative. Long Term Capital is one of many real world instances in which overbetting led to disaster. See Ziemba and Ziemba (2007) for additional examples. Thus, long term growth maximizing investors should bet Kelly or less. We call betting less than Kelly "fractional Kelly", which is simply a blend of Kelly and cash. Consider the negative power utility function ow d for 0 < O. This utility function is concave and when 0 ---+ 0, it converges to log utility. As 0 becomes more negative, the investor is less aggressive since his absolute Arrow-Pratt risk aversion index is also higher. For the case of a stationary lognormal process and a given 0 for utility function owd and ex = 1/(1 - 0) between 0 and I , they both will 3 --- ## Page 595 566 Prohability 1.0 0.8 0.6 0.4 0.2 L. C. MacLean, E. O. Thorp and W T Ziemba Optimal Kelly wager 0 .01+----r---+----r--~ 0.0 0.81 0.02 0.03 Fraction .fWealth Wagered Figure 3 Probability of doubling and quadrupling before halving and relative growth rates versus fraction of wealth wagered for Blackjack (2% advantage, p = 0.51 and q = 0.49). [Source: MacLean and Ziemba (1999)] provide the same optimal portfolio when ex is invested in the Kelly portfolio and 1 - ex is invested in cash. This handy formula relating the coefficient of the negative power utility function to the Kelly fraction is correct for lognormal investments and approximately correct for other distributed assets; see MacLean, Ziemba and Li (2005). For example, half Kelly is 6 = -1 and quarter Kelly is 6 = -3. So if you want a less aggressive path than Kelly, pick an appropriate 6. This formula does not apply more generally. For example, for coin tossing, where Pr(X = 1) = p, Pr(X - 1) = q, p + q = I, 1 1 * p'- 6 _ q~ fo = 1 1 p'- 6 + q'- 6 which is not exf*, where f* = p - q ;? 0 is the Kelly bet. We now list these and other important Kelly criterion properties, updated from MacLean, Ziemba, and Blazenko (1992), MacLean and Ziemba (1999), and Ziemba and Ziemba (2007). See also Cover and Thomas (2006, Chapter 16). The Good Properties Good Maximizing ElogX asymptotically maximizes the rate of asset growth. See Breiman (1961), Algoet and Cover (1988). Good The expected time to reach a preassigned goal A is asymptotically least as A increases without limit with a strategy maximizing ElogXN . See Breiman (1961), Algoet and Cover (1988), Browne (1997a). Good Under fairly general conditions, maximizing ElogX also asymptotically max- imizes median logX. See Ethier (1987, 2004, 2010). 4 --- ## Page 596 Good and Bad Properties of the Kelly Criterion Good The ElogX bettor never risks ruin. See Hakansson and Miller (1975). Good The absolute amount bet is monotone increasing in wealth. 567 Good The ElogX bettor has an optimal myopic policy. He does not have to con- sider prior nor subsequent investment opportunities. This is a crucially important result for practical use. Hakansson (1971) proved that the my- opic policy obtains for dependent investments with the log utility function. For independent investments and any power utility, a myopic policy is opti- mal, see Mossin (1968). In fact, past outcomes can be taken into account by maximizing the conditional expected logarithm given the past (Algoet and Cover, 1988). Good Simulation studies show that the ElogX bettor's fortune pulls ahead of other "essentially different" strategies' wealth for most reasonable-sized samples. "Essentially different" has a limited meaning. For example, g* ~ 9 but g* - 9 = E will not lead to rapid separation if E is small enough. The key again is risk. See Bicksler and Thorp (1973), Ziemba and Hausch (1986), and MacLean, Thorp, Zhao and Ziemba (2009) in this volume. General formulas are in Aucamp (1993). Good If you wish to have higher security by trading it off for lower growth, then use a negative power utility function, 6wo, or fractional Kelly strategy. See MacLean, Sanegre, Zhao and Ziemba (2004) reprinted in Part III, who show how to compute the coefficent to stay above a growth path at discrete points in time with given probability or to be above a given drawdown with a certain confidence limit. MacLean, Zhao and Ziemba (2009) add the feature that path violations are penalized with a convex cost function. See also Stutzer (2009) for a related but different model of such security. Good Competitive optimality. Kelly gambling yields wealth X* such that E ( ; . ) ~ 1, for all other strategies X. This follows from the Kuhn Tucker conditions. Thus, by Markov's inequality, Pr [X ~ tX*] ~ i, for t ~ 1, and for all other induced wealths x . Thus, an opponent cannot outperform X* by a factor t with probability greater than i. This inequality can be improved when t = 1 by allowing fair randomization U. Let U be drawn according to a uniform distribution over the interval [0,2], and let U be in- dependent of X*. Then the result improves to Pr [X ~ U X*] ~ ~ for all portfolios X. Thus, fairly randomizing one's initial wealth and then invest- ing it according to the Kelly criterion, one obtains a wealth U X* that can only be beaten half the time. Since a competing investor can use the same strategy, probability ~ is the best competitive performance one can expect. We see that Kelly gambling is the heart of the solution for this two-person zero sum game of who ends up with the most money. So we see that X* (actually U X*) is competitively optimal in a single investment period (Bell and Cover 1980, 1988). 5 --- ## Page 597 568 L. C. MacLean, E. O. Thorp and W. T Ziemba Good If X* is the wealth induced by the log optimal (Kelly) portfolio, then the expected wealth ratio is no greater than one, i.e., E( ;.) ~ 1, for the wealth X induced by any other portfolio (Bell and Cover, 1980, 1988). Good Super St. Petersburg. Any cost c for the St. Petersburg random variable X, Pr [X = 2k] = 2-k, is acceptable. But the larger the cost c, the less wealth one should invest. The growth rate G* of wealth resulting from repeated such investments is G* = max E In (1 -f + 1 x) 0";1";1 C where f is the fraction of wealth invested. The maximizing 1* is the Kelly proportion (Cover and Bell, 1980). The Kelly fraction 1* can be computed even for a super St. Petersburg random variable Pr [Y = 22k] = 2- k , k = 1,2, . . . , where E In Y = 00, by maximizing the relative growth rate 1-f+ly max Eln lIe 0";1";1 '2 + 2c:Y This is bounded for all f in [0,1] . Now, although the exponential growth rate of wealth is infinite for all proportions f and it seems that all f E [0,1] are equally good, the max- imizing 1* in the previous equation guarantees that the 1* portfolio will asymptotically exponentially outperform any other portfolio f E [0,1]. Both investors' wealth have super exponential growth, but the 1* investor will ex- ponentially outperform any other essentially different investor. The Bad Properties Bad The bets may be a large fraction of current wealth when the wager is favorable and the risk of loss is very small. For one such example, see Ziemba and Hausch (1986; 159- 160). There, in the inaugural 1984 Breeders Cup Classic $3 million race, the optimal fractional wager, using the Dr. Z place and show system using the win odds as the probability of winning, on the 3- 5 shot Slew of Gold was 64% (see also the 74% future bet on the J anuary effect in MacLean, Ziemba and Blazenko (1992) reprinted in this volume). Thorp and Ziemba actually made this place, show bet and won with a low fractional Kelly wager. Slew finished third but the second place horse Gate Dancer was disqualified and placed third. Wild Again won this race and it was the first great victory of the masterful jockey Pat Day. Bad For coin tossing, any fixed fraction strategy has the property that if the num- ber of wins equals the number of losses, then the bettor is behind. For n wins and n losses and initial wealth Wo, we have W2n = Wo(l- j2)n. 6 --- ## Page 598 Good and Bad Properties of the Kelly Criterion 569 Bad The unweighted average rate of return converges to half the arithmetic rate of return. Thus, you may regularly win less than you expect. This is a consequence of weighting equally rather than by size of the wager. See Ethier and Tavare (1983) and Griffin (1985). Some Observations • For an iid process and a myopic policy, which results from maximizing ex- pected utility in case the utility function is log or a negative power, the result is fixed fraction betting, hence fractional Kelly includes all these policies. • A betting strategy is "essentially different" from Kelly if Sn == L~l Elog(l + it X i) - L~l Elog(l + i iX i) tends to infinity as n increases. The sequence un denotes the Kelly betting fractions and the sequence Ud denotes the corresponding betting fractions for the essentially different strategy. • The Kelly portfolio does not necessarily lie on the efficient frontier in a mean- variance model (Thorp, 1971). • Despite its superior long-run growth properties, Kelly, like any other strategy, can have a poor return outcome. For example, making 700 wagers, all of which have a 14% advantage, the least of which has a 19% chance of winning, can turn $1000 into $18. But with full Kelly 16.6% of the time, $1000 turns into at least $100,000, see Ziemba and Hausch (1996). Half Kelly does not help much as $1000 can become $145 and the growth is much lower with only $100,000 plus final wealth 0.1 % of the time. For more such calculations, see Bicksler and Thorp (1973) and MacLean, Thorp, Zhao and Ziemba (2009) in this volume. • Fallacy: If maximizing E log X N almost certainly leads to a better outcome, then the expected utility of its outcome exceeds that of any other rule provided N is sufficiently large. Counter-example: u(x) = x,1/2 < p < 1, Bernoulli trials i = 1 maximizes EU(x) but i = 2p - 1 < 1 maximizes ElogXN. See Samuelson (1971) and Thorp (1971, 2006). • It can take a long time for any strategy, including Kelly, to dominate an es- sentially different strategy. For instance, in continuous time with a geometric Wiener process, suppose /.Lo: = 20%, /.L f3 = 10%, (Jo: = (Jf3 = 10%. Then in five years, A is ahead of B with 95% confidence. But if (Jo: = 20%, (Jf3 = 10% with the same means, it takes 157 years for A to beat B with 95% confidence. As another example, in coin tossing, suppose game A has an edge of 1.0% and game B 1.1%. It takes two million trials to have an 84% chance that game A dominates game B, see Thorp (2006). The theory and practical application of the Kelly criterion is straightforward when the underlying probability distributions are fairly accurately known. However, in investment applications, this is usually not the case. Realized future equity 7 --- ## Page 599 570 L. C. MacLean, E. O. Thorp and W T Ziemba returns may be very different from what one would expect using estimates based on historical returns. Consequently, practitioners who wish to protect capital above all, sharply reduce risk as their drawdown increases. Prospective users of the Kelly Criterion can check our list of good properties, bad properties and observations to test whether Kelly is well suited to their intended application. Given the extreme sensitivity of E log calculations to errors in mean estimates, these estimates must be accurate and to be on the safe side, the size of the wagers should be reduced. For long term compounders, the good properties dominate the bad properties of the Kelly criterion, but the bad properties may dampen the enthusiasm of naive prospective users of the Kelly criterion. The Kelly and fractional Kelly strategies are very useful if applied carefully with good data input and proper financial engineering risk control. Appendix In continuous time, with a geometric Wiener process, betting exactly double the Kelly criterion amount leads to a growth rate equal to the risk free rate. This result is due to Thorp (1997), Stutzer (1998), Janacek (1998), and possibly others. The following simple proof, under the further assumption of the Capital Asset Pricing Model, is due to Harry Markowitz and appears in Ziemba (2003). In continuous time 1 gp = Ep - 2Vp E p, Vp, gp are the portfolio expected return, variance and expected log, respectively. In the CAPM Vp = 0'~X2 where X is the portfolio weight and ro is the risk free rate. Collecting terms and setting the derivative of gp to zero yields X = (EM - ro)jO'~ which is the optimal Kelly bet with optimal growth rate g* = ro + (EM - ro)2 - ~[(EMro)jO'~l20'~ = ro + (EM - ro)2 jO'~ - ~(EM - ro)2 jO'~ 1 2 = ro + 2[(E - M - r))jO'Ml . 8 --- ## Page 600 Good and Bad Properties of the Kelly Criterion 571 Substituting double Kelly, namely Y = 2X for X above into 1 2 2 gp = TO + (EM - TO)y - "2aMY and simplifying yields go - TO = 2(EM - To)2/a'iI - ~(EM - To)2/a'iI = o. Hence go = TO when Y = 2S. The CAPM assumption is not needed. For a more general proof and illustration, see Thorp (2006). References Algoet, P. H. and T . Cover (1988). Asymptotic optimality and asymptotic equipartition properties of log-optimum investment. Annals of Probability, 16(2), 876- 898. Aucamp, D. (1993). On the extensive number of plays to achieve superior performance with the geometric mean strategy. Management Science, 39, 1163- 1172. Bell, R. M. and T. M. Cover (1980). Competitive optimality of logarithmic investment. Math of Operations Research, 5, 161- 166. Bell, R. M. and T. M. Cover (1988). Game-theoretic optimal portfolios. Management Science, 34(6), 724- 733. Bicksler, J. L. and E. O. Thorp (1973). The capital growth model: an empirical investiga- tion. Journal of Financial and Quantitative Analysis, 8(2) , 273- 287. Breiman, L. (1961). Optimal gambling system for favorable games. Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, 1, 63- 8. Browne, S. (1997). Survival and growth with a fixed liability: optimal portfolios in con- tinuous time. Math of Operations Research, 22, 468- 493. Chopra, V. K. (1993). Improving optimization. Journal of Investing, 2(3) , 51- 59. Chopra, V. K. and W. T. Ziemba (1993). The effect of errors in mean, variance and co- variance estimates on optimal portfolio choice. Journal of Portfolio Management, 19, 6- 11. Cover, T. and J. A. Thomas (2006). Elements of Information Theory (2 ed.). Ethier, S. (1987). The proportional bettor's fortune. Proceedings 7th International Confer- ence on Gambling and Risk Taking, Department of Economics, University of Nevada, Reno. Ethier, S. (2004). The Kelly system maximizes median fortune. Journal of Applied Probability, 41, 1230- 1236. Ethier, S. (2010). The Doctrine of Chances: Probabilistic Aspects of Gambling. New York: Springer. Ethier, S. and S. Tavare (1983). The proportional bettor's return on investment. Journal of Applied Probability, 20, 563- 573. Finkelstein, M. and R. Whitley (1981). Optimal strategies for repeated games. Advanced Applied Probability, 13, 415- 428. Griffin, P. (1985). Different measures of win rates for optimal proportional betting. Man- agement Science, 30, 1540- 1547. Hakansson, N. H. (1971). On optimal myopic portfolio policies with and without serial correlation. Journal of Business, 44, 324-334. 9 --- ## Page 601 572 L. C. MacLean, E. 0. Thorp and W. T. Ziemba Hakansson, N. H. and B. Miller (1975). Compound-return mean-variance efficient portfolios never risk ruin. Management Science, 22, 391- 400. Janacek, K. (1998). Optimal growth in gambling and investing. M.Sc. Thesis, Charles University, Prague. Kallberg, J . G. and W. T. Ziemba (1981). Remarks on optimal portfolio selection. In G. Bamberg and O. Opitz (Eds.), Methods of Operations Research, pp. 507- 520. Cambridge, MA: Oelgeschlager. Kallberg, J. G. and W. T. Ziemba (1984). Mis-specifications in portfolio selection problems. In G. Bamberg and K. Spremann (Eds.) , Risk and Capital, pp. 74- 87. New York: Springer Verlag. MacLean, L., R. Sanegre, Y. Zhao, and W. T. Ziemba (2004). Capital growth with security. Journal of Economic Dynamics and Control, 28(4), 937- 954. MacLean, L. , E. O. Thorp, Y. Zhao, and W. T. Ziemba (2009). Medium term simulations of Kelly and fractional Kelly strategies. MacLean, L. and W . T . Ziemba (1999). Growth versus security tradeoffs in dynamic in- vestment analysis. Annals of Operations Research, 85, 193- 227. MacLean, L., W. T. Ziemba, and G. Blazenko (1992). Growth versus security in dynamic investment analysis. Management Science, 38, 1562- 85. MacLean, L., W. T. Ziemba, and Li (2005) . Time to wealth goals in capital accumula- tion and the optimal trade-off of growth versus security. Quantitative Finance, 5(4) , 343- 357. MacLean, L. C., Y. Zhao, and W . T. Ziemba (2009). Optimal capital growth with convex loss penalties. Working Paper, Dalhousie University. Mossin, J. (1968). Optimal multiperiod portfolio policies. Journal of Business, 41, 215- 229. Samuelson, P. A. (1971). The fallacy of maximizing the geometric mean in long sequences of investing or gambling. Proceedings, National Academy of Science, 68, 2493-2496. Stutzer, M. (2009). On growth optimality versus security against underperformance. Thorp, E. O. (1971). Portfolio choice and the Kelly criterion. Proceedings of the Business and Economics Section of the American Statistical Association, 215-224. Thorp, E. O. (2006) . The Kelly criterion in blackjack, sports betting and the stock mar- ket. In S. A. Zenios and W. T. Ziemba (Eds.) , Handbook of Asset and Liability Management, Handbooks in Finance, pp. 385-428. Amsterdam: North Holland. Ziemba, R. E. S. and W. T. Ziemba (2007). Scenarios for Risk Management and Global Investment Strategies. NY: Wiley. Ziemba, W. T. (2003). The Stochastic Programming Approach to Asset Liability and Wealth Management. AIMR, Charlottesville, VA. Ziemba, W. T. (2005). The symmetric downside risk Sharpe ratio and the evaluation of great investors and speculators. Journal of Portfolio Management, Fall, 108-122. Ziemba, W. T. (2009). Utility theory for growth versus security. Working Paper. Ziemba, W. T. and D. B. Hausch (1986) . Betting at the Racetrack. Dr. Z. Investments, San Luis Obispo, CA. 10 --- ## Page 602 Part V Utility foundations --- ## Page 603 This page is intentionally left blank --- ## Page 604 575 40 Introduction to the Utility Foundations of Kelly The criticisms of the Kelly strategy based on utility are general. If preferences are identified with wealth characteristics such as growth and tail values, then there is a utility foundation for Kelly and fractional Kelly strategies. Hakansson and Ziemba (1995) survey their and others' works on capital growth theory. They discuss Hakansson's consumption-investment model over time in the expected log context and MacLean and Ziemba's growth and security tradeoffs and its relationship with fractional Kelly strategies. They discuss the properties of these models and relate capital growth to expected utility. Luenberger (1993) investigates the long term behavior of E log strategies. Mer- ton and Samuelson (1974) show that lim E [u(Wn )] n-HXJ is not an expected utility so the idea of using utility at a terminal time and taking the limit is not valid. Hence, the log mean-variance approach is not consistent with expected utility theory. Luenberger provides a new preference based approach to the problem assuming independent non-negative identically distributed returns. He establishes preferences on infinite sequences of wealth rather than wealth at a fixed (but later taken to the limit) terminal time. This approach avoids the limit problems and allows use of probability limit theorems. If the tail preference is defined by a simple utility function, then the utility must be the expected log of wealth. Tail preferences are appropriate if the goal of investment is long term wealth maximization. If preferences are represented by a compound utility function, then that function will be a function of the expected logarithm and the variance of the logarithm of wealth. So in that case, there is a Markowitz type efficient tradeoff frontier of E log Wand Var log W that was originally suggested by Williams (1936). Stutzer (2003) provides an alternative behavioral foundation for an investor's use of power utility and its defining risk aversion parameter. This involves an investor's desire to minimize the probability that the wealth growth rate will not exceed an exogenous target growth rate. Stutzer uses the Gartner-Ellis large deviations theory to show that this leads to an equivalent power utility function. This means that the investor's risk aversion parameter depends on the investment opportunity set contrary to typical model assumptions. Stutzer's formulation uses a wealth variable in the utility function that is the ratio of current wealth to the level of wealth growing at the constant target rate. Also the power coefficient is not specified in advance --- ## Page 605 576 L. C. MacLean, E. 0. Thorp and W T. Ziemba but rather is an output of the optimization. For ease of estimation, an independent identically distributed asset return assumption is made. Stutzer (2010) in a followup paper discusses further how to increase security against under-performance in dynamic investment analysis over finite horizons against a specified exogenous benchmark. Motivation is provided because it takes a long time to be fairly sure that Kelly strategies dominate, and the Kelly strategy has considerable risk of under-performing or having very low returns. He uses a Bayesian formulation of the Occam's Razor Principle to illustrate the unavoidable reduction of statistical testability (namely the ability to more easily falsify) inherent in objective functions that have non-directly observable parameters. The coefficient of the power utility function is endogenously determined. The setting used is the familiar blackjack example previously discussed by Thorp and by MacLean, Ziemba, and Blazenko (1992). --- ## Page 606 R. Jarrow et aI., Eds., Handbooks in OR & MS, Vol. 9 © 1995 Elsevier Science B.Y. All rights reserved Chapter 3 41 Capital Growth Theory Nils H Hakansson Walter A. Haas School of Business, University of California, Berkeley, CA 94720, U.S.A. William T. Ziemba Faculty of Commerce and Business Administration, University of British Columbia, Vancouver, B.c. V6T lY8, Canada 1. Introduction 577 Even casual observation strongly suggests that capital growth is not just a catch-phrase but something which many actively strive to achieve. It is therefore rather surprising that capital growth theory is a relatively obscure subject. For example, the great bulk of today's MBA's have had little or no exposure to the subject, having had their attention focussed almost exclusively on the single-period mean-variance model of portfolio choice. The purpose of this essay is to review the theory of capital growth, in particular the so-called growth-optimal investment strategy, its properties, its uses, and its links to betting and other investment models. We also discuss several applications that have tended to refine the basic theory. The central feature of the growth-optimal investment strategy, also known as the geometric mean model and the Kelly criterion, is the logarithmic shape of the objective function. But the power and durability of the model is due to a remarkable set of properties. Some of these are unique to the growth-optimal strategy and the others are shared by all the members of the (remarkable) small family to which the growth optimal strategy belongs. Investment over time is multiplicative, not additive, due to the compounding nature of the process itself. This makes a number of results in dynamic investment theory appear nonintuitive. For example, in the single-period portfolio problem, the optimal investment policy is very sensitive to the utility function being used; the set of policies that are inadmissible or dominated across all utility functions is relatively small. The same observation holds in the dynamic case when the number of periods is not large. But as the number of periods does become large, the set of investment policies that are optimal for current investment tends to shrink drasti- cally, at least in the basic reinvestment case without transaction costs. As we will see, many strikingly different investors will, in essence, invest the same way when the horizon is distant and will only begin to part company as their horizons near. 65 --- ## Page 607 578 N. H. Hakansson and W T. Ziemba 66 N.H. Hakansson, w.T. Ziemba It is tempting to conjecture that all long-run investment policies to which risk-averse investors with monotone increasing utility functions will flock, under a favorable return structure, insure growth of capital with a very high probability. Such a conjecture is false; many investors will, even in this case, converge on investment policies which almost surely risk ruin in the long run, in effect ignoring feasible policies which almost surely lead to capital growth. Similarly, the relationship between the behavior of capital over time and the behavior of the expected utility of that same capital over time often appears strikingly nonintuitive. Section 2 reviews the origins of the capital growth model while Section 3 contains a derivation and identifies its key properties. The conditions for capital growth are examined in Section 4. The model's relationship to other long-run investment models is studied in Section 5 and Section 6 contains its role in intertemporal investment/consumption models. Section 7 adds various constraints for accomplishing tradeoffs between growth and security, while Section 8 reviews various applications. A concluding summary is given in Section 9. 2. Origins of the model The approach to investment commonly known as the growth-optimal investment strategy has a number of apparently independent origins. In particular, Williams [1936], Kelly [1956], Latane [1959], and Breiman [1960, 1961] seem to have been unaware of each other's papers. But one can also argue that Bernoulli (1738) unwittingly stumbled on it in 1738 in his resolution of the st. Petersburg Paradox - see the 1954 translation - and Samuelson's survey [1977]. Samuelson [1971] appears to be the earliest to have related the geometric mean criterion to utility theory - and to find it wanting. The growth optimal strategy's inviolability in the larger consumption-investment context when preferences for consumption are logarithmic was first noted by Hakansson [1970]. Finally, models considering tradeoffs between capital growth and security appear to have been pioneered by MacLean & Ziemba [1986]. 3. The model and its basic properties The following notation and basic assumptions will be employed: Wt = amount of investment capital at decision point t (the end of the tth period); M t = the number of investment opportunities available in period t, where Mt:S M; St the subset of investment opportunities which it is possible to sell short in period t; nt rate of interest in period t; --- ## Page 608 Capital Growth Theory 579 Ch. 3. Capital Growth Theory 67 rit = return per unit of capital invested in opportunity i, where i = 2, . . . , Mt , in the tth period (random variable). That is, if we invest an amount () in i at the beginning of the period, we will obtain (1 + rit W at the end of that period; ZIt = amount lent in period t (negative ZIt indicate borrowing) (decision vari- able); Zit amount invested in opportunity i, i = 2, ... ,Mt at the beginning of the tth period (decision variable); Ft(Y2, Y3 ,···, YM,) == Pr {r2t S Y2, r3t S Y3, . . . , rM,t S YM,}; Zt - (Z2t, . . . , ZM,t); Zit Wt-l i = 1, .. . , Mt ; The capital market will generally be assumed to be perfect, i.e. that there are no transaction costs or taxes, that the investor has no influence on prices or returns, that the amount invested can be any real number, and that the investor has full use of the proceeds from any short sale. The following basic properties of returns will be assumed: rlt 2: 0, t = 1,2, ... E[rit] 2: 8 + rlf> 8> 0, some i, t = 1,2, . . . E[rit] S K, all i, t. (1) (2) (3) These assumptions imply that the financial market provides a 'favorable game.' We also assume that the (nonstationary) return distributions Ft are either independent from period to period or obey a Markov process and they also satisfy the 'no-easy-money condition' ! M, I M, P t; (rit - rlt Wi < 81 > 82 for all t and all (}i such that t; I(}i I = 1, and (}i 2: ° for all i ¢ St, (4) where 81 < 0, 82 > 0. Condition (4) is equivalent to what is often referred to as the no-arbitrage condition. It is generally a necessary condition for the portfolio problem to have a solution. We also assume that the investor must remain solvent in each period, i.e., that he or she must satisfy the solvency constraints Pr{Wt 2: O} = 1, t = 1,2, . .. . (5) --- ## Page 609 580 N. H. Hakansson and W T. Ziemba 68 N.H. Hakansson, w.T. Ziemba The amount invested at time t - 1 is Mt LZit = Wt-l i=l and the value of the investment at time t, broken down between its risky and riskfree components, is Mt Mt Wt = L(1 + rit)Zit + (1 + rlt)( 1 - L Zit), i=2 i=2 which together yield the basic difference equation where Mt Wt = L(rit -rlt)Zit+ Wt-l(1+rlt), i=2 M t Rt(Xt) == L(rit - rlt)xit + 1 + rlt · i=2 t = 1,2, . . . (6) t = 1,2, .. . (7) Let us now turn to the basic reinvestment problem which (ignores capital infusions and distributions and) simply revises the portfolio at discrete points in time. In view of (5), (6) may be written t Wt = Wo exp { LIn Rn(Xn)}, n=l Defining (8) becomes Wt = wo[exp{Gt«(Xt)}]t = wo(1 + gt)t, t = 1,2, .... (8) (9) (10) where gt = exp G t «(Xt) - 1 is the compound growth rate of capital over the first t periods. By the law of large numbers, under mild conditions, Thus, it is evident that for large T, Wt -+ 0 if E[Gd s 8 < 0, t 2: T, (11) --- ## Page 610 Capital Growth Theory Ch. 3. Capital Growth Theory WI -+ 00 if E[GtJ :::: 8 > 0, t:::: T gl -+ exp E[GtJ- 1. Under stationary returns and policies (XI ), (11) and (12) simplify to WI -+ 0 if E[ln Rn(xn)] < 0 WI -+ 00 if E[ln Rn (xn)] > 0 any n. 581 69 (12) (13) There is nothing intuitive that would suggest that the sign of E[ln Rn (xn)] is the determinant of whether our capital will decline or grow in the (stationary) simple reinvestment problem. What is evident is that the expected return on capital, E[Rn] - 1, is not what matters. As (6) reminds us, capital growth (positive or negative) is a multiplicative, not an additive process. To illustrate the point, consider the case of only two assets, one riskfree yielding 5% per period, and the other returning either -60% or + 100% with equal proba- bilities in each period. Always putting all of our capital in the riskfree asset clearly gives a 5% growth rate of capital. The expected return on the risky asset is 20% per period. Yet placing all of our funds in the risky asset at the beginning of each period results in a capital growth rate that converges to -1O.55%! It is easy to see this. We will double our money to 200% roughly half of the time. But we will also lose 60% (bringing the 200% to 80%) of our beginning-of-period capital about half the time, for a 'two-period return' of -20% on average, or -10.55% per period. Expected capital E[ WI]' on the other hand, has a growth rate of 20% per period. What this simple example demonstrates is that there are many investment strategies for which, as t -+ 00, E[wtJ-+ 00 Median [wtJ -+ 0 Mode [WI] -+ 0 Pr{wl < $1} -+ 1. The coexistence of the above four measures results when E[GtJ S 8 < 0 for t :::: T and a long (but thin) upper tail is generated as WI moves forward in time. In view of (7), (9) and (10), we observe that to 'maximize' the long-run growth rate g(' it is necessary and sufficient to maximize E[ GI «XI))], or Max {E[ln Rl (Xl)] + E[ln R2(X2)] + ... } (14) Whenever returns are independent from period to period or the economy obeys a Markov process 1, it is necessary and sufficient to accomplish (14) to Max E[ln RI(xl)] sequentially at each t - 1. (15) XI I Algoet & Cover [1988] show formally that the growth-optimal strategy maintains its basic properties under arbitrary returns .processes. --- ## Page 611 582 N H. Hakansson and W T. Ziemba 70 NH. Hakansson, w.T. Ziemba Since the geometric mean of Rt (Xt) = exp{E[ln Rt (Xt)]), we observe that (15) is also equivalent to maximizing the geometric mean of principal plus return at each point in time. 3.1. Properties of the growth-optimal investment strategy Since the solution (xn to (15), in view of (10) and (13), almost surely leads to more capital in the long run than any other investment policy which does not converge to it, (xn is referred to as the growth-optimal investment strategy. Existence is assured by the no-easy-money condition (4), the bounds on expected returns (1)-(3), and the solvency constraint (5). The strict concavity of the objective function in (15) implies that the optimal payoff distribution Rt (x7) is unique; the optimal policy x7 itself will be unique only if, for any security i, there is no portfolio of the other assets which can replicate the return pattern rit. It is probably not surprising that the growth-optimal strategy never risks ruin, i.e. Pr {Rt (x7) = O} = 0 - because to grow you have to survive. But this need not mean that the solvency constraint is not binding: E[ln Rt(xt)] may exist even when Rt touches 0 as long as the lower tail is very thin. The conditions (1)-(3) imply that positive growth is feasible. Another dimension of the consistency between short-term and long-term performance was observed by Bell & Cover [1988]. As shown by Breiman [1961], the growth-optimal strategy also has the property that it asymptotically minimizes the expected time to reach a given level of capital. This is not surprising in view of the characteristics noted in the previous two paragraphs. It is also evident from (15) that the growth-optimal strategy is myopic even when returns obey a Markov process (Hakansson 1971c). This property is clearly of great practical significance since it means that the investor only needs to estimate the coming period's (joint) return structure in order to behave optimally in a long-run sense; future periods' return structures have no influence on the current period's optimal decision. No other dynamic investment model has this property in a Markov economy; only a small set of other families have it when returns are independent from period to period (see Section 5). The growth-optimal strategy implies, and is implied by, logarithmic utility of wealth at the end of each period. This is because at each t - 1 Max E[ln Rt(xt)] x, ~ Max {E[ln Rt(xt) + In wt-d} x, = Max E[ln(wt-1Rt(Xt))] = Max E[ln Wt(Zt)]. ~ ~ Since every utility function is unique (up to a positive linear transformation), it also follows that the growth-optimal strategy is not consistent with any other end-of-period utility function (more on this in the next subsection). --- ## Page 612 Capital Growth Theory Ch. 3. Capital Growth Theory The relative risk aversion function wu"(w) q(w) == ----'--- u'(w) 583 71 equals 1 when u(w) = In(w) (it is 0 for a risk-neutral investor). Thus, we observe that to do 'the best' in the long run in terms of capital growth, it is not only necessary to be risk averse in each period. We must also display the 'right' amount of risk aversion. The long-run growth rate of capital will be lower either if one invests in a way which is more risk averse than the logarithmic function or relies on an objective function which is less risk averse. The growth-optimal investment strategy is not only linear in beginning-of-period wealth but proportional as well since definitionally Both of these properties are shared by only a small family of investment models. Since the growth-optimal strategy is consistent with a logarithmic end-of-period utility function only, it is clearly not consistent with the mean-variance approach to portfolio choice - which in turn is consistent with quadratic utility for arbitrary security return structures, and, for normally distributed returns, with those utility functions whose expected utilities exist when integrated with the normal distribution, plus a few other cases, as shown by Ziemba & Vickson [1975] and Chamberlain [1983]. This incompatibility is easy to understand; in solving for the growth-optimal strategy, all of the moments of the return distributions matter, with positive skewness being particularly favored. When the returns on the risky assets are normally distributed, no matter how favorable the means and variances are, the growth-optimal strategy cooly places 100% of the investable funds in the riskfree asset. The preceding does not imply that the growth-optimal portfolio necessarily is far from the mean-variance efficient frontier (although this may be the case [see e.g. Hakansson, 1971a]). It will generally be close to the MV-efficient frontier, especially when returns are fairly symmetric. And as shown in Section 8, the mean-variance model can in some cases be used to (sequentially) generate a close approximation to the growth-optimal portfolios. Other properties of the Kelly criterion can be found in MacLean, Ziemba & Blazenko [1992, table 1]. 3.2. Capital growth vs. expected utility Based on (10), the uniqueness properties implied by (15), and the law of large numbers, it is undisputable, as noted in the previous subsection, that the growth- optimal strategy almost surely generates more capital (under basic reinvestment) in the long run than any other strategy which does not converge to it. At the same time, however, we observed that the growth-optimal strategy is consistent with logarithmic end-of-period utility of wealth only. This clearly implies that there must be 'reasonable' utility functions which value almost surely less capital --- ## Page 613 584 N. H. Hakansson and W. T. Ziemba 72 N.H. Hakansson, w.T. Ziemba in the long run more than they value the distribution generated by the Kelly criterion. Consider the family 1 u(w) = -w Y , Y Y < 1, (16) to which u(w) = In(w) belongs via Y = 0, and let (Xt(Y)) be the optimal portfolio sequence generated by solving Max E [~wr] at each t - 1. x, Y For simplicity, consider the case of stationary returns. Since Xt (y) i= Xt (0) = x:' it is evident that (17) even though there exist numbers a > 1 and T(E) such that (18) for every (1 » E > O. Many a student of investment has stubbed his toe by interpreting (18) to mean that (xn generates higher expected utility than, say, (Xt(Y)) . (17) and (18) may seem like a paradox but clearly implies that the geometric mean criterion does not give rise to a 'universally best' investment strategy. The intuition behind this truth is as follows. For Y < 0 in (16), (17) and (18) occur because, despite the fact that the wealth distribution for (Xt(Y)) lies almost entirely to the left of the wealth distribution for (xn , the lower tail of the distribution for (Xt(Y)) is shorter and (imperceptibly) thinner than the (bounded) left tail of the growth-optimal distribution. Thus, for negative powers, very small adverse changes in the lower tail overpower the value of almost surely ending up with a higher compound return. Conversely, for Y > 0, it is the longer (though admittedly very thin) upper tail that gives rise to (17) in the presence of (18) even though, again, the wealth distribution for (Xt(Y)) lies almost entirely to the left of the wealth distribution for (xn. 4. Conditions for capital growth As already noted, the determinants of whether capital will grow or decline (almost surely) in the long run are given by (12) and (11). Conditions (1)- (2) insure that (12) is feasible; in the absence of (1)-(2), positive growth may be infeasible. If a positive long-run growth rate (bounded away from zero) is achievable, then the growth-optimal strategy will find it. Thus we can state: --- ## Page 614 Capital Growth Theory 585 Ch. 3. Capital Growth Theory 73 Theorem. In the absence of (1) and (2), a necessary and sufficient condition for long-run capital growth to be feasible is that the growth-optimal strategy achieves a positive growth rate, i. e. that for some E > 0 and large T (19) For y < 0, the objective functions in (16) attain long-run growth rates of capital between those of the risk-free asset and of the growth-optimal strategy. But for y > 0, the long run growth-rate may be negative. Consider for a moment the util- ity function u (w) = W 1/2, one of the most frequently cited examples of 'substantial' risk aversion since Bernoulli's time. Even this venerable function may, however, lead to (almost sure) ruin in the long-run: suppose, for example, that the riskfree asset yields 2% per period and that there is only one risky asset, which gives either a loss of 8.2% with probability 0.9, or a gain of 206% with probability 0.1. The opti- mal policy then calls for investing the fraction 1.5792 in the risky asset (by borrow- ing the fraction 0.5792 of current wealth to complete the financing) in each period. But the average compound growth rate gt in (10) will now tend to -0.00756, or -3/4%. Thus, expected utility 'grows' as capital itself almost surely vanishes. What this example illustrates is that risk aversion plus a favorable return structure [see (1)-(3)] are not sufficient to insure capital growth in the basic reinvestment case. 5. Relationship to other long-run investment models As shown in Section 3, the growth-optimal investment strategy has its traditional origin in arguments concerning capital growth and the law of large numbers. But it can also be derived strictly from an expected utility perspective - but only as a member of a small family. Let n be the number of periods left to a terminal horizon point at time O. Assume that wealth at that point, Wo, has utility Uo(wo), where U~ > 0 everywhere and U~ < 0 for large WOo Then, with one period to go, we have the single-period portfolio problem Ul(Wl) == Max E[Uo(wo(zl))] zllwl where U I (WI) is the induced, or derived, utility of wealth WI at time 1 and the difference equation (6) has been trivially modified to Mn Wn-I = L(1 + rin)Zin + wn(1 + rln), n = 1,2, .... i=2 Thus, with n periods to go, we obtain Un(Wn) == Max E[Un-l(Wn-l(Zn))], n = 1,2, ... Znlwn where (21) is a standard recursive equation. (20) (21) --- ## Page 615 586 N H. Hakansson and W. T. Ziemba 74 N.H. Hakansson, WT. Ziemba The induced utility of current wealth, Un(wn), of course, generally depends on all the inputs to the problem, that is the utility of terminal wealth Uo, the joint distribution functions of future returns Fn , ... , Fl, and the future interest rates rln, ... , rll. But there are two rather interesting special cases. The first is the case in which the induced utility functions Un(wn) depend only on the terminal utility function Uo. This occurs when the returns are independent from period to period and Uo( wo) is isoelastic, i.e. 1 Y Uo(wo) = -wo , some y < l. y As first shown by Mossin [1968], (21) now gives Un(Wn) = anUo(wn) + bn '"" Uo(wn) (where '"" means equivalent to) since an and bn are constraints with an positive. The optimal investment policy is both myopic and proportional, i.e. z7n(wn) = Xin(Y)Wn, all i where the Xin (y) are constants. The second special case obtains when returns are independent from period to period, interest rates are deterministic, and the terminal utility function reflects hyperbolic absolute risk aversion, that is (Hakansson 1971c) 1 -(wo + I/>)Y, Y < 1, Y or Uo(wo) = (I/> - wo)Y, Y > 1, I/> large; (22) or - exp{ -I/>wo} I/> > O. In the first sub case 1 ( I/»Y Un(wn) = - Wn + -------- Y (1 + rd ... (1 + rln) (23) where (23) holds globally for I/> ~ 0 and locally for I/> > 0, i.e. for Wn ::: Ln > O. The optimal investment policy is z7n(wn) = Xi/l(Y) ( Wn + (1 + rll) .~. (1 + rln») , i::: 2. In the other two subcases, a closed form solution holds only locally. But the most interesting result associated with (21) is surprisingly general. Under mild conditions on Uo(wo), and independent (but nonstationary) returns from period to period, we obtain [Hakansson, 1974; see also Leland, 1972; Ross, --- ## Page 616 Capital Growth Theory Ch. 3. Capital Growth Theory 1974; Huberman and Ross 1983]: 1 Y un(wn) -+ -wn y and, if returns are stationary, Thus, the class of utility functions 1 u(w) = -wY , y < 1, Y 587 75 (24) (25) (16) the only family with constant relative risk aversion (ranging from 0 to infinity) and exhibiting myopic and proportional investment policies, is evidently applicable to a large class of long-run investors. The optimal policies aoove are not mean- variance efficient, but for reasonably symmetric return distributions, they come close to MV efficiency. Since y = 0 in (24) corresponds to logarithmic utility of wealth, the growth- optimal strategy is clearly a member of this elite family of long-run oriented investors. In other words, the geometric mean investment strategy has a solid foundation in utility theory as well. 6. Relationship to intertemporal consumption-investment models Up to this point, we have examined the basic dynamic investment problem, i.e. without reference to cash inflows or outflows. Under some conditions, the inclusion of these factors is straightforward and does not materially affect the optimal investment policy. But a realistic model incorporating noncapital in- and outflows typically complicates the model substantially. The basic dynamic consumption-investment model incorporates consumption and a labor income into the dynamic reinvestment model. Following Fisher [1936], wealth is viewed as a means to an end, namely consumption. The basic difference equation (6) now becomes M, Wt = L(ri/ - rlt)Zi/ + (1 + rlt)(Wt-l - Ct) + Yt, t = 1, ... , T, (26) i=2 where Ct is the amount consumed in period t (set aside at the beginning of the period) and Yt is the labor income received at the end of period t. Consistent with the foregoing, the individual's objective becomes Max E[U(Cl, ... , CT)] subject to Ct :::: 0, all t --- ## Page 617 588 N. H. Hakansson and W T. Ziemba 76 NH. Hakansson, w.T. Ziemba where U is assumed to be monotone, strictly concave, and to reflect impatience, i.e. considering the two consumption streams (a, b, C3, ... , CT) (b, a, C3, .. . , CT), a > b the first is preferred to the second. In order to attain tractability, several strong assumptions are usually imposed: 1) the individual's lifetime (horizon) is known, 2) interest rates are viewed as deterministic, 3) the labor income Yt is deterministic; its present value is thus Yt YT Yt-l == - + ... + , rlt (l+rlt)···(l+rIT) 4) the utility function is assumed to be additive, i.e. U (CI, ... , CT) = UI (cl) + alu2(c2) + ... + al .. . aT-Iur(cT), (27) where u; > 0, u;' < 0, and typically at < 1, for all t, which implies that preferences are independent of past consumption. Let ft-I(Wt-l) = maximum expected utility at t -1 given Wt-I· This gives fr-l(Wt-l) = Max {Ut(Ct) + at E[ft(Wt)]), t = 1, . .. , T, (28) C"Z, where h(WT) == ° or bT(WT) subject to Ct ~ ° Pr{Wt ~ -Yrl = 1 Zit ~ 0, i 1- St (29) (30) (31) for each t, where br( WT) represents a possible bequest motive. It is apparent that ft-I (Wt-l) represents the utility of wealth and that it is induced or derived; it clearly depends on everything in the model. Solving (28) recursively, it is evident that, under our assumptions concerning labor income and interest rates, Yt can be exchanged for cash in the solution. Suppose that in (27) 1 y Ut(Ct) = -ct , Y < 1, t = 1, ... , T. Y Then [Hakansson, 1970] ft-I(Wt-l) = At-I(Wt-l + Yt-l)Y + Bt-l, c;(Wt-d = Ct(Wt-1 + Yt- I), (32) --- ## Page 618 Capital Growth Theory 589 Ch. 3. Capital Growth Theory 77 and M, Z;t(Wt-d = Wt-l - c7 - LZ7t(Wt-l), i=2 where the Ar, Br, and Ct are constants. Thus, the optimal consumption and investment policies are again proportional, not to Wt-l but to Wt-l + Yt-l. The latter quantity is sometimes referred to as permanent income. Note that when y = 0 in (32), the consumer-investor does indeed employ the growth-optimal strategy to invested funds. Finally, the model (28)-(31) has been extended in a number of directions, to incorporate a random lifetime, life insurance, a subsistence level constraint on consumption, a Markov process for the economy, and an uncertain income stream from labor - with limited success [see Hakansson 1969, 1971b, 1972; Miller, 1974]. In general, closed-form solutions do not exist when income streams, payment obligations, and interest rates are stochastic. In such cases, multi-stage stochastic programming models are helpful [see e.g. Mulvey & Ziemba, 1995]. 7. Growth vs. security Empirical evidence suggests that the average investor is more risk averse than the growth-optimal investor, with a risk-tolerance corresponding to y ~ -3 in (16) [see e.g. Blume & Friend, 1975]. While real-world investors exhibit a wide range of attitudes towards risk, this means that the majority of investors are in effect willing to sacrifice a certain amount of growth in favor of less variability, or greater 'security'. 7.1 The discrete-time case In view of the convergence results (24) and (25), it is evident that repeated employment of (16) for any y < 0 attains an efficient tradeoff between growth and security, as defined above, for the long-run investor. The concept of 'efficiency' is thus employed in a sense analogous to that used in mean-variance analysis. A number of more direct measures of the sacrifice of growth for security have also been examined. In particular, MacLean, Ziemba & Blazenko [1992] analyzed the tradeoffs based on three growth and three security measures. The three growth measures are: 1. E (Wt «Xt)], the expected wealth level after t periods; 2. E[gtJ, the mean compound growth rate over the first t periods; 3. E[t : Wt«Xt) ~ y], the mean first passage time to reach wealth level y; while the three security measures are: 4. Pr{ Wt «Xt» ~ y}, the probability that wealth level y will be reached in t periods; --- ## Page 619 590 N. H. Hakansson and W. T. Ziemba 78 N.H. Hakansson, w.T. Ziemba 5. Pr{wl«xl)) :::: hI, t = 1,2, ... }, the probability that the investor's wealth is on or above a specified path; 6. Pr{ WI «XI)) :::: y before WI «(XI)) s h, where h < Wo < y}, which includes the probability of doubling before halving. Tradeoffs were generated via fractional Kelly strategies, i.e. strategies involving (stationary) mixtures of cash and the growth-optimal investment portfolio. Applied to a stationary environment, these strategies were shown to produce effective tradeoffs in that as growth declines, security increases. However, these tradeoffs, while easily computable, are generally not efficient, i.e. do not maximize security for a given (minimum) level of growth. Other comparisons involving the growth- optimal strategy and half Kelly or other strategies may be found in Ziemba & Hausch [1986], Rubinstein [1991], and Aucamp [1993]. 7.2. The continuous-time case Since transaction costs are zero under the perfect market assumption, it is natu- ral to consider shorter and shorter periods between reinvestment decisions. In the limit, reinvestment takes place continuously. Assuming that the returns on risky assets can be described by diffusion processes, we obtain that optimal portfolios are mean-variance efficient in that the instantaneous variance is minimized for a given instantaneous expected return. The intuitive reason for this is that as the trading interval is shortened, the first two moments of the security's return become more and more dominant [see Samuelson, 1970]. The optimal portfolios also exhibit the separation property - as if returns over very short periods were normally distributed. Over any fixed interval, however, payoff distributions are, due to the compounding effect, usually lognormal. In other words, all investors with the same probability assessments, but regardless of risk attitude, invest in only two mutual funds, one of which is riskfree [Merton, 1971]. See also Karatzas, Lehoczky, Sethi and Shreve [1986] and Sethi and Taksar [1988]. In view of the above, it is evident that the tradeoff between growth and security generated by the fractional Kelly strategies in the continuous-time model when the wealth process is lognormal is efficient in a mean-variance sense. Li [1993] has addressed the growth vs. security question for the two asset case while Li & Ziemba [1992] and Dohi, Tanaka, Kaio & Osaki [1994] have done so when there are n risky assets that are jointly lognormally distributed. 8. Applications 8.1. Asset allocation In view of the myopic property of the optimal investment policy in the dynamic reinvestment problem [see (24) and (25)], it is natural to apply (15) for different values of y to the problem of choosing investment portfolios over time. In particular, the choice of broad asset categories, also known as the asset allocation --- ## Page 620 Capital Growth Theory 591 Ch. 3. Capital Growth Theory 79 problem, lends itself especially well to such treatment. Thus, to implement the growth-optimal strategy, for example, we merely solve (15) subject to relevant constraints (on borrowing when available and on short positions) at the beginning of each period. To implement the model, it is necessary to estimate the joint distribution function for next period's returns. Since all moments and comoments matter, one way to do this is to employ the joint empirical distribution for the previous n periods. This approach provides a simple and realistic means of generating nonstationary scenarios of the possible outcomes over time. The raw distribution may of course may be modified in any number of ways, for example via Stein estimators [Jorion, 1985, 1986, 1991; Grauer & Hakansson, 1995], an inflation adapter [Hakansson, 1989], or some other method. Grauer and Hakansson applied the dynamic reinvestment model in a number of settings with up to 16 different risk attitudes y under both quarterly and annual portfolio revision. In the domestic setting [Grauer & Hakansson, 1982, 1985, 1986], the model was employed to construct and rebalance portfolios composed of U.S. stocks, corporate bonds, government bonds, and a riskfree asset. Borrowing was ruled out in the first article while margin purchases were permitted in the other two. The third article also included small stocks as a separate investment' vehicle. On the whole, the growth-optimal strategy lived up to its reputation. On the basis of the empirical probability assessment approach, quarterly rebalancing, and a 32-quarter estimating period applied to 1934-1992, the growth-optimal strategy outperformed all the others - with borrowing permitted, it earned an average annual compound return of nearly 15%. In Grauer & Hakansson [1987], the model was applied to a global environment by including in the universe the four principal U.S. asset categories and up to four- teen non-U.S. equity and bond categories. The results showed that the gains from including non-U.S. asset classes in the universe were remarkably large (in some cases statistically significant), especially for the highly risk-averse strategies. With leverage permitted and quarterly rebalancing, the geometric mean strategy again came out on top, generating an annual compound return of 27% over the 1970- 1986 period. A different study examined the impact from adding three separate real estate investment categories to the universe of available categories [Grauer & Hakansson, 1994b]. Finally, Grauer, Hakansson & Shen [1990] examined the asset allocation problem when the universe of risky assets was composed of twelve equal- and value-weighted industry components of the U.S. stock market. Mulvey [1993] developed a multi-period model of asset allocation which in- corporates transaction costs, including price impact. The objective function is a general concave utility function. A computational version developed by Mulvey & Vladimirou [1992] focused on the isoelastic class of functions in which the objective was to maximize the expected utility of wealth at the end of the planning horizon. This model, like those based on the empirical distribution approach, can handle assets possessing skewed returns, such as options and other derivatives, and can be extended to include liabilities [see Mulvey & Ziemba, 1995]. Based on historical data over the period 1979 to 1988, this research, based On multi-stage --- ## Page 621 592 N. H. Hakansson and W T. Ziemba 80 NH. Hakansson, w.T. Ziemba stochastic programming, showed that efficiencies could be gained vis-a-vis myopic models in the presence of transaction costs by taking advantage of the network or linear structure of the problem. Mean-variance approximations. A number of authors have argued that, in the single period case, power function policies can be well approximated by MV policies, e.g. Levy & Markowitz [1979], Pulley [1981, 1983], Kallberg & Ziemba [1979, 1983], and Kroll, Levy & Markowitz [1984]. However, there is an opposing intuition which suggests that the power functions' strong aversion to low returns and bankruptcy will lead them to select portfolios that are not MV-efficient, e.g. Hakansson [1971a] and Grauer [1981, 1986]. It is therefore of interest to know whether the power policies differ from the corresponding MV and quadratic policies when returns are compounded over many periods. Let fLit be the expected rate of return on security i at time t and O"ijt be the covariance between the returns on securities i and j at time t. Then the MV investment problem is Max {T(1 + fLt} -10"/L XI subject to the usual constraints. The MV approximation to the power functions in (16) are obtained [Ohlson, 1975; Pulley, 1981] when 1 T=--. l-y Under certain conditions this result holds exactly in continuous time [see Merton, 1973, 1980]. With quarterly revision, the MV model was found to approximate the exact power function model very well [Grauer & Hakansson, 1993]. But with annual revision, the portfolio compositions and returns earned by the more risk averse power function strategies bore little resemblance to those of the corresponding MV approximations. Quadratic approximations proved even less satisfactory in this case. These results contrast somewhat with those of Kallberg & Ziemba [1983], who in the quadratic case with smaller variances obtained good approximations for horizons up to a whole year [see also MacLean, Ziemba & Blazenko, 1992]. 8.2. Growth-security tradeoffs The growth vs. security model has been applied to four well-known gambling- investment problems: blackjack, horse race wagering, lotto games, and commodity trading with stock index futures. In at least the first three cases, the basic invest- ment situation is unfavorable for the average player. However, systems have been developed that yield a positive expected return. The various applications use a variety of growth and security measures that appear to model each situation well. The size of the optimal investment gamble also varies greatly, from over half to less than one millionth of one's fortune. --- ## Page 622 Capital Growth Theory 593 Ch. 3. Capital Growth Theory 81 Blackjack. By wagering more in favorable situations and less or nothing when the deck is unfavorable, an average weighted edge is about 2%. An approximation to provide insight into the long-run behavior of a player's fortune is to assume that the game is a Bernoulli trial with a probability of success equal to 0.51. With a 2% edge, the optimal wager is also 2% of one's fortune. Professional blackjack teams often use a fractional Kelly wagering strategy with the fraction drawn from the interval 0.2 to 0.8. For further discussion, see Gottlieb [1985] and Maclean, Ziemba & Blazenko [1992]. Horseracing. There is considerable evidence supporting the proposition that it is possible to identify races where there is a substantial edge in the bettor's favor (see the survey by Hausch & Ziemba [1995] in this volume). At thoroughbred racetracks, one can find about 2-4 profitable wagers with an edge of 10% or more on an average day. These opportunities arise because (1) the public has a distaste for the high probability-low payoff wagers, and (2) the public is unable to properly evaluate the worth of multiple horse place and show and exotic wagers because of their complexity; for example, in a ten-horse race there are 120 possible show finishes, each with a different payoff and chance of occurrence. In this situation, interesting tradeoffs between growth and security arise as well. The Kentucky Derby represents an interesting special case because of the long distance (1 1/4 miles), the fact that the horses have not previously run this distance, and the fame of the race. Hausch, Bain & Ziemba [1995] tabulated the results from Kelly and half Kelly wagers using the system in Ziemba & Hausch [1987] over the 61-year period 1934-1994. They also report the results from using a filter rule based on the horse's breeding. Lotto games. Lotteries tend to have very low expected payoffs, typically on the order of 40 to 50%. One way to 'beat' parimutuel games is to wager on unpopular numbers - see Hausch & Ziemba [1995] for a survey. But even when the odds are 'turned' favorable, the optimal Kelly wagers are extremely small and it may take a very long time to reach substantial profits with high probability. Often an initial wealth level in the seven figures is required to justify the purchase of even a single $1 ticket. Comparisons between fractional and full Keily strategies can be found in MacLean, Ziemba & Blazenko [1992]. Commodity trading. Repeated investments in commodity trades can be modeled as a capital growth problem via suitable modifications for margin requirements, daily mark-to-the-market procedures, and other practical details. An interesting example is the turn-of-the-year effect exhibited by U.S. small stocks in January . . One way to benefit from this anomaly is to take long positions in a small stock index and short positions in large stock indices, because the transaction costs (commissions plus market impact) are less than a tenth of what they would be by transacting in the corresponding basket of securities. Using data from 1976 through January 1987, Clark & Ziemba [1987] calculated that the growth-optimal --- ## Page 623 594 N. H. Hakansson and W T. Ziemba 82 N.H. Hakansson, w.T. Ziemba strategy would invest 74% of one's capital in this opportunity. Hence fractional Kelly strategies are suggested. See also Ziemba [1994]. 9. Summary Capital growth theory is useful in the analysis of many dynamic investment situations, with many attractive properties. In the basic reinvestment case, the growth-optimal investment strategy, also known as the Kelly criterion, almost surely leads to more capital in the long run than any other investment policy which does not converge to it. It never risks ruin, and also has the appealing property that it asymptotically minimizes the expected time to reach a given level of capital. The Kelly criterion implies, and is implied by, logarithmic utility of wealth (only) at the end of each period; thus, its relative risk aversion equals 1, which makes it more risk-tolerant than the average investor. As a result, tradeoffs between growth and security have found application in a rich set of circumstances. The fact that the growth-optimal investment strategy is proportional to begin- ning-of-period wealth is of great practical value. 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Stochastic Optimization Models in Finance, Aca- demic Press, New York, N.Y. --- ## Page 628 Journal of Economic Dynamics and Control 11 (1993) 881-906. North-Holland 42 A preference foundation for log mean-variance criteria in portfolio choice problems David G. Luenberger Stanford University. Stanford. CA 94305. USA Received October 1991, final version received June 1992 599 The appropriate criterion for evaluating, and hence also for properly constructing, investment portfolios whose performance is governed by an infinite sequence of stochastic returns has long been a subject of controversy and fascination. A criterion based on the expected logarithm of one-period return is known to lead to exponential growth with the greatest exponent, almost surely; and hence this criterion is frequently proposed. A refinement has been to include the variance of the logarithm of return as well, but this has had no substantial theoretical justification. This paper shows that log mean-variance criteria follow naturally from elementary assumptions on an individual's preference relation for deterministic wealth sequences. As a first and fundamental step, it is shown that if a preference relation involves only the tail of a sequence, then that relation can be extended to stochastic wealth sequences by almost sure equality. It is not necessary to introduce a von Neumann-Morgenstern utility function or the associated axioms. It is then shown that if tail preferences can be described by a 'simple' utility function, one that is of the form lim._~p( W., II) where W. is wealth at period II, this utility must under suitable conditions be a function of the expected logarithm of return, independent of the functional form of p. Finally, 'compound' utility functions are introduced; and they are shown under suitable conditions to be functions of the expected value and variance of the logarithm of one-period return, again indepen- dently of the specific form ofthe underlying function. The infinite repetitions of the dynamic process essentially 'hammer' all utility functions into a log mean-variance form. I_ Introduction Consider an investment situation wherein a fund is initially endowed with capital and, through returns from investment, grows (or decreases) with time. After the initial endowment, no additions or withdrawals are made. The initial endowment is simply transformed by repeated chance operations of investment as controlled by the portfolio structure. The management objective in this Co"espolldellce to: David G. Luenberger, Department of Engineering-Economic Systems, Stanford University, Stanford, CA 94305-4025. USA. ·The author wishes to thank David R. Carino for many helpful suggestions on this paper. 0165-1889/93/$06.00 © 1993-Elsevier Science Publishers B.V. All rights reserved --- ## Page 629 600 D. G. Luenberger 888 D.G. Luenberger, Log mean- variance criteria in investment problems situation, roughly, is to cause the investment value to increase as rapidly as possible, subject perhaps to considerations of security. Beyond this general statement it is, at least initially, difficult to define a concrete investment criterion, Yet this situation, even though highly idealized, closely approximates many real money management situations, such as those involving university endowment funds, mutual funds, gambling, expansion of a business enterprise, and personal wealth-building. It is a situation that has received a great deal of attention in the literature. To make the situation tractable, it is usually assumed that the investment environment is stationary in the sense that there is a fixed set of investment opportunities each of which are available in every period and that the returns of these investments are independent and identically distributed between periods (although there is dependency among the different investments), With this framework, the academic literature has traditionally taken three main ap- proaches to this idealized management situation, One approach is to argue, on the basis of very strong limit properties, that the best policy is the one that maximizes Elog W", where W" denotes the wealth in the fund at some terminal period nand E denotes the expectation operator [Kelly (1956), Brieman (1961), Latane (1959)]. Because of the stationarity assumption, this policy reduces to that of successively repeating the single-period policy of maximizing E log WI' The second approach is based on adherence to the expected utility framework of von Neumann and Morgenstern and suggests that one should maximize EU( Wn ) for some utility function U, This includes U( w,,) = log W", but there is no particular rationale for selecting that over any other utility function [see Samuelson (1971)]. A third approach, also based on the expected utility frame- work, suggests that one maximize E [7= I (P)iU(Cj), where Cj is (a monetary equivalent of) consumption drawn out .of the wealth at period i, and P, o < fJ < 1, is a discount factor [Samuelson (1969)]. This last approach has good economic justification within the von Neumann-Morgenstern framework, al- though the special form of additive discounted utility is quite specialized and does not really capture the spirit of the original (vague) objective of managing money which is not to be drawn down until the (distant) future, Our objective is not to argue that one approach is superior to another, but simply to provide a new preference foundation for the E log WI approach (or a generalization of it), so that it can be addressed by economic principles, There has been considerable interest in considering the limiting behavior of the EU( WII) criterion as n -+ 00, to obtain approximate and simple results. It has been hoped that this might provide some economic justification for the expected log approach, The principal technique in such investigations has been to impose restrictions on the utility function (usually through boundedness assumptions) in order to deduce that the investment policy derived from the Elog W" criterion also maximizes EU(W,,) as n -+ 00 [cf. Markowitz (1959), Leland (1972)]. Such attempts have been of limited success, as elucidated in --- ## Page 630 A Preference Foundation f or Log Mean-Variance Criteria in Portfolio Choice Problems 601 D.G. Luenberger. Log mean- variance criteria in investment problems 889 Samuelson (1971) and Goldman (1974). No convincing argument has yet been made that reduces the general utility function approach (for a wide class of useful utility functions) to the Elog Wn (or equivalently Elog W.) approach. The E log Wn approach has been brought closer to standard economic theory by generalizing it to a criterion based on a trade-off between E log Wn and var(log w,,) [Williams (1936)]. Specifically, one defines an efficient frontier for fixed n, where var(log Wn) is minimized subject to E log Wn ~ m, for various values of m. An efficient policy corresponds to a point on this frontier. This procedure parallels the familiar mean-variance approach to single-period in- vestment [Markowitz (1952), Sharpe (1970), Tobin (1965), Lintner (1965)] and has the appeal of simplicity of that familiar procedure. Of course, this itself may not eliminate the conceptual gap between the asymptotic approach and classical utility theory, since even single-period mean- variance theory is not consistent with expected utility maximization except in special circumstances [Borch (1963)]. However, it has been shown that the log mean~variance approach I serves as an approximation to. the terminal utility approach when uncertainty is 'small' [Samuelson (1970)]. Because of its intuitive appeal and simplicity, it is natural to look for stronger justification for log mean-variance criteria, and the central limit theorem might seem to provide the basis for such a justification. The distribution of the sum of the logarithms of n independent and identically distributed returns, when its mean is subtracted and the result is normalized by dividing by n 112, converges to a normal distribution. Ignoring the required normalization, one might argue that the distribution of the sum itself is close to a normal distribu- tion for large n. Then since this normal distribution is completely characterized by its mean and variance, it is plausible to use these two parameters as determinants for an investment criterion. This idea has been explored in Hakansson (1971 a, b), and it was found that the corresponding efficient frontier degenerates to a single point as n .... 00, since the ratio of standard deviation to mean goes to zero. If the normalization is accounted for, the situation is far more complex, since there seems to be no valid limit argument. This was forcefully pointed out in Merton and Samuelson (1974) and observation of this limit fallacy led them once again to dismiss log mean-variance criteria as too simplistic. (Addi- tionally, the very idea of using utility at a terminal time and taking the limit also implicitly involves a fallacy since limn_co E{ U(Wn )} is not an expected utility.) So where does all of this leave us? The log mean criterion, or more generally, the log mean-variance approach is intuitively appealing, but apparently not consistent with standard expected utility theory. I The phrase 'log mean-variance criteria' is used throughout as shorthand for 'criteria involving only the mean and variance of the logarithm of the return'. --- ## Page 631 602 D. G. Luenberger 890 D.C. Luenberger. Log mean- l'ariance criteria in investment problems In section 3 we set the stage for a new preference-based approach to the problem. We begin by establishing preferences on infinite sequences of wealth rather than wealth at a fixed (but later taken to the limit) terminal time. This approach circumvents the limit problems encountered by the earlier ap- proaches, and in fact provides a path whereby powerful limit theorems of probability theory can be applied. The preference relations on infinite determin- istic sequences are extended to stochastic sequences, not by imposition of a von Neumann-Morgenstern expected value criterion, but instead quite naturally by an 'almost sure' criterion, which is applicable when the original preference relation depends entirely on asymptotic properties, as is appropriate for this problem. It is then shown that a wide spectrum of these asymptotic preferences reduce to log mean- variance criteria. Thus, under quite broad conditions the log mean-variance approach is identical with an asymptotic preference approach. The justification for the variance comes about, by the way, not from the central limit theorem, but from its stronger cousin, the law of the iterated logarithm. 2. The investment environment We begin by more explicitly defining the investment environment. There is an underlying vector random process Z = {Zk} defined on a probability space (D, §', Pl. Each Zk = (Z lk' Z2k> ... ,ZSk) is a vector of S random variables. An investment policy (ill' ilz, ... ) is a sequence of mappings ilk: R' -+ R. This policy defines a new process X = {Xd of random variables by X k = ilk(Zk)' The X process is the single-period return process. The X process in turn generates a wealth process W according to where Wo is the (fixed) initial wealth. The investment problem, in this general setting, is that of selecting the policy a within a class of feasible policies that leads to the most desirable W process. It should be noted that systematic withdrawals from the accumulated wealth can be accommodated in this framework by incorporating them into the definition of (Xk' For instance, a withdrawal of a fraction 13k of the wealth at period k using investment policy ilk is treated by setting X k = cxk(Zd = (I - 13da/l(Zk)' A concrete example of the general framework is where there are S securities, each of which may be purchased in any amount at the beginning of each period. If one unit of wealth (one dollar, say) is invested in security j, j = 1,2, ... ,S, during period k, its value at the end of the period is Zjb where Zjk is a non- negative random variable. The random vector Zk = (Zlk, 2 2k , ... ,2Sk ) thus describes the single-period returns for period k. At the beginning of any period k, . --- ## Page 632 A Preference Foundation for Log Mean- Variance Criteria in Portfolio Choice Problems 603 D.G. Luenberger, Log mean- variance criteria in inveslment problems 891 the investment manager apportions the current wealth Wk _ 1 among the S se- curities by choosing a weighting vector !Xk = «("ta, 0 such that C(Z/c) ~ C for all ("t and k. Primarily we shall be concerned with the processes X and W derived from Z and a policy n n-co n (3) - 1 " 1 " lim '2 L logwk ~ lim '2 L logvk' "~" n ,,7:1 (2c)n - ,,7:1 (2C)n' (5) -1 -1 lim -log WI! W,,+ 1 ~ lim -log VnV" + 1 • (6) n-+C,() n n-co n We assume an additional property that is really the heart of our analysis. (P.4) Tail Property: The preference relation depends only on the tail of the sequences in r. That is if w, De rand w'=v, then w;2:v for any W, veT that differ from wand v in at most a finite number of elements. The examples (3), (4), and (6) are tail preferences. The motivation for this assumption stems from the original proposed setting of the investment problem. The investor is assumed to care only about the long-term behavior of the investment; no finite portion is of concern. From a technical perspective the introduction of tail preferences is precisely the concept required to introduce the limit idea early, so that it need not be 5The notation lim indicates lim sup (that is, limit superior). --- ## Page 635 606 D. G. Luenberger 894 D.G. Luenberger. Log mean- variance criteria in investment problems introduced at a later stage of the logical development and then interchanged with some other operation. Preferences on stochastic processes Now let "II' be the family of stochastic wealth processes derived from the investment procedure of section 2. We wish to establish a preference relation on "II' x "11'. That is, we wish to be able to compare two different stochastic processes. This new preference relation should be an extension of the preference relation on deterministic sequences defined above. r x r is a subset of R cx:. x R cD and we let L!I be the corresponding subset of the Borel field on R en x R OO. This leads to an additional technical requirement on the preference relation. (P.5) Measurability: The subset {(w, v): w;:::v} of r x r is in 31. We now extend the preference relation to a preference relation ;:::s on stochastic wealth processes. The extension is by almost sure association. Axiom. Let W = (WI' W2 , . . • ), V = ( VI, V2 , .. • ) he stochastic processes in "11'. Then W;:::s V if W:2: Va.s. That is W;:::s V if P{ W:2: V] = 1 . (7) In other words, if for almost every realization of the processes the sequence generated by W is deterministically preferred to the sequence generated by V, then the stochastic process W is preferred to the stochastic process V. This is a very natural (and very weak) axiom. Notice that the usual axioms required to define preferences over stochastic events (through an expected utility criterion) are not imposed. Our stochastic preference criterion is not an 'ex- pected' criterion; it is an 'almost sure' criterion. Stoc~astic preference is induced by almost sure deterministic preference. In a very real sense then this preference structure is substantially weaker than that employed in traditional preference approaches to this problem. (To be sure, it is in another sense stronger because of the tail assumption.) . A potential difficulty with this axiom is that it may not make sense. It might be that P{ W:2: V} = t in which case neither W;:::s V nor V:2:s W would hold. A zero-one law insures that this is never a problem. Theorem 1. !f (P.1)-(P.5) hold, the relation ;:::s is complete, reflexive, and transitive. Proof Given Wand V in "/t', the sequence of pairs {( Jti, II;)} is a process on (Q, §, Pl. Let X = (X I, X 2, ... ) and Y = (Y I , Y2 , .. • ) be the corresponding --- ## Page 636 A Preference Foundation for Log Mean- Variance Criteria in Portfolio Choice Problems 607 D.C. Luenberger. Log mean- variance criteria in investment problems 895 single-period return processes. Under our assumptions (A.l) and (A.2) the pairs (X .... Y,,) and (X" Y1) are independent for k # t. Because Xi and Yi are uniformly bounded, all realizations of Wand V are in r. The event {W~ V} c Q is measurable by (P.5) and is a tail event on the process of pairs {( Wi, Vi)}. However, any tail event in this process is a symmetric event [see Beeiman (1968) or Shiryayev (1984)] for the process {(Xi, Yi)} since such an event is not influenced by a permutation of any finite number of elements of the processes. This latter process is independent, and hence by the Hewitt-Savage Zero-One Law [see Breiman (1968)], the probability of the event {W~ V} is either zero or one. If P{ W~ V} = 1, then W~s V. Suppose, however, that P{ W~ V } = O. We have 1 = P {{ W ~ V} u { V ~ W} } ~ P { W ~ V} + P { V ~ W} , where the first equality follows from the completeness of the order ~. Hence, in this case P{ V~ W} = 1, and V~s W. Therefore in any case either W~s V, or V~sw. Reflexivity follows immediately from the reflexivity of ~. Transitivity is fairly immediate also: Let U ~s V, V ~s W. Then U ~ V and V~ Wexcept on subsets M 1 and M 2, respectively, of Q, each of measure zero. Thus by transitivity of ~, U ~ W except on M 1 U M 2' I The key ingredient of this result is that measurable tail events on wealth processes have probability of either zero or one. It is for that reason we can define preference for stochastic wealth processes by almost sure association - the relation is always almost sure one way or the other. We exploit this property throughout the remainder of the paper. 4. Simple utility functions A particularly useful way to describe a large class of preferences is through a utility function. Again, a utility function first can be introduced on determinis- tic sequences, that is, on r. Then this utility, ifit defines a tail preference relation, can be extended by almost sure association to a utility function on the set "Ir of stochastic processes. A utility function on r is a measurable real-valued function U: r -+ R. It defines a preference relation through the definition w~v if U(w) ~ U(v). We say that a utility function is a tail utility function if U(w) depends only on the tail of w = (wt> W2 • ••• ) [that is, U(w) = U(w) if wand w differ in at most a finite number of elements]. --- ## Page 637 608 D. G. Luenberger 896 D.G. Luenberger. Log mean- variance criteria in investment problems We extend a tail utility function U from r to 1fr by use of the fact that, by the zero-one law, if WE if/, then if U (W) is finite-valued, it is a degenerate random variable; that is, it is a constant almost surely. The value of this constant is taken to be the value of U ( W). Although there is an infinite variety of possible tail utility functions, we focus first on what we call simple utility functions. These are defined in terms of limiting operations on functions of wealth at individual periods, with no interac- tion between periods. Specifically, simple utility functions have the form U(w) = lim p(wn , n). or U(w) = lim p(wn, n), " ..... ex:· or algebraic combinations of these forms. In these expressions, P is a continuous and increasing function of Wn for each n. This appears to be a very general class of tail utility functions; and if we accept the idea of basing preferences on tail events, this form of utility is very natural. We shall show, however, that this form (with some minor restrictions) leads inevitably to the expected log return criterion. We note first that without loss of generality we may write p( w, n) = p(log w, n) by applying the one-to-one transformation log w, which maps between (0, 00) and ( - 00, 00). The resulting p( ., n) is con'tinuous and increasing since Pc. n) is. Finally, by adding a constant to U, if necessary, we may assume p(l, n) = 0 or equivalently p(O, n) = O. Thus we define a simple utility function as having the form U(w) = lim p(log W n , n). (8) n- rJ:..- or U(w) = lim p(log W n , n) , (9) or combinations of the forms (8) and (9), where p has the properties stated above. We extend these to the stochastic wealth process W by use of the fact, from the proof of Theorem I, that tail events on W have probability of either zero or one. This means, in particular, that if f: r -+ R is measurable and f(w) depends only on the tail of the sequence w, then the random variable f( W) is constant almost surely. Using this fact we define U( W) = lim p(Jog Wn• n), (lO) n ........ -x; --- ## Page 638 A Preference Foundation for Log Mean- Variance Criteria in Portfolio Choice Problems 609 D.G. Luenberger. Log mean- variance criteria in investment problems 897 or U( W) = lim p(log Wit, n). (11 ) n-c.c'! The random variables U ( W) defined above are measurable as extended random variables - see Shiryayev (1984, pp. 171, 358). If U ( W) is finite it is constant a.s., and the utility is taken to be this constant. Not all functions p(.,.) will lead to meaningful utility functions. For example, if p(z, n) = z, the process Wit = c" with c > 1 will lead to U( W) = 00. In general, in order to obtain finite utility for feasible wealth processes, it is necessary that p(z, n) satisfy appropriate growth-rate conditions with respect to n. In particu- lar, p must decrease with n to counterbalance possible increases in z. The following theorem gives suitable conditions. Theorem 2. For each n, let p(z, n) be continuous and increasing in z with p(O, n) = O. Let X and W be processes satisfying (A.2). Assume E log Xl = m, and define U by (/0). (a) Iflimn~'" p(nz, n) = sgn(z) ' 00, then U( W) = { + 00, m > 0, -00, m 0, there is an N such that I Zit - ml < € for all n> N. By monotonicity of p it follows that p(n(m - €), n) ~ p(nz", n) ~ p(n(m + €), n), for all n > N. Hence lim p(n(m - €), n) ~ lim p(nzlt • n) ~ lim p(n(m + €), n). "~oo If m > O. then for sufficiently small € > 0 both m - e > 0 and m + e > O. There- fore, if the hypothesis (a) holds. it follows that limlt_ oop(nz,,; n) = 00. A similar argument shows that for m < 0 there holds limn_oop(nzn, n) = - 00. If the hypothesis in (b) holds, then lim p(nz", n) = g(m) . --- ## Page 639 610 D. G. Luenberger 898 D.G. Luenberger. Log mean- variance criteria in investment problems Finally, the conclusion follows since I I " - log Wn = - L log X k --. m a.s. n n k= 1 by the strong law of large numbers, and hence the above values hold for - - ( I ) !~_ p(log Wn, n) = !~. p n ~Iog Wm n almost surely. I The parallel of Theorem 2 holds for the form (10), with limn_ x- replaced everywhere by limn_ cx:, . It is clear that in either case g(m) is an increasing function of m. Theorem 2, and the above remarks, say roughly that, to within a monotone transformation, the only real-valued tail simple utility function is m = E log X l ' We remark again, however, that although this 'appears' to be an expecteQ value criterion, it is actually an 'almost sure' criterion. The expected value arises from the strong law of large numbers, rather than from an assumption of expected value utility. It is perhaps useful to relate the theorem more directly to our earlier discussion of other utility functions. For example, consideration of the popular utility function defined as w~ on terminal wealth leads one to set p(log W n , n) = exp( y log wn ) in (8). However, clearly p(nz, n) -+ sgn(z)' 00, leading to the infinite result of part (a) of Theorem 2. The idea can be salvaged by introduction of a I /n in the power, i.e., w~ /n or p(z, n) = e)·tln. This leads to U ( W) = e ym• which again gives the expected logarithm criterion. 5. Compound utility functions It is apparent that there is a slight gap in Theorem 2. Part (a) covers the cases m > 0 and m < 0 but not m = O. This is no real detraction from the theorem, for part (a) says that under its conditions the corresponding utility function is degenerate unless it is applied to a process with m = O. However, this gap motivates the introduction of compound utility functions; since by subtracting out the mean m from the process, there is room for other possible forms. Another motivation for compound utility functions is purely structural. We let a compound utility function have a simple utility function as an argument. --- ## Page 640 A Preference Foundation for Log Mean- Variance Criteria in Portfolio Choice Problems 611 D.G. Luenberger. Log mean- variance criteria in investment problems 899 However, by Theorem 2, we can take this simple utility function to be U s( W) = m. This motivates the specific compound forms U( W) = lim ",(log Wn - nm, m, n) a.s., (12) U( W) = lim ",(log w" - nm, m, n) a.s., ( 13) where", is continuous and increasing in its first two arguments. As before, the equalities in (12) and (13) are to be interpreted in the almost sure sense. Such a utility, depending on m, is clearly still a tail function. Once we recognize the role of the mean log return m in compound utility functions, it is not necessary to carry it along in our analysis. Instead, we consider U ( W) = lim 4>(log w", n) a.s., (14) U(W) = lim 4>(1og Wit, n) a.s., (15) where 4>(z, n) is increasing in z, for the special case where m = E log XI = o. Then, later, we can simply put m back into the utility as an additional variable, as in (12) or (13). This restriction to the study of utility functions (14) and (15) amounts, really, to a further examination of simple utility functions when it is known that m = O. A special situation is the degenerate one where Wn = 1 for all n. We wish this to have finite utility, and hence without loss of generality we may take 4>(0, n) = 0 for all n. As before the behavior of 4> as a function of n is critical. To characterize the appropriate behavior of 4> it turns out that the asymptotic behavior of 4> (Jnz, n) is critical. The following proposition and its corollary rule out a large class of functions. Lemma 1. Suppose 4>(z, n) is increasing in z for each n with 4>(0, n) = ° and Let X and W be processes satisfying (A.2) and assume E log XI = 0, var(log X d = (12 > O. Then,for U defined by (14), U( W) ~ A. --- ## Page 641 612 D, G, Luenberger 900 D,G, Lumberger. Log mean- variance criteria in investment problems Proof Since 4>(0, n) = 0, it follows that A ~ O. In order to treat the case A = oc as well as A < oc, we prove that if lim lim 4>(\/~z, n) ~ A', =- ~_ n- 'y., with A' ~ 00, then V( W) ~ A'. The lemma follows from this. Let p(z) = lim n_", 4>(Jnz, n). Select f: > O. Then there is a z such that p(z) > A' - f:. Then, by the definition of p, there is N such that for all n > N, 4>(Jnz, n) > A' - 2e. By the monotonicity of 4>(' , n) we have 4>(Jnz, n, ~ 4>(Jnz, 11) > A' - 2f:, for all z ~ i and all n > N. Thus,6 where the last equality follows from the law of the iterated logarithm, I, log Wn 1 1m = a.s., n- oo (2a2nloglogn)1 /2 which shows that limn _ ", (Iog Wn)/Jn = oc, a.s. Therefore V( W) > A' - 2e, Since this is true for all c > 0, it follows that V (W) ~ A '. I Corollary. If for all z > 0, lim 4>(Jnz, n) = ox, n- ~' then,for V defined by (14). V( W) = oc. Proof We have A = 00 in the above lemma. I Let us consider some examples. The function 4>(z, 11) = zln corresponds, according to Theorem 2 of the previous section, to the simple utility function with value V( W) = m; and hence in this context, with m = 0, V( W) = O. A = 0 for this function, so Lemma 1 gives V( W) ~ 0, The reverse inequality can be established by a symmetric argument (see Lemma 3), and together these agree with ,what is known from Theorem 2. 6 i.o, denotes 'infinitely often', --- ## Page 642 A Preference Foundationfor Log Mean-Variance Criteria in Portfolio Choice Problems 613 D.G. Luenberger. Log mean- variance criteria in investment problems 901 n==l z Fig. I. An unusual function that is decreasing in n above A. In general, if 4>(z, n) = z/q(n), then clearly q(n) must go to infinity faster than In in order that A be finite and hence for the corresponding compound utility function to be real-valued. For example, the function 4>(z, n) = z/Jn has A = co . Definition. Let A be a constant. The function of two variables 4>(Jnz, n) is said to be decreaSing with respect to n above A if 4>(Jnz, n) > A implies 4>(j;+1z, n + 1) ~ 4>(Jnz, n). The monotonicity property with respect to n is quite reasonable even if A > O. An example is shown in fig. 1. Corresponding to a function 4>(z, n) that is continuous and strictly increasing in z for each n, let f(', n): D" -+ R [where D" is the range of 4>(' , n)] be the inverse function of 4>( . , n); that is, f(4)(z, n), n) = z, for every z. This inverse exists and is also strictly increasing. We need the following result which shows that if qJ ( Vnz, n) is decreasing with respect to n, then f / Vn is increasing with respect to n. Lemma 2. Suppose 4>(j;+1 z, n + 1) ~ 4>(Jnz, n) = r. Then, f(r, n)/Jn ~f(r, n + l)/j;+1 . --- ## Page 643 614 D. G. Luenberger 902 D.G. Luenberger. Log mean- variance criteria in investment problems Proof We have (p(Jnz, n) = r, cP(Jn+jz, n + I) = s, with s ~ r. By applying f to these we have z =f(r, n)/Jn, z =f(s, n + l)/Jn+j ~f(r, n + l)/Jn+j, where the inequality follows from the monotonicity of f(· , n + 1). Therefore f(r, n)/Jn ~f(r, n + l)/Jn+j. I We are now prepared to state the basic result of this section. We show, essentially, that any compound utility function of wealth (14), derived from a process with zero mean (of the logarithm), is a function only of a 2 = var(Jog X d (or, equivalently, of a). Proposition 1. Let U be defined by (14) and assume that cP(z, n) is continuous and strictly increasing with respect to z for each n, and that cP(O, n) = 0. Suppose further that cP(j-;,z, n) is decreasing with respect to n above A as defined in Lemma 1. Then there is afunction h on R + possibly taking values - 00 or + 00 such that (i) h( a) is nondecreasing with respect to a; (ii) for any process X and W satisfying (A.2) with m = E(Iog Xl) = 0, var(log Xl) = 0'2 > 0, there holds U( W) = h(O') . Proof Defining A as in Lemma 1, we know that U( W) ~ A. For any r > A, we have U ( W) > r if and only if P{cP(log Wn , n) > r i.o.} = I. This is equivalent to p{IOg ~n >f(r,;.) i.O.} = 1. O'v'n O'v'n --- ## Page 644 A Preference Foundationfor Log Mean-Variance Criteria in Portfolio Choice Problems 615 D.G. Luenberger. Log mean- variance criteria in investment problems 903 According to the assumptions on 4> and the above lemma, Yn = f(r, n)/(11 In) is increasing for r> A. Furthermore, f(r, n) > 0 for r> A ~ O. Therefore the extended law of the iterated logarithm [Feller (1943), Breiman (1968, p. 297)] states that the probability above is either zero or one, depending on whether the sum ~ Yn -y2 /2 L- -e " n = 1 n converges or diverges, respectively. If the sum converges for a given sequence {Yn}, then it will converge also for {Y~} with Y~ > Yn' This is because (d/dx) x [xe- X2/2] = [1 - x3Je-x2/2 < 0 for x > 1. Thus, for sufficiently large n, each term in the sum for {Y~} is smaller than the corresponding term in the sum for {Yn} . Therefore, if the sum converges for a given value of r, it also converges for r' > r. It follows that In a similar way, if the series converges for a given 11, it wiJ] also converge for 11' < 11. Hence U( W) = h(l1) where h(') is nondecreasing. I A useful 4> for consideration is z 4>(z, n) = (2 I I )1 /2 . n og ogn In this case the standard law of the iterated logarithm states that . 10gWn U ( W) = hm (2 1 I ) 1/2 = 11 a.s. , n- oo n og ogn and hence this 4> gives h(l1) = 11. Proposition I shows that an investor with a utility function of the general form (14), with the lim operation, prefers increased variance. Increased variance increases the extent of upward variations, and presumably the investor values these peaks. The comparison to PrOposition 1 treats utilities of form (15), with the lim operation. In this case the iiWestor wants to increase the lowest deviations, and hence such an investor values low variances more than high variances. Specifically, we have the following results. --- ## Page 645 616 D. G. Luenberger 904 D.G. Luenberger. Log mean- variance criteria ill int'eSlmenl problems Lemma 3. Suppose 4J(z, n) is increasing in z for each n with 4J(0, n) = 0 and lim lim 4J(y'7zz, n) = B. z- - ''X:' n-::(. Let X and W be processes satis/:ving (2) and assume E log X I = 0, var(log X I) = a 2 > O. Then, for U defined hy (J 5). U (W) s B. Proposition 2. Let U be defined by (I5) alld assume that 4J( z, n) is continuous and strictly increasing with respect (0 ;:: for each n, and that ¢(O, n) = O. Suppose further that 4J('\/~z, n) is increasing with respect to n helow B as defined in Lemma 3. Then there is afunction 9 on R + possibly taking values - oc: or + 00 such that (i) g(a) is nonincreasing with respect to a; (ii) for any processes X and W satisfving (A.2) with m = E(log X I) = 0, var(log X t) = a 2 > 0, there holds U( W) = g(a). Finally, we may combine Propositions I and 2 and insert the explicit depen- dence of a general compound utility function on the mean m. We let I/I(z, m, n) be continuous and strictly increasing with respect to z and increasing with respect to m. Theorem 3. Let U be defined by either (a) U( W) = limn~ cx: I/I(log Wn - nm, m, n) a.s., or (b) U( W) = lim n _ lO 1/1 (log Wn - nm, m, n) a.s. Correspondingly, assume thaljor each m, 4Jm(z, 11) = I/I(z, m, n) satisfies the condi- tions of Propositions I or 2. Then there is a function f on R x R + possibly taking values - iX or + ,x; such that (i) f(m, a) is increasing with respect to m and either increasing or decreasing with respect to a corresponding to (a) or (b); (ii) for any processes X and W satisfying (A.2) with In = E log Xl, var(log Xl) = a 2 > 0, there holds U (W) = .f(m, (J). For a given process Z and a given family of feasible investment policies, one can define the set S of attainable (m, a) pairs (achieved as the policy ranges over all possibilities). If the set S is compact, the right frontier of the set S is the right-most boundary of this set; that is, the points (m, a) having the largest a for --- ## Page 646 A Preference Foundation for Log Mean-Variance Criteria in Portfolio Choice Problems 617 D.G. Luenberger, Log mean- variance criteria in investment problems 905 a given m. Then, according to the above theorem, an investor with a utility function of the form (a) will select a policy corresponding to a point on this frontier. Similarly, an investor with a utility function of the form (b) will select a policy corresponding to a point on the left frontier. 6. Conclusion We have approached infinite-horizon investment situations by consider- ing preference orders on infinite sequences of wealth, and we suggested that tail preferences are appropriate if the goal of investment is long-term 'wealth building'. We showed that if the tail preference takes the form of a simple utility function, then utility must be equivalent to the expected logarithm of return. If preferences are represented by a compound utility function, that function will be equivalent to a function of the expected logarithm and variance of the logarithm. Although our results require that certain technical assumptions be satisfied, they are robust enough to support, at least roughly, the statement that: a tail utility function involving the limits of total return must be equivalent to a log mean- variance criterion. We do not anticipate, however, that the con- troversy over the use of such a criterion will subside. Indeed it should not. Our result is based on the tail preference concept, which is an idealization of what is generally a complex economic problem. However, since the log mean-variance approach is so intuitively appealing, it is nice to have a framework in which it is validated. References Algoet, P.H. and T.M. Cover, 1985. Asymptotic optimality and asymptotic equipartition properties of log-optimum investment. Technical report 57 (Department of Statistics, Stanford University, Stanford, CAl. Bell, R. and T.M. Cover. 1980, Competitive optimality of logarithm investment, Mathematics of Operations Research 5. 161-166. Borch, K., 1963. A note on utility and attitudes to risk, Management Science 9. 697-701. Breiman, L., 1961, Optimal gambling systems for favourable games, Proceedings of the Fourth Berkeley Symposium 1,65-78. Breiman, L., 1968, Probability (Addison-Wesley, Reading, MA). Feller, W., 1943, The general form of the so-called law of the iterated logarithm, American Mathematical Society Transactions 54, 373-402. Goldman, M.B., 1974, A negative report on the 'near optimality' of the max-expected-Iog policy as applied to bounded utilities for long-lived programs, 10urnal of Financial Economics 1,97- 103. Hakansson, N.H., 197130 Capital growth and the mean-variance approach to portfolio selection, 10urnal of Financial and Quantitative Analysis 6, 517-557. Hakansson, N.H .• 1971 b, Multi-period mean-variance analysis: Toward a general theory of port- folio choice, Journal of Finance 26. 857-884. Kelley, J.L., Jr .. 1956. A new interpretation of information rate, Bell System Technical Journal 35, 917-926. --- ## Page 647 618 D. G. Luenberger 906 D.G. Luenberger. Log mean- variance criteria in investment problems Latani:, H., 1959, Criteria for choice among risky ventures, Journal of Political Economy 67, 144-155. Leland, H., 1972, On turnpike portfolios, in: G. Szego and K. Shell, eds., Mathematical methods in investment and finance (North-Holland, Amsterdam). Lintner, J., 1965, The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, Review of Economics and Statistics 48, 13- 37. Markowitz, H., 1952, Portfolio selection, Journal of Finance 12, 77-9\. Markowitz, H., 1959, Portfolio selection: Efficient diversification of investment (Wiley, New York, NY). Merton, R.C. and P.A. Samuelson, 1974, Fallacy of the log-normal approximation to optimal portfolio decision-making over many periods, Journal of Financial Economics I, 67-94. Samuelson, P.A., 1969, Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics 51,239-246. Samuelson, P.A., 1970, The fundamental approximation theorem of portfolio analysis in terms of means, variances, and higher moments, Review of Economic Studies 37, 537-542. Samuelson, P.A., 1971, The 'fallacy' of maximizing the geometric mean in long sequences of investing or gambling, Proceedings of the National Academy of Sciences 68, 2493-2496. Sharpe, W., 1970, Portfolio theory and capital markets (McGraw-Hill, New York, NY). Shiryayev, A.N., 1984, Probability (Springer-Verlag, New York, NY). Tobin, J., 1965, The theory of portfolio selection. in: I.H. Hahn and F.P.R. Brechling, eds., The theory of interest rates (Macmillan, New York. NY). Williams. J.B., 1936, Speculation and the carryover, Quarterly Journal of Economics 50, 436. --- ## Page 648 ELSEVIER Available online at www.sciencedirect.com aCIENCE@DIRECYO Journal of Econometrics 116 (2003) 365-386 619 JOURNAL OF Econometrics www.elsevier.com/locate/econbase 43 Portfolio choice with endogenous utility: a large deviations approach Michael Stutzer* Burridge Center for Securities Analysis and Valuation, Leeds School of Business. University of Colorado, Boulder, CO 20309-0419, USA Abstract This paper provides an alternative behavioral foundation for an investor's use of power utility in the objective function and its particular risk aversion parameter. The foundation is grounded in an investor's desire to minimize the objective probability that the growth rate of invested wealth will not exceed an investor-selected target growth rate. Large deviations theory is used to show that this is equivalent to using power utility, with an argument that depends on the investor's target, and a risk aversion parameter determined by maximization. As a result, an investor's risk aversion parameter is not independent of the investment opportunity set, contrary to the standard model assumption. © 2003 Elsevier B.V. All rights reserved. JEL classification: C4; 08; GO Keywords: Portfolio theory; Large deviations; Safety-first; Risk aversion 1. Introduction What criterion function should be used to guide personal investment decisions? Per- haps the earliest contribution was Bernoulli's critique of expected wealth maximization, which led him to advocate maximization of the expected log wealth as a resolu- tion of the S1. Petersburg Paradox. This was resurrected as a long-term investment strategy in the 1950s, and is now synonymously described as either the log optimal, growth-maximal, geometric mean, or Kelly investment strategy. As also noted in their 'Tel.: +1-319-335-1239. E-mail address: michael-stutzer@colorado.edu (M. Stutzer). 0304-4076/03/$ - see front matter © 2003 Elsevier B.Y. All rights reserved. doi:1 0.1 016/S0304-4076(03 )001 12-X --- ## Page 649 620 M Stutzer 366 M. Stutzer /Journal of Econometrics 116 (2003) 365-386 excellent survey on this portfolio selection rule, Hakansson and Ziemba (1995, pp. 65-70) argue that " .. . the power and durability of the model is due to a remarkable set of properties", e.g. that it "almost surely leads to more capital in the long run than any other investment policy which does not converge to it". 1 But even as a long-term investment strategy, the log optimal portfolio is problem- atic. It often invests very heavily in risky assets, which has led several researchers to highlight the possibilities that invested wealth will fall short of investor goals, even over the multi-decade horizons typical of young workers saving for retirement. For example, MacLean et al. (1992, p. 1564) note that "the Kelly strategy never risks ruin, but in general it entails a considerable risk of losing a substantial portion of wealth". Findings like these motivated Browne (1995, 1999a) to develop a variety of alter- native, shortfall probability-based criteria, in specific continuous-time portfolio choice problems. Browne (1999b) considers these ideas in the context of the simplest possi- ble investment decision, which will also be utilized to illustrate the criterion developed herein. Further discussion of his work is included in Section 2.3. Another similarly motivated criterion for continuous time portfolio choice is developed in Bielecki et al. (2000), which will be discussed further in Section 2.2. The problem is exacerbated when investors have specific, short to medium term val- ues for their respective investment horizons. If so, some criteria will lead to horizon- dependent optimal asset allocations, but others will not. For example, Samuelson (1969) proposes the criterion of intertemporal maximization of expected discounted, time- additive constant relative risk aversion (eRRA) power utility of consumption. He proves that when asset returns are lID, portfolio weights are independent of the hori- zon length. So in that case, long horizon investors should not invest more heavily in stocks than do short horizon investors. Samuelson (1994) provided caveats to this investor advice, citing six modifications of this specification that will result in horizon dependencie~. 2 But an investment advisor, hired to help an investor formulate asset allocation ad- vice, may have difficulty determining a specific value for the investor's horizon. The advisor may be unable to determine an'investor's exact horizon length when it ex- ists, while other investors may not have a specific investment horizon length at all. A considerable simplification results when an infinite horizon is assumed, as has also been done when deriving many, but not all, consumption-based asset pricing models. 3 An exception to the infinite horizon formulations is found in Detemple and Zapatero (1991). Of course, the cost of this simplification is the inability to model horizon dependencies. While the time horizon parameter is irrelevant for Samuelson's intertemporal power utility investor with IID returns, the optimal asset allocation is still very sensitive to the specific risk aversion parameter adopted, so an advisor would have to determine it with I See the analysis of Algoet--and Cover (1988) and the lucid exposition of Cover and Thomas (1991, Chapter 15) for more informationon the growth maximal portfolio problem. For a spirited nonnative defense of the growth maximal portfolio criterion, see Thorp (1975). 2 However not all of these modifications would support the oft-repeated advice to invest more heavily in stocks when the investor's horizon is longer. 3 For a survey, see Kocherlakota (1996). --- ## Page 650 Portfolio Choice with Endogenous Utility: A Large Deviations Approach 621 M. Stutzer/]ournal of Econometrics 116 (2003) 365-386 367 precision. An even more basic consideration is specification of the utility functional form and its argument. Should it be a power function, or an exponential function, or perhaps some function outside the HARA class? Should the argument be a function of current wealth, current consumption, or some function of the consumption path (as in habit formation models)? As a first step toward answering these questions, Section 2 of this paper develops a new criterion of investor behavior. It starts from the observation that the realized growth rate of investor wealth is a random variable, dependent on the returns to invested wealth and the time that it is left invested (i.e. the investment horizon). To obviate the need to specify a value for the latter, first assume that an investor acts as-if she wants to ensure that the (horizon-dependent) realized growth rate of her invested wealth will exceed a numerical target that she has, e.g. 8% per year. By choosing a portfolio that results in a higher expected growth rate of wealth than the target rate, the investor can ensure that the probability of not exceeding the target growth rate decays to zero asymptotically, as the time horizon T ~ 00. But the probability that the realized growth rate of wealth at finite time T will not exceed the target might vary from portfolio to portfolio. Which portfolio should be chosen? Without adopting a specific value of T, a sensible strategy is to choose a portfolio that makes this probability decay to zero as fast as possible as T ~ 00. This will ensure that the probability will be minimized for all but the relatively small values of T. In other words, the decay rate maximizing portfolio will maximize the probability that the realized growth rate will exceed the target growth rate at time T, for all but relatively small values of T. In fact, this turns out to be true for all values of T in the special lID cases studied in Sections 2.1 and 3. Calculation of the decay rate maximizing portfolio is enabled by use of a simply stated, yet powerful result from large deviations theory, known as the Gartner-Ellis Theorem. Straightforward application of it in Section 2.2 provides an expected power utility formulation of the decay rate criterion. But there are two important differences between this formulation and the standard expecte4 power utility problem. First, the argument of the utility function is the ratio of invested wealth to a level of wealth growing at the constant target rate. Second, the value of the power, i.e. the risk aversion parameter, is also determined by maximization. As a result, a decay rate maximizing investor's degree of relative risk aversion will depend on the investment opportunity set, an effect absent in extant uses of power utility. Because this endogenous degree of risk aversion is greater than 1, the third deriva- tive of the utility is positive, so there is also an endogenous degree of skewness preference. This is fortunate, as some have argued that skewness preference helps explain expected asset returns. To see why, note that in the standard CAPM, in- vestor aversion to variance makes an asset return's covariance with the market re- turn a risk factor, so it is positively related to an asset's expected return. Kraus and Litzenberger (1976) argue that investor preference for positively skewed wealth dis- tributions (ceteris paribus) should make market coskewness an additional factor, that should be negatively related to an asset's expected return. They thus generalized the standard CAPM to incorporate a market coskewness factor. The estimated model sup- ports this implication of investor skewness preference. Harvey and Siddique (2000) ex- tended this approach by incorporating conditional coskewness, concluding that "a model --- ## Page 651 622 M Stutzer 368 M. StUlzer/]ournal oj Econometrics 116 (2003) 365-386 incorporating coskewness is helpful in explaining the cross-sectional variation of asset returns". 4 The decay rate maximization criterion also nests Bernoulli's expected log maximiza- tion (a.k.a. growth optimal) criterion. An investor who has a target growth rate suitably close to the maximum feasible expected growth rate has an endogenous degree of risk aversion slightly greater than 1. As a result, the associated decay rate maximizing port- folio approaches the expected log maximizing portfolio. If the investor's target growth rate is lower, the investor uses a higher degree of risk aversion, and the associated decay rate maximizing portfolio is more conservative, with a lower expected growth rate, but a higher decay rate for the probability of underperforming that target growth rate (and hence a higher probability of realizing a growth rate of wealth in excess of that target). The (perhaps unlikely) presence of an unconditionally riskless asset, i.e. one with an intertemporally constant return, provides a floor on the attainable target growth rates. When the target growth rate is sufficiently near that floor, the investor's risk aversion will be quite high, and the associated decay rate maximizing portfolio will be close to full investment in the unconditionally riskless asset. The relationship between the target growth rate and the associated (maximum) decay rate of the prob- ability that it will not be exceeded quantifies the tradeoff between growth and shortfall risk that has concerned analysts studying the expected log criterion. Exact calculation of the decay rate (or equivalently, the expected power utility) requires the exact portfolio return process. In practice, the distribution is not exactly known. Even if its functional form is known, its parameters must still be estimated. To cope with this lack of exact knowledge, Section 3 adopts the assumption that portfolio log returns are IID with an unknown distribution, and follows Kroll et a1. (1984) in estimating expected utility by substitution of a time average for the expectation operator. The estimated optimal portfolio and endogenous risk aversion parameter are those that jointly maximize the estimated expected power utility. An illustrative application of this estimator is included, contrasting decay rate maximization to both Sharpe Ratio and expected log maximization when allocating funds among domestic industry sectors. In it, decay rate maximization selects portfolios with higher skewness than Sharpe ratio maximization does. The IID assumption that underlies the estimator also permits the use of both a relative entropy minimizing, Esscher transformed log return distribution and a cumulant expansion to help interpret the empirical findings. Section 4 summarizes the most important results, and concludes with some good topics for future research. 2. Porfolio analysis Following Hakansson and Ziemba (1995, p. 68), the wealth at time T resulting from investment in a portfolio is Wr= WO I1:=1 Rpt. where Rpl is the gross (hence positive) 4 Hence, it is possible that an asset pricing model incorporating decay rate maximizing investors could outperform the CAPM, which incorpmhtes Sharpe ratio maximizing investors. This topic is left for another paper. --- ## Page 652 Portfolio Choice with Endogenous Utility: A Large Deviations Approach 623 M. Stutzer!Journal of Econometrics 116 (2003) 365-386 369 rate of return between times t - 1 and t from a portfolio p. Note that Wr does not depend on the length of the time interval between return measurements, but only on the product of the returns between those intervals. Dividing by WQ, taking the log of both sides, mUltiplying and dividing the right-hand side by T and exponentiating both sides produces the alterative expression (1) From (1), we see that Wr is a monotone increasing function of the realized time average of the log gross return, denoted log R p, which is the realized growth rate of wealth through time T. When the log return process is ergodic in the mean, this will converge to a number denoted E[logRp), as T -> 00. Accordingly, there was early (and still continuing) interest in the portfolio choice that maximizes this expected growth rate, i.e. selects the portfolio argmaxpE[logRp], also known as the "growth optimal" or "Kelly" criterion. As noted by Hakansson and Ziemba (1995, p. 65) " . .. the power and durability of the model is due to a remarkable set of properties", e.g. that it "almost surely leads to more capital in the long run than any other investment policy which does not converge to it". 5 But maximizing the expected log return often invests very heavily in assets with volatile returns, which has led several researchers to highlight its substantial downside performance risks. Specifically, we will now examine the probability of the event that the realized growth rate of wealth log R p will not exceed a target growth rate log r specified by the investor or analyst. This is an event that will cause W T in (1) to fail to exceed an amount equal to that earned by an account growing at a constant rate log r. The following subsection uses a simple and widely analyzed portfolio problem to calculate this downside performance risk for the growth optimal portfolio and a portfolio chosen to minimize it. 2.1. The normal case A simple portfolio choice problem, used in Browne (1999b), requires choice of a proportion of wealth p to invest in single stock, whose price is lognormally distributed at all times, with the rest invested in a riskless asset with continuously compounded constant return i. In this case, 10gRpI rv IID %(E[logRp), Var[logRp)). We now com- pute the probability that 10gRp ~ logr. Because the returns are independent, 10gRp rv % (E[logRp], Var[logRp]/ T). The elementary transformation to the standard normal variate Z shows that the desired probability is __ [IOgr - E[IOgR p ]] Prob[log R p ~ log r) = Prob Z ~ . / . V Var[Rp]/ T (2) 5 See the analysis of Algoet and Cover (1988) and the lucid exposition of Cover and Thomas (1991, Chapter 15) for more information on the expected log criterion. For a spirited normative defense of this criterion, see Thorp (1975). --- ## Page 653 624 M Stutzer 370 M. StutzerlJournal of Econometrics 116 (2003) 365-386 Tn order to minimize (2), i.e. to maximize the complementary probability that 10gRp > logr, one must choose the proportion of wealth p to minimize the expres- sion on the right-hand side of (2). This is equivalent to maximizing -I times this expression. Independent of the specific value of T, this portfolio stock weight is E[log R p] - log r argmax. (3) p JVar[logR p] Portfolio (3) will differ considerably from the following growth optimal portfolio arg max E[log R p] " (4) because of the presence of the target log r in the numerator of (3) and the standard deviation of the log portfolio return in its denominator. Portfolio (3) will also differ from the following Sharpe Ratio maximizing portfolio: E[R ] - i argmax p (5) p JVar[Rp] because of the presence of the presence of the target log r in (3) in place of the riskless rate i in (5), and because of the presence of log gross returns in (3) in place of the net returns used in (5). It will soon prove useful to reformulate the rule (3) in the following way. Note that Prob[log R p ~ log r] will not decay asymptotically to zero unless the numerator of (3) is positive, so we need only consider portfolios p for which E[log R p] > log r, (6) in which case the objective in problem (3) can be equivalently reformulated by squar- ing, and dividing by 2. The result is the following criterion: ( ) 2 1 E[logR] -logr arg max D p(log r) == arg max - p p p 2 JVar[logRp] (7) Tn order to quantitatively compare criteria (4), (5), and (7), it is useful to follow Browne (1999b) in using a parametric stochastic stock price process that results in the stock price being lognormally distributed at all times t, so that 10gRpt '" IID 5(E[logRp], Var[logRp]) as assumed above. Specifically, the stock price S follows the geometric brownian motion with drift dSIS = m dt + v dW, where m denotes the instantaneous mean parameter, v denotes the instantaneous volatility parameter, and W denotes a standard Wiener process. The bond price B follows dBIB = i dt. Denoting the period length between times t and t + 1 by i::lt, Hull (1993, p. 210) shows that (8) (9) Now substitute Eqs. (8) and (9) into (7), and write down the first-order condition for the maximizing stock weight p. You can verity by substitution that the following --- ## Page 654 Portfolio Choice with Endogenous Utility: A Large Deviations Approach M. Stutzer/Journal of Econometrics 116 (2003) 365-386 p solves it: arg max D p(log r) = p 2(log r - i) v2 Using (8), the growth optimal criterion (4) yields the portfolio (m - i) argmaxE[logRp] = --2- ' P V 625 371 (10) (11 ) Using Browne's (1999b, p. 77) parameter values m = 15%, v = 30%, i = 7%, and a target growth rate log r = 8%, the outperformance probability maximizing rule (10) advocates investing a constant p=47% of wealth in the stock, while the growth optimal rule (11) advocates p=89%. Of course, (10)'s p=47% minimizes the probability that the realized growth rate log R p ~ 8%. Fig. I illustrates the phenomena, by graphing Prob[logRp ~ 8%] for the two portfolios, and a third portfolio with just 33% invested in the stock. It shows that Prob[log R p ~ 8%] decays to zero for all three portfolios, but decays at the fastest rate when (IO)'s p=47% is used. Section 2.2 will show that the rate of probability decay rate in Fig. 1 is Dp(logr) in (7). Fig. I also shows that even though investors can invest in a riskless asset earning 7%, and can try to beat the modest 8% target growth rate by also investing in a stock with an instantaneous expected return of 15%, there is still almost a 20% probability that the investor's realized growth rate of wealth after 50 years will be less than 8%! Table I contrasts performance statistics for the outperformance probability maximiz- ing portfolios and the growth optimal portfolio p = 89% over the feasible range of target growth rates log r. Because the riskless rate of interest is only 7%, the prob- ability of earning more than a target rate log r > 7% is always less than one. If the target rate log r ~ 7%, the investor could always ensure outperforming that rate by in- vesting solely in the riskless asset. Hence the lower limit of the feasible target growth rates is the 7% riskless rate. 6 Line 1 in Table 1 shows that 'in order to maximize the probability of outperforming a target growth rate one basis point higher than this, i.e. log r = 7.01 %, the investor need invest only p = 5% of wealth in the stock. As a result of this conservative portfolio, this investor will have a relatively low probabil- ity of not exceeding this target; the decay rate of the underperformance probability is max p Dp(7.01)=3.l9%. But by investing 89% of wealth in the stock, the growth opti- mal investor will have a higher probability of not exceeding this 7.01 % target, because its associated decay rate is just 0.88%. This occurs despite its much higher expected growth rate E[log R p] (10.6% vs. 7.4%) and higher expected net return J1= pm+( 1 - p)i (14.1% vs. 7.4%). Of course, the major reason for this is its higher volatility (J = pV (26.7% vs. 1.5%), which increases the probability that a bad series of returns will drive the growth optimal portfolio's realized growth rate below log r = 7.01 %. Also note in line 1 that in order to maximize the probability of outperforming the 7.01% target, the investor must choose a portfolio with a higher expected growth rate (7.4%) than the target, as explained earlier. 6 Of course, if a riskless rate does not exist, it would not provide a floor on the feasible target rates. --- ## Page 655 626 372 ~ :c til ~ 2 II.. 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 M Stutzer!Journal of Econometrics 116 (2003) 365-386 Underperformance Probabilities log r= 8% --+-p= 89% -&-p=33% -lr-p =47% M Stutzer 0.1+----+----+----+----+----+----+----+----~--~----~ 10 20 30 40 50 60 70 90 90 100 T (years) Fig. I. The probability of not exceeding the 8% target growth rate approaches zero at a portfolio dependent rate of decay. The rate of decay is highest for the portfolio with p = 47% invested in the stock. There is an important tradeoff present in Table 1. Note from columns I and 3 that investors with successively higher growth targets log r have successively lower underperformance probability decay rates max p D p(log r). This implies that investors with higher targets will be exposed to a higher probability of realizing growth rates of wealth that do not exceed their respective targets. This occurs despite the fact that they did the best they could to minimize the probability of that happening. This is a consequence of the successively more aggressive portfolios needed to ensure asymptotic outperformance of their successively higher targets. This tradeoff is analogous to the tradeoff between mean and standard deviation asso- ciated with the efficiency criterion that selects the portfolio with the smallest standard --- ## Page 656 Portfolio Choice with Endogenous Utility: A Large Deviations Approach 627 M. StutzerlJournal oj Econometrics 116 (2003) 365-386 373 Table I Perfonnance statistics for the maximum expected log portfolio and maximum decay rate portfolios associated with feasible target growth rates log r, when portfolios are fonned from a lognonnally distributed stock with m=15% instantaneous mean return and v=30% instantaneous volatility, and a riskless asset with instantaneous riskless rate i = 7% Perfonnance of portfolio (II) vs. portfolio (10) logr% Stock weight p% Dp(logr)% E[logRp]% ~% 0-% 7.01 89 (5) 0.88 (3.19) 10.6 (7.4) 14.1 (7.4) 26.7 (1.5) 7.5 89 (33) 0.66 (1.39) 10.6 (9.2) 14.1 (9.6) 26.7 (9.9) 8.0 89 (47) 0.46 (0.78) 10.6 (9.8) 14.1 (10.8) 26.7 (14.1) 8.5 89 (58) 0.30 (0.44) 10.6 (10.1) 14.1 (11.6) 26.7 (17.4) 9.0 89 (67) 0.17 (0.22) 10.6 (10.3) 14.1 (12.4) 26.7 (20.1) 9.5 89 (75) 0.08 (0.09) 10.6 (10.5) 14.1 (\3.0) 26.7 (22.5) 10.0 89 (82) 0.02 (0.02) 10.6 (10.5) 14.1 (13.6) 26.7 (24.6) 10.6 89 (89) 0.00 (0.00) 10.6 (10.6) 14.1 (14.1) 26.7 (26.7) deviation of return, once the investor fixes a mean return. Here, the criterion selects the portfolio with the highest underperformance probability decay rate, once the investor fixes a target growth rate. In this way, the tradeoff between logr and maxpDp(logr) can be thought of as an alternative efficiency frontier, which yields the growth opti- mal portfolio on one extreme and full investment in the constant interest rate (when it exists) on the other. The efficiency frontier is graphed in Fig. 2, which shows it to be a convex curve in this example. In Section 2.2, we will see that this is true more generally, i.e. with multiple risky assets, whose log returns are not necessarily normal nor IlD. Finally, there is just one risky asset used to form the optimal portfolios in Table 1, so the mean-standard deviation efficiency frontier is just swept out by varying the stock weight and calculating the mean fJ and standard deviation (J of the net returns. For comparison purposes, this is reported in the last two columns of Table 1. Reading down the last three columns of the table, note that the difference between fJ and the expected growth rate E[log R,,] of wealth grows wider as the standard deviation of portfolio returns (T gets larger. The mean return increasingly overstates the expected growth rate of wealth as portfolio volatility increases. This is due to (1); as Hakansson and Ziemba (1995, p. 69) note, " .. . capital growth (positive or negative) is a multiplicative, not an additive process".7 Here, due to lognormality, there isa precise relationship between the two: E[logRpJ = fJ- (J2 j2 (see Hull, 1993, p. 212). The following section will show that a simple, yet powerful result from large devia- tions theory permits us to rigorously characterize D p(log r) in Table 1 as the decay rate of the portfolios' underperformance probabilities graphed in Fig. 1. More importantly, the result also shows how to correctly calculate the decay rate and associated decay rate maximizing portfolios when portfolio returns are not lognormally distributed. 7 In this regard, see Stutzer (2000) for a simpler model of fund managers who use arithmetic average net returns rather than average log gross returns, under the assumption that net returns are lID. --- ## Page 657 628 374 3.5 3 2.5 0 as T --> 00. Calculation of this probabil- ity's decay rate D p(log r) is facilitated by use of the powerful, yet simple to apply, Gartner-Ellis Large Deviations Theorem, e.g. see Bucklew (1990, pp. 14-20). For a log portfolio return process with random log return 10gRpt at time t, consider the following time average of the partial sums' log moment generating functions, i.e.: ¢J(8) == lim ~ 10gE[e0L:;=1 logRp,] = lim ~ logE [( WT)O] , T->oo T T->oo T Wo (12) where the last expression is found by usipg (1) to compute WT/Wo = I1 Rpl , sub- stituting 10g(WT/ Wo) for the sum of the logs in (12), and simplifying. Hence (12) depends on the value of the random WT , and so does not depend on the particUlar discrete time intervals between the log returns 10gRpt . We maintain the assumptions that the limit in (12) exists for all 8, possibly as the extended real number +00, and is differentiable at any 8 yielding a finite limit. From the last expression in (12), these assumptions must apply to the asymptotic growth rate of the expected power of Wr/Wo. Some well-analyzed log return processes will satisfy these hypotheses, as will be demonstrated shortly by example. However, these assumptions do rule out some pro- posed stock return processes, e.g. the stable Levy processes with characteristic exponent '1. < 2 and hence infinite variance, used in Fama and Miller (1972, pp. 261-274). The calculation of the decay rate D p(log r) is the following Legendre-Fenchel trans- form of (12): Dp(log r) == max D log r - ¢J(D). ° (13 ) When log returns are independent, but not identically distributed, (12) specializes to 1 TIT ¢J(8)= lim - ~logE[eOlogR p, ]= lim -T~logE[R~I]. (14) T->oo T ~ T->oo ~ 1= 1 1= 1 When log returns are additionally identically distributed (lID), (12) simplifies to ¢J(D) = 10gE[eBIOgRp] = 10gE[R~], (15) which when substituted into (13) yields the decay rate calculation for the IID case. This result will form the basis for the empirical application in Section 3. It is known as Cramer's Theurem (Bucklew, 1990, pp. 7-9). To illustrate these calculations, let us return to the widely analyzed case where the log portfolio return log R pI is a covariance-stationary normal process with absolutely summable autocovariances. Then the partial sum of log returns in (12) is also normally --- ## Page 659 630 M Stutzer 376 M. SlUtzer /Journal of Econometrics 116 (2003) 365-386 distributed. The mean and variance of it can be easily calculated by adapting Hamilton's (1994, p. 279) calculations for the distribution of the sample mean, i.e. the partial sum divided by T. One immediately obtains T 10g(Wr/Wo) == L 10gRpI ,....,.At (TE[logRp), T Covo t=l T-l ) + ~(T-r)(COVr+COV_ r) , (16) where E[log R p) denotes the log return process' common mean and Cov, denotes its r-Iagged autocovariance. Formula (12) is the limiting time average of the log moment generating functions of these normal distributions. Now remember from elementary statistics that a normal distribution's log moment generating function is linear in its mean and quadratic in its variance. As a result, use (12) to calculate ,=+00 = E[logRp)8 + L Cov,8 2/2. (17) r=-OC) N ow substitute (17) into (13) and set its first derivative with respect to (J equal to zero to find that the maximum in (13) is attained by the following maximizer: 8max = (logr - E[logRp)) I:'J;: Cov,. (18) Substituting (18) back into (17) and rearranging yields the underperformance proba- bility decay rate (1 1 (E[IOgR p) _IOgr)2 Dp ogr) =- 2 /",=+00 C V ~,=-oo OV, (19) Note that maximization of the decay rate (19) rewards portfolios with a high expected growth rate E[logRp) (in its numerator) and a low asymptotic variance 2:::~: Cov, == limT--->oo Var[log(Wr/Wo»)/T (in its denominator). This differs from the criterion in Bielecki et al. (2000), which is approximately the asymptotic expected growth rate minus a mUltiple of the asymptotic variance. For the lID case used in Section 2.1, all covariance terms in (19) are zero except Covo == Var[logRp), so the decay rate function (19) reduces to the expression (7) used in Section 2.1 and Table 1. Fig. 3 depicts this decay rate function over a range of log r, for each of the three portfolios whose underperformance probabilities are graphed in Fig. 1. There, we see that the portfolio p = 47% from (10) does indeed have the highest decay rate when log r = 8%. --- ## Page 660 Portfolio Choice with Endogenous Utility: A Large Deviations Approach 631 M. Stutzer!Journal of Econometrics I/6 (2003) 365-386 377 Underperformance Probability Decay Rates 1.00% -+-p=89% -I!J-p=33% 0.90% -6-p=47% 0.80% 0.70% 0.30% 0.20% 0.10% 0.00% ~ ?ft ?ft ?ft ~ ~ ?ft ~ ~ ~ ~ . . . 0 '" 0 '" 0 '" 0 '" 0 '" 0 0 C\I '" '" 0 C\I '" ..... 0 N '" cO a:i cO to oj oj oj oj 0 0 g ~ Target Rate log(r) Fig. 3. The decay rate function Dp(log r) is convex, with a minimum at log r = E[log R p]. The portfolio p = 47% attains the highest decay rate when log r = 8%. Note that the decay rate function D p(log r) in (19) for a covariance stationary Gaus- sian portfolio log return process is non-negative, and is a strictly convex function of log r, achieving its global minimum of zero at the value log r = E[log R pl. These properties are true for more general processes (for a discussion, see Bucklew, 1990). As a result, remember from (6) that the decay rate criterion ranks portfolios with E[logRp] > logr, and apply the envelope theorem to the general rate function (13) to yield dDp(logr) a . d I = -::'-1 - max Blogr - c/J(B) = Bmax < 0 ogr v ogr {/ (20) --- ## Page 661 632 M Stutzer 378 M. Stutzer!Journal of Econometrics 116 (2003) 365-386 as seen in the special case (18). Now differentiate (20) to find dD~(Iog r) dtJmax --"--..:.".- = -- > 0 d log r2 d log r (21 ) due to convexity of Dp(logr). Again due to the envelope theorem, (20) and (21) con- tinue to hold for max p D p(log r) as well. Fig. 2 depicts the convexity of max p D p(log r) over the relevant range of log r in the example of Table 1. 2.3. Analogy with power utility The general decay rate criterion is a generalization of the expected power utility criterion. To uncover the generalization, substitute the right-hand side of (12) into (13), to derive max Dp(log r) == max max 810g r - lim ~ logE [( WT)II] p p () T->oo T Wo =maxmaxlogrB - lim ~logE [(WT)B] p B T-+oo T Wo = max max - lim 2. logE [( WTT)O] , P () T-+oo T Wor (22) which yields the following large T approximation: (23) where we write 8max(P) in (23) to stress dependence (through the joint maximization (22» of tJ on the portfolio p. The left-hand side of (23) increases with D p' so a large T approximation of the portfolio ranking is produced by use of the expected power utility on the right-hand side of (23). There are both similarities and differences between the right-hand side of (23) and a conventional expected power utility E[ - (WT )BJ.. From (20), 8max(P) < O. Evaluating it at the investor's decay rate maximizing portfolio p, note that the power function in (23) with the form U = -(. YJm",(P) increases toward zero as its argument grows to infinity, is strictly concave, and has a constant degree of relative risk aversion y == 1 - 8max(P) > 1. Furthermore, 8max(P) < 0 implies that the third derivative of U is positive, so the criterion exhibits positive skewness preference. But there are two important differences between the concepts. First, the argument of the power function in (23) is altered; it is the ratio of invested wealth to a "benchmark" level of wealth accruing in an account that grows at the geometric rate r. While absent from tradi- tional criteria, this ratio is also present in other non-standard criteria, such as Browne's (1999a, p. 276) criterion to "maximize the probability' of beating the benchmark by some predetermined percentage, before going below it by some other predetermined --- ## Page 662 Portfolio Choice with Endogenous Utility: A Large Deviations Approach 633 M. Stutzerllournal of Econometrics 116 (2003) 365-386 379 percentage". Browne (1999a, p. 277) notes that " ... the relevant state variable is the ra- tio of the investor's wealth to the benchmark". 8 Second, conventional portfolio theory assumes that the risk aversion parameter () is a preference parameter that is independent of the investment opportunity set. But in (23), () = ()max (p) is detennined by maximiza- tion, and hence is not independent of the investment opportunity set. Investors could utilize different investment opportunity sets, either because of differential regulatory constraints, such as hedge funds' greater ability to short sell, or because of different opinions about the parameters of portfolios' log return processes. When this happens, investors will have different decay rate maximizing portfolios p, and different degrees of risk aversion y = 1 - ()max(P), even if they have the same target growth rate logr. Assuming that asset returns are generated by a continuous time, correlated geometric Brownian process, Browne (1999a, p. 290) compares the fonnula for the optimal port- folio weights resulting from his criterion, to the fonnula resulting from conventional maximization of expected power utility at a fixed tenninal time T. In this special case, he finds that the two fonnulae are isomorphic, i.e. there is a mapping between the models' parameters that equates the two fonnulae. He concludes that "there is a con- nection between maximizing the expected utility of tenninal wealth for a power utility function, and the objective criteria of maximizing the probability of reaching a goal, or maximizing or minimizing the expected discounted reward of reaching certain goals". Connection (23) between decay rate maximization and expected power utility is quite specific, yet does not depend on a specific parametric model of the assets' joint return process. Critics such as Bodie (1995, p. 19) have argued that "the probability of a shortfall is a flawed measure of risk because it completely ignores how large the potential shortfall might be". It is possible that this is a fair assessment of expected power utility maximization of wealth at a fixed horizon date T, subject to a "Value-At-Risk" (VaR) constraint that fixes a low probability for the event that tenninal wealth could faIl below a fixed floor. This problem was intensively studied by Basak and Shapiro (2001, p. 385), who concluded that "The shortcomings ... stem from the fact that the VaR agent is concerned with controlling the probability of a loss rather than its magnitude". They proposed replacing the VaR constraint with an ad hoc expected loss constraint, resulting in fewer shortcomings. The investor's target growth rate serves a similar function in the horizon-free, unconstrained criterion (22). 3. Non-parametric implementation Tn the TID case, there is a simple, distribution-free way to estimate D p(log r) for a portfolio p. Following the comparative portfolio study of Kroll et al. (1984), we replace the expectation operator in (15) by an historical time average operator, substitute into ( 13), and numerically maximize that. 9 This estimator eliminates the need for prior 8 While Browne considers a stochastic benchmark, the constant growth benchmark here can be modified to consider an arbitrary stochastic benchmark, at the cost of fewer concrete expository results. 9 It is important to remember that the log moment generating function of the log return distribution necessarily has to exist near Ilrnax in order for this technique to work here. --- ## Page 663 634 M Stutzer 380 M. StutzerlJournal of Econometrics 116 (2003) 365-386 knowledge of the log return distribution's functional fonn and parameters. Specifically, let Rp(t)= L;=o pjRj(t) denote the historical return at time t of a portfolio comprised of n+ 1 assets with respective returns Rj(t), with constantly rebalanced portfolio weights Lj Pj = 1. The estimator is If,(log r) ~ m;x e log r - log [ ~ t (t, PjRj(')) l (24) and the optimal portfolio weights are estimated to be jJ = arg max max () log r PI ,···.P" (J (25) The maximum expected log portfolio was similarly estimat"ed, by numerically finding the weights that maximize the time average of 10gRp(t). Let us now contrast the estimated decay rate maximizing portfolio (25) to both the expected log and Sharpe ratio maximizing, constantly rebalanced portfolios fonned from Fama and French's 10 domestic industry, value-weighted assets,1O whose annual returns run from 1927 through 2000. The sample cross-correlations of the 10 indus- tries' gross returns range from 0.32 to 0.86, suggesting that diversified portfolios of them will provide significant investor benefits. The sample covariance matrix is in- vertible, pennitting estimation of the Sharpe ratio maximizing "tangency" portfolio, by mUltiplying this inverse by the vector of sample mean excess returns over a riskless rate, and then nonnalizing the result. We assume that it was possible to costlessly store money between 1927-2000, with no positive constant nominal rate riskless asset available. 11 Hence we assume a zero constant riskless rate when computing the Sharpe ratio maximizing tangency portfolio of the 10 industry assets. The results are seen in Table 2. The perfonnance statistics in Table 2 show that the Sharpe ratio maximizing portfolio has almost no skewness. But the decay rate maximizing portfolios all have a skewness of about 1, as does the expected log maximizing portfolio. This reflects the skewness preference inherent in the generalized expected power utilities with degrees of risk aversion greater than (in the log case, equal to) one. 12 In fact, these investors prefer all odd order moments and are averse to all even order moments. To see this, note that (15) is the cumulant generating function for the (assumed) TID log portfolio return 10 The data are currently available for download from a website maintained by Kenneth French at MIT. II Treasury Bills are not a constant rate riskless asset, like the one used to form portfolios in Section 2.1. A fixed percentage of wealth invested in Treasury Bills is just like any other risky asset. 12 See Kraus and Litzenberger (1976) and Harvey and Siddique (2000) for evidence that investors prefer skewness. --- ## Page 664 Portfolio Choice with Endogenous Utility: A Large Deviations Approach 635 M. Stutzerllournal of Econometrics 116 (2003) 365-386 381 Table 2 Comparison of estimated Sharpe ratio, expected log, and decay rate maximizing portfolios from Fama-French 10 industry indices, 1927-2000 Industries Asset moments Portfolio weights Max logr logr logr Max J.I. (J Skewness Sharpe 5% 10% 15% Log NoDur 0.130 0.198 -0.12 0.80 0.92 1.0 1.11 1.15 Durbl 0.166 0.328 0.86 -0.01 0.27 0.52 0.97 1.22 Oil 0.137 0.220 0.01 0.75 0.77 0.96 1.24 1.36 Chems 0.146 0.225 0.63 0.14 0.35 0.52 0.89 1.15 Manuf 0.136 0.254 0.21 0.03 -0.10 0 0.11 0.20 Telcm 0.123 0.200 0.07 0.35 0.48 0.38 0.30 0.28 Utils 0.118 0.225 0.25 0.05 -0.20 -0.34 -0.61 -0.76 Shops 0.141 0.256 - 0.25 - 0.13 - 0.44 - 0.60 - 0.96 -1.2 Money 0.142 0.245 - 0.43 -0.23 -0.20 0.07 0.48 0.70 Other 0.106 0.242 - 0.04 - 0.76 - 0.86 - 1.5 -2.52 -3.09 Performance statistics Mean 0.148 0.162 0.195 0.248 0.278 Std. dev. 0.153 0.181 0.240 0.368 0.446 Skewness -0.02 1.05 1.07 1.06 1.05 Decay rate D pClog r) 0.18 0.04 0.004 0 Risk aversion I - emax Cp) 5.3 2.5 1.3 I distribution. Substituting it into (J 3) and evaluating it at 8max (p) yields the following cumulant expansion: ~Ki i - ~ ~Umax (P)' l. i=3 (26) which uses the facts that E[log R p] is the first cumulant of the log return distribution and that Var[log R p] is its second cumulant, while Ki denotes its ith order cumulant. Because 8max(P) < 0, we see that the decay rate increases in odd-order cumulants and decreases in even-order cumulants. With normally distributed log returns, all the cumulants in the infinite sum are zero. But with non-normally distributed returns, increased skewness will increase the decay rate (due to K3)' The relative weighting of the mean, variance and skewness in (26) is determined by their sizes, the sizes of the higher order cumulants, the target growth rate log r, and the value of Umax(P) < 0 associated with log r. The top panel of Table 2 contains the 10 industry weights in each portfolio. As is typical of estimated Sharpe ratio maximizing portfolios with more than a few assets, it is heavily long in just three industries (Non-durables, Oil, and Telecommunications). The decay rate maximizing portfolio for the target growth rate log r = 0.10 is also heavily invested in these industries, but in addition it has considerable long positions in the two most positively skewed industries (Durables and Chemicals). The Sharpe --- ## Page 665 636 M Stutzer 382 M. Stutzer /Journal oj Econometrics 116 (2003) 365-386 ratio maxlmlzmg portfolio is heavily short in one industry (Other). The decay rate maximizing portfolios are heavily short in both this industry and as well as two others (Shops and Utilities). The differences between Sharpe ratio and decay rate maximizing portfolios are due to the presence of the target growth rate in decay rate maximization, its use of log gross returns rather than net returns when calculating portfolio means and variances, and the presence of higher order moments. It is difficult to assess the impact of higher order moments on the differences in portfolio weights. Bekaert et aI. (1998, p. 113) were able to produce only a two percentage point difference in an asset weight, when simulating the effects of its return's skewness over the range -1 to 2.0, on the portfolio chosen by an expected power utility maximizing agent whose degree of risk aversion was close to 10. This suggests that the use of a target growth rate, and the use of log gross returns rather than arithmetic net returns, account for most of the differences between the decay rate and Sharpe ratio maximizing portfolios' weights. The convergence of decay rate maximizing portfolios to the expected log maximizing portfolio is seen when reading across the last four columns of Table 2. The last two rows in the bottom panel of Table 2 show the relationship between the target growth rates, their respective efficient portfolios' maximum decay rates, and their respective endogenous degrees of risk aversion. Despite the fact that 8max(P) is determined by maximization in (25), we see that the degree of risk aversion 1 - f)max(P) is not unusually large in any of the decay rate maximizing portfolios tabled, 13 and converges toward 1 as log r --> maxp E[logRp). An alternative interpretation of this is enabled by computing the first order condition for 8max(P) in the IID case. To do so, substitute (15) into (13) and differentiate to find E [IOgR p :;] = logr, (27) where the Esscher transformed probability density dQ Ro,;"(P) dP = E[R~m"( ")] (28) is used to compute the expected log return (i.e. growth rate) in (27). 14 Furthermore, a result known as Kul\back's Lemma (1990) shows that the Esscher transformed density (28) is the solution to the following constrained minimization of relative entropy, whose minimized value is the decay rate, i.e. [dQ dQ] D p(log r) = min E dP log dP s.t. (27). (29) From (27), an efficient portfolio has the highest decay rate among those with a fixed transformed expected growth rate equal to logr. As logr -4 maxpE[logRp), 13 Of course, it can get unusually large when the target growth rate is unusually low, i.e. when the investor is unusually conservative. 14 See Gerber and Shiu (1994) for option pricing fonnula derivations that use the Esscher transfonn to calculate the risk-neutral density required for option pricing. --- ## Page 666 Portfolio Choice with Endogenous Utility: A Large Deviations Approach 637 M. S{u{zer /Journal of Econometrics 116 (2003) 365-386 383 0.4 Underperformance Probabilities 0.35 0.3 0.25 ~ :0 0.2 «I .J:l 0 ... Q. 0.15 0.1 Tyears Fig. 4. Bootstrap estimated underperformance probabilities for portfolios in Table 2, when log r = 10%. f}max( p) --> 0, density (28) concentrates at unity and the minimal relative entropy in (29) approaches zero, i.e. the transformed probabilities approach the actual probabilities. As a result, the transformed expected log return in (27) approaches the actual expected log return, so constraint (27) collapses the portfolio constraint set onto the log optimal portfolio. In order to determine if a decay rate maximizing portfolio in Table 2 will have lower underperformance probabilities than the Sharpe ratio and expected log maximiz- ing portfolios do, the probabilities were estimated by resampling the portfolios' log returns 5000 times for each investment horizon length T, and then tabulating the em- pirical frequency df underperformance for each T. The results for the target decay rate --- ## Page 667 638 M Stutzer 384 M. S{UlzerlJournal of Economelrics 116 (2003) 365-386 log r = 10% are graphed in Fig. 4. Fig. 4 shows that the estimated decay rate max- imizing portfolio in Table 2 had lower underperformance probabilities for all values of T. 3.1. More general estimators The empirical estimates above were made under the assumption of lID returns. There is little evidence of serially correlated log returns in many equity portfolios, and what evidence there is finds low serial correlation. Hence there is little benefit in using an efficient estimator for the covariance stationary Gaussian rate function (20), e.g. using a Newey-West estimator of its denominator. But the presence of significant GARCH (perhaps with mUltiple components) effects (see Bollerslev, 1986) in log returns, as described in Tauchen (2001, p. 58), motivates the need for additional research into efficient estimation of (12) and (13) under specific parametric process assumptions. Alternatively, it may be possible to find an efficient nonparametric estimator for (12) and (13) by utilizing the smoothing technique exposited in Kitamura and Stutzer (1997, 2002) to estimate the expectation in (12). 4. Conclusions and future directions A simple large deviations result was used to show that an investor desiring to maxi- mize the probability of realizing invested wealth that grows faster than a target growth rate should choose a portfolio that makes the complimentary probability, i.e. of wealth growing no faster than the target rate, decay to zero at the maximum possible rate. A simple result in large deviations theory was used to show that this decay rate maxi- mization criterion is equivalent to maximizing an expected power utility of the ratio of invested wealth to a "benchmark" wealth accruing at the target growth rate. The risk aversion parameter that determines the required power utility, and the investor's degree of risk aversion, is also determined by maximization and is hence endogenously dependent on the investment opportunity set. Yet it was not seen to be unusually large in the applications developed here. The highest feasible target growth rate of wealth is that attained by the portfolio maximizing the expected log utility, i.e. that with the maximum expected growth rate of wealth. Investors with lower target growth rates choose decay rate maximizing portfolios that are more conservative, corresponding to degrees of risk aversion that exceed 1. As the target growth rate falls, it is easier to exceed it, so the decay rate of the probability of underperforming it goes up. The relationship between possible target growth rates and their corresponding maximal decay rates form an efficiency frontier that replaces the familiar mean-variance frontier. An investor's specific target growth rate determines the specific decay rate maximizing portfolio chosen by her. A decay rate maximizing investor does not choose a portfolio attaining an expected growth rate of wealth equal to her target growth rate (instead it is higher than her target). But there is an Esscher transformation of probabilities, under which the transformed expected growth rate of wealth is the target growth rate. --- ## Page 668 Portfolio Choice with Endogenous Utility: A Large Deviations Approach 639 M. Stutzer /Journal of Econometrics 116 (2003) 365-386 385 Researchers choosing to work in this area may select from several interesting topics. First, it is easy to generalize the analysis to incorporate a stochastic benchmark. This would be helpful in modelling an investor who wants to rank the probabilities that a group of similarly styled mutual funds will outperform their common style benchmark. Second, one could calculate the theoretical decay rate function using a multivariate GARCH model for the asset return processes, and then estimate the reSUlting function. Third, one could extend the decay rate maximizing investment problem to the joint consumption/portfolio choice problem, enabling the derivation of consumption-based asset pricing model with a decay rate maximizing representative agent. If it is possible to construct a model like this, the representative agent's degree of risk aversion will depend on the investment opportunity set-an effect heretofore unconsidered in the equity premium puzzle. Acknowledgements Thanks are extended to Eric Jacquier, the editors, and other participants at the Duke University Conference on Risk Neutral and Objective Probability Measures, to sem- inar participants at NYU, Tulane University, University of Illinois-Chicago, Chicago Loyola University, Georgia State University, University of Alberta, Bachelier Finance Conference, Eurandom Institute, Morningstar, Inc., and Goldman Sachs Asset Manage- ment, and to Edward o. Thorp, David Bates, Ashish Tiwari, Georgios Skoulakis, Paul Kaplan, and John Cochrane for their timely and useful comments on the analysis. References Algoet, P., Cover, T.M., 1988. Asymptotic optimality and asymptotic equipartition property of log-optimal investment. Annals of Probability 16, 876-898. Basak, S., Shapiro, A., 2001. Value-at-risk based risk management: optimal policies and asset prices. Review of Financial Studies 14, 371-405. Bekaert, G., Erb, C., Harvey, C., Viskanta, T., 1998. Distributional characteristics of emerging market returns and asset allocation. Journal of Portfolio Management 24, 102-116. Bielecki, T.R., Pliska, S.R., Sherris, M., 2000. Risk sensitive asset allocation. Journal of Economic Dynamics and Control 24, 1145-1177. Bodie, Z., 1995. On the risk of stocks in the long run. Financial Analysts Journal 51, 18-22. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31 , 307-327. Browne, S., 1995. Optimal investment policies for a finn with a random risk process: exponential utility and minimizing the probability of ruin. Mathematics of Operations Research 20, 937- 958. Browne, S., 1999a. Beating a moving target: optimal portfolio strategies for outperforming a stochastic benchmark. Finance and Stochastics 3, 275-294. Browne, S., 1999b. The risk and rewards of minimizing shortfall probability. Journal of Portfolio Management 25, 76-85. Bucklew, J.A., 1990. Large Deviation Techniques in Decision, Simulation, and Estimation. Wiley, New York. Cover, T.M., Thomas, l.A., 1991. Elements of Information Theory. Wiley, New York. Detemple, J., Zapatero, F., 1991. Asset prices in an exchange economy with habit formation. Econometrica 59, 1633-1657. Fama, E., Miller, M., 1972. The Theory of Finance. Holt, Rhinehart and Whinston, New York. --- ## Page 669 640 M Stutzer 386 M. Stutzer!Journal of Econometrics 116 (2003) 365-386 Gerber, H., Shiu, E., 1994. Option pricing by Esscher Transfonns. Transactions of the Society of Actuaries 46, 99-140. Hamilton, J.D., 1994. Time Series Analysis. Princeton University Press, Princeton, NJ. Hakansson, N.H., Ziemba, W.T., 1995. Capital growth theory. In: Jarrow, R.A., Maksimovic, V., Ziemba, W.T. (Eds.), Handbooks in Operations Research and Management Science: Finance, Vol. 9. North-Holland, Amsterdam. Harvey, C., Siddique, A., 2000. Conditional skewness in asset pricing tests. Journal of Finance 40, 1263-1293. Hull, J., 1993. Options, Futures, and Other Derivative Securties. Prentice-Hall, Englewood Cliffs, NJ. Kitamura, Y., Stutzer, M., 1997. An infonnation-theoretic alternative to generalized method of moments estimation. Econometrica 65, 861-874. Kitamura, Y., Stutzer, M., 2002. Connections between entropic and linear projections in asset pricing estimation. Journal of Econometrics 107, 159-174. Kocherlakota, N.R., 1996. The equity premium: Its still a puzzle. Journal of Economic Literature 34, 42-71. Kraus, A., Litzenberger, R.H., 1976. Skewness preference and the valuation of risk assets. Journal of Finance 31, 1085- 1100. Kroll, Y., Levy, H., Markowitz, H., 1984. Mean-variance versus direct utility maximization. Journal of Finance 39, 47-61. MacLean, L.e., Ziemba, W.T., Blazenko, G., 1992. Growth versus security in dynamic investment analysis. Management Science 38, 1562-1585. Samuelson, P.A., 1969. Lifetime portfolio selection by dynamic programming. Review of Economics and Statistics 51, 239-246. Samuelson, P.A., 1994. The long-tenn case for equities. Journal of Portfolio Management 21, 15-24. Stutzer, M., 2000. A portfolio perfonnance index. Financial Analysts Journal 56, 52-61. Tauchen, G., 2001. Notes on financial economics. Journal of Econometrics 100, 57-64. Thorp, E.O., 1975. Portfolio choice and the Kelly criterion. In: Ziemba, W.T., Vickson, R.G. (Eds.), Stochastic Optimization Models in Finance. Academic Press, New York. --- ## Page 670 44 On Growth-Optimality vs. Security Against Underperformance Michael Stutzer Professor of Finance and Director, Burridge Center for Securities Analysis and Valuation, University of Colorado, Boulder, CO michael.stutzer@colorado. edu Abstract The expected log utility of wealth (i.e., the growth-optimal or Kelly) criterion has been oft-studied in the management science literature. It leads to the highest asymptotic growth rate of wealth, and has no adjustable "preference parameters" that would otherwise need to be precisely "adjusted" to a specific individual's needs. But risk-control concerns led to alternative criteria that stress security against under performance over finite horizons. Large deviations theory enables a straightforward generalization of log utility's asymptotic analysis that incorpo- rates these security concerns. The result is a power utility criterion that (like log utility) is free of an adjustable risk aversion parameter, because the latter is endogenously determined by expected utility maximization itself! A Bayesian formulation of the Occam's Razor Principle is used to illustrate the unavoidable reduction of scientific testability (i.e., the ability to more easily falsify) inherent in criterion functions that introduce additional adjustable parameters that are not directly observable. 1 A Simple Repeated Betting Problem 641 All concepts and results are illustrated using the simplest repeated betting problem analyzed in the management science literature. This is the popular "Blackjack" example, discussed by Thorp (1984) , MacLean, Ziemba, and Blazenko (1992), and MacLean and Ziemba (1999). The agent must choose a fraction p of wealth to bet on an IID Bernoulli process that has a gross return per bet Rp = 1 + P with probability Jr > 1/ 2, and Rp = 1 - p with probability 1 - Jr. In other words, the bettor either wins or loses the fraction p of accumulated wealth each try. For example, if the bettor's initial wealth is Wo = $500 and chooses p = 5%, the bettor will either win or lose $25 on the first try. If the bettor wins, the second try will either win or lose $525 * .05 = $26.25. But if the better lost the first time, the second try will either win or lose 475 * .05 = $23.75. This is isomorphic to a binomially distributed stock portfolio initially worth $500, and that returns ±5% in each subsequent time period. --- ## Page 671 642 M Stutzer The accumulated wealth after T bets is WT = Wo rr;=l Rpt. The bettor is interested in the security against underperforming some benchmark wealth path WbT = Wo rr;=l Rbt· Taking the log of both establishes that: while taking the exponentials and rearranging shows: 2:[- 1 log Rpt T WT = Woe T (2) By laws of large numbers valid for this binomial and more realistic processes, Lf-l~OgRpt T~ E[logRp]. So (2) shows that the expected growth rate E[logRp] resulting from a bet p is realized only asymptotically. Hence arg maxp E[log Rp] is the bet yielding the highest asymptotic growth rate of wealth. For illustrative purposes only, let us assume that the bettor's "edge" is 60 - 40 = 20%. It is easy to verify that betting arg maxp E[log Rp] = .20 yields the maximum asymptotic growth rate E[log R.20 ] = 0.60 log 1.20 + 0.40 log 0.80 ::::; 0.0201. As Hakansson and Ziemba (1995) noted, this maximum expected log, a.k.a. growth-optimal or "Kelly", criterion" almost surely leads to more capital in the long run than any other investment policy which does not converge to it" , and that "the growth-optimal strategy also has the property that it asymptotically minimizes the expected time to reach a given level of capital". Other interesting characterizations of the criterion are found in Bell and Cover (1988) as well as in the textbook of Cover and Thomas (1991). But some other researchers, e.g., Aucamp (1993), argued that in practice it is quite possible that the asymptotic outperformance won't be realized with high probability until after a very, very long time - perhaps hundreds of years. Over finite horizons, MacLean, Ziemba and Blazenko (1992) note that "the Kelly strategy never risks ruin, but in general it entails a considerable risk of losing a substantial portion of wealth", and define three measures of security against underperforming a benchmark. Figure 1 graphs the probability (1) of underperforming a benchmark wealth path growing at a constant target growth rate denoted log r = 1% per bet , i.e., substitute logRbt == 1% for all t in (1). Money growing at a constant rate of log r = 1 % will double every log 2/.01 ::::; 70 bets. Figure 1 shows that the probability of underperformance achieved by each of the three betting fractions p approaches zero as T ----* 00, as the law of large numbers guarantees (because E[log Rp] > 1% for all three values of p). But Figure 1 shows that the convergence to zero is quite slow, and that betting either too high a fraction of wealth (e.g., the growth-optimal p = 20% of wealth) or too low a fraction (e.g., p = 8%) results in lower security against underperformance than betting p = 14.1 % of wealth per bet. This following section will show how that fraction is found. --- ## Page 672 On Growth-Optimality vs. Security against Underperformance 45 ~------------------------------------~ 40 - ._ 35 C 30 ~ 25 - :c (II 20 .c e 15 c.. 10 5 -- ----.- O ~~~~~~~~~~~~~~~~~~~ 100 200 300 400 500 600 700 800 900 1000 T= Number of Bets ----p=8% - ---- -p = 20% - p=14.1% Figure 1 Probability of underperforming Logr = 1% benchmark. 2 Asymptotics of Benchmark Underperformance Probabilities 643 The underperformance probabilities (1) graphed in Figure 1 decay to zero exponen- tially as T -+ 00. Our emphasis on the security after suitably large T differs from the security notions in MacLean, Ziemba, and Blazenko's (1992) , who define security at either a fixed T (their definition 81) or at all T both small and large (their definition 82). But Figure 1 also suggests that the betting fraction with the relatively best large-T security will also have relatively good security when T is smaller, which is usually the case in other problems studied by the author. For each betting fraction p, large deviations theory can be used to calculate the exponential decay rate of the underperformance probability curve associated with it. The optimal betting fraction is the one associated with the most rapidly decaying curve. This criterion focuses attention on the asymptotics of the realized growth rate (i.e., Lt log Rpt/T) relative to a benchmark's realized growth rate (Lt log Rbt/T) , rather than just the asymp- to tics of the realized growth rate itself (i.e., E[log Rp). When the benchmark is the growth-optimal policy (i.e., argmaxpE[logRp]), the two criteria coincide. Hence the criterion exposited herein is a natural generalization of the asymptotic reasoning used by growth-optimal policy advocates, that additionally incorporates the objective fear of underperforming a benchmark (fixed or stochastic) over finite horizons. To find the optimal policy in our example, one must first determine the bet- ting fractions p that make those probabilities decay to zero asymptotically (i.e., E[log Rp]- log r > 0), like the three fractions in Figure 1 do when log r = 1 %. This restriction defines both minimum (denoted p) and maximum (denoted p) acceptable betting fractions. In our example, E ~ 5.9% < p < f5 ~ 33.6%. Within this range of p, it is a consequence of the Gartner-Ellis Large Deviation Theorem (e.g., see Bucklew (1990)) that under fairly general conditions, the under- --- ## Page 673 644 M Stutzer performance probabilities approach zero at an asymptotic exponential rate of decay denoted D p , computed below: (3) But there are more insightful ways of restating this criterion. It turns out that when E. < P < 15, the maximizing e < 0 in (3) (Stutzer, 2003). So without loss of generality we can replace e in (3) by "( == -e > 0, yielding the equivalent formulation (4) Using the right hand side of (1) to calculate the summed exponent in (4), a bit of algebra (Stutzer, 2003) yields the following restatements of the optimal choice problem: Popt 1 [(WT)-"I] arg max Dp == arg max max lim - - log E -- p p "1>0 T->oo T WbT (5) IID arg max max - log E [( Rp) -"I] p "1>0 r (6) arg max max E [_ (Rp) -"I] p "1>0 r (7) Blackjack [ ( 1 + p) -"I (1 -p) -"I] == argm:xr;:;tf- 0.60 -r- +0.40 -r- (8) where WbT == Wo rr;=l Rbt = WorT when the benchmark is just a constant target growth rate. Fixing the choice vector or scalar (in our example) denoted p, the inner maxi- mization over "( in (5) is the most general expression for the asymptotic decay rate of its under performance probability curve (see Figure 1), valid in both lID or non- lID situations with either constant or stochastic benchmark log returns. The inner maximization over "( in (6) is the decay rate in lID situations when the benchmark is a constant target growth rate. There are two major differences between (4) or (5) or the lID special case (7), and conventional maximization of the following expected power utility with constant relative risk aversion 1 + T (9) First, the argument in our criterion (7) is the ratio of the bet's return to the bench- mark return, rather than just the former in (9). The second major difference is that "( is not adjustable by the analyst to "fit" a bettor's supposedly fixed, exogenous degree of relative risk aversion (1 + "(). Instead, for a fixed p, "( is determined by the inner maximization, and thus varies across the values of p considered. The --- ## Page 674 On Growth-Optimality vs. Security against Underperformance 645 endogenous degree of risk aversion exhibited by the agent is 1 +,(Popt), which arises from the subsequent outer maximization over P that determines the agent's optimal choice Popt. This will depend on both the opportunity set of alternatives evaluated by the agent, and the constant target growth rate (or stochastic benchmark) that the agent wants to beat. In other words, the bettor jointly maximizes expected power utility by jointly varying both the feasible betting fraction and the utility's curvature or risk aversion "coefficient" . This agent's rank ordering of each betting fraction in the range of P whose underperformance probability curves decay to zero (i.e., 5.9% = p.. < P < p = 33.6%) is determined by numerical solution of (8). A summary of the optimization is seen in Table 1 below: Table 1 The best security against underperforming a log r = 1 % target growth rate of wealth per bet is achieved by betting Popt = 14.1 %, resulting in the fastest decay rate (0.18% per bet) of underperformance probabilities. Bettor With log r = 1.0% Per Bet Growth Target p% Value of (8) Dp % from (6) E[logRp] % Risk A version 1+, 5.9 ;::0 -1 ;::00 ;::0 1.0 ;::01 8.0 -.9994 0.06 1.28 1.45 14.1 -.9982 0 .18 1.83 1.43 20.0 -.9987 0.13 2.01 1.25 33.6 ;::0 -1 ;::00 ;::0 1.0 ;::01 Table 1 shows that the highest attainable value of the decay rate is found by solving (8), producing Popt = 14.1% and ,(14.1%) = .43. The asymptotic growth rate of wealth associated with the optimal bet is E[log R 14.1%] = 1.83%. This is higher than the agent's target growth rate of log r = 1 %, as it must be in order to maximize the probability of outperforming that target (equivalently, to minimize the probability of underperforming it) over finite horizons T, as depicted in Figure 1, and is of course lower than the 2.01% asymptotic growth rate associated with the growth-optimal P = 20%. The bettor exhibits an endogenous degree of relative risk aversion 1 +,(14.1%) = 1.43, but uses different values of, to evaluate the expected utility of different P values, listed in the last column of Table 1. The optimization problem can be reformulated as a constrained entropy prob- lem of the sort analyzed in the management science literature, e.g., Dinkel and Kochenberger (1979). Let D(Q I 7r) == EQ [lOg~ ] denote the familiar Kullback- Leibler discrimination statistic (a.k.a. relative entropy, see Cover and Thomas, 1991) measuring the difference between a variable probability measure Q and the fixed measure 7r. Here, EQ is used to denote the expectation taken with re- spect to the measure Q. There is a close connection between entropy and the large deviations rate function, so it is not surprising that the optimal betting problem can be reformulated as a constrained entropy problem. That connec- tion (e.g., Kitamura and Stutzer, 2002) shows that the optimal solution to the --- ## Page 675 646 M Stutzer Table 2 A bettor with a lower target growth rate of wealth will use different p-dependent coefficients of risk aversion to evaluate the same bets, will choose to bet less, and will exhibit a higher degree of endogenous risk aversion. Bettor With logr = 0.1% Per Bet Growth Target p% Value of (9) Dp % from (6) E[logRp] % Risk A version 1+ 1 0.507 ~ -1 ~O ~ 0.1 ~1 1.0 -.9954 0.46 0.20 10.73 4.5 -.9877 1.23 0 .79 4.53 14.1 -.9924 0.77 1.83 1.88 20.0 -.9954 0.46 2.01 1.48 33.0 -.9996 0.04 1.09 1.09 38.5 ~ - 1 ~O ~ 0.1 ~1 betting problem can also be computed by solving the constrained entropy problem: maxpminQ D(Q I 'if) s.t. EQ[logRp] <::::: logr. In the above example, this problem reduces to finding scalar values for 0 <::::: Q <::::: 1 and 0 <::::: P <::::: 1 solving max min Q log( Q / 0.60) + (1 - Q) log( (1 - Q) /0.40) p Q s.t. Q 10g(1 + p) + (1 - Q) 10g(1 - p) <::::: 0.01 It is easy to verify that a numerical solution to this problem is Popt = 14.1% and Q = 57%, so that the relative entropy D p op , = 0.18% at the saddlepoint. Fur- thermore, use of the notation D p o p , for both the optimal relative entropy and the underperformance probability decay rate is not an accident: The bold faced row in Table 1 shows that the latter (0.18% per bet) is the same as the former. The following Table 2 summarizes the decision process for a more conservative agent, who would be satisfied by beating a lower, and hence more easily beaten target growth rate of logr = 0.1% (i.e., wealth doubles only every 700 bets or so), over the (now slightly different) range of bets 0.507% = E < P < p = 38.5% that yield underperformance probabilities decaying to zero: 45 -',---------------------------------, 40 - 35 - ~ 30 - \ \ 25 - \ \ 20 - \ 15 - 10 - ~ .,., 5 - ~ "------------ ----- . o +--,--~~~~--~==r==T--~--r_~ T = Number of Bets - · _ ·p= 1% - p= 20% - p=4.5% Figure 2 Probability of underperforming Log r = 0.1 % benchmark. --- ## Page 676 On Growth-Optimality vs. Security against Underperformance 647 Table 2 shows that this agent should bet only Popt = 4.5% of wealth per bet to maximize (minimize) the probability of outperforming (underperforming) the more conservative (and hence more easily beaten) growth target logr = 0.1% per bet. Comparing D pop, in the bold faced row of Tables 2 and 1 shows that the lower target enables more rapid decay of the optimal bet's underperformance probability curve, seen by comparing the lowest curves in Figures 2 and 1. Table 2 shows that this bettor exhibits a higher (but still plausible) endogenous degree of relative risk aversion equal to 4.53. The following Table 3 shows the optimal betting fractions and degrees of relative risk aversion (solving (8)) exhibited by bettors satisfied with beating lower target growth rates than the growth-optimal 2.01 %. Finally, it is easy to show that the relationship between the first and third columns in Table 3 is convex, as it will be in other betting or investment problems. This relationship shows the tradeoff between an index of desired growth (i.e., the target growth rate) and an index of security against underperformance (i.e., the decay rate). This efficiency frontier will shift out when the betting opportunity set is more favorable (i.e., a higher probability of winning 7r), as depicted in Figure 3. Table 3 A Bettor who adopts a lower (than growth-optimal) target logr < 2.01% to attain more security against underperforming it (i.e. , a higher decay rate Dp > 0) should choose a smaller betting fraction Popt < 20% than growth-optimal in- vestors do, and will exhibit a higher endogenous degree of relative risk aversion 1 + 'Y(Popt) > 1 than growth-optimal investors do. Bettors With Other Growth Targets logr% Popt% Dpopt(logr)% E[logRpop,l% Risk A version 1 + 'Y(Popt) 2.01 20.0 0 2.01 1.0 14.1 0.18 1.83 0 .5 10.0 0.51 1.50 0.1 4.5 1.23 0.79 4 -r-------------------------~ ~ o Q. 3.5 - 3 -; 2.5 - ~ 2 - » ~ 1.5 Q) o 0.5 - 0 +------r--===-r-----,-~~_1 o 2 3 4 Benchmark Growth Rate Log r % Figure 3 Efficiency frontiers. 1 1.43 2.02 4.53 - Prob[Win)=60% - Prob[Win] = 70% --- ## Page 677 648 M Stutzer 3 A Brief Discussion of Closely Related Literature There is a management science literature advocating that the growth-optimal policy be combined with a riskless asset to strike an agent's desired tradeoff of growth for security against underperformance. Such fractional Kelly strategies have been studied by MacLean, Ziemba, and Blazenko (1992), MacLean and Ziemba (1999), Li (1993), and others. The criterion embodied in (5) permits the policy p to be any adapted process. For example, in portfolio choice over N-assets, the policy p would be a (generally time-varying) N-vector of portfolio weights varying over time. The only restriction on p is that the underperformance probabilities decay to zero as T ----> 00, in accord with the motive to minimize such probabilities. There is (as yet) no basis for restricting the strategy space to fractional Kelly strategies. Samuelson (1974, 1979) challenged the accuracy and/or relevance of some of the outperformance probability-based asymptotic reasoning advanced by those who advocated the use of log utility, i.e., the limiting case of exogenous 'Y lOin (9). But an analysis in Stutzer (2004) showed that Samuelson's critiques do not apply to the criterion developed in the previous section. Furthermore, an important risk-scaling paradox highlighted by Rabin (2000) applies to the expected concave utility function maximization (like (9)) that Samuelson preferred to outperformance probability- based criteria. The Blackjack problem can be used to illustrate the nature of Rabin's critique. The optimal betting fraction when conventionally maximizing the expected power utility E[WT] = E[-Wi'Y] with exogenous 'Y does not depend on the number of bets T. Suppose you were offered the opportunity to bet p = 10.8% of your initial wealth T = 10,000 times, the end result of which is determined in a split second by computer simulation. The same calculations used to produce Figures 1 and 2 (see footnote 3) show that after betting p = 10.8% of accumulated wealth T = 10,000 times, there is almost no chance that you could underperform a log r = 0.6% growth rate per bet. That is, there is near-certainty that your wealth would be multiplied by a factor of at least e· 006*lOOOO ~ $1026 in the split second it takes for the computer-determined outcome. Yet conventional use of expected constant relative risk aversion utility (9) - arguably the most commonly made behavioral assumption in the economics literature - implies that a bettor with an exogenous degree of risk aversion greater than just 1 + 'Y = 3.76 would rather not play at all than have to bet p = 10.8%, no matter the size of T nor the initial wealth W o, even though it is always impossible to lose more than Wo in the Blackjack problem. Nonetheless maintaining the assumption that each individual acts as-if he/she does have an exogenous constant degree of relative risk aversion, Barsky et al. (1997) designed a questionnaire to measure it, which was given to thousands of individuals in person by Federal interviewers. About 2/3 of those surveyed had a degree of relative risk aversion higher than 3.76, so their maintained assumption implies that all of them would have refused to bet p = 10.8% once, twice, ten thousand, or ten million times. Would you? Fortunately, a bettor using the criterion in Section 2, --- ## Page 678 On Growth-Optimality vs. Security against Underperformance 649 and who has the relatively modest growth target used above (i.e., log r = 0.6%), would indeed be willing to bet this much. Most importantly, Rabin (2000) also showed that analogous expected utility paradoxes are construct able when concave utility functions other than (9) are as- sumed. His concerns about expected concave utility of wealth criteria are not easily dismissed. Finally, there is of course a large literature proposing other ways to integrate shortfall probability into portfolio analysis, e.g., Browne (2000) and the other papers cited by him, or maximization of non-log expected utility subject to a "Value- At-Risk" constraint (2001). A full exploration of the connection between these alternatives and the criterion in Section 2 is too lengthy to be attempted in this note on the growth-optimal literature. 4 Occam's Razor Critique of Alternative Decision Criteria One long-held disideratum was expressed by Albert Einstein: "Everything should be made as simple as possible, but not simpler". This is the principle of parsimonious parameterization that has historically been termed Occam's (or Ockham 's) Razor. There are other ways to generalize the standard CRRA (9) and its limiting log utility criteria. But they have more adjustable parameters than the criterion described in Section 2. Each of the alternatives introduced additional adjustable parameters when modifying a conventional CRRA power utility (such as log utility), gaining flexibility that is helpful when confronting individual behavioral observations. But by impeding the ability to make as sharp a prediction about what could potentially be observed, the flexibility enabled by the additional adjustable parameters comes at a cost. This tradeoff is quantified in the Bayesian analysis of Jefferys and Berger (1992), summarized as follows: Bayesian analysis can shed new light on what the notion of the "simplest" hypothesis consistent with the data actually means ... By choosing prior probabilities of hypotheses, one can quantify the scientific judgment that simpler hypotheses are more likely to be correct. Bayesian analysis also shows that a hypothesis with fewer adjustable parameters automatically has an enhanced posterior probability, because the predictions it makes are sharp. To formalize this, suppose we are trying to assess the plausibility (a.k.a. subjec- tive probability, as defined and interpreted by Bayesians) of one hypothesized deci- sion criterion Ho with fewer adjustable parameters, e.g. , that described in Section 2, relative to another criterion HA. A Bayesian assesses prior probabilities J.L(Ho) and J.L(HA) , finds the likelihoods L of each potential individual decision under the two hypotheses, and finally calculates the following ratio of posterior probabilities (i.e., --- ## Page 679 650 M Stutzer the posterior odds in favor of Ho): Prob[Hol = L(Dabs I Ho) M(Ho) D;;l B * Prior Odds Prob[HAl L(Dabs I HA) M(HA) (10) where Dabs is the individual decision that is actually observed and B is the Bayes factor, i.e., the ratio of the actual observation's likelihood under each hypothesis. Because Ho has fewer adjustable parameters, it is possible that the likelihood of a large subset of potential observations will be smaller under Ho than under HA . But if so, the likelihood of the complementary (possibly smaller) set of potential observations will be higher under Ho . If the actually observed individual decision Dabs lies in this complementary set, the Bayes factor in (10) will favor Ho, even though its general fitting ability is lower than an alternative with more adjustable parameters. Unless there is some reason to place higher prior probability on the more profligately parameterized alternative HA, the posterior odds ratio (10) will thus favor the "simpler", sharper hypothesis Ho over the more flexible HA. The crucial claim that the Ho developed in section 2 makes a sharper prediction than the conventional (without benchmark) CRRA hypothesis HA hinges on the ability to more accurately determine an agent's benchmark than direct measure- ments can accurately determine the hypothesized exogenous curvature parameter 1 (e.g., the aforementioned problematic questionnaire used by Barsky, et al. (1997)). It is fair to question this ability in the context of our betting example, where the analyst would have had to more accurately determine a value for the bettor's target growth rate log r than a value for 1, before observing the fraction of wealth bet. But the ability to more accurately determine the benchmark is easy to envision in the more important context of investment behavior. A typical fund manager, and an investor who uses the manager, expressly try to outperform an identified benchmark portfolio tailored to the fund's style. For example, a recent advertise- ment by Standard and Poors cited independently gathered data indicating that $3.5 trillion of funds are professionally invested in attempts to beat the S&P 500 index benchmark. Hence when considering fund managers (and their voluntary clients), the benchmark required for the criterion in Section 2 is a directly observable port- folio (e.g., the S&P 500), in which case the criterion in Section 2 has no adjustable parameters, and hence makes the sharpest conceivable predictions. Notes 1 Ergodic processes that are not IID still have limiting infinite time averages, that would need to be denoted differently than E [log Rp]. 2 MacLean, Ziemba, and Blazenko (1992) assumed t hat a value of 7r = 51% might be achievable by card counting techniques. 3For each value of T and p, t he event of underperformance will occur when t here are x or fewer wins, where [x log( l +p)+(T-x) log(l -p)J/T :::; 0.01. Hence the probability of this event is computed by evaluating the cumulative binomial distribution (with probability of winning equal to 7r = 0.60) at the highest integer x satisfying this inequality. --- ## Page 680 On Growth-Optimality vs. Security against Underperformance 651 41n brief, define the partial sum Y T == I:,;=llog R pt - I:,;=l log Rbt. Define the time- averaged log moment generating function rPT((J) == ~ log E[eOYTj. Bucklew (1990) lists the following two implicit restrictions on the process needed to ensure that (3) is the aforementioned decay rate: (i) limT-+= rPT((J) exists for all (J - but where 00 is allowed both as a limit value as well as member of the sequence and (ii) this limit function is differentiable at all values of (J for which it is finite. As shown in Bucklew (1990) , these restrictions hold for a wide variety of both independent and dependent stochastic processes used in practice to model YT , including our example's IID Bernoulli return process with return Rp and constant benchmark return log r. Stutzer (1995) used a vector version of the result to characterize an asset pricing model error diagnostic. I recently discovered that Sornette (1998) applied the IID version (Cramer's Theorem) in several portfolio choice settings. With specific parametric processes, Pham (2003) found solutions for related continous time, dynamic portfolio choice problems. Dembo, Deuschel, and Duffie (2004) also used Cramer's Theorem to approximate loss probabilities for suitably large numbers of loans. 5Probably the earliest hint of a result like this is in Ferguson (1965). He studied a class of betting problems, and noted that if "the probability of ruin goes to zero exponentially as the fortune tends to infinity" , maximization of expected exponential utility would be "approximately valid" for some value of its (J coefficient "when the fortune is large". A precise finite T result of this nature was developed by Browne (1995) , for a problem where wealth is generated by a specific parametric controlled stochastic differential equation. The more general, nonparametric results (5)- (7) were enabled by formulating shortfall probabilities using (1), and then applying the powerful Gartner-Ellis Theorem to simply compute the asymptotics. 6Perhaps the first use of constrained entropy in finance was Cozzolino and Zahner (1973). 7The constraint is binding at the solution, so (in this scalar P example!) t he constraint equation can be used to write Q as a function of P in closed form. Substitute this for Q in D(Q 17r) and numerically maximize over P to find Popt = 14.1%. BSee Stutzer (2004) for a proof applicable to this and more general situations. 9For example, consider the following "habit formation" modification of the CRRA utility (9): divide Rp by t he exogenous parameter r as in (7) , but leave 'Y as an additional exogenous parameter. lOThe likelihood functions must be calculated before the individual's decision Pobs) is observed. This is to avoid the following ersatz calculation: observe an individual's Pobs, then "fit" the adjustable parameters in HA by working backwards to find adjustable parameter values that exactly "explain" Pobs, and then assert that L(Pobs 1 HA) = 1 in (1O)! This procedure would make it impossible to reject this or any other hypothesis that is flexible enough to generate all possible data that could be observed (and hence any particular Pobs . According to the widely accepted view of Popper (1977), an hypotheses that can't be rejected (in his terms, falsified) is not a scientific hypothesis. References Donald, C. A. (1993). On the extensive number of plays to achieve superior performance with the geometric mean strategy. Management Science, 39(9) , 1163- 1172. Barsky, R. B., F. T. Juster, M. S. Kimball, and M. D. Shapiro (1997). Preference pa- rameters and behavioral heterogeneity: An experimental approach in the health and retirement study. Quarterly Journal of Economics, 112(2), 537-579. --- ## Page 681 652 M Stutzer Suleiman, B. and A. Shapiro (2001) . Value-at-risk based risk management: Optimal poli- cies and asset prices. Review of Financial Studies, 14(2), 371- 405. Robert, M. B. and T. M. Cover (1988) . Game-theoretic optimal portfolios. Management Science, 34(6), 724-733. Browne, S. (1995) . Optimal investment policies for a firm with a random risk process: Ex- ponential utility and minimizing the probability of ruin. Mathematics of Operations Research, 20, 937- 958. Browne, S. (2000) . Risk-constrained dynamic active portfolio management. Management Science, 46(9) , 1188-1199. Bucklew, J. A. (1990) . Large Deviation Techniques in Decision, Simulation, and Estima- tion. New York: Wiley. Cover, T. M. and J. A. Thomas (1991) . Elements of Information Theory. New York: Wiley. Cozzolino, J. M. and M. J. Zahner (1973) . The maximum entropy distribution of the future market price of a stock. Operations Research, 23, 1200- 1211. Dembo, A., J.-D. Deuschel, and D. Duffie (2004) . Large portfolio losses. Finance and Stochastics, 8(1) , 3-16. Dinkel, J. J . and G. A. Kochenberger (1979) . Constrained entropy models: Solvability and sensitivity. Management Science, 25(6) , 555- 564. Ferguson, T. S. (1965). Betting systems which minimize the probability of ruin. SIAM Journal, 13(3) , 795- 818. Hakansson, N. H. and W. T. Ziemba (1995). Capital growth theory. In R. A. J arrow , V. Maksimovic, and W. T. Ziemba, editors, Handbooks in Operations Research and Management Science: Finance, Volume 9, pp. 65- 86. Amsterdam: North Holland. Jefferys, W. H. and J. O. Berger (1992). An application of robust Bayesian analysis to hypothesis testing and Occam's razor. Journal of the Italian Statistical Society, 1, 17- 32. Kitamura, Y. and M. Stutzer (2002). Connections between entropic and linear projections in asset pricing estimation. Journal of Econometrics, 107, 159- 174. Li, Y. (1993) . Growth-security investment strategy for long and short runs. Management Science, 39, 915- 924. MacLean, L. C. and W. T. Ziemba (1999). Growth versus security: Tradeoffs in dynamic investment analysis. Annals of Operations Research, 85, 193- 225. MacLean, L. C., W . T . Ziemba, and G. Blazenko (1992) . Growth versus security in dynamic investment analysis. Management Science, 38, 1562- 1585. Merton, R. and P. Samuelson (1974). Fallacy of the log-normal approximation to optimal portfolio decision-making over many periods. Journal of Financial Economics, 1(1) , 67- 94. Pham, H. (2003). A large deviations approach to optimal long term investment. Finance and Stochastics, 7, 169- 195. Popper, K. (1977). The Logic of Scientific Discovery. US: Routledge (14th Printing). Rabin, M. (2000) . Risk aversion and expected-utility theory: A calibration theorem. Econometrica, 68(5), 1281- 1292. Samuelson, P. A. (1979). Why we should not make mean log of wealth big though years to act are long. Journal of Banking and Finance, 3, 305- 307. Sornette, D. (1998). Large deviations and portfolio optimization. Physica A, 256, 251- 283. Stutzer, M. (1995). A Bayesian approach to diagnosis of asset pricing models. Journal of Econometrics, 68, 367- 397. Stutzer, M. (2003) . Portfolio choice with endogenous utility: A large deviations approach. Journal of Econometrics, 116, 365- 386. --- ## Page 682 On Growth-Optimality vs. Security against Underperformance 653 Stutzer, M. (2004). Asset allocation without unobservable parameters. Financial Analysts Journal, 60(5), 38- 5l. Thorp, E. O. (1984). The Mathematics of Gambling. New Jersey: Lyle Stuart. --- ## Page 683 This page is intentionally left blank --- ## Page 684 Part VI Evidence of the use of Kelly type strategies by the great investors and others --- ## Page 685 This page is intentionally left blank --- ## Page 686 45 Introduction to the Evidence of the Use of Kelly Type Strategies by the Great Investors and Others 657 We know that in the long run, full Kelly strategies dominate other strategies, but they are very risky short term. Hence, practical applications to long sequences of wagers are especially appropriate. Current hedge fund trading that enters and exits in a few seconds is such an application. Thorp (1960) coined the term Fortunes's Formula in his application of Kelly to the game of blackjack using his card counting system. The count provides an estimate of the mean so that with more favorable counts the player should wager more. Typically, blackjack players can play about 60+ hands per hour. So this application works well for full Kelly. But the risk is high that even after a few hours, with say a 1% edge, a player may be behind. Hence, blackjack teams typically use fractional Kelly strategies, with fractions of 0.2 to 0.8 being common. Gottlieb (1984, 1985) describes the early use of these fractional Kelly strategies. It is no coincidence that many of the greatest hedge fund and portfolio man- agers are former blackjack players. Billionaires like trend follower Harold McPike, bond guru Bill Gross, options trader Blair Hull, and racetrack guru Bill Benter are some examples. Other top investors like Renaissance Medallion hedge fund trader Jim Simons and Warren Buffett, the legendary greatest investor in the world, use strategies that employ Kelly-type ideas to attain long term wealth growth. So did the great economist John Maynard Keynes in managing the King's College Cam- bridge endowment. As we have seen the size of Kelly wagers is largely dependent upon the probability of losing and it is important not to overbet. Since, the effect of errors in means are risk aversion dependent, these wagers for Kelly bettors are very sensitive to good mean estimates. Otherwise, it is easy to overbet. The book Fortune's Formula written by William Poundstone (2005) brought the Kelly ideas to a wider audience. So now many investment newsletters such as Morningstar and The Motley Fool suggest Kelly strategies (see Fuller (2006) and Lee, 2006). Unfortunately, most of these sources do not fully understand the computation of Kelly strategies in the multivariate case and when there are portfolio management imperfections (such as slippage). As well, they often miss the subtleties and danger of these Kelly applications. In this final part of the book, we present applications of Kelly and fractional Kelly modeling. Hausch, Ziemba, and Rubinstein (HRZ) (1981) present a weak market efficiency anomaly in the horserace place and show markets and how it can be exploited and wagered on using the Kelly criterion. The idea is to use probabilities derived in a --- ## Page 687 658 L. C. MacLean, E. 0. Thorp and W T. Ziemba simple market (the win pool involving n horses) in a market that is more complex - the place pool involving n( n - 1) combinations and/or the show pool involving n(n - 1)(n - 2) combinations, Previous studies (see Hausch and Ziemba, 2008, for an up to date survey of results), indicate that the win market at racetracks is efficient except for a favorite-Iongshot bias, Favorites are underbet and longshots are overbet, This bias has been apparent for well over 50 years, although recent changes in the betting to allow rebates for large bettors and head to head wagering on a betting exchange have changed this shape to some extent; see graphs in Hausch and Ziemba (2008). This biases for win is in the opposite direction of the bias for place and show where favorites are overbet and longshots underbet so in the HZR analysis that was the probability of finishing one-two for place or one-two-three for show, the biases tend to cancel. So HZR assume that the probability that horse i wins is simply its betting fraction Wi/W, and the probability that i is first and j is second is the chance that i wins and j wins a race that does not contain i, namely (WdW ) { W j / W } 1-Wd W where Wi are the wagers to win on horse i and W = 2:: Wi. While its known that these Harville formulas are biased, the bias cancels in the HZR application, again see Hausch and Ziemba (2008) for current research on this bias. HZR compute the expected value of place and show bets and note that there frequently are favorable bets with expected values well above l.0 per dollar bet. Applying the Kelly criterion leads to a complex non-concave non-linear programming model to compute optimal wagers that include the transactions costs or slippage that comes from the effect of our investor's bets on the odds. They show that the system yields positive profits at several racetracks. To make the system operational, they solved thousands of Kelly models and regressed the optimal bets and expected values on the ratio of the place and show pool bet on horse i. Then at the track, one only needs to look at Wi and W and Pi and P or Si and S, where Pi are the wagers to place on horse i and Si are the wagers to show on horse i. A quick scan can indicate whether a bet is good or not. Hausch and Ziemba (1985) added further refinements to the basic system. These included equations for the optimal place and show bets for varying initial wealth and place or show betting pool size. These were subsequently used in a calculator that allows players to punch in the four parameters from above and compute quickly the expected value and Kelly bet. This includes the track take adjustment, and coupled entry adjustments if needed that Hausch and Ziemba worked out. They also looked into the effects of the track take and breakage (the rounding to 2.20 or 2.10 for example from 2.25 and 2.18 payoffs to the nearest dime per $2 or $1 bet , respectively. Finally, they computed how many players can be using the system at the same time before the market will become efficient. Ziemba and Hausch (1984, 1986, 1987) wrote two books that made the system popular and famous. Now twenty-five years late, the system is still in use although the conditions are rougher --- ## Page 688 Introduction to the Evidence of the Use of Kelly Type Strategies 659 in 2010 than in the 1980s because of the two factors mentioned above (rebates and betting exchanges) and the fact that there is so much cross track betting these days that about half the betting pool has not arrived when the horses begin running. But the clever punter can estimate this odds movement that occurs after the bets have to be made. Ziemba and Hausch (1994, 2008) show how to modify the place and show system developed in the HZR and HZ papers to a different betting system that is used in some other countries such as England. In the UK, there is no show betting, only place, but the number of horses that place is not necessarily two as in North America. Rather, it depends on the number of horses running in the race. There can be one, two, three or four horses that place when the number of horses running is at most four, 5- 7, 8- 15 or 16+, respectively. In the UK there are bookies offering fixed odds and the parimutuel tote as discussed here. In North America place and show bets pay back the stakes of the horses that place or show and then the net profit is divided equally among these two or three qualifiers. The the payoffs depend on the number of winning tickets on each qualifier. But in the UK, the entire pool, after the take, is divided into 1, 2, 3 or 4 equal portions which is then split into the individual payoffs. So you can see that with a large amount bet on one horse, the payoffs for a £1 bet can be as low as £1. With this minimum payoff there are arbitrages possible, see Hausch and Ziemba (1990) for the US and Jackson and Waldron (2003) for the UK and other locales. Ziemba and Hausch develop the expected value equation and Kelly criterion bet sizing. These equations yield simple graphs to apply the system in the UK. In all of these applications, the theory indictes that the calculations should be based, as Thorp has argued, on total liquid wealth not just the track wealth. Wilcox (2003ab, 2005) has argued for a related theory where decisions are made on excess wealth after liabilities are deducted. This is a fractional Kelly strategy. Grauer and Hakansson (1986) and in a number of other papers apply the Kelly modeling approach to asset-allocation problems. They use a simple asset return distribution procedure: it is assumed that the past will repeat itself. At the be- ginning of each period t, the investor chooses a portfolio on the basis of a utility function for gross returns, R, given by ~ { R r, for "y::;; 1. "y The data used are the total monthly and yearly returns including interest and dividends on stocks, long-term government bonds, long-term corporate bonds and small capitalized stocks for 1926- 1983 in the Ibbotson and Sinquefield CRSP file. They compute results for the no-leverage and leveraged cases with and without small capitalized stocks for "y = 1 (positive power), 3/4, 1/2, 1/4 and 0 (full Kelly) and negative powers - 1, -2, ... , -75. They did quarterly and yearly revisions. The simple historical probability approach gives good results. Small stocks generally add value. --- ## Page 689 660 L. C. MacLean, E. 0. Thorp and W T. Ziemba Mulvey, Bilgili and Vural (2010) apply Kelly type models to deal with vari- ous practical issues involving imperfections such as transaction costs, operational constraints and path dependencies, etc for use by investors such as defined ben- efit pension plans, Their model is modified to also provide downside protection in turbulent markets, They apply the theory to recent US stock market behavior including the 2008 crash period, They show how large endowments can modify their typical three step process: select benchmarks, revise portfolio allocation tar- gets and select active managers to deal with these issues. Fixed mix portfolio rates along with exchange traded ETFs provide volatility induced wealth growth. They show that gains are larger in upswings and losses lower in crash periods than the underlying indices. The increasing correlation of equity assets during crash periods is dealt with by developing a portfolio of low correlation investment strategies. Rudolf and Ziemba (2004) present a continuous time model for pension or insur- ance company surplus management over time. Such lifetime intertemporal portfolio investment models date to Ramsey (1928), Phelps (1962) and Samuelson (1969) in discrete time and Merton (1969) in continuous time. Rudolf and Ziemba use an extension of the Merton (1973, 1990) model that maximizes the intertemporal ex- pected utility of the surplus of assets net of liabilities using liabilities as a new vari- able. They assume that both the asset and the liability return follow Ito processes as functions of a risky state variable. The optimum occurs for investors holding four funds: the market portfolio, the hedge portfolio for the state variable, the hedge portfolio for the liabilities, and the riskless asset. This is a four fund CAPM, while Merton has a three fund CAPM, and the ordinary Sharpe-Lintner-Mossin CAPM has two funds. The hedge portfolio provides maximum correlation to the state vari- able, that is, it provides the best possible hedge against the variance of the state variable. In contrast to Merton's result in the assets only case, the liability hedge is independent of preferences and depends only on the funding ratio. With hyper- bolic risk aversion utility, which includes negative exponential, power and log, the investments in the state variable hedge portfolio are also preference independent, and with Kelly type log utility, the market portfolio investment depends only on the current funding ratio. Having surplus over time is what life and other insurance companies, pension funds and other organizations try to achieve. With both life insurance companies and pension funds, parts of the surplus are distributed to the clients usually once every year. Hence, optimizing their investment strategy is well represented by maximizing the expected lifetime utility of the surplus. A case study involves a US based investor investing in the stock and bond markets of the US, UK, Japan, the EMU, Canada and Switzerland. The investor faces an exposure to five foreign currencies euro, British pound, Japanese yen, Canadian dollar and Swiss franc, and can invest in eight funds. Five of them are hedge portfolios for the state variables, which are assumed to be currency returns --- ## Page 690 Introduction to the Evidence of the Use of Kelly Type Strategies 661 and the others by portfolio separation are the market portfolios, the riskless asset and the liability hedge portfolio. The holdings of the various funds depend only on the funding ratio and on the currency betas of the distinct markets. For a funding ratio of one, there is no investment in the market portfolio and only diminishing investments in the currency hedge portfolios. The portfolio betas against the five currencies are close to zero for all funding ratios. The higher the funding ratio is, the higher is the investment in the market portfolio, the lower are the investments in the liability and the state variable hedge portfolios, and the lower is the investment in the riskless fund. Negative currency hedge portfolios imply an increase of the currency exposure instead of a hedge against it. The increase of the market portfolio holdings and the reduction of the hedge portfolio holdings and the riskless fund for increasing funding ratios shows that the funding ratio is directly related to the ability to bear risk. Rather than risk aversion coefficients, the funding ratio provides an objective measure to quantify attitudes towards risk. Ziemba (2005) investigates the records of great investors and proposes a simple modification of the ordinary Sharpe ratio to evaluate them more fairly. In most cases these investors have mostly monthly or quarterly gains with few losses so the ordinary Sharpe ratio penalizes them for gains by increasing the portfolio standard deviation. The funds discussed are Windsor of George Neff, the Ford Foundation, the Harvard Endowment, Tiger of Julian Robertson, Quantum of George Soros and Berkshire Hathaway of Warren Buffett. The modified Sharpe ratio simply replaces the gains by the mirror image of the losses. So gains are not penalized but only losses. With the ordinary Sharpe ratio the Ford Foundation beats Berkshire Hathaway even though the geometric mean return is much less. So the question is does Berk- shire dominate with the modified DSSR downside symmetric Sharpe ratio. Only Berkshire improves with the DSSR but the Ford Foundation and the Harvard en- dowment are still slightly better. This is because Berkshire has the most large gains of all the funds studied but also has a lot of large losses on a monthly or quar- terly basis and more than the other funds. So Berkshire is oriented towards a full Kelly long term valuation where intermediate monthly, quarterly and even yearly returns are not the goal, rather long terms growth is the goal. So we believe Buffett acts like a full Kelly bettor in his investment strategies. These funds have DSSRs about 1.0. Princeton-Newport, run by Edward O. Thorp, a co-editor of this book, had a DSSR of 13.8 with a very smooth wealth path with only three monthly losses, and no quarterly or yearly losses, over the twenty year period 1969-1988. Thorp had similar good results in other funds run later. Renaissance Medallion, run by James Simon, arguably the current top hedge fund in the world, had, as reported by Ziemba and Ziemba (2007), an even higher 26.4 DSSR with very few monthly or quarterly losses. no yearly losses, and very high mean returns during the period January 1993 to April 2005. --- ## Page 691 662 L. C. MacLean, E. 0. Thorp and W T. Ziemba Thorp (2006) presents a comprehensive survey of his considerable use over almost fifty years of Kelly strategies in blackjack, sports betting and the stock market along with theoretical results he has found useful in practice. He traces the early first serious application of Kelly methods to blackjack bet sizing, to various sports betting applications and his great success as a hedge fund trader with the Princeton- Newport and later hedge funds. --- ## Page 692 Management Science, 27, 1435- 1452 (1985) 46 EFFICIENCY OF THE MARKET FOR RACETRACK BETTING* DONALD B. HAUSCH, t WILLIAM T. ZIEMBAt AND MARK RUBINSTEIN§ Many racetrack bettors have systems. Since the track is a market similar in many ways to the stock market one would expect that the basic strategies would be either fundamental or technical in nature. Fundamental strategies utilize past data available from radng forms, spedal sources, etc. to "handicap" races. The investor then wagers on one or more horses whose probability of winning exceeds that determined by the odds by an amount sufficient to overcome the track take. Technical systems require less information and only utilize current betting data. They attempt to find inefficiencies in the "market" and bet on such "overlays" when they have positive expected value. Previous studies and our data confirm that for win bets these inefficiencies, which exist for underbet favorites and overbet longshots, are not sufficiently great to result in positive profits. This paper describes a technical system for place and show betting for which it appears to be possible to make substantial positive profits and thus to demonstrate market inefficiency in a weak form sense. Estimated theoretical probabili- ties of all possible finishes are compared with the actual amounts bet to determine profitable betting situations. Since the amount bet influences the odds and theory suggests that to maximize long run growth a logarithmic utility function is appropriate the resulting model is a nonlinear program. Side calculations generally reduce the number of possible bets in anyone race to three or less hence the actual optimization is quite simple. The system was tested on data from Santa Anita and E)(hibition Park using ellact and approximate solutions (that make the system operational at tl:e track given the limited time available for placing bets) and found to produce substantial positive profits. A model is developed to demonstrate that the profits are not due to chance but rather to proper identification of market inefficiencies. (FINANCE-PORTFOLIO; GAMES- GAMBLING) 1. The Racetrack Market 663 For the most part,l academic research on security markets has bypassed an interesting and accessible market-the racetrack for thoroughbred horses-with its highly standardized form of security-the tote ticket. The racetrack shares many of the characteristics of the arch typical securities market in listed common stocks. Moreover the racetrack gains further interest from its significant differences, and more impor- tantly, because it is inherently a more elementary market context, lacking many of the dynamic features which complicate analysis of the stock market. The "market" at the track in North America convenes for about 20 minutes, during which participants place bets on any number of the six to twelve horses in the following race. In a typical race, participants can bet on each horse, either to win, place or show.2 The horses that finish the race first, second or third are said to finish • Accepted by Vijay S. Bawa, former Departmental Editor; received April 3, 1980. This paper has been with the authors 2 months for I revision. tNorthwestern University. tUniversity of British Columbia. I University of California, Berkeley. I For surveys, see Copeland and Weston [6], Fama [9J, [10] and Rubinstein [22J. 20ther bets such as the daily double (pick the winners in the first and second race), the quinella (pick the first two finishers regardless of order in a given race) and the e)(acta (pick the first two finishers in e)(act order in a given race) as well as various combinations are possible as well. Such bets are utilized by the public to construct low probability high payoff bets. For discussion of some of the implications of such bets see Rosett (21]. Efficiency of Racelrack Belting MarkelS 373 --- ## Page 693 664 D. B. Hausch, W T Ziemba and M Rubinstein 374 D. B. HAUSCH, W. T. ZIEMBA AND M. R. RUBINSTEIN "in-the-money." All participants who have bet a horse to win, realize a positive return on that bet only if the horse is first, while a place bet realizes a positive return if the horse is first or second, and a show bet realizes a positive return if the horse is first, second OT third. Regardless of the outcome, all bets have limited liability. Unlike most casino games such as roulette, but like the stock market, security prices (i.e. the "odds") are jointly determined by all the participants and a rule governing transaction costs (i.e. the track "take"). To take the simplest case, for win, all bets across all horses to win are aggregated to form the win pool. If Wi represents the total amount bet by all participants on horse i to win, then W = L:i W; is the win pool and WQ/ W; is the payoff per dollar bet on horse I to win if and only if horse i wins, where 1 - Q is the percentage transaction costs.3 If horse i does not win, the payoff per dollar bet is zero. The track is "competitive" in the sense that every dollar bet on horse i to win, regardless of the identification of the bettor, has the same payoff. The "state contin- gent" price of a dollar received if and only if horse i wins is Pi = (W;/ WQ) and the one plus riskless return is 1/L:iPi = Q, a number less than one. The interest rate at the track is thus negative and solely determined by the level of transactions costs. Thus, apart from an exogeneously set riskless rate, the participants at the track jointly determine the security prices. For example, by betting more on one horse than another, the state-contingent price of the first horse increases relative to the second, or alternatively the payoff per dollar bet decreases on the first relative to the second. The rules for division of the place and show pools are slightly more complex than the win pool. Let Pj be the amount bet on horse j to place and P == L:jPj be the place pool. Similarly Sk is the amount bet on horse k to show and the show pool is S == L:kSk' The payoff per dollar bet on horse j to place is I +[PQ- Pi - Pj ]!(2Pj ) if o { i is first and j is second or j is first and i is second otherwise. Thus if horses i and j are first and second each bettor on j (and also i) to place first receives the amount of his bet back. The remaining amount in the place pool, after the track take, is then split evenly between the place bettors on i and j. The payoff to horse j to place is independent of whether j finishes first or second, but it is dependent on which horse finishes with it. A bettor on horse j to place hopes that a longshot, not a favorite, will finish with it. The payoff per dollar bet on show is analogous I + [SQ - Si - ~ - Sk]!(3Sk ) if o { k is fi.Ts.t, seco~d or thir~ and fInIshes WIth i and) otherwise. In many ways the racetrack is like the stock market. A technical strategy based on discrepancies between the amounts bet on the same horses to win, place and show, is examined in this paper. Since a short position has a perfectly negative correlated outcome to the result of normal bet a given horse can be "shorted" by buying tickets on all the other horses in a race. The racetrack also differs from the stock market in important ways. In the stock market, an investor's profit depends not only on the initial price he pays for a security, lThe actual transactions cost is more complicated aoo is described below. --- ## Page 694 Efficiency of the Market for Racetrack Betting 665 EFFICIENCY OF TIlE MARKET FOR RACETRACK BETIING 375 but also on what some other investor is willing to pay him for it when he decides to sell. Thus his profit depends not only on how well the underlying firm does in terms of earnings over the time he holds its stock (i.e. supply uncertainty), but also on how other investors value that stock in the future (i.e. demand uncertainty). Given the initial price, both the nature and behavior of other market participants determine his profit. Thus current stock prices might depend not only on "fundamental" factor~ but also on market "psychology"-the tastes, beliefs, and endowments of other investors, etc. In contrast once all bets are placed at the track prior to a given race (i.e. initial security prices are given), the result of the race and the corresponding payoffs depend only on nature. There is no demand uncertainty at the track. 2. Previous Work on Racetrack Efficiency A market is efficient if current security prices fully reflect all available relevant information. If this is the case, experts should not be able to achieve higher than average returns with regularity. A number of investigators have demonstrated that the New York Stock Exchange and other major security markets are efficient and so-called experts in fact achieve returns when adjusted for risk that are no higher than those that would be received from random investments (see Copeland and Weston [6], Fama [9], [10] and Rubinstein [22] for discussion, terminology, and relevant refer- ences). For an exception see Downes and Dyckman [7]. Snyder [24] provided an investigation of the efficiency of the market for racetrack bets to win. The question Snyder poses is whether or not bets at different odds levels yield the same average return. The rate of return for odds group i is N~(O. + I) - N R. = I I I I Nj where Nj and Nj* are the number of horses, and the number who won, respectively, at odds OJ = ( WQ / Wi - I). A weakly-efficient market in Snyder's sense would set Ri = Q for all i, where I - Q is the percentage track take. His results as well as those of Fabricant [8], Griffith [12], [13], McClothlin [18], Seligman [23] and Weitzman [27] suggest that there are "strong and stable biases but these are not large enough to make it possible to earn a positive profit" [24; 1110]. In particular, extreme favorites tend to be under bet and longshots overbet. The combined results of several studies comparing over 30,000 races are summarized in Table 1. TABLE I (Snyder [24]) Rates of Return on Bets to Win by Grouped Odds, Take Added Back Midpoint of grouped odds Study 0.75 1.25 2.5 5.0 7.5 10.0 15.0 33.0 Fabricant 11.1' 9.0' 4.6' -1.4 - 3.3 - 3.7 - 8.1 - 39.5' Griffith 8.0 4.9 3.1 - 3.1 - 34.6' - 34.1' - 10.5 - 65.5' McGlothlin 8.0b 8.0' 8.0' - 0.8 - 4.6 _7.0b - 9.7 - 11.0 Seligman 14.0 4.0 - 1.0 1.0 - 2.0 -4.0 -7.8 - 24.2 Snyder 5.5 5.5 4.0 -1.2 3.4 2.9 2.4 - 15.8 Wei~man 9.0' 3.2 6.8" - 1.3 - 4.2 - 5.1 - 8.2b - 18.0' Combined 9.1' 6.4' 6.1' - 1.2 - 5.2" - 5.2' - 10.2' - 23.7" 'Significantly different from zero at 1% level or better. b Significantly different from zero at 5% level or better. --- ## Page 695 666 D. B. Hausch, W. T. Ziemba and M Rubinstein 376 D. B. HAUSCH. W. T. ZIEMBA AND M. R. RUBINSTEIN Since the track take averages abou t 18%. the net rate of return for any strategy which consistently bets within a single odds category is - 9% or less. For horses with odds averaging 33 the net rate of return is about - 42%. Conventional financial theory does not explain these biases because it is usual to assume that as nondiversifiable risk (e.g. variance) rises expected return rises as well. In the win pool expected return declines as risk increases. An explanation consistent with the expected utility hypothesis is that investors (as a composite) are risk lovers and behave as if the betting opportunities are limited to a single race. Weitzman [27] and Ali [I] have estimated such utility functions. Ali's estimated utility function over wealth w is the convex function u( x) = 1.91 W 1.1784 ( R 2 = .9981 ). which has increasing absolute risk aversion. Thus by the Arrow-Pratt (3], [19] theory investors will take more risk as their wealth declines. This explains the common phenomenon that bettors, when losing, tend to bet more and more on longer odds horses in a desperate attempt to recoup earlier losses. Moreover, since u is nearly linear for large w investors are nearly risk neutral at such wealth levels. A second explanation is that gamblers simply prefer low probability high prize combinations (i.e. longshots) to high probability low prize combinations. Besides the possible gains involved, gamblers have egos associated with analyzing racing forms and pitting one's predictions against others. Luck and entertainment as well are largely absent in betting favorites. The thrill is to successfully detect a moderate or long odds winner and thus confirm one's ability to outperform the other bettors. Such a scenario is consistent with the data and leads to the biases. Rosett [21] provides an analysis of ways to construct low probability high prize bets through parlays and other combina- tions that can be used to avoid the longshot tail bias problem and to take advantage of the favorite bias. In an effort to capitalize on this market many tracks feature such bets in the form of the daily double, the exacta and the quinella. However, none of these schemes appear to yield bets with positive net returns.4 Other studies of racetrack efficiency have been conducted by Ali [2], Figlewski [II] and Snyder [24]. Ali shows that the win market is efficient in the sense that indepen- dently derived bets with identical probabilities of winning do in fact have the same odds in a statistical sense. His analysis utilizes daily double bets and the corresponding parlays, i.e. bet the proceeds if the chosen horse wins the first race on the chosen horse in the second race, for 1089 races. Figlewski, using a multinomial logit probability model to measure the information content of the forecasts of professional handicap- pers and data from 189 races at Belmont in 1977, found that these forecasts do contain considerable information but the track odds generated by bettors discount almost all of it. Snyder provided strong form efficiency tests of the form : are there persons with special information that would allow them to outperform the general public? He found using data on 846 races at Arlington Park in Chicago that forecasts from three leading newspapers. the daily racing form and the official track handicapper did not lead to bets that outperformed the general public. 4 For an entertaining account of an " expert" who was able to achieve positive net returns over a full racing season. see Beyer (51. See also Vergin [261. --- ## Page 696 Efficiency of the Market for Racetrack Betting 667 EFFICIENCY OF THE MARKET FOR RACETRACK BETTING 377 3. Proposed Test The studies described in §2 examine racetrack efficiency with respect to the win pool only. In this paper our concern is with the efficiency of the place and show pools relative to the win pool and with the development of procedures to best capitalize on potential inefficiencies between these three "markets." We utilize the following defini- tion of weak-form efficiency: the market is weakly-dficient if no individual can earn positive profits using trading rules based on historical price information. In Baumol's [4; 46) words .. . "all opportunities for profit by systematic betting are eliminated". Our analysis utilizes the following two data sets: I) Data Set I: all dollar bets to win, place and show for the 627 races over 75 days involving 5895 horses running in the 1973/74 winter season at Santa Anita Racetrack in Arcadia, California, collected by Mark Rubinstein; and 2) Data Set 2: all dollar bets to win, place and show for the 1065 races over 110 days involving 9037 horses running in the 1978 summer season at Exhibition Park, Vancouver, British Columbia, collected by Donald B. Hausch and William T. Ziemba. In the analysis of the efficiency of the win pool one may compare the actual frequency of winning with the theoretical probability of winning as reflected through the odds. Similar analyses are possible for the place and show pools once an estimate of the theoretical probabilities of placing and showing for all horses is available. There is no unique way to obtain these estimates. However, very reasonable estimates obtain from the natural generalization of the following simple procedure. Suppose three horses have probabilities to win of 0.5, 0.3, and 0.2, respectively. Now if horse 2 wins, a Bayesian would naturally expect that the probabilities that horses 1 and 3 place (i.e. win second place) are 0.510.7 and 0.2/0.7, respectively. In general if q; (i = I, ... , n) is the probability horse i wins, then the probability that i is first and j is second is and the probability that i is first, j is second and k is third is q;qjqk/(1 - q;)(1 - q; - qj). (I) (2) Harville [14) gives an analysis of these formulas. Despite their apparent reasonableness they suffer from at least two flaws: I) no account is made of the possibility of the "Silky Sullivan" problem; that is, some horses generally either win or finish out-of-the-money-for example, see footnote 5 in (15); for these horses the formulas greatly over-estimate the true probability of finishing second or third; and 2) the formulas are not derivable from first principles involving individual horses running times; even independence of these random variables T 1, •• • , Tn is neither necessary nor sufficient to imply the formulas. In addition to assuming (I) and (2) we assume that n q; = W;/ 2: W;. i-I (3) That is, the win pool is efficient. The discussion above, of course, indicates that this assumption is suspect in the tails and this is discussed below. Table 2a-c compares the actual versus theoretical probability of winning, placing and showing for data set 2. --- ## Page 697 668 D. B. Hausch, W. T. Ziemba and M Rubinstein 378 D. B. HAUSCH, W. T. ZIEMBA AND M. R. RUBINSTEIN Similar tables for data set 1 appear in King [17]; see also Harville (14). The usual tail biases appear in Table 2a (although they are not significant at the 5% confidence level). One has reverse tail biases in the finishing second and third probabilities, see Tables 2b, c in [15]. This occurs because if the probability to win is overestimated (underestimated) and probabilities sum to one it is likely that the probabilities of finishing second and third would be underestimated (overestimated). Tables 2b, c, indicate how these biases tend to cancel when they are aggregated to form the theoretical probabilities and frequencies of placing and showing.s TABLE 2 Actual vs. Theoretical Probability of Winning. Placing and Showing: Exhibition Park 1978 (a) Theoretical Number Average Actual Estimated Probability of Theoretical Frequency Standard of Winning Horses Probability of Winning Error 0.000 -0.025 540 0.019 0.016 0.005 0.026 -0.050 1498 0.037 0.036 0.005 0.051 -0.100 2658 0.073 0.079 0.005 0.\01 -0.150 1772 0.123 0.126 0.008 0.151 -0.200 1199 0.172 0.156 0.010 0.201 -0.250 646 0.223 0.227 0.016 0.251 -0.300 341 0.272 0.263 0.024 0.301 -0.350 199 0.323 0.306 0.033 0.351 -0.400 101 0.373 0.415' 0.049 0.401+ ~ 0.450 ".469 0.055 9037 (b) Theoretical Number Average Actual Estimated Probability of Theoretical Frequency Standard of Placing Horses Probability of Placing Error 0.000 -0.025 21 0.022 0.000· 0.000 0.026 - 0.050 391 0.040 0.030 0.009 0.051 -0.100 1394 0.075 0.080 0.007 0.101 -0.150 1335 0.124 0.1 52" 0.010 0.151 -0.200 1295 0.174 0.174 0.011 0.201 -0.250 1057 0.223 0.243 0.013 0.251 - 0.300 871 0.274 0.304 0.016 0.301 -0.350 772 0.323 0.314 0.017 0.351 -0.400 580 0.373 0.3\3" 0.019 0.401 -0.450 420 0.424 0.395 0.024 0.451 -0.500 321 0.472 0.457 0.028 0.501 -0.550 202 0.523 0.415· 0.035 0.551 -0.600 149 0.573 0.483" 0.041 0.601 -0.650 114 0.623 0.570 0.046 0.651 -0.700 51 0.672 0.627 0.068 0.701 - 0.750 41 0.721 0.731 0.069 0.751 + 23 0.792 0.782 0.086 9037 S Since it is the accuracy of these probabilities rather than the q/s that is of crucial importance in the calculations and model below this canceling provides some justification for omitting the tail biases in (3). In practice. bets are only made on horses with expected returns considerably above I. e.g .• \.16 at Santa Anita. Modification of the qj to include these biases might change the 1.16 to 1.14. for example. See also the discussion below and in §§4 and 5. --- ## Page 698 Efficiency of the Market for Racetrack Betting 669 EFFICIENCY OF THE MARKET FOR RACETRACK BETTING 379 TABLE 2 (continued) Actual vs. Theoretical Probability of Winning. Placing and Showing: Exhibition Park 1978 (c) Theoretical Number Average Actual Estimated Probability of Theoretical Frequency Standard of Showing Horses Probability of Showing Error 0.000 -0.025 0 0.026 -0.050 55 0.043 0.036 0.025 0.051 -0.100 592 0.078 0.081 0.011 0.101 -0.150 895 0.125 0.165' 0.012 0.151 -0.200 909 0.175 0.195 0.013 0.201 -0.250 799 0.224 0.292' 0.016 0.251 -0.300 885 0.275 0.289 0.015 0.301 -0.350 794 0.324 0.346 0.017 0.351 -0.400 703 0.374 0.398 0.018 0.401 -0.450 655 0.425 0.433 0.019 0.451 -0.500 617 0.475 0.452 0.020 0.501 -0.550 542 0.524 0.477' 0.021 0.551 -0.600 396 0.573 0.477' 0.025 0.601 -0.650 375 0.624 0.599 0.025 0.651 -0.700 264 0.672 0.609' 0.030 0.701 -0.750 206 0.722 0.582' 0.034 0.751 -0.800 154 0.773 0.655' 0.038 0.801 -0.850 113 0.821 0.752 0.041 0.851 -0.900 53 0.873 0.830 0.052 0.901 + -1Q. 0.925 0.833 0.068 9037 • Categories when the theoretical probability and the actual frequency are different at the 5% significance level are denoted by ··s. The estimated standard error is (S2/ N)t where the actual frequency sample variance S2 = N(E(X 2) - (EX)2)/(N - I). Since the X; are either 0 or I. E(X2) - EX and s2 - N(EX - (EX)2)/(N - I). Formulas (1)-(3) can be used to develop procedures that yield net rates of return for place and show betting that are higher than expected (i.e.-18%) and indeed make positive profits. As a first step towards development of a "system" we present the results on the two data sets of $1 bets when the theoretical expected return is a for varying a. The expected return from a $1 bet to place on horse I iS6.7 EX r == ± ( q/qj ) [1 + .l INT {( Q (P + 1) - (1 + PI + Pj ) ) ( _1_) x 20} ] i-I I - ql 20 2 I + PI i*' .s ( ~ ) [ ...L {( Q( P + I) - (I + Pi + Pd ) ( _I) } ] +.~ I _ . I + 20 INT 2 I + P x 20 , ,-I ~ I i*' (4) 6The expressions (4) and (5) give the marginal expected return for an additional $1 bet to place or show on horse I. To obtain the average expected rates of return one simply replaces (I + PI) and (I + Sf) in these expressions by PI and St. respectively. From a practical point of view with usual track data these quantities are virtually identical. 7 A further complication, not reflected in (4) and (5) jJelow, is that a $2 winning bet must return at least $2.10. Hence in these "minus pools" involving an extreme favorite the track's take is less than 1 - Q. --- ## Page 699 670 D. B. Hausch, W T Ziemba and M Rubinstein 380 D. B. HAUSCH, W. T. ZIEMBA AND M. R. RUBINSTEIN where Q is I minus the track take, P == 2:.P; is the place pool, P; is bet on horse i to place and INT( Y) means the largest integer not exceeding Y. In (4) the quantities P and P; are the amounts bet before an additional $1 is bet; similarly with Sand S; in (5), below. The expressions involving INT take into account the fact that $2 bets return payoffs rounded down to the nearest $0.10. The two expressions represent the expected payoffs if I is first or second, respectively. Similarly the expected payoff from a $1 bet to show on horse I is [ I '{( Q(S+l)-(l+S,+Sj+Sk»)( I) }] X I + 20 INl 3 I + S, X 20 [ I {( Q(S+I)-(I+S;+S'+Sk»)( I) }] X I + 20 INT 3 . I + S, X 20 [ I {( Q ( S + I) - (I + S; + Sj + S, ) ) ( I) } ] X I + 20 INT 3 I + S, X 20 , (5) where S == LS; is the show pool, S; is bet on horse i to show and the three expressions represent the expected payoffs if I is first, second and third, respectively. Naturally one would expect that positive profits would not be obtained, given the inherent inaccuracies in assumptions (1)-(3), unless the theoretical expected return ex was significantly greater than I. However we might hope that the actual rate of return would at least increase with ex and be somewhat near ex. Table 3 indicates this is true for both data sets. The perverse behavior for high ex in the place pool is presumably a small sample phenomenon. Additional calculations along these lines appear in Harville [14]. 4. A Betting Model The results in Table 3 give a strong indication that there are significant inefficienCies in the place and show pools and that it is possible not only to achieve above average returns but to make substantial profits. In this section we develop a model indicating not only which horses should be bet but how much should be bet taking into account investor preferences and wealth levels and the effect of bet size on the odds. We consider an investor having initial wealth Wo contemplating a series of bets. It is --- ## Page 700 Efficiency of the Market for Racetrack Betting 671 EFFICIENCY OF THE MARKET FOR RACETRACK BETIING 381 TABLE 3 Results of Belling $1 to Place or Show on Horses with a Theoretical Expected Return of at Least a Exhibition Park Place Show Number of Total Net Net Rate Number of Total Net Net Rate a Bets Profit ($) of Return (%) Bets Profit ($) of Return (%) 1.04 225 5.10 2.3 612 33.20 5.4 1.08 126 - 10.10 -8.0 386 53.50 13.9 1.12 69 11. JO 16.1 223 40.80 18.3 1.16 40 5.10 12.8 143 26.30 18.4 1.20 18 5.30 29.4 95 21.70 22.8 1.25 II -2.70 -24.5 44 11.20 25.5 1.30 3 -3 - 100.0 27 10.80 .40.0 1.50 0 0 3 6 200.0 Santa Anita Place Show Number of Total Net Net Rate Number of Total Net Net Rate a Bets Profit ($) of Return (%) Bets Profit ($) of Return (%) 1.04 103 12.30 11.9 307 - 18.00 - 5.9 1.08 52 12.80 24.6 162 6.90 4.3 1.12 22 9.:W 41.8 89 3.00 3.4 1.16 7 2.30 32.9 46 12.40 27.0 1.20 3 -1.30 -43.3 27 6.20 23.0 1.25 0 0 9 6.00 66.7 1.30 0 0 5 5.10 102.0 1.50 0 0 0 0 natural to suppose that the investor would wish to maximize the long run rate of asset growth and thus employ the so-called Bernoulli capital growth model; see Ziemba and Vickson (28) for references and discussion of various assumptions and results. We use the following result: if in each time period t = 1,2, . .. there are J investment opportunities with returns per unit invested denoted by the random variables X/I' ... ,xrI ' where the X/j have finitely many distinct values and for distinct t the families are independent, then maximizing E log2:A,;x/j• s.t. 2:A,j < WI' A/; ;;. 0 maxi- mizes the asymptotic rate of asset growth. The assumptions are quite reasonable in a horseracing context because there are a finite number of return possibilities and the race by race returns ar~ likely to be nearly independent since different horses will be running (although the jockeys and trainers may not be). The second key feature of the model is that it considers an investor's ability to influence the odds by the size of his bets.8 This yields the following model to calculate SThe first model to include this feature seems to be Isaacs [16J. Only win bets are considered with linear utility, and he is able to determine the. el(act solution in closed form. His model may be useful in situations where the perfect market assumption (3) is violated or where special expertise leads one to believe their estimates of the q; are better than those of the other bettors. --- ## Page 701 672 D. B. Hausch, W T Ziemba and M Rubinstein 382 D. B. HAUSCH, W. T. ZIEMBA AND M. R. RUBINSTEIN the optimal amounts to bet for place and show. n Q( P + 2:1-1 PI) - (Pi + Pj + Pi + Pj ) 2 n + Wo - 2: 51- 1- 1 I".i.j.k s.t. 2: (p, + s,) <: wo, p, > O,S, > 0,/ = I, .. . , n, '-I (6) where Q = 1 - the track take, W;, lJ and Sk are the total dollar amounts bet to win, place and show on the indicated horses by the crowd, respectively, W == ~ Wi' P =="LlJ and S == "LSk are the win, place and show pools, respectively, qi == Wi / W is the theoretical probability that horse i wins, Wo is initial wealth and p, and s, are the investor's bets to place and show on horse I, respectively. The formulation (6) maximizes the expected logarithm of final wealth considering the probabilities and payoffs from all possible horserace finishes. It is exact except for the minor adjustment made that the rounding down to the nearest $0.10 for a two dollar bet, see (4) and (5), is omitted.9 For the values of a ~ 1.16 it was observed that in a given race at most three p, and three s, were nonzero. When (6) is then simplified it can be solved in less than I second of CPU time.1O A discussion of the generalized concavity properties of (6) will appear in a forthcoming paper by Kallberg and Ziemba. The results are illustrated by function 1 in Figures 1 and 2 for the two data sets using an initial wealth of $10,000. In both cases the bets produced from (6) lead to well above average returns and to positive profits. J J These results may be contrasted with random betting; function 2 in Figures 1 and 2. Intuition suggests that Santa Anita with its larger betting pools would have more accurate estimates of the qi than would be obtained at Exhibition Park. Hence positive profits would result from lower values of a. The results bear this out and only horses with a ~ 1.20 for Exhibition Park were considered for possible bets. Generally speaking the bets are usually favorites and almost always on those horses with maximum (Wj W)/(Pj P) and (Wi/ W)/(Sj S) 91t is possible to include this feature in (6) but it greatly complicates the solution procedure (e.g. differentiability is lost) with lillie added gain in accuracy. 10 All calculations were made on UBC's AMDAHL 470V6 Model II computer using a code for the generalized reduced gradient algorithm. 11 The procedure was to calculate the optimal bets to place and show in each race using (6) with the present wealth level. The results of the race and the actual payoffs that reflect the track's take and breakage are known. The payoff for our investor's bets were calculated using all the bets of the crowd plus the bets of our investor taking into account the track's take and breakage. i.e. the payoffs are thus those that would have occurred had our investor actually made his bets. The wealth level was then lIdjusted to reflect the race's gain or loss and the procedure continued for all races. --- ## Page 702 Efficiency of the Market for Racetrack Betting 673 EFFICIENCY OF THE MARKET FOR RACETRACK BETTING 383 18000 17263 17242 :I: f- ...J « w ~ Wo 3 6000 4000 2000 2500 "'- '. 2 00 10 20 30 40 SO 60 70 80 DAYS I Results from eKpected log betting to place and show when expected returns are 1.16 or better with initial wealth SIO,OOO. 2 Approximate wealth level history for random horse betting. Total dollars bet is as in system I (S 116,074). Track payback is 82.5%, therefore final wealth level is SI0,000-0.175(SI16.074)- -SIO,313 (Note: breakage is not taken into consideration) ] Results from using the Exhibition Park approximate regression scheme (with initial wealth S2,500) at Santa Anita. FIGURE 1. Wealth Level Histories for Alternative Betting Schemes; Santa Anita: 1973/74 Season. ratios. These ratios of the theoretical probability of winning to the track take unad- justed odds to place or show form a type of cost-benefit ratio that provides a first approximation to a. This is discussed further in §5, below. Most of the bets are to show and one tends to bet only about once per day. The numbers of bets and their size distribution are presented in Table 4. As expected, the influence on the odds made by our investor's bets is much greater at Exhibition Park hence the bets there tend to be much smaller than at Santa Anita. However, even there about 10% of the bets exceed $1000. The log formulation has absolute risk aversion 1/ w, which for wealth around $10,000 is virtually zero. Zero absolute risk aversion is, of course, achieved by linear utility. One then will bet on the horse (or horses) with the highest a until the influence on the odds drops this horse (or horses) below another horse's a, etc., continuing until there are no favorable bets fr the betting wealth has been fully utilized. The results of such linear utility betting With Wo = $10,000 are: at Exhibition Park with a ;;. 1.20 final wealth is $14,818; at Santa Anita with a;;' 1.16 final wealth is $10,910. Such a strategy is a very risky one and leads to some very large bets. The log function has the distinct advantage that it implies negative infinite utility at zero wealth hence bets having any significant probability of yielding final wealth near zero are avoided. --- ## Page 703 674 D. B. Hausch, W. T. Ziemba and M Rubinstein 384 D. B. HAUSCH, W. T. ZIEMBA AND M. R. RUBINSTEIN 18000 ::t: 1.6220 f- 16000 ....J 1001 35.8 65.7 57.1 85.3 11.8 51.8 9.7 39.5 n - 14 SII,932 n -77 $104, 142 n - 17 $4,954 n - 94" $33,507 ToO' pi \0' L. r Bets Bets ~"' PI"/ ~'" Show Bets Bets Total Sa~ta = $116 074 Anita Bettmgs ' 'Two of these bets had EX: ;. 1.20 and st = O. Total Exhibition = $38 461 Park Belling , --- ## Page 704 Efficiency of the Market for Racetrack Betting 675 EFFICIENCY OF THE MARKET FOR RACETRACK BETTING 385 S. Making the System Operational The calculations reported in Figures I and 2 were made under the assumption that the investor is free to bet once al1 other bettors have placed their bets. In practice one can only attempt to be one of the last bettors. 12 There is a natural tradeoff between placing a bet too early and increasing the inaccuracies and running the possibility of arriving too late at the betting window to place a bet. 13 The time just prior to the beginning of a race is crucial since many bets are typically made then including the so called "smart money" bets made very close to the beginning of the race so their impact on other bettors is minimized. It is thus extremely important that the investor be able to perform all calculations necessary to place the bet(s) very quickly. Typically, since tracks have neither public phones nor electricity, calculations can at most utilize battery operated calculators or possibly a battery operated special purpose computer. Even if computing times were negligible the very act of punching in the data needed for an exact calculation is too time-consUJping since it takes more than one minute. Therefore, in practice, approximations that utilize a limited number of input data elements are required. Several types of approximations are possible such as the tabular rules of thumb developed by King [17] or the regression procedures suggested here. Our procedure indicates whether or not a bet to place or show is warranted and at what level using the following eight data inputs: wo, Wi*' Pi*, ~*' 8;., W, P and S, where i* is an i for which (WJ W)/(PJ P) is maximized and}* IS a) for which (Uj/W)/(S/S) is maximized. It is easy to determine i· and}*, particularly since i* often equals)*, by inspection of the tote board. The approximation supposes that the only possible bets are i* to place and)· to show. The regressions, as given below, must be calibrated to a given track and initial wealth level. As a prelude to actual betting the 'regressions were calibrated at Exhibition Park for Wo = $2500. For calculated i* the expected return on a $1 bet to place is approximated by _ Wi./W EXt. = 0.39445 + 0.51338 Pi./ P , R2 = 0.776. (7) If EXt.;.. 1.20 then the optimal bet to place is approximated by Pi. = -459.32 + 1715.6qi.-0.042518qi.P -7440.1qt. + 1379Iq? +0.J0247Pi.+49.572lnwo,R2 = 0.954. (8) Similarly for}* the expected return on a $1 bet to show is approximated by _ ~*/W EX/. = 0.64514 + 0.32806 Sj./ S ' R2 = 0.650. (9) 12The model as developed in this paper utilizes an inefficiency in the place and show pools to yield positive profits. A:1 investor's bets are determined by his wealth level as well as the profitability of one or more such bets. There mayor may not exist "enough" inefficiency to provide positive profits for additional investors using a system of this nature. In "the context of the Isaac's model, for win bets with linear utility, Thrall (25) showed that if there were positive profits to be made and each new investor was aware of all previous investor's bets then the profits for these various bettors are shared and in the limit become zero. It is likely that a similar result obtains for the model discussed here. I) At some tracks, such as Santa Anita, betting ends precisely at post time when the electric totalizator machines are shut off. At other tracks including Exhibition Park. betting ends when the horses enter the starting gate which may be 2 or even 3 minutes past the post time. --- ## Page 705 676 D. B. Hausch, W. T. Ziemba and M Rubinstein 386 D. B. HAUSCH, W. T. ZIEMBA AND M. R. RUBINSTEIN If EX; ;;. 1.20 then the optimal bet to show is approximated by Sj* = - 660.97 - 867.69qj* + 0.25933qj*S + 3715.2q]. -0.19572Sj * + 77.014 In wo, R2 = 0.970. ( 10) All the coefficients in (7)-( 10) are highly significant at levels not exceeding 0.05. Utilizing the equations (7)-(10) results in a scheme in which the data input and execution time on a modern hand held programmable calculator is about 35 seconds. The results of utilizing this method on the Exhibition Park data with initial wealth of $2500 are shown in Figure 3, functions 1 and 2. Using the exact calculations yields a final wealth of $5197 while the approximation scheme has an even higher final wealth of $7698. The approximation scheme leads to 63 more bets (174 versus Ill) than the exact calculation. Since these bets had a positive net return the total profit of the approximation scheme exceeds that of the exact calculation. Thus, it is clear that one would maximize profits with a cutoff rule below the conservative level of 1.20. The size distribution of bets from the exact and approximate solutions are remarkably similar; see Table 6 in [15]. For place there were 17 bets where EXP exceeded or equalled 1.20 of which 7 (41%) were in the money. Only 2 of these 17 bets were not chosen by the regression. However 14 horses were chosen for betting by the regressions even though their "true" EXP was less than 1.20. Most of these had "true" EXP values close to 1.20 and were favorites. Four of these horses (29%) were in the money. For show there were 94 bets where EXt exceeded or equalled 1.20 of which 52 (55%) were in the money. Only 8 of these were 10000 9763 9000 :I: !:i 8000 -< UJ ~ 7000 2 6000 Wo= SOOO 4000 3000 2000 1000 0 0 10 20 30 40 SO 60 70 80 90 100 110 DAY I Results from expected log betting to place and show when expected returns are 1.20 or better with initial wealth $2,500. 2 Results from using the approximate regression scheme with initial wealth $2,500. FIGURE 3. Wealth Level Histories for Exacl and Approximate Regression Betting Schemes; Exhibition Park: 1978 Season. --- ## Page 706 Efficiency of the Market for Racetrack Betting 677 EFFICIENCY OF THE MARKET FOR RACETRACK BETTING 387 overlooked by the regressions while 59 bets with "true" EX' values less than 1.20 were chosen by the regressions. Of these 39 (66%) were in the money. The regression method is a simple procedure that seems to work well. For example using it on the Santa Anita data (without reestimating the coefficients) indicates that initial wealth of $2500 would yield a final wealth of $8104; see function 3 in Figure I. Conceivably, there are many possible refinements using fundamental information that could be added. Many of these refinements as well as discussion of the results of actual betting will appear in a forthcoming book by Ziemba and Hausch. One such refine- ment is the supposition that the win market is more efficient under normal fast track conditions and less efficient when the trac;:k is slow, muddy, heavy, wet, sloppy, etc. Bets were made on 87 of the 110 days at Exhibition Park; of these 57 days had a fast track. Using the regressions yields "fast track" bets of $32,501 returning $38,364 for a net profit of $5863. On the nonfast days the regressions suggest bets of $17,180 returning $16,515 for a loss of $665. Hence this refinement decreases the time spent at the track and increases the net profit from $5198 to $5863. 6. Implementation and Reliability of the System The model presented in this paper assumed that one can utilize the betting data that prevails at the end of the betting period, say t, to calculate optimal bets. In practice, however, even with the approximations given by (7)-(10), one requires about 1-1.5 minutes to physically calculate the optimal bets and place them. Thus one can only utilize betting information from T (~ t - 1.5). Hence bets that were optimal at T may not be as profitable at t. Some evidence by Ritter [20] seems to indicate that the odds on the expost favorite at t often decrease from T to t. In which case an optimal bet at 7' may be a poor bet at t. Ritter investigated the systems: bet on i* if (Pi*/ P /(Wi./ W) <; 0.7 and on j. if (Sj./ S)/( W)./ W) (; 0.7. (For comparison equations (7) and (9) indicate the more restrictive constraints 0.6373 and 0.5912, respectively, instead of 0.7. The less restrictive Ritter constraints yield about three times as many bets as (7) and (9) indicate.) Using a sample of 229 harness races at Sportsman's Park and Hawthorne Park in Chicago he found that with $2 bets these systems gained 24% and 16%, respectively. But a 15% advantage shrunk to -1% if one uses the 7' bets with a random shock over the 7' to t period (for the show bets for the 95 races at Hawthorne with a 0.65 cutoff). There are some difficulties with the design of Ritter's experiment, such as the small sample, the inclusion of only $2 bets and the use of a random shock rather than an estimate of usual trends from 7' to t, etc., however, his results point to the difficulty of actually making substantial profits in a racetrack setting. An attempt was made during the summer 1980 racing season at Exhibition Park to implement the proposed system and observe the "end of betting problem." Nine racing days were attended during which 90 races were run. At t - 2 minutes the win, place, and show data were recorded on any horse that, through equations (7) and (9), warranted a bet. Equations (8) and (10) were then used on this data to determine the regression estimates of the optimal bets. Updated toteboard data on these horses were recorded until the end of betting to determine if the horse remained a system bet. Results of this experiment are shown in Table 5 where: I) an initial wealth of $2500 is assumed; 2) size of bet calculations are done on data at t - 2 minutes; and 3) returns are based on final data. Twenty-two bets were made that yielded final wealth of $3716 for a profit of $1216. Actual betting was not done but by making the calculations two minutes before the end of betting allows 1.5 minutes to place the bet, more than --- ## Page 707 678 D. B. Hausch, W. T. Ziemba and M Rubinstein 388 D. B. HAUSCH, W. T. ZIEMBA AND M. R. RUBINSTEIN TABLE 5 Resulls from Summer 1980 Exhibition Park Belling Regression Regression Regression Net estimate estimate estimate of return of expected of expected optimal based on return per return per bet final data dollar, 2 dollar, at 2 minutes with minutes the end before considera tion before end of end of of our bets Final Date Race of betting betting betting Finish affecting odds wealth $2500 July 2 9 120 122 $19,SHOW ON 4 5·6·7 - $19 2481 10 120 123 72,S HOW ON 8 8·1·2 72 2553 July 9 7 121 110 292,SHOW ON I 2·7·1 131 2684 10 135 122 248,PLACE ON I 1·6·2 260 2944 July 16 6 131 122 487,PLACE ON 9} 9·S·6 536 } 682 3626 139 117 292,SHOW ON 9 146 7 125 127 7,SHOWON I 5·2·8 -7 3619 July 23 3 149 149 30,SHOWON 2 2·10·7 92 3711 4 139 134 573,SHOW ON 10 6·10-4 201 3912 July 30 8 121 III 215,PLACE ON 4 4·1-5 129 4041 9 123 125 591,SHOWON 6 8·1·5 - 591 3450 Aug 6 6 128 112 39,SHOWON 4 4·3-1 59 3509 9 124 103 51,SHOWON 2 4-1-3 - 51 3458 Aug 8 I 121 132 87,SHOWON I 1·10-4 139 3597 .. 3 127 III 635,SHOW ON 3 3·4·7 127 3724 4 126 113 126,SHOW ON 2 2·7·1 82 3806 Aug II 8 121 112 94,SHOWON 8 8·6·2 113 3919 .. 9 131 130 688,SHOW ON 5 5·3-4 138 4057 Aug 13 3 128 106 33,SHOW ON 2 1·6·7 - 33 4024 6 131 122 205,SHOW ON 5 5·8·4 144 4168 7 134 133 511,SHOW ON 6 8-5·9 - 511 3657 10 123 109 10S,SHOW ON 5 3·5·1 59 3716 enough time. Note in Table 5 that many systems bets at t - 2 were not system bets at t, however all had regression expected returns of more than one_ An important question concerns the reliability of the results: are the results true exploitations of market inefficiencies or could they be obtained simply by chance? This question is investigated utilizing a simple model which was suggested to us by an anonymous referee. The first application is concerned with an estimate of the probabil- ity that the system's theory is vacuous and indeed the observations conform to specific favorable samples from a random betting popUlation. The second application esti- mates the probability of not making a positive profit. The calculations utilize the 1980 Exhibition Park data; see Table 5. Let 'TT be the probability of winning a bet in each trial and X. = { I + w if the bet is won, I 0 otherwise, be the return from a $1 bet in trial i. In n trials, the probability of winning at least 100y% of the total bet is ( 11 ) --- ## Page 708 Efficiency of the Market for Racetrack Betting 679 EFFICIENCY OF THE MARKET FOR RACETRACK BETTING 389 Assume that the trials are independent. Since the Xi are binomially distributed (II) can be approximated by a normal probability distribution as [ r,; {y - (I + w)'IT + l} 1 I - (I + W)'IT'/(I - 77)/77 ( 12) where is the cumulative distribution function of a standard N(O, I) variable. The observed probability of winning a bet, weighted by size of bet made, yields 0.771 as an estimate of 'IT. If the systems theory was vacuous and random betting was actually being made then (I + w)'IT = 0.83 since the track's payback is approximately 83%. The 22 bets made totalled $5304 and resulted in a profit of $1216 for a rate of return of 22.9%. Using equation (12) with n = 22, gives 3 X 10- 5 as the probability of making 22.9% through random betting. Suppose that the 1980 Exhibition Park results represent typical system behavior so 'IT = 0.771 and (I + w)'IT = 1.229. In n trials the probability of making a non-positive net return is ( 13) which can be approximated as = <1>( - 0.342m). [ m(1 - (1 + W)77) 1 (I - W)'lTJ(I - 77)/7T ( 14) For n = 22 this probability is only 0.054 and for n = 50 and 100 this probability is 0.008 and 0.0003, respectively. Thus it is reasonable to suppose that the results from the 1980 Exhibition Park data (as well as the 1978 Exhibition Park and 1973/1974 Santa Anita data with their larger samples) represent true exploitation of a market inefficiency. 14 "Without implicating them we would like to thank Michael Alhadeff of Longacres Racetrack in Seallle, the American Totalizator Corporation and the staff of the Jockey Club at Exhibition Park in Vancouver for helping us obtain the data used in this study, and M. J. Brennan and E. U. Choo for helpful discussions. Thanks are also due to V. S. Bawa, J. R. Riller, and two anonymous referees for helpful comments on an earlier draft of this paper. This paper is a condensation of 1\5J. References I. Au, M. M., "Probability and Utility Estimates for Racetrack Bellors," J. Political Economy, Vol. 85 (1977), pp. 803-815. 2. --, "Some Evidence of the Efficiency of a Speculative Market," Econometrica, Vol. 47, No. 2 (1979), pp. 387-392. 3. ARROW, K. J., Aspects of the Theory of Risk Bearing, Yrjo Jahnsson Foundation, Helsinki, 1965. 4. BAUMOL, W. 1., The Stock Markel and Economic Efficiency. Fordham Univ. Press, New York, 1965. 5. BEYER, A., My $50, 000 Year althe Races, Harcourt, Brace, Jovanovitch, New York, 1978. 6. COPELAND, 1. E. AND WESTON, J. F., Financial Theory al/d Corporale Po/icy, Addison-Wesley, Reading, Mass., 1979. 7. DOWNES, D. AND DYCKMAN, T. R., "A Critical Look at the Efficient Market Empirical Research Literature as it Relates to Accounting Information," Accountil/g Rev. (April 1973), pp. 300-317. 8. FABRICANT. B. F., Horse Sense, David McKay. New York, 1965. --- ## Page 709 680 D. B. Hausch, W. T. Ziemba and M Rubinstein 390 D. B. HAUSCH, W. T. ZIEMBA AND M. R. RUBINSTEIN 9. FAiolA, E. F., "Efficient Capital Markets: A Review of Theory and Empirical Work," J. Finance, Vol. 25 (1970), pp. 383-417. 10. --, Foundations of Finance, Basic Books, New York, 1976. I\, FIGLEWSKI, S., "Subjective Information and Market Efficiency in a Betting Model," J. Political ECOIIOIl1Y, Vol. 87 (1979). pp. 75- 88. 12. GRIFFITH, R. M., "Odds Adjustments by American Horse Race Bettors," Amer. J. Psychology, Vol. 62 (1949), pp. 290-294. 13. ---, "A Footnote on Horse Race Belting," Trans. Kentucky Acad. Sci., Vol. 22 (1961), pp. 78-81. 14. HARVILLE, D. A., "Assigning Probabilities to the Outcomes of Multi-Entry Competitions," J. A mer. Statist. Assoc., Vol. 68 (1973), pp. 312-316. 15. HAUSCH, D. B., ZIEMBA, W. T. AND RUBINSTEIN, M. E., "Efficiency of the Market for Racetrack Betting," U.B.C. Faculty of Commerce, W. P. No. 712, September 1980. 16. ISAACS, R., "Optimal Horse Race Bets," Amer. Math. Monthly (1953), pp. 310-315. 17. KING, A. P., "Market Efficiency of a Multi-Entry Competition," MBA essay, Graduate School of Business, University of California, Berkeley, June 1978. 18. MCGLOTHLIN, W. H., "Stability of Choices Among Uncertain Alternatives," Amer. J. Psychology, Vol. 63 (1956), pp. 604-615. 19. PRATT:J., " Risk Aversion in the Small and in the Large," Econometrica, Vol. 32 (1964), pp. 122-136. 20. RITTER, J. R., "Racetrack Betting: An Example of a Market with Efficient Arbitrage," mimeo, Department of Economics, University of Chicago, March 1978. 21. ROSETT, R. H., "Gambling and Rationality," J. Political Economy, Vol. 73 (1965), pp. 595-607. 22. RUBINSTEIN, M., "Securities Market Efficiency in an Arrow-Depreu Market," Amer. Econom. Rev., Vol. 65, No. 5 (1975), pp. 812-824. 23. SELIGMAN, D., "A Thinking Man's Guide to Losing at the Track," Fortune, Vol. 92 (1975), pp. 81 - 87. 24. SNYDER, W. W., "Horse Racing: Testing the Efficient Markets Model," J. Finance, Vol. 33 (1978), pp. 1109-1118. 25. THRALL, R. M., "Some Results in Non-Linear Programming," Proc . • Second Sympos. in Unear Programming, Vol. 2, National Bureau of Standards, Washington, D. C., January 27- 29, 1955, pp. 471-493. 26. VERGlN, R. c., "An Investigation of Decision Rules for Thoroughbred Race Horse Wagering," Interfaces, Vol. 8, No. 1 (1977). pp. 34-45. 27. WEITZMAN, M., " Utility Analysis and Group Behaviour: An Empirical Study," J. Political Economy, Vol. 73 (1965), pp. 18-26. 28. ZIEMBA, W. T. AND VICKSON, R. G., eds., Stochastic Optimization Models in Finance, Academic Press, New York, 1975. --- ## Page 710 Management Science, 31,381- 394 (1985) 47 TRANSACTIONS COSTS, EXTENT OF INEFFICIENCIES, ENTRIES AND MULTIPLE WAGERS IN A RACETRACK BETTING MODEL* DONALD B. HAUSCH AND WILLIAM T. ZIEMBA School of Business, University of Wisconsin, Madison, Wisconsin 53706 Faculty of Commerce, University of British Columbia, Vancouver, British Columbia, Canada V6T 1 W5 In a previous paper (Management Science, December 1981) Hausch, Ziemba and Ru- binstein (HZR) developed a system that demonstrated the existence of a weak market inefficiency in racetrack place and show betting pools. The system appeared to make possible substantial positive profits. To make the system operational, given the limited time available for placing bets, an approximate regression scheme was developed for the E1thibition Park Racetrack in Vancouver for initial belting wealth between S2500 and $7500 and a track take of 17.1%. This paper: (I) extends this scheme to virtually any track and initial wealth level; (2) develops a modified system for mUltiple horse entries; (3) allows for multiple bets; (4) analyzes the effects of the track take and breakage on profits; (5) presents recent results using this system; and (6) considers the e1ttent of the inefficiency. i.e., how much can be bet before the market becomes efficient? (FINANCE-PORTFOLIO; GAMES-GAMBLING) 1. The Racetrack Market 681 The "market" at the track in North America convenes for about 20 minutes, during which participants make bets on any number of the six to twelve horses in the following race. In a typical race, participants can bet on each horse. either to win, place or show. All participants who have bet a horse to win realize a positive return on that bet only if the horse is first, while a place bet realizes a positive return if the horse is first or second, and a show bet realizes a positive return if the horse is first, second or third. Regardless of the outcome, all bets have limited liability. Unlike casino games such as roulette, but like the stock market, security prices (i.e. the "odds") are jointly determined by all the participants and the rules governing transactions costs (i.e. the track "take" and "breakage"). To take the simplest case, all bets across all horses to win are aggregated to form the win pool. If Wi represents the total amount bet by all participants on horse i to win, then W = Li Wi is the win pool and WQ/ Wi is the payoff per dollar bet on horse i to win if and only if horse i wins, where Q is the track payback proportion. The rules for division of the place and show pools are as follows. Let lj be the amount bet on horse j to place and P == LjPj be the place pool. The payoff per dollar bet on horse j to place is I +[PQ- Pi - Pj ]!(2Pj } o if I i is first and j is second or j is first and i is second, otherwise. (1) • Accepted by Donald G. Morrison as Special Departmental Editor; received December 13. 1983. This paper has been with the authors 2 months for 2 revisions. Efficiency of Racelrack Belling Markels 391 --- ## Page 711 682 D. B. Hausch and W T. Ziemba 392 D. B. HAUSCH AND W. T. ZIEMBA Thus if horses i and j are first and second each bettor on j (and also i) to place first receives the amount of his bet back. The remaining amount in the place pool, after the track take, is then split evenly between the place bettors and i and j. The payoff to horse j to place is independent of whether j finishes first or second, but it is dependent on which horse i finishes with it. A bettor on horse j (0 place hopes that a longshot with a small Pi not a favorite will finish with it. The payoff per dollar bet on horse k to show is analogous I +[SQ- Sj- Sj- Sk]/(3Sk ) o if lk is first, second or third and finishes with i and j, otherwise, (2) where Sk is the amount bet on horse k to show and the show pool is S == 2:kSk' Equations (I) and (2) are not quite correct as they do not account for "breakage". Breakage is discussed in §6; it is an extra commission resulting from payoffs being rounded down to the nearest 10¢ or 20¢ on a $2 bet. 2. Racetrack Efficiency A market is efficient, see Fama (1970), if current prices fully reflect all available relevant information. In this case experts should not be able to achieve higher than average returns with regularity. To investigate the efficiency of the racetrack's win market Snyder (J 978) tested whether or not bets at different odds levels yielded the same average return. A weakly-efficient market in Snyder's sense would have the average rate of return for each odds level equal to Q, the track payback ratio which currently varies in North America from 0.852 in Ontario to 0.779 in Saskatchewan. His results suggest there are "strong and stable biases but these are not large enough to make it possible to earn a positive profit" (Snyder 1978, p. (101). In particular, favorites tend to be underbet and longshots overbet. See Ziemba and Hausch (1984), hereafter referred. to as ZH, for a survey of the literature on this bias. To test the efficiency of the racetrack's place and show market HZR assumed: (I) If qi (i = I, .. . , n horses) is the probability that i wins, then the probability that i is first and j is second is and the probability that i is first, j is second and k is third is q/l.iqk Harville (1973) developed and analyzed these formulas. (3) (4) (2) If Wj ·is. the total amount bet on horse i to win and W == 2:7., Wi then qi = Wj W, i.e., the win market is efficient. While this assumption ignores the bias for favorites and longshots mentioned above there are reverse tail biases in the probabili- ties of finishing second and third. They occur because if the probability to win is overestimated (underestimated) and probabilities sum to one it is likely that the probabilities of finishing second and third are underestimated (overestimated). Tables 2b, c in HZR indicate how these biases tend to cancel when they are aggregated to form the theoretical probabilities of placing and showing which are used in the HZR model.' I See HZR, §§4-6. for a more in-depth discussion of the model. --- ## Page 712 Transaction Costs, Extent of Inefficiencies, Entries and Multiple Wagers 683 A RACETRACK BETTING MODEL 393 Using equations (1)-(4). the Bernoulli capital growth model (see Ziemba and Vickson 1975), and assuming initial wealth is woo the HZR model to calculate optimal amounts to bet for place (p,.1 = 1.2 .. . .. n) and show (5, . 1 = I. 2, . . . ,n) is Q(P + 'Z'i-IP,) - (Pi + Pi + Pij) 2 x [ Pi + Pi ] Pi + Pi Pi + Pi Q(S + Li_ls,) - (Si +~. + Sk + Sijk) + 3 n n + Wo - 2: s, - 2: P, '-I '-I /.".i.j.k / ... i.J n S.1. 2:(p,+S,)O. ", >0. I-l ..... n .. '-I (5) The formulation (5) maximiz.es the expected logarithm of final wealth considering the probabilities and payoffs from the possible horserace finishes. For notational simplic- ity Pij == Pi + Pj and Sijk == Si + Sj + Sk' . The generaliz.ed concavity properties of (5) are discussed in Kallberg and Ziemba (198 I). Using equations (I) and (3), the expected return on an additional $1 bet to place on horse I is: EXf==± [~+ qjql ][I+..!..INT[ Q(P+I)-(I+PI+~) X20]]. j _ 2 1 - q I I - qj 20 2( P I + 1) (6) INTI YJ means the largest integer not exceeding Y. The TNT and multiplying and dividing by 20 accounts for breakage, i.e. the payoffs on a $2 bet being rounded down to the nearest 10¢ in this instance. The ql~/(I - ql) term is the probability that I is first and) is second while the qjql/(l - qj) term is the probability that) is first and I is second. A similar equation is available for show bets (see HZR). One might hope that when EXf or EX; equals say. a, the actual average rate of return would be near a or at least increasing in a . Despite the inherent inaccuracies in assumptions (I) and (2) this is indeed the case as is borne out in Table 3 in HZR and Table 5.1 in ZH . In fact for cases of a above about 1.02, positive profits seem realiz.able. These profits are maximiz.ed when a is about 1.16. An ideal model is then: (i) for each i check EXf and EX; to decide whether or not to bet on horse i, and then (ii) solve (5) to determine the optimal bet siz.e after setting Pi and Sj to z.ero for horses you definitely do not want to bet on. This analysis presumes one can bet after all other bettors have placed their bets. In practice one can only attempt to be one of the last bettors. Thus it is extremely important that one be able to perform all calculations necessary to determine the bet(s) quickly so that the bets can be placed as close to the end of the betting period as possible. The approximations developed in HZR are regression schemes with the minimal data inputs: wo' Wi" P;., Wi" Sr' W, P and S, where i* = argmax i« Wj W)/ (Pj P» --- ## Page 713 684 D. B. Hausch and W T. Ziemba 394 D. B. HAUSCH AND W. T. ZIEMBA and J* = argmax/( Hj/ W)/(Sjl S». The ratios (Wj W)/(Pj P) and (Hj/ W: /(Sj/ S) may be thought of as simple measures of the inefficiency to place on jane show on) respectively. The regressions were calibrated for an initial wealth Wo betweer. $2500 and $7500 and a track about the size of Exhibition Park in Vancouver, B.C (daily handle about $1.2 million) with Q = 0.829. Using this regression scheme on th( Exhibition Part data with an initial wealth of $2500 and updating wealth over time resulted in a final wealth of $7698 at the end of the I IO-day season (see Figure 3 in HZR). We now extend equations (7)-(10) in HZR to account for different wealth levels, different size tracks, different Q's, and coupled entries. 3. Track Size and Wealth Level Track size and wealth level do not affect the expected return per dollar bet to place or show. However both are important factors in determining the optimal amounts to bet to place and show because the larger the betting pools at the track the less our bet TABLE I The Optimal Place Bet Jar Various Betting Wealth Levels and Place Pools Sizes Wo = 550 Wo = $500 Wo = $2,500 Wo=SIO.OOO Place Pool 261q + 256q2 + 180q' 426q + 802q2 487q + 901 q2 - 52,000 ( I 99qP, ) - qPo- 0.70P, [P2] ( 459qP, ) - qP _ 0.60P; [P5] ( 521qP, ) - qP - 0.60P, [ Place Pool 39q + 52q2 375q + 525q1 1,307 q + 1,280q2 2,497 q + 1.806q2 = 510,000 ( 25qP; ) - qP-0.75P, [PI} ( 271qP, ) - qP _ 0.70P, [P31 + 902q1 + 2,073q3 ( 993qP, ) - qP _ O,70P, [P6) _ ( 2,199qP, ) [ qP - 0.60P, Place Pool 505q + 527q2 2,386q + 2,668q2 7,072q + 10,470q: = 5150,000 ( 386qP, ) - qP _ 0.60P{ [P41 ( 1.877qP, ) - qP _ 0.60P; [P7) - [I ( 5,273qP, ) qP - 0.70P; TABLE 2 The Optimal Show Bel Jar Various Belling Wealth Levels and Show Pool Sizes Wo = 550 Wo = 5500 Wo = 52,500 Wo - SIO,OOO Show Pool 9 + 994q2 - 464q' 13 + 1.549q 2 - 901 q' - SI,2oo ( 150qS; ) - qS - 0.80S, [S2) ( 303qS; ) - qS _ 0.60S, (55) Show Pool JO + 183q2 - 135q3 86 + 1,516q2 53 +5,219q2 58 + 7,406q2 = $6,000 ( I IS ) - qS - O .~OS, [51) - 968q' -2,513q' - 4,211 q3 ( 90.7S; ) - qS - 0.85S; [S31 ( 934qS, ) - qS - 0.70S, [56) - [~ ( 1,359qS, ) qS - 0.65S; Show Pool 131 + 2,I50l 533 + 9,862q2 1,682 + 28,2ooq2 - $100,000 - 1,778q' - 7,696q' - 16,880q' ( I 50S, ) .,.. qS - 0.70S; [S4J ( 571S; ) - qS - 0.80S, [S7J _ ( 1,769S; ) [' qS - 0.85S, • --- ## Page 714 Transaction Costs, Extent of inefficiencies, Entries and Multiple Wagers 685 A RACETRACK BETTING MODEL 395 influences the odds and as our wealth increases we tend to wager more. Therefore new regressions were calculated for most reasonable track sizes and wealth levels. The data for these regressions were the true optimal bets from the NLP model (5) over a broad range of wealth levels and track sizes. The Ideal data [0 represent a broad range of track sizes would be a season's data from many different tracks. A more practical alternative was to mUltiply the Exhibition Park data by varying constants to simulate data from smaller and larger tracks. Nineteen different regressions which appear in Tables I and 2 were determined for place and show depending upon wealth and track size. These regressions labelled PI-PIO and SI-S9 give the optimal place or show bet for specific values of initial wealth and size of pool. For intermediate values of these variables one may determine accurate betting amounts by taking convex combinations of these basic regressions. For example, for a place bet with initial wealth $1000 and place pool of $5000 the optimal bet is a 2 (equation [f 2]) + a) (equation [f)]) + a 5 (equation [fsD + a6 (equation [P6 j). where 4. Track Payback Both the expected return per dollar bet and the optimal bet size are increasing functions of Q, the track's payback. Equations (7) to (10) in HZR were calculated with Q = 0.829. Modification of these equations for use at tracks with Q =1= 0.829 are now developed. 4.1. Adjustment of EX; and EX; for Q The regression equivalent for EX; when Q = 0.829 is:2 ~ Wj W EX!, = 0.319 + 0.559 Pj P . (7) How can this equation be adjusted for a track payback different from 0.829? The "true" expected return on a one dollar place bet on horse i is EXf= ±( q;qj + q;qj )(1+ QP-(P;+Pj )) j _ t I - q; I - qj 2 P; . (8) j*; Equation (7) is linear in Q with a EX; = q;P [I + 2: ( ~ )1. aQ 2f; j= I I - qj .1*' Using 124 Exhibition Park races with EXf in the range 1.10 and greater, the true aEXf/aQ was regressed against q; to give aEXf/aQ~2 .22 - 1.29q; (R 2 =0.86, SE = 0.055, both coefficients highly significant). Therefore when the track payback is Q, the expected return on a $1 place bet can be approximated by adjusting (7) to EXf == EXf + (2.22 - ) .29q;){ Q - 0.829) = 0.3) 9 + 0.559( ~:~;) + (2.22 - 1.29( ~ ) )( Q - 0.829). (9) 2Note thai the coefficienls of EXr here are different from lhose of equation (7) in HZR. In HZR only cases of expected return grealer lhan 1.16 were considered. Here we m~y wish lo hel on horses ~ilh expecled relurns as low as 1.10 lo reFlecl a high (juality track and high qualilY horses. Thus lhe_EX; had to be recalculated to he accurale in the range 1.10 to 1.16. A similar change will he noted for EX;. --- ## Page 715 686 D. B. Hausch and W T. Ziemba 396 D. B. HAUSCH AND W. T. ZIEMBA A similar analysis for show yields EX; == EX; + (3.60 - 2.13q;)( Q - 0.829) (R 2 = 0.565, SE = 0.198, both coefficients highly significant) = 0.543 + 0.369( ~:~:) + (3.60 - 2.13( ~ ))( Q - 0.829). (10) 4.2. Adjustment of ft* and s· for Q Equations (8) and (10) in HZR were calibrated for Q = 0.829. Since the true p. and s* are nondecreasing in Q which varies from track to track we must adjust ft· and s· at tracks with Q * 0.829. The exact NLP model was used on a number of Exhibition Park examples to compute the optimal place or show bets at different initial wealths, track sizes and different Q's (from 0.809 to 0.859). The results indicated that t:.p. / t:.Q and t:.s* / t:.Q are independent of Q in this range. Therefore with a t:.Q of 0.0 I, t:.p* / t:.Q and t:.s*/t:.Q were regressed on p., wO' Pi' P and s·, Wo, Si' S, respectively. The analysis showed for t:.p. / t:.Q that p. and Wo were very significant independent variables but neither Pi and P were significant; similar results for t:.s· / t:.Q were observed. Then the (t:.p. / t:.Q, p., wo) and (t:.s* / t:.Q, s*, wo) were aggregated (due to a small number of place data points) to give: [i.::j~g] =[0.0316][ ~*] + 0.000351wo (R 2 = 0.948, SE = 2.23, n = 56, both coefficients highly significant). Therefore p* and s· (i.e. ft· and s· adjusted for Q) are: P* = ft* + ( Q - 0.829),(3. 16ft* + 0.0351 wo) and s· = s· + (Q - 0.829)(3.16s· + 0.035Iwo). 4.3. Example Involving Q * 0.829 (II) ( 12) May 7. 1983-Kentucky Derby at Churchill Downs, Louisville, Kentucky. The final win and show pools and the win and show bets on # 8, Sunny's Halo, were: W = $3,143,669, S = $1,099,990, Ws = $745,524, Ss = $179,758. Using just the EX: portion of equation (10) gives EXs = 1.08, i.e. not enough to consider a bet if one is using a typical cutoff of 1.10 as recommended in ZH for a race like the Kentucky Derby. But Kentucky has Q = 0.85 and therefore it is more accurate to use equation (10) resulting in EXg = 1.14. Hence a show bet should be made. With an initial wealth of $1000, Table 2 gives the optimal show bet as S* = $48. The correction for Q'" .85 using equation (12) yields s· = $52. Sunny's Halo won the Derby and paid $4.00 per $2.00 bet to show so the $52 bet returned $104 for a $52 profit. Full details on this race appear in ZH. 5. Coupled Entries Occasionally two or more horses are run as a single "coupled entry" or simply "entry" because (I) an owner or a trainer has two or more horses in the same race, or --- ## Page 716 Transaction Costs, Extent of Inefficiencies, Entries and Multiple Wagers 687 A RACETRACK BETTING MODEL 397 (2) there are more horses than the toteboard can accommodate (commonly called a field). The entry wins, places or shows if just one of the horses wins, places or shows. If any two of the horses in the entry come first and second all the place pool goes to the place tickets on the entry. If two of three of the 'in-the-money' horses are the entry then typically two thirds of the show pool goes to the draw tickets on the entry (rather than the usual third). Suppose the coupled entry has number I and let ql = WI/ W. Then ql estimates the probability that one of the horses in the entry will win the race. Suppose further that qlA and qlB (with qlA + qlB = ql) are the correct winning probability estimates of the two horses in the entry. Using ql (i.e. thinking of the entry as a single horse) and equation (3) to calculate the probability of the entry placing gives Pr(entry I is 1st or 2nd) = ql + ± I q~qi . i=2 qi ( 13) Using qlA and qlB (i.e. thinking of the entry as two horses) and equation (3) to calculate the probability of the entry placing gives Pr(entry I is 1st and/or 2nd) = Pre I A is I st and any horse but I B is 2nd) which equals + Pr(IB is 1st and any horse but IA is 2nd) + Pre I A is 2nd and any horse but I B is 1st) + Pre I B is 2nd and any horse but IA is 1st) + Pre I A and I B are I st and 2nd in either order) = ± qlAqi + ± qlBq, + ± qlAqi i = 2 I - q I A i = 2 I - q I B i = 2 I - q, + ± qlBqi + qlAqlB + qlAqlB i = 2 I - qi I - q I A I - q I B ' which is equation (13). Hence considering the entry as two horses does not affect our estimate of the entry's probability of placing. It does, however, affect our estimate of the expected return on a dollar bet to place since the possibility of a I A -I B or I B-1 A finish exists and for those finishes the place payoff will be high since the whole place pool (net of the track take and breakage) goes to the holders of place tickets on the entry I. Thus equation (9) will underestimate EX,. For the same reason p*, from Table I, will also be underestimated. We now consider the use of Tables I and 2 and equations (9), (11), (10) and (12) on EXP, p*, EX' and s*, respectively, to account for an entry. 5.1. Adjustments of EX, and EXi for Coupled Entries Equation (6) gives the expected return on a dollar bet to place on a horse, considering the entry I as a single horse. Ignoring breakage, this expected return is --- ## Page 717 688 D. B. Hausch and W. T. Ziemba 398 D. B. HAUSCH AND W. T. ZIEMBA More properly considering entry I as two horses, I A and I B, the expected return on a one dollar bet to place is EXL.,B = ± (~+ qlAq, )(1 + QP - (PI + P,)){ ~orse.s IA and i~2 I - qlA I - q, 2PI l:place + ± (~+ qlBqi )(1 + QP - (PI + P;)){ ~orses IB and 1-2 I - qlB 1- q, 2PI I place + ( qlAqlB + qlAqlB )(1 + QP - PI ){ horses IA and I - qlA I - qlB PI I B place. Let 6 P == EXfA.IB - EXf . It is generally the case that the two horses in the coupled entry are not of equal ability. We assume that qlA = t ql and ql B = 1- ql' Using the Exhibition Park data,) 6P was regressed on W I / Wand P I / P, giving ~ WI PI 6P = 0.867 W - 0.857 P ( 14) (R 2 = 0.996, SE = 0.00267, and both coefficients highly significant). Then using equations (7) and (18) the regression approximation for expected return on a one dollar bet on the coupled entry I is ~ ~ ~ ( W I / W) WI PI EXfA .IB == EXf + 6 P = 0.319 + 0,559 P I / P + 0.867 W - 0.857 P . ( 15) The same procedure for the expected return on a one dollar show bet on the coupled entry I yields ~ ~ ~ [ WI / W ] WI S, EX~A . 'B == EX~ + 6P = 0.543 + 0.369 SI/ S + 0.842 W - 0.8 \0 S · (16) 5.2. Adjustments of p. and s· for Coupled Entries When the possible bet is on an entry, equations (7) or (9) underestimate the expected return on an additional dollar bet to place. Therefore the optimal bet from the NLP (5) will underestimate the true optimal coupled entry place bet. To understand this phenomenon many Exhibition Park examples were solved using the exact NLP (5) assuming the entry was one horse. Then the same examples were solved supposing the entry was two horses (the formulation of the NLP was adjusted to consider the possibility of the two horses finishing first and second and then receiving a high place payoff) but the win bet on the entry was lowered, using an iterative scheme, until the optimal bet was the same as the optimal bet assuming the entry was one horse. This procedure gave pairs of iif and qi' where iii was the probability" of entry i winning in part a (thinking of the entry as one horse); and q, was the adjusted probability that gave the same optimal bet when thinking of the entry as two horses as was observed when treating it as one horse. Since the regression formula (13) gives the approximate optimal place bet when the horse's probability of winning is q" then using ii; in tha t formula gives the approxi- mate optimal place bet when the coupled entry's probablity of winning is q,. 'The data used were cases where Ihe expecled return (equation (7)) was .. 1.16. i.e .. the cases of inleres!. For instances wilh a low expecled return the correction (actor is meaningless. --- ## Page 718 Transaction Costs, Extent of Inefficiencies, Entries and Multiple Wagers 689 A RACETRACK BETTING MODEL 399 The regression rela ting qf and q; is qf = 0.99Iq; + 0.137q? + 3.47 X IO- 7WO (17) (R 2 = 0.9998, SE = 0.00 161, all coefficients highly significant). The examples from which the data were derived spanned many wealth levels and pool sizes. While the wealth level was a very significant independent variable the pool size was found to be statistically insignificant. Therefore to compute the optimal place bet on coupled entry i the procedure is: (0) determine that EXP is large enough to consider a place bet, (I) set q; = WJ W, (2) determine qf, and (3) substitute qf, Wo, P, P; in Table I. The same procedure was carried out for the optimal show bet on coupled entry i: (0) determine that EX' is large enough to consider a show bet, (I) set q; = W;/ W, (2) determine q/ = 1.07q; + 4.13 X IO- 7WO - 0.00663, and (R 2 = 0.999, SE = 0.00298, all coefficients highly significant), and (3) substitute q/, wo, S, and S;, in Table 2. (18) Three additional questions to be considered are: (I) Will the results be different with a weighting other than 1/3 and 2/3 on the two horses in the entry? (2) Should there be larger adjustmertts for entries of three or more horses? and (3) Should there be adjustments of the eql\ations for a single horse running against a coupled entry? The answer to all three questions is yes, but how important is it to account for these possibilities? (I) The coupled entry adjustment appears to be fairly robust to the weighting. Also each race would require handicapping to determine its more accurate weighting. Since that goes beyond the scope of this research (in fact the system described in ZH requires absolutely no handicapping), we choose to opt for no adjustments for different weightings. In the extreme case where qlA ~ ql and qlB ~ 0 it may be better to treat the entry as a single horse and ignore the entry adjustments. (2) The additional benefits of three or more horses in an entry are small beyond accounting for the entry as two horses. Thus we suggest the simplification of no further adjustment. (3) Generally the expected return equations on a single horse running against an entry wi\! overestimate the true expected return. In most cases though the bias is small and again we suggest no further adjustment. 6. Multiple Bets Occasionally there is more than one system bet in a given race. Since the optimal bet equations in Tables I and 2 were calibrated assuming only one place bet or one show bet in a race it is not correct, in a multiple betting situation, to calculate each bet individually from the tables and then wager those amounts. Often that would result in overbetting but there are also times where, for diversification reasons, that would actually result in under betting. We have attempted to deal with the most common multiple betting situation-a place and show bet on the same horse. Ninety-eight cases of place and show system bets on the same horse were analyzed covering a wide range of track handles, q,'s, and wo's. Using the optimization model (5) resulted in the quadruples (Pt,St, P~ ,s;). The P~ is the optimal place bet supposing it is the only good bet in the race, s~ is the optimal show bet supposing it is the only good bet in the race, and (Pt, St) is the --- ## Page 719 690 D. B. Hausch and W. T. Ziemba 400 D. B. HAUSCH AND W. T. ZIEMBA ~r--------------------------------=' ... FIGUR.E I. Betting Wealth Level Histories for System Bets at the 1981-82 Aqueduct Winter Meeting Using an Eltpected Value Cutoff of 1.14 for Track Takes of 14%. 15% and 17%. optimal pair of place and show bets when they are considered together. The PA and sA are the values we should calculate from Tables 1 and 2. Then Pt was regressed on PA, SA, Wo, Pi> P and qi ' The only statistically significant independent variables were PA and sJ leading to the regression equation Pt = 1.59pA - 0.639sJ (19) (R' = 0.967, SE = 73.7, both coefficients highly significant). A similar procedure for St yields S; = 0.907s': - O.l34pA (20) (R 2 = 0.992, SE = 72.6, both coefficients highly significant). 7. Effect of the Track Take The track take is a commission of 14-22% on every dollar wagered. As mentioned in §4 a change in Q, the track payback proportion, can have a substantial effect on EX; and EX~ . A change in Q can also have a surprisingly dramatic effect on long run profit. This is illustrated in Figure 1 using data from the 1981-82 Winter Season at Aqueduct, New York4 and supposing three different track takes: 14%, 15% and 17%. In 1981-82 the track take was 15%, earlier it had been 14% and just recently it has increased to 17%. Assuming an initial wealth of $2,500 the final wealths are $7,090, $6,292 and $5,058 at track's takes of 14%, 15% and 17% respectively. The track take increasing from 14% to 15% dropped profits by $798 (17.4%). The track take increasing from 15% to 17% dropped profits by $1,234 (32.5%). 8. Effect of Breakage In addition to the track take, bettors must also pay an additional commission called breakage. This commission ·refers to the funds not returned to the betting public because 'the payoffs are rounded down to the nearest 1O¢ or 20¢ on a $2 bet. For example, a payoff net of the track take of $6.39 would pay $6.30 or $6.20, respectively. 4The data consisted of the final win. place and show mucuels [or the 43-day period December 27. 1981-March 27. 1982. During this period 3,470 horses ran in 380 races. Thanks go to Dr. Richard Van Slyke for collecting these data for us. --- ## Page 720 Transaction Costs, Extent of Inefficiencies, Entries and Multiple Wagers 'IX» A RACETRACK BETTING MODEL _"0 ~'QIo(l9. b'l'Qkoql' 01 S trnl$ 0'\ IhI- dQrtor -- or.o"'OQl'of ~Ctfll$ DrlIt". cb'1CII' 691 401 FIGURE 2. Wealth Level Hislories at Exhibition Park (1978) with Alternative Breakage Schemes. Breakage occurs in the win, place and show as well as other pools. We refer to rounding down to the nearest \O¢ on a $2 bet as "5¢ breakage" that is 5¢ per dollar and rounding down to the nearest 20¢ on a $2 bet as "IO¢ breakage". Initially most tracks utilized a 5¢ breakage procedure. However, in recent years more and more tracks have switched to 10¢ breakage. The \O¢ breakage is never less than 5¢ breakage and usuaIly is considerably more. As a percentage of the payoff breakage usually increases as the payoff becomes smaller unless the payoff is close to the breakage roundoff amount. An exception is the "minus pool" when $2. \05 m.ust be paid on a winning $2 ticket even if the payoff before breakage is only $2.08, or even $1.73. On average bettors lose about 1.79% of the total payoff to 5¢ breakage and 3.14% to 10¢ breakage on bets using the system in ZH. Adding these amounts to the track take gives the total commission. For example, at Churchill Downs in Louisville, Kentucky the 15% track take becomes about 18.1% with their 10¢ breakage, and at Exhibition Park in Vancouver, British Columbia the 15.8% track take becomes about 17.6% with their 5¢ breakage. Thus to determine the full commission at a given racetrack one must take into account the breakage as well as the track take. The fuIl extent of the effect of breakage is shown in its effect on profits. Using the 1978 system bets for Exhibition Park with an initial wealth of $2500 one would have $8319 at the end of the year without breakage. With 5¢ breakage one would have $7521 and $6918 with 10¢ breakage. The effect of breakage throughout the 1978 season is shown in Figure 2. In addition to taking money away from total wealth the breakage has the effect of lowering the bet size (because of this lower wealth) thus resulting in lower future profits. These calculations indicate that 5¢ breakage averages 13.7% of profits and \O¢ breakage averages 24.1 % of profits on system bets. These calculations indicate that breakage (especially the very common 10¢ variety) is a very substantial cost. The costs are highest when one is placing bets on short odds horses. This is, unfortunately, an unavoidable aspect of the system presented in ZH. 9. Results Using the Betting System In ZH many examples are presented showing precisely how to use the system. In Table 3 we present summary statistics on system bets made at several tracks over different seasons. The data sets for Aqueduct, Santa Anita and Exhibition Park 1978 were coIlected after their seasons finished. The Exhibition Park 1980 and Kentucky Derby Days data were collected race by race at the track. In all cases initial betting wealth is assumed to be $2,500. The different expected cutoff levels reflect the quality of the horses at the different tracks. The most common system bet is to show on a favorite. Show and place bets occur about 85% and 15% of the time, respectively. The percent of bets won is about 59% 5 In Kentucky it is $2.20. --- ## Page 721 TABLE 3 Summary Sialislics on Syslem BelS Made al Aqueduci in 1981/82. Sonia Anila in 1973/74. £xhibilion Park in 1978 and 1980. and allhe Kenlucky Derby Days 1981/82/83 wilh an Inilial Belling Weaflh 0/$2500 Number Number Percent of Average Rate of Track Number Number Expected of of Percent Bets Won Total Payout Return and of of Track Value System Bets of Bets Weighted by Money Track Total Per S2 on Bets Season Days Races Take Cutoff Bets Won Won Size of Bet Wagered Take Profits Bet Made Aqueduct 43 380 15% 1.14 124 68 55% 65% S42.686 $6.403 S3.792 $3.33 8.9% 1981/82 Santa 75 627 15% 1.14 192 114 59% 69% $51 .631 S7.745 S2.837 $3.16 5.5% Anita 1973/74 Exhibition 110 1.065 18.1% 1.20 174 97 56% 72% $49.991 $9.048 S5.198 $3.08 10.4% Park 1978 Exhihition 10 90 17.1% 1.20 22 16 73% 77% S 5.403 S 924 $1 .216 $3.18 22.5% Park 1980 Derby 3 30 15% 1.10 19 17 89% 96% $12.766 SI.915 $5.462 $2.97 42.8% Days· 1981/82/83 Totals and Weighted Averages 241 2.192 - 531 312 59% 71% SI62,477 S26,035 $18,505 $3.12 11.4% 0- '0 tv ~ !=>:I ~ 1:; () ~ ::, ;:; I:l... ~ ~ ~ "" ~ ~ ::, --- ## Page 722 Transaction Costs, Extent of Inefficiencies, Entries and Multiple Wagers 693 A RACETRACK BETTING MODEL 403 while the percent of bets won weighted by the size of the bet is 71 %. This difference is because the bets are on shorter odds horses which finish in-the-money more often. At a track such as Santa Anita with large betting pools, our bets do not affect the odds very much and the average bet is about 7% of the betting pool. Over all these thousands of races and hundreds of system bets the total amount wagered was $162,477. The track take was $26,035. Our profit was $18,505, for an 11.4% rate of return on dollars wagered. The higher rates of return were on the races where we were at the track, so we were able to skip rainy days and reject certain horses on the basis of very simple handicapping rules. The lower rates of return, as expected, were at the tracks where we had no information other than the win, place and show mutuel pools. A simple correction which has a surprisingly large effect is removing from the Exhibition Park 1978 data the days when the track was not a fast track, i.e. rainy days when the track was slow, muddy, heavy, sloppy, etc. Doing so decreases the total money wagered from $49,991 to $32,811 but increases the profit from $5,198 to $5,863. Thus the rate of return on the "fast track" days is 17.9%, up from 10.4% over all days. It also increases the rate of return over all the racetracks from 11.4% to 13.2%. Since the bets usually have a high EX; or EX; we might expect a rate of return around 16%-20%. Remember that EXf and EX; are on the first dollar bet. These values drop as we bet large amounts due to our bets affecting the odds. These profits do not consider several minor "entertainment type" costs that one must incur by actual attendance at the track to apply the system. Parking, gasoline. racing program, racing form, track admission and food amount to $3-10 or more. For example. at $10 per day Aqueduct's profits of $3792 over 43 days become $3362. Finally the average payout per $2 bet ranged from $2.97 to $3.33 at the various tracks, with an average of $3.12. This value is actually a high show return when one considers the heavy favorites the system often picks. 10. Will the Market Become Efficient? As more and more individuals use this system the markets for place and show belling will tend to become efficient. Two important questions are : (I) How many p'eople can play this system and still have it provide a return of 1O-20%? and (2) How many people can play this system before the market becomes efficient enough that expected profits are zero? To consider the first question we can determine how much additional money can be wagered on a particular horse to place or show before the expected value per dollar bet drops to the suggested cutoff for good betting opportunities. As an illustration Figure 3 indicates this amount to show for a cutoff of 1.14. These figures are based on a track take of 17.1 % so for lower track takes more can be bet and for higher track takes less can be bet. FIGURE 3. How Much Can Be Bel. 8. by System Bellors Relative to the Crowd's Show Bel. S,. on Horse i to Lower the Expected Value to Show on the Horse i from Z to 1.14. When the Track Take is 17.1%. --- ## Page 723 694 D. B. Hausch and W T. Ziemba 404 D. B. HAUSCH AND W. T. ZIEMBA An example that more directly answers questions 1 and 2 is provided below and is based on the data given in §4.3 on Sunny's Halo, the 1983 Kentucky Derby winner. Additional examples appear in ZH. a 1.10 1.06 1.02 Sunny's Halo Optimal System Bets (Using the Actual Data One Minute Before Post Time) Assuming a Betting Wealth of "'0 Total Amount That Can be Bet Before the Expected Return per Dollar Bel Drops to a $19,323 $41,175 $68,409 S200 Betting Wealth "'0 $500 $1000 $2000 $11 S31 552 $96 Number of System Bettors Needed to Drop the E"pected Return per Dollar Bet to a Assuming a Betting Wealth of $200 $500 SI000 $2000 1,757 623 372 201 3,743 1,328 792 429 6,219 2,207 1,316 713 Our results show that a ... 1.02 is a breakeven cutoff and a = 1.06 yields a rate of return of about 5-6%.6 References FAMA, E. F., " Efficient Capital Markets: A Review of Theory and Empirical Work," J . Finance, 25 (1970), 383-417. HAUSCH, D. B., W. T. ZIEMBA AND M. RUBINSTEIN, "Efficiency of the Market for Racetrack Betting," Management Sci., 27 (1981), 1435-1452. HARVILLE, D. A., "Assigning Probabilities to the Outcome of Multi-Entry Competitions," J. Amer. Statist. Assoc., 68 (1973), 312-316. KALLBERG, J. G . AND W. T. ZIEMBA, "Generalized Concave Functions in Stochastic Programming and Portfolio Theory," in Generalized Concavity in Optimiration and Economics, S. Schaible and W. T. Ziemba (Eds.), Academic Press, New York, 1981,719-767. SNYDER, W. W., "Horse Racing: Testing the Efficient Markets Model," J. Finance, 33 (1978), 1109-1118. ZIEMBA, W. T. AND D. B. HAUSCH, Beat the Racetrack, Harcourt, Brace and Jovanovich, San Diego, 1984. -- AND R. G . YICKSON, Eos., Srochost;c Oplimizalion Models in Finance, Academic. Press, New York, 1975. --- ## Page 724 Efficiency of Racetrack Betting Markets, 567- 574. World Scientific (2008) 48 The Dr.Z Betting System in England t William T. Ziemba and Donald B. Hausch Managemen/ Science Division, Faculty of Commerce, University of British Columbia, Vancouver Be, Canada V6T lZZ School of Business, University of WISconsin, Madison, WISconsin 53706 Abstract 695 The betting strategy proposed in Hausch, Ziemba and Rubinstein (1981) and Ziemba and Hausch (1984,1987) has had considerable some success in North American place and show pools. The place pool is England is very different. This paper applies a similar strategy with appropriate modifications for places bet at British racetracks. The system or minor modifications also applies in a number of other countries such as Singapore with similar betting rules. The system appears to provide positive expectation wagers. However, with the higher track take it is not known how often profitable wagers will exist or what the long run performance might be. I. IntroductIon At North American racetracks, the parimutuel system of betting utilizing electric totalizator boards in the dominant method of betting. Las Vegas and the other legal sports books may set odds on particular betting situations, but these fixed odds are not available at racetracks. In England and in other Commonwealth countries, such as Italy and France, odds betting against bookies is the dominant betting scheme. This fixed-odds system is introduced in Hausch, Lo and Ziemba (1994). They also indicate a tendency of higher (lower) returns for lower (higher) odds ranges. Thus, the favorite-Iongshot bias appears to exist in England (see e.g. Ali (1977), Busche and Hall (1988» . This paper applies the betting system proposed by Hausch, Ziemba and Rubinstein (1981) and Hausch and Ziemba (1985) in England. This strategy is also called the Dr.Z system in the trade books Ziemba and Hausch (1984,1987) who discuss it more fully. The strategy utilizes the Kelly criterion (Kelly (1956» which maximizes the expected logarithm of wealth. The Kelly criterion has several advantages. First, it maximizes the capital growth asymptotically. Second, it prevents bankruptcy. Third, the expected time to reach a specified goal is minimum when the goal increases. Fourth, it is superior to any different strategy in the long mn. These properties were proved in Breiman (1961). See McLean, Ziemba and Blazenko (1992) for discussion of these properties. n. The System Instead of the North American parimutuel system of win, place, and show, the bets in England are to win and place. By "place" the British mean "finish in the money.· This is what North Americans call show except for one important difference. The number of horses that can place in a particular race is dependent on the number of starters. t Modified from an Appendix in Ziemba and Hausch (1987). Efficiency of Racetrack Belling Markets Copyright © 1994 by Academic Press. Inc. All rights of reproduction in any form reserved. 567 --- ## Page 725 696 W T. Ziemba and D. B. Hausch 568 W. T. ZIEMBA AND D. B. HAUSCH Table 1. Relationship between number of horses that place and number of starters Number of Horses that Place Number of Starters one: the winner four or less two: winner and second five, six, or seven three: winner, second, and third eight to fifteen four: winner, second, third, and fourth sixteen or more The place pools are not shown on the tote board, but the current payoffs for place bets for each horse are flashed on the screen. Bookies, on the othet hand, simply pay a percentage of the win odds, as shown in Table 2. There are many types of exotic bets as well. The tote jackpot corresponds to what North Americans call the pick six or sweep six. The tote placepot bet has no analogue in North America. The average rates of return on various bets on and off course against a bookmaker or the tote are listed in Table 3. The track take is 5% larger in the place pool than in the win pool. The tote take is larger than what the bookies make on average, and on-course betting takes are much less than off-course takes. The races in England are on the turf for distances of generally at least a mile, except for some shorter races for two-year·-olds. The season in southern England is unique in that races are run for about three days at each race course. The jockeys, trainers, and so forth then move on to a new course. After a month or so they return to the same course. Handicapping is very sophisticated in England. It has to be, with little infonnation easily accessible (they have no analogue of the Daily Racing Fonn, although some past performances are available in newspapers) and all that moving from course to course. The method of computing the place payoffs in England differs from that used in North America. In both locales, the net pool is the total amount wagered minus the track take. In North America, the cost of the winning in-the-money tickets is first subtracted to form the profit. This profit is then shared equally among the in-tbe-money horses. Holders of winning tickets receive a payoff consisting of the Original stake plus their proportionate share of the horse's profits. This means that the amount of money wagered on the other horses in the money greatly affects the payoff. In England, the total net pool is divided equally among the horses that finish in the money. This means that the payoff on a particular horse depends upon how much is bet on this horse to place but not on how much is bet on the other horses. Since the minimum payoff is £1 per £1 wager, management is able to keep a control on betting for particular favorites. Once this minimum level is reached, it does not pay to wager on a given horse. This occurs whenever the percentage of the place pool that is bet on a given horse becomes as large as Q,. which is the track take for place, divided by m, which is the number of in-the-money horses. In a race with 8-15 starters, if Q, is about 0.735, and m=3, thejust-get-your-money-back point is reached when the bet on a particular horse to place becomes 24.5~ of the total place pool: 0.735/3 = 24 . 5~. Hence in England you will often see horses whose place payoffs are £1 or just slightly higher. This method of sharing the place pool tends to favor longer-priced horses at the expense of the favorites. --- ## Page 726 The Dr. Z Betting System in England THE DR. Z BETIING SYSTEM IN ENGLAND ~umbcr of R",nncn Two 10 five Six or seven Eighl or more Twch'e 10 tifteen Sixteen 10 twent}··one Twenty·two or more S,wm: Roduchi'c (197S). Table 2. Bookmakers payoff for place bets fnctioa of Win Odds Paid on Hones T),pe of Il>cc Place Element Rcprded No place betting Any l Finland . ~ond Anyexcepl 1 Fint, handicaps second. involving and third IWdve or morr runnen IUndlcaps : fint, second. and third Handlcaps Fint, second. third, and fourth Handicaps fint, second, third. and fourth 697 569 Table 3. Rates of return on different types of betS in England on thoroughbred and greyhound racing Type of Bcl On·course bookmaker Olf·course bookmaker Single bello " 'in " 'ith olr·course bookmaker Double bello " 'in ... ith oll'·course bookmaker Treble bet 10 "'in " 'ith off·course bookmaker lTV x"en bello .... in ... ·ith olf·course bookmaker Computer slraighl forecasl v";lh olf·course bookmaker Greyhound forecasl with olf·course bookmaker Greyhound forecast double ... ·ith olf·course bookmaker Place clement of e .. ch· ... "')" with oll'·course bookmaker ."nle·post betting .. .-Ith off·coutR bookmaker Horse race 10le ... ·in pool (on course) Horse I'3ce 10le ... ·in pool (oll' course) Horse race tOle place pool (on course) Horse race tOle place pool (off course) Horse raCe tOle daily double pool (on course) Horse race 10le daily double pool (off course) Horse race 10le da.i.l~· treble pool (on coune:) Horse racc 10le da.i.ly treble pool (olf course) Horse race tote da.i.l~· forecast pool (on course) Horse race tOle da.i.lr forecasl pool (olf course) Horse raCe lote jackpOI pool (on course) Horse raCe 10le Jackpol pool (off course) Horse race 10le pbccpol pool (on course) Horse race 10le pbccpol pool (oft' course) Greyhound lote pool betting. average S ...... ,: Ro,h"hild (l9i6). J\.\te of ReNm <" ) 90 81 85 78 72 70-75 65 76 58 80 96 80 i7 75 -:'2 7. 71 70 67 70 67 "0 67 70 67 83.5 --- ## Page 727 698 W T. Ziemba and D. B. Hausch 570 W. T. ZIEMBA AND D. B. HAUSCH The current track take to win is about 20.6% and to place is 26.5%. and the breakage is of lOc variety. or more properly lOp. for pence2. These track takes are much higher than those in North America. Since the track paybacks to win and place are different. we call the former. Q..=0.794. and the latter. Q,=0.735. It is easy to apply the Dr.Z system in Great Britain, although with its much higher track takes, there may not be many Dr.Z system bets. see Mordin (1992) for a discussion of this. We utilize the substitution that q, = QJO" where 0, are the odds to win on the horse under consideration. The expected value per pound bet to place on horse i is EXPlace - (probability of placing) (place odds) - (Prob) (PO,). (1) In (1). PO, refers to the odds to place on horse i. Prob. the probability of placing is determined as follows.' • -:We can calculate these track takes as follows: The payoff on horse i if it wins is Qw W/W, where Qw is the track payback to win. and W. and W are the bet amount of horse i and the total bet amount. respectively. So let q, = W,tW, the efficient-market assumption. Let B be the average breakage, namely, 4.5p. Since breakage can be 0.1.2 ...... 9 pence, its average is 4.5p. Then the payoff on i is Q,/q,-B. which equals the odds 0,. since the odds are based on total return (not return plus original stake as in North America). So q, = QJ(B+O,). Summing over all n horses gives A 1 - Q .. :E (B 0)' ; .. 1 + I since some horse must win. Hence 1 Q ... - -.--- For place. there are one. two. three. or four horses that are in the money. depending upon the number of starters. So m Q, - --'--- where m = 1.2.3 or 4. With the above formulas. Qw '" 0.794 and Q,. '" 0.745. 'These equations were developed using the 1981-1982 Aqueduct data to relate probability of in-the- money finishes to q. the probability of winning and n, the number of horses. Equations (2), (3) and (4) had R2 of 0.991,0.993 and 0.998, respectively. These equations are valid when q ranges from 0 to 0.6 --- ## Page 728 The Dr. Z Betting System in England 699 THE DR. Z BETTING SYSTEM IN ENGLAND 571 With n=5 to 7 horses, the first 2 horses place and Prob - 0.0667 +2.37q -1.61q2 -0.OO97n. (2) With n=8 to 15 horses, the first 3 horses place and Prob - 0.0665 + 3.44q - 3.47q2 - 0.0049n. (3) With n= 16 or more horses, the first 4 horses place and Prob - 0.0371 +4.47q -6.29q2 -O.OOI64n. (4) Figures 1,2, and 3 determine Prob directly using only 0" the win odds on the horse in question. Figure 1 applies when there are five, six or seven horses . Figure 2 corresponds to equation (3) and applies when there are eight to fifteen horses. Finally, Figure 3 corresponds to equation (4) and applies when there are sixteen or more horses. The optimal Kelly criterion bet is to wager (prob POi-I)/(POi-l) percent of your betting wealth'. We can determine the optimal fraction of your wealth to bet indicated by equation (5) using Figure 4. for (2), from 0 to 0.45 for (3), and 0 to 0.3 for (4), which should be the case in most instances. However, Figures 1,2 and 3 are valid for any q. 'In a race with n=2,3, or 4 horses, only one horse places, the winner. Such races are rare. Also, it is unlikely that the win and place pools would then become so unbalanced as to yield a Dr.Z system bet. However, one would occur when PO/Oj was at least 1.44, for a track payback of 0.794 and an expected- value cutoff of 1.14, since 1.14/0.794 is 1.44. In such a case, one would have a good bet. 'We have assumed that your bets will be small and hence will not affect the odds very much . Thus to determine the optimal bet b for betting wealth wo, you maximize Prob log(wo + (PO;- I)b] + (1 - Prob)log(woO)' whose solution is equation (5). --- ## Page 729 700 W T. Ziemba and D. B. Hausch 572 W. T. ZIEMBA AND D. B. HAUSCH 1.1)r-------------------, I) , 2 3 ? 9 Odds 10 Win on Ho,.. I Figure 1. Probabilities of placing for different odds horses when the race has five to seven starters o 3 ? 8 9 10 Figure 3. Probabilities of placing for different odds horses when the race has eight to fifteen starters --- ## Page 730 The Dr. Z Betting System in England 701 THE DR. Z BETTING SYSTEM IN ENGLAND 573 0.25 o 3 s 6 7 8 9 Odds to Win on Horu J Figure 3. Probabilities of placing (or different odds horses when the race has sixteen or more starters 1.0 1 ~ 0.91 ~ O.S 1 ~ 0.7 1 .§ 0.6 i j :: j .. 0.2 ~ .E e 0.1 0.0 Place Odds, PO, Proc • 0.95 Prob - 0.90 Prob - 0.80 Prob - 0.70 Prob - 0.60 Proo • O.SO Prob ~ 0.40 Prob - 0.30 :ure 4. Optimal bets when the probability of placing is Prob anel the place odds of the horse in question ;POj --- ## Page 731 702 W. T. Ziemba and D. B. Hausch 574 W. T. ZIEMBA AND D. B. HAUSCH References Breiman,L. (1961) ·Optimal gambling systems for favorable games.· in Proceedings of the Fourth Berkeley Symposium. Mathematical Statistics and Probability I, 65-78. University of California Press. Figgis,E.L. (1974) "Rates ofrerurn from flat race betting in England in 1973. · Sponing Life 11 (March). Hausch,D.B., Ziemba,W.T. and Rubinstein,M. (1981) "Efficiency of the market for racetrack betting.' Management Science 27, 1435-1452. Hausch,D.B. and Ziemba,W.T. (1985) "Transactions costs, extent of inefficiencies, entries and multiple wagers in a racetrack betting model." Management Science 31, 381-394. Kelly ,J .L. (1956) " A new interpretation of information rate. " Bell System Technica/Journa/ 35, 917 -926. MacLean,L.C., Ziemba,W.T. and Blazenko,G. (1992) "Growth versus security in dynamic investment analysis.· Management Science 38, 1562-1585. Mordin,N. (1992) "Grab your place in the pool!" Sponing Life. Rothschild, Lord (1978) Royal commission on gambling, Vols I and II. Presented to parliament by Command of Her Majesty (July). Ziemba,W.T. and Hausch ,D.B. (1984) Beat the racetrack. Harcourt Brace Jovanovich, San Diego. Ziemba,W.T. and Hausch,D.H. (1987) Dr.Z's beat the racetrack. Revised edition. William Morrow and Co. Inc. New York. --- ## Page 732 Journal a/Business, 592, 287- 318 (1986) 49 Robert R. Grauer Simon Fraser University Nils H. Hakansson University of California, Berkeley A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds, and Bills, with and without Small Stocks· I. Introduction In two earlier papers (Grauer and Hakansson 1982; Grauer and Hakansson 1984), we applied the multi period portfolio theory of Mossin (1968), Hakansson (1971, 1974), Leland (1972), Ross (1974), and Huberman and Ross (1983) to the construction and rebalancing of portfolios composed of U.S. stocks, corporate bonds, gov- ernment bonds, and a risk-free asset. Borrowing was ruled out in the first article, while margin purchases were permitted in the second. The probability distributions used were naively es- timated from past realized returns in Ibbotson • Presented at the Australian Graduate School of Manage- ment and at Macquarie University, both in Sydney, Southern Methodist University, Duke University, the University of South~rn California, the London Graduate School of Busi- ness, and the annual meeting of the Western Financo Associ- ation in Scottsdale, Arizona. The authors thank the partici- pants of these seminars, especially Michael Brennan and Ehud Ronn, for helpful comments. Financial support from the Social Sciences and Humanities Research Council of Canada and the Financial Research Foundation of Canada is gratefully acknowledged. The authors also wish to thank Changwoo Lee for research assistance and are greatly in- debted to Frederick Shen for computational assistance. (Journal of Business, 1986, vol. 59, no. 2, pt. 1) © 1986 by The University of Chicago. All rights reserved. 0021-9398/86/5902-0005$01 .50 287 703 This paper applies multiperiod portfolio theory to the construc- tion and rebalancing of portfolios composed of U.S. stocks, corporate bonds, government bonds, and a risk-free asset, with small stocks included as a separate investment vehicle. Probability assessments are based on the past, joint empirical distribu- tion. Our principal findings are (I) small stocks, while totally ig- nored at times, entered even the most risk- averse portfolios most of the time and (2) small stocks, when chosen, tended to re- place common stocks, except in the 1970s and early 1980s, when they were primarily held in lieu of the risk-free as- set. --- ## Page 733 704 R. R. Grauer and N. H. Hakansson 288 Journal of Business and Sinquefield (1982) and in their Center for Research in Security Prices (CRSP) 1926-83 data base, and both annual and quarterly hold- ing periods were employed from the mid-1930s forward. The results of both papers revealed that the gains from active diversification among the major asset categories were substantial, especially for the highly risk averse strategies. In addition, we found evidence of substantial use of, and gains from, margin purchases for the more risk-tolerant strate- gies from the mid-1930s to the mid-1960s. In the present paper, small stocks are included as a separate invest- ment vehicle. Thus the opportunity set is expanded to five categories: common stocks, government bonds, corporate bonds, small stocks, and a risk-free asset. The resulting sequential portfolio problem is solved both with and without leverage. When used, all borrowing is assumed to bear interest at the call money rate + 1 % and to be limited in size by applicable initial margin requirements. Our principal findings are (1) small stocks, while totally ignored at times, entered even the most risk-averse portfolios most of the time, (2) small stocks, when chosen, tended to replace common stocks, except in the 1970s and early 1980s, when they were primarily held in lieu of the risk-free asset, and (3) small stocks had a salutary effect on the geometric means of realized returns, especially from the mid-1930s to the early 1950s in the presence of margin purchases. In the 1934-47 subperiod with leverage, for example, small stocks raised the geometric mean return for the power .5 strategy from 9.09% to 25.95% while reducing variability. II. Theory Despite explosive development over three decades and extensive ap- plication to the construction of equity portfolios, modern portfolio theory has found relatively little use in the larger portfolio context, namely, the choice of the proportions to be held in the major categories of common stocks and in different types of bonds, money market in- struments, real estate, and foreign securities. There are several reasons for this. First, extant portfolio theory, being principally based on the mean-variance model, is fundamentally single-period in nature, where- as the larger problem accents the multiperiod, sequential nature of investment decisions. On top of this, since the universe of interest extends well beyond common stocks, extant betas are too narrowly defined to be useful, and the appropriate betas are not easily estimated because of data problems concerning the market weights of bonds, for example. At the other extreme, continuous-time portfolio theory is somewhat intractable in a world of nontrivial transaction costs. Fi- nally, extant models rely heavily on narrow classes of theoretical (sta- tionary) return distributions, with limited ability to capture the richness of joint, real-world stochastic processes. There is, however, a middle category of investment models, usually --- ## Page 734 A Half Century a/Returns on Levered and Unlevered Portfolios a/Stocks, Bonds, and Bills 705 Half Century of Retllrns 289 classified under the heading of discrete-time multiperiod portfolio theory, which has been largely ignored in portfolio selection applica- tions. This is so despite the fact that these models have a strong foun- dation in theory and lend themselves naturally to the problem of re- balancing portfolios over many periods (200 quarters in the present study). An additional virtue of these models is that they can handle return distributions of every conceivable shape and are unfazed by such things as non stationary returns or distributions that are just plain unknown beyond the period just ahead. To review, consider the simplest reinvestment problem, in which the market is perfect and returns are independent over time but otherwise arbitrary and not necessarily stationary. The investor has a preference function Vo (with V'o > 0, V'O < 0) defined on wealth 11'0 at some terminal point (time 0). Let Wn denote the investor's wealth with n periods to go, rin the return on asset i in period n, Z;n the amount invested in asset i in period n (with i = I being the safe asset), and V,,(wn) the relevant (unknown) utility of wealth with n periods to go. At the end of period n (time n - I), the investor's wealth is M wn-.(Zn) = I (rin - r.n)Zin + wn(l + r. n), ;=2 where Z = (Z2n , .. . , ZMIl) and M is the number of securities. Consider the portfolio problem with one period to go. The investor, with w. to invest, must now solve max E[Vo(wo(z.))] == V.(WI)' l,llI', Clearly, V.(w.) represents the highest attainable expected utility level from capital level w. at time I and thus the "derived" utility of 11' •• Employing the induced utility function V .(WI), the portfolio problem with two periods to go becomes V 2(W2) == max E[V 1(w.(Z2))]. z!lw2 Thus with n periods to go we obtain (the recursive equation) Vn(wn) = max E[Vn-.(wn-.(Zn))], n = 1,2, . .. z"! \1',, Examining the above system, it is evident that the induced utility of current wealth, Vn(wn), generally depends on "everything," namely, the terminal utility function V~, the joint distribution functions of fu- ture returns, and future interest rates. There is, however, a special case in which Vn(wn) depends only on Vo: this occurs (Moss in 1968) if and only if Vo(wo) is isoelastic; that is, if and only if I = -11'''1 "Y ' "Y < I. --- ## Page 735 706 R. R. Grauer and N. H. Hakansson 290 Journal of Business (Note that for 'Y = 0, Uo[wo] In wo.) The variable Un(wn) is now a positive linear transformation of Uo(wn); that is, we can write Un(wn) = ~ Who 'Y (1) For these preferences, the optimal investment policy z~ is proportional to wealth; that is, ' (2) where the xfn are constants. It is also completely myopic since it only depends on Uo and the current period's return structure and not on returns beyond the current period. Both of these properties hold only for family (1), which is also the only class of preferences exhibiting constant relative risk aversion. I Finally, (2) also implies that the utility of wealth relatives, V n(1 + r n), is of the same form only for this family; that is, 1 1 Un(wn) = - wh < = > Vn(1 + rn) = -(1 + rnr'· 'Y 'Y While the above properties are interesting, they are clearly rather special. However, the isoelastic family's influence extends far beyond its numbers. As shown by Leland (1972), Hakansson (1974), Ross (1974), and Huberman and Ross (1983), there is a very broad class of terminal utility functions Uo(wo) for which the induced utility functions Un converge to an isoelastic function, that is, for which Un(Wn) ~ ~ wh for some 'Y < 1. 'Y (3) Hakansson (1974) has also shown that (3) is usually accompanied by convergence in policy; that is, Thus the objectives given by (1) are quite robust and encompass a broad variety of different goal formulations for investors with inter- mediate- to long-term investment horizons.2 In particular, class (1) spans a continuum of risk attitudes all the way from risk neutrality ('Y = 1) to infinite risk aversion ('Y = _00).3 Having selected our model, we turn next to what we need to operate it. The major input to the model is an estimate of next period's joint return distribution for the various asset categories. 4 As in our previous I. This measure is defined as - wU~(w)IU~(w) and equals I - "I for the class (I). 2. The simple reinvestment formulation does ignore consumption of the course. 3. A plot of the functions (1/"1)(1 + r)' for several values of "I was given in Grauer and Hakansson (1982, p. 42). 4. For a comprehensive overview of the issues and problems associated with the estimation of return distributions, see Bawa, Brown, and Klein (1979). --- ## Page 736 A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds, and Bills 707 Half Century of Returns 291 studies, we based this estimate on the so-called simple probability assessment approach, In this approach, the realized returns of the most recent n periods are recorded; each of the n joint realizations is then assumed to have probability lin of occurring in the coming period, Thus estimates are obtained on a moving basis and used in raw form without adjustment of any kind. Since the whole joint distribution is specified and used, there is no information loss; all moments- marginal and conditional, for example-and every last shred of corre- lation are implicitly taken into account. It may be noted that the empir- ical distribution of the past n periods is optimal if the investor has no information about the form and parameters of the true distribution but believes that this distribution went into effect n periods ago.s Conse- quently, we have, as a starting point, resisted all temptations to param- eterize the input distributions. III. Calculations The model used can be summarized as follows . At the beginning of each period t, the investor chooses a portfolio, Xr, on the basis of some member, 'Y, of the family of utility functions for returns r given by V(l + r) = J.- (I + rp. (4) 'Y This is equivalent to solving the following problem in each period t: subject to where 'Y ~ 1 Xir ;;, 0, XLr;;' 0, XBr ~ 0, all i, t, ~Xir + XLr + XBr = 1, all t, i ~mitXit ~ 1, all t, i pr(l + ~xirir + xLrrLr + xBrrir ;;, 0) i a parameter that remains fixed over time; Xr - (Xlr , . . • , Xnr' XLr, XBr) ; 1, (Sa) (6) (7) (8) (9) Xir the amount invested in risky asset category i in period t as a fraction of own capital; 5. See Bawa et al. 1979, p. 100. --- ## Page 737 708 292 XLI XBr 1Trs R. R. Grauer and N. H. Hakansson Journal of Business the amount invested in the risk-free asset in period t as a fraction of own capital; the amount borrowed in period t as a fraction of own capital; the anticipated return on asset category i in period t; the return on the risk-free asset of period t; the borrowing rate at the time of the decision at the begin- ning of period t; the initial margin requirement for asset category i in period t expressed as a fraction; and the probability of event s in period t, in which case the random return 'ir will assume the value 'irs ' As noted, when 'Y equals zero, (4) reduces to the logarithmic utility function: v = In(l + ,). The equivalent of (5a) is then max E[ln(l + "'i.xir'it + XLI'LI + XBr'~r)]. (5b) x, i Constraint (6) rules out short positions, and (7) is the budget con- straint. Constraint (8) serves to limit borrowing (when desired) to the maximum permissible under the margin requirements that apply to the various asset categories. Finally, since "'i.Xil'il + XLI'LI + XBr'~r i is the (ex ante) return distribution for portfolio Xr, the expressions in parentheses in (5a) and (5b) represent the (ex ante) wealth relative in period t. On the basis of the probability estimation method described earlier, the (sequential) solution of the portfolio problem may be pictured as follows. Suppose quarterly revision is used. Then, at the beginning of quarter t, the portfolio problem (5a) or (5b) for that quarter uses the following inputs-the (observable) risk-free return for quarter t; the (observable) call money rate + 1% at the beginning of quarter t; and the (observable) realized returns for common stocks, government bonds, corporate bonds, and small stocks for the previous n quarters. Eachjoint realization in quarters t - n through t - I is given probabil- ity lin of occurring in quarter t. With these inputs in place, the portfolio weights for the various asset categories and the proportion of assets borrowed are calculated by solving system (5a)-(9) (or [5b]-[9]) via nonlinear programming methods. 6 At the end of quarter t, the realized returns on stocks, gov- 6. The nonlinear programming algorithm employed is described in Best (1975). --- ## Page 738 A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds, and Bills 709 Half Century of Returns 293 ernment bonds, corporate bonds, and small stocks are observed, along with the realized borrowing rate rBt (which may differ from the decision borrowing rate rit).7 Then, using the weights selected at the beginning of the quarter, the realized return on the portfolio chosen for quarter t is recorded. The cycle is then repeated in all subsequent quarters. 8 All reported returns are gross of transaction costs and taxes and assume that the investor in question had no influence on prices. There are several reasons for this approach. First, since this is one of the first studies in this area, we wish to keep the complications to a minimum. Second, the Ibbotson and Sinquefield series used as inputs and for comparisons also exclude transaction costs (for reinvestment of inter- est and dividends) and taxes. Third, rriany investors are tax-exempt, and various techniques are available for keeping transaction costs low. Finally, since the proper treatment of these items is nontrivial, they are better left to a later study. IV. Data The data used to estimate the probabilities of next period's returns and to calculate each period's realized returns on risky assets are the total monthly and annual returns on common stocks, long-term government bonds, long-term corporate bonds, and small stocks for 1926-83 in the Ibbotson and Sinquefield CRSP data file; both dividends and capital appreciation are taken into account. The risk-free asset used for quar- terly revision was assumed to be 90-day U.S. Treasury bills maturing at the end of the quarter; we used the Survey of Current Business and the Wall Street Journal as sources. In the annual portfolio revision case, the risk-free return was obtained from the yield, as of the begin- ning of the year, on that U. S. government obligation (note or bond) that matured on the date closest to the end of the year in question; we obtained the 1926-76 data privately from Roger Ibbotson and the re- mainder from the Wall Street Journal. Margin requirements for stocks were obtained from the Federal Re- serve Bulletin. Initial margins were set at 10% for government bonds and at 35% for corporate bonds. These levels are on the conservative side and are designed to compensate for the absence of maintenance requirements. 9 7. The realized borrowing rate rBr was calculated as a monthly average. S. Note that, if n = 32 under quarterly revision, then the first quarter for which a portfolio can be selected is the first quarter of 1934 since the period 1926-33 is required to develop the estimated return distributions used for that quarter's portfolio choice. 9. There was no practical way to take maintenance margins into account in our pro- grams. In any case, it is evident from the results that they would come into play only for the more risk tolerant strategies, and even for them only occasionally, and that the net effect would be relatively neutral. --- ## Page 739 7lO R. R. Grauer and N. H. Hakansson 294 TABLE 1 Portfolio A Portfolio B Portfolio C Portfolio 0 Portfolio E Portfolio F Portfolio G Portfolio H Portfolio I Portfolio J Composition of Fixed Weight Portfolios Proportion in: Government Corporate Stocks Bonds Bonds .40 .40 .20 .30 .30 .45 .20 .25 .55 .15 .20 .65 .10 .15 .75 .15 .90 .05 .05 .70 .05 .05 .50 .05 .05 .30 .05 .05 Journal of Business Small Risk-free Stocks Asset .20 .20 .10 .10 .10 .10 .20 .40 .60 As noted, the borrowing rate was assumed to be the call money rate + 1 %; for decision purposes (but not for rate of return calcuhltions), the applicable beginning of period rate, r'1u, was viewed as persisting throughout the period and thus as risk free. 10 For 1934-76, the call money rates were obtained from the Survey of Current Business; for later periods, the Wall Street Journal was the source. V. Results Because of space limitations, only a portion of the results can be re- ported here. However, tables 2-6 below and figures 1 and 2 provide a fairly representative sample of our findings. For comparison, we have calculated and included the returns for 10 fixed-weight portfolios. The compositions of these fixed-weight port- folios are shown in table 1. Portfolio Returns: The No-Leverage Case Table 2 compares the geometric means of the annual returns, along with the standard deviations of InO + r t), with and without small stocks for the subperiods 1936-47, 1948-65, and 1966-83 and for the full 1936-83 period, under quarterly portfolio revision (with a 40- quarter estimating period) in the absence of leverage. Note that the realized returns without small stocks differed substantially between the three subperiods. For example, the highest geometric mean was 7.75% for 1936-47, 15.69% for 1948-65,7.38% for 1966-83, and 10.11% for 10. It may be noted that the minimum differential between the borrowing and the lending rate was 1.4% (1935). while the maximum was 5.0% (1981). The differential was . 2% or less through 1952 and again in 1964-65. For most of the 1950s, 1960s, and 1970s, and again in 1983, the differential was between 2% and 3%; the exceptions were 1974 and 1980, when it was 4.5%, and, as already mentioned, 1981. --- ## Page 740 A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds, and Bills 711 Half Century of Returns 295 the whole 1936-83 period, In contrast, the realized returns were much more uniform with small stocks included: the highest geometric means were then 15.79% for 1936-47, 15.27% for 1948-65, 14.97% for 1966- 83, and 14.54% for the full period. For the 1936-47 subperiod, table 2 shows that the presence of the small stock category increased both the standard deviation and the geometric mean for each of the 16 strategies. These increases are rela- tively small for powers -75 through - 10 but quite substantial for the others, especially powers -1 and up. The situation for 1966-83 is similar but less dramatic. In the 1948-65 subperiod, on the other hand, the presence of the small stock category again led to uniformly higher standard deviations but also to uniformly lower geometric means. The differences are quite small, however. Standard deviations also in- creased across the board for the full period, as did the geometric means. Portfolio Returns: The Leverage Case Table 3 shows the geometric means of the annual returns, and the standard deviations of InO + rr) , with and without small stocks for the leverage case. This table is based on quarterly portfolio revision and a 32-quarter estimating period. Again it is evident that the presence of small stocks tended to increase both the geometric means and the standard deviations of returns. This was uniformly so in the 1966-83 subperiod as well as for the full 1934-83 period. This pattern was broken in only two cases: in the 1934-47 subperiod, when powers 0, .25, and .50 achieved lower standard deviations in the presence of the small stock category, and in the 1948-65 subperiod, when powers - 3 through 1 attained higher geometric means in the absence of small stocks. Table 3 also shows that small stocks had a rather uneven effect across the three subperiods. In the 1934-47 subperiod, they provided an enormous lift to most geometric means with very little effect on standard deviations. On the other hand, their effect in the 1948-65 subperiod was minimal. The effect in the last subperiod was fairly modest, while the effect of small stocks on returns over the full 50 years was rather substantial. Portfolio Compositions An examination of the strategies used reveals some interesting pat- terns. In the no-borrowing case, small stocks were ignored by all the active strategies, including the risk-neutral one, from the third quarter of 1955 through the first quarter of 1962, in the third quarter of 1963, and again in the first quarter of 1979. The rest of the time, however, small stocks tended to be an important investment outlet, mostly at the expense of common stocks. Even the ultraconservative power -75 --- ## Page 741 712 R. R. Grauer and N. H. Hakansson 296 Journal of Business TABLE 2 Comparison of Geometric Means and Standard Deviations of Annual Returns with and without Small Stocks, 1936-83: No-Leverage Case (Quarterly Portfolio Revision, 40-Quarter Estimating Period) A. 1936-47 Without Small Stocks With Small Stocks Geometric Standard Geometric Standard Portfolio Mean Deviation* Mean Deviation* Common stocks 6.71 22.28 6.71 22.28 Government bonds 3.46 3.62 3.46 3.62 Corporate bonds 3.25 2.26 3.25 2.26 Small stocks 14.65 41.68 Risk free .24 .18 .24 .18 Power 1 4.78 23.74 14.65 41.68 Power .75 5.19 20.98 12.64 41.72 Power .50 6.95 18.21 14.29 35.77 Power .25 7.75 14.92 15.79 30.19 Power 0 7.72 12.92 15.48 26.11 Power -1 6.18 8.31 9.88 14.29 Power -2 5.27 6.04 7.67 9.87 Power -3 4.79 4.96 6.54 7.71 Power -5 4.31 3.96 5.43 5.63 Power -7 4.03 3.39 4.94 4.66 Power -10 3.73 2.99 4.48 3.81 Power -15 3.53 2.68 4.03 3.24 Power -20 3. 11 2.49 3.50 2.92 Power -30 2.52 2.\3 2.78 2.51 Power -50 2.09 \.94 2.28 2.27 Power -75 1.74 1.65 2.07 2.14 Portfolio A 2.74 2.19 2.74 2.19 Portfolio B 3.78 5.45 3.78 5.45 Portfolio C 5.13 10.58 5. \3 10.58 Portfolio D 5.47 12.63 5.47 12.63 Portfolio E 5.75 14.71 5.75 14.71 Portfolio F 5.97 16.76 5.97 16.76 Portfolio G 6.59 20.13 6.59 20.\3 Portfolio H 8.75 23.70 Portfolio I 10.68 27.45 Portfolio J 12.36 31.32 --- ## Page 742 A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds, and Bills 713 Half Century of Returns 297 TABLE 2 (Continued) B. 1948-65 Without Small Stocks With Small Stocks Geometric Standard Geometric Standard Portfolio Mean Deviation* Mean Deviation* Common stocks 15.67 14.54 15.67 14.54 Government bonds 1.94 4.96 1.94 4.96 Corporate bonds 2.52 4.11 2.52 4.11 Small stocks 15.61 19.36 Risk free 2.29 .94 2.29 .94 Power I 15.67 14.54 14.03 17.12 Power .75 15.67 14.54 14.64 17.14 Power .50 15.67 14.54 14.70 17.16 Power .25 15.67 14.54 14.83 17.04 Power 0 15.67 14.54 14.90 16.98 Power -I 15 .69 14.54 15.27 16.21 Power -2 15.52 14.48 15.25 15.54 Power -3 15.18 14.42 14.93 14.72 Power - 5 13.65 14.45 13.24 14.60 Power -7 11.85 13.58 11.59 13.79 Power -10 9.48 10.91 9.35 11.09 Power -15 7.27 7.60 7.15 7.81 Power -20 6.12 5.68 6.06 5.86 Power -30 4.98 3.73 4.95 3.87 Power -50 3.92 2.19 3.91 2.31 Power -75 3.39 1.46 3.38 1.56 Portfolio A 2.26 3.56 2.26 3.56 Portfolio B 5.00 3.02 5.00 3.02 Portfolio C 8.41 5.99 8.41 5.99 Portfolio D 9.75 7.46 9.75 7.46 Portfolio E 11.09 9.01 11.09 9.01 Portfolio F 12.44 10.63 12.44 10.63 Portfolio G 14.37 12~0 14.37 12.90 Portfolio H 14.44 13.55 Portfolio I 14.47 14.41 Portfolio J 14.46 15.46 --- ## Page 743 714 R. R. Grauer and N. H. Hakansson 298 Journal of Business TABLE 2 (Continued) C. 1966-83 Without Small Stocks With Small Stocks Geometric Standard Geometric Standard Portfolio Mean Deviation* Mean Deviation* Common stocks 7.62 17.03 7.62 17.03 Government bonds 4.13 9.91 4.13 9.91 Corporate bonds 4.90 10.82 4.90 10.82 Small stocks 16.11 27.78 Risk free 7.37 2.86 7.37 2.86 Power I 7.38 12.03 14.97 27.29 Power .75 6.65 11 .33 13.06 25.72 Power .50 6.40 11.09 12.85 24.50 Power .25 6.34 10.85 11.86 24.30 Power 0 6.35 10.39 11.48 23.98 Power -I 6.84 9.07 10.24 22.37 Power -2 7.08 7.79 10.01 19.65 Power -3 7.18 6.22 9.89 15.98 Power -5 7.28 4.58 9.23 10.88 Power -7 7.31 3.86 8.84 8.35 Power -10 7.33 3.37 8.49 6.33 Power -15 7.35 3.05 8.17 4.74 Power -20 7.36 2.94 7.99 4.00 Power -30 7.36 2.86 7.80 3.36 Power -50 7.37 2.84 7.63 2.99 Power -75 7.37 2.83 7.55 2.88 Portfolio A 5.20 8.27 5.20 8.27 Portfolio B 6.03 7.96 6.03 7.96 Portfolio C 6.61 10.08 6.61 10.08 Portfolio D 6.92 10.99 6.92 10.99 Portfolio E 7.20 12.06 7.20 12.06 Portfolio F 7.48 13.38 7.48 13.38 Portfolio G 7.44 15.62 7.44 15.62 Portfolio H 9.41 16.86 Portfolio I 11.25 18.73 Portfolio J 12.95 21.06 --- ## Page 744 A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds, and Bills 715 Hal/Century of Returns 299 TABLE 2 (Continued) D. 1936-83 Without Small Stocks With Small Stocks Geometric Standard Geometric Standard Portfolio Mean Deviation* Mean Deviation* Common stocks 10.34 17.64 10.34 17.64 Government bonds 3.14 6.96 3.14 6.96 Corporate bonds 3.59 7.12 3.59 7.12 Small stocks 15.56 28.66 Risk free 3.64 3.40 3.64 3.40 Power I 9.74 16.69 14.54 27.97 Power .75 9.57 15.63 13.54 27.45 Power .50 9.93 14.65 13.90 24.97 Power .25 10.11 13.65 13.94 23.11 Power 0 10.11 13.03 13.75 21.78 Power -I 9.91 11 .77 12.01 18.14 Power -2 9.70 11.10 11.35 16.06 Power -3 9.50 10.56 10.89 13.93 Power -5 8.86 9.97 9.74 11.63 Power -7 8.15 9.12 8.87 10.24 Power -10 7.21 7.33 7.79 8.11 Power -15 6.35 5.33 6.74 5.92 Power -20 5.82 4.33 6.13 4.79 Power -30 5.24 3.52 5.46 3.82 Power -50 4.73 3.13 4.88 3.30 Power -75 4.44 3.06 4.59 3.16 Portfolio A 3.48 5.67 3.48 5.67 Portfolio B 5.08 5.82 5.08 5.82 Portfolio C 6.91 8.80 6.91 8.80 Portfolio D 7.60 10.19 7.60 10.19 Portfolio E 8.27 11 .70 8.27 11.70 Portfolio F 8.93 13.34 8.93 13.34 Portfolio G 9.77 15.93 9.77 15.93 Portfolio H 11.10 17.50 Portfolio I 12.30 19.51 Portfolio J 13 .37 21.84 • Standard deviation is for the variables In(l + r,). --- ## Page 745 716 R. R. Grauer and N. H. Hakansson 300 Journal of Business TABLE 3 Comparison of Geometric Means and Standard Deviations of Annual Returns with and without Small Stocks, 1934-83: Leverage Case (Quarterly Portfolio Revision, 32-Quarter Estimating Period) A. 1934-47 Without Small Stocks With Small Stocks Geometric Standard Geometric Standard Portfolio Mean Deviation* Mean Deviation* Common stocks 8.59 22.42 8.59 22.42 Government bonds 4.03 3.72 4.03 3.72 Corporate bonds 4.42 3.61 4.42 3.61 Small stocks 16.98 38.76 Risk free .24 .17 .24 .17 Power I -5.86 56.91 11.57 88.59 Power .75 5.30 40.76 19.95 48.99 Power .50 9.09 38.63 25.95 35.10 Power .25 9.19 34.75 23.43 31.01 Power 0 10.67 31.30 22.58 28.25 Power -I 10.19 22.95 17.47 24.83 Power -2 9.32 17.86 13.94 20.89 Power -3 8.83 14.91 12.33 17.41 Power -5 7.17 10.36 9.44 11.87 Power -7 6.24 7.67 7.85 8.67 Power -10 5.53 5.59 6.58 6.36 Power -15 4.85 4.43 5.69 5.06 Power -20 3.92 3.60 4.55 4.18 Power -30 2.99 2.64 3.39 3.02 Power -50 2.42 2.09 2.63 2.31 Power -75 1.86 1.64 2.29 1.98 Portfolio A 3.44 2.72 3.44 2.72 Portfolio B 4.68 5.60 4.68 5.60 Portfolio C 6.40 10.74 6.40 10.74 Portfolio D 6.83 12.80 6.83 12.80 Portfolio E 7.22 14.88 7.22 14.88 Portfolio F 7.57 16.96 7.57 16.96 Portfolio G 8.38 20.28 8.38 20.28 Portfolio H 10.63 23.01 Portfolio I 12.64 26.09 Portfolio J 1.4.40 29.40 --- ## Page 746 A Half Century a/Returns on Levered and Unlevered Portfolios a/Stocks, Bonds, and Bills 717 Half Century of Returns 301 TABLE 3 (Continued) B. 1948-65 Without Small Stocks With Small Stocks Geometric Standard Geometric Standard Portfolio Mean Deviation* Mean Deviation* Common stocks 15.67 14.54 15.67 14.54 Government bonds 1.94 4.96 1.94 4.96 Corporate bonds 2.52 4.11 2.52 4.11 Small stocks 15.61 19.36 Risk free 2.29 .94 2.29 .94 Power I 24.70 24.79 23.40 28.61 Power .75 24.70 24.79 22.92 28.01 Power .50 24.70 24.79 23.31 27.60 Power .25 24.70 24.79 23.54 27.49 Power 0 24.70 24.79 23.75 27.41 Power -I 22.79 25.18 21.93 27.25 Power -2 20.17 23.67 19.49 24.30 Power -3 17.97 22.31 17.79 22.82 Power -5 15.04 19.35 15.08 19.73 Power -7 12.38 16.07 12.49 16.28 Power - 10 9.96 12.82 10.09 13.00 Power -15 8.00 9.45 8.17 9.62 Power -20 6.73 7.16 6.81 7.37 Power -30 5.36 4.71 5.39 4.88 Power -50 4.17 2.73 4.19 2.87 Power -75 3.56 1.77 3.57 1.90 Portfolio A 2.26 3.56 2.26 3.56 Portfolio B 5.00 3.02 5.00 3.02 Portfolio C 8.41 5.99 8.41 5.99 Portfolio D 9.75 7.46 9.75 7.46 Portfolio E 11.09 9.01 11.09 9.01 Portfolio F 12.44 10.63 12.44 10.63 Portfolio G 14.37 12.90 14.37 12.90 Portfolio H 14.44 13.55 Portfolio I 14.47 14.41 Portfolio J 14.46 15.46 --- ## Page 747 718 R. R. Grauer and N. H. Hakansson 302 Journal of Business TABLE 3 (Continued) C. 1966-83 Without Small Stocks With Small Stocks Geometric Standard Geometric Standard Portfolio Mean Deviation* Mean Deviation* Common stocks 7.62 17.03 7.62 17.03 Government bonds 4.13 9.91 4.13 9.91 Corporate bonds 4.90 10.82 4.90 10.82 Small stocks 16.11 27.78 Risk free 7.37 2.86 7.37 2.86 Power I 9.22 15.82 13.26 33.56 Power .75 10.16 14.98 11.80 34.55 Power .50 9.91 13.58 10.11 32.42 Power .25 9.74 11.93 11 .54 31.12 Power 0 9.25 11.44 12.17 30.16 Power -I 9.32 10.42 10.87 25.07 Power -2 8.96 8.68 10.51 19.69 Power -3 8.41 7.24 10.33 16.11 Power -5 8.05 5.46 9.54 11.13 Power -7 7.89 4.58 9.08 8.61 Power - 10 7.75 3.92 8.66 6.58 Power - 15 7.64 3.45 8.29 4.97 Power -20 7.58 3.25 8.08 4.21 Power -30 7.51 3.07 7.86 3.53 Power - 50 7.46 2.96 7.67 3.11 Power -75 7.43 2.92 7.58 2.97 Portfolio A 5.20 8.27 5.20 8.27 Portfolio B 6.03 7.96 6.03 7.96 Portfolio C 6.61 10.08 6.61 10.08 Portfolio D 6.92 10.99 6.92 10.99 Portfolio E 7.20 12.06 7.20 12.06 Portfolio F 7.48 13.38 7.48 13.38 Portfolio G 7.44 15.62 7.44 15.62 Portfolio H 9.41 16.86 Portfolio I 11.25 18.73 Portfolio J 12.95 21.06 --- ## Page 748 A Half Century a/Returns on Levered and Unlevered Portfolios a/Stocks, Bonds, and Bills 719 Half Century of Returns 303 TABLE 3 (Continued) D. 1934-83 Without Small Stocks With Small Stocks Geometric Standard Geometric Standard Portfolio Mean Deviation* Mean Deviation* Common stocks 10.73 17.84 10.73 17.84 Government bonds 3.31 6.88 3.31 6.88 Corporate bonds 3.90 7.14 3.90 7.14 SmaJl stocks 16.17 28.22 Risk free 3.51 3.40 3.51 3.40 Power 1 9.89 35.87 16.32 52.70 Power .75 \3.74 28.00 17.98 36.62 Power .50 14.78 26.70 19.09 31.49 Power .25 14.75 24.96 19.05 29.63 Power 0 14.99 23.59 19. \3 28.44 Power -1 14.25 20.67 16.61 25.62 Power -2 12.97 18.09 14.64 21.60 Power -3 11.88 16.31 \3 .53 18.95 Power -5 10.26 \3.39 11.47 14.88 Power -7 9.01 10.87 9.95 11.87 Power -10 7.91 8.57 8.58 9.28 Power -15 6.98 6.48 7.51 6.97 Power -20 6.24 5.19 6.63 5.61 Power -30 5.45 3.97 5.71 4.23 Power -50 4.84 3.27 4.99 3.41 Power -75 4.45 3.12 4.63 3.16 Portfolio A 3.64 5.62 3.64 5.62 Portfolio B 5.28 5.81 5.28 5.81 Portfolio C 7.19 8.89 7.19 8.89 Portfolio D 7.91 10.31 7.91 10.31 Portfolio E 8.59 11.85 8.59 11.85 Portfolio F 9.27 13.50 9.27 13.50 Portfolio G 10.16 16.12 10.16 16.12 Portfolio H 11.54 17.51 Portfolio I 12.79 19.39 Portfolio J 13.90 21.60 • Standard deviation is for the variables In(l + ,,). --- ## Page 749 720 R. R. Grauer and N. H. Hakansson 304 Journal of Business strategy, for example, had an uninterrupted presence in small stocks from the beginning of 1942 through the first quarter of 1952, from the fourth quarter of 1963 through the second quarter of 1974, and again from the third quarter of 1980 on. Table 4 gives a comparison of the quarter-by-quarter portfolio com- positions and returns for powers 0 and - 15 with and without small stocks in the leverage case. For each power, the first five columns show the proportions invested in common stocks, government bonds, corporate bonds, small stocks, and the risk-free asset. The sixth col- umn reports the fraction in borrowing and the last column the port- folio 's return for the quarter. This is then repeated with small stocks excluded. Turning first to the logarithmic investor, we note that he stayed away from small stocks in the first quarter of 1934, from the beginning of 1953 through mid-1960, during the second quarter of 1961, and from the beginning of 1974 through mid-1977 with the exception of three quar- ters. During these periods, portfolio holdings and returns were thus unaffected by the opportunity to invest in small stocks. In the 1930s, small stocks tended to replace holdings in governments and, to a lesser extent, common stocks. In the 1940s and early 1950s, and again in the 1960s and early 1970s, it was principally common stocks that were being crowded out by small stocks for the logarithmic investor. In the late 1970s through the third quarter of 1982, it was primarily the risk- free asset that gave way for small stocks, followed by common stocks one more time. The logarithmic investor's use of leverage was remarkably similar with and without the small stock category. Its presence did not always increase leverage; while they markedly did so in 1982 and 1983, small stocks sharply reduced the use of leverage in 1946 and 1947. Turning now to the rather conservative power - 15 strategy, we first observe that this investor was a heavy and repeated user of the risk- free asset beginning with the third quarter of 1962. Prior to the second quarter of 1951, however, the risk-Jree asset was practically ignored. In fact, from the beginning of 1940 through the third quarter of 1946, the power - 15 strategy was a consistent user of leverage, with holdings of government bonds on margin. Small stocks first entered the power - 15 portfolio in the last quarter of 1939, growing gradually in importance to a 19% allocation in the first quarter of 1950, then declining and disappearing as of mid-1951, 47 quarters later. As table 4 shows, these holding~ were primarily at the expense of common stocks. After a nearly total hiatus during the rest of the 1950s, small stocks reappeared for 55 consecutive quarters begin- ning with the second quarter of 1960, reaching a maximum allocation of 30% in the second quarter of 1962, in a streak that lasted through 1973. These positions were also essentially at the expense of common --- ## Page 750 A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds, and Bills 72 1 Half Century of Returns 305 stocks, except in the second half of 1973, when they replaced the risk- free asset, When the power - 15 investor returned to small stocks at the beginning of 1978, they first replaced principally investments in the risk-free asset and, beginning in the last quarter of 1982, holdings in common stocks. Many investigators have noted that small stocks have, over long periods, produced what is usually referred to as excess risk-adjusted returns (e.g., Banz 1981; and Reinganum 1981). This suggests that the empirical (past) distribution used in our model should, at a minimum, have found small stocks a consistently attractive outlet. While this was obviously the case most of the time, small stocks were, as noted, totally ignored in the middle and late 1950s and for intervals in the middle 1970s even by the more risk-tolerant strategies. Other Results We also examined the case in which portfolios were revised only once a year rather than quarterly, with 8- and to-year estimating periods. As in the previous studies, the differences in returns and portfolio compo- sitions were fairly small. The use of 24-quarter and 40-quarter estimat- ing periods also led to only minor differences in the results. VI. Tests The returns under quarterly reinvestment with leverage in the presence of small stocks over the 1934-47 subperiod are shown as round dots in figure I, while the returns over the full 1934-83 period are similarly depicted in figure 2. The same figures also plot the returns for the higher powers in the absence of small stocks (see diamonds) as well as for the fixed-weight portfolios in table I (see triangles). The geometric means of the annual returns are measured on the vertical axis and the standard deviations of InO + rt ) on the horizontal. The attained returns reflect clearly the benefits from diversification among the five (or four) asset categories (represented il\ the figures by the square points RL [risk-free asset], GB [government bonds], CB [corporate bonds], CS [common stocks], and SS [small stocks]). In other words, the portfolios selected by the model have enabled inves- tors to travel in a distinctly " northwesterly" direction, confirming the results of the previous studies. Recall that terminal wealth Wo in terms of beginning wealth Wn is given by n Wn exp [~ InO + r t)} --- ## Page 751 722 R. R. Grauer and N. H. Hakansson 306 Journal of Business TABLE 4 Portfolio Composition and Realized Returns for Powers 0 and - 15 with and without Small Stocks, 1934-83 (Quarterly Revision, with Leverage, 32·Quarter Estimating Period) Power 0 With Small Stocks Without Small Stocks Inv. Fractions Inv. Fractions Period CS GB CB SS RL B " CS GB CB RL B " 1934:1 .29 4.74 1.34 -5.37 33.21 .29 4.74 1.34 -5.37 33.21 1934:2 .32 5.23 1.16 .04 -5.75 15.46 .38 5.29 1.13 -5.80 15.52 1934:3 .04 5.78 1.10 .14 - 6.07 -17.50 .25 5.98 1.01 -6.23 - 17.65 1934:4 4.39 1.53 .13 -5.05 18.89 .14 4.61 1.46 -5.21 18.56 1935:1 4.26 1.50 . 11 -4.87 14.94 .03 4.48 1.54 -5.05 17.56 1935:2 4.82 1.42 .05 -5.29 7.95 4.92 1.45 -5.37 7.62 1935:3 6.28 1.03 .03 -6.34 -5.75 6.34 1.05 -6.39 -6.27 1935:4 5.06 1.31 .08 -5.44 10.30 5.23 1.36 -5.59 7.91 1936:1 2.73 1.85 .18 -3.76 15.90 .10 3.02 1.87 -3.98 10.90 1936:2 3.34 1.59 .24 -4.18 - .71 .11 3.79 1.64 -4.53 4.03 1936:3 3.09 1.75 .17 -4.02 7.84 .08 3.42 1.77 -4.28 5.98 1936:4 4.90 1.25 .16 -5.31 15.04 .04 5.29 1.30 -5.62 12.61 1937:1 5.27 1.09 .17 -5.52 -18.69 5.69 1.23 -5.92 -24.08 1937:2 5.84 .80 .25 -5.88 -4.75 6.32 1.05 -6.37 3.19 1937:3 4.98 1.15 .18 -5.31 -2.24 5.33 1.33 -5.66 1.95 1937:4 5.60 1.03 .14 -5.77 6.40 5.88 1.18 -6.06 12.40 1938:1 4.03 1.50 .18 -4.71 -5.63 4.54 1.56 -5.10 .11 1938:2 4.47 1.53 .04 -5.05 15.33 4.59 1.54 -5.14 13.11 1938:3 4.35 1.35 .24 ... -4.93 2.90 5.07 1.41 -5.48 2.75 1938:4 4.13 1.35 .28 -4.77 12.11 4.98 1.43 -5.42 7.97 1939:1 4.16 1.22 .39 -4.77 - 1.89 .15 5.15 1.22 -5.52 9.92 1939:2 6.23 .86 .19 -6.28 14.67 6.80 .92 -6.71 15.88 1939:3 6.32 .78 .24 -6.34 - 30.65 .04 7.02 .80 -6.86 -48.79 1939:4 5.59 .47 .69 -5.75 32.70 .79 5.67 .33 -5.80 37.90 1940:1 7.28 .68 -6.96 20.44 .67 7.32 -6.99 9.62 1940:2 7.04 .74 -6.78 -28.20 .75 7.01 -6.76 - 21.87 1940:3 6.84 .79 -6.63 15.40 1.03 5.89 -5.92 16.87 1940:4 7.21 .03 .67 -6.91 20.70 .75 6.25 .21 -6.22 16.92 1941:1 6.84 .79 -6.63 -8.89 .88 6.48 -6.36 -12.80 1941 :2 6.74 .82 -6.55 12.38 1.00 6.00 -6.00 11.50 1941:3 7.88 .53 -7.41 6.58 .48 6.71 .39 -6.58 1.65 1941 :4 7.45 .64 -7.09 -22.65 .75 7.01 -6.76 -17.81 1942:1 .. . 7.98 .50 -7.49 14.96 .45 8.21 -7.66 7.46 1942:2 8.11 .47 -7.58 -.10 .41 8.13 .06 -7.61 2.65 1942:3 7.86 .54 -7.39 12.18 .60 7.60 -7.20 5.74 1942":4 7.56 .61 -7.17 5.12 .61 7.56 -7.17 6.86 1943:1 7.43 .64 -7.07 41.32 .75 6.99 -6.75 14.04 1943:2 6.28 .93 -6.21 23.69 1.12 5.53 -5.65 12.80 1943:3 6.16 .96 -6.12 -7.91 1.00 5.98 -5.99 -2.25 1943:4 6.61 .85 -6.46 -.82 .81 6.77 -6.58 -3.29 1944:1 6.93 .77 -6.69 15.42 .69 7.25 -6.93 4.74 1944:2 7.04 .74 -6.78 11.72 .68 7.27 -6.95 6.68 1944:3 6.45 .89 -6.34 1.72 .78 6.88 -6.66 1.60 1944:4 6.66 .84 -6.49 12.71 .72 7.12 -6.84 5.99 1945:1 6.68 .83 -6.51 16.37 .67 6.39 .27 -6.32 14.14 1945:2 7.08 .58 -6.66 40.15 .39 8.07 -7.45 32.01 1945:3 5.79 .56 -5.36 .54 .27 7.99 -7.26 -1.76 1945:4 4.76 .70 -4.46 33.27 .38 7.16 - 6.54 30.68 1946:1 2.83 .96 -2.79 11.22 .47 6.44 -5.91 3.77 1946:2 1.31 .87 -1.18 5.38 .37 6.26 -5.64 -6.59 1946:3 1.49 .85 -1.34 -26.30 .39 6.14 - 5.53 - 19.88 1946:4 .75 .93 - .67 1.81 .50 5.00 .. , -4.50 7.38 --- ## Page 752 A Half Century a/Returns on Levered and Unlevered Portfolios a/Stocks, Bonds, and Bills 723 Half Century of Returns 307 TABLE 4 (Continued) Power -15 With Small Stocks Without Small Stocks Inv. Fractions Inv. Fractions CS GB CB SS RL B r, CS GB CB RL B r, .15 .83 .02 5.83 .15 .83 .02 5.83 .15 .85 3.52 .15 .85 3.52 . 14 .86 -.04 .14 .86 - .04 .08 .92 3.36 .08 .92 3.36 .05 .95 3.95 .05 .95 3.95 1.00 2.68 1.00 2.68 1.00 .69 1.00 .69 1.00 1.95 1.00 1.95 1.00 2.20 1.00 2.20 1.00 1.49 1.00 1.49 1.00 1.46 1.00 1.46 1.00 1.44 1.00 1.44 1.00 -1.36 1.00 - 1.36 .05 .95 1.58 .05 .95 1.58 1.00 .47 1.00 .47 1.00 2.02 1.00 2.02 1.00 - .39 1.00 - .39 1.00 2.45 1.00 2.45 .01 .99 1.55 .01 .99 1.55 .03 .97 2.38 .03 .97 2.38 1.00 1.08 1.00 1.08 .11 .89 1.61 .11 .89 1.61 .15 .85 -3.11 .15 .85 -3.11 .14 .81 .02 .03 4.07 .03 .95 .02 3.98 2.64 .04 -1.68 2.92 2.42 - 1.42 2.17 2.70 .05 -1.75 -1.84 2.46 - 1.46 - .60 2.79 .06 -1.85 2.83 .06 2.69 -1.74 2.83 2.79 .06 -1.85 1.74 .06 2.72 -1.78 1.59 2.79 .06 -1.85 -1.17 .07 2.67 -1.74 - 1.44 2.78 .06 -1.85 4.44 .08 2.63 -1.71 4.26 2.79 .06 -1.85 4.10 .08 2.68 - 1.76 3.44 2.78 .07 -1.85 -3.92 .09 2.67 -1.75 -3.46 2.72 .06 -1.77 1.31 .05 2.43 -1.47 .45 2.69 .06 -1.74 .71 .03 2.35 -1 .39 .92 2.58 .06 -1.64 2.28 .04 2.27 - 1.31 1.39 2.70 .07 -1.77 1.40 .06 2.34 -1.40 1.49 2.61 .07 -1.68 5.53 .07 2.25 -1.33 2.43 2.54 .08 -1.62 4.52 .09 2.16 -1.25 3.28 2.51 .08 - 1.58 - .27 .09 2.13 - 1.22 .21 2.43 .07 - 1.50 -.26 .07 2.09 -1.16 - .36 2.26 .07 -1.32 2.80 .05 1.92 - .98 1.69 2.21 .06 - 1.27 1.67 .05 1.85 - .90 1.11 2.13 .07 - 1.21 1.25 .06 1.78 - .83 1.10 2.07 .07 - 1.14 4.80 .05 1.71 - .76 3.59 2. 11 .07 - 1.17 2.75 .04 1.75 - .80 2.24 2.67 .06 - 1.73 1.88 .05 2.29 -1.33 .73 .21 2.24 .08 -1.53 .31 .07 2.20 -1.27 .56 .24 2.22 .10 -1.56 6.73 .10 2.20 - 1.30 4.56 .75 1.25 .13 -1.13 3.82 .14 .26 1.68 -1 .09 3.69 .33 2.31 .16 -1.80 .21 .05 2.25 - 1.51 -.29 .21 2.35 .16 -1.71 -8.51 .20 2.28 - 1.48 -7.35 .59 .29 .12 1.41 .11 .20 .78 -.09 1.53 --- ## Page 753 724 R. R. Grauer and N. H. Hakansson 308 Journal of Business TABLE 4 (Continued) Power 0 With Small Stocks Without Small Stocks Inv. Fractions Inv. Fractions Period CS GB CB SS RL RB RP CS GB CB RL RB RP 1947: I 1.47 ... .85 -1.32 -.03 .47 5.30 -4.77 - .89 1947:2 1.33 -.33 -14.08 .96 2.78 -2.74 .30 1947:3 1.33 - .33 11.44 1.17 1.19 -1.37 1.04 1947:4 .97 1.20 -1.18 -.60 10.00 -9.00 - 45.20 1948:1 1.33 - .33 -0.61 1.33 - .33 - .46 1948:2 1.33 - .33 20.06 1.33 - .33 16.55 1948:3 1.33 - .33 - 14.46 1.33 - .33 -8.54 1948:4 1.33 - .33 -6.34 1.33 - .33 -.01 1949:1 1.33 -.33 3.81 1.33 - .33 .59 1949:2 2.00 -1.00 - 20.04 2.00 -1.00 -9.04 1949:3 2.00 -1 .00 28.94 2.00 -1.00 22.72 1949:4 2.00 -1.00 23.58 2.00 - 1.00 19.98 1950: I 2.00 -1.00 13.00 2.00 -1.00 8.80 1950:2 2.00 -1.00 -3.72 2.00 -1.00 7.66 1950:3 2.00 -1.00 34.02 2.00 - 1.00 23.20 1950:4 2.00 - 1.00 24.14 2.00 ... -1.00 15.14 1951: I 2.00 -1.00 6.77 2.00 - 1.00 11.95 1951:2 1.33 - .33 -7.01 1.33 - .33 -.77 1951:3 1.33 - .33 16.22 1.33 - .33 16.24 1951:4 1.33 - .33 -3.75 1.33 -.33 5.27 1952: I 1.33 -.33 .50 1.33 - .33 4.94 1952:2 .19 1.14 - .33 -2.14 1.33 - .33 5.24 1952:3 1.25 .09 -.33 -1.03 1.33 - .33 -1.03 1952:4 1.28 .05 - .33 12.76 1.33 - .33 12.99 1953:1 1.33 - .33 -5.14 1.33 - .33 -5.14 1953:2 2.00 -1.00 -6.9 1 2.00 - 1.00 -6.9 1 1953:3 2.00 -1 .00 -5.24 2.00 -1.00 -5.24 1953:4 2.00 -1.00 15.18 2.00 - 1.00 15.18 1954:1 2.00 -1.00 18.93 2.00 ... -1.00 18.93 1954:2 2.00 -1 .00 18.80 2.00 -1.00 18.80 1954:3 2.00 -1 .00 22.48 2.00 -1.00 22.48 1954:4 2.00 . . . -1.00 25.00 2.00 -1.00 25.00 1955:1 1.67 -.67 3.77 1.67 - .67 3.77 1955:2 1.67 -.67 21.53 1.67 - .67 21.53 1955:3 1.43 - .43 10.00 1.43 - .43 10.00 1955:4 1.43 - .43 7.15 1.43 - .43 7.15 1956:1 1.43 - .43 10.43 1.43 - .43 10.43 1956:2 1.43 - .43 -3.56 1.43 - .43 -3.56 1956:3 1.43 - .43 -4.32 1.43 - .43 -4.32 1956:4 1.43 - .43 4.94 1.43 - .43 4.94 1957:1 1.43 -.43 -7.06 1.43 - .43 - 7.06 1957:2 1.43 -.43 11.49 1.43 - .43 11.49 1957:3 1.43 -.43 -14.31 1.43 - .43 -14.3 1 1957:4 1.43 - .43 -7.31 1.43 -.43 -7 .31 1958:1 1.43 - .43 8.54 1.43 - .43 8.54 1958:2 2.00 -1.00 15.87 2.00 -1.00 15.87 1958:3 2.00 - 1.00 22.18 2.00 - 1.00 22.18 1958:4 1.43 -.43 15.59 1.43 - .43 15.59 1959:1 1.11 - .11 1.22 1.11 - .11 1.22 1959:2 1.11 -.11 6.84 1.11 -. 11 6.84 1959:3 1.11 -.11 -2.34 1.11 - .11 - 2.34 1959:4 1.11 -.11 6.71 1.11 - . 11 6.71 1960:1 1.11 - . 11 -7.72 1.11 - .11 - 7.72 1960:2 1.11 -.11 3.98 1.11 - . 11 3.98 --- ## Page 754 A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds, and Bills 725 Half Century of Returns 309 TABLE 4 (Continued) Power - 15 With Small Stocks Without Small Stocks Inv. Fractions Inv. Fractions CS GB CB SS RL B '/ CS GB CB RL B '/ .74 .15 .12 .40 .12 .38 .51 .55 .34 .52 . 14 - 1.24 .15 .15 .70 .60 .08 .78 .13 -.17 .16 .84 - 1.44 .89 .1 1 -3.21 .07 1.08 - .15 -4.16 .87 .13 1.51 . 13 .30 .58 1.30 .87 .13 1.58 .13 .11 .76 1.48 .12 .69 .18 -1.79 .25 .40 .35 - 1.49 .83 .17 1.25 .21 .29 .50 1.66 .84 .16 1.18 .. 21 .79 .78 .84 .16 - .31 .22 . 19 .59 .30 .85 .15 3.53 .20 .24 .55 3.59 .08 .78 .15 1.39 .21 .37 .42 2.29 .03 .17 .61 .19 1.78 .31 .52 .17 1.43 .20 .70 .11 .70 .37 .04 .59 1.58 .19 .70 .11 4.61 .36 .10 .53 4.70 .25 .68 .07 3.36 .37 .05 .58 3.31 .18 .71 .11 -.27 .37 .63 .67 .38 .01 .03 .01 .57 - .05 .40 .03 .02 .56 -.01 .38 .62 4.93 .38 .62 4.93 .40 .60 1.91 .40 .60 1.91 .42 .58 1.88 .42 .58 1.88 .42 .58 1.97 .42 .58 1.97 .41 .59 .05 .41 .59 .05 .41 .59 4.33 .41 .59 4.33 .41 .59 -1.18 .41 .59 - 1.18 .38 .62 - .78 .38 .62 - .78 .35 .65 - .38 .35 .65 - .38 .34 .66 2.96 .34 .66 2.96 .34 .66 3.61 .34 .66 3.61 .35 .09 .57 3.57 .35 .09 .57 3.57 .35 .25 .39 4.48 .35 .25 .39 4.48 .92 .08 11.98 .92 .08 11.98 .92 .08 2.40 .92 .08 2.40 .93 .07 12.35 .93 .07 12.35 .94 .06 6.87 .94 .06 6.87 .92 .08 4.99 .92 .08 4.99 .92 .08 7.09 .92 .08 7.09 .94 .06 - 1.97 .94 .06 - 1.97 .90 .10 -2.31 .90 .10 -2.3\ 1.00 3.86 1.00 3.86 1.00 -4.53 1.00 - 4.53 .90 .10 7.72 .90 .10 7.72 .98 .02 -9.58 1.00 -9.60 .65 .35 -2.75 .65 .35 -2.75 .61 .39 4.13 .61 .39 4.13 .69 .26 .05 6.29 .69 .26 .05 6.29 .70 .30 6.47 .70 .30 6.47 .61 .. , .39 7.18 .61 .39 7.18 .61 .39 1.02 .61 .39 1.02 .59 .41 4.02 .59 .4 1 4.02 .61 .39 - .88 .61 .39 - .88 .53 .47 3.74 .53 .47 3.74 .52 .48 -2.99 .52 .48 -2.99 .43 .04 .52 2.20 .47 .53 2.20 --- ## Page 755 726 R. R. Grauer and N. H. Hakansson 310 Journal of Business TABLE 4 (Continued) Power 0 With Small Stocks Without Small Stocks Inv. Fractions Inv. Fractions Period CS GB CB SS RL B r, CS GB CB RL B r, 1960:3 .79 .32 - .11 -5.66 1.11 -.11 - 5.93 1960:4 .90 .52 - .43 9.92 1.43 - .43 13.11 1961:1 1.13 .30 -.43 20.71 1.43 - .43 17.71 1961:2 1.43 - .43 - .48 1.43 - .43 - .48 1961:3 .88 .55 - .43 1.92 1.43 - .43 5.09 1961:4 .89 .54 - .43 11.86 1.43 - .43 10.95 1962:1 1.43 -.43 4.91 1.43 - .43 -3.59 1962:2 1.43 -.43 -34.28 1.43 -.43 - 30.05 1962:3 1.43 - .43 4.34 1.43 - .43 4.67 1962:4 2.00 -1.00 13 .26 2.00 -1.00 25. 18 1963:1 1.64 .36 -1.00 12.98 2.00 - 1.00 11.30 1963:2 1.76 .24 - 1.00 8.96 2.00 -1.00 8.64 1963:3 .34 1.66 -1.00 6.32 2.00 -1.00 6.82 1963:4 2.00 -1.00 .20 2.00 -1.00 9.84 1964:1 1.43 - .43 12.01 1.43 -.43 8.07 1964:2 1.43 - .43 5.38 1.43 - .43 5.41 1964:3 1.43 -.43 10.62 1.43 - .43 4.81 1964:4 1.43 - .43 .87 1.43 - .43 1.67 1965:1 1.43 - .43 16.55 1.43 - .43 2.75 1965:2 1.43 - .43 -7.92 1.43 - .43 -2.94 1965:3 1.43 -.43 20.17 1.43 -.43 10.40 1965:4 1.43 - .43 22.89 1.43 - .43 4.60 1966:1 1.43 - .43 11.86 1.43 -.43 -4.57 1966:2 1.43 - .43 -10.16 1.43 - .43 -6.77 1966:3 1.43 - .43 - 18.42 1.43 - .43 -13.41 1966:4 1.43 - .43 5.61 1.17 - . 17 6.68 1967:1 1.43 - .43 43.99 1.00 13.21 1967:2 1.43 - .43 16.72 1.43 -.43 1.13 1967:3 1.43 - .43 22.06 1.43 - .43 10.02 1967:4 1.43 - .43 9.98 1.43 - .43 .12 1968:1 1.43 - .43 -10.31 1.43 - .43 -8.92 1968:2 1.43 -.43 36.96 1.43 -.43 15.26 1968:3 1.25 - .25 7.14 1.25 - .25 4.39 1968:4 1.25 -.25 10.34 1.25 - .25 1.99 1969:1 1.25 - .25 -10.37 1.25 - .25 -2.38 1969:2 1.25 -.25 -8.76 1.00 -3 .00 1969:3 1.25 - .25 -8.92 1.00 -3.92 1969:4 1.25 - .25 -8.97 .72 .28 .27 1970:1 1.25 - .25 -7.12 1.00 1.98 1970:2 1.25 - .25 -41.44 .50 .50 -8.20 1970:3 1.05 -.05 29.38 .75 .25 13.01 1970:4 1.37 - .37 .66 1.00 10.43 1971:1 1.45 - .45 37.53 1.00 9.69 1971:2 1.54 - .54 - 11.34 1.54 -.54 -.61 1971:3 1.44 - .44 -4.21 1.01 -.01 -.61 1971:4 1.35 - .35 1.27 1.00 4.66 1972: I 1.47 - .47 18.42 1.19 - . 19 6.54 1972:2 1.51 -.51 -6.34 1.17 -.17 .53 1972:3 1.37 -.37 -8.46 1.00 3.91 1972:4 1.12 - .12 1.87 1.00 7.56 1973: I 1.09 -.09 -15.27 1.00 - 4.89 1973:2 .75 .25 - 11.86 .62 .38 -2.98 1973:3 .32 .68 7.21 1.00 2.00 1973:4 .48 .52 -8.20 1.00 1.79 --- ## Page 756 A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds, and Bills 727 Half Century of Returns 311 TABLE 4 (Continued) Power -15 With Small Stocks Without Small Stocks Inv. Fractions Inv. Fractions CS GB CB SS RL B r, CS GB CB RL B r, .40 .21 .15 .25 -1.90 .52 .15 .33 -2.03 .36 .39 .19 .06 4.86 .51 .30 .19 5.58 .37 .43 .19 .01 9.29 .52 .34 .14 6.95 .53 .42 .06 .58 .42 .02 .47 .40 .13 2.14 .60 .40 2.88 .47 .37 . 16 5.05 .62 .38 4.75 .41 .37 .23 1.27 .63 .36 .01 - .07 .3 1 .39 .30 -13.30 .60 .39 .01 -12.18 .10 .07 .13 .70 1.49 .24 .07 .70 1.54 .10 .13 .13 .64 2.94 .24 .13 .64 3.76 .20 . 11 .03 .66 2.21 .23 .11 .66 2.06 .22 .16 .02 .60 1.74 .24 .16 .60 1.71 .12 .09 .10 .69 1.43 .22 .10 .69 1.48 .03 .04 .17 .76 .96 .20 .04 .76 1.80 .08 .11 .80 2.19 .20 .80 1.90 .04 .03 .15 .78 1.52 .20 .03 .78 1.54 .05 .11 .16 .68 2.18 .22 .11 .67 1.54 .06 .23 .18 .53 1.06 .25 .25 .50 1.18 .03 .30 .20 .46 3.24 .25 .32 .44 1.32 .03 .23 .21 .53 -.57 .26 .22 .52 .13 .42 .22 .36 3.58 .24 .39 .37 2.20 .03 .46 .25 .26 3.73 .31 .44 .25 .67 .29 .71 3.35 .29 .71 .02 .28 .72 -1.06 .27 .73 -.29 .25 .75 -2. 15 .22 .78 - 1.02 .18 .82 1.90 .13 .87 1.96 .17 .83 6.43 .14 .86 2.84 .01 .20 .79 3.15 .19 .81 1.02 .21 .79 4.19 .17 .83 2.15 .23 .77 2.60 .19 .81 1.04 .22 .78 -.50 .16 .84 .16 .22 .78 6.74 .16 .84 2.89 .23 .77 2.44 .17 .83 1.79 .25 .75 3.15 .21 .79 1.46 .24 .76 -.77 .17 .83 1.03 .21 .79 - .13 .12 .88 .98 .18 .82 .23 .08 .92 1.28 .17 .83 .33 .05 .95 1.66 .13 .87 1.05 1.00 1.98 .13 .87 -2.87 .03 .97 1.00 .10 .90 4.15 .05 .95 2.35 .12 .88 1.43 . 11 .89 2.50 .13 .87 4.28 .15 .85 2.44 .14 .86 -.13 .18 .82 .81 .11 .89 .94 .12 .88 1.13 .11 .89 1.13 .13 .87 1.56 .12 .88 2.31 .16 .84 1.62 .12 .88 .37 .15 .85 .89 .11 .89 .27 .12 .88 1.36 .09 .91 1.24 .10 .90 I~ .08 .92 .05 .10 .90 .05 .95 .72 .04 .96 1.28 .02 .98 2.33 1.00 2.00 .03 .97 1.16 1.00 1.79 --- ## Page 757 728 R. R. Grauer and N. H. Hakansson 312 Journal of Business TABLE 4 (Continued) Power 0 With Small Stocks Without Small Stocks Inv. Fractions Inv. Fractions Period CS GB CB SS RL RB RP CS GB CB RL RB RP 1974:1 1.00 1.94 1.00 1.94 1974:2 1.00 2.06 1.00 2.06 1974:3 1.00 1.94 1.00 1.94 1974:4 1.00 1.81 1.00 1.81 1975: I 1.00 1.62 1.00 1.62 1975:2 .02 .98 1.72 1.00 1.42 1975:3 .08 .92 .68 .03 .97 1.17 1975;4 1.00 1.52 1.00 1.52 1976:1 .62 .38 3.07 .62 .38 3.07 1976:2 .23 .35 .23 .19 .27 .49 .35 .16 1.52 1976;3 .27 .73 1.48 .27 .73 1.48 1976:4 .14 .64 .22 5.55 . 14 .64 .22 5.55 1977:1 1.00 -2.31 1.00 -2.31 1977:2 1.01 -.01 2.94 1.01 -.01 2.94 1977:3 .90 .10 .99 1.00 1.09 1977;4 .85 .15 .51 1.00 - .82 1978:1 .69 .31 3.75 1.00 .03 1978:2 .36 .64 8.90 1.00 -1.08 1978:3 1.28 - .28 20.47 .36 .64 5.10 1978:4 1.00 -17.37 .14 .86 1.05 1979: I .73 .27 16.92 1.00 2.43 1979:2 .71 .29 7.19 1.00 2.37 1979:3 1.00 5.64 1.00 2. 15 1979:4 1.00 1.70 1.00 2.52 1980:1 .81 .19 -10.24 1.00 2.96 1980:2 .05 .95 4.58 1.00 3.68 1980:3 1.00 . 25. 10 1.00 1.97 1980:4 1.00 7.49 1.00 2.85 1981;1 .85 . 15 13.13 1.00 3.73 1981:2 1.08 -.08 8.72 1.00 3.24 1981 :3 1.06 -.06 -18.98 1.00 3.73 1981:4 1.00 11.29 1.00 3.77 1982:1 1.74 -.74 - 12.76 1.00 2.86 1982:2 1.11 -.11 - .86 1.00 3.46 1982:3 1.51 -.51 13.78 1.00 3.33 1982:4 2.00 -1.00 44.04 1.00 18.14 1983: I 2.00 -1 .00 36.76 1.00 10.05 1983:2 2.00 -1.00 39.57 1.25 - .25 13.25 1983:3 2.00 -1.00 -5.90 1.00 -.14 1983:4 2.00 -1.00 -7.32 I.78 - .78 -1.59 NOTE.-CS = common stocks. GB = government bonds. CB = corporate bonds. SS = small stocks. RL = risk-free asset. B = the fraction in borrowing. and r, = return for th7 quarter. --- ## Page 758 A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds, and Bills 729 Half Century of Returns 313 TABLE 4 (Continued) Power - 15 With Small Stocks Without Small Stocks Inv. Fractions Inv. Fractions CS GB CB SS RL RB RP CS GB CB RL RB RP 1.00 1.94 1.00 1.94 1.00 2.06 1.00 2.06 1.00 1.94 1.00 1.94 1.00 1.81 1.00 1.81 1.00 1.62 1.00 1.62 1.00 1.44 1.00 1.42 1.00 1.49 1.00 1.52 1.00 1.52 1.00 1.52 .04 .96 1.35 .04 .96 1.35 .02 .02 .01 .95 1.17 .03 .02 .95 1.24 .02 .98 1.33 .02 .98 1.33 .01 .04 .95 1.50 .01 .04 .95 1.50 .07 .20 .73 - .05 .07 .20 .73 -.05 .24 .76 1.56 .24 .76 1.56 .22 .78 1.21 .22 .78 1.21 .18 .82 1.05 .18 .82 1.05 .21 .01 .78 1.29 .23 .77 1.18 .06 .04 .91 1.93 .13 .87 1.26 .09 .91 3.15 .02 .09 .89 1.99 .08 .92 .44 .01 .99 1.93 .04 .96 3.31 1.00 2.43 .04 .96 2.66 1.00 2.37 .07 .93 2.38 1.00 2.15 .06 .94 2.47 1.00 2.52 .05 .95 2.15 1.00 2.96 1.00 3.74 1.00 3.68 .07 .93 3.64 1.00 1.97 .07 .93 3.18 1.00 2.85 .05 .95 4.31 1.00 3.73 .09 .91 3.73 1.00 3.24 .11 .89 1.36 1.00 3.73 .07 .93 4.30 1.00 3.77 .14 .86 1.70 1.00 2.86 .10 .90 3.08 1.00 3.46 .12 .88 4.19 1.00 3.33 .21 .79 6.56 .17 .83 4.70 .23 .77 6.16 .18 .82 3.44 .23 .77 6.45 .14 .86 3.44 .22 .78 1.42 .12 .88 1.98 .24 .76 1.16 .18 .82 1.90 --- ## Page 759 730 314 c tJ Q) ::IE ~ ~ Q) E 0 Q) " 26.00 22 .75 19.50 16.25 13.00 9.75 6.50 3.25 .-3 .-7 t1r 0-5 t1( .-10 t10 0_7 t1C -15 ~-10 -15 i t1B CB -20 GB ..... -30 I -50 -75 0.00 RL .-1 .-2 t1H 0-1 .0.25 .0 t1J 00 R. R. Grauer and N. H. Hakansson Journal of Business .0.50 .0.75 00.75 Legend • With 'malt .took. <> Without .mall .tock. 0.0 8.5 17.0 25.5 34.0 42.5 51.0 Standard Deviation FIG. I.-Geometric means and standard deviations of annual returns with and without small stocks, 1934-47 (quarterly revision, with leverage, 32- quarter estimating period). Since the returns themselves are not additive but compound multiplica- tively, we employed the paired t-test for dependent observations to the quarterly (and additive) variables In(l + rt) to test whether the pres- ence of small stocks improved returns significantly. II Thus to compare the returo series rl, ... , r~ with the return series r1, ... , ,.;, for two different strategies, we calculate the statistic t = d cr(d)IVn' II. This test was also employed by Fama and MacBeth (1974) . --- ## Page 760 A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds, and Bills 731 Half Century of Returns 20.0 17.5 15.0 12,5 c 0 CD ~ 0 ·c 10.0 ~ E 0 CD (!) 7.5 5.0 2.5 eO.75 e-l e_2000 0 0.50 0-1 0.25 I:1J e-3 -2 o 1:11 0-3 e-5 I:1H .CS e_~-5 t>G 0_71:1r e-l01:1( -10 o 1:10 e-15 0_151:1C e-20 e-30 I:1B e-50 e-75 0 0.75 01 Legend • With .mall .Iock. 315 <> WIthout small .tock. O . O+--------r------~------~--------~------,_------__, 0.0 6.5 13.0 1 •• 5 26.0 32.5 39.0 Standard Deviation FIG. 2.-Geometric means and standard deviations of annual returns with and without small stocks, 1934-83 (quarterly revision, with leverage, 32- quarter estimating period). where d I InO + rf} InO + rf) 1=1 n and a(d) is the standard deviation of the differences between In( I + r]) and InO + r:). In each case, the null hypothesis is that E[ln(l + r})] = E[lnO + d)] and the alternative hypothesis is that E[ln(l + r})] > E[ln(l + r~)]. --- ## Page 761 732 R. R. Grauer and N. H. Hakansson 316 Journal of Business TABLE 5 Paired t-Tests, 1934-47 Subperiod, with Leverage Comparison d cr(d) Power - I vs. common stocks .0196 .1267 1.16 Power 0 vs. portfolio I .0378 .1438 1.97* Power .5 with small stocks vs. power .5 without small stocks .0360 .1813 1.48 Power .25 with small stocks vs. power .25 without small stocks .0306 .1333 1.72* Power .25 with small stocks vs. power 0 without small stocks .0273 .1157 1.76* Power 0 with small stocks vs. power 0 without small stocks .0256 . \068 \.79* Power - I with small stocks vs. power - I without small stocks .0160 .0591 2.03** Power - 2 with small stocks vs. power - 2 without small stocks .0103 .0389 1.99** Power - 3 with small stocks vs. power - 3 without small stocks .0079 .0311 1.90* Power - 5 with small stocks vs. power - 5 without small stocks .0052 .0211 1.85* Power -7 with small stocks vs. power -7 without small stocks .0038 .0156 1.81 * Power - 10 with small stocks vs . power - \0 without small stocks .0025 .0104 1.79' Power - 15 with small stocks vs . power - 15 without small stocks .0020 .0073 2.06** • Significant at the 10% level. •• Significant at the 5% level. The results for selected pairs with comparable standard deviations in the 1934- 47 period (56 quarters) are given in table 5, while table 6 gives the results for the full 200-quarter 1934-83 period. Table 5 suggests that the presence of small stocks was especially significant in improving returns for the " middle" strategies (powers -15-.25) from the early 1930s to the late 1940s. Statistically, the im- provements were less impressive for the full period (fig. 2 and table 6). The active strategies performed extremely well when compared to bonds and bills, attaining highly significant improvements, through diversification, in the geometric means without increasing variability. Figures 1 and 2 and tables 5 and 6 also reveal that the active strategies performed surprisingly well when compared to the 10 fixed-weight portfolio strategies. VII. Concluding Remarks By way of summary, several conclusions emerge. First, the present study confirms that the simple probability assessment approach, which uses only the past to (naively) forecast the future, is apparently not without merit when combined with multi period investment theory. Not --- ## Page 762 A Half Century of Returns on Levered and Unlevered Portfolios of Stocks, Bonds, and Bills 733 Half Century of Returns 317 TABLE 6 Paired t.Tests, 1934-83 Period, with Leverage Comparison d cr(d) Power - 3 vs. common stocks .0062 .0758 1.16 Power - 15 vs. corporate bonds .0085 .0452 2.67*** Power - 15 vs. government bonds .0100 .0458 3.08*** Power - 50 vs. risk·free asset .0036 .0102 4.94*** Power - 10 vs. portfolio C .0032 .0419 1.09 Power - 20 vs . portfolio B .0032 .0311 1.45 Power - 20 vs. portfolio A .0071 .0356 2.82*** Power 0 with small stocks vs. power .75 without small stocks .0116 .1073 1.52 .*. Significant at the 1% level. only did it produce respectable results, but the logarithmic policy also always came out at or near the top in the geometric mean dimension, suggesting an absence of clear biases in the estimating method; the exception occurs in the last subperiod, when the approach appears to have been somewhat too conservative. 12 Second, small stocks entered even the most risk-averse portfolios most of the time. On the other hand, they were ignored by all the strategies some of the time (espe· cially in the mid- and late 1950s and mid-1970s). Third, small stocks, when chosen, tended to replace common stocks, except in the 1970s and early 1980s, when they were primarily held in lieu of the risk-free asset. This was true even when (additional) margin purchases would have been possible. Consequently, the presence of small stocks had relatively little effect on the use of leverage; in fact, the effect was not always in the positive direction. Thus the use of, and gains from, margin purchases occurred, as in our previous paper (Grauer and Hakansson 1982), primarily for the more risk-tolerant strategies from the mid-1930s to the mid-1960s. Fourth, small stocks had a notable positive effect on returns, not only for the risk-tolerant strategies but also for conservative investors. This effect was especially significant from the mid-1930s to the early 1950s in the presence of margin purchases. Fifth, the strong gains from diversification among the major asset categories reported in the previ- ous papers are, if anything, strengthened when small stocks are in- cluded. Sixth, the performances of the active strategies, when measured against those of the fixed-weight strategies, were surprisingly strong. Could it be that the naive, simple-minded empirical distribution con- tains the kind of information money managers usually pay good money for? 12. Under perfectly valid distributions, power 'I = liT, where T is the number of periods, has the highest asymptotic probability of attaining the largest geometric mean as T increases. --- ## Page 763 734 R. R. Grauer and N. H. Hakansson 318 Journal of Business Finally, one cannot escape the conclusion that U.S. financial mar- kets offered a generous environment during the half-century we studied. As a point of reference, consider the following. The maximum employee-portion of the social security contribution for an individual in 1984 was $2,532.60. The purchasing power of this amount (in Janu- ary I, 1984, dollars) at the beginning of 1934 was $328.78. Under a naive logarithmic policy in an environment with access to small stocks and leverage, this initial contribution alone would have grown to $2,078,925 as of the end of 1983. Of course, the reader should also be reminded of the limitations of the study. The model used is focused on sequential reinvestments only, without concern for intermediate consumption; even though its birth occurred in the mid-1970s, it was applied as far back as 1934. The latter statement also applies at least partially to the data base used. The joint probability estimates were based on periods ranging from the most recent 5-1 ° years only. All investors were assumed to be strict price takers. Transactions costs and taxes were ignored (as in the underlying returns series). Finally, maintenance margins were ignored whenever leverage was used. References Banz, Rolf. 198\. The relationship between return and market value of common stocks. Journal of Financial Economics 9 (March): 3-18. Bawa, Vijay; Brown, Stephen; and Klein, Roger. 1979. Estimation Risk and Optimal Portfolio Choice. Amsterdam: North-Holland. Best, Michael. 1975. A feasible conjugate direction method to solve linearly constrained optimization problems. Journal of Optimization Theory and Applications 16 (July): 25-38. Fama, Eugene, and MacBeth, James. 1974. Long-term growth in a short-term market. Journal of Finance 29 (June): 857-85. Grauer, Robert, and Hakansson, Nils. 1982. Higher return, lower risk: Historical returns on long-run, actively managed portfolios of stock, bonds and bills, 1936-1978. Finan- cial Analysts Journal 38 (March-April): 39-53. Grauer, Robert, and Hakansson, Nils. 1985. 1934-1984 returns on levered, actively managed long-run portfolios of stocks, bonds and bills. Financial Analysts Journal 41 (September-October): 24-43. Hakansson, Nils. 197\. On optimal myopic portfolio policies, with and without serial correlation of yields. Journal of Business 44 (July): 324-34. Hakansson, Nils. 1974. Convergence to isoelastic utility and policy in multi period choice. Journal of Financial Economics I (September): 201-24. Huberman, Gur, and Ross, Stephen. 1983. Portfolio turnpike theorems, risk aversion and regularly varying utility functions . Econometrica 51 (September): 1104-19. Ibbotson, Roger, and Si/lquefield, Rex. 1982. Stocks, Bonds, Bills, and Inflation: The Past and the Future. Charlottesville, Va.: Financial Analysts Research Foundation. Leland, Hayne. 1972. On turnpike portfolios. In Karl Shell and G. P. Szego (eds.), Mathematical Methods in Investment Finance. Amsterdam: North-Holland. Mossin, Jan . 1968. Optimal multiperiod portfolio policies. Journal of Business 41 (April): 215-29. Reinganum, Marc. 1981 . Misspecification of capital asset pricing: Empirical anomalies based on earnings yields and market values. Journal of Financial Economics 9 (March): 19-46. Ross, Stephen. 1974. Portfolio turnpike theorems for constant policies. Journal of Finan- cial Economics I (July): 171-98. --- ## Page 764 50 A Dynamic Portfolio of Investment Strategies: Applying Capital Growth with Drawdown Penalties Professor John M. Mulvey, Mehmet Bilgili and Taha M. Vural Department of Operations Research and Financial Engineering, Princeton University Abstract The growth optimal investment strategy has been shown to be highly effective for structured decision problems such as blackjack, sports betting, and high fre- quency trading. For securities markets, these strategies are more difficult to apply due to a variety of practical issues: structural changes in market behavior due to varying risk premium and related factors, transaction costs, operational con- straints, and path dependent risk measures for many investors, including surplus risks for a defined-benefit pension plan. In addition, the standard three step approach for institutional money management does not allow for rapid changes in asset allocation - especially needed during highly turbulent periods. We modify the growth models to address downside protection, along with applying a portfolio of investment strategies - to improve diversification of the portfo- lio. Empirical results show the benefits of the concepts during normal and crash (2008) periods. 1 Introduction 735 The growth optimal model and its siblings have been applied successfully in a wide variety of decision problems. As an early example, Thorp (1969) imple- mented card counting approximations in the area of gambling situations such as blackjack, as well as extensions in the investment domain including option pricing models. Grauer and Hakkasson (1986) evaluated asset allocation models with the iso-elastic expected utility model (including log-utility); these authors applied this framework for many years in a series of related papers. More recently, Stutzer (2003,2010), Ziemba (2005), and MacLean et al. (2004, 2009) extended the growth models to address downside protection. See Bell and Cover (1980), Kelly (1956), and Samuelson (1971, 1979) and the remainder of this book for details of growth optimal strategies. There are a number of practical issues to address when implementing growth models. For example, individual investors rarely are able to monitor their affairs on a short term basis and take proper corrective actions. In fact, individuals are often accused of rendering poor judgment during market turning points - --- ## Page 765 736 J M Mulvey, M Bilgili and T M Vural increasing risk assets at market tops, and vice versa. Individuals can be unaware of their asset allocation or surplus capital, except during occasional visits to a financial planner. In contrast, institutional investors often apply well established risk management tools via a three step process. First, a set of generic asset categories is defined, such as large-cap U.S. equities, small-cap value equities, long duration government bonds, real estate, and so on. Standard asset categories along with tracking benchmarks are provided by firms including, Russell, Standard and Poor's, MSCI, and Dow Jones. In most cases, these asset categories can be passively managed by means of index funds or exchange traded funds (ETFs); they serve as benchmarks for active portfolio managers. Second, on a semi-regular basis, the investor conducts an asset allocation or asset-liability management study to ascertain their asset proportions to best meet their goals, liabilities, and risk tolerances. For example, many uni- versity endowment or pension plan administrators carry out this task every 2 to 3 years. As a third step, the investor selects active managers to beat their respective benchmarks, or to invest in a low-cost passive index (which can be simply defined such as the usual cap-weighted index or a variant such as fundamental weighted index). The three step process has evolved to provide an implementable process for large institutions; otherwise, there can be great difficulty achieving diversification benefits and simply managing a large pool of capital. The three step process has worked reasonably well over the past twenty plus years. It is durable and has certain benefits in terms of ease of implementation and adding judgment by an advisory firm or investment management organization. Unfortunately, several issues have arisen over the past few years that have slowly undermined the approach. First, the correlation among many of the traditional style segmentation has increased over time, and reached almost unity in 2008. See Figure 1 for the average correlation among six segments along two dimensions: market capitalization (large, medium, and small), and fundamental valuation (value, growth) over the past 15 years. This situation became critical in 2008 when the contagion increased even further. The great majority of stocks performed quite poorly in 2008 - and diversification strategies failed in many cases. Second, many institutional investors have turned to alternative private markets to improve their performance. This shift has been championed by David Swensen at Yale University endowment (with his book Pioneering Portfolio Management (2000) and superb performance up until early 2008). Private markets are much harder to track since the securities are rarely traded. Also, private markets possess substantial spreads between the return of the top managers (say top decile) and the average manager in these segments. Thus, it is difficult to apply a passive index. And importantly, the return patterns can be hard to model since there are few detailed data available for analysis and issues such as leverage are not always evident to outside researchers. Conducting an asset allocation study with a majority of alternative investments is a treacherous assignment. Importantly, a portfolio of 2 --- ## Page 766 A Dynamic Portfolio of Investment Strategies " " " " , \ \ , .,. , • ICB-A ~ ST6.A ~ STI)*A .... _ ... - rum ........... OPT Figure 1 Rolling correlation of traditional equity style classification 737 private market assets is quite difficult to modify as conditions warrant due to their illiquidity. Third, the emergence of ETFs and passive mutual funds in particular segments, along with more fundamental causes, has changed the relationships of securities to each other. For example, institutional investors will trade a basket of commodities (e.g., Goldman's GSCI, or Deutsche Bank's commodity index) in large volumes - again altering the diversification benefits. Thus, in late 2008, we saw very high correlation in returns for a very wide variety of securities - equities, corporate bonds of all types, commodities, mortgage-backed securities, and so on. Even many alternative investors saw substantial losses - 20, to 30 to 40% in 2008. There was a very rapid and substantial increase in risk premium - with commensurate increases in volatility and correlation. The correlation has increased to close to one for the classical segments: Value/growth and Large/mid/small (ST6-A = style segmentation with six cate- gories, ST9-A = style segmentation with nine categories). Industry segments (ICB = Industry Classification Benchmark) and an optimal classification scheme (OPT) give better diversification. The Traditional segmentations are close to a randomly selected classification (RND). Given our knowledge about these turbulent events, investor could have benefited by applying capital growth models. A critical issue involves ensuring a consistent 3 --- ## Page 767 738 J M Mulvey, M Bilgili and T M Vural relationship between the investor's wealth and the anticipatory risks in the market. The investor must avoid taking risks (bets) above the log-optimal solution, in order to optimize the long-term consequence of their decisions. Critically, as the investor's wealth decreases, they must lower their risk threshold - by reducing committed capital. This rule is quite well known; however, it has been difficult or almost impossible for many investors to take the appropriate action (due to a number of factors including those mentioned above). In the next section, we propose an alternative to the usual three-step approach to address the abovementioned issues. In this regard, we link traditional growth models with strong downside protection. We emphasize capital and risk allocation across investment strategies - rather than individual assets (or asset categories). 2 Dynamic Portfolio Tactics This section describes the fundamental tenets of dynamic portfolio tactics (DPT). Two of the primary concepts are: protect the investor's capital (or goal capital) as a first priority - prevent large losses at all costs; and construct and revise a portfolio of investment strategies - rather than a portfolio of assets - based on dynamic risk analysis. To achieve the first concept, we employ highly liquid investments such as futures markets and high-volume exchange traded funds. In both cases, the transaction costs are quite low for most investors (except the largest institutions such as CALPERs and the Canadian Pension System). For investors possessing illiquid assets, we advocate a variation of the DPT approach for tactical asset allocation based on replication strategies (see Mulvey and Ling, 2009). In this paper, we focus on single-period, asset only investments; see Mulvey et aZ. (2003a, 2008, 2009), Zenios and Ziemba (2006, 2007), and Ziemba and Mulvey (1998) and it references for applications of asset-liability and multi-stage models. The developed approach can be extended in the natural manner. To start, we define the traditional growth model as a sequence of nonlinear stochastic optimization models. First, we are interested in a relatively long sequence of decisions occurring at equal time steps (for simplicity), t = {I, 2, ... , T} where the horizon equals time T. For our purposes in the empirical section, we will render decisions every five trading days (a balance between high frequency and mid-term models) in order to capture relative rapid changes in the investor's wealth. Roughly, the investor will makes fifty asset allocation decisions per year, or 500 per decade. Next, at time point t, we define uncertainties via a set of stochastic scenarios {S}, in which the returns for the next five days will depend upon a number of factors known at the time of the decision - including recent volatility, recent momentum returns, current risk premium, and so on. Importantly, we do not assume a sta- tionary process; instead the risk premium and volatility/correlation will encounter bursts of high turbulence (as we have seen displayed in 2008 and in several previous episodes). 4 --- ## Page 768 A Dynamic Portfolio of Investment Strategies 739 Figure 2 Rolling correlation of equity returns and government long bond returns We give a straightforward example of a changing relationship in market behav- ior (Figure 2). Here we display the rolling correlation between the return of large U.S. cap stocks (S&P 500) and the return on long-duration U.S. government bonds. Many portfolio models assume that this correlation remains constant. However, during sharp downturns, the correlation generally becomes negative - due to fun- damental factors - mostly due to a flight to quality during a recession. It is clear that the scenario generator must take these changing regimes into account (more on multiple regimes later). As compared with traditional asset allocation models, we will focus on a set of investment strategies {J}, rather than subsets of securities. Several example investment strategies will be described in the next section. We distinguish two types of investment strategies: core strategies taking capital {J1}, and overlay strategies {J2} that employ futures markets and swaps, taking risk capital. We separate these strategies in order to emphasize the linkage of overlay strategies with the core strategies, and when the core strategies cannot be easily traded, such as private equity or venture capital. We are interested in the optimal combination of core and overlay strategies that best suits the investor at a given time. The traditional growth models employ the following stochastic optimization model [CO]. [CO] Max EU(W2) Subject to WI = L Xj jEjl Xj 2': 0, and x E X and W2 ,s = L Xj * (1 + 1'j,s) for s E {S} Jl 5 --- ## Page 769 740 J M Mulvey, M Bilgili and T M Vural The decision variables, Xj, are defined over set {Jl}, in our case investment strategies rather than securities or groups of securities, except when indicated oth- erwise. We define the investor's initial wealth equal to WI, to reference the start of the planning period t = [1 to 2]. For this initial model, we greatly simplify the investor's situation by ignoring transaction costs, by assuming a static, single period planning period, by assuming no cashfiow considerations, by focusing on the assets without reference to liabilities or goals, defining a long-only perspective, and so on. In a discrete-time model such as [GO], there is an implicit assumption that the model will be executed over a short enough horizon so that the investor's wealth is not greatly affected by intermediate changes in returns. Next, we assume the standard growth-optimal utility function U (W2) = Ln( W2) within the family of iso-elastic functions U(w) = (llY) X w'Y for a single risk aversion parameter gamma. Returns are designated over a finite set of stochastic scenarios {S}. All policy and legal constraints are contained in set {X}. This model is a straightforward nonlinear program, which is concave for any risk adverse decision maker, and thus can be readily solved with a standard package (such as the solver in Excel). For the optimization, we employ versions of the iso- elastic utility function. However, the model can be readily adapted for other convex reward and risk measures. To extend the model, we allow for a borrowing variable Xb (with upper limit Ub) and a set of over lay variables - set {J2}. The first equation above is modified as: WI + Xb = L Xj and 0:::; Xb :::; Ub jEJI The revised second equation, ending wealth W2 is: W2 ,s = L Xj * (1 + Tj ,s ) - Y * (Tb) + L Xj * Tj j jEJ2 The overlay variables do not require borrowing (implicit) and do not take direct capital. Rather, the core assets {Jl} serve as the margin requirements for the overlay strategies (Mulvey et al., 2006, 2007). We assure that the total level of overlay risk capital is small enough to prevent margin calls and to maintain the investor's overall risk profile (limits in constraints X). The cost of borrow is defined by Tb ,s. This value is uncertain due to two causes: since the cost of borrowing may depend upon interest rates during the planning period, and secondly due to the variable amount of leverage that may occur during the planning period as specified by the investment strategies. The growth strategy [GO] (and variants) can be applied in a straightforward manner given a reliable and robust model for generating/forecasting the return scenarios embedded in set {S}. Unfortunately, due to the changing structure of security markets (e.g., DeBondt and Thaler, 1985, 1987) and the limited number of time junctures for most investors, there is a need for a carefully designated risk management system. A typical theoretical approach would be to increase the risk 6 --- ## Page 770 A Dynamic Portfolio of Investment Strategies 741 Figure 3 A penalty function to protect downside losses aversion parameter 'Y within the iso-elastic family - lowering risks throughout the entire planning period. Instead, we design an alternative approach in which the investor's risk aversion changes as a function of her wealth path. Accordingly, the utility function is modified to be conditional on the wealth path {wt} and especially drawdown values {dt} at time t. The following function reflects this concept: here the function p(.) penalizes investment decisions with outcomes falling below specified drawdown values. The drawdown variable is a function of the distance between the high water mark of the wealth path and the current wealth. dt = w~"i9h - Wt where w~igh = max { W T } T=l , ... ,t-l For the empirical tests, we have found that two thresholds values, Dl = 12%, and D2 = 16%, in conjunction with a piecewise approximation have proven to be a robust approach. Thus, when drawdown exceeds these thresholds, additional risk aversion is obtained by assessing penalties over the iso-elastic function. This approach is similar in spirit to Ziemba (2005) and MacLean et al. (2004,2009). Our approach allows for changing risk premium as a function of the investor's wealth and market conditions. The downside penalty function is particularly important during periods of high turbulence for assets possessing returns with a positive function of risk premium, such as equities or corporate bonds. To this point, we refer to the fall 2008 months in which drawdown became quite excessive, and risk protection was important at that time. A critical aspect of any investment model involves the forecasting system for projecting the returns for the assets and investment strategies over the upcoming period. We advocate a projection system based on multiple regimes and an embed- ded Markov switching matrix (Guidolin and Timmermann, 2007; Hamilton, 1989; Kim, 1994; and Mulvey and Bilgili, 2009). In other words, at each time period, we estimate the current regime (for example, normal, bubble, crash) by means of current conditions such as volatility and correlation over the past 5 to 10 days. Several algorithms can be applied to this problem (Ding and He, 2004; Mulvey and 7 --- ## Page 771 742 1600 1,--------, - Regime-1 ~ Regime·2 1400 - Regime-3 1200 800 600 400 200 1. M Mulvey, M Bilgili and T M Vural S&P 500 °01J05J83 03,{J6185 05.u6ta7 07105£9 09,{l48 1 11/03.193 01103.196 03,04,98 O5.1l3iOO 07t1J3,{12 (9!Ol,{)4 l1J01!08 12131.0:1 Figure 4 A time series of a triple regime analysis for U.S. equities (1/1/1983 to 1/ 1/2009) Table 1 Transition probability matrix for three regime analysis Regime 1 Regime 2 Regime 3 Number of Periods Regime 1 0.65 0.22 0.33 101 Regime 2 0.21 0.35 0.44 81 Regime 3 0.12 0.19 0.63 158 Bilgili, 2009). Once a regime is estimated, we can calculate a subset of internally consistent scenarios for each regime. As an example, suppose that we estimate 3 regimes for equities. Then, we form three sets of scenarios {Sl}, {S2}, and {S3}, with probabilities, 1f1, 1f2, and 1f3, respectively. The number of scenarios in each set depicts the relative probabilities. And the probability of moving between scenarios is indicated by the Markov transition matrix. Mulvey and Bilgili (2009) provide further details. Figure 4 and Table 1 show the path and the transition probability matrix of a three regime analysis over the period 1983 to 2009 for the U.S. equity market, respectively. Note that the down cycles correspond mostly to the third regime, and the persistence of these regimes over time. Regimes one and three are the most stable (60-65% probability of remaining in that regime for the subsequent period), whereas regime two is the least stable (35% probability). 3 Investment Strategies and Empirical Results This section takes up the proposed growth model with downside protection. The first step is to define several investment strategies that possess good long term 8 --- ## Page 772 A Dynamic Portfolio of Investment Strategies 743 Table 2 Performance of equity strategies for emerging markets (Benchmark EEM, rebalanced = equal weighted, and 130/20 strategies) e~d2d' hmldiilo/ ZQQS:JldD!i: ZQQ2 e!i:ri2sj' Ja!lldiilDr: 12~J!.1D!i: ZQW Period length: 4.42 years EEM Rebalanced 130/20 Period length: 14.42 years EEM Rebalanced 130/20 Annual Geometric Return 10.80% 12.80% 13.9QO/n Annual Geometric Return 6.61% 11.70% 13.61% Annual Standard Deviation 36.90% 33.30% 36.10% Annual Standard Deviation 25.86% 25.11% 27.71% 29.60% 32.30% " """""""." Sharpe Ratio 0.18 0.27 0.27 Sharpe Ratio 0.10 0.31 0.35 Worst Month Return -26.10% -25.70% -28.20% Worst Month Return -26.10% -25.7O"A. -28.20% ~~~.I?r~~~,~~~"w~~~~~~~~~~~~~~~~~~~ic~~~~,,§?,:~.w~~c+~~~o~~.: ,~ ,<~~+~~~~~~,~!:.~~"w~~~~c,.~~~.I?~~~~,~~r.-"w~~~~~~~~~~~~~~~~~~~~c.~~~o §?: ,~ ,<~~+~~~,§.~}~"w~+~~o §!:,,~~ ,<~~c. _~_~~~ _rr1-'~~_)(I?_ri~~~Cl:~r1 __________________ .1_~:_~?% ____ _ , ____ ?9:?~_'J_b ____ + ------ ?9:(;-~'Yci ------ -~-E!~I.J -r~/~CI-)(~-r;;I\o\I~Cl:\o\II1 _____________ _ , _____ ~9:~ ____ _ ~ ____ ~~:?_()')1, _______ ?9:}c:J%_ £~,!!~!,~!!,~,~"~,!!~,~,~~,~,~,~!~,~"",,,,,i.. """"} :,Q2"""""",, """"Q:~~ """"",,,9:,~,~ """"""",~,~,~E~!,~!!,~,~,,~!!~,~,~~,~,~,~~!~,~ """",!:~""""""""""",g:,~2"""",,~ 0.93 characteristics. We illustrate the modeling principles by reference to several strate- gies that have proven effective over long time periods. The core investment strategy employs three equity markets: (a) countries in the emerging market segment (symbol = EEM), (b) sectors of the S&P 500 index (symbol = SPY) , and (c) developed countries outside the U.S (symbol = EFA). In each case, there are cap-weighted indices that can be readily purchased (EEM, SPY, and ETA, respectively) via high volume exchange traded funds (ETFs) . These instruments are among the most highly traded in the world (over $500 million per day in mid-2009). To develop an investment strategy, we form a portfolio of ten ETFs that depict each of the three underlying markets. For example, the emerging markets can be largely covered by purchasing single country ETF (China, Brazil, India, Korea, etc.). The long portfolio will be set up by equally weighting the 10 countries (10% each), with rebalancing back to the equal weight target on a short- term basis (5 days in our tests). We can show that in many cases, a fixed-mix rebalanced portfolio will outperform a traditional buy-and-hold cap-weighted index (next subsection). Table 2 shows a comparison of the equal-weighted index versus the cap-weighted index from the emerging markets, over two periods 1/ 1/ 1995 to 7/1/2009 and 1/1/2005 to 7/1/2009. Similar results occur for EAFE and S&P 500 sectors. In fact, equal weighting is the optimal solution to a dynamic portfolio model if the returns of the individual components are equal and independent. This section illustrates the advantages of applying a multi-period policy rule. We begin with the well-known fixed-mix investment rule due to its simplicity and profitability. This policy rule serves as a benchmark both for other types of rules and for the recommendations of a stochastic programming model. 3.1 Fixed mix policy rules First, we describe the performance advantages of the fixed-mix rule over a static, buy-and-hold perspective. This rule generates greater return than the static model by means of rebalancing. The topic of re-balancing gains (also called excess growth or volatility pumping) as derived from the fixed-mix decision rule is well understood for a theoretical perspective. The fundamental solutions were developed by Merton (1969) and Samuelson (1969) for long-term investors. Further work was done by 9 --- ## Page 773 744 J M Mulvey, M Bilgili and T M Vural Fernholz and Shay (1982) . Luenberger (1998) presents a clear discussion. We illustrate how rebalancing the portfolio to a fixed-mix creates excess growth (Mulvey and Kim, 2008). Suppose that a stock price process Pt is lognormal so it can be represented by the equation (1) where ex is the rate of return of Pt and (/2 is its variance, Zt is Brownian motion with mean 0 and variance t. The risk-free asset follows the same price process with rate of return equal to r- and standard deviation equal to O. We represent the price process of risk-free asset by B t : When we integrate the equation (1) , the resulting stock price process is Pt = Poe(a-a2 /2)t+az t (2) (3) It is well documented that the growth rate "( = ex-(/2/2 is the most relevant measure for long-run performance. For simplicity, we assume equality of growth rates across all assets. This assumption is not required for generating excess growth, but it makes the illustration easier to understand. Next, let's assume that the market consists of n stocks, each with stock price processes Pl ,t, . . . , Pn,t following the lognormal process. A fixed-mix portfolio has a wealth process W t that can be represented by the equation (4) where 'TIl , . . . , 'TIn are the fixed weights given to each stock (proportion of capital allocated to each stock). In this case, the weights sum up to one n (5) The fixed-mix strategy in continuous time always applies the same weights to stocks over time. The instantaneous rate of return of the fixed-mix portfolio at anytime is the weighted average of the instantaneous rates of returns of the stocks in the portfolio. In contrast, a buy-and-hold portfolio is one where there is no rebalancing and therefore the number of shares for each stock remains constant over time. This portfolio can be represented by the wealth process W t : dWt = mldPl,t + ... + m ndPn,t where ml, . .. ,mn depicts the number of shares for each stock. (6) Again for simplicity, let's assume that there is one stock and a risk-free instru- ment in the market. This case is sufficient to demonstrate the concept of excess growth in a fixed-mix portfolio as originally presented in Fernholz and Shay (1982). 10 --- ## Page 774 A Dynamic Portfolio of Investment Strategies 745 Assume that we invest 7] portion of our wealth in the stock and the rest (1 - 7]) in the risk-free asset. Then the wealth process W t with these constant weights over time can be expressed as dWt!Wt = 7]dPt! Pt + (1 - 7])dBt! B t (7) where Pt is the stock price process and B t is the risk-free asset. When we substitute the dynamic equations for Pt and B t , we get dWt!Wt = (r + 7](a - r))dt + 7]adzt (8) For simplicity, assume the growth rate of all assets in the ideal market should be the same over long-time periods so that the growth rate of the stock and the risk-free asset are equal. Hence a - a 2/2 = r From equation (8), we can see that the rate of return of the portfolio, a w , is a w =r+7](a-r) By using (9), this rate of return is equal to a w = r + 7]a2/2 The variance of the resulting portfolio is a; = 7]2a 2 Hence, the growth rate of the fixed-mix portfolio becomes "Yw = a w - a; /2 = r + (7] - 7]2)a2/2 (9) (10) (11) (12) (13) This quantity is greater than r for 0 < 7] < 1. As it is greater than r, which is the growth rate of individual assets, the portfolio growth rate has an excess component, which is (7] -7]2)a2 /2. Excess growth is due to rebalancing the portfolio constantly to the target fixed-mix. The strategy moves capital out of stock when it performs well and moves capital into stock when it performs poorly. By moving capital between the two assets in the portfolio, a higher growth rate than each individual asset is achievable. It can be shown that the buy-and-hold investor with equal returning assets lacks the excess growth component. Therefore, buy-and-hold portfolios under-perform fixed-mix portfolios in various cases. We can easily see that the excess growth component is larger when a takes a higher value. Next, we design a simple long/short strategy based on the observation that an equal weighted portfolio will generally outperform a capitalized weighted portfolio such as the S&P 500 or the MSCI emerging market index EEM. A good compromise is 130/20 - wherein the equal weighted index is set at 130% long, and the cap- weighted index is 20% short. The portfolio is slightly levered at 110% with a commensurate cost of borrowing the extra 10%. The statistics for this variant is listed in Table 2, along with the associated statistics for the benchmarks: capital weighted index, equal weighted/5 trading days, and the 130/20 index. 11 --- ## Page 775 746 J M Mulvey, M Bilgili and T. M Vural Table 3 Performance of equity strategies wit h drawdown constraints (January 1999 to J une 2009) (Benchmarks = EEM, EAFE, and S&P500; equal weighted ; and Dynamic Portfolio Tactics DPT ) EEM 10.80% 30.60% 0.18 -8.51% 100 12.80% 29.60% -3.10"Ai 0.99 DPT 3100% 7.80% ~rnerging . l'v1iirkets m~ 11.39% 1.00 16.95% 0.97 DPT 20.38% 19.02% 8.80% 0.86 68.75% 0.76 As mentioned in the previous section, capital growth theory requires a consistent relationship between the investor's capital at time t and their risks at the same time. Thus, as capital is lost and drawdown increases, the investor must lower risks - otherwise the size of the bets will be too large in most cases (roughly speaking). The DPT model takes this into account. As mentioned, we designate two breakpoints for approximating the nonlinear objective function, Dl and D2 to reduce risks by reducing capital in a complementary fashion. The results of applying the approximation are shown in Table 3. Here we see that the overall performance has been increased, especially with regard to the worst case losses. Especially, note the worst and best case time periods. In the case of the cap-weighted index, we see that the upside periods are much larger than the upside periods for DPT. Conversely, the worst downside periods are much better with DPT than with the cap-weighted indices. As mentioned, we are willing to give up a substantial portion of the best periods, if we are able to protect the investor's capital during drawdown periods. This condition is an important criterion for achieving the capital growth path. To improve upon the core equity results (core assets), we add two strategies. These strategies are designated as overlays, i.e., they do not require any dedicated capital for their purchase (only as marginable assets). See Mulvey et al. (2004) for a well known example of a successful overlay strategy based on trend fol- lowing. Table 4 shows the historical performance of the two developed overlay strategies. 12 --- ## Page 776 A Dynamic Portfolio of Investment Strategies Table 4 Performance of two overlay strategies (DEO and PUll at 100 Period: Ju11l1999-June 2009 100% DEO 100% PUI Annual Geometric Return 8.69% 7.32% Annual Standard Deviation Max Drawdown Return/Volatility 0.53 0.95 Return/MaxDrawdown 0.61 0.90 Worst Month Return -8.58% -6.37% Best Month Return 35.59% 5.56% 747 The first overlay strategy is called Duration Enhancing Overlay - DEO. See Mulvey et al. (2006) and Zhang (2006). This strategy increases the duration of a portfolio by entering into a swap agreement (or a long future purchase of long government bonds). DEO, in general, takes a long position on the long term bond and a short position on T-bill. This strategy has a particular benefit for a defined benefit pension plan to help them in matching duration of assets and liabilities (Mulvey et al., 2006). Dynamic DEO is a strategy that uses certain signals to improve performance vis a vis returns and reduced correlation with equity markets. The signals that the strategy uses are from (1) interest rates, (2) equity market returns, and (3) the strategy's performance over recent time periods. The basic idea is to use the trend property of interest rates and the volatility level of the equity market returns. When equity market volatility increases, the strategy increases its capital commitment. In addition to these, past performance of the strategy itself is used; it is assumed that in the DEO strategy a bad day precedes a cluster of bad days. The second overlay strategy applies momentum and futures curve factors to a collection of futures markets - extending upon the traditional trend following rules developed by Dr. Frank Vannerson at Commodity Corp and later at Mt. Lucas Management, and many others (e.g., Bodie and Rosansky, 1980; Chan et al., 1996; Erb and Harvey, 2006; Mulvey et al., 2004; and Rouwenhorst, 1998). We call our strategy the Princeton University Index (PUI) . Herein, the goal is to improve performance by capitalizing on patterns occurring in commodity markets. The index is based on two ideas: the expected return of commodities futures depends upon the degree of hedging by producers or consumers - as evidenced by the shape of the futures curve for each commodity (Brennan et al., 1997). Thus, a curve with sharp contango signifies that consumers are the primary hedgers, whereas backwardation signifies that the producers are the primary hedgers. We assume, and with much evidence to support the supposition, that hedgers will pay an expected costs for their behavior (and conversely that speculators will possess a positive expected cost) over long time periods and on average. Accordingly, we take positions that correspond to positive expected values. A rebalanced portfolio of commodity futures is constructed to greatly reduce risks for the PUI strategy. 13 --- ## Page 777 748 J M Mulvey, M Bilgili and T. M Vural Table 5 Performance of portfolio of strategies - July 1, 1999 to June 30, 2009 GS DPT 100% 7bond (30 conmodity Corrbined Period' July 19!!9-June 2lXB 1CXJ%SP5(X) 1CXJ%EAFE year) AJG conmodity Index Index Strategies Amual Geometric Return ·2.26"10 1.88% 6.95% 4.02% 10.67% 25.25% Amual Standard Deviation 15.92% 1B.08'10 13.59% 17.46"10 25.05% . 14.94% 0==C====~~~~=:C~ ~~=~ ~~=~~ 0~~~~~~~-+~~~·,~~~' Max DraIAdoIMl SO.80'10 54.24% 23.77% 5450'10 61.03% 12.99"10 .Return/llolatility -0.14 0.10 0.51 Q23 0.43 1.69 !!"tlJ",fi'JIiI><~ ____________ + ______ :0:()1 ____ + _____ (l(l3 _____ 4 _________ (l~ __________________________ (lIJ7 __________________ + _____ (l}L ___ + _____ 1:~ ____ + ~"':st_IIIIJrTth_~tlJ"' ________________ ~ ____ : Hi??'~ ________ : 19:(l;~ _________ :~4:5(j"~ ______________________ :?:L_~'J> _______________ 4 ___ .:2!2?'~ __ 4 ____ :!1)g'1o _~st.IIIIJrTth .. ~turT1 .. _______________ ~ 9.93% 13.19'10 _____ 11i_(l;'J>__ 12.99"10. _______ .. ___________ 21.10'10 22.79'10 __ spSOO -- eafe 30y bond ··· ... ··· aig com ind j' -.11- gs com ind , 1; ---- PJi .,l \ --dec ,hi" \ -d comtined .,.~ ,.,. .. Figure 5 Time series performance for equity and bond benchmarks and combined DPT strategy (July 1, 1999 to June 30, 2009) As a second factor, we measure the momentum of return for each of the com- modities over the past six to twelve months. If momentum is positive, we increase our exposure to that commodity in the index. The overall commodity portfolio is rebalanced twice per month. We next combine the three strategies: core equities as discussed in the previous section, the commodity overlay via PUI, and the interest rate overlay via DEO in an integrated portfolio (called DPT). Note that we are employing the core equity strategy as traditional investable assets, along with the two overlay strategies DEO and PUr. The percentage of traditional assets assigned for the overlays is a small portion of capital ~ around 10 to 15%. This small percentage prevents margin calls and remains within the target level of risks. Also, from the standpoint of risk allocation, we designate the proper proportion of the overlays, in this case ~ 100% for PUI, and 50% for DEO. These proportions are targets for a fixed-mix rebalancing rule on a monthly basis. Turning to Table 5 and Figure 5, we see that the overall performance is enhanced by combining the three strategies in a rebalanced portfolio. In particular, the Sharpe 14 --- ## Page 778 A Dynamic Portfolio of Investment Strategies 749 ratio is improved since the strategies provide their best returns at different time conjunctures. In a similar fashion and more importantly, the return per drawdown ratios are superior for the integrated system. This performance is robust with respect to the modeling parameters since the individual strategies are designed to provide wide diversification. Also, the rebalancing gains can be significant and can be readily implemented due to the liquidity of ETFs and futures markets. 4 Conclusions This paper extends capital growth models with downside protection. To achieve the twin goals of protecting the downside while generating good returns during upswings, we penalize drawdown values in a sequential fashion, first via a stochastic nonlinear program, and second by implementing relatively simple policy rules. We show that a long-term investor can perform well by avoiding large losses, as is well known. As a corollary, we demonstrate that a portion of the upside gains can be forsaken in our quest to improve the downside protection. This latter observation is somewhat controversial since some have argued that an investor must stay fully invested in order to improve on long term performance. Our objective function has an asymmetric relationship between the upside and the downside. But the changing market structure necessitates a more conservative strategy than is the case with traditional capital growth models. We focus on the drawdown values to implement this primary concept. A second feature involves combining a set of investment strategies via dynamic portfolio tactics. Since sharp market corrections take place within the context of ris- ing volatility and correlation, we cannot depend upon diversification benefits among asset securities. There is simply too much contagion during periods of high turbu- lence. Instead we design a portfolio of strategies in which the performance of one strategy is relatively uncorrelated with the performance of neighboring strategies. Several examples of these phenomena are presented. The primary damage of the 2008 market crash was done in a quick amount of time - three months September toNovember (along with earlier losses in February). It is evident, therefore, that investors must remain vigilant and nimble if they wish to avoid large losses. The traditional asset allocation procedure possessing relatively long periods between revisions of policy have not served the investor well; as we mentioned, many investor experienced large losses (25- 30% or more) . A more dynamic framework is needed, with much shorter time intervals, in order to achieve the goals of capital growth. The investor must be careful to not only grow their capital during positive markets, but they must protect their capital during turbulent periods if they are to achieve the goals of capital growth theory. References Bell, R. M. and T. M. Cover (1980) . Competitive optimality of logarithmic investment. Math of Operations Research, 5, 161-166. 15 --- ## Page 779 750 J M Mulvey, M Bilgili and T M Vural Brennan, D., J. Williams, and B. D. Wright (1997) . 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Dynamic linear models with Markov switching. Journal of Econometric, 60, 1- 22. Luenberger, D. (1998). Investment Science. New York: Oxford University Press. MacLean, L. C., R. Sanegre, Y. Zhao, and W. T. Ziemba (2004). Capital Growth with Security. Journal of Economic Dynamics and Control, 28, 937- 954. MacLean, L. c., E. Thorp, and W. T. Ziemba (2009). Optimal Capital Growth with Convex Loss Penalties, Working paper. Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. Review Economics Statistics, 51, 247- 257. Merton, R. C. (1973). An intertemporal capital asset pricing model. Econometrica, 41 , 867- 887. Merton, R. C. and P. A. Samuelson (1974). Fallacy of the log-normal approximation to op- timal portfolio decision-making over many periods. Journal of Financial Economics, 1, 67- 94. Mulvey, J. M., M. Bilgili, and W. C. Kim (2007/2008). Dynamic investment strategies and rebalancing gains. The Euromoney Algorithmic Trading Handbook. 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Constantly rebalanced portfolio - Is mean reversion necessary? Encyclopedia of Quantitative Finance, 2, 714- 717. Mulvey, J. M., B. Pauling, and R. E. Madey (2003a). Advantages of multiperiod portfolio models. Journal of Portfolio Management, 29, 35- 45. Mulvey, J. M., K. Simsek, Z. Zhang, and F. Fabozzi (2008) . Assisting defined-benefit pension plans. Operations Research, 56, 1066- 1078. Mulvey, J. M., K. Simsek, and Z. Zhang (2006). Improving investment performance for pension plans. Journal of Asset Management, 7, 93- 108. Mulvey, J. M., C. Ural, and Z. Zhang (2007). Improving performance for long-term in- vestors: Wide diversification, leverage and overlay strategies. Quantitative Finance, 7(2) , 1- 13. Mulvey, J. M. and S. Ling (2009). Replicating private equity returns by means of sector ETFs. Working paper, Princeton University. Rouwenhorst, K. G. (1998). International momentum strategies. The Journal of Finance, 53(1) , 267-284. Samuelson, P. A. (1969). 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Series of Handbooks in Finance, Vol. 2. Amsterdam: Elsevier. Zhang, Z. (2006). Stochastic optimization for enterprise risk management. Doctoral Dis- sertation. Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ. Ziemba, W. T. and J. M. Mulvey (1998) , eds. Worldwide Asset and Liability Modeling. UK: Cambridge University Press. Ziemba, W. T. (2005). The symmetric downside risk Sharpe ratio and the evaluation of great investors and speculators. Journal of Portfolio Management, Fall, 108- 122. 17 --- ## Page 781 This page is intentionally left blank --- ## Page 782 Available online at www.sciencedirect.com SCIENCE@DIR 'ECTO JOURNAL OF Economic D)'!lamics & Control 753 ELSEVIER Journal of Economic Dynamics & Control 28 (2004) 975 - 990 www.elsevier.com/locate/econbase 51 Intertemporal surplus management Markus RudolP'*, William T. Ziembab a WHU-Ol/o Beisheil1l Graduate School ot Mal/aaement, Dresdnel" Bank chair 01 Finance, Burgplatz 2, 56179 Vallendar, Germal/y b Facility (it Commerce, University 01 BriTish Columbia, 2053 Main Mall, Vancollt;er, BC, Canada V6T 122 Abstract This paper presents an intertemporal portfolio selection model for pension funds or life in- surance funds that maximizes the intertemporal expected utility of the surplus of assets net of liabilities.' Following Merton (Econometrica 41 (1973) 867), it is assumed that both the asset and the liability retU11l follow Ito processes as functions of a state variable. The optimum occurs for investors holding four funds: the market portfolio, the hedge portfolio for the state variable, the hedge portfolio for the liabilities, and the riskless asset. In contrast to Merton's result in the assets only case, the liability hedge is independent of preferences and only depends on the funding ratio. With HARA utility the investments in the state variable hedge portfolio are also preference independent. Finally, with log utility the market portfolio investment depends only on the current funding ratio. © 2003 Elsevier B.V. All rights reserved . .! EL classification: G23; Gil Keywords: Asset; Beta; Funding ratio; HARA utility function; Hedge portfolio; Intertcmporal capital assct pricing model; Ito process; .! -li.mction: Liabilities; Log utility function: Safety first; Shortfall risk; State variable; Surplus management 1. Introduction intertemporal asset allocation models date to Merton (1969, 1971, 1973) (see also the summary in Merton, 1990) and Samuelson (1969). Merton presents a continuous time intertemporal model whereas Samuelson discusses the discrete time case. Both models • Corresponding author. WHU-Otto Beisheim Hochschule, Burgplatz 2, 56179 Vallendar, Gelmany. Tel.: + 49-261-6509421; fax: + 49-261-6509409. E-Illail addres,w!s: markus.rudolf@unisg.ch, mrudolf@whu.edu (M. Rudolf), ziemba@interchange.ubc.ca (W.T. Ziemba). URL: http://www.whu.edu/banking 0165-1889/031$ - see front matter ~) 2003 Elsevier B.Y. All rights reserved. doi: I 0.1 0 16/S0.l65-1889(03 )00058-7 --- ## Page 783 754 M Rudolf and W T. Ziemba 976 M. Rudo/I w.T. ZiembalJournal o/Economic Dynamics & Conlrol28 (2004) 975 - 990 formulate a lifetime portfolio selection problem. Merton's (1973) intertemporal capital asset pricing model derives equilibrium asset premia. In contrast to Sharpe's (1964) CAPM, Merton's CAPM is based on a three-fund theorem. Each rational investor holds the riskless asset, the market portfolio, and a hedge portfolio for a so-called state variable in order to maximize his lifetime expected utility. The state variable is a stochastic term, which affects the asset price processes. The hedge portfolio provides maximum correlation to the state variable, i.e. it provides the best possible hedge against the state variable variance. Continuous time models can be applied to surplus optimization problems, since surplus optimizers, such as pension funds or life insurers, distribute the surplus to those who are insured. Pension funds or life insurance funds are frequently broadly internationally diversified. Exchange rate movements are considered as state variables. Merton, (1993, reprinted in Ziemba and Mulvey (1998), see also Constantinides' (1993) comments on Merton's paper) addresses a problem similar to surplus optimiza- tion. He advocates the view that University endowment funds can be managed by using Merton's (1969) intertemporal portfolio selection model. His objective function is to maximize the lifetime expected utility of University activities with specific costs. The University activity portfolio includes education, training, research, storage of knowl- edge, etc. Since the activities of a University have to be optimized with respect to their costs, the activity costs are "liabilities" for universities. In contrast to our approach, Merton (1993) specifies the University's non-endowment cash flows as Ito processes dependent on the activity costs. This is the classical setting of Merton 's (1973) in- tertemporal CAPM, where the state variables are given by the activity costs. The result is that each utility maximizing University holds three funds, the market portfolio, the riskless portfolio and a hedge portfolio against fluctuations of the activity costs. The composition across those three funds depends on the risk preferences towards market and activity cost volatility. This paper provides a synthesis of the surplus management and the continuous time finance literature. Both the assets and liabilities of a pension fund are modeled as stochastic processes dependent on stochastic state variables. Only the asset mix can be influenced by the fund manager's decision. The paper provides a solution to his decision problem. In comparison to Merton's (1993) setting, the advance made by the paper is that one (or several) state variable(s) are allowed to be distinct from the liabilities. I.e. in contrast to Merton (1993), this paper explicitly examines the impact of long-term liabilities. Three important results developed are: first, if only one state variable is considered, Merton's three-fund theorem is extended to a four-fund theorem. The four distinct mutual funds are: the market portfolio, the state variable hedge port- folio, the liability hedge portfolio, and the riskless asset. Secondly and most important, the investment in the liability hedge portfolio depends only on the current funding ratio of a pension or life insurance fund and is independent of the utility function. This result differs from Merton's (1993) findings. The preference independence of the liability hedge portfolio has major implications for monitoring pension funds and life insurance companies. Thirdly, the state variable hedging policy is preference indepen- dent if hyperbolic absolute risk aversion (HARA) utility functions are assumed. With log utility, even the market portfolio investment is independent of the risk aversion --- ## Page 784 Intertemporal Surplus Management 755 lvl. RudolJ: W. T Ziemba /Journal 0/ Economic Dynamics & Con/rol28 (2004 ) 975 - 990 977 coefficient which is caused by the fact that log utility is equivalent to HARA utility with a risk tolerance coefficient of a = O. [n Section 2, the intertemporal surplus management model and a four-fund theorem are derived. In Section 3 we show that with HARA utility, the risk tolerance coefficients of the model are related to the funding ratio and to the currency betas of the portfo- lio. A k state variable case is derived in Section 4. Section 5 describes a case study for a surplus optimizer who is diversified across the stock, the bond markets, and cash equivalents of four countries (EMU countries are treated as a single country). Section 6 concludes the paper. 2. An intertemporal surplus management model-a four-fund theorem Having surplus over time is what life insurance companies and pension funds try to achieve. In both life insurance companies and pension funds, parts of the surplus are distributed to the clients usually once every year. Hence, optimizing the investment strategy of a life insurance or a pension fund is equivalent to maximizing the expected lifetime utility of the surplus. For t ~ 0 we have the stochastic processes A(t),L(t), Y(t), representing assets, liabil- ities, and a state variable Y. An extension of the setting to k state variables is possible without difficulties (see e.g. Richard (1979) and Adler and Dumas (1983» . Adler and Dumas (1983) take purchasing power parities as state variables. The surplus S(t) and the funding ratio F(t) are defined by Set) := A(t) - L(t), F(t) := A(t)/L(t). (1 ) According to Merton (1973), the state variable Y follows a geometric Brownian motion (i.e. log-normally distributed) where J.ly and O'y are constants representing the drift and volatility, and Zy(t) is a standard Wiener process. The state variable as well as the assets and the liabilities are assumed to follow the stochastic processes: dY(t) = Y(t)[J.ly dt + O'y dZy(t)], dA(t) = A(t)[J.lA(t, Y(t»dt + O'ACt, Y(t»dZACt)], dL(t) = L(t)[J.lL(t, Y(t»dt + O'L(t, Y(t»dZL(t)] , (2) where the drift and volatility parameters for the assets and liabilities are allowed to depend on both time and state variable, and where ZAt),ZL(t) are Wiener processes correlated with Z y(t) and also with each other. Hence, we have cross variations dZA(t) dZy(t) = PAY dr, dZL(t)dZy(t) = PLY dt, --- ## Page 785 756 M Rudolf and W T. Ziemba 978 M. Rudolf; W.T Ziemba/Joul'l1([/ o./Ecol1omic Dynamics & Conlro/28 (2004) 975 - 990 where PAY represents the instantaneous correlation between the Wiener processes ZAt) and Zy(t), PLY is the instantaneous correlation between ZL(t) and Zy(t), and PAL refers to the instantaneous correlation between ZAt) and ZL(t). The return processes Ry(t),RACt),RL(t) for the state variable, assets and liabilities are defined by Ry(t) := dY(t)/ Y(t), RAt) := dA(t)/A(t), RL(t) := dL(t)/L(t). Following Sharpe and Tint (1990) and according to Eq. (1), the surplus return process is defined as R ( ).= dS(t) = R (t) _ RL(t) st. A(t) A F(t) = [f1A - :C~)] dt + aA dZAt) - :C~) dZL(t). Since dS(t) = Rs(t)A(t), using Eq. (3) we have E[dS(t)] = A(t) (f1A - :C~») dt, 2 (2 az aAL ) dS(t)dS(t) =A (t) aA + pet) - 2 F(t) dt, dS(t)dY(t)=A(t)Y(t) (aAY - ;g») dt, dY(t) dY(t) = y2(t)a} dt, (3 ) (4) where aAL := PALUAuL,UAY := PAYUAuy,aLY := PLYaLay are the covariances of assets with liabilities and of the state variable with assets and liabilities, respectively. The objective is to maximize the expected lifetime utility of surplus, which implies identifying an optimum surplus strategy. I For an analysis of a situation with convex penalties for underfunding, see Carino and Ziemba (1998). The expected utility is pos- itively related to the surplus in each period. This is because positive surpluses improve the wealth position of the insurants of a pension or a life insurance fund, even if the yearly retirement benefits are over-covered by the surplus. Hence, in this interpreta- tion, the insurants are like shareholders of the fund. Then the following equation is the fund's optimization problem, where U is an additively separable, twice differentiable, I To avoid underfunding in particular periods we assume a steady state implying that the dollar value of cmployecs (insurance policies) entering equals those leaving the pension fund (insurance company) dollar value. --- ## Page 786 Intertemporal Surplus Management 757 M. RudoU: W. T Ziemba I Journal of Economic Dynamics & Control 28 (201J4 ) 975 · 990 979 concave utility function and T is the end of the fund's existence: T J(S,Y,t):= m\~xEt ([ U(S,Y,T)dT) (! I+ dt j.T ) = max Et U(S, Y, T) dT + U(S, Y, T) dT . II t .I+ dl (5) Et denotes expectation with respect to the information set at time t and the J -function is the maximum of the life insurance or pension fund's expected lifetime utility. The maximum in (5) is taken with respect to w, the vector of portfolio fractions of the risky assets. Let n be the number of portfolio assets and Wi (I ::::; i ::::; n) be the portfolio fraction of asset i, then WI := (WI, ... , wn ) . Applying the Bellman principle (see e.g. Dixit and Pindyck (1994, Chapter 4», according to Merton (1969) and Merton (1990, p. 102) and according to the mean value theorem for integrals yields J(S, Y, t) = U(S, Y, t) dt + max Et[J(S + dS, Y + dY, t + dt)]. (6) w Let Js be the first partial derivative of J with respect to S, Jss the second partial derivative of J with respect to S, Jy and J yy the first and second partial derivatives with respect to Y, and JSY the derivative of J with respect to Sand Y. Applying Ito's lemma and Eq. (4) yields (see the appendix for proof), 0= U(t,S, Y) Jt + JsA(t) (/lA - :C~») 1 2 (2 al a AL ) +JyY(t)/ly + 2. JssA (t) aA + F2(t) - 2· F(t) w (7) + max +~ J yy y2(t)a} + JsyA(t)Y(t) (aAY - ;~;») Let rnA be the vector of expected asset returns of the risky assets of dimension n, e be the n-dimensional vector of ones, V be the n x n matrix of covariances between the 11 risky assets, VAL and VAl' be the vectors of covariances between the 11 assets and the liabilities, respectively, with the state variable. It is assumed that a riskless asset with return r exists. Then rearranging (7) yields the following equation (see the appendix for proof and definitions of the matrix and the vectors): 0= U(t,S, Y) Jt+JsA(t)(w/(rnA-re)+r- :C~») I 2 (' al w' VAL) +Jy Y(t)/ly + 2. JssA (t) w Vw + F2(t) - 2 F(t) . w (8) + max 1 2 2 ( I aLl' ) +2. JyyY (t)a y +JsyA(t)Y(t) w VAY - F(t) --- ## Page 787 758 M Rudolf and W T. Ziemba 980 M. RudoU: W. T Ziemba/Journal 0/ Economic Dynamics & Control 28 (2()04) 975990 Differentiating (8) with respect to W yields 2 where Js V - Ie ) Y(t)JSY V-I V-IVAL W = - --- rnA - re - VAY + --- A(t)Jss A(t)Jss F(t) Js Y(t)JSY c =: -a --- . WM - b . Wy + -- . WL, A(t)Jss A(t)Jss F(t) V -- I (rnA - re) WM := e'V - I(rnA - re) ' V - I VAL WL := ' V - I ' e V,1 L (9) The vectors WM, Wy, and WL are of dimension n with elements that sum to I; a, b, and c are real constants. The optimum portfolio consists of four single portfolios: the market portfolio WM, the hedge portfolio for the state variable Wy, which is Merton's (J 973) state variable hedge portfolio, the hedge portfolio for the liabilities WL, and the riskless asset. The state variable hedge portfolio Wy reveals the maximum correlation with the state variable Y (see the appendix). A perfect hedge for the state variable could be achieved if the universe of n risky assets contains forward contracts on the state variable. Then the state variable hedge portfolio would consist of a single asset, which is the forward contract. The third portfolio WL is interesting. For the liabilities there exist no hedging opportunities at the financial markets (i.e. a portfolio which hedges wage increases or inflation rates). Eq. (9) shows how a liability hedge can be constructed. This is related to the problem addressed by Ezra (1991) and Black (1989) and solves it intertemporally. In the four-fund theorem, life insurance and pension funds invest in the following four funds, if they maximize their expected lifetime utility (5) based on the stochastic differential equations (2) and (3): 1. The market portfolio WM with level - a(Js/A(t )Jss ). 2. The state variable hedge portfolio Wy with level - b(Y(t)Jsr/A(t)Jss ). 3. The riskless asset with level 1+ a(Js/A(t)Jss ) + b(Y(t)Jsr/A(t)Jss ) - c/F(t). 4. Finally, the liability hedge portfolio WI. with level c/F(t). Thus, the holdings of the liability hedge portfolio are independent of preferences. The most interesting result in the portfolio selection equation (9) is that the liability hedge portfolio holdings depend only on the current funding ratio and not on the form of the utility function. In order to maximize its lifetime expected utility, each life insurance or pension fund should hedge the liabilities according to the financial endowment. The percentages of each of the three other funds differ according to the risk pref- erences of the investors. For example, -a (Js/A (t)Jss ) is the percentage invested in 2 The derivati ve of (8) with respect to w is: 0 =.Js(m,j -re)+A(/).Jss Vw - [A(t)/ F(t)J./\:\·VAL + Y(/).JSy VAl" --- ## Page 788 Intertemporal Surplus Management 759 il1. Rudolf, W. T. Ziemba /Journal of Economic Dynamics & Control 28 (2004) 975 ~ 990 981 the market p01ifolio. Since J is a "derived" utility function, this ratio is a times the Arrow/Pratt relative risk tolerance with respect to changes in the surplus. That is, the higher the risk tolerance towards market risk, the higher the fraction of the market portfolio holdings. The percentage of the state variable hedge portfolio is - be Y(t)Jsr/A(t)Jss ). Merton (1973) showed that this ratio is b times the Arrow/Pratt relative risk tolerance with respect to changes in the state variable. The percentage of the liability hedge portfolio is c/F(t). Surprisingly, and in contrast to Merton's results, this portfolio does not depend on preferences nor on a specific utility function, but only on the funding ratio of the pension fund. The lower the funding ratio, the higher the percentage of the liability hedge portfolio. This allows for a simple technique to monitor life insurance and pension funds, which extends Merton's (1993) approach. In most funding systems, funds are legally obliged to invest subject to a deterministic threshold return. Since payments of life insurance or pension funds depend on the growth and the volatility of wage rates, this is not appropriate. For instance, if the threshold return is 4% p.a. and the wages grow by more than this, the liabilities cannot be covered by the assets. Our model suggests instead that a portfolio manager of a pension fund should invest in a portfolio which smoothes the fluctuation of the surplus returns caused by wage volatility, i.e. in a liability hedge portfolio. Since the liability hedge portfolio depends only on the funding ratio, preferences of the insurants have not to be specified. 3. Risk preference, funding ratio, and currency betas Assume that the utility function is from the HARA class. Merton (1971 and 1990, p. 140) shows that this is equivalent to assuming that the J-function (6) belongs to the HARA class as well. Let rx < I be the risk tolerance coefficient. Then U(S, Y, t) C HARA q J(S, Y, t) C HARA, 1 - rx (KS )rt. J(S, Y,t) =J[S(A(y),t] = -(- -1- + 11 , eP rx - rx (10) where /( and 11 > ° are real constants and p is the utility deflator. Observe that (10) implies linear absolute risk tolerance since -Js/lss = S/ (l - rx) + I1/K. The HARA class of utility functions as defined in (10) is commonly used; it implies the negative exponential utility functions J = _e-as when rx approaches -00, 11 = 1, and p = 0. Moreover, the isoelastic power utility (see Ingersoll (1987, p. 39» J=Srt./rx is obtained for 11 = 0, P = 0, and K = (I - rx )(rt.-I )/rt. . Log utility is that member of isoelastic power utility when rx approaches ° (11 = P = 0), since by applying I'Hopital's rule to the equivalent utility function, lim J(S, Y, t) = lim Srt. - 1 = lim (SIX In S) = In S. rt.~O rt.~O rx rt.~O ( II ) --- ## Page 789 760 M Rudolf and W. T. Ziemba 982 M. Rudolf W. T Ziemba /Journal (J/ Ecol1omic Dynamics & Con/rol28 (2004) 975 990 Under the HARA utility function assumption, K (KS )~ -I Js' = - -- + 11 , . ePI I -(X n;2 (n;s )~-2 Jss =- - -- + 11 , ePI I - (X dJs dJs dS dA . dA Jsy = - = -' - . -=Jss' -' dY dS dA dY dY The percentage holdings of the market portfolio may be re-expressed as ( 12) (13 ) The market portfolio holdings thus depend on the funding ratio and on the risk aversion. The higher F, the higher the investment in the market portfolio, and the higher (X, the lower the market portfolio investment for funding ratios smaller than I. If a sufficient funding is observed (i.e. F > I), then there is a positive relationship between (X and F. If (X approaches 0 (log utility case), the coefficient for the market portfolio investment becomes I - I /F plus the constantl1/ (AK). Thus, for log utility pension or life insurance funds, risk aversion does not matter. Indeed only the funding ratio matters to determine the market portfolio investment. For either case of utility functions, if the funding ratio is 100%, there will be no investment in the risky market portfolio. Thus, the funding ratio of a fund does not only determine the capability to bear risk but also the willingness to take risk. We now consider the percentage holding of the state variable hedge portfolio. Sup- pose the state variable Y is an exchange rate fluctuation, which affects the surplus of a life insurance or pension fund. Given (12) it follows that YJSY dA/A AJss dY/Y (14 ) Hence, the weight of the state variable hedge portfolio (14) is preference independent if HARA utility is assumed. Assume the regression model RA = {3(RA' Ry )R y (asset returns are linear functions of currency returns). Hence, -RA/Ry is the negative beta of the portfolio with respect to the state variable. Using (14) it follows that YJSY - - = -f3(RA,Ry). AJss (15 ) If Y is an exchange rate, then -Ii equals the minimum variance hedge ratio for the foreign currency position. The holdings of the state variable hedge portfolio are in- dependent of preferences; they only depend on the foreign currency exposure of the portfolio. The higher the exchange rate risk in the p0!1folio, the higher the currency hedging. Hence, for the HARA utility case, only the investment in the market portfolio de- pends on the risk aversion ('I.. The investments in all other funds are preference in- dependent. They depend only on the funding ratio and on the exposure of the asset portfolio to the state variable. --- ## Page 790 Intertemporal Surplus Management 761 III. Rudolf W T Ziemba / Journal 0/ Economic Dynamics & Control 28 (201J4) 975 ,, 990 983 4. The multiple state variable case Let k be the number of foreign currencies contained in a life insurance or pension fund's portfolio, YI , ... , Yk be the exchange rates in terms of the domestic currency, and R Y" ... , R Yk the exchange rate retUll1S of the k currencies. Then YIJSY, YkJSYk - --, = -/J(R,1,Ry, ), ... , - -- = - P(R,1,R yk), AJss AJss which are the Arrow/Pratt relative risk tolerances with respect to changes in the ex- change rates for the HARA case. Let V,1y" . . . ,V,1Yk be the covariance vectors of the retUll1S of the asset portfolio with the k exchange rate retull1s, V - I V,1Y, ._ y - IVAl'k WY' := e'V- lv.1Y, ' '' ·'wyk . I' e'Y- VAYk which are the state variable hedge portfolios I-k, and bl := e'V- lvAy" ... ,b" := e'V- 1 vAYk are the coefficients of the state variable hedge portfolios. A pension or a life insurance fund with HARA utility function facing k state variables has the following investment strategy (R,: = dA,IA,: retUll1 on risky asset I, 1=1, .. . ,11): [ a ( I) all] 2: k c W = -- I - -- + - WM - bP(R4 Ry)wy + --WL 1-0: F(t) AI( . 1 . , , , F(t) , 1= 1 ( 16) where II P(R;I, Ry,) = 2: WI P(RI, Rr;). 1=1 Since the right-hand side of Eq. (16) depends on the portfolio allocation w, it is not possible to solve (16) analytically for w, However, numerical solutions can be applied. For k state variables a (k + 3 )-fund theorem thus follows. 5. Case study The following case study is based on a USD-based surplus optimizer investing in the stock and bond markets of the US, UK, Japan, the EMU countries, Canada, and Switzerland. Monthly MSCI data between January 1987 and July 2000 (163 observa- tions) are used for the stock markets. The monthly JP Morgan indices are used for the bond markets in this period (Salomon Brothers for Switzerland). The stochastic bench- mark for a surplus optimizer is the quarterly Thomson Financial Datastream index for US wages and salaries. Quarterly data are linearly interpolated in order to obtain monthly wages and salaries data. The average growth rate of wages and salaries in the U.S. between January 1987 and July 2000 was 5.7% p.a. (see Table I) with annual- ized volatility of 4.0%. Table I also contains the stock and bond market descriptive statistics in USD, and the currency betas of the indices. All foreign currencies except CAD, i.e. GBP, JPY, EUR, and CHF, have volatility of about 12% p.a., and all currencies except JPY depreciated against the USD by a little --- ## Page 791 762 M. Rudolf and W T. Ziemba 984 11I. Rudolt: W.T Ziemba/Journal oI Economic Dynamics & Conlrol28 (2004) 975 - 990 Table I Descriptive statistics Mean Volatility Beta Beta Beta Beta Bela return GBP JPY EUR CAD CHF Stocks USA 13.47 14.74 0.18 0.05 0.35 - 0.59 0.35 UK 9.97 17.96 - 0.47 - 0.36 - 0.29 - 0.48 - 0.14 Japan 3.42 25.99 - 0.61 - 1.1 I - 0.45 --0.44 - 0.43 EMU countries 10.48 15.80 - 0.32 - 0.27 - 0.26 - 0.45 - 0.11 Canada 5.52 18.07 0.05 - 0.02 0.27 - 1.44 0.34 Switzerland 11 .56 18. 17 - 0.14 - 0.32 - 0.26 0.13 - 0.32 Bonds USA 5.04 4.50 - 0.03 0.00 - 0.06 - 0.04 - 0.06 UK 6.86 12.51 - 0.92 - 0.44 --0.80 - 0.39 - 0.59 Japan 3.77 14.46 -0.53 - 1.04 - 0.75 0.09 - 0.72 EMU countries 7.78 10.57 - 0.69 - 0.42 - 0.93 -0.06 - 0.74 Canada 5.16 8.44 - 0.09 · 0.03 - 0.02 - 1.14 0.02 Switzerland 3.56 12.09 -0.67 - 0.53 - UJ3 0.19 -0.99 Exchange GBP 0.1 I 11.13 0.37 0.86 0.42 0.64 rates in JPY -2.75 12.54 0.48 0.65 0.04 0.61 USD EUR" 1.13 10.08 0.7 0.42 I 0.1 0.79 CAD 0.79 4.73 0.08 0 0.02 - 0.02 CHF 0.32 11.57 0.69 0.52 1.04 - 0.13 I Wages and salaries 5.71 4.0 0 0.01 0 - 0.01 0 The stock market data are based on MSC indices and the bond data on .IP Morgan indices (Switzerland on Salomon Brothers data). The wage and salary growth rate is from datastream. Monthly data between January 1987 and July 2000 (163 observations) is used. All coetncients are in USD. The average returns and volatilities arc in percent per annum. a ECU before January 1999. more than 0 to 1.13% per year. From a USD viewpoint, for GBP-beta is especially high (absolute value) for the UK bond market. The Japanese stock and bond market reveal a lPY -beta of - 1.11 and - 1.04, respectively, and the EMU bond market has a EUR-beta of -0.93. Furthennore, the CAD-beta of the Canadian bond market is -1.14, the CHF-beta of the Swiss bond market is -0.99. All other countries have substantially lower currency betas. Since the betas are close to zero, the wages and salaries in the U.S. obviously do not depend on currency movements. The investor faces an exposure against five foreign currencies (GBP, lPY, EUR, CAD, CHF), and has to invest into eight funds. Five of them are hedge pOlifolios for the state variables, which are assumed to be CUITency returns. The next step is to calculate the compositions of the eight funds. The results appear in Table 2. The major holdings in the market portfolio are investments in the US stock market and the EMU bond market. Substantial short positions for the tangency portfolio are obtained for the Canadian stock and the Swiss bond market. The US bond portfolio is a major part of a portfolio providing the best hedge against fluctuations in wages and salaries. More than 126% of the liability hedge portfolio are invested in the US bond market. The bond markets of the respective currencies dominate the currency hedge portfolios. --- ## Page 792 Intertemporal Surplus Management 763 I'd. RudoU: W. T ZiembaiJoul'na/ 0/ Economic Dynamics & Control 28 (2004 ) 975 990 985 Table 2 Optimum portfolios of an internationally diversified pension fund from a USD perspective Market Liability hedge Hedge Hedge Hedge Hedge Hedge poJtfolio portfolio portfolio portfolio portfolio portfolio portfolio GBP Jpy EUR CAD CHF Stocks USA 83.9 - 30.4 3.9 -4.5 -7.5 60.9 -5.1 UK - 14.8 60.6 -3 1.0 - 1.1 - 8.6 -85.3 1.9 Japan -6.7 2.7 6. 1 8.9 -2.4 - 4.0 - 0.7 EMU - 19.2 - 683 35.3 12.8 23.5 138.3 -3.8 Canada - 39.2 5.1 14.6 13.6 5.5 --2.6 - 0.9 Switzer!' 21.6 1.5 -24.8 -14.6 - 14.7 -100.7 1.0 Bonds USA 14.8 126.0 -126.4 - 28.6 - 41.2 - 627.3 -35.9 UK - 9.7 - 56.8 189.7 7.9 -3.4 97.5 -8.5 Japan 0.6 39.1 -31.2 133.9 -3.6 -30.1 6.4 EMU 138.5 9.6 -1 6.4 -38.0 97.3 - 170.1 34.0 Canada 7.9 5.0 -11.8 -32.6 - 7.9 679.6 0.9 Switzer!' -77.7 5.9 91.9 42.4 62.9 143.8 110.7 The portfolio holdings are based on Eg. (9). All portfolio fractions are percentages. A riskless rate of interest of 2%) per annum is asslImed. Table 3 Weightings of the funds due to difTerent funding ratios Funding ratio 0.9 1. 1 1.2 1.3 1.5 Market pOlttolio (%) -1 1.5 0.0 6.3 12.3 18.2 24.6 Liability hedge portfolio (% ) 14.8 13.3 12.1 11.1 10.2 8.9 I-ledge portfolio GBp (%) - 0.6 - 0.5 - 0.5 - 0.5 - 0.4 - 0.4 Hedge portfolio Jpy (%) 1.2 1.1 1.0 0.9 0.9 0.8 Hedge pOItft)lio EUR (%) 0.3 0.2 0.1 0.0 0.0 -0.1 Hedge portfolio CAD (% ) - 0.1 -0.1 -0.1 - 0.1 - 0.1 - 0.1 Hedge portfolio CHF (% ) 1.1 1.0 0.9 0.8 0.8 0.7 Riskless assets (%) 94.9 85.1 80.2 75.3 70.4 65.6 Portfolio beta against GBP - 0.01 - 0.01 - 0.01 - 0.01 - 0.0 1 -0.01 Portfolio beta against JPY 0.02 0.02 0.02 0.02 0.02 0.02 Portfolio beta against EUR 0.01 0.00 0.00 0.00 0.00 0.00 Portfolio beta against CAD - 0.02 - 0.02 - 0.01 - 0.01 - 0.01 - 0.01 Portfolio beta against CI-IF 0.02 0.01 0.0 1 0.01 0.01 0.0 1 The weightings of the portfolios according to Eg. ( 16), where C( = 0, i.e. log utility, is assllmed. As derived in Section 3, the holdings of the six funds depend only on the funding ratio and on the currency betas of the distinct markets. Thts is shown in Table 3. For a funding ratio of one, there is no investment in the market portfolio and only diminishing investments in the currency hedge portfolios. The portfolio betas against the five currencies are close to zero for all funding ratios. The higher the funding ratio, the higher the investment in the market portfolio, the lower the investments in --- ## Page 793 764 M Rudolf and W T. Ziemba 986 M. Rudolf W. T Ziemba I Journal oj' Economic Dynamics & Control 28 (2004) 975-- 990 the liability and the state variable hedge portfolios, and the lower the investment in the riskless fund. Negative currency hedge portfolios imply an increase of the currency exposure instcad of a hedge against it. The increase of the market portfolio holdings and the reduction of the hedge portfolio holdings and the riskless fund for increasing funding ratios shows that the funding ratio is directly related to the ability to bear risk. Rather than risk aversion coefficients, the funding ratio provides an objective measure to quantify attitudes towards risk. 6. Conclusions This paper derives a four-fund theorem for intertemporal surplus optimizers such as life insurance and pension funds. In addition to the three funds identified by Merton ( 1973), the expected utility maximizing portfolio contains a liability hedge portfolio, which is preference independent. Its holdings depend only on the funding ratio of a pension fund. The higher the funding ratio, the lower the necessity for liabilities hedging. As a practical consequence, the hedging policy of pension or life insurance funds could very easily be monitored by authorities. This is due to the fact that in the optimum, only the funding ratio is decisive for the liabilities hedge, and not the utility function, which hardly can be determined by law. Today's pension fund laws do not contain any rules for the treatment of stochastic wage growths. Although Merton ( 1993) addresses a similar problem of the optimal investment strategy for University endowment funds, his setting is different. He describes the "liabilities" of universities, i.e. the costs for their activities, as state variables. He obtains a preference-dependent hedge portfolio for the activity costs. In contrast, here the liability returns are specified as a part of the surplus return. In contrast to Merton's results, the hedge portfolio for the liabilities is exclusively dependent on the funding ratio (and not on risk preferences) of a life insurance or pension fund. The funding ratio is an objective measure, whereas risk preferences are hard to determine. The model provides an intertemporal portfolio selection approach for surplus opti- mizers. The intertemporal surplus management approach holds for investors who have to cover liabilities by their assets in each moment of time. Since the investment strategy of all investors is consumption orientated, it is reasonable to assume that all investors invest in order to cover their liabilities. If the growth rate of individual consumption is related to the growth of wages and salaries, the relevant benchmark for surplus op- timizers refers to the growth rate and the volatility of wages and salaries. Finally, this model suggests a new type of product for investment banks. Pension and life insurance funds need to protect themselves against unanticipated changes in the growth rates of wages and salaries. Investment banks could offer liability protection and hedge those positions by purchasing a liability hedge portfolio as is given by Eq. (9). Acknowledgements Two anonymous referees of this journal have contributed valuable comments. With- out implicating them, the authors thank the participants in a workshop at the ETH in --- ## Page 794 lntertemporal Surplus Management 765 M. Rudolf w.T. Ziemba /Joul'nal o/ Economic Dynamics & Confrol28 (201M) 975- 990 987 Zurich, May 1996, the Annual Conference of the Deutsche Gesellschaft fUr Finanzwirtschaft (DGF) in Berlin, September 1996, the EURO INFORMS Meeting in Barcelona, July 1997, the Eighth International Conference on Stochastic Program- ming, Vancouver, August 1998, the INQUIRE, Venice Meeting, October 2000, Gunter Franke, Christian Hipp, Astrid Eisenberg, Karl Keiber, Alexander Kempf, Andre Kron- imus, Matthias Muck, Stanley Pliska, Bernd Schips, and Heinz Zimmennann for helpful comments on an earlier draft. All remaining errors are ours. Appendix A A.I. Derivation of (7) The arguments of the J -function are dropped for simplicity. o( dt) summarizes all terms of higher order than I of dt. For o(dt) : limd(--->o o(dt)/dt = O. Applying Itas lemma yields J(S, Y, t) = U(S, Y, t) dt + max E([J(S + dS, Y + dY, t + dt)] w = U dt + max E([J + J( dt + Js dS + J y dY + ! Jss dS2 + ! J yy dy2 w +Jsy dS dY + o(dt)]. Transforming the expression above yields 0= U dt + max E([J( dt + Js dS + Jy dY + ! Jss dS2 + ! Jyy dy2 w +Jsy dS dY + o(dt)], 0= U dt + max[J( dt + JsE( dS) + lyE(dY) + ! Jss dS dS w + ! lyy d Y d Y + JSY dS d Y], and sub&tituting in the expressions in (4) yields (7). A.2. Derivation of (8) Denote the prices of the n risky assets by Aj(t), i = 1, . .. , n, and suppose they follow the processes dAi(t) = Ai(t)[J.li dt + ai dZi(t)], i = 1, ... n, --- ## Page 795 766 M Rudolf and W T. Ziemba 988 M. Rudol{ W. T Ziemba !Journal 0/ Economic Dynamics & Conlrol28 (2004) 975 · 99IJ which defines the vector of risky expected asset retul1ls m~ := (iJ.i, ' .. , iJ.,,). The Wiener processes Zi(t), i = 1, ... 11, are correlated according to where Pi} is the instantaneous correlation between Zi(t) and Zj(t). If the covariance between assets i and j is denoted by (Jij, then we have (Jij = (Ji(JjPij and the covariance matrix V has entries (Ju, i.e. V= Suppose the fund holds Xi(t) shares of asset i at time t, and that the cash in the asset pOltfolio at time t is B(t). Then the asset portfolio value A(t) is given by (Ai(t): price of asset i) " A(t) = I>i(t)Ai(t) + B(t). io= 1 In the next infinitesimal time interval dt, this evolves according to n II dA(t) = LXi(f)dAi(f) + dB(t) = LXi(l)dAi(t) + rB(t)dt i= 1 i=1 where r is the riskless interest rate. Define the vector of pOltfolio fractions w' = (WI, ... ,Wn ) by .( ) ._ xi(t)Ai(f) wzt.- A(t)' i = l, ... ,I1. The sum of risky and riskless investments is assumed to be I. Hence, the fraction invested in the riskless asset is B(t)/A(t) = 1 - I:~ I Wi. Therefore, the expression above for dA(t) becomes dA(f) = A(t) [t[UJi(iJ.i-r) + r]df+ tWi(JidZi(t)]. Comparing the above expression with Eq. (2), i.e. dA(t) =A(t)[iJ.A dt + (JA dZAt»), and assuming that e' := (I, ... , I) is the n-dimensional vector of ones, gives iJ.A = w'(mA - re) + r, n (JA dZAt) = L Wi(Ji dZi(t). i=1 From this follows that (J~ = w'Vw, --- ## Page 796 Intertemporal Surplus Management 767 lv1. Rudol/ W. T Ziemba / Journal 0/ Economic Dynamics & Conlrol 28 (201N j 975····990 989 where V is the covariance matrix given earlier. Furthermore, define (JiL( (JiY) as the covariance between the ith risky asset and the liabilities (state variable). Denote the vector of these covariances by V~ L:= ((JIL, ... ,(Jnd, V~y:= «(Jly, ... ,(JnY). Then the relationship (JA L = W'VAL and ClAY = W'VAY is obvious. From the last four representations of expected returns, variances, and covariances Eg. (8) follows. A.3. Discussing Eq. (9) Maximizing the covariance between the asset portfolio and the state variable, = W'VAY--> max w 'y 2 s.t. w w = ClA , where R I, ... , Rn refer to the n asset returns. Let }, be a Lagrange multiplier: L = W'VAY - A(W'YW - Cl~) => oL/ow = VAY - 2A ' .. Yw = 0 => w = y- I VAr/(2A). o2L/(owi < 0 if and only if Y is positive definite. Hence, Wy (the state variable hedge portfolio) maximizes the correlation between the asset portfolio and the state variable. Furthermore, 2}.. w'Yw = W'VAY <=} 2ACl~ = (JAY ClAY _I 2). = -2 = [JAY => Y VAY = [JAYW. (JA Multiplying the fractions of the asset portfolio by the regression coefficient [JAY provides the fractions of the hedge portfolio. References Adler, M., Dumas, B., 1983. International portfolio choice and corporation finance: a synthesis. The Journal of Finance 38 (3), 925-984. Black, F., 1989. Should you use stocks to hedge your pension liability? Financial Analysts Journal 22, 10- 12. Carino, D.R., Ziemba, W.T., 1998. Fonnulation of the Russell Yasuda-Kasai financial planning model. Operations Research 46 (4), 433- 449. Constantinides, G.M., 1993. Comment. In:.Clotfelter, c.T., Rothschild, M. (Eds.), The Economics of Higher Education. University of Chicago Press, National Bureau of Economic Research, Chicago, pp. 236-242. Dixit, A.K., Pindyck, R.S., 1994. Investment under uncertainty. Princeton University Press, New Jersey. Ezra, D.O., 1991. Asset allocation by surplus optimization. Financial Analysts Journal 24, 51 - 57. --- ## Page 797 768 M Rudolf and W T. Ziemba 990 M. Rudoll w. T Ziemba /Journal 0/ Econol11ic Dyn({l11ics & Control 28 (2004) 975 990 Ingersoll Jr., J.E., 1987. Theory of Financial Decision Making. Rowman & Littlefield, New York. Merton, R.C., 1969. Lifetime portfolio selection under uncertainty: the continuous-time case. The Review of Economics and Statistics 51, 247-257. Merton, R.C., 1971. Optimal consumption and portfolio rules in a continuous-time model. Journal of Economic Theory 3, 373-413. Merton, R.C., 1973. An intertemporal capital asset pricing model. Econometrica 41, 867-887. Merton, R.C., 1990. Continuous Time Finance. Blackwell Publishers Inc., Cambridge, MA (reprinted 1995). Merton, R.C., 1993. Optimal investment strategies for university endowment funds. In: Clotfelter, C.T., Rothschild, M. (Eds.), The Economics of Higher Education. University of Chicago Press, National Bureau of Economic Research, Chicago (reprinted in: Ziemba,W.T., Mulvey, J.M. (Eds.), (1998), Worldwide Asset and Liability Modeling, Cambridge University Press, Cambridge, MA). Richard, S.F., 1979. A generalized capital asset pricing model. In: Gruber, M.J., Elton, E.J. (Eds.), Portfolio Theory, 25 Years After. North-Holland, New York, pp. 215-232. Samuelson, P.A., 1969. Lifetime portfolio selection by dynamic stochastic programming. The Review of Economics and Statistics 51, 239-246. Sharpe, W.F., 1964. Capital asset prices: a theory of market equilibriums under conditions of risk. The Journal of Finance 19 (3), 425-442. Sharpe, W.F., Tint, L.G., 1990. Liabilities-a new approach. The Journal of Portfolio Management 17, 5-10. Ziemba, W.T., Mulvey, J.M. (Eds.) 1998. World Wide Asset and Liability Modeling. Cambridge University Press, Cambridge, MA. --- ## Page 798 Journal a/Portfolio Management, 32(1), 108- 122 (2005) 769 52 The Symmetric Downside-Risk Sharpe Ratio And the evaluation if great investors and speculators. William T. Ziemba WrLLIAM T. ZIEMBA is the alumni professor of financial modeling and stochastic optimization emeritus at the Sauder School of Business of the Univenity of British Columbia in Vancouver, Canada, and a visiting pro- fessor of finance at the Sloan School of Manage- ment of the M.assachusetts Institute of Technology in Cambridge, MA. 108 THE SYMMETRIC DOWNSIDE-RISK SHARPe RATIO T he Sharpe ratio is a very useful measure of investment performance. Because it is based on mean-variance theory, and thus is basically valid only for quadratic preferences or normal distributions, skewed investment returns can lead to mis- leading conclusions. This is especially true for superior investors with many high returns. Superior investors may use capital growth wagering ideas to implement their strategies, which produces higher growth rates but also higher variability of wealth. My simple modification of the Sharpe ratio to assume that the upside deviation is identical to the down- side risk provides a useful modification and gives more realistic results. Exhibit 1 plots wealth levels using monthly data from December 1985 through March 2000 for the Wind- sor Fund of George Neff, the Ford Foundation, the Tiger Fund of Julian Robertson, the Quantum Fund of George Soros, and Berkshire Hathaway, the fund run by Warren Buffett, as well as the S&P 500 total return index, U.S. Treasuries, and T-bills. Yearly data are shown in Exhibit 2. The means, standard deviations, and Sharpe [1966, 1994] ratios of these six funds, based on monthly, quar- terly, and yearly net arithmetic and geometric total return data, are shown in Exhibit 3. Shown as well are data on the Harvard endowment (quarterly) plus U.S. Treasuries, T-bills, and U.S. inflation, and number of negative months and quarters. The first panel in Exhibit 3 shows the data behind FALL 2005 --- ## Page 799 770 PERFORMANCE EVALUATION -----AND RISK ANALySIS---- The Symmetric Downside-Risk Sharpe Ratio WILLIAM T. ZIEMBA 108 The Sharpe ratio, a most useful measure of investment per- formance, has the disadvantage that it is based on mean- variance theory and thus is valid basically only for quadratic preferences or normal distributions. Hence skewed invest- ment returns can engender rnisleading conclusions. This is especially true for superior investors with a number of high returns. Many of these superior investors use capital growth wagering ideas to implement their strategies, which means higher growth rates but also higher variability of wealth. A simple modification of the Sharpe ratio to assume that the upside deviation is identical to the down- side risk gives more realistic results. W T. Ziemba --- ## Page 800 The Symmetric Downside-Risk Sharpe Ratio 771 EXHIBIT 1 Growth of Assets-Monthly Data December 1985-April2000 Exhibit I, which illustrates the high mean returns ofBerk- shire Hathaway and the Tiger Fund as well as that the Ford Foundation's standard deviation was about a third ofBerk- shire's. The Ford Foundation actually trailed the S&P 500 mean return. We can see the much lower monthly and quarterly Sharpe ratios compared to the annualized values based on monthly, quarterly, and yearly data. By the Sharpe ratio, the Harvard endowment and the Ford Foundation had the best performance, followed by the Tiger Fund, the S&P 500 total return index, Berk- shire Hathaway, Quantum, Windsor, and U.S. Treasuries. The basic conclusions remain the same according to monthly or quarterly data and arithmetic or geometric means. Because of data smoothing, the Sharpe ratios with yearly data usually exceed those with quarterly data, which in turn exce.ed the monthly calculations. The reason for this ranking is that the Ford Foun- dation and the Harvard endowment, \\~th less growth, also had much less variability. These endowments have different purposes, different investors, different portfolio managers, and different fees, so such differences are not surprising. Clifford, Kroner, and Siegel [2001) give us similar calculations for a larger group of funds. They also show that over July 1977-March 2000, Berkshire Hathaway's FAll 2005 D ... Sharpe ratio was 0.850, Ford's 0.765, and the S&P 500's 0.676. The geometric mean returns were: 32.07% (Buf- fett), 14.88% (Ford), and 16.71% (S&P 500). Exhibit 4 shows Buffett's returns on a yearly basis in terms of increase in per share book value of Berkshire Hathaway versus the S&P 500 total return yearly index values for the 40 years 1965--2004. Buffett's geometric and arithmetic means were 22.02% and 22.84%, respectively, versus 10.47% and 11.83% for the S&P 500. This mea- sure does not fully reflect the trading prices of Berkshire Hathaway shares and thus yearly net returns, but it does indicate that Buffett has easily beaten the S&P 500 over these 40 years with a 286,841 % increase verSUS the S&P 500's 5,371 % increase. Typically the Sharpe ratio is computed using arith- metic returns. This is because the basic static theories of portfolio investment management such as mean-variance . analysis and the capital asset pricing model are based on arithmetic means. These are static, one-period theories. For asset returns over time, however, the geometric mean is a more accurate measure of average performance because the arithmetic mean is biased upward. The geometric mean helps mitigate the autocorre- lated and time-varying mean and other statistical proper- THE JOURNAl OF PORTFOLIO MANAGEMENT 109 --- ## Page 801 772 EXHIBIT 2 Yearly Return Data (%) Berkshire S&P SOD Date Windsor Hathaway Quantum Tiger Ford Harvard Total Yearly data, 14 years Neg 2 years Dec · 20.27 14.17 42.12 26.H3 18.09 22.16 18.47 86 Dec- 1.23 4.61 14.13 728 5.20 12.46 5.23 87 Dec- 28.69 59.32 10.13 15.76 10.42 12.68 16.81 88 Dec- 15.02 84.57 35.21 24.72 22.15 15.99 31.49 89 Dec· ·15.50 ·23.05 23.80 5.57 1.96 -1.0 J -3.17 90 Dec- 28.55 35.58 50.58 37.59 22.92 15.73 30.55 91 Dec - 16.50 29.83 6.37 8.42 5.26 4.88 7.67 92 Dec- 19.37 38.94 33.03 24.91 13.07 21.73 9.99 93 Dec- -0.15 24.96 3.94 1.71 -1.96 3.71 1.31 94 Dec- 30.15 57.35 38.98 34.34 26.47 24.99 37.43 95 Dcc- 26.36 6.23 -1.50 8.03 15.39 26.47 23.07 96 Dec- 21.98 34.90 17.D9 18.79 19.11 20.91 33.36 97 Dec- 0.81 52.17 12.46 11.21 21.39 12. 14 28.58 98 Dec- 11.57 -19.86 34.68 27.44 27.59 23.78 21.04 99 ties of returns that are not independent and identically dis- tributed. If one has returns of +50% and -50"10 in two peri- ods, for example, the arithmetic mean is zero, which does not reflect the fact that 100 became 150 and then 75. The geometric mean, which is -13.7%, is the correct measure to use. For investment returns in the 10%-15% range, the arithmetic returns are about 2 percentage points above the geometric returns. But for higher returns, this approxi- mation is not accurate. Hence, I use geometric means as well as more typical arithmetic means in this article. La [2002] points out that we must take care in mak- ing Sharpe ratio estimations when the investment returns are not iid, which they are for the investors I discuss here. For dependent but stationary returns La derives a cor- rection of the Sharpe ratios that deflates artificially high values to correct values using an estimation of the corre- lation of serial returns. Miller and Gehr [1978] and Knight and Satchell [2005] derive exact statistical properties of the Sharpe ratio with normal and lognormal assets, respectively. The Sharpe ratios are almost always lower when geometric 110 TI-IE SYMMETR!C DOWNSIDE-RISK SHARPE RATIO US Teea 15.14 2.90 6.10 13.29 9.73 15.46 7.19 11.24 -5.14 16.80 2.10 8.38 10.21 -1.77 W T. Ziemba means are used rather than arithmetic means; the difference between these two measures is a function of return volatil- US US ity. The basic conclusions in this research, T· Infl such as the relative ranking of the vari- bills ous funds, are the same for the arith- metic and geometric means. Exhibit 5 shows that the Harvard 6.16 1.13 Investment Company, that great school's 5.47 4.41 endowment, had essentially the same 6.35 4.42 wealth record over time as the Ford Foundation. This conclusion is based on 8.37 4.65 quarterly data, which are all I have on 7.81 6.11 Harvard. Harvard beats Ford by the ordi- 5.60 3.06 nary Sharpe ratio, but Ford is better by 3.51 2.90 the symmetric downside risk measure I develop later. 2.90 2.75 Before evaluating positive and neg- 3.90 2.67 ative returns performances of these var- 5.60 2.54 ious funds using the Sharpe ratio and my modified ver5ion, it is useful to discuss 5.21 3.32 how these funds got their outstanding but 5.26 1.70 50metimes volatile records. 4.86 1.61 4.68 2.68 SOME GREAT INVESTORS Ideally, we would want to penalize only losses such as those shown in Exhibit 6, while rewarding positive returns. The Sharpe ratio penalizes high-return but volatile records. In the theory of optimal investment over time, it is not quadratic (the utility function behind the Sharpe ratio) but log that yields the most long-term growth. But the elegant results in the Kelly [1956] criterion, as it is known in the gambling literature, and the capital growth theory, as it is known in the investments literature, as proven rigorously by Breiman (1961] and generalized by A1goet and Cover [1988], are long-run asymptotic results (see Hakansson and Ziemba [1 995], Ziemha [2003], and MacLean and Ziemba [2005]). The Arrow-Pratt absolute risk aversion of the log utility criterion: -u"(w) u'(w) lIw is essentially zero, where u is the utility function of wealth w, and the primes denote differentiation. Hence, in the FAlL 2005 --- ## Page 802 The Symmetric Downside-Risk Sharpe Ratio EXHIBIT 3 Fund Return Data-December 1985--April2000 Berkshire Ford Windsor Hathaway Quantum Tiger Found Harvard FAl l 2005 Momhly dala, 172 months Neg 61 58 months arith 1.1 7 2.15 mean, mon sldev, 4.70 mon Sharpe, 0.157 arilh 14.10 mean sl dey 16.27 Sharpe. 0 .543 Y' gcornean, 1,06 mon gea 51 4 .70 dev,olDn Sharpe, 0.133 geo 12.76 mean. yr geo SI 16.27 de\', yr Sharpe, 0.460 Y' 7.66 0.223 25.77 26.54 0.773 1.87 7.67 0 .186 22.38 26.56 0.644 QUllrtcrly dahl, 57 qUtlrtcrs Neg 14 quarters mean, 3.55 q[!y st dey, 8.01 qtly mean, yr 14.20 SI dey. yr 16.03 Sharpe. 0.556 Y' geomean. 3.23 qtly geo 51 8.02 dev.qtly gea 12.90 mean, y r gea 51 16.04 de v, yr Sharpe, 0.475 Y' 15 6.70 14.75 26.81 29.50 0.729 5.67 14.79 22.67 29.58 0.588 Yearly Data, 14 years Neg years mean. SI d ey, Sharpe. yrly gcom mean 51 dev Sharpe 14.63 13.55 0.681 13.H3 \3.58 0.621 28.55 30.34 0.763 24.99 30.57 0.641 53 1.77 7.42 0.180 2 1.25 25 .70 0.622 1.48 7.42 0.140 17.76 25.72 0.486 16 5.70 12.67 22.79 25.33 0.691 4 .94 12.69 19.78 25.38 0.571 22.93 16.17 1.084 21.94 16.20 1.022 56 44 2.02 1.19 6.24 2,68 0.54 0.80 24.27 14.29 21 .62 9.30 0.879 0.970 1.83 1.16 6.25 2.69 0.222 0.267 21.92 13.86 2].63 9.30 0.770 0 .924 II 4.35 7.70 17.42 15.40 0.788 II 3.68 4.72 14.71 9,43 0,999 4.07 3.57 7.70 4.72 16.28 14.29 15.41 9.43 0.713 0.954 18.04 11.40 1. 109 17.54 11,41 1.064 14.79 9.38 1.001 14.43 9.39 0.962 II 3.t~6 4.72 15.44 9,45 1.074 3.75 4.73 15.01 9.45 1,029 15,47 8.52 1.181 15.17 8.'3 1.146 S&P500 Total 56 1.45 4,41 0.230 17.44 15.28 0.797 1.35 4.41 0.208 16.25 15.28 0.719 10 4.48 7.52 17.91 15.05 0.839 4.20 7.53 16.80 15.06 0.764 18.70 12.88 1.033 18.04 12.90 0.981 us nc. 54 0.63 1.32 US T- bills 0.44 0.12 0 .14 50.000 us Infl 13 0.26 0.21 0 .827 7.57 5.27 3.14 4.58 0.43 0 .74 0.50 40.000 2.865 0.62 0 .44 0 .26 1.32 0.12 0.21 0.13 90.000 0.828 7.47 5.27 3.14 4.58 0.43 0 .74 0.48 20,000 15 ).93 1.32 2.67 0.36 7.73 5.29 5.34 0.73 0.45 60,000 2.868 0 .79 0.49 3 .16 0 .97 2.188 1.90 1.32 0.79 2.67 0.36 0.49 7.59 5.29 3.16 5.34 0.73 0.97 0.43 10.000 7.97 6.59 0 .39 7.78 6.59 0 .36 5.40 1.50 00.000 5.39 1.50 20.000 2,190 3.14 1.35 1.67) 3.13 1.35 1.672 773 THE JOURNAL Or: PO R. TrOLlO M ANAGEMEN T 111 --- ## Page 803 774 EXHIBIT 4 Increase in Per Share Book Value of Berkshire Hathaway versus S&P 500 (dividends included) 1965-2004 ("!o) Year BH S&P 500 Diff Year BH S&P 500 Diff 1965 23.8 10.0 13.8 1985 48.2 31.6 16.6 1966 20.3 (11.7) 32.0 1986 26.1 18.6 7.5 1967 11.0 30.9 (19.9) 1987 19.5 5.1 14.4 1968 19.0 11.0 8.0 1988 20.1 16.6 3.5 1969 16.2 (8.4) 24.6 1989 44.4 31.7 12.7 1970 12.0 3.9 8.1 1990 7.4 (3. I) 10.5 1971 16.4 14.6 1.8 1991 39.6 30.5 9.1 1972 21.7 18.9 2.8 1992 20.3 7.6 12.7 1973 4.7 (14.8) 19.5 1993 14.3 10.1 4.2 1974 5.5 (26.4) 31.9 1994 13.9 1.3 12.6 1975 21.9 37.2 (15.3) 1995 43.1 37.6 5.5 1976 59.3 23.6 35.7 1996 31.8 23.0 8.8 1977 31.9 (7.4) 39.3 1997 34. 1 33.4 .7 1978 24.0 6.4 17.6 1998 48.3 28.6 19.7 1979 35.7 18.2 17.5 1999 .5 21.0 (20.5) 1980 19.3 32.3 (13.0) 2000 6.5 (9.1) 15.6 1981 31.4 (5.0) 36.4 2001 (6.2) (11.9) 5.7 1982 40.0 21.4 18.6 2002 10.0 (22.1) 32.1 1983 32.3 22.4 9.9 2003 21.0 28.7 (7.7) 1984 13.6 6.1 7.5 2004 10.5 10.9 (0.04) Overall Gain 286,841 5,371 Arithmetic Mean 22.84 11.83 11.01 Geometric Mean 22.02 10.47 10.07 Sources: Berkshire Hathaway 2004 annual report, Hagstrom {2004}. EXHIBIT 5 Ford Foundation and Harvard Investment Corporation Returns-Quarterly Data June 1977-March 2000 :)0. -tWvlW Foro 20 · 10 i .-.~ ~ o )4':(1 ,,1"~' '#',;. ~ ~ )~ ,,"',,~ ,,<1> "",1 ~ short run, log can be an exceedingly risky utility func- tion with wide swings in wealth values because the opti- mal bets can ·be so large. Long-run exponential growth is equivalent to maxi- mizing the one-period expected log of that period's returns. To illustrate how large Kelly (expected log) bets are, con- sider the simplest case with Bernoulli trials where you win 112 THE SYMMETRIC DOWNSIDE-RISK SHARPE RATIO W T. Ziemba with probability p and lose with probability q = 1 - p. Log utility is related to negative power utility, namdy, for awa for a < 0 since negative power converges to log when a -? O. Kelly [1956] discovered that log utility investors had the best utility function, provided they were very long-run investors. The asymptotic rate of asset growth is G = lim IOg( WN)* N ...... "" Wo where wN is period Ns wealth and Wo is initial wealth. Consider Bernoulli trials that win + 1 with proba- bility p and lose -I with probability I - p. If we win M out of N of these independent trials, the wealth after period Nis: where f is the fraction of our wealth bet in each period. Then: G(f) = lim [M log(1 + f) + N - M log(1 - f)] N~_ N N which by the strong law oflarge numbers is G(f) = plog(1 + f) + qlog(1 - f) = E(logw) Hence, the criterion of maximizing the long-run exponential rate of asset growth is equivalent to maxi- mizing the one-period expected logarithm of wealth. Hence, to maximize long-run (asymptotic) wealth, max- imizing expected log is the way to do it period by period. The optimal fractional bet, obtained by setting the derivative of G(f) to zero, is l' = p - q, which is simply the investor's edge or expected gain on the bet. If the bets are win 0 + 1 Or lose I-that is, the odds are 0 to 1 to win-the optimal Kelly bet is l' = p~ q or the ~~:. Since edge is a mean concept and odds is a risk concept, you wager more with higher mean and less with higher risk.' In continuous time: F = j.l-r = edge a' risk(odds) with optimal growth rate FALL 2005 --- ## Page 804 The Symmetric Downside-Risk Sharpe Ratio 775 EXHIBIT 6 small-cap ,tocks under Democrats Summary-Negative Observations and Arithmetic and Geometric Means had 24.5 times as much wealth as a 60-40 large-cap/bond investor. That's the idea, more or less. Berkshire Ford Number of Windsor Hathaway Quantum Tiger Found Harvard negative ... months out 6J 58 53 56 44 na of 172 ... quarters out 14 15 J6 11 10 11 of 57 ... years oul of 14 G' '21 (I'(>-r)' 1 + r = -(Sharpe Ratio)' + Risk-Free Asset 2 where /1 is the mean portfolio return, r is the risk-free return, and (5' is the porrfolio return variance. So the ordi- nary Sharpe ratio determines the optimal growth rate. Kelly bets can be large. Consider Bernoulli trials where you win 1 or lose 1 with probabilities p and 1 - p, respectively. Then: P .5 .51 .6.8 .9 .99 1 - P .5 .49 .4 .2 .1 .01 l' 0 .02 .2 .6 .8 .98 So, if the edge is 98%, the optimal bet is 98% of one's fortune. With longer-odds bets, the wagers are lower. The Kelly bettor is sure to win in the end if the hori- zon is long enough. Breiman [1960, 1961) was the first to clean up the math in Kelly's [1956J and Latane's [1959J heuristic analyses. He proves that: lim wK.(N) -700 N~ ~ w.(N) where "'KB (N) and "'B (N) are the wealth levels of the Kelly bettor and another essentially different bettor after N play. That is, the Kelly bettor wins infinitely more than bettor B and moves farther and farther ahead as the long time horizon becomes more distant.' Breiman also shows that the expected time to reach a preassigned goal is asymptotically the shortest with an expected log strategy. Moreover, the ratio of the expected log bettor's fortune to that of any other essentially differ- ent investor goes to infinity. The log investor gets all the money in the end if one plays forever. Hensel and Ziemba [2000J calculate that over 1942 through 1997 a 100% long investor solely in large-cap stocks under Republican administrations and solely in FhLL 2005 S&P US Total Treas 56 10 54 15 Keynes and Buffett are essen- tially Kelly bettors. Kelly bettors will have bumpy investment paths, but most of the time, in the end, accu- mulate more money than other investors. Ziemba and Hausch (1986) perform a simulation to show medium-run ptoperties of log utility and half-Kelly betting (using the _",- 1 utility function). Starting with an initial wealth of $1 ,000 and considering 700 independent wagers with probability of winning 0.19 to 0.57, all with expected values of$1. 14 per dollar wagered, they compute the final wealth pro- files over 1,000 simulations. These wagers correspond to odds of 1-1, 2-1, 3-1, 4-1, and 5-1. The results show that the log bettor has more than 100 times initial wealth 16.6% of the time and more than 50 times as much 30.2% of the time. This demonstrates the great power oflog betting, as the half-Kelly strategy has very few such high outcomes. Yet this high return comes at a price. Despite making 700 independent bets, each with a 14% success rate, the investor could have lost more than 98% of the initial wealth of$l,OOO. Exhibit 7 provides the simulation results. The Ziemba-Hausch simulation used the data in Exhibit 8. The edge over odds gives l' equal to between 0.14 and 0.028 for the optimal Kelly wagers for 1-1 versus 5-1 odds bets. The 18 in the first column in Exhibit 7 shows it is possible for a Kelly bettor to make 700 independent wagers, all with a 14% edge, having 19% to 57% chance of winning each wager, and still lose over 98% of one's wealth. Even with half-Kelly, the minimum starting with $1,000 was $145, or a 85.5% loss. This shows the effect of a sequence of very bad scenarios that may be unlikely but are certainly possible. The last column in Exhibit 7 shows that 16.6% of the time the Kelly bettor increases initial wealth more than 100-fold. The half-Kelly strategy is much safer, as the chance of being ahead after the 700 wagers is 95.4% ver- sus 87.0% for full-Kelly. But the growth rate is much lower, since the 16.6% chance of making 100 times initial wealth is only 0.1 % for half-Kelly wagerers. The Kelly bettor accumulates more wealth but with a much riskier time path of wealth accumulation. The Kelly bettor can take a long time to get ahead of another bettor. THE JOUR.NAL O f PORTfOLIO MAN AGEMENT 113 --- ## Page 805 776 W. T. Ziemba EXHIBIT 7 Distributions of Final Wealth-Kelly and Half-Kelly Wagers Final Wealth Strategy Number of Times Final Wealth out of 1000 Trials Was Kelly Half-Kelly EXHIBIT 8 Min Ma;( 18 483,883 145 111,770 Value of Odds on Wagers Probability of Odds Probability of Winning Being Chosen in the Simulation at Each Decision Point 0.57 I-I 0.1 0.38 2-1 0.3 0.285 3-1 0.3 0.228 4-1 0.2 0.19 5-1 0.1 EXHIBIT 9 Mean 48,135 13,069 Median 17,269 8,043 Optimal Kelly Bets Fraction of Current Wealth 0.14 om 0.047 0.035 0.028 Place-and-Show Betting on Kentucky Derby 1934-1994 .. ....... -_ .... ............. - Source: Bain, Hausch, and Ziemba 12005]. 16.861 6,945 What one wants is to have those swings in the right direction. And who better, it seems, at predicting these swings than Warren Buffett. Chopra and Ziemba (1993) show that, in portfolio problems, errors in estimating 111eans, variances, and covariances influence investment performance roughly in the ratio 20:2: 1. When risk aver- sion is lower,. as it is with log, the errors are even greater. In this case, the effect on portfolio performance of mean errors can be 100 times the errors in covariances. Who then would use a log utiliry function if it is so risky, or use a toned-down log utiliry function by mixing 114 THE SYMMETRIC DOWNSIDE-RISK SHARPE RATIO >500 >1000 >10,000 >50,000 >100,000 916 870 598 302 166 990 954 480 30 I the log utiliry investment fraction of one's wealth with cash? These so-called fractional Kelly strategies are actually mathematically equivalent to negative power utiliry when the assets are lognormally distributed and approximately equivalent otherwise; see Maclean, Ziemba, and li [2005). The difference in wealth paths between Kelly and half-Kelly strategies is illustrated in Exhibit 9, which shows the wealth level histories from place-and-show betting on the Kentucky Derby over 1934-1994 following the DrZ system. This system uses a 4.00 dosage index breeding ftl- ter rule with fitll- and half-Kelly wagering and $200 Oat bets on the favorite, with initial wealth of$2,500 (dosage is a mea- sure of a horse's speed OVer starnina ratio) . Starting with $2,500, full-Kelly yields a final wealth of$16,861, while half-Kelly, with a much smoother path, has final wealth of $6,945. These two strategies for the Kentucky Derby are winning ones compared to the losing strategy of betting on the favorite, which turns the $2,500 into $480. Some rather good investors, including four I have worked with or consulted with, have used the Kelly and fractional Kelly approach to turn humble starts into for- tunes of hundreds of millions. One was the world's most successful racetrack bettor; see Benter (1994). Another was a trend-following futures trader in the Caribbean for whom I designed a Kelly betting system for the 90 liquid futures markets he traded. The Kelly system added $9 million extra profIts per year to his already good betting system based on an ad hoc but sound probabiliry- of-success approach. Another was an options trader eking out nickels and dimes with slightly mispriced options in Chicago. The fourth was the popularizer of the Kelly approach in sports betting, Edward 0. Thorp. Thorp's Princeton- Newport Fund had a net mean return of 15.1%, when the S&P 500's was 10.2% and T-bills returned 8.1%. Interest rates were very high in 1968-1988 while Thorp was running his fund. Thorp had no losing quarters, ortly three losing months, and a most impressive yearly stan- dard deviation of 4%. To show the differences between the gamblers and other investors, Exhibit 10 shows Benter's racetrack bet- f. ... LL 2005 --- ## Page 806 The Symmetric Downside-Risk Sharpe Ratio EXHIBIT 10 Gamblers Like Smooth Wealth Paths R(SULTS (a) Hong Kong racing syndicate 19R9 to 1994. See Benter [1994J. EXHIBIT 11 Princeton-Newport Partners-Cumulative Results November 1968-December 1988 Price (U.s. doll.,,) :;-----------------" ;1 1 6 5 i 3 2 1 .' .. "" ... ''''::~~:/'~. , .:-::-:c . .,.~.-~ . O~ __ ~ ____ ~ ____ ~ ____ ~ __ ~ 11/68 IO/7~ 10/77 10/81 10/85 12/88 --PNP .......... 5&P500 --- U.S. r·BiIl ting record over 1989-1994 and 1989-2001, and Exhibit 11 shows Thorp's record over 1968-1988. Compare these rather smooth graphs with the brilliant but quite volatile record in Exhibit 12 of the eminent economist, John May- nard Keynes, who ran the King's College Chest Fund, the college's endowment, from 1927 until his death in 1945. Keyneslost more than 50% of his fortune during the difficult years of the depression around the 1929 crash. This bad start was followed by many years of outperformance on relative and absolute terms, so that by 1945 Keynes's geometric mean return was 9.12% versus the U.K. index of -0.89%. Keynes's Sharpe index was 0.385. The gamblers have several common characteristics: FALL 2005 777 f: t ! .~ ____ ~ __ ~ ____ ~ __ -, ____ ~ __ ~-J - - - -... (b) Hong Kong racing syndicate 1989 to 2001 See Benter [2001J. - - • They carefully developed anomaly systems with positive means. • They carefully developed computerized betting systems that automated the betting process. • They constantly updated their research. • They were more focused on not losing than on winning in their careful risk control. Thorp [2006] shows that the Buffett trades are actu- ally very similar to those of a Kelly trader. In Ziemba [2003J I find Keynes is well approximated as a fractional Kelly bettor with 80% Kelly and 20% cash; this is equiv- alent to the negative power utility function _W",·2S Recall log is when the power coefficient goes to zero, and that log is the most risky utility function that should ever be considered. Positive power utility has less growth and more risk, so is not an acceptable utility function. SYMMETRIC DOWNSIDE SHARPE RATIO PERFORMANCE MEASURE Now back to the records of the funds in Exhibit 1. How do we compare these various investors? And can we determine if Warren Buffett really is a better investor than the rather good funds, especially the Ford Founda- tion and the Harvard endowment? Exhibit 13 plots the Berkshire H athaway and Ford Foundation monthly returns as a histogram and shows the losing months and the winning months in a smooth curve. We want to penalize Buffett for losing but not for winning. We define the downside risk as: T HE JOURNAL OF PO ll-TFO LIO M ANACEMENT 115 --- ## Page 807 778 W. T. Ziemba EXHIBIT 12 King's College Chest Fund 1927-1945 ..... ," .. .n. • MII . .... EXHIBIT 13 Berkshire Hathaway versus Ford Foundation-Monthly Returns Distribution January 1977-April2000 <40,00 ){),OO 20.00 10.00 " ". "'" " r" __ '_" • - --1 SO 51 64 7\ 7. as 92 99 106 11) 120 12113-11<411<48 US 162 16~ ·){).oo t 16 THE SVMMEiRIC DOWNSIDE-R.ISK SHltRPE RATIO i-Berkshire Hathaway - Ford Foundation F"LL 2005 --- ## Page 808 The Symmetric Downside-Risk Sharpe Ratio EXHIBIT 14 Comparison of Ordinary and Symmetric Downside Sharpe Yearly Performance Measures Ordinary Downside Ford 0.970 0.920 Foundation Tiger Fund 0.879 0.865 S&P500 0.797 0.696 Berkshire 0.773 0.917 Hathaway Quantum 0.622 0.458 Windsor 0.543 0.495 I :-I ex, - xl: n - 1 where the benchmark x is zero, i is the index on the n months in the sample, and the xi taken are those below X, namely, those In of the n months with losses. This is the downside variance measured from zero, not the mean, so it is more precisely the downside risk. To get the total vari- ance, we use rwice the downside variance (2a ;J, so that Buffett gets the symmetric gains added, not his actual gains. Using 2a;, the usual Sharpe ratio with monthly data and arithmetic returns is: EXHIBIT 15 779 Berkshire Hathaway versus the much less volatile Ford Foundation returns. When Berkshire Hathaway had a los- ing month, it averaged -5.36% versus + 2.15% for all months. Meanwhile, Ford lost 2.44% and won, on average, 1.19%. Exhibit 15 shows the histogram of quarterly returns for all funds including Harvard (for which monthly data are not available). The plots show that the distributions of all the funds lie between those of Berkshire Hathaway, Harvard, and Ford. By the quarterly data, the Harvard endowment has a record almost as good as the Ford Foundation's; see Exhibits 16 and 17. Berkshire Hathaway made the most money but also took more risk, and by either the Sharpe or the downside Sharpe measure the Ford Foundation and the Harvard endowment had superior rewards. I first used this measure in Ziemba and Schwartz [1991] to compare the results of superior investment in Japanese small- cap stocks during the late 1980s. The choice of x ~ 0 is convenient and has a good interpre- tation. But other x are possible and might be useful in other applications. This measure is closely related to the Sortino ratio (see Sortino and van der Meer [1991] and Sortino Quarterly Returns Distributions-December 1985-March 2000 0.500 0.400 Exhibit 14 provides the results for the ordinary and symmetric 0.300 downside Sharpe ratios using monthly data and arithmetic means. My measure moves Warren Buffett higher, to 0.917, but not up to the Ford Foundation and not higher because of his high monthly losses. He does gain in the switch from ordinary Sharpe to downside symmetric Sharpe, while all the other 0.200 0.100 J - TIger l sor arvard sap Total funds drop. Ford is 0.920, and Tiger (0.100) f-f-lf-jrL-,'-----------------------I 0.865. The Berkshire Hathaway monthly losses when annualized are over 64% versus under 27% for the (0.200) H4-,.L-- ------------------ ------I Ford Foundation. Exhibit 14 shows these rather fat tails on the up side and down side of FAl.L 2005 10 20 30 40 so 60 THE JOURNAL OF POR TFOLJO M ANAGEMENT 117 --- ## Page 809 780 W. T. Ziemba EXHIBIT 16 similar rankings. Yearly Sharpe and Symmetric Downside Sharpe Ratios December 1985- In Exhibit 2, I showed the yearly returns for the various funds and their Sharpe ratios computed using arith- metic and geometric means and the yearly data. There are insufficient data to compute the downside Sharpe ratios based on yearly data. The Ford Foundation had only one losing year, and that loss was only 1.96%. Berkshire Hathaway had two losing years with losses of23.1% and 19.9%. The Ford Foundation had a higher Sharpe ratio than Berkshire Hathaway, but that was April 2000 Windsor qtly data, 57 quarters neg qts 14 mean, neg -6.69 mean, qtly 3.55 ds st dey, qt1y 10.52 mean, yr 14.20 ds sl dey. yr 21.04 ds Sharpe 0.424 geomean, qtly 3.23 ds gea sl dev,qtly 10.20 geo mean, yr 12.90 ds geo st dev. yr 20.41 ds Sharpe 0.373 Berkshire Hathaway 15 -10.50 6.70 14.00 26.81 28.00 0.769 5.67 13.49 22.67 26.97 0.644 Quantum Tiger 16 11 -7.77 -6.26 5.70 4.35 12.09 8.18 22.79 17.42 24.17 16.35 0.724 0.742 4 .94 4.07 11.80 7.71 19.78 23.60 0.614 16.28 15.43 0.712 Ford Found 10 -3.59 3.68 4.44 14.71 8.89 1.060 3.57 3.91 14.29 7.83 1.150 and Price [1994]), which considers downside risk only. That measure does not have the two-sided interpretation of my measure, and the ,J2 does not appear. The notion of focusing on downside risk is popular these days as it represents real risk better. I started using it in asset-liability models in the 1970s; EXHIBIT Harvard 11 -2.81 3.86 5.12 15.44 10.24 0.991 3.75 4.99 15.0 1 9.97 0.975 S&P Total Trea 10 15 -6.92 -1.35 4.48 1.93 9.89 1.35 17.91 7.73 19.78 2.70 0.638 0.903 4.20 1.90 9.34 1.28 16.80 7.59 18.68 2.56 0.616 0.900 exceeded by the Tiger and Quantum funds and the S&P 500. Exhibits 17 and 18 summarize the annualized results using monthly, quarterly, and yearly data. The Ford Foundation had the highest Sharpe ratio, followed by the Harvard endowment, and both of these exceeded Berkshire Hathaway. The Ford Foundation had the highest synunetric-downside Sharpe ratio, followed by Harvard, and both exceeded Berkshire Hathaway and the other funds. 17 Summary of Means (%) and Sharpe and Symmetric Downside Sharpe Ratios Annualized Berkshire Ford S&P Windsor Hathaway Quantum Tiger Found Harvard Total Trea mean (arith. rnon) 14.10 24.27 14.29 na 17.44 7.57 Shnrpe (arilh, mon) 0.543 0.879 0.970 n. 0.797 0.504 see Kallberg, White, and Ziemba [1982) and Kusy and Ziemba (1986) for early applications. In those mod- els we measure risk as the downside non-attainment of investment target goals that can be deterministic such as wealth growth over time, or stochas- tic such as the non-attainment of a portfolio of weighted benchmark returns. See Geyer et al. [2003) for an application of this to the Siemens Aus- tria pension fund. ' ds Sharpe (arith,mon) 0.495 25.77 0.773 0.917 21.25 0.622 0.458 0.865 0.920 na 0.696 0.631 Calculating the Sharpe ratio and the downside-symmetric Sharpe ratio using quarterly or yearly data does not change the results much, even though it smooths the data because individual monthly losses are combined with gains for lower volatility. The yearly data move us closer to normally distributed returns so the symmetric downside and ordinary Sharpe measures will yield mean (geom, man) Sharpe (geom. mon) ds Sharpe (geom, mon) mean (ar:ith, qtly) Sharpe (arilh, qtly) ds Sharpe (arith,qt1y) mean (geom, qtly) Sharpe (geom, qlty) ds Sharpe (gcorn, qt1y) mean (arith, yrly) Sharpe (arith, yrly) mean (geom, yrly) Sharpe (geom, yrly) 118 THE SYMMETRIC DOWNSIDE- RISK SHARPE RATIO 12.76 0.460 0.420 14.20 0.556 0.424 12.90 0.475 0.373 14.63 0.681 13.83 0.621 22.38 0.644 0.765 26.8 1 0.729 0.769 22.67 0.588 0.644 28.55 0.763 24.99 0.641 17.76 0.486 0.358 22.79 0.691 0.724 19.78 0.571 0.614 22.93 1.084 21.94 1.022 21.92 13.86 n. 16.25 7.47 0.770 0.924 na 0.719 0.482 0.758 0.876 0.628 1.053 17.42 14.71 15.44 17.91 7.73 0.788 0.999 1.074 0.839 0.456 0.742 1.060 0.991 0.638 0.903 16.28 14.29 15.0t 16.80 7.59 0.713 0.954 1.029 0.764 0.431 0.712 1.150 0.975 0.616 0.900 18.04 14.79 15.47 18.70 7.97 1.109 1.001 1.181 1.033 0.390 17.54 14.43 15.17 18.04 7.78 1.064 0.962 1.146 0.981 0.362 FIILl2005 --- ## Page 810 The Symmetric Downside-Risk Sharpe Ratio 781 EXHIBIT 18 Summary of Means and Sharpe and Downside Sharpe-Monthly, Quarterly, and Yearly Data Annualized December 1985-March 2000 lAO L20 LOO D Windsor - [] BH 0 .80 D Quantum DTiger .. Ford Found f---- . _ . - - o Harvard 0.60 Cl S&P Total DAD r--- I ~ ~~ ~ I~ 0 .20 0.00 2- 2- 2- 2- 2- 2- ,,-\, ,,-\'I ,,-\, ,,-\'I ~ <::t" <:<10 co" JJ-Q'lI ~~e.- ~'" .I- co' ~e; co' .I- $'~ ~ -z} 0"" 0"" ~'" 0"" 0"" 0"" 0"" 0"" 0"" /)" /)" EXHIBIT 19 Asset Allocation of Harvard Endowment-June 2004 Harvard's Holdings Annual Returns Weight In By Sector I-Year, % to-Year, % Endowment, % Domestic Equities 22.8 17.8 15 Foreign Equities 36.1 8.5 !O Emerging Market.s 6.6 9.7 5 Private Equity 20.8 31.5 13 Hedge Funds 15.7 n.a 12 High Yield 12.4 9.7 5 Commodities 19.7 !O.9 13 Real Estate 16.0 15.0 IO Domestic Bonds 9.2 14.9 I I Foreign Bonds 17.4 16.9 5 Inflation-Indexed Bonds 4.2 n.a. 6 Total Endowment 21.1 15.9 105%* *includes slight leverage Source: Bllrrotls, February 1, 2005. As Lawrence Siegel of the Ford Foundation privately acknowledges, some of the Ford Foundation's high Sharpe ratio results from dividing by an artificially smoothed FALL. 2005 /)" /)" standard deviation, due to that institution's high private equity allocation whose market prices do not reflect actual volatility. This is also true of Harvard. Exhibit 19 shows the Harvard endowment's asset allocation. Its ten-year 15.9% performance on a $22.6 billion portfolio was 3. 1 percentage points better than the median of the 25 largest university endowments through June 2004. Harvard has continued its good performance, with extensive use of private equity and other non-tradi- tional investments. Ford's performance in 2001 -2002 was poor, and Berkshire Hathaway has doubled in price since the 2000 lows, so the rankings based on more data may well have changed. Exhibits 6, 16-17, and 20 give a short overview of key data. The numbers in italics indicate the worst out- comes and those in bold the best. Windsor had the most negative months and negative years (tied with Berkshire Hathaway) and the lowest means. Berkshire Hathaway had the highest annual mean returns. THE JOU!"-NAL OF POR. HOllO MA NAGEMENT 119 --- ## Page 811 782 W T. Ziemba EXHIBIT 20 Yearly Sharpe and Symmetric Downside Sharpe Ratios December 1985-April 2000 (%) Berkshire Ford S&P US T- US Windsor Hathaway Quantum Tiger Neg months 61 58 53 56 aTith mean, -3,54 -5.36 -5,65 -4.41 neg arith mean, 1.17 2,15 1.77 2,02 mon ds SI dey, 5,15 6.46 10,08 6.34 mon arith mean, 14,10 25,77 21.25 24.27 yr ds 51 dey, yr 17.85 22.36 34,91 21.97 ds Sharpe 0.495 0,917 0.458 0,865 geomean, -3.62 -5.48 ·5,95 -4.53 neg gt!omean, 1.06 1.87 1.48 1.83 mon ds geo 51 5,15 6.46 10,09 6,34 dev,mon geo mean, yr 12.76 22.38 17,76 21.92 ds geo 51 17,86 22.37 34.94 21.98 dev, yr ds Sharpe 0.420 0,765 0,358 0,758 CONCLUSIONS AND CAVEATS My downside-risk Sharpe ratio measure is ad hoc, as all performance measures are, and adds to but does not close the debate on this subject. Hodges [1998J proposes a generalized Sharpe measure that eliminates some of the paradoxes the Sharpe measure leads to, It uses a constant, absolute risk-return exponential utility function and general return distributions. When the rerurns are normally distributed, this is the usual Sharpe ratio, as that portfolio problem is equivalent to a mean-variance model. A better utility function is the constant relative risk aversion negative power. Leland [1999] shows how to modifY /3s when there are fat tails into more correct {3s in a CAPM fran1ework. Goetzmann et al. [2002] and Spurgin [2000] show how the Sharpe ratio may be manipulated using option strategies to obtain what looks like a superior record to obtain more funds to manage. Managers sell calls to cut off upside variance, and use the proceeds to buy puts to cut off downside variance, leading to higher Sharpe ratios because of the reduced portfolio variance. These options transactions may actually lead to poorer investment per- 120 THE SYMMETRIC D O WNSIDE- RISK SHARPE RATIO Found Harvard Total Trea bills 1nfl 44 n. 56 54 13 -2.24 n. -3.31 -0,87 n. -0,14 1.19 n' 1.45 0,63 0.44 0,26 2,83 n' 5,05 1.06 0.00 0.19 14,29 na 17.44 7,57 5,27 3,14 9,81 n. 17.49 3,66 0.00 0.64 0,920 na 0.696 0,631 na -3.320 -2.26 na -3,38 1.16 n. 1.35 0.62 0.44 0.26 2,83 n' 5,05 0,60 0.00 0.00 13,86 n. 16,25 7.47 5,27 3,14 9,81 na 17,49 2,09 0,00 0,00 0,876 n. 0,628 1.053 formance in flnal wealth terms even with their higher Sharpe ratios. Tompkins, Ziemba, and Hodges [2003], for example, show how on average the calls sold and the puts purchased on the S&P 500 both had negative expected val- ues over 1985-2002. I have not tried to establish when the symmetric down- side-risk Sharpe ratio might give misleading results in real investment situations or to establish its mathematical and sta- tistical properties, I note only that it is consistent with an investor whose utility is based on the negative of the disutil- ity oflosses. The technique does seem to provide a simple way to avoid penalizing superior performance in order to more fairly evaluate performance. We will have to find another way to measure and establish the superiority of Warren Buffett, One likely candidate is related to the Kelly approach to evaluate investments, which looks at compounded wealth over a long period of time, which we know at the limit is attained by the log bettor (which Buffet seems to be), After 40 years, most of us believe Buffett is in the skill, not luck, category, Mter all, $15 a share in 1965 became $87,000 in June 2005, but since he is a log bettor only more time will tell, F.o.u200S --- ## Page 812 The Symmetric Downside-Risk Sharpe Ratio ENDNOTES 'If there are two independent wagers, and the size of the bets does not influence the odds, an analytic expression can be derived; see Thorp [1997, pp. 19-20J. In general, to solve for the optimal wagers when the bets influence the odds, there is depen- dence. In the case of three or more wagers, one must solve a 000- convex nonlinear program; see Ziemba and Hausch [1984, 1987J for technique. This gives the optimal wager, ta1cing into account the effect of our bets on the odds (prices). 2Bettor B must usc an essentially different strategy from our Kelly bettor for this to be true. This means that the strategies dif- fer infinitely often. For example, they are the same for the first ten years, and then every second trial is different. This is a tech- nical point to get proofS correct, but nothing much to worry about in practice since non-log strategies will differ infinitely often. ' Roy [1952], Markowitz [1959J, Mao [1970J, Bawa [1975, 1978J, BaWd and Lindenberg [1977], Fishburn [19771, Harlow and Rao [1989J, and Harlow [19911 have used downside-risk mea- sures in portfolio theories other than those based on mean-vari- ance and related analyses. REFERENCES Algoet, P., and T . Cover. "Asymptotic Optimality and Asymp- totic Equipartition Properties of Log-Optimum Investment." Annals of Probability, 16 (1988), pp. 876-898. Bain, R. , D.B. Hausch, and W .T. Ziemba. "An Application of Expert Information to Win Betting on the Kentucky Derby, 1981-2005." Working paper, University of British Columbia, 2005. !Jawa, V. "Optimal Rules for Ordering Uncertain Prospects." Journal of Financial Economics, 2 (March 1975), pp. 95-121. - -. "Safety First, Stochastic Dominance and Optimal Port- folio Choice." j ournal of Financial and Quantilative Analysis 13 Uune 1978), pp. 255-271. Bawa, V. , and E. Lindenberg. "Capital Market Equilibrium in a Mean, Lower Partial Moment Framework." Journal of Fifwn- cial Economics, 5 (November 1977), pp. 189-200. Benter, W. "Computer-Based Horse Race Handicapping and Wagering Systems: A Report. " In D.B. Hausch, V. Lo, and W.T. Ziemba, cds., Efficiency of Racetrack Belling Markets. San Diego: Academic Press, 1994. --. "Development of a Mathematical Model for Successful Horse Race Wagering." Presented at University of Nevada, Las Vegas, 2001. FAll 2005 783 Breiman, L. 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"Sample Bias and Sharpe's Perfonnance Measure: A Nocc," )OI/rnal oj Finandal and Quantitative Analy- sis, 13 (2005), pp. 943-946. 122 TI!E SYMMETRIC D OWNSIDE-RISK SHARPE RATIO W. T. Ziemba Roy, A. "Safety First and the Holding of Assets." Econometrica, 20 Guly 1952), pp. 431-449. Sharpe, W. "Mutual Fund Perfonnance."Jolirnal of Business, 39 (1966), pp. 119-138. Sharpe, W.F. "The Sharpe Ratio." TheJou,"al of Portfolio Man- agement,21 (1) (1994), pp. 49-58. Sortino, F.A. , and L.N. Price. ('Pcrfom1ance Measurement in a Downside Risk Framework." TheJou,"al of Investing, Fall 1994. Sortino, F .A., and R. van der Meer. "Downside Risk." The Jour- nal of Portfolio Management, Summer 1991. Spurgin, R.B. "How to Game Your Sharpe Ratio." TheJour- nal oj Alternative Investments 4 (3) (2000), pp. 38-46. Thorp, E.O. "The Kelly Criterion in Blackjack, Sports Bet- tingand the Stock Market." In S.A. Zenios and W.T. Ziemba, cds., Handbook of Asset Liability Managem,nt, Volume 1: Thmry and Methodology. Amsterdam: North Holland Elsevier, 2006. Tompkins, R., W. Ziemba, and S. Hodges. "The Favorite- Longshot Bias in S&P 500 Futures Options: The Return to Bets and the Cost ofInsurance." Working paper, Sauder School of Business, UBC, 2003. Ziemba, W.T. "The Stochastic Programing Approach to Asset Liability and Wealth Management." A1MR, 2003. Ziemba, W.T., and D.B. Hausch. Beat tile Racetrack, 1st cd. San Diego: Harcourt, Brace, Jovanovich, 1984. --. DrZ's Beat tlze Racetrack, 2nd cd. New York: William Morrow, 1987. --. Belling at the Racetrack. San Luis Obispo, CA: DrZ Invest- ments, Inc., 1986. Ziemba, W. T., and S.L. Schwartz. Invest Japan. Chicago: Probus, 1991. To order reprints oj this article, please WrItaa Dewey Palmieri at dpalmieri@iijournals.com or 212-224-3675. FALL 2005 --- ## Page 814 Scenarious for Risk Management and Global Investiment Strategies, 295- 298. Wiley (2007) 785 53 Appendix The Great Investors: Some Useful Books 295 POSTSCRIPT: THE RENAISSANCE MEDALLION FUND The Medallion Fund uses mathematical ideas such as the Kelly criterion to run a superior hedge fund. I The staff of technical researchers and traders, working under mathematician James Simons, is constantly devising edges that they use to generate successful trades of various durations including many short term trades that enter and exit in seconds. The fund, whose size is in the $5 billion area, has very large fees (5% management and 44% incentive). Despite these fees and the large size of the fund, the net returns have been consistently outstanding, with a few small monthly losses and high positive monthly returns; see the histogram in Figure At. Table Al shows the monthly net returns from January 1993 to April 2005. There were only l7 monthly losses in 148 months and 3 losses in 46 quarters and no yearly losses in these 12+ years of trading in our data sample. The mean monthly, quarterly and yearly net returns, Sharpe and Symmetric Downside Sharpe ratios are shown in Table A2.2 We calculated the quarterly standard deviation for the DSSR by multiplying the monthly standard deviation by sqrt(3) and multiplied it by sqrt(12) for the annual standard deviation. All calculations use arithmetic means. We know from Ziemba (2005) that the results using geometric means will have essentially the same conclusions. In Figure A3 we assumed that the fund had initial wealth of 100 dollars on Dec 31, 1992. Figures A2 and A3 show the rates of return over time and the wealth graph over time assuming an initial wealth of 100 on December 31, 2002. Medallion's outstanding yearly DSSR of 26.4 is the best we have seen even higher than Princeton Newport's 13.8 during 1969-1988. Jim Simon's Medallion fund is near or al the top of the worlds hedge funds. Indeed Simons' $1.4 billion in 2005 was the highest in 18r-----,------.------~----~-----r----_, 16 15 20 25 Figure Al Histogram of monthly returns of the Medallion Fund, January 1993 to April 2005. 1 WTZ is pleased to have had a minor role in teaching Simons about the Kelly criterion in 1992. 2 Thanks to Ilkay Boduroglu for making these calculations. --- ## Page 815 Table Al Net returns in percent of the Medallion Fund, January 1993 to April 2005, Yearly, Quarterly and Monthly 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 Annual 39.06 70.69 38.33 31.49 21.21 41.50 24.54 98.53 31.12 29.14 25.28 27.77 Quarterly Ql 7.81 14.69 22.06 7.88 3.51 7.30 (0.25) 25.44 12.62 5.90 4.29 9.03 Q2 25.06 35.48 4.84 1.40 6.60 7.60 6.70 20.51 5.64 7.20 6.59 3.88 Q3 4.04 11.19 3.62 10.82 8.37 9.69 6.88 8.58 7.60 8.91 8.77 5.71 Q4 (0.86) (1.20) 4.31 8.44 1.41 11.73 9.48 20.93 2.42 4.44 3.62 6.72 Monthly January 1.27 4.68 7.4 3.25 1.16 5.02 3.79 10.5 4.67 1.65 2.07 3.76 February 3.08 5.16 7.54 1.67 2.03 1.96 - 2.44 9.37 2.13 3.03 2.53 1.97 March 3.28 4.19 5.68 2.77 0.29 0.21 - 1.49 3.8 5.36 1.12 -0.35 3.05 April 6.89 2.42 4.1 0.44 1.01 0.61 3.22 9.78 2.97 3.81 1.78 0.86 May 3.74 5.66 5.53 0.22 4.08 4.56 1.64 7.24 2.44 1.11 3.44 2.61 June 12.78 25. 19 -4.57 0.73 1.36 2.28 1.71 2.37 0.15 2.13 1.24 0.37 July 3.15 6.59 - 1.28 4.24 5.45 -1.1 4.39 5.97 1 5.92 1.98 2.2 August -0.67 7.96 5.91 2.97 1.9 4.31 1.22 3.52 3.05 1.68 2.38 2.08 September 1.54 -3.38 -0.89 3.25 0.85 6.33 1.15 -1.02 3.38 1.13 4.18 1.33 October 1.88 -2.05 0.3 6.37 - 1.11 5.33 2.76 6.71 1.89 1.15 0.35 2.39 November - 1.51 - 0.74 2.45 5.93 -0.22 2.26 5.42 8.66 0.17 1.42 1.42 3.03 December -1.2 1.62 1.52 -3.74 2.77 3.73 1.06 4.3 0.35 1.81 1.81 1.16 2005 8.30 2.26 2.86 2.96 0.95 '" 1/ 2 and the probability oflosing is q = 1 - p. Our initial capital is Xo. Suppose we choose the goal of maximizing the expected value E (X,,) after n tria~s. How much should we bet, Bk, on the kth trial? Letting Tk = 1 if the kth trial is a win and Tk = -I ifit is a loss, then Xk = Xk- I +TkBk for k = 1, 2, 3, ... , and X" = Xo + :L%=I TkBk. Then " " Since the game has a positive expectation, i.e., p - q > 0 in this even payoff situation, then in order to maximize E(X,,) we would want to maximize E(Bd at each trial. Thus, to maximize expected gain we should bet all of our resources at each trial. Thus BI = Xo and if we win the first bet, B2 = 2X 0, etc. However, the probability of ruin is given by 1 - p" and with p < I, lim,,--+oo[ I - p"] = I so ruin is almost sure. Thus the "bold" criterion of betting to maximize expected gain is usually undesirable. Likewise, if we play to minimize the probability of eventual ruin (i.e., "ruin" occurs if Xk = 0 on the kth outcome) the well-known gambler's ruin formula in Feller (\966) shows that we minimize ruin by making a minimum bet on each trial, but this unfortu- nately also minimizes the expected gain. Thus "timid" betting is also unattractive. This suggests ·an intermediate strategy which is somewhere between maximizing E(X,,) (and assuring ruin) and minimizing the probability of ruin (and minimizing E (X,,». An asymptotically optimal strategy was first proposed by Kelly (\956). In the coin-tossing game just described, since the probabilities and payoffs for each bet are the same, it seems plausible that an "optimal" strategy will involve always wa- gering the same fraction f of your bankroll. To make this possible we shall assume from here on that capital is infinitely divisible. This assumption usually does not matter much in the interesting practical applications. If we bet according to Bi = f Xi - I, where 0 ~ f ~ I, this is sometimes called "fixed fraction" betting. Where Sand F are the number of successes and failures, respectively, in n trials, then our capital after n trials is X" = Xo(l + nS (1- nF, where S + F = n. With f in the interval 0 < f < I, Pr(X" = 0) = O. Thus "ruin" in the technical sense of the gambler's ruin problem cannot occur. "Ruin" shall henceforth be reinterpreted to mean that for arbitrarily small positive E, lim"--+oo[Pr(X,, ~ E)] = I. Even in this sense, as we shall see, ruin can occur under certain circumstances. We note that since [ X ] 1/" e"log X~ --- ## Page 822 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 793 Ch. 9: The Kelly Criterion .in) Blackjack Sporls Betting, and the Stock Market 389 the quantity [ X"]! /,, S F G,,(f) = log - = -log(1 + j) + -Iog(l - j) Xo n n measures the exponential rate of increase per trial. Kelly chose to maximize the expected value of the growth rate coefficient, g(f), where g(f) = E{ 10g[~~r /"}. = E{ ~ 10g(1 + j) + = l~g(1 - j)} = P 10g(1 + j) + q 10g(1 - j). Note that g(f) = (1 /n)E(logX,,) - (l/n)logXo so for n fixed, maximizing g(f) is the same as maximizing E log X". We usually will talk about maximizing g(f) in the discussion below. Note that , p q p-q-f g (f) = 1 + f - 1 - f = (1 + j) (1 _ j) = 0 when f = f* = p - q. Now g"(j) = -p/(1 + j)2 - q/(1 - j)2 < 0 so that g' (f) is monotone strictly decreasing on [0, 1). Also g' (0) = p - q > 0 and lim/---> !- g' (f) = -00. Therefore by the continuity of g' (j), g(f) has a unique maxie mum at f = f*, where g(f*) = p log p + q log q + log 2 > O. Moreover, g(O) = 0 and limj--->q- g(f) = -00 so there is a unique number f,. > 0, where 0 < f* < fe < 1, such that g(fc) = O. The nature of the function g(f) is now apparent and a graph of g(f) versus f appears as shown in Figure I. The following theorem recounts the important advantages of maximizing g(f). The details are omitted here but proofs of (i)-(iii), and (vi) for the simple binomial case can be found in Thorp (1969); more general proofs of these and of (iv) and (v) are in Breiman (1961). Theorem 1. (i) If g(f) > 0, then limn->oo X" = 00 almost surely, i.e., for each M, Pr[liminfn->oo XII > M] = 1; (ii) If g(j) < 0, then limll->oo X" = 0 almost surely; i.e., for each 8 > 0, Pr[limsuPn->oo X" < 8] = 1; (iii) If g(f) = 0, then lim sUPn-> 00 Xn = 00 a.s. and lim inf,,-> 00 Xn = 0 a.s. (iv) Given a strategy cJ>* which maximizes E log X" and any other "essentially different" strategy cJ> (not necessarily a fixed fractional betting strategy), then lim'l->oo Xn(cJ>*)/ X,,(cJ» = 00 a.s. (v) The expected time for the current capital X" to reach any fixed preassigned goal C is, asymptotically, least with a strategy which maximizes E log X". (vi) Suppose the return on one unit bet on the ith trial is the binomial random vari- able Ui; further, suppose that the probability of success is Pi, where 1/2 < Pi < l. --- ## Page 823 794 E. 0. Thorp 390 E.O. Thorp G(f) o f Fig. I. Then E log XII is maximized by choosing on each trial the fraction 1;* = Pi - qi which maximizes E log(1 + if Vi). Part (i) shows that, except for a finite number of terms, the player's fortune XII will exceed any fixed bound M when f is chosen in the interval (0, fe) . But, if f > f e, part (ii) shows that ruin is almost sure. Part (iii) demonstrates that if f = fe, X I1 will (almost surely) oscillate randomly between ° and +00. Thus, one author's statement that XI1 -+ Xo as n -+ 00, when f = f e, is clearly contradicted. Parts (iv) and (v) show that the Kelly strategy of maximizing E log XII is asymptotically optimal by two important criteria. An "essentially different" strategy is one such that the difference E In X,: - E In XI1 between the Kelly strategy and the other strategy grows faster than the standard deviation of In X~ - In XII' ensuring P (In X~ - In XII > 0) -+ 1. Part (vi) establishes the validity of utilizing the Kelly method of choosing 1;* on each trial (even if the probabilities change from one trial to the next) in order to maximize E log X n . Example 2.1. Player A plays against an infinitely wealthy adversary. Player A wins even money on successive independent flips of a biased coin with a win probability of p = .53 (no ties). Player A has an initial capital of Xo and capital is infinitely divisible. --- ## Page 824 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 795 Ch. 9: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 391 Applying Theorem I(vi), f* = p - q = .53 - .47 = .06. Thus 6% of current capital should be wagered on each play in order to cause XII to grow at the fastest rate possible consistent with zero probability of ever going broke. If Player A continually bets a fraction smaller than 6%, XII will also grow to infinity but the rate will be slower. If Player A repeatedly bets a fraction larger than 6%, up to the value f e, the same thing applies. Solving the equation g(f) = .5310g(l + f) + .4710g(1 - f) = 0 numerically on a computer yields fe = .11973-. So, if the fraction wagered is more than about 12%, then even though Player A may temporarily experience the pleasure of a faster win rate, eventual downward fluctuations will inexorably drive the values of XII toward zero. Calculation yields a growth coefficient of g(f*) = f(.06) = .001801 so that after n successive bets the log of Player A's average bankroll will tend to .001801n times as much money as he started with. Setting .00180 I n = log 2 gives an expected time of about n = 385 to double the bankroll. The Kelly criterion can easily be extended to uneven payoff games. Suppose Player A wins b units for every unit wager. Further, suppose that on each trial the win probability is p > 0 and pb - q > 0 so the game is advantageous to Player A. Methods similar to those already described can be used to maximize g(f) = E 10g(XIII Xo) = P log(l + bf) + q log(l - f). Arguments using calculus yield f* = (bp - q) I b, the optimal fraction of current capital which should be wagered on each play in order to maximize the growth coefficient g(f). This formula for f* appeared in Thorp (1984) and was the subject of an April 1997 discussion on the Internet at Stanford Wong's website, http://bj2I.com (miscellaneous free pages section). One claim was that one'can only lose the amount bet so there was no reason to consider the (simple) generalization of this formula to the situation where a unit wager wins b with probability p > 0 and loses a with probability q . Then if the expectation m == bp - aq > 0, f* > 0 and f* = mlab. The generalization does stand up to the objection. One can buy on credit in the financial markets and lose much more than the amount bet. Consider buying commodity futures or selling short a security (where the loss is potentially unlimited). See, e.g., Thorp and Kassouf (1967) for an account of the E.L. Bruce short squeeze. For purists who insist that these payoffs are not binary, consider selling short a binary digital option. These options are described in Browne (1996). A criticism sometimes applied to the Kelly strategy is that capital is not, in fact, infinitely divisible. In the real world, bets are multiples of a minimum unit, such as $1 or $.0 I (penny "slots"). In the securities markets, with computerized records, the minimum unit can be as small as desired. With a minimum allowed bet, "ruin" in the standard sense is always possible. It is not difficult to show, however (see Thorp and Walden, 1966) that if the minimum bet allowed is small relative to the gambler's initial capital, then the probability of ruin in the standard sense is "negligible" and also that the theory herein described is a useful approximation. This section follows Rotando and Thorp (1992). --- ## Page 825 796 E. O. Thorp 392 £.0. Thorp 3. Optimal growth: Kelly criterion formulas for practitioners Since the Kelly criterion asymptotically maximizes the expected growth rate of wealth, it is often called the optimal growth strategy. It is interesting to compare it with the other fixed fraction strategies. I will present some results that I have found useful in practice. My object is to do so in a way that is simple and easily understood. These results have come mostly from sitting and thinking about "interesting questions". I have not made a thorough literature search but I know that some of these results have been previously published and in greater mathematical generality. See, e.g., Browne (1996, 1997) and the references therein. 3. J. The probability of reaching a fixed goal on or before n trials We first assume coin tossing. We begin by noting a related result for standard Brownian motion. Howard Tucker showed me this in 1974 and it is probably the most useful single fact I know for dealing with diverse problems in gambling and in the theory of financial derivatives. For standard Brownian motion X (t), we have p(sup[X(t) - (at +b)] ~ 0, 0 ( t ( T) = N(-a - (3) + e- 2{/b N(a - (3) (3.1) where a = ag and f3 = b / g. See Figure 2. See Appendix B for Tucker's derivation of (3.1). In our application a < O,b > o so we expect limT ...... oo P(X(t) ~ at+b, 0 ( t ( T) =1. Let f be the fraction bet. Assume independent identically distributed (i.d.d.) trials Yi , i = I, ... ,n, with P(Yi = I) = p > 1/2, P(Yi = -I) = q < 1/2; also assume p < 1 to avoid the trivial case p = I. X(!) /' (0, b) slope a< ° / at+ b /' /' (0,0) T ! b (IOT ,O) Fig. 2. --- ## Page 826 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 797 Ch. 9: The Kelly Criterion in Blackjack Sports Belling, and the Stock Market 393 Bet a fixed fraction f, 0 < f < 1, at each trial. Let Vk be the value of the gambler or investor's bankroll after k trials, with initial value Vo . Choose initial stake Vo = 1 (without loss of generality); number of trials n; goal C > I. What is the probability that Vk ?': C for some k, 1 :s; k :s; n? This is the same as the probability that log Vk ?': log C for some k, I :s; k :s; n. Letting In = loge we have: k Vk = n(l + Yd) and i - I k In Vk = Lln(l + Yif) , i=1 k ElnVk = LEln(l +Yif), i=l k Var(ln Vk) = L Var(ln(l + Yi f)), i=1 E In(l + Yi f) = p In(l + f) + q In(l - f) == m == g(f), Var[ln(1 + Yif)] = p[ln(1 + f)]2 + q[ln(1 - f)]2 - m2 = (p _ p2)[ln(1 + f)]2 + (q _ q2)[ln(l _ f)]2 - 2pq In(l + f) In(l - f) = pq ([In ( 1 + f)]2 - 21n( 1 + f) In(l - f) + [InC 1 - f)]2} = pq{ln[(1 + f)/(I - f)]}2 == s2. Drift in n trials: mn . Variance in n trials: s2n. In Vk ?': InC, 1 :s; k:S; n, iff k L In(l + Yi f) ?': In C, I :s; k :s; n, iff i=1 k Sk == L[ln(l + fif) - m] ?': InC - mk, I :S; k :s; n, i=l We want Prob(Sk ?': In C - mk, I :s; k :s; n). Now we use our Brownian motion formula to approximate SII by Prob(X (t) ?': In C- mt/s2, 1 :s; t :s; s2n) where each term of SII is approximated by an XU), drift 0 and --- ## Page 827 798 E. 0. Thorp 394 E.O. Thorp variance s2 (0 :::::; t :::::; s2, s2 :::::; t :::::; 2s2 , .. . , (n - l)s2 :::::; t :::::; ns2). Note: the approximation is only "good" for "large" n. Then in the original formula (3.1): T = s2n , b = InC, a = -m/s2, (X = afl = -m..jii/s, f3 = b/fl = InC/s..jii. Example 3.1. C =2, n = 104 , p = .51, q = .49, f = .0117, m = .000165561, 052 = .000136848. Then PO = .9142. Example 3.2. Repeat with f = .02, then m = .000200013, 052 = .000399947 and PO = .9214. 3.2. The probability of ever being reduced to a fraction x of this initial bankroll This is a question that is of great concern to gamblers and investors. It is readily an- swered, approximately, by our previous methods. Using the notation of the previous section, we want P(Vk :::::; x for some k, 1 :::::; k :::::; 00). Similar methods yield the (much simpler) continuous approximation formula: Prob(·) = e2"h where a = -m/s2 and b = -Inx which can be rewritten as Probe-) = x ;\(2m/o5 2 ) where /\ means exponentiation. (3.2) --- ## Page 828 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market Ch. 9: The Kelly Criterion in Blackjack Sports Betting. and the Stock Market Example 3.3. p = .51, f = 1* = .02, 2m/s = 1.0002, Probe-) ":,, x. 799 395 We will see in Section 7 that for the limiting continuous approximation and the Kelly optimal fraction 1*, P (VI; (f*) ~ x for some k ) 1) = x. My experience has been that most cautious gamblers or investors who use Kelly find the frequency of substantial bankroll reduction to be uncomfortably large. We can see why now. To reduce this, they tend to prefer somewhat less than the full betting fraction f*. This also offers a margin of safety in case the betting situations are less favorable than believed. The penalty in reduced growth rate is not severe for moderate underbetting. We discuss this further in Section 7. 3.3. The probability of being at or above a specified value at the end of a specified number of trials Hecht (1995) suggested setting this probability as the goal and used a computerized search method to determine optimal (by this criterion) fixed fractions for p - q = .02 and various c, n and specified success probabilities. This is a much easier problem than the similar sounding in Section 3.1. We have for the probability that X (T) at the end exceeds the goal: 1 1 00 P(X(T» ) aT+b)= ~ exp/-x2/2T}dx v2rrT (/1'+1> = -- exp/-u2 / 2} du 1 1 00 .J2rrT (/1'1 /2+1>1'- 1/2 where u= x/~ so x = aT +b gives u~ = aT +b and U = aTI /2 +bT- I/2. The integral equals 1 - N(aTI /2 + bT- I/2) = N( _(aTI /2 + bT- 1/ 2)) = 1- N(a + f3) = N(-a - f3). (3.3) For example (3.1) f = .01l7 and P = .7947. For example (3.2) P = .7433. Example (3.1) is for the Hecht optimal fraction and example (3.2) is for the Kelly optimal fraction . Note the difference in P values. Our numerical results are consistent with Hecht's simulations in the instances we have checked. Browne (1996) has given an elegant continuous approximation solution to the prob- lem: What is the strategy which maximizes the probability of reaching a fixed goal C on or before a specified time n and what is the corresponding probability of success? Note --- ## Page 829 800 E. O. Thorp 396 £.0. Thorp that the optimal strategy will in general involve betting varying fractions, depending on the time remaining and the distance to the goal. As an extreme example, just to make the point, suppose n = I and C = 2. If Xo < 1 then no strategy works and the probability of success is O. But if I ~ Xo < 2 one should bet at least 2 - Xo, thus any fraction f ? (2 - Xo) / Xo, for a success probability of p. Another extreme example: n = 10, C = 2 10 = 1024, Xo = 1. Then the only strategy which can succeed is to bet f = 1 on every trial. The probability of success is pI 0 for this strategy and 0 for all others (if p < 1), including Kelly. 3.4. Continuous approximation of expected time to reach a goal According to Theorem I(v), the optimal growth strategy asymptotically minimizes the expected time to reach a goal. Here is what this means. Suppose for goal C that m(C) is the greatest lower bound over all strategies for the expected time to reach C. Suppose t*(C) is the expected time using the Kelly strategy. Then limc --,> 00 (t* (c) / m(c» = 1. The continuous approximation to the expected number of trials to reach the goal C > Xo = I is n(C, f) = (lnC)/ g(f) where f is any fixed fraction strategy. Appendix C has the derivation. Now g(f) has a unique maximum at g(f*) so n(C, f) has a unique minimum at f = f *. Moreover, we can see how much longer it takes, on average, to reach C if one deviates from f*· 3.5. Comparing fixed fraction strategies: the probability that one strategy leads another after n trials Theorem I (iv) says that wealth using the Kelly strategy will tend, in the long run, to an infinitely large multiple of wealth using any "essentially different" strategy. It can be shown that any fixed f -I- f * is an "essentially different" strategy. This leads to the question of how fast the Kelly strategy gets ahead of another fixed fraction strategy, and more generally, how fast one fixed fraction strategy gets ahead of (or behind) another. If WI/ is the number of wins in n trials and n - WI/ is the number of losses, G(f) = (WI//n) In(l + f) + (1 - WI//n) In(l - f) is the actual (random variable) growth coefficient. As we saw, its expectation is g(f) = E(G(f)) = p log(l + f) + q log(l - f) and the variance of G(f) is VarG(f) = (pq) / n)lln((l + f) / (l- f))}2 (3.4) (3.5) and it follows that G(f), which has the form G(f) = a(L Tk) / n +b, is approximately normally distributed with mean g(f) and variance VarG(f). This enables us to give --- ## Page 830 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 801 Ch. 9: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 397 the distribution of XI1 and once again answer the question of Section 3.3. We illustrate this with an example. Example 3.4. p = .51, q = .49, 1* = .02, N = 10,000 and s = standard deviation of G(f) gls 1.5 1.0 .5 1 .01 .02 .03 g .000150004 .000200013 .000149977 s .0001 .0002 .0003 Pr(G(f) ( 0) .067 .159 .309 Continuing, we find the distribution of G(h) - GUt). We consider two cases. Case 1. The same game Here we assume both players are betting on the same trials, e.g., betting on the same coin tosses, or on the same series of hands at blackjack, or on the same games with the same odds at the same sports book. In the stock market, both players could invest in the same "security" at the same time, e.g., a no-load S&P 500 index mutual fund. We find E(G(h) - G(It») = plog((1 + 12)/(1 + It») + q 10g((1- 12)/(1 - 1\») and Var( G (h) - G (ft)) = (pq In) { log [ C ~ ~~) C ~ ~: ) J r where G(h) - G(ft> is approximately normally distributed with this mean and vari- ance. Case 2. Identically distributed independent games This corresponds to betting on two different series of tosses with the same coin. E(G(h) - G(fl» is as before. But now Var(G(h) - G(fj) = Var(G(h» + Var(G(fd) because G(h) and G(fj) are now independent. Thus Let Var( G(h) - G(It») = (pq In) {[IOgC ~ ~~) r + [IOgC ~ ~: rJ}· a = log ( 1 + 1\ ) , 1 - It b = log (1 + h). 1- 12 Then in Case 1, VI = (pq In)(a - b)2 and in Case 2, V2 = (pq In)(a2 + b2) and since a, b > 0, VI . < V2 as expected. We can now compare the Kelly strategy with other --- ## Page 831 802 E. O. Thorp 398 E.O. Thorp fixed fractions to determine the probability that Kelly leads after n trials. Note that this probability is always greater than 1/2 (within the accuracy limits of the continuous approximation, which is the approximation of the binomial distribution by the normal, with its well known and thoroughly studied properties) because g(f*) - g(f) > 0 where 1* = p - q and I -=1= 1* is some alternative. This can fail to be true for small n, where the approximation is poor. As an extreme example to make the point, if n = 1, any I > f* beats Kelly with probability p > 1/ 2. If instead n = 2, I > f* wins with probability p2 and p2 > 1/ 2 if p > 1/.J2 ~ .707l. Also, I < f* wins with probability I - p2 and 1 - p2 > 1/ 2 if p2 < 1/ 2, i.e., p < 1/.J2 = .7071. So when n = 2, Kelly always loses more than half the time to some other I unless p = 1/.J2. We now have the formulas we need to explore many practical applications of the Kelly criterion. 4. The long run: when will the Kelly strategy "dominate"? The late John Leib wrote several articles for Blackjack Forum which were critical of the Kelly criterion. He was much bemused by "the long run". What is it and when, if ever, does it happen? We begin with an example. Example 4.1. p = .51 , n = 10,000, Vi and Si, i = 1, 2, are the variance and standard deviation, respectively, for Section 3.5 Cases I and 2, and R = V2/V[ = (a 2 + b2)/(a - b)2 so S2 = s[-/k Table I sum- marizes some results. We can also approximate v'"R with a power series estimate using only the first term of a and of b: a ~ 2ft, b ~ 212 so v'"R == J I? + 122/111 - 121· The approximate results, which agree extremely well, are 2.236, 3.606 and 1.581, re- spectively. The first two rows show how nearly symmetric the behavior is on each side of the optimal f* = .02. The column (g2 - gl) / s[ shows us that f* = .02 only has a .5 standard deviation advantage over its neighbors I = .0 I and I = .03 after n = 10,000 Table I Comparing strategies II .Ii g 2 - g l .1'1 C~2 - g l ) j.~ 1 ~ .01 .02 .00005001 .00010000 .50 2.236 .03 .02 .00005004 .00010004 .50 3.604 .03 .01 .00000003 .00020005 .00013346 1.581 --- ## Page 832 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 803 Ch. 9: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 399 Table 2 The long run: Cli2 - g I) /s after 11 trials II h 11 = 104 11=4 x 104 11=16 x 104 11 = 106 .01 .02 .5 1.0 2.0 5.0 .03 .02 .5 1.0 2.0 5.0 .03 .01 .000133 .000267 .000534 .001335 trials. Since this advantage is proportional to vIn, the column (g2 - gl)ISj from Table 1 gives the results of Table 2. The factor JR in Table 1 shows how much more slowly h dominates fl in Case 2 versus Case 1. The ratio (g2 - g I) 1 S2 is JR times as large so the same level of domi- nance takes R times as long. When the real world comparisons of strategies for practical reasons often use Case 2 comparisons rather than the more appropriate Case I compar- isons, the dominance of f* is further obscured. An example is players with different betting fractions at blackjack. Case 1 corresponds to both betting on the same sequence of hands. Case 2 corresponds to them playing at different tables (not the same table, because Case 2 assumes independence). (Because of the positive correlation between payoffs on hands played at the same table, this is intermediate between Cases 1 and 2.) It is important to understand that "the long run", i.e., the time it takes for f* to dominate a specified neighbor by a specified probability, can vary without limit. Each application requires a separate analysis. In cases such as Example 4.1, where dominance is "slow", one might argue that using f * is not important. As an argument against this, consider two coin-tossing games. In game 1 your edge is 1.0%. In game 2 your edge is 1.1 %. With one unit bets, after n trials the difference in expected gain is E2 - E I = .00In with standard deviation s of about..JIn hence (E2 - EI )Is ~ .001 vlnl v'2 which is 1 when n = 2 x 106 . So it takes two million trials to have an 84% chance of the game 2 results being better than the game 1 results. Does that mean it's unimportant to select the higher expectation game? S. Blackjack For a general discussion of blackjack, see Thorp (1962, 1966), Wong (1994) and Griffin (1979). The Kelly criterion was introduced for blackjack by Thorp (1962). The analysis is more complicated than that of coin tossing because the payoffs are not simply one to one. In particular the variance is generally more than I and the Kelly fraction tends to be less than for coin tossing with-the same expectation. Moreover, the distribution of various payoffs depends on the player advantage. For instance the frequency of pair splitting, doubling down, and blackjacks all vary as the advantage changes. By binning the probability of payoff types according to ex ante expectation, and solving the Kelly equations on a computer, a strategy can be found which is as close to optimal as desired. --- ## Page 833 804 E. 0. Thorp 400 E.O. Thorp There are some conceptual subtleties which are noteworthy. To illustrate them we'll simplify to the coin toss model. At each trial, we have with probability .5 a "favorable situation" with gain or loss X per unit bet such that P(X = I) = .51 , P(X = -I) =.49 and with probability.5 an unfavorable situation with gain or loss Y per unit bet such that P(Y = I) = .49 and P(Y = -1) = .51. We know before we bet whether X or Y applies. Suppose the player must make small "waiting" bets on the unfavorable situations in order to be able to exploit the favorable situations. On these he will place "large" bets. We consider two cases. Case 1. Bet fa on unfavorable situations and find the optimal f* for favorable situa- tions. We have g(f) = .5(.5110g(l + f) + .4910g(1 - f)) + .5(.4910g(l + fa) + .51 log (I - fa»). (5.1) Since the second expression in (5.1) is constant, f maximizes g(f) if it maximizes the first expression, so f* = P - q = .02, as usual. It is easy to verify that when there is a spectrum of favorable situations the same recipe, f/ = Pi - qi for the ith situation, holds. Again, in actual blackjack f * would be adjusted down somewhat for the greater variance. With an additional constraint such as fi ( kfo, where k is typically some integral multiple of fo, representing the betting spread adopted by a prudent player, then the solution is just fi ( min(f/, kfo) . Curiously, a seemingly similar formulation of the betting problem leads to rather different results. Case 2. Bet f in favorable situations and af in unfavorable situations, 0 ( a ( I. Now the bet sizes in the two situations are linked and both the analysis and results are more complex. We have a Kelly growth rate of g(f) = .5(.5110g(1 + f) + .4910g(1 - f)) + .5(.4910g(1 + af) + .5110g(l - af)) . (5.2) If we choose a = 0 (no bet in unfavorable situations) then the maximum value for g(f) is at f* = .02, the usual Kelly fraction. If we make "waiting bets", corresponding to some value of a > 0, this will shift the value of f* down, perhaps even to O. The expected gain divided by the expected bet is .02(1 - a)/ (l + a) , a ~ O. If a = 0 we get .02, as expected. If a = 1, we get 0, as expected: this is a fair game and the Kelly fraction is f* = O. As a increases from 0 to 1 the (optimal) Kelly fraction f* decreases from .02 to O. Thus the Kelly fraction for favorable situations is less in this case when bets on unfavorable situations reduce the overall advantage of the game. Arnold Snyder called to my attention the fact that Winston Yamashita had (also) made this point (March 18, 1997) on the "free" pages, miscellaneous section, of Stanford Wong's web site. --- ## Page 834 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market Ch. 9: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market a 1'* a 0 .0200 1/3 .1 .0178 .4 .2 .0154 .5 .3 .0128 .6 Table 3 f* versus a 1'* .0120 .0103 .0080 .0059 a .7 .8 .9 1.0 805 401 1'* .0040 .0024 .0011 .0000 For this example, we find the new f* for a given value of a, 0 < a < 1, by solving g' (f) = O. A value of a = 1/3, for instance, corresponds to a bet of 1/3 unit on Y and I unit on X, a betting range of 3 to 1. The overall expectation is .0 I. Calculation shows /* = .01200l. Table 3 shows how /* varies with a. To understand why Cases I and 2 have different /*, look first at Equation (5.1). The part of g(f) corresponding to the unfavorable situations is fixed when fo is fixed. Only the part of g(f) corresponding to the favorable situations is affected by varying f. Thus we maximize g(f) by maximizing it over just the favorable situations. Whatever the result, it is then reduced by a fixed quantity, the part of g containing fo. On the other hand, in Equation (5.2) both parts of g(f) are affected when f varies, because the fraction af used for unfavorable situations bears the constant ratio a to the fraction f used in favorable situations. Now the first term, for the favorable situations, has a maximum at f = .02, and is approximately "flat" nearby. But the second term, for the unfavorable situations, is negative and decreasing moderately rapidly at f = .02. Therefore, it we reduce f somewhat, this term increases somewhat, while the first term decreases only very slightly. There is a net gain so we find f* < .02. The greater a is, the more important is the effect of this term so the more we have to reduce f to get f*, as Table 3 clearly shows. When there is a spectrum of favorable situations the solution is more complex and can be found through standard multivariable optimization techniques. The more complex Case 2 corresponds to what the serious blackjack player is likely to need to do in practice. He will have to limit his current maximum bet to some multiple of his current minimum bet. As his bankroll increases or decreases, the corresponding bet sizes will increase or decrease proportionately. 6. Sports betting In 1993 an outstanding young computer science Ph.D. told me about a successful sports betting system that he had developed. Upon review I was convinced. I made suggestions for minor simplifications and improvements. Then we agreed on a field test. We found a person who was extremely likely to always be regarded by the other sports bettors as a novice. I put up a test bankroll of $50,000 and we used the Kelly.system to estimate our bet size. --- ## Page 835 806 E. O. Thorp 402 £.0. Thorp +80,000.00 +70,000.00 1- Total Profit I --------- Expected Profit I +60.000.00 +50,000.00 +40,000.00 +30.000.00 +20,000.00 ---t. II ~ . +10,000.00 +0.00 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 - 10,000.00 Fig. 3. Betting log Type 2 sports. +70.000.00 +60,000.00 H - Total Profit I ------ Expected Profit I ~ .... 1--- / .... +50.000.00 +40.000.00 +30,000.00 +20,000.00 . -~- - -.-.--.------ --- .- - --.-.---- .-.- - -- I +10,000.00 +0.00 _I, ~ 1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 8 1 85 89 93 97 101 -1 0.000.00 Fig. 4. Betting log Type I sports. We bet on 101 days in the first four and a half months of 1994. The system works for various sports. The results appear in Figures 3 and 4. After 101 days of bets, our $50,000 bankroll had a profit of $123,00Q, about $68,000 from Type I sports and about $55,000 from Type 2 sports. The expected returns are shown as about $62,000 for Type 1 and about $27,000 for Type 2. One might assign the additional $34,000 actually won to luck. But this is likely to be at most partly true because our expectation estimates from the model were deliberately chosen to be conservative. The reason is that using too large an f* and overbetting is much more severely penalized than using too small an f * and underbetting. . --- ## Page 836 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 807 efl. 9: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 403 Though $123,000 is a modest sum for some, and insignificant by Wall Street stan- dards, the system performed as predicted and passed its test. We were never more than a few thousand behind. The farthest we had to invade our bankroll to place bets was about $10,000. Our typical expectation was about 6% so our total bets ("action") were about $2,000,000 or about $20,000 per day. We typically placed from five to fifteen bets a day and bets ranged from a few hundred dollars to several thousand each, increasing as our bankroll grew. Though we had a net win, the net results by casino varied by chance from a substantial loss to a large win. Particularly hard hit was the "sawdust joint" Little Caesar's. It "died" towards the end of our test and I suspect that sports book losses to us may have expedited its departure. One feature of sports betting which is of interest to Kelly users is the prospect of betting on several games at once. This also arises in blackjack when (a) a player bets on multiple hands or (b) two or more players share a common bankroll. The standard techniques readily solve such problems. We illustrate with: Example 6.1. Suppose we bet simultaneously on two independent favorable coins with betting fractions It and 12 and with success probabilities PI and P2, respectively. Then the expected growth rate is given by gUI, h) = PIP2 In(1 + II + h) + Plq2 ln(1 + It - h) + QlP2 ln(l - It + h) + QlQ2 ln(l - II - h)· To find the optimal it and H we solve the simultaneous equations ag/alt = 0 and ilg/a12 = O. The result is I + f - PI P2 - q I q2 - I 2 - = c, PIP2 + qlq2 It - 12 = PI q2 - ql P2 == d, Plq2 + qlP2 it = (c + d)/2, H = (c - d)/2. (6.1) These equations pass the symmetry check: interchanging 1 and 2 throughout maps the equation set into itself. An alternate form is instructive. Let mi = Pi - qi, i = 1, 2 so Pi = (1 + mj) /2 and qi = (1 - mj)/2. Substituting in (6.1) and simplifying leads to: ml +m2 c=---- l+mlm2 ' * ml(l-m~) II = I 2 2 ' -m l m2 (6.2) which shows clearly the factors by which the It are each reduced from m7- Since the mj are typically a few percent, the reduction factors are typically very close to 1. --- ## Page 837 808 E. O. Thorp 404 E.O. Thorp In the special case PI = P2 = p, d = 0 and 1* = it = f; = e/2 = (p - q)/ (2(p2 + q2». Letting m = p - q this may be written 1* = m/(l + m2) as the optimal fraction to bet on each coin simultaneously, compared to f* = m to bet on each coin sequentially. Our simultaneous sports bets were generally on different games and typically not numerous so they were approximately independent and the appropriate fractions were only moderately less than the corresponding single bet fractions. Question: Is this al- ways true for independent simultaneous bets? Simultaneous bets on blackjack hands at different tables are independent but at the same table they have a pairwise correlation that has been estimated at .5 (Griffin, 1979, p. 142). This should substantially reduce the Kelly fraction per hand. The blackjack literature discusses approximations to these problems. On the other hand, correlations between the returns on securities can range from nearly -1 to nearly 1. An extreme correlation often can be exploited to great advantage through the techniques of "hedging". The risk averse investor may be able to acquire combinations of securities where the expectations add and the risks tend to cancel. The optimal betting fraction may be very large. The next example is a simple illustration of the important effect of covariance on the optimal betting fraction. Example 6.2. We have two' favorable coins as in the previous example but now their outcomes need not be independent. For simplicity assume the special case where the two bets have the same payoff distributions, but with ajoint distribution as in Table 4. Now c+m +b = (l +m)/2 so b = (l-m)/2 -c and therefore 0 ::;; e ::;; (l-m)/2. Calculation shows Var(Xi) = 1 - m 2, Cor(XI, X2) = 4c - (1 - m)2 and Cor(XI, X2) = [4c - (1 - m)2]/(l - m2). The symmetry of the distribution shows that g(fl, h) will have its maximum at fl :;= h = f so we simply need to maximize g(f) = (c + m) In(1 + 2f) + e In(l - 2f). The result is 1* = m/(2(2c + m». We see that for m fixed, as c decreases from (l - m)/2 and cor(X I , X2) = 1, to 0 and cor(X I, X2) = -(1 - m)/(1 + m), 1* for each bet increases from m/2 to 1/2, as in Table 5. Table 4 Joint distribution of two "identical" fa- vorable coins with correlated outcomes XI: X2: I -I C+1I1 h -I b c Table 5 I* increases as Cor( X I . X 2) decreases Cor(XI. X2) C I* (I - m) / 2 111/ 2 0 (I - /112)/ 4 111 / (1 +111 2 ) -(I - 111) / (1 + 111) 0 1/2 --- ## Page 838 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 809 ('II. 9: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 405 It is important to note that for an exact solution or an arbitrarily accurate numerical approximation to the simultaneous bet problem, covariance or correlation information is not enough. We need to use the entire joint distribution to construct the g function. We stopped sports betting after our successful test for reasons including: (I) It required a person on site in Nevada. (2) Large amounts of cash and winning tickets had to be transported between casinos. We believed this was very risky. To the sorrow of others, subsequent events con- firmed this. (3) It was not economically competitive with our other operations. If it becomes possible to place bets telephonically from out of state and to transfer the corresponding funds electronically, we may be back. 7. Wall street: the biggest game To illustrate both the Kelly criterion and the size of the securities markets, we return to the study of the effects of correlation as in Example 6.2. Consider the more symmetric and esthetically pleasing pair of bets VI and V2 , with joint distribution given in Table 6. Clearly 0 :::;; a :::;; 1/ 2 and Cor( VI , V2) = Cor( V I, Vz:) = 4a - I increases from -1 to 1 as a increases from 0 to 1/ 2. Finding a general solution for (ft, 1;) appears algebraically complicated (but specific solutions are easy to find numerically), which is why we chose Example 6.2 instead. Even with reduction to the special case m 1 = m2 = m and the use of symmetry to reduce the problem to finding f* = ft = f;, a general solution is still much less simple. But consider the instance when a = 0 so Cor(VI , V2) = -\. Then g(f) = In(l + 2m!) which increases without limit as f increases. This pair of bets is a "sure thing" and one should bet as much as possible. This is a simplified version of the classic arbitrage of securities markets: find a pair of securities which are identical or "equivalent" and trade at disparate prices. Buy the relatively underpric~d security and sell short the relatively overpriced security, achiev- ing a correlation of -I and "locking in" a riskless profit. An example occurred in 1983. My investment partnership bought $ 330 million worth of "old" AT&T and sold short $332.5 million worth of when-issued "new" AT&T plus the new "seven sisters" regional telephone companies. Much of this was done in a single trade as part of what was then the largest dollar value block trade ever done on the New York Stock Exchange (De- cember 1, 1983). In applying the Kelly criterion to the securities markets, we meet new analytic prob- lems. A bet on a security typically has many outcomes rather than just a few, as in 11/1 + 1 11/1 -I Table 6 Joint distribution of VI and V2 (/ 1/ 2 - ({ 11/ 2 - 1 1/ 2 - ({ ({ --- ## Page 839 810 E. O. Thorp 406 E.O. Thorp most gambling situations. This leads to the use of continuous instead of discrete prob- ability distributions. We are led to find f to maximize g(f) = E In(l + f X) = f In(l + fx) dP(x) where P(x) is a probability measure describing the outcomes. Fre- quently the problem is to find an optimum portfolio from among n securities, where n may be a "large" number. In this case x and fare n-dimension vectors and f x is their scalar product. We also have constraints. We always need I + fx > 0 so In( . ) is defined, and L fi = I (or some c > 0) to normalize to a unit (or to a c > 0) invest- ment. The maximization problem is generally solvable because g(f) is concave. There may be other constraints as well for some or all i such as fi ~ 0 (no short selling), or fi ~ Mi or fi ~ mi (limits amount invested in ith security), or L Ifi! ~ M (limits total leverage to meet margin regulations or capital requirements). Note that in some instances there is not enough of a good bet or investment to allow betting the full f* , so one is forced to underbet, reducing somewhat both the overall growth rate and the risk. This is more a problem in the gaming world than in the much larger securities markets. More on these problems and techniques may be found in the literature. 7.1. Continuous approximation There is one technique which leads rapidly to interesting results. Let X be a random variable with P(X = m + s) = P(X = m - 5) = .5. Then E(X) = m, Var(X) = 52. With initial capital Vo, betting fraction f, and return per unit of X, the result is V(f) = Vo(1 + (I - f)r + f X) = Vo(1 + r + f(X - r»), where r is the rate of return on the remaining capital, invested in, e.g., Treasury bills. Then g(f) = E(G(f») = E(ln(V(f)1 Vo)) = E In(1 + r + f(X - r») = .5In(1 + r + f(m - r + s») + .5In(1 + r + f(m - r - s»). Now subdivide the time interval into n equal independent steps, keeping the same drift and the same total variance. Thus m, 52 and r are replaced by min, 52 I nand r I n, respectively. We have n independent Xi, i = I , .. . , n, with P(Xi =mln + sn- I / 2) = P(Xi = min - sn- I / 2) = .5. Then II V,,(f)IVo = n(1 + (I - f)r + fXi). i=1 Taking E(log(·» of both sides gives g(f). Expanding the result in a power series leads to (7.1 ) where 0(n - I / 2) has the property n 1/20(n - I/ 2) is bounded as n ~ 00. Letting n ~ 00 in (7.1) we have goo(f) == r + f(m - r) - s2 f2 /2 . (7.2) --- ## Page 840 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 811 ('iI. 9: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 407 The limit V == V 00 (f) of Vii (f) as n ---+ 00 corresponds to a log normal diffusion process, which is a well-known model for securities prices. The "security" here has instantaneous drift rate m, variance rate s2, and the riskless investment of "cash" earns at an instantaneous rate r. Then goo (f) in (7.2) is the (instantaneous) growth rate of capital with investment or betting fraction f. There is nothing special about our choice of the random variable X. Any bounded random variable with mean E(X) = m and variance Var(X) = s2 will lead to the same result. Note that f no longer needs to he less than or equal to 1. The usual problems, with log(·) being undefined for negative arguments, have disappeared. Also, f < 0 causes no problems. This simply corresponds to selling the security short. If m < r this could be advantageous. Note further that the investor who follows the policy f must now adjust his investment "instantaneously". In practice this means adjusting in tiny increments whenever there is a small change in V. This idealization appears in option theory. It is well known and does not prevent the practical application of the theory (Black and Scholes, 1973). Our previous growth functions for finite sized betting steps were approximately parabolic in a neighborhood of f* and often in a range up to 0 ~ f ~ 2f*, where also often 2f* == t. Now with the limiting case (7.2), goo (f) is exactly parabolic and very easy to study. Lognormality of V (f) / Vo means 10g(V (n / Vo) is N (M, 52) distributed, with mean M = goo(nt and variance 52 = Var(Goo(f)t for any time t. From this we can de- termine, for instance, the expected capital growth and the time tk required for V (f) to be at least k standard deviations above Vo. First, we can show by our previous methods that Var(Goo(f» = s2 f2, hence Sdev(Goo(f» = sf. Solving tkgoo = kt} /2 Sdev(Goo(f» gives tkg~ hence the expected capital growth tkgoo, from which we find tk. The results are summarized in Equations (7.3). f* = (m - r)/s2, goo(f) = r + f(m - r) - s2 f2 /2, goo(f*) = (m - r)2/2s2 + r, Var(Goo(f») = s2 f2, Sdev(Goo(f») = sf, tkgoo(f) = k2s2 f2 / goo, tk = k2s2 12 / g~. (7.3) Examination of the expressions for tk goo (f) and tk show that each one increases as f increases, for 0 ~ 1 < 1+ where f + is the positive root of s2 f2 /2 - (m - r) 1 - r = 0 and 1+ > 2f*. Comment: The capital asset pricing model (CAPM) says that the market portfolio lies on the Markowitz efficient frontier E in the (s, m) plane at a (generally) unique point P = (so, mo) such that the line determined by P and (s = 0, m = r) is tangent to E (at P). The slope of this line is the Sharpe ratio 5 = (mo - ro)/so and from (7.3) Koo(f*) = 52/2 + r so the maximum growth rate goo(f*) depends, for fixed r, only on the Sharpe ratio. (See Quaife (1995).) Again from (7.3), f* = 1 when m = r + s2 in which case the Kelly investor will select the market portfolio without borrowing or lending. If m > r + s2 the Kelly investor will use leverage and if m < r + s2 he will --- ## Page 841 812 E. 0. Thorp 408 E.O. Tho!]) invest partly in T-bills and partly in the market portfolio. Thus the Kelly investor will dynamically reallocate as f* changes over time because of fluctuations in the forecast m, rand s2, as well as in the prices of the portfolio securities. From (7.3), goo(1) = m - s2/2 so the portfolios in the (s, m) plane satisfying m - s2/2 = C, where C is a constant, all have the same growth rate. In the contin- uous approximation, the Kelly investor appears to have the utility function U (s, m) = m - s2/2. Thus, for any (closed, bounded) set of portfolios, the best portfolios are ex- actly those in the subset that maximizes the one parameter family m - s2/2 = C. See Kritzman (1998), for an elementary introduction to related ideas. Example 7.1. The long run revisited. For this example let r = O. Then the basic equa- tions (7.3) simplify to r = 0: 1* = m/s2 , goo(f) = mf - s2 f2/2, goo(f*) = m 2/2s2, Var(G oo(f») = s2 f2, Sdev(Goo(f») = sf. (7.4) How long will it take for V (f*) ~ Vo with a specified probability? How about V (f* /2)? To find the time t needed for V (f) ~ Vo at the k standard deviations level of significance (k = I, P = 84%; k = 2, P = 98%, etc.) we solve for t == tk: (7.5) We get more insight by normalizing all f with f* · Setting f = c f* throughout, we find when r = 0 r = 0: f* == m/s2 , f = em/s2 , goo(c1*) = m 2(c - c2/2)/s 2 , Sdev(Goo(c1*») = cm/s, tgoo (c1*) = k 2e/(1 - e/2), t(k,ef*) =k2s2/{m 2(1_e/2)2). (7.6) Equations (7.6) contain a remarkable result: V (f) ~ Vo at the k standard deviation level of significance occurs when expected capital growth tgoo = k 2e/(1 - e/2) and this result is independent of m and s. For f = f* (e = 1 in (7.6», this happens for k = I at tgoo = 2 corresponding to V = Voe2 and at k = 2 for tgoo = 8 corresponding to V = Voe8. Now e8 === 2981 and at a 10% annual (instantaneous) growth rate, it takes 80 years to have a probability of 98% for V ~ Yo. At a 20% annual instantaneous rate it takes 40 years. However, for f = f* /2, the number for k = I and 2 are tgoo = 2/3 and 8/3, respectively, just 1/3 as large. So the waiting times for Prob(V ~ Yo) to exceed 84% and 98% become 6.7 years and 26.7 years, respectively, and the expected growth rate is reduced to 3/4 of that for f*. --- ## Page 842 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 813 ('iI. 9: The Kelly Criterion in Blackjack Sports Betting, and the Stock Marker 409 Comment: Fractional Kelly versus Kelly when r = 0 From Equations (7,6) we see that goo(cf*)/goo(f*) = c(2 - c), 0 ~ c < 00, showing how the growth rate relative to the maximum varies with c, The relative risk Sdev(Goo(cf*»/Sdev(Goo(f*» = c and the relative time to achieve the same ex- pected total growth is 1/c(2 - c), 0 < c < 2, Thus the relative "spread" for the same expected total growth is 1/(2 - c), 0 < c < 2. Thus, even by choosing c very small, the spread around a given expected growth cannot be reduced by 1/2, The cOlTesponding results are not quite as simple when r > O. 7.2. The (almost) real world Assume that prices change "continuously" (no "jumps"), that portfolios may be revised "continuously", and that there are no transactions costs (market impact, commissions, "overhead"), or taxes (Federal, State, city, exchange, etc.). Then our previous model applies. Example 7.2. The S&P 500 Index. Using historical data we make the rough estimates 111 = .11, s = .15, r = .06. The equations we need for r =1= 0 are the generalizations of (7.6) to r =1= 0 and f = cf*, which follow from (7.3): cf* = c(m - r)/s2, goo(cf*) = (m - r)2(c - c2/2))/s2 + r, Sdev(Goo(cf*») = c(m - r)/s, tgoo(cf*) = k2c2/(c - c2/2 + rs2/(m - r)2) , t(k, cf*) = k2c2( (m - r)2 /s2) / (({m - r)2/s2)(c - c2/2) + r ( (7.7) If we define iii = m - r, Goo = Goo - r, goo = goo - r, then substitution into Equations (7.7) give Equations (7.6), showing the relation between the two sets. It also shows that examples and conclusions about P (Vn > Vo) in the r = 0 case are equivalent to those about P(ln(V(t)/Vo) > rt) in the r =1= 0 case. Thus we can compare various strategies versus an investment compounding at a constant riskless rate r such as zero coupon U.S. Treasury bonds. ' From Equations (7.7) and c = 1, we find f* = 2.22, goo(f*) = .115, Sdev( Goo(f*») = .33, t = 8.32k2 years. Thus, with f* = 2.22, after 8.32 years the probability is 84% that VII > Vo and the expected value of loge VII / Vo) = .96 so the median value of VII / Vo will be about £, . ,96 = 2.61. With the usual unlevered f = 1, and c = .45, we find using (7.3) goo(1) = m - s2/2 = .09875, Sdev( Goo(l») = .15, t(k, .45f*) = 2.31k2 years. --- ## Page 843 814 E. 0. Thorp 410 E.O. Thorp Writing tgoo = h(c) in (7.7) as we see that the measure of riskiness, h(c), increases as c increases, at least up to the point c = 2, corresponding to 2f* (and actually beyond, up to 1 + 1 + (1/:~~. )1 ). Writing t(k, c1*) = t(c) as t(c) = k2«m - r)2/s2)/(m - r)2/s2)(l - c/2) + r/c2 shows that t(c) also increases as c increases, at least up to the point c = 2. Thus for smaller (more conservative) f = cf*, c ( 2, specified levels of P(VI/ > Vo) are reached earlier. For c < 1, this comes with a reduction in growth rate, which reduction is relatively small for f near 1*. Note: During the period 1975-1997 the short term T-bill total return for the year, a proxy for r if the investor lends (i.e., f < 1), varied from a low of 2.90% (1993) to a high of 14.71 % (1981). For details, see Ibbotson Associates, 1998 (or any later) Yearbook. A large well connected investor might be able to borrow at broker's call plus about 1 %, which might be approximated by T-bills plus I %. This might be a reasonable esti- mate for the investor who borrows (f > 1). For others the rates are likely to be higher. For instance the prime rate from 1975-1997 varied from a low of 6% (1993) to a high of 19% (1981), according to Associates First Capital Corporation (1998). As r fluqtuates, we expect m to tend to fluctuate inversely (high interest rates tend to depress stock prices for well known reasons). Accordingly, f* and goo will also fluctuate so the 'long term S&P index fund investor needs a procedure for periodically re-estimating and revising f* and his desired level of leverage or cash. To illustrate the impact of r" > r, where r" is the investor's borrowing rate, suppose r" in example (7.2) is r + 2% or .08, a choice based on the above cited historical values for r, which is intermediate between "good" rt, ~ r + I %, and "poor" r" ~ the pri~e rate ~ r + 3%. We replace r by rt, in Equations (7.7) and, if f* > I, f* = 1.33, goo(f*) = .100, Sdev(Goo(f*» = .20, tgoo(f*) = .4k2, t = 4k2 years. Note how greatly f* is reduced. Comment: Taxes Suppose for simplicity that aU gains are subject to a constant continuous tax rate T and that all losses are subject to a constant continuous tax refund at the same rate T. Think of the taxing entities, collectively, as a partner that shares a fraction T of all gains and losses. Then Equations (7.7) become: c1* = c(m - r) /s20 - T), goo(c.t*) = (m - r)2(c - c2/2))/s2 + r( I - T), Sdev(Goo(c.t*») = c(m - r)/s, --- ## Page 844 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 815 ( 'II. mr is likely. --- ## Page 845 816 E. 0. Thorp 412 £.0. Thorp g(f) unit ~ m'/2s' PENALTY FOR EACH m, IFYOU CHOOSE THIS r o Sf: f: 1 Sf: -- - r - - T ---- r- 2.0 1.5 1.0 0.5 0.0 -+----+------'-:--__\! !}--'-----Ht-----..---+- If: 2:0 LOSS, -0.5 - 1.0 - -- ----- --- -. - -- -- ----.-.------ - --- --.- --- --.-- Fig_ 5_ Penalties for choosing f = j;, i- I* = ji. Sports betting has much the same caveats as the securities markets, with its own differences in detail. Rules changes, for instance, might include: adding expansion teams; the three point rule in basketball; playing overtime sessions to break a tie; changing types of bats, balls, gloves, racquets or surfaces. Blackjack differs from the securities and sports betting markets in that the prob- abilities of outcomes can in principle generally be either calculated or simulated to any desired degree of accuracy. But even here ml is likely to be at least somewhat less than me. Consider player fatigue and errors, calculational errors and mistakes in applying either blackjack theory or Kelly theory (e.g., calculating f* correctly, for which some of the issues have been discussed above), effects of a fixed shuffle point, non-random shuffling, preferential shuffling, cheating, etc. (2) Subject to (I), choosing f in the range .5.ft; ~ f < f{; offers protection against g ~ 0 with a reduction of g that is likely to be no more'than 25%. --- ## Page 846 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 817 ( '/1, I > x > 0 then for /* the probability that V (t) reaches y Vo before x Vo is Prob(V (t, f*) reaches y Vo before x Vo) = (I - x)/ (I - (x / y») (7.10) and more generally, for f = cf*, 0 < c < 2, Prob(V(t, cf*) reaches y VO before xVo) = [I - / \(2 / c ~ 1)]/[1 - (x/y('(2/c - I)] (7.11 ) where 1\ means exponentiation. Clearly (7.10) follows from 0 .11) by choosing c = I. The r = 0 case of our Equation (7.8) follows from 0 .11) and the r = 0 case of our Equation (7.9) follows from (7.10). We can derive a generalization of (7.1 I) by using the classical gambler's ruin formula (Cox and Miller, 1965, p. 31, Equation (2.0» and passing to the limit as step size tends to zero (Cox and Miller, 1965, pp. 205-206), where we think of 10g(V(t, f)/ Vo) as following a diffusion process with mean goo and variance v(Goo), initial value 0, and absorbing barriers at log y and log x . The result is Prob(V(t, cf*) reaches yVO before xVo) = [I - x Aa]/[1 - (x/y) Aa] (7.12) where a = 2goo/ V (G (0) = 2M / V where M and V are the drift and variance, respec- tively, of the diffusion process per unit time. Alternatively, (7.12) is a simple restatement of the known solution for the Wiener process with two absorbing barriers (Cox and Miller, 1965, Example 5.5). As Schlesinger notes, choosing x = 1/2 and y = 2 in (7.10) gives probe V (t , /*) doubles before halving) = 2/3. Now consider a gambler or investor who focuses only on values VII = 211 Vo. n = o. ± I. ±2, . . . , multiples of his initial capital. In log space, loge VII / Vo) = n log 2 so we have a random walk on the integer multiples of log 2, where --- ## Page 848 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 819 ( "II (): The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 415 tlie probability of an increase is p = 2/3 and of a decrease, q = 1/3. This gives us a l( lItvcnient compact visualization of the Kelly strategy's level of risk. I f instead we choose c = 1/2 ("half Kelly"), Equation (7.11) gives probe V (t, !* /2) douhles before halving) = 8/9 yet the growth rate goo(f* /2) = .75goo(f*) so "half Kelly" has 3/4 the growth rate but much less chance of a big loss. A second useful visualization of comparative risk comes from Equation (7.8) which Prob(V(t,c!*)/Vo ::;;x forsomet) = xl\(2/c -l). (7.13) I'or c = 1 we had Prob(·) = x and for c = 1/2 we get Prob(·) = x 3. Thus "half Kelly" has a much lessened likelihood of severe capital loss. The chance of ever losing lialf the starting capital is 1/2 for f = !* but only 1/8 for f = !* /2. My gambling alld investment experience, as well as reports from numerous blackjack players and Il'ams, suggests that most people strongly prefer the increased safety and psychological l'Olllfort of "half Kelly" (or some nearby value), in exchange for giving up 1/4 of their gn lwth rate. X. A case study III the summer of 1997 the XYZ Corporation (pseudonym) received a substantial alllount of cash. This prompted a review of its portfolio, which is shown in Table 7 i II the column 8/17/97. The portfolio was 54% in Biotime, ticker BTIM, a NASDAQ hiotechnology company. This was due to existing and historical relationships between people in XYZ Corp. and in BTIM. XYZ's officers and directors were very knowledge- Monthly Allllual Monthly Monthly Table 7 Statistics for logs of monthly wealth relatives, 3/31/92 through 6/30/97 Berkshire Mean .0264 Standard deviation .0582 Mean .3167 Standard deviation .2016 Covariance .0034 Correlation 1.0000 BioTime .0186 .2237 .2227 .7748 -.0021 .0500 - .1581 1.0000 SP500 .0146 .0268 .l753 .0929 .0005 -.0001 .0007 .2954 -.0237 1.0000 T-bills .0035 .0008 .0426 .0028 1.2E-06 3.2E-05 5.7E- 06 6.7E-07 .0257 .1773 .2610 1.0000 --- ## Page 849 820 E. O. Thorp 416 E.O. Thorp able about BTIM and felt they were especially qualified to evaluate it as an investment. They wished to retain a substantial position in BTIM. The portfolio held Berkshire Hathaway, ticker BRK, having first purchased it in 1991. 8.1. The constraints Dr. Quaife determined the Kelly optimal portfolio for XYZ Corp. subject to certain con- straints. The list of allowable securities was limited to BTIM, BRK, the Vanguard 500 (S&P 500) Index Fund, and T-bills. Being short T-bills was used as a proxy for mar- gin debt. The XYZ broker actually charges about 2% more, which has been ignored in this analysis. The simple CAPM (capital asset pricing model) suggests that the investor only need consider the market portfolio (for which the S&P 500 is being substituted here, with well known caveats) and borrowing or lending. Both Quaife and the author were convinced that BRK was and is a superior alternative and their knowledge about and long experience with BRK supported this. XYZ Corp. was subject to margin requirements of 50% initially and 30% mainte- nance, meaning for a portfolio of securities purchased that initial margin debt (money lent by the broker) was limited to 50% of the value of the securities, and that whenever the value of the account net of margin debt was less than 30% of the value of the secu- rities owned, then securities would have to be sold until the 30% figure was restored. In addition XYZ Corp. wished to continue with a "significant" part of its portfolio in BTIM. 8.2. The analysis and results Using monthly data from 3/31/92 through 6/30/97, a total of 63 months, Quaife finds the means, covariances, etc. given in Table 7. Note from Table 7 that BRK has a higher mean and a lower standard deviation than BTIM, hence we expect it to be favored by the analysis. But note also the negative correlation with BTIM, which suggests that adding some BTIM to BRK may prove advantageous. Using the statistics from Table 7, Quaife finds the following optimal portfolios, under various assumptions about borrowing. As expected, BRK is important and favored over BTIM but some BTIM added to the BRK is better than none. If unrestricted borrowing were allowed it would be foolish to choose the correspond- ing portfolio in Table 8. The various underlying assumptions are only approximations with varying degrees of validity: Stock prices do not change continuously; portfo- lios can't be adjusted continuously; transactions are not costless; the borrowing rate is greater than the T-bill rate; the after tax return, if different, needs to be used; the process which generates securities returns is not stationary and our point estimates of the statistics in Table 7 are uncertain. We have also noted earlier that because "over- --- ## Page 850 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market ( h 1 m = r + !I(m) - r) + . .. + fll(m ll - r) - e,,(j+ - 1) - es/-. (8.4.2) 9. My experience with the Kelly approach How does the Kelly-optimal approach do in practice in the securities markets? In a little-known paper (Thorp, 1971) I discussed the use of the Kelly criterion for portfolio management. Page 220 mentions that "On November 3, 1969, a private institutional investor decided to ... use the Kelly criterion to allocate its assets". This was actually a private limited partnership, specializing in convertible hedging, which I managed. A notable competitor at the time (see Institutional Investor (1998» was future Nobel prize winner Harry Markowitz. After 20 months, our record as cited was a gain of 39.9% versus a gain for the Dow Jones Industrial Average of +4.2%. Markowitz dropped out after a couple of years, but we liked our results and persisted. What would the future bring? Up to May 1998, twenty eight and a half years since the investment program began. The partnership and its continuations have compounded at approximately 20% annu- ally with a standard deviation of about 6% and approximately zero correlation with the market ("market neutral"). Ten thousand dollars would, tax exempt, now be worth 18 --- ## Page 853 824 E. O. Thorp 420 £.0. Thorp million dollars. To help persuade you that this may not be luck, 1 estimate that dur- ing this period I have made about $80 billion worth of purchases and sales ("action", in casino language) for my investors. This breaks down into something like one and a quarter million individual "bets" averaging about $65,000 each, with on average hun- dreds of "positions" in place at anyone time. Over all, it would seem to be a moderately "long run" with a high probability that the excess performance is more than chance. 10. Conclusion Those individuals or institutions who are long term compounders should consider the possibility of using the Kelly criterion to asymptotically maximize the expected com- pound growth rate of their wealth. Investors with less tolerance for intermediate term risk may prefer to use a lesser function. Long term compounders ought to avoid using a greater fraction ("overbetting"). Therefore, to the extent that future probabilities are uncertain, long term compounders should further limit their investment fraction enough to prevent a significant risk of overbetting. Acknowledgements T thank Dr. Jerry Baese\, Professor Sid Browne, Professor Peter Griffin, Dr. Art Quaife, and Don Schlesinger for comments and corrections and to Richard Reid for posting this chapter on his website. I am also indebted to Dr. Art Quaife for allowing me to use his analysis in the case study. This chapter has been revised and expanded since its presentation at the 10th Inter- national Conference on Gambling and Risk Taking. Appendix A. Integrals for deriving moments of E 00 10(a2, /)2) = ( DO exp[- (a2x 2+b2jx2)]dx, Jo III(a2, b2) = ( DO xllexp[- (a2x2+b2 jx2)]dx . Jo Given 10 find 12 lo(a2, b2) = roo exp[- (a2x2 +I}jx2)]dX Jo = - f~ exp[ _(a 2 ju2 + h2u2)]( -du jll?) --- ## Page 854 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market I 'Ii, 0, p(X(t) ~ at + b, 0 ~ t ~ T I X(T) = s) = 101 - exp{ - ~ (aT + B - S)} if s ~ aT + b, if s > aT + B. Write this as: p(X(t) ~ at +b, 0 ~ t ~ T I X(T)) ~. 1 - exp{ -2b(aT + b - X(T))~} if X(T) ~ aT + b. Taking expectations of both sides of the above, we get p(X(t) ~ at + b, 0 ~ t ~ T) = (1 - e - 2iJ(aT+b-s)I / 1') __ e - s2/2T ds / aT+I> 1 - 00 J2nT = __ e-·I /21 ds _ __ e- (.1 - 2b) /21 ds. 1 /aT + I> .2" e - 2ab /"1' + 1> . 2" J2n T -00 J2n T -00 Hence P(X going above line at + b during [0, T)) = 1 - previous probability = __ e- I /21 ds + e - 2ab . __ e - u /21 du, I JOO .2 " I /aT - I> 2, J2nT a1'+iJ J2nT - 00 where u = s - 2b. (B.l) Now, when a = 0, b > 0, [ ] (21°O 2 ' P sup X(t) ~ b = V -;;;r- e- V /21 dv, O~ / ~ T nT I> which agrees with a known formula (see, e.g., p. 261 of Tucker (1967)). In the case a > 0, when T -+ 00, since fliT -+ 0 and fl = s.d. of X(T) , the first integral -+ 0, the second integral -+ 1, and P(X ever rises above line at + b) = e- 2"I>. Similarly, in the case a < 0, peever rises above line at + b) = 1. The theorem it comes from is due to Sten Malmquist, On certain confidence contours for distribution functions, Ann. Math. Stat. 25 (1954), pp. 523-533. This theorem is stated in S.R. Paranjape and C. Park, Distribution of the supremum of the two-parameter Yeh-Wiener process on the boundary, J. App!. Prob. 10 (1973). Letting ex = a fl, f3 = b I fl, formula (B. 1) becomes PO = N( -ex - f3) + e- 2afi N(ex - f3) where ex, f3 > 0 or --- ## Page 856 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 827 Ch. 9: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market 423 p(x(t) :( at + b, 0 :( t :( T) = 1 - PC) = N(a + fJ) - e- 2afJ N(a - fJ) for the probability the line is never surpassed. This follows from: _1_1 00 e-·\·2j2T ds = _1_1 00 e-x2j2 dx = N( -a - fJ ) .J2rrT aT+b ~ a./f+bj./f -- e- u-j2T du = N(a - fJ) 1 j aT-b ? .J2rrT - 00 where s = aT + b, x = s / .JT = a.JT + b/ .JT, a = a.JT and fJ = b/ .JT. The formula becomes: p(sup[X(t) - (at + b)] ~ 0: 0 :( t :( T) = N(-a - fJ) + e- 2a/J N(a - fJ) = N(-a - fJ) + e- 2afJ N(a - fJ), a , fJ > O. Observe that PC) < N(-a - m + N(a - fJ) = {I - N(a +fJ)} + N(a - fJ) = ja- fJ a(x) dx + 1 00 a(x) dx < I -00 a+fJ as it should be. Appendix C. Expected time to reach goal and Reference: Handbook of Mathematical Functions, Abramowitz and Stegun, Editors, N.B.S. Applied Math. Series 55, June 1964. P. 304, 7.4.33 gives with erf z == Jrr f~ e- r2 dt the integral: f exp{ _(a 2 x2 + b2 /x2) } dx = :- [e2a!Jerf(ax + b/x) + e- 211 !J erf(ax - b/x)] + C, a i- O. (C.l) Now the left side is > 0 so for real a, we require a > 0 otherwise the right side is < 0, a contradiction. We also note that p. 302, 7.4.3. gives roo exp{ -(at2 + b/ t 2)} dt = ~ ~e-2~ 10 2V~ (C.2) with ma > 0, mb > O. --- ## Page 857 828 E. 0. Thorp 424 E.O. Thorp To check (C.2) v. (C.l), suppose in (C.l) a > 0, b > 0 and find Iimx -> o and limt->oo of erf(ax + b/x) and erf(ax - b/x), lim (ax + b/x) = +00, x.j.o+ lim (ax + b/ x) = +00, . .\' ----7 00 Equation (C.l) becomes lim (ax - b/x) = -00, x.j.o+ lim (ax - b/x) = +00 . x ---+oo -Jii e-2al> [erf(oo) _ erf(-oo)] = -Jii e- 2ab2erf(00) = -Jii e- 2ab 4a 4a 2a since we know erf( 00) = 1. In (C.2) replace a by a2 , b by b2 to get which is the same. Note: if we choose the lower limit of integration to be 0 in (C. I), then we can find C: 0+ -Jii 0= [ exp{ _(a2x 2 + b2 /x2)} dx = -[ e2al>erf(00) + e- 2ah erf( -00)] + C 10 4a = -Jii [e2al> _ e-211h ] + c. 4a Whence F(x) == fox exp{ _(a2x 2 + b2 / x2)} dx = -Jii (e2"j,[erf(ax + b/x) - I] + e- 211 j,[erf(ax - b/x) + I]}. (C.3) 4a To see how (C.3) might have been discovered, differentiate: F'(x) = exp{-(a2x 2 +b2/x2)} = -Jii {e211h (a _ b/x2) erf'(ax + b/ x) 4a + e- 211h (a + b/x2) erf'(ax - b/x)}. Now erf' (z) = Jrr exp( - Z2) so 2 2 erf'(ax+b/ x) = -Jiiexp[-(ax+b/x)2] = -Jiiexp{-(a2x2+b2/x2+2ab)} 2 = -Jiie- 2a/J exp{ _(a2x 2 + b2 /x2)} , --- ## Page 858 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market Ch. 9: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market and, setting b +- -b, whence F'(x) = : {]rr(a - b/x2) + ]rr(a + b/x2)} exp{ _(a2x 2 + b2 /x2)} = ~{2a} exp{ _(a2 x2 + b2 /x2)} = exp{ _(a2x2 + b2 /x2)} . 2a Case of interest: a < 0, b > O. Expect: b > 0, a ::;; 0 ::::} F(T) t I as T -+ 00, b> 0, a > 0 ::::} F(T) t c < 1 as T -+ 00. If b > 0, a = 0: F(T) = N(-f3) + N(-fi) = 2N(-b/.Jf) t 2N(0) = 1 as T t 00. Also, as expected F(T) t I as h ~ O. If b > 0, a < 0: See below. If b > 0, a > 0: F(T) = N(-afl - h/fl) +e-2abN(a.Jf -b/.Jf) -+ N(-oo) +c 211i>N(00) = e--2,,/! < I as T i 00. This is correct. If b .= 0: F(T) = N (-(/ Jf) + N(a~) = l. This is correct. 829 425 Let F(T) = P(X (I) ;? (/1 + h for some t, 0 ::;; t ::;; T) which equals N( -a - f3) + e-2ab N(a - f3) where a = (/ Jf and fi = b/~ so ab = af3; we assume b > 0 and a < 0 in which case 0 ::;; F( T)' I and limT -+00 F(T) = I, limT -+0 F(T) = 0; F is a probability distribution funct illll: lim F(T) = N( -(0) + c 2,,/, N ( - 00) = 0, T-+O lim F(T)=N(+oo)+c 211"N(_00)=l. T-+oo The density function is a a nn = P'(T) = -(-a -/l)N'( - a - f3) +e-211"_(a - f3)N'(a - f3) . aT aT where il( y I IIF 1/ 2 Afi I .. _ = _ __ hl l / 2 ;IF 1 aT :2 --- ## Page 859 830 E. O. Thorp 426 E.O. Thorp The expected time to the goal is Eoo = Tf(T) dT = -- ' T - I / 2 exp dT, 1 00 be- IIh 100 { (a 2T + b2/T) } o ~ 0 2 = x 2b -lIh 00 2 l 2 2 e a 2 ') - 2 T I / 2 1 T - x - - - exp - - x + - x dx dT -= 2x dx - ~ 1 {[ (~) (~) ] } Now I (a 2 b2 ) = jJi e- 2111iJI so 0 , 21al lo( (~r, (~r) = :'Ie-I"hl whence 2b -IIi, ~ l e yJr - llIhl ') Eoo = -- ~-e = - a < 0, b > O. ~ ~Ial lal ' Note: f(T)=F'(T)= -- T - 3/ 2 exp - > 0 be-IIh { (a2T + b2/ T) } ~ 2 for all a, e.g., a < 0, so F(T) is monotone increasing. Hence, since IimT--.oo F(T) = 1 for a < 0 and limr--.oo F(T) < I for a > 0,0 :::;; F(T) :::;; I for all T so we have more --- ## Page 860 The Kelly Criterion in Blackjack Sports Betting, and the Stock Market Ch. 9: Th e Kelly Criterion in Blackjack Sports Belling, and the Stock Market confidence in using the formula for a < 0 too. Check: Eoo(a , b) -I- 0 as -I- -00 yes, Eoo(a, b) t as b t yes, Eoo (a , b) t aslal -I- yes, note lim Eoo(a, b) = +00 as suspected. a~O+ 831 427 This leads us to believe that in a fair coin toss (fair means no drift) and a gambler with finite capital, the expected time to ruin is infinite. This is correct. Feller gives D = z(a - z) as the duration of the game, where z is initial capital, ruin is at 0, and (/ is the goal. Then lim,,->oo D(a) = +00. Note: Eoo = b/lal means the expected time is the same as the point where aT + b crosses X(t) = O. See Figure 2. so Eoo = h/lal , (/ = _ 111/.1'", h = InA, A = C/ Xo = normali/.ed goal. m = p In(1 + fl + £I IIl( I - n == g(f), s2 = pq{ln[(l + fl /( 1 - n1l 2, Kelly fraction f* = p - £f . g(f*) = p In2p + q In2q, Form> 0, Eoo = (InA).I'2/g (f). Now this is the expected time in variance units. However s2 variance units = 1 trial Eoo InA InA n(A , f) == ~ = g(f) = --;;; is the expected number of trials. Chcck: I/(A , f) t as A t , II (A , f) ---+ 00 as A ---+ 00, II (A , f) t as m -I- 0, II(A, f) ---+ 00 as m ---+ O. N(I\\ t:1 I ) it;IS uilique maximum at g(f*) where f* = p - q, the "Kelly fraction", thcn'l, II,' 1111 , I) has a unique minimum for f = f*, Hence f* reaches a fixed goal in leas I ("I"',It'lIIIlIlC in this, the continuous case, so we must be asymptotically close to l e a~1 ( " IIt" !t." 1111lL' ill the discrete case, which this approximates increasing by well in thl' "('II',,' .. I I h,' ( 'I T (Central Limit Theorem) and its special case, the normal approx- ill);11 It III I .. 1111' hlllol11ial distribution, The difference here is the trials are asymmetric. Till' 1"1',111\ " ,lIltllll'galive step sizes are unequal. --- ## Page 861 832 E. 0. Thorp 428 £.0. Thorp References Associates First Capital Corporation, 1998. 1997 Annual Report. Associates First Capital Corporation, Dallas, TX. Black, E, Scholes, M., 1973. The pricing of options and corporate liabilities. 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Vickson (2010). Models of optimal capital accumulation and portfolio selection. 5 --- ## Page 867 This page is intentionally left blank --- ## Page 868 839 Author Index A Aase, K. 5 Abbott, D. 450,453 Adler and Dumas 755, 767 Alais, M. 23 A1chian, M. 274 Algoet and Cover 143,145,157,281, 428,623 Ali, M. 666 Amir, R. 274 Arrow, KJ. 23,119,183,474 Arrow-Pratt 7,249 Artzner, P. 265 Aurell, E.R. 428 B Baesel, 1. 824 Barron, A. 155, 156 Barron and Cover 153 Baumol, W.l. 3,11 Bell and Cover 4, 143, 144, 148,568, 582 Bellman, R. 49,60, 111, 146 Bellman and Kalaba 6, 38, 74, 80, 89 Benter, W. 776, 783 Bernoulli, D. 3, 12,38, 81, 500, 791 Bernoulli, N. 19,578 Bicksler and Thorp 248,463, 543, 555 Bicksler, 1. 89 Bielecki and Pliska 386,387,407 Black,F. 758,767 Black and Perold 308, 309, 374 Blackwell, D. 51,60, 183,209 Blume and Easley 275 Breiman, L. 6,47, 74, 75, 78, 236, 247,281,428,497,480,578, 791 Breitmeyer, C. 266, 359 Brown, G 99 Browne, S. 301, 303, 305, 373, 384, 428,620,623,649,791,795,796, 799,824 Bucklew, 1. 631,643 Buffett, W. 9, 11,510,511,515,566, 769,771,776,8l7 Burkhardt, T. 260 C Campbell, 1. 262 Carino and Ziemba 271,369,756 Chopra, V.K. 249 Chopra and Ziemba 8,9, 145,249, 369,370,563,565, 776,778 Clark, D. 817 Clark, R. 593 Clark and Ziemba 353, 593, 595 Constantinides, G.c. 409,419,425, 754, 767 Cootner, P. 90 Cover and Thomas 4, 144, 566, 623, 642 Cover, T. 181,428,520 Cox and Leland Cox and Miller 818 Culioli,l. 365 D Davis and Lleo 303, 385, 386 Dempster, M.A.H. 331, 427 --- ## Page 869 840 Dempster, Evstigneev and Schenk- Hoppe 427 Dexter, Yu and Ziemba 251, 257 Dixit and Pindyck 757, 767 Dreze, J. 91 Dubins and Savage 48, 60, 80, 316, 319 Durham, S. 246 Dybvig, P. 469 E Emory 511 Epstein, RA. 65,301 Esscher 622, 636, 638 Estes, B.S. 77 Ethier, S. 335, 515,518 Evstigneev, LV. 284 Evstigneev, Hens and Schenk-Hoppe 273,454 Ezroy, D. 758, 767 F Fama, E.F. 629, 680, 682 Farmer and Lo 284 Feller, W. 80, 301 , 338, 791, 792 Ferguson, T.S. 6, 80, 301,331 , 353 Fernholz, E.R 448 Finkelstein, M. 236 Finkelstein and Whitley 145, 209,235, 356 Fisher, I. 91, 587 Freund, RJ. 45.257 Friedman and Savage 23, 44, 539 G Gauss, K.F. 67 Geyer and Ziemba 145, 462, 512,523, 780 Goetzmann, W. 782 Gottlieb, G. 346, 593, 595, 657 Grauer, R 249,591,592,595 Grauer and Hakansson 369, 703, 704 Griffin, P. 803, 808, 824 Grinold and Kahn 373,403, 407 Author Index Gross, B. 373,515,516,657 Grossman and Zhou 360 H Hagstrom 817 Hakansson and Liu 125, 126, 134 Hakansson and Ziemba 281,284,356, 575,577,579,595,622,627,642 Hakansson, N.H. 7,8,80,81,86,89, 91,113,125,133, 414,473, 475- 479,483,484,491,578,586,595 Harville, D.A. 667, 682 Hausch and Ziemba 346,347,351, 463,543,544,567,593, 595,598, 622,658, 659, 664,667, 677, 681, 683,691, 692,694-702,775 Hausch, Lo and Ziemba 695, 783 Hausch, Ziemba and Rubinstein 657, 663 Hecht, R. 799 Hensel and Ziemba 775, 783 Hodges, S.D. 782 Hume, D. 64 J Jorion, P. 26~356 , 359 , 591,596 K Kabanov and Stricker 438 Kallberg and Ziemba 145, 249, 250, 254,257,428,462,563, 592, 597, 672, 677, 684 Karlin and Taylor 266 Kassouf, S. 90,822 Kelly, J. 5, 25,49, 236,281,331,334, 428, 471 , 578,792 Keynes, J.M. 567 Kilpatrick, A. 817 Kritzman, M. 812 Kitamura, Y. 638, 645 L Latane, H.A. 5, 8,35,82,461,471, 481, 484, 578 --- ## Page 870 Author Index Lawley and Maxwell 263 Leib, J. 512,802 Leland, H.E. 113,586,597, 782 Leland and Mossin 8 Lensberg, T. 282 Levhari and Srinivasan 470 Levy and Markowitz 593 Li, Y. 590, 597 Lintner, J. 9, 125, 134 Lowenstein, R 817 Luenberger, D. 428,448,460, 575, 599 Lv and Meister 146,285 M MacLean and Weldon 263 MacLean and Ziemba 260,271,566, 578,597,641,648 MacLean, Sanegre, Zhao and Ziemba 302,356,741,750 MacLean, Thorp and Ziemba 463, 563,576,741,750 MacLean, Thorp, Zhao and Ziemba 463, 543,564 MacLean, Zhao and Ziemba 303, 453, 454 MacLean, Ziemba and Blazenko 8, 283301,331,428,512,566,641, 643,648 MacLean, Ziemba and Li 145,259, 519,566 Markowitz, H.M. 7,8,44,78,82,85, 125,143,249,257,461,481,484, 495,511,539,583,589,592,593, 597,791,823,494,516,703,811 Markowitz and Van Dijk 516 Marschak, J. 23,44,91 Marshall, A. 44, 278 McEnally, RW. 791 Menger, K. 3-5, 11,22,23 Merton, R.C. 262,309,310, 315, 326, 332,356,373,374,376,385,386, 392,414,426,480,481,590,597, 753, 767 841 Merton and Samuelson 209,455, 500 Mielke, A. 454 Milnor and Shapley 274 Modigliani and Brumberg 92 Montmart, P. 3, 19, 21 Mordin, N. 698, 702 Morgenstern, O. 89,90,222,520 Mossin and Samuelson 83 Mossin, J. 90, 113,209,525,586,597 Mulvey, J.M. 591, 597, 735 Mulvey, Bilgili and Vural 660, 735 N Nassim, N. 512 Neave,E. 473,474,484,663 o Ordentlich and Cover 144, 145,211 P Pabrai, M. 509, 523 Pearson, K. 64 Phelps, E. 91,96, 111,455,468,473, 484 Platen, E. 303,409,410 Platen and Heath 409-412, 417, 418, 420,422,423,426 Pliska,S. 309 Poincare, H. 64 Poundstone, W. 509,518,523,543, 547,657 Proebsting, T. 521-523 Pye,G. 478,484,664 Q Quaife, A. 811,817,820,821,824 R Radner, R 273, 284, 436 Rabin, M. 648 Ramsey, F.P. 455, 473, 484 Reid, R 824 Richard, S. 755, 768 Rockafeller and Uryasev 260,271 --- ## Page 871 842 Rockafeller and Ziemba 271 Roll, R. 7,8-10,125,133, 143,4l7, 481,484,598 Ross,S.A. 409,426, 586,587,598 Rotando, L.M. 598, 795 Roy, A.D. 45, 125 Rubinstein, M. 356, 372, 680 Rudolf and Ziemba 303,455, 660, 753 S Samuelson, P.A. 5,23,82,83, 140, 148,209,453,455,460,465,466, 473-475, 484,487,488,491-493, 496,518,520,525,578,598,648, 753 Santa Fe Institute 274 Savage, L. J. 42-44 Schenk-Hoppe, K. 427 Schlesinger, D. 818,824 Shannon,C. 5,25,38, 39,428,509, 791 Shapley, L. 274 Sharpe and Tint 756, 768 Sharpe, W.P. 9, 125, l34, 373, 374, 428, 756 Shay,B. 448 Shubik and Whitt 274 Siegel, J. 555 Siegel, L. 407,781 Simons, J. 515,785, 787 Snyder, A. 804 Sommer, L. 3, 11 Soros, G. 769, 771 Spurgin, R.B. 782, 784 Stigler, G. 43 Stutzer, M. 575, 619,627,641, 648 Swensen, D. 736 T Taleb, N. 512, 523 Author Index Thorp and Kassouf 86, 795 Thorp and Whitley 248,462, 464,525, 519,539 Thorp, E.O. 6-8,61, 71, 81, 83, 143, 281,331,347,385,426,428,461, 463,480,481,484,509,521-523, 576,598,623,641,657,661,662, 789,791,800,811 Tucker, H. 796 V von Neumann J. 520 von Neumann and Morgenstern 40, III W Wald, A. 51 Walden, L. 795 Weitzman, M. 666, 680 Whitley, R. 236 Williams, J.B. 147, 152,471,575,578 Wong, S. 795, 803,804 y Yamashita, W. 804 Young and Trent 496 Z Zabel, E. 473 Zenios and Ziemba 738 Ziemba and Hausch 547,590,777 Ziemba and Mulvey 428,589,591, 738, 754, 768 Ziemba and Vickson 249,257, 473, 475,476, 485, 509, 511, 512, 583, 598,671, 675, 683,685 Ziemba, R. 511 Ziemba, W.T. 259, 351,385,431, 455, 462, 485,481, 511, 565, 594,661, 769, 785 Ziemba and Ziemba 566, 785 --- ## Page 872 843 Subject Index A Acceptable set 360, 265 Action 824 Active portfolio management 373, 427 Actuarial pricing formula 419 Admissible strategy 277, 517 AEP 160 Alpha-omega targets 403 Alternate wealth sequences 500 American Express 511 Analysis of variance 128 Approximate form of transitivity 83 Arbitrage 809 Arbitrage and growth 429 Arbitrarily small bets are allowed 521 Arithmetic mean 40 Arithmetic mean rate return 569 Arithmetic MV 7 Asset market equilibrium 277 Asset market with stationary returns 435,442 Associates First Capital Corporation 814 Asymptotic equipartition property 171 Asymptotic growth 8, 108 Magnitude 54, 236 Time 50 Asymptotic optimality principle 168, 499, 794 Asymptotically best 48 AT&T 809 Augmented filtration 358 A vailable strategies 35 Avalanche Effect 128 B Baccarat 63 Backward elimination 364 Balanced growth path 279,434 Balancing of measures 333 Bankruptcy 47,556 Basic investment problem 332 strategies 62, 276 Basic validity test 128 Bayes estimation 262, 324 empirical Bayes 262 Theorem 262, 369 Beat the dealer 509,515,516,791 Behavior of capital 108 Bellman backward induction 84,476 Benchmark 303, 373-379, 382, 384, 386 management 385 approach 409 dynamic 395 with alpha target 400 mutual fund theorem 397 Berkshire Hathaway 771,817,820 Bernoulli trials 5, 71, 72, 83 Bernoulli's logarithmitic function 470 Best portfolio in hindsight 211 Betting system in England 695 Betting your beliefs 281 Binary channel 26 Binary digital option 795 Binomial lattice price 226 Biotime 819 Black Swan 278,446,461 Blackjack 61, 344,461,515, 801,803, 816 --- ## Page 873 844 Blackjack forum 512,802 Black-Scholes 229 Bold play 316,319,323 Bookmakers 697 Borrowing 106 and lending 91 and solvency 115 limits 117 Boundedness 98 Breakage 690,691 Broker's call 814 Brownian motion 261,263 with drift 817 Buffett thinks like a Kelly investor 510 lunch 77 Businessman's risk 471 Buy low - sell high 438,477 Buy-and-hold strategy 428 C Calculus of variations 466 Capital 521 Accumulation 259 Asset pricing model 8, 570 Growth 92, 331 Growth Theory 577 Market equilibrium 125 Position 105 Casestudy 819 Cash equivalent value 251,255,563 Casino betting 817 Cauchy equations 96 Central limit theorem 59, 109,488 Certain return 56 Chance events 28 Chemin de Fer 63 Classical gambler 33 Co-existence of portfolio rules 282 Coin tossing 792 Commodity futures 795 trading 593 Common factors 261 Communication channel 25 theory 25 Subject Index Competitive optimality 147,463,567 Complete ruin 125 Completely mixed strategy 435 Compound growth 516 mean-variance efficient frontier 516,519 Computation of performance measures 336 Concavity 16 Conditional expected log return 158 Conditional Value-at-Risk, CVaR 260, 302 Conservative lower bound estimate 512 Constant proportions 316, 326, 374, 376 benchmark 398 portfolio 305 portfolio insurance (CPPI) 301, 305,310,325,326,331 strategy 428,435,443 Constant rebalanced portfolio 181 Constant returns to scale 8 Consumption hypothesis 92 Contingent claim 419 Continuous approximation 800,810 Continuous time portfolio theory 704 Continuous-time Kelly criterion 324, 325 Convertible hedge 86 security 67 Convex combinations 335 cost function 568 risk measure 369 shortfall penalty 303 Correction fund 391,397 Cost function 26 Coupled entries 687, 688 Cramer's Theorem (Bucklew) 629 Crash of October, 1987 817 Cross-sectional regression 135 Cross-sectional spread 136 Cumulative effects 37 Currency hedge portfolio 764 Currency markets 439 --- ## Page 874 Subject Index D Daily double bets 666 Darwinian theory of portfolio selection 274 Death constant 77, 89 Decay rate maximizing portfolios 627, 636 Derivative of expected growth 413 Derivative security 226 Description length 156 Difference equation 94 Dirichlet problem 309,311,313,319, 322,377,378,380,381 Discreteness of money 71 Discrimination statistic 645 Disjunctive problem 361 Distribution of returns 107 Distribution-free 212 Diversification 42, 721 theorem 423 portfolio 127,423 Doubling before halving 333 Doubling rate 153 for side information 153 Downside focus 265 Downside risk control 358 measure 260, 265, 356, 358 Sharpe ratio 782 Drawdown 359,376,511 Dynamic estimation 261 investment process 261 reallocation 812 Dynamic programming 49, 84,476, 496 Dynamic stochastic programming 465 E E.L. Bruce short squeeze 795 Economic decision making 39 Edge/ Odds 5 Effective tradeoff 302, 335 Efficient frontier 334, 516 Efficient markets 665,693 Efficient portfolio 42, 44 845 Equilibrium conditions 134 Equity market returns 555 Equivalence classes of functions 539 Equivalent strategy 247 Ergodic 157 Mode 174 Errors in estimates 249,261 in means 255 in variances/covariances 250, 255 Esscher transformation 622, 636, 638 Essentially different strategy 6,81, 461,569,794,512 Evolutionary asset market model 275 dynamics 274, 278 finance 273 stability 281, 282 Excess risk adjusted returns 721 Expected average compound growth 489 growth rate of capital 123 loss 16 return on place bet 685 return ratio 139 time to goal 6,259,303,567, 800,827 utility 36, 501 utility counterexample 569 utility maximizer 487,496 value 27 wealth ratio 268, 568 Exponential dominance 209 gain 32 growth rate 27,73,77,236,433 440 Extinction 280 Extremal strategy 219 F Factors 388 False corollary 82 Favorable investments 260,332 Favorable outcome 17 Favorableness 106 --- ## Page 875 846 Filtration 260 Financial engineering 438 Financial intermediation 361 Financial value of information 153 Finite goals 56 Finite yield assumption 120 First passage time 265, 333 First-order conditions 127 Fixed fraction strategy 48, 235, 238, 278,333,792 Fixed rebalance times 264 Fixed-mix investment strategies 282, 427,438 in currency markets 440 Fixed-weight portfolios 710 Flexibility condition 342 Flexible process 341 Ford Foundation 769, 771 Foreign currencies 761, 762 Fortune's formula 461,509,518,543 Four fund separation theorem 758, 764 Fractional Kelly strategies 283, 269, 301,331,335,390,397,517,565, 593,815,821 versus Kelly 813 Fractionalobjective 374 Fund separation theorem 398, 399 Fundamental problem 267, 365 Funding ratio 760, 763, 764 G Gambler 26 Gambler's fallacy 442 ruin 71,792 capital 27 liability 237 Games absolutely fair 16 fair 42,418 favourable 6,47,240 dynamic 276, 278 identically distributed independent 801 asset market 280, 283 Subject Index alternate sequence 497 competitive investment 150 of survival 274 repeated 91 sequence 181 stochastic dynamic 264 two-person zero-sum 61,807 Gamma function 224 Gartner-Ellis Theorem 621, 643 GEICO 511 Generalized concavity 683 Generic programming 283 Geometric Brownian motion 305,309, 324,327,377,379,380,382 random walk 443 Wiener process 213 Geometric mean 38,40,358,471 Geometric mean fallacy 487 Geometric MV 7 Goal utility 48 Good and bad properties of Kelly 459, 563,566,568 Growth - security 8,331,355,578,589 profile 333 strategy 265, 268 tradeoffs 592 monotonicity 302 Growth and decay 555 Growth condition 263 Growth in stationary markets 428,429, 431 Growth measures 301,331 Growth optimality 413 optimal portfolio 414 model 4,125,130,265 Growth rate 157,436 of balanced strategies 436 of constant proportions strategies 444 of wealth 435 Growth under transaction costs 450 Growth-optimum theory 126 strategy 324 never risks ruin 582 --- ## Page 876 Subject Index H Hamilton-Jacobi-Bellman Partial Differential Equation (HJB PDE) 390 Harvard endowment 769,771,781 Hedge portfolio 763 Hedging 240 High probability of low prize combinations 666 Hindsight allocation option 211, 226 Holding time 267 Horseracing 346 Hotellings T**2 128 I Ibbotson Associates 814, 821 Identical risks 38 Income stream 7 Incomplete mean 360 Incompleteness of Markowitz theory 79 Increased security 568 Incremental Kelly Criterion 521 Independent trials 40 Index fund 303 Individual as a piece of capital equipment 89 Individual sequence 213 Individual's impatience 105 Infinite horizon investment strategy 216 Infinitely divisable capital 237, 521 Infinitely wealthy adversary 794 Information rate 25 Innovations process 263 Inside information 34 Insurance explained 89 Insurance of commodities 17 Interior maximum 466 Intermediate strategy 792 Intertemporal hedging portfolio 303 hedging term 391 surplus management 753 In-the-money 664, 687 847 Intuition on volatility 429 Inverse cumulative distribution 550 Inversely proportional 41 Investment for long run 494 Investment opportunities 95 Investment strategy 276 Investor's borrowing rate 814 risk tolerance 249 Iroquois 198 Ito process 753 J James-Stein estimate 263 Jensen's inequality 247,437,445 Joint distribution functions 120 K Kaufman and Broad common stock and warrants 87 Kelly bettors: Keynes and Buffett 775 Kelly capital growth criterion 5, 324, 356,563 misunderstandings 82 portfolio 391, 397, 511, 820 strategy 266,280,281,543 strategy gets ahead 281 Kelly fraction 331 Kelly model with transactions costs 671,681 Kelly-Breiman criterion 543 Kentucky Derby 239, 266, 593, 800 Keynes geometric mean return 686 Kin Ark Corp. 691,776 King's College Chest Fund 777 Kullback Leibler information number 153, 198 Kullback's Lemma 778 L Lagrangian 153 Large deviations 636, 645 Large drawdowns 363 Large fixed goal 619 Large risks 641 --- ## Page 877 848 Large stock index 520 Largest dollar value block rate 515 Las Vegas 40 Latent factors 351, 809 Law of large numbers 34, 38, 260, 263,515 Law of the minimal price 47 Leib's Paradox 236 Lending and borrowing rates 580 Lengendre-Fenchel transform 418 Level of consumption 512 Leverage 106, 107, 629 Leverage case 704 Levered and unlevered strategies 711, 720, 817 Liability hedge portfolio 268, 703 Lifetime sequence of bets 551 Limited liability 758 Linear risk tolerance 763 Little Caesar's 77 Loading matrix 490 Logarithm 3, 8, 126, 148 Logarithmitic approximation 148 curve 110, 122, 134 investor 15 Lognormal 41, 517 diffusion process 134 wealth 236 Log-optimum investment 331,357, 549 Long run 157 growth rate 125, 428,512, 824 revisited 5 long run 706 Long-lived assets 332 Lotto games 415, 812 Low probability of high prize combinations 281 Lucrativity 347 M Margin 593 of safety 666 requirements 44 Subject Index Market at the track 799 Market diversification 820 portfolio 43, 131, 663 return 758, 763 Markov process 812 Markowitz efficient frontier 9 Martingale 119, 127,811 Theorem 51 Matched strategies 245 Matching heuristic 246 Maximize expected utility 266, 267 Maximum chance 125 chance analysis 792 expected log (MEL) 36 growth rate 39 security 55 payout 301 value 43 Max-min ratio 494, 811 Mean accumulated wealth 27 growth rate 215 log wealth 332 reversion 109 Mean squared error 492 Mean value theorem 491 Mean-risk problem 438 Mean-variance 262 analysis 244 approximations 260 efficiency 260, 461 , 592, model 7, 125 optimality 517 Measure of value 125 Measurement of risk 704 Median log wealth 516 MEL 36 Merton model 11, 502, 567 Minimal time 392 Minimize probability of eventual ruin 753, 755 time to a goal 47 Minimum variance 792 hedge ratio 71 Min-max criterion 325 --- ## Page 878 Subject Index Mix of risky investments 44 Mixed strategy 72, 760 Modem portfolio theory 106 Moral expectation 222 Morningstar 704 Motley fool 22 Multiperiod investment theory 89 Multiperiod model of asset allocation 509 Multiple betting situations 732 Multi-stage stochastic programming models 591 Mutual fund 589 theorem 516, 689 Mutual information 390, 396 Mutually optimal to trade 361 Myopic policy 5,8,89, 153 N Nevada roulette wheels 261 Newtonian method 822 No easy money condition 65, 228, 460,519,582 No-arbitrage price 7, 579 No-leverage case 228 Nonanticipating investment strategy 91,93,211,218,710 N ondecline of capital 218 Non-degenerate asset prices 158 (?) Nonlattice random variables 108 Nonlinear programming optimization 436 Nonmyopic nature 444 Nonstationary 53 Nonterminating strategy 512 Non-transitive dice 115 (?) Nonuniform strategy 110 Normally distributed 54 Normative 83 Numeraire portfolio 109, 410,412 o Occam's Razor 82 Odds 303,520,649 849 Off-the-run 80 year Treasury bonds 50 On the Accuracy of Economic Observations 236 Opportunity costs 511 Opportunity set 520 Optimal capital accumulation 110 Optimal consumption function 510 Optimal gambling system 704 Optimal growth 473,525 portfolio 57 ratio 510 policies 47,105,305,376 Optimal investment 305 Optimal investment fractions 325 Optimal place bets 376 Optimal portfolio fractions 91 policy 127 strategies 684, 467 Optimality condition 381 Options 374 Overbetting 382 Over-levered strategies 468 Oversized bets 86 p Pacific Investment Mangement Company PIMCO 547 Parimutuel betting market 560 Partial myopia 568,516 Path independence 279 Payoff function 117, 362 to show 333 to place 222 matrices 222 Perfect liquidity 664 Performance measures 664 record 35 weighted 114,338 Permanent income hypothesis 86 Perron-Frobenius theorem 182 Personal savings 182 PIMCO 105 Place and show betting system return 441 --- ## Page 879 850 inefficiencies 91 pools 515,692 system 664,670,681 Place payoffs in England 695 formulas 699 bookmakers payoffs 697 number of horses that place 696 optimal bets 701 pools 695 probability of placing 699-701 rates of return on different bets 697 track take 698 Planning portfolio models 677 Portfolio comparisons 263 insurance 147 management 711 rebalancing 310 risk 383 rule 37 selection 369 weights 133 with many securities 276 Positive linear transformation 83, 85, 276 Positive probability of total loss 4 Power series approximation 89 Predictive 547 Present value 85 Price 387 of universality 82, 104 process with trend 12 relative 212, 442 Principle of optimality 181,814 Probabilistic uncertainty 181 Probability 114 Probability assessment approach 94, 114 beliefs of analysis 36 of exceedence 707 of ruin 732 of shortfall 84 conditional and joint 9 Subject Index Process control methodology 46, 373 Productive investments 28 Proebstings paradox 265,268 Professional blackjack teams 93 Profit and safety 462 Profit expectation 521 Propensity to consume 593 Proportional strategies 42 Q Quantity of goods 547 R Race track utility function 821 Racetrack as a stock market 134 Ramsey model 3 Ramsey-Phelps problem 666 Random rebalance times 466 Randomization in the competitive investment game 277, 469 Rate of capital growth 148, 264 of return distribution 567 of transmission 48, 240, 358 profile 25 of return on English bets 28 Rational choice 30, 338, 697 Real world pricing formula 45 Rebalancing 41 point 35 strategies 419 Recursive optimality equation 261 Reinvestment problem 446, 451,468 Relation to the Markowitz theory 42 Relative dividends 705 Relative entropy 85, 105, 153, 279 Relative impact of estimation errors 622 Renaissance Medallion Fund 515 net returns 249 Repeated reinvestment 119 Results for a real institutional portfolio 786 Returns to scale 235 Risk measures 86 --- ## Page 880 Subject Index Risk neutral pricing 114 Risk preference 260, 359 premia 409 sensitive asset management 44 sensitivity 139 tolerance 303,386,471 Risk-aversion 237,250 Arrow-Pratt risk aversion 7, 8, 266,463,563 absolute risk aversion index 7, 96 decreasing absolute risk aversion 474 relative risk aversion 721 relative risk aversion index 645, 646 Riskless asset 260 Risky propositions 461 Risky ventures 45 RMF - wealth over time 758 Robust 14,35 Roulette 787 Roulette wheel tilted 181 Ruin almost sure 181 s S & P 500 64, 65, 792 Safety first 817 Safety preference rates 813 Safety zone investors 820 Salad oil scandal 45 Sample covariances 301 Samuelson's objections 511 Sandwich argument 82, 129, 157, 160, 171,801 Sawdust joint 178 Scale function 178 Scenario selection 157 elimination 807 Secured annual drawdown 317, 364 Security 365 measures 302, 367 Self-financing strategy 260, 301 Selling short 331 Selling warrants short 427 Sensitivity matrix 86,434 Separation theorem 68 851 Serial correlation of yields 126, 496 Shannon entropy 113, 119, 211, 504 Shannon-McMillan-Breiman theorem 211,212 Sharpe ratio 212 Sharpe-Lintner equilibrium 160 Sharpe-Lintner model 160 Short run 133, 811 Short sales 130, 151 Shortfall 93,114,151 Short-lived assets 373 Siemens Austria pension fund 379 Silky Sullivan 382 Simple strategy 282 Simulations 780 Singularity 667 Size of total investment 282 Slope coefficient bias 543 Small stock index 107 Small stocks 136 Solution approximation 43 Solvency 351 constraint 703 Source rate 242 Sports betting 117, 118 Sports book 30 St Petersburg paradox 3-5, 19,21,77, 151, 488,801,805, 816 generalized 82 modified by Kelly 366 Super 260 Standard Brownian motion 245 State-dependent 469 Static or "desert island" strategy 791 Stationary asset market 796 asset prices 122 asset returns 69 Statistical transducer 433 Statistically independent 436 Steward Enterprises 443 Stochastic control theory 26, 122 Stochastic dominance 303,481,509 --- ## Page 881 852 Stochastic programming 268 problem 560 Stochastically risky alternative 274 Stock market 462, 467 Stopping rule profile 466 Stopping rules 75 Strictly concave 801 Strong arbitrage 341 Structural model 260 Subgoal 96 Subjective utility 262,421 Subsistence wealth 36 Subtangent 36 Successive approximations 40 Superior properties are asymptotic 15 Supermartingale 99, 159,512,568 Surplus management 159 Surplus optimizers 245 Survival strategy 412, 754, 764 Symmetric downside risk Sharpe ratio 769, 785 System 769 Systematic outperformance 785 T Tail estimates 788 Tampering 278 Target growth rate 417 Taxes 59, 268 T-bill 788,814 interest rate 619 total return 84 Telescope 812,814 Temporal dependence 820 Temporally averaged means 185 Temporaryequilibrium 185 Theory of Games and Economic Behavior 512 Theory of Interest 65 Three mutual fund theorems 520 Tilted wheel 643 Time dependence 91 Timid betting 65, 110,303 Track take 31, 316, 323, 666, 792 Subject Index Tradeoff 550, 690 index 692 theorem 698 Trajectories 335 Transactions costs 342 Transistor timing device 301 Trapping state 549 Turn of the year effect 84 Twenty-one 66 Two independent favorable coins 109 Two parameter models 351 U Unbounded growth 126 U nderperformance probabilities 147 Unending games 439 Uneven payoff games 643 Uniform returns 646 Universal data compression 500, 795 Universal portfolio 215,551 Unrestricted borrowing 215 Upper and lower goals 181 Utility 268, 519 bounded 489 concave 525, 526 CRRA 264 equivalent 181,820 exponential 319 everyman's 43 HARA 387 induced short run 114 induced 705 inequivalent 82, 529 inversely proportionate 13 isoelastic 705 logarithmic 16, 33, 113,263, 720,807 marginal 84 negative power 89, 113,356 of an infinite capital sequence 526 one period 64 normative 489 quadratic 106 --- ## Page 882 Subject Index V power 113, 118 slope of 242 terminal 126 theories 12 Value 539 Value line index 12,82, 184,480 Value-at-risk (VaR) 184, 186 constraint 260 control 302 Variance errors 358 Volatility 253,791 Volatility and constancy of asset proportions 452 Volatility as energy 447 Volatility induced growth 304,450 Volatility pumping 447 Volatility-growth relationship 232, 366,444,449 W Wald's formula 446,450 Wall street 429 Warrant hedge 51, 67, 809 Warrants 48, 68 Weak law oflarge numbers 86 Weakly-efficient market 77 Wealth dynamics of constant proportions strategies 509, 665 Wealth goals 428,445,494 losses 259 path 264 profile 260 Wheel of fortune 302, 340, 460 Win bets 66 Win pool efficiency 65, 533 Withdrawals 664, 667 Worst sequence 37,667 853 --- ## Page 883 World Scientific Handbook in Financial Economic Series - Vol. 3 ---------- THE --------------------------------------------- KELLY CAPITAL GROWTH INVESTMENT CRITERION "Utility functions and marginal utility, a gift from mathematics to economics, helped to decisively shape economic thinking over the past centuries. Logarithmic utility played a central role from the beginning. Now often named after the seminal work of John Kelly Jr. in 7956, it maximizes the geometric growth rote of a portfolio. This handbook presents the classical papers and highlights from modem research, along with limitations on the use of the Kelly criterion .• Walter Schachermayer Fakultiit fur Mathematik, Universitiit Wien 7he tragicolly short-lived genius John Kelly Jr. is best remembered for one of the most original and far-reaching ideas in modem finance. The Kelly criterion con be described as a gambling "system" that really works, in that it achieves the maximum long-term return from a favorable speculation. Kelly's idea has long had a cult following among people ranging from hedge fund managers to blackjack players. Far those who have heard of the Kelly mythos and want to explore the science behind it, this book will be an instant classic. The editors have collected all the pivotal original papers, spanning centuries and the rarely bridged gulf between theory and practice. This book is indispensable for anyone interested in Kelly's legacy. " William Poundstone The author of "Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street" "Edward O. Thorp and the Kelly criterion have been a lighthouse for risk management for me and PIMCO for over 45 years. First at the blackjack tables, and then in portfolio management, the Kelly system has helped to minimize risk and maximize return for thousands of PIMCO clients. " William H. Gross, Managing Director, PIMCO About the Book This volume provides the definitive treatment of fortune's formula or the Kelly capital growth criterion as it is often called. The strategy is to maximize long run wealth of the investor by maximizing the period by period expected utility of wealth with a logarithmic utility function. Mathematical theorems show that only the log utility function maximizes asymptotic long run wealth and minimizes the expected time to arbitrary large goals. In general, the strategy is risky in the short term but as the number of bets increase, the Kelly bettor's wealth tends to be much larger than those with essentially different strategies. So most of the time, the Kelly bettor will have much more wealth than these other bettors but the Kelly strategy can lead to considerable losses a small percent of the time. There are ways to reduce this risk at the cost of lower expected final wealth using fractional Kelly strategies that blend the Kelly suggested wager with cash. The various classic reprinted papers and the new ones written specifically for this volume cover various aspects of the theory and practice of dynamic investing. Good and bad properties are discussed, as are fixed-mix and volatility induced growth strategies. The relationships with utility theory and the use of these ideas by great investors are featured. World Scientific www.worldscientific.com 7598 he ISBN-13 978-981 -4293-49-5 9 J]Ji!~{~JiUII