Handbook of the Fundamentals of Financial Decision Making - Part 1 9789814417372

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HANDBOOK OF

THE FUNDAMENTALS OF

FINANCIAL

DECISION MAKING Part I

World Scientific Handbook in Financial Economic Series (ISSN: 2010-1732) Series Editor:

William T. Ziemba University of British Columbia, Canada (Emeritus); ICMA Centre, University of Reading, UK; Visiting Professor at The Korean Advanced Institute for Science and Technology and Sabanci University, Istanbul

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

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 of a 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 O. Thorp & William T. Ziemba

Vol. 4

Handbook of the Fundamentals of Financial Decision Making (In 2 Parts) edited by Leonard C. MacLean & William T. Ziemba

Forthcoming The World Scientific Handbook of Insurance (To be announced) The WSPC Handbook of Futures Markets edited by A. G. Malliaris & William T. Ziemba

World Scientific Handbook in Financial Economic Series — Vol. 4

HANDBOOK OF

THE FUNDAMENTALS OF

FINANCIAL

DECISION MAKING Part I

Editors

Leonard C MacLean Dalhousie University, Canada (Emeritus)

William T Ziemba University of British Columbia, Canada (Emeritus);  ICMA Centre; University of Reading, UK; Visiting Professor at The Korean Advanced Institute for Science and Technology and Sabanci University, Istanbul

World Scientiic NEW JERSEY

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LONDON

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BEIJING

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SHANGHAI

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TA I P E I

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CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Handbook of the fundamentals of financial decision making (in 2 Parts) / edited by Leonard C. MacLean and William T. Ziemba. v. cm. -- (World scientific handbook in financial economic series, ISSN 2010-1732 ; v. 4) Includes bibliographical references and index. ISBN 978-9814417341 (Set) ISBN 978-9814417372 (Part I) ISBN 978-9814417389 (Part II) 1. Investments--Decision making. 2. Finance--Decision making. 3. Risk management. 4. Uncertainty I. MacLean, L. C. (Leonard C.) II. Ziemba, W. T. HG4515.H364 2013 332--dc23 2012037307

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright © 2013 by World Scientific Publishing 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 photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

In-house Editor: Alisha Nguyen

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Printed in Singapore.

This part is dedicated to the memory of Professor Paul Anthony Samuelson, arguably the most influential economist of the 20th century and a major contributor to the topics in this handbook. Paul was born in Gary Indiana on May 15, 1915 and died in Belmont, Massachusetts on December 13, 2009. He was Institute Professor of Economics and Gordon Y Billard Fellow at the Massachusetts Institute of Technology.

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Preface

The theory and practice of decision making in finance has a long and storied history. Many of the fundamental concepts in economics as well as the models in stochastic optimization have their origins in finance. In this handbook, a selection of the key papers in financial decision making are brought together to provide a comprehensive picture of the components and methods. The handbook is composed of two parts. Part I consists of a collection of reprints of seminal papers on the foundations of decision making in finance. Part II contains mostly original papers which explore aspects of decision models. A special emphasis is placed on models which optimize capital growth. In Section A of Part I, the concept of arbitrage is presented in a set of papers. Arbitrage is an opportunity for risk free gains, and an absence of arbitrage is a condition for a stable financial market. The seminal paper is from Ross (1976), where the Arbitrage Pricing Theory (APT) is introduced. He defines the prices on assets by a linear relation to common factors (latent random variables) each with expectation zero. After accounting for risk there is no price premium, that is, arbitrage does not exist. Another way of stating this condition is that after adjusting for risk, the price process is a martingale. The Fundamental Theorem of Asset Pricing states that “no arbitrage” is equivalent to the existence of an equivalent “martingale measure.” The papers by Schachermayer (2010a, 2010b) put the fundamental theorem in a general setting. There exists a martingale measure if and only if the price process satisfies a no free lunch with vanishing risk condition. The martingale measure is a risk-neutral measure and all assets have the same expected value (the risk-free rate) under the risk-neutral measure. Kallio and Ziemba (2007) use Tucker’s theorems of the alternative from mathematical programming to establish the existence of risk neutral probabilities for the discrete time and discrete space price process. The equivalence between no arbitrage and the existence of a martingale measure is extended to markets with various imperfections. The concept of utility as an expression of preference is developed in Section B. Fishburn’s paper (1969) gives a succinct presentation of expected utility theory. With a set of assumptions over decision maker preferences, Fishburn establishes the existence of a utility function and subjective probability distribution such that rational individuals act as though they were maximizing expected utility. Expected utility allows for the fact that many individuals are risk averse, meaning that the individual would refuse a fair gamble (a fair gamble has an expected value of zero). vii

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In expected utility theory, the utilities of outcomes are weighted by their probabilities. Machina (2004) in his paper describes non expected utility. It has been shown that people overweight outcomes that are considered certain relative to outcomes which are merely probable. There have been a variety of proposals for dealing with the violation of the independence axiom and the linearity in probabilities. One approach is Prospect Theory proposed by Kahneman and Tversky (1979). The value function is S-shaped, being convex for losses (x < 0) and concave for gains (x > 0). Prospect Theory has its critics. The papers by Levy and Levy (2004) cast doubt on the S-shaped value function, based on experimental results. The basis of their analysis is stochastic dominance which orders random variables. The paper by Wakker (2003) reinforces the importance of probability weighting as well as the S-shaped value in prospect theory. He re-examines the experiments, showing that the S-shape is compatible with the data when probability weighting is used correctly. Baltussen et al. (2006) provide experimental results supporting the usual risk averse (concave) function over either the S-shape or reverse S-shape. In financial decision making, the (investment, consumption) process is dynamic, with the alternatives consisting of investment and consumption decisions at points in time. Kreps and Porteus (1979) consider the inter-temporal consistency of utility for making dynamic decisions. Preferences at different times are linked by a temporal consistency axiom, so that decisions are consistent in the sense that revealed outcomes at a later time would not invalidate the optimality of earlier decisions. The paper by Epstein and Zin (1989) builds on the temporal utility form to define a recursive utility which incorporates inter-temporal substitution and risk aversion separately. The Epstein–Zin utility contains many popular utility functions such as power utility as special cases. In Section C, the preference for random variables with the order relations of stochastic dominance are considered. Hanoch and Levy (1969) show that stochastic dominance can be defined by classes of utilities as characterized by higher order derivatives of the utility, with u(k) being the k th derivative. The dominance described is for a single period, usually the final period of accumulated capital. However, the capital is accumulated from investments over time and the trajectory to final wealth could be important in the assessment of utility. Levy and Paroush (1974) extend the notions of stochastic dominance to the multi-period case. With additive utility the stochastic dominance result is extended to multiple periods. Levy and Paroush also consider the geometric process where the final return is the product of period returns and give necessary conditions for first order dominance. Efficient investment strategies, where risk is minimized for specified return, are the topic of Section D. A major advantage of Markowitz’ mean-variance analysis is the relative ease of computing optimal strategies and as such it is a practical technique. The paper by Ziemba, Parkan and Brooks-Hill (1974) considers the general risk-return investment problem for which they propose a 2-step approach: the first step is to find the proportions invested in risky assets using a fractional program; the second step is to determine the optimal ratios of risky to non-risky assets.

Preface

ix

Ziemba (1975) considers the computation of optimal portfolios when returns have a symmetric stable distribution. The normal is a stable distribution when the variance is finite, but the stable family is more general with dispersion replacing variance and has four rather than two parameters. The typical utility function in the investment models is concave to reflect risk aversion. There are aspects of risk aversion which are not captured by concavity. In the paper by Pratt (1964), an additional property of utility, decreasing absolute risk aversion, is introduced. Rubinstein (1973) develops a measure of global risk aversion in the context of a parameter-preference equilibrium relationship. Some properties of this measure in the context of risk aversion with changing initial wealth levels appear in Kallberg and Ziemba (1983). Kallberg and Ziemba establish an important property of the global risk measure: Investors with the same risk have the same optimal portfolios. Chopra and Ziemba (1993) consider the relative impact of estimation errors at various degrees of risk aversion on portfolio performance, with the estimate of mean return being most important. MacLean, Foster and Ziemba (2007) use a Bayesian framework to include the covariance in an estimate of the mean. In essence, the return on one asset provides information about the return on related assets, and the sharing of information through the covariance improves the quality of estimates. Expected utility may not be the right theory for many risk attitudes, and does not explain the modest-scale risk aversion observed in practice. Rabin and Thaler (2001) in their paper propose that the right explanation incorporates the concepts of loss aversion and mental accounting. They argue that loss aversion and the tendency to isolate each risky choice must both be components of a theory of risk attitudes.

Part II The basic components of risk are (i) the chance of a potential loss and (ii) the size of the potential loss. F¨ ollmer and Knipsel (2012) view the financial risk of X as the capital requirement ρ(X) to make the position X acceptable, with the corresponding acceptance set of positions for measure ρ. With reference to acceptance sets, Rockafeller and Ziemba (2000) have established the following result: There is a one-to-one correspondence between acceptance sets Aρ and the risk measures ρ. The concept of capital requirement to cover the losses from investment captures the financial risk idea, but the probability of loss is not taken into account. Details on measures which use the distribution are provided in the papers of Krokhmal et al. (2011) and F¨ ollmer and Knipsel (2012). Deviation measures are a generalization of variance. There is a one-to-one relationship between averse risk measures and deviation risk measures. A variation is to consider the average of the values at risk (AVaR). F¨ollmer and Knipsel show how AVaR is a building block for law-invariant risk measures. Since convexity is the desired property of a risk measure, the class of convex risk measures is considered in F¨ollmer and Knispel (2012), following Rockafeller and Ziemba (2000). F¨ ollmer and Knispel also raise the issue of model uncertainty

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or model ambiguity. They discuss a robustification where the probability measure P is a member of a class ℘. Markowitz’s mean-variance analysis approach is generalized in Krokhmal et al. (2011). Their decision problem is a trade-off, where a weighted combination of risk and reward is optimized. The weight is viewed as a penalty on risk. An application with the use of the penalty parameter approach is provided by the financial planning model InnoALM for the Austrian pension fund of the electronics firm Siemens, developed in Geyer and Ziemba (2008). The model uses a multiperiod stochastic linear programming approach and scenarios dependent correlation matrice. There are numerous models for investment choice in a stochastic dynamic environment and some are presented in Section F. Campbell et al. (2003) propose a standard first order autoregressive model for log-returns, with the inclusion of state variables in the dynamics. The the Epstein–Zin recursive utility is used. With a log-linearization, the approximate optimal portfolio rule is the sum of two terms: a myopic component from the vector of excess returns (log optimal solution), and an inter-temporal hedging component, which accounts for the fact that asset returns are state dependent and thereby time varying. The returns from securities are related to the efficient operation of firms. The efficiency of a firms operation can be analyzed from financial ratios, and firm efficiency related to market returns. Edirisinghe et al. (2012) in their paper look at firm input dimensions and output dimensions from an efficiency perspective — maximum output for a given level of input. A Data Envelopment Analysis calculates relative efficiency scores for firms from a linear programming model, and these scores are used in selection of securities for investment In a series of papers, Browne (1999a, 1999b, 2000) considers a number of investment problems involving benchmarks. He determines a proportional strategy which maximizes the probability of exceeding a deterministic target. The strategy is equivalent to buying a European digital option with a particular strike and payoff. In the more general case of a stochastic benchmark, the optimal proportional strategy is composed of the benchmark and a hedging component. If the benchmark is the optimal growth strategy (Kelly), Browne solves the probability maximizing problem for fixed T , finding a strategy that will beat the Kelly by an arbitrary percentage. It is well known that features such as volatility of returns are time dependent. However, time periods can be segmented into regimes so that within a regime the distribution characteristics are stable. For example, the price and wealth dynamics may be defined by geometric Brownian motion within each regime, with the parameters being regime dependent. In this setup, the maximizing of the growth rate is analytic. These and other variations on the log utility model are described in MacLean and Ziemba (2012). Some of the strongest properties of the optimal growth strategy relate to its evolutionary performance in an equilibrium capital market. In a frictionless market (no transactions costs) Bashoun et al. (2012) show that the Kelly rule is globally evolutionary stable, meaning that any other essentially different strategy will become extinct with probability 1.

Preface

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Davis and Lleo (2012) consider a variety of alternative models for price dynamics and determine the strategy which maximizes the power utility of wealth. The first variation is an inter-temporal asset pricing model. The optimal portfolio invests in a fractional Kelly portfolio and cash. The fractional Kelly fund is a blend of funds: a time dependent Kelly portfolio and an inter-temporal hedging portfolio. Davis and Lleo also consider the pricing model where the diffusion is augmented by shocks, defined by a homogeneous Poisson process. They show that a model with shocks in the dynamics of returns and a power utility function of final wealth has an optimal portfolio strategy which has an option component. Browne shows this is also true when the objective is maximizing the probability of beating a stochastic benchmark. Thorp and Mizusawa (2012) consider the option component in two models for asset returns: log-normal, and fat tails. The strategies are a blend of stock and T-Bills versus the same blend of an option on stock and T-Bill.

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Contents

vii

Preface About the Editors

xvii

List of Contributors

xix

Acknowledgments

xxi

Part I: Decision Making Under Uncertainty

1

Section A. Arbitrage and Asset Pricing

3

1. 2.

3. 4.

Ross, SA (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13(3), 341–360.

11

Schachermayer, W (2010a). The fundamental theorem of asset pricing. In R Cont (Ed.), Encyclopedia of Quantitative Finance, 2, 792–801. New York: Wiley.

31

Schachermayer, W (2010b). Risk neutral pricing. In R Cont (Ed.), Encyclopedia of Quantitative Finance, 4, 1581–1585. New York: Wiley.

49

Kallio, M and WT Ziemba (2007). Using Tucker’s theorem of the alternative to provide a framework for proving basic arbitrage results. Journal of Banking and Finance, 31, 2281–2302.

57 79

Section B. Utility Theory 5.

6. 7. 8.

Fishburn, P (1969). A general theory of subjective probabilities and expected utilities. Annals of Mathematical Statistics, 40(4), 1419–1429.

87

Kahneman, D and A Tversky (1979). Prospect theory: An analysis of decisions under risk. Econometrica, 47(2), 263–291.

99

Levy, M and H Levy (2002). Prospect theory: Much ado about nothing? Management Science, 48(10), 1334–1349.

129

Wakker, PP (2003). The data of Levy and Levy (2002) “Prospect theory: Much ado about nothing?” Actually support prospect theory. Management Science, 49(7), 979–981.

145

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Contents

Levy, M and H Levy (2004). Prospect theory and mean-variance analysis. Review of Financial Studies, 17(4), 1015–1041.

149

10. Baltussen, G, T Post and PV Vliet (2006). Violations of cumulative prospect theory in mixed gambles with moderate probabilities. Management Science, 52(8), 1288–1290.

177

11. Kreps, DM and EL Porteus (1979). Temporal von Neumann–Morgenstern and induced preferences. Journal of Economic Theory, 20(1), 81–109.

181

12. Epstein, LG and SE Zin (1989). Substitution, risk aversion and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica, 57(4), 937–969.

207

13. Rabin, M (2000). Risk aversion and expected-utility theory: A calibration theorem. Econometrica, 68(5), 1281–1292.

241

14. Machina, M (2004). Non-expected utility theory. In JL Teugels and B Sundt (Eds.), Encyclopedia of Actuarial Science, 2, 1173–1179. New York: Wiley.

253

15. Tversky, A and D Kahneman (1974). Judgment under uncertainty: Heuristics and biases. Science, 185(4157), 1124–1131.

261

16. Kahneman, D and A Tversky (1984). Choices, values, and frames. American Psychologist, 39(4), 341–350.

269

Section C. Stochastic Dominance

279

17. Hanoch, G and H Levy (1969). The efficiency analysis of choices involving risk. Review of Economic Studies, 36(3), 335–346.

287

18. Levy, H (1973). Stochastic dominance, efficiency criteria, and efficient portfolios: The multi-period case. American Economic Review, 63(5), 986–994.

299

Section D. Risk Aversion and Static Portfolio Theory

309

19. Pratt, JW (1964). Risk aversion in the small and in the large. Econometrica, 32(1–2), 122–136.

317

20. Li, Y and WT Ziemba (1993). Univariate and multivariate measures of risk aversion and risk premiums. Annals of Operations Research, 45, 265–296.

333

21. Chopra, VK and WT Ziemba (1993). The effect of errors in means, variances, and co-variances on optimal portfolio choice. Journal of Portfolio Management, 19, 6–11.

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22. Ziemba, WT, C Parkan and R Brooks-Hill (1974). Calculation of investment portfolios with risk free borrowing and lending. Management Science, 21(2), 209–222.

375

23. Kallberg, JG and WT Ziemba (1983). Comparison of alternative utility functions in portfolio selection problems. Management Science, 29(11), 1257–1276.

389

24. Li, Y and WT Ziemba (1989). Characterizations of optimal portfolios by univariate and multivariate risk aversion. Management Science, 35(3), 259–269.

409

25. Ziemba, WT (1975). Choosing investment portfolios when the returns have stable distributions. In WT Ziemba and RG Vickson (Eds.), Stochastic Optimization Models in Finance, 243–266. San Diego: Academic Press.

421

26. MacLean, LC, ME Foster and WT Ziemba (2007). Covariance complexity and rates of return on assets. Journal of Banking and Finance, 31(11), 3503–3523.

445

27. Rabin, M and RH Thaler (2001). Anomalies: Risk aversion. Journal of Economic Perspectives, 15(1), 219–232.

467

Part II: From Decision Making to Measurement and Dynamic Modeling

481

Section E. Risk Measures

483

28. Geyer, A and WT Ziemba (2008). The innovest Austrian pension fund planning model InnoALM. Operations Research, 56(4), 797–810.

491

29. Rockafellar, RT and WT Ziemba (2000). Modified risk measures and acceptance sets.

505

30. F¨ ollmer, H and T Knispel (2012). Convex risk measures: Basic facts, law-invariance and beyond, asymptotics for large portfolios.

507

31. Krokhmal, P, M Zabarankin and S Uryasev (2011). Modeling and optimization of risk. Surveys in Operations Research and Management Science, 16(2), 49–66.

555

Section F. Dynamic Portfolio Theory and Asset Allocation

601

32. Edirisinghe, C, X Zhang and S-C Shyi (2012). DEA-based firm strengths and market efficiency in US and Japan.

611

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33. MacLean, LC and WT Ziemba (2012). The Kelly strategy for investing: Risk and reward.

637

34. Browne, S (1999a). Reaching goals by a deadline: Digital options and continuous-time active portfolio management. Advances in Applied Probability, 31, 551–577.

683

35. Browne, S (1999b). Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark. Finance and Stochastics, 3, 275–294.

711

36. Browne, S (2000). Stochastic differential portfolio games. Journal of Applied Probability, 37(1), 126–147.

731

37. Davis, M, and S Lleo (2012). Fractional Kelly strategies in continuous time: Recent developments.

753

38. Bahsoun, W, IV Evstigneev and MI Taksar (2012). Growth-optimal investments and numeraire portfolios under transactions costs.

789

39. Campbell, JY, YL Chan and LM Viceira (2003). A multivariate model of strategic asset allocation. Journal of Financial Economics, 67, 41–80.

809

40. Thorp, EO, and Mizusawa, S (2012). Maximizing capital growth with black swan protection.

849

Author Index

873

Subject Index

877

About the Editors

Leonard C MacLean is Professor Emeritus in the School of Business Administration at Dalhousie University in Halifax, Canada. Dr. MacLean has held visiting appointments at Cambridge University, University of Bergamo, University of British Columbia, Simon Fraser University, Royal Roads University, University of Zimbabwe, and University of Indonesia. From 1989 to 1995, he served as Director of the School of Business Administration at Dalhousie University. His research focuses on stochastic models in finance, and models for repairable systems in aviation. This work is funded by grants from the Natural Sciences and Engineering Council of Canada and the Herbert Lamb Trust. Professor MacLean teaches in the areas of statistics and operations management. William T Ziemba is the Alumni Professor (Emeritus) of Financial Modeling and Stochastic Optimization in the Sauder School of Business, University of British Columbia where he taught from 1968–2006 and Professor at the ICMA Centre, University of Reading. His PhD is from the University of California, Berkeley. He has been a Visiting Professor at Cambridge, Oxford, London School of Economics, and Warwick in the UK, at Stanford, UCLA, Berkeley, MIT, University of Washington and Chicago in the US, Universities of Bergamo, Venice and Luiss in Italy, the Universities of Zurich, Cyprus, Tsukuba (Japan), KAIST (Korea) and the National University of Singapore. He has been a consultant to a number of leading financial institutions including the Frank Russell Company, Morgan Stanley, Buchanan Partners, RAB Hedge Funds, Gordon Capital, Matcap, Ketchum Trading and, in the gambling area, to the BC Lotto Corporation, SCA Insurance, Singapore Pools, Canadian Sports Pool, Keeneland Racetrack and some racetrack syndicates in Hong Kong and Australia. His research is in asset-liability management, portfolio theory and practice, security market imperfections, Japanese and Asian financial markets, hedge fund strategies, risk management, sports and lottery investments and applied stochastic programming. He has published widely in journals such as Operations Research, Management Science, Mathematics of Operations Research, Mathematical Programming,

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About the Editors

American Economic Review, Journal of Economic Perspectives, Journal of Finance, Journal of Economic Dynamics and Control, and Journal of Financial and Quantitative Analysis and in many books and special journal issues. Recent books include Applications of Stochastic Programming with SW Wallace (SIAM-MPS, 2005), Stochastic Optimization Models in Finance, 2nd edition with RG Vickson (World Scientific, 2006) and Handbook of Asset and Liability Modeling, Volume 1: Theory and Methodology (2006) and Volume 2: Applications and Case Studies (2007) with SA Zenios (North Holland), Scenarios for Risk Management and Global Investment Strategies with Rachel Ziemba (Wiley, 2007), Handbook of Investments: Sports and Lottery Betting Markets, with Donald Hausch (North Holland, 2008), Optimizing the Aging, Retirement and Pensions Dilemma with Marida Bertocchi and Sandra Schwartz and The Kelly Capital Growth Investment Criterion, with legendary hedge fund trader Edward O Thorp and Leonard C MacLean (World Scientific, 2010). His co-written practitioner paper on the Russell–Yasuda model won second prize in the 1993 Edelman Practice of Management Science Competition. He has been a futures and equity trader and hedge fund and investment manager since 1983. He is the series editor for North Holland’s Handbooks in Finance, World Scientific Handbooks in Financial Economics and Books in Finance, and previously was the CORS editor of INFOR and the department of finance editor of Management Science, 1982–1992. He has continued his columns in Wilmott and has prepared the book Investing in the Mondern Age with Rachel Ziemba on the 2007–2012 columns plus other materials for World Scientific. Ziemba, along with Hausch, wrote the famous Beat the Racetrack book (1984) (which was revised into Dr Z’s Beat the Racetrack (1987), which presented their place and show betting system and the Efficiency of Racetrack Betting Markets (1994, 2008) — the so-called bible of racetrack syndicates. Their 1986 book Betting at the Racetrack extends this efficient/inefficient market approach to simple exotic bets. Ziemba is revising Betting at the Racetrack into Exotic Betting at the Racetrack which adds Pick3,4,5,6, etc. and updates a few things for World Scientific to be out in Spring 2013.

List of Contributors

Bahsoun, W (Loughborough University) Baltussen, G (Erasmus School of Economics and Tinbergen Institute) Brooks-Hill, FJ (University of British Columbia) Browne, S (Columbia University) Campbell, JY (Harvard University) Chan, YL (Hong Kong University of Science and Technology) Chopra, V (Frank Russell Company) Davis, M (Imperial College London) Edirisinghe, C (University of Tennessee) Epstein, LG (University of Rochester) Evstigneev, IV (University of Manchester) Fishburn, PC (Research Analysis Corporation) F¨ ollmer, H (Humboldt University) Foster, M (Dalhousie University) Geyer, A (Vienna University of Economics and Business) Hanoch, G (Hebrew University of Jerusalem) Kahneman, D (Princeton University) Kallberg, JG (New York University) Kallio, M (Helsinki School of Economics) Knispel, T (Leibniz University) Krokhmal, P (University of Iowa) Levy, H (Hebrew University of Jerusalem) Levy, M (Hebrew University of Jerusalem) Li, Y (State University of California, Fullerton) Lleo, S (University of Reims) Machina, M (University of California, San Diego) MacLean, LC (Dalhousie University) Mizusawa, S (Edward O Thorp and Associates) Parkan, C (University of Calgary) Post, T (Erasmus School of Economics) Pratt, JW (Harvard University) Rabin, M (University of California, Berkeley) Rockafellar, RT (University of Washington) Ross, SA (MIT) Schachermayer, W (University of Vienna) Taksar, MI (University of Missouri) Thorp, EO (Edward O Thorp and Associates) xix

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List of Contributors

Tversky, A (Stanford University) Uryasev, S (University of Florida) Viceira, L (Harvard University) Vliet, PV (Robeco) Wakker, PP (University of Amsterdam) Zabarankin, M (Stevens Institute of Technology) Zhang, X (Austin Peay State University) Ziemba, WT (University of British Columbia, ICMA Centre and University of Reading) Zin, SE (Carnegie Mellon University)

Acknowledgments

Thanks go to the following journals and publishers for permission to reproduce the papers and book chapters listed below. Academic Press Ziemba, WT (1975). Choosing investment portfolios when the returns have stable distributions. In WT Ziemba and RG Vickson (Eds.), Stochastic Optimization Models in Finance, 243–266. San Diego: Academic Press. Advances in Applied Probability Browne, S (1999a). Reaching goals by a deadline: Digital options and continuoustime active portfolio management. Advances in Applied Probability, 31, 551–577. American Economic Review Levy, H (1973). Stochastic dominance, efficiency criteria, and efficient portfolios: The multi-period case. American Economic Review, 63(5), 986–994. American Psychologist Kahneman, D and A Tversky (1984). Choices, values and frames. American Psychologist, 39(4), 341–350. Annals of Mathematical Statistics Fishburn, P (1969). A general theory of subjective probabilities and expected utilities. Annals of Mathematical Statistics, 40(4), 1419–1429. Annals of Operations Research Li, Y and WT Ziemba (1993). Univariate and multivariate measures of risk aversion and risk premiums. Annals of Operations Research, 45, 265–296. Econometrica Epstein, LG and SE Zin (1989). Substitution, risk aversion and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica, 57(4), 937–969. Pratt, JW (1964). Risk aversion in the small and in the large. Econometrica, 32(1–2), 122–136. xxi

xxii

Acknowledgments

Rabin, M (2000). Risk aversion and expected utility theory: A calibration theorem. Econometrica, 68(5), 1281–1292. Kahneman, D and A Tversky (1979). Prospect theory: An analysis of decisions under risk. Econometrica, 47(2), 263–291. Finance and Stochastics Browne, S (1999b). Beating a moving target: Optimal portfolio strategies for outperforming a stochastic benchmark. Finance and Stochastics, 3, 275–294. Journal of Applied Probability Browne, S (2000). Stochastic differential portfolio games. Journal of Applied Probability, 37(1), 126–147. Journal of Banking and Finance Kallio, M and WT Ziemba (2007). Using Tucker’s theorem of the alternative to provide a framework for proving basic arbitrage results. Journal of Banking and Finance, 31, 2281–2302. MacLean, LC, ME Foster and WT Ziemba (2007). Covariance complexity and rates of return on assets. Journal of Banking and Finance, 31(11), 3503–3523. Journal of Economic Perspectives Rabin, M and RH Thaler (2001). Anomalies: Risk aversion. Journal of Economic Perspectives, 15(1), 219–232. Journal of Economic Theory Kreps, DM and EL Porteus (1979). Temporal von Neumann–Morgenstern and induced preferences. Journal of Economic Theory, 20(1), 81–109. Ross, SA (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13(3), 341–360. Journal of Financial Economics Campbell, JY, YL Chan and L Viceira (2003). A multivariate model of strategic asset allocation. Journal of Financial Economics, 67, 41–80. Journal of Portfolio Management Chopra, VK and WT Ziemba (1993). The effect of errors in mean and co-variance estimates on optimal portfolio choice. Journal of Portfolio Management, 19, 6–11. Management Science Baltussen, G, T Post and PV Vliet (2006). Violations of cumulative prospect theory in mixed gambles with moderate probabilities. Management Science, 52(8), 1288–1290.

Acknowledgments

xxiii

Kallberg, JG and WT Ziemba (1983). Comparison of alternative utility functions in portfolio selection problems. Management Science, 29(11), 1257–1276. Levy, M and H Levy (2002). Prospect theory: Much ado about nothing? Management Science, 48(10), 1334–1349. Li, Y and WT Ziemba (1989). Characterizations of optimal portfolios by univariate and multivariate risk aversion. Management Science, 35(3), 259–269. Wakker, PP (2003). The data of Levy and Levy (2002) “Prospect theory: Much ado about nothing?” actually support prospect theory. Management Science, 49(7), 979–981. Ziemba, WT, C Parkan and R Brooks-Hill (1974). Calculation of investment portfolios with risk free borrowing and lending. Management Science, 21(2), 209–222. Operations Research Geyer, A and WT Ziemba (2008). The Innovest Austrian pension fund planning model InnoALM. Operations Research, 56(4), 797–810. Review of Economic Studies Hanoch, G and H Levy (1969). The efficiency analysis of choices involving risk. Review of Economic Studies, 36(3), 335–346. Review of Financial Studies Levy, M and H Levy (2004). Prospect theory and mean-variance analysis. Review of Financial Studies, 17(4), 1015–1041. Science Tversky, A and D Kahneman (1974). Judgement under uncertainty: Heuristics and biases. Science, 185(4157), 1124–1131. Surveys in Operations Research and Management Science Krokhmal, P, M Zabarankin and S Uryasev (2011). Modeling and optimization of risk. Surveys in Operations Research and Management Science, 16(2), 49–66. Wiley Machina, M (2004). Non-expected utility theory. In JL Teugels and B Sundt (Eds.), Encyclopedia of Actuarial Science, 2, 1173–1179. New York: Wiley. Schachermayer, W (2010a). The fundamental theorem of asset pricing. In R Cont (Ed.), Encyclopedia of Quantitative Finance, 2, 792–801. New York: Wiley. Schachermayer, W (2010b). Risk neutral pricing. In R Cont (Ed.), Encyclopedia of Quantitative Finance, 4, 1581–1585. New York: Wiley.

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Part I

Decision Making Under Uncertainty

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Section A

Arbitrage and Asset Pricing

4

Section A. Arbitrage and Asset Pricing

The theme of this handbook is financial decision making. The decisions are the amount of investment capital to allocate to various opportunities in a financial market. The opportunity set can be very complex, with sets of equities, bonds, commodities, derivatives, futures, and currencies having trading prices changing stochastically and dynamically over time. To consider decisions in a complex market, it is necessary to impose structure. In the abstract, the assets and the participants buying and selling them are parts of a system with underlying economic states. The system’s dynamics and the factors defining the states in the system have been studied extensively in finance and economics. The dynamics of the market and the behavior of participants determine the trading prices of the various assets in the opportunity set. A simplifying assumption is that the financial market is perfectly competitive. There are conditions that must be present for a perfectly competitive market structure to exist. There must be many participants in the market, none of which is large enough to affect prices. Individuals should be able to buy and sell without restriction. All participants in the market have complete information about prices. In the competitive market, investors are price takers. These assumptions are strong, and in actual financial markets they are not exactly satisfied. However, with the assumed structure an idealized market can be characterized and that provides a standard by which existing practice can be measured. If investors are price takers, then a fundamental component of financial decision making is asset pricing. A common approach to asset pricing is to derive equilibrium prices for assets in a competitive market. This can be achieved with a model mapping the abstract states defined by a probability space into prices of assets such as equities and bonds. The Capital Asset Pricing Model (CAPM) developed independently by Sharpe (1964), Lintner (1965), Mossin (1966) and Treynor (1961, 1962) is a standard for pricing risky assets. Some clarification is provided in Fama (1968). The model proposes that the expected excess return of a risky asset over a riskless asset is proportional to the expected excess return of the market over the riskless asset. The returns on assets are assumed to be normally distributed. In this setting the financial market is in competitive equilibrium. Consistent with this structure, the optimal investment decisions are determined from the mean-variance approach developed by Markowitz (1952, 1959). The CAPM is the theoretical basis for much of the sizable index fund business. Dimension Fund Advisors alone manages $250 billion, most of which is passive. The CAPM model has a single explanatory variable, the market portfolio, in a simple linear regression. This model has been extended to include other market variables in a multivariate linear regression. For example, following Rosenberg (1974), Rosenberg, Reid and Lanstein (1985), Fama and French (1992) have added two explanatory variables: (i) small minus large capitalization; and (ii) high minus low book to market ratio. The equilibrium pricing in the CAPM type models implies that no arbitrage opportunities exist. An arbitrage is a transaction that involves no negative cash

Section A. Arbitrage and Asset Pricing

5

flow at any probabilistic or temporal state and a positive cash flow in at least one state; in simple terms, it is the possibility of a risk-free profit at zero cost. The Arbitrage Pricing Theory (APT) for asset pricing following from an assumption of no arbitrage was developed by Ross (1976). This theory defines the expected returns on assets with a linear factor model. The theory linking arbitrage to the factor model are presented in the paper “The Arbitrage Theory of Capital Asset Pricing.” The Ross argument considers a well-diversified portfolio of risky assets which uses no wealth (free lunch). The portfolio is essentially independent of noise. If the portfolio has no risk, then the random return is certain and to avoid disequilibrium the certain return must be zero. This no arbitrage condition implies that the returns on the assets are defined by a linear relation to a set of common random factors with zero expectation. This type of equilibrium arbitrage argument follows the famous Modigliani and Miller paper (1958) and is part of the reasoning in the Black–Scholes option pricing (1973) model. There are a number of differences between the CAPM and APT theories. The most significant distinction is the “factors.” In CAPM the factors/independent variables are manifest market variables (e.g., market index). With APT the factors are intrinsic (not manifest) variables, whose existence follows from diversification and no arbitrage. It is not required that the APT factors have clear definitions as entities. The APT factors are structural, without implied causation. That is, CAPM: factors → returns; APT: factors ↔ returns. So the factor model in APT is really a distributional condition on prices following from no arbitrage. The essence of arbitrage is captured in Ross’ theory. There are no assumptions in APT about the distribution of noise, whereas CAPM assumes normality. However, the use of the factor model in empirical work on pricing does use algorithms which sometime assume normality of the factors and returns. The statistical estimation would also suggest definitions/entities for the intrinsic factors, which could further link the CAPM and APT models. Factor models have been used in practice by many analysts (see Jacobs and Levy (1988) and Ziemba and Schwartz (1991), and Schwartz and Ziemba (2000)). Companies such as BARRA sell these models. The APT does not assume the existence of a competitive equilibrium. Disequilibrium can exist in the theory, but it is assumed that in aggregate the returns are uniformly bounded. The no arbitrage assumption is a natural condition to expect of a stable financial market. The existence of arbitrage free prices for assets is linked to the probability measure on which the stochastic process of prices is defined. The Fundamental Theorem of Asset Pricing states that If S = {St, t ≥ 0} are asset prices in a complete financial market, then the following statements are equivalent (i) S does not allow for arbitrage (ii) There exists a probability measure which is equivalent to the original underlying measure and the price process is a martingale under the new measure.

6

Section A. Arbitrage and Asset Pricing

A martingale is a stochastic process where the conditional expected value for the next period equals the current observed value, and does not depend on the history of the process. So a martingale is a model for a fair process and it is not surprising that the fairness of no arbitrage can be characterized by a martingale measure. Indeed the Ross (1976) argument establishes the link between arbitrage and a martingale measure using the famous Hahn–Banach theorem. The assumptions used by Ross on the underlying measure were some what limiting. In the case of an infinite probability space, the Ross result only applies to the sup norm topology. For finite dimensional space, it is not clear that the martingale measure is actually equivalent. These limitations were considered by Harrison and Kreps (1979) and Harrison and Pliska (1981). They extended the Fundamental Theorem of Asset Pricing in several ways: (i) If the price process is defined on a finite, filtered, probability space, then the market contains no arbitrage possibilities if and only if there is an equivalent martingale measure. (ii) If the price process is defined on a continuous probability space and the market admits “no free lunch,” then there exists an equivalent martingale measure. (iii) If the price process is defined on a countably generated probability space, taking values in Lp space, then the “no free lunch” condition is satisfied if and only if there is an equivalent martingale measure satisfying a q moment condition, where p1 + 1q = 1. Although the work of Kreps and colleagues made significant contributions to the theory of arbitrage pricing, there were still assumptions which limited the applicability. Ideally a more economically natural condition could replace the moment condition on the martingale measure. Delbaen and Schachermayer (2006) discuss many open questions. One particular advance links the existence of an equivalent martingale measure in processes in continuous time or infinite discrete time to a condition of “no free lunch with bounded risk.” Unfortunately, this result does not hold for price processes which are semi-martingales. Furthermore, there are strong mathematical and economic reasons to assume that the price process is a semimartingale. In that setting the no free lunch with bounded risk is replaced by a no free lunch with vanishing risk, where risk disappears in the limit. The latter is stronger than the former, but is weaker than a no arbitrage condition. So Schachermayer (2010), and Delbaen and Schachermayer (2006) have a general statement of the fundamental theorem: Assume the price process is a locally bounded real-valued semi-martingale. There is a martingale measure which is equivalent to the original measure if and only if the price process satisfies the no free lunch with vanishing risk condition. Yan (1998) brought the results even closer to the desired form. The concept of allowable trading strategies was introduced, where the trader remains liquid during

Section A. Arbitrage and Asset Pricing

7

the trading interval. The Yan formulation yields the result: Let the price process be a positive semi-martingale. There is a martingale measure which is equivalent to the original measure if and only if the price process satisfies the no free lunch with vanishing risk condition with respect to allowable trading strategies. Another term for an equivalent martingale measure is a risk-neutral measure. Prices of assets depend on their risk, with a premium required for riskier assets. The advantage of the equivalent martingale or risk-neutral measure is that risk premia are incorporated into the expectation with respect to that measure. Under the riskneutral measure all assets have the same expected value — the risk-free rate. The stock price process discounted by the risk-free rate is a martingale under the riskneutral measure. This simplification is important in the valuation of assets such as options and is a component of the famous Black–Scholes (1973) formula. Of course, the risk-neutral measure is an artificial concept, with important implications for the theory of pricing. The actual risk-neutral measure used for price adjustment must be determined from economic reasoning. The separating hyperplane arguments underlying the results linking arbitrage and no free lunch to martingale measures have an analogy in theorems of the alternative for discrete time and discrete space arbitrage pricing models. In theorems of the alternative competing systems of equlities/inequalities are posed, with only one system having a solution. A famous such theorem is due to Tucker (1954). Kallio and Ziemba (2007) used Tucker’s Theorem of the Alternative to derive known and some new arbitrage pricing results. The competing systems define arbitrage on the one hand and the existence of risk-neutral probabilities on the other hand. For a frictionless market the Fundamental Theorem of Asset pricing is established using matrix arguments for the discrete time and discrete space price process: If at each stage an asset exists with strictly positive return (there exists a trading strategy), then arbitrage does not exist if and only if there exists an equivalent martingale measure. Although the discrete time and space setting is limiting, it is used in practice as an approximation to the continuous process. Obviously there are considerable computational advantages with a discrete process, and assumptions required for its implementation are few. In the general setting the fundamental theorem posits the existence of a risk-neutral measure. Actually finding such a measure requires additional assumptions. In the discrete setting, the equations for calculating the probabilities in the measure can be solved. This is analogous to the option pricing models, where in the Black–Scholes approach strong distribution assumptions are required to get the pricing formula, but the binomial lattice approach obtains option prices with a linear programming algorithm. Even from a theory perspective, the discrete time and space extension to more complex financial markets is feasible since the mathematics is based on systems of equations. In Kallio and Ziemba (2007)

8

Section A. Arbitrage and Asset Pricing

the equivalence between no arbitrage and the existence of a martingale measure is extended to markets with various imperfections. The no arbitrage condition is fundamental to much of the theory of efficient capital markets. However, it is important to acknowledge the existence of arbitrage opportunities in actual markets. Examples are the Nikkei put warrant arbitrage discussed in Shaw, Thorp and Ziemba (1995), and the race track arbitrages discussed by Hausch and Ziemba (1990a, 1990b). Investors exhibit behavioral biases which can lead to mispricing and arbitrage. Usually over/under pricing is temporary, but correctly identifying those events and using them for financial advantage has attracted attention.

Readings Black F and M Scholes (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654. Delbaen, F and W Schachermayer (2006). The Mathematics of Arbitrage. New York: Springer. Fama, EF (1968). Risk, return and equilibrium: Some clarifying comments. Journal of Finance, 23(1), 29–40. Fama, EF and F French (1992). The cross-section of expected stock returns. Journal of Finance, June, 427–466. Harrison, JM and DM Kreps (1979). Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20, 381–408. Harrison, JM and SR Pliska (1981). Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications, 11, 215–260. Hausch, DB and WT Ziemba (1990a). Arbitrage strategies for cross track betting on major horseraces. Journal of Business, 63, 61–78. Hausch, DB and WT Ziemba (1990b). Locks at the racetrack. Interfaces, 20(3), 41–48. Jacobs, BL and KN Levy (1988). Disentangling equity return regularities: New insights and investment opportunities. Financial Analysts Journal, 44, 18–43. Kallio, M and WT Ziemba (2007). Using Tucker’s Theorem of the alternative to provide a framework for proving basic arbitrage results. Journal of Banking and Finance, 31, 2281–2302. 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, 47(1), 13–37. Markowitz, H (1952). Portfolio selection. Journal of Finance, 7, 77–91. Markowitz, H (1959). Portfolio Selection: Efficient Diversification of Investments. New York: Wiley. Modigliani, F and M Miller (1958). The cost of capital, corporation finance and the theory of investment. American Economic Review, 48(3), 261–297. Mossin, J (1966). Equilibrium in a capital asset market. Econometrica, 34(4), 768–783. Rosenberg, B (1974). Extra-market components of covariance in securities markets. Journal of Financial and Quantitative Analysis, 263–274. Rosenberg, B, K Reid and R Lanstein (1985). Persuasive evidence of market inefficiency. Journal of Portfolio Management, 11(3), 9–16. Ross, SA (1976). The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13(3), 341–360. Schachermayer, W (2010). The fundamental theorem of asset pricing. In R Cont (Ed.), Encyclopedia of Quantitative Finance, 2, 792–801. New York: Wiley.

Section A. Arbitrage and Asset Pricing

9

Schachermayer, W (2010). Risk neutral pricing. In R Cont (Ed.), Encyclopedia of Quantitative Finance, 4, 1581–1585. New York: Wiley. Schwartz, SL and WT Ziemba (2000). Predicting returns on the Tokyo stock exchange. In DB Keim and WT Ziemba (Eds.), Security Market Imperfections in Worldwide Equity Markets, 492–511. Cambridge University Press. Sharpe, WF (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19(3), 425–442. Shaw, J, EO Thorp and WT Ziemba (1995). Convergence to efficiency of the Nikkei put warrant market of 1989–1990. Applied Mathematical Finance, 2, 243–271. Treynor, JL (1961). Market Value, Time, and Risk. Unpublished manuscript. Treynor, JL (1962). Toward a theory of market value of risky assets. In RA Korajczyk (Ed.) Asset Pricing and Portfolio Performance: Models, Strategy and Performance Metrics. London: Risk Books. Tucker, A (1956). Dual systems of homogeneous linear relations. In H Kuhn and A Tucker (Eds.), Linear Inequalities and Related Systems, Annals of Mathematics Studies. Princeton: Princeton University Press. Yan, JA (1998). A new look at the fundamental theorem of asset pricing, 35, 659–673. Ziemba, WT and SL Schwartz (1991). Invest Japan: The Structure, Performance and Opportunities of Japan’s Stock, Bond and Fund Markets. Chicago: Probus Publishing.

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JOURNAL OF ECONOMIC THEORY

13, 341-360 (1976)

Chapter 1 The Arbitrage Theory of Capital Asset Pricing STEPHEN

A. Ross*

Departments of Economics and Finance, University of Pennsylvania, The Wharton School, Philadelphia, Pennsylvania 19174

Received March 19, 1973: revised May 19, 1976

The purpose of this paper is to examine rigorously the arbitrage model of capital asset pricing developed in Ross [13, 14]. The arbitrage model was proposed as an alternative to the mean variance capital asset pricing model, introduced by Sharpe, Lintner, and Treynor, that has become the major analytic tool for explaining phenomena observed in capital markets for risky assets. The principal relation that emerges from the mean variance model holds that for any asset, ;, its (ex ante) expected return (l)

where p is the riskless rate of interest, .\ is the expected excess return on the market, Ern - p, and

is the beta coefficient on the market, where arn2 is the variance of the market portfolio and aim is the covariance between the returns on the ith asset and the market portfolio. (If a riskless asset does not exist, p is the zero-beta return, i.e., the return on all portfolios uncorrelated with the market portfolio.)1 The linear relation in (I) arises from the mean variance efficiency of the market portfolio, but on theoretical grounds it is difficult to justify either the assumption of normality in returns (or local normality in Wiener diffusion models) or of quadratic preferences to guarantee such efficiency, and on empirical grounds the conclusions as well as the *Professor of Economics, University of Pennsylvania. This work was supported by a grant from the Rodney L. White Center for Financial Research at the University of Pennsylvania and by National Science Foundation Grant GS-35780. 1 See Black [2] for an analysis of the mean variance model in the absence of a riskless asset.

341 Copyright Ｇｾ＠ 1976 by Academic Press, Inc. All rights of reproduction in any form reserved.

11

12

SA Ross

342

STEPHEN A. ROSS

assumptions of the theory have also come under attack. 2 The restrictiveness of the assumptions that underlie the mean variance model have, however, long been recognized, but its tractability and the evident appeal of the linear relation between return, E;, and risk, bi, embodied in (I) have ensured its popularity. An alternative theory of the pricing of risky assets that retains many of the intuitive results of the original theory was developed in Ross [13, 14]. In its barest essentials the argument presented there is as follows. Suppose that the random returns on a subset of assets can be expressed by a simple factor model X; == E;

+ (3/j + €; ,

(2)

where 8 is a mean zero common factor, and i!i is mean zero with the vector (€) sufficiently independent to permit the law of large numbers to hold. Neglecting the noise term, i!i, as discussed in Ross [14] (2) is a statement that the state space tableau of asset returns lies in a twodimensional space that can be spanned by a vector with elements 88 , (where () denotes the state of the world) and the constant vector, e ｾ＠ (1, ... , 1). Step I. Form an arbitrage portfolio, TJ, of all the n assets, i.e., a portfolio which uses no wealth, 7Je = 0. We will also require 7J to be a well-diversified portfolio with each component, TJi , of order 1/n in (absolute) magnitude. Step 2. By the law of large numbers, for large arbitrage portfolio 7JX

+ (7]{3) 8 + YJE TJE + (7Jf3) 8.

= TJE RoO

n

the return on the

(3)

ln other words the influence on the well-diversified portfolio of the independent noise terms becomes negligible. Step 3. If we now also require that the arbitrage portfolio, 7], be chosen so as to have no systematic risk, then 7Jf3

=

0,

and from (3) 2 See Blume and Friend [3] for a recent example of some of the empirical difficulties faced by the mean variance model. For a good review of the theoretical and empirical literature on the mean variance model see Jensen [6].

Chapter 1. The Arbitrage Theory of Capital Asset Pricing CAPITAL ASSET PRICING

13

343

Step 4. Using no wealth, the random return T)X has now been engineered to be equivalent to a certain return, T)E, hence to prevent arbitrarily large disequilibrium positions we must have TJE = 0. Since this restriction must hold for all T) such that T)e ］Ｍｾ＠ TJf3 =o= 0, E is spanned by e and f3 or

E, = p

+ /o.f3,

(4)

for constants p and /o.. Clearly if there is a riskless asset, p must be its rate of return. Even if there is not such an asset, p is the rate of return on all zero-beta portfolios, ex, i.e., all portfolios with exe = 1 and exf3 = 0. If ex is a particular portfolio of interest, e.g., the market portfolio, cx 111 , with £, = ex 111 E, (4) becomes

E;

=

p

+ (E"'- p) {3;.

(5)

Condition ( 5) is the arbitrage theory equivalent of (I), and if f5 is a market factor return then /3 1 will approximate b, . The above approach, however, is substantially different from the usual mean-variance analysis and constitutes a related but quite distinct theory. For one thing, the argument suggests that (5) holds not only in equilibrium situations, but in all but the most profound sort of disequilibria. For another, the market portfolio plays no special role. There are, however, some weak points in the heuristic argument. For example, as the number of assets, n, is increased, wealth will, in general, also increase. Increasing wealth, though, may increase the risk aversion of some economic agents. The law of large numbers implies, in Step 2. that the noise term, T)E, becomes negligible for large n, but if the degree of risk aversion is increasing with n these two effects may cancel out and the presence of noise may persist as an influence on the pricing relation. In Section I we will present an example of a market where this occurs. Furthermore, even if the noise term can be eliminated, it is not at all obvious that (5) must hold, since the disequilibrium position of one agent might be offset by the disequilibrium position of another. 3 In Ross [13], however, it was shown that if (5) holds then it represents an E or quasi-equilibrium. The intent of this paper is to supply the rigorous analysis underlying the stronger stability arguments above. In Section 11 we will present some weak sufficient conditions to rule out the above exceptions (and the example of Section l) and we will prove a general version of the arbitrage result. Section II also includes a brief argument on the empirical practicality of the results. A mathematical appendix 3 Green has considered this point in a temporary equilibrium model. Essentially he argues that if subjective anticipations differ too much, then arbitrage possibilities will threaten the existence of equilibrium.

14

SA Ross

344

STEPHEN A. ROSS

contains some supportive results of a somewhat technical and tangential nature. Section l II will briefly summarize the paper and suggest further generalizations.

I. A

COUNTEREXAMPLE

In this section we will present an example of a market where the sequence of equilibrium pricing relations does not approach the one predicted by the arbitrage theory as the number of assets is increased. The counterexample is valuable because it makes clear what sort of additional assumptions must be imposed to validate the theory. Suppose that there is a riskless asset and that risky assets are independently and normally distributed as

x;

E;

=

+ i;,

(6)

where E{ii}

=

0,

and

The arbitrage argument would imply that in equilibrium all of the independent risk would disappear and, therefore, Ei ﾷｾ＠

(7)

P·

Assume, however, that the market consists of a single agent with a von Neumann-Morgenstern utility function of the constant absolute risk aversion form, U(z)

=

-exp( -Az).

(8)

Letting w denote wealth with the riskless asset as the numeraire, and ex the portfolio of risky assets (i.e., ex; is the proportion of wealth placed in the ith risky asset) and taking expectations we have E{U[w(p

+ a[x- p · e])]}

=

-exp( -Awp) E{exp( -Awcx[x- p · e])}

=

-exp( -Awp){exp( -Awcx[E- p · e]

+ (a /2)(Aw)2(cx'cx))}. 2

(9)

The first-order conditions at a maximum are given by a 2 (Aw) ｾｘ Ｑ＠

= E; - p.

(10)

Chapter 1. The Arbitrage Theory of Capital Asset Pricing

15

345

CAPITAL ASSET PRICING

If the riskless asset is in unit supply the budget constraint (Walras' Law

for the market) becomes n

1\'

=

I

cx;w

+l

n

= (1/Aa 2 )

i -1

L (E; -

p)

+ l,

(11)

i=l

The interpretation of the budget constraint (11) depends on the particular market situation we are describing. Suppose, first, that we are adding assets which will pay a random total numeraire amount, ci . If Pt is the current numeraire price of the asset then

Normalizing all risky assets to be in unit supply we must have

=

Pt

CX;W,

and the budget constraint simply asserts that wealth is summed value, n

I

\\' =

Pt

+ l.

ｩｾｬ＠

If we let

c; denote the mean of c; and c2, its variance, then (10) can be

solved for p1 as

As a consequence, the expected returns,

will be unaffected by changes in the number of assets, n, for i < n, and need bear no systematic relation to p as n increases. This is a violation of the arbitrage condition, (7). Notice, too, that as long as c1 is bounded above Ac2 , wealth and relative risk aversion, Aw, are unbounded in n. An alternative interpretation of the market situation would be that as 11 increases the number of risky investment opportunities or activities is being increased, but not the number of assets. In this case wealth, w, would simply be the number of units of the riskless asset held and would remain constant as 11 increased. The quantities cx,:W now represent the amount of the riskless holdings put into the ith investment opportunity and for the market as a whole we must have n

I

i=l

CX;

< I.

16

SA Ross

346

STEPHEN A. ROSS

Furthermore, if the random technological activities are irreversible, then each rx; ;;;: 0. From (10) it follows that E;-

p;;;: 0

and n

I

n

E, -· p =

n

I

I E;

·

pI =

a 2(Aw)

i=l

i=l

I

rx;


U(p),

but by concavity hence lim E{U[p

+ Xn]}

= U(p). Q.E.D.

A problem arises when U(-) is unbounded from below. About the weakest condition which assures convergence is uniform integrability (U.I.):

r

lim sup n ., Ｈｌｾｯ＠

Q.,

0 rr

I U(p

= {I U(p

+ Xn)l dr;n

= 0,

+ Xnll > c-.:},

where YJn is the distribution function of X, .

•

Chapter 1. The Arbitrage Theory of Capital Asset Pricing

29

359

CAPITAL ASSET PRICING

A number of familiar conditions imply U.l. If the sequence U(p + Xn) is bounded below by an integrable function the Lebesque convergence theorem can be invoked or if (38 > 0) sup £{1 U(p n

+ X nW' < 8}

oo,

then the sequence is U.I. In general, then, the convergence criterion will depend on both the utility function and the random variables. It is possible, however, to find weak sufficient conditions on the random variables alone, by taking advantage of the structure of .Yn, but the condition that X,.= (1/n) Li €;; al uniformly bounded and E; , Ej independent is not su:fficient. 9 In the text, it is assumed that all sequences satisfy the U.l. condition, and therefore (q.m.) will imply that

REFERENCES

1. P. BILLINGSLEY, "Convergence of Probability Measures," Wiley, New York, 1968. 2. F. BLACK, Capital market equilibrium with restricted borrowing, J. Business 45 (1972), 444-455. 3. M. BLUME AND I. FRIEND, A new look at the capital asset pricing model, J. Finance (March 1973), 19-33. 4. I. FRIEND, Rates of return on bonds and stocks, the market price of risk, and the cost of capital, Working Paper No. 23-73, Rodney L. White Center for Financial Research, 1973. 5. J. GREEN, Preexisting contracts and temporary general equilibrium, in "Essays on Economic Behavior under Uncertainty" (Balch, McFadden, and Wir, Eds.), North-Holland, Amsterdam, 1974. 6. M. ｊｅｎｓｾＬ＠ Capital markets: theory and evidence, Bell. J. Econ. and Management Science 3 (1972), 357-398. 7. J. LINTNER, The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets, Rev. Econ. Statist. (February 1965), 30--55. 8. M. LOEVE, "Probability Theory," Van Nostrand, Princeton, N. J., 1963. 9. S. MYERS, A reexamination of market and industry factors in stock price behavior, J. Finance (June 1973), 695-705. 10. J, PRATT, Risk aversion in the small and in the large, Econometrica 32 (1964), 122-137. 11. H. RAIFFA, "Decision Analysis," Addison-Wesley, Reading, Mass., 1968. ' It is not difficult to construct counterexamples by having U(o) go to - oo rapidly enough as x approaches its lower bound.

30

SA Ross

360

STEPHEN A. ROSS

12. S. Ross, Comment on "Consumption and Portfolio Choices with Transaction Costs," by E. Zabel and R. Multherjee, in "Essays on Economic Behavior under Uncertainty" (Balch, McFadden, and Wir, Eds.), North-Holland, Amsterdam, 1974. 13. S. Ross, Portfolio and capital market theory with arbitrary preferences and distributions-The general validity of the mean-variance approach in large markets, Working Paper No. 12-72, Rodney L. White Center for Financial Research, 1971. 14. S. Ross, Return, risk and arbitrage, in "Risk and Return in Finance" (Friend and Bicksler, Eds.), Ballinger, Cambridge, Mass., forthcoming. 15. W. SHARPE, Capital asset prices: A theory of market equilibrium under conditions of risk, J. Finance (September 1964), 425-442. 16. .T. TREYNOR, Toward a theory of market value of risky assets, unpublished manuscript, 1961.

Chapter 2 In R Cont (Ed.), Encyclopedia of Quantitative Finance, 2, 792-801. New York: Wiley.

The Fundamental Theorem of Asset Pricing 'Walter Schachermayer Univer-sity of Vienna, Austr-ia

The subsequent theorem is one of the pillars supporting the modern theory of Mathematical Finance. Fundamental Theorem of Asset Pricing: The following two statements are essentially equivalent for a model S of a financial market: (i) S does not allow for arbitrage (NA) (ii) There exists a probability measure Q on the underlying probability space (0, :F, lP'), which is equivalent to lP' and under which the process is a martingale (EMM). \Ve have formulated this theorem in vague terms which will be made precise in the sequel: we shall formulate versions of this theorem below which use precise definitions and avoid the use of the word essentially above. In fact, the challenge is precisely to turn this vague "meta-theorem" into sharp mathematical results. The story of this theorem started - like most of modern Mathematical Finance - with the work of F. Black, M. Scholes [3] and R. :VIerton [25]. These authors consider a model S = (S1 )o 0), and such that its price 1r( m) is less than or equal to zero. The question now arises whether it is possible to extend 1r : AI ---+ JR. to a non-negative, continuous linear functional1r* : X ---+ JR.. \Vhat does this have to do with the issue of martingale measures'? This

3

34

W Schachermayer

theme was developed in detail by M. Harrison and D. Kreps [14]. Suppose that X = LP(fl, :F, IP') for some 1 ｾ＠ p < oo, that the price process S = (St)o 0 such that 11* = ｌｾ］ｬ＠ f-Ln"hn converges to a probability measure in (X, T)* = Lq(n, F, lP'), where ｾ＠ + = 1. This yields the desired extension 11* of 11 which is strictly positive on X+ \{0}. vVe still have to specify the choice of (lvf0 , 11). The most basic choice is to take for givenS= (St)o 0. jESk

Hence, LsEs,Ys

=

1, n5 /nk

=

yjR{, and Theorem 2 implies

Theorem 3 (Risk neutral valuation). If a single period risk free asset exists at each node

k r;t T, then Arbitrage A does not exist if and only probabilities 'lk > 0, for all k > 0, such that

if there

exist conditional risk neutral

(6) For all assets i, nodes k rj T, and s E S 10 denote the total return by R; = ｐｪｾ＠ from node k to s. Let Rk = (RD denote an n vector of the total returns. Then equivalently with (6), for all i,

R{ = LYsR:

'c/k r;f T.

sr;::-Sk

The risk neutral representation was first employed by Cox and and Ross (1976).

(7)

Chapter 4. Using Tucker’s Theorem of the Alternative

63

ｐｾ＠

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2287

If for each node k >t T, there is an asset (a numeraire) with a strictly positive price > 0, for all s E Sk, then PZ > 0 in (5). Hence, we may define normalized prices Pk = ｐｫＯｾ＠ and P, = ｐＬＯｾ＠ and parameters CJ, = ＨｮＬｐｾＩＯｫ＠ such that (5) becomes Pk = LsESk CJsPs. As LsESk CJk = 1, and CJ,. > 0, for all s E Sk, we interpret parameters CJ, as conditional probabilities. Theorem 2 implies the following result stating that the normalized price process under no arbitrage is a martingale. Theorem 4 (Martingale valuation). If at each stage an asset exists with a strictly positive return, then Arbitrage A does not exist if and only if there exist conditional probabilities CJk > 0, for all k > 0, under which the normalized price process is a martingale; i.e.,

pk =

L

CJsPs

\:/k

>t T.

(8)

sESk

This result in a much more general framework, including vector ditlusion processes for the price vector, was developed in Harrison and Kreps (1979). 3.2. Valuation of a contingent claim

Consider a contingent claim u = (vk), where vk, fork> 0, is the cash flow provided by the claim, and for the root node, -v0 = Vis the purchase price. As an example, in case of a European put option allowing but not forcing the owner to sell a given asset at a given time and at a prespecified price, such a stochastic cash flow stream is contingent on the evolution the price of the underlying asset. For k > 0, vk is the payoff provided by the option at node k, and v0 = -Vis the price paid. A price or value of a contingent claim is admissible, if an arbitrage opportunity is not introduced while adding the claim with this price to the investment possibilities. We determine an admissible price V based on a no arbitrage assumption. Let ｾ＠ denote the amount invested in the contingent claim. Then the net cash flow vector is c = Cx + ｶｾ＠ and an arbitrage exists, if there is x and ｾ＠ such that c ? 0 and c i= 0. By Stiemke's theorem, there is no arbitrage if and only if the set

(9)

Il={nlnC=O, n>O, no= I}

of state price vectors n is non-empty and v0 is chosen such that nv Hence, under a no arbitrage assumption, for n E II

=

0, for some n

E II.

(10) Computations for V can be done recursively as well. For terminal nodes k E T, define vk = 0. For other nodes k, define vk = L,·ES,, (n,/nk)(v, + V,), or equivalently using risk neutral valuation (6), Vk = (1/ROL,EsJ,(v, + V,). Then the value in (10) is V= V0 . In special cases, duplicated computations can be avoided by applying the recursion in a lattice rather than in a tree; see for instance the option pricing by Cox et al. (1979). If u = Cx for some x and v0 , then v is replicated by the portfolio strategy x. Let Ck denote row vector k of C, for k = 0, 1, ... Hence C0 is the first row in C. Then v = Cx and n E II implies nv = nCx = 0, and V = -v 0 = - C0 x. Hence, if replication is possible and there is no arbitrage, then the value V = -C0 x is the initial investment needed for replication and this value is the same for all replicating portfolio strategies x.

64

M Kallio and WT Ziemba M. Kallio, W T Ziemba I Journal of Banking & Finance 31 ( 2007) 2281-2302

2288

The market is complete, if any contingent claim v can be replicated. The following observation follows from elementary linear algebra applied to nC = 0; i.e. Lk>onkCk = -no Co. Lemma 2. Assume the set II in (9) is non-empty. Then

II consists of a single state price vector n

vectors

c," .fln· k > 0, are linearly independent

the market is complete.

As a special case, if a non-singular matrix results from C when the row vector C0 of the root node is omitted, then the state price vector n E II is unique, and the market is complete. However, it is possible, that a contingent claim v can be replicated even if the market is incomplete as the following result shows. Theorem 5. If no arbitrage opportunities exist, then any admissible price V of a contingent claim u satisfies v- ｾ＠ v ｾ＠ v+, where v- and v+ are obtainedfrom the linear programming problems

L nkVk, v- =mil]. L nkvh v+

=

ｭｾ＠

nEil

nEn

( 11)

k>ll

(12)

k>O

where TI is the closure of IT. If v can be replicated, then

v- = v+ = V

The replication principle for valuation was first employed by Black and Scholes ( 1973) for option pricing. By Lemma 2, if the market is complete, replication is possible and there is a single price vector in II. In case v- < v+, any price V E ( v-, V+) is admissible. However, the end points v- and v+ may or may not be admissible; see Karatzas and Kou (1996). The duals of (11) and (12) are (13) v+ =min{ -c0 lc = Cx,ck ? vk'Vk > 0}, X

v- =max{ -c0 lc =

Cx, ck ｾ＠

vk'Vk

> 0}.

(14)

X

Hence, the bound v+ is equivalently given by the smallest initial investment -c0 of a portfolio strategy x for which the net cash flow c superreplicates the contingent claim; i.e. ck ? Vto for all k > 0. Similarly, a subreplicating portfolio strategy with ck ｾ＠ vk defines v- in (14). Bertsimas et al. (2001) proposed an E-arbitrage approach for approximate replication, which can provide option prices free of arbitrage in an incomplete market. 3.3. General option 1:aluation

Consider an option allowing choices a in a feasible set A. If a strategy a E A is chosen for executing the option, then a contingent claim v(a) = (vk(a)) is obtained. Assume, that

Chapter 4. Using Tucker’s Theorem of the Alternative

65

M. Kallio, W T. Ziemba I Journal of Banking & Finance 31 (2007) 2281-2302

2289

is the cash flow at the set of feasible contingent claims v(a) is compact. For k > 0, ｶｾ｣Ｈ｡Ｉ＠ node k, and for the root node, V0 = -v0 , independent of a, is the price of the option. For equilibrium prices, besides no-arbitrage, we also assume that each buyer of the option is not willing to pay more than the smallest initial investment needed in a portfolio strategy whose cash flow c superreplicates any contingent claim v(a) (i.e., ｣ｾ＠ ｾ＠ ｶｾ｣Ｈ｡ＩＬ＠ fork> 0), for a E A; A price higher than such value does not necessarily provide an arbitrage opportunity, because the seller may not know, which strategy a E A the buyer chooses. Suppose that v(a) can be replicated, for all a E A; e.g., the market is complete. Then the value of the option is uniquely determined with any n E II as

2.:= nkvk(a).

Vo = ｾｸ＠

(15)

k>O

To see this, recall that Lk>onkvk(a) is the investment needed to replicate v(a), and let a* E A denote a maximizing strategy in (15). If the option price is less than V0 , then buying one option and selling short a replicating portfolio of v(a*) provides an arbitrage opportunity. On the other hand, by our assumption, no agent is willing to pay a price higher than V0 . If v( a) cannot be replicated, for all a E A, then upper and lower bounds Vri and V 0, respectively, are obtained for V0 (16)

and (17)

The arguments are as above. Let a* E A denote a maximizing strategy in (17). By ( 12) and (14), V 0 is the maximal investment subreplicating v(a*). Hence, if the option price is less than V0, then selling short such a portfolio subreplicating v(a*) and buying the option leads to an arbitrage. Furthermore, no agent is willing to pay a price higher than Vri, because any contingent claim z;(a), for a E A, can be superreplicated at a cost not exceeding Limits analogous to (16) and (17) were first proposed by Harrison and Kreps (1979). The upper limit in (16) is found by optimization, while the lower limit in (17) is obtained as a saddle value, provided one exists (Rockafellar, 1974, p. 3).

vt.

3.4. Impact ofperiodic dividends and interest payments

For each node k tf. T and for all s E ｓｾ｣Ｌ＠ let D, denote a vector of dividends per share and interest payments. Then the net cash flow vector c = Ｈ｣ｾＩ＠ is defined by - Poxo

=

+ Dk)xk (Pk + Dk)xk (Pk

for k

c0 =

ck

=

0,

\fk E T,

- Pkxk = ck

otherwise

or by (4) with matrix C including dividends and interest payments; see Fig. 3. Theorem 2, with definition of Arbitrage A, remains valid with Eq. (5) replaced by nkPk = z=n,(D, +P,) ｳｅｓｾｲ＠

\fk

tf. T.

(18)

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M Kallio and WT Ziemba M. Kallio, W. T. Ziemha I Journal of Banking & Finance 31 (2007) 2281-2302

2290

-Po D, +P1

C=

-P, D3+P3

Fig. 3. The matrix C in (4) with dividends and interest payments for the scenario tree of Fig. I.

4. Imperfect markets

This section introduces various imperfections in the model. The first subsection discusses proportional transaction costs for selling and buying. Other imperfections might be studied similarly one by one. However, to shorten the discussion, we directly study the simultaneous impacts of several market frictions. Taxes are not discussed. Our simple activity analysis does not apply to account for taxes in a serious way. There are several reasons for this: rules of tax laws may involve nonlinearities, discontinuities etc.; taxation is country (region) dependent; and taxation rules change occasionally over time. 4.1. impact of transaction costs

In this subsection, we assume proportional transaction costs of buying and selling. For simplicity, transaction costs of closing terminal positions (converting terminal wealth into cash) are omitted. Such an omission may be justified by using a utility function accounting for terminal wealth. The impact of relaxing this assumptions is discussed in Section 4.2. Furthermore, transaction costs of short selling any given quantity are assumed the same as transaction costs of selling the same quantity to reduce long position. Similarly, transaction costs of reducing short positions by any given quantity are assumed the same as transaction costs of buying the same quantity to increase long positions. The vector of prices at node k is Pk. Let the vector Yt ): 0 be the quantities bought and the vector Yi: ): 0 be the quantities sold at node k, for all k if_ T. The quantity Yt can be interpreted as an increase in long position or as a reduction in short position. Similarly, the quantity Yi: can be interpreted as a reduction in long position or as an increase in short position. It will become clear shortly, that the outcome of our analysis is independent of such alternative interpretations. For all nodes k and instruments i, assume that the fractional shares c);k and A;k, with 0 :::_: 6;10 A;k < 1, are incurred in transaction costs while buying and selling, respectively. Defining diagonal scaling matrices

yields the price vectors at node k, including transaction costs, for buying and selling, respectively, as

(19) The net cash flow at node k is Ck

= -Ptyt

+ Piji:

Vk if_ T,

(20)

Chapter 4. Using Tucker’s Theorem of the Alternative M. Kallio, W. T. Ziemba I Journal of Banking & Finance 31 (2007 j 2281-2302 -

p_0

P.+ 0

-P{

p-

p3 p4

-P3 -P•

1

-

C=

p3 p4 Ps p6

-P3 -P• -Ps -P6

p,+ 2

p5 p6

67 2291

p,2

-Ps -P6

Fig. 4. The matrix C in (22) with transaction costs for the scenario tree of Fig. I.

and the wealth at a terminal node k is Ck = pk

2)Yt- y;:)

(21)

Vk E T,

kEBk

where Bk is the set of nodes preceding k E Tin the path from terminal node k to the root node 0. The net cash flows ck both in (20) and in (21) are the same irrespective of quantities y;: being interpreted as short sales or not. In matrix notation, the net cash flow vector c having one component for each node is (22)

Cy=c,

where y = (y;:, y;:) ｾ＠ 0 is a vector with a pair y;, y;: of subvectors for each k ｾ＠ T, and Cis a matrix defined by (20) and (21); see Fig. 4. Arbitrage B. An arbitrage opportunity exists if c ｾ＠ 0 and c =f. 0 in (22), for some vector y ｾ＠ 0. If Llk = Ak =I, then Arbitrage B is equivalent to Arbitrage A. To study the case with transaction costs we employ the following special case of Tucker's Theorem 1 with A 2 =I and A 3 void. Lemma 3. For each given matrix A 1, either Atz ｾ＠

0,

Atz

ic 0

and z

ｾ＠

0 has a solution

or nA 1

ｾ＠

0 and n > 0 has a solution

but never both.

If an Arbitrage B does not exist, then Lemma 3 implies existence of a state price vector > 0, with n0 = 1, such that nC ｾ＠ 0. Taking into account the structure of C in (22), yields n = (nk)

nkP;: ｾ＠

L n,P, ｾ＠

nkPt

Vk ｾ＠ T,

(23)

tETk

where Tk c T is the set of terminal nodes succeeding node k. Hence, Theorem 6. Arbitrage B does not exist k, with n 0 = 1, ｳ｡ｴｩｾｦｹｮｧ＠ (23).

if and only if there exists a node price ｮｾ｣＠ > 0, for

all

This result is shown by Naik (1995) assuming that a risk free asset exists and there are no transaction costs on this asset. If b;k = ).;k = 0, for all i; i.e. P;: = Pt = Pk, then (23) is equivalent with recursive application of (5). For transaction cost parameters b;10 A;k > 0, Theorem 6 involves a set of positive state price vectors n that can be used for valuation

68 2292

M Kallio and WT Ziemba ,71,[, Kallio,

W: T. Ziemha I Journal of Banking & Finance 31 (2007) 2281-2302

of contingent claims employing standard software; see e.g., Murtagh and Saunders (1978). Valuation formulas (ll) and (12) and (16) and (17) for contingent claims and general options in Section 3.2 and Section 3.3 remain valid if the set of state price vectors II in (9) is replaced by the set of vectors n satisfying (23) with n > 0, and n 0 = 1. Similarly as in (13) and (14), duality can be applied to obtain bounds v- and v+ on the price V of a contingent claim with a cash flow vector v = (vk). In this case, v+ is the smallest initial investment needed in an investment strategy defined by y > 0 satisfying (22) such that c superreplicates v; i.e. ck > v"' for all k > 0. The lower bound v- is the largest initial net cash flow c0 created by such an investment strategy y such that -c subreplicates v; i.e. -ck :( Vk, for all k > 0. For Arbitrage B, we obtain the following discrete equivalent of the result of Jouini and Kallal (1995a). While Jouini and Kallal characterize valuation of contingent claims using martingale measures, they do not explicitly consider evaluation.

?g

> O,for all k, Proposition 1. If there is an asset (a numeraire) with a strictly positive price then the absence of Arbitrage B is equivalent to the existence of an equivalent probability measure that transforms some process Pk between the bid and ask price processes PJ: and Pt of traded securities (after normalization) into a martingale. Proof. For a state price vector n > 0 with n 0 = 1 satisfying (23) define a price process Pk as follows. For each terminal node k E T, h = Pk. For all other nodes k, define Pk such that ｾｫｰ＠ = 'E_,.Es,n,P,. Recursively we obtain nkPk = 'E_,ET,n,P,. Hence, by (23) PJ: :( Pk :( PJ:. Repeating the arguments for Theorem 4, given there is an asset (a numeraire) with a strictly positive price ｐｾ＠ > 0, for all k, define normalized prices Pk = ｐｫＯｾ＠ and parameters u, = ＨｮＬｐｾＩＯｫＮ＠ Then Pk = Lscs,usPs, Lscs,uk = 1, and u,>O. Hence, the normalized price process Pk is a martingale. Consequently, if an Arbitrage B does not exist, then there is a price process F\ with PJ: :( F\ :( PJ:, such that the normalized price process Pk is a martingale. Conversely, suppose that a price process Pk and the probabilities 1'55 > 0 exist such that PJ: :( Pk :( Pt, ｐｾ＠ > 0 for the numeraire process, and the normalized price process Pk is a martingale. Then forward recursion ns = ｵＬｮｫｐｚＯｾ＠ for s E Sk, with n 0 = 1 yields state prices nk > 0 satisfying (23). D

4.2. Combined impact of impe1jections

Next, consider simultaneously several market imperfections: (i) transaction costs as above, (ii) an interest rate spread between borrowing and lending, (iii) charges for short positions, and (iv) restricted short selling. As in Napp (2003), additional restrictions on the net positions can be easily imposed. For example, limits may be given on the value of net positions of some class of assets relative to the value net positions in another class. Such constraints arc discussed but excluded from the model, because of their subjective nature. King (2002) studies a pricing model of imperfect markets with proportional transaction costs, restricted short selling and charges for shorting. However, periodic holding costs for short positions or benefits (interest and dividend payments) for long positions are excluded. We discuss in detail the imperfect market model on King in Appendix. Edirisinghe et a!. ( 1993) study superreplication of a contingent claim under imperfect markets in a binary tree framework involving a risky asset and a riskless bond. They propose a

Chapter 4. Using Tucker’s Theorem of the Alternative M. Kallio, W T. Ziemba I Journal of Banking & Finance 31 (2007) 2281-2302

69 2293

dynamic programming approach allowing nonlinear transaction costs, lot-size constraints, and position limits for trading. As in Section 4.1, we assume proportional transaction costs of buying and selling. Transaction costs of short selling are assumed the same as the transaction costs of reducing long positions, and transaction costs of reducing short positions are assumed the same as transaction costs of buying. However, we now apply an additional holding cost for short positions and restrict short positions relative to long positions. Unlike in Arbitrage B, for the definition of an arbitrage, all wealth at terminal nodes may be converted into cash taking into account transaction costs. We also include dividends and periodic interest payments for lending and borrowing as in Section 3.4. Independence of imperfections is not assumed. Again, for each node k, Pk is the vector of prices at node k. Price vectors Pt and F;, including transaction costs, are defined by (19) such that Pf: ? Pf:. The vector Yt ? 0 denotes the quantities bought and the vector y;: ? 0 the quantities sold at node k, for all k rf_ T. Again, the vector Yt is interpreted as an increase in long positions or as a reduction in short positions and yf: is a reduction in long positions or as an increase in short positions. It is unnecessary to keep track of these alternative interpretations. Let xk = xt - x;: denote the vector of quantities held in each instrument at node k, with xt ,x;: ? 0 referring to long and short positions, respectively. For each k rf_ T, and for each and D; as follows. For long positions x;:, ? 0 is the vector of s E S,o define vectors cash value per share provided by all instruments. This vector consists of dividends and interest, and if s is a terminal node, then also the selling price P; is included inn;. For short positions x;:, D; is composed of dividends, interest, shorting costs. For terminal nodes, the buying price is also included. The borrowing and lending components in D; and D; are interpreted as interest payments. Interest rates for short positions are assumed to be at least as large as for long positions. The shorting cost can be infinite, if the short position is prohibited. Hence, for all s =I= 0, assume that

n;

n;

P;

(24) The net cash flow at the root node is (25)

For nodes k > 0 and k rf_ T ck =

-Ptyt + Pf:yf: + Dtxt - Df:xf: .

Finally, for terminal nodes k ck =

E

(26)

T

Df:xt - Df:xf: .

(27)

Hence, the net cash flow vector c having one component for each node is

Ax+By = c,

(28)

where x = (xt, x;:) ? 0 is a vector with a pair x;:, x;: of subvectors for each k rf_ T, y = (yt,y;:) ? 0 is a vector with a pair yt,y;: of subvectors for each k rf_ T, and A and B are matrices defined by (25)-(27).

The quantities held at the root node 0 satisfy

-xt + xo +Yo -Yo

=

o,

(29)

70

M Kallio and WT Ziemba M. Kallio, W T Ziemha I .Journal of Banking & Finance 31 (2007) 2281 2302

2294

and for all other nodes k

-xt + xT;

Ft T

+ xt - xT; + Yt - Yi = 0.

(30)

ln matrix notation, Cx+Dy=O,

(31)

with C and D defined by (29) and (30). For nodes k Ft T, a margin requirement limits short positions PtxT: to a share p::;; 1 of long positions PT;xt; i.e.

>o

pPT;xt- Ptxz

'ik

et T.

(32)

For p = 1, (32) requires that the total liquidation value of all assets and liabilities is nonnegative; i.e., (32) represents a solvability condition. ln matrix notation (32) is Ex ): 0.

(33)

Obviously, redefining the matrix E in (33), we can account for additional restriction on the positions xt and xz. For example, as in Napp (2003), we may restrict positions to be nonnegative or non-positive, the value of net positions in some class of assets (such as bonds) may be restricted by a given share of total net value, the net number of some asset in short position (e.g., a European call) can be restricted by the net number of another asset in long position (e.g., the underlying stock). While the consequence of such modifications in Theorem 7 below is straightforward, it is omitted because the data involved in such constraints is largely subjective. The structure of the model of (28)-(33) is shown in Fig. 5. Arbitrage C. An arbitrage opportunity exists if c ): 0 and c =I= 0 in (28), for vectors x ): 0 andy ): 0 satisfying (33) and (31). ln a perfect market Arbitrage C is equivalent to Arbitrage A. At each node k, (31) allows long and short positions to be incremented by the same quantity, but we must ensure that both xt and X/: remain non-negative. Reasonable interpretations exist. For example, one might simultaneously reduce both long and short position without transaction costs. On the other hand, we might consider a revised model (28), (31) and (33), defining the increments in long and short positions by separate vectors. Such a revised model, which is shown in Appendix, exhibits an arbitrage if and only if the original model does.

x()

xt D+1 D+2

-I I I pPo

xi

x;-

xi

X2

Yt Yo -Po

-Dl -

-

D-2

Yi Y2

p+ p1 1 -

D+ -D:J 3 D+ -D4 4

I -I -I

Yi Y!

Po

p,2

I

-I

D+5 -D[; D+6 -D(;

I -I

p,+ 2

I

-I I

-I

I

pP2-

_p2+

-I

-Po pP,-

-P,+

Fig. 5. The structure of the model of (28), (33) and (31) for the scenario tree of Fig. I.

Chapter 4. Using Tucker’s Theorem of the Alternative M. Kallio. WT. Ziemba I Journal of Banking & Finance 31 (2007) 2281-2302

71 2295

To analyze the case with mixed imperfections, we employ another special case of Tucker's Theorem 1. Lemma 4. Let A 1 , A 2 and A 3 be given matrices, with A 1 non-vacuous. Then Atz;? 0,

Atz

# 0,

A2z;? 0,

z ;? 0,

and A3z = 0 has a solution z

or nA 1 + vA 2 + pA 3 ｾ＠

0,

n

> 0,

v ;? 0 has a solution n, v, p

but never both.

If Arbitrage C does not exist, then Lemma 4 with A 1 = [A, B], A 2 = [E, OJ and A 3 = [C,D] with submatrices A, B, C, D and E in (28), (33) and (31), implies existence of a state price vector n = (nk) > 0, with n 0 = 1, a vector v = (vk) ;? 0, and a vector p = (!lk), with components vk and subvectors Ilk. for all k (j!. T, such that nA

+ vE + pC ｾ＠

nB+ pD

ｾ＠

(34)

0,

(35)

0.

Taking into account the structure of matrices A, B, C, D and E, (34) becomes (36) with Ps

=0, for s

nkYi: ｾｉｬｫ＠

ｾ＠

E

T, and (35) becomes

nkPt

Vk (j!. T.

(37)

Hence, we conclude with the following new result: Theorem 7. Arbitrage C does not exist if and only if there exists a node price vector n v ;? 0 and p satisfying (36) and (37), with n0 = 1.

> 0,

We discuss valuation procedures shortly. To help understand and interpret Theorem 7, we first discuss a number of special cases, starting with the most simple ones. In these cases the margin requirement is omitted, so that vk is zero in (36). (a) The basic model (Section 3.1). We haver;= Pt = Pk. for all k, D'i: = Dt = 0, for = Pk, for k E T. The condition (37) implies flk = nkPk so (36) k (j!. T and DJ; =

n;

yields the basic state price valuation result (5). (b) The model with dividends and interest payments (Section 3.4). Again PJ; = Pt = Pk, for all k. For the cash payments, DJ; = Dt = Dk. fork (j!. T, and DJ; = Dt = Pk + Dk, fork E T. Then, condition (37) implies flk = nkh· Hence, (36) yields the result (18). (c) The model with transaction costs (Section 4.1). Let PJ; ｾ＠ Pk ｾ＠ Pt and DJ; = Dt = 0, fork (j!. T, and PJ; = Pt = Pk =Dr; = Dt, for terminal nodes k E T. Then, fork such that Sk c T, condition (36) implies Jlk = ｾｳｆＬ＠ n,P,.. For all other nodes k (j!. T, (36) so that with backward recursion p, = ｾｴｅｔｫ＠ n 1P,. This with implies Ilk = ｾｳｅｫｬＬ＠ (37) implies (23). Related models have been studied by Jouini and Kallal (1995a) (for more details, see Section 4.1 above), Leland (1985), Zhao and Ziemba (2007), Boyle and Vorst (1992) and Gilster and Lee (1984).

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(d) Restricted short and long positions with interest rate spread. This model is a discretized version of the model of J ouini and Kallal ( 199 5b). Consider a finite set of marketed instruments i with prices ｐｾＮ＠ There are no transaction costs so that Pi = Y;; =Ph for all k. Instrument i = 0 refers to lending and borrowing. We set PZ = 1, for all k. Fork rf. T, the interests of lending and borrowing are paid in cash and given by DZ I = rt and = rr;' respectively. For k E T, the face value is included as well, so that ｄｾＫ＠ = 1 + ｲｾ＠ and = 1 + rr;. To avoid arbitrage, ｲｾ＠ :( rr;. For all other instruments i, either short positions are prohibited with ｄｾＭ = oo or long positions are prohibited with ｄｾＫ＠ = -oo. In the former case we set ｄｾＫ＠ = 0, for k rf. T, and ｄｾＫ＠ = ｐｾＬ＠ for k E T, and in the latter case ｄｾＭ = 0, for k rf. T, and ｄｾＭ =Pi, fork E T. lfboth short and long positions are possible for some securities, such instruments are duplicated to fit in the model. lf there is no Arbitrage C, then condition (37) implies Jlk = nkPk, for all k rf. T. For i = 0, (36) yields ｾｳｅｓＬ＠ n,(l + r;) :( nk :( ｾｳｅ Ｑ Ｌ＠ n,(l + r;), for all k rf. T. Hence, for some interest rate process h, such that rt :( h :( rr;, we have nk = ｾｳｅｓ｣ｮＬＨｬ＠ + r,), or equivalently, 1 ::::' ｾｳｅＬ＠ O"" where O", = ( 1 + rs)n,/"!.k > 0. Define recursively a numeraire process Rk as follows. For the root node, R 0 = 1, and for other nodes, Rk = (1 + rk)Rk , where k_ denotes the predecessor node of node k. Then the nor= oo then malized price process is Pk = Pk/Rk. For instruments i =I= 0, if ｄｾＭ nkPi ;? ｾｳｅｓ｣＠ n,P; by (36), or equivalently, ｐｾ＠ ;? ｾｳｅｓＬ＠ ｏＢＬｐｾ［＠ i.e., ｐｾ＠ is a supermartingale. Similarly, if ｄｾＫ＠ = -oo, then ｮｫｐｾ＠ :( ｾｳ･｣ｓ＠ ｮＬｐｾ＠ and Pic :( ｾｳ･｣Ｌ＠ ｏＢＬｐｾ＠ so that P/, is a submartingale. It is straightforward to show the converse. Hence in conclusion, an Arbitrage C does not exist if and only if there is some interest rate process h, with rt :( h :( rr;, an associated numeraire process Rk. and an equivalent probability measure such that the normalized price processes ｐｾＬ＠ fori =I= 0, are supermartingales or submartingales. This is the main result in Jouini and Kallal (1995b). (e) The model with holding costs and benefits; no transaction costs. Assume Pi. = Pi = Pk and DJ. ;? Di, for all k. In this case, different interest rates for lending and borrowing as well as costs for short positions may be included in the model. Such a case in continuous time has been studied by Bergman (1995). Condition (37) implies Jlk = nkPk. Hence, for k such that sk c T, (36) implies ｾｳｅｓｫｮＬｄＭＧ［＠ :( nkPk :( ｾｳｅｓＬｮｄＮ＠ For other nodes k rf. T, (36) implies ｾｳｅｓ｣ｮＬＨｄＭＧ［＠ +P,) :( nkPk :( ｾｳｅｓＬ＠ ｮＬＨｄｾ＠ + P,). For terminal nodes k E ｾ｢ｯｴｨ＠ Di and DJ. ｩｮｾｬｵ､･＠ the price comｰｯｮ･ｾ＠ Pk. Hence, there exists a process Dk such ｴｾ｡＠ n,; :( Dk :( DJ., for k rf. T, D,; :( Dk + Pk :( DJ., for k E T, and nkPk = ｾｳ･｣＠ n,(D, + P,), for k rf. T. The latter equation provides a frictionless condition (18) for the absence of Arbitrage C. For further discussion, see case (f) where (e) is a special case. (f) The model with transaction costs, and holding costs and benefits. Assume PI_ :( Pk :( Pi and n-;; ;? DJ., for all k. ln this case, define l\ such that nkPk = JLk, fork rf. T, and Pk = Pk. fork E T. Then condition (37) and our assumption imply Pi. :( Pk :(Pi, for all k. Hence, for k such that Sk c T, (36) implies ｾｳｅＬｮｬＮ｟ＡＭＧ［＠ :( nJ) :( ｾｳｅｓ｣ｮｄ［Ｎ＠ F£r other nodes k rf. T, (36) implies ｾｳｅｩｨｮＨｄＭＧ［＠ + Ps) :( nkPk :( ｾｳｅｓＬ＠ ｮＬＨｄｾ＠ + P,). Consequently, absence of Arbitrage C implies existence of a pro-

nz-

cess jjk with Di :( jjk :( Dt;, for k

nz-

rf. T, Dt;

:( jjk

+ h :( Dt;, fork E

T, such that

Chapter 4. Using Tucker’s Theorem of the Alternative M. Kallio, W. T Ziemha I Journal

o/ Banking & Finance 31 (2007) 2281-2302

73 2297

This equation is a frictionless condition (18) for the absence of Arbitrage C One may readily check, that the converse is true with flk = nJ; b for k rf_ T, in Theorem 7, In conclusion, an Arbitrage C does not exist if and only if there is n > 0 with n 0 = 1, and a price process F\ with PJ; ｾ＠ F\ ｾ＠ Pt satisfying (38) for some process lh such that Dt ｾ＠ lh ｾ＠ DJ;, for k rf_ T, and Dt ｾ＠ lh + i\ ｾ＠ DJ;, for k E T Of course, interpretation of (38) in terms of risk neutral or martingale valuation is straightforward. Hence our conclusion extends earlier results. We now return to the general case of Theorem 7, Similarly as before, valuation formulas (11), (12) and (16), (17) for contingent claims and general options in Section 3.2 and Section 3,3 remain valid if the set of state price vectors II in (9) is replaced by the set of vectors n satisfying (36) and (37) with n > 0, n 0 = 1, and v ;_, 0; for an illustration, see Example 1 below. Optimization codes, such as Minos (Murtagh and Saunders, 1978), can compute bounds v- and vt on the price of a contingent claim. Because II is a convex set, every price outside the interval [ v-, 0] leads to an Arbitrage C, and there is no Arbitrage C for price levels in the interior of this interval; see also Karatzas and Kou ( 1996), As in ( 13) and (14 ), duality can be applied to obtain bounds v and vt on the admissible prices V of a contingent claim with a cash flow vector v = ( ｶｾ｣ＩＬ＠ The upper limit 0 is the smallest initial investment needed in an investment strategy defined by x ;_, 0 and for all y ;_, 0 satisfying (28), (33) and (31) such that c superreplicates v; i.e. ck ;_, ｶｾｯ＠ k > 0. The lower limit v- is the largest initial net cash flow c0 created by such an investment strategy x andy such that -c subreplicates v; i.e. -ck ｾ＠ ｶｾ｣Ｌ＠ for all k > 0. However, the replication result of valuation in frictionless markets is no longer valid; a result first pointed out by Bensaid et al. (1992). Example 2 demonstrates the case. We now show that a frictionless price is in the price interval obtained with frictions, Suppose an Arbitrage C does not exist in a frictionless case (b), including (a) as a special case, Consider n > 0 with n 0 = 1 and Jlk in Theorem 7 satisfying (36) and (37), Then Jlk = nkPk by (37) and nkPk = LsES, n,(D, + P,) by (36), fork rf_ T Next, consider the case (f) with frictions such that PJ; ｾ＠ Pk ｾ＠ Pt, for all k, Dt ｾ＠ Dk ｾ＠ DJ;, for k rf_ T, and Dt ｾ＠ Dk + Pk ｾ＠ DJ;, fork E T Then nkPJ; ｾ＠ nkPk = flk ｾ＠ nkPt so that (37) holds for case (f). To check that (36) holds as well, we obtain for k such that ｓｾ｣＠ C T, LsESkn.,D-'; ｾ＠ LsESk ns(D, + Ps) = nkPk = flk ｾ＠ LsES, nsD;, and for all other k rf_ T, LsES, ( nsD; + Jl,) ｾ＠ LsES; n,(D, + P,) = nkPk = flk ｾ＠ LsES, (n,D; + flsl· Hence, if n and Jlk for case (b) satisfy (36) and (37) in Theorem 7, then they apply for case (f) as well. Consequently, the values (or the value, in case it is unique) obtained for a contingent claim in the frictionless case are within the bounds v and 0 obtained with transaction costs and holding costs; see Jouini and Kallal (1995a) and Karatzas and Kou ( 1996). Example 1. Consider an example of Hull (2002, pp. 395-396). A 5-month American put

option on a non-dividend paying stock, with current stock price 50 and annual volatility 40%, has an exercise price 50. The risk ti-ee annual interest rate is 10°1 v+. There is a superreplicating strategy for which the initial investment vt is less than the initial investment in any replicating strategy. Consequently, any price V based on replication results in existence of arbitrage. 5. Conclusions

Tucker's theorem of the alternative provides a simple tool to prove arbitrage results in frictionless cases and when there are transaction costs and other frictions in the finite case. The approach yields known results directly and some new results and is a useful approach for further research. Appendix. An alternative model of Section 4.2

Consider a revised model (28), (31) and (33) of Section 4.2. and define the increments in long and short positions by separate vectors. For each node k, let P; denote the price vector for shorting. As in King (2002), P; may be obtained from the selling price vector PJ: by subtracting some extra charge for shorting. Hence, we have P: ,;::; PJ: ,;::; Pt for shorting, selling and buying prices. However, if shorting charges only include holding costs of short for s E Sk), then ｐｾ＠ = PJ: as positions in each period (included in holding cost vectors ｄｾＬ＠ in Section 4.2. Hence, in this case shorting charges are modeled similarly as interest payments. For the revised model, the solvability conditions (33) remain unchanged, while the cash balance Eq. (28) and position dynamics Eq. (31) are reformulated as follows. For each k rt T, Yt ? 0 andy;: ? 0 denote the increase and decrease in long positions, respectively. For short positions, such increments are denoted by st ? 0 and sJ; ? 0. After such increments, ? 0 and x;: ? 0 are the long and short positions, respectively. Eq. (28) is replaced by cash balance equations, for all k, as follows

xt

(39)

xt_ x

For the root node 0, initial positions are = 0 = 0, and for terminal nodes k E T, positions are closed so that xt = x;: = 0, y;: = xt , s;: = x;:_ and Yt = st = 0. Eq. (31) is replaced by separate equations for long and short positions. For all nodes k we have

xt - xt + Yt - Yi:

=

0

(40)

Chapter 4. Using Tucker’s Theorem of the Alternative

75 2299

M. Kallio, W T. Ziemha I Journal of Banking & Finance 31 (2007) 2281-2302 x;i

X

-

xi

x;-

x;-

xi

y+

s;;-

y-

s+

-Po

-Po

Po

P.* 0

yj -

p-

_p+

p-

-PJ

3

4

p5 p-

I

p+ 1

p-

P.* 1

1

Yt

s2

-Pi -Pi

Y2

si

p-

P.* 2

2

-I I

-I I

-I I

I

-I I

-I I

-I I

-I pPo

sj

_p't

I -I

-

Y1

_p+

-I I

p+ 1

s;-

-I I

-I

-Po

pP1-

-P{ pP2-

-Pi

Fig. 6. The structure of the revised model of King (2002) for the scenario tree of Fig. I.

and

-x-;; + xf: +sf: - st For nodes k

=

0.

(41)

rf. T, the margin requirement for short positions is (42)

For this model, consider the case with P: = Pf: and periodic short pos1t1on costs included in holding cost vectors Df:. If an arbitrage opportunity exists in the revised model, then such an arbitrage strategy immediately defines an arbitrage opportunity in the original model of Section 4.2. On the other hand, suppose that an arbitrage opportunity exists in the original model. Then, because Pt ;;;, Pf:, Df: ;;;, Dt, for all k, and p ｾ＠ I, it is straightforward to check that reducing at each node k both positions xt and x-;; by max{xt,x;;} (the maximum component-wise), results in another arbitrage strategy. In the latter strategy, no asset is held simultaneously long and short, and hence this arbitrage provides an arbitrage in the revised model. In conclusion, the revised model, exhibits an arbitrage if and only if the original model does. Finally, we consider the model of King (2002, Section 7). Assume that asset i = 0 is a numeraire (bank account) with a strictly positive price, for all k. Suppose that all prices PZ, Pf: and Pt are given in terms of the buying price of the numeraire (using this price as the monetary unit). Additionally, assume that there are no transaction costs for the numeraire, but there may be a shorting charge. Hence, the scaled prices of the numeraire satisfy ｐｾＪ＠ ｾ＠ ｰｾＭ = ｰｾＫ＠ = l, for all k. All shorting charges are taken into account in the shorting prices so that P; ｾ＠ Pf: ｾ＠ Pt and periodic shorting costs are excluded. There are no dividend nor interest payments so that Dt = Df: = 0, for all k rf. T. For terminal nodes k E T, prices applied for closing positions define Dt = Pf: and Df: = Pt. The structure of this model is shown in Fig. 6. For this model, by Lemma 4, there is no arbitrage if and only if there exists a state prices nk > 0 for cash balance equations, for all k, with n 0 = 1, parameters vk ;;;, 0 for solvability requirements, row vectors flt for long positions, and row vectors flk' for short positions such that, for all k rf. T,

76

M Kallio and WT Ziemba M. Kallio, W T. Ziemha I Journal of Banking & Finance 31 (2007) 2281-2302

2300

(43)

fl"k ｾ＠

v"Pt

+ ＲＩｮｳｦｬｾ＠

+ ｻｬｾＩＬ＠

(44)

sESk

with {t.;

= !(

=0, for s

E T,

and

nkP;; ｾ＠ fli ＠ｾ nkPi, nkP; ｾ＠ tt."k ｾ＠ nkPi.

(45) (46)

Hence, conditions (43)-(46) are now in place of conditions (36) and (37) of Theorem 7. As in the discussion following Theorem 7, we provide the following interpretation for conditions (43)-(46). F_£r all k rf_ T, ､ｾｦｩｮ･＠ hand Pk such that flt = nkPk and fll: = nkPk. ｾｯｲ＠ all k E T, define Pk = PT; and Pk = PT;. Then, with (45) and (46). we ｨ｡ｶ･ｲ［ｾ＠ Pk ｾ＠ and ｐｾ＠ ｾ＠ Pk ｾｐｩＬ＠ for all k. Given that Di = D;; = 0, for all k rf_ T, and Di = PT; = Pk and DT; = Pt = 7\, fork E T, conditions (43) and (44) are restated as

?.;;

nkPk ;::, pvkP"J;

+L

nsP"

(47)

sESk

and

nkPk ｾ＠

vkPi

+L

nsPs.

(48)

ｳｾｓｫ＠

Let Q denote the (real) probability measure associated with the scenario tree, and for all k, let qk > 0 be the probability of node k. For all k, define parameters 'rk such that nk = qkYk· At time t, let y, denote the random parameter with realizations ｙｾｯ＠ for all k at stage t. To interpret ｙｾｯ＠ consider a bond paying 1 (the value of the numeraire) in each node at time t. Then the (scaled) value of the bond at time 0 is ｾｫｮ＠ = EQ[yJ Hence, ;, may be interpreted as a stochastic discounting factor for scaled prices. Employing the definition of Ylo (47) and (48) become

qk·rkPk

> pv"P;; + L

qdi"

(49)

sESk

and

qk'fkl\ ｾ＠

vkPt

+L

qsysPs.

(50)

sESk

In conclusion, if there is no arbitrage, then there exists price processes Pk E [P;;, Pi] and E [P;, PtJ, a stochastic discounting factory, and parameters vk ;::, 0 such that (49) and (50) hold. It is straightforward to show the converse: starting with Pk. Pk, Yk and vk, define nk = qk'Yio flt = nkPk and fll: = nkPk to satisfy conditions (43) and (46). These conclusions correspond to Theorem 5 in King (2002). To interpret this result, consider the case with solvability conditions omitted. In this case, corresponding to Theorem 4 of King (2002), the process y}, E [y,P;, y,P7J is a supermartingale and the process y,P, E [y,P;, ;,Pj] is a submartingale, where for each t, P; ｾ＠ P, ｾ＠ Pi and P; ｾ＠ P, ｾ＠ Pi denote random price vectors with realizations PJ; ｾ＠ Pk ｾ＠ Pt and P; ｾ＠ Pk ｾ＠ P;;, for k at stage t. For the numeraire, ｰｾＭ = ｐｾ＠ = ｰｾＫ＠ = 1 and ｐｾＪ＠ ｾ＠ ｐｾ＠ ｾ＠ ｰｾＫ＠ = 1, for all k. Hence, (49) and (50) yield qkyk > ｾｳｅｓＬｱｹ＠ and qkYkPZ ｾ＠

7\

Chapter 4. Using Tucker’s Theorem of the Alternative M. Kallio. WT. Ziemba I Journal of Banking & Finance 31 (2007) 2281-2302

ｱｳｹｐｾＮ＠

77 2301

Consequently, the discounting process ·y 1 is a supermartingale, y0 = n0/q 0 = ｾ＠ ＲＺｳｅｓｯｱＬｹｦＵｾ＠ ｾ＠ ｌＺ［ｫｅｔｱｹｐｾ＠ ｾ＠ L:;kETqkyk = EQ[; 1 ]. Hence, EQ[YtJ is noincreasing in t, and ｐｾＪ＠ ｾ＠ EQ[·;,] ｾ＠ 1. Consider a bond paying 1$ at time t and let /1 1 denote the inverse of the $-price of the numeraire at time t with realizations f3k for nodes k at t. Then the scaled value of the bond at t is {J 1, and the value of the bond at time 0 is V, = L:;knkfJk = EQ[{J,y 1]. If /3 1 is deterministic, then l/{3 1 is the (total) risk free return from time 0 to t, and V 1 = /3 1EQ[y 1] is a risk free discounting factor, the value of the risk free bond. Due to friction costs, V1 ｾ＠ {J 1, because EQ[y 1] ｾ＠ 1. Following Ingersoll (2006), V1 is interpreted as a subjective risk free discounting factor. I:sES

I and

ｐｾＪ＠

ｾ＠ ｐｾ＠

References Arrow, K., 1970. Essays in the Theory of Risk Bearing. North-Holland, London. Bensaid, B., Lesne, J., Pages, H., Scheinkman, J., 1992. Derivative asset pricing with transaction costs. Mathematical Finance 2, 63-86. Bergman, Y., 1995. Option pricing with differential interest rates. Review of Financial Studies 8, 475500. Bertsimas, D., Kogan, L., Lo, A.W., 2001. Hedging derivative securities and incomplete markets: An c-arbitrage approach. Operations Research 49, 327-397. Boyle, P., Vorst, T., 1992. Option replication in discrete time with transaction costs. Journal of Finance 47,271293. Bjork, T., 1998. Arbitrage Theory in Continuous Time. Oxford University Press, Oxford. Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637-654. Constantinides, G., 1976. Multiperiod consumption and investment behavior with convex transaction costs. Management Science 25, 1127-1137. Cox, J., Ross, S., 1976. The valuation of options for alternative stochastic processes. Journal of Financial Economics 3, 145-166. Cox, J., Ross, S., Rubinstein, M., 1979. Option pricing: A simplified approach . .lollrnal of Financial Economics 7, 229-264. Cvitanic, J ., Karatzas, 1., 1996. Hedging and portfolio optimization under transaction costs: A martingale approach. Mathematical Finance 6. 133-165. Cvitanic, J., 2001. Theory of portfolio optimization in markets with frictions. Tn: Jouini, E., Cvitanic, J., Musiela, M. (Eds.), Option Pricing Interest Rates and Risk Management, Handbooks in Mathematical Finance. Cambridge University Press. Davis, M., Norman, A., 1990. Portfolio selection with transaction costs. Mathematics of Operations Research 15, 676-713. Debreu, G., 1959. Theory of value. Tn: Cowles Foundation Monograph, vol. 17. Yale University Press, New Haven. Dermody, J.C., Prisman, E.Z., 1993. No arbitrage and valuation in markets with realistic transactions costs. Journal of Financial and Quantitative Analysis 28, 65-80. Dybvig, P.H., Ross, S.A., 1992. Arbitrage. In: Newman, P., Milgate, M., Eatwell, J. (Eds.), The New Palgrave Dictionary of Money and Finance. Palgrave Macmillan, London. Duffie, D., 2001. Dynamic Asset Pricing Theory, third cd. Princeton University Press, Princeton. Edirisinghe, C., Naik, V., Uppal, R., 1993. Optimal replication of options with transaction costs and trading restrictions. Journal of Financial and Quantitative Analysis 28, 117-138. Garman, M.B., Ohlson, J.A., 1981. Valuation of risky assets in arbitrage free economies with transaction costs. Journal of Financial Economics 9, 271-280. Gilster, J., Lee, W., 1984. The effect of transaction costs and different borrowing and lending rates on the option pricing model. Journal of Finance 39, 1215-1221. Harrison, J.M., Kreps, D.M., 1979. Martingales and arbitrage in multi-period securities markets. Journal of Economic Theory 20, 381-408. Hull, J., 2002. Options, Futmes and Other Derivatives, fifth ed. Prentice Hall.

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Ingersoll, J.E., 2006. The subjective and objective evaluation of incentive stock options. Journal of Business 79, 453-487. Jouini, E., Kallal, H., 1995a. Martingales and arbitrage in securities markets with transaction costs. Journal of Economic Theory 66, 178 197. Jouini, E., Kallal, H., 1995b. Arbitrage in securities markets with short-sales constraints. Mathematical Finance 5. 197-232. Karatzas, 1., 1996. Lectures on the Mathematics of Finance. American Mathematical Society, Providence. Karatzas, 1., Kou, S.G., 1996. On the pricing of contingent claims under constraints. The Annals of Applied Probability 6, 321 369. King, A., 2002. Duality and Martingales: A stochastic programming perspective on contingent claims. Mathematical Programming, Series B 91, 543-562. King, A., Koivu, M., Pennanen, T., 2002. Calibrating Option Bounds, Working Paper, Helsinki School of Economics. Luenberger, D.G., 1998. Investment Science. Oxford University Press, New York. Leland, H., 1985. Option pricing and replication with transaction costs. Journal of Finance 40, 1283-1301. Magill, M., Constantinides, G., 1976. Portfolio selection with transaction costs. Journal of Economic Theory 13, 264-271. Mangasarian, O.L., 1969. Nonlinear Programming. McGraw-Hill, New York. Merton, R., 1973. Theory of rational option pricing. Bell Journal of Economics and Management Science 4, 141-183. Merton, R., 1992. Continuous Time Finance, seconded. Basil Blackwell, Oxford and Cambridge. Murtagh, B., Saunders. M., 1978. Large-scale linearly constrained optimization. Mathematical Programming 14. 41-72. Naik, V., 1995. Finite state securities market models and arbitrage. In: Jarrow, R.A., Maksimovic, V., Ziemba, W.T. (Eds.), Finance, Handbooks in Operations research and management Science, vol. 9. Elsevier, pp. 31-64. Napp, C., 2001. Pricing issues with investment flows, applications to market models with frictions. Journal of Mathematical Economics 35, 383-408. Napp, C., 2003. The Dalang-Morton-Willinger theorem under cone constraints. Journal of Mathematical Economics 39, 111-126. Pliska, S.R., 1997. Introduction to Mathematical Finance: Discrete Time Models. Blackwell Publishers, Oxford. Prisman, E.Z., 1986. Valuation of risky assets in arbitrage free economies with frictions. Journal of Finance 41, 545-557. Rockafellar, R.T., 1974. Conjugate Duality and Optimization. Society for Industrial and Applied Mathematics, Philadelphia. Ross, S.A., 1976. The arbitrage theory of capital asset pricing. Journal of Economic Theory 13, 341-360. Ross, S.A., 1977. Return, risk, and arbitrage. In: Friend, 1., Bicksler, J.L. (Eds.), Risk and Return in Finance. Ballinger Publishing company, Cambridge, MA. Ross, S.A., 1978. A simple approach to the valuation of risky streams. Journal of Business 51, 453-475. Ross, S.A., 1987. Arbitrage and martingales with Laxation. Journal of Political Economy 95, 371-393. Stiemke, E., 1915. Uber positive Liisungen homogener linearer Gleichungen. Mathematische Annalen 76, 340342. Tucker, A., 1956. Dual systems of homogeneous linear relations. In: Kuhn, H., Tucker, A. (Eds.), Linear Inequalities and Related Systems, Annals of Mathematics Studies, vol. 38. Princeton University Press, Princeton. Zhao, Y., Ziemba, W.T., 2003. On Leland's option pricing and hedging strategy with transaction costs. Working Paper, Nanyang Business School, Nanyang Technological University. Zhao, Y., Ziemba, W.T., 2007. On Leland's option pricing and hedging strategy with transactions costs. Finance Research Letters 4, 49-58.

Section B

Utility Theory

80

Section B. Utility Theory

Decision makers in financial markets are faced with a variety of opportunities and must decide how much capital to allocate to the various assets in the opportunity set at points in time. Investing in assets produces returns (gains/losses), resulting from the changes in trading prices. The returns on a unit of capital invested in assets are uncertain, that is, the return vector is a random variable. The basic information input to the decision on how much to invest in each asset is the distribution for the return vector. In Section A, some models for the dynamics of the stochastic return vector were considered. Assuming that the return distribution is known, the investment decisions are based on preferences for changes in wealth or accumulated wealth. To structure the decision process, a theory of preferences is required. The theory of preferences concerns the ability to represent a preference structure with a real-valued function. This has been achieved by mapping it to the mathematical index called utility. To put the preference relation for an individual into a theory of utility the following axioms were proposed by Von Neumann and Morgenstern (1944) and Savage (1954). Let S be the set (possibly infinite) of alternatives for a system each having a monetary payoff with a known probability. There are four axioms of the expected utility theory that define a rational decision maker. They are completeness, transitivity, independence and continuity. Completeness: For any two alternatives A and B in S, either A is preferred to B or B is preferred to A or there is indifference between the alternatives. Transitivity: For alternatives A, B, and C, if A is preferred to B and B is preferred to C, then A is preferred to C. Continuity: For alternatives A, B and C, if A is preferred to B is preferred to C, then there exists a probability π such that B is as good as (indifferent to) πA + (1 − π)C. Independence: For alternatives A, B and C, with A preferred to B, for α ∈ (0, 1], αA + (1 − α)C is preferred to αB + (1 − α)C. If the four preference axioms are satisfied then the preference relationship can be expressed in terms of a utility function u(X(A)), where X(A) is the random monetary payoff from an alternative A and FX(A) is the distribution for X(A). Expected Utility Theorem: For any two alternatives A and B in S, A is preferred to B if and only if Eu(X(A)) > Eu(X(B)) and there is indifference iff Eu(X(A)) = Eu(X(B)). A general proof of the expected utility theorem is provided in the article by Fishburn (1969). The fact that preferences can be defined by a utility function is very useful for the analysis of preferences and the decision problem of choosing the best alternative. In following this approach it is important to keep in mind that it is assumed that decision makers satisfy the axioms in stating their preferences. This is referred to as rational decision making (Savage, 1954).

Section B. Utility Theory

81

Expected utility theory implies that rational individuals act as though they were maximizing expected utility, and that allows for the fact that many individuals are risk averse, meaning that the individual would refuse a fair gamble (a fair gamble has an expected value of zero). If X(A) is a random outcome and X(B) is a random outcome with distribution equal to that of X(A) + ε, where ε is uncorrelated noise, then X(A) is preferred to X(B) by any risk inverse
αaverter.  −1 With the  cumulative −1 −1 −1 and FX(B) , let T (α) = 0 FX(A) (p) − FX(B) (p) dp = the area distributions FX(A) between the distributions in the α tail. Then T (α) ≥ 0, 0 ≤ α ≤ 1, T (1) = 0, which follows from the greater variability of X(B) around the same mean as X(A). So there is a class of decision makers who are averse to risk as characterized by greater uncertainty. Going back to the expected utility theorem, this class has a particular type of utility function. A utility function u for which:  α  −1  −1 FX(A) (p) − FX(B) (p) d ≥ 0, 0 ≤ α ≤ 1, implies Eu(X(A)) ≥ Eu(X(B)) 0

is a concave function (Rothschild and Stiglitz, 1970). The area/integration condition implies that the utility has decreasing first derivatives (negative second derivatives). So the risk averter has a concave utility. In fact the degree of aversion at an outcome level is defined by the size of the second derivative relative to the first derivative (Pratt, 1964; Arrow, 1965). The risk aversion implied by expected utility theory has a shortcoming in that it does not provide a realistic description of risk attitudes to modest stakes. To have realistic risk aversion for large stakes produces virtual risk neutrality for moderate ones. Rabin (2000) presents a theorem that calibrates a relationship between risk attitudes over small and large stakes. The theorem shows that, within the expected-utility model, anything but virtual risk neutrality over modest stakes implies manifestly unrealistic risk aversion overlarge stakes. For example, a person who would for any initial wealth turn down 50–50 lose $1,000/gain $1,050 bets would always turn down 50–50 bets of losing $20,000 or gaining any sum. With utility function u and initial wealth of $20,000, the first bet implies on a wager of $1,000 that 0.5u(19,000) + 0.5u(21,050) < u(20,000) and therefore on a wager of $20,000, it follows that 0.5u(0) + 0.5u(220,000) < u(20,000). In this sense, expected utility theory can be misleading when analyzing situations involving modest stakes. A decision that maximizes expected utility also maximizes the probability of the decision’s consequences being preferable to some uncertain threshold (Castagnoli and LiCalzi, 1996; Bordley and LiCalzi, 2000; Bordley and Kirkwood, 2004). In the absence of uncertainty about the threshold, expected utility maximization simplifies to maximizing the probability of achieving some fixed target. If the uncertainty is uniformly distributed, then expected utility maximization becomes expected value maximization. Intermediate cases lead to increasing risk-aversion above some fixed threshold and increasing risk-seeking below a fixed threshold. There are examples of choice problems where preferences do not satisfy the axioms. The Allias paradox (Allias, 1952), the Ellsberg paradox (Ellsberg, 1961), and the Bergen paradox (Allias and Hagen, 1979) provide well known contradictions

82

Section B. Utility Theory

to the expected utility theorem. The Allias paradox has received particular attention. From the independence axiom, for alternatives A, B and C, with A preferred to B, for α ∈ (0, 1] it follows that αA + (1 − α)C is preferred to αB + (1 − α)C. However, it has been experimentally demonstrated that the preference for αA+(1−α)C over αB + (1 − α)C can be reversed depending on the value of α. The independence axiom is the key to the linearity in probabilities of the expected utility and that property is violated in the Allias paradox. In expected utility theory, the utilities of outcomes are weighted by their probabilities. It has been shown that people overweight outcomes that are considered certain relative to outcomes which are merely probable, a phenomenon which is labelled the certainty effect. There have been a variety of proposals for dealing with the violation of the independence axiom and the linearity in probabilities. One approach is Prospect Theory (PT) proposed by Kahneman and Tversky (1979). The bilinear form of expected utility is retained, but probabilities and outcomes are transformed. The value of a prospect, denoted V , is expressed in terms of two scales, π and v. The first scale, π, associates with each probability p a subjective decision weight π(p), which reflects the impact of p on the over-all value of the prospect. The second scale, v, assigns to each outcome x a number, v(x), which reflects the subjective value of that outcome. The outcomes are defined relative to a reference point, which serves as the zero point of the value scale. Hence, v measures the value of deviations or changes from that reference point. The value function is S-shaped, being convex for losses (x < 0) and concave for gains (x > 0). This idea dates to Markowitz (1952) who commented on the Friedman–Savage (1948) utility functions leading to the S-shape. The basic equation of the theory describes the manner in which π and v are combined to determine the over-all value of regular prospects. In a simple case, if (x, p; y, q) is a prospect then the value of the prospect is V (x, p; y, q) = π(p)v(x) + π(q)v(y), where v(0) = 0, π(0) = 0, and π(1) = 1. This equation generalizes expected utility theory by relaxing the expectation principle. An axiomatic analysis of this representation is provided in Kahneman and Tversky (1979). Prospect Theory has its critics. Levy and Levy (2002, 2004) cast doubt on the S-shaped value function, based on experimental results. The class of all prospect theory value functions are S-shaped with an inflection point at x = 0. Thus, v ′ > 0, v(x)′′ > 0 for x < 0, and v(x)′′ < 0 for x > 0. This is contrasted with the class of all Markowitz (1952) value functions which are reverse S-shaped with an inflection point at x = 0. Thus, v ′ > 0, v ′′ (x) > 0 for x > 0, and v ′′ (x) < 0 for x < 0. Markowitz’s function, like the prospect theory value function, depends on change of wealth. Levy and Levy (2002) define Prospect Stochastic Dominance (PSD) and the Markowitz Stochastic Dominance (MSD). If Prospect F dominates Prospect G by PSD, then F is preferred over G by any prospect theory S-shaped value function. Markowitz stochastic dominance rule (MSD) corresponds to all reverse S-shaped value functions. PSD and MSD are opposite if the two distributions have the same

Section B. Utility Theory

Figure 1.

83

Non-concave utilities.

mean: Let F and G have the same mean. Then F dominates G by PSD if and only if G dominates F by MSD. Levy and Levy (2002) conducted a set of decision making experiments to determine if the decision behavior of subjects conforms to prospect theory. The focus of their analysis is the S-shape of the value function in prospect theory and contrasted that function with the Markowitz reverse S-shape. With the stochastic dominance approach they take the weighting function as π(p) = p so the probabilities are not transformed. In each of the Levy and Levy experiments the subjects decision behavior supported the reverse S-shape value function and they considered that as evidence against prospect theory, or at least the S-shape value function. By defining stochastic dominance (PSD, MSD) based on the value function while retaining the original probability distribution, the weighting function of prospect theory is not a factor. In Kahneman and Tversky, the bi-criteria (value, probability) are both transformed. That is, the decision maker distorts both dimensions. To illustrate this affect, Wakker (2003) analyzes the Levy and Levy experiments using decision weights to transform probabilities based on assumptions of Tversky and Kahneman. The result is that the observed decision behavior in the experiments is consistent with the S-shape of prospect theory. In the experiments conducted by Levy and Levy neither of the competing gambles were preferred by second order stochastic dominance (SSD). Baltussen et al. (2006) augmented the Levy and Levy tasks with a third gamble which by SSD is preferred to either original gamble, with all gambles having the same expectation. The empirical evidence supports SSD as opposed to either PT or MSD. The support for the S-shaped function as obtained by Kahneman and Tversky (1979) is actually due to the certainty effect. Obviously, different individuals have

84

Section B. Utility Theory

Figure 2.

Risk Matrix.

different preferences. Classes of functions: concave, S-shaped, reverse S-shaped, represent contrasting perspectives on risk. There is evidence to support different utilities for gains and losses, which are defined with respect to a reference point. When considering gains and losses separately the utility is concave for gains and convex for losses (Abdellaoui, 2000, 2007) as implied by prospect theory. However, evidence indicates individuals are less sensitive to probability differences when choosing among mixed gambles than when choosing among either gain or loss gambles (Wu and Markle, 2008). Interesting perspective on risk assessment comes from the “risk matrix” used in reliability engineering, where value and probability are on a log scale (Figure 2). The classification of risk is based on the combination of outcome and probability. There are criteria for a consistent classification defined in Cox (2008). The stochastic dominance definitions provide a way to characterize consistent preferences for risk. A standard technique for defining risk preferences is meanvariance analysis (Markowitz, 1952). Assuming a normal distribution for outcome returns, the mean-variance rule is consistent with expected utility and second order stochastic dominance (concave utility). So MV and Prospect theory are not compatible when considering the preferences between two alternatives. However, Levy and Levy (2004) establish that the efficient sets (un-dominated) from second order dominance (MV) and prospect stochastic dominance are almost identical when dealing with a mixture of sets of alternatives (diversified portfolios). The value and utility functions to this point have been atemporal, being concerned with preferences between alternatives. In financial decision making the system is dynamic and the alternatives consist of outcomes such as consumption at points in time. For example, in discrete time the outcome could be the consumption stream C = (c1 , . . . , ct , . . . , cT ). The standard approach to valuing C is to

Section B. Utility Theory

85

T have V (C) = t=1 ρt−1 u(ct ), for the discount factor ρ. With this form of intertemporal additive and homogeneous utility, the expected utility theory translates readily. However, the problems giving rise to non-expected utility apply to this intertemporal format. An additional problem is that the two distinct aspects of preference, inter-temporal substitutability and relative risk aversion, are intertwined; indeed the elasticity of substitution and the risk aversion parameter are reciprocals of one another. Epstein and Zin (1989) consider the inter-temporal utility issues. They define a general class of preferences which is sufficiently flexible to permit those two aspects of preference to be separated. The utility V is recursive, so that V satisfies the following equation: V (c1 , . . . , cT ) = W (c1 , µ(c2 , . . . , cT )), where W is an increasing aggregator function and µ is a certainty equivalent. They require that the aggregator function have a constant elasticity of substitution. The certainty equivalent can be any member of a broad class of mean value functionals. A form of continuity is required of the functional. Epstein and Zin (1989) establish the existence of the inter-temporal utility V following from the aggregator W and functional µ.

References Abdellaoui, M (2000). Parameter-free elicitation of utility and probability weighting functions. Management Science, 46(11), 1497–1512. Abdellaoui, M, H Bleichrodt and P Corina (2007). Loss aversion under prospect theory: A parameter-free measurement. Management Science, 53(10), 1659–1674. Allais, M (1953). Le comportement de l’homme rationnel devant le risque: Critique des postulatset axiomes de l’´ecole Am´ericaine. Econometrica, 21(4), 503–546. Allais, M and O Hagen (1979). Expected Utility Hypotheses and the Allias Paradox. Dordrecht, Holland: Reidel. Arrow, KJ (1965). The theory of risk aversion. In YJS Helsinki (Ed.), Aspects of the Theory of Risk Bearing. Baltussen, G, T Post and PV Vliet (2006). Violations of cumulative prospect theory in mixed gambles with moderate probabilities. Management Science, 52(8), 1288–1290. Bordley, R and C Kirkwood (2004). Preference analysis with multi-attribute performance targets. Operations Research, 52, 823–835. Bordley, RF and M LiCalzi (2000) Decision analysis using targets instead of utility functions. Decisionsin Economics and Finance, 23(1), 53–74. Castagnoli, E and M LiCalzi (1996). Expected utility without utility. Theory and Decision, 41, 281–301. Cox, LA (2008). What’s wrong with risk matrices? Risk Analysis, 28(2), 497–512. Ellsberg, D (1961). Risk, ambiguity, and the savage axioms. Quarterly Journal of Economics, 75(4), 643–669. Epstein, LG and SE Zin (1989). Substitution, risk aversion and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica, 57(4), 937–969. Fishburn, F (1969). A general theory of subjective probabilities and expected utilities, Annals of Mathematical Statistics, 40(4), 1419–1429.

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Friedman, M and LJ Savage (1948). Utility analysis of choices involving risk. Journal of Political Economy, 56(4), 279–304. Kahneman, D and A Tversky (1979). Prospect theory: An analysis of decisions under risk. Econometrica, 47(2), 263–291. Kahneman, D and A Tversky (1984). Choices, values, and frames. American Psychologist, 39(4), 314–350. Levy, M and H Levy (2002). Prospect theory: Much ado about nothing? Management Science, 48(10), 1334–1349. Levy, M and H Levy (2004). Prospect theory and mean-variance analysis. Review of Financial Studies, 17(4), 1015–1041. Pratt, JW (1964). Risk aversion in the small and in the large. Econometrica, 32, 122–136. Rabin, M (2000). Risk aversion and expected utility theory: A calibration theorem. Econometrica, 68(5), 1281–1292. Rothschild, M and J Stiglitz (1970). Increasing risk: 1. A definition. Journal of Economic Theory, 3(2), 225–243. Savage, LJ (1954). The Foundations of Statistics. New York: Wiley. Tversky, A and D Kahneman (1974). Judgment under uncertainty: Heuristics and biases. Science, 185(4157), 1124–1131. Von Neumann, J and O Morgenstern (1944). Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press. Wakker, P (2002). The Data of Levy and Levy (2002) “Prospect theory: Much ado about nothing?” Actually support prospect theory. Management Science, 49(7), 979–981. Wu, G and M Alex (2008). An empirical test of gain-loss separability in prospect theory. Management Science, 54(7), 1322–1335.

Chapter 5 T.he Annals of Mathematical Statiatics 1969, Vol. 40, No. 4, 1419-1429

A GENERAL THEORY OF SUBJECTIVE PROBABILITIES AND EXPECTED UTILITIES

BY

PETER

C.

FISHBURN

Research Analysis Corporation 1. Introduction. The purpose of this paper is to present a general theory for

the usual subjective expected utility model for decision under uncertainty. With a set S of states of the world and a set X of consequences let F be a set of functions on S to X. F is the set of acts. Under a set of axioms based on extraneous. measurement probabilities, a device that is used by Rubin [14], Chernoff [3], Luce and Rai_ffa [9, Ch. 13], Anscombe and Aumann [1], Pratt, Raiffa, and Schlaifer [11], Arrow [2], and Fishburn [5], we shall prove that there is a realvalued function u on X and a finitely-additive probability measure P* on the set of all subsets of S such that, for all j, g E F, (1)

f

ｾ＠

g

if and only if E[u(f(s)), P*] ｾ＠

E[u(g(s)), P*].

In (1 ), ｾ＠ ("is not preferred to") is the decision-maker's binary preferenceindifference relation and E (y, z) is the mathematical expectation of y with respect to the probability measure z. Because we shall use extraneous measurement probabilities, (1) will be extracted from the more involved expression (2) that is presented in the next section. The axioms we shall use to derive (2) imply that P* is uniquely determined and that u is unique up to a positive linear transformation. u may or may not be bounded: however, it is bounded if there is a denumerable partition of S each element of which has positive probability under P*. Our theory places no restrictions on S and X except that they be nonempty sets with X containing at least two elements. X may or may not have a least (most) preferred consequence. In addition, no special restrictions are placed on P*. For example, if S is infinite, it may or may not be true that P* (A) = 1 for some finite subset A ｾ＠ S, and if p* (A) < 1 for every finite A ｾ＠ S it may or may not be true that S can be partitioned into an arbitrary finite number n of subsets such that P* = 1/n for each subset. Finally, no special properties will be implied for u apart from its uniqueness and its boundedness in the case noted above. To indicate briefly how this differs from other theories, we note first that the theories of Chernoff [3], Luce and Raiffa [9], Anscombe and Aumann [1], Pratt, Raiffa, and Schlaifer [11], and Fishburn [5] assume that S is finite. The ｴｨ･ｯｾｹ＠ presented here is a generalization of a theory in Fishburn [5]. The theory of Davidson and Suppes [4] assumes that X is finite and implies that, if x < y and z < w and there is no consequence between x and y or between z and w then u (y) - u (x) = u (w) - u (z). The theories of Ramsey [13] and Suppes [16] Received 19 August 1968; revised 6 March 1969.

87

88

PC Fishburn 1420

PETER C. FISHBURN

place no special restrictions on S but they imply that X is infinite and that if u (x) < u(y) then there is a zc X such that u(z) = .5u (x) + .5u(y). On the other hand Savage [15] does not restrict X in any unusual way, but his theory requires S to be infinite and implies that, for any positive integer n, there is an n-part partition of S such that P* = 1/n on each part of the partition. Arrow [2) also assumes this property for p*. 2. Definitions and notation. u(w). That is, a prospect is acceptable if the utility resulting from integrating the prospect with one's assets exceeds the utility of those assets alone. Thus, the domain of the utility function is final states (which include one's asset position) rather than gains or losses. Although the domain of the utility function is not limited to any particular class of consequences, most applications of the theory have been concerned with monetary outcomes. Furthermore, most economic applications introduce the following additional assumption. (iii) Risk Aversion: u is concave (u" < 0). A person is risk averse if he prefers the certain prospect (x) to any risky prospect with expected value x. In expected utility theory, risk aversion is equivalent to the concavity of the utility function. The prevalence· of risk aversion is perhaps the best known generalization regarding risky choices. It led the early decision theorists of the eighteenth century to propose that utility is a concave function of money, and this idea has been retained in modern treatments (Pratt [33], Arrow [4]). In the following sections we demonstrate several phenomena which violate these tenets of expected utility theory. The demonstrations are based on the responses of students and university faculty to hypothetical choice problems. The respondents were presented with problems of the type illustrated below. Which of the following would you prefer? A:

50% chance to win 1,000,

B: 450 for sure.

50% chance to win nothing; The outcomes refer to Israeli currency. To appreciate the significance of the amounts involved, note that the median net monthly income for a family is about 3,000 Israeli pounds. The respondents were asked to imagine that they were actually faced with the choice described in the problem, and to indicate the decision they would have made in such a case. The responses were anonymous, and the instructions specified that there was no 'correct' answer to such problems, and that the aim of the study was to find out how people choose among risky prospects. The problems were presented in questionnaire form, with at most a dozen problems per booklet. Several forms of each questionnaire were constructed so that subjects were exposed to the problems in different orders. In addition, two versions of each problem were used in which the left-right position of the prospects was reversed. The problems described in this paper are selected illustrations of a series of effects. Every effect has been observed in several problems with different outcomes and probabilities. Some of the problems have also been presented to groups of students and faculty at the University of Stockholm and at the

Chapter 6. Prospect Theory: An Analysis of Decision Under Risk

101

265

PROSPECT THEORY

University of Michigan. The pattern of results was essentially identical to the results obtained from Israeli subjects. The reliance on hypothetical choices raises obvious questions regarding the validity of the method and the generalizability of the results. We are keenly aware of these problems. However, all other methods that have been used to test utility theory also suffer from severe drawbacks. Real choices can be investigated either in the field, by naturalistic or statistical observations of economic behavior, or in the laboratory. Field studies can only provide for rather crude tests of qualitative predictions, because probabilities and utilities cannot be adequately measured in such contexts. Laboratory experiments have been designed to obtain precise measures of utility and probability from actual choices, but these experimental studies typically involve contrived gambles for small stakes, and a large number of repetitions of very similar problems. These features of laboratory gambling complicate the interpretation of the results and restrict their generality. By default, the method of hypothetical choices emerges as the simplest procedure by which a large number of theoretical questions can be investigated. The use of the method relies on the assumption that people often know how they would behave in actual situations of choice, and on the further assumption that the subjects have no special reason to disguise their true preferences. If people are reasonably accurate in predicting their choices, the presence of common and systematic violations of expected utility theory in hypothetical problems provides presumptive evidence against that theory. Certainty, Probability, and Possibility In expected utility theory, the utilities of outcomes are weighted by their probabilities. The present section describes a series of choice problems in which people's preferences systematically violate this principle. We first show that people overweight outcomes that are considered certain, relative to outcomes which are merely probable-a phenomenon which we label the certainty effect. The best known counter-example to expected utility theory which d:ploits the certainty effect was introduced by the French economist Maurice Allais in 1953 [2]. Allais' example has been discussed from both normative and descriptive standpoints by many authors [28, 38]. The following pair of choice problems is a variation of Allais' example, which differs from the original in that it refers to moderate rather than to extremely large gains. The number of respondents who answered each problem is denoted by N, and the percentage who choose each option is given in brackets. PROBLEM

A:

1: Choose between 2,500 with probability

.33,

2,400 with probability

.66,

Owithprobability

.01;

N=72

[18]

B:

2,400 with certainty.

[82]*

102

D Kahneman and A Tversky

D. KAHNEMAN AND

266

A.

TVERSKY

PROBLEM 2: Choose between C:

2,500 with probability

.33,

0 with probability

.67;

D:

2,400 with probability

.34,

0 with probability

.66.

[83]*

N=72

[17]

The data show that 82 per cent of the subjects chose Bin Problem 1, and 83 per cent of the subjects chose C in Problem 2. Each of these preferences is significant at the .01level, as denoted by the asterisk. Moreover, the analysis of individual patterns of choice indicates that a majority of respondents (61 per cent) made the modal choice in both problems. This pattern of preferences violates expected utility theory in the manner originally described by Allais. According to that theory, with u(O) = 0, the first preference implies u(2,400) > .33u(2,500) + .66u(2,400) or .34u(2,400) > .33u(2,500) while the second preference implies the reverse inequality. Note that Problem 2 is obtained from Problem 1 by eliminating a .66 chance of winning 2400 from both prospects under consideration. Evidently, this change produces a greater reduction in desirability when it alters the character of the prospect from a sure gain to a probable one, than when both the original and the reduced prospects are uncertain. A simpler demonstration of the same phenomenon, involving only twooutcome gambles is given below. This example is also based on Allais [2]. PROBLEM 3: A:

(4,000,.80),

N=95

or

B:

(3,000). [80]*

[20]

PROBLEM 4: C: N

(4,000,.20), =

95

[65]*

or

D:

(3,000,.25). [35]

In this pair of problems as well as in all other problem-pairs in this section, over half the respondents violated expected utility theory. To show that the modal pattern of preferences in Problems 3 and 4 is not compatible with the theory, set u(O) = 0, and recall that the choice of B implies u(3,000)/u(4,000)>4/5, whereas the choice of C implies the reverse inequality. Note that the prospect C = (4,000, .20) can be expressed as (A, .25), while the prospect D = (3,000, .25) can be rewritten as (B,.25). The substitution axiom of utility theory asserts that if B is preferred to A, then any (probability) mixture (B, p) must be preferred to the mixture (A, p). Our subjects did not obey this axiom. Apparently, reducing the probability of winning from 1.0 to .25 has a greater effect than the reduction from

Chapter 6. Prospect Theory: An Analysis of Decision Under Risk PROSPECT THEORY

103 267

.8 to .2. The following pair of choice problems illustrates the certainty effect with non-monetary outcomes.

PROBLEM

5:

A:

50% chance to win a threeweek tour of England, France, and Italy;

N=72

B:

A one-week tour of England, with certainty.

[78]*

[22]

PROBLEM6:

C:

5% chance to win a threeweek tour of England, France, and Italy;

N=72

D:

[67]*

10% chance to win a oneweek tour of England.

[33]

The certainty effect is not the only type of violation of the substitution axiom. Another situation in which this axiom fails is illustrated by the following problems. PROBLEM7:

A:

(6,000, .45),

B:

N =66 [14] PROBLEM

C:

(3,000, .90).

[86]*

8: (6,000, .001),

N = 66 [73]*

D:

(3,000, .002).

[27]

Note that in Problem 7 the probabilities of winning are substantial (.90 and .45), and most people choose the prospect where winning is more probable. In Problem 8, there is a possibility of winning, although the probabilities o( winning are minuscule (.002 and .001) in both prospects. In this situation where winning is possible but not probable, most people choose the prospect that offers the larger gain. Similar results have been reported by MacCrimmon and Larsson [28]. The above problems illustrate common attitudes toward risk or chance that cannot be captured by the expected utility model. The results suggest the following empirical generalization concerning the manner in which the substitution axiom is violated. If (y, pq) is equivalent to (x, p ), then (y, pqr) is preferred to (x,pr),O (3,000, .25). [35] > (6,000, .45). [14] < (6,000, .001). [73]*

Negative prospects

Problem 3': N=95 Problem 4': N=95 Problem 7': N=66 Problem 8': N=66

(-4,000, .80) [92]* (-4,000, .20) [42] (-3,000, .90) [8] (-3,000, .002) [70]*

>

(-3,000). [8] < ( -3,000, .25). [58] < (-6,000, .45). [92]* > (-6,000, .001). [30]

In each of the four problems in Table I the preference between negative prospects is the mirror image of the preference between positive prospects. Thus, the reflection of prospects around 0 reverses the preference order. We label this pattern the reflection effect. Let us turn now to the implications of these data. First, note that the reflection effect implies that risk aversion in the positive domain is accompanied by risk seeking in the negative domain. In Problem 3', for example, the majority of subjects were willing to accept a risk of .80 to lose 4,000, in preference to a sure loss of 3,000, although the gamble has a lower expected value. The occurrence of risk seeking in choices between negative prospects was noted early by Markowitz [29]. Williams [48] reported data where a translation of outcomes produces a dramatic shift from risk aversion to risk seeking. For example, his subjects were indifferent between (100, .65; -100, .35) and (0), indicating risk aversion. They were also indifferent between (-200, .80) and (-100), indicating risk seeking. A recent review by Fishburn and Kochenberger [14] documents the prevalence of risk seeking in choices between negative prospects. Second, recall that the preferences between the positive prospects in Table I are inconsistent with expected utility theory. The preferences between the corresponding negative prospects also violate the expectation principle in the same manner. For example, Problems 3' and 4', like Problems 3 and 4, demonstrate that outcomes which are obtained with certainty are overweighted relative to uncertain outcomes. In the positive domain, the certainty effect contributes to a risk averse preference for a sure gain over a larger gain that is merely probable. In the negative domain, the same effect leads to a risk seeking preference for a loss

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that is merely probable over a smaller loss that is certain. The same psychological principle-the overweighting of certainty-favors risk aversion in the domain of gains and risk seeking in the domain of losses. Third, the reflection effect eliminates aversion for uncertainty or variability as an explanation of the certainty effect. Consider, for example, the prevalent preferences for (3,000) over (4,000, .80) and for (4,000, .20) over (3,000, .25). To resolve this apparent inconsistency one could invoke the assumption that people prefer prospects that have high expected value and small variance (see, e.g., Allais [2]; Markowitz [30]; Tobin [41]). Since (3,000) has no variance while (4,000, .80) has large variance, the former prospect could be chosen despite its lower expected value. When the prospects are reduced, however, the difference in variance between (3,000, .25) and (4,000, .20) may be insufficient to overcome the difference in expected value. Because (-3,000) has both higher expected value and lower variance than (-4,000, .80), this account entails that the sure loss should be preferred, contrary to the data. Thus, our data are incompatible with the notion that certainty is generally desirable. Rather, it appears that certainty increases the aversiveness of losses as well as the desirability of gains. Probabilistic Insurance

The prevalence of the purchase of insurance against both large and small losses has been regarded by many as strong evidence for the concavity of the utility function for money. Why otherwise would people spend so much money to purchase insurance policies at a price that exceeds the expected actuarial cost? However, an examination of the relative attractiveness of various forms of insurance does not support the notion that the utility function for money is concave everywhere. For example, people often prefer insurance programs that offer limited coverage with low or zero deductible over comparable policies that offer higher maximal coverage with higher deductibles-contrary to risk aversion (see, e.g., Fuchs [16]). Another type of insurance problem in which people's responses are inconsistent with the concavity hypothesis may be called probabilistic insurance. To illustrate this concept, consider the following problem, which was presented to 95 Stanford University students. PROBLEM 9: Suppose you consider the possibility of insuring some property against damage, e.g., fire or theft. After examining the risks and the premium you find that you have no clear preference between the options of purchasing insurance or leaving the property uninsured. It is then called to your attention that the insurance company offers a new program called probabilistic insurance. In this program you pay half of the regular premium. In case of damage, there is a 50 per cent chance that you pay the other half of the premium and the insurance company covers all the losses; and there is a 50 per cent chance that you get back your insurance payment and suffer all the losses. For example, if an accident occurs on an odd day of the month, you pay the other half of the regular premium and your losses are covered; but if the accident

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occurs on an even day of the month, your insurance payment is refunded and your losses are not covered. Recall that the premium for full coverage is such that you find this insurance barely worth its cost. Under these circumstances, would you purchase probabilistic insurance:

N=95

Yes, No. [20] [80]*

Although Problem 9 may appear contrived, it is worth noting that probabilistic insurance represents many forms of protective action where one pays a certain cost to reduce the probability of an undesirable event-without eliminating it altogether. The installation of a burglar alarm, the replacement of old tires, and the decision to stop smoking can all be viewed as probabilistic insurance. The responses to Problem 9 and to several other variants of the same question indicate that probabilistic insurance is generally unattractive. Apparently, reducing the probability of a loss from p to p/2 is less valuable than reducing the probability of that loss from p/2 to 0. In contrast to these data, expected utility theory (with a concave u) implies that probabilistic insurance is superior to regular insurance. That is, if at asset position w one is just willing to pay a premium y to insure against a probability p of losing x, then one should definitely be willing to pay a smaller premium ry to reduce the probability of losing x from p to (1- r)p, 0 < r < 1. Formally, if one is indifferent between ( w - x, p; w, 1- p) and ( w - y ), then one should prefer probabilistic insurance ( w- x, (1- r)p; w- y, rp; w - ry, 1- p) over regular insurance ( w- y ). To prove this proposition, we show that

pu(w -x)+(l-p)u(w) = u(w- y) implies (1- r)pu(w -x)+ rpu(w- y)+(l- p)u(w- ry) > u(w- y).

Without loss of generality, we can set u ( w - x) = 0 and u ( w) = 1. Hence, u ( wy) = 1- p, and we wish to show that

rp(1-p)+(l-p)u(w -ry)> 1-p

or

u(w-ry)>1-rp

which holds if and only if u is concave. This is a rather puzzling consequence of the risk aversion hypothesis of utility theory, because probabilistic insurance appears intuitively riskier than regular insurance, which entirely eliminates the element of risk. Evidently, the intuitive notion of risk is not adequately captured by the assumed concavity of the utility function for wealth. The aversion for probabilistic insurance is particularly intriguing because all insurance is, in a sense, probabilistic. The most avid buyer of insurance remains vulnerable to many financial and other risks which his policies do not cover. There appears to be a significant difference between probabilistic insurance and what may be called contingent insurance, which provides the certainty of coverage for a

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specified type of risk. Compare, for example, probabilistic insurance against all forms of loss or damage to the contents of your home and contingent insurance that eliminates all risk of loss from theft, say, but does not cover other risks, e.g., fire. We conjecture that contingent insurance will be generally more attractive than probabilistic insurance when the probabilities of unprotected loss are equated. Thus, two prospects that are equivalent in probabilities and outcomes could have different values depending on their formulation. Several demonstrations of this general phenomenon are described in the next section. The Isolation Effect In order to simplify the choice between alternatives, people often disregard components that the alternatives share, and focus on the components that distinguish them (Tversky [44]). This approach to choice problems may produce inconsistent preferences, because a pair of prospects can be decomposed into common and distinctive components in more than one way, and different decompositions sometimes lead to different preferences. We refer to this phenomenon as the isolation effect. PROBLEM 10: Consider the following two-stage game. In the first stage, there is a probability of .75 to end the game without winning anything, and a probability of .25 to move into the second stage. If you reach the second stage you have a choice between (4,000, .80)

and

(3,000).

Your choice must be made before the game starts, i.e., before the outcome of the first stage is known. Note that in this game, one has a choice between .25 x .80 = .20 chance to win 4,000, and a .25 x 1.0 = .25 chance to win 3,000. Thus, in terms of final outcomes and probabilities one faces a choice between (4,000, .20) and (3,000, .25), as in Problem 4 above. However, the dominant preferences are different in the two problems. Of 141 subjects who answered Problem 10,78 per cent chose the latter prospect, contrary to the modal preference in Problem 4. Evidently, people ignored the first stage of the game, whose outcomes are shared by both prospects, and considered Problem 10 as a choice between (3,000) and (4,000, .80), as in Problem 3 above. The standard and the sequential formulations of Problem 4 are represented as decision trees in Figures 1 and 2, respectively. Following the usual convention, squares denote decision nodes and circles denote chance nodes. The essential difference between the two representations is in the location of the decision node. In the standard form (Figure 1), the decision maker faces a choice between two risky prospects, whereas in the sequential form (Figure 2) he faces a choice between a risky and a riskless prospect. This is accomplished by introducing a dependency between the prospects without changing either probabilities or

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3000

0 4000

FIGURE

FIGURE

1.-The representation of Problem 4 as a decision tree (standard formulation).

2.-The representation of Problem 10 as a decision tree (sequential formulation).

outcomes. Specifically, the event 'not winning 3,000' is included in the event 'not winning 4,000' in the sequential formulation, while the two events are independent in the standard formulation. Thus, the outcome of winning 3,000 has a certainty advantage in the sequential formulation, which it does not have in the standard formulation. The reversal of preferences due to the dependency among events is particularly significant because it violates the basic supposition of a decision-theoretical analysis, that choices between prospects are determined solely by the probabilities of final states. It is easy to think of decision problems that are most naturally represented in one of the forms above rather than in the other. For example, the choice between two different risky ventures is likely to be viewed in the standard form. On the other hand, the following problem is most likely to be represented in the sequential form. One may invest money in a venture with some probability of losing one's capital if the venture fails, and with a choice between a fixed agreed return and a percentage of earnings if it succeeds. The isolation effect implies that the contingent certainty of the fixed return enhances the attractiveness of this option, relative to a risky venture with the same probabilities and outcomes.

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The preceding problem illustrated how preferences may be altered ｾｹ＠ different representations of probabilities. We now show how choices may be altered by varying the representation of outcomes. Consider the following problems, which were presented to two different groups of subjects. PROBLEM 11: In addition to whatever you own, you have been given 1 ,000. You are now asked to choose between A:

(1,000, .50),

N=70

and

B:

(500).

[84]*

[16]

PROBLEM 12: In addition to whatever you own, you have been given 2,000. You are now asked to choose between C:

(-1,000, .50),

N = 68

and

D:

[69*]

(-500). [31]

The majority of subjects chose B in the first problem and C in the second. These preferences conform to the reflection effect observed in Table I, which exhibits risk aversion for positive prospects and risk seeking for negative ones. Note, however, that when viewed in terms of final states, the two choice problems are identical. Specifically, A= (2,000, .50; 1,000, .50)= C,

and

B = (1,500) =D.

In fact, Problem 12 is obtained from Problem 11 by adding 1,000 to the initial bonus, and subtracting 1,000 from all outcomes. Evidently, the subjects did not integrate the bonus with the prospects. The bonus did not enter into the comparison of prospects because it was common to both options in each problem. The pattern of results observed in Problems 11 and 12 is clearly inconsistent with utility theory. In that theory, for example, the same utility is assigned to a wealth of $100, 000, regardless of whether it was reached from a prior wealth of $95,000 or $105,000. Consequently, the choice between a total wealth of $100,000 and even chances to own $95,000 or $105,000 should be independent of whether one currently owns the smaller or the larger of these two amounts. With the added assumption of risk aversion, the theory entails that the certainty of owning $100,000 should always be preferred to the gamble. However, the responses to Problem 12 and to several of the previous questions suggest that this pattern will be obtained if the individual owns the smaller amount, but not if he owns the larger amount. The apparent neglect of a bonus that was common to both options in Problems 11 and 12 implies that the carriers of value or utility are changes of wealth, rather than final asset -positions that include current wealth. This conclusion is the cornerstone of an alternative theory of risky choice, which is described in the following sections.

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3.

THEORY

The preceding discussion reviewed several empirical effects which appear to invalidate expected utility theory as a descriptive model. The remainder of the paper presents an alternative account of individual decision making under risk, called prospect theory. The theory is developed for simple prospects with monetary outcomes and stated probabilities, but it can be extended to more involved choices. Prospect theory distinguishes two phases in the choice process: an early phase of editing and a subsequent phase of evaluation. The editing phase consists of a preliminary analysis of the offered prospects, which often yields a simpler representation of these prospects. In the second phase, the edited prospects are evaluated and the prospect of highest value is chosen. We next outline the editing phase, and develop a formal model of the evaluation phase. The function of the editing phase is to organize and reformulate the options so as to simplify subsequent evaluation and choice. Editing consists of the application of several operations that transform the outcomes and probabilities associated with the offered prospects. The major operations of the editing phase are described below. Coding. The evidence discussed in the previous section shows that people normally perceive outcomes as gains and losses, rather than as final states of wealth or welfare. Gains and losses, of course, are defined relative to some neutral reference point. The reference point usually corresponds to the current asset position, in which case gains and losses coincide with the actual amounts that are received or paid. However, the location of the reference point, and the consequent coding of outcomes as gains or losses, can be affected by the formulation of the offered prospects, and by the expectations of the decision maker. Combination. Prospects can sometimes be simplified by combining the probabilities associated with identical outcomes. For example, the prospect (200, .25; 200, .25) will be reduced to (200, .50). and evaluated in this form. Segregation. Some prospects contain a riskless component that is segregated from the risky component in the editing phase. For example, the prospect (300, .80; 200, .20) is naturally decomposed into a sure gain of 200 and the risky prospect (100, .80). Similarly, the prospect (-400, .40; -100, .60) is readily seen to consist of a sure loss of 100 and of the prospect (-300, .40). The preceding operations are applied to each prospect separately. The following operation is applied to a set of two or more prospects. Cancellation. The essence of the isolation effects described earlier is the discarding of components that are shared by the offered prospects. Thus, our respondents apparently ignored the first stage of the sequential game presented in Problem 10, because this stage was common to both options, and they evaluated the prospects with respect to the results of the second stage (see Figure 2). Similarly, they neglected the common bonus that was added to the prospects in Problems 11 and 12. Another type of cancellation involves the discarding of common constituents, i.e., outcome-probability pairs. For example, the choice

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between (200, .20; 100, .50; -50, .30) and (200, .20; 150, .50; -100, .30) can be reduced by cancellation to a choice between (100, .50; -50, .30) and (150, .50; -100, .30). Two additional operations that should be mentioned are simplification and the detection of dominance. The first refers to the simplification of prospects by rounding probabilities or outcomes. For example, the prospect (101, .49) is likely to be recoded as an even chance to win 100. A particularly important form of simplification involves the discarding of extremely unlikely outcomes. The second operation involves the scanning of offered prospects to detect dominated alternatives, which are rejected without further evaluation. Because the editing operations facilitate the task of decision, it is assumed that they are performed whenever possible. However, some editing operations either permit or prevent the application of others. For example, (500, .20; 101, .49) will appear to dominate (500, .15; 99, .51) if the second constituents of both prospects are simplified to (1 00, .50). The final edited prospects could, therefore, depend on the sequence of editing operations, which is likely to vary with the structure of the offered set and with the format of the display. A detailed study of this problem is beyond the scope of the present treatment. In this paper we discuss choice problems where it is reasonable to assume either that the original formulation of the prospects leaves no room for further editing, or that the edited prospects can be specified without ambiguity. Many anomalies of preference result from the editing of prospects. For example, the inconsistencies associated with the isolation effect result from the cancellation of common components. Some intransitivities of choice are explained by a simplification that eliminates small differences between prospects (see Tversky [43]). More generally, the preference order between prospects need not be invariant across contexts, because the same offered prospect could be edited in different ways depending on the context in which it appears. Following the editing phase, the decision maker is assumed to evaluate each of the edited prospects, and to choose the prospect of highest value. The overall value of an edited prospect, denoted V, is expressed in terms of two scales, 1r and v. The first scale, 1r, associates with each probability p a decision weight 1r(p ), which reflects the impact of p on the over-all value of the prospect. However, 1r is not a probability measure, and it will be shown later that 1r(p) + 1r(l- p) is typically less than unity. The second scale, v, assigns to each outcome x a number v (x ), which reflects the subjective value of that outcome. Recall that outcomes are defined relative to a reference point, which serves as the zero point of the value scale. Hence, v measures the value of deviations from that reference point, i.e., gains and losses. The present formulation is concerned with simple prospects of the form (x, p; y, q), which have at most two non-zero outcomes. In such a prospect, one receives x with probability p, y with probability q, and nothing with probability 1- p- q, where p + q ｾ＠ 1. An offered prospect is strictly positive if its outcomes are all positive, i.e., if x, y > 0 and p + q = 1; it is strictly negative if its outcomes

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are all negative. A prospect is regular if it is neither strictly positive nor strictly negative. The basic equation of the theory describes the manner in which 71' and v are combined to determine the over-all value of regular prospects. If (x, p; y, q) is a regular prospect (i.e., either p + q < 1, or x ｾ＠ 0 ｾ＠ y, or x ｾ＠ 0 ｾ＠ y), then (1)

V(x, p; y, q)

= 7r(p)v(x)+7r(q)v(y)

where v(O) = (}, 7r(O) = 0, and 71'(1) = 1. As in utility theory, V is defined on prospects, while v is defined on outcomes. The two scales coincide for sure prospects, where V(x, 1.0) = V(x) = v(x). Equation (1) generalizes expected utility theory by relaxing the expectation principle. An axiomatic analysis of this representation is sketched in the Appendix, which describes conditions that ensure the existence of a unique 71' and a ratio-scale v satisfying equation (1). The evaluation of strictly positive and strictly negative prospects follows a different rule. In the editing phase such prospects are segregated into two components: (i) the riskless component, i.e., the minimum gain or loss which is certain to be obtained or paid; (ii) the risky component, i.e., the additional gain or loss which is actually at stake. The evaluation of such prospects is described in the next equation. If p + q = 1 and either x > y > 0 or x < y < 0, then (2)

V(x, p; y, q)

= v(y)+7r(p)[v(x)-v(y)].

That is, the value of a strictly positive or strictly negative prospect equals the value of the riskless component plus the value-difference between the outcomes, multiplied by the weight associated with the more extreme outcome. For example, V(400, .25; 100, .75)= v(100)+7r(.25)[v(400)-v(100)]. The essential feature of equation (2) is that a decision weight is applied to the value-difference v (x)- v (y ), which represents the risky component of the prospect, but not to v (y ), which represents the riskless component. Note that the right-hand side of equation (2) equals 7r(p)v(x)+[1-7r(p)]v(y). Hence, equation (2) reduces to equation (1) if 7r(p) + 71'(1- p) = 1. As will be shown later, this condition is not generally satisfied. Many elements of the evaluation model have appeared in previous attempts to modify expected utility theory. Markowitz [29] was the first to propose that utility be defined on gains and losses rather than on final asset positions, an assumption which has been implicitly accepted in most experimental measurements of utility (see, e.g., [7, 32]). Markowitz also noted the presence of risk seeking in preferences among positive as well as among negative prospects, and he proposed a utility function which has convex and concave regions in both the positive and the negative domains. His treatment, however, retains the expectation principle; hence it cannot account for the many violations of this principle; see, e.g., Table I. The replacement of probabilities by more general weights was proposed by Edwards [9], and this model was investigated in several empirical studies (e.g.,

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[3, 42]). Similar models were developed by Fellner [12], who introduced the concept of decision weight to explain aversion for ambiguity, and by van Dam [46] who attempted to scale decision weights. For other critical analyses of expected utility theory and alternative choice models, see Allais [2], Coombs [6], Fishburn [13], and Hansson [22]. The equations of prospect theory retain the general bilinear form that underlies expected utility theory. However, in order to accomodate the effects described in the first part of the paper, we are compelled to assume that values are attached to changes rather than to final states, and that decision weights do not coincide with stated probabilities. These departures from expected utility theory must lead to normatively unacceptable consequences, such as inconsistencies, intransitivities, and violations of dominance. Such anomalies of preference are normally corrected by the decision maker when he realizes that his preferences are inconsistent, intransitive, or inadmissible. In many situations, however, the decision maker does not have the opportunity to discover that his preferences could violate decision rules that he wishes to obey. In these circumstances the anomalies implied by prospect theory are expected to occur.

The Value Function An essential feature of the present theory is that the carriers of value are changes in wealth or welfare, rather than final states. This assumption is compatible with basic principles of perception and judgment. Our perceptual apparatus is attuned to the evaluation of changes or differences rather than to the evaluation of absolute magnitudes. When we respond to attributes such as brightness, loudness, or temperature, the past and present context of experience defines an adaptation level, or reference point, and stimuli are perceived in relation to this reference point [23]. Thus, an object at a given temperature may be experienced as hot or cold to the touch depending on the temperature to which one has adapted. The same principle applies to non-sensory attributes such as health, prestige, and wealth. The same level of wealth, for example, may imply abject poverty for one person and great riches for another-depending on their current assets. The emphasis on changes as the carriers of value should not be taken to imply that the value of a particular change is independent of initial position. Strictly speaking, value should be treated as a function in two arguments: the asset position that serves as reference point, and the magnitude of the change (positive or negative) from that reference point. An individual's attitude to money, say, could be described by a book, where each page presents the value function for changes at a particular asset position. Clearly, the value functions described on different pages are not identical: they are likely to become more linear with increases in assets. However, the preference order of prospects is not greatly altered by small or even moderate variations in asset position. The certainty equivalent of the prospect (1,000, .50), for example, lies between 300 and 400 for most people, in a wide range of asset positions. Consequently, the representation

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of value as a function in one argument generally provides a satisfactory approximation. Many sensory and perceptual dimensions share the property that the psychological response is a concave function of the magnitude of physical change. For example, it is easier to discriminate between a change of 3° and a change of 6° in room temperature, than it is to discriminate between a change of 13° and a change of 16°. We propose that this principle applies in particular to the evaluation of monetary changes. Thus, the difference in value between a gain of 100 and a gain of 200 appears to be greater than the difference between a gain of 1,100 and a gain of 1,200. Similarly, the difference between a loss of 100 and a loss of 200 appears greater than the difference between a loss of 1,100 and a loss of 1,200, unless the larger loss is intolerable. Thus, we hypothesize that the value function for changes of wealth is normally concave above the reference point (v"(x) < 0, for x > O) and often convex below it (v"(x) > 0, for x < 0). That is, the marginal value of both gains and losses generally decreases with their magnitude. Some support for this hypothesis has been reported by Galanter and Pliner [17], who scaled the perceived magnitude of monetary and non-monetary gains and losses. The above hypothesis regarding the shape of the value function was based on response& to gains and losses in a riskless context. We propose that the value function which is derived from risky choices shares the same characteristics, as illustrated in the following problems. PROBLEM

13: (6,000, .25),

N=68 PROBLEM

or

(4,000, .25; 2,000, .25). [82]*

[18]

13': (-6,000, .25),

N =64

[70]*

or

(-4,000, .25; -2,000, .25). [30]

Applying equation 1 to the modal preference in these problems yields 1T(.25)v(6,000) < 1T(.25)[v(4,000) + v(2,000)]

and

1T(.25)v( -6,000) > 1T(.25)[v (-4,000) + v( -2,000)].

Hence, v(6,000) < v(4,000) + v(2,000) and v(-6,000) > v(-4,000) + v(-2,000). These preferences are in accord with the hypothesis that the value function is concave for gains and convex for losses. Any discussion of the utility function for money must leave room for the effect of special circumstances on preferences. For example, the utility function of an individual who needs $60,000 to purchase a house may reveal an exceptionally steep rise near the critical value. Similarly, an individual's aversion to losses may increase sharply near the loss that would compel him to sell his house and move to

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a less desirable neighborhood. Hence, the derived value (utility) function of an individual does not always reflect "pure" attitudes to money, since it could be affected by additional consequences associated with specific amounts. Such perturbations can readily produce convex regions in the value function for gains and concave regions in the value function for losses. The latter case may be more common since large losses often necessitate changes in life style. A salient characteristic of attitudes to changes in welfare is that losses loom larger than gains. The aggravation that one experiences in losing a sum of money appears to be greater than the pleasure associated with gaining the same amount [17]. Indeed, most people find symmetric bets of the form (x, .50; -x, .50) distinctly unattractive. Moreover, the aversiveness of symmetric fair bets generally increases with the ·size of the stake. That is, if x > y ;30, then (y, .50; -y, .50) is preferred to (x, .50; -x, .50). According to equation (1), therefore, v(y) + v(-y) > v(x) + v( -x)

and

v(-y)- v( -x) > v(x)- v(y ).

Setting y=O yields v(x) p for low probabilities, there is evidence to suggest that, for all O 1T(p 1 )v(x) + 1T(q 1 )v(y ),

or 7T(p)-7r(pr)

v(y)

--''--'-----'=--'-> - - .

1T(q )-1T(q) 1

v(x)

Hence, as y approaches x, 1r(p) -7T(p 1 ) approaches 1r(q 1 ) -1r(q). Since p- p' = qr- q, 1r must be essentially linear, or else dominance must be violated. Direct violations of dominance are prevented, in the present theory, by the assumption that dominated alternatives are detected and eliminated prior to the evaluation of prospects. However, the theory permits indirect violations of dominance, e.g., triples of prospects so that A is preferred to B, B is preferred to C, and C dominates A. For an example, see Raiffa [34, p. 75]. Finally, it should be noted that the present treatment concerns the simplest decision task in which a person chooses between two available prospects. We have not treated in detail the more complicated production task (e.g., bidding) where the decision maker generates an alternative that is equal in value to a given prospect. The asymmetry between the two options in this situation could introduce systematic biases. Indeed, Lichtenstein and Slavic [27] have constructed pairs of prospects A and B, such that people generally prefer A over B, but bid more for B than for A. This phenomenon has been confirmed in several studies, with both hypothetical and real gambles, e.g., Grether and Plott [20]. Thus, it cannot be generally assumed that the preference order of prospects can be recovered by a bidding procedure. Because prospect theory has been proposed as a model of choice, the inconsistency of bids and choices implies that the measurement of values and decision weights should be based on choices between specified prospects rather than on bids or other production tasks. This restriction makes the assessment of v and 1r more difficult because production tasks are more convenient for scaling than pair comparisons.

4.

DISCUSSION

In the final section we show how prospect theory accounts for observed attitudes toward risk, discuss alternative representations of choice problems induced by shifts of reference point, and sketch several extensions of the present treatment.

Risk Attitudes The dominant pattern of preferences observed in Allais' example (Problems 1 and 2) follows from the present theory iff 7T(.33) 7T(.34)

--->

v(2,400) 7r(.33) > --'---'-v (2,500) l-7T(.66)'

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Hence, the violation of the independence axiom is attributed in this case to subcertainty, and more specifically to the inequality 7T(.34) < 1- 1r(.66). This analysis shows that an Allais-type violation will occur whenever the v-ratio of the two non-zero outcomes is bounded by the corresponding 1r-ratios. Problems 3 through 8 share the same structure, hence it suffices to consider one pair, say Problems 7 and 8. The observed choices in these problems are implied by the theory iff 7T(.001) 7T(.002)

--->

v (3,000) 7T(.45) >---. v (6,000) 7T(.90)

The violation of the substitution axiom is attributed in this case to the subproportionality of 7T. Expected utility theory is violated in the above manner, therefore, whenever the v- ratio of the two outcomes is bounded by the respective 7T-ratios. The same analysis applies to other violations of the substitution axiom, both in the positive and in the negative domain. We next prove that the preference for regular insurance over probabilistic insurance, observed in Problem 9, follows from prospect theory-provided the probability of loss is overweighted. That is, if (- x, p) is indifferent to (-y ), then (-y) is preferred to (-x, p/2; -y, p/2; -y/2, 1-p). For simplicity, we define for f(x) = - v( -x). Since the value function for losses is convex, f is a concave x ｾＰＬ＠ function of x. Applying prospect theory, with the natural extension of equation 2, we wish to show that 1r(p )f(x) = f(y)

implies

f(y),;;, f(y/2) + 7T(p/2)[f(y)-f(y/2)] + 7T(p/2)[f(x)- f(y/2)]

= 7T(p/2)f(x) + 7T(p/2)f(y) + [1- 27T(p/2)Jf(y/2). Substituting for f(x) and using the concavity off, it suffices to show that 7T(p/2) f(y),;;, 7T(p) f(y) + 7T(p/2)f(y) + f(y )/2- 7T(p/2)f(y)

or 7T(p )/2,;;, 7T(p/2),

which follows from the subadditivity of 7T.

According to the present theory, attitudes toward risk are determined jointly by

v and 7T, and not solely by the utility function. It is therefore instructive to examine the conditions under which risk aversion or risk seeking are expected to occur. Consider the choice between the gamble (x, p) and its expected value (px ). If x > 0, risk seeking is implied whenever 1r(p) > v (px )/ v (x ), which is greater than p if the value function for gains is concave. Hence, overweighting (1r(p) > p) is necessary but not sufficient for risk seeking in the domain of gains. Precisely the same condition is necessary but not sufficient for risk aversion when x < 0. This analysis restricts risk seeking in the domain of gains and risk aversion in the domain of losses to small probabilities, where overweighting is expected to hold.

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Indeed these are the typical conditions under which lottery tickets and insurance policies are sold. In prospect theory, the overweighting of small probabilities favors both gambling and insurance, while the S-shaped value function tends to inhibit both behaviors. Although prospect theory predicts both insurance and gambling for small probabilities, we feel that the present analysis falls far short of a fully adequate account of these complex phenomena. Indeed, there is evidence from both experimental studies [37], survey research [26], and observations of economic behavior, e.g., service and medical insurance, that the purchase of insurance often extends to the medium range of probabilities, and that small probabilities of disaster are sometimes entirely ignored. Furthermore, the evidence suggests that minor changes in the formulation of the decision problem can have marked effects on the attractiveness of insurance [37]. A comprehensive theory of insurance behavior should consider, in addition to pure attitudes toward uncertainty and money, such factors as the value of security, social norms of prudence, the aversiveness of a large number of small payments spread over time, information and misinformation regarding probabilities and outcomes, and many others. Some effects of these variables could be described within the present framework, e.g., as changes of reference point, transformations of the value function, or manipulations of probabilities or decision weights. Other effects may require the introduction of variables or concepts which have not been considered in this treatment. Shifts of Reference

So far in this paper, gains and losses were defined by the amounts of money that are obtained or paid ｷｨ･ｾ＠ a prospect is played, and the reference point was taken to be the status quo, or q_ne's current assets. Although this is probably true for most choice problems, there are situations in which gains and losses are coded relative to an expectation or aspiration level that differs from the status quo. For example, an unexpected tax withdrawal from a monthly pay check is experienced as a loss, not as a reduced gain. Similarly, an entrepreneur who is weathering a slump with greater success than his competitors may interpret a small loss as a gain, relative to the larger loss he had reason to expect. The reference point in the preceding examples corresponded to an asset position that one had expected to attain. A discrepancy between the reference point and the current asset position may also arise because of recent changes in wealth to which one has not yet adapted [29]. Imagine a person who is involved in a business venture, has already lost 2,000 and is now facing a choice between a sure gain of 1,000 and an even chance to win 2,000 or nothing. If he has not yet adapted to his losses, he is likely to code the problem as a choice between (-2,000, .50) and (-1,000) rather than as a choice between (2,000, .50) and (1,000). As we have seen, the former representation induces more adventurous choices than the latter. A change of reference point alters the preference order for prospects. In particular, the present theory implies that a negative translation of a choice

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problem, such as arises from incomplete adaptation to recent losses, increases risk seeking in some situations. Specifically, if a risky prospect (x, p; - y, 1-p) is just acceptable, then (x- z, p; - y- z, 1- p) is preferred over (- z) for x, y, z > 0, with x>z. To prove this proposition, note that V(x,p; y, 1-p)=O

iff

1r(p)v(x) = -7T(1-p)v(-y).

Furthermore, V(x-z,p;-y-z, 1-p)

= 1r(p )v(x- z)+ 7T(1- p)v(- y- z) > 1r(p )v(x) -7r(p)v(z) + 7T(1- p )v(- y) +7T(1-p)v(- z)

by the properties of v,

= -7T(1- p )v(- y) -1r(p )v(z) + 7T(1- p)v( -y)

+7T(1- p)v(-z)

by substitution,

= -1r(p )v(z) + 7T(l- p )v( -z)

> v(- z )[1r(p) + 7T(1- p )] >v( -z)

since v(- z) < -v(z ),

by subcertainty.

This analysis suggests that a person who has not made peace with his losses is likely to accept gambles that would be unacceptable to him otherwise. The well known observation [31] that the tendency to bet on long shots increases in the course of the betting day provides some support for the hypothesis that a failure to adapt to losses or to attain an expected gain induces risk seeking. For another example, consider an individual who expects to purchase insurance, perhaps because he has owned it in the past or because his friends do. This individual may code the decision to pay a premium y to protect against a loss x as a choice between (-x + y, p; y, 1- p) and (0) rather than as a choice between (-x, p) and (-y ). The preceding argument entails that insurance is likely to be more attractive in the former representation than in the latter. Another important case of a shift of reference point arises when a person formulates his decision problem in terms of final assets, as advocated in decision analysis, rather than in terms of gains and losses, as people usually do. In this case, the reference point is set to zero on the scale of wealth and the value function is likely to be concave everywhere [39]. According to the present analysis, this formulation essentially eliminates risk seeking, except for gambling with low probabilities. The explicit formulation of decision problems in terms of final assets is perhaps the most effective procedure for eliminating risk seeking in the domain of losses.

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Many economic decisions involve transactions in which one pays money in exchange for a desirable prospect. Current decision theories analyze such problems as comparisons between the status quo and an alternative state which includes the acquired prospect minus its cost. For example, the decision whether to pay 10 for the gamble (1,000, .01) is treated as a choice between (990, .01; -10, .99) and (0). In this analysis, readiness to purchase the positive prospect is equated to willingness to accept the corresponding mixed prospect. The prevalent failure to integrate riskless and risky prospects, dramatized in the isolation effect, suggests that people are unlikely to perform the operation of subtracting the cost from the outcomes in deciding whether to buy a gamble. Instead, we suggest that people usually evaluate the gamble and its cost separately, and decide to purchase the gamble if the combined value is positive. Thus, the gamble (1,000, .01) will be purchased for a price of 10 if 7r (.01)v(l,OOO) + v( -10) > 0. If this hypothesis is correct, the decision to pay 10 for (1,000, .01), for example, is no longer equivalent to the decision to accept the gamble (990, .01; -10, .99). Furthermore, prospect theory implies that if one is indifferent between (x(1p ), p; -px, 1- p) and (0) then one will not pay px to purchase the prospect (x, p ). Thus, people are expected to exhibit more risk seeking in deciding whether to accept a fair gamble than in deciding whether to purchase a gamble for a fair price. The location of the reference point, and the manner in which 'choice problems are coded and edited emerge as critical factors in the analysis of decisions. Extensions In order to encompass a wider range of decision problems, prospect theory should be extended in several directions. Some generalizations are immediate; others require further development. The extension of equations (1) and (2) to prospects with any number of outcomes is straightforward. When the number of outcomes is large, however, additional editing operations may be invoked to compound simplify evaluation. The manner in which complex options, ･Ｎｧｾ＠ prospects, are reduced to simpler ones is yet to be investigated. Although the present paper has been concerned mainly with monetary outcomes, the theory is readily applicable to choices involving other attributes, e.g., quality of life or the number of lives that could be lost or saved as a consequence of a policy decision. The main properties of the proposed value function for money should apply to other attributes as well. In particular, we expect outcomes to be coded as gains or losses relative to a neutral reference point, and losses to loom larger than gains. The theory can also be extended to the typical situation of choice, where the probabilities of outcomes are not explicitly given. In such situations, decision weights must be attached to particular events rather than to stated probabilities, but they are expected to exhibit the essential properties that were ascribed to the weighting function. For example, if A and B are complementary events and neither is certain, 7r(A) + 7r(B) should be less than unity-a natural analogue to subcertainty.

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The decision weight associated with an event will depend primarily on the perceived likelihood of that event, which could be subject to major biases [45]. In addition, decision weights may be affected by other considerations, such as ambiguity or vagueness. Indeed, the work of Ells berg [10] and Fellner [12] implies that vagueness reduces decision weights. Consequently, subcertainty should be more pronounced for vague than for clear probabilities. The present analysis of preference between risky options has developed two themes. The first theme concerns editing operations that determine how prospects are perceived. The second theme involves the judgmental principles that govern the evaluation of gains and losses and the weighting of uncertain outcomes. Although both themes should be developed further, they appear to provide a useful framework for the descriptive analysis of choice under risk.

The University of British Columbia and Stanford University Manuscript received November, 1977; final revision received March, 1978.

APPENDIX2 In this appendix we sketch an axiomatic analysis of prospect theory. Since a complete self-contained treatment is long and tedious, we merely outline the essential steps and exhibit the key ordinal properties needed to establish the bilinear representation of equation (1), Similar methods could be extended to axiomatize equation (2). Consider the set of all regular prospects of the form (x, p; y, q) with p + q < 1. The extension to regular prospects with p + q = 1 is straightforward. Let 2: denote the relation of preference between prospects that is assumed to be connected, symmetric and transitive, and let = denote the associated relation of indifference. Naturally, (x, p; y, q) = (y, q; x, p). We also assume, as is implicit in our notation, that (x, p; 0, q) = (x, p; 0, r), and (x, p; y, 0) = (x, p; z, 0). That is, the null outcome and the impossible event have the property of a multiplicative zero. Note that the desired representation (equation (1)) is additive in the probability-outcome pairs. Hence, the theory of additive conjoint measurement can be applied to obtain a scale V which preserves the preference order, and interval scales f and g in two arguments such that V(x, p; y, q) = f(x, p)+ g(y, q).

The key axioms used to derive this representation are: Independence: (x, p; y, q) 2: (x, p; y'q') iff (x', p'; y, q) 2: (x', p'; y', q'). Cancellation: If (x, p; y' q') 2: (x', p'; y, q) and (x', p'; y", q") 2: (x", p"; y', q'), then (x, p; y", q") 2: (x", p"; y, q). Solvability: If (x, p; y, q) 2: (z, r)2: (x, p; y' q')for some outcome z and probability r, then there exist y", q" such that (x,p; y"q")=(z, r).

It has been shown that these conditions are sufficient to construct the desired additive representation, provided the preference order is Archimedean [8, 25]. Furthermore, since (x, p; y, q) = (y, q; x, p ), f(x, p) + g(y, q) = f(y, q) + g(x, p ), and letting q = 0 yields f = g. Next, consider the set of all prospects of the form (x, p) with a single non-zero outcome. In this case, the bilinear model reduces to V(x, p) = 1r(p)v(x). This is the multiplicative model, investigated in [35]

and [25]. To construct the multiplicative representation we assume that the ordering of the probability-outcome pairs satisfies independence, cancellation, solvability, and the Archimedean axiom. In addition, we assume sign dependence [25] to ensure the proper multiplication of signs. It should be noted that the solvability axiom used in [35] and [25] must be weakened because the probability factor permits only bounded solvability. 2 We are indebted to David H. Krantz for his help in the formulation of this section.

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Combining the additive and the multiplicative representations yields V(x, p; y, q) = f[ 1T(p)v(x)] + f[ 1T(q)v(y)]. Finally, we impose a new distributivity axiom: (x, p; y, p) = (z, p)

iff

(x, q; y, q) = (z, q).

Applying this axiom to the above representation, we obtain

f[ 1T(p) v (x)] +f[ 1T(p )v (y )] = f[ 1T(p )v(z)] implies f[ 1T(q)v(x )] + f[ 7T(q)v(y )] = f[ 1T(q )v (z)]. Assuming, with no loss of generality, that 7T(q) < 1T(p), and letting a= 7T(p)v(x), (3 = 1T(p)v(y), y= 17 (p)v(z), and 0=7T(q)/7T(p), yields f(a)+f((3)=f(y) implies f(Oa)+f(0(3)=f(Oy) for all 0:: 0 for x > 0, and V" : >: 0 for x < 0. Notice that since the value function is nondecreasing (V' : >: 0), V KT is a subset of U 1 . We call V a value function, rather than a utility function, to be consistent with the terminology of Kahneman and Tversky. In the discussion of Markowitz utility functions we confine ourselves to the range between the two extreme inflection points of the Markowitz utility function (Points A and B in Figure 1c), which are expected to be at extreme wealth levels (see Markowitz 1952b). Thus we define: VM-the class of all Markowitz utility functions (where the subscript M denotes Markowitz): functions which are reverse S-shaped with an inflection point at x=O. Thus, VEVM if V' ::>::Oforallx,PO/ V" ::>::0 for x > 0, and V" ::0: 0 for x < 0. Notice that since the value function is nondecreasing (V' ::>:: 0), VM is also a subset of U 1• As Markowitz's function, like the prospect theory value function, depends on change of wealth, we denote it by VM, rather than UM. We turn now to a few decision rules which are known as stochastic dominance rules. As explained above, because all of these rules (not only FSD as in the example above) are invariant to the initial wealth, they can be written in terms of the change of wealth x. We briefly describe the well-known FSD and SSD rules. We then present the prospect stochastic dominance rule (PSD), which corresponds to all S-shaped value functions, and the Markowitz stochastic dominance rule (MSD), which corresponds to all reverse S-shaped value functions. All these rules will be examined in the experiment. We focus on PSD and MSD, which may indicate whether the prospect theory value function is valid in the general case, where the subjects have to make choices between mixed prospects with negative as well as positive outcomes, and in which there is no certainty effect. FSD (First-Degree Stochastic Dominance). Let F and G be two distinct prospects with cumulative distributions F and G, respectively. Then, F(x) ::0: G(x) for all x {}

EFU(x) "':: EGU(x) for all

u E ul,

(1)

As in the case of 5-shaped functions, in the general case we allow for V' (O) to be nonexistent.

7

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October 2002

where there is a strict inequality for some x = x0 , and a strict inequality for some UQ E U 1 . SSD (Second-Degree Stochastic Dominance). Define F and G as above. Then, {)c(t)-F(t)]dt::>::O forallx {}

EFU(x) : >: EGU(x) for all U E U2 ,

(2)

where there is a strict inequality for some x = x0 , and a strict inequality for some UQ E U2 • For proof of the FSD and SSD rules, see Hadar and Russell (1969), Hanoch and Levy (1969), and Rothschild and Stiglitz (1970). For a survey on stochastic dominance rules, see Levy (1992). PSD (Prospect Stochastic Dominance). Define F and G as above. Then F dominates G for all S-shape utility /value functions, V E V KT• if and only if

jy

0

[G(t)- F(t)] dt : >: 0 for ally ::0:0 and

f

[G(t)- F(t)] dt ::>:: 0 for all x : >: 0.

(3)

(Once again, we require a strict inequality for some pair (y0 , x0 ) and for some V0 E VKr·) A proof of PSD and more detail can be found in Levy (1998), and Levy and Wiener (1998). MSD (Markowitz Stochastic Dominance). Define F and G as above. Then F dominates G for all reverse S-shaped value functions, V E VM, if and only if

L:

[G(t)- F(t)] dt : >: 0 for ally:::: 0 and £oo!G(t)-F(t)]dt::>::O forallx::>::O,

(4)

(with at least one strict inequality). We call this dominance relation MSD-Markowitz Stochastic Dominance. PROOF. See Appendix A. Although Conditions (3) and (4) look similar, we will show below that it is possible that F dominates G by PSD, but G dominates F by MSD. While the intuitions of FSD and SSD are relatively straightforward, the intuition for PSD and MSD is less transparent. The intuition for SSD and the riskseeking dominance (RSD) rule (presented below) are important in particular because both PSD and MSD

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preferences contain both risk-averse and risk-seeking segments. Therefore, once we understand the intuition for SSD and RSD, the intuition for PSD and MSD becomes straightforward. FSD. F(x) :S G(x) for all x implies that 1- F(x) ::: 1- G(x) for all x, or PrF(X::: x)::: Pre( X::: x) for all x; Thus, the probability of getting x or more is higher under F than under G for all x; consequently; any individual who prefers more money over less money will prefer F. SSD. For any utility function the difference in expected utility between prospects F and G is given by: !J.=EFU(x)-EGU(x)= /_:[G(t)-F(t)]U'(t)dt. (5)

Equation (5) is obtained by integration by parts of the difference in expected utility of the two prospects, see Hadar and Russell (1969), Hanoch and Levy (1969), Rothschild and Stiglitz (1970), and Levy (1998). The condition for SSD dominance of F over G is that foo[G(t) -F(t)] dt::: 0 for all x. Graphically, this means that the area enclosed between G and F from -oo to any value x is positive. Thus, the SSD condition states that for any negative area (F > G) there is a larger preceding positive area (G >F). In calculating !J. these areas are multiplied by U' (see Equation (5)), and as U' is a declining function for risk averters (U" :S 0) the SSD condition ensures that the positive contribution to !J. of the positive areas is greater than the negative contribution to !J. of the negative areas, and therefore !J. :;:0. RSD. The dominance condition ofF over G for riskseeking investors (U'::: 0, U"::: 0 in the whole domain of x) is that fxoo[G(t)- F(t)] dt:;: 0 for all x (see Levy 1998, pp. 33Q-332). Graphically this means that the last area (largest x) enclosed between G and F is positive (which is not required by SSD). There may be a negative area before the last positive area, but it must be smaller than the last positive area, and so forth-for each negative area there is a larger positive area which follows (at larger x). The explanation for the RSD criterion is that we have, as before, !J. = EFU(x)- EGU(x) = ｦｾｯ｛ｇＨｴＩ＠ -F(t)]U'(t) dt (see Equation (5)), but this time U' is an increasing function (U"::: 0). If the RSD criterion holds, the contribution of the positive areas to !J. is larger than the negative contribution of the negative areas, and we have !J. ::: 0 or

1340

EFU(x) ::: EGU(x) for all risk-seeking investors. Having the intuitive explanation for risk-averse and riskseeking dominance, the intuition of PSD and MSD becomes straightforward. PSD. In this case we have U" ::: 0 for x < 0, and U" :S 0 for x > 0, and the condition for the dominance ofF over G by PSD is: fox[G(t)- F(t)]dt::: 0 for all x > 0 and f:[G(t) -F(t)]dt::: 0 for ally< 0 (see Equation (3)). The intuition for this condition relates directly to the preceding explanations about SSD and RSD. We want F to be preferred over G by any S-shaped utility function. Consider an 5-shaped utility function which is almost linear and with a very small slope for y < 0 (i.e., U' "" 0 for y < 0). For such a function the contribution of the negative domain to !J. of Equation (5) is negligible, and we should only consider the risk-averse condition for the positive range, which is f 0x[G(t) -F(t)]dt:;: 0 for all x > 0. Similarly, considering an S-shaped function which is almost linear and flat in the positive range yields the risk-seeking condition for the negative range, which is f:[ G(t)- F(t)] dt::: 0 for ally < 0. Therefore, for F to dominate G by PSD for all 5-shaped functions, both of these conditions must hold. MSD. Similar to the logic of the PSD intuition, consider a reverse 5-shaped function which is almost linear and flat in the negative domain. For such a function the contribution of the negative domain to !J. is negligible, and we should only consider the riskseeking condition for the positive range, which is fxoo[G(t)- F(t)] dt::: 0 for all x > 0. A reverseS-shaped function which is almost linear and flat in the positive domain dictates the risk-averse condition for the negative range, which is f!oo[G(t) -F(t)]dt::: 0 for all y :S 0. As we encompass all reverse 5-shaped functions by MSD, the MSD rule is a combination of these two conditions (for formal proof of the MSD rule, see Appendix A). MSD is generally not "the opposite" of PSD. In other words, if F dominates G by PSD, this does not necessarily mean that G dominates F by MSD. This is easy to see, because having a higher mean is a necessary condition for dominance by both rules. 8 Therefore, if F dominates G by PSD, and F has a higher To see this, choose y = -oo and x = +oo in Equation {3), and y = x = 0 in Equation (4), and employ Equation (5) with U(x) = x.

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mean than G, G cannot possibly dominate F by MSD. However, as Corollary 1 below states, PSD and MSD are opposite if the two distributions have the same mean. CoROLLARY 1. Let F and G have the same mean. Then F dominates G by PSD if and only if G dominates F by MSD.

PROOF. From the relation EF(x) -Ec(x) = f"oo[G(t)F(t)] dt (Equation (5) with U(x) = x) we know that if F and G have the same mean, then f_:"oo[G(t)F(t)] dt = 0, or ｦｾｯ｛ｇＨｴＩＭ F(t)] dt + f 000 [G(t) -F(t)] dt = 0. If F dominates G by PSD, then each of these two terms must equal zero. To see this claim, recall that by the equal-mean condition, both terms cannot be negative. If one is positive, by the equal mean condition the other must be negative, and Equation (3) does not hold, so F cannot dominate G by PSD. By the same argument, if G dominates F by MSD we must also have too[G(t)- F(t)] dt = fooo[G(t)- F(t)] dt = 0. Thus, if one distribution dominates the other by either PSD or MSD, this implies too[G(t)- F(t)] dt = 0 and f 000 [G(t)- F(t)] dt = 0. Now, let us prove the claim of Corollar6. 1. F dominates by PSD only if for every y < O,JY[G(t) -F(t)]dt ::0:0. But because in the case of equal means we have too[G(t)- F(t)] dt = 0, this implies that f!oo[G(t) -F(t)]dt :0:0 for every y < 0, or J!oo[F(t)- G(t)] dt ::0: 0 for every y < 0, which is the condition for the MSD dominance of G over F in this range (see Equation (4)). Similarly, f 0x[G(t)- F(t)] dt ::0: 0 for all x > 0 implies fxoo[G(t) -F(t)] dt :0:0 for all x > 0, which in turn leads to froo[F(t)- G(t)] dt ::0:0 for all x > 0. Thus, for F and G with equal mean, F dominates G by PSD if and only if G dominates F by MSD. D

3. The Experiments and the Results The subjects in our experiments were students, university professors, and practitioners. The students are graduate and undergraduate students in the business schools at UCLA, the University of Washington, and the Hebrew University. The university professors are faculty in the business and economics departments at Baruch College, UCLA, and the Hebrew University. The practitioners are financial analysts, mutual funds managers, and portfolio managers.

MANAGEMENT SCIENCE/Yo!. 48, No. 10, October 2002

The subjects were asked to answer a questionnaire with one or more tasks. In each task the subjects had to choose between two investments denoted by F and G. We have conducted three experiments, with questionnaires as given in Tables 1, 2, and 3. All probabilities given in the experiments are relatively large (p ::0: 0.25),9 hence it is unlikely that subjective probability distortion plays an important role in the decision-making process. 10 Since we have no certain outcome, the certainty effect is neutralized. Experiment 1: Design Experiment 1 is designed to investigate which type of preference class (risk aversion, 5-shaped functions, or reverse S-shaped functions) best describes individual choice under risk. Subjects were presented with four tasks, as given in Table 1. In Task I, G dominates F by PSD, but F dominates G by MSD, as shown by Figure 3. Thus, G dominates F for all value functions V E VKr because Condition (3) holds with F and Gin switched roles. At the same time, Condition (4) holds, and hence F dominates G for all reverse S-shaped functions V E VM. Thus, Task I is a head-to-head competition between these two alternative theories. Note that neither prospect dominates the other by SSD. Task IV is designed to test whether risk aversion provides a good description of behavior. In this task, G dominates F by SSD. In the remaining two tasks, Tasks II and III, one prospect dominates the other by FSD. These tasks are "controls" designed to test whether the subjects comprehend the experimental setup, and whether they are rational in the sense of preferring more money to less money. The subjects participating in Experiment 1 were 132 students, 66 professors, and 62 practitioners, for a total of 260 subjects. Experiment 1: Results The results of the experiment are given in Table lb. As there are no significant differences between the results Except in Task 3 of Experiment 1, which tests for FSD, and is a "control" task in our experiment.

9

It is found experimentally that small probabilities, e.g., those related to the chance of winning a lottery, are subjectively distorted by subjects. However, this distortion typically takes place for extreme probabilities, and not for the moderate probabilities employed here. 10

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Table 1a

Experiment 1: The Choices Presented to the Subjects

Table 1b

Suppose that you decided to invest $10,000 enher in stock For in stock G. Which stock would you choose, F, or G, when it is given that the dollar gain or loss one month from now will be as follows. TASK 1:

G Gain or loss

Probability

-3.000 4,500

112

Please write F for G:

1/2

Gain or loss

Probability

-6,000 3,000

114 3/4

G Gain or loss

-500 +2,500

Probability

Gain or loss

Probability

1/3 213

-500 2,500

112 112

o

TASK Ill: Which would you prefer, F or G, if the dollar gain or loss one month from now will be as follows:

G Gain or loss

Probability

Gain or loss

Probability

+500 +2.000 +5,000

3/10 3/10 4/10

-500 0 +500 +1.000 +2.000 +5.000

1/10 1/10 1/10 2110 1110 4/10

Please write For G:

o

TASK IV: Which would you prefer, F or G, if the dollar gain or loss one month from now will be as follows:

G Gain or loss

Probability

-500 +500 +1,000 +2.000

1/4 1/4 1/4 114

Please write F or G: 0

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I (G >-pso F, F '""""G) II (F >-FSo G) ill (F >-FSo G) IV (G >-sso F)

G 71

27

96 82 47

18 51

Indifferent

Total

100 100 100 100

Number of subjects: 260 •Numbers in the tables are in percent. rounded to the nearest integer. The notations,...... >-sso· '""""'and '""•so indicate dominance by FSD, SSD, PSD, and MSD, respectively.

o

TASK II: Which would you prefer, F or G. if the dollar gain or loss one month from now will be as follows:

Please write F for G:

The Results of Experiment 1•

Task No.

Gain or loss

+1.500

obtained in different institutions, or across the subject categories, we report only the aggregate results. In Tasks II and III F dominates G by FSD because F(x) ::: G(x) for all values x. Indeed, in all groups of subjects we find a strong preference for F, the FSD dominating prospect (see Task II and Task III). In Task II there is an obvious FSD superiority of F over G, and therefore we are not surprised with the experimental findings. In Task III the FSD preference of F over G is a little less transparent in comparison to Task II; still, almost all subjects identified the superior FSD investment in this task. Given that investors are rational, and always prefer more money to less money, the findings in Tasks II and III are expected regardless of the investors' specific preferences (at least with our sophisticated group of subjects, where we do not expect to find many errors in the decision making). This is because FSD is a decision rule which corresponds to all types of nondecreasing utility or value functionsY The findings of Tasks II and III neither confirm nor reject prospect theory, since there is FSD dominance In their 1979 paper, Kahneman and Tversky propose possible distortion in probabilities by the decision makers. Such a distortion

11

Probability

may imply that an option which is inferior to the other by FSD (with the objective probabilities) may be selected and the supe-

1/2

rior FSD option may be rejected. Kahneman and Tversky realize that FSD dominance is a cornerstone for the decision-making process, hence suggested in their 1992 paper the cumulative version of prospect theory such that if F dominates G with the objective probabilities, the dominance will not be violated even after the probabil-

112

ity distortion (see also Quiggin 1982 and Yaari 1987). Indeed, FSD is sometimes used as one of the axioms of expected utility theory (see Fishburn 1982 and Levy 1998).

MANAGEMENT ScJENCE/Vol. 48, No. 10, October 2002

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Figure 3

Task I of Experiment 1- G Dominates F by PSD, F Dominates G by MSD

1.00

ｾ＠

-

0.75

:0

as .c a.

1500x0.5

e CD

.ｾ＠

.....oF

0.50

ro :;

+

+

3000x0.25

3000x0.25

E :::J

u

G

0.25

-

3000x0.25

0.00 -6000

•

!

4000 -3000 -2000

0

2000 3000 40004500

6000

payoff($}

of F over G, and therefore a subject with any nondecreasing utility or value function should prefer F to G, whether he has a "standard" concave utility function, an 5-shaped value function, or a reverse 5-shaped function. Thus, any choice of G over F is due to irrationality or due to a possible human error. Fortunately, we did not find many such errors, indicating that the subjects were not answering the questionnaire irrationally or arbitrarily. This point is important for the interpretation of the results of Tasks I and IY, which are the focus of this experiment. In Task I neither F nor G dominates the other by F5D or by 550 (see Figure 3). However, G dominates F by PSD, thus we would expect any subject with 5-shaped preferences to select Prospect G in Task I. At the same time, F dominates G by M5D. Thus Task I constitutes a head-to-head race between two competing theories about preferences: the prospect theory 5-shaped function and the Markowitz reverse 5-shaped function. The striking result is that the majority of subjects (71%) preferred prospect F, strongly contradicting the 5-shaped function suggested by prospect theory, and supporting the reverse 5-shaped function hypothesis. It is interesting to note that the preference of F over G in Task I is even stronger among the more sophisti-

MANAGEMENT Sc!ENCE/Vol. 48, No. 10, October 2002

cated investors-practitioners (79%) and faculty (86%) (for brevity's sake not reported in the table). The highest proportion rejecting at least one segment of the 5-shaped value function in the Kahneman and Tversky experiment is 20% (see Kahneman and Tversky 1979, p. 268). We have on the aggregate 71% of the choices rejecting the 5-shaped function. A statistical test reveals that the difference in the proportion of subjects rejecting the 5-shaped function in these two studies is highly significant with a z-statistic of 8.59. 12 12

The sample statistic is:

z=

PEI- fiKr p•q•

+ p•q•

NEt

NKT

where PEt and fin are the sample mean proportions rejecting the Sshaped function in Experiment 1 (Task I} and the Kahneman and Tversky experiment, respectively, NEt and NKT are the number of observations in these two experiments, p* is the overall sample proportion: * PEl NEt +PKTNKT p = NEl+NKT 1 and q* = 1 - p". For Task I of Experiment 1 we obtain: Z=

0.71-0.20

=8.59. J0.57 ·0.43 0.57. 0.43 ----uo+ _9_5_

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Chapter 7. Prospect Theory: Much Ado About Nothing?

139

LEVY AND LEVY

Prospect Theory Revisited

The results in Task I constitute a rejection of the S-shaped function because they imply that at least 71% of the subjects do not have this type of preference.13 This result lends support to the Markowitz reverse S-shaped function hypothesis, yet proving this hypothesis is much more difficult than rejecting the S-shape hypothesis. This is because one could claim that individuals are driven to choose F by preferences other than a reverse S-shaped function. For example, it is possible that subjects are risk averse, and while F does not dominate G by SSD, it could be that most subjects have higher expected utility under F. One could also suspect that the subjects (explicitly or implicitly) employ the Markowitz (1952a, 1959, 1987) well-known mean-variance rule, because,

hence, F is slightly preferable to G by the meanvariance rule. Thus, the subjects may employ the mean-variance rule even if it is not justified in this specific case because the distributions are not normal; see Tobin (1958) and Hanoch and Levy (1969). Task IV is designed to address these issues. In Task IV, there is no FSD and no PSD or MSD dominance, but G dominates F by SSD. Thus, if subjects are risk-averse we would expect strong support for G. We find that only about half of the subjects selected Prospect G. 14 Thus, almost half of the subjects made choices contradicting risk aversion, casting grave doubt on the ability of preferences which are concave throughout (as in Figure 1a) to describe the subjects' behavior. This contradiction of risk aversion is consistent with the findings of Levy and Levy (2001). The results of Task IV also casts doubt on the hypothesis that the subjects employed the meanvariance rule in their decision making. We tend to reject this hypothesis for the following reason: In 13 We stress "at least" because any individual with S-shaped preferences should prefer G, but choosing G does not imply that preferences are necessarily 5-shaped.

71% of the faculty preferred G in Task IV. It is interesting that while the faculty (most of whom are conscious of the normative theory) chose G, most students (59%) who decide based on "gut feeling" prefer F. 14

1344

Task IV, Prospect G dominates prospect F not only by SSD but also by the mean-variance rule: EF(x)

= Ec(x) = 750

O'F(x)

= 901 >

O'c(x)

= 750.

Moreover, the mean-variance dominance of G over F is much more pronounced in Task IV than the mean-variance dominance of F over G in Task I. If the subjects employ the mean-variance rule (even if there is no theoretical justification to employ this rule here, as suggested above), we would expect to find a strong preference for Prospect G in Task IV. This did not occur, and approximately half of the subjects preferred Prospect F in Task IV. Thus, the decision making of the subjects is not driven by the mean-variance rule. Experiment 2: Design Experiment 2 is designed to create another headto-head competition between the prospect theory S-shaped function and the Markowitz reverse S-shaped function. The subjects in this experiment were 84 business school students. Table 2a provides the task in this experiment. It is straightforward to verify that F dominates G by PSD, but G dominates F by MSD. The task in this experiment is different than Task I of Experiment 1 in two respects: First, it is more complex. In this experiment there are four possible outcomes for each prospect, rather than the two outcomes of Task I in Experiment 1. Second, in Experiment 2 all the outcomes are equally likely. This is an attractive feature, because it makes any subjective probability distortion very unlikely. 15 Experiment 2: Results The results of Experiment 2 are given in Table 2b. The table reveals that 62% of the subjects chose Prospect G, which is inferior by PSD but dominant by MSD. Thus, we can state that at least 62% of the choices are inconsistent with prospect theory. While this result again rejects the S-shaped function and This point is made by Quiggin (1982). In addition, any subjective transformation performed directly on the probabilities (as in prospect theory) will still attach an equal probability weight to each outcome. This is also true of Viscusi's (1989) prospective reference theory with a symmetric reference point. 15

MANAGEMENT SCJENCE/Vol. 48, No. 10, October 2002

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Prospect Theory Revisited

Tabla 2a

The Choices Presented to the Subjects

Suppose that you decided to invest $10,000 etther in stock For in stock G. Which stock would you choose, F, or G, when it is given that the dollar gain or Joss one month from now will be as follows:

G Gain or loss

Probability

Gain or loss

Probability

-1,600 -200 1,200 1,600

1/4 1/4 1/4 1/4

-1,000 -800 800 2,000

1/4 1/4 1/4 1/4

Please write For G: 0 Tabla 2b

The Results of Experiment 2•

(F >-pso G, G >-MSo F)

38%

G

Indifferent

Total

62%

0%

100%

Number of subjects: 84. •Numbers in the tables are in percent, rounded to the nearest integer. The notations >-pso and >-MSo indicate dominance by PSD and MSD, respectively.

prospects. By this procedure we can pinpoint whether having prospects with mixed outcomes is of crucial importance. Experiment 3: Design The tasks in Experiment 3 are given in Table 3a. Tasks I and II are precise replications of the tasks in the Kahneman and Tversky (1979) experiment (see p. 268). Task I compares a positive prospect with a certain positive outcome, and Task II compares a negative prospect with a certain negative outcome. Task III involves two uncertain prospects with mixed outcomes. In this third task, Prospect G dominates by P5D, but Prospect F dominates by M5D. The subjects participating in this experiment were 51 practitioners and 129 business school students.

Tabla 3a

lends support to the reverse 5-shaped function, it is somewhat less dramatic than the 71% who chose the PSD-inferior (and M5D-dominant) prospect in Task I of Experiment 1. This difference may be due to the increased complexity of the task in Experiment 2 relative to Task I of Experiment 1. Indeed, a similar drop (from 96% to 82%) is observed in the choices conforming with F5D as the complexity of the tasks was increased (see Tasks II and III in Experiment 1). Yet the proportion rejecting the 5-shaped function here is significantly larger than in the Kahneman and Tversky (1979) experiment, with a z-value of 5.72 (see Footnote 12). The strong results rejecting the prospect theory 5-shaped function that was revealed in Experiments 1 and 2 raise the following questions: How do these results fit with the wide support documented for prospect theory? Is there perhaps some special attribute to the subject population in our experiments which systematically biases our results? To answer these questions, we have conducted a third experiment, in which the same group of subjects is presented with tasks identical to the original Kahneman and Tversky certainty equivalent tasks, and also with a task involving mixed and uncertain

MANAGEMENT SciENCE/Vol. 48, No. 10, October 2002

The Choices Presented to the Subjects

Task 1: Imagine that you face the following two alternatives and you must choose one of them. Which one would you select?

G Gain or loss

Probability

Gain or loss

4,000

0.80 0.20

3,000

0

Probability

Please write F for G: 0 TASK II: Imagine that you face the following two alternatives and you must choose one of them. Which would you select?

G Gain or loss

Probability

Gain or loss

-4,000 0

0.80 0.20

-3,000

Probability

Please write F for G: 0 TASK Ill: Suppose that you decided to invest $10,000 either in stock F or in Stock G. Which stock would you choose, F, or G, when it is given that the dollar gain or foss one month from now will be as follows:

G Gain or loss

Probability

Gain or loss

Probability

-1,500

1/2 1/2

-3,000 3,000

3/4

4,500

1/4

1345

Chapter 7. Prospect Theory: Much Ado About Nothing?

141

LEVY AND LEVY Prospect Theory Revisited

Table 3b

The rasuiiS ol Experiment3

Task

Ill (G >-pso F, F >-Mso G)

19% 69% 76%

G

Indifferent

Total

81% 30% 23%

0%

100% 100% 100%

1% 1%

Number of subjects: 180. Numbers in the tables are in percent, rounded to the nearest integer. The notations >-pgo and >-Mso indicate dominance by PSD and MSD, respectively.

Experiment 3: Results Table 3b reports the results of Experiment 3. Because there are no significant differences between the subject populations, we report only the aggregate results. In Task I we find that 81% of the choices are consistent with risk aversion for gains. In Task II, 69% of the choices are consistent with risk seeking for losses. Thus, the results in these two tasks are very similar to Kahneman and Tversky' s results, albeit our results in the negative domain are somewhat weaker (Kahneman and Tversky obtain 80% and 92%, respectively, see Kahneman and Tversky 1979, Table 1). The striking result is related to Task ill. Here, 76% of the subjects chose F, the prospect which is inferior by PSD! 16 The same subjects which seem to support risk aversion for gains and risk seeking for losses when they are confronted with nonmixed gambles in the certainty equivalent framework, reject the S-shaped function when mixed and uncertain prospects are compared. Only 17% of the subjects made choices consistent with the S-shaped function throughout: a choice of G in Task I, F in Task II, and G in Task III. This leads us to believe that the support found in previous studies for the S-shaped function may be due to the use of nonmixed gambles and the certainty equivalent framework, with one certain outcome. As these constitute very special and atypical situations, we believe that the results in these tasks may be biased due to their special setup or to the well-known "certainty effect". Note that in Task ill, F dominates G by MSD. Thus, when the two competing theories of the S-shaped function and the reverse S-shaped function race head This is significantly larger than the proportion of subjects rejecting the S-shaped value function in the Kahneman and Tversky experiment, with a z-value of 8.92

16

1346

to head, 76% of the choices conform with MSD and contradict PSD.

4.

Concluding Remarks

Prospect theory is a paradigm challenging expected utility theory. The four main components of prospect theory are: Individuals make decisions based on change of wealth rather than total wealth; preferences are described by an S-shaped value/utility function, V(x), with an inflection point at x = 0; individuals distort small probabilities; and the "framing" of alternatives affects individuals' choices. The main justification for prospect theory is the results of experimental studies in which the subjects had to declare the certainty equivalent of either negative or positive bets, bets which virtually do not exist in practice, and certainly are not common in the financial markets. This unrealistic "framing" of the bets, and possible biases induced by the "certainty effect" could account for the support for the S-shaped value function which is found in previous experiments, and which is rejected in our experiment. In our experiment, we employ stochastic dominance investment criteria. Because the stochastic dominance criteria are invariant to the initial wealth, we can focus on change of wealth rather than total wealth. We deal with relatively large probabilities (p 2:: 0.25), hence it is very unlikely that probability distortion plays an important role in our experiment. Furthermore, in our study we have prospects with no certain outcome, hence the certainty effect is irrelevant here. Neutralizing these three factors allows us to focus on the preference and to examine whether it is indeed 5-shaped, as claimed by prospect theory. In the previous experiments the certainty equivalent approach was employed, hence, in order to characterize the properties of the value function there was no practical choice but to use nonmixed bets (either negative or positive). The stochastic dominance approach allows us to employ mixed bets, which are much more typical of real investment situations. For the first time, we use a recently developed investment criterion called Prospect Stochastic Dominance (PSD), and a criterion which is developed here, called Markowitz Stochastic Dominance (MSD).

MANAGEMENT SCIENCE/Yo!. 48, No. 10, October 2002

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M Levy and H Levy LEVY AND LEVY

Prospect Theory Revisited

Using the PSD criterion and mixed bets, based on the experimental results of three distinct groups of subjects (students, university professors, and practitioners), we conclude that the S-shaped preference is rejected. Moreover, employing the MSD criterion reveals support for the reverse S-shaped value function, which was suggested by Markowitz (1952b). To be more specific, in tasks where one prospect dominates the other by PSD but an opposite dominance holds for MSD, three experiments revealed that 71%, 62%, and 76% of the subjects preferred the MSD dominating prospect. Moreover, this preference is even stronger with the more sophisticated investors: 79% of the practitioners and 86% of the professors preferred F over G, where G is the PSD dominating option! Thus, we reject the prospect theory S-shaped function. It is interesting to note that when faced with outcomes restricted either to the positive domain or to the negative domain, as in the certainty equivalent approach, most subjects make choices according to prospect theory. However, the same subjects reject the PSD superior option with mixed bets. This leads us to believe that the support for the S-shaped function as obtained by Kahneman and Tversky is actually due to the well-known certainty effect and does not represent investors' preferences in a realistic setting of mixed bets. Obviously, different individuals may have preferences of different types, i.e., some may be characterized by risk aversion, others by an S-shaped value function, etc. Some individuals may not even be consistent in their decision making. Thus, one cannot hope to find a preference class which perfectly characterizes all individuals. However, our experimental evidence indicates that the best characterization of subjects' behavior is given by the class of reverse Sshaped functions, as suggested by Markowitz, which is just the opposite of the 5-shaped function advocated by prospect theory. Acknowledgments The authors acknowledge the helpful comments of the associate editor and the two anonymous referees. This study was financially supported by the Krueger Center of Finance and by the Zagagi Fund.

MANAGEMENT ScrENCE/Vol. 48, No. 10, Cktober 2002

Appendix A. A Proof of the Markowitz Stochastic Dominance (MSD) Criterion THEOREM 1. Let V(x) E VM where VM is the class of all continuously and twice differentiable Milrkowitz utility functions such that V' 0> 0 for all x, with V" :50 for x < 0 and V" 2: 0 for x > 0. Then F dominates G for all V E VM if and only if:

/_: [G(t)- F(t)] dt 0> 0 for ally< 0 and ioo[G(t)-F(t)]dto-0 forallx>O. PROOF. Let us first formulate our proof in terms of change of wealth, x, rather than total wealth, w + x, and then show that the dominance is invariant to the value w. For convenience, assume that the outcomes of Prospects F and G have lower and upper bounds a and b. One can later generalize the result to the case of unbounded prospects by taking the limits ｡ｾ＠ -oo b ｾ＠ oo. Define:

b.= E, V(x)- Ec V(x) =f.

b

V(x) dF(x)-

f.

b

V(x) dG(x).

Integrating by parts, we have

b.= [F(x)- G(x)]V(x{- J.' (F(x)- G(x))V'(x) dx. As F(b) = G(b) =I and F(a) = G(a) = 0, we have:

b.= J.'[G(x)-F(x)]V'(x)dx = f[G(y)-F(y)]V'(y)dy+ {[G(x)-F(x)]V'(x)dx,

where we use the notation y for variable values in the negative domain. Integrating once again by parts, the tvvo terms on the righthand side yield:

b.= V'(y) f.

y

10

(G(t)-F(t))dt,-

f. 0V"(y) f.

y

(G(t)-F(t))dtdy

+V'(x) {(G(t)-F(t))dtl'- { V"(x) {(G(t)-F(t))dtdx. 0

0

0

0

As some of the terms (i.e., the cases y =a, and x = 0) are equal to zero, A can be rewritten as:

b.= V'(O) f[G(t)-F(t)]dt-

t

V"(y) t(G(t)-F(t))dtdy

+ V'(b) {[G(I)-F(t)]dt- { V"(x) [(G(t)-F(t))dtdx. Because V' 0> 0 and V" :0 0 for y < 0, the condition J;'[G(t)F(t)]dt 0> 0 ensures that the first two terms on the right-hand side of A are nonnegative. (Note that we assume that the utility function is twice differentiable, and that V' 2: 0 for all x. If the utility function is not differentiable at a given point x0 , approximations can be used without altering the results.) One is tempted to believe that for

1347

Chapter 7. Prospect Theory: Much Ado About Nothing?

143

LEVY AND LEVY Prospect Theory Reuisited

x > 0, the condition for dominance should be fo'[G(t)-F(t)]dt : 0, the conditions of the theorem guarantee that ｾＢＰＮ＠

Finally, note that if the utility function is V ( w + x) and the inflection point is at x = 0, the proof is kept unchanged because F(w+x) and G(w+x) are simply shifted to the right by w with no change in the area enclosed between F and G. Necessity. It can be easily shown that if 0 [G(t)- F(t)]dt < 0 for some x 0 < 0, then there is some V E V M for which .6. 0 < 0. To show this, employ the same necessity proof of Hanoch and Levy (1969) for second-degree stochastic dominance. By a similar argument, one can show that J:[G(t)- F(t)]dt > 0 for x > 0 is also a necessary condition for MSD dominance.

J..'

References Allais, M. 1953. Le Comportement de l'homme rationnel devant le risque: Critique des postulates et axioms de I'ecole Americaine. Econometrica 21 503-546.

1348

ｂ｡ｲ｢･ｩｳｾ＠

N., M. Huang, T. Santos. 2001. Prospect theory and asset prices. Quart. J. Econom. 116 1-53. Benartzi. S., R. Thaler. 1995. Myopic loss aversion and the equity premium puzzle. Quart. J. Econom. 110(1) 73-92. - , - - 1999. Risk aversion or myopia? Choices in repeated gambles and retirement investments. Management Sci. 45(3) 364-381. Edwards, K. D. 1996. Prospect theory: A literature review. Internat. Rev. Financial Anal. 5(1), 13-38. Fishburn, P. C. 1982. Nontransitive measurable utility. J. Math. Psych. 26 31-67. Friedman, M., L. J. Savage. 1948. The utility analysis of choices involving risk. J. Political Econom. 56 279-304. Hadar, J., W. Russell. 1969. Rules for ordering uncertain prospects. Amer. Econom. Rev. 59 25-34. Hanoch, G., H. Levy. 1969. The efficiency analysis of choices involving risk. Rev. Econom. Stud. 36 335-346. Kahneman, D., A. Tversky. 1979. Prospect theory of decisions under risk. Econometrica 47(2) 263-291. Levy, H. 1992. Stochastic dominance and expected utility: Survey and analysis. Mimagement Sci. 38(4) 555-593. - - . 1998. Stochastic Dominance: Investment Decision Making Under Uncertainty. Kluwer Academic Publishers, - - . 2000. Cumulative prospect theory and the CAFM. Working paper, Hebrew ｕｮｩｶ･ｲｳｴｹｾ＠ Jerusalem, Israel. - , Z. Wiener. 1998. Stochastic dominance and prospect dominance with subjective weighting functions. ]. Risk Uncertainty 16 147-163. Levy, M., H. Levy. 2001. Testing for risk-aversion: A stochastic dominance approach. Econom. Lett. 71 233-240. - - , - - , S. Solomon. 2000. Microscopic Simulation of Financial Markets: From Investor Behavior to Market Phenomena. Academic Press, San Diego, CA. Markowitz, H. M. 1952a. Portfolio selection. f. Finance 7 77-91. - - . !952b. The utility of wealth. J. Political Econom. 60 151-156. - . 1959. Portfolio Selection. John Wiley and Sons, New York. - - . 1987. Mean Variance Analysis, Portfolio CJwice and Capital Markets. Basil Blackwell, New York. Quiggin, J. 1982. A theory of anticipated utility, J. Econom. Behavior Organ. 3 323-343. - - . 1993. Generalized Expected Utility Theory: The Rilnk Dependent Model. Kluwer Academic Publishers, Boston, MA. Rothschild, M., J. Stiglitz. 1970. Increasing risk I. A definition. J. Econom. Theory 2 225-243. Shefrin, H., M. Statman 1993. Behavioral aspect of the design and marketing of financial products. Financial Management. 22(2) 123-134. Swaim, R. 0. 1966. Utility theory-Insights into risk taking. Harvard Bus. Rev. 44 123-136. Thaler, R. 1985. Mental accounting and consumer choice. Marketing Sci. 4(3) 199-214. Tobin, J. 1958. Liquidity preferences as behavior toward risk. Rev. Econom. Stud. 25 65-86.

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Prospect Theory Revisited

Tversky, A., D. Kahneman. 1981. The framing of decisions and the psychology of choice. Science 211 453-480. - - , - - . 1986. Rational choice and the framing of decision. J. Bus. 59(4) 251-278. - , - - . 1992. Advances in prospect theory: Cumulative representation of uncertainty. ]. Risk Uncertainty 5 297-323.

Viscusi, W. K. 1989. Prospective reference theory: Toward an Explanation of the paradoxes.]. Risk Uncertainty. 2 235-264.

von Neuman, J., 0. Morgenstern. 1944. Theory of Games and Economic Behavior. Princeton University Press, Princeton, NJ. Yaari, M. 1987. The dual theory of choice under risk. Econometrica 55(1) 95-115.

Accepted by Phelim. P. Boyle; received February 2001. This paper was with the authors 2 months for 2 revisions.

MANAGEMENT SciENCE/Vol. 48, No. 10, October 2002

1349

Chapter 8

The Data of Levy and Levy (2002) "Prospect Theory: Much Ado About Nothing?" Actually Support Prospect Theory Peter P. Wakker Department of Economics, University of Amsterdam, Roetersstraat 11, Amsterdam 1018 WB, The Netherlands [email protected]

L

evy and Levy (Management Science 2002) present data that, according to their claims, violate prospect theory. They suggest that prospect theory's hypothesis of an S-shaped value function, concave for gains and convex for losses, is incorrect. However, all the data of Levy and Levy are perfectly consistent with the predictions of prospect theory, as can be verified by simply applying prospect theory formulas. The mistake of Levy and Levy is that they, incorrectly, thought that probability weighting could be ignored.

(Prospect Theory; Stochastic Dominance; Utility; Probability Weighting; Inverses)

Levy and Levy (Management Science 2002, henceforth LL) present data that, according to their claims, violate the S-shaped value function posited by prospect theory. This comment will show, however, that LL's data are in perfect agreement with prospect theory. Following LL, we will throughout restrict attention to outcomes that are not very extreme, say between $6,000 and -$6,000. The classical views on risk attitudes assumed universal risk aversion. Empirical studies have revealed a more complex, fourfold pattern in behavior. For gains, people are mostly risk averse; but for specific prospects, yielding a best outcome with a low probability (below 1/3), we often find risk-seeking behavior, as observed in gambling for instance. The pattern for losses is less clear but seems to be reversed. People are mostly risk seeking, but for prospects yielding a worst outcome with a low probability (below 1 /3), risk aversion can occur, as observed for instance in insurance. This fourfold pattern is based on extensive empirical evidence (reviewed in Starmer 2000 and Luce 2000) and entails extremity-orientedness whereby the best and worst

outcomes of a prospect are overweighted and middle outcomes are underweighted. Prospect theory models the fourfold pattern. We focus on the most recent, cumulative version of prospect theory (Tversky and Kahneman 1992). It assumes a utility or value Junction v(x) that isS-shaped: increasing for all amounts, concave for gains, and convex for losses. This assumed shape reflects the psychological phenomenon of diminishing sensitivity as one moves away from the "reference point." The reference point divides gains from losses, and is taken to be zero in this comment. Consider a prospect (or gamble) with outcomes x, :S · · · :S x1 :S 0 :S xk+l :S · · · :S x, having probabilities p 1 , . . . , p,. Prospect theory predicts that people will choose prospects according to the value given by k

I: 7T;Av(x;) + I:

v(x;),

7T1

(1)

j=k+l

where A > 0 is a loss-aversion parameter, and the 7TS are decision weights that are calculated based on the "cumulative" probabilities associated with the

0025-1909/03/4907 I 0979$05.00

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1526-5501 electronic TSSN

145

146

PP Wakker WAKKER

Data of Levy and Levy (2002) Jlctually Support Prospect Theory

outcomes. The details of this model are provided in the appendix. Following Markowitz (1952), LL posit that the value function is convex for gains and concave for losses, implying a reverse 5-shape-the opposite of that assumed by prospect theory. In their theoretical analysis, LL assume expected utility theory and, thus, assume that the decision weights in Equation (1) are simply equal to the probabilities associated with the outcomes. This coupled with their assumption about the form of the value function implies risk aversion (and, more strongly, secondorder stochastic dominance) for losses and risk seeking (and reversed second-order stochastic dominance) for gains, in contrast to the more complex fourfold pattern of observed behavior. LL develop a stochastic dominance rule that they call Markowitz stochastic dominance. Given expected utility theory, this rule allows them to show that some gambles are preferred to others for all value functions having the reverse 5-shape. LL test their hypothesis about the shape of the value function through three choice exercises that they refer to as head-to-head competitions, and in which they claim that an S-shaped value function would lead to one choice and a reverse S-shaped value function to another. In all three experiments, the majority choice corresponds to the choice that supposedly supports the reverse 5-shape, and LL interpret this as evidence contradicting prospect theory. Simple calculations show, however, that prospect theory with the functional forms and parameter estimates of Tversky and Kahneman (1992) correctly predicts the majority choice in all head-to-head competitions; these calculations are displayed in Table 1, with explanations given in the appendix. This finding is contrary toLL's claims (p. 1344, "Thus, we can state that at least 62% of the choices are inconsistent with prospect theory"). We conclude that LL's data actually support prospect theory. The error in LL's analysis is that they neglect the probability weighting function of prospect theory. They argue, "All probabilities given in the experiments are relatively large (p ::> 0.25), hence it is unlikely that subjective probability distortion plays an important role in the decision-making

980

Table 1

Prospects Yield Outcome

x with Probability p.

(a) LL's Experiment 1, Task 1 Prospect

p

0.50 0.50 0.25 0.75

X

-3,000 4,500 -6,000 3,000

(b) 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25

-1,600 -200 1,200 1,600 -1,000 -800 800 2,000

0.50 0.50 0.25 0.75

-1,500 4,500 -3,000 3,000

ｌｾｳ＠

0.45 0.42 0.29 0.57

-1,148 1,640 -2,112 1,148

PT

Choice

-483

71%

-743

27%

-216

38%

-138

62%

53

76%

-106

23%

Experiment 2 0.29 0.16 0.13 0.29 0.29 0.16 0.13 0.29

-660 -106 512 660 -437 -359 359 803

(c) LL's Experiment 3, Task 3 0.45 0.42 0.29 0.57

-624 1,640 -1,148 1,148

Note: Participants in Levy and Levy (2002) chose between the head-to-head prospect pairs F and G. The PT column gives the values of the prospects according to prospect theory, using the parameters estimated by Tversky and Kahneman (1992). Each " is the decision weight of outcome x, and v its utility/value. Bold printing indicates the majority choice, which is always the option preferred according to prospect theory because it has the higher value under that theory.

process" (p. 1341; a similar statement appears on p. 1346) 1 As Table 1b shows, the extreme outcomes of F and G in Experiment 2 have decision weights about twice as much as the intermediate outcomes 1 ln addition to this argument, LL suggest (p. 1344) that taking the outcomes to be equally likely in their experiment 2 "makes

any subjective probability distortion very unlikely." In their Footnote 15, they expand on this and say "This point was made by Quiggin (1982). Tn addition, any subjective transformation performed directly on the probabilities (as in prospl'ct thl'ory) will still attach an equal probability weight to each outcome." While in the original version of prospect theory (Kahneman and Tvcrsky 1979) the transformation was performed directly on the probabilities alone, this is not so in the current version and equally likely outcomes can be weighted differently. Contrary to "'What LL claim, Quiggin (1982) also argued that equally likely outcomes may be weighed differently (end of §1 ).

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Data of Levy and Levy (2002) Actually Support rrospect Theory

under prospect theory (weights 0.29 versus 0.13 or 0.16), which deviates considerably from the equal weighting assumed by LL. While these calculations are based on the specific parameter assumptions suggested by Tversky and Kahneman (1992), LL's experimental results are also qualitatively consistent with the extremity-orientedness predicted by prospect theory. In each of LL's three head-to-head competitions, the majority chose the gamble that had both the best maximal outcome and the best minimal outcome, as would be done if only the extreme outcomes mattered. In conclusion, the data of LL support the predictions of Tversky and Kahneman's (1992) prospect theory. The incorrect claims of LL are mostly due to their overlooking the crucial role of probability weighting in prospect theory. While the data could also be consistent with other theories, it is extremely misleading to interpret this as evidence against prospect theory or to suggest that prospect theory is "much ado about nothing." In particular, the results of LL do not provide new insights into the shape of the value/utility function. Their hypothesis of convex utility for gains is contrary to the diminishing marginal utility assumed in classical analyses, the diminishing sensitivity assumed in prospect theory, and virtually all empirical findings of the vast literature on this topic.

Acknowledgments The author thanks John Payne, Thierry Post, Pim van Vliet, George Wu, and three anonymous referees for helpful comments, and the department editor for numerous thoughtful and constructive suggestions.

Appendix:

Prospect Theory

Prospect theory assumes, besides the utility or value function and the loss-aversion parameter, a probability weighting function w+: [0, 1J ---+ [0, 1] for gains, and a probability weighting jimction w [0, 1] -+ [0, 1] for losses. The

decision ·weights 7T in Equation (1) are defined as follows. If k ::: 1 then 7T1 = ur(p 1), and 7T; = w-(p 1 + · · · + p,) -w-(p1 +···+PH) for 2 :S i :S k. If k < n then 7T11 = w-(Pn) and 7Ti = w+(pll +···+pi)- zu+(p 11 +· · · +pw for n-1 ｾ＠ j > k. Tversky and Kahneman (1992) estimated the following parametric form: v(x) = for x ｾ＠ 0, v(x) = -(-x) 0 -Rs for x :S 0, 1)

ｾ＠

po'' /(po'' + (1 - p)"')''06'), w-(p) ｾ＠

pooo /(po"+

For prospect F in Table 1 b, the decision ,.veights are 7T1 = w-(0.25) = 0.29 for outcome x1 = -1,600, 7T2 = w-(0.50)w-(0.25) ｾ＠ 0.16 for outcome ｾ＠ -200, 1r, ｾ＠ w-(0.50)- w+(0.25) ｾ＠ 0.13 for outcome x3 = 1,200, and 7T-t = w+(0.25) = 0.29 for outcome x4 =1,600. The value ofF is 7T 1Av(-1,600)+7T 2 Av(-200)+ 7T3 v(1,200) + 7T4 v(1,600) = -215.70. The other prospects are evaluated similarly. A program to calculate prospect-theory values, -.;;vritten by Veronika KObberling, is available at http://wwwl. fee.uva.nl/ creed/-o;;'\rakker / miscella/ calculate.cpt.kobb / index.htm.

References Kahneman, Daniel, Amos Tversky. 1979. Prospect theory: An analysis of decision under risk. Econometrica 47 263-291. Levy, Moshe, Haim levy. 2002. Prospect theory: Much ado about nothing. Management Sci. 48 1334-1349. Luce, R. Duncan. 2000. Utility of Cains and Losses: MeasurementTheoretical and Experimental Approaches. Lawrence Erlbaum Publishers, London, UK. Markowitz, Harry M. 1952. The utility of wealth. [. Political Econom. 60 151-158. Quiggin, John. 1982. A theory of anticipated utility. J. Econom. Behaviour Organ. 3 323-343. Starmer, Chris. 2000. Developments in non-expected utility theory: The hunt for a descriptive theory of choice under risk. }. Econom. Literature 38 332-382. Tversky, Amos, Daniel Kahneman. 1992. Advances in prospect theory: Cumulative representation of uncertainty. J. Ri:;k Uncertainty 5 297-323.

Note from the Editor-in-Chief. Levy and Levy, the authors of the paper "Prospect Theory: Much Ado About Nothing," also wrote the following paper: "Experimental Test of the Prospect Theory Value Function: A Stochastic Dominance Approach" (Orgnnizatio11 Behavior and Huma11 Decision Processes 89, 2002, pp. 1058-1081). The two papers present very similar experiments and results. The failure of Levy and Levy to cross-cite these papers is a violation of proper scholarly practice and may have contributed to the controversy surrounding their work.

Accepted by James E. Smith; received Nmxmbcr 27, 2002. This paper ·was 'With the author 1 month for 2 revisions.

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Chapter 9

Prospect Theory and Mean-Variance Analysis Haim Levy Hebrew University

Moshe Levy Hebrew University The experimental results of prospect theory (PT) reveal suggest that investors make decisions based on change of wealth rather than total wealth, that preferences are S-shaped with a risk-seeking segment, and that probabilities are subjectively distorted. This article shows that while PT's findings are in sharp contradiction to the foundations of mean-variance (MV) analysis, counterintuitively, when diversification between assets is allowed, the MV and PT-efficient sets almost coincide. Thus one can employ the MV optimization algorithm to construct PTefficient portfolios.

The Markowitz (1952a)-Tobin (1958) mean-variance (MV) rule is probably the most popular investment decision rule under uncertainty in economics and in finance, and it is widely employed by both academics and practitioners. The strength of the MV analysis is that in the case of normal return distributions the choice of any expected utility maximizing risk-averse individual will be according to the MV rule. 1 The MV framework is the foundation of the Sharpe (1964)-Lintner (1965) capital asset pricing model (CAPM), which is a cornerstone of modern finance. Moreover, the MV framework provides a very important practical procedure for the construction of efficient portfolios. While standard economic theory and, in particular, MV analysis assume expected utility maximization and risk aversion, in a breakthrough article Kahneman and Tversky (1979) show that the actual behavior of individuals systematically and consistently violates these We are grateful to Campbell Harvey aud to the auouymous referee of this jourual for their mauy helpful commeuts aud suggestions. Fiuaucial support for this study was provided by the Krueger Fuud aud by the Zagagi Fund. Address correspondence to Haim Levy, Jerusalem School of Business Administration, The Hebrew University, Jerusalem 91905, Israel, or e-mail: [email protected]. 1

Formally, when the returu distributions are normal, the MV rule coincides with second-degree stochastic dominance (SSD) [see Hanoch and Levy (1969) and Levy (1998)]. The MV rule is also valid if one replaces the assumption of normality with the assumption of quadratic utility [Tobin (1958)]. However, this latter assumption is problematic as it implies increasing absolute risk aversion and utility which is decreasing beyond a certain wealth level [see Arrow (1971)]. In contrast, the assumption of normal returu distributions is approximately valid for investment horizons longer than one month but shorter than several years. [For very short horizons, the returu distribution is "fat tailed," as shown by Fama (1963), Mandelbrot (1963), Mantegna and Stanley (1995), and others]. The Review of Financial Studies Vol. 17, No. 4 © 2004 The Society for Financial Studies; all rights reserved. doi:l0.1093/rfs/hhg062 Advance Access publication October 15, 2003

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H Levy and M Levy The Review of Financial Studies I v 17 n 4 2004

assumptions. The alternative framework they present is well known as prospect theory (PT). The three main findings of PT are Individuals make decisions based on change of wealth rather than the total wealth, which is in direct contradiction to expected utility theory. (b) Risk aversion does not globally prevail-individuals are riskseeking regarding losses. (c) Individuals distort objective probabilities and subjectively transform them in a systematic way. Tversky and Kahneman (1992) extend PT and suggest cumulative prospect theory, in which the transformation is of the cumulative probability rather than the probability itself. The transformation of cumulative probability allows cumulative prospect theory to avoid violations of firstdegree stochastic dominance (FSD). In this study we refer to the cumulative version of PT. (a)

Many studies support the above findings of PT. 2 Only fairly recently, however, researchers have begun to explore the implications of PT for economics and finance. 3 This article investigates the implications of PT to Markowitz's portfolio theory. It is clear that each ofPT's main elements is in sharp contradiction to the assumptions of MV analysis: the MV framework assumes expected utility maximization and risk aversion while the above-mentioned elements (a) and (b) are at odds with both of these assumptions. Furthermore, if individuals subjectively distort probabilities, the MV analysis seems inappropriate because, even if the objective probability distributions are normal, the subjective distributions are not. Thus it would seem that the MV framework is completely incompatible with PT. Does PT imply that the MV framework, which is so central in finance, should be abandoned? Can any of the effective MV machinery (e.g., the efficient diversification algorithm it provides) be employed to benefit investors with PT preferences? This article addresses these issues. We employ stochastic dominance rules to prove our main claim, which is that the PT and MV efficient sets almost coincide. To be more specific, assume e 1 and e 2 are two random variables. Then the theory of stochastic 2

For an extensive review of studies supporting PT, see Edwards (1996). A number of studies reject PT or some of its elements; see, for example, Battalio, Kagel, and Jiranyakul (1990), Casey (1994), Harless and Camerer (1994), and Luce (2000). In two recent experimental studies, Levy and Levy (2002a,b) reject the PT S-shape value function and find support for the Markowitz (1952b) value function, which has a reverse S-shape. It is interesting to note the main results presented here hold also for the set of reverse S-shape value functions. The proof of this statement is available from the authors upon request.

3

For example, Shefrin and Statman (1985) and Ferris, Haugen, and Makhija (1988) show that PT can explain the empirically observed "disposition effect"- the disposition to sell winning stocks too early and to ride losing stocks too long. Thaler (1985) investigates the implications of PT to marketing. Benartzi and Thaler (1995) show that PT may explain the equity premium puzzle. Gomes (2000) investigates the implication ofPT to trading volume, and Barberis, Huang, and Santos (2001) and Levy, DeGiorgi, and Hens (2003) investigate the consequences of PT for asset pricing.

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dominance is concerned with the conditions on the CDFs of e 1 and e 2 under which E[u(e 1 )] < E[u(e 2 )]

for some set u E U. The standard concepts of stochastic dominance refer to three situations: (a) U is the set of all increasing functions (firstdegree stochastic dominance, FSD); (b) U is the set of increasing and concave functions (second-degree stochastic dominance, SSD); (c) U is the set of increasing and concave functions with a positive third derivative, u"' > 0, which reflects a preference for positive skewness (third-degree stochastic dominance, TSD). Recently the theory of stochastic dominance has been extended to the set of functions that are S-shaped, and that include the family of prospect theory value functions (prospect stochastic dominance, PSD), which have been suggested by Kahneman and Tversky (1979) as a better description of individuals' decisions under uncertainty. In this article we link the standard and familiar concepts of stochastic dominance to the new concept of prospect stochastic dominance. In particular, what are the implications of PSD for the traditional meanvariance analysis of Markowitz? It is well known that SSD is equivalent to mean-variance efficiency when the CDFs are normal. We show that in this case PT efficiency is "almost" identical to mean-variance efficiency. We focus on the canonical case of normal distributions, and offer two results: First, a characterization of PSD efficiency versus MV efficiency when the initial CDFs are not subjectively transformed and the focus on the first two moments is warranted. Second, we characterize PSD efficiency versus MV efficiency when we allow for subjective transformations of the CDFs that preserve FSD. Thus, while in most empirical studies in finance and economics probability distortion is ignored, in this article we analyze MV and PSD efficiency once with no probability distortion and once with probability distortion exactly as suggested by CPT. Note that in this second case the transformed CDFs are not necessarily normal, and may have nonzero skewness and higher odd moments. We show that in both cases the PSD-efficient set is a subset of the (objective) MV -efficient frontier, and it typically almost completely coincides with the MV -efficient set. The implication of this result is that one can employ the well-known MV optimization techniques to derive the efficient set for PT investors. Both frameworks, MV analysis and PT, are strengthened by this result: the MV framework is shown to be valid for a broader class of preferences, and PT is provided with an algorithm for the construction of PT -efficient portfolios. Thus, despite the sharp contradiction between the assumptions in the PT and MV frameworks, MV analysis turns out to be very central in portfolio optimization even under PT preferences.

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The focus of this article is on the canonical case of normal return distributions. However, as long-horizon empirical return distributions tend to be positively skewed, we also analyze the case of lognormal return distributions. The results for the lognormal case, which are derived in Appendix B, are very similar to those of the normal case. Thus it seems that the results of this article are quite robust and do not hinge on specific assumptions regarding the functional form of the return distribution. The structure of this article is as follows: In Section 1 we provide a review of the various decision rules employed in the article, and in particular the prospect stochastic dominance rule. Section 2 contains the theoretical results. Section 3 provides an empirical analysis. Section 4 concludes. 1. Investment Decision Rules

This section briefly reviews the main investment decision rules discussed in the article, with a special emphasis on the recently developed and less familiar PSD rule, which is a criterion for dominance by all S-shaped value functions. 1.1 The MV rule

Let F and G be two investments with stochastic outcomes. Investment F dominates G by MV iff ILF ;:::: !La

and

£Tp

:s; £Ta,

(1)

with at least one strict inequality, where /LF and /LG denote the expected values of investments F and G, and £Tp and £Ta denote the respective standard deviations. Under normal distributions and risk aversion the MV rule coincides with EU maximization [see Markowitz (1952a) and Tobin (1958)]. 1.2 First-degree stochastic dominance

Let F and G be two stochastic investments with cumulative distributions F(x) and G(x), and let U 1 denote the set of all nondecreasing utility functions. Then, F dominates G for all utility functions U E U 1 (i.e., EUF;:::: EUa) iff F(x) :s; G(x)

for all x,

(2)

with at least one strict inequality. When the two distributions are normal and £T 1 = £T2 , a dominance by the MV rule coincides with FSD dominance, namely MV {c} FSD. Note that expected utility can be expanded to

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a Taylor series such that choices depend on all of the distribution moments, for example, mean, variance, skewness, etc. However, if F(x) :s; G(x) for all x, then F is preferred over G by all nondecreasing utility functions, regardless of preference for variance, skewness, and higher moments. 1.3 Second-degree stochastic dominance Let F and G be two stochastic investments with cumulative distributions F(x) and G(x), and let U 2 denote the set of all nondecreasing riskaverse utility functions (U';:::: 0, U" < 0). Then, F dominates G for all U EU 2 , iff

lssn(x) =

1:

[G(z)- F(z)]dz;:::: 0 for all values x,

(3)

with at least one strict inequality. When the distributions are normal, Markowitz's MV rule coincides with SSD, namely SSD {c} MV. In other words, in the EU framework with normal distributions and risk aversion, MV is an optimal rule. For proofs and discussion ofFSD and SSD, see Fishburn (1964), Hadar and Russell (1969), Hanoch and Levy (1969), and Rothschild and Stiglitz (1970). For a survey of SD rules, see Levy (1998). While the above rules are widely used, the recently developed PSD rule [Levy and Wiener (1998)] is still well known. We therefore discuss this rule in some detail below. 1.4 Prospect stochastic dominance Let F and G be two stochastic investments with cumulative distributions F(x) and G(x), and let Us denote the set of all S-shaped utility (or value) functions (U';:::: 0 for all x # 0, U" > 0 for x < 0, and U" < 0 for x > 0). F dominates G for all U E Us iff

lpsn= 1x[G(z)-F(z)]dz;:::O

ｦｯｲ｡ｬｾＺｳ［ＮｸＬ＠

(4)

with at least one strict inequality. Graphically F dominates G by PSD if and only if the area enclosed between the two cumulative distributions G and F is positive for any range [x, x] with x :s; 0 and x;:::: 0. For a discussion and proof of the PSD rule, see Levy (1998) and Levy and Wiener (1998). To illustrate the PSD rule, consider the two prospects given in Table 1. Which of these two prospects will various individuals with S-shaped value functions prefer? Tversky and Kahneman (1992) suggest the following

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Table 1 Two alternative investments F

Probability

G

Outcome

Probability

Outcome

-12 3.5 10 14

1/3 1/3 1/3

-10 5 15

1/4 1/4 1/4 1/4

specific form for the value function: V(x)

x" { -A(

=

-xl

if X;:::: 0 if X< 0'

(5)

and they experimentally estimate the parameters of the "typical" individual as a= 0.88, f3 = 0.88, and A= 2.25. It is straightforward to calculate the expected value of the two prospects in Table 1 for such an individual: 4 =

ｾＨＭＲＮＵＩＱ

ＰＸ＠

Ｘ＠ Ｘ＠ Ｘ＠ + ｾＳＮＵﾰ + ｾＱＰﾰ + ｾＱＴﾰ EVa= t(-2.25)(-(-10))088 + t5o88 + t15o88 = -0.703.

EVF

=

0.190

Thus, as EVF >EVa, this implies that an individual with the S-shaped preferences of Equation (5) with the above parameters would prefer prospect F over G (see footnote 4). What about individuals with different parameters, or with an altogether different functional form of the Sshaped value function? Would they all prefer F, or would some individuals with S-shaped value functions prefer prospect G? The PSD criterion provides an answer. Figure 1 depicts the cumulative distributions of the two prospects in Table 1. The numbers in the figure denote the areas enclosed between the two cumulative distributions. It is easy to verify from this figure that Equation (4) holds, and therefore F dominates G by PSD. This implies that for these two specific prospects, individuals with any S-shaped function, and not only the specific function given by Equation (5), would prefer F over G. As in the other stochastic dominance rules, the strength of the PSD rule is that it provides a criterion for the preference of all individuals with S-shaped preferences, with no need to know the specific parameters or the exact functional form of the value function. This 4

For simplicity, in this example we employ the objective probabilities. The possible distortion of probabilities is discussed in the next section, where our general results are derived.

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1. 0

ｾ＠

0.8

Fr - - - -

:0 Ill .0 0

c.

-0.666

G 0.6

... o.a33

Ql

ｾ＠

>

:::J

E

f F, and the"-" areas are those where the opposite holds. The PSD condition [Equation (4)] states that F dominates G by PSD iff the area enclosed between G and F ｦｲｯｭｾ＠ to x is positive for any ｾＭ＼Ｚ＠ 0- £T0 ). 6 In a portfolio context, however, when one can diversify across assets, surprisingly, it turns out that there is a very close relationship between the MYand PSD-efficient sets. 7 The two theorems below state this relationship. Both theorems are based on the following three standard assumptions: Assumption 1. Returns are normally distributed. Assumption 2. Portfolios can be formed without restrictions. Assumption 3. No two assets are perfectly correlated, IRiil < 1 for all i,j. 6

Conflicting dominance, that is, F dominates G by MV and G dominates F by PSD, can only be obtained in the special case of /.LF = /.LG· This is because /.LF > /.LG is a necessary condition for the dominance ofF over G by both rules. /.LF :> /.LG is obviously a necessary condition for MV dominance. It is also necessary for PSD dominance, because /.Lp- !La= J0000 [G(z)- F(z)[dz [see Hanoch and Levy (1969)]. Taking -oo and x oo in Equation (4), we see that /.LF :> /.LG is a necessary condition for the PSD dominance ofF over G.

"'= 7

=

Strictly speaking, PT as postulated by Kahneman and Tversky refers only to "one-shot" lotteries and is silent about individuals' behavior when confronted with a simultaneous series of lotteries, as in the portfolio context. Here we assume that the investor is faced with a universe of portfolios (e.g., those offered by a multitude of mutual funds) and individuals assets alike, and treats each of these as a lottery according to PT.

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(a) : F dominates G by MV , but not by PSD 1.0

cr,=0.4

f1, =0. 1 flG=0.05

g

crG=0 .8 - area

0.8

:0

ｾ＠ FsD G =? T(F) >FsD T(G); see Quiggin (1982) and Levy and Wiener (1998)]. Indeed, the extension of PT called cumulative prospect theory (CPT) employs a transformation of the cumulative distributions which maintains FSD [see Tversky and Kahneman (1992)]. Theorem 2 states the relationship between the MV- and PSD-efficient sets when probability distortion takes place.

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(a) F' dominates F by PSD (and by FSD) 10

q .0

08

cu

.0 0

ct

Q)

06

>

'@ (ij

E ::0

04

0

Outcome

(b) 0' dominates 0 by PSD (but not by SSD) 1.0

0

q

0.8

:0

cu

.0 0

ct

0.6

Q)

ｾ＠

>

(ij

E ::0

0.4

0

0.2

Outcome Figure 4 Graphical explanation of Theorem 1 (a) Depicts the CDFs ofF' and F, where F has the same standard deviation as F', but a higher mean. In this case F dominates F' by PSD (and actually also by FSD). (b) Shows the CDFs of portfolios 0 and 0', where 0 is the minimum variance portfolio and O' is a portfolio slightly above it on the MY-efficient frontier. O' has a higher mean, but only a slightly higher standard deviation compared with 0, and therefore O' dominates 0 by PSD (but not by SSD).

Theorem 2. Suppose that the objective probabilities are subjectively distorted by any transformation that does not violate FSD, for example, the cumulative prospect theory transformation. Then, the PSD-efficient set is a subset of the MV-efficient set.

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Proof The proof that the PSD-efficient set is a subset of the MY-efficient frontier is as in Theorem 1. No portfolio interior to the MY-efficient frontier, such as portfolio F' in Figure 3, can be PSD efficient, because it is FSD dominated by portfolio F on the frontier (see the proof of Theorem 1 in the appendix). As we are considering FSD-maintaining probability transformations, the FSD dominance of F over F' with the objective probabilities implies that F dominates F' for every individual with an increasing utility/value function, even if he subjectively distorts the probability distributions. Namely, if F dominates F' by FSD, then also T(F) dominates T(F') by FSD. As FSD =? PSD, portfolio F' is dominated by portfolio F, also by PSD. Thus the PSD-efficient set is a • subset of the MY -efficient set.

Theorem 2 proves that the PSD-efficient set cannot be larger than the MY-efficient set, but like in Theorem 1, it could be strictly smaller. However, unlike the case of Theorem 1, where the objective probabilities are employed, when probabilities are subjectively distorted we cannot generally restrict the location of the PSD-inefficient portfolios to a specific segment of the MY-efficient frontier, that is, the set of MY-efficient portfolios that are not PSD efficient cannot be characterized neatly as a segment between the minimum variance portfolio and the tangency point from the origin. The reason for this is that the probability transformation may change the perceived portfolio mean and standard deviation. Therefore the intuition behind result (ii) of Theorem 1 does not hold, because the perceived efficient frontier is not necessarily smooth and with a diminishing slope. As a result, some portfolios that are located on the MY-efficient frontier and which are PSD efficient with the objective probabilities may become PSD inefficient when the probabilities are transformed. Similarly some portfolios that are located on the MYefficient frontier and are PSD inefficient with the objective probabilities may become PSD efficient with the transformed probabilities. As this claim is not obvious, let us demonstrate it with an example. Consider two portfolios, F and G, located on the MY-efficient frontier, with normal CDFs and the following parameters: /LF = 0.25, O"p = 0.50; /La= 0.23, O"a = 0.05. First, note that when the objective probabilities are employed, F is PSD efficient. This follows from the fact that F is necessarily on the segment of the MY-efficient frontier, which is to the right of the point of tangency from the origin to the frontier (point a in Figure 3), because F is on the segment of the frontier ｷｨ･ｲｾ＠ is diminish< ｾﾷ＠ (IfF and G were located below point a ing, that is, /LF >/La ｡ｮ､ｾＺ＠ than we would have ｨ｡､ｾＺ＠ > ｾＬｷｨｩ｣＠ does not hold for the parameters in this example.) Thus F is PSD efficient with the objective probabilities. (G may be PSD efficient or inefficient, depending on its location on the frontier.) However, when a subjective probability transformation is

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employed, F may become PSD inefficient. For example, consider the two CDFs, F and G, with objective probabilities as given in Figure Sa, and the monotonic (and FSD maintaining) probability transformation T(F) = F 0 · 2 . With this transformation F becomes PSD inefficient-G dominates it. This dominance is revealed in Figure 5b, which depicts the transformed CDFs. The figure shows that with the transformed cumulative distributions F is PSD dominated by G: T(F) is above T(G) for x < 0, and for x > 0 the"+" area is larger than the"-" area [see Figure 5b and Equation (4) with T(F) and T(G) replacing F and G]. Note that as we have a PSD dominance of Gover F with the transformed probabilities, it must be that G's perceived mean is larger than the perceived mean of F, (/LT(F) < fLT(G)) (see footnote 6). Thus when subjective probability transformation is employed, portfolios that are PSD efficient with the objective probabilities may become inefficient, and the segment of the MY-efficient frontier that is PSD inefficient is generally no longer necessarily restricted to be between the minimum variance portfolio and the point of tangency from the origin. 2.1 The role of skewness We focus in this article on the MY frontier. Some researchers emphasize the importance of skewness and advocate the mean-variance skewness (MYS) criterion [see, e.g., Arditti (1967), Rubinstein (1973), Levy and Sarnat (1984), Harvey and Siddique (2000), and Harvey et al. (2002)]. This section discusses the role of skewness in our analysis. The article discusses three cases: normal distributions (Theorem 1), transformed normal distributions (Theorem 2), and lognormal distributions (both transformed or not; Theorem 3 in Appendix B). Let us deal with each case separately. 2.1.1 Normal distributions. Here there is no skewness. Theorem 1 proves that all PT investors will choose a portfolio from the MY-efficient frontier. Moreover, in the case of normal distributions, the FSD-, SSD-, and TSD-efficient sets exactly coincide with the MY -efficient frontier: Theorem 1 first shows that the FSD-efficient set is a subset of the MY frontier, and as FSD =? SSD =? TSD, the SSD- and TSD-efficient sets are also subsets of the MY frontier. Second, the fact that, for any two normal distributions with fLl > fL 2 and £T 1 > £T 2 , there is no FSD, SSD, or TSD dominance implies that all portfolios on the MY efficient frontier are FSD, SSD, and TSD efficient. Thus the FSD-, SSD-, TSD-, and MY -efficient sets are exactly identical. 2.1.2 Transformed normal distributions. This case is dealt with in Theorem 2. The theorem proves that all portfolios located below the MY -efficient frontier are inefficient by FSD even if the distributions are

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(a) : Objective Probabilities - No PSD

ｾ＠

:.0

0.8

rn

.0

e0..

0.6

"S

E

:l

u

0.4

0

-1

outcome (b): Transformed Probabilities - G dominates F by PSD

1.0

0.8

ｾ＠

:.0

T(F)

rn

.0

e0..

0.6

＠ｾ "S E

0.4

::J

u

0. 2

0.0 -1

0

outcome Figure 5

Cumulative probability transformation and PSD A portfolio that is PSD efficient when the objective probabilities are employed may become inefficient when the transformed probabilities are employed. The figure depicts the CDFs of two hypothetical portfolios on the MY-efficient frontier, F and G, where /LF = 0.25, ap = 0.50; /LG = 0.23, a a = 0.05. (a) Depicts the objective probabilities, where no PSD exists. (b) Shows the transformed distributions with the monotonic transformation T(F) = F 0 ·2 , T(G) = G 0 ·2 While F is PSD efficient with the objective probabilities (as shown in the text), when the transformed probabilities are employed, G dominates F by PSD: Equation (4) holds with the reversed roles ofF and G (i.e., the"+" area in the figure is larger than the "-" area). Thus, with the transformed probabilities, F is PSD inefficient.

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transformed. As FSD =? SSD =? TSD, TSD investors will also not choose a portfolio that is not on the MY-efficient frontier. However, in this case the transformed distributions may be skewed and MYS and MY investors may choose different portfolios on the frontier. We cannot tell which portfolios will be chosen unless we know the exact preferences as well as the precise parameters of the transformed distributions (i.e., skewness, which is determined by the specific transformation employed). Moreover, suppose that all investors "switch" from quadratic preferences to cubic preferences. This will effect portfolio demands and the equilibrium prices, and thus the shape of the frontier, but still all investors will choose their portfolios from the MY frontier. In this case it is possible that the SSD-, and in particular the TSD-efficient sets, will be different from the MYefficient frontier, but Theorem 2 shows that these efficient sets are always subsets of the objective MY-efficient frontier. 2.1.3 Lognormal distributions. Here skewness prevails with the objective distributions, and of course may also prevail with the transformed distributions. If investors have a preference for skewness, skewness may be very important and will be priced. However, Theorem 3 shows that when distributions are lognormal the FSD-efficient set is a subset of the MYefficient frontier (this result holds both with and without probability transformation; see Appendix B). Hence no investor with increasing utility will choose an MY-inefficient portfolio. In particular, all MYS investors will choose their portfolios from the MY-efficient frontier. To sum up, in all of the above three cases, MYS and PT investors will generally choose different portfolios, but all of them will choose their portfolios from the objective MY-efficient frontier. Theorem 1 states that at most the lower part of the MY -efficient frontier (segment Oa in Figure 3) is PSD inefficient. Theorem 2 states that the PSD-efficient set is a subset of the MY-efficient frontier, but it does not state which parts of the frontier are PSD inefficient. The important result of these theorems is that in the PT framework, with or without probability transformation, one will always choose his optimal portfolio from Markowitz's MY-efficient set. In the next section we numerically investigate which part of the MY -efficient set is PT efficient by employing the empirical MY frontier and the specific probability transformation suggested by Tversky and Kahneman (1992).

3. The Empirical PT-Efficient Set Theorems 1 and 2 state that the PT -efficient set is a subset of the MYefficient frontier, but they provide only partial information (in the case that the objective probabilities are employed) or no information (in the case of probability distortion) regarding the exact part(s) of the MY-efficient

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frontier that is PSD inefficient. In this section we numerically find the empirical PSD-efficient set with and without probability distortion. We select a random sample of 50 stocks from the Center for Research in Security Prices (CRSP) data set, and employ their monthly rates of return over the period January 1980 to January 2000 to estimate the means and the variance-covariance matrix. These parameters, in turn, are employed to derive the MY-efficient set, as shown in Figure 6. To find out which portfolios on the frontier are PSD inefficient, we go over each portfolio (with discrete increments of 0.0001 in f.L) and check its efficiency by comparing it with all portfolios located on the efficient frontier, and numerically testing whether it is PSD dominated. 9 We conduct this analysis twice: once assuming that the return distributions are normal, and once by transforming the normal return distributions with the subjective probability transformation suggested by Tversky and Kahneman (1992): T(F)F" - (F" + (1- F)") 1/r

(6)

with y experimentally estimated as 0.6. 10 The results are shown in Figure 6. In the case where the objective probabilities are employed, the PSD-efficient set is the MY-efficient frontier except segment Oc. In the case where the PT probability transformation of Equation (6) is employed, the PSD-efficient set is the MY-efficient frontier except segment Ob. In both cases, only a small lower part of the MY -efficient frontier is relegated to the PSD-inefficient set. It is perhaps surprising that when the probabilities are distorted, the part of the MY frontier that is PSD inefficient is even smaller than when the objective probabilities are employed, that is, empirically the probability transformation actually makes the MY- and PSD-efficient sets even more similar. 11 It is important to point out that while the main focus of this article is on the standard case of normal return distributions, our theoretical and empirical results can be extended to the case of lognormal distributions

9

The numerical testing of PSD is greatly simplified in the case of normal CDFs (see the lemma in the Appendix). As two normal CDFs with different standard deviations cross exactly once, and as Equation (6) is monotonic, the subjectively transformed cumulative probability distributions also cross exactly once (and at the same x 0). Thus we can employ Equations (A. I) and (A.2) of Appendix A to check for PSD between the transfonned distributions without worrying about 1nultiple crossings of the distributions.

10

Tversky and Kahneman estimate y separately for gains and losses. For gains they estimate y = 0.61, while for losses they estimate y = 0.69. Camerer and Ho (1994) estimate the parameter y as about 0.6.

11

A possible explanation for this may be as follows: PSD relegates the low-mean, low-variance part of the MY-efficient frontier to the inefficient set. The PT probability transformation of Equation (6) has the effect of increasing the perceived likelihood of low-probability extreme events [see Tversky and Kahneman (1992)]. Thus this transformation shifts probability weight to the extremes and therefore tends to increase the perceived variance. As a result, the range of perceived low-mean, low-variance portfolios becomes smaller under this probability transformation.

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0.030

0.025

0.020 -

Inefficient with objective probabilities

Inefficient with transformed probabilities

0.015 +------.-------,---------r------.-0.030 0.035 0.040

a Figure 6 The empirical MV- and PSD-efficient sets The MY-efficient set is derived for a random sample of 50 stocks from the CRSP monthly return dataset. The PSD-efficient set is found by employing Equations (A.!) and (A.2). 0 is the minimum variance portfolio. When the objective probabilities are employed, the PSD-efficient set is the MY-efficient frontier excluding segment Oc. This is the same as Baumel's efficient set with k = 1.5. When the transformation of cumulative probability, as suggested by cumulative PT, is employed [Equation (6)], the PSD-efficient set is the MY-efficient frontier excluding segment Ob.

as well (see Appendix B). The results for the lognormal case are very similar to those obtained in the normal case, which suggests that the results are robust to the exact mathematical form of the return distributions. Finally, it is interesting to note that as early as 1963, Baumol suggested an intuitive investment criterion that does not rely on expected utility maximization. Baumol's criterion asserts that F dominates G iff

f.Lp- kcrp ;:::: f.La- kcra,

where k is a subjective positive value related to Chebechev's inequality. It turns out that Baumol's efficient set is a subset of the MY-efficient set, relegating to the inefficient set the lower part of the MV -efficient set (like segments Oc or Ob in Figure 6). The size of the efficient set relegated to the inefficient set by Baumol's criterion depends on the value k one selects: the larger k, the smaller Baumol's inefficient set; fork---+ oo, the Baumol and the MV -efficient sets coincide. With the actual empirical data corresponding to Figure 6, we investigate what value k is needed in Baumol's criterion

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such that Baumol's efficient set and the PSD-efficient set with no probability distortion coincide. As we can see from Figure 6, the two efficient sets coincide fork ｾ＠ 1.5, namely Baumol's lower bound is 1.5 standard deviations from the mean. 12 Of course, this particular value of k is for illustration purposes only, as k is in general a function of the particular dataset under consideration. Thus we see some similarity between Baumol's and the PSD-efficient sets, where both relegate to the inefficient set portfolios with low mean and low variance. However, recall that there is one important difference; while Baumol's rule has no theoretical justification, the PSD-efficient set is based on the foundations of PT. 4. Conclusion

The MV rule is probably the most common investment rule under uncertainty among both academics and practitioners. Moreover, the MV rule is very important as it is the foundation of the cornerstone Sharpe-Lintner capital asset pricing model (CAPM) and other theoretical models. Kahneman and Tversky (1979), and Tversky and Kahneman (1992), based on experimental results, suggest PT as an alternative to the EU paradigm. According to PT, the value function is S-shaped with a riskseeking segment, it is a function of change of wealth rather than total wealth, and in addition, individuals subjectively distort probabilities by a transformation of the cumulative probability. These experimental results seem to be fatal to EU in general and to the MV framework in particular. First, employing change of wealth rather than total wealth contradicts the EU paradigm, and thus casts doubt on the validity of the MV analysis, which is derived in the EU framework. Second, the risk-seeking segment of the PT value function contradicts the risk-aversion assumption, which is the foundation of the MV rule. Finally, when individuals distort probabilities, the perceived distributions are generally not normal, even if the objective distributions are, which again seems to be fatal to the MV analysis. We show in this study that, in general, when considering two alternative investments F and G, there is no relationship between the PT- and MYefficient sets: F may dominate G by PSD (which corresponds to PT) and not by MV, and vice versa. This is true even when normality is assumed, let alone in the general case when no restrictions are imposed on the distributions. However, in a portfolio context when diversification between assets is allowed, as is common in the security market, and as suggested by Markowitz (1952a), and when individuals face portfolios 12

For any given value of k, Baumel's efficient set is the segment of the MY-efficient frontier to the right of the point where a straight line of slope k is tangent to the hyperbola. For the PSD-efficient set obtained with the transformed probability distributions we obtain a slightly higher value of k (k = 1.7).

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(mutual funds) as well as individual assets, we find that the PT-efficient set is a subset of the MY -efficient frontier. This result holds both for the case where objective probabilities are employed, as well as for the case where probabilities are subjectively transformed, as suggested by cumulative PT. When the objective probabilities are employed, we theoretically show that at most a small segment of low-mean low-variance portfolios is relegated to the PSD-inefficient set. With probability transformation the segment of the MY-efficient set relegated to the PT-inefficient set cannot be determined in general. However, empirical analysis reveals that the PSD rule (with and without probability distortion) relegates to the inefficient set a very small segment of the MY-efficient set corresponding to very riskaverse investors. Thus, surprisingly, even with the S-shaped value function, which contains a risk-seeking segment and which is a function of change of wealth rather than total wealth, and even with probability distortion, the PT- and MY-efficient sets almost coincide. It is interesting to note that PT relegates to the inefficient set some segment of the MY-efficient set in a similar way to the relegation of portfolios to the inefficient set by Baumol's (1963) intuitive criterion. In Baumol's criterion risk is measured by E-klr, where k is a positive constant reflecting the safety requirement of the individual investor. We find in the empirical analysis that the PSD rule relegates to the efficient set the same segment relegated by Baumol's criterion corresponding to the value of approximately k= 1.5. Note that the similarity of the MY- and PT-efficient sets does not necessarily imply that the CAPM equilibrium holds when investors have PT preferences. In particular, for some PT preferences the risk-seeking segment of the utility function may imply infinite borrowing, which contradicts the notion of market equilibrium. The main results of this article are derived for the canonical case of normal return distributions. However, it is well known that for long investment horizons the return distributions are typically positively skewed. To what extent do the results of this article hold when the return distributions are positively skewed? In order to answer this question we analyze the PT-efficient set in the case of lognormal return distributions (see Appendix B). We find that even in this case the PT- and MY-efficient sets almost coincide. Thus while skewness may be priced (and may perhaps be even more important in the case of PT investors with high sensitivity to losses), the set of PSD-cfficicnt portfolios is very similar to the MY -efficient frontier. In conclusion, the cumulative PT- and the well-known MY-efficient sets almost coincide. What seems a priori to be a severe contradiction between these two cornerstone paradigms, turns out to be a minimal one. Thus one can employ the existing MY diversification algorithm to construct the PTefficient set. This is an important step toward bridging between subjects'

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observed behavior, as manifested by PT, and the standard MV portfolio theory. Appendix A: Proof of Theorem 1 Before we turn to the proof of Theorem 1, consider the following lemma, which is employed in the proof. Lemma. Assume that both F and G are normal CDFs, such that dominates G by PSD iff the following two conditions hold:

f.LF

xo

-oo

[F(z)- G(z)]dz::.;

f.LG,

o-p >

o-G

Then F

(A.1)

xo

10 xo

[G(z)- F(z)]dz,

(A.2)

where x 0 is the crossing point ofF and G (i.e., F(x 0) = G(x0), recall that two normal CDFs with different variances cross exactly once). In other words, for normal distributions, Equation (4) is equivalent to Equations (A.1) and (A 2). Proof Necessity Equation (A.1) is required for the PSD dominance ofF over G. To see this recall that for normal distributions with o-p > o-G, F crosses G from above (as in Figure 2b). If x 0 2:0 (unlike what is shown in Figure 2b), then lpsn ｊｾ Ｐ ｛ｇＨｺＩＭ F(z)]dz < 0, in violation of the PSD criterion of Equation ( 4). Equation (A.2) also follows directly from Equation ( 4) by taking .r=-oo and x = 0, because when x 0 a-G). x 0 < 0 ensures that (i) ｊｾ｛ｇＨｺＩＭ F(z)]dz > 0 for all x > 0 (because G >Fin the positive range; see 0 Figure 2b). Equation (A.2) ensures that (ii) .fx [G(z)- F(z)]dz > 0 for all.r < 0, because this expression is minimal for .r = -oo, and Equation (A.2) states that it is positive even in this case. Taken together, (i) and (ii) imply Equation (4). •

Note that even though the PSD condition of Equation (4) requires verification for all 0::.; x, in the case of normal CDFs with f.LF > f.LG and o-p > o-G, this boils down to the verification of only the two conditions [Equations (A.1) and (A.2)]. This result can be extended to the case of any two CDFs that cross only once. Proof of Theorem 1. Let us first prove that the PSD-efficient set is a subset of the MY-efficient set [claim (a)], and then go on to derive the segment of the MY-efficient frontier which may be PSD inefficient [claim (b)].

MY-efficient set (a) PSD-efficient ｳ･ｴｾ＠ The MY-efficient set is well known to have the shape of a hyperbola, which is called the efficient frontier [see Roy (1952), Merton (1972), and Roll (1977)]. We claim that any portfolio that is interior to the MY-efficient frontier (i.e., which is MY inefficient) is also PSD inefficient. To see this, consider any portfolio F' that is MY inefficient (see Figure 3). This portfolio is FSD dominated by portfolio F, which has the same standard deviation, but higher mean (see the FSD condition, [Equation (2)] for the case of normal distributions). As FSD implies dominance by all individuals with increasing utility/value functions, it also implies preference by all PT investors (FSD PSD). Hence all PT investors prefer F over

'*

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F', and therefore F' is PSD inefficient. As the same argument holds for all portfolios located below the MY-efficient frontier, we conclude that all the portfolios below the MY-efficient frontier are PSD inefficient. (b) The segment of the MY frontier that may be PSD inefficient While the MY-efficient frontier coincides with the FSD-efficient set for normal distributions, some portfolios located on the M V-etlicient frontier may be relegated to the PSDinefficient set, because the PSD-efficient set is by definition a subset of the FSD-efficient set. We show below that the PSD-efficient set is the entire MY-efficient frontier excluding at most the segment between the minimum variance portfolio and the point of tangency from the origin to the frontier (segment Oa in Figure 3). To see this, consider any two portfolios F and G on the efficient frontier, and assume without loss of generality that f.LF > f.LG, O"p > O"Q. Figure A.l depicts such a situation. As mentioned above, any two cumulative normal distributions with O"p # D"G cross exactly once. Denote the crossing point ofF and G by x 0 (see Figure 2). G cannot dominate F by PSD because f.LF > f.LG (see footnote 6). By the above lemma, F dominates G by PSD iffEquations (A.l) and (A.2) hold. The condition x 0 < 0 has a direct geometric interpretation in the MY plane. For two normal distributions the intersection point x 0 of the cumulative distributions is given by the solution of the following equation: Xo- f.Lp

Xo- f.LG

O"p

O"G

(A.3)

:::t

0 Figure A.l PSD inefficiency and the MV frontier Necessary conditions for the PSD dominance ofF over G are that F has a higher mean than G, and that the slope from the origin (0,0) to F is higher than the slope to G. Thus all portfolios on the MV frontier to the right of point a are PSD efficient: for any such portfolio there is no other portfolio with both higher mean and higher slope. For example, in the figure neither F nor G dominate the other by PSD.

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[see, e.g., Hanoch and Levy (1969)]. Thus x 0 is given by xo

The condition tion), or

x 0

=

< 0 therefore implies

f.LG(J"F- f.LF(J"G

f.LG(J"F- f.LF(J"G

< 0 (recall that

(A.4) (J"F

> (J"G by construc(A.5)

Thus, Equation (A.5) can replace Equation (A.l) as a condition for the PSD dominance ofF over G. ｇ･ｯｭｴｲｩ｣｡ｬｹｾ＠ is the slope from the origin to portfolio F in the MV plane. Thus a necessary condition for the PSD dominance ofF over G is that the slope from the origin to F is higher than the slope toG (see FigureA.l). For the specific F and G depicted in Figure A.l there is no PSD dominance, because the slope ofF is smaller than the slope ofG (and G also cannot dominate because f.LF > f.La). In contrast, F may dominate portfolio G' (see Figure A.l), because F has both higher mean and higher slope [however, this dominance is not guaranteed because one still has to check whether Equation (A.2) holds]. In general, all the portfolios on the Markowitz efficient frontier to the right of the tangency point from the origin (point a in Figure A.l) are PSD efficient. Consider, for example, portfolio F: this portfolio is not dominated by any portfolio to its left, because F has a higher mean. It is also not dominated by any portfolio to its right, because F has a higher slope. Thus portfolio F, and all other portfolios to the right of point a, are PSD efficient. PSD relegates to the PSD • inefficient set at most segment Oa of the efficient frontier.

Appendix B: The PSD- and MY-Efficient Sets in the Case of Lognormal Distributions We show in this appendix that when the distributions are lognormal, the FSD- (and hence the PSD-) efficient set is a subset of the MY-efficient set, despite of the fact that the lognormal distributions are skewed. Let us elaborate. In the text we analyze the relationship between MV analysis and PT for the case of normal distributions. It is well known, however, that for long investment horizons the return distributions are positively skewed, and the lognormal distribution is a better approximation for the empirical return distribution. A drawback of the lognormal distribution framework is that the sum of two random variables, which are lognormally distributed, is not precisely lognormal. However, Lintner (1972) shows that the sum of two such lognormal random variables is almost lognormal. He concludes, "the approximation to lognormally distributed portfolio outcomes of lognormally distributed stocks is sufficiently good that theoretical models based on these twin premises should be useful in a wide range of applications and empirical investigations." [See also Ohlson and Ziemba (1976), and Dexter, Yu, and Ziemba (1980)]. In Theorem 3 we extend our main theoretical results given in the text to the case of lognormal distributions. Theorem 3. Consider n assets with lognormal return distributions and correlations IRyl < 1 for all i, j. Portfolios can be formed without restrictions, and all portfolio returns are also lognormally distributed. Objective probabilities may be subjectively distorted by any transformation which does not violate FSD. Then, the PSD-efficient set is a subset of the MVefficient frontier, excluding at least the segment between the minimum variance portfolio and the point of tangency from coordinate (0, -1) in the MV plane to the frontier (segment Oa' in Figure A.2).

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0.0 /

/

-1.00

Moreover,

UQ"D(b) is dense in D(b).

The interpretation of the Theorem is evident in light of the preceding discussion. (The difference between M(D(b)) and M(D(b)) is clarified in Appendix 1 following the proof of the theorem.) 3. RECURSIVE UTILITY

All utility functions are defined on D( b) and are recursive there. It may be helpful, therefore, to begin by considering briefly the structure of recursive utility over deterministic consumption streams. If V is such a utility function, then (Koopmans (1960)) in the obvious notation (3.1)

V(c 0 ,c1 , ••• ) = W(c 0 , V(c1 ,c 2 , ••. )),

for some function W. This structure has been explored also by Lucas and Stokey (1984) and Boyd (1987) where W is termed an aggregator, as it combines current consumption and future utility to determine current utility. In contemplating an extension of (3.1) to the stochastic case we note that future utility is random. It seems natural, in that case, to compute a certainty equivalent for random future utility and then to combine the certainty equivalent utility level with c0 via an aggregator. Thus we are led to consider certainty equivalent (or generalized mean value) functionals p,. Each such mean value is a map, p,: dom p, c.;;;, M(R+)-+ R+,

which is consistent with first and second degree stochastic dominance and satisfies

(3.2)

p,(8x)=x 'VX ER+,

i.e., if a gamble yields the outcome x with certainty, then x is the certainty equivalent of the gamble. Given a utility function V: D(b)-+ R+ and (c 0 , m) E D(b), denote by V[m] the probability measure for future utility implied by V and m E M( D( b)) c

Chapter 12. Substitution, Risk Aversion and the Temporal Behavior of Consumption TEMPORAL BEHAVIOR OF CONSUMPTION

215

945

temporal lottery d, rna and mb eM(D)

temporal lottery d,

ma ana mbe M(D)

Consistency:

i=a,b

=> d>; d FIGURE

2

M(D(b)), i.e., (3.3)

V[m](Q)=m{dED: V(d)EQ},

QEB(R+).

The utility function V is called recursive if it satisfies the following equation on its domain:

for some increasing aggregator function W: Ri-+ R+ and some certainty equivalent IL· This relation is the cornerstone of our analysis. Of course, it generalizes the more familiar structure (3.1). Note also that the recursive structure immediately implies the intertemporal consistency of preference (in the sense of Johnsen and Donaldson (1985) or Figure 2) and the stationarity of preference (in the sense of Koopmans (1960), for example).

216

LG Epstein and SE Zin

946

LARRY G. EPSTEIN AND STANLEY E. ZIN

The question which immediately arises is whether and under what circumstances there exist utility functions V satisfying (3.4). 3 To answer this question we restrict the admissible aggregators and certainty equivalents. First, we require that W have the CES form (3.5)

W(c,z)=[cP+,BzP] 11 P,

O*p is continuous, increasing, concave, and f/>(1) = 0. Note that if (3.12)

4J(x)=(xa-1)ja,

O*a are possible. For example, one that is investigated empirically in Epstein and Zin (1989) is (3.13)

f/>(x)=(xa-1)/a+a(x-1),

O*a that -xf/>"(x)/f/>'(x) is (strictly) decreasing. This restriction can be readily incorporated into specifications for 4>, e.g., it is satisfied by (3.13) if a> 0. The above examples do not exhaust the class of recursive intertemporal utility functions covered by Theorem 3.1. Other specifications for J.L, taken from the atemporal non-expected utility literature for example, could be adopted if the seemingly mild continuity condition MV.1 is satisfied. Thus Theorem 3.1 should permit the integration into a temporal setting of a substantial portion of the non-expected utility literature. 4. SUBSTITUTION, RISK AVERSION, AND TIMING

The key properties of recursive utility functionals will be discussed here. It has already been noted that the specification (3.5) for the aggregator implies that deterministic consumption sequences are ranked by an intertemporal CES utility function with elasticity of substitution o = (1 - p) -l. Thus we interpret p as a parameter reflecting substitutability. Next tum to risk aversion and in particular to comparative risk aversion. Let V and V* be two recursive utility functions with possibly distinct aggregators W and W* conforming to (3.5). 7 We wish to define what it means for V* to be more risk averse than V. To do that, define c = 'A(m) for any (c 0 , m) by V(c 0 , m) = V(c 0 , c, c, ... ). 7 The ensuing discussion could be carried out without restricting aggregators to conform with (3.5). But such generality would complicate the exposition somewhat for the following reason: If V is functional p., then any monotouic transform of recursive with aggregator W and certainty ･ｱｵｩｾ｟Ａ＠ V, say h(V), is also recursive with aggregator W, W(c, z) = hW(c, h- 1(z)) and certainty equivalent functional Ji., Ji.(Px) = h( u(Ph-'(x))}. Thus a given preference ordering of temporal lotteries can be represented by many ( W, p.) pairs. By fixing the representation (3.5) for the aggregator, we avoid the need to refer to the entire class of "equivalent" ( W, p.) pairs.

220

LG Epstein and SE Zin

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LARRY G. EPSTEIN AND STANLEY E. ZIN

Interpret the "nearly" constant and deterministic path ( c0 , c, c, ... ) which is indifferent to ( c0 , m) as the certainty equivalent for the latter. (Note that c depends only on m and not on c0 .) The path ( c0 , A*( m ), A*( m ), ... ) is defined analogously given V*. It is natural to say that V* is more risk averse than V if and only if A*(m) ｾ＠ A(m) for all (c0 , m) in some common domain for V and V*. Evidently, if V and V* are comparable in the above sense, then they must rank nonstochastic consumption programs identically, i.e., W = W* or equivalently, p = p* and /3 = /3*. Moreover, V* is more risk averse than V if and only if W= W* and (4.1)

ｰＮＪＨﾷＩｾ＠

on the appropriate domain. (Necessity of these conditions is obvious. For sufficiency, note that they imply, given the construction in the proof of Theorem 3.1, that V*( ·) ｾ＠ V( · ).) Thus the certainty equivalent functional determines the degree of risk aversion of the corresponding intertemporal utility function, at least for comparative purposes. Further support for this interpretation of p. is provided at the end of Section 5. Since, by assumption, mean value functionals p. * exhibit risk aversion in the sense of second degree stochastic dominance, it follows that ｰＮＪＨﾷＩｾｅＬ＠

where E( ·) denotes the expected value operator. Thus the least risk averse intertemporal utility function is the one for which p.( ·) = E( ·).Moreover, there is a sense in which the latter specification implies risk neutrality, e.g., in the context of timeless wealth gambles or in the portfolio choice context described at the end of Section 5. It is apparent, therefore, that "low" or "moderate" risk aversion can coexist with a small elasticity of substitution, which is impossible in the expected utility specification. a. In the case of KP functionals (3.8), the condition (4.1) is equivalent to ｡Ｊｾ＠ Thus we interpret a as a measure of risk aversion for comparative purposes with smaller a's indicating greater risk aversion. A separation between the risk aversion parameter (a) and the substitution parameter ( p) is achieved. For the Chew-Dekel class based on (3.11), (4.1) is equivalent to

4>*"(. )/4>*'(-)

ｾ＠

4>"(. )/4>'(.).

a and ｡Ｊｾ＠ a. This is satisfied in the parametric class defined by (3.13) if ｡Ｊｾ＠ A comparable separation between risk aversion and substitution does not appear possible within the expected utility model. To see this, consider consumption programs (c 0, m), with mE M(Y(b; /)) c M(R"':.) and let

V( c0 , m) =Em{ cgjp +

ｾＯＳ Ｑ ｣ｦ＠

/p},

which is ordinally equivalent to (3.6). The general multicommodity analysis of Kihlstrom and Mirman (1974) suggests that we take a monotonic transform h of

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the von Neumann-Morgenstem utility index and define

Then V* is more risk averse than V in the sense defined above, or equivalently in the sense of Kihlstrom and Mirman (1974), if and only if h is concave. At first glance, therefore, it would seem that comparative risk aversion analysis is feasible within an expected utility framework. But in a temporal setting, this familiar approach encounters serious difficulties. To see this, consider an individual with the utility function (4.2) who arrives at period T and contemplates the remaining future. If past consumption levels were c0 , ... , Cr_ 1, then the utility function for the remaining future is presumably 8

Thus the preference ordering at T depends upon past consumption values unless h has constant absolute risk aversion. Dependence upon the past is in principle sensible but the form which this dependence takes above is implausible, since (4.3) and 0 < {3 < 1 imply that the dependence on past consumption is greater as the past becomes more distant. (For example, denote by H( c0 , ... , Cr_ 1 ; cr, cr+ 1, ... ) the von Neumann-Morgenstern utility index on the right side of (4.3). If derivatives are evaluated at a point where c0 = · · · = cr_ 1, then

Thus the risk premium for a small gamble in period (T + 1) consumption is affected more by a small change in c0 than by a small change in Cr_ 1.) On the other hand, if h(z) = -exp( -Az), A> 0, then the nonstationarity of preferences is implied as period T preferences are represented by vr( cr, m) = Em exp(- Afjru( Cr, Crn ... )), where u is the additive functional f. 0{3 1cfl p. Declining risk aversion with T is imposed a priori. Though such a "changing tastes" specification may be appropriate in some modelling exercises, it would appear to be a hindrance rather than a help to exploring the questions outlined in the introduction and in Section 7. More importantly, if attitudes towards future gambles are changing with the passage of time as above, then plans will generally 8 Otherwise tastes are changing through time. This is unappealing as an a priori specification and, moreover, implies that preferences are intertemporally inconsistent. Hall's (1981) specification suffers from these problems.

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d, eED

early resolution

d

(co, '\c 1,m))

m =nod+ (1-n)oe e M(D)

late resolution FIGURE

3

not be intertemporally consistent. The above unappealing features of the familiar Kihlstrom and Mirman approach to comparative risk aversion are not restricted to the case where the von Neumann-Morgenstem utility index is additively separable. For example, they may be confirmed also for the nonadditive indices axiomatized in Epstein (1983) which feature variable discount rates. Finally, with regard to attitudes towards the timing of the resolution of uncertainty, see Figure 3. The two temporal lotteries portrayed there differ precisely in the timing of the resolution of uncertainty as defined by KP (1978). A recursive utility function is indifferent to the timing of resolution (in all such pairs of lotteries) if and only if it is an expected utility functional such as (3.6). (This follows by a straightforward extension of the finite horizon arguments in Chew and Epstein (1989).) In particular, for the KP class, the curvature of H( c0 , ·) defined in (3.9) is the determinant of attitudes towards timing with indifference towards timing prevailing only if H(c 0 , ·)is linear (KP (1978)). We can conclude, in fact, that given (3.8) early (late) resolution is preferred if a )p. For more general recursive utility functions, we have not found a characterization in terms of W and p. of the conditions under which early or late resolution is preferred. But the characterization for the KP class raises an issue which we suspect is relevant more generally and which calls for some attention. We have interpreted a as a risk aversion parameter. But with p fixed, a reduction in a not only increases risk aversion but also may transform a preference for late resolution into a preference for early resolution. One is left wondering how to interpret the comparative statics effects of a change in a. Similarly, a change in p for given a affects both substitutability and attitudes towards timing. Thus the latter aspect of preference seems intertwined with both substitutability and risk aversion.

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We offer three comments in response. First, from the perspective of potential empirical applications, the specifications (3.8) and a fortiori (3.4) are still more flexible than the common expected utility functional form. Second, the behavioral analysis in the next section will provide further support for our interpretation of a, or more generally p., as a risk aversion parameter since a or p. will determine the degree of risk taking in certain portfolio choice problems. Finally, we suspect that the lack of separation noted above reflects the inherent inseparability of these three aspects of preference rather than a deficiency of our theoretical framework. Further study of this issue is required. To conclude this section, we observe that attitudes towards timing can be used to distinguish, within the family of recursive utility functions, each of the three subclasses defined in Section 3. It has already been pointed out that timing indifference implies an expected utility ordering. Next consider the Chew-Dekel subclass. Suppose that V is such that the lotteries in Figure 3 are indifferent to one another whenever V( d) = V( e); that is, the timing of resolution is a matter of indifference if the two future prospects regarding which information is being provided, are themselves indifferent. Refer to this property as quasi-timing indifference (QTI). A straightforward extension of the finite horizon arguments in Chew and Epstein (1989) shows that the only recursive utility functions satisfying QTI are those based on (3.10). If an appropriate homotheticity assumption is imposed on V, then p. must satisfy MV.2 and (3.11) is obtained. (See Chew and Epstein (1989) for the basis for a comparable argument for the KP class (3.8) and for discussion of QTI. An alternative basis for an axiomatization of KP preferences may be found in KP (1978).) Thus a theoretical case can be made for interest in the KP and Chew-Dekel subclasses of recursive utility functions. Accordingly, we do not apologize for the fact that some of the discussion of the asset pricing implications of our framework is limited to these subclasses. 5. THE REPRESENTATIVE AGENT

The remainder of this paper derives relations between aggregate consumption and real rates of return which must hold in a competitive equilibrium. The procedure adopted is that of the rational expectations literature on aggregate consumption (Hall (1978)). In this section we determine the optimal consumption and portfolio behavior of an individual who faces exogenous rates of return to saving. Then, in the next section, we take the individual to be a representative agent in the economy so that the Euler equations corresponding to his intertemporal plan define relations between aggregate consumption and rates of return that must hold in equilibrium. We deviate from earlier literature in the specification of a recursive (but not necessarily expected utility) specification for preferences. Consequently, the derivation of the Euler equations is nonstandard. Our representative agent operates in a standard environment. There are K assets. The gross return to holding the kth asset between t and (t + 1) is described by the random variable rkl' - 00 < t < 00, where each rkt has support

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"=

in [r, r], r > 0. Let (111 , ••• ,fK1 ). We assume that (", ｺ Ｑ ＩＧｾ＠ oo is a stationary stochastic process. The role of z 1 E R 2 is to provide information regarding the future. It is assumed that ( "' 1 ) is observed at the start of period ( t + 1), just prior to the time at which period (t + 1) consumption and portfolio decisions are made. Without loss of generality we may take the underlying probability space to be (fl, B(fl), >¥) where fl is the set of doubly infinite sequences with tth component (rP z 1 ) and B(fl) is the (product) Borel a-algebra. The state of the world at time t is defined by the existing wealth level x 1 and by the history of realized past values of ( r;, z;)' s. Thus let

z

t-l

fl 1 =[0,oo)x

X ([r,r]KxR 2 )=[0,oo)xii' i= -oo

where the (product) Borel a-algebra is adopted for I 1 and fl 1 . A consumptionportfolio plan (from t = 0 onwards) is a sequence (h 0 , ••• , hn ... ) of measurable functions ht: [lt ｾ＠ [0, oo) X sK, where sK is the unit simplex in RK. The interpretation of ht(xl' It)= (ct, w1 ) is that given period t wealth xt and history It E It, the agent consumes ct and invests the proportion wkt in the k th asset, wt = ( wln ... ' wKt). A plan is homogeneous if "iit>O and "ii(x,It)Eflt,hl1,It)=(cl'wJ=> h l x, It) = ( c1 x, wt ). Because of the homotheticity of preferences and the linearity of "technology," it is natural to restrict oneself to homogeneous plans which are henceforth simply plans. A plan is stationary if 3h such that ht = h "iit. Finally, a plan is feasible if "iit ｾ＠ 0 and "ii(xf' I 1 ) E fln c1 ｾ＠ xt where ct is the first component of ht(xn IJ and where wealth evolves according to (5.1)

X1

= (x 1 _ 1 -

C1 _ 1

)w/_ 1"_ 1 ,

t ｾ＠

1,

x 0 > 0 given. 9 Each plan implies an infinite probability tree in consumption levels. Moreover, if the plan is feasible, then the corresponding probability tree itself corresponds to a temporal lottery in D(r; x 0 ) c D(r). Formally, denote by FP the set of feasible plans for a given (x 0 , I 0 ). Appendix 4 shows that FP can be embedded in a "natural fashion" in D(r) by the map e. Moreover, D(r) is a subspace of D(b) for any b > r. Thus, if Vis defined on D(b), the problem (5.2)

J(I0 , x 0 )

=sup{V(d): dE e(FP)}

is well-defined if r < b. The conditions under which the supremum is attained are specified in the next theorem. THEOREM 5.1: Let V be the recursive utility function constructed in Theorem 3.1, defined on D( b) and having aggregator (3.5) and a mean value functional satisfying 9 This budget constraint excludes exogenous sources of income and labor income. This exclusion is important for the homogeneity property (5.4) below and subsequently for our derivation of Euler equations. Some discussion of the modifications necessary to accommodate these sources of income may be found in Epstein and Zin (1989).

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MV.1 and MV.2. Then (5.2) possesses a maximum, achieved by a stationary and homogeneous plan if (a) p > 0, and r < b, or (b) p < 0 and f3rP < 1. In either case, J>O.

Note that the homogeneity of JL (MV.2) has been added as an assumption. Also note that the strict positivity of the value function of J is used below. We now turn to the implications of optimality and more particularly to the appropriate set of Euler equations for interior optima. The recursive structure of utility functions immediately implies the "Bellman equation" (5.3)

J(Io,Xo)=

max c0 ;;.0, "'JES

Jct;+f3JLP[PJ(i,,(x 0 -c0 )w0r0 )/lolr/p'

where the argument of JL is the probability measure for ｊＨｾＬｸ Ｐ Ｍ c0 )w0r0 ) conditional on 10 . Moreover, the maximizing values of c0 and w0 correspond to the utility maximizing plan in the customary fashion. It is evident from the homotheticity of utility, that J can be expressed in the form

(5.4)

J(I,x)=A{I)x.

Thus (5.3) can be written (5.5)

A{I0 )x 0 =

max ｣ Ｐ ｾＰ＠

Jct;+f3(x 0 -c0 )PJLp[PA(i,)woro/Io]r 1P.

w0 ES

An immediate implication is the portfolio separation property and more particularly that the portfolio decision is determined by the solution to

(5.6)

maxKJL [PA(i,)w6ro/Io] · w0 ES

Write c6 = a 0 x 0 , where an asterisk denotes the maximizing value. Substitution into (5.5) yields

AP{I0 ) =at;+ /3(1- a 0 )PJL *P, and the first order condition for consumption in (5.5) yields

ag- 1 =

(5.7)

(1- a 0 )p- 1f3JL*P.

These last two equations can be combined to yield A(I0 ) = a&p- 1)/p = (c6 jx 0)-'l/> ·(F,- ;,)jfolｾ＠

0,

k=2, ... ,K.

=

10 LetS be a set of probability measures and J.L* max{J.L(p): pES} where J.L is given by (3.10). Then J.L* =J.L(p*) =>p* solves max(EpF(·,J.L*): pES}. See Epstein (1986, Section 3).

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Also, substitute (5.9) into (3.11) to deduce

Equations (6.2) and (6.3) constitute the Euler equations for the model based on Chew-Dekel preferences. The former reveal more clearly than (5.10) the joint role played by consumption and the market return in determining systematic risk. A further specialization to KP preferences is obtained by imposing (3.12). The resulting Euler equations take the form (k = 2, ... , K),

( 6.4)

Alternatively, these equations are equivalent to (6.6)

- ]a(p-1)/p /] f3EP!a [[ :: MJ/'-p)/Pfko Io = 1

(k=1, ... ,K).

If the further specialization a = p is adopted, then we obtain

(6.7)

(k=1, ... ,K),

which are the familiar Euler equations of the expected utility model (see Hansen and Singleton (1983), for example). Earlier we pointed out that one consequence of the generalization from expected utility to recursive utility is the emergence of the market return as a factor in explaining excess mean returns. The significance of the market return is apparent from (6.4), which can be rewritten

Thus both consumption and the market return enter into the covariance that defines systematic risk. The consumption-CAPM is obtained if o: = p. But if the substitution o: = 0 is adopted instead, then covariance with the market return

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alone determines systematic risk which is the prediction of the static CAPM. 11 (Though a = 0 lies outside the scope of our formal analysis, one can show that the a = 0 version of (6.4) is the model of mean excess returns that corresponds to the a= 0 version of KP preferences.) Another specialization of (6.4) which yields the market based CAPM model is p = 1, or infinite elasticity of substitution. This case was excluded from the preceding analysis because it would generally rule out interior optima for consumption (see (5.5)). But the appropriate form of (6.4) is still valid. Intuitively, the emergence of the static CAPM here is presumably due to the perfect substitutability of consumption across time. We have described some relations between aggregate consumption and asset returns which must hold in a competitive equilibrium, but we have not demonstrated the consistency of our analysis with a general equilibrium framework such as Lucas' (1978) stochastic pure endowment economy. Such an extension would need to confront the questions of existence and uniqueness of equilibrium asset prices. Moreover, Lucas' contraction mapping techniques would not suffice for the same reasons that those techniques were inadequate in establishing Theorem 3.1. Thus we leave such an extension to a separate paper. However, we conclude this section by describing some asset pricing implications of our analysis which are valid for any general equilibrium extension. For simplicity, consider KP preferences, though comparable formulae may be derived for Chew-Dekel preferences. Consider an asset which pays the dividend ij1 in period t. The real gross return to holding the asset during period 0 is + ij1 ) /P0 , where P0 and i\ denote the current and random period 1 prices respectively. Then substitution into (6.6) implies that the asset price satisfies the recursive relation

0\

which has a solution

if the right-hand side is finite. Price equals the discounted expected value of 11 The p = 0 version of the expected utility model, i.e., logarithmic within period utility function, leads to E[M0 1(i'ko- i'10 )jl0 ] = 0 Vk, a CAPM model. But then it is also true that

so that the consumption-CAPM also applies. In contrast, our specialization a = 0 implies that only the market return determines systematic risk. A related observation is that we generate a CAPM model without restricting the intertemporal elasticity of substitution in consumption. See Grossman and Laroque (1987) for an alternative intertemporal model which leads to the static CAPM.

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future dividends where the discount factors involve both consumption and market returns. In the case of a one-period pure discount bond, ij1 = 1 and ij1 = 0 for t > 1,

Given fixed marginal distributions for i\ and M0 , the bond price increases with the covariance between fi(p-l)/p and M/)"--P)IP. The two exponents have the same signs if and only if 0 < a < p or p < a < 0. In those cases, the bond price increases as consumption c1 and the market return M0 become more correlated. 12 Otherwise, the bond price falls in response to an increase in correlation. 7. CONCLUDING REMARKS

The intertemporal utility functions we have formulated have three very appealing features: (1) intertemporal substitution and risk aversion are disentangled; (2) they integrate atemporal non-expected utility theories into a temporal framework; and (3) they generate implications for the temporal behavior of consumption and asset returns. Moreover, these implications may be investigated empirically by existing econometric techniques as demonstrated in Epstein and Zin (1989). Some empirical work is done in the latter paper but further empirical investigation, exploring alternative data sets and functional forms, would be worthwhile. A promising application on the theoretical front is to recursive dynamic GE modelling (Sargent (1987)). For example, we have already mentioned the need to integrate our model into a general equilibrium framework such as Lucas' (1978). Because of the inseparability of substitution and risk aversion in his expected utility model, Lucas is unable to provide a clear interpretation for some of his comparative statics results. Our utility functions should clarify those results and thus provide a clearer understanding of the determinants of asset prices. (See Epstein (1988) for such a comparative statics analysis in a stochastic pure endowment economy where endowments are i.i.d.). In addition, the separation which they provide should make them useful in exploring the role played by differences in risk aversion in influencing the distribution of wealth across agents. Such an investigation would complement existing theories of distribution that are based on differences in time preference (Epstein (1987)). In these and other theoretical developments, the specific functional form (3.5) for the aggregator could be useful in providing some initial insights. Indeed many of the early multiperiod expected utility models of consumption/portfolio behavior are based on the homogeneous parameterization (3.6). But it would clearly be desirable to generalize to a larger class of recursive utility functions. It is hoped that several elements of this paper, such as the proofs of Theorems 3.1 and 5.1 12 A

notion of "greater correlation" that-suffices here is described in Epstein and Tanny (1980)_

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and especially our formalization of the space of infinite horizon temporal lotteries, will be useful in developing such generalizations. Department of Economics, University of Toronto, Toronto, Ontario, Canada M5S JAJ and Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, PA 15213, U.S.A. Manuscript received July, 1987; final revision received September, 1988.

APPENDIX I In this appendix we prove Theorems 2.1 and 2.2. The notation of Section 2 is adopted. LEMMA ALl: The function f defined in (2. 7) is continuous.

PROOF: By Parthasarathy (1967, Theorem 6.1, p. 40) it suffices to show that m.--+ m

=

f(m.)(B)-+ f(m)(B)

'T/ BE 8( R";) such that j(m)( a B)= 0. But the latter condition implies v { y E R";: ( c0 , y) E aB}

= 0 a.e. [m] and hence (Billingsley (1968, Theorem 5.5)) that T0 is continuous a.e. [m]. By Billingsley (1968, Theorem 5.1), Em,,Tn-> EmTB as desired. Q.E.D.

LEMMA AL2: For all t;;. 1 the functions ft and g, defined in (2.8) are continuous and satisfy (2.9).

PROOF: Continuity follows from Lemma Au and Billingsley (1968, p. 29). Condition (2.9) is readily verified. Q.E.D. Recall the projection maps .,, defined in Section 2. LEMMA A1.3: (a) For each t ;;.1 and B, E 8" .,,- 1(B,) t;;. L {c) Uj"'.,,- 1(81 ) is an algebra. (d) 8 = ＨｕｦＧＮＬＭｯ ur.,,-1 d, then m;-> m, 'T/t;;. 1 and so by (ALl) m"(A)-> m(A) for any A EUj"'.,,- 1(8,) such that m(aA)=O.

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But the algebraUi.,.- 1(B,) is convergence determining in the sense of Billingsley (1968, exercise 7, p. 22) and thus m"-> min M(D). This proves that e is a homeomorphism. D is a metric space by construction. It is also a closed subspace of the separable space Xj'D, by the continuity of the g,'s. Thus D is separable. Finally, the desired denseness is immediate. Q.E.D. PROOF OF THEOREM 2.2: One can show using Billingsley (1968, Theorems 6.2 and 6.4) that D(b; l) from (2.5) is a compact metric space and hence separable. Moreover, D(b) =U 1D(b; l) and so D(b) is separable. It remains to prove the asserted homeomorphism. Let be the map constructed in the proof of Theorem 2.1. We need to show that

e

B(D(b))

ｾ＠

R+X M(D(b)),

or equivalently, that mE M(D) defined in (ALl) satisfies m(D(b)) = 1. Let d = (d1 , ... , d,, . .. ) E D(b), d, = (c0 , m 1 ) 'Vt. It suffices to show that

(A1.2) Since dE D( b), we know that

(A1.3) Prove the t

=

1 version of (A1.2), i.e.,

(A1.4) Of course, m 2 such that

E

M(R+x M(R";)) is given. By (A1.3) and the definition (2.7) of j, there exists ｬｾ＠

f(m 2 )(Y(b;l))=1

0

=>

v{yER";:(c,y)EY(b;/)}=1 a.e.[m 2 ]

=>

m2 {(c,v)ER+XM(R";): v{yER";: (c,y)E Y(b;l)} =1} =1

=>

m2 {(c,v)ER+XM(R";):v(Y(b;bl))=1}=1·= m 2 (R+xM(Y(b;bl)))=1

=>

(A.1.4).

In a similar fashion (A1.2) can be proven by induction. Finally, note that the subspace M( D( b)) in the statement of the theorem satisfies

B(D(b)) = R+x M(D(b)).

Q.E.D.

To clarify the difference between M(D(b)) and M(D(b)), consider the following: Let D0 (b) = {(co, y): Co> 0, y E u/>0 Y(b; l)}, which qm be identified with a ｾｵ｢ｳｰ｡｣･＠ of D(b). Then a given mE M(D0 (b)) c M(D(b)) will ｡ｬｾｯ＠ lie in M(D(b)) if and only if 3[> 0 for which the support of m lies in Y(b; l), i.e., mE M(Y(b; l)). That will not generally be the case for the same reason that U 1 > 0 M(Y(b; l)) is a proper subset of M(U 1 > 0 Y(b; l)).

APPENDIX2

Two lemmas are provided here. The first is used in the proof of Theorem 3.1 and the second establishes that the Chew-Dekel mean value (3.10) satisfies MV.l. LEMMA A2.1: Let X be a metric space and m n X-> R u.s.c. and bounded above.

->

m in M( X). Then lim sup ff dm n

.;;

ff dm for all f:

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PROOF: When f is bounded below, see Billingsley (1968, exercise 7, p. 17). In general, for each integer K < 0, define fK(x) =max{f(x),

K}.

Then fK is bounded and u.s.c. Thus (by above) q,K; M(X)-+ R is u.s.c. where q,K(p) fK(x) hf(x), (p) = ffdp = inf q,K(p) and so is u.s.c.

= ffK dp. Since Q.E.D.

LEMMA A2.2: The implicit weighted functional p. in (3.10) satisfies MV.l. PROOF: (a) Let P.n =p.(p"), p.=p.(p). The p./s and p. all lie in [0, a]. Suppose (for some subsequence) P.n-+ 8. Then since F is uniformly continuous on [0, 1] 2 ,

Since F( x, z) is increasing in x,

It follows that 0=/F(·,p.")dpn-+fF(·,o)dp, which implies 0=p.n-+fF(·,8)dp. But then 8=p. since F(x, z) is decreasing in z. Similarly for (b). Q.E.D.

APPENDIX3 We prove Theorem 3.1 regarding the existence of recursive utility functions. Denote by s+ ( D( b)) the set of functions from D( b) into R +. Let h E s+ ( D( b)) be strictly positive and define sh+(D(b)) = { vE s+(D(b)): llvllh

=supv(d)/h(d) < oo }.

s,; (

s,; (

With the norm ll·llh, D( b)) is a complete metric space. A transformation T: D( b)) -+ sh+(D(b)) is a strict contraction if IITv- Tullh ｾ＠ euv- ullh withe< 1. Every strict contraction on a complete metric space has a unique fixed point. The following is an immediate corollary which is adapted from Boyd. WEIGHTED CONTRACTION MAPPING THEOREM (WCMl): LetT: Sh+(D(b))-+ s+(D(b)) be such T(u) + E>Ah that (WCM.1) u ｾ＠ v => T(u) ｾ＠ T(v); (WCM.2) T(O) E sh+ (D); (WCM.3) T(u +A ｨＩｾ＠ for some constant e < 1 and 'ItA > 0. Then T has a unique fixed point v*. Moreover, TNO-+ v* in sh+ (D). We are able to base the following proof on WCMT in the KP case when a and p have identical signs. An advantage of such a proof is that it leads to the uniqueness of the solution to (3.4) and also to stronger continuity properties for V than described in the theorem. Moreover, it facilitates the proof of existence of optimal plans. But we could not find a way to apply WCMT to the remaining cases for the KP functional or to more general specifications for p.. Thus we use it below only where absolutely necessary, namely in Case 1, and otherwise we present a shorter, simpler argument. PROOF OF THEOREM 3.1: CASE 1: KP functional, 0 < p ｾ｡］＠ by h(d)

= [ 1 + c0 +

ｅｭ Ｑ ｾｘＧＨ＠

c,/b')

r

1, {fbP < 1. Define h: D(b)-+ R++

where X E (0,1) is to be determined below and d= (c0 , m), m 1 = f(P2 m). (See (2.3) or (2.7) and (2.11); m 1 is the "atemporal" probability measure on future consumption induced by m.) The above expected value exists since 'i,'f)N(c,/b') is bounded on each Y(b; l) and since m1 EU1 > 0 M(Y(b; !)).

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Rewrite (3.8) in terms of U = VP j

(A3.1)

cg

H(c 0 ,z)=-+{3z, p

p

as follows:

y=ljp;;.l.

Use (A3.1) to define the transformation T: sh+(D(b))-+ s+(D(b)), where

(Note that [ EmuY( · )] 1/y.;;; llullh · [ EmhY( · )] 1/y

if u E sh+ ( D( b)). Moreover, 00

EmhY( ·)

=

1 + c0

+ Em1 LXc,jb' 1

which is finite for the same reasons provided above for the finiteness of h(d).) We wish to prove that T has a fixed point and so we verify the conditions of WCMT. The first two conditions are immediate. For the third, note that T(u +Ah)(c0 , m)

=

H( c0 , [ Em(u( ·) +Ah(·))Y] 1

.;;; H( co, ( EmuY( · )j 11Y)

h)

+ f3A(EmhY( ·)flY (by Minkowski's inequality and y;;, 1)

=Tu(c0 ,m)+f3A[Emg(-)fh

(g=hY)

bP

=

Tu( c0 , m)

+ f3 ;\_P A(g( c0 , m)] 1h

( g ( c, ·) satisfies the independence axiom on M ( D (b))) =

Tu( c, m)

+ f3bPAh( c0 , m)jV.

Thus WCM.3 is satisfied with e = f3bP jV if ;\. is any number such that {3bP < V. By WCMT, rNo-+ U= VP jp in the ll·llh topology on st(D(b)). Moreover, his bounded on any D(b; /). Thus sup { iTNO(d)- U(d)l: dE D(b; /)}-+ 0 as N-+ oo. Each TNO( ·) is continuous on D( b; /). Thus U and V are continuous there. It can be shown that D(b; /)is compact. Thus (A3.2)

max{V(d): dED(b;l)} < oo,

a fact which is used below. CASE 2: General p., p > 0, {3bP < 1. Refer to the previous case and let v• be the corresponding utility functional which satisfies (A3.3) Define T: sh+ ( D( b)) -+ Tv (Co ' m)

s+ ( D( b)) by =

w( Co' IL ( v [ m])) .

(See the notation introduced in (3.3).) Then TNv•;;, 0 so the sequence {TNv*(c0 , m)} is bounded

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TEMPORAL BEHAVIOR OF CONSUMPTION

below. It is also declining in N:

W( c0 , Emv* ( ·)) (since p. is risk averse)

Tv* ( c0 , m) .,;

=v*(c0 ,m)

(by(A3.3)).

By induction, TNv*(c 0 , m).,; TN- 1v*(c0 , m) Ｇｖｎｾ＠ It remains to show that V solves (3.4 ). Since TNv•( Co, m)

=

1. Define V(c0 , m) =lim TNv*(c0 , m).

w( Co, JL( TN-lv•[m l))

and the left side converges to V(c0 , m), it suffices to show that JL(TN- 1v*[ml) ...... JL(V[m]).

But this is true by (A2.2), MV.1, and the monotone convergence theorem (Billingsley (1979, p. 179)), the latter of which implies

JI ( T"- v* (-)) dm (') ...... JI ( V (')) dm (') 1

for all increasing

f.

Since u* was shown to be continuous, on D( b; I) each T"v* is continuous there by induction. Thus, as the infimum of continuous functions, V is u.s.c. CASE 3: General JL, p < 0. Define the sequence { u"} of nonpositive extended real valued functions on D( oo) inductively: u1 ( c0 , m) = c81P.

u"(c0 ,m)=c81p+

ＨｉｾｐＩｊｌｰｵＢＧ

Ｑ

ＩＱＯｐ｛ｭｊＬ＠

n;;.2.

(Recall that (pu" -I ) 11P[m] denotes the distribution of ( pu"- 1 ( · )) 1/P induced by m. Define JL = oo if there is positive probability that (pu"- 1 ) 1/P = oo .) Then u".,; u"- I So the extended real valued function U, U(d) =lim u"(d), is well-defined. Let V solve VP = pU. That V solves (3.4) is demonstrated as in Case 2. Each u" is u.s.c. on D(b; /)by recursive applications of Lemma A2.1 and MV.1(b). Thus the same is true of the infimum. Q.E.D.

APPENDIX 4

This appendix defines the map e employed in (5.2) and then proves Theorem 5.1. Denote by F;+ the 0'-algebra generated by (1,. i, )'{'.We assume that associated with the stochastic process ( r,, i, )o 0, B E F; + and I, E / 1 , (A4.1)

.Y (·I I,) is a probability measure on (fl. F; +) and

(A4.2)

.Y( Bl ·)is f;_-measurable.

The fact that .Y does not vary with t reflects the assumed stationarity of the process. Each plan, in conjunction with the wealth accumulation equation, defines a random variable (r.v.) _)i: !? ...... Y( b)= U, > 0 Y( b; I) such that .Y = ( E0 , ... , E,, ... ) and E, is measurable with respect to f;_, the For given I0 and x 0 , E0 is nonstochastic and is written simply c0 . 0'-algebra generated by (1,. ｩ［ＩＧ｟ＭｾＮ＠ Feasibility implies that oc

(A4.3)

LE,/r'.,; x 0 a.e. ['1']. 0

For given I 0 and x 0 , we associate with each feasible r.v. consumption program y a temporal lottery d = (d1 , ... , d,, ... ), d, = (c0 , m,), m, E M(D,_ d for t > 2 and m 1 E M(Y(i'; x0 )). Each such dE D(i'; x 0 ) c D(b; x 0 ) c D(b) 'Vb > r. Roughly speaking, the association from .Y to dis as follows. The measure m 1 is that implied by the function .Y and the probability distribution for ( i, )g' by treating all of the latter as though they were realized at t = 1. In this way one obtains a consumption Similarly, if we lottery in which all uncertainty is resolved by t = 1, i.e., m 1 E M(Y(r; x0 )) c ｍＨｒｾＩＮ＠

r,,

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LARRY G. EPSTEIN AND STANLEY E. ZIN

recognize that (r0 , z0 ) is realized at r = 1 and regard (T,, z,), t ;;.1, as being realized at t- 2, we obtain m 2 e M(R+x M(Ro;')) = M(D1 ); and so on for the other m,'s. These measures differ only in the way the (common) uncertainty regarding future consumption is resolved through time. Thus an element de D is obtained. Formally, define m1 by

m1 (Q/I0 )='1i{{we0: Y(w)eQ}/I0 ), QeB(Ro;'). By (A4.3), m 1 e M(Y(r; x 0 )). Suppose that we have constructed m 1 (·/l1 )EM(D,_I) for some r;;.l. (Actually, for t=1, m1 (·/l0 ) e M(Ro;').) Then m1+ 1(·/l) is defined by m,+l(Q/l) =

>Y( {we 0: {c1 (w ), m,( ·j(I, "'o)}) e

Q} ji).

Here, "'o = (r0 , z0 ) and (I, "'o) represents realized history at t = 1. In this way d = (( c0 , mi), ... , ( c0 , m, ), ... } e D{r; x 0 )

is constructed. The desired map e: FP .... D(r; x 0 ) is the composition: plan-+ r.v. consumption prograni-+ temporal lottery. PROOF OF THEOREM 5.1: Define Cases 1-3 as in the proof of Theorem 3.1. In those cases where a contraction mapping is applicable, existence follows in the customary fashion. For the remaining cases we employ parallel arguments to those in the proof of Theorem 3.1. An alternative route would be to prove the compactness of e(FP) and apply the u.s.c. of utility. But we were able to show only that the closure of e(FP) is compact and not that e(FP) is closed. Thus only the finiteness of the supremum in (5.2) could be inferred in this way. CASE 1: Since WCMT applies, the existence of a maximum follows from Denardo (1967). That it can be achieved by a stationary plan follows from Sobel (1975). Oearly, J > 0 since c0 = x 0 is feasible and yields utility= x0 . CASE 2: Let v* and J* be the utility and value functions respectively for Case 1. Because of the homotheticity of preferences we may write J*(l, x) = A*(l)x. A*(·) is bounded above since v* is bounded on D(r; x 0 ) (see (A3.2)). Moreover, we have just seen that J*;;. x 0 . We conclude that (A4.4)

1.;>A*(·).;>a

max

w( a,(1-a)E[A•(i;,)w'r0 ji]),

aE[O,l], wESK

which in tum equals A*(l0 ) by the definition of A*. Thus (FA*)(l) .;>A*(l).

By induction, prove that FNA*(l)J, in N. Since the sequence is bounded below by 0, A{l)= lim FNA*(l) N-oo

is well-defined. We now show that A(l0 )x0 is at least as ｬ｡ｲｾ･＠ as the supremum in (5.2): For any feasible lottery (c0 , m), v*(c0 , m) .;> A*(l0 )x0 • By induction, T v*(c0 , m) .;> FNA*(l0 )x0 , where Tis defined following (A3.3). Thus taking limits yields V(c0 , m) .;> A(l0 )x0 • We need to show that the supremum is attained. For any plan h=(h 0 , ... ,h1, ... ) denote by e(h; [0 , x 0 ) the temporal lottery in D(b), where e is defined above and its dependence on I0 and x 0 is made explicit. Then V(e(h; l 0 , x0 )) is the utility of the plan h given initial conditions (l0 , x 0 ). By

Chapter 12. Substitution, Risk Aversion and the Temporal Behavior of Consumption TEMPORAL BEHAVIOR OF CONSUMPTION

237

967

homogeneity, (A4.6) Let h** be any (not necessarily feasible) plan such that e(h**; 10 , x 0 )

E

D(b; I) for some I> 0 and

(Note that h** is not the optimal plan of Case 1 since the utility function Vis not the utility function of Case 1.) For example, h** could be the plan in which c0 = A*(/0 )x 0 and consumption is set equal to zero in all subsequent periods. Denote by ( a0 (!; H), w0 (!; H)) the solution to (A4.5) when A*(·) in the maximand is replaced by H( · ). Define the transformation of plans G by G(h 0 , ... , h,, ... ) = ( h 0 , h 0 , .. . , h,, ... ),

where h0 (l, x) = (xa 0 (!; H), w0 (!; H)) and His defined in (A4.6). Consider the sequence of plans { GN(h**); N ｾ＠ 1 }. Following is a list of facts regarding this sequence. Write GN(h**) = ＨｨｾＬ＠ ... , h{', ... ) and h{' =(he{', hp,N). FACT 1: V(e(GN(h**): 10 , x 0 )) = FNA*(/0 ) · x 0 ｾ＠ A(l0 )xo. FACT 2: For each t and N ｾ＠ t + 1, he{'(·) is increasing in N if p > 0 and decreasing if p < 0. In either case, he, ( ·) = lim he{' ( ·) exists. FACT 3: For each I, [and N ｾｉＫ＠ l,

FACT 4: The plan h = (( hc0 , hp0 ), ... , (he" hp1), ... ) is feasible and stationary. FACT 5: e(GN(h**); 10 , x0 )--> e(h; 10 , x0 ) in D(b; I). (To prove the latter use the fact that a.e. pointwise convergence of a sequence of random variables implies the weak convergence of the corresponding sequence of probability measures.) Combine Facts 1, 5, and the u.s.c. of utility to cpnclude that

But A (!0 )x0 is no smaller than the utility supremum over feasible paths. Thus by Fact 4, equality must prevail and the supremum is achieved. The positivity of J is clear as in Case l. The argument for Case 3 is sintilar to that for Case 2. To prove positivity when p < 0, note that there exists i\ ｾ＠ 1 and a sufficiently small e > 0 such that the plan of consunting e · (r/i\) 1 in each t (and making any portfolio decisions whatsoever) is feasible. This plan yields utility

Q.E.D.

REFERENCES BILLINGSLEY, P. (1979): Probability and Measure. New York: John Wiley. - - (1968): Convergence of Probability Measures. New York: John Wiley. BoYD, J. H. (1987): "Recursive Utility and the Ramsey Problem," University of Rochester, ntimeo. CHEW, S. H. (1983): "A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox," Econometrica, 51, 1065-1092. - - (1989): "Axiomatic Utility Theories with the Betweenness Property," Annals of Operations Research: Choice Under Uncertainty, forthcoming.

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CHEW, S. H., AND L. G. EPSTEIN (1987): "Non-Expected Utility Preferences in a Temporal Framework with an Application to Consumption-Savings Behavior," University of Toronto, Working Paper 8701. To appear in Journal of Economic Theory. - - (1989): "The Structure of Preferences and Attitudes Towards the Timing of the Resolution of Uncertainty," University of Toronto, Working Paper 8706, forthcoming in the International Economic Review, 30, 103-117. CHEW, S. H., L. G. EPSTEIN, AND I. ZILCHA (1988): "A Correspondence Theorem Between Smooth Utility and Expected Utility," Journal of Economic Theory, 46, 186-193. DEKEL, E. (1986): "An Axiomatic Characterization of Preferences under Uncertainty," Journal of Economic Theory, 40, 304-318. DENARDO, E. V. (1967): "Contraction Mappings in the Theory Underlying Dynamic Programming," SIAM Review, 9, 165-177. EPSTEIN, L. G. (1988): "Risk Aversion and Asset Prices," Journal of Monetary Economics, 22, 179-192. - - (1987): "Global Stability and Efficient Allocations," Econometrica, 55, 329-355. - - (1986): "Implicitly Additive Utility and the Nature of Optimal Economic Growth," Journal of Mathematical Economics, 15, 111-128. - - (1983): "Stationary Cardinal Utility and Optimal Growth Under Uncertainty," Journal of Economic Theory, 31, 133-152. EPSTEIN, L. G., AND S. M. TANNY (1980): "Generalized Increasing Correlation: A Definition and Some Economic Consequences," Canadian Journal of Economics, 13, 16-34. EPSTEIN, L. G., AND S. E. ZIN (1989): "Substitution, Risk Aversion and the Temporal Behavior of Consumption and Asset Returns II: An Empirical Analysis," Working Paper 8718, University of Toronto, revised. FARMER, R. E. A. (1987): "Closed-Form Solutions to Dynamic Stochastic Choice Problems," University of Pennsylvania. GROSSMAN, S., AND G. LAROQUE (1987): "Asset Pricing and Optimal Portfolio Choice in the Presence of Illiquid Durable Consumption Goods," mimeo. GROSSMAN, S., AND R. SHILLER (1981): "The Determinants of the Variability of Stock Market Prices," American Economic Review, 71, 222-227. HALL, ROBERT E. (1978): "Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence," Journal of Political Economy, 86, 971-987. - - (1981): "Intertemporal Substitution in Consumption," NBER Working Paper 720. - - (1985): "Real Interest and Consumption," NBER Working Paper 1694. HANSEN, L., AND K. SINGLETON (1983): "Stochastic Consumption, Risk Aversion, and the Temporal Behavior of Asset Returns," Journal of Political Economy, 91, 249-265. - - (1982): "Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models," Econometrica, 50, 1269-1286. JOHNSEN, T. H., AND J. B. DONAWSON (1985): "The Structure of Intertemporal Preferences under Uncertainty and Time Consistent Plans," Econometrica, 53, 1451-1458. KIHLSTROM, R. E., AND L. J. MIRMAN (1974): "Risk Aversion with Many Commodities," Journal of Economic Theory, 8, 361-88. KOOPMANS, T. C. (1960): "Stationary Ordinal Utility and Impatience," Econometrica, 28, 287-309. KREPS, D. M., AND E. L. PORTEUS (1978): "Temporal Resolution of Uncertainty and Dynamic Choice Theory," Econometrica, 46, 185-200. - - (1979a): "Temporal von Neumann-Morgenstem and Induced Preferences," Journal of Economic Theory, 20, 81-109. - - (1979b): "Dynamic Choice Theory and Dynamic Programming," Econometrica, 47, 91-100. LUCAS, R. E. (1978): "Asset Prices in an Exchange Economy," Econometrica, 46, 1426-1445. LUCAS, R. E., AND N. STOKEY (1984): "Optimal Growth with Many Consumers," Journal of Economic Theory, 32, 139-171. MACHINA, M. J. (1982): "'Expected Utility' Analysis Without the Independence Axiom," Econometrica, 50, 277-323. - - (1984): "Temporal Risk and the Nature of Induced Preference," Journal of Economic Theory, 33, 199-231. MEHRA, R., AND E. PRESCOTT (1985): "The Equity Premium: A Puzzle," Journal of Monetary Economics, 15, 145-161. MERTENS, J. F., AND S. ZAMIR (1985): "Formalization of Bayesian Analysis for Games with Incomplete Information," International Journal of Game Theory, 14, 1-29.

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MYERSON, R. (1985): "Bayesian Equilibrium and Incentive Compatability: An Introduction" in Social Goals and Social Organizations, Essays in Memory of Elisha Pazner, ed. by L. Hurwicz, D. Schmeidler, and H. Sonnenschein. Cambridge: Cambridge University Press. PARTHASARATHY, K. R. (1967): Probability Measures on Metric Spaces. New York: Academic Press. SARGENT, T. (1987): Dynamic Macroeconomic Theory. Cambridge, Mass.: Harvard University Press. SELDEN, L. (1978): "A New Representation of Preference over 'Certain X Uncertain' Consumption Pairs: The 'Ordinal Certainty Equivalent' Hypothesis," Econometrica, 46, 1045-1060. SOBEL, M. J. (1975): "Ordinal Dynamic Programming," Management Science, 21, 967-975. WEIL, P. (1987a): "Asset Prices with Non-Expected Utility Preferences," Harvard University, mimeo. - - (1987b): "Non-Expected Utility in Macroeconomics," Harvard University, mimeo. YAARI, M. E. (1987): "The Dual Theory of Choice Under Risk," Econometrica, 55, 95-115. ZIN, S. (1987): "Intertemporal Substitution, Risk and the Time Series Behavior of Consumption and Asset Returns," Ph.D. Thesis, University of Toronto.

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Econometrica, Vol. 68, No. 5 (September, 2000), 1281-1292

NOTES AND COMMENTS

Chapter 13 RISK AVERSION AND EXPECTED-UTILITY THEORY: A CALIBRATION THEOREM

BY

MATIHEW RABIN 1

1. INTRODUCTION USING EXPECTED-UTILITY THEORY, economists model risk aversion as ansmg solely because the utility function over wealth is concave. This diminishing-marginal-utility-ofwealth theory of risk aversion is psychologically intuitive, and surely helps explain some of our aversion to large-scale risk: We dislike vast uncertainty in lifetime wealth because a dollar that helps us avoid poverty is more valuable than a dollar that helps us become very rich. Yet this theory also implies that people are approximately risk neutral when stakes are small. Arrow (1971, p. 100) shows that an expected-utility maximizer with a differentiable utility function will always want to take a sufficiently small stake in any positive･ｸｰ｣ｴ､Ｍｶｾｬｵ＠ bet. That is, expected-utility maximizers are (almost everywhere) arbitrarily close to risk neutral when stakes are arbitrarily small. While most economists understand this formal limit result, fewer appreciate that the approximate risk-neutrality prediction holds not just for negligible stakes, but for quite sizable and economically important stakes. Economists often invoke expected-utility theory to explain substantial (observed or posited) risk aversion over stakes where the theory actually predicts virtual risk neutrality. While not broadly appreciated, the inability of expected-utility theory to provide a plausible account of risk aversion over modest stakes has become oral tradition among some subsets of researchers, and has been illustrated in writing in a variety of different contexts using standard utility functions. 2 In this paper, I reinforce this previous research by presenting a theorem that calibrates a relationship between risk attitudes over small and large stakes. The theorem shows that, within the expected-utility model, anything but virtual risk neutrality over modest stakes implies manifestly unrealistic risk aversion over 1 Many people, including David Bowman, Colin Camerer, Eddie Dekel, Larry Epstein, Erik Eyster, Mitch Polinsky, Drazen Prelec, Richard Thaler, and Roberto Weber, as well as a co-editor and two anonymous referees, have provided useful feedback on this paper. I thank Jimmy Chan, Erik Eyster, Roberto Weber, and especially Steven Blatt for research assistance, and the Russell Sage, MacArthur, National Science (Award 9709485), and Sloan Foundations for financial support. I also thank the Center for Advanced Studies in Behavioral Sciences, supported by NSF Grant SBR-960123, where an earlier draft of the paper was written. 2 See Epstein (1992), Epstein and Zin (1990), Hansson (1988), Kandel and Stambaugh (1991), Loomes and Segal (1994), and Segal and Spivak (1990). Hansson's (1988) discussion is most similar to the themes raised in this paper. He illustrates how a person who for all initial wealth levels is exactly indifferent between gaining $7 for sure and a 50-50 gamble of gaining either $0 or $21 prefers a sure gain of $7 to any lottery where the chance of gaining positive amounts of money is less than 40%-no matter how large the potential gain is.

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large stakes. The theorem is entirely "nonparametric," assuming nothing about the utility function except concavity. In the next section I illustrate implications of the theorem with examples of the form "If an expected-utility maximizer always turns down modest-stakes gamble X, she will always turn down large-stakes gamble Y." Suppose that, from any initial wealth level, a person turns down gambles where she loses $100 or gains $110, each with 50% probability. Then she will turn down 50-50 bets of losing $1,000 or gaining any sum of money. A person who would always turn down 50-50 lose $1,000/gain $1,050 bets would always turn down 50-50 bets of losing $20,000 or gaining any sum. These are implausible degrees of risk aversion. The theorem not only yields implications if we know somebody will turn down a bet for all initial wealth levels. Suppose we knew a risk-averse person turns down 50-50 lose $100/gain $105 bets for any lifetime wealth level less than $350,000, but knew nothing about the degree of her risk aversion for wealth levels above $350,000. Then we know that from an initial wealth level of $340,000 the person will turn down a 50-50 bet of losing $4,000 and gaining $635,670. The intuition for such examples, and for the theorem itself, is that within the expected-utility framework, turning down a modest-stakes gamble means that the marginal utility of money must diminish very quickly for small changes in wealth. For instance, if you reject a 50-50 lose $10/gain $11 gamble because of diminishing marginal utility, it must be that you value the eleventh dollar above your current wealth by at most (10/11) as much as you valued the tenth-to-last-dollar of your current wealth. 3 Iterating this observation, if you have the same aversion to the lose $10/gain $11 bet if you were $21 wealthier, you value the thirty-second dollar above your current wealth by at most (10/11)X(10j11)"'(5/6) as much as your tenth-to-last dollar. You will value your two-hundred-twentieth dollar by at most (3 /20) as much as your last dollar, and your eight-hundred-eightieth dollar by at most (1/2,000) of your last dollar. This is an absurd rate for the value of money to deteriorate-and the theorem shows tAe rate of deterioration implied by expected-utility theory is actually quicker than this. Indeed, the theorem is really just an algebraic articulation of how implausible it is that the consumption value of a dollar changes significantly as a function of whether your lifetime wealth is $10, $100, or even $1,000 higher or lower. From such observations we should conclude that aversion to modest-stakes risk has nothing to do with the diminishing marginal utility of wealth. Expected-utility theory seems to be a useful and adequate model of risk aversion for many purposes, and it is especially attractive in lieu of an equally tractable alternative model. "Extremely-concave expected utility" may even be useful as a parsimonious tool for modeling aversion to modest-scale risk. But this and previous papers make clear that expected-utility theory is manifestly not close to the right explanation of risk attitudes over modest stakes. Moreover, when the specific structure of expected-utility theory is used to analyze situations involving modest stakes-such as in research that assumes that large-stake and modest-stake risk attitudes derive from the same utility-for-wealth 3 My wording here, as in the opening paragraph and elsewhere, gives a psychological interpretation to the concavity of the utility function. Yet a referee has reminded me that a common perspective among economists studying choice under uncertainty has been that the concavity of the utility function need be given no psychological interpretation. I add such a psychological interpretation throughout the paper as an aid to those readers who, like me, find this approach to be the natural way to think about utility theory, but of course the mathematical results and behavioral analysis in this paper hold without such interpretations.

Chapter 13. Risk Aversion and Expected-Utility Theory RISK AVERSION

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function-it can be very misleading. In the concluding section, I discuss a few examples of such research where the expected-utility hypothesis is detrimentally maintained, and speculate very briefly on what set of ingredients may be needed to provide a better account of risk attitudes. In the next section, I discuss the theorem and illustrate its implications.

2.

SOME CALIBRATIONS BASED ON A THEOREM

Consider an expected-utility maximizer over wealth, w, with Von Neumann-Morgenstem preferences U( w ). Assume that the person likes money and is risk-averse: For all w,U(w) is (strictly) increasing and (weakly) concave. Suppose further that, for some range of initial wealth levels and for some g > l > 0, she will reject bets losing $1 or gaining $g, each with 50% chance. 4 From the assumption that these bets will be rejected, the theorem presented in this paper places an upper bound on the rate at which utility increases above a given wealth level, and a lower bound on the rate at which utility decreases below that wealth level. Its proof is a short series of algebraic manipulations; both the theorem and proof are in the Appendix. Its basic intuition is straightforward, as described briefly in the introduction. The theorem handles cases where we know a person to be averse to a gamble only for some ranges of initial wealth. A simpler corollary, also in the Appendix, holds when we know a lower bound on risk aversion for all wealth levels. Table I illustrates some of the corollary's \implications: Consider an individual who is known to reject, for all initial wealth levels, 50-50, lose $100/gain g bets, for g = $101, $105, $110, and $125. The table presents implications of the form "the person will turn down 50-50 gambles of losing L and gaining G," where each L is a row in the table and the highest G (using the bounds established by the corollary) making the statement true is the entry in the table. 5 All entries are rounded down to an even dollar amount. So, for instance, if a person always turns down a 50-50 lose $100jgain $110 gamble, she will always turn down a 50-50 lose $800jgain $2,090 gamble. Entries of oo are literal: Somebody who always turns down 50-50 lose $100/gain $125 gambles will turn down any gamble with a 50% chance of losing $600. This is because the fact that the bound on risk aversion holds everywhere implies that U( w) is bounded above. 4 The assumption that U is concave is not implied by the fact that an agent always turns down a given better-than-fair bet; if you know that a person always turns down 50-50 lose $10jgain $11 bets, you don't know that her utility function is concave everywhere-it could be convex over small ranges. (For instance, let U(w) = 1- C±)w for w $ (19,20), but

1 ) 19 + U(w)=1- ( 2

[( 1 ) 19 - ( 1 ) 20] (w-19) 2 2

2

for wE (19, 20).) Concavity is an additional assumption, but I am confident that results hold approximately if we allow such small and silly nonconvexities. 5 The theorem provides a lower bound on the concavity of the utility function, and its proof indicates an obvious way to obtain a stronger (but uglier) result. Also, while the theorem and applications focus on "S0-50 bets," the point is applicable to more general bets. For instance, if an expected-utility maximizer dislikes a bet with a 25% chance of losing $100 and a 75% chance of winning $100, then she would turn down 50-50 lose $100jgain $300 bets, and we could apply the theorem from there.

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MATTHEW RABIN TABLE I IF AVERSE TO 50-50 LOSE $100 I GAIN g BETS FOR ALL WEALTII LEVELS, WILL TuRN DowN 50-50 LosE L 1 GAIN G BETS; G's ENTERED IN TABLE. g

L

$400 $600 $800 $1,000 $2,000 $4,000 $6,000 $8,000 $10,000 $20,000

$101

$105

$110

$125

400 600 800 1,010 2,320 5,750 11,810 34,940

420 730 1,050 1,570

550 990 2,090

1,250 00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

The theorem and corollary are homogeneous of degree 1: If we know that turning down 50-50 lose ljgain g gambles implies you will turn down 50-50 lose Ljgain G, then for all x > 0, turning down 50-50 lose xl I gain xg gambles implies you will turn down 50-50 lose xLjgain xG. Hence the L = $10,000, g = $101 entry in Table I tells us that turning down 50-50 lose $10jgain $10.10 gambles implies you will turn down 50-50 lose $1,000/gain co gambles. The reader may worry that the extreme risk aversion shown in Table I relies heavily on the assumption that the person will turn down the given gamble for all initial wealth levels. It doesn't. While without knowing a global lower bound on a person's modest-stakes risk aversion we cannot assert that she'll turn down gambles with infinite expected return, Table II indicates that the lack of a lower bound does not salvage the plausibility of expected-utility theory. Table II shows calibrations if we know the person will turn down 50-50 lose $100jgain g gambles for initial wealth levels less than $300,000, indicating which gambles she'll turn down starting from initial wealth level of $290,000. Large entries are approximate. TABLE II TABLE I REPLICATED, FOR INITIAL WEALTH LEVEL $290,000, WHEN ilg BEHAVIOR IS ONLY KNOWN TO HoLD FOR W :5:$300,000. g L

$400 $600 $800 $1,000 $2,000 $4,000 $6,000 $8,000 $10,000 $20,000

$101

$105

$110

$125

400 600 800 1,010 2,320 5,750 11,510 19,290 27,780 85,750

420 730 1,050 1,570 69,930 635,670 1,557,360 3,058,540 5,503,790 71,799,110

550 990 2,090 718,190 12,210,880 60,528,930 180,000,000 510,000,000 1,300,000,000 160,000,000,000

1,250 36,000,000,000 90,000,000,000 160,000,000,000 850,000,000,000 9,400,000,000,000 89,000,000,000,000 830,000,000,000,000 7, 700,000,000,000,000 540,000,000,000,000,000,000

Chapter 13. Risk Aversion and Expected-Utility Theory

245

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RISK AVERSION TABLE III

IF A PERSON HAs CARA UTILITY FuNCTION AND Is AvERSE TO 50150 LosE$/ 1 GAIN $g BETS FOR ALL WEALTH LEVELS, THEN (i) SHE HAs COEFFICIENT OF ABSOLUTE RISK AVERSION No SMALLER THAN p AND (ii) INVESTS $X IN THE STOCK MARKET WHEN STOCK YIELDS ARE NORMALLY DISTRIBUTED WITH MEAN RE/\L RETURN 6.4% i\ND STANDi\RD DEVIATION 20%, AND BONDS YIELD A RISKLESS RETURN OF 0.5%. ljg

$1001$101 $1001$105 $1001$110 $1001$125 $100j$150 $1,0001$1,050 $1,0001$1,100 $1,0001$1,200 $1,0001$1,500 $1,0001$2,000 $10,0001$11,000 $10,0001$12,000 $10,0001$15,000 $10,0001$20,000

f>

X

.0000990 .0004760 .0009084 .0019917 .0032886 .0000476 .0000908 .0001662 .0003288 .0004812 .0000090 .0000166 .0000328 .0000481

$14,899 $3,099 $1,639 $741 $449 $30,987 $16,389 $8,886 $4,497 $3,067 $163,889 $88,855 $44,970 $30,665

If we only know that a person turns down 50-50 lose $100jgain $125 bets when her lifetime wealth is below $300,000, we also know she will turn down a 50-50 lose $600/gain $36 billion bet beginning from lifetime $290,000. 6 The intuition is that the extreme concavity of the utility function between $290,000 and $300,000 assures that the marginal utility at $300,000 is tiny compared to the marginal utility at wealth levels below $290,000; hence, even if the marginal utility does not diminish at all above $300,000, a person will not care nearly as much about money above $300,000 as she does about amounts below $290,000. The choice of $290,000 and $300,000 as the two focal wealth levels is arbitrary; all that matters is that they are $10,000 apart. As with Table I, Table II is homogeneous of degree 1, where the wedge between the two wealth levels must be multiplied by the same factor as the other entries. Hence-multiplying Table II by 10-if an expected-utility maximizer would turn down a 50-50 lose $1,000/gain $1,050 gamble whenever her lifetime wealth is below $300,000, then from an initial wealth level of $200,000 she will turn down a 50-50 lose $40,000/gain $6,356,700 gamble. While these "nonparametric" calibrations are less conducive to analyzing more complex questions, Table III provides similar calibrations for decisions that resemble realworld investment choices by assuming conventional functional forms of utility functions. It shows what aversion to various gambles implies for the maximum amount of money a person with a constant-absolute-risk-aversion (CARA) utility function would be willing to

6 Careful examination of Tables I and II shows that most entries that are not oo in Table I show up exactly the same in Table II. The two exceptions are those entries that are above $10,000-since Table II implicitly makes no assumptions about concavity for gains of more than $10,000, it yields lower numbers.

246

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keep invested in the stock market, for reasonable assumptions about the distribution of returns for stocks and bonds. Hence, an expected-utility maximizer with CARA preferences who turns down 50j50 lose $1,000/gain $1,200 gambles will only be willing to keep $8,875 of her portfolio in the stock market, no matter how large her total investments in stocks and bonds. If she turns down a 50j50 lose $100/gain $110 bet, she will be willing to keep only $1,600 of her portfolio in the stock market-keeping the rest in bonds (which average 6% lower annual return). While it is widely believed that investors are too cautious in their investment behavior, no one believes they are this risk averse.

3. DISCUSSION AND CONCLUSION

Expected-utility theory may well be a useful model of the taste for very-large-scale insurance. 7 Despite its usefulness, however, there are reasons why it is important for economists to recognize how miscalibrated expected-utility theory is as an explanation of modest-scale risk aversion. For instance, some research methods economists currently employ should be abandoned because they rely crucially on the expected-utility interpretation of modest-scale risk aversion. One example arises in experimental economics. In recent years, there has been extensive laboratory research in economics in which subjects interact to generate outcomes with payoffs on the order of $10 to $20. Researchers are often interested in inferring subjects' beliefs from their behavior. Doing so often requires knowing the relative value subjects hold for different money prizes; if a person chooses $5 in event A over $10 in event B, we know that she believes A is at least twice as likely as B only if we can assume the person likes $10 at least twice as much as $5. Yet economic theory tells us that, because of diminishing marginal utility of wealth, we should not assume people like $10 exactly twice as much as $5. Experimentalists (e.g., Davis and Holt (1993, pp. 472-476)) have developed a clever scheme to avoid this problem: Instead of prizes of $10 and $5, subjects are given prizes such as 10% chance of winning $100 vs. 5% chance of winning $100. Expected-utility theory tells us that, irrespective of the utility function, a subject values the 10% chance of a prize exactly twice as much as the 5% chance of winning the same prize. The problem with this lottery procedure is that it is known to be sufficient only when we maintain the expected-utility hypothesis. But then it is not necessary-since expected-utility theory tells us that people will be virtually risk neutral in decisions on the scale of laboratory stakes. If expected-utility theory is right, these procedures are at best redundant, and are probably harmful. 8 On the other hand, if we think that subjects in experiments are risk averse, then we know they are not expected-utility maximizers. Hence the lottery procedure, which is motivated solely by expected-utility theory's assumptions that preferences are linear in probabilities and that risk attitudes come only 7 While there is also much evidence for some limits of its applicability for large-scale risks, and the results of this paper suggest an important flaw with the expected-utility model, the specific results do not of course demonstrate that the model is unuseful in all domains. 8 If expected-utility theory explained behavior, these procedures would surely not be worth the extra expense, nor the loss in reliability of the data from making experiments more complicated. Nor should experimentalists who believe in expected-utility theory ever be cautious about inferences made from existing experiments that don't use the lottery methods out of fear that the results are confounded by the subjects' risk attitudes.

Chapter 13. Risk Aversion and Expected-Utility Theory RISK A VERSION

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1287

from the curvature of the utility-of-wealth function, has little presumptive value in "neutralizing" risk aversion. Perhaps there are theories of risk attitudes such that the lottery procedure is useful for neutralizing risk aversion-but expected-utility theory isn't one of them. 9 A second example of problematic research methods relates to the logic underlying the theorem: Expected-utility theory makes wrong predictions about the relationship between risk aversion over modest stakes and risk aversion over large stakes. Hence, when measuring risk attitudes maintaining the expected-utility hypothesis, differences in estimates of risk attitudes may come from differences in the scale of risk comprising data sets, rather than from differences in risk attitudes of the people being studied. 10 Data sets dominated by modest-risk investment opportunities are likely to yield much higher estimates of risk aversion than data sets dominated by larger-scale investment opportunities. So not only are standard measures of risk aversion somewhat hard to interpret given that people are not expected-utility maximizers, but even attempts to compare risk attitudes so as to compare across groups will be misleading unless economists pay due attention to the theory's calibrational problems. The problems with assuming that risk attitudes over modest and large stakes derive from the same utility-of-wealth function relate to a long-standing debate in economics. Expected-utility· theory makes a powerful prediction that economic actors don't see an amalgamation of independent gambles as significant insurance against the risk of those gambles; they are either barely less willing or barely more willing to accept risks when clumped together than when apart. This observation was introduced in a famous article by SamueJson (1963), who showed that expected-utility theory implies that if (for some sufficiently wide range of initial wealth levels) a person turns down a particular gamble, then she should also tum down an offer to play n > 1 of those gambles. Hence, in his example, if Samuelson's colleague is unwilling to accept a 50-50 lose $100/gain $200 gamble, then he should be unwilling to accept 100 of those gambles taken together. Though Samuelson's theorem is "weaker" than the one in this paper, it makes manifest the fact that expected-utility theory predicts that adding together a lot of independent risks should not appreciably alter attitudes towards those risks.

9 Indeed, the observation that diminishing marginal utility of wealth is irrelevant in laboratory experiments raises questions about interpreting experimental tests of the adequacy of expectedutility theory. For instance, while showing that existing alternative models better fit experimental data than does expected-utility theory, Harless and Camerer (1994) show that expected-utility theory better fits experimental data than does "expected-value theory"-risk-neutral expected-utility theory. But because expected-utility theory implies that laboratory subjects should be risk neutral, such evidence that expected-utility theory explains behavior better than expected-value theory is evidence against expected-utility theory. 10 Indeed, Kandel and Stambaugh (1991, pp. 68-69) discuss how the plausibility of estimates for the coefficient of relative risk aversion may be very sensitive to the scale of risk being examined. Assuming constant risk aversion, they illustrate how a coefficient of relative risk aversion needed to avoid predicting absurdly large aversion to a 50j50 lose $25,000jgain $25,000 gamble generates absurdly little aversion to a 50/50 lose $400/gain $400 gamble. They summarize such examples as saying that "Inferences about [the coefficient of relative risk aversion] are perhaps most elusive when pursued in the introspective context of thought experiments." But precisely the same problem makes inferences from real data elusive.

248

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M Rabin MAITHEW RABIN

Yet virtually everybody would find the aggregated gamble of 100 50-50 lose $100/gain $200 bets attractive. It has an expected yield of $5,000, with negligible risk: There is only a (1/700) chance of losing any money and merely a Cl/25,000) chance of losing more than $l,OOO.U While nobody would turn down this gamble, many people, such as Samuelson's colleague, might reject the single 50-50 lose $100jgain $200 bet. 12 Hence, using expected-utility theory to make inferences about the risk attitudes towards the amalgamated bet from the reaction to the one bet-or vice versa-would be misleading. What does explain risk aversion over modest stakes? While this paper provides a "proof by calibration" that expected-utility theory does not help explain some risk attitudes, there are of course more direct tests showing that alternative models better capture risk attitudes. There is a large literature (see Machina (1987) and Camerer (1992) for reviews) of formal models of such alternatives. Many of these models seem to provide a more plausible account of modest-scale risk attitudes, allowing both substantial risk aversion over modest stakes and nonridiculous risk aversion over large stakes, and researchers (e.g., Segal and Spivak (1990), Loomes and Segal (1994), Epstein and Zin (1990)) have explicitly addressed how non-expected-utility theory can help account for small-stake risk aversion. Indeed, what is empirically the most firmly established feature of risk preferences, loss aversion, is a departure from expected-utility theory that provides a direct explanation for modest-scale risk aversion. Loss aversion says that people are significantly more averse to losses relative to the status quo than they are attracted by gains, and more generally that people's utilities are determined by changes in wealth rather than absolute levels.U Preferences incorporating loss aversion can reconcile significant small-scale risk aversion with reasonable degrees of large-scale risk aversion. 14 A loss-averse person will, for instance, be likely to turn down the one 50/50 lose $100/gain $200 gamble Samuelson's colleague turned down, but will surely accept one hundred such gambles pooled' together. Variants of this or other models of risk attitudes can provide useful alternatives to

II The theorem in this paper predicts that, under exactly the same assumptions invoked by Samuelson, turning down a 50-50 lose $100jgain $200 gamble implies the person turns down a 50-50 lose $200jgain $20,000 gamble. This has an expected return of $9,900-with zero chance of losing more than $200. IZ As Samuelson noted, the strong statement that somebody should turn down the many bets if and only if she turns down the one is not strictly true if a person's risk attitudes change at different wealth levels. Indeed, many researchers (e.g., Hellwig (1995) and Pratt and Zeckhauser (1987)) have explored features of the utility function such that an expected-utility maximizer might take a multiple of a favorable bet that they would turn down in isolation. But characterizing such instances isn't relevant to examples of the sort discussed by Samuelson. We know from the unwillingness to accept a 50j50 lose $100jgain $200 gamble that Samuelson's colleague was not an expected-utility maximizer. 13 Loss aversion was introduced by Kahneman and Tversky (1979) as part of the more general "prospect theory," and is reviewed in Kahneman, Knetch, and Thaler (1991). Tversky and Kahneman (1991) and others have estimated the loss-aversion-to-gain-attraction ratio to be about 2:1. I 4 While most formal definitions of loss aversion have not made explicit the assumption that people are substantially risk averse even for very small risks (hut see Bowman, Minehart, and Rabin (1999) for an explicit treatment of this issue), most examples and calibrations of loss aversion imply such small-scale risk aversion.

Chapter 13. Risk Aversion and Expected-Utility Theory

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RISK AVERSION

expected-utility theory that can reconcile plausible risk attitudes over large stakes with nontrivial risk aversion over modest stakes. 15 Dept. of Economics, University of California-Berkeley, Berkeley, CA 94720-3880, U.S.A.; rabin@econ. berkeley. edu; http: I I emlab. berkeley. edul usersI rabin1index.html Manuscript received October, 1997; final revision received May, 1999.

APPENDIX:

THE THEOREM AND A CoROLLARY

THEOREM: Suppose that for all w, U(w) is strictly increasing and weakly concave. Suppose that there exists w > }:!:', g >I> 0, such that for all wE[):!:', w], .5U(w -I)+ .5U(w +g)< U(w ). Then for all WE[!:!:', w], for all X> 0, (i) if g :s; 21, then

(ii)

U(w +x)- U(w) :s;

Ｈ

ｫｲ｟ｸＩＨｾｩｷ＠

if x:s;w-w, ｩｾｏ＠

.

g

.

k"'(W) ( [ )'

ｩｾｏ＠

L

-

g

( [

r(w)+[x-w]-

)k'"(W)

r(w)

g

where, letting int(y) denote the smallest integer less than or equal toy, k*(x) int((xjg) + 1), and r(w) U(w)- U(w -I).

=

if

X

2: W-

W,

=int(xj2/), k**(x) =

15 But Kahneman and Lovallo (1993), Benartzi and Thaler (1995), and Read, Loewenstein, and Rabin (1999) argue that an additional departure from standard assumptions implicated in many risk attitudes is that people tend to isolate different risky choices from each other in ways that lead to different behavior than would ensue if these risks were considered jointly. Samuelson's colleague, for instance, might reject each 50/50 lose $100/gain $200 gamble if on each of 100 days he were offered one such gamble, whereas he might accept all of these gambles if they were offered to him at once. Benartzi and Thaler (1995) argue that a related type of myopia is an explanation for the "equity premium puzzle"-the mystery about the curiously large premium on returns that investors seem to demand to compensate for riskiness associated with investing in stocks. Such risk aversion can be explained with plausible (loss-averse) preferences-if investors are assumed to assess gains and losses over a short-run (yearly) horizon rather than the longer-term horizon for which they are actually investing.

250

M Rabin

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MATIHEW RABIN

PROOF OF PART (i) OF THEOREM: For notational ease and without loss of generality, let r( w) = U( w) - U( w -f) = 1. Then clearly U(w -f) - U(w - 2/) ?= 1, by the concavity of UO. Also, since 2/ > g > l, we know that w- 21 + g E ( w -I, w ), and by the concavity of UO we know that

U( w - 21 +g) - U( w -I) ?= Ｍｾ

g-l

g

=

I -

1.

Hence, g g U(w- 21 +g)- U(w- 2/) >- -1 + 1 = - .

- I

l

Hence, if w- 21 ＿］ｾＮｷ･＠ know that U(w- 2/)- U(w- 3/) ?= (gjl) since by assumption, U(w2l-l)+U(w-2l+g),s,2u(w-2l). By concavity, we also know that U(w-31)-U(w-4/)2 (gjl). More generally, if w - 2kl 2 ｾＮ＠ then U(w- (2k -l)l)- U(w- 2kl) 2 U(w- 2(k -1)1)- U(w- (2k- l)l)

=

g

U(w- 2kl +g)- U(w- 2kl) ?= l[U(w- 2(k -1)1)- U(w- (2k -l)l)] g

= U(w- 2kl)- U(w- (2k + l)l) 2/[U(w- 2(k -l)l)- U(w- (2k -1)1)]. These lower bounds on marginal utilities yield the lower bound on total utilities U(w)- U(w- x) in part (i) of the Theorem. 16

PROOF OF PART (ii) OF THEOREM: Again let r( w) = U( w) - U( w -I) = 1. Then U( w +g) - U( w) ,s,l. By the concavity of U, U(w+g)-U(w+g-l),s,(ljg). But if w+g,s,w, this implies by assumption that U(w + 2g)- U(w +g) ,s, (/jg) (since U(w + g -I)+ U(w + 2g) ,s, 2U(w +g)). More generally, we know that if w + mg ,s, w, then U(w +mg +g)- U(w +mg) 2 (/jg)[U(w + mg)- U(w +mg- g)].

These upper bounds on marginal utilities yield the upper bound on utilities U( w + x) - U( w) in part (ii) of the Theorem.

16

The theorem is weaker than it could be. If we observe, for all m such that 2 ,s, m ,s, w -

w+ 1,

that U( w - m) - U( w - m - 1) 2 U( w - m

+(

+ 1) -

T- 1)

U( w - m)

[U(w- m + 2)- U( w- m + 1)],

we can prove a stronger (but far messier) theorem. (The current theorem merely invokes U(w- m) - U(w- m -l) 2 U(w- m + 1)- U(w- m) for even m's.)

Chapter 13. Risk Aversion and Expected-Utility Theory

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RISK AVERSION

COROLLARY: Suppose that for all w, U(w) is strictly increasing and weakly concave. Suppose there exists g > 1 > 0 such that, for all w, .5U( w -I) + .5U( w +g) < U( w ). Then for all positive integers k, for all m < m(k), .S(w- 2kl) + .5U(w + mg) < U(w ), where

(fr]

ln[1- (1-ＩＲＬｾ＠ ｾ＠ m(/c)

l In-

=

-1

(g)' I > o,

if 1 -

(1 - :gl) 2 ｾ＠

if 1 -

(1 - :g1) 2 ｾ＠ (g)' I

k

g

k

: 0.10U($5M) + 0.89U($1M) + 0.01 U ($0), whereas the second ranking implies the inconsistent inequality O.IOU($5M) + 0.90U($0) > 0.11 U($1M) + 0.89U($0). By setting x1 = $0, x2 = $1M, and x3 = $5M, the four prospects can be plotted in the probability triangle as in Figure 2, where they are seen to form a parallelogram. The typical preference for a 1 over a 2 and a 3 over a 4 suggests that indifference curves depart from the expected utility property of parallel straight lines in the direction of 'fanning out', as in the figure. The Allais Paradox is merely one example of a widely observed phenomenon termed the Common Consequence Effect,

Increasing preference

0

Figure 2 Allais Paradox choices and indiflerence curves that 'fan out'

3

which states that when P* is riskier than P, individuals tend to depart from the Independence Axiom by preferring aP* + (1 - a)P** to aP + (1- a)P** when P** involves low payoffs, hut reverse this preference when P** instead involves high payoffs. (In the Allais example, P =($1M,l), P* =($5M, 10/11;$0,1/11), a= 0.11, and P** is ($1M,l) in the first pair and ($0,1) in the second pair.) The second broad class of systematic violations, also originally noted by Allais, can he illustrated by the following example from Kahneman and Tversky [18]: b . { 0.45 chance of 1' 0.55 chance of versus b . { 0.90 chance of 2· 0.10 chance of

$6000 $0 $3000 $0

b . { 0.00 I chance of $6000 3·

0.999 chance of $0 versus b . { 0.002 chance of $3000 4' 0.998 chance of $0 Most individuals express a preference for b 2 over b 1 and for b3 over b4 , which again violates expected utility theory since they imply the inconsistent inequalities 0.45U($6000) + 0.55U($0) < 0.90 U($3000) + O.IOU($0) and O.OOIU($6000) + 0.999U($0) > 0.002U($3000) + 0.998U($0). Setting x1 = $0, x2 = $3000 and x3 = $6000, the prospects can be plotted as in Figure 3, where the above preference rankings again suggest that indifference curves depart from expected utility by fanning out. This is one example of a widely observed phenomenon termed the Common Ralio Effect, which states that for positive-outcome prospects P* which is risky and P which is certain, with P** being a sure chance of $0, individuals tend to depart from the Independence Axiom by preferring aP* + (1 - a )P** to aP + (1 - a)P'* for low values of a, but reverse this preference for high values of a. Kahneman and Tversky [18] observed that when the positive outcomes $3000 and $6000 are replaced by losses of $3000 and then $6000 to create the prospects b'1, bS, b; and ｢ｾＬ＠ preferences typically 'reflect', to prefer b; over bS and ｢ｾ＠ over b;. Setting x1 = -$6000, x2 = -$3000 and x3 = $0 (to preserve the ordering x 1 < x2 < x3 ) and plotting as in Figure 4, such preferences again

256

MJ Machina

4

Nonexpected Utility Theory Most of these forms have been axiomatized, and given proper curvature assumptions on their component functions v(-), r(·), G(-), K(·, ·), ... , most are capable of exhibiting first-order stochastic dominance preference, risk aversion, and the common consequence and common ratio effects. The rankdependent expected utility form, in particular, has been widely applied to the analysis of many standard questions in economic choice under uncertainty.

Increasing preference

P3

Generalized Expected Utility Analysis An alternative approach to nonexpected utility, which yields a direct extension of most of the basic results

0

Figure 3 Common ratio effect and indifference curves that fan out

7 suggest that indifference curves in the probability triangle fan out. Further descriptions of these and other violations of the expected utility hypothesis can be found in [6, 18, 21, 23, 24, 34, 35, 38],

Increasing preference

b'1

Nonexpected Utility Functional Forms Researchers have responded to the above types of departures from linearity in the probabilities in two manners. The first consists of replacing the expected utility preference function VEU (-) by some more general functional form for V (P) = V (xt, Pt; ... ; X 11 , p 11 ), such as the forms in the Table below.

0

Figure 4 Couuuon ratio etlect for negative payotls and inditlerence cnrves that fan out

ｉＺ［Ｇｾ

'Prospect theory' ＨｌＺ［Ｇｾ

f

Moments of utility

Ｑ＠ v(xi)p;, ｉＺ［Ｇｾ ｉＺ［Ｇｾ

Weighted utility

ｉＺ［Ｇｾ

ｉＺ［Ｇｾｬ＠

Dual expected utility Rank-dependent expected utility Quadratic in probabilities Ordinal independence

Ｑ＠ v(x;):n:(p;)

ｉＺ［Ｇｾ

Ｑ＠

X; [

G

ＨｉＺｾ＠

v(x;) [ G ( ｉＺＧｩｾｬ＠

[11, 12, 18]

Ｑ＠ v(x;) 2p;, .. .)

Ｑ＠ v(x;)p; Ｑ＠ T(x;)p;

I PJ)- G ＨＲＺｾ

Pi) - G ( ｉＺＧｩｾ｜＠

[17]

Ｑ

[7]

Ｑ＠

PJ)]

Pi)]

[39]

[28]

[8]

[33]

Chapter 14. Nonexpected Utility Theory Nonexpected Utility Theory of expected utility analysis, applies calculus to a general 'smooth' nonexpected utility preference function V(P) = V(x1, PI; ... ;xn,JJn), concentrating in particular on its probability derivatives av (PJ I a prob (x ). This approach is based on the observation that for the expected utility function VEu (x 1, PI; ... ; x,, Pn) = U(xt)p 1 + · · · + U(x,)pn, the value U(x) can be interpreted as the coefficient of prob(x), and that many theorems relating to a linear function's ｣ｯｾｦﾭ ficients continue to hold when generalized to a nonlinear function's derivatives. By adopting the notation U(x;P) = iJV(P)/iJ prob(x) for the probability derivative, the above expected utility characterizations of first -order stochastic dominance preference, risk aversion, and comparative risk aversion generalize to any smooth preference function V (P) in the following manner: V(·) will exhibit first-order stochastic dominance preference if and only if at each lottery P, U (x ;P) is an increasing function of x

V(·) will exhibit risk aversion (an aversion to all meanpreserving increases in risk) if and only if at each lottery P, U (x ;P) is a concave function of x V*(-) will be at least as risk averse as V(·) if and only if at each lottery P, U*(x;P) is a concave transformation of U (x ;P) It can be shown that the indifference curves of a smooth preference function V (·) will fan out in the probability triangle and its multiple-outcome generalizations if and only if U (x;P') is a concave transformation of U (x;P) whenever P* first order stochastically dominates P; see [3, 9, 19, 22, 23, 37] for the development and further applications of generalized expected utility analysis.

Applications to Insurance The above analysis also permits the extension of many expected utility-based results in the theory of insurance to general smooth nonexpected utility preferences. The key step in this extension consists of observing that the formula for the expected utility outcome derivative -upon which most insurance results are based - is linked to its probability coefficients via the relationship iJVEu(P)jiJx = prob(x)·U'(x), and that this relationship generalizes to any smooth nonexpected utility preference function V (·),where it takes the form iJV(P)jiJx = prob(x)·U'(x;P) (where U'(x)

257 5

and U'(x;P) denote derivatives with respect to the variable x ). One important expected utility result that does not directly extend is the equivalence of risk aversion to the property of outcome-convexity, which states that for any set of probabilities (PI, ... , p, ), if the individual is indifferent between the lotteries (x1, P1: .. . ;x,, p,) and , p1; ... ;x,;, p,) then they will strictly prefer any outcome mixture of the form (f3xt +(I- f3)x1, p1; ... ［ｦＳｸＬｾ＠ +(I- f3)x1, Pnl (for 0 < f3 < I). Under nonexpected utility, outcomeconvexity still implies risk aversion but is no longer implied by it, so for some nonexpected utility insurance results, both risk aversion and outcome-convexity must be explicitly specified. (Nevertheless, the hypothesis that the individual is risk averse and outcomeconvex is still wealcer than the hypothesis that they are a risk averse expected utility maximizer.) One standard insurance problem - termed the demand for coinsurance - involves an individual with initial wealth w who faces a random loss i. with probability distribution (1 1. p 1; ... ; 1,, p,) (/; :;. 0), who can insure fraction y of this loss by paying a premium of .Ay E[i.], where E[i.] is the expected loss and the load factor A equals or exceeds unity. The individual thus selects the most preferred option from the family of random variables {w - ( 1 y)i.- AyEliJIO :5: y :5: 1]. Another standard problem- termed the demand for deductible insurance involves fully covering any loss over some amount d, so the individual receives a payment of max{£d, 0), and paying a premium of AElmax{i.- d, O]j. The individual thus selects the most preferred option from the family of random variables {w- min{£, d)- AEfmax{i.- d, OJ lid:;. 0]. In each case, insurance is said to be actuarially }i1ir if the load factor A equals unity, and actuarially unj(zir if A exceeds unity. Even without the assumption of outcomeconvexity, the following results from the classic expected utility-based theory of insurance can be shown to extend to any risk averse smooth nonexpected utility preference function V (-): V ( ·) will purchase complete coinsurance (will choose to set the fraction y equal to one) if and only if such insurance is actuarially fair V 0 will purchase complete deductible insurance (will choose to set the deductible d equal to zero) if and only such insurance is acluarially fair If V*(-) is at least as risk averse as V (·),then whenever they each have the same initial wealth and

258

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MJ Machina Nonexpected Utility Theory

face the same random loss, V '(-) will purchase at least as much coinsurance as V ( ·) If V*(-) is at least as risk averse as V(·), then whenever they each have the same initial wealth and face the same random loss, V * ( ·) choose at least as low a deductible level as V(·) If we do assume outcome-convexity, many additional results in the theory of insurance and optimal risk-sharing can be similarly extended from expected utility to smooth nonexpected utility preferences; see [25] for the above and additional extensions, and [10, 14, 16, 20, 32, 40] as well as the papers in [15], for additional results in the theory of insurance under nonexpected utility risk preferences.

References [I]

[2]

[3]

[4] [5]

[6]

[7]

[8] [9]

[10]

[II] [ 12]

Allais, M. (1953). Le Comportement de !'Homme Rationnel devant le Risque, Critique des Postulats et Axiomes de I'Rcole Am6ricaine, Rconometrica 21, 503-546. Allais, M. & Hagen, 0., eds (1979). Expected Utility Hypotheses and the Allais Paradox. D. Reidel Publishing, Dordrecht. Allen, B. (1987). Smooth preferences and the local expected utility hypothesis, Journal of Economic Themy 41. 340-355. Arrow, K. (1963). Comment, Review of' Economics and Statistics 45, (Supplement), 24-27. Bernoulli, D. (1738). Specimen Theoriae Novae de Mensura Sortis, Commentarii Academiae Scientiarum lmperialis Petropolitanae V, 175-192; English translation (1954). Exposition of a New theory on the measurement of risk, liconometrica 22, 23-36. Camerer, C. (1989). An experimental test of several generalized utility theories, Journal r4Risk and Uncertainty 2, 61-104. Chew, S. (1983). A generalization of the quasilincar mean with applications to the measurement of income inequality and decision theory resolving the allais paradox, Econometrica 51, 1065-1092. Chew, S., Epstein, L. & Segal, U. (1991). Mixture symmetry and quadratic utility, Econometrica 59, 139-163. Chew, S., Epstein, L. & Zilcha, I. ( 1988). A correspondence theorem between expected utility and smooth 'theory 46, 186-193. utility, Journal ｯｬｾＧ｣ｮｭｩ＠ Doherty, N. & Eeckhoudt, L. (1995). Optimal insurance without expected utility: the dual theory and the linearity of insurance contracts, Journal ｲｾｦＢ＠ Ri.•Jk and Uncertainty 10, 157-179. Edwards, W. (1955). The prediction of decisions among bets, Journal ofExperimental Psychology 50, 201-214. Edwards, W. ( 1962). Subjective probabilities inferred from decisions, Psychological Revinv 69, I 09-135.

[13]

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[311 [32j

Friedman, M. & Savage. L. (1948). The utility analysis of choices involving risk, Journal ofPoUt;caf Economy 56, 279-304. Gollier, C. (2000). Optimal insurance design: What can we do without expected utility, in Handbook £?!" Insurance, G. Dionne, ed., Kluwer Academic Publishers, Boston. Gallier, C. & Machina, M., eds (1995). Non-Expected UtWty and R;sk Management, Kluwer Academic Publishers, Boston. Gallier, C. & Schlesinger, H. (1996). Arrow's theorem on the optimality of deductibles: a stochastic dominance approach, Economic Theory 7, 359-363. Hagen, 0. (1979). Towards a Positive Theory of Preferences Under Risk, in [2]. Kahneman, D. & Tversky, A. (1979). Prospect theory: an analysis of decision under risk, Econometrica 47, 263-291. Karni, E. (1989). Generalized expected utility analysis of multivariate risk aversion, International Economic Review 30, 297-305. Konrad, K. & Skaperdas, S. (1993). Self-insurance and self-protection: a nonexpected utility analysis, Geneva Papers on Risk and Insurance Theor.v 18, 131-146. MacCrimmon, K. & Larsson, S. (1979). Utility Theory: Axioms Versus "Paradoxes", in [2]. Machina, M. (1982). "Expected Utility" analysis without the independence axiom, Econometrica 50, 277-323. Machina, M. (1983). Generalized expected utility analysis and the nature of observed violations of the independence axiom, in Foundations of Utility and Ri.•Jk Theor.v with Applications, B. Stigum & F. Wenstop, eds, D. Reidel Publishing, Dordreeht. Machina, M. (1987). Choice under uncertainty: problems solved and unsolved, Journal of Economic Perspectives 1, 121-154. Machina, M. (1995). Non-expected utility and the robustness of the classical insurance paradigm, Geneva Papers on Risk and Insurance Theory 20, 9-50. Markowitz, H. (1952). The utility of wealth, Journal of Political Economy 60, 151-158. Pratt, J. (1964). Risk aversion in the small and in the large, Econometrica 32, 122-136. Quiggin, .1. (1982). A theory of anticipated utility. Journal ql Economic Behavior and Organization 3, 323-343. Rothschild, M. & Stiglitz, J. ( 1970). Increasing risk: I. A definition, Journal of Economic Theory 2, 225-243. Rothschild, M. & Stiglitz, J. (1971 ). Increasing risk: II. Its economic consequences, Journal of Economic Theory 3, 66-84. Savage, L. (1954). The Foundations of Statistics, John Wiley & Sons, New York. Schlesinger, H. (1997). Insurance demand without the expected utility paradigm, Journal of Risk and Insurance 64, 19-39.

Chapter 14. Nonexpected Utility Theory

259

Nonexpected Utility Theory [33]

[34]

[351

[36]

[37]

Segal, U. (1984). Nonlinear Decision Weights with the Independence Axiom, Manuscript, University of California, Los Angeles. Starmer, C. (2000). Developments in non-expected utility theory: the hunt for a descriptive theory of choice under risk, Journal o.f EconomJc Literature 38, 332-382. Sugden, R. (1986). New developments in the theory of choice under uncertainty, Bulletin of Economic Research

38. 1-24. von Neumann, J. & Morgenstern, 0. (1947). Theory of' Games and Economic Behavior, 2nd Edition, Princeton University Press, Princeton. Wang, T. (1993). Lp-Frechet differentiable preference and "local utility" analysis. Journal of Economic Theon-' 61, 139-159.

[38]

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Weber, M. & Camerer, C. (1987). Recent developments in modeling preferences under risk, OR Spektrum 9, 129-151. Yaari, M. (1987). The dual theory of choice under risk, Hconometrica 55, 95- 115. Young, V. & Browne, M. (2000). Equilibrium in competitive insurance markets under adverse selection and Yaari's dual theory of risk, Geneva Papers on Risk and Insurance Theory 25, 141-157.

(See also Risk Measures) MARK

J.

MACHINA

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Chapter 15 Science,Vol.I85,No.4157, 1974,1124-1131

Judgment under Uncertainty: Heuristics and Biases Biases in judgments reveal some heuristics of thinking under uncertainty. Amos Tversky and Daniel Kahneman

Many decisions are based on beliefs concerning the likelihood of uncertain events such as the outcome of an election, the guilt of a defendant, or the future value of the dollar. These beliefs are usually expressed in statements such as "I think that . . . ," "chances are . . . , " "it is unlikely that . . . ," and so forth. Occasionally, beliefs concerning uncertain events are expressed in numerical form as odds or subjective probabilities. What determines such beliefs? How do people assess the probability of an uncertain event or the value of an uncertain quantity? This article shows that people rely on a limited number of heuristic principles which reduce the complex tasks of assessing probabilities and predicting values to simpler judgmental operations. In general, these heuristics are quite useful, but sometimes they lead to severe and ｾｹｳｴ･ｭ｡ｩ｣＠ errors. The subjective assessment of probability resembles the subjective assessment of physical quantities such as distance or size. These judgments are all based on data of limited validity, which are processed according to heuristic rules. For example, the apparent distance of an object is determined !n part by its clarity. The more sharply the object is seen, the closer it appears to be. This rule has some validity, Ｍ｢･｣｡ｵｳｾ＠ in any given scene the more distant objects are seen less sharply than nearer objects. However, the reliance on this rule leads to systematic errors in the estimation of distance. Specifically, distances are often overestimated when visibility is poor because the contours of objects are blurred. On the other hand, distances are often underestiThe authors are members of the department of psychology at the Hebrew University, Jerusalem, Tsrael.

mated when visibility is good because the objects are seen sharply. Thus, the reliance on clarity as an indication of distance leads to common biases. Such biases are also found in the intuitive judgment of probability. This article describes three heuristics that are employed to assess probabilities and to predict values. Biases to which these heuristics lead are enumerated, and the applied and theoretical implications of these observations are discussed.

Representativeness Many of the probabilistic questions with which people are concerned belong to one of the following types: What is the probability that object A belongs to class B? What is the probability that event A originates from process B? What is the probability that process B will generate event A? In answering such questions, people typically rely on the representativeness heuristic, in which probabilities are evaluated by the degree to which A is representative of B, that is, by the degree to which A resembles B. For example, when A is highly representative of B, the probability that A originates from B is judged to be high. On the other hand, if A is not similar to B, the probability that A originates from B is judged to be low. For an illustration of judgment by representativeness, consider an individual who has been described by a former neighbor as follows: "Steve is very shy and withdrawn, invariably helpful, but with little interest in people, or in the world of reality. A meek and tidy soul, he has a need for order and structure, and a passion for detail." How do people assess the probability that Steve is engaged in a particular

occupation from a list of possibilities (for example, farmer, salesman, airline pilot, librarian, or physician)? How do people order these occupations from most to least likely? In the representativeness heuristic, the probability that Steve is a librarian, for example, is assessed by the degree to which he is representative of, or similar to, the stereotype of a librarian. Indeed, research with problems of this type has shown that people order the occupations by probability and by similarity in exactly the same way (J). This approach to the judgment of probability leads to serious errors, because similarity, or representativeness, is not mftuenced by several factors that should affect judgments of probability. Insensitivity to prior probability of outcomes. One of the factors that have no effect on representativeness but should have a major effect on probability is the prior probability, or base-rate frequency, of the outcomes. In the case of Steve, for example, the fact that there are many more farmers than librarians in the population should enter into any reasonable estimate of the probability that Steve is a librarian rather than a farmer. Considerations of base-rate frequency, however, do not affect the similarity of Steve to the stereotypes of librarians and farmers. If people evaluate probability by representativeness, therefore, prior probabilities will be neglected. This hypothesis was tested in an experiment where prior probabilities were manipulated (1). Subjects were shown brief personality descriptions of several individuals, allegedly sampled at random from a group of 100 professionals-engineers and lawyers. The subjects were asked to assess, for each description, the probability that it belonged to an engineer rather than to a lawyer. In one experimental condition, subjects were told that the group from which the descriptions had been drawn consisted of 70 engineers and 30 lawyers. In another condition, subjects were told that the group consisted of 30 engineers and 70 lawyers. The odds that any particular description belongs to an engineer rather than to a lawyer should be higher in the first condition, where there is a majority of engineers, than in the second condition, where there is a majority of lawyers. Specifically, it can be shown by applying Bayes' rule that the ratio of these odds should be (.7/.3)2, or 5.44, for each description. In a sharp violation of Bayes' rule, the subjects in the two conditions produced essenSCIENCE, VOL, 185

1124

261

262

A Tversky and D Kahneman

tially the same probability judgments. Apparently, subjects evaluated the likelihood that a particular description be-

sample size. Indeed, when subjects assessed the distributions of average height for samples of various sizes,

longed to an engineer rather than to a

they produced identical distributions. For example, the probability of obtaining an average height greater than 6 feet was assigned the same value for samples of 1000, 100, and 10 men (2).

lawyer by the degree to which this description was representative of the two stereotypes, with little or no regard

for the prior probabilities of the categories.

The subjects used prior probabilities correctly when they had no other information. In the absence of a personality sketch, they judged the probability that an unknown individual is an engineer

to be .7 and .3, respectively, in the two base-rate conditions. However,

prior

probabilities were effectively ignored when a description was introduced, even when thjs description was totally uninformative. The responses to the

following description illustrate this phenomenon: Dick is a 30 year old man. He is married with no children. A man of high ability and high motivation, he promises

to be quite successful in his field. He is well liked by his colleagues. This description was intended to convey no information relevant to the question of whether Dick is an engineer or a

lawyer. Consequently, the probability that Dick is an engineer should equal the proportion of engineers in the group, as if no description had been given. The subjects, however, judged the probability of Dick being an engineer to be .5 regardless of whether the stated proportion of engineers in the

group was .7 or .3. Evidently, people respond differently when given no evidence and when given worthless evidence. When no specific evidence is given, prior probabilities are properly utilized; when worthless evidence is given, prior probabilities are ignored (I). Insensitivity to sample size. To eval-

uate the probability of obtaining a particular result in a sample drawn from

a specified population, people typically apply the representativeness heuristic.

That is, they assess the likelihood of a sample result, for example, that the average height in a random sample of ten men will be 6 feet (180 centimeters), by the similarity of this result to the corresponding parameter (that is, to the average height in the population of men). The similarity of a sample statistic to a population parameter does not depend on the size of the sample. Consequently, if probabilities are assessed by representativeness, then the judged probability of a sample statistic will be essentially independent of 21 SEPTEMBER 1974

In this problem, the correct posterior odds are 8 to 1 for the 4 : 1 sample and 16 to 1 for the 12 : 8 sample, assuming equal prior probabilities. However, most people feel that the first sample provides much stronger evidence

for the hypothesis that the urn is predominantly red, because the proportion

of red balls is larger in the first than in

Moreover, subjects failed -to appreciate the role of sample size even when it was emphasized in the formulation of

judgments are dominated by the sample

the problem. Consider the following

proportion and are essentially unaffected

question: A certain town is served by two hospitals. In the larger hospital about 45 babies are born each day, and in the smaller hospital about 15 babies are born each day. As you know, about 50 percent of all babies are boys. However, the exact percentage varies from day to day. Some· times :i.t may be higher than 50 percent, sometimes lower. For a period of 1 year, each hospital recorded the days on which more than 60 percent of the babies born were boys.

Which hospital do you think recorded more such days? ｾ＠ ｾ＠

The larger hospital (21) 'Jlhe smaller hospital (21)

,.. About the same (that is, within 5

percent of each other) (53) The values in parentheses are the ｮｵｭｾ＠ ber of undergraduate students who chose each answer.

Most subjects judged the probability of obtaining more than 60 percent boys to be the same in the small and in the large hospital, presumably because these events are described by the same statistic and are therefore equally representative of the general population. In contrast, sampling theory entails that the expected number of days on which more than 60 percent of the babies are boys is much greater in the small hospital than in the large one, because a large sample is less likely to stray from 50 percent. This fundamental notion of statistics is evidently not part of people's repertoire of intuitions. A similar insensitivity to sample size

has been reported in judgments of posterior probability, that is, of the probability that a sample has been drawn from one population rather than from another. Consider the following example: Imagine an urn fiiled with ｢｡ｬｳｾ＠ of which % are of one color and lh of another. One individual has drawn 5 balls .from the urn, and found that 4 were red and 1 was white Another individual has drawn 20 balls and found that 12 were red and 8 were white. Which of the two individuals should feel more confident that the urn contains 2h red balls and V3 white balls, rather than the opposite? What odds should each individual give?

the second sample. Here again, intuitive

by the size of the sample, which plays a crucial role in the determination of

the actual posterior odds (2). In addition, intuitive estimates of posterior odds are far less extreme than the cor· rect values. The underestimation of the impact of evidence has been observed repeatedly in problems of this type (3, 4). It has been labeled "conservatism." Misconceptions of chance. People exM pect that a sequence of events generated

by a random process will represent the essential characteristics of that process even when the sequence is short. In considering tosses of a coin for heads

or tails, for example, people regard the sequence H-T·H-T-T-H to be more likely than the sequence H-H-H-T-T-T, which does not appear random, and

also more likely than the sequence H-HH-H-T-H, which does not represent the fairness of the coin (2). Thus, people expect that the essential characteristics

of the process will be represented, not only globally in the entire sequence, but also locally in each of its parts. A locally representative sequence, how· ever, deviates systematically from chance expectation: it contains too many al· ternations and too few runs. Another

consequence of the belief in local representativeness is the well-known gam·

bier's fallacy. After observing a long run of red on the roulette wheel. for example, most people erroneously be-

lieve that black is now due, presumably because the occurrence of black will result in a more representative sequence

than the occurrence of an additional red. Chance is commonly viewed as a self-correcting -process in which a devi· ation in one direction induces a deviation in the opposite direction to restore the equilibrium. In fact. deviations are not "corrected" as a chance process

unfolds, they are merely diluted. Misconceptions of chance are not

limited to naive subjects. A study of the statistical intuitions of experienced

research psychologists (5) revealed a lingering belief in what may be called the "law of small numbers," according

to which even small samples are highly 1125

Chapter 15. Judgment Under Uncertainty: Heuristics and Biases representative of the populations from which they are drawn. The responses of these investigators reflected the expectation that a valid hypothesis about a population will be represented by a statistically significant result in a sample-with little regard for its size. As a consequence, the researchers put too much faith in the results of small samples and grossly overestimated the replicability of such results. In the actual conduct of research, this bias leads to the selection of samples of inadequate size and to overinterpretation of findings. Insensitivity to predictability. People are sometimes called upon to make such numerical predictions as the future value of a stock, the demand for a commodity, or the outcome of a football game. Such predictions are often made by representativeness. For example, suppose one is given a description of a company and is asked to predict its future profit. If the description of the company is very favorable, a very high profit will appear most representative of that description; if the description is mediocre, a mediocre performance will appear most representative. The degree to which the description is favorable is unaffected by the reliability of that description or by the degree to which it permits accurate prediction. Hence, if people predict solely in terms of the favorableness of the description, their predictions will be insensitive to the reliability of the evidence and to the expected accuracy of the prediction. This mode of judgment violates the normative statistical theory in which the extremeness and the range of predictions are controlled by considerations of predictability. When predictability is nil, the same prediction should be made in all cases. For example, if the descriptions of companies provide no information relevant to profit, then the same value (such as average profit) should be predicted for all companies. If predictability is perfect, of course, the values predicted will match the actual values and the range of predictions will equal the range of outcomes. In general, the higher the predictability, the wider the range of predicted values. Several studies of numerical prediction have demonstrated that intuitive predictions violate this rule, and that subjects show little or no regard for considerations of predictability (I). In one of these studies, subjeczts were presented with several paragraphs, each describing the performance of a stu1126

dent teacher during a particular practice lesson. Some subjects were asked to evaluate the quality of the lesson described in the paragraph in percentile scores, relative to a specified population. Other subjects were asked to predict, also in percentile scores, the standing of each student teacher 5 years after the practice lesson. The judgments made under the two conditions were identical. That is, the prediction of a remote criterion (success of a teacher after 5 years) was identical to the evaluation of the information on which the prediction was based (the quality of the practice lesson). The students who made these predictions were undoubtedly aware of the limited predictability of teaching competence on the basis of a single trial lesson 5 years earlier; nevertheless, their predictions were as extreme as their evaluations. The illusion of validity. As we have seen, people often predict by selecting the outcome (for example, an occupation) that is most representative of the input (for example, the description of a person). The confidence they have in their prediction depends primarily on the degree of representativeness (that is, on the quality of the match between the selected outcome and the input) .with little or no regard for the factors that limit predictive accuracy. Thus, people express great confidence in ·the prediction that a person is a librarian when given a description of his personality which matches the stereotype of librarians, even if the description is scanty, unreliable, or outdated. The unwarranted confidence which is produced by a good fit between the predicted outcome and the input information may be called the illusion of validity. This illusion persists even when the judge is aware of the factors that limit the accuracy of his predictions. It is a common observation that psychologists who conduct selection interviews often experience considerable confidence in their predictions, even when they know of the vast literature that shows selection interviews to be highly fallible. The continued reliance on the clinical interview for selection, despite repeated demonstrations of its inadequacy, amply attests to the strength of this effect. The internal consistency of a pattern of inputs is a major determinant of one's confidence in predictions based on these inputs. For example, people express more confidence in predicting the final grade-point average of a student

263

whose first-year record consists entirely of B's than in predicting the gradepoint average of a student whose firstyear record includes many A's and C's. Highly consistent patterns are most often observed when the input variables are highly redundant or correlated. Hence, people tend to have great confidence in prediciions based on redundant input variables. However, an elementary result in the statistics of correlation asserts that, given input variables of stated validity, a prediction based on several such inputs can achieve higher accuracy when they are independent of each other than when they are redundant or correlated. Thus, redundancy among inputs decreases accuracy even as it increases confidence, and people are often confident in predictions that are quite likely to be off the mark (I) . Misconceptions of regression. Suppose a large group of children has been examined on two equivalent versions of an aptitude test. If one selects ten children from among those who did best on one of the two versions, he will usually find their performance on the second version to be somewhat disappointing. Conversely, if one selects ten children from among those who did worst on one version, they will be found, on the average, to do somewhat better on the other version. More generally, consider two variables X and Y which have the same distribution. If one selects individuals whose average X score deviates from the mean of X by k units, then the average of their Y scores will usually deviate from the mean of Y by less than k units. These observations illustrate a general phenomenon known as regression toward the mean, which was first documented by Galton more than 100 years ago. In the normal course of life, one encounters many instances of regression toward the mean, in the comparison of the height of fathers and sons, of the intelligence of husbands and wives. or of the performance of individuals on consecutive examinations. Nevertheless, people do not develop correct intuitions about this phenomenon. First, they do not expect regression in many contexts where it is bound to occur. Second, when they recognize the occurrence of regression, they often invent spurious causal explanations for it (I). We suggest that the phenomenon of regression remains elusive because it is incompatible with the belief that the predicted outcome should be maximally SCIENCE, VOL. 185

264 representative of the input, and, hence, that the value of the outcome variable should be as extreme as the value of the input variable. The failure to recognize the import of regression can have pernicious consequences, as illustrated by the following observation ( 1). In a discussion of flight training, experienced instructors noted that praise for an exceptionally smooth landing is typically followed by a poorer landing on the next try, while harsh criticism after a rough landing is usually followed by an improvement on the next try. The instructors concluded that verbal rewards are detrimental to learning, while verbal punishments are beneficial, contrary to accepted psychological doctrine. This conclusion is unwarranted because of the presence of regression toward ,the mean. As in other cases of repeated examination, an improvement will ｵｳｾ＠ ally follow a poor performance and a deterioration will usually follow an outstanding performance, even if the instructor does not respond to the trainee's achievement on the first attempt. Because the instructors had praised their trainees after good landings and admonished them after poor ones, they reached the erroneous and potentially harmful conclusion that punishment is more effective than reward. Thus, the failure to understand the effect of regression leads one to overestimate the effectiveness of punishment and to underestimate the effectiveness of reward. In social interaction, as well as in training, rewards are typically administered when performance is good, and punishments are typically administered when performance is poor. .By regression alone, therefore, behavior is most likely to improve after punishment and most likely to deteriorate after reward. Consequently, the human condition is such that, by chance alone, one is most often rewarded for punishing others and most often punished for rewarding them. People are generally not aware of this contingency. In fact, the elusive role of regression in determining the apparent consequences of reward and punishment seems to have escaped the notice of students of this area.

Availability

There are situations in which people assess the frequency of a ciass or the probability of an event by the ease with 27 SEPTEMBER 1974

A Tversky and D Kahneman which instances or occurrences can be brought to mind. For example, one may assess the risk of heart attack among middle-aged people by recalling such occurrences among one's acquaintances. Similarly, one may evaluate the probability that a given business venture will fail by imagining various difficulties it could encounter. This judgmental heuristic is called availability. Availability is a useful clue for assessing frequency or probability, because instances of large classes are usually recalled better and faster than instances of less frequent classes. However, availability is affected by factors other than frequency and probability. Consequently, the reliance on availability leads to predictable biases, some of which are illustrated below. Biases due to the retrievability of ｩｮｾ＠ stances. When the size of a class is judged by the availability of its instances, a class whose instances are easily retrieved will appear more numerous than a class of equal frequency whose instances are less retrievable. In an elementary demonstration of this ･ｦｾ＠ feet, subjects heard a list of well-known personalities of both sexes and were subsequently asked to judge whether the list contained more names of men than of women. Different lists were presented to different groups of subjects. In some of the lists the men were relatively more famous than the women, and in others the women were relatively more famous than the men. In each of the lists, the subjects erroneously judged that the class (sex) that had the more famous personalities was the more numerous (6).

In addition to familiarity, there are other factors, such as salience, which affect the retrievability of instances. For example, the impact of seeing a house burning on the subjective probability of such accidents is probably greater than the impact of reading about a fire in the local paper. Furthermore, recent occurrences are likely to be relatively more available than earlier occurrences. It is a common experience that the subjective probability of traffic accidents rises temporarily when one sees a car overturned by the side of the road. Biases due to the effectiveness of a search set. Suppose one samples a word (of three letters or more) at random from an English text. Is it more likely that the word starts with r or that r is the third letter? People approach this problem by recalling words that

begin with r (road) and words that have r in the third position (car) and assess the relative frequency by the ease with which words of the two types come to mind. Because it is much easier to search for words by their first letter than by their third letter, most people judge words that begin with a given consonant to be more numerous than words in which the- same consonant apM pears in the third position. They do so even for consonants, such as r or k, that are more frequent in the third position than in the first (6). Different tasks elicit different search sets. For example, suppose you are asked to rate the frequency with which abstract words (thought, love) and concrete words (door, water) appear in written English. A natural way to answer this question is to search for contexts in which the word could appear. It seems easier to think of contexts in which an abstract concept is mentioned (love in love stories) than to think of contexts in which a concrete word (such as door) is mentioned. If the frequency of words is judged by the availability of the contexts in which they appear, abstract words will be judged as relatively more numerous than concrete words. This bias has been observed in a recent study (7) which showed that the judged frequency of occurrence of abstract words was much higher than that of concrete words, equated in objective frequency. Abstract words were also judged to appear in a much greater variety of contexts than concrete words. Biases of imaginability. Sometimes one has to assess the frequency of a class whose instances are not stored in memory but can be generated according to a given rule. In such situations, one typically generates several instances and evaluates frequency or probability by the ease with which {he relevant instances can be constructed. However, the ease of constructing instances does not always reflect their actual frequency, and this mode of evaluation is prone to biases. To illustrate, consider a group of I 0 people who form committees of k members, 2 :=:;; k ::;; 8. How many different committees of k members can be formed? The correct answer to this problem is given by the binomial coefficient (lg) which reaches a maximum of 252 for k = 5. Clearly, the number of committees of k members equals the number of committees of (10- k) members, because any committee of k 1127

Chapter 15. Judgment Under Uncertainty: Heuristics and Biases members defines a unique group of (10- k) nonmembers. One way to answer this question without computation is to mentally construct committees of k members and to evaluate ,their number by the ease with which they come to mind. Committees of few members, say 2, are more available than committees of many members, say 8. The simplest scheme for the construction of committees is a partition of the group into disjoint sets. One readily sees that it is easy to construct five disjoint committees of 2 members, while it is impossible to generate even two disjoint committees of 8 members. Consequently, if frequency is assessed by imaginability, or by availability for construction, the small commiltces will appear more numerous than larger committees, in contrast to the correct bell-shaped function. Indeed, when naive subjects were asked to estimate the number of distinct committees -of various sizes, their estimates were a decreasing monotonic function of committee size (6). For example, the median estimate of the number of committees of 2 members was 70, while the estimate for committees of 8 members was 20 (the correct answer is 45 in both cases). Tmaginability plays an important role in the evaluation of probabilities in reallife situations. The risk involved in an adventurous expedilion, for example, is evaluated by imagining contingencies with which the expedition is not equipped to cope. If many such difficulties are vividly portrayed, the expedition can be made to appear exceedingly dangerous, although the case with which disasters are imagined need not reflect their actual likelihood. Conversely, the risk involved in an undertaking may be grossly underestimated if some possible dangers are either difficult to conceive of, or simply do not come to mind. Illusory correlation. Chapman and Chapman (8) have described an interesting bias in the judgment of the frequency with which two events co-occur. They presented naive judges with information concerning several hypothetical mental patients. The data for each patient consisted of a clinical diagnosis and a drawing of a person made by the patient. Later the judges estimated the frequency with which each diagnosis (such as paranoia or suspiciousness) had been accompanied by various features of the drawing (such as peculiar eyes). The subjects markedly overestimated the frequency of co-occurrence of H28

natural associates, such as suspiciOusness and peculiar eyes. This effect was labeled illusory correlation. In their erroneous judgments of the data to which they had been exposed, naive subjects "rediscovered" much of the common, but unfounded, clinical lore concerning the interpretation of the draw-aperson test. The illusory correlation effect was extremely resistant to contradictory data. It persisted even when the correlation between symptom and diagnosis was actually negative, and it prevented the judges from detecting relationships that were in fact present. Availability provides a natural account for the illusory-correlation effect. The judgment of how frequently two events co-occur could be based on the strength of the associative bond between them. When the association is strong, one is likely to conclude that the events have been frequently paired. Consequently, strong associates will be judged to have occurred together frequently. According to this view, the illusory correlation between suspiciousness and peculiar drawing of the eyes, for example, is due to the fact that suspiciousness is more readily associated with the eyes than with any other part of the body. Lifelong experience has taught us that, in general, instances of large classes are recalled better and faster than instances of less frequent classes; that likely occurrences are easier to imagine than unlikely ones; and that the associative connections between events are strengthened when the events frequently co-occur. As a result, man has at his disposal a procedure (the availability heuristic) for estimating the numerosity of a class, the likelihood of an event, or the frequency of co-occurrences, by the ease with which the relevant mental operations of retrieval, construction, or association can be performed. However, as the preceding examples have demonstrated, this valuable estimation procedure results in systematic errors.

Adjustment and Anchoring

In many si-tuations, people make estimates by starting from an initial value that is adjusted to yield the final answer. The initial value, or starting point, may be suggested by the formulation of the problem, or it may be the result of a partial computation. In either case, adjustments arc typically insufficient (4).

265

That is, different starting points yield different estimates, which are biased toward the initial values. We call this phenomenon anchoring. Insufficient adjustment. In a demonstration of the anchoring effect, subjects were asked to estimate various quantities, stated in percentages (for example, the percentage of African countries in the United Nations). For each quantity, a number between 0 and 100 was determined by spinning a wheel of fortune in the subjects' presence. The subjects were instructed to indicate first whether that number was higher or lower than the value of the quantity, and then to estimate the value of the quantity by moving upward or downward from the given number. Different groups were given different numbers for each quantity, and these arbitrary numbers had a marked effect on estimates. For example, the median estimates of the percentage of African countries in the United Nations were 25 and 45 for groups that received 10 and 65, respectively, as starting points. Payoffs for accuracy did not reduce the anchoring effect. Anchoring occurs not only when the starting point is given to the subject, but also when the subject bases his estimate Dn the result of some incomplete computation. A study of intuitive numerical estimation illustrates this effect. Two groups of high school students estimated, within 5 seconds, a numerical expression that was written on the blackboard. One group estimated the product 8X7X6X5X4X3X2Xl

while another product

group

estimated the

IX2>O for some u in U. In this section, U is the class of all non-decreasing functions (assumed to have finite values for any finite value of x). 1 The variables X and Y 2 are defined here as the money payoffs of a given venture, which are additions (or reductions) to the individual's (constant) wealth, so that no restriction to positive values of X or Y is necessary. (The utility function u depends, of course, on the individual's initial wealth position). A sufficient condition for FDG (or X D Y) is an efficiency criterion. 3 The set of all risks which are not dominated by another risk according to the given criterion, is an efficient set. The weaker the sufficient condition, the smaller the efficient set. The minimal efficient set E is generated when the criterion is optimal, that is, when the condition for FDG is both sufficient and necessary. If the class of admissible utility functions is restricted to a proper subset of U, the minimal efficient set may be reduced to a subset of E, and the corresponding optimal efficiency criterion may be weakened. Before proceeding to state and prove the general criterion, we need the following: Lemma 1. Let G, F, be two (cumulative) distributions, and u(x) a non-decreasing function, with finite values for any finite x; then

t1Eu = EFu(x)-Eau(x) = Proof By definition, 5

t1Eu =

=

[f

f

d(u.F)-

[G(x)-F(x)]du(x). 4

f f f [J f

t1Eu =

Integrating (1) by parts gives:

f

udF-

Fdu]-

d[u .(F-G)]+

...(1)

udG.

d(u.G)-

f

Gdu]

(G-F)du.

In order to show that the first term on the right vanishes, we define a sequence of functions u.(x), converging to u(x): u( -n) for x< -n { u.(x) = u(x) -n ｾ＠ x ｾ＠ n u(n)

x>n.

I Quirk and Saposnik [18] proved a similar criterion for the discrete, finite case, and assumed a strictly monotone utility. They also sketched a proof for the (pure) continuous case, requiring that u(x) be bounded and piecewise differentiable (p. 144). All these requirements are not necessary. Similar comments apply to Hadar and Russel [7), who assumed u(x) to be twice continuously differentiable, and restricted to a finite range. Cf. also Hammond [8]. 2 In much of the following discussion, we do not distinguish in notation between the variables X and Y, restricting our attention to the two distributions, F(x) and G(x). 3 An additional requirement is the transitivity of the sufficient condition; but this is obviously assured by the given definition of dominance, since EFu-EGu ｾ＠ 0 and EGu-EHu ｾ＠ 0, imply EFu-EHu ｾ＠ 0, andFDH. 4 The integrals throughout are Stieltjes-Lebesgues integrals, ranging on all real values of x, unless specified otherwise; cf. Cramer [4), p. 62. 5 The arguments in the functions appearing in integrals are omitted in cases where no misunderstanding should arise.

Chapter 17. The Efficiency Analysis of Choices Involving Risk

EFFICIENCY ANALYSIS OF CHOICES INVOLVING RISK

289

337

But now,

f

d[u(F-G)] =

Ｎｾ＠

f d[u.(F-G)]

=lim {u(n)[F(oo)-G(oo)]-u(-n)(F(-oo)-G(-oo)]}

= Hence t.Eu

lim {u(n).O-u( -n).O}

= 0.

= f (G- F)du, if the integral exists. 1 Q.E.D.

The optimal criterion for FDG is given in Theorem 1.

Theorem 1. Let F, G and u be as in Lemma 1. A necessary and sufficient condition for FDG is: F(x)

ｾ＠

G(x)for every x, and F(x 0 )0, due to the right-continuity ofF and G, there is an interval x 0 ｾｸ＼＠ x 0 +[3 where G(x)-F(x)>O. To show that there exist some u 0 for which t.Eu 0 >0, choose u0 (x) as follows: U0 (x) =

{

Xo

ｘｾ＠

X

X 0 ｾｘ＠

Xo X 0 +f3

x 0 +f3 X'?;._ x 0 +f3.

Then

Uo E

u, and f

(G-F)du =

r:+/l

(G-F)dx>O.

(b) The necessity is proved similarly. If, for some xi, G(xi)-F(x 1) xi +e), where G(x)-F(x)O, and another function v where ilEv Ea Y, i.e., a larger mean value of X, is a necessary condition for dominance of X. The variance, however, plays no direct role in the efficiency criterion. For example, consider two random variables with rectangular distributions, i.e. X has a constant probability density in the range x ｾ＠ x 2 , and Y a constant density in the range y 1 ｾ＠ y ｾ＠ Yz. 1 If x 1 > y 1 and x 2 > y 2 , X 1 ｾ＠ X is preferred to Y with every possible utility function. The variance of X (which equals / 2 (x 2 - x 1) 2 ) may be much larger than var Y, and the degree of risk aversion implied by u(x) may be as high as one wants to assume, still X is better than Y, if individuals do not prefer less money to more money (i.e., if u(x) is non-decreasing). If one wishes to consider utility functions which vary only in a bounded range, 2 it can easily be verified that the optimal criterion for dominance of F over G, is that F(x) ｾ＠ G(x) for all x and F(x 0 )E6 (Y)

=

s:

(G-F)dt

0 0·21 0·20

0·25,

varp (X)= 0·4475>var6 (Y) = 0·4075, but X is preferred to Y for any concave utility function. Since F(l)>G(1), the general criterion of theorem 1 is not satisfied. Indeed, choosing a convex u(x) = l0X 2 , we get Y preferred to X: !J.Eu = Epu(x)-E0 u(y) ｾ＠ 1003-1101O], exceeds the area to the right, where (G-F)p,2 ; For all x, x-p, 1 < x-p, 2 ; thus F(x) ｾ＠ G(x) 0' 0' for all x, and there is no intersection point. F dominates G by Theorem 1. If p, 1 = p, 2 , F and G are identical. ·

If u 1 #= u2 , we have an intersection point at x 0 , where

(a) If u1 >u 2 , then for xx0 ,

x-p,

1 -0'1

ｾ＠

x-p,2

--,

x-p,

1 -0'1

and F(x)

0'2

ｾ＠

x-p,

> - -2 , and ｆＨｸＩｾｇ＠ 0'2

(F>G for x1); and

G(x). Thus, the condition of Theorem 3 is not

satisfied, and F cannot dominate G. (b) If u 1 x-p,2, and F ｾ＠ G. Since FO, cf>"(x)O, cf>"(x) 0.

A. General Efficiency Criterion

In the most general case, where investors are assumed to have no systematic preferences with respect to risk (i.e., no restrictions are placed on investors' utility function beyond the assumption that their first derivatives be nonnegative), it has been proved (see articles by James Quirk and Rubin Saposnik, Joseph Hadar and William Russell, and Giora Hanoch and Levy) that option F eliminates option G from the efficient set (i.e., F dominates G, or F DG) for all (utility) functions cf>(x) (where

f_:

[G(I) -

F(t)]q,'(t)dt?:: 0

functions, is: 8 F;DGi (i= I, 2), where F, and G; denote the single period distributions. PROOF: Let x 1 and x2 be two continuous random variables for periods 1 and 2, respectively, and x = x 1x 2 is the random variable (return) for the two periods. Since F,DG; (i= 1, 2), it is given that: for every x 1

and F 1 (x 10 )

for every nondecreasing concave ¢(x).

II. Multi-Period Dominance in the Case of an Unrestricted Utility Function

In the rest of the paper we shall prove several theorems, first for the two-period horizon, and then the results will be generalized for an n-period horizon. Like Tobin, we assume independence over time. More precisely, we accept the Random Walk Theory (see, for example, Paul Cootner). Let R; denote the return on one dollar invested in period i. The final wealth for an n-period horizon is therefore given by 1 R ］ｉｔｾ＠ (l+R;). Denoting 7 1+R;=x;, the joint density function ｯｦｬｾ＠ X; isf(x,, ... , Xn)- Accepting the Random Walk hypothesis, the joint density function can be written as f(x,, ... , Xn-t)f(xn), since f(xn) is independent of f(x 1, . . . , x,_1). For the twoperiod case, independence implies f(x 1 , x2 ) = f(x,)f(x,).

+

THEOREM 1: A sufficie,nt condition for a two-period risk F(x) to dominate another twoperiod risk G(x), for all nondecreasing utility

( 4)

F 2 (x 2)

< G1 (x10 )

for some x 10

G,(x,)

for every x 2

:::;

and F 2 (x2 ,) < G2 (x2 ,)

for some value x20

Assuming independence over time, the distribution function of the two-period return is given by: (5)

F(x)

P,(x :::; x)

=

i"' fo

=

x/xr[t(t,)j,(t,)dt,dt,

Jo"' F 2 (xjt 1 )f1 (t 1 )dt 1

=

Similarly, the distribution function of option G is given by:

(6)

G(x)

=

i"'

G 2 (x/t 1 )g 1 (t 1 )dtt,

where f;, g; stand for the density functions of the two risks in period i. We must prove that the two-period distribution F, dominates the two-period distribution G, i.e., F(x) :s;G(x) for every x, and F(xo) 0 for some value (x 0/t 10), and hence there is a strict inequality in (7) at least for one value. Thus, F(x) S G(x) and F(x 0 ) < G(x 0) for some value x 0 which implies that the two-period distribution F dominates the two-period distribution G.

COROLLARY 1: If, in addition to the assumption of independence, we also assume stationarity over time of the distributions under consideration, one may conclude that for any two horizons, n 1 and n 2 (n, > n 1), the number of options in the e_flicient set which is appropriate to horizon n 2 is not larger than the number of options in the efficient set which is appropriate to horizon n 1, where the e_flicient sets are defined for all nondecreasing utility 0 Note that in Theorem 1 we did not assume stationarity over time of the distributions. The ｣ｯｭｰ｡ｲｩｾｮ＠ of the pair (Fn-I, Fn) with the pair (Gn-l, Gn) does not violate the conditions under which Theorem 1 holds.

Chapter 18. Stochastic Dominance, Efficiency Criteria, and Efficient Portfolios 990

THE AMERICAN ECONOMIC REVIEW

junctions. T/u:s conclusion stems directly from Theorem 1. Suppose we have two options, F and G.

The stationarity assumption implies that, F1=F2 ... =Fn and G1=G2= ... Gn. For the sake of simplicity, suppose that n 1 = 1 and n 2 =n. If F 1DG1, then, by the stationarity assumption, Fi also dominates G, for all other periods i = 2, ... , n. 10 Hence, by Theorem 1, F" dominates Gn. To sum up, an option which is eliminated from the one-period efficient set, is also eliminated from the n-period efficient set. Hence, the number of options in the efficient set of the long horizon cannot exceed the number of options in the efftcient set of the short horizon. It is interesting to note that one can find numerical examples for which the number of elements in the efficient set is striclly decreasing when the horizon increases. Consider the following two-period example, where F1 = F2 and G1 = G2. Example 1 G1 (or G2)

F, (or F,)

Outcome

Probability

1 4

1/4

2

3/4

10

Outcome

Probability

1/2 1/2

G'

II. Multi-period Dominance in the Case of Risk Aversion

It has been shown that for the one-period investment case, a necessary and sufficient condition for a risk with cumulative probability distribution function F 1 to dominate one with Gt(F1DG1) as its cumulative distribution function, for all nondecreasing concave utility functions, is:

for every x, and GJ7"'Ft for some x 0 , where the subscript 1 indicates the distribution for period one. (See articles by Hadar and Russell, Hanoch and Levy, and Rothschild and Stiglitz.) We shall now prove that if F1DGt and F,DG2, then, F dominates G, where F and G are the two-period distributions. Before proceeding to state and prove a theorem for the multiperiod risk averswn case we need the following: LEMMA: Let T(x, It)= fJF(t/lt)dt, where F is some ( cumulatit•e) distribution function. Then, T' (x, It) :::; 0

at!

a T(x, 1 2

F'

Probability

Outcome

Probability

1 4 16

1/16 6/16 9/16

4 20 100

1/4 1/2 1/4

Since F 1 and G1 (or F 2 and G2) intersect, both of them are included in the one-period efficient set. But F 2 (x) :s;G 2 (x) for every x (with a strict inequality for some values) and, hence, F 2 DG 2 • This numerical example shows that portfolios which are not eliminated from the one-period efficient set might

10 The stationarity assumption implies that the number of elements in the efficient sets constructed for each single period (e.g., year) is the same.

DECEMBER 197 J

be eliminated from the multi-period efficient set. 11

Assuming independence over time, the twoperiod distributions F 2 and G 2 are:

Outcome

303

1)

---(att) 2

T"(x, It)

PROOF:

a

Jo

x

(12) =

J

x (

o

and

2: 0

F(l/tl)dl

att aF(I/

11l) dt

att

Let j(l/11) be the density function of F(l/11 ). Then, 11 The numerical example confirms that the conditions for dominance given in Theorem 1 are sufficient, but not necessary.

304

H Levy 991

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VOL. 63 NO.5

F DC, for every nondecreasing concare utility function (cj/(x) 2_0, q:,"(x):::; 0) is:

since the integrand is always nonnegative. The proof that T"(x, t1 )2_0 is somewhat more involved:

(16)

fox; [Gi(ti)- Fi(ti)]dti

2. 0 (i

=

1, 2)

for every value x;, with a strict inequality for at least one mlue.

PROOF: We have to prove that (17) holds: (17)

(with a strict inequality for some value x 0). Using (5) and (6), equation (17) can be rewritten:

Hence, (13)

T"(x,

t

1) = -

[f)'(t/t

1)(

ＭｴＯｾＩＨ､＠

(18)

+

2. fox fo"' F ,(t/tt)ft(tt)dt,dt By condition (16), for the second-period distribution we have:

c/ ＯｴｾＩ､＠ =fox

(19)

｛ｪＧＨｴＬＩＱＮＯｾ｝､＠

Integrating by parts (with respect to x), we get:

(14)

=

ｻ｛ｦＨｴｪＬＩＯＺｊｬｾＭ f(x/t 1 )x

f 2

/1 13

-

f

X

0

0

fox G,(t/tt)dl

=fox F 2 (t/t 1 )dt

+ o(x/t 1)

where ii(x/lt)::::: 0 and o(xo/ltn) > 0 for at least one value (x 0/tt 0). Substituting (19) in (18), we obtain:

J:x [J'(t/t,)(1/l,) ·t 2 /t:]dl

=

fox fo"' G2(tjt,)g 1(t 1)dt 1dt

f)wt,)C -2t/t:)dt]

Let us expand the fust term on the right side of (13): fo xj'(tjt,).

J:x [G(t) - F(t)]dt :;::,_ 0

xf(tjt,)21/t:dt 3

f(t/t 1 )2t/t 1 dt

Substituting these results in (13), we get:

We can now prove the following theorem:

THEROEM 3: Let F and G be the cumulatit'e distributions for the two- period horizon of two alternative risks. A sufficient condition for

Since the second term is nonnegative, it will suffice to prove that the f1rst term in (20) is nonnegative. Integrating the first term in (20) by parts, with respect tot, we obtain:

305

Chapter 18. Stochastic Dominance, Efficiency Criteria, and Efficient Portfolios THE AMERICAN ECONOMIC REVIEW

992

-f

J

a

00

[(Gl(ti)-Fl(t,)]

Since

the

forward; since pn-l dominates cn-l, one can conclude that

"F,(t/tl)dt

､ｴＱｾＰ＠

0

｡ｴｾ＠

o

DECEMBER 1973

first

term equals zero, and t,) (by definition), it remains to prove that the following term is nonnegative.

f 0'F,(t/t2)dt== T(x,

(22)

fo

oo

[G1(t1)- F 1 (t 1 )]dt 1

X;

i=l

0

Xote that ＯｾＧ｛ｇｴＨｬＩＭｆ for Ｑ Ｉ｝､ｩｴｾｏ＠ every x 1 (by (16)), and -T(x, 11) is nondecreasing concave (utility) function in 11 (bY the Lemma). Applying (2') to this case we get,

Jon

II

Since by assumption of the theorem,

j[G,(t,) - FtCtt)]}

( - aT(x, It)\ dt 1 ｾ＠ at1 )

(23)

7t-l

for every

ｾ＠

for every x,, we may again employ Theorem 3 1 ' and prove that n

f ,_

II

_,

Xi

1

[G"(t) - F"(t)]dt

ｾ＠

0

Il?-1

0

Hence inequality (23) holds. We also have in (20) at least one strict inequality, since for some value x 0 jl1" IJ(xo/t,.,) > 0. Hence, F DC.

THEOREM 4: Let Fn and G" be the cumulath•e distributions of two n-period risks. A sufficient condition for F" DC" for all nondecreasing conca1•e utility functions ｩｳＯｾＧ＠ [G J!J ＭｆＬＨｴｬＩ｝､［ｾｏ＠ for every x, (with a strict inequality for some X; 0 ) , where i=l, 2, ... , n. PROOF: The proof is by induction and similar to that given in Theorem 2. For n= 2, it has been proved in Theorem 3 that ｝ｾＧ＠ [G;(I;) - F.,(t;) ｝､ｴ［ｾ＠ 0 (for every X; where i = 1, 2) implies that F 2 dominates G2, for all risk averters. Assume, that the statement of the theorem holds for n -1 period, i.e., fo' [G;(l,) Ｍｆ［ＨｴＩ｝､Ｌｾｏ＠ (for every X; where i=1, 2, ... , n- 1) implies that Fn-l dominates G"- 1 • vVe have to prove that the theorem also holds for n-periods. This is straight-

for all values X;, i.e., pn dominates Gn for all risk averters. COROLLARY 2 If we are prepared to assume stationarity mer time, in addition to independence of the returns, i.e., F 1 =F.= ... =Fn and G 1 =G 2 = ... =G,, we may conclude that t!ze number of elements in the efficient set which is constructed for all risk a1•erters does not increase when the im·estment horizon increases. This conclusion follows from Theorem 3. Suppose that in period 1, F 1 dominates G1 , i.e., Ｏｾ Ｑ ｛ｇ Ｑ ＨＱＩＭｆ for all values Ｑ ＨＱＩ＠ ｝､ｴｾｏ＠ x 1 , then, by the stationarity assumption, F; dominates G; for all other periods i (i = 1, 2, ... , n), and by Theorem 3, F" dominates G". In other words, any option which is eliminated from the risk averters' oneperiod efficient set is also eliminated from the n-period efficient set. The efficient set in each single period is identical, and the nperiod efficient set is no greater than the single period efficient set. Again, as in the previous case, we can find examples where the efficient set is strictly decreasing when 12 Note again that, by assumption, Xn and IJt.:i Xi are independent, i.e., by looking at the results of the first n-1 periods one cannot improve prediction with respect to the nth period.

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the investment horizon increases. 13 \Ve shall again use Example 1 above to illustrate this argument: F 1 does not dominate Gt (i.e., F 1 DG1) for all risk averters since,

But, also, G1 docs not dominate F 1

Hence, F 1 and G1 are included in the ｯｮ･ｾ＠ period efficient set. Assuming stationarity over time F 1 = F 2 , G1 = G2, and hence also F 2 DG2 , G2 DFz, which means that Fz and Gz are included in the second period efficient set. Looking back at Example 1, we can see that, in the ｴｷｯｾｰ･ｲｩ､＠ horizon, F 2 (x) ::; G2 (x), for every x and for some value x a strict inequality holds. Consequently ft(G 2 (t)-F 2 (t)dt]2:0 (with a strict ｩｮｾ＠ equality for some value x 0 ) and, hence, F 2 dominates G2 . Therefore, in this example the risk averters' efficient set decreases when the investment horizon increases. IV. Concluding Remarks

This study investigated the role of the investment horizon in determining the ･ｦｩｾ＠ cient set of portfolios. We found that, unlike the ｯｮ･ｾｰｲｩ､＠ case, the efficiency criteria for the ｭｵｬｴｩｾｰ･ｲｯ､＠ case are sufficient but not necessary for dominance. If a portfolio is eliminated from each single period efficient set it will also be eliminated from the ｭｵｬｴｩｾｰ･ｲｯ､＠ efficient set, but the ｩｮ｣ｬｵｾ＠ 13 Thi;; means that the longer the horizon the smaller the efficient set. Similar results have been obtained by Nils Hakansson, who assumes that the objective ｦｵｮ｣ｾ＠ tion of the investors is to maximize the geometric average return over time. In Hakansson's analysis, when n-> oo, the efficient set includes only one portfolio, which is also the optimal one. It is of interest to note that Menachem Yaari, who analyzes the individual's multi-period investment-consumption decision, obtains similar results: the longer the horizon, the more certain the optimal consumption and when n-+"", the ｩｮ､ｶｾ＠ ual ｩｮｶ･ｳｴｭｾ｣ｯｵｰ＠ decisions are delern1ined as with certainty.

sian of a portfolio in each single period efficient set does not guarantee that this portfolio will be included in the multi-period efficient set. Thus, an investment consultant who screens all the available portfolios should construct the efficient set for each group of investors according to their investment horizon. However, the consultant may construct the efficient set for the shortest relevant horizon, knowing that the efficient set which is appropriate for longer horizons must be a subset of (or identical to) the efficient set which has been obtained for the short horizon. N everthelcss, one should keep in mind that the suggested procedure is valid only under the case in which probability functions are stationary over time. Otherwise, the consultant can not limit his attention only to the first period distributions. The magnitude of the decrease in the efficient set resulting from an increase in the investment horizon is left to be investigated either empirically or by simulation. REFERENCES K. Arrow, Aspects of the Theory of Risk Bearing, Helsinki 1965. P. H. Cootner, The Random Character of Stock Market Prices, rev. ed., Cambridge, Mass. 1967. J. Hadar and W. R. Russell, "Rules for Ordering Uncertain Prospects," Amer. Econ. Rev., Mar. 1969, 59, 25-34. N. H. Hakansson, "Multi-Period :\1ean-Variance Analysis: Toward a General Theory of Portfolio Choice," J. Finance, Sept. 1971, 26, 85 7-84. G. Hanoch and H. Levy, "The Efficiency Analysis of Choices Involving Risk." Rev. Econ. Stud., July 1969, 36, 335-46. H. Levy and M. Sarnat, hn,estment and Portfolio Analysis, New York 1972. ---and-·--, "International Diversification of Investment Portfolios," Amer. Econ. Rev., Sept. 1970, 60, 668-7 5. J. Lintner, "Security Price, Risk, and Maximal Gains from Diversification," J. Finance, Dec. 1965, 20, 587-615. H. M. Markowitz, Portfolio Selection, New York 1959.

Chapter 18. Stochastic Dominance, Efficiency Criteria, and Efficient Portfolios 994

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THE AMERICAN ECONOMIC REVIEW

Mossin, "Optimal iVIulti-Period Portfolio Policies," f. Bus., Univ. Chicago, Apr. 1968, 41, 215-29. J. Pratt, "Risk Aversion in the Small and in the Large," Econometrica, Jan. 1964, 32, 122-36. J. P. Quirk and R. Saposnik, "Admissibility and Measurable Utility Functions," Rev. Econ. Stud., Feb. 1962, 29, 140-46. M. Rothschild and J. E. Stiglitz, "Increasing Risk: I. A Definition," f. Econ. Tlzeor., Sept. 1970, 2, 225-43.

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DECEMBER 1973

W. F. Sharpe, "Capital Assets Prices: A Theory of Market Equilibrium Under Conditions of Risk," f. Finance, Sept. 1964,19, 42 5-42. J. Tobin, "The Theory of Portfolio Selection," in F. H. Hahn and F. P.R. Brechlings, eds., The Theory of Interest Rates, London 1965. M. E. Yaari, "A Law for Large Numbers in the Theory of Consumer's Choice Under Uncertainty," working pap. no. CP-330, Berkeley Center Res. Manage. Sci., Univ. California, Mar. 1971.

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Section D

Risk Aversion and Static Portfolio Theory

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Section D. Risk Aversion and Static Portfolio Theory

The components of the financial decision problem have been presented: assets, prices, utilities, and efficiency. However, there are important considerations in the decision models and the data inputs which affect the quality of investment decisions. One significant issue is the computability of an efficient investment strategy. A major advantage of Markowitz’ mean-variance analysis is the relative ease of computing optimal strategies and as such it is a practical technique. Ziemba, Parkan and Brooks-Hill (1974) consider the general problem max Eu(R′ x),

st . e′ x = 1, x ∈ K,

where E represents expectation with respect to the uncertainty in R′ = (R1 , . . . , Rn ), the vector of risky asset returns. K is a convex set that represents additional constraints on the choice of x. It is assumed that R is joint normal and that Tobin’s riskless asset exists, so that borrowing and lending can occur at the risk free rate r. Ziemba et al. propose a 2-step approach based on Tobin’s (1958) result that the distribution of proportions x∗ in the risky assets is independent of the utility function assuming it is concave. So the first step is to find the proportions invested in risky assets using a fractional program:
 (R − re)′ x − r ′ max |e x = 1, x E K . 1 x (x′ Σx) 2 The fractional program always has a unique solution and it may be solved via Lemke’s algorithm for the linear complementarity problem or as a quadratic program. The second step is to determine the optimal ratios of risky to non-risky assets. ′ The return on the risky portfolio is R∗ ∼ N (µ′ x∗ , x∗ Σx∗ ), where (µ, Σ) are the (mean, variance) of R. The optimal mix of R∗ and the risk free asset may be found by solving max

−∞ 0. If UD is the class of non-decreasing, concave, decreasing absolute risk version utilities, then UD ⊂ U3 . Vickson (1975) considered stochastic dominance tests for decreasing absolute risk aversion. The class UD is difficult to characterize. Pratt defines a measure of relative risk xu(2) (x) , and identifies the class of constant relative risk aversion ρ∗ (x) = xρ(x) = − (1) u (x) aversion utilities (CRRA). If UC denotes the CRRA family of utilities, then UC ⊂ xu(2) (x) = α can be solved for the function u, UD . Furthermore, the equation − (1) u (x) 1 with the solution: u(x) = 1−α x1−α , α < 1. When α = 1 the utility is u(c) = log(c). The CRRA class of utilities is often used in the analysis of investment decisions. The parameter α captures the aversion to risk, and the impact of risk aversion on investment decisions can be analyzed analytically. The Arrow–Pratt risk aversion measure is local, that is, it depends on the level of wealth. Rubinstein (1973) developed a measure of global risk aversion in the context of a parameter-preference equilibrium relationship: R(x0 ) =

−x0 E{u(2) (x)} E{u(1) (x)}

where x0 is the initial wealth level and expectation is with respect to the distribution of wealth. Some properties of this measure in the context of risk aversion with changing initial wealth levels appear in Kallberg and Ziemba (1983). This measure is less tractable and less familiar than the Arrow–Pratt measure. In contrast to the Arrow–Pratt measure which is a function of wealth x, R is a constant. Except for a few special cases, R does not have a simple form. For quadratic utility, R(1) = ρ(¯ x); thus the Rubinstein measure is the Arrow–Pratt measure evaluated at the expected final wealth level. For the optimal investment problem, Kallberg and Ziemba establish an important property of the global risk measure. The investor’s problem is max Eu(R′ x),

s.t . e′ x = 1, x ∈ K,

where E represents expectation with respect to the randomness in R′ = (R1 , . . . , Rn ), the vector of risky asset returns. K is a convex set that represents additional constraints on the choice of x. Optimality of Rubinstein’s Risk Aversion Measure. Investors with the same R have the same optimal portfolios.

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313

They also show with empirical studies that investors with similar average ArrowPratt risk aversion have similar portfolios, regardless of their utility function. The results also yield the following conclusions: • The special exponential and negative power utility functions yield very risk averse portfolios, while the positive power (CRRA) utility function which yields highly risky portfolios and moderately risk averse portfolios for different values of α. • The arctangent utility function usually yields highly risky portfolios. • The quadratic, exponential and logarithmic utility functions yield the largest range in variation of ρ and yield the safest and riskiest portfolios. • The quadratic utility function may well play a useful role as a computational surrogate for more plausible utility functions when the number of possible investment securities is large, say n > 50. • With horizons of a year or less one can substitute easily derived surrogate utility functions that are mathematically convenient for more plausible but mathematically more complicated utility functions and feel confident that the errors produced in the calculation of the optimal portfolios are at most of the order of magnitude of the errors in the data. • Calculations indicate that the maximum expected utility and optimal portfolio composition are relatively insensitive to errors in estimation of the variancecovariance matrix. However, errors in estimating the mean return vector do significantly change these quantities. Chopra and Ziemba (1993), following the earlier papers of Kallberg and Ziemba (1981, 1984), consider the relative impact of estimation errors and the impact of risk aversion on portfolio performance. Mean percentage cash equivalent loss due to errors in means, variances and covariances in a mean-variance model are found to be in relative terms roughly 20:2:1 times as important, respectively. The error depends on the risk tolerance, the reciprocal of the Arrow–Pratt risk aversion ρ(x). With low risk aversion, like log, the ratios can be 100:2:1. So good estimates of asset return distribution moments are the most crucial aspect for successful application of a mean-variance analysis, and in all other stochastic modeling approaches. The sensitivity of the mean carries into multiperiod models. There the effect is strongest in period 1 then less and less in future periods, see Geyer and Ziemba (2008). The impact of modeling and estimation errors on forecasts for securities prices and the resulting effect on portfolio decisions and capital accumulation have been considered in other studies. Pastor and Stambaugh (1999) conclude that model error is less important than estimation error; see also Kallberg and Ziemba (1981) who conclude the same. With regard to estimation error, alternative estimates for the mean return have been considered in a long series of asset prices (Grauer and Hakansson, 1995), with improved results from shrinkage (Stein) estimators. MacKinlay and Pastor (2000) use a restriction, which incorporates the covariance of returns to calculate an estimate of expected returns which is superior to the shrinkage estimator.

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MacLean, Foster and Ziemba use a Bayesian framework to include the covariance in an estimate of the mean. In essence, the return on one asset provides information about the return on related assets, and the sharing of information through the covariance improves the quality of estimates (Jones and Shanken, 2003). The significance of quality estimates for model parameters is highlighted when those values become inputs to portfolio decisions and the accumulation of wealth over time. This process is illustrated for the portfolio selection problem with constant relative risk aversion utility. That utility enables a closed form solution which depends on the conditional expected return. With the CRRA utility, the loss in wealth resulting from estimation errors can be mitigated with the risk aversion parameter. In the decision rule, the risk aversion parameter α defines a fraction 1 of capital invested in the optimal portfolio. When α < 0, the control of deci1−α sion risk also reduces the impact of estimation error. Correspondingly, when α > 0, the over investment increases the effect of estimation error. Expected utility theory certainly captures some of the intuition for risk aversion over very large stakes. But the theory is manifestly not close to the right explanation for most risk attitudes, and does not explain the modest-scale risk aversion observed in practice. Rabin and Thaler (2001) think that the right explanation incorporates two concepts: loss aversion and mental accounting. Loss aversion is the tendency to feel the pain of a loss more acutely than the pleasure of an equal-sized gain. Loss aversion is incorporated in and Tversky and Kahneman’s prospect theory (1979), which models decision makers who react to changes in wealth, rather than levels, and are roughly twice as sensitive to perceived losses than to gains. Mental accounting, which refers to the tendency to assess risks in isolation rather than in a broader perspective. If investors focus on the long-term returns of stocks they would recognize how little risk there is, relative to bonds, and would be happy to hold stocks at a smaller equity premium. Instead, they consider short-term volatility, with frequent mental accounting losses, and demand a substantial equity premium as compensation. Rabin and Thaler argue that loss aversion and the tendency to isolate each risky choice must both be key components of a good descriptive theory of risk attitudes.

References Arrow, K (1965). The theory of risk aversion. In YJS Helsinki (Ed.), Aspects of the Theory of Risk Bearing, Chicago: Markham Publishing. Brandt, M and P Santa-Clara (2006). Dynamic portfolio selection by augmenting the asset space. Journal of Finance, 61, 2187–2217. Chopra, V and WT Ziemba (1993). The effect of errors in mean and co-variance estimates on optimal portfolio choice. Journal of Portfolio Management, 19, 6–11. Fama, EF and F French (1992). The cross-section of expected stock returns. Journal of Finance, 47(2), 427–466.

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Geyer, A and WT Ziemba (2008). The innovest Austrian pension fund financial planning model InnoALM. Operations Research, 56, 797–810. Grauer, RR and NH Hakansson (1995). Stein and CAPM estimators of the means in asset allocation. International Review of Financial Analysis, 4, 721–739. Hanoch, G and H Levy (1969). The efficiency analysis of choices involving risk. Review of Economic Studies, 36(3), 335–346. Jones, CS and J Shanken (2003). Mutual fund performance with learning across funds. Journal of Financial Economics, 78(3), 507–552. Kahneman, D and A Tversky (1979). Prospect theory: An analysis of decisions under risk. Econometrica, 47(2), 263–291. Kallberg, JG and WT Ziemba (1983). Comparison of alternative utility functions in portfolio selection problems. Management Science, 29, 1257–1276. Kelly, JR (1956). A new interpretation of the information rate. Bell System Technical Journal, 35, 917–926. Li, Y and WT Ziemba (1989). Characterizations of optimal portfolios by univariate and multivariate risk aversion. Management Science, 35, 259–269. Li, Y and WT Ziemba (1993). Univariate and multivariate measures of risk aversion and risk premiums. Annals of Operations Research, 45, 265–296. MacKinlay, AC and L Pastor (2000). Asset pricing models: Implications for expected returns and portfolio selection. The Review of Financial Studies, 13(4), 883–916. MacLean, LC, M Foster and WT Ziemba (2007). Covariance complexity and rates of return on assets. Journal of Banking and Finance, 31(11), 3503–3523. Markowitz, HM (1952). Portfolio selection. Journal of Finance, 7, 77–91. Markowitz, HM (1959). Portfolio Selection: Efficient Diversification of Investments. New York: John Wiley & Sons. Markowitz, HM (1987). Mean-Variance Analysis in Portfolio Choice and Capital Markets. Cambridge, MA: Basil Blackwell. Markowitz, HM and E Van Dijk (2006). Risk-return analysis. In SA Zenios and WT Ziemba (Eds.), Handbook of Asset and Liability Analysis, Volume 1. Amsterdam: Elsevier. Pastor, L and RF Stambaugh (1999). Costs of equity capital and model mispricing. Journal of Finance, 54, 67–121. Pratt, JW (1964). Risk aversion in the small and in the large. Econometrica, 32, 122–136. Rabin, M and RA Thaler (2001). Anomalies: Risk aversion. Journal of Economic Perspectives, 15(1), 219–232. Rosenberg, B, K Reid and R Lanstein (1985). Persuasive evidence of market inefficiency. Journal of Portfolio Management, 11(3), 9–16. Rosenberg, B (1974). Extra-market components of covariance in securities markets. Journal of Financial and Quantitative Analysis, 9(2), 263–274. Rubinstein, ME (1973). The fundamental theorem of parameter-preference security valuation. Journal of Financial and Quantitative Analysis, 8, 61–70. Tobin, J (1958). Liquidity preference as behavior towards risk. Review of Economic Studies, 25, 65–86. Vickson, RG (1975). Stochastic dominance for decreasing absolute risk aversion. Journal of Financial and Quantitative Analysis, 10(5), 799–811. Vickson, RG (1977). Stochastic orderings from partially known utility functions. Mathematics of Operations Research, 2(3), 244–252. Ziemba, WT, C Parkan and FJ Brooks-Hill (1974). Calculation of investment portfolios with risk free borrowing and lending. Management Science, 21, 209–222. Ziemba, WT (1975). Choosing investment portfolios when the returns have stable distributions. In WT Ziemba and RG Vickson (Eds.), Stochastic Optimization Models in Finance. San Diego: Academic Press.

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Chapter 19 Econometrica, Vol. 32, No. 1-2 (January-April, 1964)

RISK AVERSION IN THE SMALL AND IN THE LARGEl BY JOHN

w. PRATT

This paper concerns utility functions for money. A measure of risk aversion in the small, the risk premium or insurance premium for an arbitrary risk, and a natural concept of decreasing risk aversion are discussed and related to one another. Risks are also considered as a proportion of total assets.

1.

SUMMARY AND INTRODUCTION

LET u(x) BE a utility function for money. The function r(x)= -u"(x)fu'(x) will be interpreted in various ways as a measure oflocal risk aversion (risk aversion in the small); neither u"(x) nor the curvature of the graph ofu is an appropriate measure. No simple measure of risk aversion in the large will be introduced. Global risks will, however, be considered, and it will be shown that one decision maker has greater local risk aversion r(x) than another at all x if and only if he is globally more risk-averse in the sense that, for every risk, his cash equivalent (the amount for which he would exchange the risk) is smaller than for the other decision maker. Equivalently, his risk premium (expected monetary value minus cash equivalent) is always larger, and he would be willing to pay more for insurance in any situation. From this it will be shown that a decision maker's local risk aversion r(x) is a decreasing function of x if and only if, for every risk, his cash equivalent is larger the larger his assets, and his risk premium and what he would be willing to pay for insurance are smaller. This condition, which many decision makers would subscribe to, involves the third derivative of u, as r' ｾＰ＠ is equivalent to u'"u'?;,u" 2 • It is not satisfied by quadratic utilities in any region. All this means that some natural ways of thinking casually about utility functions may be misleading. Except for one family, convenient utility functions for which r(x) is decreasing are not so very easy to find. Help in this regard is given by some theorems showing that certain combinations of utility functions, in particular linear combinations with positive weights, have decreasing r(x) if all the functions in the combination have decreasing r(x). The related function r*(x)=xr(x) will be interpreted as a local measure of aversion to risks measured as a proportion of assets, and monotonicity of r*(x) will be proved to be equivalent to monotonicity of every risk's cash equivalent measured as a proportion of assets, and similarly for the risk premium and insurance. These results have both descriptive and normative implications. Utility functions for which r(x) is decreasing are logical candidates to use when trying to describe the behavior of people who, one feels, might generally pay less for insurance against 1 This research was supported by the National Science Foundation (grant NSF-G24035). Reproduction in whole or in part is permitted for any purpose of the United States Government. 122

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a given risk the greater their assets. And consideration of the yield and riskiness per investment dollar of investors' portfolios may suggest, at least in some contexts, description by utility functions for which r*(x) is first decreasing and then increasing. Normatively, it seems likely that many decision makers would feel they ought to pay less for insurance against a given risk the greater their assets. Such a decision maker will want to choose a utility function for which r(x) is decreasing, adding this condition to the others he must already consider (consistency and probably concavity) in forging a satisfactory utility from more or less malleable preliminary preferences. He may wish to add a further condition on r*(x). We do not assume or assert that utility may not change with time. Strictly speaking, we are concerned with utility at a specified time (when a decision must be made) for money at a (possibly later) specified time. Of course, our results pertain also to behavior at different times if utility does not change with time. For instance, a decision maker whose utility for total assets is unchanging and whose assets are increasing would be willing to pay less and less for insurance against a given risk as time progresses if his r(x) is a decreasing function of x. Notice that his actual expenditure for insurance might nevertheless increase if his risks are increasing along with his assets. The risk premium, cash equivalent, and insurance premium are defined andrelated to one another in Section 2. The local risk aversion function r(x) is introduced and interpreted in Sections 3 and 4. In Section 5, inequalities concerning global risks are obtained from inequalities between local risk aversion functions. Section 6 deals with constant risk aversion, and Section 7 demonstrates the equivalence of local and global definitions of decreasing (and increasing) risk aversion. Section 8 shows that certain operations preserve the property of decreasing risk aversion. Some examples are given in Section 9. Aversion to proportional risk is discussed in Sections 10 to 12. Section 13 concerns some related work of Kenneth J. Arrow. 2 Throughout this paper, the utility u(x) is regarded as a function of total assets rather than of changes which may result from a certain decision, so that x = 0 is equivalent to ruin, or perhaps to loss of all readily disposable assets. (This is essential only in connection with proportional risk aversion.) The symbol ｾ＠ indicates that two functions are equivalent as utilities, that is, ｵ Ｑ ＨｸＩｾｵ means there exist Ｒ ＨｸＩ＠ constants a and b (with b>O) such that u 1 (x)=a+buz(x) for all x. The utility functions discussed may, but need not, be bounded. It is assumed, however, that they are sufficiently regular to justify the proofs; generally it is enough that they be twice continuously differentiable with positive first derivative, which is already re2 The importance of the function r(x) was discovered independently by Kenneth J. Arrow and by Robert Schlaifer, in different contexts. The work presented here was, unfortunately, essentially completed before I learned of Arrow's related work. It is, however, a pleasure to acknowledge Schlaifer's stimulation and participation throughout, as well as that of John Bishop at certain points.

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quired for r(x) to be defined and continuous. A variable with a tilde over it, such as

z, is a random variable. The risks z considered may, but need not, have "objective" probability distributions. In formal statements, z refers only to risks which are not degenerate, that is, not constant with probability one, and interval refers only to an interval with more than one point. Also, increasing and decreasing mean nondecreasing and nonincreasing respectively; if we mean strictly increasing or decreasing we will say so. 2.

THE RISK PREMIUM

Consider a decision maker with assets x and utility function u. We shall be interested in the risk premium n such that he would be indifferent between receiving a risk z and receiving the non-random amount E(z) -n, that is, n less than the actuarial value E(z). If u is concave, then ｮｾ＠ 0, but we don't require this. The risk premium depends on x and on the distribution of z, and will be denoted n(x,z). (It is not, as this notation might suggest, a function n(x,z) evaluated at a randomly selected value of z, which would be random.) By the properties of utility, (1)

u(x+E(z)-n(x,z))=E{u(x+z)}.

We shall consider only situations where E{u(x+z)} exists and is finite. Then n(x,z) exists and is uniquely defined by (1), since u(x+E(z)-n) is a strictly decreasing, continuous function of n ranging over all possible values of u. It follows immediately from (1) that, for any constant p., (2)

n(x,z)=n(x+ll, z-ll).

By choosing p.=E(z) (assuming it exists and is finite), we may thus reduce consideration to a risk z- p. which is actuarially neutral, that is, E(z- p.) = 0. Since the decision maker is indifferent between receiving the risk z and receiving for sure the amount na(x,z) = E(z) -n(x,z), this amount is sometimes called the cash equivalent or value of z. It is also the asking price for z, the smallest amount for which the decision maker would willingly sell z if he had it. It is given by (3a)

u(x+na we have (3c)

nix,z)= -na(x,z)=n(x,z)-E(z).

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If z is actuarially neutral, the risk premium and insurance premium coincide. The results of this paper will be stated in terms of the risk premium n, but could equally easily and meaningfully be stated in terms of the cash equivalent or insurance premium.

3.

LOCAL RISK AVERSION

To measure a decision maker's local aversion to risk, it is natural to consider his risk premium for a small, actuarially neutral risk z. We therefore consider n(x, z) for a risk z with E(z)=O and small variance a;; that is, we consider the behavior of n(x,z) as a;-+0. We assume the third absolute central moment of z is of smaller order than a;. (Ordinarily it is of order a;.) Expanding u around x on both sides of (1), we obtain under suitable regularity conditions 3 (4a)

u(x-n)=u(x)-nu'(x)+O(n 2 ) ,

(4b)

E{u(x +z)} =E{u(x) + zu'(x)+!z 2 u"(x)+O(z 3)} = u(x) +ta; u"(x) + o(a;) .

Setting these expressions equal, as required by (1), then gives

where (6)

r(x) = -

u"(x) d u'(x) = - dx log u'(x).

Thus the decision maker's risk premium for a small, actuarially neutral risk z is approximately r(x) times half the variance of z; that is, r(x) is twice the risk premium per unit of variance for infinitesimal risks. A sufficient regularity condition for (5) is that u have a third derivative which is continuous and bounded over the range of all z under discussion. The theorems to follow will not actually depend on (5), however. If z is not actuarially neutral, we have by (2), with p. = E(z), and (5):

Thus the risk premium for a risk z with arbitrary mean E(z) but small variance is approximately r(x + E(z)) times half the variance of z. It follows also that the risk premium will just equal and hence offset the actuarial value E(z) of a small risk (z); that is, the decision maker will be indifferent between having z and not having it when the actuarial value is approximately r(x) times half the variance of z. Thus r(x) 3 In expansions, O( ) means "terms of order at most" and o( ) means "terms of smaller order than."

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may also be interpreted as twice the actuarial value the decision maker requires per unit of variance for infinitesimal risks. Notice that it is the variance, not the standard deviation, that enters these formulas. To first order any (differentiable) utility is linear in small gambles. In this sense, these are second order formulas. Still another interpretation of r(x) arises in the special case z= ±h, that is, where the risk is to gain or lose a fixed amount h > 0. Such a risk is actuarially neutral if +h and -hare equally probable, so P(z=h)-P(z= -h) measures the probability premium ofz. Letp(x,h) be the probability premium such that the decision maker is indifferent between the status quo and a risk z = ± h with (8)

P(z=h)-P(z=-h)=p(x,h).

Then P(z=h)=t[I +p(x,h)], P(z= -h)=t[l-p(x,h)], and p(x,h) is defined by (9)

u(x)=E{u(x + z)} =t[l + p(x,h)] u(x+h)+![l- p(x,h)]u(x -h).

When u is expanded around x as before, (9) becomes (10)

u(x)=u(x)+hp(x,h)u'(x)+th 2 u"(x)+O(h 3 ) .

Solving for p(x,h), we find (11)

p(x,h)=thr(x)+O(h2 ) .

Thus for small h the decision maker is indifferent between the status quo and a risk of ±h with a probability premium of r(x) times th; that is, r(x) is twice the probability premium he requires per unit risked for small risks. In these ways we may interpret r(x) as a measure of the local risk aversion or local propensity to insure at the point x under the utility function u; -r(x) would measure locally liking for risk or propensity to gamble. Notice that we have not introduced any measure of risk aversion in the large. Aversion to ordinary (as opposed to infinitesimal) risks might be considered measured by n(x,z), but 1t is a much more complicated function than r. Despite the absence of any simple measure of risk aversion in the large, we shall see that comparisons of aversion to risk can be made simply in the large as well as in the small. By (6), integrating - r(x) gives log u'(x) + c; exponentiating and integrating again then gives e"u(x)+d. The constants of integration are immaterial because ･ＢｵＨｸＩＫ､ｾＮ＠ (Note e">0.) Thus we may write (12)

u ｾ＠ Je-fr ,

and we observe that the local risk aversion function r associated with any utility function u contains all essential information about u while eliminating everything arbitrary about u. However, decisions about ordinary (as opposed to "small") risks are determined by r only through u as given by (12), so it is not convenient entirely to eliminate u from consideration in favor of r.

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127

CONCAVITY

The aversion to risk implied by a utility function u seems to be a form of concavity, and one might set out to measure concavity as representing aversion to risk. It is clear from the foregoing that for this purpose r(x)= -u"(x)ju'(x) can be considered a measure of the concavity of u at the point x. A case might perhaps be made for using instead some one-to-one function of r(x), but it should be noted that u"(x) or - u"(x) is not in itself a meaningful measure of concavity in utility theory, nor is the curvature (reciprocal of the signed radius of the tangent circle) u"(x)(l + [u'(x)F)- 3 12 • Multiplying u by a positive constant, for example, does not alter behavior but does alter u" and the curvature. A more striking and instructive example is provided by the function u(x)= -e-x. As x increases, this function approaches the asymptote u=O and looks graphically less and less concave and more and more like a horizontal straight line, in accordance with the fact that u'(x)=e-x and u"(x)= -e-x both approach 0. As a utility function, however, it does not change at all with the level of assets x, that is, the behavior implied by u(x) is the same for all x, since u(k+x)= Ｍ･ｊ｣ｸｾｵＨＩＮ＠ In particular, the risk premium n(x,z) for any risk z and the probability premium p(x,h) for any h remain absolutely constant as x varies. Thus, regardless of the appearance of its graph, u(x) = -e-x is just as far from implying linear behavior at x = oo as at x = 0 or x = - oo. All this is duly reflected in r(x), which is constant: r(x)= -u"(x)ju'(x)= 1 for all x. One feature of u"(x) does have a meaning, namely its sign, which equals that of -r(x). A negative (positive) sign at x implies unwillingness (willingness) to accept small, actuarially neutral risks with assets x. Furthermore, a negative (positive) sign for all x implies strict concavity (convexity) and hence unwillingness (willingness) to accept any actuarially neutral risk with any assets. The absolute magnitude of u"(x) does not in itself have any meaning in utility theory, however.

5.

COMPARATIVE RISK AVERSION

Let u 1 and u2 be utility functions with local risk aversion functions r 1 and r 2 , respectively. If, at a point x, r 1 (x)>rz(x), then u 1 is locally more risk-averse than u 2 at the point x; that is, the corresponding risk premiums satisfy n 1 (x,z)> nz(x,z) for sufficiently small risks z, and the corresponding probability premiums satisfy p 1 (x,h)>pix,h) for sufficiently small h>O. The main point of the theorem we are about to prove is that the corresponding global properties also hold. For instance, if r 1 (x) > r 2 (x) for all x, that is, u 1 has greater local risk aversion than u 2 everywhere, then n 1 (x,z)>n 2 (x,z) for every risk z, so that u 1 is also globally more risk-averse in a natural sense. It is to be understood in this section that the probability distribution of z, which determines n 1 (x,z) and nz(x,z), is the same in each. We are comparing the risk

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JOHN W. PRATT

premiums for the same probability distribution of risk but for two different utilities. This does not mean that when Theorem 1 is applied to two decision makers, they must have the same personal probability distributions, but only that the notation is imprecise. The theorem could be stated in terms of n 1 (x,z 1 ) and n 2 (x,z 2 ) where the distribution assigned to z1 by the first decision maker is the same as that assigned to z2 by the second decision maker. This would be less misleading, but also less convenient and less suggestive, especially for later use. More precise notation would be, for instance, n 1(x, F) and nix, F), where F is a cumulative distribution function. THEOREM 1: Let r;(x), n;(x,z), and p;(x) be the local risk aversion, risk premium, and probability premium corresponding to the utility function U;, i=l,2. Then the following conditions are equivalent, in either the strong form (indicated in brackets), or the weak form (with the bracketed material omitted). (a) r 1(x);;:;rix)for all x [and> for at least one x in every interval). (b) n 1 (x,z);;:; [> ]nix,z)for all x and z. (c) p 1(x,h);;:;[> )pix,h)for all x and all h>O. (d) u 1 (u; 1 (t)) is a [strictly] concave function oft.

(e)

ｵＱ＿ＩＭｾｸ＠

s;: [ O. The same equivalences hold if" increasing" is substitutedfor "decreasing" throughout and/or attention is restricted throughout to an interval, that is, the requirement is added that x, x+z, x+h, and x-h all lie in a specified interval.

PROOF: This theorem follows upon application of Theorem 1 to u 1 (x)=u(x) and uix)=u(x+k) for arbitrary x and k. It is easily verified that (a') and hence also (b') and (c') are equivalent to (d') u'(u- 1(t)) is a [strictly] convex function oft. This corresponds to (d) of Theorem I. Corresponding to (e) of Theorem and (20)-(22) is (e') u'(x)u"'(x);;::;(u"(x))Z [and > for at least one x in every interval]. The equivalence of this to (a')-(c') follows from the fact that the sign of r'(x) is the same as that of(u"(x))Z-u'(x)u"'(x). Theorem 2 can be and originally was proved by way of(d') and (e'), essentially as Theorem I is proved in the present paper.

8. OPERATIONS WHICH PRESERVE DECREASING RISK AVERSION We have just seen that a utility function evinces decreasing risk aversion in a global sense if an only if its local risk aversion function r(x) is decreasing. Such a utility function seems of interest mainly if it is also risk-averse (concave, r;;::;O). Accordingly, we shall now formally define a utility function to be [strictly] decreasingly risk-averse if its local risk aversion function r is [strictly] decreasing and nonnegative. Then by Theorem 2, conditions (i) and (ii) of Section 7 are equivalent to the utility's being strictly decreasingly risk-averse. In this section we shall show that certain operations yield decreasingly riskaverse utility functions if applied to such functions. This facilitates proving that functions are decreasingly risk-averse and finding functions which have this property and also have reasonably simple formulas. In the proofs, r(x), r 1 (x), etc., are the local risk aversion functions belonging to u(x), u 1 (x), etc. THEOREM 3: Suppose a>O: u 1 (x)=u(ax+b) is [strictly] decreasingly risk-averse for ｸ Ｐ ｾｸ Ｑ＠ if and only ifu(x) is [strictly] decreasingly risk-averse for ｡ｸ Ｐ Ｋ｢ｾ＠ ｸｾ｡ Ｑ＠ +b. PROOF: This follows directly from the easily verified formula: (26)

r 1 (x)=ar(ax+b).

THEOREM 4: If u 1 (x) is decreasingly risk-averse for ｸ Ｐ ｾｸ Ｑ Ｌ＠ and uz(x) is decreasingly risk-averse for ｵ Ｑ Ｈｸ Ｐ Ｉｾｸｵ Ｑ Ｈｸ Ｑ ＩＬ＠ then u(x)=uz(u 1 (x)) is decreasingly

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132

risk-averse for x 0 ｾｸ Ｑ Ｌ＠ and strictly so unless one ofu 1 and u2 is linear from some x on and the other has constant risk aversion in some interval. PROOF: We have log u'(x) =log u;(ul_(x)) +log (27)

ｲＨｸＩ］

Ｒ Ｈｵ

Ｑ ＨｸＩｵｾＫｲ

ｵｾＨｸＩＬ＠

and therefore

Ｑ ＨｸＩＮ＠

The functions r 2 (u 1 (x)), ｵｾＨｸＩＬ＠ and r 1 (x) are ;;;;o and decreasing, and therefore so is r(x). Furthermore, ｵｾＨｸＩ＠ is strictly decreasing as long as r 1 (x)>0, so r(x) is strictly decreasing as long as r 1 (x) and r 2 (u 1(x)) are both >0. If one of them is 0 for some x, then it is 0 for all larger x, but if the other is strictly decreasing, then so is r. THEOREM 5: If u 1 , • . . , un are decreasingly risk-averse on an interval [x 0 , xtJ, and c 1 , ••• , c" are positive constants, then u = ｌｾ＠ C;U; is decreasingly risk-averse on [x 0 ,xtJ, and strictly so except on subintervals (if any) where all u; have equal and constant risk aversion. PROOF: The general statement follows from the case u=u 1 +u 2 • For this case (28)

(29)

ｵｾＧ＠

Ｋｵｾ＠

r=- - - - = ｵｾ＠

Ｋｵｾ＠

ｵｾ＠

---1'1

ｵｾ＠

+u;

+ - -ｵｾ＠ - 1 ' 2 ; ｵｾ＠

Ｋｵｾ＠

r'= ｵｾｲ＠

+u;r; ｵｾ＠

+u;

We have ｵｾ＾ｏＬ＠ u;>O, ｲｾｏＬ＠ and ｲ［ｾｯＮ＠ Therefore ｲＧｾｏＬ＠ and ｲｾ＠ = r; = 0. The conclusion follows.

and r'±oo

= 0,

(2.1)

wheref(y) is the density of y, given by

f(y)

=

1

r;c_

v27ray

exp

(

f.l-yf) ,

(yＭｾ＠ 2cry

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then Cov(x,g(ji))

= (Cov(x,ji))E[g'(ji)].

(2.2)

The proof of this lemma can be found in Stein [25], or Rubinstein [23, 24]. We now develop a new formula, called the "second-order covariance operator".

LEMMA 2 (SECOND-ORDER COVARIANCE OPERATOR)

Suppose that x, ji have a bivariate normal distribution and g : lR --+ lR is twice differentiable. If E[g'(ji)], E[g"(ji)] exist and lim (y- f.Lp)g(y)f(y)

y-+±oo

lim g'(y)f(y)

y-+±oo

= 0,

= 0,

(2.3) (2.4)

wheref(y) is the density of ji, given by

f(y)

1

= ../firup exp

(

-

(y- f.1.·)2) 2u( '

then Cov ((x- f.Lx) 2 ,g(ji))

= (Cov (x,ji)) 2E[g"(ji)].

(2.5)

The proof of this lemma is given in appendix A. Since the first-order and second-order covariance operators will be applied to u' and u" where u is a utility function, the following definition is given to ease the representations of several results.

REGULARITY CONDITIONS

A utility function u(x) satisfies the (single) regularity condition if u' (x) satisfies the condition for the first-order covariance operator. Moreover, u(x) satisfies the regularity conditions if in addition, u" (x) satisfies the first and second-order covariance operator conditions.

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LEMMA3

Assuming that u(x) is twice continuously differentiable, then u(x) satisfies the (single) regularity condition if and only if2 lim ｵＧＨｸＩｦ･ｂｾ＠

=

x-+±oo

0.

(2.6)

Assuming that u(x) is continuously differentiable of order 4, then u(x) satisfies the regularity conditions if and only if in addition to (2.6), the following two conditions also hold: lim ｸｵＢＨＩｦ･ｂｾ＠

x-+±oo

= 0,

(2.7)

0,

(2.8)

and lim ｵ

x-+±oo

where B = 1/(2ai) and

Ｑ

ＨｸＩｪ･ｂｾ＠

=

oi is the variance of a normal random variable x.

This lemma is a direct consequence of the definition of the regularity conditions and (2.1), (2.3) and (2.4). Thus the regularity conditions for u(x) will hold as long as u'(x), xu"(x) and does as x tends to infinity. Therefore, u 111 (x) do not approach infinity as fast as ･ｂｾ＠ the regularity conditions are the boundary conditions for the utility functions. They are satisfied by exponential, quadratic, logarithmic, power and many other utility functions. We now formally derive Rubinstein's measure of absolute risk aversion from the concept of risk premium in the two-risk situation. Suppose that a decision maker's preferences can be represented by a utility function u(x). He has a random initial wealth x and faces some other risky. He can insure against risky, but not x. Then the risk premium 7r(x, y) is the value satisfying E[u(x + .Y)]

= E[u(x + E.Y- 7r(x,y))].

(2.9)

When a risky is added to some non-random initial wealth x, a decision maker is risk averse if and only if 7r(x, y) > 0. However, if the risk y is added to some random initial wealth x, since y may be negatively correlated with x an4 then have a hedging effect on the initial random wealth, the corresponding risk premium 7r(x,y) may be negative. If x and yare bivariate normal random variables, the sign 2A

similar version of this condition was given by Chipman [3].

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of the risk premium can be determined based on the Markowitz [17, 18] mean-variance criterion: THEOREM I

Suppose that .X, ji have a bivariate normal distribution, and 1r(.X, ji) is the risk premium for a decision maker with utility function u(x). If the decision maker is risk averse, that is, u" < 0, then

(i)

2 2 1r(x,ji) > 0 1'f O'x+.Y > crx;

(ii)

7r(.X,ji) = 0 if oi+y = ｏＧｾｪ＠

(iii)

7r(.X,ji) < 0 if oi+.Y < ＰＧｾＮ＠

Proof

Without loss of generality, assume that E[ji] = 0. We only prove (i) since the proofs of (ii) and (iii) are analogous. If 1r(.X,ji)::::; 0, then

E[x + .Y] ::::; E[x- 1r]. Suppose that var (.X+ ji)

> var (.X),

so that var (.X+ ji) > var (.X- 1r). By [7, theorem 4],

E[u(x + ji)] < E[u(x- 1r)] for all u" < 0. But this contradicts the definition of 1r(x, ji) in (2.9).

D

This result shows that the sign of the risk premium for random wealth depends only on the difference between the variance of the initial random wealth accompanied by the additional random variation and the variance of the initial wealth alone. If the additional random wealth ji were to increase the total variance, a risk averse individual with random wealth .X plus a random variation ji would be indifferent between the status quo and a reduction in wealth by the amount of risk premium coupled with an elimination of the wealth variation due

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271

to y. Conversely, if the additional random wealth y were to reduce the total variance, a risk averse individual would have to be compensated by an amount equal to the hedging premium, that is, the money value of the risk premium when the additional wealth were to be withdrawn. Although the sign of the risk premium can be determined by the comparison of variances, its magnitude would be expected to relate to the appropriate measure of a decision maker's risk aversion. We now derive an approximate expression for the risk premium. We still assume that the initial wealth and risk have a bivariate normal distribution and the utility function is well-behaved. For simplicity, we also assume that the risk has a zero mean. THEOREM2

Let x,y have a bivariate normal distribution such that E[y] = 0, and u(x) be an increasing utility function. Assume that u(x) satisfies the regularity conditions. Then the risk premium 1r(x, y) for the given x can be approximated by - -

7r(x,y) ｾ＠

2)

2

1

where the approximation is of the order ｯＨＭｾＫｙ＠ utility u(x) = - exp (-,Bx),

7r(x,y)

E[u"(x)J

-2(o-.H.Y- O"x E[u'(x)] ,

ｯＭｾＩＮ＠

In particular, for exponential

2 E[u"(x)] o-x) E[u'(x)] .

= ＭｾＨｯＫＮｙ＠

(2.10)

(2.11)

The proof of the theorem appears in appendix A. DEFINITION 1

Let x be a random variable and u(x) be an increasing utility function. Assume that u(x) is twice differentiable and E[u'(x)], and E[u"(x)] exist. Rubinstein's measure of absolute risk aversion of utility function u(x) at xis R(x)

=

E[u"(x)] E[u'(x)] .

Thus the risk aversion R(x) can then be interpreted as twice the risk premium a decision maker requires per unit of incremental variance for infinitesimal risk. If xis non-random and equal to the constant x, then (2.10) becomes

1r(x,y) ｾ＠ ｾｯＭｪｲＨｸＩＬ＠

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where the approximation is of order o(uj). Thus Rubinstein's measure of risk aversion is a generalization of the Arrow-Pratt measure. If x and ji are independent, (2.10) becomes

1r(x,ji) ｾ＠ ｾｵｪｒＨｸＩＬ＠ where the approximation is of order uj. This result is the case of a risk aversion measure with independent wealth and risk obtained by Kihlstrom et al. [13] and Ambarish and Kallberg [1]. For given bivariate normal random variables x and ji, a decision maker with larger Rubinstein's measure of risk aversion than another would require a higher risk premium in the absolute sense if the incremental variance is small. But if two individuals have constant risk aversion then this property holds for any bivariate normal random variables x and ji. COROLLARY l

Let x,ji be bivariate normal random variables and u;(x) -e-/3;x be exponential utility functions for i = 1, 2. Let R; (.X) and 11"; (x, ji) be Rubinstein's measures of absolute risk aversion and risk premiums for utility function u; (x) (i= 1,2), respectively. If

then

Equivalently,

(.X, ji) > 1rz (x, ji) > 0 if ｵｾＫｦ＠

> ｵｾ［＠

if ｵｾＫｦ＠

= ｵｾ［＠

1r, (.X, ji) < 1rz (x, ji) < o if ｵｾＫｙ＠

< ｵｾＮ＠

(i)

7rt

(ii)

1r,(x,ji) = 1r2 (x,ji)

(iii)

0

This result follows immediately from theorem 2. For the exponential utility function, the Arrow-Pratt measure of absolute risk aversion is identical to Rubinstein's measure of absolute risk aversion. In general case, the two measures have the following relationship: THEOREM 3

A decision maker with an increasing utility function u(x) is risk averse, that is, r(x) > 0 for all x, if and only if R(x) > 0 for any random variable .X.

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Proof R(x) > 0 for any random variable implies that R(x) > 0 for any degenerated random variable x x. But in this degenerated case, R(x) = r(x). Thus the sufficient condition is proved. The necessary condition is true since u is increasing, thus r(x) > 0 for any x implies the concavity of u, namely, u" ｾ＠ 0. Hence, by the D definition of R(x), R(x) > 0 for any random variable x.

=

In the preceding analysis, the main results are based on the assumption that the initial random wealth and risk have a bivariate normal distribution. We now show that for the class of quadratic utility functions, all of the previous results hold for any random initial wealth and risk with an arbitrary bivariate distribution. Since the mean-variance criterion is also valid for the class of quadratic utility functions, we have the following result which is analogous to theorem 1: THEOREM4

Let U(x) = x- f3x 2 be a quadratic utility function on (-oo, 1/2{3) where f3 > 0, then for any random variables x and y on ( -oo, 1/2{3), the following properties hold: (i)

2 2 1r(x,y) > o 1.f ai+ji >ax;

(ii)

1r(x,y)

(iii)

1r(x,y) < o

=o

2. if clx+ji ax, .f 2 2 1 ai+ji 0. Let R(x) and 1r(x, y) be the risk aversion measure and risk premium of utility function u(x). Assume that x and y are in the suitable range such that R(x) > 0, and R(x- 1r) > 0, then 1r(x,y) ｾ＠ ＡＨ｡ｾＫｹＭ

｡ｾＩｒＨｸＬ＠

(2.12)

where the approximation is of the order o(oi+ji- oi). The proof appears in appendix A. As shown in corollary 1, under the assumption that two random variables have a bivariate normal distribution and the

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decision makers have constant absolute risk aversion, one decision maker is more risk averse than the other if and only if the absolute value of the risk premium for the former is larger than that for the latter. Similarly, for the class of quadratic utility functions, the above result holds for random variables with arbitrary bivariate distributions. COROLLARY2

Let .X,ji be any two random variables and u1(x), u2 (x) be quadratic utility functions given by

Let Rlx) and 1r/x, y) be the risk aversion measures and risk premiums for utility function u;(x), i = 1, 2, respectively. Assume that R1 (.X)

> R2(i) > 0;

then

Equivalently,

(i)

1rJ (.X, ji) > 7!"2 (i,ji) > 0 if ｏＢｾＫｪｩ＠

(ii)

1rJ(i,y)

(iii)

1r) (i,ji) < 7r2(i,ji) < 0 if ｏＢｾＫｪｩ＠

=

7r2(i,ji)

0

if ｏＢｾＫｪｩ＠

> ＰＢｾ［＠ ｏＢｾ［＠

=

< ＰＢｾＮ＠

So far we have been concerned with risks and risk premiums in absolute terms. We now view them as the proportions of some fixed value w0 • Specifically, for two random variables .X and y, we consider w0i and w0 ji. The relative risk premium is defined as follows: DEFINITION 2

Let u(x) be an increasing utility function, and i, y be bivariate random variables. For a constant w0 , the relative risk premium 1r* (w0; .X, y) satisfies

E[u(w 0x + w0 ji)]

= E[u(w0i + w0 (Ey- 1r*))].

(2.13)

In portfolio selection problems, if w0 is the initial dollar investment, .X and y as the dollar returns on two risky assets for each dollar's investment, then the

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relative risk premium 1r* can be interpreted as the reduction in or compensation for the return on the second asset the investor would accept if that asset were to be replaced with a risk-free asset. DEFINITION 3

Let u(x) be an increasing utility function, and x be a random variable. For a constant w0 , Rubinstein's measure of relative risk aversion R*(w0 , x) is defined to be * _ E[u"(w0 x)J R (w0 , x) = -w0 E[u'(wo·x)] .

(2.14)

From (2.13) and (2.14), it follows that 1r*(l;x,y) = 1r(x,y)

R*(l,x) = R(x).

and

This implies that Rubinstein's measure of absolute risk aversion and absolute risk premium can be recovered from Rubinstein's measure of relative risk aversion and relative risk premium by taking the constant w0 as unity. The following theorem shows that Rubinstein's measure of relative risk aversion can also be justified by the associated relative risk premium in the local sense. THEOREM 6

Let u(x) be an increasing utility function, and x,y be bivariate random variables such that E[y] = 0. If either of the following conditions holds: (i) (ii)

x andy have a bivariate normal distribution, and u(x) satisfies the regularity conditions, u(x) is a quadratic utility function,

then for a constant w0 , the relative risk premium 1r* (w0 ; x, y) can be approximated by - -) ｾ＠ 7r*( wo;x,y

where the approximation is of order ｯＨＭｾＫｙ

2 21(D"x+yD"x2)R*( Wo,x-) ｯＭｾＩＮ＠

Proof Based on the given conditions, from theorems 2 or 5, it follows that

(2.17)

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or, -

7r ( WoX,

2 WoY-) -_ 2I Wo2( C1x+ji

Comparing (2.9) with (2.13) yields

that is, -1 7 r ( WoX,

wo

WoY-) .

(2.15)

Similarly, comparing definitions 1 with 3 yields (2.16)

Dividing the above equation by w0 , from (2.15) and (2.16), it follows that

D From (2.17), the interpretation of R* ( w0 , .X) is similar to that of R( x). If x is non-random (x = 1), from (2.14) and (2.15), then R*(w 0 , 1) = r*(w0 )

and 1r*(w0 ; 1,y)

= 1r*(w0 ,y),

where r*(x) and 1r*(x) are the Arrow-Pratt measure of relative risk aversion and risk premium. Thus Rubinstein's measure of relative risk aversion R* ( w0 , x) is a generalization of the Arrow-Pratt measure of relative risk aversion r* (x), and has the corresponding local properties of the latter. 3.

Multivariate measures of risk aversion

We first present the key tools for the derivation of these multivariate measures of risk aversion. The first-order covariance operator developed by Stein [25) and Rubinstein [23, 24] has been generalized to a multivariate version by Gassmann [6]. LEMMA 4 (FIRST-ORDER MULTIVARIATE COVARIANCE OPERATOR)

Let X, Y be multivariate random vectors on IR.m and IR.n, respectively, with joint distribution

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Let g : ｒＮｮｾＭＫｫ＠ be di.fferentiable with Jacobian matrix lg of dimension k x n. Assume that Cov(X,g(Y)) and E[lg(Y)] exist. If lim g(y0 + ty)fy(Yo + ty) = 0

(3.1)

t--->±00

for ally, y 0 E R.n, wherefy is the probability density of Y, given by

then Cov (X,g(Y))

= Cov (X, Y)E[lg(Y)f

(3.2)

The proof can be found in Gassmann [6]. Now we develop a new formula which will be called "Second-Order Multivariate Covariance Operator". This operator is the multivariate version of secondorder covariance operator in the univariate case developed earlier in this section. LEMMA 5 (SECOND-ORDER MULTIVARIATE COVARIANCE OPERATOR)

Let X, Y be multivariate random vectors on R.n, with joint distribution

be twice continuously differentiable with the Jacobian matrix of Let H : ｒＮｮｾＭＫｸ＠ its ith row H; being J[H;(y)] of dimension n x n. If lim (Yo+ ty)H(Yo + ty)fy(Yo + ty)

t--->±00

lim J[H;(Yo

t->±oo

= 0,

+ ty)Jfy(Yo + ty) = 0,

(3.3) (3.4)

for i = 1, 2, ... , n, and y 0, y E R.n, where fy is the density of Y, then

E[(X

J-Lx)T H(Y)(X -J-Lx)J

= tr(:ExE[H(Y)]) + O(tr(Ej,y)).

(3.5)

The proof of this lemma is outlined in appendix B. Suppose that the preferences of a decision maker can be represented by a multi-attributed increasing concave utility function u(x) with xT = (x 1, ••• , xn)· We may interpret x as a wealth vector representing a decision maker's wealth measured at different points of time, or the wealth held in different assets at the same point of time.

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Consider that a decision maker has two sources of risk in his wealth holdings. He has an initial wealth X which is a random vector. He also faces another risk Y which affects the wealth vector. These risks may be li:Orrelated. Assume that risk Y can be insured against but the initial wealth risk is uninsurable. We are interested in determining the amount of risk premium that a decision maker is willing to pay to make him indifferent between having the other risk with random wealth X+ Y and eliminating that risk by paying the premium 1r with remaining wealth X 1r. This model was proposed by Ambarish and Kallberg [1] and was developed from Ross's [22] partially insurable lottery and from the random initial wealth model of Kihlstrom et al. [13]. We now formally state the model's assumptions. A decision maker's preferences for wealth are represented by a von Neumann and Morgenstern utility function which is a mapping from the wealth space into the real line. The wealth variables are allowed to be negative. U :

Rnt-tlR.

(Al) The gradient of u at x,

'\7Tu(x)

(8u(x)j8x 1 , ... , 8u(x)j8xn),

satisfies

8u(x)j8xi > 0 for all x E IR.n and all i. (A2) The Hessian of u at x, H(x), is an n x n symmetric, negative semidefinite matrix with (i, j)th element lfu(x) / 8xi 8xj. (A3) u is continuously differentiable on Rn of order 4 and all of the relevant

integrals are finite. Let X, Y be two random vectors on Rn, and u(x) be a utility function satisfying (Al)-(A3). The risk premiums 1r are any constant vector on Rn such that E[u(X + Y)J

= E[u(X + EY- 1r)].

(3.6)

The left-hand side of (3.6) represents the expected utility of wealth when the decision maker has two types of risks; and the right-hand side represents the expected utility after one of the risks is eliminated. The following lemma is a result of Ambarish and Kallberg [1] with minor different assumptions. LEMMA6

Given that (Al)-(A3) are valid and E[Y] E['\lTu(X)]1r

= 0, then

= -E['\7T u(X) Y]- (1/2)E[YT H(X) Y] + o(trEx).

(3.7)

Chapter 20. Univariate and Multivariate Measures of Risk Aversion

347

279

Y. Li, W. T. Ziemba/ Univariate and multivariate measures

This result is derived from the Taylor expansion of both sides of (3.6). The proof appears in Ambarish and Kallberg [1]. For the derivation of the risk aversion measure with a correlated random wealth vector and a risk vector, it is assumed that (A4) X, Yare multivariate joint normal random vectors on lRn, with distribution

For simplicity and without loss of generality, assume that E[Y] = 0. MULTIVARIATE REGULARITY CONDITIONS

Given that (Al)-(A4) are valid, u satisfies the regularity condition if the gradient of u, 'Vu(x) in place of g satisfies condition (3.1); and u satisfies the regularity conditions if u satisfies the regularity condition and the Hessian of u, H(x) satisfies conditions (3.3) and (3.4) where all of the variables are replaced by x. (A5) u satisfies the regularity conditions. THEOREM 7

Given that (Al)-(A5) are valid, an approximation of order o(tr(Ex+Y- Ex)) is

(3.8) where (ET'Vu(X))- is a generalized inverse of E['Vu(X)]. The proof of this result is provided in appendix B. For any m x n matrix A, the n x m matrix A- is a generalized inverse if AA-A= A (21]. Thus the solutions for 1r are not unique. Hence, the risk premiums can be paid in different ways to induce the same expected utility. Let the ijth element of EX+Y- Ex be dij, and ui 8uj8xi, and uij = Hij. Written in summation form, (3.8) is then 1 n

n

I:: E[ui(X)]1ri ｾ＠

I:: dijE[uij(X)], 2i,j=l

-

i=l

or,

ｾｅ｛ｵｩＨｘＩ｝Ｑｲ＠

n

ｾ＠

Ｒ

1

ｾ＠

n

(

[Eui(X)] ｾ､ｩｪＨＭｅ｛ｵｘＩ｝Ｏ＠

n

)

.

(3.9)

348

Y Li and WT Ziemba

280

Y. Li, W.T. Ziemba/ Univariate and multivariate measures

Let R(X)

= [R!i(X)] = (

E[u!i(X)]) E[ui(X)] '

then (3.9) can be written as

(3.10) where diag (A) is a diagonal vector of the matrix A. Therefore, (3.11) is a typical approximation of the risk premium. The other approximations are located on a hyperphase passing through it and orthogonal to E[\i'u(X)]. DEFINITION 4

Given that (Al}-(A5) are valid. The absolute risk aversion matrix is R(X)

=(

E[u!i(X)]). E[ui(X)]

If the dimension of the wealth vector is one, then this matrix measure is the same as Rubinstein's measure and we have the same approximate relationship between risk aversion and risk premium as obtained in theorem 2. This matrix measure is thus a multivariate generalization of Rubinstein's absolute risk aversion measure. If the wealth vector X and risk vector Yare uncorrelated, from (3.11), we have 0 1r

ｾＡ＠

diag(EyR(X)).

(3.12)

This relationship has been obtained by Ambarish and Kallberg [1]. In particular, if X is non-random, that is X= x, then

This is the matrix measure of risk aversion for non-random wealth developed by Duncan [4]. The approximate risk premium in (3.11) suggests that the multivariate formulation of risk aversion is important when either (Ex+Y- Ex) or R(X) is

Chapter 20. Univariate and Multivariate Measures of Risk Aversion

Y. Li, W. T. Ziemba/ Univariate and multivariate measures

349

281

not diagonal. In the .first case, the correlations between different dimensions of wealth would be unchanged if the risk premium were paid. In the second case, the utility function is additive, thus the risks in different dimensions of wealth are mutually independent. To further understand the meaning of the matrix measure of risk aversion, we introduce the directional risk premium and the directional matrix measure of risk aversion. We assume that the risk premiums can be paid only in one of the directions of the wealth vector. Then the risk premium in kth direction 1r(k) is the kth element of the risk premium vector 1r = (1rj) satisfying 1!'·

J

= { 1r(k) l 0,

if j = k; ifj # k

for any k 1, ... , K. From (3.8), the directional risk premiums can be approximated by (3.13) DEFINITION 5

The matrix measure of risk aversion in the kth direction is

fork= 1, ... ,n.

The directional matrix measures of risk aversion are positive definite since E[H(X)] is negative definite and E[uk(X)] > 0 for all of k. Comparing this definition with the definition of absolute risk aversion matrix yields (3.14)

where Diag ＨｾＩ＠ for a ｶ･｣ｴｯｲｾ＠ means the matrix with diagonal elements ｯｦｾＮ＠ Since the elements in the diagonal matrix are the marginal rates of substitution of utility in one direction for the other, the directional matrix measures of risk aversion are determined once the matrix measure of risk aversion R(X) and these marginal rates of substitution are known. On the other hand, since the kth row of R(X) is identical to that of R(k)(X), the matrix measure of risk aversion is a combination index of the set of directional matrix measures of risk aversion. The approximation of the risk premium in the kth direction is 7r(k). From (3.13), it follows that (3.15)

350

Y Li and WT Ziemba

282

Y. Li, W.T. ZiembafUnivariate and multivariate measures

THEOREMS

Given that (Al)-(A5) are valid,

(i)

ir(k)

> 0 if EX+Y- Ex is positive definite;

(ii)

ir(k)

=0

(iii)

ir(k)

< 0 if Ex+Y- Ex is negative definite,

for all k

if Ex+Y = Ex;

= 1,2, ... ,K.

The proof is given in appendix B. It is well known that if all individuals' preferences are represented by uni-

dimensional increasing concave utility functions, and the random variables have a joint normal distribution, the risks of the random variables can be completely represented by the variances. With the assumption that all individuals have multiattributed concave utility functions, and the random vectors have a joint multivariate normal distribution, the risks of the random vectors can analogously be represented by the covariance matrices. One random vector can be said to be more (less) risky than the other if the difference between the first covariance matrix and second covariance matrix is positive (negative) definite. The results in theorem 8 can be interpreted as follows: the directional risk premium is positive if the risk with two components of random wealth is higher than that with only one component; the directional risk premium is negative if the risk with two components of random wealth is lower than that with only one component. Therefore, the directional risk aversion matrices can yield the desirable results with multi-attributed utility functions and joint normally distributed wealth vectors. Appendix A: The univariate case

The proof of lemma 2 is based on:

LEMMA A

g : lR ｾ＠

Let Let y be a normal random variable 'Yith mean P,ji and variance ｵｾＮ＠ lR be differentiable of order n. If E [g(' >(y)] exists for all i = 1, 2, ... , n and lim (y- J.Lyt-jg(j-I)(y)f(y)

y-+±oo

=0

for all}= 1,2, ... ,n,

Chapter 20. Univariate and Multivariate Measures of Risk Aversion

Y. Li, W.T. Ziemba/ Univariate and multivariate measures

351

283

wheref(y) is the density of y, then

To,g

E[g(y)J,

T1,g

= ojE[g'(y)], E[(y ＭｾｴｹＩ

T2,g Tn,g

= (n-

Ｒ

ｧＨｹＩ｝＠

= ＱＷｾｅ｛ｧＨｹＩｊ＠

ＱＩＷｾｔｮＭＲＬｧ＠

+ 17fE[g"(y)],

+ (n- 2)17fTn-3,g' + 17fTn-2,g"

for n 2: 3,

where Tn,g = E[(y ＭｾｴＮｙｧＨｹＩ｝＠

Proof Sincef(y) is the density of the normal random variable y, 1

f(y) = vf21r !'(y)

=

.Y

17

ＨｹｾｐＬＩ＠

exp

(

(y

/Ly)2) ,

2o}

f(y), y

ＱＷｾ＠

f(y) = - _Y_j'(y). y -tLy

(A. I)

Then for n 2: 3,

l

+oo

-oo

ＭＱＷｾ＠

(y- IL.Ytg(y)J(y) dy

1-:

00

= ＭｬＷｾＨｹ＠

ＭｾｴｹｲｬｧＨＩｊＺ＠

- ｊ｟ｾ＠ ＱＷｾＨａ＠

(y- ILyr-lg(y)J'(y) dy

f(y)[(n

l)(y ＭｾｴｹＲｧＨＩ＠

+ (y -tLy)n-lg'(y)] dy)

+ B +C).

By assumption,

A= 0,

B = (n- l)Tn-2,g·

352

Y Li and WT Ziemba

284

Y. Li, W.T. Ziemba/ Univariate and multivariate measures

Substituting (A.l) into C yields

1:

00

c

-O'j

(y- J.Ljit- 2g'(y)f'(y) dy

-O'j((y- ｊＮｌｪｩｴＭ

-1:

00

ｵｾＨａｬ＠

Ｒ

ｧＧＨｹＩｦｩＡｾ＠

f(y)[(n- 2)(y- f.Lyt- 3g'(y)

+ (y- f.Lyt- 2g"(y)J dy)

+ Bl + Cl).

By assumption, AI = 0,

Bl = (n- 2)Tn-3,g''

Cl = Tn-2,g''·

Therefore, for n :2: 3,

Tn,g

= (n-

l)O'jTn-2,g + (n- 2)ujTn-3,g'

+ O'jTn-2,g''·

Hence,

= E[g(y)J, T1,g = ujE[g'(y)J,

To,g

T2,g = E[(y- f.Ly) 2g(y)J

= ujE[g(y)J + O'jE[g"(y)]. Proof of lemma 2

Cov ((.X- J.L.x) 2,g(y)) = E[(x- f.L.x) 2g(y)]- E[(x- J.L.x) 2]E[g(y)] =

E[(x- J.L.x) 2g(y)J- ｏＧｾｅ｛ｧＨｹＩｊＮ＠

The conditional distribution of .X given

and variance

y is normal with mean

0

Chapter 20. Univariate and Multivariate Measures of Risk Aversion

Y. Li, W. T. Ziemba/ Univariate and multivariate measures

353

285

From the conditional mean,

ks. Alternatively considering u for g fixed the investor must receive more return for the same risk as k increases because - 2kw 2 dominates ( (e + l)w ). Hence for a given g ask decreases the investor becomes more risk averse and, as would be expected, he increases the proportion of the riskless asset in his portfolio.

The Exponential Utility Function The exponential utility function

u(w)

(19)

=

1 -e

-aw

,

is concave if and only if a :i;; 0 and it has constant absolute risk aversion equal to a at all wealth levels. Freund [7] has shown that maximizing E 1[1 - e-•!'"], where ｾＬＮ＠ N L) is equivalent to maximizing x - ax' L x/2. Letting x = (a, 1 - a)', ｾ＠ = ＨｾｯＬ＠ R )' and

a,

t

means that one may solve (5) by solving lmax,P.(a)

= ｡ｾｯＫ＠

(1- a)R- a((l- a) 2 rrR 2)/2 -oo 0 and {j ｾ＠ 0. The function u is defined only for w ｾ＠ - {j/'Y and tends to - oo as w ----> - {j/'Y. As in the case of the power utility function it is appropriate to append a linear segment to u for w ｾ＠ w0 where Wo > -{j/'Y. Then

(23)

u(w) = u1(w) = u2(w)

Since w

= ｡ｾｯ＠

+ (1

loge (w - wo) ')'Wo

== 'Y {j + == log

([j

- a)R, w ｾ＠

+

+ log

([j

+

')'Wo)

f'W)

w0 if and only if R ｾ＠

ifw if w

(w 0 - ｡ｾ Ｐ ＩＯＨＱＭ

ｾ＠ ｾ＠

Wo Wo.

a) ==A. Thus

Chapter 22. Calculation of Investment Portfolios CALCULATION OF INVESTMENT PORTFOLIOS

387

221

08

05

QUARTERLY DATA l'o=0.93 w. =1.0)

0.4

0.2

00

0.1

0.3

0.5

0.4

0.7

-02

-0.4

VEARLV DATA ('b=0.95 Wo

Wo

=1.0)

= 1.0)

-0.6

-0.5

-0.8 6

FIGURE 5. Power Utility Function-Quarterly and Yearly Data: a* vs. {3.

FIGURE 6. Logarithmic Utility FunctionQuarterly and Yearly Data: a* vs. {3.

to calculate a* one must

max_.,., 0 and that the relevant expectations are finite. Computing the partial derivatives of the objective of (2) with respect to the portfolio proportions yields

aZ(x)jax;

ｅｾｻ＠

= =

ｵＧＨｷ

Ｐ ｾ［ｽ＠

Ｐ ･ｸＩｷ

｣ｯｶＨｷｾ［＠

ＬｵＧＨｷ

Ｐ ｾｔｸＩ＠

+ wJ;E 0 and d 2p.j da 2 > 0, i.e., the indifference curves of portfolio mean and variance are convex to the a-axis. It can be shown, see e.g., Rubinstein [19], under usual two parameter capital market equilibrium assumptions that ap.jaa = aR.

392

JG Kallberg and WT Ziemba

1260

J. G. KALLBERG AND W. T. ZIEMBA

4.

Empirical Results

A major conclusion of this study is that "similar" RA values yield "similar" optimal portfolios, regardless of the functional forms of the utility functions concerned. The empirical results are outlined after defining these measures of similarity. An appropriate measure of the proximity of portfolios x 1 and x 2 is the percentage cash equivalent difference; see Dexter, Yu, and Ziemba [4]: u- 1 {Z(x 1) } - u- 1 {Z(x 2)} X 100

u- 1 { Z(x 1)}

(5)

where Z(xi) = E {u(exi)}, i = 1,2. Recall that u- 1{Z(xi)} is the cash equivalent value of the random return ｾｔｸｩ＠ generated by portfolio xi. To formalize the concept of similar RA, let u be the base utility function to which comparisons of the other ui will be made. If f3 = ｾ＠ and x* solves (2), the average risk a version of u is

(6) We say ui has the same risk aversion as u if its average where ｷＭｎＨｾｔｸＪＬＲＺＩＮ＠ risk aversion is the same as u's over the distribution of (u-optimal) final wealth, i.e.,

(7) The solution of (7) is the appropriate value of f3i to compare ui with u. To simplify the solution4 of (7), we let exponential utility (u 2) be the base utility function (since with constant absolute risk aversion, the right-hand side of (7) is simply {3 2 , the parameter of the exponential utility function). This choice also simplifies the calculation of the percentage error because

see Freund [7]. The expected utility 5 may now be computed explicitly using only the 4 Equation (7) was solved using a hybrid bisection-secant algorithm. Difficulties in solving this equation 6 so that motivated the exponential extension described in footnote 5. In all cases we used a tolerance of the {31 chosen were within ± of the true {31 • See [II] for further details. 5 0f the seven utility functions used in this study only the exponential is well behaved over the whole real line as required for normally distributed final wealth. The other utility functions for certain values of w may be undefined (e.g., log, special exponential, power and negative power), may have decreasing marginal utility (e.g., quadratic) or may be convex (e.g., arctan). A common method of modifying the domain utilizes a linear segment near the boundary of the problem area, see [25] for technique. For the data and utility functions used here this method (usually) leads to a jump discontinuity in the RA(w) function. This may lead to either instability of the optimal portfolio weights and/or difficulties in solving (7). Since RA = 0 at w = 0 for the arctan function the linear segment suffices since it also has RA = 0 at w = 0. For the log, special exponential, power and negative power we appended an exponential segment at the points w 0 = 0.01, 0.25, 0.8, and 0.6, respectively, to yield the extended function

w-

w-•

u(w)

=

-

{

u(w) u(wo)

+ K(l-

e-RX(w-w))

where

Ro = A-

-

u"(wo) u'(wo)

and

u'(w 0 ) K=--.

Ro A

Using this u(w) guarantees that u, u', u" and hence RA are continuous. We did not append any segment onto the quadratic utility since u' < 0 only above {3 1 ""0.5. However, the u' ""0 behavior near 0.5 yields flat behavior in this region, see e.g., Figure Ia.

Chapter 23. Comparison of Alternative Utility Functions

393

1261

ALTERNATIVE UTILITY FUNCTIONS IN PORTFOLIO SELECTION TABLE I

Security Means and Variances Mean I. Cunningham Drug Stores 2. National Cash Register 3. Metro-Goldwyn-Mayer 4. Gillette Co. 5. Household Finance Corp. 6. H. J. Heinz Co. 7. Anaconda Co. 8. Kaiser Al. & Chern. 9. Maytag Co. 10. Firestone Tire and Rubber

Monthly Variance

1.0194 1.0173 1.0162 1.0137 1.0128 1.0112 l.Olll 1.0095 1.0084 1.0064

0.0105 0.0050 0.0148 0.0044 0.0052 0.0049 0.0081 0.0073 0.0048 0.0036

Mean

Yearly Variance

1.2852 1.2549 1.1819 l.l694 l.l484 l.l608 1.1711 1.0861 l.l292 1.0626

0.3276 0.2152 0.1711 0.0666 0.0517 0.1413 0.1865 0.0324 0.1676 0.0155

TABLE2 Monthly Security Correlation Matrix

2 I 2 3 4 5 6 7 8 9 10

1.000

0.2843 1.0000

4 0.1313 0.1333 1.0000

0.3049 0.3859 0.2882 1.0000

Security Number 5 6 0.1739 -0.0101 0.1251 0.2063 1.0000

0.3457 0.2334 0.3243 0.2750 0.2847 1.0000

7 0.4556 0.1880 0.2974 0.4110 0.2757 0.3031 1.0000

0.3033 0.4155 0.2010 0.3557 0.2670 0.1259 0.4546 1.0000

9

10

0.4229 0.2680 0.1762 0.2088 0.2528 0.3178 0.3445 0.3315 1.0000

0.2822 0.1332 0.1272 0.2616 0.3167 0.2501 0.3020 0.1968 0.1898 1.0000

given means and variances. Furthermore, since

the calculation of the cash equivalent is simplified. The data were obtained from the CRSP tape and consisted of monthly observations of ten well-known U.S. securities. These were selected from a group of 60 to form a sample that provided ample diversification and smooth transitions in means and variances (so that no securities were obviously dominant or dominated). Ten was chosen as the number of securities because it appears to be the optimum number when one balances the criteria of ease of exposition, interpretation of results, computational considerations, and diversification properties. (Evans and Archer [5] found that about 10-12 securities provide ample diversification for individual investors.) The relatively stable period January 1965-December 1969 was used in an attempt to satisfy the stationarity assumption made. The price relatives were adjusted to reflect investment of dividends, stock splits, etc. The 60 observations were used to obtain maximum likelihood estimates of the means and covariances of monthly returns. Tables I and 2 summarize these statistics. 6 6 Quarterly and yearly means and variances estimates were obtained in the usual way after computing the corresponding compound rates of return. The variances were then combined with the monthly correlation coefficients to generate the quarterly and yearly covariances using the formula aij(k) = pija;(k)aj(k), where k =quarterly and yearly, and i, j = I, ... , 10 with i 7" j. Note that these are no longer MLE. See [10] for more details.

394 t:::J

0\

N

ｾ＠

;J>

t"" t"" to

Corresponding Range Utility

Function

u(w)

Absolute Risk Aversion. RA

Theoretical Parameter

Parameter

Theoretical Range

Range

of RA with fixed

of RA with fixed w = I with Parameter

Range

Used

w=!

Range Used

trj

)0

Remarks u 1 begins with the highest mean return portfolio

Quadratic

w- /3w 2

2/3, 2/3 1w)

(I-

which is also a high variance portfolio. Then as {3 1 __,.0.50. RA __,. oo which yields the minimum 0-->w/2

0.....,.0.50

0-->

00

0->

00

variance portfolio. This is consistent with the theory that increasing RA yields more risk averse behavior, see Theorem I in Pratt [18].

u2 is quite similar to u 1 except that the final portfolio (when /3 2 = 10) has a low variance but

is not a minimum variance portfolio. There is a Exponential

Logarithmic

I - e- fl?"'

log( /3 3 + w)

/3,

I

(/3,+ w)

0--'>

00

0--'JoiO

-0.80--71.2

0-->

00

00

-->0

0---?10

5.0--70.45

gradual shift for increasing /3 2 out of the risky security I into the safer securities ＲｾＵＮ＠ The largest change occurs in the 0 < /3 2 = RA < 3 range. When /32 = RA ;;. 3 and there is little change in behavior. As u3 ranges from -0.60 to 1.2 there is ｲ･ｬ｡ｾ＠ tively little change in the optimal portfolio composition except for a gradual switch out of the safer securities 3, 5 into the riskier security I. Security 2 with its high mean and low variance always forms about half of the portfolio.

Cl

ｾ＠

0

ｾ＠

"ｾ＠ N

to

;J>

JG Kallberg and WT Ziemba

,... 0

TABLE 3 Monthly Empirical Test Data

TABLE 3 (continued) Monthly Empirical Test Data Corresponding Range Utility

Function

Special

e/34/w

Exponential

Power

(w- w 0)Ps w0 =

Negative Power

0.75

- (w- w 0)-lh

w0

=

0.50

Absolute Risk Aversion, RA

Parameter Range

Theoretical Range of RA with fixed

w = 1 with Parameter

Range

Used

w=l

of RA with fixed

Range Used

2w+ {34 0.10---+4.1

max(0,2w) 0--> 00

ｾ＠

2-->

2.1---+6.1

00

(I- /3,)

(w- w0 )

I+ [36 (w- w0 )

ｯｾｴＮ＠

0.05----* 1.0

0-->

00

0.10--72.1

4-->0

2-->

00

3.8 --+0

ＲＮｾＶ＠

Remarks

Arctan

tan -l(w + /37)

Ｍｷｾ｣ｯ＠

I +(w+/37) 2

ＰＮＱｾＴ＠

0-->0 max= 1.0

atfh=O

:;:: tn c -1

E

::;! '"Il

c z

g 0

z

"'z

identical to that of u 4 and indeed the optimal portfolio propol-tions are also nearly identical.

ｾ＠

fh E [0.10, 4.1]

and always yields high variance portfolios. As

1.0--+0.38

ｾ＠ ｾ＠

u5 initially yields low variance portfolios when {3 5 = 0.05 and RA = 3.8. However, aS /3 5 ___, I.O, R A drops sharply to 0 and the corresponding optimal portfolios have high variances. There is a gradual shift out of the safer securities 5, 7, 9 into the high return (and also high variance) Security I. u 6 has an RA range of 2.2--+ 6.2 which is nearly

u7 has low risk aversion for all 2(w+/37)

ｾ＠

There is relatively little variation in the optimal portfolio composition because RA ranges from 2.1 to 6.1 and hence never has low values. There is a gradual shift into the lower variance securities 4, 5 at the expense of the higher variance security 1 and security 2 (because of the diversification effects, in particular 2 and 5 are negatively correlated).

/h increases the portfolios become more and more risky since RA is decreasing.

t'"'

0

"'tn

ｾ＠

0 z

Chapter 23. Comparison of Alternative Utility Functions

u(w)

Theoretical Parameter

N

a-,

395

""

396

JG Kallberg and WT Ziemba

1264

QUADRATIC MONTHLY

Vl

z

0

i= c.: 0 a. 0 c.: a.

5

·' 10

3

9

6

ＫＭｲｾＬＮＱＳ＠ 05

35

25

-1

FIGURE

,g

Ia.

t.s

s(31

1.

1·2

Quadratic Monthly.

EXPONENTIAL MONTHLY

.a

Vl

ｾ＠

.6

1-

ｾ＠

a.

.s

0

c.: ·' a.

.2

FIGURE

lb.

Exponential Monthly.

,g

LOG MONTHLY .a

Vl

z 2

·6

1-

ｾ＠

5

a. 0

a: a. .' .3

.2

-. 8

-. 6

-. 4

FIGURE

I c.

·'

Log Monthly.

.a

Chapter 23. Comparison of Alternative Utility Functions ALTERNATIVE UTILITY FUNCTIONS IN PORTFOLIO SELECTION .g

·•

SPECIAL EXP MONTHLY

·1

Ul .•

z

Q

:ro .sr----Q.

0

a:

·4

Q.

.3

.2

.1

.s

.g

1·3

FIGURE ld.

2-5

17

2.9

3 7

4 1

Special Exponential Monthly.

.7

POWER

MONTHLY

.6

.5

Vl

z

0

i=

a:

0 0.. 0

a:

0..

.2

·1

.LｯＺＵ］ｾＮ＠ ＱｾＵ］［Ｎ＠

Ｔ］ＵＮＺ［ｾ＠

2:::5::::.;:35==;.

;;;,;;;;.6;:5;;;;;;;;;:;:7o:5=•ar5="'."\9r5--,,.o5

FIGURE le.

Power Monthly.

NEG. POWER MONTHLY

Vl

z

·4

0

f-

a:

.J

0 0.. 0

a:

·2

0..

·' FIGURE If.

Negative Power Monthly.

ｾ＠ s

397

1265

398

JG Kallberg and WT Ziemba

1266

J. G. KALLBERG AND W. T. ZIEMBA

ARCTAN MONTHLY

Vl

ｾ＠

i= ｾ＠

3

0.. 0

a:

2

0..

., ＫＬＭ｟ｲＵｧＮＳｾＱＷＲＹｔＴ＠

ｾ Ｗ＠ FIGURE

lg.

Arctan Monthly.

The seven utility functions used in this study along with the ranges of the various parameter values are described in Table 3. This table also gives the corresponding ranges of the absolute risk aversion and describes the qualitative behavior of the portfolio composition for changing parameter values. For comparison, the theoretically largest ranges of the parameters and corresponding risk aversion are also given. Plots of the optimal proportions of the investor's initial wealth to be invested in each security over various parameter values using monthly data appear in Figures la-lg. (Quarterly and yearly plots, omitted here because of space considerations, may be found in [9].) The optimal portfolio weights were calculated at 21 discrete values and the plots were drawn accordingly. Note the similarity between Figures ld (special exponential) and If (negative power); this occurs because the RA ranges are virtually identical 2.1---* 6.1 and 2.2---* 6.2, respectively. Table 4 summarizes the solutions to the portfolio problems7 classified by RA = 0.5, I, 2, 4, and 10. For each value of RA the portfolio weights across various utility functions are very close and in all cases the percentage error is less than 10- 4 % (recorded as zero). Table 5 provides portfolio The portfolio weights using yearly data for the various utility functions and ｾＮ＠ weights for different utility functions while not as close as with the monthly data are still reasonably close. They tend in the yearly as well as monthly data to be closer for smaller values of RA because at the higher level of risk aversion the resulting utility function equivalents have more "curvature" hence approximations are less accurate. The percentage error with the yearly data is still quite small and is always less than I%. Similar tables for quarterly data appear in [9], [ 10] and give results intermediate to the monthly and yearly values; the maximum percentage error is less than 0.2%. Of importance is the significance of particular RA values for the composition of the optimal portfolio. The results in Figures la-lg, and Tables 4 and 5, provide guidance regarding the significance of the (changes in) the magnitude of RA. For example, RA values greater than about 4 always yield very risk averse portfolios with low variance, and there is little change in the optimal composition even for large changes in RA. The range 2 < RA < 4 yields moderately risk averse portfolios with a modest degree of change in the optimal portfolio composition when RA changes. The range 0 < RA < 2 yields risky portfolios and there are dramatic changes in the optimal portfolio composition for even small changes in RA . 7 Additional calculations for different Kallberg [9].

RA

values, omitted here because of space considerations, appear in

ｾ＠

TABLE4 Optimal Portfolio Weights for Alternative Utility Functions and RA Values: Monthly Data

Parameter

Exponential

Quadratic

Log

Power

Arctan

Exponential

Quadratic

Log

Power

0.5

0.165483

0.984259

0.879183

2.714659

1.0

0.247237

-0.013835

0.750988

I

0.618500

0.618500

0.621151

0.611083

0.626174

0.414406

0.418291

0.418190

0.410246

2

0.381500

0.381500

0.378849

0.388917

0.373826

0.506770

0.508365

0.508198

0.507673

0.078824

0.073344

0.073612

0.082082

3

ｾ＠ ｾ＠

;:; § t: ::! '"11

4 Security

ｾ＠

5 6

B

7

Negative

Quadratic

Special Exponential

0.440442

8.156772

4.109086

I

0.179684

0.177566

0.169275

0.177531

0.118851

0.095302

0.079358

0.080020

2

0.438363

0.447755

0.440878

0.436866

0.342845

0.364688

0.336837

0.333334

3

0.086301

0.088783

0.089433

0.085550

0.062053

0.068848

0.053716

0.053053

4

0.051708

0.045061

0.054509

0.054076

0.120721

0.129129

0.129992

0.130616

5

0.243944

0.240835

0.245978

0.245978

0.264604

0.278670

0.259364

0.260044

0.044528

0.043953

0.062782

0.062214

6

I

Power

tr1

§ t: ｾ＠

'11

c

z

B 0

7

z "'z

8 9

0.008197

0.009482 i

0.646398

0.019411

0.069754

0.071237

Mean

1.016303

1.016335

1.016260

1.016282

1.015080

1.015229

1.014559

1.014531

Variance

0.002509

0.002525

0.002487

0.002498

0.002113

0.002141

0.001994

0.001989

Expected Utility

0.982493

0.982493

0.942493

0.982493

0.999957

0.999957

0.999957

0.999957

0

0

0

0

0

0

0

"'

10

%Error

ｾ＠

t:

-

-

1 Only those utility functions for which (7) has a solution (i.e., such that RA is attainable) are included for each value of J[A. ' Zeros and blanks indicate values less than w- 6 • 2 For comparison the riskiest portfolio (i.e. highest variance) has mean and variance of (1.016216, 0.014840) and the equal investment portfolio x = (0.1, ... , 0.1) has mean and variance of (1.012607, 0.002232).

ｾ＠

0

z

Chapter 23. Comparison of Alternative Utility Functions

Utility Function

N

o--

\0

401

402

-

!j 0

TABLES Optimal Portfolio Weights for Alternative Utility Functions and RA Values: Yearly Data

Parameter

Exponential

Quadratic

Log

Power

Arctan

Exponential

Quadratic

Log

Power

0.5

0.148687

0.821859

0.907691

2.498551

1.0

0210483

-0.112980

0.765732

0.502147

0.480496

0.509709

0.342082

0.381592

0.287004

0.245520

9

0.497853

0.509913

0.490291

0.421457

0.454136

0.373936

0.339830

0.146545

0.146148

0.163999

0.141238

ｾ＠

I

0.511442

0.518551

2

0.488558

0.481449

3

0.009590

4 Security

!-


JG Kallberg and WT Ziemba

Utility Function

Utility Function

Special

Parameter

I

Security

Negative Power

Exponential

Quadratic

Log

2.0

0.275157

-0.587446

0.350128

0.413450

0.213501

0.174457

0.208418

0.116236

0.149547

0.081275

0.120428

Exponential

Power

2

0.262884

0.300979

0.202592

0.240565

0.167586

0.205169

3

0.126734

0.133008

0.115598

0.129158

0.099687

0.113806

4

0.094320

0.048109

0.159015

0.116267

0.184367

0.157239

5

0.341605

0.309487

0.406559

0.364464

0.417025

0.403389

6

ｾ＠

§

ｾ＠

'"11

ｾ＠

B ｾ＠

7

"'z

8 9 0.050061

10 Mean

1.206489

1.214433

1.193087

1.201245

1.180277

1.193834

Variance

0.053081

0.061813

0.041833

0.048167

0.035125

0.042347

Expected Utility

0.900422

0.900265

0.899992

0.900356

0.898764

0.900039

0.0684

0.1868

0.0286

0.7158

%Error

-

ｾa＠ I""'

ｾ＠

0.1665

I

Chapter 23. Comparison of Alternative Utility Functions

ｾ＠

TABLE 5 (continued) Optimal Portfolio Weights for Alternative Utility Functions and RA Values: Yearly Data

t::i

-...(

403

404

N -..! N

TABLE 5 (continued)

Optimal Portfolio Weights for Alternative Utility Functions and RA Values: Yearly Data Utility Function

Quadratic

Log

Special Exponential

Negative Power

4.0

0.351447

-0.832954

2.884400

1.443557

1

0.088239

0.082991

0.046975

0.021224

0.0476ll

Parameter

Exponential

10.0

Quadratic

0.423749

0.031553

Negative Power

4.935791 0.017099

2

0.169455

0.165982

0.11622

0.185274

0.112794

0.075474

0.050898

3

0.106894

0.106663

0.080160

0.104064

0.079600

0.056237

0.050898

0.041265

4

0.194026

0.198830

0.161247

0.048522

0.154474

O.lll908

0.094011

0.054%6

5

0.441385

0.445533

0.343318

0.44ll82

0.328958

0.237341

0.190958

0.135844

0.044526

0"" ｾ＠ >

E ｾ＠

6 7 8

0.087302

0.143023

0.184158

0.400185

0.470213

0.522144

9 0.252077

lO

0.199733

0.258232

Mean

l.l86170

1.185175

l.l51634

l.l58397

l.l49527

1.125199

1.108230

l.l01730

Variance

0.037743

0.037247

0.024382

0.027802

0.023756

0.017605

0.014485

0.013892

Expected Utility

0.988236

0.988236

0.987863

0.987589

0.987821

0.999969

0.999968

0.999%7

0.7030

0.7099

0.7827

0.2261

0.4728

%Error

-

0

-

----------- ------------

those utility functions for which (7) has a solution (i.e., such that RA is attainable) are included for each value of RA. Zeros and blanks indicate values less than 10- 6 . 2 For comparison the riskiest portfolio (i.e. highest variance) has mean and variance of (1.285196, 0.327557) and the equal investment portfolio x = (0.1, ... , 0.1) has mean and variance of (1.164958, 0.042719). 10nly

Cl

'

ｾ＠

0

ｾ＠ :-l

ｾ＠

N

> ""

JG Kallberg and WT Ziemba

Security

Exponential

Chapter 23. Comparison of Alternative Utility Functions ALTERNATIVE UTILITY FUNCTIONS IN PORTFOLIO SELECTION

405

1273

To illustrate the relative "riskiness" of the various optimal portfolios, consider the monthly data with an exponential utility function. Then the variances at RA = (0.5, I, 2,4, 10) are (0.55,0.4!,0.31,0.24,0.20) times the variance at RA = 0 (i.e., the highest return portfolio). This is further illustrated in Figure 2 which shows the percentage of maximum variance as a function of RA for monthly, quarterly, and yearly data. The results also yield the following conclusions: I. The special exponential and negative power utility functions yield very risk averse portfolios. This is in contrast with the positive power utility function which yields highly risky portfolios and moderately risk averse portfolios for different values of f3 5 • 2. The arctangent utility function usually yields highly risky portfolios and has risk aversion equal to one or less. 3. The quadratic, exponential and logarithmic utility functions yield the largest range in variation of RA and yield the safest and riskiest portfolios for extreme values of /3. Extremely low values of RA are easily implementable for all three utility functions. However, extremely high values of RA are not implementable for the exponential or logarithmic functions because one must either have /3 2 ｾ＠ oo or /3 3 ｾＭ w (which causes domain problems). Thus the quadratic utility function despite its two well-known limitations (u' < 0 for large w and ｒｾ＠ > 0) may well play a useful role as a computational surrogate for more plausible utility functions when the number of possible investment securities is large, say n ｾ＠ 50. 4. With horizons of a year or less one can substitute easily derived surrogate utility functions that are mathematically convenient for more plausible but mathematically more complicated utility functions and feel confident that the errors produced in the calculation of the optimal portfolios are at most of the order of magnitude of the errors in the data. Moreover there is a fairly well-defined tradeoff of computational accuracy versus computing costs. For rough calculations one can use the measure w0RA (w) and obtain reasonable results for small variance problems (e.g., monthly data). For the most accurate results one uses Rubinstein's measure in (7). The most cost effective measure is w 0 RA. 5. An implication of the results for econometric research is that based only on observed portfolio data it will be extremely difficult to discriminate amongst alternative models since these models will be observationally practically equivalent with different parameter values. 1 0

.g

w z

.?

0

4

;::[

ｾ＠

X 11.

'

0

yearly

FIGURE 2.

Percentage of Maximum Variance vs. RA.

406

JG Kallberg and WT Ziemba

1274

J. G. KALLBERG AND W. T. ZIEMBA

6. The results suggest that one can derive an analytically tractable utility function whose use will generate portfolio allocations nearly indistinguishable from those of a given risk averse utility function given a known distribution of returns. However if we substitute for u when the distribution is F(g) are the results robust if the distribution is really ｆＨｾＩ＿＠ Calculations in Kallberg and Ziemba [10] indicate that the maximum expected utility and optimal portfolio composition are relatively insensitive to errors in estimation of the variance-covariance matrix. However, errors in estimating the mean return vector do significantly change these quantities. 8

u

Appendix:

Optimality of Rubinstein's Risk Aversion Measure

THEOREM I. Suppose ｾ＠ E R" has a multivariate normal distribution with finite means and variances and that u; is twice continuously differentiable with u; > 0 and u(' < 0 for i = I, 2. Consider the following two portfolio problems (Pl)

{ｭ｡ｸｅＬｵ

Ｑ Ｈｷ { ｭ｡ｸｅＬｵ

(P2)

Ｑ ｾｔｸＩ＠

I e'x =I, x > 0}, and

Ｒ Ｈｷ

Ｒ ｾｔｸＩ＠

I e'x =I, x

> 0},

where W; refers to investor i' s initial wealth. Suppose that x* is an optimal solution to (PI) and that ｷ Ｑ ｅ＼ｵｾＨｷ

Ｑ ｾｔｸＪＩ＠

w2E 0,

and

0,

where i

=

0,

I, ... , n.

we have

Now cov(x, g,(y))

=

E(g'(y))cov(x,y),

whenever g is continuously differentiable, all expectations are finite and x andy have

8 This research was partially supported by National Research Council Grant No. 67-7147 and Canada Council Grant No. S75-l307·Rl. Without implicating them, we wish to thank M. J. Brennan, S. Ginsburg, R. C. Grinold, M. J. Gruber, D. Hausch, A. Kraus, M. Kusy, M. E. Rubinstein, G. Sick, and the referees for helpful discussions and comments on earlier drafts of this paper. Special thanks go to M. E. Rubinstein for formulating Theorem l for us and allowing us to publish this joint result here. Rubinstein devised a proof for the case when nonnegativity constraints are absent. The proof given here applies for the nonnegativity case. Cases when some or all of the assets may be sold short or long follow directly from this result.

Chapter 23. Comparison of Alternative Utility Functions

407

ALTERNATIVE UTILITY FUNCTIONS IN PORTFOLIO SELECTION

1275

a bivariate normal distribution. Thus cov( ｷ Ｑ ｾ［Ｌ＠

u;(w 1Vx))

= ｅｾＨ＠

ｵ［ＧＨｷ

Ｑ ｾｔｸＩ｣ｯｶＨｷ

Ｑ ｾ［Ｌ＠

ｷｴｅｾＨｵ［Ｇ

=

ｷ Ｑ ｾｔｸＩ＠

Ｑ ｖｸＩＲＺ｣ｯｶＨｾ［ＧｩＬ＠

(10) J

since

cov(u,Lv;) = Lcov(u,v;). Using (9) and (10) in (AI) yields A+ JL; = ｷ Ｑ ｾ［ｅＨ＠

u;( ｷ Ｑ ｾｔｸＩ＠

+ ｷｩｅｾＨ＠

u!(w 1Vx)) 2: ｣ｯｶＨｾ［＠

Ｇｾｩ＠

)xi.

J

The Kuhn-Tucker conditions for (P2) are then

"A+ (i;

(A2)

=

ｷ Ｒ ｾ［ｅＨ＠

u2(w2Vx)) + wiE 0,

=

0, and

where i

=

I, ... , n.

Now (AI) and (A2) are equivalent to:

(A+

/l;)/( w1E0 dwo

-

as

dR*(wo, Z*) 0, and R*(w0 , Z*) > 0, the result is immediate.

D

THEOREM 3. Under the hypotheses of Theorem 2, the mean return and variance of the optimal portfolio per dollar invested are increasing, constant, or decreasing functions ofinitial wealth as an investor's Rubinstein's measure ofrelative risk aversion is decreasing, constant, or increasing in his initial wealth. That is, (Var (Z*)) > 0

as

dR*(wo, Z*) d < 0; Wo

d (Var (Z*)) = 0 Wo

as

dR*(w0 , Z*) = 0; dwo

as

dR*(wo, Z*) > 0 dwo

(E[Z*]) > 0,

dd

_:!__

(E[Z*]) = 0,

d

/

(E[Z*J) < 0,

dd

(a)

/

(b) (c)

Wo

dwo Wo

Wo

Wo

(Var (Z*)) < 0

where w0 is the investor's initial wealth, Z * is the return per dollar invested and R * (w0 , Z *) is his Rubinstein's measure of relative risk aversion. · PROOF. Taking the derivative of E[Z * J with respect to w0 yields d E[Z*] dwo

dA* (E[X]- E[Y]). dwo

=-

Taking the derivative ofVar (Z*) with respect to w0 yields d -d Var (Z*)

=

Wo

dA* 2 - d (A* Var (X- Y)

+ Cov (X- Y, Y))

Wo

=

2 dA* E[X]- E[Y] dwo R*(wo, Z*)

using (2.5). Since E[X] > E[Y], and R*(w0 , Z*) > 0, by Theorem 2, the proof is complete. For the exponential utility function u(x) = -e-#x (fl > 0), R*(wo, Z*)

=

wofl.

=

f1 > 0.

D

Thus dR*(w 0 , Z*) d Wo

Therefore, Theorems 2 and 3 imply that if an investor has an exponential utility function,

Chapter 24. Characterizations of Optimal Portfolios CHARACTERIZATIONS OF OPTIMAL PORTFOLIOS

415

265

as his initial wealth increases, the optimal allocation of his portfolio with two risky investments have bivariate normally distributed returns to the investment with the higher mean return will decrease. Thus the mean return and the portfolio variance fall. For other utility functions, it is usually difficult to determine the increasing, constant or decreasing property of Rubinstein's measure of relative risk aversion in initial wealth. Fortunately, however, as Kallberg and Ziemba ( 1983) illustrated, for small variance problems (e.g., daily, monthly, or quarterly return data), Rubinstein's measure ofrelative risk aversion can be approximated by the Arrow-Pratt measure of relative risk aversion to obtain very accurate optimal portfolio weights and virtually identical expected utility within the limits of data estimation errors. 6 Thus in these cases, Rubinstein's measure of relative risk aversion can be expected to have the same increasing, constant or decreasing property in initial wealth as the Arrow-Pratt measure of relative risk aversion for any particular utility function. For example, as the logarithmic utility function u(x) = log (x + {3) has decreasing, constant or increasing Arrow-Pratt measure of relative risk aversion as f3 < 0, f3 = 0, or f3 > 0, respectively, Rubinstein's measure of relative risk aversion will also have these properties for the portfolio problems with investment horizons one quarter in the future or less. Therefore, for investors with the logarithmic utility functions, the wealth effect on optimal mixtures and risk-return characteristics of portfolios can be determined by Theorems 2 and 3, if the portfolios consist of two investments with bivariate normally distributed returns and the investment horizons are in a suitable range. 3. Risk Aversion Matrix and Optimal Portfolio

We start with a description of the portfolio model with a multi-attributed utility function. Assume that there are two risky investments with nominal returns X and Y and K factors FT = (Fh F 2 , ••• , FK) which affect the real value ofwealth. 7 The two returns and K factors are assumed to have a joint normal distribution. Consider that an investor is to allocate his initial wealth w0 between the two investments. We assume that in the existence of the uncertain factors, a risk-averse investor's preference for the real value of wealth can be represented by a multi-attributed increasing concave utility function u( w, F), where w represents the nominal value of the final wealth, and F is the vector of factors affecting the real value of wealth. Assume also that the investor's objective is to maximize the expected utility of the real value of his portfolio at the end of his investment horizon. Let a be the amount of dollar allocation of the initial wealth w0 to the investment with return X. Then his decision problem is max V(a) == E[u(a(X- Y)

+ w0 Y,

F)].

(3.1)

O.:o:;aswo

This model is an extension of the portfolio selection model in §2 and is developed from a generalized CAPM model in Losq and Chateau ( 1982). Suppose that the optimal allocation of the portfolio is a non-trivial allocation of investments. Differentiating ( 3.1) with respect to a yields the first-order condition V'(a) = E[uw(W*, F)(X- Y)] = 0

(3.2)

6 The approximation error depends nonlinearly on the variance of final wealth. When the variance of final wealth approaches zero, the average of the Arrow-Pratt measures converges to Rubinstein's measures. Although Kallberg and Ziemba (1983) have given empirical estimates of this error, theoretical approximations and/or bounds have not been developed. 7 For example, consider a portfolio problem with a mixture of investments in a mutual fund and a money market account with interest-rate risk, where the future inflation rate could be one of the factors.

416

Y Li and WT Ziemba

YUMING Ll AND WILLIAM T. ZIEMBA

266

=

where W* a* (X- Y) =Bu(w, F)jaw. Since

+ w0 Y is the terminal

value of the optimal portfolio, and u"'

Cov (A, B)= E[AB]- E[A]E[B] for random variables A and B, (3.2) can be written as Cov (uw(W*, F), X- Y)

=

-E[uw(W*, F)]E[X- Y].

(3.3)

We now present the key tool for the analysis in the multivariate case, which is the multivariate version of the Stein/Rubinstein covariance operator generalized by Gassmann (1987). LEMMA 4 (Multivariate Covariance Operator). Let X, Y be multivariate random vectors on R"' and R", respectively, with joint distribution

Let g: R" ｾＭＫ＠ Rk be differentiable with Jacobian matrix Jg of dimension k X n. Assume that Cov (X, g(Y)) and E[Jg(Y)] exist. If lim g(yo

+ ty).fv(Yo + ty)

=

(3.4)

0

t-±w

for all y, Yo E R", where fv is the probability density of Y, given by

JiY ( y) -_ ( 2 ·n-)" /2 1I Lv I 1/2 exp( -

1

2 ( y - !LY)

7

-1

Lv ( y - !LY ) ) '

then Cov (X, g(Y))

=

Cov (X, Y)E[Jg(Y)]T.

(3.5)

MULTIVARIATE REGULARITY CONDITION. Let u(x): R" ｾＭＫ＠ R be a multi-attributed utility function. Then u is said to satisfy a multivariate regularity condition if the conditions of multivariate covariance operator are satisfied with the gradient of u, \lu in place of g. LEMMA 5. Assume that the bivariate random variables X, Y, and the random vector F of dimension K have ajoint normal distribution. Also assume that u(w, F): RK+I ｾＭＫ＠

R 1 is an increasing concave utility function which is twice continuously differentiable and satisfies the multivariate regularity condition. If 0 < a* < w0 , then the optimal allocation of the portfolio a* satisfies

+ w0 Cov (Y, X-

Var (X- Y)a*

Y)

E[X -* Y] Rww(W , F) where (R"'"'' RwF1 , aversion 8 R(W*

'

••• ,

ｫｾｬ＠

f Cov

(Fk. X- Y) RwFk(W:, F)

(3.6)

Rww(W , F)

RwFx) is the first row of the matrix measure of absolute risk

F)=(- E[uu(W*, F)]) E[u;(W*, F)]

for

i,j = W, Fl' ... 'FK,

U; is the first-order partial derivative of u with respect to the ith dimension of u and uu is the second-order partial derivative of u with respect to the ith and jth dimensions.

8

See Li and Ziemba ( 1987) for the derivation and the properties of this measure.

Chapter 24. Characterizations of Optimal Portfolios

417

CHARACTERIZATIONS OF OPTIMAL PORTFOLIOS

267

PROOF. Applying the multivariate covariance operator to ( 3.3) and rearranging terms yields ( 3.6). The proof appears in Appendix. D Consider the portfolio problems for two investors with the same initial wealth w0 • Assume that u 0. Now (2) is a parametric concave program which would generally be difficult to solve, for all fJ > 0. It is easier to bypass the calculation of the J-t-d curve if the risk-free asset is not available (see Section IV). However, when the risk-free asset does exist, as we are assuming, it is convenient to consider points that are linear combinations of x 0 = 1 (total investment in the riskfree asset) and points that lie on cp({J). These combinations correspond to straight lines in J.l-d space as well because the mean is linear and the a-dispersion measure is positively homogeneous [i.e., f(U) = J...f(x) for all A. ;;:; OJ in the X;. Clearly the best points will lie on the line L that is a support to the concave function cp ({J) (proof below); see Fig. 1. Such heuristic arguments indicate that the J-t-d efficient surface is now L, and one gets the analog of the Tobin separation theorem for normal distributions which has the important implication that x;* JL.'J= 1 x/, i-# 0, is independent of u and of initial wealth. The result may be stated as the Separation Theorem (2')

max('x,

Let the efficiency problem be

s.t. f(x)

ｾ＠

fJ,

e' x = w,

i ;;:; 0,

x 0 unconstrained.

In (2') f is the a-dispersion measure, w is the initial wealth, and explicit consideration of the risk-free asset (i = 0) is allowed. Suppose that asset i = 0 has no dispersion and that borrowing or lending any amount is possible at the fixed rate ( 0 . Assume thatfis convex and homogeneous of degree I and that u is concave and nondecreasing. 7 Samuelson [29] analyzes a problem similar to (2) in which one minimizes dispersion subject to the mean equalizing a given parameter. He notes that his problem is a convex program and he analyzes it using the Kuhn-Tucker conditions. His paper also contains some illustrative graphical results for some special cases. See also Fama [5] for a similar analysis, presented in the context of the Sharpe-Markowitz diagonal model, that is particularly concerned with diversification questions.

Chapter 25. Choosing Investment Portfolios

429

CHOOSING INVESTMENT PORTFOLIOS

a -dispersion

Fig. I

(a) Total separation: If x 0 * =1= 0, the relative proportions invested in the risky assets, namely x;*/L.j= 1 x/, i =I= 0, are independent of u and initial wealth w. (b) Partial separation: If x 0 * = 0, all investment is in the risky assets and L.}= 1 x/ = w and the x/ are independent of u. Proof (a) Suppose x 0 * =I= 0. Since x* solves (2') it must satisfy the Kuhn-Tucker conditions: (i) (ii) (iii)

ｦＨｾＩ＠

ｾ＠

ｾ＠

fJ,

ｾ＠

0,

ｾ［ＭｽＮＨｯｦＯｸＩ

(a) (b) ｾｯ＠

- J1

= 0,

ｸ［ＨｾＭａＮｯｦｬＩｊＬ＠

e'x = w, Jl ｾ＠ 0, i = l, ... ,n. }.. ｾ＠ 0, = o, i = 1, ... ,n.

It will suffice to show that for all u > 0 there exists a y =I= 0 such that x** = (yx 0 *,ux 1 *, ... ,ux.*) also solves the Kuhn-Tucker conditions (which are necessary and sufficient) for all P** ｾ＠ fJu. Condition (i) is satisfied because ｦＨｾＪＩ＠

ｾ＠ ｾＪ＠

ｾ＠

p

=>

0

=>

= uf(x*) ｦＨｵｾＪＩ＠ ＰＧｾＪ＠

ｾ＠

+I X;*= i= I

ufJ

ｾ＠

0,

n

Xo*

ｾ＠

n

=>

YXo * + 0'

I

i=l

X;*

= 1

P**,

430

WT Ziemba W. T. ZIEMBA

Now ofjox; is homogeneous of degree 0, hence i=.

iff ｾＮ＠ - Aof(x*) < ｾ＠

_A of(ux*) :5 e

.,,

"' uX;

-

0, (iii) is equivalent to (ii(a)) which is satisfied for u.X* iff .X* satisfies (ii(a)). By Theorem 2 an optimal solution to (2') must solve (1); hence Li= 1 x;* = w and the x;* are independent of the u. The proof and statement of the separation theorem given here are similar in spirit to that given by Breen [2]. Breen considered an efficiency problem in which one minimizes ｣ｸｾ､ｩｳｰ･ｲｯｮ＠ given that expected return is a stipulated level as well as some alternative assumptions regarding the risk-free asset. For the analysis indicated in Fig. 1 to be valid it is necessary that the aＱ dispersion measuref(x) {2:1= 1 ｓ［ｸＢｽ ｾ＠ be convex and that cp be a concave function of {3. The function f is actually strictly convex as we now establish using

=

Theorem 3

(Minkowski's inequality) Mr(Y)

Suppose

={(1/n) it y{rr.

n

1

ｾ＠

r > 1,

n


Mr(a+b).

Proof See, e.g., Hardy et a/. [13, p. 30].

Let P = {.X IＮｘｾ＠ Lemma 1 f(x)

2

ｾ＠

IX

0, e'x = 1}.

= {2:7= 1 S;xt} 11a

is a strictly convex function of .X on P if

> 1 and S; > 0, i = 1, ... , n.

Proof Let a; = AS; x/ and b; = (1- A) S; x/, i = 1, ... , n, where .X 1 =1=

and 0 < A < 1. Then by Minkowski's inequality

{o!n) it1 ｡［Ｂｲｾ＠

+ {o!n)

it1 ｢［Ｂｲｾ＠

>

{o!n) Jl Ｈ｡［Ｋｨｴｲｾ＠

x2 ,

Chapter 25. Choosing Investment Portfolios

431

CHOOSING INVESTMENT PORTFOLIOS Ｑ Hence ｾｦＨｸ Ｉ＠ + (l-'A)f{x 2 ) > f{'Ax 1 + (l-'A)x 2 }. Thus f is strictly convex unless a and b are proportional. But this requires that there exist constants q 1 and q 2 not both zero such that q 1 a= q2 b or that

q 1 A.x/ =q 2 (1-A.)x/

=>

S; > 0)

(since

or that x/ and x/ are proportional since q 1 ' = q 1 ). and q 2 ' = q 2 (1- 'A) are both not zero. However, the condition that L:7= 1 x/ and L:7= 1 x/ = 1 means that x1 and x2 cannot be proportional unless x/ = x/ for all i, which is a contradiction. Remark f is convex but not strictly convex on M = is linear on every ray that contains the origin.

{xI x ｾ＠

0} because f

Lemma 2 qJ is a concave function of f3 > f3L, where f3L > 0 is defined below, if 2 ｾ＠ rx > 1 and S; ｾ＠ 0, i = 1, ... , n.

Proof Let Kp ={xI e'x = 1, x ｾ＠ 0, j{x) ｾ＠ /3}. Let PL> 0 be the smallest f3 for which Kp # 0. Clearly PL ｾ＠ f>, where ｦ＾ｾ＠ =min; S;, and Kp # 0 if and only if f3 ｾ＠ PL· Choose /3 1 ｾ＠ PL and /3 2 ｾ＠ f3L· Since (2) has a linear objective function and a nonempty compact convex feasible region there is an optimal solution for all f3 ｾ＠ PL· Let optimal solutions when f3 equals /3 1 and /3 2 be x1 and x2 , respectively. Consider

/3;. = x;. =

0

(1-A.)/31 +A.f3z,

ｾＩＮ＠

ｾ＠

1,

(l-A.)x 1 + AX 2.

Now X;. is feasible when f3 = By the concavity of h(x) = ｾＧｸＬ＠

/3;.

because the constraint set of (2) is convex. (i)

But h(x;)

since

x; is optimal,

i

=

(ii)

qJ (/3;)

= 1, 2, and (iii)

since

({J(/3;.)

denotes the maximum when f3=

qJ ({3 ;.) ｾ＠

(1- A.) qJ (/31)

/3;..

Combining (i)-(iii) gives

+ AqJ (f3z).

432

WT Ziemba

W. T. ZIEMBA

One may find the slope of Land the point M (see Fig. 1) by maximizing

(3) g (xA)

ｾ［ｘＭｯ＠ S ·"} 11,. , = ｻｾ＠ Li'=t L..•=l ;X,

By letting ｾ＠

=

ｾＭ

0

"

s.t.

A

;;;;

,

Ｐ Ｌ＠ i = 1, ... ,n, and utilizing g(x) =

{

ｾ＠

n

ｾ［ｸ＠

_

}

I{ ｾ＠

n

= 1. ･Ｇｾ＠

e'x = \ 1/IZ

S;x;"'J

1,

.

We will assume that ｾＧｸ＠ > 0 for all "interesting" feasible x. This is a very minor assumption since the fact that the x ;;; 0 and e'x = 1 implies that there always exists an x such that ｾＬ＠ x > 0 unless all ｾ＠ ｾ＠ 0 in which case it is optimal to invest entirely in the risk-free asset. It will now be shown that (3) has a unique solution. A differentiable function O(x): A-> R is said to be strictly pseudo-concave on A c Rn if for all x, .X E A, x #- .X (x-x)'VO(.X)

ｾ＠

B(x) < O(.X)

=

0

[V denotes the gradient operator so VB= (88j8xt> ... ,aejaxn)']. Geometrically functions are strictly pseudo-concave if they strictly decrease in all directions that have a downward- or horizontal-pointing directional derivative. A normal distribution is such a function in R. Theorem 4 Suppose the differentiable functions '¥ and 1/J are defined on A, a convex subset of Rn, and that'¥ > 0 is concave and 1/J > 0 is strictly convex. Then () = '¥ 11/1 is strictly pseudo-concave on A. Proof VO = {(1/!V'¥- 'I'VI/1)/1/1 2 }. Let .X EA. Thus = {1/J(.X)V'I'(.X)-'I'(.X)VI/I(.X)}'( --) < O V()( X-)'( X --) X [I/J(x)] 2 X X = ,

X E

A,

X#-

X

= {1/!(.X)V'I'(.X)-'I'(.X)Vt/J(.X)}'(x-x)

ｾ＠

0

1/J (.X)['¥ (x)- '¥ (x)J - ['¥ (x) Vt/J (x)]' (x- x) ｾ＠

1/J(x)['l'(x)-'l'(x)J

+ 'l'(x)[I/J(x)-1/J(x)J
0 which implies that

0) ·

since 1/1 (x) > ( andlj!(x)>O

Chapter 25. Choosing Investment Portfolios CHOOSING INVESTMENT PORTFOLIOS

Lemma 3 Suppose/: A--+ R, where A is a convex subset of Rn, is differentiable at x, and there is a direction d such that Vf(x)'d > 0. Then a u > 0 exists such that for all r, u ｾ＠ r > O,f(x + rd) > f(x). Proof See Zangwill ([32, p. 24]. Theorem 5 Suppose O(x): A--+ R, where A is a convex subset of Rn, is strictly pseudo-concave. Then the maximum of 8, if it exists, is attained at most at one point x E A. Proof Case I: 3x E A such that VO(x) = 0. Then by the strict pseudoconcavity of(), i.e., V()(x)'(y-x) ｾ＠ 0 ｾ＠ ()(y) < O(x) for ally E A, x is the unique maximizer. Case 2: ｾｸ＠ E A such that VO(x) = 0. Suppose x maximizes e over A. Then Vy =1- x, V8(x)'(y-x) ｾ＠ 0 ｾ＠ ()(y) < ()(x) and x is clearly the unique maximum unless 3y E A such that V()(x)'(y-x) > 0. But by Lemma 3 that would imply that there exists a r > 0 such that O(x+r(y-x)) > 8(x), which contradicts the assumed optimality of x. (Note that the point [x+r(y- x)] = [ry+(l-r)x] E A, by the convexity of A.) Theorem 6 (3) has a unique solution. Proof The function g may be seen to be strictly pseudo-concave on P by letting \f'(.X) =''.X, which is positive and concave, and tf;(.X) = {L7=t S;xt} 11 a, which is positive and strictly convex. Hence by Theorem 5, it has at most one maximizer. But P is compact and g is continuous; hence g is maximized at a unique point x* E P.

Suppose that the fractional program (3) has been solved to obtain .X*. The problem then is to determine the optimal ratios of risky to nonrisky assets. The risky asset is

R

］ｾＧＮｘＪＢ＠

ｆＨｾＧｸＪ［＠

t

ｾＧｸＪＬ＠

S;(.X;*Y,O,IX) =: F(R; R,SR,0,1X),

•=1

=

where ｾ＠ ¢ 1 , ... , ¢., and the best combination of Rand the risk-free asset may be found by solving max ｜ｦＧＨａＮＩ］ｅｒｵ｛ｾ

(4) Ｍ｣ｯ＼ＮｾＡ＠

Ｐ ＫＨＱＭａＮＩｒ｝＠

Problem (4) is a stochastic program with one random variable (R) and one decision variable (A.). Since u is concave in w it follows that \f' is concave in A.. Under the assumptions of Theorem 1, \f' (A.) and d\f' (A.)/dA. will be bounded.

433

434

WT Ziemba W. T. ZIEMBA

Now (5)

'P(.A.) = =

Ｐ ＫＨＱＭＮａＩｒ｝ｦ＠

J_'Xloo ｵ｛Ｎａｾ

t:

dR

ｵ｛ＮａｾｯＫＨｬＭＩｻｒｓ Ｑ

｡＠

R.}]j(R.) dR.,

where the standardized stable variate., 1{

R-R =(SR)lfa ｾ＠

F(R.; 0, l,O,cc).

The continuous derivative of 'I' is (6)

Joo go-R-(SR)lfa R.} -oo

d'P(.A.) = d).

X

､ｵ｛ＮａｾｯＫＨｬＭＩｻｒｓｦ｡＠

R.}] J(R) dR..

dw Problems having the general form of (4) are generally easy to solve by combining a univariate numerical integration scheme with a search procedure that uses function or derivative evaluations (see Ziemba [35]). The difficulty here is that the density j(R.) is not known in closed form except for very special cases such as the normal (ri = 2) and Cauchy (ri = 1) distributions. However, Fama and Roll [7] have utilized series expansions ofj(R.), due to Bergstrom, to tabulate approximate values of j(R.) and F(R.). The tables consider \i = 1.0, 1.1, ... ,2.0 and R. = ±0, 0.05, ... , 1.0, 1.1, ... ,2.0, 2.2, ... ,4.0, 4.4, ... , 6.0, 7.0, 8.0, IO.O, 15.0, and 20.0 (a grid of 50 points). One may then utilize the tables to get a good approximation to (5) of the form (7)

m

Lｐ

'P(.A.) ｾ＠ ｊｾｬ＠

Ｑ

Ｑ

Ｐ ＫＨＱＭＮａＩｻｒｓ ｵ｛Ｎａｾ

where ｐ Ｑ ］ｐｲｻｒＮ Ｑ ｐ Ｑ ］ＱＬ＠ Ｑ ｽ＾ＰＬｌＺｪｾ symmetric). The approximation to (6) is (

8)

d'I'(.A.) ｾ＠ d A1 -

ｦ｡ｒ｟Ｉ｝Ｌ＠

and m=lOO (recall that

ｾ＠ P.{fl -R-(S )lfaR.} ､ｵ｛ａｾｯＫＨｬＭＮＩｻｒｓｦ｡｟ｊｽ｝＠

Ｑｾ＠

.L...

1

R

'>O

d

1

W

R

1s

•

One may then obtain an approximate solution .A.a to (5) via a golden section search using (7) or a bisecting search using (8) (see Zangwill [32] for details on these search methods.) The approximate optimal portfolio is then 8

Xo 0 = A0 ｷｨｯｳ･ｾ［Ｚ＠

8

and

xt =

ＨｬＭａ Ｐ

ＩｾｪＬ＠

i = 1, ... ,n.

Since the ,; 1 are independent it is known (see Samuelson [28]) that x1* > 0 for each i ｭｩｮｬＺｊＧＢｾﾷ＠

Chapter 25. Choosing Investment Portfolios CHOOSING INVESTMENT PORTFOLIOS

ID. The Dependent Case

=

A set of random variables r (r 1 , •.. ,r.) is said to be multivariate stable if every linear combination of r is univariate stable. The characteristic function of r is t/l,(t)

=t/l,(tl> ... , t.) =

Eit'r

=

s_: . .

J_""oo eit'r g(r) dr 1 ... dr.,

where g is the joint density of (r 10 ... , r.). If g(r) is symmetric, then the log characteristic function (see Ferguson [10]) is ln t/J,(t) = ib (t)- y(t), where the dispersion measure y(A.t 1 , ... , A.t.) = lA. Ia y (t 1 , ... , t.), i.e., positive homogeneous of degree a, y > 0, and the central tendency measure 0 if ＧｉＨｾＩ］ｻＡ＠ l:j= 1 ＨｾＧｮ Ｑ ｸＩＧ ＱＲ ｽ Ｑ ＢＧ＠ is a convex function of x. The pertinent facts relating to the convexity of 'I' are summarized in Theorem 7 (a)

(Independent case)

Suppose m =nand thejjth element of

n1 is w1 > 0 (and all other elements are zero). Then 'I' is strictly convex on P.

(b) (Totally dependent case) Suppose m = 1 and 0 1 is positive definite. Then/is strictly convex on any convex subset of R". (c) (General case) Suppose n G; m G; 1, each n x n matrix is positive semidefinite, and for at least one j, x'0.1 .X > 0 if x ｾ＠ 0. Thenfis convex on M. 9 Press [23] also proves that Ar+b is multivariate symmetric stable of form (9) if A is any m x n constant matrix and b is any n vector of constants.

Chapter 25. Choosing Investment Portfolios CHOOSING INVESTMENT PORTFOLIOS

Proof (a)

Let S1 > 0 be defined by w1 'P(x)

= {-t ｪｾＯＮｸＧｮ

Ｑ ｸＩＧ ＱＲ

ｲ

= (IJ/l)Sj1 2 • Then Ｑ＠

..

=

ｊｾ＠

s1 x/,

which was shown to be strictly convex on P in Lemma I. (b)

Hence '¥ will be strictly convex if