Stochastic Optimization Models in Finance

STOCHASTIC OPTIMIZATION MODELS IN FINANCE 2006 Edition illiam T. Ziemba • Raymond G. Vickson editors STOCHASTIC OPTIM...

Author: William T. Ziemba | Raymond G. Vickson

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STOCHASTIC OPTIMIZATION MODELS IN FINANCE 2006 Edition

illiam T. Ziemba • Raymond G. Vickson editors



editors

William T. Ziemba University of British Columbia, Canada

Raymond G. Vickson University of Waterloo, Canada

\[p World Scientific NEW JERSEY

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• SINGAPORE

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• SHANGHAI

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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 Stochastic optimization models in finance (2006 Edition) / edited by William T. Ziemba and Raymond G. Vickson. p. cm. Originally published: New York: Academic Press, 1975, in series: Economic theory and mathematical economics. Includes bibliographical references and index. ISBN 981-256-800-X (alk. paper) 1. Finance. 2. Mathematical optimization. 3. Stochastic processes. I. Ziemba, W. T. II. Vickson, R. G. HG106.S75 2006 332.01'51922-dc22

2006042611

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

Copyright © 2006 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.

Printed in Singapore by B & JO Enterprise

Dedicated to the memory of my father, William Ziemba, 1910-1967, from whom I inherit my interest in gaming and financial problems. W.T.Z. To Lynne, who displayed infinite patience. R.G.V.

CONTENTS

Preface and Brief Notes to the 2006 Edition

xiii

Preface in 1975 Edition

xxxi

Acknowledgments

xxxv

PART I.

MATHEMATICAL TOOLS

Introduction 1. Expected Utility Theory

3 11

A General Theory of Subjective Probabilities and Expected Utilities Peter C. Fishburn The Annals of Mathematical Statistics 40, 1419-1429 (1969) 2. Convexity and the Kuhn Tucker Conditions

/1 23

Pseudo-Convex Functions O. L. Mangasarian Journal ofSIAM Control A3, 281-290 (1965)

23

Convexity, Pseudo-Convexity and Quasi-Con vexity of Composite Functions O. L. Mangasarian Cahiers du Centre d'Etudes de Recherche Operationelle 12,114-122(1970) 3. Dynamic Programming

33 43

Introduction to Dynamic Programming W. T. Ziemba

43

Computational and Review Exercises

57

Mind-Expanding Exercises

67

PART II.

QUALITATIVE ECONOMIC RESULTS

Introduction

81

1. Stochastic Dominance

89

The Efficiency Analysis of Choices Involving Risk G. Hanoch and H. Levy The Review of Economic Studies 36, 335-346 (1969)

89

A Unified Approach to Stochastic Dominance S. L. Brumelle and R. G. Vickson

101

2. Measures of Risk Aversion

115

Risk Aversion in the Small and in the Large John W. Pratt Econometrica 32, 122-136 (1964)

115

3. Separation Theorems

131

The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets John Lintner The Review of Economics and Statistics 47, 13-37 (1965)

131

Separation in Portfolio Analysis R. G. Vickson

157


171


183

PART III.

STATIC PORTFOLIO SELECTION MODELS

Introduction

203

1. Mean-Variance and Safety First Approaches and Their Extensions

215

The Fundamental Approximation Theorem of Portfolio Analysis in Terms of Means, Variances and Higher Moments Paul A. Samuelson The Review of Economic Studies 37, 537-542 (1970)

215

The Asymptotic Validity of Quadratic Utility as the Trading Interval Approaches Zero James A. Ohlson vm

221 CONTENTS

Safety-First and Expected Utility Maximization in Mean-Standard Deviation Portfolio Analysis David H. Pyle and Stephen J. Turnovsky The Review of Economics and Statistics 52, 75-81 (1970)

235

Choosing Investment Portfolios When the Returns Have Stable Distributions W. T. Ziemba "Mathematical Programming in Theory and Practice," P. L. Hammer and G. Zoutendijk, eds., pp. 443-482. North-Holland, Amsterdam (1974) 2. Existence and Diversification of Optimal Portfolio Policies

243 267

On the Existence of Optimal Policies under Uncertainty Hayne E. Leland Journal of Economic Theory 4, 35-44 (1972)

267

General Proof That Diversification Pays Paul A. Samuelson Journal of Financial and Quantitative Analysis 2, 1-13(1967) 3. Effects of Taxes on Risk Taking

277 291

The Effects of Income, Wealth, and Capital Gains Taxation on Risk-Taking J. E. Stiglitz Quarterly Journal of Economics 83, 263-283 (1967)

291

Some Effects of Taxes on Risk-Taking B. Naslund The Review of Economic Studies 35, 289-306 (1968)

313


331


343

PART IV.

DYNAMIC MODELS REDUCIBLE TO STATIC MODELS

Introduction

367

1. Models That Have a Single Decision Point

373

Investment Analysis under Uncertainty Robert Wilson Management Science 15, B-650-B-664 (1969) CONTENTS

3 73 IX

2. Risk Aversion over Time Implies Static Risk Aversion

389

Multiperiod Consumption-Investment Decisions Eugene F. Fama The American Economic Review 60, 163-174 (1970)

389

3. Myopic Portfolio Policies

401

On Optimal Myopic Portfolio Policies, with and without Serial Correlation of Yields Nils H. Hakansson The Journal of Business of the University of Chicago 44, 324-334 (1971)

401


413


417

PART V. DYNAMIC MODELS Introduction

429

Appendix A. An Intuitive Outline of Stochastic Differential Equations and Stochastic Optimal Control R. G. Vickson

1. Two-Period Consumption Models and Portfolio Revision

453

459

Consumption Decisions under Uncertainty Jacques H. Dreze and Franco Modigliani Journal of Economic Theory 5, 308-335 (1972)

459

A Dynamic Model for Bond Portfolio Management Stephen P. Bradley and Dwight B. Crane Management Science 19, 139-151 (1972)

2. Models of Optimal Capital Accumulation and Portfolio Selection

487

501

Multiperiod Consumption-Investment Decisions and Risk Preference Edwin H. Neave Journal of Economic Theory 3, 40-53 (1971)

501

Lifetime Portfolio Selection by Dynamic Stochastic Programming Paul A. Samuelson The Review of Economics and Statistics 51, 239-246 (1969) x

517 CONTENTS

Optimal Investment and Consumption Strategies under Risk for a Class of Utility Functions Nils H. Hakansson Econometrica 38, 587-607 (1970)

3. Models of Option Strategy The Value of the Call Option on a Bond Gordon Pye Journal of Political Economy 74, 200-205 (1966)

525

547

547

Evaluating a Call Option and Optimal Timing Strategy in the Stock Market Howard M. Taylor Management Science 14, 111-120 (1967)

553

Bond Refunding with Stochastic Interest Rates Basil A. Kalymon Management Science 18, 171-183 (1971)

563

Minimax Policies for Selling an Asset and Dollar Averaging Gordon Pye Management Science 17, 379-393 (1971)

4. The Capital Growth Criterion and Continuous-Time Models

577

593

Investment Policies for Expanding Businesses Optimal in a Long-Run Sense Leo Breiman Naval Research Logistics Quarterly 7, 647-651 (1960)

593

Portfolio Choice and the Kelly Criterion Edward O. Thorp 599 Business and Economics Statistics Section. Proceedings of the American Statistical Association 215-224 (1971). Optimum Consumption and Portfolio Rules in a Continuous-Time Model Robert C. Merton Journal of Economic Theory 3, 373^113 (1971)

621


663


677

Bibliography

701

Index

715

CONTENTS

xi

PREFACE AND BRIEF NOTES TO THE 2006 EDITION Over the years we have been pleased that Stochastic Optimization Models in Finance has stood the test of time in being a path breaking book concerned with optimizing models of financial problems that involve uncertainty. The book has been well known and respected for its excellent fundamental articles that are reprinted and the several new articles specifically written for the volume, as well as additionally for its large collection of computational and review, and mind expanding exercises. All of these appear in five separate parts with introductions that tie the articles in this book to the problems, and these are further elaborated on in the exercise source notes that discuss the literature further. Many of the mind expanding exercises presaged important articles that later appeared in the financial economics literature. However, in recent years, the book has been hard for researchers, graduate students and professors to locate. Numerous publishers were interested in the problems as a separate book or a new fully reworked edition. We felt though that the book was a classic and should remain that way. Hence, we are extremely pleased that World Scientific is publishing this 2006 edition with the only addition being this new preface. The fundamentals in the 1975 Academic Press edition in Karl Shell's series of books on mathematical economics and econometrics are still fundamental, especially in static and dynamic portfolio theory, some thirty years later. However, the world in 2006 in academic finance, mathematical finance, financial derivatives, financial econometrics, stochastic optimization, and multiperiod stochastic programming and stochastic control optimization in finance and other areas has expanded enormously with much activity in leading trading and investment centers such as London, New York, Tokyo, Paris, Chicago, Boston, Sydney, Singapore and countless other places. New fields have evolved and many books and articles have been written in the large number of outstanding journals now publishing papers. These brief notes try to put Stochastic Optimization Models in Finance (2006 Edition) in some perspective emphasizing areas I am familiar with including some of my own work and how I am Xlll

using or others could use this book in courses, seminars and workshops that deal with useful theory for real investment problems. The 1975 edition was used by many students and faculty and helped evolve the then young field of optimization under uncertainty models of theory and economic results. Part I on mathematical tools focuses on expected utility theory, convexity and the Kuhn-Tucker conditions and dynamic programming. Each of these areas was well developed in 1975; however, progress has continued in all three areas. Fishburn's comprehensive theory of subjective probabilities and expected utilities still provides an excellent introduction to utility theory. A notable subsequent area of expected utility theory is the prospect theory of Kahneman and Tversky (1979, 1982), work for which Kahneman won the Nobel prize in economics in 2003. They devised a utility theory to analyze the notion that individuals typically fear loss greater than they enjoy gains, that there is framing of decisions with different decisions made in identical situations except the way it is presented and that low probability events are typically overestimated and high probability events underestimated. These ideas are used in many places in mainstream academic finance, financial engineering, fund management and other areas, and some date from much earlier. For example, articles on the favorite-longshot bias, related to the third Kahneman and Tversky area, were originally published in 1949 and 1956. They are reprinted in the racetrack efficiency studies volume of Hausch, Lo and Ziemba (1994). The bias there is that high probability events have somewhat higher expected average returns and low probability events are greatly overestimated. The advent of betting exchanges such as Betfair and rebate betting has flattened this bias somewhat but it is still strong; see Hausch and Ziemba (2007). Hausch, Ziemba and Rubinstein (1981) introduced the notion that biases might be exploitable in racetrack betting if you use probabilities from simple markets in complex markets. That paper and Hausch and Ziemba (1985) introduced Kelly log utility betting into racetrack betting models. This area has grown immensely as well and a recent survey with updated results is in Hausch and Ziemba (2007). Tompkins, Ziemba and Hodges (2003) show that biases are similar in the S&P500 and the FTSEIOO stock indices as well. The two papers by Mangasarian on pseudo-convex and composite functions remain classics to this day and are constantly used in portfolio theory by those who know about them. Dynamic programming remains an active field and it is frequently used in economic and financial studies. My (Ziemba) paper is still a useful introduction to the theory and concepts. See Bertsekas (2005) for a through treatment, in discrete and continuous time. Campbell and Viceira (2002) is a recent example

XIV

PREFACE AND BRIEF NOTES TO THE 2006 EDITION

and shows that current research in continuous time dynamic programming focuses on computational strategies and qualitative economic results. Cox and Huang (1989), Brennan and Xia (2000, 2002), Zhao et al. (2003) and others have used various types of dynamic programming analyses to study financial problems. These include a martingale decomposition of Harrison and Kreps (1979) and Harrison and Pliska (1981). There is currently a revival of interest in dynamic programming because of increased capabilities of computer facilities. Part II discusses stochastic dominance, measures of risk aversion and portfolio separation theorems. The 1969 Hanoch-Levy stochastic dominance paper is still the starting point in this area and provides graduate students and others with a good understanding of when distribution X is preferred to distribution Y. Besides having the basic first and second order stochastic dominance results, it contains the single crossing result that if distributions cross only once then the one with the higher mean has higher expected utility for all concave risk averse investors. This result shows when expected utility theory with risk averse preferences and Markowitz (1952) mean variance analysis coincide; see also the Samuelson and Ohlson papers in Part III on this. In my lectures I also refer to Rothschild and Stiglitz (1970) which uses the notion of using mean preserving spreads to show that more weight in the tails is equivalent to the second order stochastic dominance results. These economic results date back to mathematical treatments of Blackwell and Girchik (1954) and Hardy, Littlewood and Polya (1934), and Diamond and Stiglitz (1974) who show that concave in parameters investors bet less with increasing risk. Brumelle and Vickson (written for this book) go deeply into the mathematics of these concepts. The 1964 Pratt paper is still the fundamental one in risk aversion. It suggests good utility functions for use by individuals and organizations with decreasing absolute risk aversion, concavity and monitonicity. These include the double exponential (see Chapter 1 of Ziemba, 2003, for a worked out example) and log and negative power (see also Bell, 1995, for the case of linear plus exponential which has additional good theoretical properties). Rabin (2000) and Rabin and Thaler (2001) show that non-acceptance of small positive gambles implies non-acceptance of very favorable large gambles under the Pratt concave expected utility frame work. Less well known is Rubinstein's risk aversion measure which has optimality in financial equilibrium (Rubinstein, 1976) and portfolio theory (Kallberg and Ziemba, 1983) but is difficult to estimate. However, in practice, the ArrowPratt measure, as it is known (see Arrow, 1970), is more useful since it can be estimated in various ways and used in static as well as dynamic models such PREFACE AND BRIEF NOTES TO THE 2006 EDITION

xv

as Carifio and Ziemba (1998), Carino et al. (1994, 1998) and Geyer et al. (2005). This area also led to the modern notion of risk measures which were developed using an Arrow-like axiomatic system in Artzner et al. (1999). Subsequent work such as in Rockafellar and Ziemba (2000) and especially in Follmer and Scheid (2002a, 2002b) rationalized the convex risk measures used in the stochastic programming literature starting with Kusy and Ziemba (1986); see also Acerbi (2004). These risk measures are theoretical improvements on the value at risk (VaR). With VaR, one presupposes a cutoff loss level and a confidence level such that one will not lose more than this amount with that probability; see Duffie and Pan (1997) and Jorion (2000) for surveys. Hence it is like a chance constraint and has the same drawbacks. For example, the penalty is independent of the loss. Rockafellar and Uryasev (2000, 2002) show that CVaR, which is linear in the penalty approximation of VaR, can be computed endogenously and exogenously via a linear program. Our exercise ME-26 shows this for the much simpler exogenous case, where the constraint confidence level is specified in advance. The Lintner and Vickson papers (written for this book) in Section 3 of Part II discuss extensions of Tobin's 1958 separation theorem that evolves when there is, as Tobin suggested, a risk free asset. So does Ziemba's stable distribution paper in Part III that further extends the idea to fat tailed stable distributions and Ziemba et al. (1974) which specifically shows how to compute the two parts of the separation in Tobin's normal distribution world: the mutual fund (i.e., market index) that is independent of the investor's concave utility function and the optimal balance of cash and this mutual fund for any given utility function. The first problem is deterministic in n variables and solved as a linear complementary or quadratic programming problem. The second problem is stochastic but has only one variable, the percent cash. Ross (1978) approaches the analysis in a more general way than the normal, stable or other distributional form and obtains theoretical conditions for separation but it is not clear on how to find the separated portfolios. Part III deals with static portfolio selection models and begins with papers by Samuelson and Ohlson (written for this book) that show when meanvariance analysis is optimal in cases other than normal distribution and quadratic utility but with special distributions that converge properly. These results extend to symmetric distributions. Pyle and Turnovsky's analysis of safety first followed Roy's 1952 Econometrica paper that was a close alternative to Markowitz's (1952) famous portfolio theory paper that ushered in modern investment management and gave Markowitz the Nobel prize in economics in

XVI


1990. This area presages the CVaR literature which has become very extensive since 2000 with contributions including the Journal of Banking and Finance special issue; see Krokhmal et al. (2006) and the volume edited by Szego (2004). Ziemba's stable distribution paper uses generalized concave mathematical programming ideas to generalize the normal distribution Markowitz and Tobin portfolio theory to the infinite variance fat-tailed stable distribution case. Applications of stable modeling are still less common in the literature but many real asset prices such as Japanese golf course prices fit such distributions; see Stone and Ziemba (1993). See Rachev and Mittnik (2000) for a thorough treatment of stable distribution theory and applications. Section 2 of Part III contains papers by Leland and Samuelson on the existence and diversification of optimal portfolio policies. Leland derives conditions under which an important class of financial economic problems will have a finite optimal policy even when the feasible region is unbounded. But it does require that the utility function is either bounded above or that marginal utility goes to zero as wealth goes to infinity. Samuelson discusses diversification with fully symmetric asset returns showing that equal investments in all assets is optimal. An independent asset with a higher mean than the other assets must enter the optimal portfolio. These results and his other extensions provide a useful starting point for diversification theories. Popular modern investment texts such as Brealey, Myers and Allen (2006) and Bodie, Kane and Marcus (2005) generally present investment theory in its pure form. But in reality, taxes and transaction costs are crucial much of the time. The Stiglitz and Naslund papers in Section 3 of Part III were early fundamental contributions to the theory of the effects of taxes on risk taking behavior. This area is a rich one with many subsequent practical studies such as the volumes edited for the NBER by Poterba (2001, 2002), and theoretical studies by Constantinides (1983, 1984), Davis and Norman (1990), and Davis etal. (1993) and others. One of the greatest changes in applied research since the 1975 edition of Stochastic Optimization Models in Finance in 1975 is the emphasis and computations concerning dynamic models with uncertainty. Given much more powerful personal computers, we now have publicly available codes to solve multiperiod stochastic programming models with millions of variables and scenarios. Ten such systems are described in Wallace and Ziemba (2005); see also Birge and Louveaux (1997), Censor and Zenios (1997), Kail and Mayer (2005) and Zenios (2007).


xvn

Parts IV and V of the volume discuss dynamic models with uncertainty taken into account in the decision-making process. Some models can be reduced to or analyzed as static models in some way and the papers by Wilson, Fama and Hakansson discuss three such ways to do this. Wilson's model has a single decision point. Fama shows the result that many dynamic stochastic models are equivalent to a static model where the future period's random variables have been eliminated through the expectation operator and the future period's decision variables have been optimized out. This notion has been used in the stochastic programming literature since the early paper of Dantzig (1955) and is standard in the current theoretical literature, see e.g. Birge and Louveaux (1997) or Wets and Ziemba (1999). This analysis assumes that the distributions of the random variables are independent of the decisions made. The results show that concavity is preserved over maximization and expectation but strict concavity is not preserved without additional assumptions; see Ziemba (1974, 1977) on this latter point. Mossin (1968) showed that for essentially unconstrained dynamic investment problems with power utility functions, the optimal policy was myopic if the asset returns were intertemporally independent. Hakansson's paper showed that if the objective function was logarithmic, then an optimal myopic policy existed for general dependent assets. This paper and those by Breiman and Thorp in Section 4 of Part V concern the extremely important but little used (except by some rather talented people discussed below) Kelly (1956) capital growth theory. Section V concerns dynamic models and begins with Vickson's intuitive discussion of the ltd calculus for stochastic differential equations and stochastic optimal control used in continuous time finance. The only real continuous time paper in the volume is Merton's in Section IV. Two major areas have evolved out of continuous time finance whose father is Robert C. Merton, the Nobel prize winner in economics in 1997 for his work related to the Black-Scholes (1973) and Merton (1973) option pricing model. The area has exploded with thousands of articles and hundreds of books on option pricing models and applications published since these fundamental contributions were made. Our book just touches the surface of such work so we refer readers to such option pricing and derivative security books as Rubinstein (1998), Duffie (2001), Shreve (2004), Hull (2006), and Wilmott (2006). In this area, the continuous time modeling is crucial to the option pricing. One area of use is in hedge fund mispricing that is "buy A and sell A*", a close substitute but which is more expensive. Then wait in this risk arbitrage until the prices converge within a

xvin


transaction cost band. Most such modeling is confidential and kept secret but one such application to the 1990 Nikkei put warrant market is described in Shaw, Thorp and Ziemba (1995). The other area which is well explained in Merton's (1992) book is the whole notion of using continuous time finance in a vast number of other applications. One example in asset-liability management which extends the Tobin (1958) separation theorem to multiple aggregated assets as Merton did in his book can be found in Rudolf and Ziemba (2004). There is a group of aggregated mutual funds such that investment by all concave risk averse investors is optimal across just these mutual funds with the weights depending on the specific utility function. Merton shows this for the assets-only case and we add the liabilities mutual fund. The well known and successful New York based hedge fund DE Shaw uses the Rudolf-Ziemba (2004) model. The above are aggregated assets or funds and it is optimal for all concave risk investors to choose among them. Then, once the investor and his utility function are specified, optimal portfolio weights in the funds may be determined. One of the drawbacks of continuous time models is their great sensitivity to parameter uncertainty. So in ALM stochastic programming models such as those in Ziemba and Mulvey (1998), Wallace and Ziemba (2005) and Zenios and Ziemba (2006, 2007) are generally preferred by pensions, insurance companies, wealthy individuals and hedge funds; see Ziemba (2003). One advantage of these scenario based models is that the parameters are not assumed to be known but are scenario dependent, hence they are uncertain. Also they have discrete portfolio revision periods as are used in most investment processes. Dreze and Modigliani present a two-period model of Irving Fisher's theory of savings under uncertainty where the main issues center on the tradeoff of current versus future consumption. This tradeoff is modeled so that risk aversion affects the current choice and the resulting future portfolio choices. The economic results obtained would generalize to what one might obtain in a multiperiod version which current technology could now easily analyze. Bradley and Crane present a multistage decision tree model for bond portfolio management. A novel feature was its ability to trace the bond movements from interest rate changes over time. This model is one of dynamic programming rather than stochastic programming; hence its size grows faster than the latter with more periods and scenarios. Kusy and Ziemba (1986) compare their stochastic programming model from the Vancouver Savings Credit Union with that of Bradley-Crane and argue for the SP model on computational and performance grounds. Both of these models are now easily solved with current PREFACE AND BRIEF NOTES TO THE 2006 EDITION

xix

technology as in Wallace and Ziemba (2005). The Bradley and Crane model ushered in a whole literature in bond portfolio management and the management of fixed income securities; see, for example, the early book of Dempster (1980) from the first international conference on stochastic programming and the papers by Mulvey and Zenios (1994), Golub et al. (1995), Zenios et al. (1998) and Bertocchi and Dupacova in Zenios and Ziemba (2006, 2007) for the current state of this literature which has moved more to the SP modeling approach using complex bond pricing which Bradley and Crane initiated in this paper. The Kusy and Ziemba approach was the first of many aggregated ALM models which include the Russell-Yasuda model (see Carino and Ziemba, 1998; and Carino et al, 1994, 1998) and the Siemens Austria pension fund model (see Geyer et al, 2005) and the several papers on related applications that are collected in Ziemba and Mulvey (1998), Wallace and Ziemba (2005), the Zenios and Ziemba ALM Handbook (2006, 2007), and Zenios (2007). A significant contribution of this line of research, that has been made possible due to the flexibility of stochastic optimization models to integrate through scenarios diverse multi-dimensional risk factors, is towards enterprise wide risk management. Significant improvements in the risk-reward profile of an institution can be achieved with integrative risk management models using stochastic programming; see e.g. Babbel and Staking (1991), Consiglio, Cocco and Zenios (2001), Dempster (2002) and Zenios and Ziemba (2006, 2007). Section 2 of Part V deals with models of optimal capital accumulation and portfolio selection. Neave presents conditions for a consumer's multiperiod utility function to exhibit both decreasing absolute and increasing relative risk aversion. He shows that these properties are preserved through maximization and expectation operations over time. Although some generalizations are possible, Neave essentially solves this problem. Samuelson and Merton in a pair of companion articles in 1969 devised the discrete time and continuous time dynamic portfolio consumption-investment models, respectively, which are used frequently in current research. Merton's paper and his 1971 paper in Section 4 of Part V are highly related to current continuous time models such as those described in his book (Merton, 1992) and in the strategic asset allocation work of Brennan and Schwartz (1998), Brennan et al. (1997) and Campbell and Viceira (2002), as well as the Black and Scholes (1973) and Merton (1973) option pricing models. For option pricing, the continuous time model is essential but for practical asset-liability modeling the discrete time multiperiod stochastic programming models are more practical as they allow modeling more of the real constraints and preferences xx


through targets and are less sensitive to parameter errors than the continuous time models whose optimal weights change dramatically with the arrival of new information; see e.g. Brennan and Schwartz (1998), and Rudolf and Ziemba (2004). Hakansson's 1970 paper along with Breiman's 1960 and Thorp's 1971 papers concern the Kelly criterion as it is called in the gambling literature or the capital growth criterion in the finance and economics literature. See also Williams (1936) and Latane (1959) for early related treatments of this subject. Hakansson's 1971 paper in Part IV showed that, with log utility, one has optimal myopic behavior in dynamic investment/consumption models. In this paper, Hakansson obtains closed form optimal consumption, investment and borrowing strategies for constant relative or absolute Arrow-Pratt risk aversion indices which includes various power, log and exponential utility functions. Breiman and in his more detailed paper (1961) show the powerful long run properties of log utility investing. These include the fact that the log investor will, as time increases without limit, have more wealth — in fact arbitrarily more wealth than any other investor as long as the strategies differ infinitely often. Moreover, the log investor will achieve sufficiently large goals faster than any other such investor. Thorp further expands the theory but more importantly uses it in sports betting and in hedge fund risk arbitrage in the financial markets such as in warrant trading. He also points out that Breiman's results can be generalized and the papers of Algoet and Cover (1988), Thorp (2006) and MacLean and Ziemba (2006) do that and survey other aspects of the theory and practice of log betting. Early on MacLean and I worked on the marriage of growth and security through fractional Kelly strategies; see e.g. MacLean, Ziemba and Blazenko (1992). With negative power utility functions, it is shown that growth is less but so is wealth variability. Fractional Kelly and negative power are 1:1 related for lognormally distributed assets and otherwise they are approximately related; see MacLean et al. (2005). Since the risk aversion of log utility is one divided by wealth or essentially zero, log is an exceedingly risky utility function in the short run. And this is doubly so if there are parameter errors since the errors in the mean which normally are 20:2:1 as important as those of the variances and co-variances become 100:2:1 as important as with low risk aversion; see Chopra and Ziemba (1993) which updated and added risk aversion to the earlier Kallberg and Ziemba (1981, 1984) studies. Hence, over betting is very dangerous with log utility. Markowitz in a private communication with me proved the result that MacLean, Ziemba and Blazenko (1992) observed empirically that betting double PREFACE AND BRIEF NOTES TO THE 2006 EDITION

xxi

the Kelly fraction (which maximizes the long run growth rate) actually makes the growth rate zero plus the risk free rate; see Ziemba (2003) for more discussion and the proof. Indeed, this is one explanation for part of the demise of Long Term Capital Management in 1998 from over betting. Thorp and I are proponents of the Kelly and fractional Kelly approach and observe that many of the world's greatest investors like Warren Buffett, John Maynard Keynes, Bill Benter (in horseracing in Hong Kong) and Thorp himself used such strategies. I personally consulted for six such individuals who turned zero into hundreds of millions or even billions (in the case of Jim Simons of Renaissance who made US$1.4 billion just in 2005). Ziemba (2005) and MacLean and Ziemba (2006) study many of these investors and Thorp (2006) discusses his use of the Kelly approach and that of Buffett, who acts as if he was a full Kelly bettor. Samuelson (see his article in Section 2 of Part V), however, is not a log utility supporter. His objections are recorded in the conclusion to his article and in Samuelson (1979). He argues that maximizing the geometric mean rather than the arithmetic mean maximizes expected utility only for log utility. Indeed it can be argued that log is the most risky utility function one should ever consider in the short run, since growth decreases and risk increases for any convex risk measure with higher than log utility wagers. Donald Hausch and I did a simulation (see Ziemba and Hausch, 1986), which is reproduced in MacLean and Ziemba (2006) to understand this better. We take an investor who bets $1000 with log and half Kelly (-w1) seven hundred times with five possible wagers with probability of winning 0.19 to 0.57 corresponding to 1-1, 2-1, ..., 5-1 odds with a 14% expected value advantage. The bets are independent. There are 1000 trials. In 166 of the 1000 trials, the final wealth for log is greater than 100 times the initial $1000. With half Kelly it is only once this large. But half Kelly provides higher probability of being ahead, etc. So there is a growth-security tradeoff. However, it is possible to make 700 independent bets all with a 14% advantage and still lose 98% of one's wealth. Half Kelly is not much better. You can still lose 86% of your initial wealth. So the conclusion is that log is short term risky and wins you the most money long term. The long term can be very long, however; see Thorp (2006) for some discussion and calculations. One compromise (see MacLean et al, 2004) is to choose at discrete intervals the fractional Kelly that would keep you above a wealth path with high probability. This I have just implemented in a London hedge fund but with a convex penalty for falling below the path. Log is certainly the most interesting and controversial utility function for investment. It is rarely taught, however, in most university investment courses. Despite their original publish-

XXll


ing dates, the papers by Breiman, Hakansson, Samuelson and Thorp in the volume well represent a very good beginning for students of growth-security portfolio analysis. Section 3 of Part V has various models of option strategy prior to BlackScholes (1973) and Merton (1973). Pye and Taylor present specific models to value call options on bonds and stocks. Kalymon formulates a bond refunding problem where future interest rates are Markovian. He can then determine policies that minimize expected total discounted costs. Finally, Pye shows that dollar cost averaging is a minimax strategy rather than an expected utility maximizing strategy for any strictly concave utility function. I would like to thank Michael Brennan, George Constantinides, Darrell Duffie, Jitka Dupacova, Chanaka Edirisinghe and Stavros Zenios for their useful comments on an earlier draft of this Preface. Lastly, references to papers and books not appearing in this volume that are useful in courses and for further study using this volume follow. William T. Ziemba Vancouver, July 2006


xxin

References Acerbi, C. (2004). "Coherent representations of subjective risk-aversion", in G. Szego (ed.), Risk Measures for the 21s'Century, pp. 147-207. Wiley. Algoet, P. and T. Cover (1988). Asymptotic optimality and asymptotic equipartition properties of log-optimum investment. Annals of Probability 16, 876-898. Arrow, K.J. (1970). Essays in the Theory of Risk Bearing. North Holland. Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath (1999). Coherent measures of risk, Mathematical Finance, 9, 203-227. Babbel, D.F. and K.B. Staking (1991). It pays to practice ALM, Best's Review, 92(1), 1-3. Bell, D. (1995). Risk, return and utility, Management Science, 41, 23-30. Bertsekas, D.P. (2005). Dynamic Programming and Optimal Control, 3rd Edition. Athena Scientific. Birge, J. and F. Louveaux (1997). Stochastic Programming. Springer. Black, F. and M.S. Scholes (1973). The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637-654. Blackwell, D. and M.A. Girchik (1954). Theory of Games and Statistical Decisions. New York: Wiley. Bodie, Z., A. Kane, and Marcus (2005). Investments, 6lh Edition. Irwin McGraw. Brealey, R.A., S.C. Myers, and F.A. Allen (2006). Principles of Corporate Finance, 8lh Edition. Irwin/McGraw-Hill. Breiman, L. (1961). "Optimal gambling system for favorable games", in J. Neyman (ed.), Proceedings of the 4 Berkeley Symposium on Mathematics, Statistics and Probabilities, Vol 1, 63-68. Brennan, M.J. and E. S. Schwartz (1998). "The use of Treasure bill futures in strategic asset allocation programs", in W.T. Ziemba and J.M. Mulvey (eds.), World Wide Asset and Liability Modeling, pp. 205-228. Cambridge: Cambridge University Press. Brennan, M.J., E.S. Schwartz, and R. Lagnado (1997). Strategic asset allocation, Journal of Economic Dynamics and Control, 21, 1377-1403. Brennan, M.J. and Y. Xia (2000). Stochastic interest rates and bond-stock mix, European Finance Review, 4, December, 197-210. Brennan, M.J. and Y. Xia (2002). Dynamic asset allocation under inflation, Journal of Finance, 57, June, 1201-1238. Campbell, J.Y. and L. Viceira (2002). Strategic Asset Allocation. Oxford University Press.

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Carino, D. and W.T. Ziemba (1998). Formulation of the Russell-Yasuda Kasai Financial Planning Model, Operations Research, 46(4), July/August, 433^149. Carino, D., D. Myers, and W.T. Ziemba (1998). Concepts, technical issues and uses of the Russell-Yasuda Kasai Financial Planning Model, Operations Research, 46(4), July/ August, 450^162. Carino, D. et al. (1994). The Russell-Yasuda Kasai Model: An asset/liability model for a Japanese insurance company using multistage stochastic programming, INTERFACES, January-February (Edelman Prize issue), 29-49. Censor, Y. and S.A. Zenios (1997). Parallel Optimization: Theory, Algorithms, and Applications, Series on Numerical Mathematics and Scientific Computation. New York, NY: Oxford University Press. Chopra, V. and W.T. Ziemba (1993). The effect of errors in the mean, variance, and covariance estimates on optimal portfolio choice, Journal of Portfolio Management, Winter, 6—11. Consiglio, A., F. Cocco, and S.A. Zenios (2001). The value of integrative risk management for insurance products with guarantees, Journal of Risk Finance, Spring, 1-11. Constantinides, G.M. (1983). Capital market equilibrium with personal tax, Econo-metrica, 51, May, 611-36. Constantinides, G.M. (1984). Optimal stock trading with personal taces: implications for prices and the abnormal January returns, Journal of Financial Economics, 13, March, 65-89. Cox, J. and C.-F. Huang, (1989). Optimal consumption and portfolio polices when asset prices follow a diffusion process, Journal of Economic Theory, 49, 33-83. Dantzig, G.B. (1955). Linear programming under uncertainty, Management Science, 1, 197-206. Davis, M.H.A. and A.R. Norman (1990). Portfolio selection with transaction costs, Mathematics of Operations Research, 15, 676-713. Davis, M.H.A., V.G. Panas, and T. Zariphopoulou (1993). European option pricing with transaction costs, SIAM Journal of Control and Optimization, 31, 470^193. Dempster, M.A.H. (ed.) (1980). Stochastic Programming. Academic Press. Dempster, M.A.H. (ed.) (2002). Risk Management: Value at Risk and Beyond. Cambridge University Press. Diamond, P. and J.E. Stiglitz (1974). Increases in risk and in risk aversion, Journal of Economic Theory, 8, 337-350.


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Duffie, D. (2001). Dynamic Asset Pricing Theory, 3r Edition. Princeton University Press. Duffie, D. and J. Pan (1997). An overview of value at risk, Journal of Derivatives, Spring, 7^49. Follmer, H. and A. Scheid (2002a). Convex measures of risk and trading constraints, Finance and Stochastics, 6(4), 429-447. Follmer, H. and A. Scheid (2002b). Stochastic Finance. deGruyter. Geyer, A., W. Herold, K. Kontriner, and W.T. Ziemba (2005). The Innovest Austrian Pension Fund Planning Model InnoALM. Working Paper, Sauder School of Business, UBC. Golub, B , M. Holmer, R. McKendall, L. Pohlman, and S.A. Zenios (1995). Stochastic programming models for money management, European Journal of Operational Research, 85, 282-296. Hardy, G.H., J.E. Littlewood, and G. Polya (1934). Inequalities. MA: Cambridge University Press. Harrison, M. and D. Kreps (1979). Martingale and multiperiod securities markets, Journal of Economic Theory, 20, 382^-08. Harrison, M. and S. Pliska (1981). Martingales and stochastic integrals in the theory of continuous trading, Stochastic Process Appl., 11, 215-260. Hausch, D.B., V. Lo, and W.T. Ziemba (eds.) (1994). The Efficiency of Racetrack Betting Markets. Academic Press. Hausch, D.B. and W.T. Ziemba (1985). Transactions costs, entries and extent of inefficiencies in a racetrack betting model, Management Science, XXXI, 381— 394. Hausch, D.B. and W.T. Ziemba (eds.) (2007). Handbook of Sports and Lottery Investments. Elsevier, in press. Hausch, D.B., W.T. Ziemba, and M. Rubinstein (1981). Efficiency of the market for racetrack betting, Management Science, XXVII, 1435-1452. Hull, J. (2006). Options, Futures and Other Derivatives, 6th Edition. NJ: Prentice-Hall. Jorion, P. (2000). Value-at-Risk: The Benchmark for Controlling Market Risk, 2nd Edition. NJ: McGraw-Hill. Kahneman, D. and A. Tversky (1979). Prospect theory: an analysis of decision under risk, Econometrica, 47, March, 263-291. Kahneman, D. and A. Tversky (1982). The psychology of preferences, Scientific American, 246, February, 167-173. Kail, P. and J. Mayer (2005). Stochastic Linear Programming: Models, Theory and Computation. Springer Verlag, International Series.

XXVI


Kallberg, J.G. and W.T. Ziemba (1981). Remarks on optimal portfolio selection. Methods of Operations Research, AA, 507-520. Kallberg, J.G. and W.T. Ziemba (1983). Comparison of alternative utility functions in portfolio selection problems, Management Science, XXIX, 1257— 1276. Kallberg, J.G. and W.T. Ziemba (1984). Mis-specification in portfolio selection problems, in G. Bamberg and K. Spreemann (eds.), Risk and Capital, pp 7487. Springer-Verlag. Kelly, J. (1956). A new interpretation of information rate, Bell System Technology Journal, 35, 917-926. Krokhmal, P., R.T. Rockafellar, and S. Uryasev (2006). Special issue on risk management and optimization in finance, Journal of Banking and Finance, February. Kusy, M.I. and W.T Ziemba (1986). A bank asset and liability management model, Operations Research, XXXIV, 356-376. Latane, H. (1959). Criteria for choice among risky ventures. Journal of Political Economy, 38, April, 144-155. MacLean, L.C., R. Sanegre, Y. Zhao, and W. T. Ziemba (2004). Capital growth with security. Journal of Economic Dynamics and Control, 28, 937-954. MacLean, L.C. and W.T. Ziemba (2006). "Capital growth: theory and practice" in S.A. Zenios and W.T. Ziemba (eds.), Handbook of Asset and Liability Modeling, Vol. 1: Theory and Methodology, pp. 429-473. North Holland. MacLean, L.C, W.T. Ziemba, and G. Blazenko (1992). Growth versus security in dynamic investment analysis, Management Science, Special Issue on Financial Modelling, 38, November, 1562-1585. MacLean, L.C, W.T. Ziemba, and Y. Li (2005). Time to wealth goals in capital accumulation and the optimal trade-off of growth versus security, Quantitative Finance, 5(4), 343-357. Markowitz, H.M. (1952). Portfolio selection, Journal of Finance, 7(1), 77-91. Merton, R.C. (1973). The theory of rational option pricing, Bell Journal of Economics and Management, A, 141-183. Merton, R.C. (1992) Continuous Time Finance, 2nd Edition. Maiden, MA: Blackwell Publishers, Inc. Mossin, J. (1968). Optimal multiperiod portfolio policies, Journal of Business, 41(2), 215-229. Mulvey, J.M. and S.A. Zenios (1994). Capturing the correlations of fixedincome instruments, Management Science, 40, 1329—1342.


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Poterba, J. (2001). "Taxation and portfolio structure: Issues and implications", in L. Guiso, M. Haliassos, and T. Jappelli (eds.), Household Portfolios, pp. 103-142. MIT Press. Poterba, J. (2002). "Taxation, risk-taking, and portfolio behavior", in A. Auerbach and M. Feldstein, Handbook of Public Economics: Volume 3, pp. 1109— 1171. North Holland. Rabin, M. (2000). Risk aversion and expected-utility theory: A calibration theorem, Econometrica, 68(5), September, 1281-1292. Rabin, M. and R. Thaler (2001). Anomalies: Risk aversion, Journal of Economic Perspectives, 15(1), Winter, 219-232. Rachev, S. and S. Mittnik (2000). Stable Paretian Models in Finance. Wiley, Chichester. Rockafellar, R.T. and S. Uryasev (2000). Optimization of conditional value-atrisk, Journal of Risk, 2, 21-41. Rockafellar, R.T. and S. Uryasev (2002). Conditional value-at-risk for general loss distributions, Journal of Banking and Finance, 26, 1443-1471. Rockafellar, R.T. and W.T. Ziemba (2000). Modified Risk Measures and Acceptance Sets. Working Paper, Sauder School of Business, Vancouver. Ross, S.A. (1978). Mutual fund separation in financial theory — the separating distributions, Journal of Economic Theory, 17, 254-286. Rothschild, M. and J.E. Stiglitz (1970). Increasing risk I: A definition, Journal of Economic Theory, 2, 225-243. Roy, A.D. (1952). Safety-first and the holding of assets, Econometrica, 20, July. Rubinstein, M. (1976). The valuation of uncertain income streams and the pricing of options, Bell Journal of Economics, 7, 407^125. Rubinstein, M. (1998). Derivatives: A Power Point Picture Book, Volume 1: Futures, Options and Dynamic Strategies, Self Published. CA: Haas School of Business, University of California, Berkeley, Berkeley. Rudolf, M. and W.T. Ziemba (2004). Intertemporal asset-liability management, Journal of Economic Dynamics and Control, 28(4), 975-990. Samuelson, P.A. (1979). Why we should not make mean log of wealth big though years to act are long, Journal of Banking and Finance, 3, 305-307. Shaw, J., E.O. Thorp, and W.T. Ziemba (1995). Convergence to efficiency of the Nikkei Put Warrant Market of 1989-1990, Applied Mathematical Finance, 2,243-271. Shreve, S.E. (2004). Stochastic Calculus for Finance, Volume II: Continuous Time Models. Springer-Verlag.

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Stone, D. and W.T. Ziemba (1993). Land and stock prices in Japan, Journal of Economic Perspectives, Summer, 149-165. Szego, G. (2004). Risk Measures for the 21s' Century. Wiley. Thorp, E.O. (2006). "The Kelly criterion in blackjack, sports betting and the stock market", in S.A. Zenios and W.T. Ziemba (eds.), Handbook of Asset and Liability Modeling, Volume I: Theory and Methodology, pp. 385^128. North Holland, Amsterdam. Tobin, J. (1958). Liquidity preference as behavior towards risk, Review of Economic Studies, 26, February, 65-86. Tompkins, R.G., W.T. Ziemba, and S.H. Hodges (2003). The Favorite-longshot Bias in S&P500 Futures Options: The Return to Bets and the Cost of Insurance. Working Paper, Sauder School of Business, UBC. Wallace, S.W. and W.T. Ziemba (eds.) (2005). Applications of Stochastic Programming. SIAM — Mathematical Programming Series on Optimization. Wets, R.J.B. and W.T. Ziemba (eds.) (1999). Stochastic Programming— State of the Art 1998 (main lectures VIII International Conference on Stochastic Programming), Baltzer Science Publishers BV (Special Issue Annals of Operations Research). Williams, J.B. (1936). Speculation and the carryover, Quarterly Journal of Economics, 50, May, 436-455. Wilmott, P. (2006). Paul Wilmott on Quantative Finance, Vols. 1-3, 2nd Edition. Wiley. Zenios, S.A. (2007). Practical Financial Optimization: Decision Making for Financial Engineers. Oxford, UK: Blackwell Publishing, forthcoming. Zenios, S.A. and W.T. Ziemba (eds.) (2006, 2007). Handbook of Asset and Liability Modeling (Vol. 1 on theory and methodology and Vol. 2 on applications and case studies). North Holland-Elsevier, in press. Zenios, S.A., M. Holmer, R. McKendall, and C. Vassiadou-Zeniou (1998). Dynamic models for fixed-income portfolio management under uncertainty, Journal of Economic Dynamics and Control, 22, 1517-1541. Zhao, Y., U. Haussmann, and W.T. Ziemba (2003). A dynamic investment model with a minimum attainable wealth requirement, Mathematical Finance, 13, October, 481-501. Ziemba, W.T. (1974). Note on the behavior of a firm subject to stochastic regulatory review, Bell Journal of Economics and Management Sciences, 5(2), 710-712. Ziemba, W.T. (1977). Multiperiod consumption-investment decisions: Further comments, American Economic Review, LXVII, 766-767'.


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Ziemba, W.T. (2003). The Stochastic Programming Approach to Asset Liability and Wealth Management. Charlottesville, VA: AIMR. Ziemba, W.T, C. Parkan, and F.J. Brooks-Hill (1974). Calculation of investment portfolios with risk free borrowing and lending, Management Science, XXI, 209-222. Ziemba, W.T. and D.B. Hausch (1986). Betting at the Racetrack, SelfPublished. San Luis Obispo, CA: Dr Z Investments, Inc. Ziemba, W.T. and J.M. Mulvey (eds.) (1998). Asset and Liability Management from a Global Perspective. Cambridge University Press.

XXX


PREFACE IN 1975 EDITION

There is no adequate book for an advanced course concerned with optimizing models of financial problems that involve uncertainty. The numerous texts and edited collections of articles related to quantitative business finance are largely concerned with deterministic models or stochastic models that are directly reducible to deterministic models. Our intention is that this present book of readings will partially fill this gap by providing a source that gives a reasonably thorough account of the mathematical theory and economic results relating to these problems. It is hoped that the volume can serve the dual role of text and research reference. The literature in this area is large and expanding rapidly. Hence to make the volume manageable we had to place severe constraints on our coverage. First and foremost we have only considered material relevant for optimizing models that explicitly involve uncertainty. As a result no material on statistical estimation procedures is included. Secondly, the concern is with the mathematical and economic theories involved and not with their application in practice and associated institutional aspects. However, our coverage does place emphasis on results and methods that can and have been utilized in the analysis of real financial problems. With some reluctance we also limited ourselves to models involving a single decision maker. Hence, material relating to gaming problems, market equilibrium, and other multiperson or multifirm problems is not included here. There is also very little material concerned with specific numerical algorithms or ad hoc solution approaches and that included is intended to be representative and not comprehensive. The major criterion for inclusion of articles in the collection is that they present a significant methodological advance that is of lasting interest for teaching and research in finance. Since a major goal of the volume is to provide a source for students to "get quickly to the frontiers," preference was given to papers in areas where research is currently active. We have also tried to choose contributions that were well written and did not appear to have major gaps or errors in their presentation. Papers were also given lower priority if they have appeared in other books of readings or if similar material is available in such collections. This accounts for the very limited coverage of capital budgeting XXXI

models. In several areas we felt it appropriate to provide new papers that summarize and extend existing results in the literature. Credit and blame for biases introduced in the selection of articles and their layout are due to WTZ. The five parts of this work present material that is intended to be read in a sequential fashion part by part. Part I is concerned with mathematical tools and expected utility theory. The treatment focuses on convexity and the Kuhn-Tucker conditions and the methods of dynamic programming. The next part is concerned with qualitative economic results, and particular attention is given to results relating to stochastic dominance, measures of risk aversion, and portfolio separation theorems. Part HI is concerned with static models of portfolio selection. Particular attention is placed on the mean-variance and safety-first approaches and their extensions along with their relation to the expected utility approach. The questions of existence and diversifications of optimal portfolio policies and the effects of taxes on risk taking are also dealt with here. The fourth part is concerned with dynamic models that are in some sense reducible to static models. Part V is concerned with dynamic models that are most properly analyzed by dynamic methods. Particular attention is placed on models of portfolio revision, optimal capital accumulation, option strategies, and portfolio problems in continuous time. Each part begins with an introduction that attempts to summarize and make cohesive the material that follows. This is followed by several articles that present major methodological advances in that area. At the end of each part there are numerous problems which form an integral part of the text and are subdivided into computational and review, and mind-expanding exercises. The computational and review exercises are intended to test the understanding of the preceding readings and how they relate to previous material. Some exercises present, hopefully, straightforward extensions, to new problems, of the methodological material in the readings. Other exercises fill gaps in the presentation of the reprinted articles. The mind-expanding exercises, on the other hand, are intended to traverse new problem areas whose analysis requires different techniques or nonstraightforward extensions of the material in the reprinted articles. Many of these exercises are quite difficult, and some present unsolved problems and conjectures. We have starred all exercises whose solution is unknown or in doubt to us. Some mind-expanding exercises describe previously unpublished problems and solutions. They also serve to survey, to some extent, many important papers that could not be included because of space limitations. Indeed, many exercises present major results from one or more published papers. The sources from which such exercises were adapted are indicated in the "Exercise Source Notes" at the end of each exercise section. We have tried to present the exercises in such a way that the reader can obtain maximum benefit per unit input of time spent in their solution. In xxxn


particular results are generally stated and the reader is asked to verify them. Hence, the student should be able to understand the major results and conclusions of a particular exercise even if he cannot solve all its parts. This collection of articles and exercises has not undergone extensive classroom testing. However, one of us (WTZ) has used some of the materials in courses at the University of British Columbia (in 1971) and at the University of California, Berkeley (in 1972). The typical students who would take a course based on this book are masters and doctoral level students in management science, operations research, and economics and doctoral level finance students. The ideal prerequisites consist of a course in elementary business finance using say Lusztig and Schwab (1973), Mao (1969), or Van Home (1968), a course in nonlinear programming, using say Luenberger (1973) or Zangwill (1969), a course in probability theory using say Feller (1962) or Thomasion (1969), and a course in microeconomic theory using say Henderson and Quandt (1958) or Samuelson (1965). A course in intermediate capital theory using Fama and Miller (1972) or Mossin (1973) would also be of great value to the reader. Minimum prerequisites consist of a careful reading of Part I and the working of the accompanying computational and review exercises (perhaps supplemented by reading the first three chapters of Zangwill (1969) and Chapter Two of Arrow (1971), or by working the mind-expanding exercises in Part I); a calculus course using say Thomas (1953) or Allen (1971) and a course in intermediate probability theory using say Lippman (1971) or Hadley (1967). These minimum prerequisites should be adequate for the understanding of most of the articles in this volume if the reader has developed sufficient mathematical maturity. The book is intended for a two-semester course. It can also be used for a one-semester course related to static models (Parts I (1, 2), II, and III) or to dynamic models (Parts I (3), II (2), IV, and V).


XXX11I

ACKNOWLEDGMENTS

We would first like to thank the respective authors and publishers for their permission to reproduce the articles in this collection. Thanks are due to Professors Nils H. Hakansson, William F. Sharpe, and Karl Shell and Dr. Gordon B. Pye for initial encouragement to undertake the development of the book in its present form. Professor Shelby L. Brumelle participated in many discussions with us during the 1971-1972 academic year. Due to his appointment to a major administrative post at the University of British Columbia, he was unable to be a co-editor with us. In addition to his co-authored article with Vickson in Part II we would like to thank Professor Brumelle for contributing several problems. We would also like to thank the following colleagues and friends who read and commented on various parts of the book and contributed valuable problems: Professors Michael J. Brennan, W. Erwin Diewert, Richard C. Grinold, Loring G. Mitten, James A. Ohlson, and Cary A. Swoveland and Dr. Gordon B. Pye and Mr. Stig Larsson. Various portions of the manuscript were ably typed by Ms. Marlies Adamo, Ms. Sandra L. Schwartz, and Mrs. Ikuko Workman. The clerical assistance of Mr. Jupian Leung and Mr. Johnny Yu is gratefully acknowledged. Special thanks go to Mr. Amin Amershi, Ms. Trudy A. Cameron, Mr. Martin T. Kusy, and Mr. Katsushige Sawaki for their help in reading the proofs. Mr. Kusy also helped in the preparation of the index. Thanks are also due to the National Research Council of Canada for their continuing support of Professor Ziemba's research on stochastic programming and its applications under contract NRC-67-7147.

XXXV

Part I Mathematical Tools

INTRODUCTION The first part of this book is devoted to technical prerequisites tor the study of stochastic optimization models. We have selected articles and included exercises that appear to provide the necessary background for the study of the specific financial models discussed in the remainder of the book. The treatment in this part is, however, necessarily brief because of space limitations ; hence the reader may wish to consult some of the noted additional references on some points. The prerequisites for an in-depth study of stochastic optimization models are expected utility theory, convex functions and nonlinear optimization methods, and the embedded concepts of dynamic programming.

I. Expected Utility Theory Fishburn's article presents a concise proof of a general expected utility theorem. He considers a set of outcomes and probability distributions over outcomes. The latter are termed horse lotteries. Given a set of reasonable assumptions concerning the decision-maker's preferences over alternative horse lotteries, Fishburn demonstrates the existence of a utility function and a subjective probability measure over the states of the world such that the decision-maker acts as if he maximized expected utility. The utility function is continuous and is uniquely defined up to a positive linear transformation (see Exercise CR-2 for examples). In some cases the utility function is bounded. However, most utility functions are unbounded in at least one direction. For such utility functions horse lotteries can be constructed that have arbitrarily large expected utility and an arbitrarily small probability of receiving a positive return. Such examples are called St. Petersburg paradoxes. They are illustrated in Exercises CR-4 and ME-2. Further restrictions on the utility function result if one makes additional assumptions concerning the decisionmaker's behavior. For example, an investor who never prefers a fair gamble to the status quo must have a concave utility function (Exercise CR-1). Such investors will not simultaneously gamble and purchase insurance. Exercise CR-6 is concerned with this classic Friedman-Savage paradox; see also the papers by Yaari (1965a) and Hakansson (1970).1 If the investor's preferences for horse lotteries are independent of his initial wealth level, then the utility function must be linear or exponential (Exercise CR-5). Similarly, if the investor's preferences for proportional gambles are independent of initial 1

Throughout this book references cited by date will be found in the Bibliography at the end of the book. INTRODUCTION

3

wealth, then his utility function must be a logarithmic or power function (Exercise CR-17). Exercise ME-3 investigates the behavioral significance of assuming that a utility function defined over several commodities is separable or that its logarithm is separable. The calculation of an allocation vector that maximizes expected utility may proceed via a stochastic programming or nonlinear programming algorithm as discussed in Part III. However, when the utility function is concave one can easily determine bounds on the maximum value of expected utility (Exercise ME-1). The lower bound is a consequence of Jensen's inequality (Exercise CR-7) which states that the expected value of a concave function is never exceeded by the value of the function evaluated at its mean. Some sharper Jensen-like bounds applicable under more restrictive assumptions appear in Ben-Tal and Hochman (1972) and Ben-Tal, Huang, and Ziemba (1974). Exercise CR-11 introduces the notion of a certainty equivalent, that is, a fixed vector whose utility equals the expected utility. The expected utility approach provides a natural framework for the analysis of financial decision problems. However, there are several alternative approaches which are of particular interest in certain instances. Most notable are the approaches that trade off risk and return, and various safety-first and related chance-constrained formulations. These approaches and their relations with the expected utility approach are considered in some detail in Part III. See also Exercise CR-18 for an approach that augments a decisionmaker's returns and costs in such a way that gambling and insurance conclusions that normally hinge on the shape of the utility function may be obtained for linear utility functions. Fishburn's article provides a succinct but brief presentation of an expected utility theory. The mathematical level in this article is perhaps higher than in nearly all of the other material in this book. For this reason some readers may wish to consult the less general but more lucid developments by Arrow (1971), Jensen (1967a, b), and Pratt et al. (1964). A much fuller treatment (and comparison) of many expected utility theories may be found in the work of Fishburn (1970). See also Fishburn's paper (1968) for a concise survey of the broad area of utility theory.

II. Convexity and the Kuhn-Tucker Conditions Many of the functions involved in financial optimization problems are convex (or concave, the negative of a convex function). Convex functions are defined on convex sets, which are sets such that the entire closed line segment joining any two points in the set is also in the set. Exercise ME-15 outlines some important properties of convex sets. Convex functions have 4

PART I

MATHEMATICAL TOOLS

the property that linear interpolations never underestimate the functions. Alternatively, linear supports never overestimate the functions. These and related characterizations are illustrated in Exercise ME-8. There are two important properties possessed by problems of minimizing convex functions subject to constraints having the form that a set of convex functions not exceed zero. First, the set of feasible points determined by the constraints is a convex set. Second, local minima are always global minima. For applications, certain generalizations of the convexity concept are needed. Mangasarian's first article introduces pseudo-convex functions. These functions are differentiable and possess both of the properties mentioned above, yet are not necessarily convex. They are essentially defined by the property that if a directional derivative is positive (points up), then the function continues to increase in the given direction. An even wider class of functions, called quasi-convex, is defined by retaining only the first property. There are several alternative ways to define quasi-convex functions and they are explored in Exercise ME-11. These generalized functions along with other related functions are discussed in Exercises CR-15 and ME-9. In Exercises CR-8 and 9 the reader is asked to investigate the convexity and generalized convexity properties of simple functions in one and several dimensions, respectively. Exercise ME-12 shows how one can verify whether or not a given function is a convex or generalized convex function by examining a Hessian matrix of second partial derivatives or a Hessian matrix bordered by first partials. The most useful results concerning minimization problems are surely the Kuhn-Tucker necessary and sufficient conditions. In Mangasarian's article it is shown that the Kuhn-Tucker conditions are sufficient if the objective is pseudo-convex and the constraints are quasi-concave. The conditions are necessary if a mild constraint qualification is met. This is discussed in Exercise ME-13, where a proof of their necessity that utilizes Farkas' lemma is outlined. Exercises CR-10 and 13 illustrate how the Kuhn-Tucker conditions may be used- to solve simple mean-variance tradeoff models and other optimization problems, respectively. The determination of the correct sign of the multipliers is considered in Exercise CR-16. The Kuhn-Tucker conditions are intimately related to the Lagrange function. Exercise ME-10 illustrates the relationship between saddle points of the Lagrangian and solutions of the minimization problem. The primal problem may be considered as a minimax of the Lagrangian where the min is with respect to original (primal) variables, and the max is with respect to Lagrange (dual) variables. A dual problem results when one maximins. When the functions are convex and differentiable one obtains the Wolfe dual. Mangasarian presents some results related to this dual problem in his first paper; other results are developed in Exercise ME-14. The reader is asked to determine the actual dual problems in some special INTRODUCTION

5

cases in Exercise CR-12. The relationship between the dual variables and right-hand-side perturbations of the constraints as they relate to the optimal primal objective value is developed in Exercise ME-6. Many of the functions that arise in financial optimization problems are composites of two or more simpler functions. In Mangasarian's second paper he presents a general theorem that relates the convexity and generalized convexity of composite functions to those of the simpler functions. In particular, convex functions of linear functions are convex, and convex nondecreasing functions of convex functions are convex. Such results are also useful in the analysis of the convexity properties of products, ratios, reciprocals, and so forth, of convex and related functions. Exercise ME-9 illustrates the use of such results on a class of convex-linear composite functions. The convexity and pseudo-convexity relationships between functions and their logarithms are considered in Exercise ME-4. Exercise ME-18 concerns the strict concavity of composite functions. Exercises CR-14 and ME-17 discuss properties of sums and integrals of convex and related functions. The monotonicity and convexity properties of functions defined by a minimization operation are considered in Exercise ME-7. Exercise ME-5 discusses a useful linearization result that provides the basis for the Frank Wolfe algorithm (Exercise III-ME-14). The material presented related to convexity and minimization of constrained functions provides a basis for further study and contains most of the results needed in the sequel. The following sources provide additional results, methods, and references. The standard reference work on convexity is by Rockafellar (1970). Additional material may be found in the work of Berge (1963), Mangasarian (1969), Newman (1969), and Stoer and Witzgall (1970). The relationships between quasi-convex and convex functions are explored by Greenberg and Pierskalla (1971) (see also Ginsberg, 1973). For a simplified presentation of the Wolfe duality theory and its economic interpretation consult Balinsky and Baumal (1968). See the work of Mangasarian (1970) for the most general presentation of the Wolfe duality theory. A more general duality theory based on earlier work by Fenchel (1953) is given by Rockafellar (1970). A simplified presentation of the Rockafellar duality theory and its economic interpretation appears in Williams' paper (1970) (see also Cass, 1974). An extremely lucid presentation and comparison of several duality theories has been given by Geoffrion (1971). For a fuller treatment of the theory of nonlinear programming the reader is referred to textbooks by Lasdon (1970), Luenberger (1969, 1973), Mangasarian (1969), and Zangwill (1969) and the edited conference proceedings by Abadie (1967, 1970), Fletcher (1969b), Hammer and Zoutendijk (1974), and Rosen et al. (1970). The last five of these books provide excellent source material and reference for the study of specific algorithms. 6

PART I

MATHEMATICAL TOOLS

III. Dynamic Programming Many of the financial problems considered in this book may be best analyzed in a dynamic context. For example, a consumer's consumption is best thought of as occurring either continuously in time or during certain intervals of time. In such problems the decision maker will generally make a sequence of decisions, each one in turn dependent upon past data and its impact on the future. Naturally today's decision must be made in a way that reflects the fact that tomorrow's decision will be chosen to be the best one possible given the data available tomorrow. Thus intuitively it must be true that "an optimal set of decisions has the property that whatever the first decision is, the remaining decisions must be optimal with respect to the outcome which results from the first decision." This is Bellman's famous principle of optimality which may be proved via the contradiction argument that if this statement were not true, then certainly the policy under consideration could not be optimal. Given that there is an optimal policy, the principle of optimality provides a scheme for the numerical determination of such a policy. Moving forward in time the policy indicates the optimal decisions to make in all periods given all possible past data. That is, regardless of the state in which the decision maker finds himself, the policy must indicate his optimal action. Moving backward in time generally provides a mechanism for determining an optimal policy. Suppose there are n time periods and that all the possible states that might obtain in period n — 1 are known. Then it is a relatively simple calculation to determine the best decision to make given that the individual finds himself in any one of these states. Suppose the states for period n — 2 are given and that the payoffs are additive. Given that the decision maker is in any particular state in period n — 2, an optimal decision is one which maximizes the sum of the immediate payoff in period n — 2 plus the payoff that results from the optimal action made in period n — 1. Now the optimal actions in period n— 1 were previously calculated. Hence continuing period by period an optimal decision in any state in period / is one which maximizes the sum of the immediate return in period / plus the return from the optimal policy used in period t+ 1. One thus obtains a functional equation that may be solved sequentially to determine an optimal decision in each state in each period. The calculations are generally much simpler than an exhaustive enumeration because only those policies that involve optimal policies for periods t+l,...,n need be considered in determining optimal decisions for period t. The article by Ziemba presents an introduction to the methods and uses of dynamic programming. His presentation is an adaptation of an earlier article by Denardo and Mitten; it is primarily concerned with discrete time terminating processes which include problems having finite time horizons. INTRODUCTION

7

However, some extensions to infinite horizon and continuous-time problems are made in the article and in Exercises ME-19 and 20. The basic elements of all dynamic programming problems are identified as the stages, states, decisions, transitions, and returns. Identification of these elements facilitates the formulation and solution of any particular problem. For terminating processes it is easy to prove directly the existence of an optimal policy under a mild monotonicity assumption using rather simple mathematics. The monotonicity assumption requires that if two policies a and b contain the same decision at some state x and if policy a produces at least as large a return as policy b for each state reachable from x, then it must follow that policy a produces at least as large a return as b starting from x. The reader is asked to investigate in Exercise CR-19 whether or not some common preference functions satisfy the monotonicity assumption. This approach to the study of sequential decision problems leads to an algorithm based on the functional equation of dynamic programming. There are a number of important financial problems in which it is appropriate to consider a nonterminating model, such as problems whose horizon is either subject to chance or is infinite. If it is assumed that there exists an optimal policy, then, under the monotonicity assumption, it follows that the functional equation exists and can be utilized to compute an optimal policy. Some particular nonterminating models are considered in Exercises ME-19 and 20. Exercise ME-19 is concerned with the development of necessary and sufficient conditions for the validity of the functional equation for Markovian processes. In general the functional equation is valid if one is minimizing nonnegative costs or maximizing nonnegative returns. The functional equation is not generally valid in the reverse cases, i.e., when one is minimizing nonpositive costs or maximizing nonpositive returns. However, validity does obtain in some special cases. Exercise ME-20 extends these results to the case of bounded costs in a discounted expected cost minimization problem. This exercise also provides justifications for Howaid's policy improvement and Manne's linear programming solution approach to Markovian sequential decision problems. Exercises CR-20 and 21 illustrate how one determines such a functional equation and performs the recursive calculations on simple one- and twodimensional allocation problems, respectively. Exercise CR-26 considers an allocation problem over time and the effects that different constraints and horizons have on the optimal solution. Exercise CR-21 also illustrates how a Lagrange multiplier technique may be utilized to reduce memory requirements in the recursive scheme. Exercise CR-22 is concerned with an economic planning model and illustrates how the calculations and analysis may be modified to handle uncertainty. Exercise CR-23 concerns itself with a production problem that may be solved as a linear program as well as a dynamic program. It also involves uncertainty and the reader is asked to comment on the relative 8

PART I

MATHEMATICAL TOOLS

merits of the two solution approaches for various versions of the problem. Exercise CR-24 illustrates a simple way of deducing the functional equation via a decomposition scheme based on certain separability and monotonicity assumptions. The reader is asked in Exercise CR-25 to verify two results stated in the "Introduction to Dynamic Programming" by Ziemba. The reader may consult Beckmann (1968), Bellman (1961), Bellman and Dreyfus (1962), Gluss (1972), Howard (1960), Jacobs (1967), and Nemhauser (1966) for introductory treatments of dynamic programming. More advanced treatments are given by Bellman (1957), Blackwell (1962), Dreyfus (1965), Howard (1972), Kaufmann and Cruon (1967), and Ross (1970). Exercises ME-19 and 20 present some of the pertinent results for the study of.Markovian decision problems when the objective is the maximization of discounted expected return. A lucid presentation of additional results, including the study of alternative preference functions, may be found in the work of Ross (1970). See also Blackwell (1970) for a presentation of weak assumptions that lead to the optimality of stationary strategies. Veinott (1969) has given a unified presentation of the current state of this field. Many interesting gambling problems, whose solution and spirit of formulation are not unlike many of the problems in Part V of this book, may be posed as dynamic programming problems. The interested reader may consult Breiman (1964), Dubins and Savage (1965), Epstein (1967), Ross (1972), and Thorp (1969). The Epstein book provides an extremely lucid introduction to gambling problems. It is often advisable to develop special computational schemes to reduce the memory requirements needed to solve actual dynamic programming problems. An elementary treatment of some of these schemes may be found in the work of Bellman and Dreyfus (1962), Nemhauser (1966), and Wilde and Beightler (1967). For an advanced treatment of one such scheme consult Larson (1968). Continuous-time models pose special difficulties because one must generally utilize discrete time approximations to develop a dynamic programming functional equation. Such an approach is discussed and utilized in Merton's paper in Part V. See Kushner (1967) or Dreyfus (1965) for general discussions of such problems.

INTRODUCTION

9

1. EXPECTED UTILITY THEORY The Annals of Mathematical Statistics 1969, Vol. 40, No. 4, 1419-1429

A GENERAL THEORY OF SUBJECTIVE PROBABILITIES AND EXPECTED UTILITIES B Y 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 Raiffa [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 « o n I and a finitely-additive probability measure P on the set of all subsets of S such that, for all /, g e F, (1)

/ < S i f a n d o n l y i f £ [ u ( / ( s ) ) , P * ] g E[u(g(s)),

P%

In (1), < ("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 Q S, and if P (A) < 1 for every finite i C S it may or may not be true that 5 can be partitioned into an arbitrary finite number n of subsets such that P* - \/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 theory 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.

1.

EXPECTED UTILITY THEORY

11

1420

PETER C. FISHBTJBN

place no special restrictions on S but they imply that X is infinite and that if u(x) < u(y) then there is a z E 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. (P is the set of all simple probability measures (gambles) on X, so that if P E (P then P(Y) = 1 for some finite Y included in X. The probabilities used in (P are extraneous measurement probabilities. They can be associated with outcomes of chance devices such as dice and roulette wheels. With P, Q e. Under this interpretation, (P is a mixture set. By Herstein and Milnor's [7] definition, a mixture set is a set M and an operation that assigns an element aa + (1 — a)b in M to (a, b) £ M x M and a £ [0, 1] in such a way that (l)o + (0)6 = a aa + (1 — a)b = (1 — a)b + aa a{0a

+ (1 - 0)6) + (1 - a)b = (a/3)a + (1 -

a0)b

for all a,b EM and a, f$ E [0, 1]. 3C is the set of all functions on S to (P. With P E 3C, P(S) is the gamble in
(*)(»(*)) = 1 for all s £ S . When P ( s ) ( / ( s ) ) = l f o r a l l s e S , B[fi(u, ? ( • ) ) , P * ] - ^ [ u ( f ( » ) ) , P*J3. Axioms and summary theorem. In addition to the structural assumptions of the preceding section ((P is the set of simple probability measures on JC, 3C is the set of all functions on S to

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