ECONOMIC THEORY AND MATHEMATICAL ECONOMICS Consulting Editor: Karl Shell UNIVERSITY OF PENNSYLVANIA PHILADELPHIA, PENNSYLVANIA Franklin M. Fisher and Karl Shell. The Economic Theory of Price Indices: Two Essays on the Effects of Taste, Quality, and Technological Change Luis Eugenio Di Marco (Ed.). International Economics and Development: Essays in Honor of Raul Presbisch Erwin Klein. Mathematical Methods in Theoretical Economics: Topological and Vector Space Foundations of Equilibrium Analysis Paul Zarembka (Ed.). Frontiers in Econometrics George Horwich and Paul A. Samuehon (Eds.). Trade, Stability, and Macro- economics: Essays in Honor of Lloyd A. Metzler W. T. Ziemba and R. G. Vickson (Eds.). Stochastic Optimization Models in Finance Steven A. Y. Lin (Ed.). Theory and Measurement of Economic Externalities David Cass and Karl Shell (Eds.). The Hamiltonian Approach to Dynamic Economics R. Shone. Microeconomics: A Modern Treatment C. W. J. Granger and Paul Newbold. Forecasting Economic Time Series Michael Szenberg, John W. Lombardi, and Eric Y. Lee. Welfare Effects of Trade Restrictions: A Case Study of the U.S. Footwear Industry Haim Levy and Marshall Sarnat (Eds.). Financial Decision Making under Uncertainty In preparation Yasuo Murata. Mathematics for Stability and Optimization of Economic Systems Alan S. Blinder and Philip Friedman (Eds.). Natural Resources, Uncertainty, and General Equilibrium Systems: Essays in Memory of Rafael Lusky FINANCIAL DECISION MAKING UNDER UNCERTAINTY Edited by Haim Levy and Marshall Sarnat University of Florida The Hebrew University Gainesville, Florida of Jerusalem and Israel The Hebrew University of Jerusalem Israel ACADEMIC PRESS New York San Francisco London 1977 A Subsidiary of Harcourt Brace Jovanovich, Publishers COPYRIGHT © 1977, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road. London NW1 Library of Congress Cataloging in Publication Data Main entry under title: Financial decision making under uncertainty. (Economic theory and mathematical economics) Includes bibliographies and index. 1. Finance-Mathematical models. 2. Investments- Mathematical models. 3. Risk-Mathematical models. I. Levy, Haim. II. Sarnat, Marshall. HG174.F48 658.l'5 77-4572 ISBN 0-12-445850-5 PRINTED IN THE UNITED STATES OF AMERICA LIST OF CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors' contributions begin. Michael Adler (205), Graduate School of Business, Columbia University, New York, New York Fred D. Arditti (137, 237), School of Business Administration, University of Florida, Gainesville, Florida Bernard Dumas (205), E.S.S.E.C., Cergy, France Edwin J. Elton (265), Department of Finance, Graduate School of Business, New York University, New York, New York Irmn Friend (65), Rodney L. White Center for Financial Research, The Wharton School, University of Pennsylvania, Philadelphia, Pennsyl- vania Myron J. Gordon (83), Faculty of Management Studies, University of Toronto, Toronto, Ontario, Canada Martin J. Gruber (265), Department of Finance, Graduate School of Business, New York University, New York, New York Nils H. Hakansson (165), University of California, Berkeley, California D. Kir a (151), The University of British Columbia, Vancouver, British Columbia, Canada H aim Levy (137), The Hebrew University, Jerusalem, Israel Harry M. Markowitz (3), IBM Thomas J. Watson Research Center, York- town Heights, New York Arie Melnik (251), Faculty of Industrial and Management Engineering, Technion—Israel Institute of Technology, Haifa, Israel MertonH. Miller (95), Graduate School of Business, University of Chicago, Chicago, Illinois Yoram C. Peles (237), The Hebrew University, Jerusalem, Israel ix X List of Contributors Moshe A. Pollatschek (251), Faculty of Industrial and Management Engineering, Technion—Israel Institute of Technology, Haifa, Israel Mark Rubinstein (11), Graduate School of Business Administration, Univer- sity of California, Berkeley, California William F. Sharpe (127), Graduate School of Business, Stanford University, Stanford, California Charles W Upton (95), Graduate School of Business, The University of Chicago, Chicago, Illinois W. T. Ziemba (151), Faculty of Commerce and Business Administration, The University of British Columbia, Vancouver, British Columbia, Canada PREFACE The two and one-half decades that have elapsed since Harry Markowitz first applied the von Neumann-Morgenstern theory of expected utility to financial analysis have proved to be a period of great intellectual ferment. Accepted ideas and theories were critically examined, refined, extended, or, what is much more difficult, on occasion even discarded. But perhaps the salient feature of these years has been the ever increasing application of the new approach to financial decision making under uncertainty to a myriad of practical problems, many of which had hitherto escaped rigorous analysis. Although one score and five is a relatively short period of time in which to assess the impact of any theory, it does seem adequate to support a "progress report" of sorts. This book brings together a number of papers that we feel can provide some fundamental insight into the current state of the art. As befits a young and dynamic subject, we try to achieve this goal, not through learned reviews of what already has been accomplished, but rather by presenting the reader with a representative cross section of papers by leading scholars dealing with a wide variety of actual problems. This reflects our own personal bias that the best way to learn any pro- fession, e.g., carpentry, is to observe a master craftsman in action and then go out and try to emulate his work. Thus, this book should be considered as a first step in the two-stage process of "learning by doing," and as such is intended to serve as supplementary reading for a graduate course (or advanced undergraduate seminar) in financial theory. Much of the responsibility for the book lies with a remarkable team of people in Jerusalem who organize the Israel Scientific Research Conferences for the National Council of Research and Development. In March 1975 an international group of scholars met at the invitation of the National Council at Ein Bokek on the shores of the Dead Sea to discuss the theory of xi Xll Preface financial decision making under uncertainty with special emphasis on its application in practice. Ein Bokek lies 400 meters below sea level, the lowest spot on earth, and has witnessed many dramas throughout history. To the south, a ruined Roman fortress was within walking distance, while 15 kilometers northward the participants could visit Herod's great strong- hold of Massada, or go on to the beautiful nature reserve of Ein Gedi, the spot where David sought refuge from the anger of King Saul. The conference was designed to encourage the free exchange of ideas, and at times the animated discussion and debate almost seemed to recall the "wrath of Saul." This book provides a more tangible momento of what was, for us at least, a very memorable experience. The publication of these papers also provides a very welcome opportunity to express, on behalf of all the participants, our gratitude to Dr. Schabtai Gairon, the Coordinator of ISRACON, and to Linda Cohen and Joy Lipson of his staff, as well as to Avraham Beja, Arie Melnik, and Meyer Ungar who served on the organizing committee. Finally, we wish to thank Marcia Don who very capably handled the not inconsiderable administrative task of seeing the manuscript through its publication. AN ALGORITHM FOR FINDING UNDOMINATED PORTFOLIOS Harry M. Markowitz IBM Thomas J. Watson Research Center I. Introduction Let r be the return on the ith security in period t\ i = 1 to JV, t = 1 to T. it " t " may represent a historical period or a sample from a complex model of joint returns (e.g., "complex" in the sense that the problem of finding undom- inated portfolios cannot be solved analytically). Let X be the amount f invested in the ith security. The return on the portfolio as a whole in period t is Λ,= Σ *<'.·.. f = i t or (i) 1=1 While the matrix of returns (r ) is in fact a sample—from history or from a it model—we shall henceforth treat the columns of this matrix as the entire (finite) population of possible joint returns, each column having the same probability of occurrence. This paper presents an algorithm for finding port- The author is indebted to Haim Levy for posing the problem presented here, for helping with its solution, and for comments on the first draft of this paper. Subsequent to the presentation of this paper, an alternate and perhaps superior algorithm has been designed and programmed by Bezalel Gavish of the Israel Scientific Center, IBM Israel Ltd., Haifa. 3 4 Harry M. Markowitz folios which are second-order stochastically undominated in the sense of Hadar and Russell [1], Hammond [2], and Hanoch and Levy [3]. Let Ζ = Min R, Z = Min R + Rj γ t 2 t and, in general, Z = Min R + R + ··· + R for K = 1 to T (2) K h i2 iK where the minimization is over sets of distinct indices (i, i , ..., i ) such l 2 K that /· ^ i provided j ψ k. Writing Z = 0, we define 7 k 0 V = Z - Z_ for t = 1 to T (3a) t t t 1 The inverse relationship is Z, = £ V for ί = 1 to T (3b) t The cumulative probability distribution of returns provided by a portfolio is a step function F(R) with steps at V < V < V < · · · < V . If all the p; are x 2 3 T distinct, then each step is of size l/T. If K -1 < K = w +1 = * " = K + n -1 ^^ K + w then F(R) has a step of size n/T &t X = V = V = --. t t + l A portfolio is undominated in the "first-order" sense if there is no other portfolio with V* > V for some t and V* > V for all t t t where the V* define the cumulative distribution for "the other" portfolio. We show in the next section that a portfolio is undominated in the " second order" sense if there is no other portfolio with Z* > Z, for some t and Z* > Z for all t (4a) t An investor who maximizes the expected value of a concave utility function will choose some second-order undominated portfolios. Such portfolios form a subset of the first-order undominated portfolios. If ß > 0 for t = 1 to T t are given constants, then a portfolio which maximizes Φ=Σβ,Ζ, (4b) i=l must satisfy condition (4a) (else Z* = (Zf,..., Z\) would have a greater value of φ), and hence must be undominated in the second-order sense. Thus, if we An Algorithm for Finding Undominated Portfolios 5 maximize (4b) with all ß > 0, we obtain (if the maximum exists) a second- t order undominated portfolio. A different undominated portfolio may result if we change the /Ts and maximize again. We shall present below an economical algorithm for maximizing φ for any given positive /Ts, subject to (1), (2), and constraints of the form X a Xj = b i = 1 to M (5) u i9 χ. > 0, j = 1 to N (6) Constraints of types (5) and (6) are those of the mean-variance portfolio analysis of Markowitz [4]. They may be used to include borrowing and/or lending, upper bounds on the amounts invested in industries or individual stocks, etc. A major advantage of mean-variance analysis is that the set of (£, V) efficient portfolios may be drawn as a curve on a two-dimensional surface. In contrast, the set of undominated (Z Z , ..., Z ) may be a l9 2 T (T — l)-dimensional surface in a Γ-dimensional space. Thus if the condi- tions for using mean-variance are met (see Markowitz [4, Chapters 6 and 13]; Young and Trent [5]), then the latter is more convenient in practice than an analysis of undominated portfolios. If the aforementioned conditions are not met, however, an exploration of the space of undominated portfolios may be appropriate. For a portfolio which maximizes φ, the given /Fs are essentially the econo- mist's marginal rates of transformation among the Z's at a particular point on the efficient Z-surface, and imply marginal rates of transformation among the Ks [by substitution of (3b) into (4b)]. The consumer of the analysis, upon being presented with the maximizing distribution, may find that he would like to trade some K/s for other V's at the rates available; and hence t may want to alter his /Ts. This process could proceed most conveniently if the algorithm described below were coded for an interactive computer system. II. An Alternate Criterion for Dominance We shall write F,.(K), V Z , Î = 1, 2 ti9 fi for two distributions of the form described in the preceding section. We also define H,(X)=[* dF,(R) (7)