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W. T. ZIEMBA 25. PRESS, S. J., "A Modified Compound Poisson Process with Normal Compounding." Journal of the American Statistical Association 60 (1968), 607-613. 26. PRESS, S. J., "A Compound Poisson Process for Multiple Security Analysis." In Random Counts in Scientific Work (G. Patil, ed.). Pennsylvania State Univ. Press, University Park, Pennsylvania, 1970. 27. ROLL, R., The Behavior of Interest Rates. Basic Books, New York, 1970. 28. SAMUELSON, P. A., "General Proof that Diversification Pays." Journal of Financial and Quantitative Analysis 2 (1967), 1-13. 29. SAMUELSON, P. A., "Efficient Portfolio Selection for Pareto-Levy Investments." Journal of Financial and Quantitative Analysis 2 (1967), 107-122. 30. SHARPE, W. F., "A Simplified Model for Portfolio Analysis." Management Science 9 (1963), 277-293. 31. TOBIN, J., "Liquidity Preference as Behavior Towards Risk." Review of Economic Studies 25 (1958), 65-86. 32. ZANGWILL, W. I., Nonlinear Programming: A Unified Approach. Prentice-Hall, Englewood Cliffs, New Jersey, 1969. 33. ZIEMBA, W. T., "Solving Nonlinear Programming Problems with Stochastic Objective Functions." Journal of Financial and Quantitative Analysis 7 (1972), 1809-1827. 34. ZIEMBA, W. T., "Computational Aspects of Optimal Portfolio Construction: Relationship and Comparison of Mean-Dispersion and SD Approaches." In Stochastic Dominance: An Approach to Decision Making Under Risk (G. A. Whitmore and M. C. Findlay, eds.). To be published. 35. ZIEMBA, W. T., PARKAN, C, and BROOKS-HILL, F. J., "Calculation of Investment Portfolios with Risk Free Borrowing and Lending." Management Science 21 (1974), 209-222. 266 PART III STATIC PORTFOLIO SELECTION MODELS
2. EXISTENCE AND DIVERSIFICATION OF OPTIMAL PORTFOLIO POLICIES JOURNAL OF ECONOMIC THEORY 4, 35-44 (1972) On the Existence of Optimal Policies under Uncertainty HAYNE E. LELAND* Stanford University, Stanford, California 94305 Received August 10, 1970 I. INTRODUCTION Many economic theories presume that a feasible policy set contains an optimal policy. In portfolio theory, for example, theorists often presume that first order maximizing conditions are satisfied by a feasible portfolio, without consideration of whether the existence of an optimal policy is implied by, or even consistent with, the basic set of assumptions. Theorists who have taken care to ensure the existence of solutions commonly have resorted to introducing explicit bounds on a closed policy or action set (in R"), thereby rendering it compact. Given a continuous objective function, Weierstrass' Theorem can then be used to prove the existence of an optimal action. Hakansson [2], for example, bounds the feasible set of his portfolio model by requiring that terminal wealth be nonnegative with probability one. But costs may be incurred by bounding the action set. Bounds often are difficult to justify on economic grounds. 1 Or they are so loose (e.g., "use of resources must not exceed world supply") that the question of existence is transformed to the question of whether the bounds are tight or slack at the optimum. But the answer to the latter question conditions the predictions of the theory as critically as the answer to the former. To escape Scylla and Charybdis, some theorists introduce constraints to ensure existence, and then assume these constraints are slack at the optimum! Clearly, this is little better than to have ignored the original existence questions. * This work was supported by National Science Foundation Giant GS-2530 at the Institute for Mathematical Studies in the Social Sciences at Stanford University. •Hakansson's requirement that final wealth be nonnegative, for example, seems rather restrictive. If an asset had normally distributed returns, no investment would occur no matter how large the expected return and how small the variance, as there always is a nonzero probability of an arbitrarily large loss. © 1972 by Academic Press, Inc. 2. EXISTENCE AND DIVERSIFICATION OF OPTIMAL PORTFOLIO POLICIES 267
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STOCHASTIC OPTIMIZATION MODELS IN F
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STOCHASTIC OPTIMIZATION MODELS IN F
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Dedicated to the memory of my fathe
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PART II. QUALITATIVE ECONOMIC RESUL
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2. Risk Aversion over Time Implies
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PREFACE AND BRIEF NOTES TO THE 2006
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and shows that current research in
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1990. This area presages the CVaR l
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transaction cost band. Most such mo
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through targets and are less sensit
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ing dates, the papers by Breiman, H
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Carino, D. and W.T. Ziemba (1998).
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Kallberg, J.G. and W.T. Ziemba (198
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Stone, D. and W.T. Ziemba (1993). L
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PREFACE IN 1975 EDITION There is no
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particular results are generally st
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Part I Mathematical Tools
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wealth, then his utility function m
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cases in Exercise CR-12. The relati
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However, some extensions to infinit
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1. EXPECTED UTILITY THEORY The Anna
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SUBJECTIVE PROBABILITIES ANT EXPECT
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SUBJECTIVE PROBABILITIES AND EXPECT
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282 O. L. MANGA3ARIAN" for every X
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284 O. L. MANGASARIAN We have from
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280 O. L. MANQASARIAN For the case
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288 O. L. MANGASAMAN Proof. Conside
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290 O. L. MANGASARIAN Second Berkel
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denotes the /w-dimensional vector o
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Proof (I) We shall prove this part
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(Ill) have not, to the author's kno
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for the case (3); and for the case
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3. DYNAMIC PROGRAMMING Introduction
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INTRODUCTION TO DYNAMIC PROGRAMMING
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COMPUTATIONAL AND REVIEW EXERCISES
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(d) xxx2 on E+ 2 = {x\xt = 0, x2 S
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Show that the Kuhn-Tucker condition
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(a) Show that/, (b, c) = gt (Z>, c)
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(c) Show that the monotonicity assu
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An individual's preferences are sai
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12. (Hessians, bordered Hessians, c
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The problem in (b)-(c) hints that t
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(1) Suppose/is twice continuously d
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(f) Assume that R(i,a) g 0 and the
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(j) Show that an optimal policy Va
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INTRODUCTION In the second part of
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andom variables when the utility fu
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detailed analysis of the qualitativ
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separation in a local sense (for al
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1. STOCHASTIC DOMINANCE G. Hanoch a
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But now, EFFICIENCY ANALYSIS OF CHO
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EFFICIENCY ANALYSIS OF CHOICES INVO
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A Unified Approach to Stochastic Do
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A UNIFIED APPROACH TO STOCHASTIC DO
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RISK AVERSION 123 a given risk the
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RISK AVERSION 125 If z is actuarial
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RISK AVERSION 127 4. CONCAVITY The
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To show that (a) implies (d), note
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RISK AVERSION 131 (b') The risk pre
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9.2. Example 2. If (30) u'(x)=(x°
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RISK AVERSION 135 12. INCREASING AN
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ADDENDUM In retrospect, 1 wish foot
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14 THE REVIEW OF ECONOMICS AND STAT
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Separation in Portfolio Analysis R.
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SEPARATION IN PORTFOLIO ANALYSIS an
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SEPARATION IN PORTFOLIO ANALYSIS II
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SEPARATION IN PORTFOLIO ANALYSIS Th
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SEPARATION IN PORTFOLIO ANALYSIS Ca
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SEPARATION IN PORTFOLIO ANALYSIS It
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SEPARATION IN PORTFOLIO ANALYSIS th
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7. Consider an investor having the
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Then eliminate p from (2) to obtain
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(a) Show that an indifference curve
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(d) Develop a similar result to (c)
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Exercise Source Notes Exercise 1 wa
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(f) Show that k > 0. [Note (Exercis
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(c) Interpret (b). (d) Illustrate s
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9. Let the partial relative risk-av
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Condition (iv) is called the "no-ea
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Then * ( 8xj \ V - s u = \-r-\ -ax,
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exhibiting decreasing absolute and
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(a) Determine necessary and suffici
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Let the random variables Og^Sl and
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Part III Static Portfolio Selection
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given parameter. Exercise ME-30 dev
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utility function. However, many con
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the characteristic exponent is 2 (i
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proportions to maximize expected ut
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investment returns are always nonne
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case to a case of one riskless and
Page 252 and 253: 538 REVIEW OF ECONOMIC STUDIES Howe
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Page 256 and 257: 542 REVIEW OF ECONOMIC STUDIES expa
Page 258 and 259: JAMES A. OHLSON that quadratic util
Page 260 and 261: JAMES A. OHLSON Assumptions A1-A3 a
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Page 264 and 265: JAMES A. OHLSON A number of interes
Page 266 and 267: JAMES A. OHLSON for all y,teSx'3C{Q
Page 268 and 269: Since Z"=o^( w »0 = 0 and/(«,«)
Page 270 and 271: JAMES A. OHLSON analysis here is "o
Page 272 and 273: 76 THE REVIEW OF ECONOMICS AND STAT
Page 279 and 280: Choosing Investment Portfolios When
Page 281 and 282: CHOOSING INVESTMENT PORTFOLIOS univ
Page 283 and 284: CHOOSING INVESTMENT PORTFOLIOS The/
Page 285 and 286: CHOOSING INVESTMENT PORTFOLIOS Tobi
Page 287 and 288: CHOOSING INVESTMENT PORTFOLIOS a-di
Page 289 and 290: CHOOSING INVESTMENT PORTFOLIOS Henc
Page 291 and 292: CHOOSING INVESTMENT PORTFOLIOS Lemm
Page 293 and 294: CHOOSING INVESTMENT PORTFOLIOS ID.
Page 295 and 296: CHOOSING INVESTMENT PORTFOLIOS Proo
Page 297 and 298: CHOOSING INVESTMENT PORTFOLIOS is c
Page 299 and 300: CHOOSING INVESTMENT PORTFOLIOS Proo
Page 301: CHOOSING INVESTMENT PORTFOLIOS 3. C
Page 305 and 306: ON THE EXISTENCE OF OPTIMAL POLICIE
Page 313 and 314: Reprinted from Journal of Financial
Page 315 and 316: Remarks: Differentiability assumpti
Page 317 and 318: distributed from the rest. Then an
Page 319 and 320: x * "22 "12 1 " (o22- a12) + (0n -
Page 321 and 322: to decline — as if having more mo
Page 323 and 324: solved for as a function 8.(y,0"),
Page 325: FOOTNOTES 1. H. Makower and J. Mars
Page 328 and 329: 264 QUARTERLY JOURNAL OF ECONOMICS
Page 338 and 339: 274 QUARTERLY JOURNAL OP ECONOMICS
Page 344 and 345: 280 QVABTBRLY JOURNAL OF ECONOMICS
Page 349 and 350: Reprinted from THE REVIEW OF ECONOM
Page 351 and 352: (1.1) and (1.2) give SOME EFFECTS O
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SOME EFFECTS OF TAXES ON RISK-TAKIN
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REFERENCES SOME EFFECTS OF TAXES ON
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and a gamble is offered that return
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among two risky assets and a riskle
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Consider problem (1) and suppose th
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21. Let pi,...,p„ be joint-normal
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Consider the following four utility
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4. In most stochastic optimizing mo
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(g) Discuss the use of semivariance
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(a) Suppose that the investor alloc
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The linear programming subproblems
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(f) Show that (j> is proportional t
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(e) Show that X has mean and varian
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(i) Show that for all Aru ...,km, t
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Some properties of the expected val
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(i) Suppose that V0 is positive def
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where V is the gradient operator an
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and where Z, Ylt...,Ym are independ
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INTRODUCTION This part of the book
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some qualitative characteristics of
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function of wealth. Hence the optim
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INVESTMENT ANALYSIS UNDER UNCERTAIN
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P if and only if INVESTMENT ANALYSI
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INVESTMENT ANALYSIS UNDEB UNCERTAIN
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2. RISK AVERSION OVER TIME IMPLIES
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ables at t, and the primed variable
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ehavior in choosing among timeless
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is to asume that the consumer behav
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The function t/,(C,_,, wt|a,, ft) i
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sumption, given state j3,+i at r+l,
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3. MYOPIC PORTFOLIO POLICIES Reprin
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326 THE JOURNAL OF BUSINESS this op
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328 THE JOURNAL OF BUSINESS with th
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330 THE JOURNAL OF BUSINESS In sum
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332 THE JOURNAL OF BUSINESS subject
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334 THE JOURNAL OF BUSINESS myopic
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where and k" 1 f (Mt-Ui) 2 4k {Oi 1
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MIND-EXPANDING EXERCISES 1. Conside
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(i.e., partial separability) and #,
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(c) Discuss the solution properties
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(d) By differentiating (cl) and (c2
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Let U be the utility function for t
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INTRODUCTION I. Two-Period Consumpt
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asked to develop explicit solutions
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that the property of nonincreasing
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The Hakansson paper studies a gener
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In the Samuelson-Hakansson additive
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techniques. The article by Breiman
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enously). These requirements are to
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in general, a monotonic function of
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ergodic theory, the book by Breiman
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or less than u; either do nothing o
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ME-24 the reader is invited to cons
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In discrete time, a nonnegative lin
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R. G. VICKSON This must not be inte
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This gives with R. G. VICKSON t(t)
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TWO-PERIOD CONSUMPTION MODELS AND P
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310 DRfeZE AND MODIGLIANI second or
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312 DREZE AND MODIGLIANI Had the sa
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314 DRfezE AND MODIGLIANI (— U^U^
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316 DREZE AND MODIGLIANI ure for te
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318 DREZE AND MODIGLIANI income sid
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320 DREZE AND MODIGLIANI (3.5) The
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322 DREZE AND MODIGLIANI Consumptio
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324 DREZE AND MODIGLIANI in (cj, c2
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326 DREZE AND MODIGLIANI Since d^Uj
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328 DREZE AND MODIGLIANI On the rig
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330 DRfeZE AND MODIGLIANI and its p
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332 DREZE AND M0DIGLIAN1 LEMMA C.2.
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334 DRfeZE AND MOD1GLIANI if the sm
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MANAGEMENT SCIENCE Vol. 19, No. 2,
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A DYNAMIC MODEL FOR BOND PORTFOLIO
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Maximize A DYNAMIC MODEL FOE BOND P
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MULTIPERIOD RISK PREFERENCE 41 mode
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MULTIPERIOD RISK PREFERENCE 43 The
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MULT1PERI0D RISK PREFERENCE between
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MULTIPERIOD RISK PREFERENCE 47 (ii)
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MULTIPERIOD RISK PREFERENCE 49 that
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or if s"(s' MULTIPERIOD RISK PREFER
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MULTIPERIOD RISK PREFERENCE 53 5. C
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Reprinted from The Review of Econom
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the risky asset, with 1 — w, goin
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w*t = glW,;Z,-1,...,Z0] = gT_i[W>]
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Since proof of the theorem is strai
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E C O N O M E T R I C A VOLUME 38 S
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OPTIMAL INVESTMENT 589 M: the numbe
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OPTIMAL INVESTMENT 591 For comparis
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(ii) in the case of Model III, 1 1
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Let Then OPTIMAL INVESTMENT 595 *.=
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where OPTIMAL INVESTMENT 597 (41) T
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where (54) T(x) = sup \ log c + log
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where b(v*) is the greatest lower b
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OPTIMAL INVESTMENT 603 noncapital i
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OPTIMAL INVESTMENT 605 Let us now t
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OPTIMAL INVESTMENT 607 REFERENCES [
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present value. Let ( be time, r the
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constant or decrease at a constant
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Denote the vector in the parenthese
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112 HOWARD M. TAYLOB easy to pictur
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114 HOWARD M. TAYLOR market and ign
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116 HOWARD M. TAYLOR the investor w
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118 HOWARD M. TAYLOR option which a
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120 HOWARD M. TAYLOR 3. BR.VDA, J.
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172 BASIL A. KALYMON more general p
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174 BASIL A. KALYMON The three case
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176 BASIL A. KALYMON an additional
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178 BASIL A. KALYMON (24) d\ = r -
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180 BASIL A. KALYMON The above resu
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182 BASIL A. KALYMON PROOF. First w
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MANAGEMENT SCIENCE Vol. 17, No. 7,
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M1NIMAX POLICIES FOB SELLING AN ASS
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MINIMAX POLICIES FOB SELLING AN ASS
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MINIMAX POLICIES FOR SELLING AN ASS
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MINIMAX POLICIES TOR SELLING AN ASS
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MINIMAX POLICIES FOB SELLING AN ASS
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648 L. BREIMAN We have discussed th
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650 L. BREIMAN Hence, it is suffici
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(a.s.) on the set lim XN < oo > u <
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EDWARD O. THORP method to this arti
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EDWARD O. THORP drastic simplificat
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EDWARD O. THORP us is that the prop
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EDWARD O. THORP Let the range of Rj
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EDWARD O. THORP is approximately on
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EDWARD O. THORP choosing P in repea
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EDWARD O. THORP return from the hed
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EDWARD O. THORP it is plausible to
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EDWARD O. THORP is maximization of
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EDWARD O. THORP 6. BREIMAN, L., "Op
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Reprinted from JOURNAL OF ECONOMIC
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CONSUMPTION AND PORTFOLIO RULES 375
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and CONSUMPTION AND PORTFOLIO RULES
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CONSUMPTION AND PORTFOLIO RULES giv
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Yin CONSUMPTION AND PORTFOLIO RULES
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Y(t) Y CONSUMPTION AND PORTFOLIO RU
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(d) Suppose n > r2. Formulate the o
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(j) Show that the absolute risk-ave
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The optimal first-period decision a
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(b) Show that the optimal policy in
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the game is stationary and favorabl
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(v) UHEV-E^CV-Af- 1 )^ w, V&0, w>0,
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MIND-EXPANDING EXERCISES 1. Conside
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Note the following distinction betw
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(e) Discuss the computational aspec
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(a) Show that U obeys the functiona
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(g) If ;ij(0) > 0 (1 g; g N), show
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Note that problem P2 has 3(N- 1)+1
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Note that (f) and (g) imply the exi
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P = (Pu), where pu = P[X„+l =j\ X
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(a) Show that {X„} is a stationar
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(b) Show that T T u = a + b Y[ c\'
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(b) Interpret this assumption. Is i
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(d) Show that (c) may be written as
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Bibliography Journal Abbreviations
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BHARUCHA-REID, A. T. (1960). Elemen
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FIACCO, A. V., and MCCORMICK, G. P.
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IGLEHART, D. (1965). "Capital accum
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MANGASARIAN, O. L. (1966). "Suffici
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PORTEUS, E. L. (1972). "Equivalent
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THOMASION, A. J. (1969). The Struct
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A Absolute risk aversion, 84, 85, 9
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Logarithmic utility functions, see
Page 755 and 756:
V Value of information, 462-467, 68
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