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592 NILS H. HAKANSSON Pratt [11] notes that (16H18) are the only monotone increasing and strictly concave utility functions for which the relative risk aversion index is a positive constant and that (19) is the only monotone increasing and strictly concave utility function for which the absolute risk aversion index is a positive constant. 6 THEOREM 1: Let u{c), a, y, r, {/?,}, and Ybe defined as in Section 2. Then, whenever u(c) is one of the functions (16H18) and ky < 1/a in Model I, a solution to (11) subject to (12)-(14) exists for x > — Y and is given by (22) (23) (24) (25) f(x) = Au(x + Y) + C, c*(x) = B(x + Y), z\(x) = (1 - B)(l - v*)(x + Y) -- Y, z'(x) = (1 - B)v](x + Y) where the constants u* (u* = £" 2 v]) and k are given by (26) k = £r«(E'(ft-r)i>: + r)] subject to (27) D.-SSO, ifS, and (28) Prl^(pt-r)vi + rô\ = \, and the constants A, B, and C are given by (i) in the case of Models /-//, A = (1 -(a/cy) 1 '"-")'- 1 , (29) B= 1 -(ofcy) 1 " 1 "". C = 0; (i = 2,...,M) 6 The underlying mathematical reason why solutions are obtained in closed form (Theorems 1 and 2) for the utility functions (16H19) is that these functions are also the only (monotone increasing and strictly concave utility function) solutions (see [8]) to the functional equations u(xy) = v(x)w(y), u(xy) = v(x) + w{y), u(x + y) = v(x)w{y), and u(x + y) = v(x) + w(y)t which are known as the generalized Cauchy equations [1, p. 141]. 530 PART V. DYNAMIC MODELS
(ii) in the case of Model III, 1 1 - a (30) B = 1 - a, C = 1 log 1 - « + Tp^-3 + 7. rj- 1 - a (1 - a)- 1 (1 — aY Furthermore, the solution is unique. OPTIMAL INVESTMENT 593 In proving this theorem, we shall make use of the following lemma and corollaries. LEMMA: Let u(c), {ft}, and r be defined as in Section 2 and let v = {v2,..., vM) be a vector of real numbers. Then the function (31) h{v2,v3,...,vu) = E subject to the constraints (27) D.-SSO, i ol = 1 where v[ 5= 0 for all i 4 S. Now consider the point v l = v° + Xv' = Xv' where A Js 0 and o' is one of the points in this neighborhood. Let b{v) be the greatest lower bound on b such that Prj I 0?i - r)f, < b\ > 0. 7 The author gratefully acknowledges a debt to Professor George W. Brown for several valuable suggestions concerning the proof of the closure and the boundedness of D. OPTIMAL CAPITAL ACCUMULATION AND PORTFOLIO SELECTION
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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
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538 REVIEW OF ECONOMIC STUDIES Howe
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540 REVIEW OF ECONOMIC STUDIES wher
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542 REVIEW OF ECONOMIC STUDIES expa
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JAMES A. OHLSON that quadratic util
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JAMES A. OHLSON Assumptions A1-A3 a
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JAMES A. OHLSON (i) snpl/ieSi\U (3)
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JAMES A. OHLSON A number of interes
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JAMES A. OHLSON for all y,teSx'3C{Q
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Since Z"=o^( w »0 = 0 and/(«,«)
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JAMES A. OHLSON analysis here is "o
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Choosing Investment Portfolios When
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CHOOSING INVESTMENT PORTFOLIOS univ
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CHOOSING INVESTMENT PORTFOLIOS The/
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CHOOSING INVESTMENT PORTFOLIOS Tobi
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CHOOSING INVESTMENT PORTFOLIOS a-di
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CHOOSING INVESTMENT PORTFOLIOS Henc
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CHOOSING INVESTMENT PORTFOLIOS Lemm
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CHOOSING INVESTMENT PORTFOLIOS ID.
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CHOOSING INVESTMENT PORTFOLIOS Proo
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CHOOSING INVESTMENT PORTFOLIOS is c
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CHOOSING INVESTMENT PORTFOLIOS Proo
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CHOOSING INVESTMENT PORTFOLIOS 3. C
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2. EXISTENCE AND DIVERSIFICATION OF
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ON THE EXISTENCE OF OPTIMAL POLICIE
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Reprinted from Journal of Financial
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Remarks: Differentiability assumpti
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distributed from the rest. Then an
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x * "22 "12 1 " (o22- a12) + (0n -
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to decline — as if having more mo
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solved for as a function 8.(y,0"),
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FOOTNOTES 1. H. Makower and J. Mars
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264 QUARTERLY JOURNAL OF ECONOMICS
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274 QUARTERLY JOURNAL OP ECONOMICS
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280 QVABTBRLY JOURNAL OF ECONOMICS
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Reprinted from THE REVIEW OF ECONOM
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(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 (— UÛ^
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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 dÛj
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Page 517 and 518: 330 DRfeZE AND MODIGLIANI and its p
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Page 525 and 526: A DYNAMIC MODEL FOR BOND PORTFOLIO
Page 531 and 532: Maximize A DYNAMIC MODEL FOE BOND P
Page 535: A DYNAMIC MODEL FOR BOND PORTFOLIO
Page 538 and 539: MULTIPERIOD RISK PREFERENCE 41 mode
Page 540 and 541: MULTIPERIOD RISK PREFERENCE 43 The
Page 542 and 543: MULT1PERI0D RISK PREFERENCE between
Page 544 and 545: MULTIPERIOD RISK PREFERENCE 47 (ii)
Page 546 and 547: MULTIPERIOD RISK PREFERENCE 49 that
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Page 553 and 554: Reprinted from The Review of Econom
Page 555 and 556: the risky asset, with 1 — w, goin
Page 557 and 558: w*t = glW,;Z,-1,...,Z0] = gT_i[W>]
Page 559 and 560: Since proof of the theorem is strai
Page 561 and 562: E C O N O M E T R I C A VOLUME 38 S
Page 563 and 564: OPTIMAL INVESTMENT 589 M: the numbe
Page 565: OPTIMAL INVESTMENT 591 For comparis
Page 569 and 570: Let Then OPTIMAL INVESTMENT 595 *.=
Page 571 and 572: where OPTIMAL INVESTMENT 597 (41) T
Page 573 and 574: where (54) T(x) = sup \ log c + log
Page 575 and 576: where b(v*) is the greatest lower b
Page 577 and 578: OPTIMAL INVESTMENT 603 noncapital i
Page 579 and 580: OPTIMAL INVESTMENT 605 Let us now t
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Page 584 and 585: present value. Let ( be time, r the
Page 586 and 587: constant or decrease at a constant
Page 588 and 589: Denote the vector in the parenthese
Page 590 and 591: 112 HOWARD M. TAYLOB easy to pictur
Page 592 and 593: 114 HOWARD M. TAYLOR market and ign
Page 594 and 595: 116 HOWARD M. TAYLOR the investor w
Page 596 and 597: 118 HOWARD M. TAYLOR option which a
Page 598 and 599: 120 HOWARD M. TAYLOR 3. BR.VDA, J.
Page 600 and 601: 172 BASIL A. KALYMON more general p
Page 602 and 603: 174 BASIL A. KALYMON The three case
Page 604 and 605: 176 BASIL A. KALYMON an additional
Page 606 and 607: 178 BASIL A. KALYMON (24) d\ = r -
Page 608 and 609: 180 BASIL A. KALYMON The above resu
Page 610 and 611: 182 BASIL A. KALYMON PROOF. First w
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Page 615 and 616: 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
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V Value of information, 462-467, 68
show all

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