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JAMES A. OHLSON Assumptions A1-A3 are extremely stringent; Al requires that all moments are finite, and, as will be shown in subsequent analysis, this is quite unnecessary. Perhaps more important, even if all moments are finite and satisfy Al, it does not follow that A2 and A3 are always satisfied for a utility function which is everywhere differentiable an infinite number of times. In fact, not even A3 is implied by A2. As a first illustration, suppose U{W) = \ogW, and Pr{K> W> 2} > 0 for every t, no matter how small. Although Al may well be satisfied, it is still true that j = o is not defined for any t > 0. Clearly, A2 is a nontrivial restriction. An example where A2 is true but A3 is not can also be easily provided. Consider the case in which U(W) = — exp{ — cW), c> 0, and where F(W) is a distribution with finite moments. Obviously, if F(0) = 0, then E \ U(W)\ < oo and A2 is satisfied. However, there are distributions which yield quadratic utility as t -* 0, but lim„âo|£/ ( " ) (l)£(»'-l)7/i!| = oo so the infinite series in (5) will not converge. 4 From both of these examples it is apparent that assumptions A1-A4 are not particularly useful even for the cases in which U(W) or W or both are bounded. In motivating the development of a set of weaker assumptions than those given so far (A1-A4), it is instructive to analyze the remainder term in expression (3), EQ3. Now, for any n jg 3, EQ3 = E Xc/ (0 (i) ((**--i)7n) + e„+i i=3 Assuming the first two moments are of order 0(t), one weak necessary condition for asymptotic quadratic utility is that E(W— 1)" = o(t) for some n ;> 3; if E(W-1) 4 = 0(t), say, it is then hardly plausible that EQ3 = o{t). More generally, it is desirable to state assumptions about F, such that these assumptions imply that no "information" is contained beyond the first two moments as / -> 0. The characteristic function of a random variable determines the distribution uniquely; it is therefore appealing to develop conditions on the moments such that the characteristic function is asymptotically determined by the first two moments alone. Preferably, the latter should hold even though higher order moments may not be finite. The following result provides some intuitively obvious and useful conditions. Theorem I Suppose £(^-1) and E{W-\) 2 are of order 0(0, and * See Section II in this paper; there it is assumed the returns are log normally distributed. 224 PART III STATIC PORTFOLIO SELECTION MODELS
THE ASYMPTOTIC VALIDITY OF QUADRATIC UTILITY E\W—1| 3 = o(t). The characteristic function of W— 1 is then given by £eiu(w-i) = j + iuE(W_ j) _ %uiE(W-1) 2 + o(0- Proof The characteristic function of (W— 1) = Z is defined as Ee'" z = £COS(MZ) + iEsin(uZ). Applying Taylor's theorem for cos(-) and sin(-), the characteristic function is also equal to where E\ Z —77- + ^„(") > (iuZ) n Rn( u ) = —r~ [cos(u
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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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Page 209 and 210: 7. Consider an investor having the
Page 211 and 212: Then eliminate p from (2) to obtain
Page 213 and 214: (a) Show that an indifference curve
Page 215 and 216: (d) Develop a similar result to (c)
Page 217: Exercise Source Notes Exercise 1 wa
Page 220 and 221: (f) Show that k > 0. [Note (Exercis
Page 222 and 223: (c) Interpret (b). (d) Illustrate s
Page 224 and 225: 9. Let the partial relative risk-av
Page 226 and 227: Condition (iv) is called the "no-ea
Page 228 and 229: Then * ( 8xj \ V - s u = \-r-\ -ax,
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Page 232 and 233: (a) Determine necessary and suffici
Page 234 and 235: Let the random variables Og^Sl and
Page 237: Part III Static Portfolio Selection
Page 240 and 241: given parameter. Exercise ME-30 dev
Page 242 and 243: utility function. However, many con
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Page 246 and 247: proportions to maximize expected ut
Page 248 and 249: investment returns are always nonne
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Page 258 and 259: JAMES A. OHLSON that quadratic util
Page 262 and 263: JAMES A. OHLSON (i) snpl/ieSi\U (3)
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/
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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 and 302: CHOOSING INVESTMENT PORTFOLIOS 3. C
Page 303 and 304: 2. EXISTENCE AND DIVERSIFICATION OF
Page 305 and 306: ON THE EXISTENCE OF OPTIMAL POLICIE
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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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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
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V Value of information, 462-467, 68
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