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384 GORDON PTB utility function is the realized outcome given in (2). The optimal, nonsequential, expected utility maximizing policy will be the solution to the problem in (7). (7) Ma,x,E{U(po+ Ei'ApizOI 0 S zT g zr-i S , • • • , S zi =S 1. Let the expected utility of the objective in (7) be denoted by V(z) where z is the vector, z = (zi, Zi, • • • , zT). The value of V is invariant to any renumbering of the Zt in z. This follows because given that the Ap, are independently and identically distributed and that the argument of U is additive in the ApiZt, any renumbering of the zt can he written as a renumbering of the variables of integration (i.e., the Ap,). The latter renumbering has no effect on the value of the expected utility integral. Furthermore, if U is strictly concave, V is a strictly concave function of z. The proof is similar to one provided in [9, p. 112]. It will now be shown that the solution to (7) must have all the values of the zt equal to each other. Suppose that z = z maximizes V(z) for all z such that 0 ^ z S 1. Suppose, furthermore, that at least two of the elements of z are not equal to each other. Select two such elements and interchange their values giving z". Since z is a renumbering of z it follows that V(z") = V(z') and also that 0 ±5 z" S 1. Consider the vector z — \z + (1 — X)z" where 0 < X < 1. It follows that 0 S Xz + (1 — X)z" g 1. Also, since z ^ z", it follows from the strict concavity of V{z) that V(Xz' + (1 - X)z") > XF(z') + (1 - X)F(z") = V(z'). Thus, Xz + (1 — X)z gives a higher value to V(z) than z . Thus, z cannot maximize V(z) on 0 £ z g 1 unless its elements are all equal to each other. Having all the zt equal satisfies the additional constraints in (7) that no value of z, have a larger value than its predecessor. The maximum of V{z) subject to these additional constraints must therefore also have the values of the Zt equal to each other. This is a necessary condition on the solution to (7). The argument holds for any strictly concave utility function and any arithmetic random walk. Since the values of the zt must be equal for an optimum, dollar averaging cannot be an optimal strategy in a multiperiod case. Dollar averaging requires that the zt strictly decrease over time. Dollar averaging could occur only in a one period case, and then only if by coincidence the value of zi turned out to be §. On the basis of this evidence it seems safe to conclude that dollar averaging is a multiperiod, nonsequential strategy based on hedging against large regrets rather than on risk averse expected utility maximization. 3. Sequential Policies In general, of course, the seller is perfectly free to let his future decisions depend on the prices which will be observed before the decision actually has to be made. It is therefore of interest to develop the sequential minimax policies. It seems to be common experience to have second thoughts once one has embarked on a nonsequential policy of the dollar averaging type. If the price rises subsequently one has a strong inclination to sell less, while if the price falls one wants to sell more. This might be explained on the basis of extrapolative expectations. Interestingly enough, however, it is also in accord with a sequential minimax policy. Let p, be defined as the maximum price which occurs up through time t (i.e., p, = Maxogigi [p,]). In terms of the notation of the last section then, pT* = p,.. Let n, be defined as the difference between the largest price which has occurred up 582 PART V. DYNAMIC MODELS
MINIMAX POLICIES FOR SELLING AN ASSET AND DOLLAR AVERAGING 385 through t and the price at time t (i.e., re, = p* — p,). Using this notation, the maximum price pT* can be written as in (8) (8) v* = V * + Max [0, -re, + £{+, APi (t + 1 S i S T)]. Subtracting (2) from (8) gives the regret in (9): (9) R = p* — Pa — Si ApiZi + Max [0, - n, + S'+i Aft (( + 1 g t g D] - S^+i Ap
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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 (— 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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Page 569 and 570: Let Then OPTIMAL INVESTMENT 595 *.=
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Page 573 and 574: where (54) T(x) = sup \ log c + log
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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
Page 617: MINIMAX POLICIES FOB SELLING AN ASS
Page 621 and 622: MINIMAX POLICIES FOR SELLING AN ASS
Page 623 and 624: MINIMAX POLICIES TOR SELLING AN ASS
Page 625 and 626: MINIMAX POLICIES FOB SELLING AN ASS
Page 627: MINIMAX POLICIES FOR SELLING AN ASS
Page 630 and 631: 648 L. BREIMAN We have discussed th
Page 632 and 633: 650 L. BREIMAN Hence, it is suffici
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Page 636 and 637: EDWARD O. THORP method to this arti
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Page 640 and 641: EDWARD O. THORP us is that the prop
Page 642 and 643: EDWARD O. THORP Let the range of Rj
Page 644 and 645: EDWARD O. THORP is approximately on
Page 646 and 647: EDWARD O. THORP choosing P in repea
Page 648 and 649: EDWARD O. THORP return from the hed
Page 650 and 651: EDWARD O. THORP it is plausible to
Page 652 and 653: EDWARD O. THORP is maximization of
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Page 657 and 658: Reprinted from JOURNAL OF ECONOMIC
Page 659 and 660: CONSUMPTION AND PORTFOLIO RULES 375
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and CONSUMPTION AND PORTFOLIO RULES
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and CONSUMPTION AND PORTFOLIO RULES
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CONSUMPTION AND PORTFOLIO RULES 389
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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
Page 747 and 748:
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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