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additions to his wealth from sources exogenous to his investment choice (Exercise ME-6). III. Myopic Portfolio Policies The paper by Hakansson and several of the exercises are concerned with the problem of optimal portfolio myopia; that is, the determination of cases when the investor can make optimal decisions by considering only a oneperiod horizon. Such a result obtains if the derived utility function in each period is equivalent to the investor's utility function over final wealth. The utility functions are equivalent if they are the same except for a positive linear transformation. Exercise ME-5 considers this question for concave utility functions when there is one risky and one risk-free asset and it is assumed that the maximum in each period occurs at an interior point. Myopia results obtain whenever the reciprocal of the Arrow-Pratt absolute risk aversion index is linear in wealth. When the intercept of the linear function is zero (i.e., the utility function has constant relative risk aversion), or if the risk-free asset is money (i.e., its net return is zero), there is true myopia. In the general case, there is partial myopia in the sense that the derived utility function is equivalent to the investor's utility function if one "pretends" that all the investor's wealth is invested in the risk-free asset. Exercise CR-3 illustrates the myopia and partial myopia results for power utility functions. In this case, myopia obtains when the investor does not receive additions to his wealth from sources exogenous to his investment choice; otherwise, there is partial myopia. Hakansson's paper considers the more realistic case when it is not permissible to assume that there are always interior maxima. If the investment returns are intertemporally independent, then the logarithmic and power utility functions induce true myopia. However, if the returns are dependent, then only the logarithmic function is myopic. He also shows that the partial myopia results and myopic results for the case when the interest rate is zero, as developed in Exercise ME-5, are true only in a highly restricted sense when there are constraints on the allowable investment allocations. Brennan and Kraus (1974) investigate the close relationship between these conditions for myopia and the Cass- Stiglitz conditions needed for portfolio separation as discussed in Vickson's paper in Part II. Exercise ME-7 considers the myopia problem when the investment horizon becomes more and more distant. Turnpike-type results that induce true myopia are obtained when the reciprocal of the risk-aversion index is linear in wealth. Additional results obtain in the more general case when the reciprocal of the risk-aversion index equals a term proportional to wealth plus a bounded 370 PART IV DYNAMIC MODELS REDUCIBLE TO STATIC MODELS
function of wealth. Hence the optimal portfolio choice depends on the current distribution of the random assets, and optimal asset holdings are proportional to current wealth when the utility function has "almost" constant relative risk aversion for large wealth. However, see Ross (1974a) for an example that illustrates how the turnpike result can fail for utility functions that are close to inelastic. For further results on turnpike portfolios, see Ross (1974a, b) and Hakansson (1974). For additional information and results concerning myopic policies, the reader may consult Arrow (1964), Dirickx and Jennergren (1973), Ignall and Veinott (1969), and Marglin (1963). INTRODUCTION 371
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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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COMPUTATIONAL AND REVIEW EXERCISES
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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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Page 365 and 366: REFERENCES SOME EFFECTS OF TAXES ON
Page 367 and 368: COMPUTATIONAL AND REVIEW EXERCISES
Page 369 and 370: and a gamble is offered that return
Page 371 and 372: among two risky assets and a riskle
Page 373 and 374: Consider problem (1) and suppose th
Page 375 and 376: 21. Let pi,...,p„ be joint-normal
Page 377: Consider the following four utility
Page 380 and 381: 4. In most stochastic optimizing mo
Page 382 and 383: (g) Discuss the use of semivariance
Page 384 and 385: (a) Suppose that the investor alloc
Page 386 and 387: The linear programming subproblems
Page 388 and 389: (f) Show that (j> is proportional t
Page 390 and 391: (e) Show that X has mean and varian
Page 392 and 393: (i) Show that for all Aru ...,km, t
Page 394 and 395: Some properties of the expected val
Page 396 and 397: (i) Suppose that V0 is positive def
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Page 400 and 401: and where Z, Ylt...,Ym are independ
Page 403 and 404: INTRODUCTION This part of the book
Page 405: some qualitative characteristics of
Page 410 and 411: INVESTMENT ANALYSIS UNDER UNCERTAIN
Page 414 and 415: P if and only if INVESTMENT ANALYSI
Page 416 and 417: INVESTMENT ANALYSIS UNDEB UNCERTAIN
Page 425 and 426: 2. RISK AVERSION OVER TIME IMPLIES
Page 427 and 428: ables at t, and the primed variable
Page 429 and 430: ehavior in choosing among timeless
Page 431 and 432: is to asume that the consumer behav
Page 433 and 434: The function t/,(C,_,, wt|a,, ft) i
Page 435 and 436: sumption, given state j3,+i at r+l,
Page 437 and 438: 3. MYOPIC PORTFOLIO POLICIES Reprin
Page 439 and 440: 326 THE JOURNAL OF BUSINESS this op
Page 441 and 442: 328 THE JOURNAL OF BUSINESS with th
Page 443 and 444: 330 THE JOURNAL OF BUSINESS In sum
Page 445 and 446: 332 THE JOURNAL OF BUSINESS subject
Page 447: 334 THE JOURNAL OF BUSINESS myopic
Page 450 and 451: where and k" 1 f (Mt-Ui) 2 4k {Oi 1
Page 453 and 454: MIND-EXPANDING EXERCISES 1. Conside
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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
Page 741 and 742:
FIACCO, A. V., and MCCORMICK, G. P.
Page 743 and 744:
IGLEHART, D. (1965). "Capital accum
Page 745 and 746:
MANGASARIAN, O. L. (1966). "Suffici
Page 747 and 748:
PORTEUS, E. L. (1972). "Equivalent
Page 749 and 750:
THOMASION, A. J. (1969). The Struct
Page 751 and 752:
A Absolute risk aversion, 84, 85, 9
Page 753 and 754:
Logarithmic utility functions, see
Page 755 and 756:
V Value of information, 462-467, 68
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