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16 THE REVIEW OF ECONOMICS AND STATISTICS With respect to an investor's criterion for choices among different attainable combinations of assets, we assume that (3) if any two mixtures of assets have the same expected return, the investor will prefer the one having the smaller variance of return, and if any two mixtures of assets have the same variance of returns, he will prefer the one having the greater expected value. Tobin [21, pp. 75-761 has shown that such preferences are implied by maximization of the expected value of a von Neumann-Morgenstem utility function if either (a) the investor's utility function is concave and quadratic or (b) the investor's utility function is concave, and he has assigned probability distributions such that the returns on all possible portfolios differ at most by a location and scale parameter, (which will be the case if the joint distribution of all individual stocks is multivariate normal). Alternative Proofs of the Separation Theorem Since the interest rates on risHess savings bank deposits ("loans to the bank") and on borrowed funds are being assumed to be the same, we can treat borrowing as negative lending. Any portfolio can then be described in terms of (i) the gross amount invested in stocks, (ii) the fraction of this amount invested in each individual stock, and (iii) the net amount invested in loans (a negative value showing that the investor has borrowed rather than lent). But since the total net investment (the algebraic sum of stocks plus loans) is a given amount, the problem simply requires finding the jointly optimal values for (1) the ratio of the gross investment in stocks to the total net investment, and (2) the ratio of the gross investment in each individual stock to the total gross investment in stock s. It turns out that although the solution of (1) depends upon that of (2), in our context the latter is independent of the former. Specifically, the separation theorem asserts that: Given the assumptions about borrowing, lending, and investor preferences stated earlier in this section, the optimal proportionate composition of the slock (risk-asset) portfolio (i.e. the solution to sub-problem 2 above) is independent of the ratio of the gross investment in stocks to the total net investment. Tobin proved this important separation theorem by deriving the detailed solution for the optimal mix of risk assets conditional on a given gross investment in this portfolio, and then formally proving the critical invariance property stated in the theorem. Tobin used more restrictive assumptions that we do regarding the available investment opportunities and he permitted no borrowing. 11 Under our somewhat broadened assumptions in these respects, the problem fits neatly into a traditional Fisher framework, with different available combinations of expected values and standard deviations of return on alternative stock portfolios taking the place of the original "production opportunity" set and with the alternative investment choices being concurrent rather than between time periods. Within this framework, alternative and more transparent proofs of the separation theorem are available which do not involve the actual calculation of the best allocation in stocks over individual stock issues. As did Fisher, we shall present a simple algebraic proof 12 , set out the logic of the argument leading to the theorem, and depict the essential geometry of the problem. 13 As a preliminary step, we need to establish the relation between the investor's total investment in any arbitrary mixture or portfolio of individual stocks, his total net return from all his investments (including riskless assets and any borrowing), and the risk parameters of his investment position. Let the interest rate on riskless assets or borrowing be r*, and the uncertain return (dividends plus price appreciation) per dollar invested in the given portfolio of stocks be r. Let w represent the ratio of gross investment in stocks to "Tobin considered the special case where cash with no return was the only riskless asset available. While he formally required that all assets be held in non-negative quantities (thereby ruling out short sales), and that the total value of risk assets held not be greater than the investment balance available without borrowing, these non-negativity and maximum value constraints were not introduced into his formal solution of the optimal investment mix, which in turn was used in proving the invariance property stated in the theorem. Our proof of the theorem is independent of| the programming constraints neglected in Tobin's proof. Later in this section we show that when short sales are properly and explicitly introduced into the set of possible portfolios, theiresulting equations for the optimum portfolio mix are identical to those derived by Tobin, but that insistence on no short sales results in a somewhat more complex programming problem (when covariances are non-zero), which may however, be readily handled with computer programs now available. "An alternative algebraic proof using utility functions explicitly is presented in the appendix, note I. "LockwoodRainhard, Jr. hasalso independently developed and presented a similar proof of the theorem in an unpublished seminar paper. 134 PART II. QUALITATIVE ECONOMIC RESULTS
total net investment (stock plus riskless assets minus borrowing). Then the investor's net return per dollar of total net investment will be (1) y =(1 -w)r*+wt =r*+w(f—r*); Ogw< *, where a value of w < 1 indicates that the investor holds some of his capital in riskless assets and receives interest amounting to (l-w)r*; while w> 1 indicates that the investor borrows to buy stocks on margin and pays interest amounting to the absolute value of (1—w)r*. From (1) we determine the mean and variance of the net return per dollar of total net investment to be: (2a) y=r*+w(f—r*), and (24)
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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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Page 125 and 126: 1. STOCHASTIC DOMINANCE G. Hanoch a
Page 127 and 128: But now, EFFICIENCY ANALYSIS OF CHO
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Page 137 and 138: A Unified Approach to Stochastic Do
Page 139 and 140: A UNIFIED APPROACH TO STOCHASTIC DO
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Page 152 and 153: RISK AVERSION 123 a given risk the
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Page 156 and 157: RISK AVERSION 127 4. CONCAVITY The
Page 158 and 159: To show that (a) implies (d), note
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Page 166 and 167: ADDENDUM In retrospect, 1 wish foot
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Page 193 and 194: Separation in Portfolio Analysis R.
Page 195 and 196: SEPARATION IN PORTFOLIO ANALYSIS an
Page 197 and 198: SEPARATION IN PORTFOLIO ANALYSIS II
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Page 201 and 202: SEPARATION IN PORTFOLIO ANALYSIS Ca
Page 203 and 204: SEPARATION IN PORTFOLIO ANALYSIS It
Page 205 and 206: SEPARATION IN PORTFOLIO ANALYSIS th
Page 207 and 208: COMPUTATIONAL AND REVIEW EXERCISES
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
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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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76 THE REVIEW OF ECONOMICS AND STAT
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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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COMPUTATIONAL AND REVIEW EXERCISES
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and a gamble is offered that return
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among two risky assets and a riskle
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Consider problem (1) and suppose th
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21. Let pi,...,p„ be joint-normal
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Consider the following four utility
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4. In most stochastic optimizing mo
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(g) Discuss the use of semivariance
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(a) Suppose that the investor alloc
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The linear programming subproblems
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(f) Show that (j> is proportional t
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(e) Show that X has mean and varian
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(i) Show that for all Aru ...,km, t
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Some properties of the expected val
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(i) Suppose that V0 is positive def
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where V is the gradient operator an
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and where Z, Ylt...,Ym are independ
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INTRODUCTION This part of the book
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some qualitative characteristics of
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function of wealth. Hence the optim
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INVESTMENT ANALYSIS UNDER UNCERTAIN
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P if and only if INVESTMENT ANALYSI
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INVESTMENT ANALYSIS UNDEB UNCERTAIN
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2. RISK AVERSION OVER TIME IMPLIES
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ables at t, and the primed variable
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ehavior in choosing among timeless
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is to asume that the consumer behav
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The function t/,(C,_,, wt|a,, ft) i
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sumption, given state j3,+i at r+l,
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3. MYOPIC PORTFOLIO POLICIES Reprin
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326 THE JOURNAL OF BUSINESS this op
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328 THE JOURNAL OF BUSINESS with th
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330 THE JOURNAL OF BUSINESS In sum
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332 THE JOURNAL OF BUSINESS subject
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334 THE JOURNAL OF BUSINESS myopic
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where and k" 1 f (Mt-Ui) 2 4k {Oi 1
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MIND-EXPANDING EXERCISES 1. Conside
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(i.e., partial separability) and #,
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(c) Discuss the solution properties
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(d) By differentiating (cl) and (c2
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Let U be the utility function for t
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INTRODUCTION I. Two-Period Consumpt
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asked to develop explicit solutions
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that the property of nonincreasing
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The Hakansson paper studies a gener
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In the Samuelson-Hakansson additive
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techniques. The article by Breiman
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enously). These requirements are to
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in general, a monotonic function of
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ergodic theory, the book by Breiman
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or less than u; either do nothing o
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ME-24 the reader is invited to cons
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In discrete time, a nonnegative lin
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R. G. VICKSON This must not be inte
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This gives with R. G. VICKSON t(t)
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TWO-PERIOD CONSUMPTION MODELS AND P
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310 DRfeZE AND MODIGLIANI second or
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312 DREZE AND MODIGLIANI Had the sa
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314 DRfezE AND MODIGLIANI (— U^U^
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316 DREZE AND MODIGLIANI ure for te
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318 DREZE AND MODIGLIANI income sid
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320 DREZE AND MODIGLIANI (3.5) The
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322 DREZE AND MODIGLIANI Consumptio
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324 DREZE AND MODIGLIANI in (cj, c2
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326 DREZE AND MODIGLIANI Since d^Uj
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328 DREZE AND MODIGLIANI On the rig
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330 DRfeZE AND MODIGLIANI and its p
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332 DREZE AND M0DIGLIAN1 LEMMA C.2.
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334 DRfeZE AND MOD1GLIANI if the sm
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MANAGEMENT SCIENCE Vol. 19, No. 2,
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A DYNAMIC MODEL FOR BOND PORTFOLIO
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Maximize A DYNAMIC MODEL FOE BOND P
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MULTIPERIOD RISK PREFERENCE 41 mode
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MULTIPERIOD RISK PREFERENCE 43 The
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MULT1PERI0D RISK PREFERENCE between
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MULTIPERIOD RISK PREFERENCE 47 (ii)
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MULTIPERIOD RISK PREFERENCE 49 that
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or if s"(s' MULTIPERIOD RISK PREFER
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MULTIPERIOD RISK PREFERENCE 53 5. C
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Reprinted from The Review of Econom
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the risky asset, with 1 — w, goin
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w*t = glW,;Z,-1,...,Z0] = gT_i[W>]
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Since proof of the theorem is strai
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E C O N O M E T R I C A VOLUME 38 S
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OPTIMAL INVESTMENT 589 M: the numbe
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OPTIMAL INVESTMENT 591 For comparis
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(ii) in the case of Model III, 1 1
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Let Then OPTIMAL INVESTMENT 595 *.=
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where OPTIMAL INVESTMENT 597 (41) T
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where (54) T(x) = sup \ log c + log
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where b(v*) is the greatest lower b
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OPTIMAL INVESTMENT 603 noncapital i
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OPTIMAL INVESTMENT 605 Let us now t
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OPTIMAL INVESTMENT 607 REFERENCES [
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present value. Let ( be time, r the
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constant or decrease at a constant
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Denote the vector in the parenthese
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112 HOWARD M. TAYLOB easy to pictur
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114 HOWARD M. TAYLOR market and ign
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116 HOWARD M. TAYLOR the investor w
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118 HOWARD M. TAYLOR option which a
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120 HOWARD M. TAYLOR 3. BR.VDA, J.
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172 BASIL A. KALYMON more general p
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174 BASIL A. KALYMON The three case
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176 BASIL A. KALYMON an additional
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178 BASIL A. KALYMON (24) d\ = r -
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180 BASIL A. KALYMON The above resu
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182 BASIL A. KALYMON PROOF. First w
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MANAGEMENT SCIENCE Vol. 17, No. 7,
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M1NIMAX POLICIES FOB SELLING AN ASS
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MINIMAX POLICIES FOB SELLING AN ASS
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MINIMAX POLICIES FOR SELLING AN ASS
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MINIMAX POLICIES TOR SELLING AN ASS
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MINIMAX POLICIES FOB SELLING AN ASS
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648 L. BREIMAN We have discussed th
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650 L. BREIMAN Hence, it is suffici
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(a.s.) on the set lim XN < oo > u <
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EDWARD O. THORP method to this arti
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EDWARD O. THORP drastic simplificat
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EDWARD O. THORP us is that the prop
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EDWARD O. THORP Let the range of Rj
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EDWARD O. THORP is approximately on
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EDWARD O. THORP choosing P in repea
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EDWARD O. THORP return from the hed
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EDWARD O. THORP it is plausible to
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EDWARD O. THORP is maximization of
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EDWARD O. THORP 6. BREIMAN, L., "Op
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Reprinted from JOURNAL OF ECONOMIC
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CONSUMPTION AND PORTFOLIO RULES 375
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and CONSUMPTION AND PORTFOLIO RULES
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CONSUMPTION AND PORTFOLIO RULES giv
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Yin CONSUMPTION AND PORTFOLIO RULES
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Y(t) Y CONSUMPTION AND PORTFOLIO RU
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(d) Suppose n > r2. Formulate the o
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(j) Show that the absolute risk-ave
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The optimal first-period decision a
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(b) Show that the optimal policy in
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the game is stationary and favorabl
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(v) UHEV-E^CV-Af- 1 )^ w, V&0, w>0,
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MIND-EXPANDING EXERCISES 1. Conside
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Note the following distinction betw
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(e) Discuss the computational aspec
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(a) Show that U obeys the functiona
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(g) If ;ij(0) > 0 (1 g; g N), show
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Note that problem P2 has 3(N- 1)+1
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Note that (f) and (g) imply the exi
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P = (Pu), where pu = P[X„+l =j\ X
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(a) Show that {X„} is a stationar
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(b) Show that T T u = a + b Y[ c\'
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(b) Interpret this assumption. Is i
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(d) Show that (c) may be written as
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Bibliography Journal Abbreviations
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BHARUCHA-REID, A. T. (1960). Elemen
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FIACCO, A. V., and MCCORMICK, G. P.
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IGLEHART, D. (1965). "Capital accum
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MANGASARIAN, O. L. (1966). "Suffici
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PORTEUS, E. L. (1972). "Equivalent
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THOMASION, A. J. (1969). The Struct
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A Absolute risk aversion, 84, 85, 9
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Logarithmic utility functions, see
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V Value of information, 462-467, 68
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