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146 STEPHEN P. BRADLEY AND DWIGHT B. CRANE Note that if security class k is purchased at the start of period 2 its purchase price and income yield are conditional upon the random event which occurred during period 1. Thus, a subproblem is defined for each commodity and each sequence of random events which precede the purchase date of that commodity. The inventory balance equations of type (2.6) define the subproblems for each commodity purchased at the start of period 2 and each preceding event sequence. As would be expected, these equations have no variables in common in equations (2.4) and (2.5), since each set of inventory balance equations simply keeps track of the holdings of each commodity. The commodity definition adopted leads to a relatively large number of subproblems. The rationale, however, is that the state variable of each subproblem is then one dimensional, i.e., the amount of the commodity held. We will in discussing the model present the results in terms of only three time periods; however, this is only for ease of exposition as the results hold for an arbitrary number of time periods. The constraints for the subproblems corresponding to the k security classes that may be purchased in period one are \fk as follows: -&i* + si.s(ei) + h k i,2(e,) = 0, Vei € £,, (3.1) -tf.5(e,) + s{.a(ei, e2) + ti*(ei, e2) = 0, V(e, , e2) 6 E1 X E, 6i* S 0, *;,a(ei) S 0, /rf,,(e,) £ 0, *J.,(e,, e2) £ 0, M.3(e,, e2) £ 0. The constraints for the subproblems corresponding to the k security classes that may be purchased in period two, conditional on events in period one, are \fk and Vei £ E\ as follows: (3.2) — b2 k (ei) + **.s(ei, e2) + /i2,3(ei, e2) = 0, Vea € E2, &2*(ei) g 0, 4,s(ei, e2) £ 0, /s2,3(ei, e2) £ 0. For eight security classes, three time periods, and three uncertain events in each period there are thirty-two subproblems. There are eight subproblems of the form (3.1) corresponding to the eight commodities available for purchase at the start of period 1 and twenty-four of the form (3.2). The basic structure of the subproblems for an arbitrary number of time periods and uncertain events should be evident, since the subproblem constraints merely reflect the holdings of each commodity for each possible sequence of uncertain events. Note that each commodity is purchased only once and then sold or held during subsequent periods. The interesting point to note about the subproblem constraints is that they are homogeneous systems of equations (i.e., zero right-hand sides). In decomposition the fundamental theorem employed is that a convex polyhedral set may be represented by a convex combination of its extreme points plus a nonnegative combination of its extreme rays [10]. The subproblems of (3.1) and (3.2) have only one extreme point, all decision variables equal to zero. For any nonzero point satisfying the subproblem constraints, a scalar times that point also satisfies the constraints and hence with a linear objective function there exists an associated unbounded solution. As a result we need only consider the extreme rays of the subproblems. It is critical for an effective procedure to be able to construct these extreme rays in an efficient manner. In the next section we show how to do this. The restricted master for the decomposition scheme defining subproblems for each commodity and event sequence is given in Figure 1. Note that the usual convex combination constraints (e.g., 23 ta = 1) ar e not present, since we are only dealing with nonnegative combinations of extreme rays. Further, there are no subproblem con- 494 PART V. DYNAMIC MODELS
Maximize A DYNAMIC MODEL FOE BOND PORTFOLIO MANAGEMENT 147 E* lE» [E>i«P( e i , et)(y{ + Di.i(ei , ei))Ai.i.«(ei , e«)]X*.» + E«i E» lE« P(e> , es)(i/2*(ei) + »i.i(ei , e!))/i!,i.»(ei , eJ)]\l»(e1) + E«i« P( e i , e») (!/J*(«I , ea) + t)'.»(ei , et))bi{ei , d)\ subject to r,:E*(E.[6{.nlx{..| =/i, »«(«•): E* lE» [-»i'S?.. - (1 + fft..(ei))s{...»(«.)]xj.» + E.,E» [5J.»(ei)]x5,»((«! , e2)]/h.2(ei) + £.,., [(1 + ffi.8(ei, ea))ir,(ei, e2) + ff*.,a(ei, e2)jr,(ei, ej)]s{.s(ei, e2) • £«,., [P(«i. e2)(2/2*i. e z) + »2.»(ei - «2))] /i 2.3( e " . e *) 1. TWO-PERIOD CONSUMPTION MODELS AND PORTFOLIO REVISION 495
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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
Page 479 and 480: in general, a monotonic function of
Page 481 and 482: ergodic theory, the book by Breiman
Page 483 and 484: or less than u; either do nothing o
Page 485 and 486: ME-24 the reader is invited to cons
Page 487: In discrete time, a nonnegative lin
Page 490 and 491: R. G. VICKSON This must not be inte
Page 492 and 493: This gives with R. G. VICKSON t(t)
Page 495 and 496: TWO-PERIOD CONSUMPTION MODELS AND P
Page 497 and 498: 310 DRfeZE AND MODIGLIANI second or
Page 499 and 500: 312 DREZE AND MODIGLIANI Had the sa
Page 501 and 502: 314 DRfezE AND MODIGLIANI (— U^U^
Page 503 and 504: 316 DREZE AND MODIGLIANI ure for te
Page 505 and 506: 318 DREZE AND MODIGLIANI income sid
Page 507 and 508: 320 DREZE AND MODIGLIANI (3.5) The
Page 509 and 510: 322 DREZE AND MODIGLIANI Consumptio
Page 511 and 512: 324 DREZE AND MODIGLIANI in (cj, c2
Page 513 and 514: 326 DREZE AND MODIGLIANI Since d^Uj
Page 515 and 516: 328 DREZE AND MODIGLIANI On the rig
Page 517 and 518: 330 DRfeZE AND MODIGLIANI and its p
Page 519 and 520: 332 DREZE AND M0DIGLIAN1 LEMMA C.2.
Page 521 and 522: 334 DRfeZE AND MOD1GLIANI if the sm
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Page 525 and 526: A DYNAMIC MODEL FOR BOND PORTFOLIO
Page 529: A DYNAMIC MODEL FOR BOND PORTFOLIO
Page 535: A DYNAMIC MODEL FOR BOND PORTFOLIO
Page 538 and 539: MULTIPERIOD RISK PREFERENCE 41 mode
Page 540 and 541: MULTIPERIOD RISK PREFERENCE 43 The
Page 542 and 543: MULT1PERI0D RISK PREFERENCE between
Page 544 and 545: MULTIPERIOD RISK PREFERENCE 47 (ii)
Page 546 and 547: MULTIPERIOD RISK PREFERENCE 49 that
Page 548 and 549: or if s"(s' MULTIPERIOD RISK PREFER
Page 550 and 551: MULTIPERIOD RISK PREFERENCE 53 5. C
Page 553 and 554: Reprinted from The Review of Econom
Page 555 and 556: the risky asset, with 1 — w, goin
Page 557 and 558: w*t = glW,;Z,-1,...,Z0] = gT_i[W>]
Page 559 and 560: Since proof of the theorem is strai
Page 561 and 562: E C O N O M E T R I C A VOLUME 38 S
Page 563 and 564: OPTIMAL INVESTMENT 589 M: the numbe
Page 565 and 566: OPTIMAL INVESTMENT 591 For comparis
Page 567 and 568: (ii) in the case of Model III, 1 1
Page 569 and 570: Let Then OPTIMAL INVESTMENT 595 *.=
Page 571 and 572: where OPTIMAL INVESTMENT 597 (41) T
Page 573 and 574: where (54) T(x) = sup \ log c + log
Page 575 and 576: where b(v*) is the greatest lower b
Page 577 and 578: OPTIMAL INVESTMENT 603 noncapital i
Page 579 and 580: OPTIMAL INVESTMENT 605 Let us now t
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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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