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178 BASIL A. KALYMON (24) d\ = r - fa = di - fa and supposing that d, = d, — fa, then (25) d ,_i = dj_i(d j, (p — hi) + ij) = d,_i(d> — fa, p — fa + ij) = dy — h if (dy — fci) — (p — fa + ij) < Sj-i = p — fa + ij if (d,- - fa) - (p - fa + ij) 6 «;-i = dj-i — fci. Thus, (23) holds for all j g i. By (23), we see that for all j g » — 1 (26) i(d'j - d'y+1) = «(d; - di+i) and (27) d'ii = diX — fax. But (28) /,_,(r _ ft, | p - ft, + €i) = D5-1 |8/B[«(d'i -
BOND REFUNDING WITH <strong>STOCHASTIC</strong> INTEREST RATES 179 LEMMA 2. For the general Case (iii), the optimal cost function /,-, fl(_,| S D,) + + «,£,,,,,/,_,(( - 1, i, r \ q) is an increasing function of x, by (33) and the induction hypothesis. Also, it is immediate that /,((, x, r | p, YR) 'defined by Mt, x,r\p, YR) = Ri + ax + min„,, {K< + diy + pUmy + hi + (Di - y \ p) (36) - k,(y - Di)+ + aiEîfiîm -l,y,r\q)\ = C& + (constant) is an increasing function for all x since Ci S 0. From these facts we have that, for t > 0, (37) Mt, x,r\p) = min {Mt, x,r\p, YK), Mt, x,r\p, YK)\ is an increasing function of x for x S Z),. For t = 0, (38) /i(0, r, x | p) = fi{t, r,x\ p, YK) — (ft, + ax) = (constant with respect to x) and thus is increasing in x. This completes the induction and proves the lemma. || The following theorem establishes the main result used in computing optimal debt levels. Note that the theorem assumes that some term m is being considered. The determination of an optimal term ro would require the consideration of some set of alternative feasible terms, with the one producing the minimum total cost being chosen. THEOREM 3. Without loss of optimality i/,(m), satisfying the equation [Ki +\di + Pi.m)Vi(m) + hi + (Di - Vi{m) | p) - fc,-(s/,-(m) - Z>,) + (39) + aiE^.ifi^m - 1, yt(m), Pi,m | q)] = min, \Kt + {di + pUm)y + hi + {Di ~y\p) - Hy - Dd + + ^.^.(m - 1, Vi{m), p,,„ | q)\, can be restricted to the set of values (0, Di, Di-i, • • • , Z),_m). 3. MODELS OF OPTION STRATEGY 571
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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 (— UÛ^
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316 DREZE AND MODIGLIANI ure for te
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318 DREZE AND MODIGLIANI income sid
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320 DREZE AND MODIGLIANI (3.5) The
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322 DREZE AND MODIGLIANI Consumptio
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324 DREZE AND MODIGLIANI in (cj, c2
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326 DREZE AND MODIGLIANI Since dÛj
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328 DREZE AND MODIGLIANI On the rig
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330 DRfeZE AND MODIGLIANI and its p
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332 DREZE AND M0DIGLIAN1 LEMMA C.2.
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334 DRfeZE AND MOD1GLIANI if the sm
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MANAGEMENT SCIENCE Vol. 19, No. 2,
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A DYNAMIC MODEL FOR BOND PORTFOLIO
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Maximize A DYNAMIC MODEL FOE BOND P
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MULTIPERIOD RISK PREFERENCE 41 mode
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MULTIPERIOD RISK PREFERENCE 43 The
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MULT1PERI0D RISK PREFERENCE between
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MULTIPERIOD RISK PREFERENCE 47 (ii)
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MULTIPERIOD RISK PREFERENCE 49 that
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or if s"(s' MULTIPERIOD RISK PREFER
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MULTIPERIOD RISK PREFERENCE 53 5. C
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Reprinted from The Review of Econom
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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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 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 and 618: MINIMAX POLICIES FOB SELLING AN ASS
Page 619 and 620: MINIMAX POLICIES FOR 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
Page 634 and 635: (a.s.) on the set lim XN < oo > u <
Page 636 and 637: EDWARD O. THORP method to this arti
Page 638 and 639: EDWARD O. THORP drastic simplificat
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
Page 654 and 655: 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
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
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A Absolute risk aversion, 84, 85, 9
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Logarithmic utility functions, see
Page 755 and 756:
V Value of information, 462-467, 68
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