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EDWARD O. THORP choosing P in repeated independent trials will outperform any strategy limited to choosing efficient portfolios. In addition, the optimal Kelly strategy gives positive growth rate, yet some of the Markowitz-efficient strategies give negative growth rate and ruin after repeated trials. Thorp [26] gave such an example and another appears in the work of Hakansson [11]. See also Hakansson [11, pp. 553-554] for further discussion of defects in the Markowitz model. V. The Theory in Action: Results for a Real Institutional Portfolio The elements of a practical profitable theory of convertible hedging were published by Thorp and Kassouf [27]. Thorp and Kassouf indicated an annualized return on investments of the order of 25% per year. Since then the theory has been greatly extended and refined [26] with most of these new results thus far unpublished. The historical data which have been used to develop the theory include for warrants monthly observations for about 20 years, averaging about 20 warrants, or about 4800 observations, plus weekly observations for three years on an average of about 50 warrants, another 7500 observations. Each of these more than 12,000 observations is an n-tuple including price of common, price of warrant, dividend, dilution, time to expiration, and several other quantities. The studies also used weekly data for three years on an average of 400 convertible bonds and 200 convertible preferreds, or about 90,000 observations. Including still other data, well over 100,000 observations have been incorporated into the study. A convertible hedge transaction generally involves two securities, one of which is convertible into the other. Certain mathematical price relationships exist between pairs of such securities. When one of the two is underpriced, compared to the other, a profitable convertible hedge may be set up by buying the relatively underpriced security and selling short an appropriate amount of the relatively overpriced security. The purpose of selling short the overpriced security is to reduce the risk in the position. Typically, one sells short in a single hedge from 50 to 125% as much stock (in "share equivalents") as is held long. The exact proportions depend on the analysis of the specific situation; the quantity of stock sold short is selected to minimize risk. The risk (i.e., change in asset value with fluctuations in market prices) in a suitable convertible hedge should be much less than in the usual stock market long positions. 610 PART V DYNAMIC MODELS
PORTFOLIO CHOICE AND THE KELLY CRITERION Date 11-3-69 12-31-69 9-1-70 12-31-70 3-31-71 6-30-71 9-30-71 12-31-71 3-31-72 6-30-72 9-30-72 12-31-72 3-31-73 6-30-73 9-30-73 12-31-73 Chg. to Date CO CO + 4.0 + 14.0 + 21.0 + 31.3 + 39.8 + 49.4 + 61.3 + 69.5 + 75.0 + 78.9 + 84.5 + 90.0 + 91.6 + 100.3 +102.9 Growth rate to date* + 26.8 + 17.0 + 17.7 + 21.2 + 22.3 + 23.3 + 24.7 + 24.4 + 23.3 + 22.1 + 21.3 + 20.7 + 19.4 + 19.4 + 17.7 TABLE I PERFORMANCE RECORD Elapsed time (months) 0 2 10 14 17 20 23 26 29 32 35 38 41 44 47 52 Closing DJIA C 855 800 758 839 904 891 887 890 940 934 960 1020 957 891 947 851 DJIA Chg. (%)» 0.0 -6.3 -11.3 -1.8 + 5.8 + 4.2 + 3.7 + 4.1 + 9.9 + 9.3 + 12.3 + 19.3 + 11.9 + 4.2 + 10.7 -0.5 " Round to nearest 0.1%. b Compound growth rate, annualized. c Round to nearest point; DJIA = Dow Jones industrial average. Starting even with DJIA C 855 889 974 1034 1123 1196 1278 1379 1449 1496 1526 1577 1625 1638 1713 1734 Gain over DJIA (%)• 0.0 + 10.3 + 25.3 + 22.8 + 25.5 + 35.6 + 45.7 + 57.2 + 59.6 + 65.7 + 66.6 + 65.2 + 78.1 + 87.4 + 89.6 +103.4 The securities involved in convertible hedges include common stock, convertible bonds, convertible preferreds, and common stock purchase warrants. Options such as puts, calls, and straddles may replace the convertible security. For this purpose, the options may be either written or purchased. The reader's attention is directed to three fundamental papers on the theory of options and convertibles which have since been published by Black and Scholes [3a, 3b] and by Merton [16b]. The theory of the convertible hedge is highly enough developed so that the probability characteristics of a single hedge can be worked out based on an assumption for the underlying distribution of the common. (Sometimes even this can almost be dispensed with! See Thorp and Kassouf [27, Appendix C].) A popular and plausible assumption is the random walk hypothesis: that the future price of the common is log-normally distributed about its current price, with a trend and a variance proportional to the time. Plausible estimates of these parameters are readily obtained. Furthermore, it turns out that the 4. THE CAPITAL GROWTH CRITERION AND CONTINUOUS-TIME MODELS 611
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STOCHASTIC OPTIMIZATION MODELS IN F
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STOCHASTIC OPTIMIZATION MODELS IN F
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Dedicated to the memory of my fathe
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PART II. QUALITATIVE ECONOMIC RESUL
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2. Risk Aversion over Time Implies
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PREFACE AND BRIEF NOTES TO THE 2006
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and shows that current research in
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1990. This area presages the CVaR l
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transaction cost band. Most such mo
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through targets and are less sensit
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ing dates, the papers by Breiman, H
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Carino, D. and W.T. Ziemba (1998).
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Kallberg, J.G. and W.T. Ziemba (198
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Stone, D. and W.T. Ziemba (1993). L
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PREFACE IN 1975 EDITION There is no
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particular results are generally st
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Part I Mathematical Tools
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wealth, then his utility function m
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cases in Exercise CR-12. The relati
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However, some extensions to infinit
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1. EXPECTED UTILITY THEORY The Anna
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SUBJECTIVE PROBABILITIES ANT EXPECT
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SUBJECTIVE PROBABILITIES AND EXPECT
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282 O. L. MANGA3ARIAN" for every X
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284 O. L. MANGASARIAN We have from
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280 O. L. MANQASARIAN For the case
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288 O. L. MANGASAMAN Proof. Conside
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290 O. L. MANGASARIAN Second Berkel
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denotes the /w-dimensional vector o
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Proof (I) We shall prove this part
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(Ill) have not, to the author's kno
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for the case (3); and for the case
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3. DYNAMIC PROGRAMMING Introduction
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INTRODUCTION TO DYNAMIC PROGRAMMING
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COMPUTATIONAL AND REVIEW EXERCISES
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(d) xxx2 on E+ 2 = {x\xt = 0, x2 S
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Show that the Kuhn-Tucker condition
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(a) Show that/, (b, c) = gt (Z>, c)
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(c) Show that the monotonicity assu
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An individual's preferences are sai
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12. (Hessians, bordered Hessians, c
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The problem in (b)-(c) hints that t
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(1) Suppose/is twice continuously d
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(f) Assume that R(i,a) g 0 and the
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(j) Show that an optimal policy Va
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INTRODUCTION In the second part of
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andom variables when the utility fu
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detailed analysis of the qualitativ
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separation in a local sense (for al
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1. STOCHASTIC DOMINANCE G. Hanoch a
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But now, EFFICIENCY ANALYSIS OF CHO
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EFFICIENCY ANALYSIS OF CHOICES INVO
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A Unified Approach to Stochastic Do
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A UNIFIED APPROACH TO STOCHASTIC DO
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RISK AVERSION 123 a given risk the
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RISK AVERSION 125 If z is actuarial
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RISK AVERSION 127 4. CONCAVITY The
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To show that (a) implies (d), note
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RISK AVERSION 131 (b') The risk pre
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9.2. Example 2. If (30) u'(x)=(x°
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RISK AVERSION 135 12. INCREASING AN
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ADDENDUM In retrospect, 1 wish foot
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14 THE REVIEW OF ECONOMICS AND STAT
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Separation in Portfolio Analysis R.
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SEPARATION IN PORTFOLIO ANALYSIS an
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SEPARATION IN PORTFOLIO ANALYSIS II
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SEPARATION IN PORTFOLIO ANALYSIS Th
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SEPARATION IN PORTFOLIO ANALYSIS Ca
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SEPARATION IN PORTFOLIO ANALYSIS It
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SEPARATION IN PORTFOLIO ANALYSIS th
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7. Consider an investor having the
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Then eliminate p from (2) to obtain
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(a) Show that an indifference curve
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(d) Develop a similar result to (c)
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Exercise Source Notes Exercise 1 wa
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(f) Show that k > 0. [Note (Exercis
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(c) Interpret (b). (d) Illustrate s
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9. Let the partial relative risk-av
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Condition (iv) is called the "no-ea
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Then * ( 8xj \ V - s u = \-r-\ -ax,
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exhibiting decreasing absolute and
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(a) Determine necessary and suffici
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Let the random variables Og^Sl and
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Part III Static Portfolio Selection
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given parameter. Exercise ME-30 dev
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utility function. However, many con
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the characteristic exponent is 2 (i
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proportions to maximize expected ut
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investment returns are always nonne
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case to a case of one riskless and
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538 REVIEW OF ECONOMIC STUDIES Howe
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540 REVIEW OF ECONOMIC STUDIES wher
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542 REVIEW OF ECONOMIC STUDIES expa
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JAMES A. OHLSON that quadratic util
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JAMES A. OHLSON Assumptions A1-A3 a
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JAMES A. OHLSON (i) snpl/ieSi\U (3)
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JAMES A. OHLSON A number of interes
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JAMES A. OHLSON for all y,teSx'3C{Q
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Since Z"=o^( w »0 = 0 and/(«,«)
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JAMES A. OHLSON analysis here is "o
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Choosing Investment Portfolios When
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CHOOSING INVESTMENT PORTFOLIOS univ
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CHOOSING INVESTMENT PORTFOLIOS The/
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CHOOSING INVESTMENT PORTFOLIOS Tobi
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CHOOSING INVESTMENT PORTFOLIOS a-di
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CHOOSING INVESTMENT PORTFOLIOS Henc
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CHOOSING INVESTMENT PORTFOLIOS Lemm
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CHOOSING INVESTMENT PORTFOLIOS ID.
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CHOOSING INVESTMENT PORTFOLIOS Proo
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CHOOSING INVESTMENT PORTFOLIOS is c
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CHOOSING INVESTMENT PORTFOLIOS Proo
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CHOOSING INVESTMENT PORTFOLIOS 3. C
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2. EXISTENCE AND DIVERSIFICATION OF
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ON THE EXISTENCE OF OPTIMAL POLICIE
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Reprinted from Journal of Financial
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Remarks: Differentiability assumpti
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distributed from the rest. Then an
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x * "22 "12 1 " (o22- a12) + (0n -
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to decline — as if having more mo
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solved for as a function 8.(y,0"),
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FOOTNOTES 1. H. Makower and J. Mars
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264 QUARTERLY JOURNAL OF ECONOMICS
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274 QUARTERLY JOURNAL OP ECONOMICS
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280 QVABTBRLY JOURNAL OF ECONOMICS
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Reprinted from THE REVIEW OF ECONOM
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(1.1) and (1.2) give SOME EFFECTS O
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SOME EFFECTS OF TAXES ON RISK-TAKIN
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REFERENCES SOME EFFECTS OF TAXES ON
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and a gamble is offered that return
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among two risky assets and a riskle
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Consider problem (1) and suppose th
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21. Let pi,...,p„ be joint-normal
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Consider the following four utility
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4. In most stochastic optimizing mo
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(g) Discuss the use of semivariance
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(a) Suppose that the investor alloc
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The linear programming subproblems
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(f) Show that (j> is proportional t
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(e) Show that X has mean and varian
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(i) Show that for all Aru ...,km, t
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Some properties of the expected val
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(i) Suppose that V0 is positive def
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where V is the gradient operator an
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and where Z, Ylt...,Ym are independ
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INTRODUCTION This part of the book
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some qualitative characteristics of
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function of wealth. Hence the optim
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INVESTMENT ANALYSIS UNDER UNCERTAIN
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P if and only if INVESTMENT ANALYSI
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INVESTMENT ANALYSIS UNDEB UNCERTAIN
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2. RISK AVERSION OVER TIME IMPLIES
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ables at t, and the primed variable
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ehavior in choosing among timeless
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is to asume that the consumer behav
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The function t/,(C,_,, wt|a,, ft) i
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sumption, given state j3,+i at r+l,
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3. MYOPIC PORTFOLIO POLICIES Reprin
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326 THE JOURNAL OF BUSINESS this op
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328 THE JOURNAL OF BUSINESS with th
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330 THE JOURNAL OF BUSINESS In sum
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332 THE JOURNAL OF BUSINESS subject
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334 THE JOURNAL OF BUSINESS myopic
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where and k" 1 f (Mt-Ui) 2 4k {Oi 1
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MIND-EXPANDING EXERCISES 1. Conside
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(i.e., partial separability) and #,
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(c) Discuss the solution properties
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(d) By differentiating (cl) and (c2
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Let U be the utility function for t
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INTRODUCTION I. Two-Period Consumpt
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asked to develop explicit solutions
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that the property of nonincreasing
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The Hakansson paper studies a gener
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In the Samuelson-Hakansson additive
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techniques. The article by Breiman
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enously). These requirements are to
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in general, a monotonic function of
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ergodic theory, the book by Breiman
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or less than u; either do nothing o
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ME-24 the reader is invited to cons
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In discrete time, a nonnegative lin
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R. G. VICKSON This must not be inte
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This gives with R. G. VICKSON t(t)
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TWO-PERIOD CONSUMPTION MODELS AND P
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310 DRfeZE AND MODIGLIANI second or
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312 DREZE AND MODIGLIANI Had the sa
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314 DRfezE AND MODIGLIANI (— U^U^
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316 DREZE AND MODIGLIANI ure for te
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318 DREZE AND MODIGLIANI income sid
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320 DREZE AND MODIGLIANI (3.5) The
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322 DREZE AND MODIGLIANI Consumptio
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324 DREZE AND MODIGLIANI in (cj, c2
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326 DREZE AND MODIGLIANI Since d^Uj
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328 DREZE AND MODIGLIANI On the rig
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330 DRfeZE AND MODIGLIANI and its p
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332 DREZE AND M0DIGLIAN1 LEMMA C.2.
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334 DRfeZE AND MOD1GLIANI if the sm
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MANAGEMENT SCIENCE Vol. 19, No. 2,
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A DYNAMIC MODEL FOR BOND PORTFOLIO
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Maximize A DYNAMIC MODEL FOE BOND P
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MULTIPERIOD RISK PREFERENCE 41 mode
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MULTIPERIOD RISK PREFERENCE 43 The
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MULT1PERI0D RISK PREFERENCE between
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MULTIPERIOD RISK PREFERENCE 47 (ii)
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MULTIPERIOD RISK PREFERENCE 49 that
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or if s"(s' MULTIPERIOD RISK PREFER
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MULTIPERIOD RISK PREFERENCE 53 5. C
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Reprinted from The Review of Econom
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the risky asset, with 1 — w, goin
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w*t = glW,;Z,-1,...,Z0] = gT_i[W>]
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Since proof of the theorem is strai
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E C O N O M E T R I C A VOLUME 38 S
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OPTIMAL INVESTMENT 589 M: the numbe
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OPTIMAL INVESTMENT 591 For comparis
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(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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Page 598 and 599: 120 HOWARD M. TAYLOR 3. BR.VDA, J.
Page 600 and 601: 172 BASIL A. KALYMON more general p
Page 602 and 603: 174 BASIL A. KALYMON The three case
Page 604 and 605: 176 BASIL A. KALYMON an additional
Page 606 and 607: 178 BASIL A. KALYMON (24) d\ = r -
Page 608 and 609: 180 BASIL A. KALYMON The above resu
Page 610 and 611: 182 BASIL A. KALYMON PROOF. First w
Page 613 and 614: MANAGEMENT SCIENCE Vol. 17, No. 7,
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 648 and 649: EDWARD O. THORP return from the hed
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Page 657 and 658: Reprinted from JOURNAL OF ECONOMIC
Page 659 and 660: CONSUMPTION AND PORTFOLIO RULES 375
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Page 665 and 666: CONSUMPTION AND PORTFOLIO RULES giv
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CONSUMPTION AND PORTFOLIO RULES 413
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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
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Logarithmic utility functions, see
Page 755 and 756:
V Value of information, 462-467, 68
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