on dynamic investment strategies - directory.umm.ac.iddirectory.umm.ac.id/data...

*Corresponding author.qEditor's note: Although presented in the privately circulated Proceedings of the Seminar on the

Analysis of Security Prices, Center for Research in Security Prices, University of Chicago, this paperhas remained unpublished for a variety of reasons. For almost 20 years it has been frequentlyreferred to and used by many "nancial economists, in areas as diverse as portfolio insurance and riskpremium analysis. I am certain that its publication will ensure its wider availability and furtherresearch into fundamental issues.

Journal of Economic Dynamics & Control24 (2000) 1859}1880

On dynamic investment strategiesq

John C. Cox!, Hayne E. Leland",*!Sloan School, MIT, 50 Memorial Drive, Cambridge, MA 02139, USA

"Haas School of Business, University of California at Berkeley, Berkeley, CA 94720, USA

Accepted 30 November 1999

Abstract

This paper presents a new approach for analyzing dynamic investment strategies.Previous studies have obtained explicit results by restricting utility functions to a fewspeci"c forms; not surprisingly, the resultant dynamic strategies have exhibited a verylimited range of behavior. In contrast, we examine what might be called the inverseproblem: given any speci"c dynamic strategy, can we characterize the results of followingit through time? More precisely, can we determine whether it is self-"nancing, yieldspath-independent returns, and is consistent with optimal behavior for some expectedutility maximizing investor? We provide necessary and su$cient conditions for a dy-namic strategy to satisfy each of these properties. ( 2000 Published by ElsevierScience B.V. All rights reserved.

JEL classixcation: D81; D91; G11; G12; G13

Keywords: Intertemporal portfolio theory; Investment strategies; Path independence;Self-"nancing

0165-1889/00/$ - see front matter ( 2000 Published by Elsevier Science B.V. All rights reserved.PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 9 5 - 0

1For example, see Hakansson (1970), Levhari and Srinivasan (1969), Merton (1971), Phelps(1962), and Samuelson (1969).

2For example, see Merton (1969) and Merton (1971). See also Black and Scholes (1973) and muchof the extensive literature based on their article.

3See Merton (1971) and Ross (1978).

1. Introduction

This paper presents a new approach for analyzing dynamic investmentstrategies. Previous studies have obtained explicit results by restricting utilityfunctions to a few speci"c forms; not surprisingly, the resultant dynamic strat-egies have exhibited a very limited range of behavior.1 In contrast, we examinewhat might be called the inverse problem: given any speci"c dynamic strategy,can we characterize the results of following it through time? More precisely, canwe determine whether it is self-"nancing, yields path-independent returns, and isconsistent with optimal behavior for some expected utility maximizing investor?We provide necessary and su$cient conditions for a dynamic strategy to satisfyeach of these properties.

Our results permit assessment of a wide range of commonly used dynamicinvestment strategies, including &rebalancing', &constant equity exposure', &port-folio insurance', &stop loss', and &dollar averaging' policies.

Indeed, any dynamic strategy that speci"es the amount of risky investment orcash held as a function of the level of investor wealth, or of the risky asset price,can be analyzed with our techniques.

We obtain explicit results for general dynamic strategies by assuming a speci-"c description of uncertainty. We consider a world with one risky asset (a stock)and one safe asset (a bond). We assume that the bond price grows deterministi-cally at a constant interest rate and that the stock price follows a multiplicativerandom walk that includes geometric Brownian motion as a limiting specialcase. This limiting case, which implies that the price of the risky asset hasa lognormal distribution, has been widely used in "nancial economics.2 Therestriction to a single risky asset involves no signi"cant loss of generality, since itcan be taken to be a mutual fund.3 Furthermore, our basic approach can beapplied to other kinds of price movements. We assume that the risky asset paysno dividends. We explain later why this too involves no real loss of generality.We allow both borrowing and short sales with full use of the proceeds. Wefurther assume that all markets are frictionless and competitive.

We wish to make full use of the tractibility that continuous time provides forcharacterizing optimal policies. However, we appreciate the view that theeconomic content of a continuous-time model is clearer when it is obtained asthe limit of a discrete-time model. Accordingly, we always establish our results ina setting in which trading takes place at discrete times and the stock price

1860 J.C. Cox, H.E. Leland / Journal of Economic Dynamics & Control 24 (2000) 1859}1880

4For an example of the use of a discrete random walk in an investment problem, see Gri!eath andSnell (1974). Our subsequent arguments draw on an application of the multiplicative random walkinitiated by Sharpe (1981) and developed further in an expository article by Cox et al. (1979).

5See Knight (1962). In particular, consider a sequence of random walks and a limit Brownianmotion de"ned on the same probability space. It can be shown that the sample paths of the randomwalks converge uniformly with probability one to the corresponding paths of the Brownian motion.The random walks thus provide a constructive de"nition of the path functions, "eld of measurablesets, and probability measure of Brownian motion.

follows a multiplicative random walk.4 In the statement of our propositions andin our examples, we emphasize the limiting form of our results as the tradinginterval approaches zero. It is well known that the (logarithms of the) approxi-mating random walks so obtained provide a constructive de"nition of Brownianmotion.5 Alternatively, our basic approach can be formulated directly incontinuous time.

To address the issues, we shall need the following de"nitions. An investmentstrategy speci"es for the current period and each future period:

(a) the amount to be invested in the risky asset,(b) the amount to be invested in the safe asset, and(c) the amount to be withdrawn from the portfolio.

In general, the amounts speci"ed for any given period can depend on all of theinformation that will be available at that time. A feasible investment strategy isone that satis"es the following two requirements. The "rst is the self-xnancingcondition: the value of the portfolio at the end of each period must always beexactly equal to the value of the investments and withdrawals required at thebeginning of the following period. The second is the nonnegativity condition:the value of the portfolio must always be greater than or equal to zero. Feasibleinvestment strategies are thus the only economically meaningful ones that canbe followed at all times and in all states without being supplemented orcollateralized by outside funds. A path-independent investment strategy is onefor which the controls (amounts (a)}(c)) can be written as functions only oftime and the price of the risky asset. Hence, the value of the portfolio at anyfuture time will depend on the stock (and bond) price at that time, but it will notdepend on the path followed by the stock in reaching that price. We shall seethat path independence is necessary for expected utility maximization. Further-more, portfolio managers may "nd path independence to be very desirable evenwhen they are not acting as expected utility maximizers. For example, withouta path-independent strategy, a portfolio manager could hold a long positionthroughout a rising market yet still lose money because of the particular price#uctuations that happened to occur along the way.

In Section 2, we establish some preliminary results that will be needed in latersections. We consider the case where the controls of an investment strategy are

J.C. Cox, H.E. Leland / Journal of Economic Dynamics & Control 24 (2000) 1859}1880 1861

6 In a related paper, Dybvig (1982) shows that our path-independence condition has importantimplications for the recoverability of individual preferences from observed actions.

given as functions of time and the value of the risky asset. This situation oftenarises in the valuation of contingent claims, so our results will also be of someuse there. We "nd necessary and su$cient conditions for the investment strategyto be feasible. These conditions take the form of a set of linear partial di!erentialequations that must be satis"ed by the amounts invested. In other words, if theseequations are not satis"ed, then the strategy cannot always be maintained; ifthey are satis"ed, then it can be. By construction, the policy in this case is pathindependent.

In the third section, we consider the case where the controls depend on timeand the value of the portfolio (wealth). Here we "nd that feasibility places nosubstantive restrictions on the investment strategy, but path independence does.We show that a given investment strategy will be path independent if and only ifthe amounts satisfy a particular nonlinear partial di!erential equation.

Finally, in Section 4 we develop necessary and su$cient conditions for a giveninvestment strategy to be consistent with expected utility maximization for somenondecreasing concave utility function. It turns out that these conditions areclosely related to the results of Section 3.6

We shall use the following notation:

S(t) stock price at time tu one plus the rate of return from an upward moved one plus the rate of return from a downward mover the one period interest rate (in the limiting cases, r stands for the

continuous interest rate)q probability of an upward move

p(1#r)!d

u!d

m local mean of the limiting lognormal processp2 local variance of the limiting lognormal processG(S(t), t) portfolio control specifying the number of dollars invested in stock

at time t as a function of the stock price at time tH(S(t), t) portfolio control specifying the number of dollars invested in bonds

at time t as a function of the stock price at time tK(S(t), t) portfolio control specifying the number of dollars withdrawn from

the portfolio at time t as a function of the stock price at time t=(t) wealth at time tA(=(t), t) portfolio control specifying the number of dollars invested in stock

at time t as a function of wealth at time t


B(=(t), t) portfolio control specifying the number of dollars invested in bondsat time t as a function of wealth at time t

C(=(t), t) portfolio control specifying the number of dollars withdrawn fromthe portfolio at time t as a function of wealth at time t

Subscripts on A, B, C, G, H, and K indicate partial derivatives. We shall saythat the functions A, B, and C are diwerentiable if the partial derivativesA

t, A

W, A

WW, B

t, B

W, B

WW, and C

Ware continuous on [t,¹)](z,R) for a

speci"ed z and ¹. A similar de"nition applied for G, H, and K.With this notation, the multiplicative random walk can be speci"ed in the

following way: for each time t, the stock price at time t#1 conditional on thestock price at time t will be either uS(t) with probability q or dS(t) withprobability 1!q. To rule out degenerate or pathological cases, we require that1'q'0 and u'1#r'd.

2. Controls that are functions of the value of the risky asset and time

In this section, we establish some preliminary results that will be needed later.Consider the case where the controls are speci"ed as functions of the value of therisky asset (the stock price) and time. Such a strategy is obviously path indepen-dent. Furthermore, the nonnegativity requirement can be veri"ed directly frominspection of the functions G(S(t), t) and H(S(t), t): G#H must be greater than orequal to zero for all s(t) and t. The self-"nancing requirement is discussed below.

Proposition 1. Necessary and suzcient conditions for the diwerentiable functionsG(S(t), t),H(S(t), t) and K(S(t), t) to be the controls of a self-xnancing investmentstrategy are that G,H, and K satisfy

12p2S2G

SS#rSG

S#G

t!rG#SK

S"0 (1)

12p2S2H

SS#rSH

S#H

t!rH#K!SK

S"0 (2)

S(GS#H

S)"G (3)

for all S(t) and t.

Proof. We "rst show that necessary and su$cient conditions for self-"nancingin the discrete-time model are

pG(uS(t), t#1)#(1!p)G(dS(t), t#1)!(1#r)G(S(t), t)

#[(1#r)/(u!d )] [K(uS(t), t#1)!K(dS(t), t#1)]"0, (4)


pH(uS(t), t#1)#(1!p)H(dS(t), t#1)!(1#r)H(S(t), t)

#pK(uS(t), t#1)#(1!p)K(dS(t), t#1)

![(1#r)/(u!d )] [K(uS(t), t#1)!K(dS(t), t#1)]"0, (5)

[G(uS(t), t#1)!G(dS(t), t#1)]#[H(uS(t), t#1)!H(dS(t), t#1)]

#[K(uS(t), t#1)!K(dS(t), t#1)]"(u!d )G(S(t), t). (6)

By de"nition, an investment strategy is self-"nancing if and only if

uG(S(t), t)#(1#r)H(S(t), t)"G(uS(t), t#1)#H(uS(t), t#1)

#K(uS(t), t#1), (7)

dG(S(t), t)#(1#r)H(S(t), t)"G(dS(t), t#1)#H(dS(t), t#1)

#K(ds(t), t#1). (8)

The equivalence of (4)}(6) and (7)}(8) follows directly from a series of straight-forward operations.

Suzciency: Eqs. (4)}(6) imply self-"nancing. Let F(S(t), t) denote G(S(t), t)#H(S(t), t)#K(S(t), t). Eq. (6) can then be written as

F(uS(t), t#1)!F(dS(t), t#1)"(u!d )G(S(t), t). (9)

Eqs. (4) and (5) imply that

pF(uS(t), t#1)#(1!p)F(dS(t), t#1)

"(1#r)[G(S(t), t)#H(S(t), t)]. (10)

Solving (9) and (10) for F(uS(t), t#1) and F(dS(t), t#1) in terms of G(S(t), t) andH(S(t), t) gives (7) and (8).

Necessity: Self-"nancing implies (4)}(6). Subtracting each side of (8) from thecorresponding side of (7) gives

G(S(t), t)"[F(uS(t), t#1)!F(dS(t), t#1)]/(u!d ), (11)

which gives (6). To see that self-"nancing implies (4) and (5), add p times (7) to(1!p) times (8) and rearrange to get

pF(uS(t), t#1)#(1!p)F(dS(t), t#1)

!(1#r)F(S(t), t)#(1#r)K(S(t), t)"0. (12)

Eq. (11) implies that

G(uS(t), t#1)!G(dS(t), t#1)

"[F(u2S(t), t#2)!2F(udS(t), t#2)#F(d2S(t), t#2)]/(u!d ). (13)


By using (12) to express the right-hand side of (13) in terms of F(uS(t), t#1) andF(dS(t), t#1) and rearranging, we obtain (4). Eq. (5) follows from using thede"nition of F to express (4) in terms of H and K and then applying (12).

We conclude the proof by outlining the argument showing that (4)}(6)converge to (1)}(3) as the trading interval approaches zero. Consider dividingeach trading period into n periods, each of length d. To obtain convergence togeometric Brownian motion, let u, d and q depend on d in the following way:

u"1#pJd, d"1!pJd, q"1

2C1#m

pJdD. (14)

To maintain the appropriate interest and withdrawal #ow, let the interest rateand withdrawal amount for each new period be d times the correspondingamounts for the original period (of unit length). Now note that (4) (and analog-ously (5) and (6)) can be rewritten using (14) as

1

2p2S2dC

G(S#g, t)!2G(S, t)#G(S!g, t)

g2 D#rSdC

G(S#g, t)!G(S!g, t)

2g D#dC

G(S, t)!G(S, t!d)

d D!rdG(S, t!d)

"![d#rd2]SCK(S#g, t)!(K(S!g, t)

2g D , (15)

where g"pSJd. The limits of the terms in the square brackets as d approacheszero are the corresponding derivatives in (1). h

In the presentation of the continuous-time case in Merton (1969), it is implicitthat necessary and su$cient conditions for self-"nancing are

12p2S2F

SS#rSF

S#F

t!rF#K"0, (16)

G"S(GS#H

S). (17)

Our analysis thus far develops this conclusion in a simple and explicit way andgives separate necessary conditions on each component of the portfolio.

Example 1. Consider the control G(S(t), t)"S(t)[1!exp(!j(t)S(t))], where j isa function of time. This is an apparently reasonable plan that increases thenumber of shares held to one as the stock price rises and decreases the number ofshares held to zero as the stock price falls. However, it can be veri"ed bydirect substitution that with K"0 this control does not satisfy the necessary


condition on G and hence there is no accompanying H(S(t), t) for which theoverall strategy will be feasible.

A few other comments are worth mentioning. When there are no withdrawals,both the amount invested in stock and the amount invested in bonds satis"es thesame equation as the total value of the portfolio. Consequently, Eqs. (1) and (2)can be thought of as value conservation relations on the components of theportfolio. Even though the portfolio composition is changing dynamically overtime, there can be no anticipated shifts in value between the stock and bondcomponents. The current value of the random amount that will be held in stockon any future date must be the amount held in stock today and the same is truefor bonds.

We conclude this section by citing a result that will be needed frequently inSection 4. Consider a hypothetical stock price process having the same samplepaths as the true process but with a di!erent probability measure de"ned onthese paths. Let this probability measure be chosen so that the expected rate ofreturn on the hypothetical process is in each period equal to the interest rate. Inour model, this is accomplished by letting q"p. Let E denote expectation withrespect to this measure conditional on the current true stock price.

Lemma 1. Consider a claim paying the amount U(S(¹)) at time ¹, where U isa known function. Then

(i) the payow to this claim can be replicated by a controlled self-xnancing portfolioof stock and bonds

(ii) the value of the claim at time t is the value of the replicating portfolio and isgiven by

F(S(t), t)"G(S(t), t)#H(S(t), t)"e~r(T~t)EU(S(¹)). (18)

Proof. See Harrison and Kreps (1979). h

3. Controls that are functions of the value of the portfolio and time

Now consider the case where the controls are speci"ed as functions of thevalue of the portfolio (wealth). Here the necessary and su$cient condition forself-"nancing is obvious: at each time t, the sum of the amount invested in stock,the amount invested in bonds, and the amount withdrawn from the portfoliomust equal the value of the portfolio,=(t). As the trading interval approacheszero, the withdrawal #ow C becomes negligible relative to the stocks A and B,and the self-"nancing condition reduces to A(=(t), t)#B(=(t), t)"=(t).

The conditions for nonnegativity are given below in Lemma 3. The con-clusions of this lemma are readily apparent and widely known; our purpose is to


summarize them in our terms. We then derive the main result of this section, thecondition for path independence.

Lemma 2. Necessary and suzcient conditions for the functions A(=(t), t),B(=(t), t)"=!A(=(t), t), and C(=(t), t) to be the controls of a nonnegativeinvestment strategy for all nonnegative initial values of wealth are

A(0, t)"0, C(0, t)"0 (19)

for all t.

Proof. In the discrete case, a su$cient condition for wealth to be nonnegativeis [d!(1#r)]A(=(t), t)#(1#r)[=(t)!C(=(t), t)]50 for A'0 and[u!(1#r)]A(=(t), t)#(1#r)[=(t)!C(=(t), t)]50 for A40. This can bewritten as

A(=(t), t)"0 and C(=(t), t)"0 for =(t)"0 (20)

C(1#r)[=(t)!C(=(t), t)]

(1#r)!d D5A(=(t), t)

5C!(1#r)[=(t)!C(=(t), t)]

u!(1#r) D for =(t)'0. (21)

In the limiting continuous case described in the proof of Lemma 1, the left andright sides of inequality (21) go to #R and !R, respectively, leaving (20) asthe e!ective condition. This condition is obviously necessary as well if non-negativity is to hold for all possible nonnegative initial values. h

For some strategies, (19) will not be necessary if initial wealth is restricted tolie in a speci"ed interval. For example, if there is a function g(t)50 such thatA(g(t), t)"0 and (1#r)g(t)!C(g(t), t)"g@(t) and initial wealth=(q) is greaterthan or equal to g(q), then=(t) will be greater than or equal to g(t) for all t.

Example 2. Consider the "xed strategy in which the initial portfolio consists ofthe amount ¸ in bonds and the di!erence, =(q)!¸, in stock. Subsequently,there are no changes in or withdrawals from the portfolio. At time t'q,=(t)"[(=(q)!¸)/S(q)]S(t)#¸er(t~q), so A(=(t), t)"=(t)!¸er(t~q). If =(q)5¸50, then the value of the portfolio can never become negative, since S(t)50.In this case, which corresponds to a simple buy-and-hold strategy,g(t)"¸er(t~q). However, A(=(t), t) does not satisfy (19), so this policy cannot beconsistent with=(t)50 for all=(q)50. If ¸'=(q)'0 or if ¸(0, the valueof the portfolio can become negative for su$ciently low stock prices.


In the "rst section, the restrictions imposed by feasibility are substantive,while those imposed by path independence are trivial. Now just the opposite istrue: the restrictions imposed by feasibility are relatively trivial, but thoseimposed by path independence are substantive.

Proposition 2. A necessary and suzcient condition for the diwerentiable functionsA(=(t), t),B(=(t), t)"=!A(=(t), t), and C(=(t), t) to be the controls of a path-independent investment strategy is that A and C satisfy

12p2A2A

WW#(r=!C)A

W#A

t!(r!C

W)A"0 (22)

for all =(t) and t.

Proof. We "rst establish that a necessary and su$cient condition for pathindependence in the discrete-time model is that A and C satisfy

pA(=(t#1) D u, t#1)#(1!p)A(=(t#1) D d, t#1)!(1#r)A(=(t), t)

#[(1#r)/(u!d )][C(=(t#1) D u, t#1)!C(=(t#1) D d, t#1)]"0,

(23)

where=(t#1) D u(=(t#1) D d ) denotes wealth at time t#1 conditional on thestock price moving up (down).

Suzciency: Eq. (23) implies

A([u!(1#r)]A(=(t), t)#(1#r)=(t)!(1#r)C(=(t), t), t#1)

][d!(1#r)]#(1#r)[[u!(1#r)]A(=(t), t)#(1#r)=(t)

!(1#r)C(=(t), t)]!(1#r)C([u!(1#r)]A(=(t), t)

#(1#r)=(t)!(1#r)C(=(t), t), t#1)

"A([d!(1#r)]A(=(t), t)!(1#r)=(t)

!(1#r)C(=(t), t), t#1)[u!(1#r)]

#(1#r)[[d!(1#r)]A(=(t), t)#(1#r)=(t)!(1#r)C(=(t), t)]

!(1#r)C([d!(1#r)]A(=(t), t)#(1#r)=(t)

!(1#r)C(=(t), t), t#1)). (24)

This follows from multiplying (23) by (d!u), adding (1#r)[(1#r)=(t)!(1#r)C(=(t), t)!(1#r)A(=(t), t)] to each side, and then rearranging to get(24). The left (right) hand side of (24) is the value of the portfolio at time t#2given that wealth at time t is=(t) and that the stock goes up (down) in the "rstperiod and down (up) in the second period. Consequently, (23) implies that the


stock price sequence SPuSPudS leads to the same wealth at time t#2 as thesequence SPdSPudS. This means that over an arbitrary number of periodsany two paths that di!er only in the interchanging of an adjoining u and d resultin the same wealth. Furthermore, all paths leading to a given "nal stock pricecan be obtained from any single path by repeated interchanging of adjoining u'sand d's. Hence, (23) implies that there is a single level of wealth associated witheach "nal stock price.

Necessity: If A(=(t), t), B(=(t), t), and C(=(t), t) are the controls of a self-"nancing path-independent investment strategy, then there exists a functionF(S(t), t) satisfying

A(F(S(t), t), t)[u!(1#r)]#(1#r)F(S(t), t)!(1#r)C(F(S(t), t), t)

"F(uS(t), t#1), (25)

A(F(S(t), t), t)[d!(1#r)]#(1#r)F(S(t), t)!(1#r)C(F(S(t), t), t)

"F(dS(t), t#1), (26)

for all S(t) and t. Eqs. (25) and (26) imply the following two equations:

pF(uS(t), t#1)#(1!p)F(dS(t), t#1)!(1#r)F(S(t), t)

#(1#r)C(F(S(t), t), t)"0, (27)

A(F(S(t), t), t)"S(t)CF(uS(t), t#1)!F(dS(t), t#1)

(u!d )S(t) D"G(S(t), t). (28)

By using Lemma 1, we obtain

pA(F(uS(t), t#1), t#1)#(1!p)A(F(dS(t), t#1), t#1)

!(1#r)A(F(S(t), t), t)

"pG(uS(t), t#1)

#(1!p)G(dS(t), t#1)!(1#r)G(S(t), t)

"![(1#r)/(u!d )][C(F(uS(t), t#1), t#1)

!C(F(dS(t), t#1), t#1)], (29)

which can be rewritten as (23),

pA(=(t#1) D u, t#1)#(1!p)A(=(t#1) D d, t#1)!(1#r)A(=(t), t)

#[(1#r)/(u!d )][C(=(t#1) D u, t#1)!C(=(t#1) D d, t#1)]"0.


To verify that (23) converges to (22), consider "rst the case C"0. With u andd speci"ed as in (14), (25) can be written as

1

2p2A2(=(t), t)dC

A(=K (t)#g, t)!2A(=K (t), t)#A(=K (t)!g, t)

g2 D#rA(=(t), t)dC

A(=K (t)#g, t)!A(=K (t)!g, t)

2g D#dC

A(=K (t), t)!A(=K (t)!e, t!d)

d D"rdA(=K (t)!e, t!d), (30)

where g"pA(=(t), t)Jd, e"r[=(t)!A(=(t), t)]d, and =K (t)"=(t)#e. Thelimits of the terms in square brackets as d approaches zero are the correspond-ing derivatives in (22). The case CO0 follows in the same way.

Example 3. Consider the controls A(=(t), t)"j(t!q), B(=(t), t)"=(t)!j(t!q), and C(=(t), t)"0, where j is a constant. This is a type of &dollaraveraging' strategy in which the amount invested in stock is being increased bya constant dollar #ow. Trying this A as a solution to (22) shows that theequation is not satis"ed. Consequently, A(=(t), t)"j(t!q) cannot be thecontrol of a path-independent portfolio. However, A(=(t), t)"jer(t~q) doessatisfy the equation, so path independence will hold if the dollar amountinvested in the stock grows geometrically at the riskless interest rate rather thanlinearly. Hence it is possible to represent the value of the portfolio as a functionof the stock price. It can be veri"ed by direct substitution into the continuous-time limits of (27) and (28) that for this new choice of A the value of the portfolioat time t, given an initial investment of=(q) at time q, is

F(S(t), t)"er(t~q)[=(q)#j log(S(t)/S(q))#j(12p2!r)(t!q)]. (31)

Although A(=(t), t)"jer(t~q) satis"es condition (22), it does not satisfy thenonnegativity condition (19). The consequences are obvious from (31): forsu$ciently low stock prices, the portfolio can take on arbitrarily large negativevalues. Hence, this policy is path independent but not feasible.

Example 4. Now consider the investment strategy given by A(=(t), t)"j=(t), B(=(t), t)"(1!j)=(t), and C(=(t), t)"i=(t), where j and i are con-stants. This &rebalancing' strategy is widely used in institutional portfolios.Unlike the previous example, this strategy is completely feasible, and sinceA and C satisfy (22), it is also path independent. By proceeding as before, we canverify that the value of the portfolio at time t, given an initial investment of=(q)at time q, is

F(S(t), t)"=(q)(S(t)/S(q))j exp[(1!j)(r#(1/2)p2j)!i](t!q). (32)


An examination of (32) shows that F(S(t), t) is an increasing convex function ofS(t) if j'1, an increasing concave function if 04j41, and a decreasingconvex function if j(0.

One further case is of some practical interest. This is the situation where thewithdrawal #ow and one control are speci"ed as functions of the stock price andtime, with the other control passively absorbing the remainder of wealth. Wediscuss this case here in Section 3 because path independence rather thanself-"nancing is the substantive issue, but it is the results of Section 2 that are ofimmediate relevance. For suppose the controls speci"ed as functions of S(t) andt fail to meet condition (1) or (2). Then from Proposition 1 we know that noself-"nancing strategy can express the other control as a function of S(t) andt alone. Thus, any self-"nancing strategy cannot be path independent. Conse-quently, condition (1) or (2) is necessary for path independence in this case. Anargument very similar to that used to establish Proposition 2 can be used toshow that it is also su$cient. That is, if the withdrawal #ow and one controlsatisfy (1) or (2), with the other control absorbing the remainder of wealth, thenthe resultant strategy will be path independent.

Any investment strategy constructed in this way is obviously self-"nancing,but nonnegativity must be checked indirectly by expressing the value of theportfolio, =(t), as a function of the stock price, G(S(t), t)#H(S(t), t). Since theresultant strategy is path independent, for any speci"ed G(S(t), t) or H(S(t), t)such a function always exists and can be found by using (1)}(3) and the conditionG(S(q), q)#H(S(q), q)"=(q), where q is the time at which the portfolio isinitiated and=(q) is initial wealth. Having obtained G#H, one can then verifyimmediately whether or not it is nonnegative for all times and stock prices, asshown in Example 5.

Example 5. Suppose that the amount to be invested in stock is speci"ed as

G(S(t), t)"er(t~q)[logS(t)#(12p2!r)(t!q)],

the withdrawal #ow is speci"ed as zero, and the remainder of the portfolio is tobe invested in bonds. This choice of G satis"es (1), so the strategy is pathindependent. By using (1)}(3) and the condition G(S(q), q)#H(S(q), q)"=(q), we"nd that

=(t)"G(S(t), t)#H(S(t), t)

"12er(t~q)[(logS(t)#(1

2p2!r)(t!q))2!p2(t!q)

!(logS(q))2#2=(q)],

which can clearly be negative under some conditions. We conclude that thisinvestment strategy is self-"nancing and path independent but not nonnegative.


7See Harrison and Pliska (1981) for details on the completion of markets and Leland (1980) fora discussion of the equivalence of the two optimization problems.

4. A characterization of optimal consumption and portfolio strategies

In this section, we examine the optimal behavior of a single individual in themarket environment described in Section 1. In Proposition 3, we consider thecase of an individual maximizing his expected utility of wealth at a terminaltime ¹. In Proposition 4, we extend our results to include maximization of theexpected utility of lifetime consumption as well as terminal wealth.

Since we are taking the market environment as given, we could considerseveral alternative speci"cations of the opportunities available to an individualwith zero wealth. For example, we could assign utility values to negative levelsof terminal wealth and then allow an individual with zero wealth to borrow andcontinue investing. However, there is little point in considering situations thatallow an individual to get something for nothing. Consequently, we do not allowan individual to incur negative wealth. An individual with zero wealth is thusprecluded from any subsequent investment or consumption. Similarly, we ruleout negative consumption, which has no economic meaning.

Our auguments in this section draw heavily on the correspondence (in theabsence of arbitrage opportunities) between the optimal behavior of an expectedutility maximizer in a world with complete markets and in a world in whichmarkets can be completed by sequential trading (such as the setting we considerhere).7 We also make frequent use of Lemma 1 and Proposition 2 and theirproofs. The results of Section 3 are relevant here because the distribution of therate of return on the stock, and hence the individual's opportunity set, does notdepend on the stock price. As a result, the optimal controls for an expectedutility maximizer will depend on wealth but not on the stock price, which is thesituation discussed in Section 3.

For future reference, we cite several properties of concave functions. A real-valued nondecreasing concave function V de"ned on I"[0,R) is lowersemicontinuous. At every interior point of I, V is continuous and has botha right-hand derivative V@̀ and a left-hand derivative V@

~. Furthermore, if

x'y are two points interior to I, then V@̀ (x)4V@~

(x)(V@̀ (y). Conversely, ifV is a lower semicontinuous function de"ned on I, is continuous in the interiorof I, has a right-hand derivative V@̀ at every point in the interior of I, andV@̀ is nonnegative and nonincreasing, then V is nondecreasing and concave.When we subsequently refer to a nondecreasing concave utility function, it is tobe understood that we exclude the trivial case of a constant function.

Proposition 3. Necessary and suzcient conditions for the diwerentiable functionsA(=(t), t) and B(=(t), t)"=(t)!A(=(t), t) to be optimal controls for some


nondecreasing concave utility of terminal wealth function are that A satisfy

12p2A2A

WW#r=A

W#A

t!rA"0, (33)

A(0, t)"0, (34)

A(=(t), t)50 (A(=(t), t)40) for all =(t)'0

and A(=(t), t)'0 (A(=(t), t)(0) for some =(t)'0

when m'r (m(r), (35)

for all =(t)'0 and all t(¹.

Proof. Suzciency: We begin with some de"nitions. Let the states i denote thepossible paths followed by the stock price from the current time t to the terminaltime ¹. Let P

i(t) denote the current state price for state i and let q

i(t) denote

the current probability of state i. Let q be the probability of an upward move inany period and, as before, p"((1#r)!d )/(u!d ). Let =

ibe the terminal

wealth in state i resulting from having initial wealth=(t) and following a policysatisfying the discrete versions of Eqs. (33)}(35) (i.e., (20), (21), (23), and (35)with C"0).

We need to show that (20), (21), (23) and (35), together with our otherassumptions, imply that there exists a nondecreasing concave function V anda nonnegative constant j such that

V@̀ (=i)4j

Pi(t)

qi(t)

for all =i"0 (36)

V@̀ (=i)4j

Pi(t)

qi(t)

4V@~

(=i) for all =

i'0 (37)

=i50 for all =

i(38)

+i

Pi(t)=

i"=(t), (39)

where we follow the convention of replacingV@̀ (0) with#R for V discontinu-ous at the origin. Conditions (20) and (21) imply that=

i50 for all=

iand the

absence of arbitrage opportunities implies that +iP

i(t)=

i"=(t). Thus, if there

is a nondecreasing concave function V satisfying (36) and (37), then the alloca-tion given by A is optimal for V.

The "nal stock price will be completely determined by the number of upwardmoves (and the complementary number of downward moves) occuring in the(say) n periods form t to ¹. Consequently, all paths leading to the same "nalstock price will have the same probability and thus the same state price. Thisfollows from Lemma 1, which implies that the state prices are the discountedprobabilities for q"p. Furthermore, (23) implies that all such paths will lead to


the same "nal wealth. Group the states i according to the "nal stock price, andlet State J denote the occurrence of j upward moves and n!j downward moves.The probability of State J is then q

J"( n

j)qj(1!q)n~j and the state price of State

J is then PJ"( n

j)p j(1!p)n~j/(1#r)n. Let=

Jbe the (unique) level of wealth in

State J.Conditions (23) and (35) imply that =

Jis a nondecreasing (nonincreasing)

function of j when q'p (q(p).Furthermore,

Pj(t)

qj(t)

"Ap(1!q)

q(1!p)Bj

A(1!p)

(1!q)(1#r)Bn

(40)

is a decreasing (increasing) function of j if q'p (q(p). (If q"p, PJ/q

Jis

a constant and optimality requires the same wealth in all states; i.e., no invest-ment in the risky asset.) Thus, for qOp, as P

J/q

Jdecreases (increases),

=J

increases or remains constant (decreases or remains constant). Conse-quently, there exists a nondecreasing concave functionV, de"ned on all levels of"nal wealth accessible from =(t) using A, and a nonnegative constant j (de-pending on =(t)) satisfying (36) and (37) for initial wealth =(t). That thisremains true in the limiting case follows from the fact that P

J/q

Jis then

a continuous function of S(¹) and =(S(¹)) has only isolated discontinuities.We still must show that this V continues to satisfy (36) and (37) for all the

allocations produced by A from other levels of initial wealth. Consider the levelof initial wealth=K (t) such that a downward move by the stock in the "rst periodwill lead to the same level of wealth at the end of the period as did an upwardmove from initial wealth=(t). Path-independence implies that when there arej upward moves from t to ¹, the initial wealth=K (t) produces the same level of"nal wealth as the initial wealth =(t) did with j#1 upward moves (i.e.,=K

i"=

i`1). Consequently, we want to show that there is a nonnegative

constant jK such that

V@̀ (=i)"V@̀ (=

i`1)4j

Pi`1

(t)

qi`1

(t)"jK

Pi(t)

qi(t)

for all =Ki"=

i`1"0

(41)

V@̀ (=i)"V@̀ (=

i`1)4j

Pi`1

(t)

qi`1

(t)"jK

Pi(t)

qi(t)

4V@~

(=Ki)"V@

~(=

i`1)

for all =Ki"=

i`1'0. (42)

Since

Pi`1

(t)

qi`1

(t)"A

p(1!q)

q(1!p)BP

i(t)

qi(t)

, (43)


there is such a jK and it is equal to [ p(1!q)/q(1!p)]j. Our earlier results implythat the extension of V to the (single) level of wealth that is accessible from=K (t)but not=(t) is nondecreasing and concave. For our purposes, it is su$cient toconsider successively higher (or lower) wealth levels in the same way.

Necessity: We want to show that if the di!erentiable functions A andB"=!A are the optimal controls for some nondecreasing concave utility ofterminal wealth function, then A satis"es (20), (21), (23), and (35). For anynondecreasing concave utility of wealth function V an optimal portfolio alloca-tion, if it exists, satis"es (36)}(39). We thus need to show that if A is di!erentiableand leads to an allocation satisfying (36)}(39), then A satis"es (20), (21), (23),and (35).

Conditions (36) and (37) imply that any two states having the same ratio ofP

ito q

imust have the same=

i. All paths leading to the same "nal stock price

will have the same Piand q

i, hence they must have the same=

i. Thus, utility

maximization implies path independence, which in turn implies (23). Condition(38) requires that =

i50 for all i, which means that the dynamically revised

portfolio duplicating this allocation must have a nonnegative value at time ¹.The absence of arbitrage opportunities then implies that the value of theportfolio must be nonnegative for all t4q4¹, which gives (20) and (21). Recallthat P

J/q

Jis a decreasing (increasing) function of j when q'p (q(p).

Since V@̀ is a nonincreasing function satisfying (36)}(37),=J

is a nondecreas-ing (nonincreasing) function of J, which in turn implies (35). h

For a given twice di!erentiable strictly concave utility of terminal wealthfunction<, there is a simple relation between the terminal condition for (33) andthe absolute risk tolerance function. In that case, for A to be optimal it isnecessary and su$cient that A satis"es (33)}(35) and, for=(¹)'0, the terminalcondition

A(=(¹),¹)"Am!r

p2 BA<@(=(¹))

!<A(=(¹))B . (44)

To see that (44), together with (33)}(35), implies that V"<, note that sinceV satis"es (41) for all J such the=

J'0, then

V@(=J`1

)!V@(=J)

=j`1

!=J

"

AP

J`1qJ`1

!

PJ

qJBj

=J`1

!=J

"

Ap(1!q)

q(1!p)!1BV@(=

J)

A(=J(¹!1),¹!1)(u!d )

(45)

which in the limit becomes

VA(=(¹))"!Am!r

p2 BA1

A(=(¹),¹)BV@(=(¹)). (46)


However, (44) says that

<A(=(¹))"!Am!r

p2 BA1

A(=(¹),¹)B<@(=(¹))

for all=(¹)'0, so (to a linear transformation) V"<. By a similar argument,if A is optimal for a given twice di!erentiable strictly concave <, then (44) musthold.

Now consider the situation where the individual also receives utility fromconsumption during his lifetime. Let the utility function be of the additive form

PT

t

;(C(s), s) ds#<(=(¹)) ;

in the discrete version, the integral is replaced with the sum of the one-periodutility of consumption functions. We assume that ; is a piecewise continuousfunction of s. This case leads to the following generalization of Proposition 3.

Proposition 4. Necessary and suzcient conditions for the diwerentiable functionsA(=(t), t), B(=(t), t)"=(t)!A(=(t), t), and C(=(t), t) to be optimal controls forsome nondecreasing concave utility of consumption and terminal wealth functionsare that A and C satisfy

12p2A2A

WW#(r=!C)A

W#A

t!(r!C

W)A"0, (47)

A(0, t)"0 and C(0, t)"0, (48)

A(=(t), t)50 (A(=(t), t)40) for all =(t)'0 and

A(=(t), t)'0 (A(=(t), t)(0) for some =(t)'0

when m'r (m(r), (49)

CW

(=(t), t)50 for all =(t)'0 and CW

(=(t), t)'0

for some =(t)'0 and all t(¹. (50)

Proof. The proof is very similar to that of Proposition 3, so we shall give onlya summary that stresses the important di!erences.

Suzciency: Consider the consumption and "nal wealth allocation resultingfrom following a given feasible investment policy satisfying the discrete versionsof (47)}(50) (i.e., (20), (21), (23), (49), and (50)). We need to show that there existnondecreasing concave functionsU(C

is, s) and V(=

i) for each s from t to ¹ such

that this consumption and wealth allocation satis"es the optimality conditions(36)}(38) supplemented with

U@̀ (Cis, s)4j

Pis(t)

qis(t)

for Cis"0, (51)


U@̀ (Cis, s)4j

Pis(t)

qis(t)

4U@~

(Cis, s) for C

is'0, (52)

Cis50 for all C

is(53)

and with (39) replaced by

+i,s

(Pis(t)C

is#P

i(t)=

i)"=(t), (54)

where Cis(t),P

is(t), and q

is(t) are the consumption, state price, and probability at

time t for state i at time s and U@̀ (Cis, s) is the right-hand derivative of U(C

is, s)

with respect to Cis.

Conditions (20), (21), (49) and (50) imply that=i50 for all i and that C

is50

for all i and s. Eq. (54) follows from the absence of arbitrage opportunities.Condition (23) implies path independence, which in turn implies that all pathswith the same P

is(t)/q

is(t) have the same wealth. Since C is a function of wealth,

they must also have the same consumption. As before, (23) and (49) imply that=

Jis a nondecreasing (nonincreasing) function of j and P

J(t)/q

J(t) is a decreas-

ing (increasing) function of j when q'p (q(p). Condition (50) implies that thesame is true for consumption for all s. All of this together implies that there existnondecreasing, concave functions U(s) and V satisfying (36)}(38) and (51)}(52)for all s for initial wealth level=(t). The extension to other wealth levels followsalong the same lines as before.

Necessity: Let U(s) and V be nondecreasing concave utility functions forwhich the di!erentiable controls A, B, and C are optimal and hence satisfy(36)}(38) and (51)}(54). We need to show that these controls satisfy (20), (21), (23),(49), and (50).

Clearly, the optimality conditions again imply path independence, which inturn implies (23). In the same way as before, the nonnegativity conditions (38)and (53) imply (20) and (21). Conditions (37) and (52) imply that for q'p (q(p)consumption at time q is a nondecreasing (nonincreasing) function of S(q) andthat=(¹) is a nondecreasing (nonincreasing) function of S(¹). Wealth at time q,which is the value at time q of future consumption and terminal wealth, is thusa nondecreasing (nonincreasing) function of S(q). Hence, we have conditions (49)and (50). h

Furthermore, suppose that C(=(t), t)'0 for all =(t)'0. Then a necessaryand su$cient condition for A and C to be optimal for a given twice di!erentiablestrictly concave utility of terminal wealth function <(=(T)) and some twicedi!erentiable strictly concave utility of consumption having a given risk toler-ance function

R(C(=(t), t), t)"!;@(C(=(t), t), t)/;A(C(=(t), t), t)


is that A and C satisfy (48)}(50) and

1

2p2A2A

WW#(r=!C)A

W#A

t!rA#A

m!r

p2 BA;@

!;AB"0 (55)

with, for=(¹)'0, the terminal condition

A(=(¹),¹)"Am!r

p2 BA<@(=(¹))

!<A(=(¹))B. (56)

To see this, note that (52) implies that for each s and for all J such thatC

J`1'0,

;@(CJ`1

)!;@(CJ)

CJ`1

!CJ

"

AP

J`1qJ`1

!

PJ

qJBj

AC

J`1=

J`1

!

CJ=

JB (=

J`1!=

J)

"

Ap(1!q)

q(1!p)!1B;@(C

J)

AC

J`1!C

J=

J`1!=

JB (=

J`1!=

J)

, (57)

which in the limit becomes

CW

(=(t), t)A(=(t), t)"Am!r

p2 BA;@(C(=(t), t), t)

!;A(C(=(t), t), t)B. (58)

Combining (58) with (47) gives (55). If C(=(t), t)"0 for some=(t)'0, then theutility function must be restricted to those for which risk tolerance goes to zeroas consumption goes to zero.

Note that (55) and (56) give a much weaker result than was obtained with (33)and (44) for the terminal wealth problem. There are two reasons for this. The"rst is that ; is a function of two variables, so it determines but is notdetermined by its risk tolerance function. (Consider, for example, utility func-tions of the separable form;(C, t)"f (t);(C(t))). Consequently, (55) and (56) arenecessary but not su$cient for A, B"=!A, and C to be optimal for a givenutility function;(C(t), t). The restriction that R(0, t) must equal zero gives a hintof the second reason. This is a result of the fact that di!erentiability of C forcesC

W(=M (t), t) to be equal to zero for all positive=M (t) such that C(=M (t), t) equals

zero. This suggests that utility functions that do not satisfy the conditionR(0, t)"0 will have optimal consumption strategies that are not di!erentiableat the largest level of wealth for which consumption equals zero. In some relatedwork, we verify this and completely characterize the class of continuous utilityfunctions for which an optimal strategy is di!erentiable. The results of Proposi-tion 4 do not apply to these cases.


5. Concluding remarks

In this paper, we have examined several issues in intertemporal portfoliotheory. In a speci"c setting, we have provided complete answers to the followingquestions: Can a given investment strategy be maintained under all possibleconditions, or are there instead some circumstances in which it will have to bemodi"ed or abandoned? How is the portfolio value resulting from followinga given investment strategy until any future date related to the prices of theunderlying securities on that date? Is a given investment strategy consistent withexpected utility maximization?

The setting that we have chosen is the most widely used speci"c model of assetprice movements. However, it does have two important features that should notbe overlooked. The multiplicative structure of the geometric random walk orBrownian motion greatly simpli"es our results. Although our general approachcan be applied to many other descriptions of uncertainty, the results willinevitably be more complicated. In addition, our model provides a setting inwhich contingent claims can be valued by arbitrage methods, and this propertyplays a crucial role in some of our arguments. Many other possible descriptionsof uncertainty also have this property, but our procedures will not directly applyto those that do not.

One specialization of our model is, however, essentially trivial. We assumedthat the risky asset pays no dividends. If dividends are allowed, the geometricrandom walk or Brownian motion would apply to the value of an investment inthe risky asset with reinvestment of dividends. It is easy to verify that this wouldleave Propositions 2}4 completely unchanged. Proposition 1 would have to bemodi"ed slightly for dividends, but it seems reasonable that most investorswould in fact not want their controls to be a!ected by price changes causedsolely by dividends. Instead, they would like to condition their controls on thereturns performance of the stock. In that case, Proposition 1 would remain validwhen the stock price is replaced by the value that an investment in one share ofstock would have if all dividends were reinvested.

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Harrison, J.M., Pliska, S.R., 1981. A stochastic calculus model of continuous trading: completemarkets. Northwestern University, Center for Mathematical Studies in Economics and Manage-ment Science, Discussion Paper No. 489, July 1981.

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