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

This paper presents a new approach for analyzing dynamic investment strategies. Previous studies have obtained explicit results by restricting utility functions to a few speci"c forms; not surprisingly, the resultant dynamic strategies have exhibited a very limited range of behavior.1 In contrast, we examine what might be called the inverse problem: given any speci"c dynamic strategy, can we characterize the results of following it through time? More precisely, can we determine whether it is self-"nancing, yields path-independent returns, and is consistent with optimal behavior for some expected utility maximizing investor? We provide necessary and su$cient conditions for a dynamic strategy to satisfy each of these properties. Our results permit assessment of a wide range of commonly used dynamic investment strategies, including &rebalancing', &constant equity exposure', &portfolio insurance', &stop loss', and &dollar averaging' policies. Indeed, any dynamic strategy that speci"es the amount of risky investment or cash 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 deterministically at a constant interest rate and that the stock price follows a multiplicative random walk that includes geometric Brownian motion as a limiting special case. This limiting case, which implies that the price of the risky asset has a lognormal distribution, has been widely used in "nancial economics.2The restriction to a single risky asset involves no signi"cant loss of generality, since it can be taken to be a mutual fund.3 Furthermore, our basic approach can be applied to other kinds of price movements. We assume that the risky asset pays no 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. We further assume that all markets are frictionless and competitive. We wish to make full use of the tractibility that continuous time provides for characterizing optimal policies. However, we appreciate the view that the economic content of a continuous-time model is clearer when it is obtained as the limit of a discrete-time model. Accordingly, we always establish our results in a setting in which trading takes place at discrete times and the stock price follows a multiplicative random walk.4 In the statement of our propositions and in our examples, we emphasize the limiting form of our results as the trading interval approaches zero. It is well known that the (logarithms of the) approximating random walks so obtained provide a constructive de"nition of Brownian motion.5 Alternatively, our basic approach can be formulated directly in continuous time. To address the issues, we shall need the following de"nitions. An investment strategy 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 the information that will be available at that time. A feasible investment strategy is one that satis"es the following two requirements. The "rst is the self-xnancing condition: the value of the portfolio at the end of each period must always be exactly equal to the value of the investments and withdrawals required at the beginning of the following period. The second is the nonnegativity condition: the value of the portfolio must always be greater than or equal to zero. Feasible investment strategies are thus the only economically meaningful ones that can be followed at all times and in all states without being supplemented or collateralized by outside funds. A path-independent investment strategy is one for which the controls (amounts (a)}(c)) can be written as functions only of time and the price of the risky asset. Hence, the value of the portfolio at any future time will depend on the stock (and bond) price at that time, but it will not depend on the path followed by the stock in reaching that price.We shall see that path independence is necessary for expected utility maximization. Furthermore, portfolio managers may "nd path independence to be very desirable even when they are not acting as expected utility maximizers. For example, without a path-independent strategy, a portfolio manager could hold a long position throughout 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 later sections. We consider the case where the controls of an investment strategy are given as functions of time and the value of the risky asset. This situation often arises in the valuation of contingent claims, so our results will also be of some use there. We "nd necessary and su$cient conditions for the investment strategy to be feasible. These conditions take the form of a set of linear partial di!erential equations that must be satis"ed by the amounts invested. In other words, if these equations are not satis"ed, then the strategy cannot always be maintained; if they are satis"ed, then it can be. By construction, the policy in this case is path independent. In the third section, we consider the case where the controls depend on time and the value of the portfolio (wealth). Here we "nd that feasibility places no substantive restrictions on the investment strategy, but path independence does. We show that a given investment strategy will be path independent if and only if the amounts satisfy a particular nonlinear partial di!erential equation. Finally, in Section 4 we develop necessary and su$cient conditions for a given investment strategy to be consistent with expected utility maximization for some nondecreasing concave utility function. It turns out that these conditions are closely related to the results of Section 3.6 We shall use the following notation: S(t) stock price at time t u one plus the rate of return from an upward move d one plus the rate of return from a downward move r 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 process p2 local variance of the limiting lognormal process G(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 t H(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 t K(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 t A(=(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 bonds at time t as a function of wealth at time t C(=(t), t) portfolio control specifying the number of dollars withdrawn from the 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 say that the functions A, B, and C are diwerentiable if the partial derivatives At , AW, AWW, Bt , BW, BWW, and CW are 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 the stock price at time t will be either uS(t) with probability q or dS(t) with probability 1!q. To rule out degenerate or pathological cases, we require that 1'q'0 and u'1#r'd.

In this paper, we have examined several issues in intertemporal portfolio theory. In a speci"c setting, we have provided complete answers to the following questions: Can a given investment strategy be maintained under all possible conditions, or are there instead some circumstances in which it will have to be modi"ed or abandoned? How is the portfolio value resulting from following a given investment strategy until any future date related to the prices of the underlying securities on that date? Is a given investment strategy consistent with expected utility maximization? The setting that we have chosen is the most widely used speci"cmodel of asset price movements. However, it does have two important features that should not be overlooked. The multiplicative structure of the geometric random walk or Brownian motion greatly simpli"es our results. Although our general approach can be applied to many other descriptions of uncertainty, the results will inevitably be more complicated. In addition, our model provides a setting in which contingent claims can be valued by arbitrage methods, and this property plays a crucial role in some of our arguments. Many other possible descriptions of uncertainty also have this property, but our procedures will not directly apply to those that do not. One specialization of our model is, however, essentially trivial. We assumed that the risky asset pays no dividends. If dividends are allowed, the geometric random walk or Brownian motion would apply to the value of an investment in the risky asset with reinvestment of dividends. It is easy to verify that this would leave Propositions 2}4 completely unchanged. Proposition 1 would have to be modi"ed slightly for dividends, but it seems reasonable that most investors would in fact not want their controls to be a!ected by price changes caused solely by dividends. Instead, they would like to condition their controls on the returns performance of the stock. In that case, Proposition 1 would remain valid when the stock price is replaced by the value that an investment in one share of stock would have if all dividends were reinvested.