T OMORROW’S H ARDEST P ROBLEMS COMPUTATIONAL CHALLENGES IN PORTFOLIO MANAGEMENT The authors describe a relatively simple problem that all investors face—managing a portfolio of financial securities over time to optimize a particular objective function. They show how complex such a problem can become when real-world constraints are incorporated into its formulation. he financial industry is one of the considerations factor into its formulation. We fastest-growing areas of scientific present the basic dynamic portfolio optimization computing. Two decades ago, terms problem and then consider three aspects of it: such as financial engineering, com- taxes, investor preferences, and portfolio con- putationalT finance, and financial mathematics did straints. These three issues are by no means ex- not exist in common usage. Today, these areas haustive—they merely illustrate examples of the are distinct and enormously popular academic kinds of challenges financial engineers face to- disciplines with their own journals, conferences, day. Examples of other computational issues in and professional societies. portfolio optimization appear elsewhere.1,2 One explanation for this area’s remarkable growth and the impressive array of mathe- maticians, computer scientists, physicists, and The portfolio optimization problem economists that are drawn to it is the formida- Portfolio optimization problems are among ble intellectual challenges intrinsic to financial the most studied in modern finance,3 yet they markets. Many of the most basic problems in fi- continue to occupy the attention of financial aca- nancial analysis are unsolved and surprisingly re- demics and industry professionals because of silient to the onslaught of researchers from di- their practical relevance and their computational verse disciplines. intractabilities. The basic dynamic portfolio op- In this article, we hope to give a sense of these timization problem consists of an individual in- challenges by describing a relatively simple prob- vestor’s decisions for allocating wealth among lem that all investors face when managing a port- various expenditures and investment opportuni- folio of financial securities over time. Such a ties over time, typically the investor’s expected problem becomes more complex once real-world lifetime. The prices and price dynamics of goods and financial securities he or she purchases and any constraints such as tax liabilities, loan re- payment provisions, income payments, and 1521-9615/01/$10.00 © 2001 IEEE other cash inflows and outflows determine the investor’s overall budget. MARTIN B. HAUGH AND ANDREW W. L O For expositional clarity, we start with a simple MIT Sloan School of Management and Operations Research Center framework in which there are only two assets 54 COMPUTING IN SCIENCE & ENGINEERING available to the investor: a bond that yields a risk- portfolio immediately after $1.21 less rate of return and a stock that yields a ran- date t. Of course, xt and yt may $1.1 dom return of either 10 percent or –10 percent only depend on the informa- with probability p and 1 – p, respectively. We de- tion available at time t, a re- $.99 note the prices of the bond and stock at date t striction that we impose $1 $.99 with Bt and St, respectively. Without loss of gen- throughout this article. $.9 erality and for notational simplicity, we assume The requirement that xt that S0 = $1 and the riskless rate of interest is 0 and yt are nonnegative means ≥ $.81 percent, so Bt = 1 for all t 0. Finally, suppose that borrowing and short that the investor’s horizon spans only three dates, sales are not allowed, a con- t = 0 t = 1 t = 2 t = 0, 1, and 2, so that the possible paths for the straint that many investors evolving stock price St are given as in Figure 1. face. That xt and yt are re- Figure 1. Stock price evolution. Of course, in practice, an investor has many as- quired to be integers simply sets to choose from over many dates, and the reflects the fact that it is not possible to purchase price of each asset can take on many values. For a fractional number of stocks or bonds. The con- illustrative purposes, though, this simpler speci- straint in Equation 3b states that Wt+1 is equal to fication is ideal because it contains all the essen- Wt multiplied by the portfolio’s gross return be- tial features of the dynamic portfolio optimiza- tween dates t and t + 1. tion problem in a basic setting. Nevertheless, We can easily solve this problem numerically even in this simple framework, it will become ap- by using the standard technique of stochastic dy- parent that practical considerations such as taxes, namic programming.4 In particular, because investor preferences, and portfolio constraints V2(W2) = u(W2), we can compute V1(W1) using can create surprisingly difficult computational the Bellman equation so that challenges. =+ Let Ct denote the value of the investor’s con- VW11()Max{} uC () 1 E 122[] VW () (4) C sumption expenditures at date t, and let Wt de- 1 note the investor’s wealth just prior to date-t consumption. We assume that the investor has subject to the constraints in Equation 3. An as- a lifetime utility function U(C0, C1, C2) defined pect of Equation 4 that makes it particularly easy • over each consumption path {C0, C1, C2} that to solve is the fact that the value function V1( ) summarizes how much he or she values the en- depends on only one state variable, W1. This en- tire path of consumption expenditures. Then, ables us to solve Equation 4 numerically without ignoring market frictions and assuming that the too many computations. Suppose, for example, ∈ ޚ+ investor’s utility function is time-additive and that W0 = $1,000. Then, because xt, yt , there time-homogenous—for example, are only 1,100 possible values that W1 can take, so the right side of Equation 4 must be evaluated U(C0,C1, C2) = u(C0) + u(C1) + u(C2), (1) for only these 1,100 values. In contrast, when there are market frictions or when the investor the investor’s dynamic portfolio optimization has a more complex utility function, the compu- problem at t = 0 is given by tational requirements increase dramatically, re- flecting Bellman’s “curse of dimensionality.”4 =++ V00() WMax E 0012[] uC () uC ()() uC (2) CCC012,, Taxes subject to Most seasoned investors are painfully aware of the substantial impact that taxes can have on the W – C = x S + y B , t = 0, 1, 2 (3a) t t t t t t performance of their investment portfolio, so Wt+1 = xtSt+1 + ytBt+1, t = 0, 1 (3b) taxes play a major role in most dynamic portfolio ≥ 5–7 Ct 0, t = 0, 1, 2 (3c) optimization problems. ∈ ޚ+ To see how taxes can increase the computa- xt, yt , t = 0, 1, 2 (3d) tional complexity of such problems, let trading x = y = 0 (3e) 2 2 profits in the stock be subject to a capital gains tax ޚ+ where denotes the nonnegative integers, and xt in the portfolio optimization problem described and yt are the number of shares of stocks and bonds, earlier. Because this model has only two future respectively, that the investor holds in his or her periods, we do not distinguish between short- MAY/JUNE 2001 55 term and long-term capital gains. For expositional see that the presence of taxes has made the dy- simplicity, we also assume that we cannot use cap- namic portfolio optimization problem consider- ital losses from one period to offset gains from a ably more difficult. Specifically, in solving Equa- • later one. Although these simplifying assumptions tion 4 numerically, V1( ) is computed for only make the problem easier to solve, it will be ap- 1,100 possible values of W1. In contrast, solving parent that the computations are considerably Equation 8 numerically requires the evaluation • more involved than in the no-tax case. of V1( ) for all possible combinations of {W1, S1, To solve the dynamic portfolio optimization N0,0}, of which there are 1,001,000! problem with taxes, we use dynamic program- (We can verify this by noting the one-to-one ming as we did earlier. However, the value func- correspondence between {W1, S1, N0,0} and {C0, S1, tion is now no longer a function of wealth only, N0,0}, and counting the possible combinations of but it also depends on past stock prices and the {C0, S1, N0,0}. Assuming, as before, that W0 = number of shares purchased at each of those $1,000, we see that there are 1,001 possible choices prices. In other words, the value function is now for C0. If C0 = i, then there are 1,000 – i possible path-dependent. If we use Ns,t to denote the num- values for N0,0. For each combination of (C0, N0,0), ber of shares of stock purchased at date s ≤ t and there are two possible values of S . This means that 1000, 1 that are still in the investor’s portfolio immedi- in total, there are 2∑ (, 1 000−=i ) 1 , 001 , 000 combinations i=0 ately after trading at date t, then we can express of {W1, S1, N0,0}.) the portfolio optimization problem as Even in a simple two-period two-asset model, =++the portfolio optimization problem with taxes be- V00() WMax E 0012[] uC () uC ()() uC (5) CCC012,, comes considerably more complex. Indeed, in the subject to T-period N-asset case, it is easy to see that by t −1 date T, there are O(NT) state variables, and if τ −− = Wt – Ct – Max 0,∑ (SSNt s )( st,,−1 N st ) each state variable can take m distinct values, then = s 0 there will be O(mNT) possible states at date T.