Challenges in Financial Computing y Martin B. Haugh and Andrew W. Lo This Draft: March 18, 2001 Abstract One of the fastest growing areas of scienti c computing is in the nancial industry. Manyof the most basic problems in nancial analysis are still unsolved, and are surprisingly resilient to the onslaught of legions of talented researchers from many diverse disciplines. In this article, we hop e to give readers a sense of these challenges by describing a relatively simple problem that all investors face|managing a p ortfolio of nancial securities over time to optimize a particular ob jective function|and showing how complex such a problem can b ecome as real-world considerations such as taxes, preferences, and p ortfolio constraints are incorp orated into its formulation. This researchwas partially supp orted by the MIT Lab oratory for Financial Engineering. y MIT Sloan Scho ol of Management and Op erations ResearchCenter. Corresp onding author: Andrew W. Lo, 50 Memorial Drive, E52-432, Cambridge, MA 02142{1347, (617) 253{0920 (voice), (617) 258{5727 (fax), [email protected] (email). Contents 1 Intro duction 1 2 The Portfolio Optimization Problem 1 3 Taxes 4 4 Preferences 6 5 Portfolio Constraints 7 6 Possible Solution Techniques 8 References 11 1 Intro duction One of the fastest growing areas of scienti c computing is in the nancial industry. Two decades ago, terms such as \ nancial engineering", \computational nance", and \ nancial mathematics" did not exist in common usage, yet to day these areas are regarded as distinct and enormously p opular academic disciplines, with their own journals, conferences, and professional so cieties. One explanation for the remarkable growth in this area and the impressive array of mathematicians, computer scientists, physicists, and economists that have been drawn to quantitative nance is the formidable intellectual challenges that are intrinsic to nancial markets. Many of the most basic problems in nancial analysis are still unsolved, and are surprisingly resilienttothe onslaught of legions of researchers from diverse disciplines. In this article, wehopetogive readers a sense of these challenges by describing a relatively simple problem that all investors face|managing a p ortfolio of nancial securities over time to optimize a particular ob jective function|and showing how complicated such a problem can b ecome as real-world considerations are incorp orated into its formulation. We presentthe basic dynamic p ortfolio optimization problem in Section 2, and then consider three asp ects 1 of the problem in Sections 3{5: taxes, investor's preferences, and p ortfolio constraints. We conclude in Section 6 by reviewing one of the most promising class of metho ds for solving these typ es of problems. 2 The Portfolio Optimization Problem Portfolio optimization problems are among the most well-studied problems in mo dern - 2 nance, yet they continue to occupy the attention of nancial academics and industry pro- fessionals, b oth b ecause of their practical relevance and their computational intractabilities. The basic dynamic p ortfolio optimization problem consists of an individual investor's deci- sions for allo cating wealth among various exp enditures and investment opp ortunities over time so as to maximize some ob jective function|typically the investor's exp ected lifetime 1 These three issues are by no means exhaustive, but are merely illustrative examples of the kinds of challenges faced by nancial engineers to day. See Haliassos and Michaelides (2000) and Haugh and Lo (2001) for examples of other computational issues in p ortfolio optimization. 2 See for example Merton (1990). 1 utility|given the prices and price dynamics of go o ds and nancial securities he purchases, and any constraints such as tax liabilities, loan repaymentprovisions, income payments, and other cash in ows and out ows that determine the investor's overall budget. For exp ositional clarity, we start with a simple framework in which there are only two assets available to the investor: a b ond that yields a riskless rate of return and a sto ck that yields a random return of either 10% or 10% with probability p and 1 p, resp ectively. Denote by B and S the prices of the bond and sto ck at date t, resp ectively. Without loss t t of generality and for notational simplicity, we assume that S =$1 and the riskless rate of 0 interest is 0% hence B =1 for all t 0. Finally, supp ose that the investor's horizon spans t only three dates, t =0; 1; and 2, so that the p ossible paths for the sto ck price pro cess S are t given in Figure 1. Of course, in practice, an investor has many assets to cho ose from over many dates and where the price of each asset can take on many values. But for illustrative purp oses, this simpler sp eci cation is ideal because it contains all the essential features of the dynamic p ortfolio optimization problem in a very basic setting. Nevertheless, even in this simple framework, it will b ecome apparent that practical considerations such as taxes, investor preferences, and p ortfolio constraints can create surprisingly dicult computational challenges. $1:21 $1:1 a a a a a a a a $1 $:99 H H ! $:99 H ! ! H ! H ! H ! $:9 H ! ! H H H H H H H H H $:81 t =0 t =1 t =2 Figure 1: Sto ck Price Evolution 2 Let C denote the consumption exp enditures of the investor at date t and let W denote t t the investor's wealth just prior to date-t consumption. We assume that the investor has a lifetime utility function U (C ;C ;C ) de ned over each consumption path fC ;C ;C g 0 1 2 0 1 2 that summarizes how much he values the entire path of consumption exp enditures. Then, absent market frictions and assuming that the investor's utility function is time-additive and time-homogeneous, i.e., U (C ;C ;C ) = u(C ) + u(C ) + u(C ) (1) 0 1 2 0 1 2 the investor's dynamic p ortfolio optimization problem at t =0 is given by: V (W ) = Max E [u(C )+u(C )+u(C )] (2) 0 0 0 0 1 2 C ;C ;C 0 1 2 sub ject to W C = x S + y B ; t =0; 1; 2 (3a) t t t t t t W = x S + y B ; t =0; 1 (3b) t+1 t t+1 t t+1 C 0 ; t =0; 1; 2 (3c) t + x ; y 2 Z ; t =0; 1; 2 (3d) t t x = y = 0 (3e) 2 2 + where Z denotes the non-negative integers, and x and y are the number of shares of t t sto cks and b onds, resp ectively, that the investor holds in his p ortfolio immediately after 3 date t. The requirement that x and y are non-negative means that b orrowing and short t t sales are not allowed, a constraint that manyinvestors face. That x and y are required to b e t t non-negative integers simply re ects the fact that it is not p ossible to purchase a fractional number of sto cks or b onds. The constraint (3b) states that W is equal to W multiplied t+1 t by the gross return on the p ortfolio b etween dates t and t +1. This problem is easily solved numerically using the standard technique of sto chastic 3 Of course x and y may dep end only on the information available at time t, a restriction that we imp ose t t throughout this article. 3 4 dynamic programming. In particular, since V (W )=u(W ), we can then compute V (W ) 2 2 2 1 1 using the Bellman equation so that V (W ) = Max u(C )+E [V (W )] (4) 1 1 1 1 2 2 C 1 sub ject to the constraints in (3). An asp ect of (4) that makes it particularly easy to solveis the fact that the value function V () dep ends on only one state variable, W . This enables 1 1 us to solve (4) numerically without to o many computations. Supp ose, for example, that + W = $1;000. Then, since x ; y 2 Z , there are only 1;100 p ossible values that W can 0 t t 1 take, so the right side of (4) must b e evaluated for only these 1;100 values. In contrast, when there are market frictions or when the investor has a more complex utility function, we will see how the computational requirements increase dramatically, re ecting Bellman's \curse of dimensionality". 3 Taxes Most seasoned investors are painfully aware of the substantial impact that taxes can have on the p erformance of their investment p ortfolio, hence taxes play a ma jor role in most 5 dynamic p ortfolio optimization problems. To see how taxes can increase the computational complexity of such problems, let trading pro ts in the sto ck b e sub ject to a capital gains tax in the simple mo del of Section 2. Because this mo del has only two future p erio ds, we do not distinguish between short-term and long-term capital gains, and for exp ositional simplicity we also assume that capital losses from one p erio d cannot be used to o set gains from a later period. Even though these simplifying assumptions do make the problem easier to solve, nevertheless, it will b e apparent that the computations are considerably more involved than in the no-tax case.