Evaluating Portfolio Policies: A Duality Approach

Transcription

1 OPERATIONS RESEARCH Vol. 54, No. 3, May June 26, pp issn 3-364X eissn informs doi /opre INFORMS Evaluaing Porfolio Policies: A Dualiy Approach Marin B. Haugh Deparmen of Indusrial Engineering and Operaions Research, Columbia Universiy, New York, New York 127, mh278@columbia.edu Leonid Kogan Sloan School of Managemen, Massachuses Insiue of Technology, Cambridge, Massachuses 2142, lkogan@mi.edu Jiang Wang Sloan School of Managemen, Massachuses Insiue of Technology, Cambridge, Massachuses 2142, and CCFR and NBER, wangj@mi.edu The performance of a given porfolio policy can in principle be evaluaed by comparing is expeced uiliy wih ha of he opimal policy. Unforunaely, he opimal policy is usually no compuable, in which case a direc comparison is impossible. In his paper, we solve his problem by using he given porfolio policy o consruc an upper bound on he unknown maximum expeced uiliy. This consrucion is based on a dual formulaion of he porfolio opimizaion problem. When he upper bound is close o he expeced uiliy achieved by he given porfolio policy, he poenial uiliy loss of his policy is guaraneed o be small. Our algorihm can be used o evaluae porfolio policies in models wih incomplee markes and posiion consrains. We illusrae our mehodology by analyzing he saic and myopic policies in markes wih reurn predicabiliy and consrains on shor sales and borrowing. Subjec classificaions: finance: porfolio; opimal conrol: applicaions. Area of review: Financial Engineering. Hisory: Received July 23; revision received Sepember 24; acceped March Inroducion Opimal porfolio choice is one of he cenral subjecs in finance (see, e.g., Meron 199). However, analyical soluions are known only in a few special cases, under resricive assumpions on he marke srucure and/or he invesor s uiliy funcion. Mos problems of pracical ineres canno be solved in closed form and mus be analyzed using approximaions, bu how cosly are he approximaion errors of a candidae porfolio policy o he invesor? To answer his quesion, one mus compare he expeced uiliy under he candidae policy wih he expeced uiliy under he opimal policy. Such a direc comparison is impossible when he opimal policy canno be compued explicily. In his paper, we consruc an upper bound on he expeced uiliy under he opimal policy based on he dual formulaion of he porfolio opimizaion problem developed by Cvianic and Karazas (1992). When he upper bound is close o he expeced uiliy achieved by a candidae porfolio policy, he uiliy cos of choosing such a subopimal policy is guaraneed o be small. Our mehod applies o models wih incomplee markes and posiion consrains. The mos general class of racable problems are hose where he financial marke is dynamically complee. Marke compleeness allows one o use he maringale echniques o resae he porfolio choice problem in an equivalen saic form. This was firs shown by Cox and Huang (1989), Karazas e al. (1986), and Pliska (1986). The saic problem ofen leads o analyical soluions and is also amenable o simulaion echniques (see, e.g., Deemple e al. 23). When he financial marke is incomplee due o posiion consrains or nonspanned risks, explici soluions have been obained for only a few special cases. For example, Kim and Omberg (1996) solve he porfolio choice problem assuming an affine nonspanned process for he marke price of risk and no porfolio consrains (wih power uiliy funcion). Liu (25) exends heir resuls o mulivariae problems. The maringale approach has been generalized by a number of researchers using sochasic dualiy heory o allow for porfolio consrains and nonspanned risks. They include Cuoco (1997), Cvianic and Karazas (1992), He and Pearson (1991), and Karazas e al. (1991), among ohers. However, explici soluions are rare. Noable excepions are problems wih logarihmic preferences, or problems wih consan relaive risk aversion (CRRA) preferences and deerminisic invesmen opporuniy se (see, for example, Karazas and Shreve 1998, 6.6). For mos porfolio choice problems, explici soluions are no available, and one mus rely on numerical approximaions. Various ypes of approximae mehods have been used in he lieraure. They include he log-linear analyical approximaions of Campbell and Viceira (1999, 22) (see also Campbell e al. 22, Chacko and Viceira 25), finie-difference PDE soluion mehods (e.g., Brennan 1998, Brennan e al. 1997, Brennan and Xia 22, Xia 21), Markov-chain approximaions (e.g., Munk 2), 45

2 46 Operaions Research 54(3), pp , 26 INFORMS numerical dynamic programming based on sae-space discreizaion for discree-ime problems (e.g., Ai-Sahalia and Brand 21, Balduzzi and Lynch 1999, Brand 1999, Barberis 2, Dammon e al. 2, Lynch 21), and he approximae dynamic programming approach of Brand e al. (25). Oher echniques for solving dynamic opimizaion problems ha are applicable o porfolio choice problems are described in Judd (1996) and Rus (1996). An imporan limiaion of he approximae soluion mehods is ha i is usually difficul o evaluae he accuracy of he obained approximaions. Judd (1996) poins ou ha here are few heoreical resuls on he convergence properies of he numerical algorihms used in economic applicaions, and even when such resuls are available, asympoic convergence does no guaranee he accuracy of any given approximaion. Moreover, some of he exising approaches are based on a single approximaion, raher han a sequence of successive approximaions (e.g., he log-normal approximaion mehod used by Campbell and Viceira 1999), and for hese he noion of asympoic convergence is no even applicable. Den Haan and Marce (1994) and Judd (1996) sugges procedures for esing how accuraely he firs-order opimaliy condiions are saisfied. Such ess, however, provide lile informaion abou he economic impac of he approximaion errors. The mehod we propose allows us o evaluae he qualiy of approximaions for porfolio choice problems by providing an upper bound on he uiliy loss associaed wih an approximae soluion. If a sequence of progressively refined approximae porfolio policies was generaed using an ieraive numerical algorihm, hen our mehod could be used o deermine a wha poin he qualiy of he approximaion becomes accepable. A ha poin he algorihm could hen be erminaed. Because our approach does no depend on he paricular mehod used o consruc he approximae policy and value funcion, i can be applied o any approximae soluion mehod. Moreover, in principle our mehodology could be used o guide he numerical algorihm as well. In his paper, however, we will focus on evaluaing he qualiy of a fixed porfolio policy and no aemp o consruc and evaluae a sequence of approximaions. Our approach is based on he dualiy formulaion of he porfolio choice problem proposed by Cvianic and Karazas (1992). For a given porfolio choice problem, which is subjec o posiion consrains, we can define a ficiious, unconsrained problem wih modified price processes such ha he maximum expeced uiliy achieved in he ficiious problem is a leas as high as in he original problem. Under cerain assumpions, he consruced upper bound can be shown o be igh, i.e., o coincide wih he maximum expeced uiliy for he original problem. Because he ficiious problem is unconsrained, we can solve i explicily using he well-known maringale echniques of Cox and Huang (1989). The soluion gives an upper bound on he maximum expeced uiliy (he value funcion) of he original problem. (Haugh and Kogan 24 develop a similar algorihm for he problem of pricing American opions, which is an opimal sopping problem. As in his paper, he upper bound on he value funcion is consruced using a dual formulaion of he original problem.) To evaluae an approximaion for he opimal policy and value funcion, we can use his upper bound when he value funcion iself is no available. The goal is o make he upper bound igh by properly choosing he ficiious problem, in paricular, he ficiious marke wih he modified price processes. Our key insigh is o perform such a modificaion using he informaion conained in he approximae soluion o he problem. In paricular, one mus firs specify he approximae porfolio policy and he value funcion. I would be naural o compue he value funcion implied by he approximae porfolio policy, bu i is no necessary o do so. In our numerical examples below, we specify an approximae value funcion direcly in a paricularly simple way and use i ogeher wih differen porfolio policies o esablish bounds on he rue value funcion. The approximaion o he value funcion is hen used o define he candidae reurn processes for all he asses in he ficiious financial marke, as implied by he opimaliy condiions of he original porfolio choice problem. The implied reurn processes in general do no qualify o be he reurn processes in he ficiious problem. Our nex sep is o choose he qualified ficiious reurn processes ha are closes o he candidae processes. We hen solve he unconsrained opimizaion problem in he ficiious marke. The value funcion of his ficiious problem gives an upper bound on he value funcion of he original problem. We illusrae our mehod wih a few simple numerical examples. These are no designed o be fully realisic, bu raher serve o demonsrae he poenial of our algorihm. Because i is no he purpose of his paper o develop new schemes for compuing approximae porfolio policies, o simplify implemenaion, we adop very simple, analyical approximaions o he porfolio policies. In paricular, we evaluae he qualiy of saic and myopic porfolio policies in a marke wih reurn predicabiliy under consrains on shor sales and borrowing. A saic porfolio policy is obained by ignoring ime variaion in he invesmen opporuniy se and fixing he insananeous momens of sock reurns a heir long-run average values. To obain a myopic policy, we assume a any poin in ime ha he invesmen opporuniy will remain consan hereafer. By comparing he expeced uiliy under he saic sraegy wih our upper bound, one can measure he economic imporance of reurn predicabiliy. A small gap beween he upper bound and he value funcion under he saic policy would indicae ha reurn predicabiliy is no economically imporan for he problem a hand. Similar logic can be applied o he myopic policy. A small gap under he myopic policy indicaes ha he hedging componen of he opimal porfolio policy has lile

3 Operaions Research 54(3), pp , 26 INFORMS 47 impac on he expeced uiliy. The analysis of he myopic policy is paricularly relevan because he exising lieraure analyzing he paerns of predicabiliy in sock reurns commonly ignores he hedging componen of porfolio policies (see, for example, Johannes e al. 22, Kandel and Sambaugh 1996, Pasor 2, Pasor and Sambaugh 2, Sambaugh 1999). This is moivaed by he fac ha for mos realisic problems he opimal porfolio policy canno be compued numerically due o he curse of dimensionaliy and analyical soluions are only available for highly resricive seings, which ypically rule ou porfolio consrains and limi he range of possible specificaions for he reurn processes. Several auhors have argued in he conex of specific applicaions ha hedging demand is relaively small (e.g., Ai-Sahalia and Brand 21, Ang and Bekaer 22, Brand 1999, Chacko and Viceira 25, Gomes 24). Based on his, one may be emped o assume ha for similar problems, hedging demand can be safely ignored. However, we know ha his canno be he case in general, and for imporan classes of problems hedging demand is crucial. These classes include longhorizon problems (Barberis 2; Campbell and Viceira 1999, 22) and problems wih nonlinear dynamics or nonsandard preferences (Chan and Kogan 22). Using our mehod, one can verify ha for a paricular problem under consideraion, ignoring he hedging componen of he opimal porfolio policy has lile effec on he expeced uiliy. The res of his paper is organized as follows. Secion 2 formulaes he porfolio choice problem. Secion 3 reviews he dualiy heory, while 4 describes he algorihm for consrucing he upper bound on he opimal expeced uiliy. Secion 5 illusraes our approach for several ypes of porfolio choice problems and approximae sraegies. We conclude in The Model We now sae a porfolio choice problem under incomplee markes and porfolio consrains. We formulae our problem in coninuous ime and assume ha sock prices follow diffusion processes. The Invesmen Opporuniy Se There are N socks and an insananeously risk-free bond. The vecor of sock prices is denoed by P = P 1 P N and he insananeously risk-free rae of reurn on he bond is denoed by r. Wihou loss of generaliy, we assume ha he socks pay no dividends. The insananeous momens of asse reurns depend on he M-dimensional vecor of sae variables X : r = r X dp = P P X d + P X db dx = X X d + X X db (1a) (1b) (1c) where P = X =, B = B 1 B N is a vecor of N independen Brownian moions, P and X are N - and M-dimensional drif vecors, and P and X are diffusion marices of dimension N by N and M by N, respecively. We assume ha he diffusion marix of he sock reurn process P is lower riangular and nondegenerae: x P P x x 2 for all x and some >. Then, one can define a process, given by = 1 P P r In a marke wihou porfolio consrains, corresponds o he vecor of insananeous marke prices of risk of he N socks (see, e.g., Duffie 21, 6.G). We adop a sandard assumpion ha our reurn generaing process is sufficienly well behaved, so ha he process is square inegrable: [ T ] E 2 d < Porfolio Consrains A porfolio consiss of posiions in he N socks and he risk-free bond. We denoe he proporional holdings of risky asses in he oal porfolio value by = 1 N.We require he porfolio policy o saisfy a square inegrabiliy condiion: T 2 d < almos surely. The value of he porfolio changes according o dw = W {[ r + P r ] d + P db } (2) We assume ha he porfolio shares are resriced o lie in a closed convex se K, conaining he zero vecor: K (3) For example, if shor sales are no allowed, hen he consrain se akes he form K =. If in addiion o prohibiing shor sales borrowing is no allowed, hen K = 1 1, where 1 = 1 1. In our analysis, we ake he se K o be consan, bu i can be allowed o depend on ime and he values of he exogenous sae variables. The Objecive Funcion We assume ha he porfolio policy is chosen o maximize he expeced uiliy of wealh a he erminal dae T, E U W T. The funcion U W is assumed o be sricly monoone wih posiive slope, concave, and smooh. Moreover, i is assumed o saisfy he Inada condiions a zero and infiniy: lim W U W = and lim W U W =. In our numerical examples, we use he uiliy funcion wih consan relaive risk aversion (CRRA) so ha U W = W 1 / 1. In summary, he porfolio choice problem is o solve for V sup E U W T subjec o (1), (2), and (3) ( ) where V denoes he value funcion a.

4 48 Operaions Research 54(3), pp , 26 INFORMS In he seing we adop, wih coninuous-ime and diffusion processes for sock prices, he financial marke has he propery ha i can be made dynamically complee by inroducing a small number of addiional securiies. Our analysis primarily relies on his propery, and hus can be exended o discree-ime and discree-sae models, using he resuls in Pliska (1997). However, our approach would no be direcly applicable o models ha do no possess his propery, for insance, o models wih coninuously disribued jumps in sock prices. 3. Review of he Dualiy Theory In his secion, we briefly review he dualiy approach o he consrained porfolio opimizaion problem, following Cvianic and Karazas (1992) and Schroder and Skiadas (23). We build on hese heoreical resuls o consruc bounds on he performance of porfolio policies in 4 below. Saring wih he porfolio choice problem ( ), we can define a ficiious problem ( ), based on a differen financial marke and wihou he porfolio consrains. Firs, we define he suppor funcion of K, N,by sup x (4) x K The effecive domain of he suppor funcion is given by K < (5) Because he consrain se K is convex and conains zero, he suppor funcion is coninuous and bounded from below on is effecive domain K. We hen define he se D of F -adaped N -valued processes o be { D T K [ T ] [ T ] } E d + E 2 d < (6) For each process in D, we define a ficiious marke M, in which he N socks and he risk-free bond are raded. The diffusion marix of sock reurns in M is he same as in he original marke. However, he risk-free rae and he vecor of expeced sock reurns are differen. In paricular, he risk-free rae process and he marke price of risk in he ficiious marke are defined, respecively, by r = r + (7a) = + 1 P (7b) where is he suppor funcion defined in (4). We assume ha is square inegrable. Following Cox and Huang (1989), he sae-price densiy process in he ficiious marke is given by ( = exp r s ds 1 2 s s ds ) s dbs (8) and he vecor of expeced reurns is given by P = r + P The dynamic porfolio choice problem in he ficiious marke wihou posiion consrains can be equivalenly formulaed in a saic form (e.g., Cox and Huang 1989; Karazas and Shreve 1998, 3): V sup W T subjec o E U W T E [ T W T ] W ( ) Due o is saic naure, problem is easy o solve. For example, when he uiliy funcion is of he CRRA ype wih relaive risk aversion so ha U W = W 1 / 1, he corresponding value funcion in he ficiious marke is given explicily by = W 1 [ 1 E V T 1 / ] (9) I is easy o see ha for any admissible choice of D, he value funcion in (9) gives an upper bound for he opimal value funcion of he original problem. In he ficiious marke, he wealh dynamics of he porfolio are given by dw so ha dw W = W dw W [( r + P ) d + ] PdB (1) = [( r ) r + ( )] P d = [ + ] d The las expression is nonnegaive according o (4) because K. Thus, W W T and V V (11) In his paper, we use his propery of he ficiious problem ( ) o consruc an upper bound on he value funcion of he original problem ( ). Resuls in Schroder and Skiadas (23) imply ha if he original opimizaion problem has a soluion, hen he upper bound is igh, i.e., he value funcion of he ficiious problem ( ) coincides wih he value funcion of he original problem ( ): V inf V = V (12) (see Schroder and Skiadas 23, Proposiion 3(b), Theorems 7 and 9). The above equaliy holds for all imes, and no jus a ime, i.e., V = V. Cvianic and Karazas (1992) have shown ha he soluion o he original problem exiss under addiional resricions on he uiliy funcion, mos imporanly ha he relaive risk aversion does no exceed one. Cuoco (1997) proves a more general exisence resul, imposing minimal resricions on he uiliy funcion.

5 Operaions Research 54(3), pp , 26 INFORMS The Performance Bound The heoreical dualiy resuls of 3 sugges ha one can consruc an upper bound on he value funcion of he porfolio choice problem ( ) by compuing he value funcion of any ficiious problem ( ). The ficiious marke is defined by he process as in (7). Of course, one can pick any ficiious marke from he admissible se D o compue an upper bound. Such a bound is uninformaive if i is oo loose. Because our objecive is o evaluae a paricular candidae policy, we can consruc a process based on such a policy o obain igher bounds. The soluion o he porfolio choice problem under he ficiious marke defined by hen gives us a performance bound on he candidae policy. To consruc he ficiious marke as defined by,we firs use he soluion o he dual problem (which also gives he soluion o he original problem) o esablish he link beween he opimal policy and value funcion V, and he corresponding ficiious asse price processes, as defined by. No knowing he opimal porfolio policy and value funcion, we insead use heir approximaions o obain he candidae process for, which is denoed by. This candidae process in general does no belong o D and canno be used o define a ficiious problem. Insead, we search for a qualified process in D, which is closes o. We hen use as an approximaion o o define he ficiious problem in ( ). Because D, he soluion o he corresponding unconsrained problem in M provides a valid performance bound for he candidae policy. According o Cox and Huang (1989), V = V [ T subjec o E = sup E U W T W T W T ] W The firs-order opimaliy condiions hen imply U W T = T /, where is he Lagrange muliplier on he ime- budge consrain. On he oher hand, according o he envelope condiion, he parial derivaive of he value funcion wih respec o he porfolio value saisfies V / W = (see Karazas and Shreve 1998, 3.7, Theorem 7.7 for a formal proof). Thus, T V / W, and hence, s = V s/ W s s V / W The above equaliy implies ha / = U W T / d ln = d ln V (13) W In paricular, he sochasic par of d ln is equal o he sochasic par of d ln V / W.IfV is smooh, Io s lemma and Equaions (8) and (2) imply ha ( ) 2 V = W / W 2 P V / W ( ) 1 ( V 2 V X W W X ) (14) where denoes he opimal porfolio policy for he original problem. In he special bu imporan case of a CRRA uiliy funcion (which we analyze in more deail below), he expression for simplifies. In paricular, he firs erm in (14) becomes equal o P, where is he relaive risk-aversion coefficien of he uiliy funcion, and one only needs o compue he firs derivaive of he value funcion wih respec o he sae variables X o evaluae he second erm in (14). This simplifies numerical implemenaion, because i is generally easier o esimae firs-order han second-order parial derivaives of he value funcion. Given an approximaion o he opimal porfolio policy, one can compue he corresponding approximaion o he value funcion, V, defined as he condiional expecaion of he uiliy of erminal wealh, under he porfolio policy. We can hen consruc a process as an approximaion o, using (14). Approximaions o he porfolio policy and value funcion can be obained using a variey of mehods (e.g., Brand e al. 25). In his paper, we ake as given and use i o consruc an upper bound on he unknown rue value funcion V. Assuming ha he approximae value funcion V is sufficienly smooh, we can replace V and in (14) wih V and and obain ( 2 V = W / W 2 V / W ( 2 V X W X ) ( ) V 1 P W ) (15) We hen define as a soluion o (7b). Obviously, is a candidae for he marke price of risk in he ficiious marke. However, here is no guaranee ha and he corresponding process belong o he feasible se D defined by (6). In fac, as we illusrae below, for many imporan classes of problems he suppor funcion may be infinie for some values of is argumen. Thus, we look for a price-of-risk process D ha is close o. We choose a Euclidean norm as our measure of disance beween he wo processes o make he resuling opimizaion problem racable. To guaranee ha D wihou sacrificing racabiliy, we replace he original inegrabiliy consrains defining he se D wih a se of igher uniform bounds, A 1 A 2 (16a) (16b) where A 1 and A 2 are posiive consans ha can be aken o be arbirarily large. Condiion (16a) implies ha he process is square inegrable, because we have assumed ha is square inegrable and 2 = 1 P 1 P A 2 for some A>. If he process, corresponding o he opimal porfolio policy, does no saisfy consrains (16a) and (16b), imposing such consrains on and would

6 41 Operaions Research 54(3), pp , 26 INFORMS increase he upper bound. This is he price one mus pay for a compuaionally racable descripion of a feasible se of and. For he examples we consider below, one can verify direcly ha he consruced processes and are feasible, and herefore consrains (16a) and (16b) do no need o be imposed. In general, because he choice of consans A 1 and A 2 is somewha arbirary, one may wan o experimen wih differen values. Consrains (16a) and (16b) may bind because he opimal process T is no uniformly bounded, or because he approximaion error is no uniformly bounded. In he former case, one would like o make he consans as large as possible, while in he laer case large values of A 1 and A 2 could increase he upper bound. We now define and as a soluion of he following problem: min 2 (17) subjec o = + 1 P (18a) < (18b) A 1 (18c) A 2 (18d) The objecive is o minimize he Euclidean disance beween he process, used o compue he upper bound, and he candidae process for he marke price of risk,. The firs consrain simply relaes he marke price of risk o he process. The second consrain requires he suppor funcion o be finie. This is a general form of a consrain, which, depending on he specific choice of he se K, represens a se of consrains on. For he examples we consider below, he requiremen ha < resuls in a se of linear consrains on. Finally, he las wo consrains in (18) are he uniform bounds guaraneeing ha D, as we discussed above. The value of and can be compued quie easily for many imporan classes of porfolio choice problems. In paricular, we consider he hree examples menioned in 2. Incomplee Markes Assume ha only he firs L socks are raded. The posiions in he remaining N L socks are resriced o zero. In his case, he se of feasible porfolio policies is given by K = i = for L<i N (19) and hence he suppor funcion is equal o zero if i =, 1 i L, and is infinie oherwise. Thus, as long as i =, 1 i L, consrain (16b) does no need o be imposed explicily. To find and, we mus solve min 2 (2) subjec o = + 1 P i = 1 i L 2 A 2 1 The diffusion marix P is lower riangular and so is is inverse. Using his, he soluion can be expressed explicily as i = i 1 i L j = j + a j j = P L<j N where [ ( A 2 ) 1/2 ] 1 a=min 1 2 L L = L 2 i L i=1 Noe ha when he consan A 1 is very large, i will invariably be he case ha a = 1, and herefore j = j for L< j N. Incomplee Markes and No Shor Sales Consider he marke in which only he firs L socks can be raded and no shor sales (of he socks) are allowed. The se of admissible porfolios is given by K = i = for L<i N (21) and he suppor funcion is equal o zero if i, i = 1 L, and is infinie oherwise. As in he previous case, consrain (16b) is auomaically saisfied. Thus, can be deermined as a soluion of a sandard quadraic programming problem: min 2 (22) subjec o = + 1 P i 1 i L 2 A 2 1 Incomplee Markes, No Shor Sales, and No Borrowing Consider he same marke as in he previous case, bu no shor sales and borrowing are allowed. Then, he se of admissible porfolios is given by K = 1 1 i = for L<i N (23) The suppor funcion is given by = max 1 L, which is finie for any vecor. Because in his case

7 Operaions Research 54(3), pp , 26 INFORMS 411, he relaion = P P A 1 (see (16a) and (16b)) implies ha as long as A 2 is sufficienly large compared o A 1, one only needs o impose (16a), and (16b) is redundan. Thus, one needs o solve he following problem: min 2 (24) subjec o = + 1 P 2 A 2 1 Then, he ficiious marke is described by ( ) A = + min 1 1 = P Again, by seing A 1 o be sufficienly large, we will invariably obain =. 5. Numerical Experimens The mehod described in he previous secion can be applied o various problems of porfolio choice. In his secion, we illusrae is performance based on several numerical experimens, bu firs we summarize our algorihm as a sequence of four basic seps: Sep 1. Sar wih an approximaion o he opimal porfolio policy of he original problem, and he corresponding approximaion o he value funcion, which can be obained using a variey of mehods. In his paper, we simply assume ha such approximaions are available. Sep 2. Use he approximae porfolio policy and parial derivaives of he approximae value funcion o consruc a process according o he explici formula (15). The process is a candidae for he marke price of risk in he ficiious marke. Sep 3. Consruc a process ha is close o and saisfies he condiions for he marke price risk of a ficiious marke in he dual problem. This involves solving he quadraic opimizaion problem (17). Sep 4. Compue he value funcion from he saic problem ( ) in he resuling ficiious marke defined by he marke price of risk process. This can be accomplished efficienly using Mone Carlo simulaion. This resuls in an upper bound on he value funcion of he original problem. The lower bound on he value funcion is obained by simulaing he erminal wealh disribuion under he approximae porfolio sraegy. Calibraion In our numerical experimens, we assume ha he uiliy funcion is of he consan relaive risk aversion (CRRA) ype so ha U W = W 1 / 1. We consider hree values for he relaive risk-aversion parameer: = 1 5, 3, and 5. We also consider wo values for he problem horizon: T = 5 and T = 1 years. Our coninuous-ime model is very similar o he discree-ime marke model in Lynch (21), and we use he parameers esimaed by Lynch o guide our calibraion. He considers a financial marke wih hree raded risky asses and a single sae variable. To represen he same marke in our framework, we assume ha here is a oal of four risky asses, he firs hree of which are raded, and a single sae variable, i.e., we consider an incomplee marke wih N = 4, M = 1, and L = 3, meaning ha he firs hree socks can be raded. The dynamics of asse reurns is given by r = r (25a) dp = P + X 1 d + P db (25b) dx = kx d + X db (25c) The diffusion marix X is of size 1 by 4 and we assume ha he fourh row of he marix P coincides wih X. This assumpion is made for convenience, and i economizes on he number of parameers o be repored. Noe ha because he fourh risky asse is no raded, one can specify he fourh row of P arbirarily, as long as he marix P saisfies he necessary regulariy condiions. We se he iniial value of he sae variable o zero, X =, in all numerical examples below. Lynch (21) considers wo choices for he sae variable X: he dividend yield and he erm spread. The dividend yield capures he rae a which dividends are paid ou, as a fracion of he oal sock marke value. Specifically, Lynch uses a coninuously compounded 12-monh yield on he value-weighed NYSE index. The erm spread is he difference in yields beween 2-year and one-monh Treasury securiies. Boh of hese predicive variables are normalized o have zero mean and uni variance. He also considers wo ses of risky asses: hree porfolios obained by soring socks on heir size or on heir book-o-marke raio. Deails of he procedure are repored in Lynch (21); here we are ineresed only in he parameers of he sock reurn process, as summarized in Tables 1 and 2 of Lynch. Thus, by considering wo choices of risky asses and wo choices of he predicive variable, we have four ses of calibraed parameer values. These are repored in Table 1. We se he risk-free rae a r = 1 hroughou. Approximae Policies As in Lynch (21), we consider wo ypes of porfolio policies. The firs policy, which we call saic, ignores predicabiliy of sock reurns. I is defined using he uncondiional average reurns insead of he ime-varying condiional expeced reurns on he socks. Specifically, saic = arg max r 1 2 P P subjec o K (26)

8 412 Operaions Research 54(3), pp , 26 INFORMS Table 1. Calibraed model parameers. k 1 P Parameer se Parameer se Parameer se Parameer se Noes. The four ses of parameers correspond o: (1) size-sored porfolios and he dividend yield as a sae variable; (2) size-sored porfolios and he erm spread as a sae variable; (3) book-omarke sored porfolios and he dividend yield as a sae variable; (4) book-o-marke sored porfolios and he erm spread as a sae variable. Parameer values are based on he esimaes in Tables 1 and 2 of Lynch (21). This is an opimal porfolio policy in a dynamic model wih a consan invesmen opporuniy se and cone consrains on porfolio posiions, e.g., Karazas and Shreve (1998, 6.6). As we discussed in he inroducion, a small gap beween he upper bound and he expeced uiliy under he saic policy indicaes ha reurn predicabiliy is no economically imporan for he problem a hand. The reverse, however, is no necessarily rue when he upper bound is significanly higher han he lower bound, we canno formally conclude ha he saic porfolio policy is significanly inferior o he opimal sraegy. We discuss hese issues in more deail in he conex of he numerical examples below. The second porfolio policy we consider is he myopic policy. I is defined in he same way as a saic policy, excep he insananeous momens of asse reurns are fixed a heir curren values, as opposed o heir long-run average values: myopic = arg max P r 1 2 P P subjec o K (27) The approximae porfolio policy in (27) ignores he hedging componen of he opimal rading sraegy. By comparing he corresponding expeced uiliy wih an upper bound, we will be able o evaluae he economic imporance of hedging as a componen of he porfolio policy. In a porfolio choice problem similar o ours, Lynch (21) finds a significan hedging demand for some of he parameer values he considers. In paricular, for he hird parameer se when here are no porfolio consrains, Lynch finds ha he opimal policy is very differen from he myopic approximaion. Below we will show ha even hough he myopic sraegy may be far from opimal in erms of acual porfolio holdings, he uiliy loss due o using i may no be very large. As we previously emphasized, he advanage of our mehod is ha such resuls can be obained wihou knowing he opimal sraegy, which would be he case for mos realisic applicaions. Our algorihm for evaluaing porfolio sraegies relies on an approximae value funcion, V. In general, one has o esimae V as he condiional expecaion of he uiliy of erminal wealh under he approximae sraegy. This could be done, for example, using sandard regression mehods. Insead, we choose a paricularly sraighforward closed-form expression for V. This simplifies compuaions, while sill allowing us o illusrae he concepual seps involved in esimaing he upper bound on he value funcion. The cos of using a simplified closed-form formula for V is a poenially wider gap beween he upper bound and he rue value funcion. Because he uiliy funcion is of CRRA ype, he rue value funcion has a homoheic funcional form, V = g X W 1. As a drasic simplificaion, we ignore he dependence of he value funcion on he sae variable and se V = g W 1. The exac form of he funcion g does no affec he specificaion of he arificial marke below. Noe ha removing he dependence of he approximae value funcion on X does no eliminae i from he price of risk in he ficiious marke, given by (14). This is because he leading erm in ha expression depends on X hrough he approximae porfolio policy,. We will see in he numerical examples below ha even such a simplisic approximaion o he value funcion can lead o informaive bounds on he rue value funcion. However, in realisic pracical applicaions, one should use more accurae approximaions o he value funcion o achieve igher bounds whenever possible. Simulaion For each of he approximae porfolio sraegies, we esimae he corresponding expeced uiliy of erminal wealh, as well as he upper bound on he rue value funcion, using Mone Carlo simulaion. The expeced uiliy under he subopimal porfolio policy provides a lower bound on he rue value funcion of he problem. We esimae he former by simulaing independenly 1, or 1,, pahs of sock prices and sae variables according o (25a) (25c). We used 1, pahs o esimae he bounds for he problems wih consrains on shor sales only, as he corresponding quadraic subproblem in ha case needs o be solved

9 Operaions Research 54(3), pp , 26 INFORMS 413 numerically. We used 1,, pahs for all oher cases. (The problems were solved using MATLAB, so he compuaional imes for he various problems are no paricularly informaive.) We compue he porfolio value along each pah under he approximae porfolio sraegy and record he uiliy of he erminal wealh. We hen average over he simulaed pahs o esimae he expeced uiliy. In simulaions, we discreize he coninuous-ime diffusion processes using a sandard Euler scheme wih a ime sep = 1 years. For he case of incomplee markes and he case of no shor sales and no borrowing, quadraic subproblems given by Equaions (17) and (18) can be solved in closed form. For he case of no borrowing, we solved each of he quadraic subproblems using he MATLAB opimizer. Each of he subproblems was in hree dimensions wih hree inequaliy consrains. The oal number of subproblems was equal o 1 7 T. As we showed in 4, he rue value funcion is bounded from above by he value funcion in he ficiious marke, which in urn is given by = W 1 [ 1 E V T 1 / ] (28) We simulae he process T by discreizing he inegrals in (8) wih he same ime sep = 1. The marke price of risk and he risk-free rae in he ficiious marke are compued according o he procedure described in 4. In our simulaions, we do no impose consrain (16a), because one can direcly verify, under our assumpion abou he reurn-generaing process and he approximae rading sraegy, ha he process is square inegrable. The second consrain (16b) does no need o be imposed explicily for he problems we are considering below, as discussed in 4. Incomplee Markes We firs consider he incomplee marke model, in which porfolio posiions in radable asses are unconsrained. Table 2 repors he esimaes of he expeced uiliy boh under he saic and myopic porfolio sraegies in Equaions (26) and (27), which provide lower bounds on he rue value funcion, as well as he corresponding esimaes of he upper bound. Expeced uiliy is repored as a coninuously compounded cerainy equivalen reurn, R. The value of R corresponding o he value funcion V is defined by U W e RT = V. Alernaively, one could repor he uiliyequivalen wealh W, defined by U W = V. We express our resuls in erms of he cerainy equivalen reurn o faciliae comparison across differen problem horizons T. We also repor he exac value funcion, which can be compued explicily using he resuls in Kim and Omberg (1996) and Liu (25). Table 2 conains resuls for four parameer ses, hree values of he risk aversion parameer, and wo values of he economy horizon T. One can see ha here is a sizeable gap beween he expeced uiliy achieved by he saic sraegy and he corresponding upper bound on he rue value funcion. For mos of he parameer combinaions considered, he difference, expressed as annualized cerainy equivalen reurn, is in he range of 1% 3%. The difference beween he upper bound and he lower bound can be decomposed ino a sum of wo erms: UB LB = V LB + UB V (29) The firs erm is he uiliy loss of he approximae porfolio sraegy under consideraion. I capures he uiliy loss due o using a subopimal porfolio sraegy. The second erm is he difference beween he upper bound and he rue value funcion, and i depends on how igh he esimaed upper bound is. Ideally, we would like o know he firs erm, bu because he rue value funcion is no known, we can only esimae he sum of wo erms, i.e., he difference UB LB. This difference is large because eiher or boh of he wo erms on he righ-hand side are large. In paricular, he magniude of he second erm depends on he accuracy of he approximae porfolio policy and value funcion used o esimae he upper bound. Table 2 indicaes ha in his paricular case, he magniude of he second erm is relaively small, as can be seen from he difference beween he upper bound, UB s, and he exac soluion, V u. (The exac soluion is available for his special case from Liu (25). This will no be he case once we impose porfolio consrains below.) This is somewha surprising, because he difference beween he lower and he upper bound suggess ha he saic policy is quie far from being opimal. To summarize, he above resuls do no allow us o conclude ha predicabiliy can be safely ignored for he problems under consideraion. However, wihou knowing he exac soluions, we could no argue ha he economic value of predicabiliy is large eiher. More generally, one should no view he difference UB LB we consruc as a formal es saisic for opimaliy of a given porfolio policy, because we have no esablished ha he difference UB LB mus be small when a porfolio policy used o consruc he upper bound is close o being opimal. One can, however, draw definiive conclusions from he cases when he difference UB LB is small, which implies ha he considered subopimal porfolio policy has a small uiliy cos. We now urn o he corresponding resuls for he myopic sraegy, also shown in Table 2. As we have discussed in he inroducion, using our mehod, one can verify ha for a paricular problem under consideraion, ignoring he hedging componen of demand has lile effec on he expeced uiliy. For he problems wih a five-year horizon, he difference beween he poin esimaes of UB and LB ends o be in he 1% 2% range, excep for he hird parameer se, where i reaches approximaely 6%. Thus, we conclude ha for mos of he parameer ses, he hedging componen of demand is no very significan economically wih a five-year model horizon.

10 414 Operaions Research 54(3), pp , 26 INFORMS Table 2. Incomplee markes. T = 5 T = 1 = 1 5 = 3 = 5 = 1 5 = 3 = 5 Parameer se 1 LB s UB s LB m UB m V u Parameer se 2 LB s UB s LB m UB m V u Parameer se 3 LB s UB s LB m UB m V u Parameer se 4 LB s UB s LB m UB m V u Noes. The four parameer ses are defined in Table 1. All resuls are compued for he iniial value of he sae variable X =. The rows marked LB s and LB m repor he esimaes of he expeced uiliy achieved by using he saic and myopic porfolio sraegies, respecively. The esimaes are based on 1,, independen simulaions. Expeced uiliy is repored as a coninuously compounded cerainy equivalen reurn. Approximae 95% confidence inervals are repored in parenheses. The rows marked UB s and UB m repor he analogous resuls for he upper bound on he rue value funcion compued according o he procedure described in 4. The row marked V u repors he opimal value funcion for he problem. When we increase he problem horizon o 1 years, he gap UB LB increases. For he second and fourh parameer ses he difference remains small, under 2%, bu for he hird parameer se i now reaches as much as 1 5% for he case of = 5. These observaions are consisen wih our inuiion ha hedging demand ends o be more imporan for problems wih longer horizons. I is well known ha he myopic policy is opimal for = 1 in his case, and herefore i is no surprising ha i performs well for values of close o one.

11 Operaions Research 54(3), pp , 26 INFORMS 415 No Shor Sales and No Borrowing We now consider he marke in which boh shor sales and borrowing are prohibied, i.e., porfolio policies mus saisfy (23). Table 3 repors he esimaes of he expeced uiliy boh under he saic and myopic porfolio sraegies and he corresponding esimaes of he upper bound. We also repor he value funcion obained by relaxing he porfolio consrains on he hree risky asses, which is he same as he values repored in Table 2. Table 3 conains resuls for four parameers ses, hree values of he risk-aversion parameer, and wo values of he economy horizon T. As in he case of incomplee markes and no posiion consrains, here is a significan gap beween he expeced uiliy achieved by he saic sraegy and he corresponding upper bound on he rue value funcion. As before, we decompose he difference beween he upper bound and he lower bound as in (29). Table 3 indicaes ha he magniude of he second erm is subsanial, a leas for some of he parameer ses. In paricular, for = 5 he esimaed upper bound ofen exceeds he value funcion of he unconsrained problem, which in urn is greaer han or equal o he rue value funcion. As in he incomplee marke case, we canno conclude ha predicabiliy can be safely ignored for he problems under consideraion. The resuls for he myopic sraegy are similar o hose in Table 2. For he problems wih a five-year horizon, he difference beween he poin esimaes of UB and LB ends o be in he.1%.2% range, excep for he hird parameer se, where i reaches approximaely.5%. Thus, we conclude ha for mos of he parameer ses he hedging componen of demand is no very significan economically wih a five-year model horizon. A closer examinaion of he porfolio composiion under he myopic policy (no repored here) suggess ha our conclusions may be oo conservaive, a leas for he hird parameer se. Because of he binding shor-sale consrains, he myopic policy is likely o be close o he opimum mos of he ime. Thus, if we were o use a more accurae approximaion o he approximae value funcion V, our upper bound would likely be igher han he values repored in Table 3. As we poined ou earlier, our objecive here is no o maximize he accuracy of compuaions, bu raher o illusrae he conceps, and herefore we do no pursue his issue any furher. No Shor Sales In he numerical experimens above, we assumed ha neiher borrowing nor shor sales are allowed. As we showed in 4, in his case he ficiious marke used o esimae he upper bound is paricularly sraighforward o define. We now consider he case when shor sales are prohibied, bu borrowing is allowed. Then, o define he ficiious marke one mus solve he quadraic opimizaion problem (22). Table 4 repors he resuls, and o reduce he compuaional effor, we only consider he case of T = 5 years. Unlike he previous wo cases, when we impose no shorsales consrain, we have o solve a quadraic opimizaion problem numerically for every poin in ime on each simulaed pah. This increases he compuaional cos. To reduce he oal compuaion ime, we limi he model horizon o T = 5 years and only use 1, simulaed pahs. A comparison wih Table 3 reveals a peculiar regulariy. Because he borrowing consrain rarely binds for he myopic policy when = 3 and = 5, he corresponding expeced uiliy, LB, is similar o he values repored in Table 3. However, he upper bound ends o be much igher for hese cases compared o Table 3, even hough he rue value funcion for he problem considered in Table 4 exceeds he one for he problem corresponding o Table 3 (because he laer problem is a more consrained version of he former). The siuaion is reversed for = 1 5, bu in ha case he saic policy calls for borrowing, hence he expeced uiliies canno be direcly compared across he wo ables. Neverheless, one can observe ha for = 1 5, he difference UB LB is significanly larger han for = 3 and = 5. For he myopic sraegy, he difference UB LB is very similar o he corresponding values in Table 3. I is as high as 4% for he hird parameer se and is less han 2% for all oher parameer ses, demonsraing ha he hedging componen of demand is no crucial when borrowing is allowed. A comparison wih he resuls for he saic sraegy shows ha he gap UB LB for he saic sraegy is primarily due o he difference beween he rue value funcion and he lower bound, and he corresponding upper bounds are relaively igh. 6. Conclusion In his paper, we developed a mehod for evaluaing he qualiy of approximaions for porfolio choice problems by providing an upper bound on he uiliy loss associaed wih an approximae soluion. Our algorihm relies on he dualiy heory for consrained porfolio opimizaion. Saring from an approximae porfolio policy, we consruc a ficiious financial marke wihou porfolio consrains in which he opimizaion problem can be easily solved. The resuling value funcion is esimaed using Mone Carlo simulaion and provides an upper bound on he (unknown) value funcion of he original problem. Our mehod is independen of he naure of he original approximaion and can be used in combinaion wih various algorihms for approximaing opimal porfolio sraegies and value funcions. The algorihm can also be applied o verify ha an approximae sraegy is in fac close o he opimum in erms of expeced uiliy. This applies no only o problems ha can be poenially handled by radiional numerical mehods, bu also o large-scale problems, because he compuaional ime required by our simulaionbased mehod does no grow exponenially wih he dimension of he sae space or he number of risky asses. In our analysis, we consider problems wih he objecive defined over he erminal wealh. Exending our resuls o

12 416 Operaions Research 54(3), pp , 26 INFORMS Table 3. No shor sales and no borrowing. T = 5 T = 1 = 1 5 = 3 = 5 = 1 5 = 3 = 5 Parameer se 1 LB s UB s LB m UB m V u Parameer se 2 LB s UB s LB m UB m V u Parameer se 3 LB s UB s LB m UB m V u Parameer se 4 LB s UB s LB m UB m V u Noes. The four parameer ses are defined in Table 1. All resuls are compued for he iniial value of he sae variable X =. The rows marked LB s and LB m repor he esimaes of he expeced uiliy achieved by using he saic and myopic porfolio sraegies, respecively. The esimaes are based on 1,, independen simulaions. Expeced uiliy is repored as a coninuously compounded cerainy equivalen reurn. Approximae 95% confidence inervals are repored in parenheses. The rows marked UB s and UB m repor he analogous resuls for he upper bound on he rue value funcion compued according o he procedure described in 4. The row marked V u repors he opimal value funcion for he model wih incomplee markes and wihou he porfolio consrains.