Hedging Demands under Incomplete Information

Transcription

1 Hedging Demands under Incomplee Informaion Jorge F. Rodriguez Firs Draf: January 2002 This Version: Ocober 6, 2002 Absrac I presen a model of consumpion and porfolio choice under marke incompleeness and imperfec informaion regarding he invesmen opporuniy se. I solve analyically he consumpion and porfolio choice problem for an invesor learning abou he rue sae of he economy. When prices are he only observaions, he previously unspanned sae variables are spanned by he marke securiies under he opimal inference/learning process. The marke is observaionally complee for he imperfecly informed invesor. I show how learning affecs boh he covariance and he duraion componen of he hedging porfolio. I apply he model o he case where he Sharpe raio is mean revering. For he parameers presened in Wacher (2002), I show a reducion in hedging demands due o imperfec informaion. I solve in closed-form for he model implied R 2 for he reurn forecas regression. I discuss he relaionship beween he reducion in hedging demands and he reducion in he model implied R 2 for he reurn forecas regression. Ph.D. Candidae, Sloan School of Managemen, Massachuses Insiue of Technology. Corresponding Address: MIT Sloan School of Managemen; 50 Memorial Drive, E52-458; Cambridge, MA jfr@mi.edu. I hank my advisors Jonahan Lewellen and Seve Ross for heir guidance. This paper benefied from suggesions by John Campbell, George Chacko, Robin Greenwood, Jonahan Lewellen, Leonid Kogan, Savros Panageas, Anna Pavlova, Seve Ross, Dimiri Vayanos, Luis Viceira, and Joshua Whie. I graefully acknowledge he financial suppor of he General Elecric Faculy for he Fuure Fellowship. All errors are mine. Updaed versions of he paper are available in my websie: 1

2 1 Inroducion This paper sudies consumpion and porfolio choice when markes are incomplee and invesors canno observe variables which deermine he invesmen opporuniy se. I esablish condiions under which he invesor s opimizaion problem under incomplee markes can be ransformed ino a complee markes problem. Invesors use prices as noisy signals o infer he value of he unobservable sae variables. The esimaion process allows he agen o projec he dynamics of he unobserved variables ino he space of he securies. From he invesors poin of view markes are complee since he inferred processes for he sae variables are spanned by he marke securiies. This allows me o apply maringale mehods presened in Karazas, Lehoczky and Shreve (1987) and Cox and Huang (1989) and solve he consumpion and porfolio choice problem analyically. I apply he model o he case where excess expeced reurns on he risky asse are mean revering and unobservable. This seup is moivaed by he sandard assumpion ha excess reurns o risky asses are a funcion of he volailiy and he marke price of risk. Several empirical sudies, paricularly Meron (1980), have shown volailiy is easily esimaed. Under his assumpion we should care abou parameer uncerainy for he marke price of risk. Hence he problem boils down o uncerainy regarding he curren Sharpe raio. In his case, when he invesors inference does no reduce he esimaion error for he unobservable variable, closed-form soluions are obained. My resuls are novel in wo dimensions. Firs, I show how parameer uncerainy, a reasonable assumpion o make given he empirical evidence, can help us simplify he consumpion and porfolio choice problem. Second, I can analyically show he role of imperfec informaion in he duraion and covariance componen of hedging demands. I calibrae o he implied model of mean revering reurns in Wacher (2002) based on he analysis in Barberis (2000). I find imperfec informaion reduces he hedging demand duraion, he sensiiviy of hedging demand o changes in he sae variable, as well as he covariance componen of hedging demand. The reducion in boh componens is due o he variance of he esimaion error. For he calibraion, he variance of he esimaion error has a differen sign han he covariance beween he shocks o he sae variable and he risky asse. Therefore, he variance of esimaion error has a empering effec in he hedging demand of he invesor. I relae he changes in hedging demand o he model implied R 2 when he invesor accouns for incomplee informaion. I find imperfec informaion reduces he model implied R 2 for fuure reurns a any horizon. I find he reducion in he model implied 2

3 R 2 is also linked o he variance of he esimaion error. Evidence of predicabiliy in asse markes has revived he consumpion and porfolio choice lieraure. Recenly, economiss have focused on quanifying hedging demands due o changes in he invesmen opporuniy se. Meron (1971) derives he exisence of a hedging porfolio ha accouns for changes in variables deermining he araciveness of fuure invesmen opporuniies. A he ime, he empirical evidence was unable o rejec he hypohesis ha asse prices followed a random walk. Wihou ime varying reurns, i followed naurally ha porfolio choice should be enirely myopic and hus heir would be no hedging componen o he opimal asse allocaion policy. Poerba and Summers (1988), Campbell and Shiller (1988) and Fama and French (1989) find evidence of predicabiliy in he ime series of asse prices. Lewellen (2001a) shows mean reversion in sock reurn may be even sronger han previously perceived. He shows ha mean revering componen comprises more han 25% of sock reurns. Wih abundan evidence of ime-varying expeced reurns, Kim and Omberg (1996) sudy he role reurn predicabiliy on he opimal asse allocaion problem, finding closed form soluions for he hedging demands. Recenly, Brennan (1998), Brennan, Schwarz, and Lagnado (1997), Campbell and Viceira (2002), Chacko and Viceira (2001), Liu (2001), and Wacher (2002) exend his work in a variey of direcions. All of he papers menion above assume he curren value in he esimae of expeced reurns is observable. Given he amoun of evidence regarding predicabiliy in asse prices and he difficulies associaed wih deermining such predicabiliy, any reasonable normaive model of porfolio choice mus hence acknowledge a role for parameer uncerainy and incomplee informaion. Bawa and Klein (1976) and Bawa, Brown, and Klein (1979) sudy he role of uncerainy in asse allocaion. Kandel and Sambaugh (1996) exend he heory o consider uncerainy abou he predicabiliy in asse prices. They find ha he predicive relaion beween reurns and he dividend o price raio, alhough saisically weak, is economically significan even in he presence of esimaion risk. In oher words, invesors should accoun for predicabiliy in he porfolio decision, hence i would be subopimal for he invesor o inves under he assumpion of a random walk process for asse prices and ignore he role predicabiliy should playinasseallocaionevenwhenheevidenceof predicabiliy is saisically weak. The works of Deemple (1986), Dohan and Feldman (1986), and Gennoe (1986) lay he foundaion of he porfolio choice problem under incomplee informaion. They show ha he op- 3

4 imizaion problem where some parameers are unknown can be ransformed ino an opimizaion problem using he esimaes of he unknown parameers and he price and sae variable dynamics obained by he inference problem. In coninuous ime, porfolio choice under incomplee informaion can hen be solved in wo seps. Firs, unobservable parameers are esimaed by filering signals from he observable daa. Second, he invesor chooses opimal consumpion and porfolio policies given hese esimaes. My paper adops heir seup and considers he opimal porfolio sraegy when he curren values of he sae variables which define he invesmen opporuniy se are no observable. 1 relaed work, Barberis (2000) and Xia (2001), consider uncerainy regarding he relaion beween sock reurns and he sae variables. Unlike Barberis and Xia, I do no focus on he possibiliy ha asses migh no be predicable, insead, I focus on how uncerainy regarding he curren value predicive variable changes he composiion of he invesor s consumpion and porfolio choice. One inerpreaion of he model is ha business cycles, as seen by ime-varying expeced reurns, do occur in he economy, bu we are unable o pinpoin where he business cycle currenly sands. The assumpion of unobservable sae variables also proxies for he inabiliy of invesor o accuraely measure he effec of macroeconomics changes in he level of sock reurns. Secion 2 discusses he srucure of he economy and solves he opimizaion problem of he agen in a parially observable economy in a general seing. I provide a simple applicaion of he separaion heorem, he filering heory of Lipser and Shiryayev (2001), and he complee markes porfolio choice mehods of Cox and Huang (1989) as i applies o my model. In In Secion 3, I sudy sock price predicabiliy under he assumpion ha he insananeous Sharpe raio is no observable and solve for he opimal consumpion and porfolio policies. In Secion 4, I calibrae he model o he VAR(1) specificaion of Barberis (2000). 2 I compare my resuls o Wacher (2002) where he invesor assumes complee markes and show incomplee informaion has a srong effec in he porfolio choice of he agen. In Secion 5, in he conex of he example considered in 1 Recen aricles in operaions research address some of he issues raised in his paper. Lakner (1995,1998), Karazas and Zhao (2001), and Rishel (1999) sudy he asse allocaion problem under incomplee informaion. These papers do no consider he consumpion aspec of an invesor s sraegic asse allocaion problem. 2 Campbell, Chacko, Rodriguez, and Viceira (2002) show, in he conex of a consumpion and porfolio choice model, how o correcly relae he discree-ime model of ime-varying expeced reurns by Campbell and Viceira (2002) o he coninuous-ime models in order o obain he correc parameer values for he coninuous-ime model. 4

5 Secion 3, I simulae how an invesor, wih a given prior variance for he esimaion error of he unobserved variable, learns abou he variable and how he variance of he esimaion error changes wih each new observaion. I show, given he amoun of daa available o he invesor, changes in he variance of he esimaion error are negligible, such ha assuming seady sae in he inference process is no as srong an assumpion as migh be iniially expeced. 3 Secion 6 derives he model implied R 2 and heir link o he observed reducion in hedging demand. Secion 7 concludes and offers a variey of exensions for he mehodology presened in his paper including exensions for oher asse allocaion models and derivaive replicaion sraegies under imperfec informaion. 2 The Model I develop a model of consumpion and porfolio choice when markes are incomplee and heir is uncerainy regarding he curren value of he sae variables. As shown by Meron (1971), sae variables deermine he invesmen opporuniy se faced by he invesor and he opimal porfolio policy conains a componen o hedge he risks associaed wih hose changes. I assume he invesor canno accuraely forecas he curren value of hose variables, bu has informaion o form an esimae of he value. Once he invesor deermines he forecas of he sae variables and he esimaion error, he marke is complee under he informaion se of he invesor. Marke compleeness under he subjecive measure of he invesor allows us o apply maringale mehods and obain analyical, exac soluions o he consumpion and porfolio choice problem. Consider a finie horizon invesor wih horizon T. Assume he exisence of a single consumpion good and assume he consumpion good is he numeraire. Uncerainy is represened by a probabiliy space (Ω, F, P) on which we define a d Z dimensional orhogonal Brownian Moion Z and a d W -dimensional orhogonal Brownian Moion W. Le F denoe he filraion generaed by he Brownian Moions (Z, W). Assume he filraion is righ-coninuous and he probabiliy space is complee. Assume he exisence of a d Z dimensional orhogonal Brownian Moion Z and a d W - dimensional orhogonal Brownian Moion W on he probabiliy space such ha F is he sandard filraion generaed by Z and W. The Brownian Moions Z and W are assumed o be orhogonal 3 Even under seady sae inference, he invesor does no observe he unobservable variable because under seady sae inference he variance of he esimaion error is posiive. Even in he seady sae he invesor is no able o precisely esimae he unobserved variable. 5

6 o each oher. For all Io processes in his paper assume all drif coefficiens are defined in L 1 and all diffusion erm coefficiens are defined in L Securiies Marke and Sae Variables The securiies marke consiss of a riskless asse, he money marke accoun, which pays he locally riskless raes a all imes, and N risky securiies which span Z, he Brownian moion relaed o shocks in asse prices. The money marke accoun grows a he riskless rae of reurn. The price of money marke accoun saisfies where r is he locally riskless rae of reurn. db = r B d, (1) The prices for he risky securiies follow he mulidimensional Io process ds = diag (S ) µ {z } {z} S d + σ {z} S dz, (2) N N N 1 N d Z where µ S L 1 N and σs L 2 N d Z. Assume he dimension of he Z is equal o he rank of σ S almos surely. The drif componen represens he insananeous expeced reurn for he asse, while he diffusion is defined as he volailiy of he asse. Changes in he invesmen opporuniy se of he agen are represened by a vecor X of sae variables. The sae variables saisfy he following mulidimensional Io process: where µ X L 1 S, σx L 2 S d Z,andσ W L 2 S d W dx = µ {z} X d + σ X dz {z} + σ W dw {z}, (3) S 1 S d Z S d W and he Brownian moion vecors Z and W are orhogonal. The marke is incomplee as long as he dimension of W is greaer han zero. Some of he sae variables migh no be observable. I will assume ha he number of unobservable parameers is equal o he difference beween he oal number of shocks and he number of shocks spanned by marke securiies. In oher words, he rank of σ W is equal o d W. 4 Assume he following definiion for he ses described in he paper hold: n L 1 = X L : R o T X 0 d < a.s., n L 2 = X L : R o T 0 X2 d < a.s.. 6

7 2.0.2 Invesors Preferences and Budge Consrain The invesor s preferences are assumed o saisfy he sandard consan relaive risk aversion, power uiliy funcion: u (C )=e φ C1 d, (4) 1 where is he coefficien of relaive risk aversion and φ is he agen s discoun rae. Denoe α as he vecor of porfolio weigh for he invesor s opimal invesmen sraegy in he risky asses. The invesors budge saisfies: dw = W r + α 0 (µ S r ι) d + α 0 σ S dz ª C d (5) and he invesor is subjec o a non-negaive wealh consrain. 2.1 Soluion for he Model This secion presens he soluion for he invesor s opimizaion problem. The agen opimizaion problem is o maximize (4) subjec o (5) and he non-negaive wealh consrain under he filered processes. Similar o Deemple (1986), Dohan and Feldman (1986), and Gennoe (1986), he invesor s consumpion and porfolio choice problem follows wo seps: (1) an inference problem in which he invesor updaes his or her esimae of he unobservable sae variables, (2) an opimizaion problem in which he invesor chooses her opimal consumpion and porfolio policies under he new esimae for he unobservable sae variables.in his secion I solve he invesor s inference problem and opimizaion problem. A second soluion mehod is provided in he Appendix Inference Problem Assume he drifs of he sock price processes in (2) is given by and he drifs of he sae variables processes in (3) saisfy µ S = β {z} 0 + β X X {z}, (6) N 1 N S µ X = a {z} 0 + a X X {z}. (7) S 1 S S 7

8 Equaions (6) and (7) represen an economy where reurns are ime-varying. Equaion (6) assume a linear relaion modeled beween expeced reurns and he predicive variable. Since some of he sae variables are no observable, he insananeous expeced reurn is no direcly observable. One inerpreaion of he model is o hink of he asses in his economy as being eiher good or bad.invesmens, depending on wheher heir curren expeced reurn is above or below heir long run expeced reurn, bu he invesor canno deermine exacly he curren expeced reurn of he asses. The inference problem is solved wih filering mehods covered in Lipser & Shiryayev (2001). I follow heir reamen as i applies o our model. Assume he invesor observes insananeous reurns o he money marke accoun (1) and he equiy (2). Assume he invesor also knows σ S,σ X,σ Y,β 0,β X,a 0,a 1. However he invesor does no observe he curren sae of X.In oher words, prices are he only signals invesors have regarding he invesmen opporuniy se. If he invesor commis o high-frequency rading, prices serve as he naural choice for informaion regarding he invesmen opporuniy se. 5 Le X 0 be he invesor s prior, such ha X 0 N bx0,v 0,wherev 0 represens he invesors prior variance-covariance marix for he sae variables. In erms of he filering lieraure, equaions (1) and (2) are he observaion equaions and (3) are he sysem equaions. The filering heory for coninuous ime developed by Lipser and Shiryayev, allows us o describe he dynamics of he mean and he variance of he disribuion of he unobservable sochasic process X. The insananeous changes in he drif and he variance-covariance marix of X are given by: d b X = i ha 0 + a X X b d + σ X σ 0 S + v β 0 X σs σ 0 1 S hdiag S 1 i ds β 0 + β XX b d (8), dv d = a X v + v a 0 X + σ X σ 0 X + σ W σ 0 W σ X σ 0 S + v β 0 X σs σ 0 S 1 σx σ 0 S + v β X 0.(9) where X b is he invesor s esimae of he unobservable sae variable and v represens he variance of he esimaion error for he unobservable sae variable a ime. When he agen has incomplee informaion, he agen s porfolio hedging demand needs o accoun for he unobserved sae variables, bu also for he reducion in variance he esimaion error as new observaions come abou. I assume inference has reached a seady sae. In oher 5 An excepion o he low frequency issues wih predicive variable is rading volume. Recenly Cremers (2002), considers he role of rading volume as a predicive variable. 8

9 words, he variance of he disribuion for he esimaed parameer does no change wih each new observaion. Thus dv =0,andv does no need o be considered a sae variable in he consumpion and porfolio choice problem. Denoe he seady sae variance marix as v ss. From he definiion of seady sae variance and equaion (9), v ss is a posiive definie marix such ha 0=a X v ss + v ss a 0 X + σ X σ 0 X + σ W σ 0 W σ X σ 0 S + v ss β 0 X σs σ 0 S 1 σx σ 0 S + v ss β X 0. In Secion 5, I discuss he meris of he seady sae learning assumpion and show ha wih a reasonable amoun of daa, he variance of he esimaion error is very close o he variance implied by he seady sae resuls. The new innovaion process, defined as he normalized deviaion of he reurn from is condiional esimaed mean is given by σ S d b Z = h µ S β 0 + β XX i b d + σ S dz (10) Alhough Z is no observable, he innovaion process b Z is derived from observable processes and is hus observable. The process (10) implies ha he risky securiies reurns (2) are observable under he form ds = S h β 0 + β X b X d + σ S d b Z i (11) The dynamics for he sae variables also become observable under he new innovaion process. The sae variables dynamics are given by he equaion d b X = i h i ha 0 + a X X b d + σ X σ 0 S + v ss β 0 X σ S σ 0 S 1 σs dz b (12) As long as he securiies span he rank of Z, b he invesor s own sae price densiy is uniquely defined. I is his resul which will allow us o ackle he opimizaion problem wih maringale mehods. The assumpion of seady sae variance allows us o reduce he sae variable space considerably and in some cases solve he opimizaion problem in closed form. The assumpion of seady sae variance formalizes he decision no o have he variance of he esimaion error as a sae variable. 6 6 Barberis (2000) also reduces he sae space by assuming he invesor s learning does no reduce he variance of he esimaion error once he invesor sars invesing. 9

10 2.1.2 Opimizaion Problem As far as he invesor is concerned, he sochasic changes o he price and he sae variable are perfecly correlaed because he price serves as he signal of he sae variable. Afer filering he unobservable processes and assuming he variance of he esimaes of he sae variables reaches is seady sae, he securiies span he number of observable Brownian Moions. The invesor has a uniquely deermined sochasic discoun facor, herefore I can apply maringale mehods developed in Cox-Huang (1989) o solve for he agen s opimal consumpion and porfolio choice. The agen assumes he prices of he money marke accoun and he risky securiies are given by he equaions db = B [r d], (13) h ds = S bµ S d + σ S dz b i, (14) where bµ S is chosen o mach equaion (11). Also, he invesor assumes he sae variables saisfy he following equaion where bµ X and bσ X are chosen o mach equaion (12). d b X = bµ X d + bσ X d b Z, (15) Alhough he marke is incomplee, a unique sochasic discoun facor can be defined for he invesor. Under he informaion se of he invesor, he sae variables are spanned by he securiies, herefore he invesor can define a sochasic discoun facor. Le M be he sochasic discoun facor, he process for M mus saisfy he following condiion: M B and M S are maringales. I define a sochasic discoun facor ha saisfies he maringale properies under he informaion se of he invesor. Denoe by c M he sochasic discoun facor under he invesor s informaion se such ha c M B and c M S are maringales. Assuming ha c M follows an Io process, an applicaion of Io s Lemma given (13) and (14) yields he following process for he sochasic discoun facor: d c M cm = r d bη d b Z, c M0 =1 (16) where bη = σ 0 Sσ S 1 σ 0S (bµ S r ι) and bη saisfies Novikov s Condiion and ι represens a vecor of ones. Noe bη is he invesor s esimae of he Sharpe raio and a affine funcion of b X. Equaion (16) can be solved o obain c M 10

11 in is exponenial form: ½ cm =exp Z 0 r s ds Z 0 bη s d b Z s 1 2 Z 0 ¾ kbη s k 2 ds. Similar o he incomplee markes consumpion and porfolio choice model of He and Pearson (1991), he invesor in my model is able o deermine a unique sochasic discoun facor, bu unlike He and Pearson, he sochasic discoun facor for he invesor is sraighforward o obain and does no require he use of a dual problem. Alhough he invesor has a unique sochasic discoun facor, his does no imply i is he unique discoun facor for he economy. Basak (2000) sudies a dynamic equilibrium model of heerogeneous beliefs and finds individual-specific Arrow- Debreu prices can differ. Similar o Basak, he invesor has a uniquely define sochasic discoun facor based on heir beliefs on cerain parameers in he economy. Therefore even when markes are incomplee, he consumpion and porfolio choice problem of he invesor can be solved wih maringale mehods. Le he superscrip I denoe operaions aken under he informaion se of he invesor. Given he process governing he dynamics of he sochasic discoun facor, he agen s opimizaion problem can be solved wih maringale mehods. As saed previously in his secion, he agen s opimizaion problem is o maximize he expeced lifeime uiliy of consumpion J where " Z # T J = sup E I φ(s ) C1 s e {α s, C s } 1 ds (17) subjec o he dynamic budge consrain under he esimaed processes for he securiies and a non-negaive wealh consrain.the exisence of he sochasic discoun facor allows us o wrie he agen s dynamic budge consrain as a saic budge consrain given by Z T cm s C s ds W 0. (18) E I s where he expecaion is defined under he invesor s informaion se as represened by he resuls of he inference process described previously. Equaion (18) saes he agen s expeced consumpion sream in he fuure appropriaely discouned will be less han or equal o his curren wealh. The invesor s problem can now be solved as a saic opimizaion problem as described in Cox and Huang (1989) and Karazas and Shreve (1998, Chaper 3). Inuiively, since he sochasic discoun facor is well defined for he invesor, he invesor can dynamically rade he long-lived 11

12 securiies o obain he opimal consumpion profile in a manner similar o ha of an invesor wih access o complee markes. Thus, here is no uncerainy regarding he consumpion and porfolio choice of he agen condiional of knowing he sae, he only uncerainy ha remains is he realizaion of a given sae. The invesor s porfolio allocaion changes accordingly wih he opimal consumpion choice. The firs order condiion for uiliy maximizaion under he budge consrain is given by C s = λ e φ(s ) cm s 1 cm, (19) where λ is he Lagrangian muliplier. Noice also λ 1 represens he choice of consumpion a ime given he informaion of he agen a ime. Subsiuing he firs order condiion for consumpion (19) ino he saic budge consrain (18) yields he following expression for he saic budge consrain: W = E I Z T cm s λ Ms c cm 1 cm e φ (s ) ds. (20) Equaion (20) saes ha wealh is a funcion of he sochasic discoun facor and he processes ha drive he disribuion of he sochasic discoun facor. As shown in (16), he only processes ha maer for he disribuion of he sochasic discoun facor are he ineres rae and he Sharpe raio.i assume he sharpe raio is a funcion of he sae variable vecor b X, herefore he curren values for boh he sochasic discoun facor and he esimaed sae variable deermine he informaion se he agen uses in forming condiional expecaions. 7 Thewealhaimes> following he opimal policy is given by Z T W s = Es I cm u C u du cm s = s λ e φ(s ) c Ms 1 cm E I s Z T e φ (u s) Mu c s 1 1 cm s du (21) Define he funcion F s,u as follows F s,u = E I s cmu 1 cm s 1 (22) F s,u is he agen s expecaion of how he invesmen opporuniy se is going o look like a ime u, wih a weighing funcion relaed o he risk aversion of he invesor. Noice ha for =1, 7 This is due o he fac ha boh M and bx are Markov processes. 12

13 he expecaion yields a consan, regardless of he value of he fracion.. As will be shown in he porfolio choice of he invesor, F s,u is direcly linked o he hedging demand componen of he invesor s porfolio. Define G s,t as G s,t = Z T s e φ (u s) F s,u du (23) such ha G s, is also funcion of b Xs. G s, weighs he invesor reacion o changes in he fuure invesmen opporuniy se by he invesor s impaience and risk aversion. If he invesor discouns fuure uiliy heavily, hen he invesor assigns greaer weigh o he shor-erm fuure invesmen opporuniy se. Highly risk averse invesors will penalize longer horizon opporuniy se changes less since hey care abou mainaining a low variance in heir consumpion. Le Q denoe he invesor specific risk-neural measure. The wealh of he invesor under he risk-neural measure is given by W s = λ e φ(s ) c Ms 1 cm Z T e φ (u s) 1 1 u (Rs r vdv) E I,Q cmu s s cm s Under he invesor specific risk-neural measure, he rae of reurn o wealh is equal o he insananeous rae of reurn for he money marke accoun, hus he drif of he wealh process, as obained by he applying Io s Lemma, mus equal he locally riskless rae imes he agen s curren wealh. The equaion above implies he following parial differenial equaion is solved by 1 du he agen s wealh funcion: r s W s = λ e φ(s ) c Ms 1 cm + W s s " W s 2 M c cm 2 s 2 s bη 0 sbη s r bσ Xs bσ 0 Xs W µ s M c Ms c r s + bη0 sbη s s 2 # 2 W s b X 2 s + W s b X s a Q 0s + aq Xs b X s 2 W s c M s b X s c Ms bσ Xs bη s (24) where a Q.s are he coefficiens of he drif for he sae variables under he invesor specific risk neural measure and r (.) is he race funcion. Equaion (24) is solved by W s = λ e φ(s ) c Ms where he boundary condiion for (24) is given by 1 cm G s,t. (25) G s,s =1 (26) 13

14 A funcional form o he invesor s wealh can be obained by aemping o solve (24) or alernaively, equaion (25) can be simplified furhermore as shown in he Appendix. One can also use he inuiion in Wacher (2002) o solve he consumpion and porfolio problem by considering he porfolio problem for each period separaely and scaling he soluion wih he Lagrange muliplier obained from he firs order condiion for consumpion. 2.2 Porfolio Choice Since he invesor beliefs he markes are complee, I follow Cox-Huang (1989) o find he opimal asse allocaion sraegy for he invesor. In complee markes, he porfolio allocaion has o be such ha he magniude and direcion sochasic changes in he wealh process are hedged by he porfolio allocaion. The invesor s percenage allocaion of wealh o he risky asses is given by α 0 M = c W W M c σs σ 0 1 S σs bη {z } myopic demand + 1 W σs σ 0 1 S σs bσ 0 X. (27) W bx {z } hedging demand The porfolio choice of he agen can be decomposed ino is myopic demand, he demand due o he curren sae of he economy, and he hedging demand, he demand due o expeced changes in he invesmen opporuniy se. In he model, he hedging demand is due o he sochasic naure of he esimaed sae variables. Boh he myopic and hedging componens are subjec o he esimaion risk due o he unobserved sae variables. The myopic demand of he agen is affeced by he esimaed sae variables by how hose esimaes change he invesors percepion of he curren invesmen opporuniy se as proxied by he Sharpe raio. The hedging demand of he agen is affeced by he invesor s percepion of he diffusion for he esimaed sae variables (how he invesmen opporuniy se changes wih ime) as well as by he curren esimae of he unobserved sae variables. Given (27) and he second boundary condiion in (25), wrie he invesors porfolio as α 0 = 1 σs σ 0 1 S σs bη + 1 G G X b σs σ 0 1 S σs bσ 0 X (28) The funcion G which deermines he magniude of he hedging demand is he agen s curren wealh o consumpion raio. As in he complee markes framework, he relaion beween he agen s curren consumpion relaive o expeced fuure consumpion is relaed o how he agen 14

15 deermines o hedge changes in he invesmen opporuniy se. Alhough our model is one of incomplee markes, invesors, via he inference problem, are able o obain individual-specific Arrow-Debreu prices, herefore heir behavior maps o ha of a complee markes invesor. Wrie he hedging demand componen of he agen s porfolio choice as α hedging = G X G σs σ 0 1 S σs bσ 0 X (29) The magniude of he hedging demand is given by he sensiiviy of he wealh o consumpion raio o he sae variables. The duraion of he hedging componen will change relaive o he perfec informaion case due o he difference beween he esimae and he rue value of he variable and because of he esimaion error. The hedging demand also depends on he perceived covariance beween he sae variables and he sock prices since σ S bσ 0 X = σ X σ 0 S + v ss β 0 0 X he covariance componen of he hedging demand will also change due o he variance of he esimaion error. The value funcion can be used o obain he opimal consumpion policy o obain he opimal porfolio policy of he agen. Define J as he indirec uiliy funcion, he indirec uiliy funcion obained via he opimal consumpion and porfolio policy solves " Z # T J = sup E I φ(s ) C1 s e {C,α } 1 ds I subsiue consumpion in (30) by he firs order condiion (19) o obain From (20), J = 1 1 EI = λ1 1 1 = λ1 1 Z T 1 G,T hen he value funcion can be saed as Z T 1 e φ(s ) λ e φ(s ) cm s ds cm e φ (s ) E I λ 1 J = G,T cms 1 cm = W G,T W ds (30) 15

16 where G,T is defined as in he previous secion. The opimal porfolio allocaion o risky asses is given by J W α = σs σ 0 1 S σs bη W J J WX σs σ 0 1 S σs bσ 0 WW W J X WW = 1 σs σ 0 1 S σs bη + 1 G,T G,T X b σs σ 0 1 S σs bσ 0 X where, by Leibniz s Rule, Z G T,T X b = e φ (s ) F,s X b ds. (31) Since F,s is a funcion of he raio of he sae price densiy a ime s relaive o he sae price densiy a ime, equaion (31) formalizes he relaion beween he hedging demand and changes in he invesmen opporuniy se. Also, noice he hedging demand is a weighed funcion of he expeced changes in he invesmen opporuniy se for all horizons up o reiremen. weighing funcion is relaed negaively o he invesor s impaience and posiively o is relaive risk aversion. Therefore, he more impaien invesors care more abou hedging demand in a shorer horizon, while more risk averse invesor care abou longer horizon consumpion needs. The assumpion seady sae variance is no necessary o obain an expression for he opimal porfolio policy since he Cox-Huang mehodology would sill apply even if he diffusion componen of he sae variable decays deerminisically. In hose cases where an analyical soluion does no obain, he invesor could use he Mone Carlo mehods of Deemple, Garcia, and Rindisbacher (2003) or Cvianic, Goukasian, and Zapaero (2002) o obain a numerical soluion. Boh mehods require marke compleeness which is saisfied under he informaion se of he invesor. The 2.3 Consumpion o Wealh Raio The consumpion o wealh raio is easily obain by applying some algebra o equaions (19) and (25) or C W = λ 1 λ 1 G,T, (32) C W = G 1,T. (33) 16

17 An applicaion of equaion (33) o he porfolio hedging demand (29) yields he following expression for he invesor s hedging demand: α hedging = C W C W X b σs σ 0 1 S σs bσ 0 X. (34) Equaion (34) shows he link beween fuure expeced consumpion and he hedging sraegy of he invesor. When markes are complee, he invesor essenially can plan he consumpion sraegy for each possible oucome a each possible horizon, equaion (34) shows how he invesor changes he porfolio sraegy o mainain he desired consumpion plan. 3 Porfolio Choice wih Unobservable Time-Varying Expeced Reurns A useful example of he srengh of our echnique is o analyze he consumpion and porfolio choice problem when he Sharpe raio is mean revering. Liu (2001) and Wacher (2002) finds a closed-form soluions o he consumpion and porfolio choice where he predicive variable is fully observable and markes are complee. In order o solve he model, Wacher assumes he marke is complee and he shocks o he proxy for he predicive variable and he sock price are perfecly negaively correlaed. The assumpion of perfec negaive correlaion does no seem conroversial given ha he empirically esimaed correlaion for he shocks o he dividend price raio and he sock price is Accouning for parameer uncerainy grealy decreases he demand of he risky asse due o hedging for changes in he invesmen opporuniy se. In his secion of he paper, I exend Wacher o accoun for incomplee informaion in he agen s opimizaion problem. Unlike Wacher, I will no assume marke compleeness. Insead, I assume uncerainy regarding he curren value of he predicive process. Noe he assumpion regarding seady-sae esimaion does no allow us o sudy he role of he variance of he esimaion error for he unobservable parameers as a sae variable in he agen s policies. 8 In his model he predicive relaion is known, since he predicive relaion in our model is given by he sandard deviaion of he risky asse. 8 In a relaed paper, Lewellen and Shanken (2002) sudy he equilibrium effecs of learning on asse prices. They find mean reversion in asse prices can be explained by he learning of he agens regarding he dividend process. Xia (2001) solves a similar model where learning plays a role in he hedging demand of he invesor. 17

18 Assume he exisence of a money marke accoun where he risk free rae is consan and he exisence of one risky securiies whose price process saisfies ds =(r + σ s η S ) d + σ s dz S, (35) such ha he Sharpe raio, η, is mean revering, and saisfies dη = κ (θ η ) d + σ η dz x. (36) Assume he correlaion beween shocks o he sock price and shocks o he Sharpe raio are imperfecly correlaed. The correlaion coefficien is denoed by ρ. The imperfec correlaion beween (35) and (36) implies he marke is incomplee. Ye, when he Sharpe raio is no observable and under assumpions explained in Secion 2.1.2, he opimizaion problem can be resaed in a complee markes framework. 3.1 Inference Problem I apply he filering mehods of Lipser and Shiryayev (2001) o find a observaionally equivalen economy under he subjecive measure of he invesor. Applying he resuls of secion o he curren problem yields he following processes for he sock price and he sae variable dynamics respecively: ds S = (r + σ S bη ) d + σ S dz b S, (37) dbη = κ (θ bη ) d + ε η dz b S, (38) where and ε η = v ss + ρσ η d b Z S =[(η bη ) d + dz S ]. (39) The measuremen error (variance) of he Sharpe raio solves he following Riccai Equaion dv d = 2κv + σ 2 η [v + ρσ η ] 2. (40) Equaion (40) can be solved following he appendix of Deemple (1986). 18

19 Following he mehodology presened in Secion 2, when compuing for he opimal consumpion and porfolio policies, assume learning has reached a seady sae in which new daa and esimaion does no reduce he measuremen error of he Sharpe raio. 9 Le v ss denoe he variance of he esimaion error under he seady sae. 10 By applying he definiion of seady sae filering o (40), v ss is deermined by he quadraic equaion 0= 2κv ss + σ 2 η [v ss + ρσ η ] 2. (41) The resuling variance will be he posiive roo of he quadraic equaion obained from our assumpion in (40). If wo posiive roos are obained, I sudy boh cases: he high-prior equilibrium and he low-prior equilibrium. 3.2 Consumpion and Porfolio Choice Afer he invesor solves he inference problem and esimaes he Sharpe raio, he esimaed processes for he sock price and he Sharpe raio are perfecly correlaed. The invesor sees his processes as perfecly correlaed because he inference problem essenially projecs he unobservable variable (he Sharpe raio) ino he space of he signal (he sock price), hus he source of uncerainy for boh processes afer he inference is he same. As seen in (38) he rue correlaion is accouned for in he diffusion coefficien for he esimaed Sharpe raio. In his secion I show he main seps and resuls of he consumpion and porfolio problem. The deails of he derivaion are provided in he Appendix. I derive he agen s porfolio choice by applying (28) o he model. Le α be he proporion of wealh allocaed o he risky asse. The porfolio choice of he agen can be decomposed ino is myopic and hedging componen. where α = α myopic α myopic + α hedging (42) = 1 bη σ S, (43) 9 A closed form soluion is obainable for equaion (40). Please refer o Deemple (1986, Appendix) for deails. 10 Barberis (2000) also reduces he sae space by no considering variance of esimaion error, bu he does no assume his occurs due o a seady-sae in he equaion deermining he variance of he esimaes. In he Barberis model, seady sae learning occurs when parameer uncerainy disappears. My seup allows for he separaion of parameer uncerainy and learning abou he variance of he esimaion error. 19

20 and α hedging = ε R T η (B (s )+C (s ) bη ) H,s ds R T (44) σ S H,s ds The hedging demand of he invesor has he usual properies found for hedging demand in he presence of excess reurn predicabiliy. The Sharpe raio does no only come ino play in he assigning of relaive weigh for he hedging demand via he funcion H, he wealh o consumpion raio, i also comes ino play linearly as a measure of marke iming. As was shown in (29), when he soluion o H is of he exponenial form, he sensiiviy of he log wealh o consumpion raio o he sae variable deermines he relaive weigh each period in he agen s horizon has on he hedging sraegy. The consumpion o wealh raio for he agen is given by µz C T 1 = H,s ds. W The duraion or sensiiviy of he wealh o consumpion raio relaive o changes in he invesmen opporuniy se is given by C W R T C (B (s )+C (s ) bη = ) H,s ds W bη R, (45) T H,s ds as shown generally in Secion 2.3, (45) esablishes he relaionship beween he agen s hedging demand and he sensiiviy of he agen s consumpion and savings decision o changes in he invesmen opporuniy se. This relaion is sraighforward due o marke compleeness under he filered processes and he inexricable link beween he agen s hedging demands and he expeced consumpion in he fuure. 4 Calibraion and Resuls Campbell and Viceira (1999) sudy opimal consumpion and porfolio choice when expeced reurns are mean revering. They assume he riskless rae of reurn is consan and he log excess reurn for socks is given by he following VAR(1) specificaion: log S n + = r f + x n + ε n +, (46) x n+ = (1 φ) µ + φx n + η n +. 20

21 Campbell and Viceira use he dividend o price raio as a proxy for changes in he invesmen opporuniy se. They derive parameers for (46) from quarerly daa. I adop he resuls from Wacher (2002), which give he monhly parameers for he models by Campbell and Viceira (1999) and Barberis (2000). As explained previously, I mainain imperfec correlaion beween he sae variable and sock reurns and se he correlaion o Table II consider he porfolio choice of he invesor wih incomplee informaion under various assumpions for he curren esimae of he Sharpe raio. The myopic and he hedging demand of he invesor seems o increase monoonically wih increases in he Sharpe raio. Ye, he percenage of he porfolio dedicaed o hedging changes in he invesmen opporuniy se decreases monoonically wih increases in he Sharpe raio. The resul is highly inuiive: When he invesor esimaes a low value for he Sharpe raio, he invesor is more willing o ime he marke because he expecs he reurns o be higher in he fuure due o he mean reversion in he parameer. This effec is also sronger when he invesmen horizon is longer.as η increases so does he myopic and hedging demand of he agen, bu here is a reducion in he amoun of he porfolio allocaed o hedging changes in he invesmen opporuniy se. For each panel, he hedging demand of he invesor increases wih he ime horizon and decreases wih respec o relaive risk aversion. Ye, he percenage of wealh held in he risky asse due o hedging demand increases wih boh he ime horizon and risk aversion. This resul implies more risk averse invesor reduce heir exposure o risky asses, bu increase he amoun of he exposure ha is due o changes in he invesmen opporuniy se. Compared o Campbell and Viceira (1999) and Wacher (2002), hedging demands for invesors considering he role of parameer uncerainy are lower. Table III compares he consumpion and porfolio sraegy of an invesor which esimaes he curren Sharpe raio and compues his or her sraegy according o he mehods in his paper agains an invesor wih perfec informaion abou he economy under he assumpion of perfec negaive correlaion beween sock reurns and he predicive process, he mean-revering Sharpe raio. The second invesor ype corresponds o he model presened in Wacher (2002). In Table III I assume he Sharpe raio, as esimaed by he firs invesor and observed by he second invesor, is he long run value. The comparison in Table III allows us o concenrae on he role of parameer uncerainy in he hedging demand of he invesor and does no accoun for he possibiliy of furher differences in he consumpion and he porfolio sraegies of boh invesors due o he 21

22 incomplee informaion srucure. In oher words, I do no accoun for furher differences due o differences in each agen s belief of he curren value of he Sharpe raio. As expeced, he differences in he proporion of wealh allocaed o sock in boh examples is due o differences in he hedging demand. The hedging demand for he invesor wih incomplee informaion is much lower. Inhecasewhereheinvesorhasacoefficien of relaive risk aversion of 5 and a 30-year invesmen horizon, he difference in he percenage of wealh allocaed o he risky asse is 24%. As expeced, he percenage of he porfolio dedicaed o hedging demands is lower when accouning for parameer uncerainy. A major driver in he difference in his resuls is he perceived volailiy of he sae variable. For he calibraion he sandard deviaion of he predicive variable is ε η = v ss + ρσ η (47) = ( 0.93) = As expeced, his is smaller han he sandard deviaion ha would be obained under he assumpion of complee markes and full informaion. The reducion of he volailiy of he predicive variable is he main reason why hedging demands under parameer uncerainy are considerably smaller han hose obained under he complee markes assumpion. Alhough in our calibraion parameer uncerainy reduces he magniude of he hedging demand, i is possible ha under a differen calibraion he hedging demand of he invesor would be high in magniude. The invesor akes a negaive posiion in he risky asse due o he measuremen error. This can be clearly seen by subsiuing (47) in he hedging demand (44): α hedging = ε η C W C σ S W = v ss C σ S W bη W C bη {z } <0 + ρσ η C W σ S W C bη {z } >0 Had he negaive hedging componen due o esimaion error dominaed he posiive hedging componen due o changes in he invesmen opporuniy se, he resuls could have been differen. A second lever in which he perceived volailiy deermines he hedging demand is in he value of he funcions B (.) and C (.). Under he complee markes assumpion, Wacher (2002) shows B (.) < 0 22

23 and C (.) < 0. Under incomplee informaion he perceived covariance is smaller, bu he funcions B (.) and C (.) are sill negaive. I follows ha anoher possible reason for he reducion in he hedging demand of he invesor is he reducion in he duraion of he hedging demand as measure by he sensiiviy of he consumpion o wealh raio o changes in he sae variable. In Table IV, I compare changes in hedging demand due o changes in he duraion measure and changes in he perceived volailiy of he sae variable. I do he comparison via he following decomposiion α hedging,cm α hedging = σ η Dur CM Dur σ S {z } Duraion Effec ε η Dur σ S σ {z S } Covariance Effec µ ση + where he superscrip CM represens he complee markes invesor assuming perfec negaive correlaion beween he sae variable and sock reurns and Dur CM and Dur is given by (48) Dur CM = Dur = R T R T B CM (s )+C CM (s ) η H CM,s ds R T H,s CMds, (B (s )+C (s ) bη ) H,s ds R. T H,s ds The soluions o he funcions A CM (.),B CM (.), and C CM (.) are provided in Wacher (2002). To concenrae on he role of he perceived volailiy in he hedging demand, I assume he curren Sharpe raio and he esimae of he second invesor for he Sharpe raio are equal. I analyze equaion (48) in Table IV for various assumpions of he curren Sharpe raio, relaive risk aversion, and invesmen horizon.the covariance effec seems o dominae he change in he hedging demand for shor-horizon invesors. 11 In general, for he longer-horizon invesors, i is he duraion effec ha carries he mos weigh in he change in hedging demands. For he low risk aversion invesor, hecovarianceeffec dominaes. As risk aversion increases, he covariance effec is reduced and he duraion effec increases. Using he decomposiion of he porfolio demand ino is marke iming and non-marke iming componens allows he comparison of he marke iming aggressiveness implied by he Wacher 11 Barberis (2000) also menions he reducion in sensiiviy o he sae variable in he porfolio demand when parameer uncerainy is aken ino accoun. Ye, his model does no allow for he separae analysis of he reducion in he hedging demand as i perains o changes in he preceived covariance and changes in he duraion, he sensiiviy of hedging demands o changes in he sae variable. 23

24 model relaive o my model. The agen s porfolio in erms of is iming componen and is noniming componen are R T α NMT θ = + ε η (B (s )+C (s ) θ) H,s ds σ S σ S R, (49) T H,s ds " α MT 1 = + ε R T # η C (s ) H,s ds σ S σ S R (bη T θ). (50) H,s ds Similar o Campbell and Viceira (1999) equains (49) is he inercep of he porfolio demand and he coefficien in (50) is he slope of he porfolio. Le he coefficien of he marke iming componen divided by he non-marke iming componen be he measure of marke iming aggressiveness for he invesor. For various assumpions of he degree of relaive risk aversion and reiremen horizon, Table V shows he marke iming and non-marke iming componen of he wo models when he expeced reurn is equal o he long-run reurn. The able also shows he measure of marke iming aggressiveness. As expeced from he previous discussion abou he hedging demand differences, Table V shows a reducion in he slope and inercep of he porfolio allocaion when parameer uncerainy is considered. Our measure of marke iming aggressiveness shows he parameer uncerainy invesor o have a higher slope (when normalized o is non-marke iming demand) han he complee markes invesors. The resuls follows from he subjecive disribuion aribued o he predicive variable by he invesor accouning for parameer uncerainy. The invesor has a more precise signal han he complee markes invesor due o how he variance of he esimaion error reduces he implied variance of he sae variable. Therefore, he invesor is willing o marke ime more aggressively han an invesor who does no consider he role of parameer uncerainy. 5 The Longeviy of Learning A crucial assumpion made o obain closed-form soluion o he invesors consumpion and invesmen problem is ha learning has reached a seady-sae process. In oher words, any new observaion of he securiies will be accouned for by he agen is his new esimaes of he unobservable parameers, bu he new observaion will no conribue in reducing he esimaion risk, he variance of he esimaes. This assumpion begs wo quesions: (1) How quickly would an agen on average, regardless of prior, reach he seady sae in he learning process? (2) Can he esimaion risk in he seady sae significanly change he invesmen sraegy of he agen? This 24

25 secion provides answers o boh quesions in he conex of he model presened in his secion. To answer he firs quesion, I consruc a simulaion of how he esimaion risk of he agen changes afer each observaion hrough ime. For he case of sock price predicabiliy, I firs obain he seady sae variance of he measuremen error and he simulae he learning of he agen under he assumpion of priors ha are muliples of he seady-sae esimaion risk. I assume changes in he variance of he esimaion error follow (40). I assume ha new observaions are made every 1/10h of a quarer. As expeced, from Meron (1980), our resuls are no sensiive o he assumpion of he sampling frequency. Figure 1 shows how he agen s esimaion risk under he assumpion ha he prior is wo-imes, five-imes, en-imes, and weny-imes he seady sae value. Noice ha by he ime 15 years (180 monhs) pass by, all variance esimaes, regardless of prior, are lower han even wo-imes he seady sae variance. By he 30 year mark (360 monhs), he agen is no significanly far away from he seady sae regardless of he assumpion of he prior. The figure provides srong evidence ha our assumpion of seady-sae learning is no ou of line and makes sense given he amoun of daa he agen has available o esimae hese parameers. Assume he agen has access o he CRSP daabase, hen i is fair o sae agens have abou 40 years (480 monhs) of daily daa and abou 75 years (900 monhs) of monhly daa o earn from before deciding on heir consumpion and porfolio sraegies, hus i is quie believable ha a raional agen would achieve a level of learning such ha he seady sae assumpion is innocuous. In order o undersand how he esimaion error is reduced wih each new observaion, I check he magniude in which he esimaion error variance is reduced wih each new observaion. Figure 2 plos he insananeous reducion in variance for a given poin in ime. By he ime he agen has observed 15 years (180 monhs) of daa, he reducion in he variance of he esimaes of he unobservable variables are negligible. This implies, he learning effec should be negligible in he hedging componen of he agen s porfolio for our model. Our resuls imply parameer uncerainy, no learning, drives he changes in he porfolio composiion in comparison o he porfolio model under perfec observabiliy of all processes. 25