Asset Pricing in General Equilibrium with Constraints

Transcription

1 Asse Pricing in General Equilibrium wih Consrains Georgy Chabakauri London Business School Insiue of Finance and Accouning Regen s Park London NW1 4SA Unied Kingdom Tel: (44) Fax: (44) gchabakauri.phd2004@london.edu Job Marke Paper November 2008 I am graeful o Suleyman Basak, Francisco Gomes, Chrisopher Hennessy, Igor Makarov, Anna Pavlova, Raman Uppal and seminar paricipans a London Business School for helpful commens. All errors are my responsibiliy.

2 Asse Pricing in General Equilibrium wih Consrains Absrac We sudy dynamic equilibrium in a coninuous-ime economy wih one consumpion good and wo heerogeneous invesors facing porfolio consrains. Despie numerous applicaions, porfolio consrains are nooriously difficul o incorporae ino dynamic equilibrium analysis unless consrained invesors are assumed o have logarihmic preferences. Our soluion mehod yields new insighs on he impac of consrains on price-dividend raios, sock reurn volailiies and oher parameers wihou relying on his assumpion. We recover expressions for ineres raes and marke prices of risk for general preferences in erms of empirically observable alhough endogenous quaniies. Based on hese resuls we compue he equilibrium in specific economic seings where boh invesors have (idenical for simpliciy) CRRA preferences and one of hem faces porfolio consrain while he oher is unconsrained. Firs, we look a he consrain ha imposes an upper bound on he proporion of wealh invesed in socks, ypical for cerain pension and hybrid muual funds, and demonsrae ha igher consrains increase price-dividend raios and decrease sock-reurn volailiies. Moreover, we show ha he effec of consrains is sronger in bad imes when he dividend growh decreases due o adverse shocks. Nex, when invesors disagree on mean dividend growh raes and he pessimis faces shor-sale consrains in proporion o wealh we show ha price-dividend raios increase wih igher consrains while sock reurn volailiies can go in eiher direcion bu move closer o he volailiy of dividend growh. Journal of Economic Lieraure Classificaion Numbers: D52, G12. Keywords: asse pricing, dynamic equilibrium, heerogeneous invesors, porfolio consrains, risk sharing, sock reurn volailiy.

3 1. Inroducion Porfolio consrains and marke fricions have long been considered among key conribuors owards undersanding he invesor behavior and equilibrium asse prices. In paricular, dynamic equilibrium models wih heerogeneous invesors facing porfolio consrains have exensively been employed by financial economiss o confron a wide range of phenomena such as he equiy premium puzzle, mispricing of redundan asses, role of arbirageurs, impac of heerogeneous beliefs on asse prices, and sock comovemens (e.g., among ohers, Deemple and Murhy, 1997; Basak and Cuoco, 1998; Basak and Croioru, 2000, 2006; Kogan, Makarov and Uppal, 2007; Gallmeyer and Hollifield, 2008; Pavlova and Rigobon, 2008). However, racable characerizaions of equilibria are only obained assuming ha a consrained invesor has logarihmic preferences which simplifies he analysis a he cos of invesor s myopia. 1 Despie recen developmens in porfolio opimizaion, such as dualiy mehod of Cvianic and Karazas (1992), porfolio consrains are nooriously difficul o incorporae ino general equilibrium analysis as well as porfolio choice when consrained invesors have more general preferences inducing hedging demands which affec equilibrium parameers. The assumpion of logarihmic preferences is no innocuous and impedes he evaluaion of he impac of consrains on sock prices and sock reurn volailiies, which is of paricular imporance during he imes of financial crises when he regulaors may impose addiional porfolio resricions on insiuional invesors such as pension funds in an aemp o limi heir risk exposure. Thus, in economic seings wih wo logarihmic invesors and single consumpion good (e.g., Deemple and Murhy, 1997; Basak and Cuoco, 1998; Basak and Croioru, 2000, 2006) sock prices and hence sock reurn volailiies are unaffeced by consrains since he income and subsiuion effecs perfecly offse each oher. When he consrained invesor is logarihmic, he volailiy effecs can only be sudied in specific seings where he oher (unconsrained) invesor has differen preferences (e.g., Gallmeyer and Hollifield, 2008), which requires furher jusificaion. To our bes knowledge, our paper is he firs o sudy he effec of consrains on sock reurn volailiy in a coninuous-ime economy wihou relying on he assumpions ha consrained invesor is logarihmic and here are oher sources of invesors heerogeneiy apar from porfolio consrains. In his paper, we solve for he equilibrium in a coninuous-ime pure exchange economy wih one consumpion good and wo heerogeneous invesors facing porfolio consrains. Our soluion mehod yields new insighs on he impac of consrains on sock reurn volailiies and oher equilibrium parameers. Firs, for general preferences and consrains we provide a characerizaion of ineres raes and marke prices of risk which highligh he role of consrains 1 The assumpion ha one invesor has logarihmic preferences is also commonly made for racabiliy in models wih unconsrained invesors who differ in risk aversions. Thus, Dumas (1989) sudies dynamic equilibrium in a producion economy, where one invesor has logarihmic while he oher general CRRA preferences. Wang (1996) sudies an exchange economy where one invesor has logarihmic while he oher square-roo preferences. One noable excepion is Bhamra and Uppal (2008), who sudy he effec of inroducing non-redundan securiies on he volailiies of asse reurns in an exchange economy wih CRRA invesors no resriced o being logarihmic. 1

4 and risk sharing, and in specific economic seings can explicily be characerized in erms of empirically observable quaniies such as sock reurn and consumpion volailiies. Based on hese resuls, we specialize o seings wih wo CRRA invesors one of whom is unconsrained while he oher faces porfolio consrains. Specifically, we firs derive he equilibrium when he consrained invesor faces an upper bound on he proporion of wealh invesed in socks, which is ypical for some pension and hybrid muual funds. 2 Then, we sudy he impac of shor-sale consrains on equilibrium when invesors have differen beliefs abou mean dividend growh. The mehodological conribuion of he paper is a soluion mehod ha allows he compuaion of equilibrium in economies wih consrains. Specifically, we derive sock price-dividend raios, sock reurn volailiies and oher parameers in erms of wealh-consumpion raios ha can be compued numerically via a simple ieraive procedure wih fas convergence. A he firs sep of our analysis when we allow for general preferences, we demonsrae ha he riskless raes and marke prices of risk are given by riskless raes and marke prices of risk in an unconsrained economy plus addiional erms ha capure he effecs of consrains and risk sharing. Moreover, in specific seings we obain he expressions for ineres raes and marke prices of risk in erms of inuiive and empirically observable parameers such as sock reurn and consumpions volailiies. The racabiliy of our resuls allows o compare ineres raes in consrained and unconsrained economies for a given allocaion of consumpion among invesors and demonsrae ha for various consrains hey will be lower in consrained economies whenever boh invesors have he same prudence-risk aversion raios. Using he insighs from he case wih general preferences we show ha when invesors have (idenical for simpliciy) CRRA preferences, one of hem faces an upper bound on he proporion of wealh invesed in socks, and dividends follow a geomeric Brownian moion, he ineres raes and marke prices of risk can explicily be expressed in erms of marginal uiliy raios, heir volailiies and he volailiies of sock reurns. We compleely characerize he equilibrium by compuing hese volailiies numerically for relaive risk aversions greaer han uniy in order o generae an empirically plausible range of ineres raes and marke pries of risk. While in models wih boh invesors being logarihmic price-dividend raios and sock reurn volailiies are deerminisic funcions of ime, in our seing hese parameers depend on consrained invesor s consumpion share which evolves sochasically. We demonsrae ha igher consrains increase price-dividend raios and decrease sock reurn volailiies. One implicaion of his resul is 2 Srinivas, Whiehouse and Yermo (2000) in a survey of pension fund regulaions show ha limis on boh domesic and foreign equiy holdings of pension funds are in place in a number of OECD counries such as Germany (30% on EU and 6% on non-eu equiies), Swizerland (30% on domesic and 25% on foreign equiies) and Japan (30% on domesic and 30% on foreign equiies), among ohers. Even hough US and UK pension funds are no subjec o equiy holdings resricions here is a growing indusry of hybrid muual funds ha commi o mainaining a significan fracion of heir wealh in bonds (e.g., Comer, 2006). Moreover, our approach allows o sudy he impac of passive invesors ha hold a fixed fracion of heir wealh in socks. Samuelson and Zeckhouser (1988) documen he populariy of his sraegy using as an example he paricipans of popular TIAA/CREF reiremen plan who choose a fracion of wealh o be invesed in socks and rarely change i due o saus quo bias. Imporan special case is sock marke non-paricipaion which in year 2002 accouned for 50% of U.S. households (e.g., Guvenen, 2006). 2

5 ha imposing limis on he equiy holdings of insiuional invesors by regulaors reduces he sock reurn volailiy, which is paricularly high during financial crises and recessions (e.g., Schwer, 1989). Moreover, due o he dominance of income effec over subsiuion effec he price-dividend raios urn ou o be increasing while sock reurn volailiies decreasing funcions of he consrained invesor s share in aggregae consumpion. The insananeous changes in his share of consumpion are negaively correlaed wih he insananeous dividend growh in he economy. Hence, our model implies ha he consrains affec equilibrium parameers in an asymmeric way and decrease sock reurn volailiies in bad imes more han in good imes. We also demonsrae ha ineres raes are decreasing while marke prices of risk are increasing funcions of he consrained invesor s share in aggregae consumpion which evolves sochasically in our model. Posiive shocks o his consumpion share make he consrained invesor more willing o smooh consumpion and hence, o lend a lower raes which causes ineres raes o fall. Moreover, when one invesor is consrained, for he sock marke o clear he oher one should hold more socks han in an unconsrained economy which causes marke prices of risk o increase o compensae her for excessive risk aking. We demonsrae ha due o addiional risk aking unconsrained invesor s consumpion is more volaile han ha of a consrained invesor, consisenly wih he lieraure (e.g., Mankiw and Zeldes, 1989; Malloy, Moskowiz, and Vissing-Jorgensen, 2008). Moreover for a given consrained invesor s consumpion share igher consrains imply lower ineres raes and higher marke prices of risk since he smoohing and risk-aking effecs become more pronounced. To evaluae he economic impac of he consrained invesor in he long run we compue probabiliy densiy funcions for her consumpion share for differen ime horizons and demonsrae ha for plausible parameers her consumpion share, and hence he marke impac, slowly declines in he course of ime bu is raher significan even afer hundred years. Finally, we exend our baseline analysis o economic seings wih heerogeneous beliefs and muliple socks. In boh cases, for general preferences we derive expressions for ineres raes and marke prices of risk similar o hose in he baseline model. In he case of heerogeneous beliefs we solve for equilibrium in a model where wo invesors have he same CRRA uiliies and disagree abou he growh of dividends in he economy. The opimis is unconsrained while he pessimis faces consrain on he proporion of wealh ha can be invesed in shor posiions in socks. We demonsrae ha igher shor-sale consrains imply higher price-dividend raios since hey increase he consrained invesor s demand for socks. We also find ha he volailiy of sock reurns in consrained economy can be boh higher or lower han he volailiy in an unconsrained economy depending on wheher he laer is higher or lower han he volailiy of dividend growh. This is because he shor-sale consrains do no allow he invesor o rade on her pessimism making her sockholding closer o wha i would be in he case of homogeneous beliefs, and hence, he sock reurn volailiy shifs owards volailiy in an unconsrained homogeneous economy, given by he volailiy of dividends. Our soluion mehod is based on he combinaion of he dualiy approach and dynamic pro- 3

6 gramming. Firs, following Cvianic and Karazas (1992) we derive opimal consumpions in erms of he sae price densiies in an equivalen unconsrained ficiious economies in which ineres raes and marke prices of risk are given by hose in he original economy plus adjusmen parameers ha accoun for he difference in he invesors behavior in consrained and unconsrained economies. Then, marke clearing for consumpion yields expression for equilibrium parameers in erms of he adjusmens ha can be derived in erms of insananeous volailiies of sock reurns and he raios of marginal uiliies of he wo invesors. Nex, hese volailiies and hence all he equilibrium parameers are explicily characerized in erms of invesors wealh-consumpion raios ha saisfy a sysem of nonlinear Hamilon-Jacobi-Bellman equaions of dynamic programming. We solve his sysem of equaions numerically via a simple ieraive procedure ha requires solving a simple sysem of linear equaions a each sep. There is a growing lieraure sudying he dynamic equilibrium in coninuous-ime economies wih heerogeneous invesors and porfolio consrains assuming ha consrained invesors have logarihmic preferences. Basak and Cuoco (1998) consider a model in which one invesor is unconsrained and guided by general CRRA uiliy while he oher canno inves in sock marke and has logarihmic preferences. They derive riskless raes and marke prices of risk in his economy and characerize all he equilibrium parameers explicily when boh invesors are logarihmic. Deemple and Murhy (1997), Basak and Croioru (2000, 2006) presen equilibrium models wih wo logarihmic invesors, heerogeneous beliefs and porfolio consrains o sudy various economic phenomena. Hugonnier (2008) considers a similar model and shows ha under resriced paricipaion he sock prices implied by marke clearing may conain a bubble and exceed prices given by presen-value formula while in he seing wih muliple socks he equilibrium migh no be unique. In conras o our work all he above papers do no find he impac of consrains on sock prices and heir momens. Dumas and Maenhou (2002) develop an approach wih wo cenral planners for solving incomplee-marke equilibrium. However, in heir analysis he variance-covariance marix of reurns is aken as given and hence hey do no sudy he impac of consrains on volailiy. Kogan, Makarov and Uppal (2007) derive equilibrium parameers in an economy wih borrowing consrains when one invesor is logarihmic while he oher has general CRRA uiliy and find ha all he momens of asse reurns are deerminisic and sock reurn volailiies are unaffeced by consrains. When lile borrowing is permied hey numerically find ineres raes and marke prices of risk as funcions of wealh disribuions bu do no consider he volailiies of sock reurns. Gallmeyer and Hollifield (2008) sudy he asse pricing wih shor-sale consrains in he presence of heerogeneous beliefs when he pessimis and opimis have logarihmic and CRRA uiliies respecively. They sudy equilibrium parameers by employing Mone-Carlo simulaions and derive condiions for sock reurn volailiies o be larger or lower han in he unconsrained case assuming ha invesors have he same share of aggregae wealh a he iniial dae. In conras o heir work, in our heerogeneous-beliefs seing consrains are more general and allow he invesor o shor a cerain proporion of wealh and he volailiies are derived as 4

7 funcions of consrained invesor s consumpion share. Bhamra (2007) analyzes he effec of liberalizaion on emerging markes cos of capial in a model wih wo logarihmic invesors, wo socks and one consumpion good. Pavlova and Rigobon (2008) and Schornick (2008) consider models wih consrained logarihmic invesors and wo consumpion goods in inernaional finance framework and derive various asse-pricing implicaions assuming ha invesors face preference shocks. Longsaff (2008) wo-asse economy where one of he asses is non-radable for a cerain period and logarihmic invesors are heerogeneous in ime discoun parameer. There are a number of papers ha solve models wih heerogeneous invesors and porfolio consrains numerically in discree ime. Cuoco and He (2001) consider a model wih general uiliies and derive equilibrium asse prices in erms of sochasic weighs of a represenaive invesor s uiliy which are obained numerically from a nonlinear sysem of equaions. Guvenen (2006) solves numerically a model wih resriced marke paricipaion when invesors are guided by recursive uiliies. Chien, Cole and Lusig (2008) also in a discree-ime framework consider a model wih non-paricipans, passive and acive invesors guided by CRRA preferences, where passive invesors hold fixed porfolios while acive ones adjus hem each period. Gomes and Michaelides (2008) sudy numerically he equilibrium wih incomplee markes and invesors subjec o fixed cos of sock marke paricipaion and by calibraion generae high equiy premium and mach observed marke paricipaion rae. These works do no provide expressions for equilibrium parameers in erms of observable quaniies as we do in his paper by employing considerable flexibiliy of coninuous-ime mehods. The remainder of he paper is organized as follows. In Secion 2, we derive ineres raes and marke prices of risk for general uiliy under he assumpion ha he dual opimizaion problem has a soluion and discuss heir properies. In Secion 3 we illusrae our soluion mehod by compuing he equilibrium in a model wih wo CRRA uiliies when one invesor is unconsrained while he oher faces an upper bound on he fracion of wealh invesed in socks. Secion 4 exends our baseline analysis o he seings wih heerogeneous beliefs and muliple socks. We also solve for equilibrium in a model wih heerogeneous beliefs in which one of he invesors faces shor-selling consrains. Secion 5 concludes, Appendix A provides he proofs and Appendix B provides furher deails for our numerical mehod. 5

8 2. General Equilibrium wih Consrains 2.1. Economic Seup We consider a coninuous-ime economy wih one consumpion good and infinie horizon. The uncerainy is represened by a filered probabiliy space (Ω, F, {F }, P ), on which is defined a Brownian moion w. All he sochasic processes ha appear in he paper are adaped o {F, [0, )}, he augmened filraion generaed by w. The invesors rade coninuously in wo securiies, a riskless bond in zero ne supply wih insananeous ineres rae r and a sock in a posiive ne supply, normalized o one uni. The sock is a claim o an exogenous sricly posiive sream of dividends δ following he dynamics dδ = δ [µ δ d + σ δ dw ], (1) where he dividend mean-reurn, µ δ, and volailiy, σ δ, are sochasic processes. The dividend process (1) and is momens are assumed o be well-defined, wihou explicily saing he regulariy condiions. We consider equilibria in which bond prices, B, and sock prices, S, follow processes db = B r d, (2) ds + δ d = S [µ d + σ dw ], (3) where he ineres rae r, he sock mean reurn µ and volailiy σ are sochasic processes deermined in equilibrium, and bond price a ime 0 is normalized so ha B 0 = 1. There are wo invesors in he economy. Invesor 1 is endowed wih s unis of sock and b unis of bonds, while invesor 2 is endowed wih 1 s unis of sock and b unis of bond. The invesors choose consumpion, c i, and an invesmen policy, {α i, θ i }, where α i and θ i denoe he fracions of wealh invesed in bonds and socks, respecively, and hence, α i + θ i = 1. Invesor i s wealh process W evolves as ( ) ] dw i = [W i r + θ i (µ r ) c i d + W i θ i σ dw, (4) and her invesmen policies are subjec o porfolio consrains θ i Θ i, i = 1, 2, (5) where Θ i = [θ i, θ i ]. We also assume ha iniial endowmens of socks are such ha θ i a ime 0 belong o ses Θ i. Thus, he financial marke in our economy is incomplee due o he presence of porfolio consrains (5). Each invesor i (i = 1, 2) is guided by an expeced uiliy over a sream of consumpion c. In paricular, her dynamic opimizaion is given by [ ] max E e ρ u i (c i )d, (6) c i, θ i 0 6

9 subjec o he budge consrain (4), no-bankrupcy consrain W 0 and porfolio consrains (5), for some discoun parameer ρ > 0. The uiliy funcions u i (c) are assumed o be increasing, concave, hree imes coninuously differeniable, saisfying Inada s condiions lim u c 0 i(c) =, lim u c i(c) = 0, i = 1, 2. (7) By A i and P i we denoe absolue risk aversion and prudence parameers of invesor i, given by A i = u i (c) u i (c), and assume ha boh are sricly posiive for each invesor. P i = u i (c) u, (8) (c) Nex, we define an equilibrium in his economy as a se of parameers {r, µ, σ } and of consumpion and invesmen policies {c i, α i, θ i }2 i=1 such ha consumpion and invesmen policies solve dynamic opimizaion problem (6) for each invesor, given price parameers {r, µ, σ }, and consumpion and financial markes clear, i.e., i c 1 + c 2 = δ, α 1 W 1 + α 2 W 2 = 0, θ 1 W 1 + θ 2 W 2 = S, (9) where W1 and W 2 denoe opimal wealhs of invesors 1 and 2 under opimal consumpion and invesmen policies Characerizaion of Equilibrium This Secion characerizes he parameers of equilibria and sudies heir properies in economies wih consrained invesors. In paricular, by employing he dualiy mehod of Karazas and Cvianic (1992), we recover expressions for ineres raes and marke prices of risk in equilibrium in erms of he parameers of equivalen ficiious unconsrained economies. These expressions are inuiive and highligh he impac of risk-sharing and aiude owards risk on equilibrium parameers. Moreover, hey form a basis for an efficien mehodology for compuing equilibria, which we develop in Secion 3. We sar by noing ha since he marke is incomplee due o he presence of porfolio consrains, a Pareo opimal allocaion may no be feasible and hence, he raio of he marginal uiliies of consumpion of he invesors follows a sochasic process. This raio can be inerpreed as a sochasic weigh in he consrucion of a represenaive-invesor preferences in an equivalen economy, and serves as a sae variable in erms of which he equilibrium can be characerized (e.g., Basak and Cuoco, 1998; Cuoco and He, 2001). By employing he mehodology of Cvianic and Karazas (1992) we obain opimal consumpions and hen derive he equilibrium parameers from marke clearing condiions. This approach is similar o he approach in Basak (2000), who 7

10 characerizes an equilibrium in an economy where invesors have heerogeneous beliefs, bu in conras o our work are unconsrained. Cuoco (1997) sudies consumpion-porfolio choice of consrained invesors, mainly a a parial equilibrium level, and exends he resuls of Cvianic and Karazas o he case of more general uiliy funcions and forms of marke incompleeness. He derives a CAPM in an economy wih porfolio consrains bu does no sudy ineres raes and oher parameers of equilibrium. Hugonnier (2008) characerizes equilibrium in he model where one invesor has general uiliy and is unconsrained, while he second invesor has logarihmic uiliy and faces porfolio consrains. We sar by characerizing opimal consumpions of consrained invesors in a parial equilibrium in which he invesmen opporuniies are aken as given and hen obain he ineres rae, r, and he marke price of risk, κ, from he consumpion clearing condiion. For each invesor i, following he approach of Cvianic and Karazas (1992), we characerize he opimaliy condiions for consumpion by embedding our parial equilibrium economy ino an equivalen ficiious complee-marke economy wih adjused riskless rae r i and marke price of risk κ i and sae prices ξ i evolving as dξ i = ξ i [r i d + κ i dw ]. (10) As demonsraed in Cvianic and Karazas, he riskless rae and he marke price of risk in a ficiious economy are given by r i = r + f i (ν i), κ i = κ + ν i σ, (11) where κ = (µ r)/σ is a marke price of risk in consrained economy, f i (ν) are suppor funcions for he ses of porfolio consrains Θ i, defined as f i (ν) = sup θ Θ i ( νθ), (12) ν1 and ν 2 solve so called dual opimizaion problem, defined in Cvianic and Karazas (1992), and lie in he effecive domains for suppor funcions, given by Υ i = {ν R : f i (ν) < }. (13) Throughou his Secion we assume ha he soluions o dual opimizaion problems exis and since he ficiious economies are complee, he marginal uiliies of opimal consumpion are given by e ρ u i(c i) = ψ i ξ i, i = 1, 2, (14) for some consans ψ i > 0. The firs order condiions (14) and sae prices (10) demonsrae ha consumpion and invesmen decisions of he consrained invesor are equivalen o hose of an unconsrained one, which faces ineres raes and marke prices of risk adjused o accoun for he consrains. Moreover, opimaliy condiions in (14) allow o express consumpions c i in erms of sae prices in ficiious economies as follows: c i = I i (ψ i e ρ ξ i ), i = 1, 2, (15) 8

11 where I i ( ) denoe inverse funcions for marginal uiliies u i ( ). The expressions for marginal uiliies in (14) also imply ha he raio of invesors marginal uiliies, defined as λ = u 1 (c 1 ) u (16) 2 (c 2 ), is sochasic, and hence, he resuling allocaion of aggregae consumpion beween invesors is no Pareo-opimal in general. Basak and Cuoco (1998) and Cuoco and He (2001) demonsrae ha he process λ serves as a convenien sae variable in erms of which he equilibrium parameers can be expressed. Moreover, in an equivalen complee-marke economy wih a represenaive invesor, parameer λ can be inerpreed as a sochasic weigh in he uiliy u(c; λ) of a represenaive invesor, given by and follows a sochasic process u(c; λ) = max u 1(c 1 ) + λu 2 (c 2 ), (17) c 1 +c 2 =c dλ = λ [µ λ d + σ λ dw ]. (18) The parameers µ λ and σ λ are deermined in equilibrium and quanify he violaion of Pareoopimaliy in he economy. ν i Nex we characerize he parameers of our economy in equilibrium in erms of adjusmens from he marke clearing in consumpion. To deermine he ineres rae r and marke price of risk κ we subsiue opimal consumpions (15) ino consumpion clearing condiion in (9), apply Iô s Lemma o boh sides and recover equilibrium parameers by maching he drif and volailiy erms. Similarly, from opimaliy condiions (14), by applying Iô s Lemma o equaion (16) for λ and comparing he resul wih he process for λ in (18) we recover parameers µ λ and σ λ. The following proposiion summarizes our resuls. Proposiion 1. If here exiss an equilibrium, he riskless ineres rae r, marke price of risk κ, drif µ λ and volailiy σ λ of weighing process λ ha follows (18) are given by r = r A f 1 (ν A 1) A f 2 (ν 1 A 2) A3 (P 1 + P 2 ) 2 2A 2 1 A2 2 σ 2 λ A3 ( P1 P ) 2 δ σ δ σ λ,(19) A 1 A 2 A 1 A 2 κ = κ A A 1 ν 1 σ A A 2 ν 2 σ, (20) µ λ = A δ σ δ σ λ + f 1 (ν 1) f 2 (ν 2) A A 1 σ 2 λ, σ λ = ν 1 ν 2 σ, (21) where r and is he riskless rae and κ is he marke price of risk in an unconsrained economy, given by r = ρ + A δ µ δ A P 2 δ2 σ 2 δ, κ = A δ σ δ (22) 9

12 A i, P i, and A and P are absolue risk aversions and prudence parameers of invesor i and a represenaive invesor wih uiliy (17), respecively. 3 Opimal consumpions c i, wealhs W i, sock S and opimal invesmen policies θ i are given by c i = g i (δ, λ ), (23) Wi = 1 [ ] E ξ is c ξ isds, (24) i 0 S = W 1 + W 2, (25) θ i = 1 σ ( Wi ( κ + ν ) i + φ ) i, (26) σ ξ i where funcions g i (δ, λ ) are such ha c 1 and c 2 saisfy consumpion clearing in (9) and equaion (16) for process λ, sae prices ξ i follow processes (10) and φ i are such ha M i E [ 0 ] ξ is c isds = M i0 + 0 φ is dw s. Iniial value λ 0 is such ha budge consrains a ime 0 are saisfied: s i S 0 + b i = W i0, (27) where s 1 = s, s 2 = 1 s, b 1 = b and b 2 = b. Moreover, adjusmens νi saisfy complemenary slackness condiion f i (ν i) + θ iν i = 0. (28) ν i Proposiion 1 provides he characerizaion of equilibrium parameers in erms of adjusmens in ficiious economy. Expression (19) decomposes ineres raes r ino groups of erms ha separae he effecs of consrains and he inefficiency of risk sharing. The firs erm in (19) is he riskless rae in he unconsrained economy wih a represenaive invesor. The nex wo erms capure he effec of binding consrains on ineres raes and end o increase or decrease hem depending on he sins of suppor funcions f i (ν). In paricular, hese erms are posiive in economic seings wih binding porfolio consrains when invesors buy more bonds. This is due o he fac ha he invesors behave as if heir subjecive ineres raes r i in heir ficiious economy were higher han in he real one, and hence posiive adjusmens f i (νi ). Finally, he las wo erms in expression (19) capure he effec of risk sharing, quanified by volailiy σ λ. The weigh λ acs as a sae variable ha gives rise o specific hedging demands ha can push ineres raes in eiher direcion. 3 As demonsraed in Basak (2000), he risk aversion, A, and prudence, P, of he represenaive invesor can be obained from he following expressions: 1 A = 1 A A 2, P A 2 = P1 A P2. A

13 Similarly, he expression (20) for he marke price of risk is comprised of he marke price of risk in an unconsrained economy (firs erm in (20)) and he effecs of consrains (second and hird erms in (20)). Expressions for he drif µ λ and volailiy σ λ parameers of he sochasic weighing process λ in (21) demonsrae ha his process, in general, is no longer a local maringale as in works assuming logarihmic consrained invesor (e.g., Basak and Cuoco, 1998; Gallmeyer and Hollifield, 2008; Pavlova and Rigobon, 2008). Finally we observe ha opimal consumpions, wealhs, sock prices and invesmens can be obained from expressions (23) (26) when he parameers of equilibrium, and hence all sae prices, are known. The resuls in Proposiion 1 can also be used o compue he equilibrium parameers numerically. On one hand, Proposiion 1 expresses equilibrium parameers and invesmen policies in erms of adjusmens νi, and on he oher, he adjusmens can be obained from he complemenary slackness condiion (28). Thus, finding adjusmens becomes essenially a fixed poin problem. Moreover, as demonsraed in Huang and Pages (1992), under cerain condiions opimal wealhs (24) saisfy linear PDEs wih coefficiens deermined by equilibrium parameers while opimal policies (26) can be expressed in erms of derivaives of wealhs Wi. Hence, he adjusmens can be expressed in erms of derivaives of W i from condiions (28) and subsiued back ino he PDE for opimal wealhs. Thus, he characerizaion of equilibrium reduces o solving a sysem of quasilinear PDEs which, as we demonsrae in Secion 3, can efficienly be solved numerically for specific consrains Furher Properies of Equilibrium We here explore he implicaions of Proposiion 1 by noing ha in various economic seings he signs of adjusmens νi and suppor funcions f i (ν) can easily be deermined explicily from he definiions of suppor funcions and effecive domains in (12) and (13). Moreover, he ineres raes r and marke prices of risk κ can be expressed in erms of empirically observed quaniies, such as sock and consumpion volailiies, hus providing empirical implicaions of he model. Table 1 presens he effecive domains and he signs of he suppor funcions for plausible consrains and allows o analyze heir effec on equilibrium parameers. For example, when invesors face consrains on he proporion of wealh invesed in socks (case (d) in Table 1) he resuls in Proposiion 1 and Table 1 imply ha hese consrains end o decrease he ineres raes and increase he marke prices of risk relaive o an unconsrained model if sock volailiy σ is sricly posiive. Hence, hese consrains work in he righ direcion for explaining he equiy premium puzzle (e.g., Mehra and Presco, 1985). The overall effec of consrains on ineres raes is convolued by he effec of risk sharing capured by he las wo erms in he expression for ineres raes (19). The following Corollary o Proposiion 1 esablishes simple sufficien condiions under which he ineres rae r will be lower han he ineres rae r in a 11

14 Table 1 Effecive Domains and Suppor Funcions Case Consrain Υ f(ν) (a) θ R 0 0 (b) θ = 0 R 0 (c) θ θ θ, θ 0 R + (d) θ θ, θ > 0 ν 0 + (e) θ θ, θ < 0 ν 0 + (f) θ θ, θ > 0 ν 0 represenaive-invesor unconsrained economy. Corollary 1. If he uiliy funcions and he allocaion of consumpion are such ha P 1 /A 1 = P 2 /A 2 and he ses of porfolio consrains have posiive suppor funcions f i (ν) hen he ineres rae in a consrained economy, r, is lower han in an unconsrained one, r, and he following upper bound for rae r holds: r r A3 (P 1 + P 2 ) 2A 2 σ 1 A2 λ 2. (29) 2 The Corollary demonsraes ha he inabiliy o share risks conribues o he decrease of ineres raes by creaing hedging needs agains flucuaing raios of marginal uiliies λ. The condiion ha invesors have he same prudence-risk aversion raio is in paricular saisfied when boh invesors have idenical HARA preferences. 4 In he case of wo logarihmic invesors when one of hem is unconsrained he resul in Corollary 1 has also been poined ou in he lieraure (e.g., Basak and Cuoco (1998)). Convenienly, in various economic seings ineres raes and marke price of risk can be expressed only in erms of he parameers of uiliy funcions and empirically observed parameers. For example, when invesor 1 is unconsrained and invesor 2 faces a consrain allowing her o inves in sock no more han a cerain fracion of wealh (case (d) of Table 1), i can be observed ha parameers r and κ are given by: r = r A θσ σ λ A3 (P 1 + P 2 ) σλ 2 A A3 ( P1 P ) 2 δ σ δ σ λ, 2 A 1 A 2 A 1 A 2 2A 2 1 A2 2 κ = κ + A A 2 σ λ, (30) where sock reurn volailiy σ can easily be obained from he daa, while he weighing process volailiy σ λ can be obained in erms of uiliy parameers and he parameers of he consumpion 4 For HARA uiliy funcion absolue risk aversion is given by u (c)/u (c) = γ/(γ 0 + c). Differeniaing boh sides of his expression and hen dividing by u (c)/u (c) we obain ha P i/a i = 1 + γ, and hence, he prudencerisk aversion raio is he same for boh invesors. 12

15 processes for each invesor. In paricular, assuming ha he consumpion processes c i for each invesor follow Iô s processes dc i = c i [µ ci d + σ ci dw ], (31) applying Iô s Lemma o he definiion of weighing process λ in (16) we find ha σ λ = A 1 c 1 σ c1 A 2 c 2 σ c2. (32) In specific frameworks he volailiies of consumpion growh can be esimaed from he daa. In paricular, for he model wih resriced paricipaion ( θ = 0) Malloy, Moskowiz and Vissing- Jorgensen (2008) esimae consumpion volailiies of sock marke paricipans and non-paricipans o be 3.6% and 1.4% respecively, while Mankiw and Zeldes (1991) and Guvenen (2006) show ha he share of consumpion of non-paricipans in aggregae consumpion is As a resul, he expressions for r and κ in(30) can poenially be used for idenifying he parameers of he uiliy funcions of invesors as well as for quanifying he impac of risk sharing inefficiencies on he ineres raes and marke prices of risk. 3. Equilibrium wih Proporional Consrains This Secion applies he resuls of Secion 2 o compue and analyze he equilibrium in a specific economic seing in which invesor 1 is unconsrained while invesor 2 faces a consrain allowing her o inves in sock no more han a cerain fracion of wealh, ypical for pension and hybrid muual funds. For simpliciy we assume ha dividends follow a geomeric Brownian moion and boh invesors have idenical CRRA preferences. Using he resuls of Secion 2, in Secion 3.1 we presen a simple soluion mehod for finding an equilibrium in his economy, and in Secion 3.2 we sudy he impac of consrains on he equilibrium. In our seing wih fully raional invesors we also sudy he survival of consrained invesors in he long run and demonsrae ha i akes a long ime o eliminae heir impac on financial markes Characerizaion and Compuaion of Equilibrium In his Secion we presen a soluion mehod which allows o compue he equilibrium in an efficien way. This mehod does no rely on a widely used assumpion of a logarihmic consrained invesor (e.g., Deemple and Murhy, 1997; Basak and Cuoco 1998; Basak and Croioru, 2000, 2006; Kogan, Makarov and Uppal, 2003; Bhamra, 2007; Gallmeyer and Hollifield, 2008; Hugonnier, 2008; Pavlova and Rigobon, 2008; Schornick, 2008), which allows o derive he adjusmens νi in ficiious economy explicily, a he cos of invesor s myopia inheren in logarihmic preferences. In discree ime, Couco and He (2001), Guvenen (2006), Chien, Cole and Lusig (2008) and Gomes and Michaelides (2008) sudy he models wih consrained heerogeneous invesors numerically wihou assuming ha consrained invesor is logarihmic. In conras o hese works, by employing he flexibiliy of coninuous-ime finance we recover ineres raes and 13

16 marke prices of risk in erms of empirically observable parameers, which are furher expressed in erms of wealh-consumpion raios of invesors in an inuiive way. Finding an equivalen unconsrained economy is a challenging problem which so far has only been solved for logarihmic invesors (e.g., Cvianic and Karazas, 1992; Karazas and Shreve, 1998) or CRRA invesors bu assuming consan invesmen opporuniy ses (Tepla, 2001). We ackle his problem by firs expressing he parameers of he ficiious economy in erms of he sochasic weighing process λ, and he volailiies of λ and sock reurns, which hen are obained in erms of he wealh-consumpion raios of invesors ha solve Hamilon-Jacobi-Bellman equaions. Even hough in equilibrium he coefficiens of HJB equaions hemselves depend on he sensiiviies of wealh-consumpion raios wih respec o parameer λ, we demonsrae ha he ime-independen soluions can easily be obained via an ieraive procedure ha a each sep requires solving a simple sysem of linear algebraic equaions. 5 moion Throughou Secion 3 we assume for simpliciy ha dividends follow a geomeric Brownian dδ = δ [µ δ d + σ δ dw ], (33) boh invesors have CRRA uiliies wih relaive risk aversion parameer γ, given by 6 u i (c) = c1 γ 1, i = 1, 2, (34) 1 γ and solve opimizaion problem in (6) subjec o budge consrain (4), no-bankrupcy consrain W 0, and porfolio consrain θ θ for invesor 2, while invesor 1 is unconsrained. By J i (W, λ, ) we denoe he indirec uiliy funcion of invesor i. For convenience, we solve he opimizaion problem of a consrained invesor 2 in an equivalen ficiious unconsrained economy in which she maximizes her objecive funcion (6) subjec o budge consrain dw 2 = ( ) ] [W 2 r + f 2 (ν2) + θ 2 (µ r + νi) c 2 d + W 2 θ 2 σ dw, (35) where ν2 and f 2(ν2 ) are adjusmens o sock mean reurns and riskless raes respecively. By applying dynamic programming we find ha he indirec uiliy funcions should saisfy he following HJB equaions: { 0 = max e ρ c1 γ i c i,θ i 1 γ + J ( ) ] i + [W r + f i (ν i) + θ i (µ r + νi) Ji c i W J i λ µ λ + 1 [ W 2 θ 2 λ 2 iσ 2 2 J i W 2 2 J i 2W θ i λ σ σ λ + λ 2 σλ 2 W λ 2 J ]} i λ 2, 5 The dualiy approach offers convenien and inuiive framework for solving equilibrium models wih consrains. However, he sufficien condiions for he solvabiliy of dual problems given in Cvianic and Karazas (1992) are difficul o saisfy in various framework. The foonoe in he proof of Proposiion 2 in Appendix A poins ou ha he resuls of his Secion can alernaively be obained wihou relying on he dualiy approach by direcly working wih he HJB equaion for he consrained invesor in he original consrained economy. 6 The assumpion ha invesors have idenical risk aversions is made for simpliciy. More general case can be considered along he same lines. (36) 14

17 wih ransversaliy condiion E [J it ] 0 as T, which guaranees he convergence of he inegral in invesors opimizaion (6). We nex obain expressions for νi and f i (νi ) wihou solving he dual problem by noing ha since invesor 1 is unconsrained ν1 = 0 (case (a) in Table 1) while ν2 can be obained from equilibrium expression for σ λ in (21), and hence, ν 1 = 0, f 1 (ν 1) = 0, ν 2 = σ σ λ, f 2 (ν 2) = θσ σ λ. (37) The HJB equaions in (36) are sandard excep for he fac ha he equaion for invesor 2 is in erms of parameers of ficiious economy, which allows o formulae her problem as an unconsrained one. We conjecure ha he indirec uiliy funcions are given by ρ W 1 γ J i (W, λ, ) = e 1 γ H i(λ, ) γ, i = 1, 2. (38) Then, from he firs order condiions wih respec o consumpion we obain c i = W i H i, i = 1, 2, (39) where H i is a shorhand noaion for H i (λ, ), and hence, funcions H i can be inerpreed as wealh-consumpion raios of invesors 1 and 2. By subsiuing indirec uiliy funcions (38) ino HJB equaions i can be verified ha wealh-consumpion raios saisfy he following PDEs: H i + λ2 σλ 2 2 H ( i 2 λ 2 λ µ λ + 1 γ ) γ κ Hi ( 1 γ ) Hi iσ λ + λ 2γ κ2 i+(1 γ)r i ρ +1 = 0, i = 1, 2, γ (40) where r i and κ i denoe riskless rae and price of risk in a ficiious economy and are defined in (11) in erms of adjusmens given in (37). Moreover, opimal invesmen policies for invesors 1 and 2 are given by θ i = 1 ( H i λ ) κ i γσ λ, i = 1, 2. (41) γσ λ H i Since he horizon is infinie we will look for ime-independen and bounded soluions of equaions (40). Moreover, hroughou his Secion we assume ha θ 1. We noe ha if invesor 2 faces borrowing consrain, i.e. θ 1, he equilibrium coincides wih he equilibrium in an unconsrained economy in which he invesors, being idenical, opimally choose θ i = 1. When θ < 1 he consrain is always binding since oherwise, having idenical preferences, boh invesors should find opimal o inves θ i < 1 which conradics marke clearing condiions. 7 Convenienly, since he ficiious economy is complee, he equaions for wealh-consumpion raios in (40) are linear if volailiies σ λ and σ are known. However, in equilibrium hese parameers hemselves depend on funcions H i. In paricular volailiy σ λ can be obained from he fac ha he consrained invesor s consrain always binds, giving rise o equaion θ 2 = θ, 7 Formally, if he consrain does no bind, from he complemenary slackness condiion (28) i follows ha ν 2 = 0. Hence, σ λ = 0 and µ λ = 0 and he economy will permanenly remain in a Pareo-efficien unconsrained equilibrium. As a resul, since he invesors have idenical preferences, hey will choose θ i = 1 which violaes consrain θ 2 θ < 1 and leads o conradicion. 15

18 while he sock reurn volailiy σ can be obained by applying Iô s Lemma o sock price S = R δ, where R is a shorhand noaion for he sock price-dividend raio which can be expressed in erms of wealh-consumpion raios from he marke clearing condiions in (9). The following Proposiion 2 summarizes our resuls and provides a characerizaion of equilibrium in erms of wealh-consumpion raios. Proposiion 2. If here exiss an equilibrium, he riskless ineres rae r, marke price of risk κ and drif µ λ of weighing process λ ha follows (18) are given by r = r λ1/γ θσ σ λ 1 + γ λ 1/γ 2γ ( ) 2 σ2 λ, (42) κ = κ + λ1/γ µ λ = γσ δ σ λ σ λ, (43) 1 σ 2 λ, (44) where r and is he riskless rae and κ is he marke price of risk in an unconsrained economy, given by γ(1 + γ) r = ρ + γµ δ σδ 2 2, κ = γσ δ. (45) Opimal consumpions c i, wealhs W i, sock price-dividend raio R and opimal invesmen policies θ are given by c 1 = 1 W 1 = H 1 1 R = H 1 1 δ, c 2 = λ1/γ δ, W2 λ 1/γ = H 2 λ 1/γ + H 2 δ, (46) δ, (47), (48) θ1 = 1 ( H 1 λ ) κ γσ λ, θ2 = γσ λ H θ, (49) 1 while he volailiies of he sock reurns, σ, and weighing process, σ λ, are given by σ = σ δ σ λ R λ λ R, σ λ = 1 1+λ 1/γ (1 θ)γσ δ + γ H 2 λ λ R λ H 2 θγ λ R, (50) where wealh-consumpion raios H 1 and H 2 saisfy equaions (40). Moreover, he iniial value λ 0 for he weighing process (18) solves equaion (1 θ)h 2 (λ 0, 0) 16 λ 1/γ 0 0 δ 0 = b. (51)

19 The expressions for riskless rae r and price of risk κ in Proposiion 2 are in erms of he of volailiies σ and σ λ, as well as parameer λ 1/γ which in our economic seing can be inerpreed as he raio of consumpions of invesors 2 and 1, as i follows from he expressions in (46). As in he general case in Proposiion 1, ineres raes are comprised of hree erms, where he firs erm is a riskless rae in an unconsrained economy, while he second and hird erms highligh he impac of consrains and risk sharing. Moreover, he effec of risk sharing, as capured by volailiy σ λ, can be expressed in erms of consumpion volailiies. In paricular, from expression (32) i follows ha σ λ = γ(σ c1 σ c2 ). (52) I will be demonsraed laer ha volailiy σ λ is posiive in equilibrium since invesor 1 is more exposed o risk and hence her consumpion growh is more volaile. Proposiion 2 also provides expressions for equilibrium volailiies σ and σ λ in erms of he elasiciies of wealh-consumpion and price-dividend raios wih respec o weighing process λ, given by ɛ H 2 = H 2 λ λ H 2, ɛ P = R λ λ R. (53) From he expression for he volailiy σ λ in (50) i follows ha σ λ is decreasing in elasiciy ɛ H 2 and increasing in ɛ P. The effec of elasiciies in (53) on volailiy σ λ hen deermines heir impac on all he oher parameers in equilibrium. To undersand he effec of hese elasiciies on volailiy σ λ we observe ha elasiciy ɛ is H 2 proporional o he sock hedging demand of invesor 2 given by he second erm in he expression for opimal policy (41). Moreover, since σ λ is posiive, i follows from his expression ha higher elasiciy ɛ ends o decrease opimal invesmen in sock. Thus, higher ɛ H 2 H2 makes he sock less aracive and he invesor s ideal, alhough infeasible, unconsrained sockholding decreases and moves closer o θ which causes σ λ o fall since he risks are shared in a more opimal way. Moreover, as follows from he expressions for volailiies (50) he increase in elasiciy ɛ P ends o decrease sock volailiy σ since he dividends and weighing process are negaively correlaed. Hence, if volailiy σ decreases, he sock becomes more aracive for boh invesors. However, since invesor 2 is consrained, her ideal unconsrained holding moves furher away from her consrained holding θ and hence he risks are shared in a less opimal way and σ λ increases. Proposiion 2 also allows o explicily idenify he coefficiens of PDEs (40) for wealhconsumpion raios H i, which depend on equilibrium parameers idenified in expressions (42) (50). Moreover, i appears ha he coefficiens hemselves depend on raios H i and hence, we obain a sysem of quasilinear PDEs he soluions o which compleely characerize he equilibrium. We nex solve for ime-independen soluions of PDEs (40) which correspond o he infinie horizon case. To solve he equaions (40), we firs fix a large horizon parameer T, choose a saring value for H i (λ, T ) and hen solve he equaion backwards using a modificaion of Euler s finie-difference mehod unil he soluion converges o a saionary one. This approach is 17

20 similar o he subsequen ieraions mehod for solving Bellman equaions in discree ime (e.g., Ljungqvis and Sargen, 2004) when a a disan ime in he fuure he value funcion is se equal o some funcion (usually zero) and hen he value funcions a earlier daes are obained by solving equaions backwards. Since weigh λ varies from zero o infiniy, we firs perform a change of variable and rewrie he PDEs (40) as well as he equilibrium parameers in Proposiion 2 in erms of consrained invesor s share in aggregae consumpion, given by y = λ1/γ. (54) Variable y akes values in he inerval [0, 1] and provides one-o-one mapping o variable λ. The soluion of PDEs in erms of new variable we label as H i (y, ). Assuming ha he soluions o new PDEs are coninuous and wice coninuously differeniable, seing in hose equaions y = 0 and y = 1 we recover boundary condiions for H i (y, ). Nex, we replace he derivaives by heir finie-difference analogues leing he ime and sae variable incremens denoe T/M and y 1/N, where M and N are ineger numbers. Solving he equaion backwards, siing a ime we compue he coefficiens of finie-difference analogues of PDEs (40) using he soluions H i (y, + ) obained from he previous sep +. As a resul, he coefficiens of equaions for H i (y, ) are known a ime and hence H i (y, ) can be found by solving a sysem of linear finie-difference equaions wih hree-diagonal marix. Appendix B provides furher deails of he numerical algorihm. The wealh-consumpion raios hen allow us o derive all he parameers of equilibrium. Remark 1 (Bond prices). Proposiion 2 allows o derive bond prices from he condiion ha consrained invesor s oal invesmen in socks and bonds should equal her wealh a all imes, ha is, θw 2 + bb = W2. Hence, he resuls in Proposiion 2 imply ha B = 1 θ b λ 1/γ H 2 Remark 2 (Exisence of Equilibrium). Second panel in he second row of Figure 2 shows ha wealh-consumpion raio H 2 is a monoone funcion of y, and hence, also of λ 1/γ, and is bounded by posiive consans from below and from above. As a resul, he funcion on he lefhand side of he equaion for λ 0 in (51) is monoone and maps he inerval [0, ) ino [C 0, C 1 ), where C 0 and C 1 are some consans, and hence, if b [C 0, C 1 ) here always exiss he unique soluion λ 0 ha saisfies he equaion. Given he exisence of λ 0 and he soluions o HJB equaions (40), expressions (42) (50) fully characerize he equilibrium in he economy. 8 8 We also assume here ha he iniial endowmens are such ha invesor 2 binds on her consrain a ime 0. If he endowmens are such ha θ 2,0 < θ hen he consrain will sar o bind over nex insan. δ. 18

21 35 θ = 0 Sock Price Dividend Raios 1.5 Raios of Sock Reurn and Dividend Growh Volailiies θ = 0 θ = 0.4 θ = 0.4 θ = θ = θ = 1 1 θ = 1 R σ/σδ y y Figure 1: Price-Dividend Raios and Raios of Sock Reurn and Dividend Growh Volailiies. The figure plos he price-dividend raios R and he raios of sock reurn and dividend growh volailiies σ/σ δ as funcions of consrained invesor s consumpion share, y. Dividend mean growh rae µ δ = 1.8% and volailiy σ δ = 3.2% are aken from he esimaes in Campbell (2003), based on consumpion daa in , while risk aversion and ime discoun are se o γ = 3 and ρ = 0.01 respecively Analysis of Equilibrium We now sudy he impac of consrains on price-dividend raios, sock reurn volailiies, ineres raes, marke prices of risk and wealh-consumpion raios. Imporan implicaion of our model is ha in conras o models wih logarihmic invesors he consrains do affec he price-dividend raios and sock reurn volailiies. Figure 1 presens price-dividend raios and he raios of sock reurn and dividend growh volailiies while Figure 2 shows ineres raes, marke prices of risk and wealh-consumpion raios under plausible parameers. 9 The graphs on Figure 1 demonsrae he predicion of our model ha igher consrains (lower θ) increase he price-dividend raios R and decrease sock reurn volailiies σ. The pracical implicaion of his resul is ha imposing limis on equiy holdings of insiuional invesors by regulaors reduces he sock reurn volailiy which is paricularly high during financial crises and recessions (e.g., Schwer, 1989). Moreover, since pension funds in a number of OECD counries face equiy holding resricions (e.g., Srinivas, Whiehouse and Yermo, 2000) our model predics ha he counries wih igher resricions should have smaller raio of he sock marke volailiy and he dividend volailiy. Figure 1 also shows ha raios R are increasing while volailiies σ are decreasing funcions of consrained invesor s consumpion share y, and hence, in our model he price-dividend raios 9 In paricular, he parameers for he dividend process (µ δ = 1.8%, σ δ = 3.2%) are aken from he esimaes in Campbell (2003), based on consumpion daa in years, and he discouning parameer is se o ρ = We choose he relaive risk aversion parameer γ = 3 o generae a plausible range for riskless raes and marke prices of risk. 19