Equilibrium in Securities Markets with Heterogeneous Investors and Unspanned Income Risk

Transcription

1 Equilibrium in Securiies Markes wih Heerogeneous Invesors and Unspanned Income Risk July 9, 29 Peer Ove Chrisensen School of Economics and Managemen, Aarhus Universiy, DK-8 Aarhus C, Denmark Kasper Larsen Deparmen of Mahemaical Sciences, Carnegie Mellon Universiy, Pisburgh, PA Claus Munk a School of Economics and Managemen & Dep. of Mahemaical Sciences, Aarhus Universiy, DK-8 Aarhus C, Denmark cmunk@econ.au.dk a Corresponding auhor. Full address: School of Economics and Managemen, Aarhus Universiy, Barholin s Alle 1, Building 1322, DK-8 Aarhus C, Denmark.

2 Equilibrium in Securiies Markes wih Heerogeneous Invesors and Unspanned Income Risk Absrac: We provide he firs closed-form soluion in he lieraure for he equilibrium risk-free rae and he equilibrium sock price in a coninuousime economy wih heerogeneous invesor preferences and unspanned income risk. We show ha lowering he fracion of income risk spanned by he marke produces a lower equilibrium risk-free rae and a lower sock marke Sharpe raio, parly due o changes in he aggregae consumpion dynamics. If we fix he aggregae consumpion dynamics, he Sharpe raio is he same as in an oherwise idenical represenaive agen economy in which all risks are spanned, whereas he risk-free rae and he expeced sock reurn is lower in he economy wih unspanned income risk due o an increased demand for precauionary savings. The reducion in he risk-free rae is highes when he more risk-averse invesors face he larges unspanned income risk. In numerical examples wih reasonable parameers, he risk-free rae is reduced by several percenage poins. Our closed-form soluion hinges on negaive exponenial uiliy and normally disribued dividends and income bu, neverheless, our resuls show ha unspanned income risk may in more general seings play an imporan role in explaining he so-called risk-free rae puzzle. Keywords: Unspanned income, heerogeneous preferences, coninuous-ime equilibrium, risk-free rae puzzle, equiy premium, incomplee markes, Brownian moion JEL-Classificaion: G12, G11, D53

3 Equilibrium in Securiies Markes wih Heerogeneous Invesors and Unspanned Income Risk 1 Inroducion Labor income and income from nonraded asses are imporan sources of wealh for mos individuals, cf. Heaon and Lucas 2 and Campbell 26, wih a poenially large impac on heir consumpion and porfolio decisions and, consequenly, on he equilibrium securiies prices. If income risk can be compleely hedged by appropriae financial asses, i is sraighforward o include income risk in sandard models of individual consumpion and porfolio choice and in equilibrium asse pricing models. However, a ypical individual s labor income is only weakly correlaed wih raded financial asses and, hus, i has a large unspanned and unhedgeable componen, which complicaes hese models remendously. This paper provides he firs closed-form soluion in he lieraure for he equilibrium riskfree rae and he equilibrium sock price in a coninuous-ime economy wih heerogeneous invesor preferences as well as unspanned income risk. The equilibrium has he following properies. Lowering he fracion of income risk which is spanned by he marke leads o a lower covariance beween he dividends of he sock and aggregae consumpion and, consequenly, a lower sock marke Sharpe raio. The equilibrium risk-free rae decreases due o a higher demand for precauionary savings. If we fix he aggregae consumpion dynamics, he Sharpe raio is he same while he risk-free rae and he expeced sock reurn is lower han in an oherwise idenical represenaive agen economy in which all risks are spanned. The reducion in he riskfree rae depends on he magniude of all invesors unspanned income risk and heir risk aversion. The reducion is highes when he more risk-averse invesors face he larges unspanned income risk. Even hough our closed-form soluion hinges on very specific assumpions abou he preferences, he dividend process, and he income processes, our resuls sugges ha unspanned income risk may in more general seings play an imporan role in explaining he risk-free rae puzzle produced by he sandard represenaive agen models. Our economy is formulaed in a coninuous-ime, finie ime horizon model wih a single consumpion good, a risk-free asse and a single risky asse. The risky asse is a claim o an exogenously given dividend sream represened by a Gaussian sochasic process an arihmeic Brownian moion. There is a finie number of consumer-invesors maximizing ime-addiive negaive exponenial uiliy of consumpion, wih heerogeneiy in heir subjecive ime preference raes and heir absolue risk aversion. Each invesor receives an exogenously given income sream represened by anoher Gaussian process, which is 1

4 imperfecly correlaed wih he sock s dividend process and, herefore, also imperfecly correlaed wih he sock s price process. In oher words, each invesor s income process conains an unhedgeable risk componen. Our analysis is performed by firs conjecuring he form of he equilibrium risk-free rae and of he equilibrium sock price, and hen we solve for each invesor s opimal consumpion and invesmen sraegy. Secondly, by aggregaing over invesors we find an equilibrium consisen wih our conjecure. In paricular, he equilibrium risk-free rae, sock price volailiy, and Sharpe raio are all deerminisic processes. Firs, we solve he uiliy maximizaion problem of each invesor assuming a deerminisic risk-free rae, sock price volailiy, and Sharpe raio. This resul makes a conribuion o he lieraure on opimal consumpion and porfolio choice. As long as income risk is spanned by he raded asses, he inroducion of income risk does no complicae he soluion of muli-period uiliy maximizaion problems, cf. e.g., Meron 1971 and Bodie, Meron, and Samuelson However, when he income process is parially unspanned, he invesor s opimizaion problem becomes remendously more complicaed. To he bes of our knowledge, explici soluions have only been found in wo such cases, boh involving negaive exponenial uiliy, a normally disribued income sream, a consan risk-free rae, and a consan drif and volailiy of he sock price. Svensson and Werner 1993 solve for he opimal consumpion and porfolio sraegies in an infinie ime horizon seing, whereas Henderson 25 assumes a finie horizon and uiliy of erminal wealh only. 2 Henderson conjecures [las paragraph of her concluding remarks ha if we were o incorporae consumpion, i is unlikely ha he model could be solved analyically. Our resuls do indeed generalize her findings o he case of consumpion over a finie lifeime. We show ha he opimal consumpion a any dae is equal o wha he invesor could ge by spreading ou evenly annuiizing he invesor s perceived oal wealh over he remaining consumpion period. The perceived oal wealh is he sum of he financial wealh, correced upwards due o he fuure invesmen opporuniies, and an appropriae measure of human wealh. Any idiosyncraic income shock, posiive or negaive, leads o an immediae idenical change in consumpion. The opimal sock invesmen is a combinaion of he sandard myopic erm and an income hedge erm which basically undoes he sock-like risk inheren in he income process. The opimal invesmen in he risk-free asse is increasing in he magniude of he unspanned income risk componen due o precauionary savings. We do no impose running non-negaiviy consrains on financial wealh, bu in order o ensure ha he invesor does no end up indebed a he erminal dae, admissible 2 Henderson 25 also finds near-explici soluions for more general income processes. Duffie and Jackson 199 and Teplá 2 derive similar soluions for invesors receiving an unspanned income only a he erminal dae. 2

5 sraegies are defined so ha financial wealh is non-negaive a he erminal dae. As a consequence, he proof of he opimal sraegies involves a mahemaically ineresing Ornsein-Uhlenbeck ype bridge process. I seems difficul if no impossible o move beyond he assumpions of negaive exponenial uiliy and a Gaussian income process and sill obain closed-form soluions o he invesor s uiliy maximizaion problem wih unspanned income risk. Several recen papers have numerically solved for opimal consumpion and porfolio sraegies in more general seings, see e.g., Cocco, Gomes, and Maenhou 25, Koijen, Nijman, and Werker 29, Lynch and Tan 29, and Munk and Sørensen 29. Obviously, i is no possible o derive he equilibrium asse prices in closed form if he individual invesors uiliy maximizaion problems can only be solved numerically. Secondly, we aggregae over he individual consumer-invesors and impose he marke clearing condiions. We derive simple explici soluions in erms of he primiives of he economy for he equilibrium sock price and risk-free rae confirming he conjecure we used in he individual uiliy maximizaion problems, i.e., he risk-free rae, he Sharpe raio, and he volailiy of he sock are all deerminisic processes. We show ha a decrease in he fracion of income risk which is spanned by he sock marke implies a lower sock marke Sharpe raio and a lower risk-free rae. We also compare our equilibrium o he equilibrium in a similar economy wih all income risk being spanned, bu he same heerogeneous preferences and he same aggregae consumpion dynamics. In such an economy, a represenaive agen exiss, which immediaely leads o he equilibrium risk-free rae and sock marke Sharpe raio. In boh economies, he Sharpe raio is he produc of he volailiy of aggregae consumpion and he reciprocal of he aggregae absolue risk olerance, he laer being lower relaive o an economy wih homogeneous risk aversions. In conras, he risk-free rae is markedly differen and is always lower in he economy wih unspanned income risk, as described in he opening paragraph. In boh economies, he equilibrium risk-free rae consiss of hree erms: The firs erm is a weighed average of he invesors subjecive ime preference raes, where he weigh for a given invesor is he raio of her absolue risk olerance o he aggregae absolue risk olerance. This weighed average may be lower or higher han he non-weighed average ime preference rae. This phenomenon is also discussed by Gollier and Zeckhauser 25, who consider an economy wih heerogenous invesors bu no uncerainy. The second erm is he produc of he expeced growh rae of aggregae consumpion and he reciprocal of he aggregae risk olerance. Neiher he firs nor he second erm are affeced by he presence of unspanned income risk. The hird erm is negaive and due o he demand for precauionary savings. This is he channel hrough which unspanned income risk affecs he equilibrium risk-free rae. The demand for precauionary savings is higher when income risk canno be hedged, which in urn produces a lower equilibrium risk-free rae. Oher hings being equal, he reducion in he risk-free rae is higher, 3

6 he larger he unspanned componen of he income shocks is. The heerogeneiy in risk aversion is crucial for he magniude of he reducion in he risk-free rae. The reducion is highes when he more risk-averse invesors face he larges unspanned income risk. In numerical examples wih reasonable parameers, we find a reducion in he risk-free rae of several percenage poins. Since he sock marke Sharpe raio is he same, he equilibrium expeced reurn on he sock is also lower han in he economy wih idenical aggregae consumpion dynamics bu fully spanned income risk. Only a few exising papers on equilibrium asse pricing ake labor income ino accoun a all. I is sraighforward o include labor income in a represenaive agen modeling framework. Sanos and Veronesi 26 consider a represenaive agen economy in which consumpion is he sum of a given labor income sream and dividends from financial asses. The relaive weighs of income and dividend in consumpion vary over ime, which lead o ime-varying covariances beween he sock marke s dividend process and he sae prices. Consequenly, he sock marke risk premium is ime-varying in line wih he evidence from empirical sudies, cf. Cochrane 25, Ch. 2. However, he represenaive agen framework requires complee or effecively complee markes, which is no he case when invesors have non-raded income sreams. Telmer 1993 considers a model wih wo invesors who have idenical preferences and iniial endowmens, bu laer heir incomes may diverge due o ransiory idiosyncraic shocks. The invesors can only rade in a risk-free asse and, hence, hey are unable o hedge he income shocks. Telmer solves numerically for he equilibrium and finds ha he equilibrium sae-price deflaor is only weakly affeced by he unhedgeable income shocks. The invesors can self-insure agains adverse income shocks by buffer savings and, herefore, he equilibrium ineres rae is lower, bu for reasonable parameerizaions he reducion is small. This small impac may parly be due o he assumpion ha shocks have no persisen impac on income. Consaninides and Duffie 1996 assume ha he invesors have idenical and sandard preferences bu are subjec o persisen non-hedgeable shocks. Their model can generae virually any paern of sock and bond prices, as long as he dispersion of income shocks across invesors is se appropriaely. In paricular, if his cross-secional dispersion is couner-cyclical and sufficienly large, he model prices are consisen wih he observed high equiy premium. However, Cochrane 25, Ch. 21 argues ha cross-secional income daa do no show such large dispersion bu agrees ha persisen idiosyncraic income shocks may have a subsanial impac on equilibrium asse prices. In our model, income shocks are persisen and can produce a significan reducion of he equilibrium risk-free rae. The reducion in he equilibrium risk-free rae ha we find wih unspanned income risk can help explain he so-called risk-free rae puzzle, which was firs idenified by Weil The risk-free rae puzzle is based on he observaion ha he hisorical risk-free rae is smaller han he risk-free rae prediced by simple consumpion-based represenaive agen models. In order o mach he observed equiy premium in hese models, he risk aversion parameer has o be very high, bu hen he risk-free rae becomes oo high, 4

7 unless we also allow for a negaive ime preference rae. Only few papers have offered explanaions of he risk-free rae puzzle in raional asse pricing models. In a calibraed version of a relaively simple overlapping generaions model, Consaninides, Donaldson, and Mehra 22 presen numerical resuls showing ha borrowing consrains resraining young individuals from borrowing agains fuure labor income can significanly reduce he risk-free rae and parly explain he risk-free rae puzzle, bu hey do no discuss he role of unspanned income risk in he deerminaion of he risk-free rae. Bansal and Yaron 24 consider an economy wih long-run consumpion risk and a represenaive agen displaying Epsein-Zin recursive uiliy. The abiliy o disenangle he risk aversion from he elasiciy of ineremporal subsiuion adds imporan flexibiliy in maching he low hisorical risk-free rae. Bansal and Yaron 24 do no include labor income explicily and by virue of heir represenaive agen framework, here is no room for unspanned income risk. The presen paper seems o be he firs o idenify unspanned income risk as a poenial explanaion of he risk-free rae puzzle. The paper proceeds as follows. Secion 2 presens he economy, i.e., he individuals preferences and endowmens as well as he radeable asses. Secion 3 solves for any given invesor s opimal consumpion and invesmen sraegy under he assumpion ha he risk-free rae, he excess expeced reurn and he volailiy of he sock are deerminisic funcions of ime. Secion 4 hen shows by aggregaion over invesors ha he conjecured form of he equilibrium is correc and pins down he precise equilibrium risk-free rae and sock price dynamics. Finally, we conclude in Secion 5. 2 The economy We consider a coninuous-ime economy over he ime inerval [, T, T >. The economy offers a single consumpion good, which is he numéraire. Two financial asses are available for rading hroughou he ime inerval. The firs asse is a risk-free asse wih B denoing is ime price iniialized a B = 1. The second asse is a risky asse, which is a claim o a coninuous exogenously given dividend process D = D evolving as dd = µ D d + σ D dw, [, T, D R, 1 for wo deerminisic funcions of ime µ D and σ D. In 1, he process W = W is a sandard Brownian moion, i.e., he dividend process D is an arihmeic Brownian moion. We assume ha he risky asse is in uni ne supply and denoe is ime price by S. We laer specify he individuals such ha, in equilibrium, he risk-free asse provides a deerminisic rae of reurn, i.e., db = rb d, [, T, 5

8 where r is a deerminisic funcion of ime. The equilibrium price dynamics of he risky asse will have he form ds = S r + µ S D d + σs dw, [, T, 2 for some deerminisic funcions of ime µ S and σ S. Here, µ S denoes he oal expeced excess reurn over he risk-free rae and σ S is he absolue price volailiy. The raio λ S = µ S /σ S is he marke price of risk associaed wih he Brownian moion W and is also idenical o he Sharpe raio of he risky asse S. For laer use, we inroduce he noaion β, s = exp ru du, s T, which denoes he ime price of a zero-coupon bond wih a uni paymen a ime s. Furhermore, he process A = β, u du, [, T, denoes he annuiy facor for [, T, i.e., he ime value of a coninuous paymen sream wih a uni paymen per period unil he erminal dae T. Noe ha A as T and, herefore, A 1 for T. The economy is populaed by I consumer-invesors all living on he ime inerval [, T and all having ime-addiive negaive exponenial uiliy of consumpion. The invesors may have differen degrees of risk aversion as well as differen ime preference raes. Invesor i is hus maximizing E[ U is, c s ds, where U i, c = e δ i e a ic, c R, [, T, 3 and δ i and a i are he ime preference rae and he absolue risk aversion coefficien of invesor i, respecively. Each invesor i receives income according o an exogenously given income rae process Y i = Y i, i.e., he cumulaive income up o ime [, T is given by Y iu du. We assume ha Y i has he dynamics dy i = µ Y i d + σ Y i ρ i dw + 1 ρ 2 i dz i, [, T, 4 where µ Y i and σ Y i are deerminisic funcions of ime, and where Z i = Z i is a sandard Brownian moion so ha W, Z 1,..., Z I are independen. 3 The consan ρ i [ 1, +1 conrols he income-sock correlaion. Hence, our model allows for boh common income 3 The analysis goes hrough when Z 1,..., Z I are allowed o be correlaed, bu a he expense of increased noaional complexiy. 6

9 risk which is hedgeable via rading in he risky asse and for idiosyncraic unspanned income risk ha canno be hedged. Noe ha all shocks o he income rae are persisen. We conclude his secion by making wo assumpions on he marke srucure. The firs assumpion is made on he primiives of he economy and is herefore compleely exogenous. Assumpion 1. The deerminisic funcions σ Y i, σ D, µ Y i, µ D, i = 1,..., I, are coninuous and finiely valued on he inerval [, T. The second assumpion is made on he price dynamics and, hus, has o be verified by he equilibrium parameer processes. Assumpion 2. The ineres rae r is a deerminisic and coninuous funcion on [, T which saisfies lim s T A 1 d = +. The deerminisic funcions µ S, σ S are coninuous on he inerval [, T and are such ha he marke price of risk λ S = µ S /σ S is well-defined and belongs o L p for all p 1, i.e., λs p d <. As we shall see, Assumpion 1 implies ha he equilibrium parameers r, µ S and σ S auomaically saisfy Assumpion 2. Indeed, Assumpion 1 leads o a bounded deerminisic ineres rae and such raes saisfy he inegrabiliy requiremen: if r r for all [, T for some consan r, we have β, s = exp ru du e rs, s T. Consequenly, A 1 1 r 1 e rt, and he righ-hand-side does no have a finie inegral over [, T. 3 The individual invesor s problem In his secion, we solve he uiliy maximizaion problem of each invesor. In order o simplify he noaion, we suppress he i subscrips idenifying he invesor hroughou his secion. 7

10 3.1 Admissible sraegies The invesor has o choose a consumpion process c = c and an invesmen process θ = θ, where θ represens he number of unis of he risky asse owned by he invesor a ime for [, T. The remaining wealh is invesed in he risk-free asse. Given a consumpion and invesmen sraegy c, θ, we le X c,θ denoe he invesor s financial wealh a ime and le x R be he invesor s iniial wealh. For [, T, we define he self-financing wealh dynamics by dx c,θ = = X c,θ θ S r d + θ ds + D d + Y c d X c,θ r + Y c d + θ µ S d + θ σ S dw, wih X c,θ = x. Noe ha he dividend process D does no appear in he wealh dynamics. In order o ensure ha X c,θ is well-defined, we require ha c L 1 and θσ S L 2, i.e., we require ha he following wo inegrabiliy condiions hold P-a.s. c d <, 5 θ σ S 2 d <. 6 Under Assumpion 2, we see ha θσ S L 2 and Cauchy-Schwarz s inequaliy imply ha θµ L 1 and as a consequence, he wealh dynamics given by 5 is well-defined. We place wo addiional requiremens on he invesor s possible choices. The firs is he naural economic requiremen ha a he end of he ime horizon, he invesor has no remaining deb obligaions and, hus, we require he processes c, θ o be such ha P X c,θ T = 1. 7 As we shall see, he opimal sraegies ĉ, ˆθ have he propery ha X ĉ,ˆθ T =, P-almos surely. The second requiremen is purely echnical and is an arifac of our coninuousime seing: in order o provide rigorous proofs, we need a cerain degree of regulariy of he invesor s possible choices. To sae his regulariy condiion, we define he auxiliary deerminisic funcion g = δ r λ S 2 + a µ Y ρσ Y λ S ρ2 a 2 σ Y 2, 8 for [, T. Given his definiion, we can define he funcion V, x, y = A exp { aa 1 x ay A 1 β, s } gu du ds, 9 for [, T and x, y R. We denoe by V, V x and V y he respecive parial derivaives of V. Finally, we can define he admissibiliy concep adoped in his paper as follows: 8

11 Definiion 1. A pair c, θ of progressively measurable processes saisfying 6 and 7 as well as ensuring ha he following sochasic inegrals for [, T e δu V x u, X u c,θ, Y u θ u σ S u dw u and e δu V y u, X u c,θ, Y u σ Y u dz u, 1 are genuine maringales, are deemed admissible. In his case, we wrie c, θ A. I is a consequence of 6 and Iô s lemma ha he sochasic inegrals 1 are always well-defined local maringales. In order o rigorously prove our main exisence resul saed in he nex secion, we insis on only allowing he invesor o use hose sraegies ha produce genuine maringales in 1. For square inegrable inegrands, he sochasic inegrals are indeed maringales, see e.g., Proer 24, Ch. IV The opimal sraegies The invesor is assumed o maximize expeced uiliy of running consumpion and herefore seeks a pair ĉ, ˆθ A aaining he following maximum [ E e δs e acs ds, X c,θ = x. sup c,θ A We do no require he invesor s [ sraegies o produce finie expecaion, i.e., we allow T for sraegies such ha E e acs ds = +. Of course, he opimal sraegies ĉ, ˆθ produce a finie expecaion. As usual, in order o apply he dynamic programming principle, he problem is embedded ino a family of problems, and since X c,θ, Y form a Markovian sysem, we can define [ J, x, y = E e δs e acs ds, [, T, X c,θ = x, Y = y. 11 sup c,θ A We refer o J as he value funcion or he indirec uiliy funcion. The explici soluion o he invesor s problem is saed in he following heorem, and he proof can be found in he Appendix. Theorem 1. When Assumpions 1 and 2 are saisfied, he value funcion J defined in 11 is idenical o he funcion V defined in 9, i.e., J = V. Furhermore, he opimal consumpion and invesmen sraegies are given by ĉ, x, y = A 1 x + y + 1 a A 1 β, s gu du ds, 12 λs ˆθ, x, y = A aσ S ρσ Y, 13 σ S for x, y R and [, T, where he deerminisic funcion g is defined by 8. 9

12 As shown in he Appendix, he opimal sraegies are such ha he dynamics of he opimal financial wealh ˆX = X ĉ,ˆθ is given by d ˆX = [m A 1 r ˆX d + A [ a 1 λ S ρσ Y dw, 14 for he mean reversion level m defined by m = Aλ S [a 1 λ S ρσ Y a 1 A 1 β, s gududs. 15 In oher words, he financial wealh is an Ornsein-Uhlenbeck ype process wih imedependen coefficiens. As T, he speed of mean reversion, A 1 r, converges o infiniy, while boh m and he volailiy of ˆX converge o zero. As we prove in he Appendix, his forces he opimal erminal wealh o become zero a ime T, i.e., we have ˆX T = almos surely and, in paricular, 7 is saisfied. In oher words he opimal financial wealh is given by an Ornsein-Uhlenbeck ype bridge process. 3.3 Discussion We can measure how he invesor values he income sream by he exra iniial wealh which is needed o compensae he invesor for he loss of he enire income sream. For he case of no income a all, he value funcion is given by where V, x = A 1 exp { aa 1 x A 1 β, s ḡ = δ r λ S 2 = g a } ḡu du ds, µ Y ρσ Y λ S a 2 1 ρ2 σ Y 2. The wealh equivalen of he income sream, L, y, is hen defined by V, x + L, y = V, x, y, which implies ha L, y = Ay + β, s µ Y u ρσ Y uλ S u a 2 1 ρ2 σ Y u 2 du ds. 16 The firs erm is he presen value of a consan paymen sream a he rae y, whereas he second erm correcs for income growh, covariance wih he sock price, and unspanned income risk. The sum of he wo erms is equal o he expecaion of he discouned fuure income under an appropriaely risk-adjused measure Q. The measure Q is defined by applying he marke-given price of risk λ S for he risk represened by W and he 1

13 agen-specific price of risk λ Y = 2 a 1 ρ 2 σ Y associaed wih he unspanned income risk represened by Z. The income dynamics can be wrien as dy = µ Y ρσ Y λ S a 2 1 ρ2 σ Y 2 d + σ Y ρ dw Q + 1 ρ 2 dz Q, where W Q and Z Q are independen Q-Brownian moions. This produces he expecaion [ β, sy s ds = L, Y. where E Q The value funcion 9 can be rewrien as V, x, y = A exp { aa 1 [x + H + L, y }, H = a 1 β, s δ ru + 12 λ Su 2 du ds can be inerpreed as a correcion of curren wealh due o he fuure invesmen opporuniies offered by he sock. The opimal consumpion 12 can be expressed as ĉ, x, y = A 1 [x + H + L, y. Hence, he opimal consumpion a ime is equal o he consan consumpion he invesor can ge by annuiizing he perceived oal wealh X +H+L, Y over he remaining lifeime. As perceived oal wealh changes and he remaining ime period shorens, he opimal consumpion changes. Noe ha he invesor wih iniial wealh x and he given income sream consumes exacly as an invesor wih iniial wealh x + L, y and no income. Given he dynamics of financial wealh in 14, he dynamics of he sum of financial wealh and he presen value of fuure income is of he form d ˆX + L, Y =... d + Aa 1 λ S dw + A 1 ρ 2 σ Y dz. Therefore, he opimal invesmen sraegy wih unspanned income risk is such ha he oal wealh has he same sensiiviy owards he hedgeable shocks represened by dw as he financial wealh has in he case wihou unspanned income risk. The presence of he unhedgeable shock represened by dz does no affec he desired exposure o he hedgeable shock. The dynamics of he opimal consumpion ĉ = ĉ, ˆX, Y becomes dĉ = µ c d + a 1 λ S dw + 1 ρ 2 σ Y dz, for some deerminisic funcion of ime µ c. Noe ha any idiosyncraic income shock posiive or negaive leads o an immediae idenical change in consumpion. 11

14 Theorem 1 shows ha he opimal number of socks held by he invesor is deerminisic and, in paricular, independen of wealh and income. The amoun opimally invesed in he sock a ime is λs ˆθ, x, ys = A ρσ Y, aσ S /S σ S /S where σ S /S is he relaive volailiy of he sock. The opimal sock posiion is a combinaion of he sandard myopic or speculaive erm and a erm correcing for he sock-like componen of he income process an income hedge erm. This is in line wih he sandard resuls for dynamic porfolio problems, cf., e.g., Meron 1969, 1971 and Bodie, Meron, and Samuelson Henderson 25 maximizes negaive exponenial uiliy of erminal wealh assuming a consan ineres rae r so ha A = 1 e rt /r, a consan Sharpe raio λ S of he sock, and a Gaussian income like 4 bu wih consan drif and volailiy. Henderson finds ha he amoun opimally invesed in he sock a ime is θ Hend rt λ, x, ys = e A ρσ Y. aσ S /S σ S /S Henderson assumes ha he relaive volailiy of he sock, σ S /S, is consan, bu his has no impac on he srucure of he opimal porfolio. 4 The appropriae weigh on he speculaive invesmen is e rt wih uiliy of erminal wealh, bu A in our case wih uiliy of inermediae consumpion. In Henderson s case, he weigh is he presen value of geing a uni paymen a he erminal dae. In our case, he weigh is he presen value of geing a uni coninuous paymen over he remaining lifeime. Furhermore, noe ha if we le T our resuls become similar o hose found in he infinie ime horizon seing of Svensson and Werner Finally, we consider he opimal amoun invesed in he risk-free asse, i.e. ˆX ˆθ S. The unspanned componen of he income dynamics affecs he wealh dynamics in 14 via he mean reversion level m in 15. More precisely, he variance of he unspanned income shock decreases he funcion g in 8 and, hus, increases he mean reversion level of wealh via he las erm in 15. The impac of unspanned income risk on m increases proporionally wih he degree of absolue risk aversion a. Therefore, he unspanned income risk leads o an increase in financial wealh and, since sock holdings are unaffeced, he wealh increase comes via an increase in he posiion in he risk-free asse. This is he precauionary savings caused by risk aversion and unhedgeable persisen income shocks. 4 Our expression for he opimal porfolio generalizes o he case wih any sochasic volailiy as long as he risk-free rae and he Sharpe raio of he sock are deerminisic, see e.g., Deemple, Garcia, and Rindisbacher 23 and Nielsen and Vassalou 26. S 12

15 4 Equilibrium 4.1 Derivaion of he equilibrium We derive he equilibrium risk-free rae and sock price in he economy by aggregaion over he invesors. We le τ i = 1/a i denoe he absolue risk olerance of invesor i and τ Σ = I i=1 τ i he aggregae absolue risk olerance in he economy. As shown in Theorem 1, invesor i opimally invess in λ S ˆθ i = A τ i σ S ρ iσ Y i, [, T, 17 σ S unis of he sock. To ensure ha he sock marke clears, we need ˆθ i i = 1, which implies ha he equilibrium Sharpe raio of he sock or he marke price of risk is [ λ S = 1 A 1 σ S + ρ i σ Y i, 18 τ Σ i where he volailiy σ S is sill o be idenified. I follows from 18 ha he Sharpe raio increases wih he income-sock correlaions ρ i. The larger he income-sock correlaions are, he more negaive he invesors income hedge erms are. Therefore, in order o mainain marke clearing, he Sharpe raio λ S of he sock has o increase as well. The sock price S is given as he appropriaely risk-adjused expecaion of he discouned fuure dividends. The adjusmen for he risk affecing he dividend sream 1 is given by he marke price of risk λ S. Under our conjecure ha he risk-free rae is deerminisic, he value of he discouned dividends is no affeced by he idiosyncraic income shocks. We wish o show ha he sock price defined as S = E Qmin [ β, ud u du, [, T, is he equilibrium sock price. Here Q min is he minimal maringale measure defined by he Radon-Nikodym derivaive dq min dp The dividend dynamics 1 can be rewrien as { = exp 1 } λ S 2 d λ S dw. 2 dd = [ µ D λ S σ D d + σ D dw min, 13

16 where W min = W + λ Su du defines a sandard Brownian moion under Q min. By means of Fubini s heorem, we now find ha S = = = AD + β, ue Qmin [D u du β, u D + u β, u µd s λ S sσ D s ds du u µd s λ S sσ D s ds du. 19 In paricular, he absolue volailiy of he sock price is σ S = Aσ D. By subsiuing his expression for σ S ino 18, we obain he following expression for he Sharpe raio λ S in erms of only exogenous quaniies: λ S = 1 τ Σ [ σ D + i ρ i σ Y i, [, T. 2 Therefore, we see ha Assumpion 1 ensures ha λ S given in 2 saisfies he second par of Assumpion 2. The bond marke clearing condiion is equivalen o he condiion ha he oal financial wealh is invesed in he sock, i.e., S = ˆX i i for all [, T where ˆX i denoes invesor i s opimal wealh process given by 14. We nex show ha S given by 19 saisfies his relaionship. Since ˆX it = for all i, i is immediae ha he relaion holds a = T and, hence, o show he validiy a ime, < T, i suffices o verify ha ds = i d ˆX i. From 5 and 12, he opimal wealh dynamics of invesor i can be wrien as d ˆX i = ˆXi r + Y i ĉ i d + ˆθ i µ S d + ˆθ i σ S dw [ r = A 1 ˆXi A 1 β, s τ i g i u du ds d + ˆθ i µ S d + ˆθ i σ S dw. By summing up and using ˆθ i i = 1, we obain [ r d ˆX i = A 1 ˆX i + µ S i A 1 β, s i i τ i g i u du ds d + σ S dw. 14

17 We rewrie he sock price dynamics 2 as ds = [ r A 1 S + µ S + A 1 S D d + σs dw, o see ha ds = i d ˆX i if, and only if, S = AD β, s By comparing 19 and 21, we conclude ha 21 is equivalen o i τ i g i u du ds. 21 µ D λ S σ D = i τ i g i, [, T. Therefore, by subsiuing in g i from 8 and rearranging, we find rτ Σ = τ i δ i + µ D + i i [ λ S σ D + i µ Y i ρ i σ Y i 1 2 τ Σλ S 1 a i 1 ρ 2 i σ Y i 2. 2 i Finally, he expression 2 for λ S can be used. following heorem. We summarize our findings in he Theorem 2. Given Assumpion 1, he equilibrium risk-free rae is r = i 1 2τ 2 Σ τ i δ i + 1 [ µ D + τ Σ τ Σ i [ σ D + i µ Y i 2 1 ρ i σ Y i a i 1 ρ 2 i σ Y i 2, 2τ Σ i 22 and he equilibrium sock price S is S = AD + β, u u [ µ D s τ Σ σ D s σ D s + i ρ i σ Y i s ds du. 23 This leads o a sock price volailiy of σ S = Aσ D and he Sharpe raio λ S = 1 τ Σ [ σ D + i ρ i σ Y i, [, T. 24 In paricular, r saisfies he firs par and λ S he second par of Assumpion 2. 15

18 Noe ha he Sharpe raio increases when more of he income risk is spanned. Higher ρ i σ Y i leads o a larger negaive income hedge demand for he sock, cf. 17, and, consequenly, a lower oal sock demand. To mainain marke clearing, a higher Sharpe raio is necessary in order o increase he speculaive demand. The invesors risk aiudes deermine how much he Sharpe raio has o increase. An increase in ρ i σ Y i does no affec he absolue volailiy of he sock, bu i lowers he sock price and, hus, increases he relaive volailiy σ S /S. The expeced excess rae of reurn on he sock, µ S /S = λ S σ S /S, is herefore also increasing in ρ i σ Y i. The equilibrium risk-free rae depends on he degree of spanning, i.e., he income-sock correlaions, via he las wo erms in 22. For simpliciy, assume for a momen ha a i = a = 1/τ, ρ i = ρ, and σ Y i = σ Y for all I invesors, and le σ D = σ D /I be he absolue volailiy of per capia dividends. In his case, he las wo erms in 22 add up o 1 σ 2τ 2 Y 2 1 [ σ 2τ 2 D 2 + 2ρ σ D σ Y, which is decreasing in he income-dividend correlaion. Inuiively, a higher income-dividend correlaion induces invesors o shif funds from he risky o he risk-free asse and, herefore, in order o mainain marke clearing, he equilibrium risk-free rae has o decrease. 4.2 Comparison wih a represenaive agen equilibrium Aggregae consumpion in our economy is C Σ = i ĉi = D + i Y i wih dynamics dc Σ = [ µ D + i [ µ Y i d+ σ D + i ρ i σ Y i dw + i 1 ρ 2 i σ Y i dz i. 25 If we vary he income-sock correlaions, he aggregae consumpion dynamics changes, which in iself causes changes in asse pricing momens. To ge a clearer undersanding of he impac on unspanned income risk, we nex compare he equilibrium in our seing o he equilibrium in an economy wih he same aggregae consumpion dynamics, same dividend process, and same preferences, bu where all income risks are spanned. In his complee marke economy, a represenaive agen exiss and maximizes ime-addiive expeced uiliy E[ exp{ δ}uc Σ d. If we wrie he aggregae consumpion dynamics as dc Σ = µ C d + σc db, where B is a possibly muli-dimensional sandard Brownian moion, i is well-known see, e.g., Breeden 1986 ha he equilibrium risk-free rae is r REP = δ + U C Σ U C Σ µ C 1 U C Σ 2 U C Σ σ C 2, 26 where denoes he sandard Euclidian norm, and ha he Sharpe raio of he risky asse is λ REP = U C Σ U ρ CD σ D σ C, 27 C Σ 16

19 where ρ CD is he dividend-consumpion correlaion. I is also well-known ha when a complee or effecively complee marke economy is populaed by I invesors having he negaive exponenial uiliy 3 wih differen ime preference raes δ i and differen absolue risk olerances τ i = 1/a i, hen a represenaive agen exiss and has negaive exponenial uiliy wih ime preference rae i τ iδ i /τ Σ and absolue risk olerance τ Σ, cf. Huang and Lizenberger 1988, Sec The risk-free rae in 26 is hen r REP = i τ i δ i + 1 µ C 1 τ Σ τ Σ 2τ 2 σ C 2 Σ and he Sharpe raio in 27 is λ REP = 1 τ Σ ρ CD σ D σ C. Consider firs he Sharpe raio. If he aggregae consumpion dynamics in he corresponding represenaive agen economy is given by 25, he Sharpe raio in ha economy is idenical o he Sharpe raio in our seing. By Jensen s inequaliy, we have 1/τ Σ i a i, which renders he Sharpe raio lower in our heerogeneous economy han in a homogeneous economy in which all invesors are equipped wih an absolue risk aversion equal o he average risk aversion in he heerogeneous economy. Consider nex he risk-free rae. If he aggregae consumpion dynamics is given by 25, he equilibrium risk-free rae in he corresponding represenaive agen economy becomes deerminisic and is equal o r REP = i 1 2τ 2 Σ τ i τ Σ δ i + 1 τ Σ [ µ D + i [ σ D + i µ Y i 2 ρ i σ Y i + 1 ρ 2 i σ Y i 2. In he absence of idiosyncraic income risk, i.e., when ρ 2 i = 1 for all invesors i, our economy does allow for a represenaive agen and, consequenly, he wo expressions 22 and 28 are idenical. In general, a represenaive agen does no exis in our framework and, hus, we canno hope for an expression of he ype 26. The wo firs erms in he expressions 22 and 28 are idenical and hus no affeced by unspanned income risk. The firs erm is a weighed average of he individual ime preference raes, where he weigh for invesor i equals her share of he aggregae absolue risk olerance. The ime preference raes of he more risk oleran invesors have he highes weighs in deermining he common ime preference rae. This is consisen wih Gollier and Zeckhauser 25, who consider how a group of heerogeneous invesors share an exogenous and deerminisic consumpion process. They show for general ime-addiive 17 i 28

20 uiliy funcions ha he collecive decisions correspond o he decisions of a represenaive agen equipped wih a ime preference rae equal o a risk olerance weighed average of he individual ime preference raes. In general, he weighs and, hence, he represenaive ime preference rae vary over ime as he individual consumpion raes vary. However, in our seing wih negaive exponenial uiliy, he weighs are consan over ime. If he more risk oleran invesors end o have he highes [lowes ime preference raes, he common ime preference rae is higher [lower han he average ime preference rae wih he obvious consequence for he equilibrium risk-free rae. The averaging of he ime preference raes is independen of he risk srucure in he economy and, hus, he same in our model as in he corresponding represenaive agen model. The second erm in he risk-free rae is he produc of he expeced growh rae of aggregae consumpion and he reciprocal of he aggregae risk olerance. As for he Sharpe raio, his erm is smaller in our heerogeneous economy compared o a homogeneous economy in which all invesors are equipped wih an absolue risk aversion equal o he average risk aversion in he heerogeneous economy. The unspanned income risk affecs he equilibrium risk-free rae via he precauionary savings moive as refleced by he difference in he las erm on he righ-hand sides of 22 and 28. Risk-averse invesors exposed o unhedgeable income shocks increase heir demand for he risk-free asse, which leads o a higher equilibrium price and a lower equilibrium risk-free rae. Oher hings being equal, more risk-averse invesors increase heir demand for he risk-free asse by more. Therefore, in deermining he equilibrium risk-free rae i is naural ha he unspanned income risk of each invesor is scaled by he invesors s risk aversion. Indeed, his is refleced by our soluion for he opimal consumpion and invesmen sraegies as discussed in he final paragraph of Secion 3. Wih idenical aggregae consumpion, he difference beween he risk-free rae in he economy wih unspanned income risk and in he represenaive agen economy is r r REP = 1 a i 1 1 ρ 2 i σ Y i τ Σ τ Σ i This difference is negaive whenever ρ 2 i < 1 and σ Y i > for some i, since a i = 1/τ i > 1/ i τ i = 1/τ Σ. We herefore have he following corollary: Corollary 1. When Assumpion 1 is saisfied, σ Y i >, and ρ 2 i < 1, he equilibrium risk-free rae in he economy wih unspanned income risk is smaller han he equilibrium risk-free rae in he corresponding complee marke economy wih idenical aggregae consumpion dynamics. As explained in he inroducion, he presen paper seems o be he firs o sugges and quanify he role of unspanned income risk in he deerminaion of he equilibrium riskfree rae and as a poenial explanaion of he risk-free rae puzzle. Wih a lower risk-free 18

21 rae and he same Sharpe raio, i is clear ha he expeced reurn on he sock keeping he volailiy fixed is also lower in our seing han in he corresponding represenaive agen seing wih idenical aggregae consumpion dynamics. In order o assess he quaniaive impac of unspanned income risk, suppose ha all I invesors have he same absolue risk aversion a i = a, he same consan income volailiy σ Y i = σ Y, and he same income-sock correlaion ρ i = ρ. Then τ Σ = I/a and we have r r REP = 1 2 a2 1 1 I 1 ρ 2 σ 2 Y 1 2 a2 1 ρ 2 σ 2 Y, for I. Various sudies repor ha individuals ypically seem o have a relaive risk aversion around 2 see e.g., Szpiro 1986; Guo and Whielaw 26; Paiella and Aanasio 27, labor income wih a percenage volailiy of around 1%, and a correlaion wih he sock marke near zero see e.g., Gourinchas and Parker 22; Cocco, Gomes, and Maenhou 25; Davis, Kubler, and Willen 26. If we hink of wealh and income being fairly evenly disribued and iniial wealh having he same magniude, he above limiing ineres rae difference becomes =.2, i.e., he model wih idiosyncraic income risk produces a risk-free rae which is 2 percenage poins lower han in he corresponding represenaive agen model. This difference is increasing in and highly dependen on he risk aversion level and he income volailiy level, bu less dependen on he income-sock correlaion as long as i is near zero. For a risk aversion of 3, he risk-free rae reducion is 4.5 percenage poins. In order o ge a feeling of he imporance of heerogeneiy in risk aversion and income risk, suppose ha one hird of all invesors has a risk aversion of 1 and an income volailiy of.5, anoher hird has a risk aversion of 2 and an income volailiy of.1, whereas he las hird has a risk aversion of 3 and an income volailiy of.15. Then we ge τ Σ = 11/18I and leing I, he ineres differenial r r REP approaches approximaely.245. Conversely, if he invesors wih a risk aversion of 1 have an income volailiy of.15 and he invesors wih a risk aversion of 3 have an income volailiy of.5, whereas he invesors wih risk aversion 2 sill have an income volailiy of.1, hen he ineres rae differenial becomes approximaely.136. These examples illusrae ha he quaniaive impac of idiosyncraic income risk on he equilibrium risk-free rae depends on he cross-secional relaion beween he invesors risk aversion and income risk. In paricular, he model does no allow aggregaion in he sense ha he equilibrium asse prices are independen of he allocaion of unspanned income risk across invesors wih heerogenous risk aversion. 5 Then a 2 σ 2 Y = γ/w 2 v Y Y 2 = γ 2 v 2 Y, where γ is he relaive risk aversion and v Y is he percenage income volailiy. 19

22 5 Conclusion This paper has offered he firs closed-form soluion for he equilibrium risk-free rae and he equilibrium sock price in a dynamic economy in which invesors have heerogeneous preferences and unspanned income risk. The degree of spanning of he individual income risks has a clear impac on he equilibrium consumpion and asse prices. A smaller fracion of income risk being spanned produces a lower equilibrium risk-free rae and a lower sock marke Sharpe raio, parly due o changes in aggregae consumpion dynamics. On he oher hand, if we keep he aggregae consumpion dynamics fixed, he Sharpe raio is he same while he risk-free rae and expeced sock reurn are boh lower in he economy wih unspanned income risk han in he corresponding represenaive agen economy wih all income risks being spanned. The reducion in he risk-free rae depends on he magniude of all invesors unspanned income risk as well as of heir risk aversion coefficiens, and he reducion is highes when he more risk-averse invesors face he larges unspanned income risk. In a numerical example wih reasonable parameers, we illusraed ha he reducion in he risk-free rae and, hus, he expeced sock reurn for a fixed volailiy can be several percenage poins. Our model herefore suggess ha unspanned income risk may play an imporan role in explaining he risk-free rae puzzle. Obviously, our assumpions of negaive exponenial uiliy and normally disribued incomes and dividends are unrealisic. I seems impossible o obain closed-form soluions under more appropriae assumpions, and even he fundamenal heoreical quesion of exisence of an equilibrium is currenly unclear. A naural nex sep is o perform a numerical analysis of such more realisic seings. This may shed furher ligh on he quaniaive effecs of unspanned income risk on he equilibrium prices of financial asses. 2

23 A Proofs Proof of Theorem 1 The Hamilon-Jacobi-Bellman HJB equaion associaed wih he problem 11 is { δv = sup e ac { + V + V x rx c + y + θµs } + 1 c,θ R 2 V xxθ 2 σ S 2 + V y µ Y + 1 } 2 V yyσ Y 2 + V xy θρσ S σ Y, [, T, x, y R, wih subscrips on V indicaing parial derivaives. Since our invesor only receives uiliy of running consumpion, he erminal condiion is given by V T, x, y =, x, y R, 3 where he requiremen x is due o our admissibiliy requiremen 7. The proof proceeds hrough he usual seps required in order o apply he dynamic programming principle. Sep I: Explici soluion o he HJB equaion. The firs-order condiions in he HJB equaion lead o he opimal conrol candidaes ĉ = a 1 log a 1 V x, ˆθ = V x V xx µ S σ S 2 V xy V xx ρσ Y σ S. 31 By subsiuing hese expressions ino he HJB equaion, we obain he PDE δv = V + V x rx + y a 1 + a 1 V x log a 1 1 V x λ S 2 2 V xx V 2 xy 1 ρ 2 σ Y 2 + V y µ Y V xx 2 V yyσ Y 2 V xv xy ρσ Y λ S, [, T, x, y R. V xx We ry a soluion of he form V, x, y = exp{ Hx Gy F } for smooh deerminisic funcions F, G and H. The opimal conrol candidaes can be expressed as ĉ = a 1 log a 1 H Hx Gy F, 1 ˆθ = ρσ Y G λ S. Hσ S Formally insering hese expressions ino he PDE gives us he wealh dynamics see 5 dx ĉ,ˆθ = X ĉ,ˆθ { r a 1 H } { + Y 1 a 1 G } + a 1 log a 1 H F + λ S H λ S Gρσ Y d + 1 λ S ρσ Y G dw. H V 2 x 21

24 Based on he below lemma, we see ha o ensure admissibiliy of he opimal conrol candidaes, we wish o have lim H = +, lim T G = a, lim T m = lim T T where we have defined he mean reversion funcion m by m = a 1 log a 1 H F 1 ρσ Y G λ S =, H + λ S λ S Gρσ Y, [, T. H To compue F, G and H explicily, we compue he involved derivaives and inser hem ino he above PDE. Afer dividing hrough wih he exponenial erm, we collec he x and y-erms and hereby obain he following coupled sysem of ODEs x erms : H = a 1 H 2 Hr, HT = +, y erms : G = a 1 HG H, GT = a. Direc calculaions show ha G = a for [, T whereas he Ricai equaion for H has he soluion H = aa 1, [, T, where A is he annuiy facor. To ge he erminal condiion for F, we inser hese expressions ino he mean reversion funcion m m = a 1 log A + F + A a 1 λ S ρσ Y. [, T, From his expression, i follows ha F s erminal condiion is log A + F =. lim T Since A for T, i mus be he case ha F for T, and his implies ha he boundary condiion 3 is saisfied for x and y R. To ge he explici form for F, we collec he remaining erms of he PDE and find he linear ODE F = A 1[ log A F g r, where we have defined he deerminisic funcion g = δ r λ S 2 + a µ Y ρσ Y λ S 1 2 a2 1 ρ 2 σ Y 2, [, T. By using Leibniz s rule, we can verify ha F is given by F = e Au 1 du gs ds log A = F log A. 22

25 Furhermore, since A /A r = A 1, we have Au 1 du = A u Au du ru du = ln As ln A ru du, and, hus, exp Au 1 du = A 1 Asβ, s. By subsiuing his ino F and inegraing by pars, we ge F = A 1 Asβ, sgs ds = A 1 β, s This proves ha V defined by 9 solves he HJB equaion. gu du ds. Sep II: Admissibiliy of he opimal conrol candidaes. We hen urn o show ha he opimal candidae policies indeed are admissible in he sense of Definiion 1. The candidae for he opimal invesmen sraegy is given by λs ˆθ = A aσ S ρσ Y. σ S Since A converges o zero, Assumpions 1 and 2 ensure he needed inegrabiliy of ˆθ. The candidae process for opimal consumpion is given by ĉ, x, y = A 1 x + y + 1 a A 1 β, s Here he dynamics of X ĉ,ˆθ is given by dx ĉ,ˆθ gu du ds. = [m + r A 1 X ĉ,ˆθ d + A[a 1 λ S ρσ Y dw, where m is he he above deerminisic funcion, i.e., m = Aλ S [a 1 λ S ρσ Y a 1 F. The las erm in ĉ is equal o a F which converges o zero as T and is herefore inegrable. The second erm is Y which is an OU process wih nicely behaved mean and variance funcions, cf. Assumpion 1. The firs erm, A 1 X ĉ,ˆθ, is more complicaed because Au for u T. By means of Io s lemma we find he dynamics [ d A 1 X ĉ,ˆθ [ A = X ĉ,ˆθ 1 2 ra 1 d + A 1 dx ĉ,ˆθ = A 1 m d + [a 1 λ S ρσ Y dw. 23

26 From his and Assumpions 1 and 2 i follows ha he variance funcion is finie 2d a 1 λ S ρσ Y <. However, he mean funcion is more complicaed. By inegraing, we find he mean funcion o be [ ψ = E A 1 X ĉ,ˆθ = A 1 X ĉ,ˆθ + λ S u[a 1 λ S u ρσ Y u du a 1 F u Au du. We wish o show ha ψ is a coninuous funcion on [, T and ha ψ does no blow up a T. This follows if we show ha he raio F u/au has a finie limi when u T because in ha case, F /A is a coninuous funcion on he compac inerval [, T. By he rules of Leibniz and L Hopial as well as he definiion of F we find lim u T F u Au = lim u T F uau Au 2 = lim u T ru F uau guau 2Au[ruAu 1 ru = lim F u gu = 1 gt, u T 2[ruAu 1 2 which is finie by Assumpions 1 and 2. Therefore, F /A is a coninuous funcion on [, T and as a consequence, he mean funcion ψ is also coninuous on [, T. In conclusion, he needed inegrabiliy condiion 6 is saisfied. Lemma 1 saed and proved below shows ha ĉ and ˆθ saisfy X ĉ,ˆθ T = and, herefore, we only we need o check he maringaliy of he wo sochasic inegrals in 1: e δu V x ˆθu σ S u dw u and e δu V y σ Y u dz u. To do so, i suffices o show square inegrabiliy of he inegrands, i.e., we need [ [ 2 T 2 E e δu V x ˆθu σ S u du < and E e δu V y σ Y u du <. By using he explici form of V and he opimal conrol candidaes ĉ, ˆθ we have V x ˆθu σ S u = a exp aau 1 X u ĉ,ˆθ ay u F λs u u Au ρσ Y u, a V y σ Y u = aau exp aau 1 X u ĉ,ˆθ ay u F u σ Y u. 24