Optimal portfolio choice and stochastic volatility

Transcription

1 Research Artcle Receved 5 October 21, Revsed 4 March 211, Accepted 6 March 211 Publshed onlne n Wley Onlne Lbrary (wleyonlnelbrary.com) DOI: 1.12/asmb.898 Optmal portfolo choce and stochastc volatlty Anne Gron a, Bjørn N. Jørgensen b and Ncholas G. Polson c In ths paper we examne the effect of stochastc volatlty on optmal portfolo choce n both partal and general equlbrum settngs. In a partal equlbrum settng we derve an analog of the classc Samuelson Merton optmal portfolo result and defne volatlty-adjusted rsk averson as the effectve rsk averson of an ndvdual nvestng n an asset wth stochastc volatlty. We extend pror research whch shows that effectve rsk averson s greater wth stochastc volatlty than wthout for nvestors wthout wealth effects by provdngfurther comparatve statc results on changes n effectve rsk averson due to changes n the dstrbuton of volatlty. We demonstrate that effectve rsk averson s ncreasng n the constant absolute rsk averson and the varance of the volatlty dstrbuton for nvestors wthout wealth effects. We further show that for these nvestors a frst-order stochastc domnant shft n the volatlty dstrbuton does not necessarly ncrease effectve rsk averson, whereas a second-order stochastc domnant shft n the volatlty does ncrease effectve rsk averson. Fnally, we examne the effect of stochastc volatlty on equlbrum asset prces. We derve an explct captal asset prcng relatonshp that llustrates how stochastc volatlty alters equlbrum asset prces n a settng wth multple rsky assets, where returns have a market factor and asset-specfc random components and multple nvestor types. Copyrght 211 John Wley & Sons, Ltd. Keywords: portfolo choce; stochastc volatlty; rsk averson; CAPM; Sten s lemma 1. Introducton Ths paper examnes the effect of stochastc volatlty on portfolo choce and equlbrum asset prces. Asset prcng models, both theoretcal and emprcal, routnely nclude stochastc volatlty. However, t s dffcult to evaluate what dfferent volatlty assumptons mply about economc behavor. In ths paper we derve a stochastc volatlty analog of the classc Samuelson [1] Merton [2] partal equlbrum optmal portfolo result, wth an explct expresson for the effectve rsk averson under stochastc volatlty. We refer to ths measure as the volatlty-adjusted rsk averson. In ths partal equlbrum settng we provde comparatve statcs on how volatlty-adjusted rsk averson changes wth the dstrbuton of the unknown volatlty. We also examne the effect of stochastc volatlty n a general equlbrum settng. We derve a stochastc volatlty analog of the captal asset prcng model (CAPM) and llustrate how the dstrbuton of volatlty affects equlbrum asset prces through ts effect on volatlty-adjusted rsk averson n a settng wth multple rsky assets and nvestor types. To perform our analyss we derve an extenson of Sten s lemma [3] that allows us to separate utlty effects from random varable covarance effects. Rubnsten [4] and Huang and Ltzenberger [5] llustrate the use of Sten s lemma for portfolo choce problems and Constantndes [6] provdes a further dscusson. The stochastc volatlty verson of Sten s lemma dffers from the standard result by addng a proportonalty factor based on a sze-based change of measure for the unobserved stochastc volatlty. Ths enables us to explctly defne volatlty-adjusted rsk averson whch we call VARA. It dffers from the standard rsk averson term by requrng an expectaton taken over ths sze-based measure. Usng our extenson of Sten s lemma we are able to show that the optmal rsky asset allocaton s separable as the expected excess return scaled by the return varance and VARA, analogous to the standard problem analyzed by Samuelson and Merton. Our research contrbutes to the lterature on optmal portfolo allocaton by examnng precsely how changes n stochastc volatlty affect portfolo choce. Our work s most closely related to Coles et al. [7], who compared optmal portfolo choces and equlbrum payoffs n the presence of stochastc volatlty to those wthout. They establsh that ndvduals wth exponental utlty are effectvely more rsk averse and asset prces are lower n the presence of stochastc volatlty. Our a NERA, Chcago, IL, U.S.A. b Unversty of Colorado, Boulder, CO, U.S.A. c Unversty of Chcago, Chcago, IL, U.S.A. Correspondence to: Ncholas G. Polson, Unversty of Chcago, Chcago, IL, U.S.A. E-mal: ngp@chcagobooth.edu Copyrght 211 John Wley & Sons, Ltd. Appl. Stochastc Models Bus. Ind. 211

2 approach, wth an explct expresson for VARA, allows for further comparatve statcs on how changes n the dstrbuton of the unknown varance change the effectve rsk averson under stochastc volatlty. In partcular, we show that a secondorder stochastc domnatng shft n the volatlty dstrbuton ncreases effectve rsk averson, whle ths s not guaranteed for a frst-order stochastc domnatng shft n the volatlty dstrbuton. Our approach dffers from recent work analyzng portfolo choce n contnuous tme, such as Lu [8], Chacko and Vcera [9], and Xa [1]. Our sngle perod settng allows us to focus on the effects of stochastc volatlty on effectve rsk averson whereas n a contnuous tme settng the level of current volatlty and the uncertanty of future volatlty are the prmary determnants of portfolo choce. Our results also contrbute to research that relates observatons about asset markets to ndvdual nvestor s behavor. One lne of ths research nvestgates the effect of asset return predctablty on optmal portfolo allocaton (see, for example, [11--14]). A related lterature examnes other emprcal regulartes, such as the equty premum puzzle, and nfers the ndvdual characterstcs that would be consstent wth these observatons. Our results allow research n these areas to derve an analytc soluton for the optmal portfolo allocaton decson wth parameter uncertanty n the covarance matrx wth a separable, well-defned volatlty-adjusted rsk averson parameter. Ths allows researchers to assess the economc behavor underlyng dfferent models of stochastc volatlty and to nvestgate the senstvty of the results to the specfcaton of the volatlty dstrbuton. The paper proceeds as follows. Secton 2 ntroduces stochastc volatlty and our measure of volatlty-adjusted rsk averson. We then examne the effect of stochastc volatlty n three dfferent settngs. Frst, we consder a partal equlbrum settng where a margnal nvestor takes the stock prce as gven. We derve an extenson of Sten s lemma to stochastc volatlty dstrbutons and use t to fnd a general analog to the classc Samuelson Merton optmal portfolo result. Second, n Secton 3, we consder an nvestor who has constant absolute rsk averson, thus wthout complcatng wealth effects. In ths case we derve comparatve statcs for the effect of changes n the dstrbuton of stochastc volatlty on effectve rsk averson of a representatve nvestor. Fnally, n Secton 4, we endogenously derve market-clearng equlbrum stock prces when frms future cash flows are affected by a sngle factor that exhbts stochastc volatlty. We verfy that the resultng equlbrum stock returns adhere to CAPM even when there s stochastc volatlty. Secton 5 provdes a summary and dscusson. All proofs are n the appendx. 2. Asset allocaton and stochastc volatlty In ths secton we derve a closed-form expresson for the optmal asset allocaton under stochastc volatlty. To do so, we present our verson of Sten s lemma that apples to random varables wth stochastc volatlty. We begn, however, wth the standard problem wthout stochastc volatlty to ntroduce notaton and provde a reference pont. In the ndvdual s portfolo problem, an agent has ntal wealth that can be nvested n a rsk-free bond or n a rsky stock. The bond yelds a known gross return of R f at the end of the perod. The rsky stock has a stochastc gross return of R. We normalze the ntal wealth to unty and denote the fracton of ntal wealth nvested n the rsky stock as ω. Then fnal wealth s W =(1 ω)r f +ωr. The agent s problem s to choose ω to maxmze the expected utlty of fnal wealth, or max ω E[U(W )]. If U s twce dfferentable, ncreasng and strctly concave n ω, the optmal allocaton s characterzed by the frst-order condton: Applyng the defnton of covarance yelds E[U (W )(R R f )]=. Cov[U (W ), R R f ]+ E[U (W )]E[R R f ]=. When the random varable R s normally dstrbuted, Sten s lemma can be appled to derve a more nformatve expresson. Sten s lemma equates the covarance of a functon of normal random varables to the underlyng covarance tmes a proportonalty constant. More precsely, let X denote a normal random varable, X N(μ,σ 2 ) wth mean μ and varance σ 2 and let g(x) be the dfferentable functon of X such that E[ g (X) ]<. ThenCov[g(X), X]= E[g (X)]σ 2. In the bvarate case for normal random varables (X,Y ) Sten s lemma becomes Cov[g(X),Y ]= E[g (X)]Cov[X,Y ], see Sten [3]. In contnuous tme the nstantaneous volatlty s known wth certanty and only future volatlty s stochastc. Thus the portfolo allocaton problem n contnuous tme s drven prmarly by the level of current volatlty and the uncertanty of future volatlty. In that our dscrete tme, sngle perod approach abstracts from nter-temporal substtuton and hedgng demands, our settng s more approprate to stuatons where the nvestment horzon s long relatve to the change n volatlty. Copyrght 211 John Wley & Sons, Ltd. Appl. Stochastc Models Bus. Ind. 211

3 Applyng ths dentty to the frst-order condton yelds ωe[u (W )]Var[R]+ E[U (W )](E[R] R f )=. Hence, the optmal allocaton ω s ω = 1 γ ( E[R] R f Var[ R] ), (1) where γ s the agent s global absolute rsk averson: γ= E[U (W )]/E[U (W )]. Ths s the well-known Samuelson [1] and Merton [2] result that the optmal portfolo weght s multplcatvely separable n rsk averson and the market prce of rsk Sten s lemma for stochastc volatlty We extend ths result to the case where the dstrbuton of the rsky asset return has stochastc volatlty. To do so we now develop an extenson of Sten s lemma that apples to random varables wth stochastc volatlty. In general, a random varable X whose volatlty s drawn from a probablty densty functon p(v ) s sad to exhbt stochastc volatlty, V,f we can wrte X V N(μ,σ 2 V )wherev p(v ), V, and σ 2 >. The dstrbuton of outcomes X s p(x)= p(x V )p(v )dv. Wth stochastc volatlty, the margnal dstrbuton of X has heavy tals. Sten s lemma equates the covarance of a functon of normal random varables to the underlyng covarance tmes a proportonalty factor. The crtcal dfference between the Sten s lemma and our dervaton s that the proportonalty factor undergoes a change of measure that we denote by q(v ). Ths densty comes from sze-basng the densty of the volatlty and s defned as q(v )= Vp(V ) E[V ], (2) where we assume that < E[V ]<.Wedefneq(X) to be the nduced margnal dstrbuton of X gven by ths measure q(v ) on the volatlty, namely q(x)= p(x V )q(v )dv. (3) In general, sze-basng n ths way causes the densty q(x) to have heaver tals than the orgnal densty p(x). Theorem 1 (Sten s lemma for stochastc volatlty) Let X be a random varable wth a stochastc volatlty so that X V s dstrbuted N(μ,σ 2 V )andv has densty p(v )thats non-negatve only for V. Suppose that < E[V ]<. Then, we have Cov[g(X), X]= E Q [g (X)]Var[X]. Further, f (X,Y V ) are bvarate Normal random varables then Cov[g(X),Y ]= E Q [g (X)]Cov[X,Y ], where E Q s the expectaton taken under the measure nduced by sze-basng q(v )= Vp(V )/E[V ]. Proof See Appendx A. There s a slght abuse of notaton as we use V to denote a random varable and p(v ) ts densty. The class of stochastc volatlty dstrbutons has a number of applcatons n fnancal economcs, manly because t allows for heavy tals. The famly of dstrbutons ncludes the well-used tν-dstrbuton, Exponental power [15], Stable [16], ellptcal dstrbutons [17], varance gamma dstrbutons [18], and the Double Exponental and logstc dstrbutons [19, 2]. Copyrght 211 John Wley & Sons, Ltd. Appl. Stochastc Models Bus. Ind. 211

4 2.2. Asset allocaton under stochastc volatlty We can now derve the optmal portfolo choce when gross returns, R, exhbt stochastc volatlty. Thus R V s dstrbuted normally and V p(v ), V. Applyng our theorem to the portfolo choce problem wth stochastc volatlty yelds the dentty Cov[U (W ), R R f ]= E Q [U (W )]Var[R]. Ths can be substtuted nto the frst-order condton, and proceedng as before we fnd that the optmal share of wealth allocated to the rsky asset s ω SV = E[U ( ) (W )] E[R] R f E Q [U. (4) (W )] Var[ R] We defne the volatlty-adjusted rsk averson, Γ,as Γ= E Q [U (W )] E[U, (5) (W )] where E Q s the expectaton taken under the measure nduced by sze-basng the volatlty dstrbuton q(v )= Vp(V )/E[V ]. Usng our defnton of VARA, we can smplfy our expresson for the optmal portfolo weght as ω SV = 1 ( ) E[R] R f. (6) Γ Var[ R] By applyng our extenson of Sten s lemma we fnd that the optmal portfolo weght wth stochastc volatlty s agan separable nto rsk averson and market prce of rsk. We defne Γ as the volatlty-adjusted rsk averson, VARA, that characterzes the nvestor s rsk preferences under stochastc volatlty. 3. Absolute volatlty-adjusted rsk averson We now examne the effects of stochastc volatlty on VARA n a partal equlbrum model where we assume that the share nvested n the rsky asset s exogenously gven. To solate rsk averson effects we consder an nvestor wth exponental utlty of fnal wealth, U(W )= e γw,whereγ>. In the absence of stochastc volatlty the nvestor has constant absolute rsk averson, namely U (W )/U (W )=γ. However, when we account for stochastc volatlty, nvestors act as f ther rsk averson s gven by the VARA measure. To be clear about ths partcular settng, we refer to VARA n ths specal case as absolute VARA. Our frst set of results compare effectve rsk averson wth and wthout stochastc volatlty. One result restates a result of Coles et al. [7] wthn our framework, namely that effectve rsk averson s greater n the presence of stochastc volatlty than wthout. The other result provdes comparatve statcs on the effect of margnal changes n the degree of stochastc volatlty, somethng that s dffcult to do wthout an explct form for VARA. We show that absolute VARA s ncreasng n both constant absolute rsk averson and n the non-stochastc varance parameter. Next we examne the effects of changes n the dstrbuton of absolute VARA, specfcally a frst- and second-order stochastc domnatng shft n volatlty. Fnally, we fnsh ths secton wth three applcatons llustratng absolute VARA wth three commonly used volatlty dstrbutons. In our analyss we make use of the fact that under exponental utlty VARA can be expressed as the product of the constant absolute rsk averson γ and a functon of the volatlty dstrbuton. To llustrate, we specfy termnal wealth as W = R f +ω(r R f ), where R exhbts stochastc volatlty as before. We can wrte expected utlty E[U(W )] as an terated expectaton E[U(W )]= E V {E[U(W ) V ]} and by drect calculaton we have E[U(W ) V ]= E[ e γw V ]= e γe[w V ]+ 1 2 γ2 Var[W V ]. Snce E[W V ]= E[W ]=(1 ω)r f +ωe[r]andvar[w V ]=ω 2 σ 2 V, we get that By a smlar argument as above, E[U(W ) V ]= e γ{(1 ω)r f +ωe[r]} e 1 2 γ2 ω 2 σ 2V. E[U (W )]= γe γe[w] E[e 1 2 γ2 ω 2 σ 2V ] and E Q [U (W )]=γ 2 e γe[w] E[V e 1 2 γ2 ω 2 σ 2V ]. E[V ] Usng these results, we can now show that the absolute VARA s ndependent of the expected returns: Γ CARA = E Q [U (W )] E[U (W )] E[V e 1 2 γ2 ω 2 σ 2V ] =γ E[e 1 2 γ2 ω 2 σ 2V ]E[V ]. Copyrght 211 John Wley & Sons, Ltd. Appl. Stochastc Models Bus. Ind. 211

5 For the representatve nvestor where ω=1, we can smplfy ths expresson to Γ CARA E[V e 1 2 γ2 σ 2V ] =γ E[e 1 2 γ2 σ 2V ]E[V ] =γ E T [V ] E[V ], where the superscrpt T ndcates that expectatons are taken under the exponentally tlted dstrbuton T (V )= e 1 2 γ2 σ 2V p(v )/E[e 1 2 γ2 σ 2V ]. Ths representaton of absolute VARA through the tlted dstrbuton facltates the comparatve statc results that follow. To provde nsght nto the role of the tlted dstrbuton on portfolo choce, consder the representaton of the optmal portfolo weght. Under constant absolute rsk averson we replace VARA, Γ, wth the specalzed absolute VARA, Γ CARA, yeldng ω SV = 1 ( ) E[R] R f Γ CARA. Var[ R] Usng the defnton of absolute VARA above, ths smplfes to ω SV = 1 ( E[R] R f γ Var T [R] ), (7) where Var T [R]=σ 2 E T [V ]. These two equatons hghlght two ways of thnkng about the effect of stochastc volatlty on the optmal portfolo weght. In the top equaton, one can thnk of nvestors usng the expected return and varance of R and then behavng as f they have a rsk averson parameter that dffers from ther utlty functon rsk averson parameter γ. On the other hand, we can thnk of nvestors as havng the utlty functon rsk averson parameter γ, but nstead of usng the varance of returns under stochastc varance to make decsons, Var[R]=σ 2 E[V ], nvestors apply the larger varance evaluated under the tlted dstrbuton, Var T [R], as n the bottom equaton. Thus the tlted dstrbuton can be thought of as nducng the equvalent certan varance that leads the nvestor to the same portfolo allocaton as under stochastc volatlty. We now turn to our frst set of results on the propertes of absolute VARA. Theorem 2 Absolute VARA Propertes I 1. Absolute VARA s greater than the constant absolute rsk averson, that s, Γ CARA γ. 2. Absolute VARA, Γ CARA, s ncreasng n constant absolute rsk averson γ and varance parameter, σ 2,thats Proof See Appendx B. Γ CARA γ >, ΓCARA σ 2 >. As noted above, the frst result restates a pror fndng wthn our framework. The second result, that an ncrease n the representatve nvestor s constant rsk averson or n the non-stochastc varance ncreases the effectve rsk averson, s new because t requres a measure of effectve rsk averson such as the one we develop here. Next, we examne the effect of a frst- and second-order stochastc domnatng change n the volatlty dstrbuton on absolute VARA. We fnd that a frst-order stochastc domnatng shft n the volatlty dstrbuton does not necessarly ncrease absolute VARA, whereas a second-order stochastc domnatng shft n the volatlty dstrbuton does ncrease absolute VARA. Theorem 3 Absolute VARA Propertes II 1. Consder two ndependent dstrbutons wth stochastc volatltes, V 1 and V 2, and assocated absolute VARA, Γ CARA 1 and Γ CARA 2.LetV = V 1 +V 2.ThenΓ CARA =πγ CARA 1 +(1 π)γ CARA 2,whereπ= E[V 1 ]/E[V 1 +V 2 ]. 2. Consder two dstrbutons wth stochastc volatltes, V and V 1, and assocated absolute VARA, Γ CARA and Γ CARA 1. Suppose that V = V 1 + Z,whereZ s ndependent of V 1,andE[Z]=, then Γ CARA Γ CARA 1. Proof See Appendx C. The frst result appears surprsng ntally. One mght expect that a frst-order stochastc domnant shft n the dstrbuton of volatlty would result n a greater effectve rsk averson. Whle a frst-order stochastc domnatng shft n the dstrbuton Copyrght 211 John Wley & Sons, Ltd. Appl. Stochastc Models Bus. Ind. 211

6 of the varance ncreases the expected varance and the varance of the varance, absolute VARA s a rato of the expectaton of the second dervatve of utlty to the expectaton of the margnal utlty. The dversfcaton effect of takng expectatons over the sum of ndependent dstrbutons partally offsets the effect of the greater mean and varance of the volatlty dstrbuton. Thus a frst-order stochastc domnatng shft n the varance dstrbuton leaves the resultng absolute VARA n between the two varance components and absolute VARA s a sub-addtve rsk measure, snce V = V 1 +V 2 mples that Γ CARA 1 +Γ CARA 2 Γ CARA. Theorem 3 also demonstrates the effect of a second-order stochastc domnatng shft n the dstrbuton of volatlty. An ncrease n the uncertanty about the varance leads to a hgher absolute VARA. In ths case there s no dversfcaton effect to offset the greater varance of the volatlty dstrbuton. The VARA representaton allows us to explctly examne how absolute VARA changes wth partcular assumptons. Consder the mplcatons on absolute VARA of three volatlty dstrbutons of nterest n fnancal economcs. All the three are members of the generalzed nverse Gaussan famly: the exponental, the nverse gamma, and a specal case of the generalzed nverse Gaussan dstrbuton. Frst, suppose that V s dstrbuted exponentally wth densty p(v )=λe λv, a smple and parsmonous choce. The nverse of the parameter λ represents both the expected volatlty and the standard devaton of volatlty. Hence, the rato of the expected volatlty to the varance of volatlty equals the constant λ. For suffcently low rsk averson, γ< 2λ/σ,the absolute VARA s fnte and can be computed as Γ CARA γ = ( ). λ γ2 σ 2 2 Note that Γ CARA / λ<. That s, an ncrease n the stochastc volatlty scaled by the expected volatlty ncreases absolute VARA. Ths s consstent wth the effect of an ncrease n the known varance, σ 2, shown n Theorem 2. Combnng ths expresson wth our pror result on the optmal portfolo share for the rsky asset, one can assess the effect of dfferent volatlty assumptons gven measures of γ, rsk prema, and volatlty. Not all dstrbutons of asset returns result n a well-defned portfolo allocaton problem, and volatlty dstrbutons assocated wth these dstrbutons of asset returns wll therefore have undefned VARA for every strctly postve constant absolute rsk averson γ. For example, the nverse gamma dstrbuton wth parameters p(v )=Γ(α) 1 α λ V λ 1 e α/v could be an attractve choce for the dstrbuton of V. When volatlty s dstrbuted nverse gamma the dstrbuton of returns s a t-dstrbuton whch arses naturally as the predctve dstrbuton of returns for a Bayesan nvestor learnng the return dstrbuton parameters as n [11, 12, 21]. However, as these researchers and others have dscussed, the expected utlty of termnal wealth under exponental utlty and t-dstrbuted asset returns s undefned. Therefore the nverse gamma dstrbuton for volatlty cannot be used to evaluate absolute VARA and portfolo allocaton. Both of these dstrbutons are part of a famly of volatlty dstrbutons known as the Generalzed Inverse Gaussan (GIG) famly. Whenever volatlty s dstrbuted n the GIG famly, the dstrbuton of returns follows a hyperbolc dstrbuton. Barndorff-Nelsen and Shephard [22] dscuss addtonal fnancal econometrc applcatons for ths famly of dstrbutons. Another member of the GIG famly that s more flexble than the exponental but stll results n a well-defned absolute VARA s the case where p(v )sgvenby p(v α,λ)= I(α,λ) 1 V 3 2 e 1 2 (αv 1 +λv ) for a sutable normalzaton constant I(α,λ). Here the expected volatlty s gven by E[V ]= α/λ and the varance of volatlty s Var[V ]= α/λ 3. For suffcently low constant absolute rsk averson, γ< λ/σ, absolute VARA s gven by Γ CARA =(γ 2 λ 1 σ 2 ) 1 2. See Appendx D for detaled proofs. Note that Γ CARA depends only on the market prce of volatlty λ= E(V )/Var(V )and the known volatlty σ 2 as n the case of exponental volatlty. In ths case, t s straghtforward to check ther comparatve statcs that Γ CARA / λ<and Γ CARA / α=. Smlar to the exponental dstrbuton, absolute VARA s affected only by the constant rato of the expected volatlty to the varance of volatlty λ. 4. Equlbrum prces under stochastc volatlty In ths secton we consder the equlbrum asset prce mplcatons when multple nvestors choose optmal portfolos over one safe and multple rsky assets ncorporatng stochastc volatlty effects. We endogenously derve the market clearng equlbrum stock prces when frm s future cash flows are affected by a sngle factor that exhbts stochastc volatlty. The resultng equlbrum stock returns adhere to a CAPM relatonshp when there s stochastc volatlty. Whle the fact Copyrght 211 John Wley & Sons, Ltd. Appl. Stochastc Models Bus. Ind. 211

7 that a CAPM relatonshp holds for stochastc volatlty s well known we provde somethng addtonal an explct mean-varance result n terms of volatlty-adjusted rsk averson wth endogenous equlbrum prces. Consder I nvestor types ndexed by =1,..., I and J frms ndexed by j =1,..., J. Investors dffer n ther ntal wealth, W, ther termnal wealth, W, and ther utlty functons over termnal wealth, U (W ). The termnal value of each frm j s stock s a random varable, X j, wth three elements: a frm-specfc mean, μ j, market-wde random factor, F,and a frm specfc, dosyncratc shock, ε j. Thus we wrte X j =μ j +b j F +ε j, where b j s the senstvty of frm j s termnal value to the market-wde factor and ε j N(,σ 2 ε ). Stochastc volatlty enters the problem n the dstrbuton of the market factor, p(f), wth p(f V ) N(, V σ 2 F ) and V p(v ). The condtonal dstrbuton of the value of stock j s then p(x j V ) N(μ j, Vb 2 j σ2 F +σ2 ε ). Gven that all parameters and dstrbutons are common knowledge, each nvestor maxmzes expected utlty of termnal wealth subject to the budget constrant by choosng the dollars to nvest n bonds, B, and the dollars to nvest n frm j.we wrte the dollars nvested n frm j as ω, j P j,whereω, j the fracton of the frm that the nvestor purchases at a prce of P j. Thus each nvestor solves max B,ω, j E[U (W )] and fnal wealth s equal to end of perod value of the nvestment, subject to W B + j ω, j P j W = B R f + ω, j X j. j Our man result n ths secton s that the market clearng equlbrum prces under stochastc volatlty follow an explct CAPM relatonshp as descrbed below. Theorem 4 (Equlbrum prces under stochastc volatlty) Let W m = W be aggregate fnal wealth, P m = j P j be the ntal value of all stocks, and X m = j X j be the end of perod aggregate value of all stocks. Then the ntal equlbrum prce of stock j for ndvdual nvestors wth end of perod utlty U (W )s P j = R 1 f (E[X j ] Γ m Cov[W m, X j ]) j, where Γ m =( Γ 1 ) 1 s the aggregate volatlty-adjusted rsk averson and Γ s the VARA of nvestor.furthermore,a CAPM prcng equaton holds E[R j ] R f =β j (E[R m ] R f ), where the beta, β j =Cov[R j, R m ]/Var[ R m ] s the rsk premum. Proof See Appendx E. The frst part of Theorem 4 establshes that volatlty-adjusted rsk averson plays a role n determnng rsk prema n equlbrum prces. In ths case nvestors rsk aversons toward stochastc volatlty s aggregated nto prces n an ntutve manner. The rsk assocated wth stochastc volatlty that s prced derves from how well a stock provdes a hedge aganst movements n the economy-wde wealth. Ths s captured by the covarance between the future payoff from the stock and the future economy-wde wealth. Further, the effect of ndvdual nvestors rsk preferences regardng stochastc volatlty materalzes n equlbrum prces only through the approprately aggregated VARA, Γ m. Ths aggregaton of rsk preferences has an analog n Wlson s [26] characterzaton of Pareto optmal rsk sharng. The second part of Theorem 4 establshes the role of stochastc volatlty n equlbrum returns. Compared to a settng wth certan varance, the presence of stochastc volatlty affects the dstrbuton of returns on ndvdual stocks and returns Chamberlan [17] and Owen and Rabnovtch [23] show that ellptcal dstrbutons, and hence stochastc volatlty dstrbutons, lead to a CAPM. For CAPM results wthout stochastc volatlty, see Ingersoll Jr [24], Huang and Ltzenberger [5], and Cochrane [25]. Copyrght 211 John Wley & Sons, Ltd. Appl. Stochastc Models Bus. Ind. 211

8 on the market portfolo, whch n turn affects the CAPM beta. In addton, the expected excess return on ndvdual stocks and on the market portfolo are affected by stochastc volatlty through equlbrum prces. In contrast to the effect on equlbrum prces, however, the calculaton of the systematc rsk factor can be undertaken wthout knowledge of ndvdual nvestors rsk preferences. In a sense the calculaton of the beta, namely β j =Cov[R j, R m ]/Var[ R m ], s standard. It s affected by stochastc volatlty solely from the fact that the dstrbuton of returns depends on p(v )vap(r j )= p(r j V )p(v )dv. However, gven prces, there s no effect from sze-basng Q(V ) or from tltng the volatlty dstrbuton T (V ). Ths s n contrast to the calculaton of CAPM betas when nvestors face asymmetrc parameter uncertanty that does not materalze n realzed returns. In that case, Coles and Loewensten [27] and Coles et al. [7] show that an adjustment s requred n the calculaton of the betas. Our fnal result reconsders the case where all nvestors have constant absolute rsk averson and so stochastc volatlty appears through absolute VARA. We show that n ths case stochastc volatlty affects the rsk premum n prces only through the varance calculated under the tlted dstrbuton. Theorem 5 (Equlbrum prces under CARA) Consder a representatve nvestor wth constant absolute rsk averson, γ m =( γ 1 ) 1 n the settng of Theorem 4 who chooses between one rsky and one safe asset for a sngle perod nvestment horzon. Then the equlbrum prce of the stock s gven by P = R 1 f (E[X] Γ CARA m where T s tlted dstrbuton for γ m defned as before. Proof See Appendx F. Var[ X])= R 1 f (μ γ m Var T [X]), As expected, the equlbrum prce separates nto a mean effect from the dscounted expected future payouts and a varance effect due to nvestor s rsk averson. Theorem 5 expresses the varance effect n two ways. Frst, the relatonshp between prces and varances s characterzed by the aggregate absolute VARA, Γ CARA m. Alternatvely, the rsk premum n prces can be convenently expressed from the nvestors aggregate rsk averson, γ m, by usng the varance calculated under the tlted dstrbuton, nstead of the varance calculated under the true probablty measure. The approprate tlted dstrbuton uses the constant absolute rsk averson of the representatve nvestor, γ m. Whle each nvestor s absolute VARA can be expressed based on the tlted probablty measure, we have dentfed a change n probablty measure that converts an economy wth stochastc volatlty nto an equvalent economy wthout stochastc volatlty. Ths change of measure s a natural analog to the standard change to the equvalent martngale measure, whch converts prces arsng n an economy wth rsk-averse nvestors nto the prces that would have prevaled n an equvalent economy wth rsk neutral nvestors. The applcaton of the latter, standard change of probablty measure s apparent after applcaton of the tlted probablty measure n ths paper. As we rely on the market-clearng condton, our constructon of the ttled measure s unque. In contrast, the constructon of the martngale measure from no arbtrage arguments need not be unque wth an ncomplete stock market whose securtes payoffs do not span all Arrow Debreu securtes. 5. Conclusons In ths paper we explore the effects of stochastc volatlty on ndvdual nvestment behavor, equlbrum prces, and market portfolos. We propose VARA as the approprate measure of effectve rsk averson when nvestors face asset returns wth stochastc volatlty. To establsh that nvestors are effectvely more rsk averse when subject to stochastc volatlty we summarze our results n the specal case wthout wealth effects. We fnd that nvestors wth exponental utlty prefer constant varance of a gven magntude to stochastc volatlty wth expected varance equal to the constant varance. We also fnd that nvestor s trade-off ncreases n the level of varance for reductons n the varance of the volatlty dstrbuton. Further, we show that results such as the optmal share of wealth to nvest n a rsky asset and CAPM have analogous relatonshps n the stochastc volatlty settng, but the parameters must be adjusted to take account of stochastc volatlty. In order to develop our analyss, we extend Sten s lemma to the case of normal dstrbutons wth stochastc volatlty. Ths allows us to separate utlty and covarance effects and provdes tractable, nterpretable results. Whle we derve Sten-lke relatonshps for dstrbutons wth stochastc volatlty, t s mportant to note that the orgnal Sten s lemma does not hold Ths suffces when two fund separaton holds and nvestors behave as f they purchased a mx of the rsk-free bond and a sngle stock portfolo. Copyrght 211 John Wley & Sons, Ltd. Appl. Stochastc Models Bus. Ind. 211

9 n these stuatons. In each case, the new dentty ncludes a dfferent proportonalty factor whch has mportant behavoral mplcatons. For example, we show that wth partcular objectve functons, stochastc volatlty causes decson makers to act more rsk averse provded that the rsk averson s approprately adjusted through the change of measure. Therefore, when uncertanty n the form of stochastc volatlty changes over tme, observed behavor should change as well. Our work s related to the lterature that examnes the effect of estmaton rsk on portfolo choce. Our analyss prmarly examnes the effect of estmaton rsk n the volatlty whle exstng research has also evaluated the effect of estmaton rsk n the mean. Ths leads to a number of avenues for future research, for example, combnng the effects of estmaton rsk n both the mean and the volatlty; extendng the results to a multvarate stochastc volatlty settng usng the technques derved n [28]; and fnally consderng nter-temporal portfolo problems wth stochastc volatlty along the lnes of Balvers and Mtchell [29]. Appendx A: Proof of Theorem 1 Frst consder Cov[g(X), X]. Usng the Law of Iterated Expectatons Cov[g(X), X] = E[g(X)(X E[X])]= E V {E X V [g(x)(x E[X])]} = E V {E X V [g(x)(x E[X V ])]}. The result now follows from applyng Sten s lemma condtonally as X V s normally dstrbuted as N(μ,σ 2 V ). Hence we have Cov[g(X), X]= E V {E X V [g (X)]Var[X V ]}. Now Var[X V ]=σ 2 V and so { } Cov[g(X), X]= E V {E X V [g (X)]σ 2 V }=E g V (X) σ 2 E[V ], E[V ] whch mples that Cov[g(X), X]= E Q [g (X)]σ 2 E[V ]. (A1) Second, consder Cov[g(X),Y ]. In the bvarate case, we can wrte Y =a+bx+ε, where Cov[X,ε]= and b =[Y, X]/Var[ X]. Note that Cov[g(X),ε]= by constructon. Moreover, we can wrte Var[X]= E V {Var[ X V ]}+ Var V {E[X V ]} where Var[X V ]=σ 2 V and Var V {E[X V ]}=. Hence Var[X]=σ 2 E[V ]. Hence from the unvarate verson of Sten s Lemma above we have Cov[g(X),Y ]= E Q [g (X)]Cov[X,Y ]. Appendx B: Absolute VARA propertes For these proofs t s useful to use the followng smplfed presentaton of Γ CARA.LetT(V)=(e 2 1 γ2 σ 2V /E[e 1 2 γ2 σ 2V ])p(v ) and let E T denote the expectaton under ths exponentally tlted volatlty dstrbuton. Then Γ CARA can be represented as Γ CARA =γ E T 1 [V ] E[V ] =γ E[V e 2 σ2 γ 2V ] E[e 1 2 σ2 γ 2V. ]E[V ] As expected ths s clearly unaffected by expected wealth, μ due to the choce of exponental utlty. Ths can be decomposed further by usng the defnton of covarance and wrtng Hence E[V e 1 2 γ2 σ 2V ]= E[V ]E[e 1 2 γ2 σ 2V ]+Cov[V,e 1 2 γ2 σ 2V ]. Γ CARA =γ+γ Cov[V,e 1 2 γ2 σ 2V ] E[e 1 2 γ2 σ 2V ]E[V ]. The frst term s the coeffcent of absolute rsk averson whereas the second term s postve and Γ CARA γ. To see ths note that snce e 2 1 γ2 σ 2V s ncreasng n the varance, V, ts covarance wth the varance s also non-negatve. Copyrght 211 John Wley & Sons, Ltd. Appl. Stochastc Models Bus. Ind. 211

10 Appendx C: Proof of VARA domnance results Proof of Theorem 2 By dfferentaton, we fnd that Γ CARA E[V e 1 2 σ2 γ 2V ] = γ E[e 1 2 σ2 γ 2V ]E[V ], Γ CARA σ 2 = γ2 2E[V ] E[V 2 e 1 2 γ2 σ 2V ( ] E[V e 1 2 γ2 σ 2V ] E[e 1 2 γ2 σ 2V ] E[e 1 2 γ2 σ 2V ] Recognzng the exponental tlt dstrbuton T (V )=e 1 2 γ2 σ 2V p(v )/E[e 1 2 γ2 σ 2V ] we obtan Γ CARA σ 2 = γ2 2E[V ] {E T [V 2 ] (E T [V ]) 2 }= Hence Γ CARA / σ 2 as clamed. Smlarly, dfferentaton wth respect to γ yelds Γ CARA = ΓCARA + γσ2 E[V 2 e 2 1 γ2 σ 2V ( ] E[V e 1 2 γ2 σ 2V ] γ γ E[V ] E[e 1 2 γ2 σ 2V ] E[e 1 2 γ2 σ 2V ] γ2 ) 2. 2E[V ] {VarT [V ]}. ) 2 = ΓCARA γ + γσ2 E[V ] VarT [V ]. Proof of Theorem 3 Consder Γ CARA for the volatlty dstrbuton V = V 1 +V 2,whereV 1 and V 2 are ndependent. Then Γ CARA E[V e 2 1 γ2 σ 2V { } ] =γ E[e 1 2 γ2 σ 2V =γ E[(V 1 +V 2 )e 1 2 γ2 σ 2 (V 1 +V 2 ) ] = γ E[V 1 e 1 2 γ2 σ 2 V 1 ] + E[V 2e 1 2 γ2 σ 2 V 2 ]. ]E[V ] E[e 1 2 γ2 σ 2 (V 1 +V 2 ) ]E[V ] E[V ] E[e 1 2 γ2 σ 2 V 1 ] E[e 1 2 γ2 σ 2 V 2 ] Defne the tlted dstrbutons T 1 (V 1 )=e 1 2 γ2 σ 2 V 1 p(v 1 )/E[e 1 2 γ2 σ 2 V 1 ]andt 2 (V 2 )=e 1 2 γ2 σ 2 V 2 p(v 2 )/E[e 1 2 γ2 σ 2 V 2 ]. Hence Γ CARA = γ E[V ] {E T 1 [V 1 ]+ E T 2 [V 2 ]}= 1 E[V ] {E[V 1]Γ CARA 1 + E[V 2 ]Γ CARA 1 }. Defnng π= E[V 1 ]/E[V 1 +V 2 ] the frst result now follows: Γ CARA =πγ CARA 1 +(1 π)γ CARA 2. A smlar argument holds f we defne Z = V 2,whereE[Z]= and E[V ]= E[V 1 ]. In that case, E T 2[V 2 ] and Γ CARA Γ CARA 1 as clamed. Appendx D: Proof for generalzed nverse Gaussan-dstrbuted volatlty Consder the Generalzed Inverse Gaussan (GIG) dstrbuton defned by p(v α,λ)= V 2 3 I(α,λ) e 1 2 (αv 1 +λv ), where the normalzaton constant, I(α,λ), ensures that p(v ) s a densty. We wll prove the results stated n the paper through a sequence of lemmas. In our proofs, the followng dentty from Andrews and Mallows [19, p. 1] wll be useful e 1 2 (a2 u 2 +b 2 u 2) ( π ) 1 du = 2 2a 2 e ab. (D1) Lemma D1 The normalzaton constant I(α,λ)= 2π/αe αλ and for <γ< λ/σ E[e 1 2 γ2 σ 2V ]=e α(λ γ 2 σ 2) e αλ. (D2) Copyrght 211 John Wley & Sons, Ltd. Appl. Stochastc Models Bus. Ind. 211

11 Proof of Lemma D1 [ ] E e 1 2 γ2 σ 2 V = = 2 I(α,λ) e 1 2 γ2 σ 2V p(v )dv = e 1 2 γ2 σ 2 V V 3 2 I(α,λ) e 1 2 (αv 1 +λv ) dv e 1 2 (αv 1 +(λ γ 2 σ 2 )V ) ( 1 2 V 3 2 )dv, where I(α,λ) s a normalzaton constant. Changng of varables u = V 2 1 yelds E[e 1 2 γ2 σ 2V ]= 2 e 2 1 (αu2 +(λ γ 2 σ 2 )u 2) du. I(α,λ) Provded that (λ γ 2 σ 2 ), we can apply (D1) and fnd that E[e 1 2 γ2 σ 2V ]= 2 π I(α,λ) 2α e α(λ γ 2 σ 2). In the specal case where γ= t follows that (D3) or 1= 2 π αλ I(α,λ) 2α e I(α,λ)= V e 2 (αv 1 +λv 2π ) dv = αλ α e. The second part of the lemma now follows from substtuton back nto (D3). Lemma D2 For any constant γ such that <γ< λ/σ, E[V e 1 2 γ2 σ 2V α ]= (λ γ 2 σ 2 ) e α(λ γ 2 σ 2) e Proof of Lemma D2 αλ. (D4) E[V e 1 2 γ2 σ 2V ] = = 2 I(α,λ) Agan change of varables u = V 1/2 yelds V e 1 2 γ2 σ 2V p(v )dv = E[V e 1 2 γ2 σ 2V ]= 2 I(α,λ) Assumng that (λ γ 2 σ 2 ), t follows from (D1) that E[V e 1 2 γ2 σ 2V ]= 2 I(α,λ) Substtutng for I(α,λ) yelds (D4) mmedately. e 1 2 (αv 1 +(λ γ 2 σ 2 )V ) ( 1 2 V 1 2 )dv. V e 1 2 γ2 σ 2 V V 3 2 I(α,β) e 1 2 (αv 1 +λv ) dv e 1 2 (αu 2 +(λ γ 2 σ 2 )u 2) du. π 2(λ γ 2 σ 2 ) e α(λ γ 2 σ 2). Lemma D3 The expected and varance of stochastc volatlty are E[V ]= α/λ,var[v ]= α/λ 3. Copyrght 211 John Wley & Sons, Ltd. Appl. Stochastc Models Bus. Ind. 211

12 Proof of Lemma D3 Frst note that E[V ] arses as the specal case of Lemma D2, where λ=. Second note that E[V 2 ]= Integraton by parts gves V 2 p(v )dv = E[V 2 ]= 1 λi(α,λ) { We now apply Lemma D2 wth γ=andfnd V 2 V 3 2 I(α,λ) e 1 2 (αv 1 +λv ) dv = 1 I(α,λ) V 1 2 e 1 2 (αv 1 +λv ) dv +α E[V 2 ]= 1 λ {E[V ]+α}= α λ 3 + α λ, { V 1 2 e α 2 V 1} e λ 2 V dv. } V 3 2 e 1 2 (αv 1 +λv ) dv. where the last equalty follows from E[V ]. The result follows from the defnton of varance α Var[V ]= E[V 2 ] (E[V ]) 2 = λ 3 + α ( ) 2 α α λ = λ λ 3. Lemma D4 Consder a representatve nvestor wth constant absolute rsk averson, γ, that s below the constant λ/σ.ifv s dstrbuted wth the GIG dstrbuton specfed above, then VARA becomes ) 1 Γ CARA = (γ 2 σ2 2. λ Proof of Lemma D4 From Secton 3, the volatlty-adjusted rsk averson can be expressed as Γ CARA E[V e 1 2 γ2 σ 2V ] =γ E[e 1 2 γ2 σ 2V ]E[V ]. Usng the prevous lemmas, we calculate the terms n Γ CARA as follows. Snce E[V e 1 2 γ2 σ 2V ] E[e 1 2 γ2 σ 2V ] = α (λ γ 2 σ 2 ). It follows that Γ CARA E[V e 1 2 γ2 σ 2V ] =γ E[e 1 2 γ2 σ 2V ]E[V ] as requred. The comparatve statcs follow mmedately. =γ α (λ γ 2 σ 2 ) λ α = (γ 2 σ2 λ ) 1 2 Appendx E: Proof of Theorem 4 We frst derve the market clearng prces and then prove that CAPM holds. Equlbrum prces Assumng that the budget constrant s bndng, termnal wealth s gven by ( ) W = W ω, j P j R f + ω, j X j = W R f + j j j ω, j (X j P j R f ). Hence, each nvestor chooses {ω, j j =1,..., J} to maxmze expected utlty E[U (W )]. The frst-order condtons can then be wrtten as = E[U (W )(X j P j R f )] j. Copyrght 211 John Wley & Sons, Ltd. Appl. Stochastc Models Bus. Ind. 211

13 Proceedng as n secton 2 we use the defnton of covarance for the frst-order condton From Sten s lemma wth stochastc volatlty, we get Cov[U (W ), X j ]= E[U (W )]E[(X j P j R f )] j. Cov[U (W ), X j ]= E Q [U (W )]Cov[W, X j ]. Substtutng ths nto the above equaton yelds E[U (W )]E[(X j P j R f )]= E Q [U (W )]Cov[W, X j ], j or where Γ 1 E[(X j P j R f )]=Cov[W, X j ], j (E1) Γ = E Q [U (W )] E[U (W )] s the absolute VARA of nvestor. Snce U s strctly ncreasng and concave, takng expectatons under the sze-based measure preserves that Γ >. Summng on both sdes of (E1) over all nvestors n the market Γ 1 E[(X j P j R f )]= Cov[W, X j ] j and usng the lnearty of the covarance, we get Γ m E[(X j P j R f )]=Cov[( W ), X j ] j, where Γ m =( Γ 1 ) 1 s the volatlty-adjusted aggregate absolute rsk averson of the economy. The relaton to economy-wde termnal wealth, W m, economy-wde ntal wealth, Wm, and the rsky return on the market portfolo, R m, can be summarzed as ( ( ) ( ) W m = W )= B R f + Wm B R m, where R m = X m /( j P j )andx m = j X j. By substtuton, the excess return on stock j can be wrtten as E[(X j P j R f )]=ΓCov[R m, X j ] j, (E2) where Γ=Γ m ( W m B ) measures the volatlty-adjusted aggregate relatve rsk averson of the economy. Alternatvely, the prce of stock j can be wrtten as P j = R 1 f (E[X j ] Γ 1 m Cov[W m, X j ]) j. (E3) Captal asset prcng model (CAPM) Dvdng (E2) by the ntal market value of stock, P j,wefndthat E[(R j R f )]=Γ m Cov[R m, R j ] j, where R j =(X j /P j ) s the return of stock j. Multplyng both sdes by the fracton of stock j n the market portfolo, ω m, j = P j / k P k = P j /(W m B ), E[ω m, j (X j P j R f )]=Γ m Cov[X m,ω m, j X j ] j (E4) Copyrght 211 John Wley & Sons, Ltd. Appl. Stochastc Models Bus. Ind. 211

14 and summng (E4) over all stocks E[ j ω m, j (X j P j R f )]=Γ m Cov[X m, j ω m, j X j ] yelds E[X m ] ( j P j )R f =Γ m Var[ X m ]. Dvdng by the ntal nvestment n the market portfolo yelds E[R m ] R f =Γ m ( j P j )Var[R m ], where, as before, R m s the return on the market portfolo. Ths reduces to Γ m ( j P j )= E[R m] R f Var[ R m ]. Hence, the presence of stochastc volatlty does not change the basc nsght that n the aggregate the rsk premum for market rsk s strctly postve. By substtuton of Γ m the excess return on stock j can be wrtten as the famlar CAPM form. E[(R j R f )]=( E[R m] R f Var[ R m ] )Cov[R m, R j ] j, Appendx F: Proof of Theorem 5 Each nvestor wth exponental utlty buys the fracton ω of the stock and nvests the remander n the safe bond. Formally, nvestor solves or snce X V N(μ,σ 2 V ) The frst-order condtons are max ω E[U(W )]= γ 1 e γ W R f E[e γ ω (X PR f ) ] mn γ ω (ω (μ PR f ))+ln(e V [e 1 2 γ2 ω2 σ2v ]). = γ (μ PR f )+γ 2 ω σ 2 E V [V e 2 1 γ2 ω2 σ2v ] E V [e 1 2 γ2 ω2 σ2v ] and the assocated second-order condtons for ω are γ 2 E σ2 V [V e 1 2 γ2 ω2 σ2v ] +γ 2 E V [e 1 2 γ2 ω2 σ2v ω2 σ2 Var T [V ] γ. ] Rearrangng terms, we fnd that each nvestor s mplct demand functon for stock s ω (P)= 1 ( ) μ PRf Γ CARA E[V ]σ 2, (F1) where Γ CARA E V [V e 1 2 γ2 ω2σ2v ] =γ E V [e 1 2 γ2 ω2 σ2v ]E[V ] s the absolute VARA defned n Secton 3. Proceedng as n the proof of Theorem 4, we sum across all nvestors n (F1), defne Γ CARA m =( (ΓCARA ) 1 ) 1, and apply the market clearng condton that ω (P)=1. Ths yelds the equlbrum prce: P = R 1 f (μ Γ CARA m E[V ]). Copyrght 211 John Wley & Sons, Ltd. Appl. Stochastc Models Bus. Ind. 211

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