Asset Prices, Funds Size and Portfolio Weights in Equilibrium with Heterogeneous and Long-Lived Funds

Transcription

1 Asset Prces, Funds Sze and Portfolo Weghts n Equlbrum wth Heterogeneous and Long-Lved Funds Jakša Cvtanć Semyon Malamud November 19, 2008 Abstract We perform a detaled asymptotc analyss of the equlbrum behavor of the asset prces, wealth sze and portfolo weghts n complete markets equlbra, wth long-lved funds In equlbrum, the fund wth the closest to) log preference wll domnate the other funds n sze, n the long-run, wth probablty one On the other hand, two funds on the opposte sdes of the log preference wll never domnate each other n expected sze In the very long run, the prce behavor of the rsky asset wll be determned solely by the fund closest to the log preference However, the prce drft and volatlty stll are affected by hgher rsk aversons, and the optmal portfolo weghts contan a hedgng component, postve negatve) for the rsk aversons hgher lower) than log The hedgng component s monotone ncreasng n rsk averson for the tmes further away from the termnal horzon, but t may become monotone decreasng closer to the termnal horzon For earler, but stll asymptotcally nfnte tmes, the prce behavor s mpacted also by the funds wth rsk aversons greater than one Sellng short s never optmal There are dstnct ncreasng tme perods such that the prce has the same asymptotc behavor n each perod The long-run per-perod return gets lower wth tme The research of J Cvtanć was supported n part by NSF grants DMS and DMS The research of S Malamud was supported n part by the Natonal Centre of Competence n Research Fnancal Valuaton and Rsk Management NCCR FINRISK), Project D1 Mathematcal Methods n Fnancal Rsk Management) We are very grateful to Julen Hugonner, Elyes Joun, Saša Parad, and to semnar partcpants at U of Southern Calforna and U of Texas at Austn for helpful comments Exstng errors are our sole responsblty Caltech, Dvson of Humantes and Socal Scences, M/C , 1200 E Calforna Blvd Pasadena, CA Ph: 626) E-mal: cvtanc@hsscaltechedu ETH Zuerch and Swss Fnance Insttute E-mal: semka@mathethzch

2 Keywords: Asset prcng; equlbrum; heterogeneous agents JEL classfcaton: D53, G11, G12 2

3 1 Introducton Consder the smplest possble fnancal market: a complete market wth ratonal traders consumng only at the end of the tme horzon There are exstence results for equlbrum n such a market, 1 and CAPM type results under well known specfc condtons However, not much s known about specfc equlbrum propertes In partcular, n ths paper we consder the analogue of the benchmark partal equlbrum dynamc model, a la Merton 1971), wth the traders havng generalzed CRRA preferences, and the dvdends followng a geometrc Brownan moton We fnd sharp bounds for the asset prces, wealth sze and portfolo weghts n equlbrum n such fnancal markets, and study ther asymptotc behavor, when the tme horzon becomes large We nterpret our traders as long-lved funds whch only care about the fnal consumpton The man conclusons we reach are the followng: the fund wth the preference closest to logarthmc wll domnate n sze other CRRA-lke funds n the long-run, wth probablty one 2 However, n terms of expected relatve sze, two funds, wth preferences separated by the log preference, wll never domnate each other In the very long run, the prce of the rsky asset s mpacted solely by the fund closest to the log preferences Interestngly, ts drft and volatlty stll are mpacted by a hgher rsk averson For asymptotcally large tmes, but bounded away from the fnal horzon, the prce behavor s affected by funds more rsk averse than log There are dstnct ncreasng tme perods such that the prce has the same asymptotc behavor n each perod The long-run per-perod return gets lower wth tme We also fnd smple bounds for the portfolo weghts of all the funds, whch show that sellng short never occurs, and that the hedgng component s negatve postve) for the rsk aversons smaller larger) than one Interestngly, the hedgng component may become monotone decreasng n rsk averson at the tmes closer to the end of the nvestment horzon For the rsk aversons larger than one, we provde explct lmt formulas for the portfolo weghts Studyng equlbrum wth heterogeneous funds s hard n general, even n complete markets, as the equlbrum relatonshp between the market s stochastc dscount factor SDF) and the exogenous total value of the rsky assets s very nonlnear A very nce contrbuton has been accomplshed by Kogan, Ross, Wang and Westerfeld 2006) henceforth KRWW 2006): they consder a market wth a CRRA fund and an rratonal fund that has ncorrect belefs about the drft of the rsky asset there s only one rsky asset n the economy) In ths framework they show that t s possble to compute the equlbrum analytcally The man message from that paper s that even when the rratonal fund does not survve n the 1 See Karatzas and Shreve 1998) 2 The domnance of the log-fund was also studed n Blume and Easley 1992), and Evstgneev, Hens and Schenk-Hoppe 2006) 1

4 long run, t stll wll have a prce mpact over a substantal fracton of tme, even wth very small wealth sze, contrary to the common wsdom that there s no long run mpact on the prce from the funds that vansh We show that the same effect happens when all the funds are ratonal even though only the one closest to the log preference survves n the very long run, the prce mpact of other funds can be sustaned for a long tme Ths reconfrms the KRWW 2006) message that wealth sze s not the same as prce mpact, the fact llustrated n our case wth only ratonal traders present n the market 3 Our results are also related to the so called portfolo turnpkes theorems, whch dentfy condtons, albet n partal equlbrum, under whch the portfolo of an nvestor converges to a CRRA type portfolo at long horzons The papers Dybvg, Rogers and Back 1999) and Huang and Zarphopoulou 1999) provde general results of ths nature, and further references A recent paper Yan 2009) studes survval n a model wth ntermedate consumpton, heterogeneous belefs and heterogeneous dscount factors In partcular, he fnds that, wth heterogenety only n rsk averson, the agent wth the smallest rsk averson wll domnate n the long run f the economy s growng Smlarly, Berrada 2008) studes the valdty of KRWW 2006) fndngs n a model wth ntermedate consumpton and fnte tme horzon In contrast to KRWW 2006), Berrada 2008) shows that survval and prce mpact may, n fact, be related mpact on the prces s ncreasng n the agents consumpton share when the agents have logarthmc preferences However, when rsk averson s larger than one, that paper s unable to obtan defnte conclusons Whle some results n Berrada 2008) are obtaned n analytc form, the paper mostly has to rely on numercal smulaton On the other hand, we obtan a very precse analytc pcture of what happens n our model: we fnd, for the case of lognormal cumulatve wealth, explct expressons for the ntervals whch span the total tme horzon, and n each of whch the prce has the same asymptotc behavor, whch we also compute explctly In each of the ntervals, the asset prce process s asymptotcally equal to a Black-Scholes, geometrc Brownan moton prce process The equlbrum prce s asymptotcally mpacted only by traders wth rsk averson no less than one In terms of methodologcal contrbutons, we show that a lot of sharp results can be obtaned wthout hghly sophstcated mathematcs whle our proofs are long and techncal, they use mostly well known results on nequaltes nvolvng expected values In the process of provng our man theorems, we obtan a lot of auxlary results, fndng bounds on values of expected utlty and margnal utlty of the traders, on expected values nvolvng wealth sze, and on expected values nvolvng the stochastc dscount factor In ths latter regard, some of the results are n the sprt of Hansen-Jagannathan 1991) bound on the standard devaton of the SDF 4 3 Wang 1996) also studes a model wth only ratonal traders, n a smpler settng 4 However, our bounds depend on the market portfolo, and not on ndvdual portfolos 2

5 We also get a smple bound on the asset prce volatlty: ts value has to be larger than the volatlty of the total fnal dvdend process, and less than the dvdend volatlty ncreased by a factor dependng on the dfference between the hghest and the lowest rsk averson In other words, n equlbrum, volatlty may rse not only due to the volatlty of the dvdend output, but also due to the dfference n rsk preferences of the traders Moreover, we provde exact formulas for the lmts of the drft and the volatlty of the prce, as well as for the optmal portfolo weghts All of these depend on rsk aversons larger than one, even at very far away tme ponts Moreover, wth a suffcently dense set of rsk aversons, the optmal portfolo weght, for the rsk aversons larger than one, and n the lmt, contans a postve hedgng component even n the very long run On the other hand, the hedgng component for the rsk aversons less than one s no larger than zero It was also observed n KRWW 2006) that even at tme ponts when the stock prce s close to ts statonary lmt, the portfolo weght s not equal to the correspondng myopc strategy However, n ther framework wth two traders, for the tme ponts far enough nto the future, the hedgng component vanshes Ths s not always the case n our model For example, f the set of the agents rsk aversons s not suffcently sparse, wth hgh enough heterogenety there s always a non-zero hedgng component Thus, some of the phenomena we fnd are completely new and depend on the relatve values of the rsk aversons of the agents, and hence havng many agents n the economy can result n a very dfferent hedgng behavor from havng two agents only, the case whch s usually studed n the lterature We descrbe the setup n Secton 2, prove the almost sure domnance of the closest to the log fund n Secton 3, provde results on expected relatve sze n Secton 4, compute the long term prce behavor n Secton 5, and the long term volatlty and portfolo behavor n Secton 6 Secton 7 concludes, whle the longer proofs are presented n Appendx 2 Setup and notaton 21 The Model We consder a standard settng smlar to that of Wang 1996) The economy has a fnte horzon and evolves n contnuous tme Uncertanty s descrbed by a one-dmensonal, standard Brownan moton B t, t [0, T ] on a complete probablty space Ω, F T, P ), where F s the augmented fltraton generated by B t There s a sngle share of a rsky asset n the economy, the stock, 5 whch pays a termnal dvdend payment D = D T = e ρt + σb T 5 The results of the frst part of the paper are easly extended to the case of multple assets and any complete market model 3

6 We also assume that a zero coupon bond wth nstantaneous constant rsk free rate r s avalable n zero net supply 6 The prce of the stock at tme t s denoted by S t The nstantaneous drft and volatlty of the stock prce S t are denoted by µ t and σ t respectvely, S 1 t ds t = µ t dt + σ t db t There are K compettve agents n the economy, whch we wll also call funds, who behave ratonally, and are heterogeneous n rsk preferences Fund k s ntally endowed wth ψ k shares of the stock at tme zero, ψ k = 1 k Fund k s tradng strategy π kt, the portfolo weght n the rsky asset, s assumed to satsfy the standard ntegrablty condton T Then, the wealth W kt of fund k evolves as 0 π 2 ktσ 2 t dt < dw kt = W kt rdt + π kt St 1 ds t rdt)) Fund k chooses portfolo strategy π kt as to maxmze the expected utlty E[u k W kt )] of ts fnal wealth W kt Utlty u k s assumed to be concave, ncreasng and satsfyng the standard Inada condtons 22 The Equlbrum lm x 0 u kx) = +, lm u x kx) = 0 Defnton 21 We say that the market s n equlbrum f the funds behave optmally and both the rsky asset market and the rsk-free market clear It s well known that the above fnancal market s complete, f the volatlty process σ t of the stock prce s almost everywhere strctly postve 7 When the market s complete, there exsts a unque stochastc dscount factor SDF) M = M T such that the stock prce s gven by S t = e rt T ) E t[md] E t [M] 6 The assumpton of constant r s ntroduced only for smplcty of exposton 7 Ths follows for example from the results of Anderson and Ramondo 2008) In the CRRA settng, t wll follow from Proposton 61 below 4

7 Snce u k satsfes Inada condtons, the nverse of the margnal utlty I k x) := u k) 1 x) 1) s strctly postve and t s well known see, for example, Cvtanć and Zapatero 2004)) that the optmal termnal wealth s then of the form W kt = I k λ k M) where λ k s determned va the budget constrant E[I k λ k M) M] = W k0 = ψ k S 0 = ψ k E[DM] Because of the market completeness, equlbrum allocaton s Pareto-effcent and can be characterzed as an Arrow-Debreu equlbrum See, eg Duffe 1986), Wang 1996) Because the endowments are co-lnear all agents hold shares of the same sngle stock), the equlbrum s n fact unque, up to a multplcatve factor, and unque f we fx the rsk-free rate See, eg, Dana 1995), Dana 2001) 8 We formalze ths n Proposton 21 The equlbrum allocaton s gven by W kt = I k λ k M), E[I k λ k M) M] = ψ k E[DM] and equlbrum SDF M solves the equaton I k λ k M) = D 2) The followng lemma s a drect consequence of 2) Lemma 21 For any fund k, the equlbrum SDF satsfes k Proof We have M max k {λ 1 k u kd)} 3) D = k I k λ k M) > I k λ k M) and the clam follows by applyng u k to both sdes of the nequalty Ths lemma wll be crucal for most bounds derved n ths paper Note that λ 1 k u k D) s, up to a constant, the SDF n the economy populated by fund k only In fact, for a partcular choce of the rsk-free rate, t wll be equal to the SDF In ths regard, Lemma 21 may be nterpreted as sayng that the prce of a state n the heterogeneous economy domnates the prce n partcular homogeneous economes 8 Snce the endowment s nether bounded away from zero nor from nfnty, some addtonal care s needed to show the exstence of equlbrum See, eg, Dana 2001) and Malamud 2008a) 5

8 3 Almost sure extncton relatve to the log fund 31 Almost sure convergence of relatve fund sze It s well known that, n typcal fnancal market models, the fund that maxmzes the logutlty from fnal wealth also maxmzes the long-run growth rate It s then to be expected that the log-fund wll domnate the market n the long-run Ths s the case n KRWW 2006), where the other fund s not ratonal, but t s also the case n our market, wth ratonal funds and a standard choce for D, as we wll now show Introduce γ k x) := x u k x) u k x) the relatve) rsk averson of fund k, and denote by b k x) := 1 γ k x) ts relatve rsk tolerance Intutvely, the only thng that matters for the long run behavor of the optmal wealth s the rsk atttude of the fund for very small and very large wealth If the rsk averson starts oscllatng as the wealth approaches zero/nfnty, the optmal behavor of a fund may become qute strange To avod ths, we mpose the followng natural Assumpton 31 The lmts γ 0 := lm x 0 γ x), γ := lm x γ x) exst and are strctly postve That s, for very small and very large wealth, the fund behaves as a CRRA nvestor, but otherwse we allow for a wealth-dependent rsk averson The class of utltes, satsfyng Assumpton 31 s qute large For techncal reasons, we also mpose the followng assumpton for each fund, whch supposes consstency n rsk averson across wealth sze relatve to the log preference: Assumpton 32 For every fund n the economy, ether γ 0, γ > 1 or γ 0, γ < 1 or γ 0 = γ = 1 We expect the fund whch s the closest to the log preference to have the hghest long term growth rate and domnate all others n the long run The followng theorem makes ths ntuton rgorous 6

9 Theorem 31 Assume Assumptons 31 and 32 and D = e ρt +σb T Suppose that there exsts a unque fund = 0 closest to beng a log fund, that s, we assume: Then, n equlbrum, max{ γ 0 0 1, γ 0 almost surely for any 0 1 } < mn 0 mn{ γ0 1, γ 1 } W T lm = 0 T W 0T Assumpton 31 s related to the portfolo turnpke theorems; see, eg, Dybvg, Rogers and Back 1999) and Huang and Zarphopoulou 1999) They show that, f the rsk averson s asymptotcally constant at large wealth levels and the stock prces are growng, then the optmal portfolo wll converge at a long horzon to that of the nvestor wth a CRRA utlty In a sense, Theorem 31, and further results below, can be vewed as equlbrum analogs of the turnpke theorems The proof of ths ntutve result s rather non-trval and s based on several useful bounds for the funds wealth The most mportant one see Lemma A3 n Appendx) shows that, n equlbrum, for fund and for some constants K 1, K 2 ndependent of T and ndependent of the aggregate endowment D, we have K 1 E[W T u W T )] E[Du D)] K 2 Because the market s complete, the margnal utlty u W T ) of fund s wealth s n fact proportonal to the unque SDF The quantty E[W T u W T )] s thus proportonal to) the value of fund s wealth On the other hand, E[D u D)] s proportonal to) the value of the aggregate endowment n an artfcal economy, populated only by fund The above bounds show that, n equlbrum, the two values have a comparable behavor When the fund has a CRRA utlty, the nequalty shows that ts expected utlty s bounded from below and from above by a multple of the expected utlty of a fund that consumes the total dvdend output D n the economy We have obtaned also results on the almost sure convergence of the relatve sze of any two funds n Proposton 42 below We present that result later because ts proof requres some addtonal estmates from the next secton 4 Expected relatve sze of funds 41 Bounds on expected relatve sze of funds The man result we want to show n ths subsecton s 7

10 Proposton 41 Let, j be funds wth rsk aversons γ x), γ j x) Then, ) If ether γ x) γ j y) 1 for all x, y or γ x) γ j y) 1, that s, fund j s closer to beng log fund than fund, then [ ] WT E W jt W 0 W j0 ) If γ j x) 1 then E [ WT W jt ] = W 0 W j0 ) If ether γ x) 1 γ j x) or γ j x) 1 γ x) for all x then [ ] WT E W 0 W j0 W jt Result ) seems surprsng at frst, gven that n the prevous secton we showed that the closest to) log fund wll domnate n the long run wth probablty one Here, however, the expected relatve wealth of other funds relatve to the log fund remans constant: log fund wll always own on average the same fracton of the economy at tme T as at tme zero Ths s actually well known, and s no longer surprsng f we recall that the log fund s wealth process s proportonal to the nverse 1/M t of the SDF process, and that W jt M t s a martngale for any wealth process W jt For example, f the rato of the wealth of the two funds s proportonal to the martngale exp{ tσ 2 /2 + σb t }, then the expected value s constant, but ths random varable converges to zero as t, wth probablty one From ) we see that f the rsk averson of fund s on the other sde of value 1 the log preference) than the fund j s rsk averson, then the rato of the two funds wealth wll not decrease n expected value as T, no matter how close one of the funds s to beng a log fund Expected value of the relatve wealth szes of two funds separated by the log fund s bounded away from zero The key to the result of Proposton 41 s the followng Lemma 41 We have [ ] W T W 1 0 E W jt W 1 j0 1 = 1 ) WT Cov, M W jt W 0 W jt Proof of Lemma 41 The clam follows by drect calculaton from the denttes W 0 = E[M W T ], W j0 = E[MW jt ] 8

11 Ths result has a clear rsk premum structure The left-hand sde s the expected return of fund over [0, T ] relatve to the return of fund j and Lemma 41 shows that the excess relatve return s gven by the covarance of the relatve return wth the SDF-weghted wealth of fund j For example, f s less rsk averse, the dependence of W T on M s steeper and hence the rato W T /W jt s decreasng n M Dependng on whether j s more or less rsk averse than log, MW T wll be ether ncreasng or decreasng n M, determnng the sgn of the covarance A detaled proof of Proposton 41 s provded n Appendx and s also based on the followng known Lemma 42 If both gx) and hx) are ncreasng or both decreasng) then E[gZ)]E[hZ)] E[gZ)hZ)] If both g, h are strctly ncreasng or both strctly decreasng), then the nequalty s also strct unless Z s constant almost surely If one functon s ncreasng and the other s decreasng, then the nequalty reverses 42 Convergence of expected relatve sze From now on we assume that all funds have constant relatve rsk averson CRRA) utltes 9 The man result of ths subsecton s the followng Theorem 41 If ether γ < γ j < 1 or γ > γ j > 1 then E[W T /W jt ] 0 Thus, as long as two CRRA funds are on the same sde of the log preference, the one closer to the log wll domnate n expectaton n the long run Snce the wealth s nonnegatve, standard Chebyshev nequalty mples that t also converges to zero n probablty When rsk averson s homogeneous across agents, the equlbrum SDF s explctly determned by D γ /E[D γ ] But, when rsk averson s heterogeneous, SDF s a soluton of a hghly non-lnear equaton and no explct soluton s possble, except for some very specal values of rsk averson; see, for example, Wang 1996) In the lemma below, we establsh strong bounds on the equlbrum SDF 9 Most of the analyss can be extended to more general utlty functons as n the prevous sectons, but t becomes overly techncal 9

12 Lemma 43 Let Γ 1 be such that Γ b > 1 for all and γ 1 be such that γb 1 for all Then, D γ /γ ψ E[DM]/E[M 1 b ] ) γ /γ ) γ M D γ /Γ ψ E[DM]/E[M 1 b ] ) Γ γ /Γ) 4) The quantty D γ ψ E[DM]/E[M 1 b ] ) γ can be vewed as the ndvdual SDF n an economy populated only by fund It s known that, when rsk averson s heterogeneous, the equlbrum SDF can be represented as a generalzed weghted Hölder average of the ndvdual SDFs see, eg, Malamud 2008a), Malamud 2008b), Joun and Napp 2008), Shefrn 2005)) Lemma 43 shows that M can be estmated from both below and above by Hölder averages wth dfferent exponents γ and Γ The weghts ψ E[DM]/E[M 1 b ] ) γ are not drectly helpful for gettng good bounds for M The followng useful lemma allows us to obtan unform bounds for these weghts It has a very clear economc meanng: the maxmal utlty of a fund s larger than the utlty from smply consumng ts endowment, and s smaller than the utlty from consumng the aggregate endowment of the economy Lemma 44 Let M be the equlbrum SDF If γ < 1 then 1 E[DM]1 γ E[M 1 b ] γ E[D 1 γ ] ψ γ 1 If γ > 1 then ψ γ 1 E[DM]1 γ E[M 1 b ] γ E[D 1 γ ] 1 Proof: The utlty of fund s optmal wealth s gven by 1 1 γ E[W 1 γ T ] = 1 ψ 1 γ 1 γ ) 1 γ E[DM] E[M 1 b ] E[M 1 b ] 1 = ψ 1 γ E[DM] 1 γ E[M 1 b ] γ 5) 1 γ The utlty from just consumng hs endowment the termnal dvdend of hs ntal portfolo) s 1 1 γ E[ψ D) 1 γ ] = 1 1 γ ψ 1 γ E[D 1 γ ] Furthermore, by defnton, n equlbrum we must have W T 1 1 γ E[ψ D) 1 γ ] 1 E[W 1 γ T ] 1 γ 10 D and therefore 1 1 γ E[D 1 γ ]

13 Multplyng both sdes by 1 γ and usng 5), we get the result Lemmas 43 and 44 together allow us to obtan good bounds on the rato W T /W jt and prove Theorem 41 The detals are contaned n Appendx 421 Almost sure convergence of relatve sze We go back now to the almost sure convergence of relatve sze of two gven funds By Theorem 31, the fund closest to log domnates all others almost surely One could then expect that Theorem 31 can be extended along the lnes of Theorem 41: f γ m s closer to one than γ m, then W mt /W kt mght be expected to converge to zero almost surely However, n general, the opposte may be true n case γ k < 1 < γ m, as the second part of the followng proposton shows Proposton 42 Suppose that one of the followng condtons holds true: γ < γ j < 1; γ > γ j > 1; γ < 1 < γ j and 1 γ ) 2 b 1 γ j ) 2 b j 1 γ k ) 2 b b j ) > 0 6) for at least one k In partcular, ths s the case f 1 γ > γ j 1, by settng k = of k = j) Then, W T /W jt 0 almost surely On the other hand, f γ < 1 < γ j and 1 γ ) 2 b 1 γ j ) 2 b j 1 γ k ) 2 b b j ) < 0 7) for all k, then, W T /W jt almost surely even f γ may be closer to one than γ j ) KRWW 2006) show that, n the case of two funds, the fund wth preferences closest to log wll domnate the other fund almost surely Proposton 42 shows that ths s not any more true when there are more than two agents: t mght happen that an agent wth preferences closer to log but less than one) wll experence extncton relatve to a fund whch s further away from log 5 Long-term prce behavor In the prevous two sectons, we have obtaned a detaled pcture of the asymptotc behavor of the wealth of dfferent funds In partcular, we have seen that the fund closest to the log domnates n the long run and owns the whole economy Intutvely, one expects that the long run behavor of the stock prce s also determned solely by the fund closest to the log As we wll see n ths secton, ths s only true to some extent We wll need the followng 11

14 Assumpton 51 The funds have CRRA preferences and there exsts a unque fund, denoted by 0, the rsk averson of whch s closest to one Let γ 0 be the rsk averson closest to one Let us reorder the rsk aversons of funds n the economy as, for some fxed k, l, γ k < γ k+1 < < γ 1 < γ 0 < γ 1 < < γ l Note that we have γ 1 < 1, otherwse 1 γ 1 < γ 0 would mply that γ 1 s closer to 1 Smlarly, γ 1 > 1, otherwse γ 0 < γ 1 < 1 would mply that γ 1 s closer to one If γ 0 < 1 then, by defnton 1 γ 0 < 1 γ 1 1 γ 0 < γ 1 1 γ 0 + γ 1 > 2 Throughout ths secton we wll let t = λt for some fxed λ, so that when we let T, then also t We use the notaton X t Y t to mean that X t /Y t 1 as t The man result of ths secton s Theorem 51 Assume D = e ρt +σb T Defne the ntervals Π l = ) 2 0, γ l + γ l 1, Π 0 = ) 2, 1 γ 0 + γ 1 and Π = ) 2 2, γ + γ +1 γ 1 + γ for = 1,, l 1 Fx {1,, l} and λ Π Then, there exsts a determnstc functon f T ) satsfyng for all T and such that ψ 1 mn{ψ 0, ψ γ 0 0 } f T ) ψ γ max{ψ 0, ψ γ 0 0 } S t P t := e r1 λ 1 )t f tλ 1 ) e ρtλ 1 e σ1 γ 0+γ ) B t e 1 2 σ2 tλ 1 1 γ 0 ) 2 γ 2 )1 λ)+1 γ ) 2 1 γ 0 ) 2) If λ Π 0, that s then λ > 2 γ 0 + γ 1 S t P 0t := D t e r1 λ 1 )t e ρtλ 1 1) e 1 2 σ2 tλ 1 1 2γ 0 )1 λ) 12 8) = e rt T ) E t[d 1 γ 0 ] E t [D γ 0 ] 9)

15 Furthermore, the prce process satsfes P,λ T e rλ 1 1)T )/P +1,λ+1 T e rλ )T ) + as T for any λ, λ +1 Π, Π +1, respectvely In partcular, P,λ T )/P +1,λ+1 T ) + Equaton 9) shows that n the very long run, that s, for t suffcently close to T, and T, there wll be no longer any prce mpact from other funds, and the prce wll be completely determned as f there s only the closest to) log fund n the market By Theorem 31, all funds except for closest to) log fund vansh n the long run and therefore the prce mpact of these funds should also vansh Theorem 51 shows that ths vanshng takes place gradually Agent wth rsk averson above one mpacts the prce precsely n the nterval Π and hs prce mpact vanshes precsely when t/t crosses from Π to Π 1 Moreover, the prce behavor changes dramatcally when we go from Π to Π 1 A smlar phenomenon has been dscovered by KRWW 2006) n the context of heterogeneous belefs They show that, even f the rratonal trader dsappears n the long run, he may generate prce mpact for a substantal fracton of tme The gradual structure of multple ntervals that we obtan here, wth the agents mpact vanshng one after another, seems to be new n the lterature Surprsngly, the agents wth rsk averson below one have no prce mpact at all, asymptotcally One reason may be due to the fact that these agents over-leverage, as dscussed next: For long horzons, the closest to) log fund smply nvests all ts wealth nto the rsky asset see Theorem 62 below) Not surprsngly as shown by Proposton 62 below), the agents wth rsk averson smaller than one nvest more nto the rsky asset than the log fund, that s, they borrow va the rsk-free market from the agents wth rsk averson above one In order that the leveraged agents can repay the debt to the agents wth rsk averson above one, as ndcated by Proposton 42 ther termnal consumpton may become small at a faster rate than for the other agents, and they do not generate any prce mpact We do not know the exact asymptotc behavor of the functons f T ) However, snce we know that they are bounded, we can explctly calculate the long run per-perod stock return Corollary 51 Let λ Π The long run per-perod expected return s gven by Rλ) := lm T T t) 1 log ) Et [D] S t = 1 2 σ2 + r 1 2 σ2 1 λ) 1 1 γ 0 ) 2 γ 2 )1 λ) + 1 γ ) 2 1 γ 0 ) 2 ) 10) In partcular, Rλ) s monotone decreasng and contnuous n λ Fgure 1 shows the graph of the expected return for the remanng perod at tmes t = λt Even though the graph looks qute smooth, there are actually knks at the boundares between ntervals Π, whch would be vsble f that porton of the graph was enlarged 13

16 Long run expected return Rlambda) lambda Fgure 1: Long run expected return at tmes =λt, wth σ=03 and γ_0=1 We complete the secton wth some nterestng global bounds for the stock prce Recall that γ k and γ l are the mnmal and maxmal rsk averson respectvely Note that n the homogeneous economy populated by only one fund wth rsk averson γ the stock prce s gven by S t = e rt T ) E t[d 1 γ ] E t[d We now show that n the mult-fund economy the prce s γ ] bounded between such quanttes correspondng to the lowest and the hghest rsk averson: Proposton 51 We have e rt T ) E t[d 1 γ l ] S E t [D γ t e rt T ) E t[d 1 γ k ] l ] E t [D γ k ] Ths result has an nterestng connecton wth the noton of bubbles and pancs, ntroduced by Cao and Ou-Yang 2005) By ther defnton, a bubble panc) occurs f the prce s above below) the maxmal mnmal) ndvdual prces S t = e rt T ) E t [D 1 γ ] E t [D γ ] Proposton 51 shows that nether bubbles nor pancs occur n our settng However, nterestngly, for models wth consumpton, the stuaton s dfferent, and both bubbles and pancs may occur; see Malamud 2008b) 6 Portfolo Strateges In ths secton we analyze the equlbrum drft and volatlty of the rsky asset, as well as the optmal portfolos, and study ther asymptotc behavor 14

17 61 Equlbrum bounds on portfolo weghts Denote by π log t the portfolo weght of the log fund whch s not necessarly present n the economy) and, as before, by π t the portfolo weght of fund The frst man result n ths subsecton s: Proposton 61 The prce volatlty s always larger than the dvdend volatlty, and bounded from above as follows, for all t: σ σ t σ1 + max γ mn γ ) 11) The nstantaneous Sharpe rato satsfes σ mn γ µ t r σ t σ max γ 12) and therefore mn γ π logt = µ t r 1 + max γ mn γ σt 2 max γ The bounds 12) are nstantaneous analogs of the bounds of Proposton 51 Namely, n the ndvdual economy of agent, the Sharpe rato s gven by σγ, whle, when the rsk aversons are heterogeneous, t stays between the mnmal and the maxmal rsk averson multpled by σ As we wll see below Theorem 61), both bounds 11) and 12) are asymptotcally sharp when rsk averson s larger than one The bounds 11) have drect mplcatons for the well known volatlty puzzle, that s, that, emprcally, the volatlty of the stock prces s sgnfcantly hgher than the volatlty of the dvdends When nequaltes 11) become sharp, they show that heterogeneous rsk averson drves the prce volatlty up The sze of the rato σ t /σ s then determned by the sze max γ mn γ of heterogenety Recall that, when both drft and volatlty of S t are constant, the optmal partal equlbrum portfolo s myopc e, t only depends on nstantaneous stock returns), and s gven by π myopc t = µ t r = γ 1 γ σt 2 π logt When rsk averson s heterogeneous, both drft and volatlty of S t are stochastc and t s generally not possble to fnd closed form solutons for π t The next proposton provdes bounds for equlbrum optmal portfolos Proposton 62 We have: 1) f γ > γ j > 1 then π t < π jt ; 15

18 2) n general, π logt < π t < π myopc t f γ < 1 13) π logt > π t > π myopc t f γ > 1 14) The fact that the stock holdng s monotone decreasng n rsk averson s ntutvely clear Interestngly enough, the proof of ths result s non-trval and we do not know whether ths monotoncty also holds for rsk aversons below one We can always decompose π t = π myopc t + π hedgng t, where π hedgng t s the hedgng component, arsng because the nvestment opportunty set s stochastc Proposton 62 shows that the hedgng component π hedgng s always postve negatve) for rsk averson above below) one Furthermore, by Proposton 61, π logt > 0 and therefore 13)-14) mply that there s no short-sellng by any fund and moreover π t max{max γ, max γ / mn γ } To prove the above results, we wll use the noton of a representatve agent Snce the market s complete, t s well known see, eg, Cvtanć and Zapatero 2004) that the prces n our heterogeneous economy concde wth those n an artfcal economy, populated by a sngle, representatve fund wth a utlty functon U, and the equlbrum stochastc dscount factor equals the margnal utlty of the representatve fund, evaluated at the aggregate endowment, M = U D) 15) That s, the functon U x) s the unque soluton to the equaton ψ E[DM] E[M 1 b ] 1 U x)) b = x 16) Let γ U x) = x U x) U x) be the rsk averson of the representatve fund Proposton 63 The relatve rsk averson γ U x) s monotone decreasng n x and satsfes max γ = lm x +0 γu x) γ U x) lm γ U x) = mn γ 17) x In the case of a one perod economy, Proposton 63 was proved by Bennnga and Mayshar 2000) The proof for our contnuous tme economy s analogous to thers, and we present t n Appendx for the reader s convenence It turns out that the stock volatlty and the optmal portfolo weghts can be expressed n terms of condtonal expected values nvolvng M, D and γ U D), as follows: 16

19 Proposton 64 The drft and volatlty of the stock prce are gven by µ t = r + σ E t[mγ U D)] σ t, σ t E t [M] = σ and the optmal portfolo of agent s gven by 1 E t[mdγ U D)] + E t[mγ U D)] E t [MD] E t [M] ) π t = b 1) Et[M 1 bγu D)] E t [M 1 b ] 1 E t[mdγ U D)] E t [MD] + Et[MγU D)] E t[m] + E t[mγ U D)] E t [M] Proposton 61 follows drectly from Proposton 64 and 17) Furthermore, by Proposton 63, the representatve fund has the decreasng relatve rsk averson DRRA) property Ths fact plays a crucal role n the proof of the followng Lemma 61 We have Furthermore, s monotone decreasng n x E t [Mγ U D)] E t [M] E t[mdγ U D)] E t [MD] E t [M 1 x γ U D)] E t [M 1 x ] Now, Proposton 62 follows by combnng Lemma 61 and Proposton 64 See Appendx for the remanng proofs 62 Long run drft and volatlty of the stock prce In Theorem 51, equaton 8), we have found the approxmate behavor of the stock prce S t at t = λt as T In ths secton we fnd the exact lmt of the drft and the volatlty of the prce and show that t may dffer substantally from what could be expected from Theorem 51 Denote by Jγ) the fund the rsk averson of whch s the closest to γ We assume that the set of rsk aversons γ j s such that the values Jγ) for γ used below are unque Theorem 61 The stock prce satsfes the SDE where, for any λ Π j, and S 1 t ds t = µ t dt + σ t db t lm σ t = σ1 + γ J1+γj ) γ 0 ) T lm µ t = r + γ J1+γj )1 + γ J1+γj ) γ 0 )σ 2 T 17

20 Remark 61 By Theorem 51, the behavor of the stock prce S t changes drastcally as λ crosses the boundary of one of the ntervals Π j The drft of S t may depend on the behavor of f tλ 1 ), but 8) suggests that perhaps the volatlty σ t s gven by σ1 γ 0 + γ j ) for λ Π j Equaton 9) suggests that for λ Π 0, e, the nterval closest to the horzon T, we should have µ t γ 0 σ 2, σ t of J1 + γ j ), we always have γ J1+γj ) and, for λ Π 0, σ Theorem 61 shows that ths s not true! By the defnton γ j and therefore lm σ t σ1 γ 0 + γ j ) T lm µ t γ 0 σ 2 T Let, for example, γ 0 = 1 and γ J2) = 2 Then, for λ Π 0, we get σ t 2σ and µ t 4σ 2 Thus, even though the prce n that nterval corresponds to a market wth only the log fund present, as shown n Theorem 51, ts drft and volatlty are affected also by the rsk averson closest to 2 In essence, there s a delayed effect by a more rsk averse fund on the mean return rate and the volatlty KRWW 2006) also fnd that the prce volatlty may dffer from the one conjectured from the asymptotc behavor of the stock prce However, KRWW 2006) show that, for λ suffcently close to one, both drft and volatlty are determned solely by the survvng agent Our Theorem 61 mples that non-survvng agents can generate mpact on drft and volatlty for all λ 0, 1) Ths s a new and surprsng phenomenon The followng result s a drect consequence of Theorem 61 Corollary 61 In the lmt T, the nstantaneous drft, the volatlty and the Sharpe rato of the stock are monotone decreasng n t = λt By Proposton 64, the sze of drft and volatlty are determned by the representatve agent s rsk averson γ U x) of the economy As tme goes by, the contrbuton to γ U of the agents wth large rsk averson becomes smaller Theorem 51 ndcates that ther prce mpact vanshes gradually) and the average rsk averson should decrease and drve the stock characterstcs down Next, also by Proposton 64, t s clear that for gettng results on the portfolo weghts we must understand the asymptotc behavor of γ U x) The followng non-trval lemma s crucal for the proof of Theorem 61 Lemma 62 Let ether t = λt for some λ [0, 1) or t be fxed Then, lm T E t [γ U D)D α ] E t [D α ] = γ J1 α) 18

21 Lemma 62 shows that any rsk averson n the economy can be generated by approprately choosng α For example, for α = 0, we get the very ntutve result: lm E t[γ U D)] = γ 0 T As agent 0 domnates n the long run, the average rsk averson of the economy converges to γ 0 However, for α suffcently dfferent from 0 we may get a very dfferent behavor Usng Theorem 51 and Lemma 62, t s possble to prove Proposton 65 1) For any λ 0, 1], we have E t [MDγ U D)] lm T E t [MD] = γ 0 2) For any λ Π j, E t [Mγ U D)] lm T E t [M] = γ J1+γj ) See Appendx for detals Theorem 61 s a drect consequence of Propostons 64 and Lmt portfolos Let > 0, γ > 1, and defne the ntervals Θ 0,, Θ 1 as follows: we set ) 2 Θ γ0 + γ 1 )b 0 = γ 0 + γ 1 )1 b ), 1 and, for j {1,, 2}, Θ j = ) 2 γj + γ j+1 )b γ j + γ j+1 )1 b ), 2 γ j 1 + γ j )b γ j 1 + γ j )1 b ) and, fnally, We start wth the followng Θ 1 = 0, ) 2 γ 1 + γ )b γ 1 + γ )1 b ) Lemma 63 Fx an such that γ 1 Then, for any λ Θ j, E t [M 1 b γ U D)] lm = γ T E t [M 1 b J1+γj 1 b ] )) Proof: Drectly from Lemmas 62 and D3 n Appendx The man result n the subsecton s the followng lmtng behavor of the portfolo weghts of the funds more rsk averse than log: 19

22 Theorem 62 Let t = λt and fx an such that γ 1 Then, for any λ Θ j Π m, lm π t = π j, m) = b 1)γ J1+γj 1 b )) + γ J1+γm) T 1 γ 0 + γ J1+γm ) If γ 0 1 then, for any λ except, maybe, for a fnte set of values, lm π 0t = 1 T Proof: Drectly from Propostons 64, 65 and Lemma 63 We can decompose the lmt portfolo π j, m) = π j, m) myopc + π j, m) hedgng By Theorem 61, the myopc component of the lmt portfolo s gven by π j, m) myopc µ t r = lm T γ σt 2 and therefore, the hedgng component s gven by = b γ J1+γm ) 1 γ 0 + γ J1+γm ) π j, m) hedgng = 1 b ) γ J1+γ m ) γ J1+γj 1 b )) 1 γ 0 + γ J1+γm ) We know by Proposton 62 that the hedgng component s nonnegatve Ths can also be seen drectly In fact, a drect calculaton shows that 2 γ j + γ j+1 )b γ j + γ j+1 )1 b ) < 2 γ j + γ j+1 18) Consequently, f the ntersecton Θ j Π m s non-empty, then necessarly γ j γ m and therefore J1 + γ m ) J1 + γ j ) J1 + γ j 1 b )), provng the hedgng component s no less than zero Actually, f the set of rsk aversons n the economy s suffcently dense, t s seen from the above formula that the hedgng component wll be strctly postve In other words, the presence of agents wth varyng rsk aversons nduces hedgng n the optmal portfolo strateges Note that 18) mples, n partcular, Π 0 Θ 0 We analyze ths specal and mportant case of λ Π 0 n a separate Proposton 66 For λ Π 0, π 0, 0) hedgng = 1 b ) γ J1+γ 0 ) γ J1+γ0 1 b )) 1 γ 0 + γ J1+γ0 ) In partcular, f J1 + γ 0 ) = J1 + γ 0 1 b )) then π 0, 0) hedgng = 0, otherwse t s strctly postve 20

23 For λ 0, 1) but not n Π 0, that s, for the ntervals before the last, the presence of the hedgng component s not surprsng: the funds antcpate changes n the future nvestment opportunty set, n partcular those when tme reaches the last nterval Π 0, and hedge aganst those future changes KRWW 2006) were the frst to dscover ths remarkable phenomenon: the hedgng component may stay non-zero long after the prce reaches a regme of constant drft and volatlty nterval Π 0 n our settng) However, they also fnd that there s always a fnte number λ < 1 such that the hedgng component vanshes for λ > λ Proposton 66 shows that the behavor n our model wth many agents may be dfferent: ether the hedgng component s dentcally zero for all λ Π 0, or t s strctly postve for all λ Π 0 Even for the tmes arbtrarly close to T, agents may contnue hedgng aganst stochastc drft and volatlty As above, when the rsk aversons of the agents cover a suffcently dense set of values, there wll be a postve hedgng component n the optmal portfolo strateges n the very long run, although the prce drft and volatlty are close to beng at ts constant lmt Ths phenomenon s drectly related to Theorems 51 and 61 Even for λ Π 0, the stock prce stll has a stochastc drft and volatlty Even though the drft and volatlty converge to constant values asymptotcally, there are small probablty events on whch they devate from ther lmts and force the prce tself to behave as f t had another drft and volatlty, as ndcated by Theorem 51 These small probablty events are precsely the events on whch non-survvng agents generate drft and volatlty mpact and force other agents to hedge aganst these events 10 On the other hand, for a suffcently rsk averse fund wth b close to zero, the hedgng component wll be close to zero n the very long run It s the funds wth lower rsk aversons but hgher than one), whch engage n substantal hedgng n the long run For rsk aversons less than log, we don t have the exact lmt, but the followng bounds for the portfolo weghts: Proposton 67 If γ < 1, then for all λ Π m, b 1)γ J1+γl 1 b )) + γ J1+γm) 1 γ 0 + γ J1+γm) lm nf T π t The result follows from Proposton 64 and Lemma 64 If α > 0 then lm sup π t b 1)γ J1+γ k 1 b )) + γ J1+γm) T 1 γ 0 + γ J1+γm ) E t [D γ kα γ U D)] E t [D γ kα ] and the nequaltes reverse for α < 0 E t[m α γ U D)] E t [M α ] E t[d γ lα γ U D)] E t [D γ lα ] 10 Such an mpact on the state-prce densty by vanshng agents was present also n the model of KRWW 2006), where t was possble to compute t explctly 21

24 64 Contnuous rsk averson lmt The lmt expressons n the prevous secton are dffcult to dgest because of ther dscrete nature In partcular, snce the rsk aversons take dscrete values, the behavor of the functon γ Jα) becomes complcated On the other hand, f the rsk averson s contnuous, γ Jα) = α and many expressons start lookng much cleaner In ths secton we study the case when the rsk aversons cover suffcently densely an nterval of the form [1, Γ] for large enough Γ In that case, ntervals Π become very small and λ Π means that λ 2 γ + γ +1 γ 1 γ λ 1 As the rsk aversons become more and more dense n the contnuous rsk averson lmt, ths approxmaton becomes exact We cannot make the dstrbuton of rsk averson truly contnuous, because then our convergence results do not hold any more, but when the rsk aversons are suffcently dense, the approxmaton errors become neglgble 11 Thus, we have the followng analog of Corollary 51: Corollary 62 In the contnuous rsk averson lmt, the long run per-perod expected return s gven, for λ > Γ 1, by Rλ) := lm T T t) 1 log ) Et [D] S t = 1 2 σ2 1 + λ 1 ) + r Snce, n the contnuous lmt, γ J1+γj ) = 1 + γ j and γ j = λ 1, Theorem 61 takes the form Proposton 68 In the contnuous rsk averson lmt, for λ > Γ 1, σλ) := lm T σ λt = σ1 + λ 1 ), µλ) := lm T µ λt = r λ 1 ) 2 σ 2 Smlarly, n the contnuous rsk averson lmt, λ Θ j Π m means that λ γ 1 j 1 γ 1 γ 1 m, that s γ j Thus, Theorem 62 takes the form γ 1 λγ 1) + 1, γ m λ 1 11 It s also possble to prove our results drectly for a contnuum of agents when the rsk averson dstrbuton s suffcently smooth 22

25 Proposton 69 Let λ > Γ 1 and π γ λ) = lm T π γ, λt be the lmt portfolo of the agent wth rsk averson γ Then, n the contnuous rsk averson lmt, In partcular, s ndependent of λ and s monotone decreasng n λ π γ λ) = λ2 γγ 1) + 2γ 2 γ) + 1)λ + γ 2 γ 2 γ 1)λ + 1)λ + 1) π myopc γ λ) = 1 γ πγ hedgng γ 1)λ + γ λ) = γ 1) γ 2 γ 1)λ + 1)λ + 1) There are several surprsng features here The most nterestng s that the formulae are ndependent of the rsk averson dstrbuton It does not matter what the cross-sectonal dstrbuton of rsk averson n the populaton s, t only matters that t covers all possble values suffcently densely Furthermore, we see that n that case the hedgng component s monotone decreasng wth tme Ths has a clear economc meanng: the closer we are to the termnal horzon T, the less stochastc fluctuatons are left n future, aganst whch the agent needs to hedge However, nterestngly enough, the hedgng demand never vanshes completely More precsely, we have π hedgng γ Γ 1 ) > πγ hedgng λ) > πγ hedgng 1) = 2γ2 3γ + 1 2γ 3 At the very begnnng, when λ s small, the agent expects a lot of fluctuatons n the stochastc nvestment opportunty set and nvests a large fracton nto the hedgng portfolo, that may approach one as hs rsk averson γ becomes suffcently large In partcular, for suffcently large Γ, s monotone ncreasng n rsk averson π γ Γ 1 ) 1 1 γ On the other hand, as tme approaches T, the hedgng demand decreases A drect calculaton shows that πγ hedgng 1) s monotone ncreasng on γ [0, 3 + 3)/2] and decreasng for larger values Thus, we dscover the followng unexpected phenomenon: for tme horzons suffcently close to T and large enough rsk averson the hedgng demand s monotone decreasng n rsk averson 23

26 65 The case of two funds In ths secton we consder the case of an economy populated by two funds wth rsk aversons γ 1 > γ 0 > 1 In ths case, Π 0 = ) 2, 1 γ 0 + γ 1, Π 1 = 0, 2 γ 0 + γ 1 ) For agent 1, we need to consder the ntervals ) Θ 1 γ 1 γ 0 0 = γ 0 + γ 1 )γ 1 1), 1, Θ 1 1 = 0, ) γ 1 γ 0 γ 0 + γ 1 )γ 1 1) Then, the nterval 0, 1) wll be parttoned nto three ntervals ) 2 Λ 1 = Θ 1 1, Λ 2 = Θ 1 0 Π 1, Λ 3 = Π 0 =, 1 γ 0 + γ 1 gven by 0, ) γ 1 γ 0 γ 0 + γ 1 )γ 1 1), ) γ 1 γ 0 γ 0 + γ 1 )γ 1 1), 2 γ 0 + γ 1, ) 2, 1 γ 0 + γ 1 Let ) πγ hedgng 0 λ), πγ hedgng 1 λ)) = lm T πhedgng γ 0, λt, lm T πhedgng γ 1, λt be the two-dmensonal vector of the lmt hedgng components of the portfolos of agents 0 and 1 We summarze our fndngs n Proposton 610 We have 1) If γ 1 < 2 then the long run drft and volatlty lm σ t = σ1 + γ 1 γ 0 ), lm µ t = r + γ γ 1 γ 0 ) σ 2, 19) T T as well as the hedgng components of the portfolos πγ hedgng 0 t), πγ hedgng 1 t)) = γ 1 γ γ 1 γ 0 1 b 0 ), 0 ) are ndependent of λ 2) If 2 < γ 1 < 2 + γ 0 then the long run drft and volatlty are gven by 19), but the hedgng components are gven by πγ hedgng 0 t), πγ hedgng 1 ) = γ 1 γ γ 1 γ 0 1 b 0 ), 0 ) λ Λ 1 ; 1 b 0 ), 1 b 1 ) ) λ Λ 2 Λ 3 24

27 3) If γ 1 > 2 + γ 0 then the long-run drft and volatlty are gven by lm σ 1 + γ 1 γ 0 ) σ, σ 2 γ 1 ) λ Λ 1 Λ 2 ; t, µ t r) = T σ, σ 2 γ 0 ) λ Λ 3, 20) and the hedgng components of the portfolos are gven by πγ hedgng 0 t), πγ hedgng γ 1 γ 1 b 0 ), 0 ) λ Λ 1 ; 0 1 t)) = 0, 1 b 1 + γ 1 γ 1 ) ) λ Λ 2 ; 0 0, 0) λ Λ 3 Because of the sparse dstrbuton of rsk averson only two values), there are several phenomena wth no analogs n the contnuous rsk averson settng of the prevous secton For example, the hedgng portfolo of agent 0 s always decreasng n λ But, the hedgng portfolo of agent 1 s ncreasng n λ n the case 2) and has a knk n case 3) The structure of the hedgng portfolo appearng n case 3) s smlar to those appearng n KRWW 2006) However, the structure arsng n cases 1) and 2) s new In case 1), agent 1 chooses a tme-ndependent, myopc portfolo, but the less rsk averse agent 0 chooses a substantal hedgng portfolo, even though the asymptotc drft and volatlty are ndependent of λ The reason s that the drft and volatlty may devate from ther lmt values and, by Theorem 51, for λ Π 0 = Λ 3, the stock prce behaves asymptotcally as f t had a drft and volatlty σ, r + γ 0 σ 2 ) A postve hedgng component ndependent of λ s new and dffers from KRWW 2006) In case 2), the asymptotc drft and volatlty are stll tme-ndependent and agent 0 holds the same portfolo On the other hand, for small λ, when there are lot of possble fluctuatons n the future nvestment opportunty set, agent 1 nevertheless decdes not to hedge at all However, for λ Π 0 e, λ > 2/γ 0 + γ 1 )), he chooses a non-trval hedgng portfolo Ths monotone ncreasng structure of the hedgng portfolo s dfferent from the one dscovered by KRWW 2006) Fnally, n case 3), agent 0 does not hedge for λ Λ 2 Λ 3, whereas agent 1 does not hedge at the begnnng, hedges n the mddle, and then agan stops hedgng 7 Conclusons We show that, wth long-lved funds, the equlbrum n complete market models lead to Merton-Black-Scholes MBS) type prces and portfolo strateges, asymptotcally However, the prce parameters change as tme goes by, leadng to asymptotcally lower and lower returns, and the portfolo weghts are not myopc as n the partal equlbrum MBS model 25

28 The sze of the prce volatlty s nfluenced by the overall dvdend volatlty and the dfference between the hghest and the lowest rsk averson Eventually, the fund closest to the log preferences s the only one that survves, but prce mpact s felt for a long tme also from other funds, more rsk averse than log The drft and the volatlty of the prce, as well as the optmal portfolo weghts, depend on rsk aversons hgher than one even at the tmes closest to the fnal tme horzon The portfolos nclude a non-zero hedgng component, even n the very long run The hedgng component s postve for the rsk aversons hgher than one, and negatve for those smaller than one, and t may be monotone decreasng n rsk averson at further away tmes In terms of the expected value of the relatve wealth sze, the log fund wll not domnate even n the long run It would be of sgnfcant nterest to show how much our results reman vald when there s also ntermedate consumpton of the funds The optmal fnal consumpton wll stll have the same form, so t s possble that some results wll qualtatvely survve Moreover, t would be desrable to see what happens to the prce and portfolo behavor n the presence of more than one rsky asset, and wth funds havng dfferent tme horzons The pcture then would lkely be more dynamc, wth funds and/or assets enterng and leavng the market as tme goes by Fnally, once the behavor for standard ratonal agents s well understood, one can see how the equlbrum dffers wth non-standard preferences and behavor In partcular, n our model leveragng s low, as the funds wealth needs to stay postve, and the probablty of bankruptcy s zero We leave the possblty of extendng our analyss to these nterestng ssues for future research Appendx A Proofs for Secton 3 We wll do the proofs of ths secton under a strengthened form of Assumpton 31 Throughout ths secton we assume a power rate of convergence for the rsk aversons: Assumpton A1 There exst constants K, a > 0 such that γ x) γ 0 K x a and γ x) γ Kx a 21) for all Condton 21) s not necessary, but the proofs are much smpler than under Assumpton 31 26