A Dynamc Heterogeneous Belefs CAPM Carl Charella School of Fnance and Economcs Unversty of Technology Sydney, Australa Roberto Dec Dpartmento d Matematca per le Scenze Economche e Socal Unversty of Bologna, Italy Xue-Zhong He School of Fnance and Economcs Unversty of Technology Sydney, Australa May 6, 2006 Abstract We reconsder the dervaton of the tradtonal captal asset prcng model (CAPM) n the dscrete tme settng for a portfolo of one rsk-free asset and many rsky assets. In contrast to the standard settng we consder heterogeneous agents whose expectatons of future returns based on statstcal propertes of past returns nduce expectatons feedback. We assume that agents formulate ther demands based upon heterogeneous belefs about condtonal means and covarances of the rsky asset returns, and that the rsky returns evolve over tme va the determnaton of market clearng prces, under a Walrasan auctoneer scenaro. In ths framework we frst construct a consensus belef (wth respect to the means and covarances of the rsky asset returns) to represent the aggregate market belef and derve a heterogeneous CAPM whch relates aggregate excess return on rsky assets wth aggregate excess return on the market portfolo va an aggregate beta coeffcent for rsky assets. We then adopt ths perspectve to establsh a market fracton model n whch agents are grouped accordng to ther belefs. The mpact of dfferent belefs on the market equlbrum returns and the beta-coeffcents s analysed. In partcular, we focus on some classcal heterogeneous agents types - fundamentalsts, trend followers and nose traders - and nvestgate how the key behavoral parameters affect the tme varyng behavour of the aggregate beta coeffcent. Introducton
... Secton 2 derves equlbrum CAPM-lke relatonshps for asset returns n the case of heterogeneous belefs and relates a consensus belef about the expected excess return on each rsky asset to a consensus belef about expected market return, va aggregate beta coeffcents. Secton 3 dscusses further the wealth dynamcs and the beta coeffcents, and relates them to the heterogeneous belefs about the returns on the rsky assets. Secton 4 consders explctly the supply of the rsky securtes, ntroduces market clearng condtons to derve equlbrum prces, and relates the aggregate beta coeffcents to the market clearng prces. Secton 5 combnes ths setup wth a dynamc, market fracton mult-asset framework wth heterogeneous groups of agents, whch generalzes earler contrbutons by Brock and Hommes (998) and Charella and He (200a,b), and hghlghts how the aggregate beta coeffcents may vary over tme once agents belefs are assumed to be updated dynamcally at each tme step as a functon of past realzed returns. Ths framework s then specalzed to the case of nteracton between fundamentalsts, trend followers and nose traders n Secton 6, whch reports the results of numercal smulatons and nvestgates how the dynamcs of market returns and aggregate beta coeffcents s affected by the key behavoural parameters. Secton 7 concludes. 2 Dervaton of Heterogeneous Belefs CAPM n terms of asset returns The present secton generalzes the dervaton of CAPM relatonshps - as carred out for nstance by Huang and Ltzenberger (988) Secton 4.5 - to the case of nvestors wth heterogeneous belefs about asset returns. Some of the deas contaned n the present secton are adapted from Lntner (969), where aggregaton of ndvdual assessments about future payoffs sperformedna mean-varance framework. Assume that there exsts a rskless asset wth a rate of return r f and that the rates of return er j, j =, 2,..., N, of the rsky assets are multvarate normally dstrbuted. Assume that nvestor, =, 2,..., I, has a concave and strctly ncreasng utlty of wealth functon u ( ). Followng Huang-Ltzenberger (Secton 4.5) the optmally nvested random wealth of nvestor (at the end of the perod) satsfes E hu 0 ( W f h ) E [er j r f ]= E u 00 ( W f ) Cov ( W f, er j ) () for any j =, 2,..., N, where NX fw = W0 +r f + w j (er j r f ) and w j s the fracton of wealth of agent nvested n the rsky asset j. By j= 2
defnng the -th nvestor s global absolute rsk averson E hu 00 ( fw ) θ := E hu 0 (f W ) condton () becomes θ E [er j r f ]=Cov ( f W, er j ) j =, 2,..., N (2) Note that à N! # Cov ( W f X NX, er j )=Cov "W 0 w k er k, er j = W0 w k Cov (er k, er j ) k= It follows that the condtons (2) can be rewrtten wth vector notaton as θ E [er r f ]=W0Ω w (3) where er =[er, er 2,..., er N ] >, =[,,..., ] >, w =[w,w 2,...,w N ] >,andω = [σ,jk ], j, k =, 2,..., N, where σ,jk := Cov (er j, er k ). Equaton (3) can be rewrtten as and summng across k= W0w = θ Ω E [er r f ] (4) W0w = = = θ Ω E [er r f ] We remark that the optmal nvestment polcy of agent s only mplctly defned by (4), because n general θ = θ (w ) wll depend, n ts turn, on w. Nevertheless, at ths stage we are nterested n equlbrum relatonshps nvolvng aggregate belefs, whch do not requre w to be made explct. By defnng the vector of the aggregate wealth proportons nvested n the rsky assets w a := W0 W w (5) m0 = where W m0 := P I = W 0 represents total ntal wealth, one obtans w a = W m0 = θ Ω E [er r f ] (6) An aggregate varance/covarance matrx Ω a, whch represents a consensus belef about the covarance structure of returns, can be defned as a weghted harmonc mean of the ndvdual covarance matrces Ω, such that Ω a = Θ = θ Ω (7) 3
³ PI. where Θ := = θ Thentfollowsfrom(6)that w a = = W m0 " = Ω a ΘW m0 θ Ω " ΘΩ a # E (er) Θ Ω a r f = = θ Ω E (er) r f # from whch Ω a w a = ΘW m0 " ΘΩ a = θ Ω E (er) r f # (8) Note that ΘW m0 can be nterpreted as the aggregate relatve rsk averson of the economy n equlbrum. Smlarly to Huang and Ltzenberger (988), Secton 4.5, we can defne the random market return er m as the one whch satsfes fw m := = fw = W m0 ( + er m ).e. er m = f W m W m0 where f W m represents the random end-of-perod wealth n the market. Note that er m can also be rewrtten n terms of aggregate wealth proportons as er m := r f + w > a (er r f ) The quantty σ 2 a,m := w a > Ω a w a can be nterpreted as the aggregate consensus belef about the varance of the market return, gven by " # σ 2 a,m := w > a Ω a w a = ΘW m0 w > a ΘΩ a = θ Ω E (er) w a > r f Next, defne the aggregate expected returns on the rsky assets E a (er) (consensus belefs about expected returns) as a weghted average of the ndvdual expected returns E (er), such that from whch = θ Ω E (er) = E a (er) =ΘΩ a = = θ Defne also the aggregate expected market return Ω E a (er) (9) θ Ω E (er) (0) E a (er m ):=r f + w > a (E a (er) r f ) () 4
Equaton (9) becomes σ 2 a,m = ΘW m0 w > a [E a (er) r f ] ª (2) whch mples that [E a (er m ) r f ]=ΘW m0 σ 2 a,m (3).e. the aggregate expected market rsk premum s proportonal to the aggregate relatve rsk averson of the economy. It follows from (8), (0) and (3) that σ 2 Ω a w a = [E a (er) r f ] a,m E a (er m ) r f.e. [E a (er) r f ]= E a(er m ) r f σ 2 Ω a w a (4) a,m The entres of Ω a w a represent the aggregate covarances between the return on each rsky asset and the market return,.e. Ω a w a =[σ a,jm ] j =, 2,..., N where σ a,jm := Cov a (er j, er m ), so that (4) can be rewrtten component by component as E a (er j ) r f = σ a,jm σ 2 [E a (er m ) r f ] j =, 2,..., N (5) a,m where σ a,jm /σ a,jm := β a,jm represents the aggregate beta coeffcent of the j-th rsky asset. Equaton (5) s the CAPM relaton, n terms of returns, generalzed to the case of heterogeneous belefs. The vector β a,m := [β a,m, β a,2m,..., β a,nm ] > of the aggregate beta coeffcents n (4) s thus gven by β a,m = σ 2 Ω a w a (6) a,m 3 Heterogeneous belefs, wealth dynamcs and beta coeffcents It s convenent to rewrte the beta coeffcents (6) n a dfferent form, whch stresses the way they depend on agents aggregate belefs about random returns and ntal wealth. Defne ζ := w W0 = θ Ω E [er r f ] as the vector of optmal dollar nvestment n each rsky asset by agent, and ζ := w W0 = = = 5 θ Ω E [er r f ] (7)
the vector whch collects the aggregate dollar demands for each rsky asset. Usng (0), ζ can be rewrtten as a functon of the aggregate belefs, as follows: ζ = Θ Ω a E a [er r f ] (8) Note also that: ζ = w a W m0 (9) where w a s the vector of aggregate wealth proportons nvested n the rsky assets, defned by (5). We now focus on the wealth dynamcs. Denote by W f m := P I f = W the random end-of-perod wealth n the market. Usng (8), the optmally nvested fnal wealth s gven by fw m = W m0 ( + r f )+ W0w > (er r f )=W m0 ( + r f )+ζ > (er r f )= = = W m0 ( + r f )+Θ [E a (er) r f ] > Ω a (er r f ) and the aggregate expectaton and varance/covarance belefs about the excess market payoff f W m W m0 ( + r f ) are E a [ W f m W m0 ( + r f )] = ζ > [E a (er) r f ] = Θ [E a (er) r f ] > Ω a [E a (er) r f ] (20) Var a [ W f m W m0 ( + r f )] = Var a [ W f m ]=ζ > Ω a ζ = Θ 2 [E a (er) r f ] > Ω a [E a (er) r f ] (2) from whch one gets Θ = E a[ W f m W m0 ( + r f )] Var a [ W f m W m0 ( + r f )] = E a (er m ) r f W m0 σ 2 a,m whch s obvously equvalent to (3), snce E a (er m ) r f = W m0 E a [ f W m W m0 ( + r f )] σ 2 a,m := Var a (er m )= Wm0 2 Var a [ W f m W m0 ( + r f )] Usng (20) and (2) the latter equatons can be rewrtten, respectvely, as E a (er m ) r f = σ 2 a,m := Var a (er m )= ΘW m0 [E a (er) r f ] > Ω a [E a (er) r f ] (22) (ΘW m0 ) 2 [E a(er) r f ] > Ω a [E a (er) r f ] (23) 6
Therefore, usng (9), (8), and (23), the beta coeffcents (6) can be rewrtten as ΘW m0 β a,m = [E a (er) r f ] > Ω a [E a (er) r f ] [E a(er) r f ] (24) An mportant remark about Equaton (24) s that t provdes the beta coeffcents as functons of the belefs about returns. However, equaton (24) also depends on the ntal wealth level. Note that we have not made any assumpton up to now about the net supply of assets n the market, and therefore about the composton of the total market wealth. The usual assumpton n the CAPM lterature s that the rskless asset s n zero net supply n the market, whch can be formalzed as NX =0.e. W m0 := W0 = = W0 = = j= j= w j à NX! W0w j = W0w > = ζ > where ζ s the dollar demand vector for the rsky assets, defned by (7). Under these assumptons, the total current market wealth s exactly equal to the total wealth nvested n the rsky assets,.e. W m0 = ζ > = Θ [E a (er) r f ] > Ω a (25) whereas the fnal wealth W f m s gven by fw m = ζ > (+er) =Θ [E a (er) r f ] > Ω a (+er) (26) As a consequence, the rate of return on total market wealth er m := W f m /W m0 can be rewrtten as er m = [E a(er) r f ] > Ω a er [E a (er) r f ] > Ω (27) a and represents the return on the market portfolo of the rsky assets, whch has weghts summng up to unty, gven by w a := W W 0w, w a > = m0 = Usng (25), the beta coeffcents wth respect to the market portfolo of the rsky assets can thus be rewrtten as a functon of the belefs about the returns of the rsky assets, as follows β a,m = = [E a (er) r f ] > Ω a [E a (er) r f ] > Ω a [E a (er) r f ] [E a(er) r f ] (28) Note that ths s not restrctve under CARA utlty - whch wll be assumed throughout thenextsectons-becausenthscasetheamountofwealthnvestedntherskyassetss ndependent on the wealth level, and thus n a dynamc settng the total amount of wealth nvested n the rskless asset has no effect on the prce and the return dynamcs of the rsky securtes. 7
4 Equlbrum prces In the followng we assume that agents have CARA utlty of wealth functons, h so that the global absolute rsk averson of agent, θ = E hu 00 (f W ) /E u 0 (f W ), and the aggregate rsk averson Θ, areconstant. Note that n ths case (3) allows to obtan explctly the optmal demand of each agent w = W0 θ Ω E [er r f ] (29) Denote now by p 0 =[p 0,p 02,..., p 0N ] > the vector of current prces, to be determned by the market clearng condtons under a Walrasan auctoneer scenaro, and by z :=[z,z 2,..., z N ] T the (postve) supply vector (number of shares). Let Z := dag[z,z 2,..., z N ] a (N N) dagonal matrx whose entres are the elements of z. Smultaneous market clearng for all the rsky assets mples ζ = Zp 0 (30).e. p 0 = Z Θ Ω a [E a (er) r f ] (3) Note that we have extracted market clearng prces by takng the vew that durng the process of market equlbraton, the nvestors belefs about the jont dstrbuton of rates of returns are fxed,.e. ndependent of current market prces. Of course Equaton (3) can be expressed also n terms of the ndvdual belefs and atttudes of the I agents: p 0 = Z = θ Ω E [er r f ] (32) Note that by usng (3) the beta coeffcents can also be expressed n terms of market clearng prces, as follows β a,m = p > 0 z p > 0 ZΩ azp 0 Ω a Zp 0 5 A dynamc market fracton mult-asset model The present secton sets up a dynamc heterogeneous belefs model whch extends to a mult-asset framework earler contrbutons developed by by Brock and Hommes (998), Charella and He (200a,b) n the smple case of a sngle rsky securty, and thus ncorporates nto a dynamc setup the CAPM-lke return relatonshps dscussed n the prevous secton n a statc framework. To ths extent we frst rewrte our aggregaton relatonshps n terms of market fractons of heterogeneous agents types, by ntroducng some slght changes of notaton. Assume that the I nvestors can be grouped nto a fnte number of agent-types, ndexed by h H, where the agents wthn the same group are homogeneous 8
n terms of belefs E h (er) and Ω h,aswellasrskatttutesθ h. Denotng by I h, h H, the number of nvestors n group h, wedenotebyn h := I h /I the fracton of agents of type h. Wethendenotebys := (/I)z the supply of shares per nvestor. Note that, nstead of usng the aggregate rsk averson coeffcent Θ := follows ³ PI = θ t s convenent to defne the average rsk averson θa as à X θ a := h H n h θ h! where obvously θ a = IΘ. Note also that the aggregate belefs about expected payoffs and varances/covarances can be rewrtten, respectvely, as follows Ω a = θ a à X h H E a (er) =θ a Ω a X h H n h θ h Ω h n h θ h Fnally the equlbrum prces are rewrtten as or n terms of the belefs of each group Ω h! E h(er) p 0 = S θ a Ω a [E a (er) r f ] p 0 = S X h H n h θ h Ω h [E h (er) r f ] We now turn to the process of formaton of heterogeneous belefs and equlbrum prces n a dynamc settng, from tme t to tme t +.Indongths,we take the vew that agents belefs about the returns n the tme nterval (t, t+), er t+, whch are formed before dvdends at tme t are realzed and prces at tme t are revealed by the market, determne the aggregate demand for each rsky assetattmet, whch n turns produces the equlbrum prces at tme t, p t,va the market clearng condtons. Of course, once prces and dvdends at tme t are realzed, then also the returns r t become known. More precsely, we assume that ) nvestors assessments of the end-of-perod jont dstrbuton of the returns er t+ are formed at tme t before the equlbrum prces at tme t are determned, and 2) these belefs reman fxed whle the market fnds ts equlbrum vector of current prces, p t. In partcular, assumpton ) mples that heterogeneous agents assessments about er t+ are functons of the nformaton up to tme t. Weassumen partcular that these belefs can be expressed as functons of the realzed returns r t, r t 2,...,.e., for any group, or belef-type h H Ω h,t := [Cov h,t (er j,t+, er k,t+ )] = Ω h (r t, r t 2,...) (33) 9
E h,t (er t+ )=f h (r t, r t 2,...) (34) where obvously smlar representatons hold also for the aggregate belefs Ω a,t := [Cov a,t (er j,t+, er k,t+ )] and E a,t (er t+ ). The market clearng prces at tme t become p t = S X h H n h θ h Ω h,t [E h,t(er t+ ) r f ] (35) (where Ω h,t and E h,t (er t+ ) are defned by (33) and (34), respectvely), or n terms of the consensus belefs: p t = S θ a Ω a,t[e a,t (er t+ ) r f ] (36) Next, note that the return r j,t on asset j, realzed over the tme nterval (t,t) s gven by r j,t = p j,t + d j,t p j,t where d j,t denotes the realzed dvdend per share of asset j, j =, 2,..., N. One can rewrte realzed returns wth vector notaton, as follows r t = P t (p t + d t ) (37) where d t :=[d,t,d 2,t,..., d N,t ] >,andp t := dag(p,t,p 2,t,..., p N,t ). Equaton (37), va the market clearng condtons (35) and the belefs updatng equatons (33) and (34), gves the return r t as a functon of r t, r t 2,... and of the realzed dvdends d t (whch wll be assumed to follow an exogenous nose process) and thus determne dynamcally nevolvng o prces and returns. In the followng we wll assume an..d. process edt for the dvdends, wth d := E( d e t ). We summarze below the dynamcal system whch descrbes the market fracton mult-asset model n terms of returns, where the market clearng prces are used as auxlary varables: r t = F(r t, r t 2,...; e d t )=P t (p t + e d t ) where p t = S X h H n h θ h Ω h,t [E h,t(er t+ ) r f ] P t := dag(p,t,p 2,t,..., p N,t ) Ω h,t = Ω h (r t, r t 2,...) E h,t (er t+ )=f h (r t, r t 2,...) Moreover, at the begnnng of each tme nterval (t, t +) the aggregate belefs about returns (based on nformaton up to tme t ) satsfy a CAPMlke equaton of the type E a,t (er t+ ) r f = β a,mt [E a,t (er m,t+ ) r f ] 0
where er m,t+ = [E a,t(er t+ ) r f ] > Ω a,t er t+ [E a,t (er t+ ) r f ] > Ω a,t denotes the random return on the market portfolo of the rsky assets, whereas the aggregate beta coeffcents are gven by [E a,t (er t+ ) r f ] > Ω a,t β a,mt = [E a,t (er t+ ) r f ] > Ω a,t[e a,t (er t+ ) r f ] [E a,t(er t+ ) r f ] Note that the aggregate betas are tme varyng due to tme varyng belefs about both the second moment and the frst moment of the returns dstrbuton. Remark The dynamcal system (37), (36) s not determnstc. However, assumng that dvdends evolve over tme as an..d. process, t can be useful to wrte down the determnstc system obtaned by replacng d e t wth d := E( d e t ).One can easly see from (37), (36) that n order the system to be at a steady state, the statonary prces and returns p and r must satsfy p = S X h H n h θ h Ω h [f h r f ] (38) where Ω h := Ω h (r, r,...), f h := f h (r, r,...), and r = P d (39) where P := dag(p, p 2,..., p N ) (note that P p = ). Equaton (39) can be rewrtten n the equvalent form p = R d (40) where R := dag(r, r 2,..., r N ), or component by component as p j = d j r j j =, 2,..., N whch gves the equlbrum prces through the usual dscounted dvdend formula va the approprate rates of returns for each asset. Substtuton of (40) nto (38) yelds R d = S X n h θ h Ω h [f h r f ] h H where Ω h := Ω h (r, r,...), f h := f h (r, r,...).e. a system of equlbrum equatons n the returns r, r 2,..., r N,fromwhch n general the statonary returns (and then prces) can be solved for.
6 An example In ths secton we provde a specfc example of nteracton of dfferent belefs types and analyze the resultng dynamcs for market returns and aggregate beta coeffcents. Ths example, whch s smlar to Charella Dec and He (2006) consders two types of agents, fundamentalsts, who have some nformaton on the fundamentals of the rsky asset and who beleve that prces wll be drven back to fundamentals n the future, and trend followers, who may have no nformaton on the fundamentals and who extrapolate the past returns nto future returns. These two types of agents are the most common and popular ones n the lterature on heterogeneous agent based models. In addton, we consder a thrd type of agents - nose traders - whose demand for each rsky asset s treated as an exogenous random dsturbance, descrbed by an..d. process wth zero mean. 6. Fundamentalsts and trend followers The example follows the heterogeneous agent model wth multple rsky assets studed n Charella Dec and He (2006) and s related to earler work by Charella and He, 200b. We consder two types of agents, fundamentalsts and trend followers. Fundamentalsts expect the future asset returns wll be drven back towards some exogenously gven levels ρ =[ρ, ρ 2,..., ρ N ] >, whch are assumed to depend on fundamental varables. Ths can be expressed as E f,t (er t+ )=r t + α(ρ r t ) where α represents the expected speed of mean reverson. For the sake of smplcty, we assume that the fundamentalsts have constant belefs about the varance/covarance structure of returns determned - for nstance - by the varance/covarance of the dvdends Ω f,t = Ω f The trend followers are assumed to compute the expected returns accordng to E c,t (er t+ )=r t + γ(r t u t ) where γ 0 s the extrapolaton parameter and u t s a vector of sample means of past realzed returns r t, r t 2,... Smlarly to Charella Dec and He (2006), we assume that u t s computed recursvely as u t = δu t 2 +( δ)r t (4) The varance/covarance matrx Ω c,t s assumed to consst of a constant component Ω c, and by a tme-varyng component λv t, λ 0. Ω c,t = Ω c + λv t 2
where V t s updated recursvely as a functon of past devatons from sample average returns, as follows V t = δv t 2 + δ( δ)(r t u t 2 )(r t u t 2 ) > (42) Note that (4) and (42) can be consdered as lmtng cases of geometrc decay processes, when the memory lag length tends to nfnty. We denote by θ f and θ c the rsk averson coeffcents of the two agent-types, by n f and n c = n f ther market fractons, and by θ a = ³ n f θ f + n c θ c the average rsk averson. Then the varances/covarances and expected excess returns are gven, respectvely, by µ Ω a,t = θ nf a Ω f + n µ c Ω nf c,t = + n µ c nf Ω f + n c Ω c,t θ f θ c θ f θ c θ f θ c nf E a,t (er t+ ) = θ a Ω a,t Ω f E f,t (er t+ )+ n c Ω θ f θ c µ nf r t + Ω f + n c αnf Ω c,t θ f θ c c,t E c,t (er t+ ) θ f = Ω f (ρ r t )+ γn c Ω c,t (r t u t ) θ c The general dynamc model gven by (36) and (37) thus specalzes to the followng nosy nonlnear dynamcal system µ p t = S nf Ω f + n c Ωc,t (r t r f )+ αn f Ω f (ρ r t )+ γn c Ω c,t (r t u t ) θ f θ c θ f θ c where r t = P t (p t + e d t ) Ω c,t = Ω c + λv t and u t and V t are updated as follows u t = δu t +( δ)r t V t = δv t + δ( δ)(r t u t )(r t u t ) > 6.2 The effect of nose traders The demand for the rsky assets (number of shares) from the nose traders at tme t s descrbed by the random vector e ξ t := [ e ξ,t, e ξ 2,t,..., e ξ N,t ] >,wherethe e ξj,t are assumed..d. wth E( e ξ j,t )=0, j =, 2,..., N. We also assume, for the sake of smplcty, that the standard devaton of the nose traders demand for each asset s proportonal to the supply of the same asset n the market,.e. Var( e ξ j,t )=q 2 s 2 j, whle demands for dfferent assets are not correlated, E( e ξ j,t, e ξ k,t )=0, j, k =, 2,..., N. 3
Set e Ξ t := dag( e ξ,t, e ξ 2,t,..., e ξ N,t ). The market clearng condtons n the presence of nose traders thus become θ a Ω a,t[e a,t (er t+ ) r f ]+ Ξ e t p t = Sp t and the market clearng prces thus become p t =(S Ξ e t ) θ a Ω a,t[e a,t (er t+ ) r f ] where E a,t (er t+ ) and Ω a,t are as n the prevous secton. Note that the ntroducton of nose traders s formally equvalent to assumng a nosy supply vector es t = s e ξ t. 6.3 Smulaton results... 7 Dscusson... References Böhm, V., Charella, C., 2005. Mean varance preferences, expectatons formaton and the dynamcs of random asset prces. Mathematcal Fnance 5(), 6-97. Brock, W.A., Hommes, C.H., 998. Heterogeneous belefs and routes to chaos n a smple asset prcng model. Journal of Economc Dynamcs and Control 22, 235-274. Charella, C., He, X.-Z., 200a. Heterogeneous belefs, rsk and learnng n a smple asset prcng model. Computatonal Economcs 9 (), 95-32. Charella, C., He X.-Z., 200b. Asset prce and wealth dynamcs under heterogeneous expectatons. Quanttatve Fnance (5), 509-526. Charella, C., Dec, R., He, X.-Z., 2006. Heterogeneous expectatons and speculatve behavor n a dynamc mult-asset framework. Journal of Economc Behavor and Organzaton, n press. Huang, C., Ltzenberger R.H., 988. Foundatons for fnancal economcs. North-Holland, Amsterdam. Lntner, J., 969. The aggregaton of nvestor s dverse judgements and preferences n purely compettve securty markets. Journal of Fnancal and Quanttatve Analyss 4 (4), 347-400. 4