Modern Dynamic Asset Pricing Models

Transcription

1 Modern Dynamic Asse Pricing Models Teaching Noes 6. Consumpion, Porfolio Allocaion and Equilibrium wih Consrains 1 Piero Veronesi Universiy of Chicago CEPR, NBER 1 These eaching noes draw heavily on Cuoco (1997) and Karazas and Shevre (1999, Chaper 6). They are inended for sudens of Business 3597 only. Please, do no disribue wihou my prior consen.

2 Piero Veronesi Modern Dynamic Asse Pricing Models page: 2 Inroducion Solving a porfolio problem wih consrains is difficul and in his eaching noes my inen is o give only a broad undersanding of he new echniques ha have recenly been inroduced. The aricles ha progressed our undersanding of hese problems are lised in he syllabus. In his eaching noes we will follow he approach of Cuoco (1997), who also gives a lucid review of he main poins on he previous lieraure.

3 Piero Veronesi Modern Dynamic Asse Pricing Models page: 3 The (Usual) Seup As before, fix a complee probabiliy space (Ω, F, P) on which a n-dimensional Brownian moion B is defined along wih is (augmened) filraion {F }. The consumpion space L + is he se of non-negaive, adaped and inegrable consumpion processes. Le here be n risky asses and a riskless asse S = S + β = e r u du I S µ u du + I S σ u db u where again I S is he n n diagonal marix wih S i on he ii-h posiion. Assume ha he process r u is uniformly bounded and ha he usual regulariy (inegrabiliy) condiions hold for he asse prices. Assume ha σ is inverible almos everywhere and define he marke price of risk vecor ν = σ 1 (µ r 1 n )

4 Piero Veronesi Modern Dynamic Asse Pricing Models page: 4 Noice ha σ inverible makes markes complee if here are no consrains. Assume also ha he Novikov s condiion is saisfied ( ( 1 T )) E exp ν 2 ν d < As in TN1, we assume ha invesor s preferences are represened by he uiliy funcion [ T ] U (c) = E u (c, ) d Some resricions on he uiliy funcions are imposed bu I refer you o he original works (see e.g. Cuoco (1997) or Karazas and Shreve (1998)). Here, Condiion A in TN1 should suffice for he resuls. Finally, invesors are endowed wih an iniial wealh w and a sochasic endowmen process e such ha for some K e we have T β 1 e K e

5 Piero Veronesi Modern Dynamic Asse Pricing Models page: 5 Porfolio Consrains We now assume ha invesors canno rade freely as in previous cases, bu hey are consrained in keeping heir sraegies in some se A. For modeling purposes, i is convenien o define a rading sraegy in dollar erms raher han unis as we did so far. Given a rading sraegy ( θ, θ ) we shall concenrae on he dollar equivalens ( ϕ, ϕ ) defined as ϕ = θ β and ϕ i = θi S Clearly, he rading sraegy ( ϕ, ϕ ) R n+1. Hence, we can now define in general erms a porfolio consrain se A R n+1 such ha he dollar posiions ( ϕ, ϕ ) are consrained o lie in A raher han he whole R n+1. We shall assume ha A is non-empy, closed and convex. Examples: 1. No Consrains: A = R n Non-radeable asses (incomplee markes): A = {( ϕ, ϕ ) R n+1 : ϕ k = for k = m + 1,..., n }

6 Piero Veronesi Modern Dynamic Asse Pricing Models page: 6 3. Shor-Sale Consrains: A = {( ϕ, ϕ ) R n+1 : ϕ k for k = m + 1,..., n } 4. Buying Consrains: A = {( ϕ, ϕ ) R n+1 : ϕ k for k = m + 1,..., n } 5. Minimum Capial Requiremens: { (ϕ A =, ϕ ) } n R n+1 : ϕ + ϕ k K for K. A special case is he porfolio insurance program, where K >. Oher consrains can also be inroduced (see Cuoco (1997)). k=1

7 Piero Veronesi Modern Dynamic Asse Pricing Models page: 7 Given a consrain se A, a consumpion plan is said o be A feasible if here exiss a rading sraegy ( ϕ, ϕ ) such ha ϕ + ϕ = w + ϕ + ϕ K ϕ T + ϕ T = ϕ u r u + ϕ u µ u du + (c u e u ) du ϕ u σ u db u and ( ϕ, ϕ ) A for all [, T]. Before urning o he soluion of hese problems, recall he approach in complee markes. We have he following ingrediens 1. A Maringale ( ξ = exp ν u db u ) ν u ν udu 2. A (uniquely defined) sae-price densiy π = β 1 ξ

8 Piero Veronesi Modern Dynamic Asse Pricing Models page: 8 3. A saic budge consrain [ T E ] π (c e )d w We hen find a saddle poin of he Lagrangian [ T ] [ T ] L (c, λ) = E u (c, )d λe π (c e )d w The problem is ha while wih complee markes and no consrains, here is a unique sae price densiy π ha is consisen wih no-arbirage (see TN1), when here are consrains here are infiniely many sae-price densiies ha are consisen wih no-arbirage. As a consequence, we no longer have only one saic budge consrain, bu a whole family of budge consrains ha mus be saisfied o ensure ha he porfolio policies lie wihin he consrain se A. This clearly makes he problem more difficul o solve. In order o sae more generally he problem, we mus inroduce some more noaion.

9 Piero Veronesi Modern Dynamic Asse Pricing Models page: 9 Suppor Funcions Le he consrain se A R n+1 be given. For any vecor v = ( v,v 1) R R n, le δ (v) = sup ( ϕ v + ϕ v 1) (ϕ,ϕ) A This is called suppor funcion of A. This funcion can easily reach + and hence i is imporan o define is effecive domain à = { v R n+1 : δ (v) < } Some resricions on δ and à are generally imposed. Noice in paricular ha à is a closed, convex cone. The cases discussed above ypically saisfy hese condiions. In hose examples, we have: 1. No Consrains: à = {} and δ (v) = for all v Ã

10 Piero Veronesi Modern Dynamic Asse Pricing Models page: 1 2. Non-radeable asses (incomplee markes): à = { v R n+1 : v k = for k =,..., m } δ (v) = for v à 3. Shor-Sale Consrains: A = { v R n+1 : v k = for k =,..., m v k for k = m + 1,..., n δ (v) = for v à } 4. Buying Consrains: A = { v R n+1 : v k = for k =,..., m v k for k = m + 1,..., n δ (v) = for v à } 5. Minimum Capial Requiremens: à = {k1 n : k } δ (v) = Kv for v Ã

11 Piero Veronesi Modern Dynamic Asse Pricing Models page: 11 A Family of Unconsrained Markes Given a consrain se A, one can consider he se N of Ã-valued processes v = ( v,v 1) saisfying [ T ] E v v d < Le v be given and define now he following processes ( ) β v = exp r u + vu du ( ν v = σ 1 µ + v 1 ( ) r + v) 1n ( ξ v = exp ν v u db u 1 ) ν v u 2 νv u du π v = (β v ) 1 ξ v These processes are well defined, and ξ v is a local maringale. Consider he resriced class N such ha ξ v is in fac a maringale. For each process v N, we can inerpre all hese processes as represening an economy under one paricular probabiliy measure Q v which is equivalen o P.

12 Piero Veronesi Modern Dynamic Asse Pricing Models page: 12 Consider he discouned price process Ŝ v = (β v ) 1 S By Io s Lemma, we have dŝv = ( r + v )Ŝv d + (β v ) 1 µ I S d + (β v ) 1 I S σ db By Girsanov s Theorem, we know we can define a Brownian moion under Q v by d B = db + ν v d Hence, by subsiuing back, we have dŝv = ( r + v)ŝv d + (β v ) 1 µ I S d (β v ) 1 I S σ ν v + (β v ) 1 I S σ d B = ( r + v)ŝv d + (β v ) 1 µ I S d (β v ) 1 ( I S µ + v 1 ( ) ) r + v 1n + (β v ) 1 I S σ d B = (β v ) 1 I S v 1 d + (βv ) 1 I S σ d B

13 Piero Veronesi Modern Dynamic Asse Pricing Models page: 13 Noice an imporan difference compared o he case of no consrains. Now, he discouned price process is no longer (necessarily) a maringale. However, given he characerizaion of he processes v 1, ha by definiion mus lie in Ã, we can sill say somehing abou he properies of he discouned price processes. Le us denoe by µ v he drif of he discouned processes. We hen have a relaionship beween he consrain ses A, and he class of equivalen maringale measures Q v ha characerize he sock price process. These properies will become useful o solve he consumpion problem. 1. No Consrains: Ã = {} and hence µ v maringales. =. Hence, he discouned price processes are (local) 2. Non-radeable asses (incomplee markes): Ã = { v R n+1 : v k = for k =,..., m } Hence µ v k, = for k = 1,.., m. In his case he se {Qv : v N } correspond o he se of probabiliy measures equivalen o P under which he radeable asses have sock prices ha are indeed local maringale. 3. Shor-Sale Consrains: Ã = { v R n+1 : v k = for k =,..., m v k for k = m + 1,..., n Hence µ v k, = for k = 1,.., m and µ v k, for k = m + 1,..., n. In his case he se {Q v : v N } corresponds o he se of Q v under which he unconsrained discouned }

14 Piero Veronesi Modern Dynamic Asse Pricing Models page: 14 sock prices are local maringales and he consrained discouned sock prices are local super maringales. 4. Buying Consrains: Ã = { v R n+1 : v k = for k =,..., m v k for k = m + 1,..., n Hence µ v k, = for k = 1,.., m and µv k, for k = m + 1,..., n. Hence, he unconsrained discouned asses are local maringales and he consrained discouned asses are local submaringales. 5. Minimum Capial Requiremens: Ã = {k1 n : k } Hence, µ v k, so ha he discouned sock price process are local supermaringale, wih drif proporional o he sock price. Anoher way of inerpreing each v N is ha i generaes he unique sae price densiy π v in a ficiuous unconsrained economy where he parameers characerizing he economy are given by } r v = r + v µ v = µ + v 1 For each of hese problems, we know how o solve he maximizaion problem.

15 Piero Veronesi Modern Dynamic Asse Pricing Models page: 15 In addiion, i urns ou ha a his opimum for he ficiuous uncosrained economy he agen is happy o be consrained, in he sense ha he consrain is non-binding in his ficiuous economy. One way o ackle he consrained maximizaion problem is o solve he ficiious unconsrained maximizaion problem wih he differen sae-price densiy and hen maximize he over he se of uiliies so obained. The following resul shows ha indeed, each sae price densiy π v wih v N consiues an arbirage free sae-price densiy in he original economy wih consrains and ha he saisfacion of a saic budge consrain wih respec o all hese sae price densiies is also sufficien o guaranee A feasibiliy. Proposiion 1: A consumpion process c is A feasible if and only if for all v N [ T ] [ T ] E Qv (β v ) 1 (c e ) d w + E Qv (β v ) 1 δ (v ) d Proof: See e.g. Cuoco (1997). The mehod used o prove his heorem is similar, albei more elaborae, o he one used o show a similar proposiion in TN1. I is good o go hrough i once in life, bu we won do i here. Hence, we can now reformulae he maximizaion problem of he invesor as max c C U (c) (1)

16 Piero Veronesi Modern Dynamic Asse Pricing Models page: 16 subjec o E Qv [ T ] (β v ) 1 (c e δ (v )) d w for all v N (2) and where C is a se of feasible consumpion plans (see resricions in Cuoco (1997)). Noice ha for mos cases, we have ha δ (v ) =, which implies an idenical budge consrain as he one we looked a in TN2. However, now here is a coninuum of budge consrains. I urns ou ha his messes up hings a lo. Some key properies of he space of inegrable funcions c (L 1 spaces) o admi a soluion o he convex problem (1) are missing.

17 Piero Veronesi Modern Dynamic Asse Pricing Models page: 17 Convex Dualiy Approach (Cvianic and Karazas (1992)) Cvianic and Karazas (1992) used a convex dualiy approach o solve he problem (1) and were able o prove exisence of a consumpion plan under somewha resricive assumpions on he uiliy funcions and on he endowmen process e. Given is applicaions in he lieraure, we review here he main poins. We sill follow Cuoco (1997) main argumens. Le c be he opimal consumpion plan and assume ha here exiss a v N, such ha [ T ( ) ] 1 E Qv (c e δ (v ))d = w (3) β v Since he se {π v : v N } is a convex se, here should be a sae price densiy π v Lagrangian muliplier λ such ha (c, λ,v ) is a saddle poin of he funcion [ T ] L (c, λ,v) = U (c) λe π v (c e δ (v )) d w and a (4) where one maximizes over c and minimizes over (λ,v).

18 Piero Veronesi Modern Dynamic Asse Pricing Models page: 18 Noice ha we can rewrie [ T L (c, λ,v) = E [ T = E ] u (c, ) λ (π v (c e δ (v ))d w ) ( T )] (u (c, ) λπ v c )dλ w + π v (e + δ (v )) d Noice ha c only eners in he erm (u (c, ) λπ v c ). Hence, we can define he funcion ũ (z, ) = sup c [u (c, ) zc] (5) This funcion ũ (z, ) is called convex conjugae of u ( c, ). If he Inada Condiions are saisfied, he problem (5) is solved by c = I u (z, ) where I u (., ) is he inverse of he marginal uiliy u c (c, ). If we firs maximize over c, we can now define he minimizaion program [ T ( T )] min J (λ,v) = E ũ (λπ v, )d + λ w + π v (e + δ (v )) d λ,v (6)

19 Piero Veronesi Modern Dynamic Asse Pricing Models page: 19 The program (6) is called Dual (Shadow Sae-Price) Problem of he original problem (1). The key resul is he following (see Cuoco (1997)) Proposiion 1. Suppose ha he uiliy funcion u (c, ) saisfies he Inada Condiions and here exiss a consan β (, 1) and γ (, ) such ha for all (c, ) (, ) [, T] βu c (c, ) u c (γc, ) (7) If hese exiss a soluion (λ,v ) o he dual problem (6), and [ T ( ) ) E π v (I u λ π v, e δ (v) ] d hen here exiss a consrained opimal consumpion plan c such ha < u c (c, ) = λ π v (8) holds and E [ T π v ( ) ) ] (I u λ π v, e δ (v ) d = w (9) Conversely, if (8) and (9) hold for some (λ,v ) (, ) N and some A-feasible consumpion plan c, hen (λ,v ) solves he dual problem.

20 Piero Veronesi Modern Dynamic Asse Pricing Models page: 2 This proposiion gives us a ool o solve some ineresing siuaions. We will use i several imes. Noice ha condiion (7) is saisfied by all he CRRA (Consan Relaive Risk Aversion) uiliy funcions u (c, ) = e φ c 1 b 1 b wih b (, ]. In fac, u c (γc, ) = e φ (γc) b = γ b e φ c b = 1 γ bu c (c, ) Hence. we can find β (, 1) and γ (, ) such ha he condiion is saisfied.

21 Piero Veronesi Modern Dynamic Asse Pricing Models page: 21 Example: Log Uiliy To illusrae he above findings wihin he conex of an example, consider he opimal consumpion and invesmen sraegy of a log-uiliy invesor in he presence of shor-sale consrains or incomplee markes. Suppose ha he consrain applies o all he securiies (for simpliciy). In he case of incomplee markes i means ha he invesors canno inves in he sock marke (we shall use he resuls obained here in TN4). Finally, assume no sochasic income: e =. From above, we have ha he effecive domain and he suppor funcions are à = { v R n+1 : v = and v 1 } δ (v) = for shor sale consrains and à = { v R n+1 : v = } δ (v) = for incomplee markes.

22 Piero Veronesi Modern Dynamic Asse Pricing Models page: 22 In addiion, we also have ha I u (z, ) = e φ z 1 which implies ũ (z, ) = u (I u (z, ),) zi u (z, ) = e φ log ( z 1 e φ) e φ = e φ (1 + φ + log (z)) Hence, we obain ha he J (λ,v) in (6) is given by [ T ] J (λ,v) = E e φ (1 + φ + log (λπ v )) d + λw T = e φ (1 + log (λ))d [ T ] E e φ log (π v )d T e φ φd + λw = (1 + log (λ)) e φt 1 + e φt φt + e φt 1 + λw φ φ [ T ( ( +E e φ ν v u db u + r u + vu + 1 ) 2 νv u νv u = (2 + log (λ)) e φt 1 + e φt T + λw φ T ( ( + e φ E r u + vu + 1 ) ) 2 νv u νv u du d ) ] du d

23 Piero Veronesi Modern Dynamic Asse Pricing Models page: 23 where we used he fac ha π v = (β v ) 1 ξ v ( = exp ( r u + vu + 1 ) ) 2 νv u νv u du ν v u db u Hence, he J (λ,v) is addiively separable in he wo argumens and we can minimize one a he ime. Wih respec o λ, we find ha J λ (λ,v) = implies 1 e φt 1 λ φ + w = which yields λ = 1 w 1 e φt φ We now have o minimize he second erm T ( ( e φ E r u + vu + 1 ) 2 νv u ν v u ) du d wih respec o v Ã. Recall firs ha we mus have v u =, by definiion of Ã.

24 Piero Veronesi Modern Dynamic Asse Pricing Models page: 24 The minimum is obained by poinwise minimizing he expression ν v ν v = ( ) µ + v 1 r 1 n (σ σ ) 1 ( ) µ + v 1 r 1 n = ν ν + v 1 (σ σ ) 1 v 1 + 2v 1 (σ σ ) 1 (µ r 1 n ) Recall ha v 1 are consrained o lie in Ã, hough. The shor-sale consrain implies ha v 1, which if (µ r 1 n ) > implies ha v = obains he minimum. To be more specific, suppose here is only one risky asse. Then, he unconsrained minimizaion would be v 1 = (µ r ) Hence, he consrained opimum is v 1 = max ( (µ r ),) This is inuiive: If µ r >, a log uiliy invesor (who is myopic) will inves a posiive amoun in he sock, hence he shor-sale consrain is no binding and v 1 =. In his case, since also =, we have ha he sae-price densiy is he same as in he unconsrained case v π v = β 1 ξ (if µ r > )

25 Piero Veronesi Modern Dynamic Asse Pricing Models page: 25 If insead (µ r ) <, i would be profiable for a myopic invesor o shor he risky asse, bu here he consrain kicks in and he ficiuous marke price of risk becomes ν v = σ 1 ( µ + v 1 r ) = Hence, he ficiuous sae price densiy becomes π v = β 1 ξ v = β 1 Tha is, i is as if here are no oher asses in he ficiuous economy! And for our invesor his is exacly he case because of he shor-sale consrain. A similar reasoning goes for he invesor always barred o inves in he sock marke. In his case we have ha v 1 is unconsrained (while v = ) which implies ha i mus be given by v 1 = (µ r 1 n ) Subsiuing his ino he sae price densiy, we obain (again) ha π v = β 1 ξ v = β 1

26 Piero Veronesi Modern Dynamic Asse Pricing Models page: 26 The ficiuous economy is made up only by he risk-free asse and he invesor opimally chooses here only. In summary, we see ha raher han solving he consrained maximizaion problem direcly, he approach aken here is o change he sae-price densiy π v of he invesor in such a way ha an unconsrained agen facing his paricular sae-price densiy would never hi he consrains and, in addiion, his/her opimal choices are idenical o he one ha he consrained agen would choose. As a final sep, given he resuls above abou he opimal v and λ, we can solve for he opimal consumpion and asse allocaion. Tha is, we have jus o use he usual resuls abou opimal allocaion ha we derived in previous eaching noes, bu use he sae-prices π v. The opimal consumpion is c = e φ 1 λ π v = e φ w φ (1 e φt ) π v

27 Piero Veronesi Modern Dynamic Asse Pricing Models page: 27 The process for wealh is W = 1 π v = 1 π v = 1 π v = 1 π v [ T E [ T E ] π v τ c τdτ π v τ e φτ w φ (1 e φt )π v τ w φ e φ e φt (1 e φt ) φ w ( e φ e φt) (1 e φt ) Noice ha we can rewrie he consumpion process as c = φ ( ) W 1 e φ(t ) ] dτ Finally, he opimal rading sraegy is ϕ = max (, (µ r )W ) if here are shor-sale consrains, or ϕ = if (clearly) he agen is forbidden from invesing in he marke.

28 Piero Veronesi Modern Dynamic Asse Pricing Models page: 28 Direc Approach (Cuoco (1997)) The condiions o prove exisence in he Dual problem (6) are very resricive. One needs o have δ =, e = and a relaive risk aversion always below 1. Cuoco (1997) ackled direcly he problem (1) by using a echnique called of relaxaion projecion, and could prove exisence of a soluion under much more general condiions (see Theorem 2). In paricular, all he uiliy funcions of he HARA (Hyperbolic Absolue Risk Aversion) class saisfy hese condiions ( u (c, ) = beφ αc ) 1 b 1 b b + β wih b (, ], α > and β, φ wih β = 1 if b =. Raher han going over he exisence resul, we pass direcly o characerize he opimal consumpion policies and obain implicaions for sock reurns. In he unconsrained case we could make use of he Lagrange muliplier o show ha he opimal consumpion equal he inverse of he uiliy funcion defined a a poin proporional o he curren sae price densiy.

29 Piero Veronesi Modern Dynamic Asse Pricing Models page: 29 Tha is, we recall ha in he unconsrained case we obained (in TN3) ha c = I u (λπ, ) where π is uniquely defined. The nex proposiion generalizes his resul: (Cuoco 1997) Proposiion 2: Le c be he opimal consumpion plan and assume ha c. Suppose ha here exiss a λ such ha [ T ] E u c (λc, ) c d < (1) Then here exiss a sequence (ψ n π v n ) wih ψ n > and v n N such ha almos everywhere (u c (c (ω), ) ψ nπ v n (ω)) c (ω) (11) If in addiion hen (11) holds wih ψ n = ψ for all n. [ T inf E v n N π v c ] < (12)

30 Piero Veronesi Modern Dynamic Asse Pricing Models page: 3 An implicaion of his is ha if c > (and his happened under condiion A in TN1), we hen have Corollary 1: If c > almos everywhere, and (1) and (12) hold, hen here exiss ψ > and a sequence of sae-price densiies {π v n } wih v n N such ha u c (c (ω),) = lim n ψπv n (ω) for almos all (, ω) [, T] Ω. To recapiulae, wha his corollary shows is ha in he presence of porfolio consrains (and incomplee markes), here are a coninuum of sae-price densiies ha are consisen wih he noion of no-arbirage. However, here exiss a sequence of sae price densiies converging (sae-by-sae) o some sae price densiy π v such ha he opimal consumpion sill saisfies he u c (c, ) = ψπ v as i was rue for he unconsrained case.

31 Piero Veronesi Modern Dynamic Asse Pricing Models page: 31 A Consrained C-CAPM Resul We end hese eaching noes wih an addiional resuls on he naure of he equilibrium when here are consrains in he economy. As in TN2, suppose here are m agens, each endowed wih an insananeous uiliy funcion u i (c) and a common discoun rae φ. We say ha he agen is a a regular opimum if he marginal uiliy process a he opimum consumpion plan is proporional o a generalized sae price densiy Ψβ 1 ξ v, where Ψ is a non-increasing process and v N. Tha is, we assume ha for all i we have ( ) u i c c i = e φ Ψ i β 1 ξ v i where v i N and Ψ i is a non-increasing process. Noice ha he necessiy o inroduce a process Ψ i sems from he fac ha in he ficiuous economy for agen i, we sill need o discoun fuure cash flows a he economy-wide ineres rae r. In oher words, he sae-price densiy for each agen we analyzed in he previous secions was π v = (β v ) 1 ξ v

32 Piero Veronesi Modern Dynamic Asse Pricing Models page: 32 where ( ) ( ) β v = exp r u + vu du = β exp vu du Since across agens we mus have he same bond price β, we now need o inroduce a new process ha ransforms he consrained opimizaion problem of each agen ino a ficiuous unconsrained problem. Le he aggregae endowmen be denoed as m m e = e i = i=1 where ĉ i is he opimal consumpion of agen i. We hen have he following ineresing resul (by Cuoco (1997)) Proposiion 3: Under regulariy condiions on he uiliy funcion, he equilibrium risk premia are deermined by [ ] ( ) ds i E ds i S i r = Γ (c ) Cov S i, de (13) m 1 Γ (c ) a j (ĉ ) j ( ) v i j, vj, (14) j=1 i=1 ĉ i

33 Piero Veronesi Modern Dynamic Asse Pricing Models page: 33 where I recall from TN5 ha a j (ĉ j ) = u j cc u j c ( ĉ j ( ĉ j ) ) is he absolue risk aversion coefficien of agen j, and Γ (c ) = 1 ( ) 1 m j=1 aj ĉ j is he coefficien of absolue risk aversion of he marke iself. Equaion (13) is he consrained version of he C-CAPM. The firs erm is he usual risk premium coefficien ha is generaed by covariance of he reurn of asse i wih he aggregae endowmen. The second componen in line (14) sems from he consrains imposed on he opimal porfolio and/or he incompleeness of markes. In fac, he laer elemen is a weighed average of he processes v j ha idenify he shadow sae-prices for agen j in he economy, when he/she is subjec o consrains. Noice ha we can see immediaely from he form of à wha are he implicaions for he C-CAPM when we impose various consrains ha are homogenous across individuals.

34 Piero Veronesi Modern Dynamic Asse Pricing Models page: 34 For example, in he case of incomplee markes, we recall ha we have v i = for i =,..., m (where m is he number of raded asses) which implies immediaely ha for he asses ha are raded, he C-CAPM holds. Similarly, for he case of shor-sale consrains, we know ha v i = for i =,..., m for he m unconsrained asses while v i for he consrained asse. This implies ha for he unconsrained asses he C-CAPM works while for he consrained asses he C-CAPM over-predics he reurns. Similar implicaions can be derived in he oher examples.

35 Piero Veronesi Modern Dynamic Asse Pricing Models page: 35 References Cuoco (1997) Opimal Consumpion and Equilibrium Prices wih Porfolio Consrains and Sochasic Income Journal of Economic Theory, 72, Cvianic and Karazas (1992) Convex Dualiy in Consrained Porfolio Opimizaion, Annals of Applied Probabiliy, 2,