Teaching Notes #2 Equilibrium with Complete Markets 1
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1 Teachng Noes #2 Equlbrum wh Complee Markes 1 Pero Verones Graduae School of Busness Unversy of Chcago Busness Sprng 2005 c by Pero Verones Ths Verson: November 17, These eachng noes draw heavly on Duffe 1996, Chapers 9 and 10 and Karazas and Shevre 1999, Chaper 3 and 4. They are nended for sudens of Busness only. Please, do no dsrbue whou my pror consen.
2 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 65 1 Compeve Equlbrum We now use he resuls n TN 1 o deermne he compeve equlbrum. The noon of equlbrum n hs se up s as follows: 1. There are m agens n he economy, each endowed wh a sream of consumpon good; 2. The consumpon good s mmedaely pershable, so ha mus be consumed mmedaely; 3. Agens can rade her endowmens, by sellng/buyng fnancal secures; 4. All fnancal secures are n zero-ne supply: For every buyer here mus be a seller. Ths s he sandard, general equlbrum noon of a pureexchange economy. Noce n parcular ha here s no producon. 1.1 Prmves Le B = B 1,..., B d be a d-dmensonal Brownan moon defned on a complee probably space Ω, F,Pandle {F } denoe he sandard flraon of B. Le us fx a horzon T and le he consumpon space be he se L of adaped processes such ha E T 0 c 2 d <. Suppose here are m agens, ndexed by =1,.., m. Each agen receves an endowmen { e } L+
3 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 66 Each agen has a srcly ncreasng uly funcon U : L + R. All agens have a common dscoun rae φ. We shall assume consan, bu could also be a funcon of me. Each agen wll maxmze U c = E 0 [ T 0 e φ u c d ] Noce ha we assume here no uly from fnal wealh, alhough could have been nsered whou any rouble. We shall assume everywhere ha condon A n TN1 hold. 1.2 Fnancal Markes We assume complee markes. Wh no loss of generaly, le here be d rsky secures wh prce processes ds = I S μ d + I S σ db where I S s he dagonal marx wh S on he h elemen, and μ and σ are adaped processes n L and L 2. Marke compleeness s acheved by assumng ha σ s nverble almos everywhere. Also, here s a rsk-free secury, wh shor rae process r and prce β = e 0 r u du
4 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 67 As n he prevous noes, defne he marke prce of rsk process ν = σ 1 μ r 1 d Assume ha he Novkov s condon E exp 1 T 2 0 ν ν d < s sasfed and le ξ =exp 0 ν db u ν uν u du We recall ha from Novkov s heorem, ξ s a P -marngale. Fnally, defne he sae-prce densy process π = β 1 ξ 1 We saw already ha π s such ha S π = Sπ s a marngale. As usual, a radng sraegy θ 0, θ s a vecor process n H 2 S, ha s, a space wh suffcen negrably condons o rule ou doublng sraeges. For convenence, le θ = θ 0, θ, S =β,s. A radng sraegy θ = θ 0, θ fnances a consumpon process c gven ncome e f θ S = θ 0 u d S u + e 0 u c u du 0 2 θ T S T = 0 3
5 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 68 Noce ha he wealh W = θ S grows also because of he addonal endowmen ha has o be aken no accoun. The las equaly means ha no oblgaons are lef a horzon. Each agen hen faces he problem sup U c 4 c,θ 0,θ Λ where Λ = c, θ 0, θ L + H 2 S such ha θ 0, θ fnances c gven e A secury-spo marke equlbrum s a collecon of prce processes β,s, consumpon processes c m =1 and radng sraeges θ m such ha gven β, S, each agen solves =1 4 and markes clear: m =1 θ =0and m =1 c e =0 Noce ha hs s an endowmen economy, so ha he aggregae consumpon s jus generaed by he aggregae endowmen. In equlbrum, agens rade her own endowmens by sellng and buyng fnancal secures.
6 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 69 2 The Indvdual Agen Opmzaon Problem We frs look a he opmzaon problem of one gven agen. Ths s assumed o be small n he sense of akng he prce processes as gven and opmze hs/her neremporal uly gven he prces. As one can guess, hs problem s he same as he one we solved for n TN1. The only dfference s ha now our nvesor s no endowed wh an nal wealh w bu wh an endowmen process e. However,wecanusehesameechnqueusedearleromake sac he budge consran 2-3. Once hs s accomplshed, s nuve ha he resulng opmal sraegy would be smlar. Under he assumpons above, le Q be he equvalen marngale measure defned by ξ T =exp T 0 ν db u 1 T 2 0 ν uν u du Le he dscouned fuure endowmen be denoed by: w = E Q [ T 0 β 1 e ] d Noce ha he expecaon s under he Q measure.
7 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 70 We recall ha for gven nal wealh w,hesac budge consran ha we obaned n TN1 was E Q [ T 0 β 1 c ] d w Hence, n analogy wh wha found n TN1, we have ha he dynamc budge consran 2-3 can be equvalenly expressed as E Q [ T 0 β 1 c ] d E Q [ T 0 β 1 e ] d 5 The way o prove hs s o go hrough he same seps as n Proposon 4 n TN1 and defne c = c e and le w =0. I s mmedae ha one ges 5. Fnally, usng he same mehod as n proposon 4 n TN1 one obans: Corollary 1: The sac budge consran 5 s equvalen o [ T E π 0 c ] [ T d E π 0 e ] d 6 where π s he sae prce densy defned n 1. Noce ha he expecaon s under he orgnal probably measure P.
8 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: Opmal Consumpon From he resul of TN1, we hen oban he followng resul. Le Iu : R R be he nverse of he nsananeous margnal uly funcon u c, ha s, s such ha for every x we have u c x = x. I u Proposon 1: Le he prce process β,sbe gven and assume ha condon A s sasfed for agen. Then, here exss a soluon o he ndvdual nvesor s problem wh c = I u λ e φ π where λ solves [ T E π 0 Iu λ e φ ] π d = E [ T 0 π e d ] 7 The analogy wh he resul n secon n TN1 s he followng: 1. The nverse margnal uly funcon: In secon we had u c,, wh ncluded n he uly funcon. Hence, he relaonshp s u c c, =e φ u c c Hence, f x = u c c,, s nverse s I u x, =I u e φ x Ths explans why we have he erm e φ nsde he nverse uly funcon.
9 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: We defned he funcon [ ] T w λ =E π 0 I u λ π, d and we mposed w λ =w. Clearly, equaon 7 s he same condon. 2.2 Opmal Porfolo Weghs As we poned ou earler, wh complee markes s necessary only o fnd he opmal consumpon. We can fnd he opmal sraegy ha fnances consumpon as a resdual. We recall he mehod here agan. From he proofs n TN1, we found a few mporan relaonshps ha we mus recall frs. For convenence, defne he wealh a me as W = θ0, β + θ S and he dscouned wealh as W = β 1 W = θ 0, + θ S β 1 = θ 0, + θ Ŝ where we recall ha Ŝ = S β 1 s a marngale under Q. Hence dŝ = IŜσ d B
10 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 73 where B s a Brownan moon under Q generaed by Grsanov s heorem hrough he formula B = B + ν 0 udu From he dynamc budge consran we also have W = θ 0, + θ Ŝ 8 = 0 θ u dŝu + 0 β 1 u e u c u du 0 9 so ha dŵ = β 1 = β 1 e c d + θ dŝ 10 e c d + θ IŜσ d B 11 By defnng c, = c e and seng w = 0, proposons 4 and 5 n TN 1 mply ha he curren dscouned wealh s jus equal o he expeced dscouned value of fuure consumpon mnus endowmen under Q: W = EQ T β 1 u c, u du = E Q T β 1 u c u e u du Ths equaly s due o he assumpon of complee markes: The opmal consumpon sream can be hough of as a secury, and Ŵ as s prce a me. To revew how o ransform hese expecaons under Q no expecaons under P, recall ha he measure Q s defned
11 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 74 hrough he Radon-Nkodym dervave dq dp = ξ T and hence ha we can use he propery ha for any random varable Z such ha E Q Z < we oban E Q Z F = E ξ TZ F E ξ T F We hen have he followng chan of equales W = E Q = E = E T β 1 u T ξ T β 1 u T ξ u β 1 u c u e u du = E ξ c u e u du c u e u du ξ = E = E ξt T β 1 u E ξ T T E u ξ T β 1 u c u e u du ξ T π u c u e u du ξ c u e u du For noaonal convenence, we can defne JT = T π 0 u c u e u du so ha we can rewre he dscouned wealh as W = 1 E J ξ T J 1 = M J ξ 12 wherewedefnedhep marngale M as M = E J T 13
12 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 75 From he Marngale Represenaon Theorem, here exss a d valued process η L 2d such ha M = M0 + 0 η udb u 14 wherewenowsem 0 =0. Recall now ha dξ = ξ ν db Hence, from Io s Lemma M J dŵ = = + ξ 2 M J ξ 3 M J + ξ M J ξ dξ + 1 ξ dm dj dξ 2 1 ξ 2 dξ dm dj ν db + 1 ξ ν ν d + 1 ξ ν η d = Ŵ ν db + 1 ξ η db β 1 +Ŵ ν ν d + 1 ξ ν η d η db π c e d c e d Hence, usng agan B = B 0 ν u du we oban dŵ = β 1 c e 1 d + η ξ + Ŵ ν d B 15
13 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 76 Comparng 10 wh 15, we hen mus have θ IŜσ = 1 η ξ + Ŵ ν η = + ξ Ŵ ν Mulplyng boh sdes by β we fnally oban θ I Sσ = η π + W ν 16 where W = 1 π E T π u c u e u du 17 Usng he defnon W = θ0, β + θ S we also fnd he allocaon n bonds θ 0, = β 1 W θ S
14 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 77 3 Equlbrum and he Represenave Agen So far we merely repeaed he exercse n TN1. Now, we mpose marke clearng condons and oban he equlbrum resuls In hs secon we are gong o skp even more of he deals. Le he aggregae endowmen be denoed as e = m e =1 From he resuls n he prevous secon, we hen have he followng Corollary 1: In any equlbrum, we mus have e = m Iu =1 λ e φ π 18 where λ sasfy he sysem of equaons [ T E 0 I u λ e φ ] π e d = 0 19 The converse s also rue: If here exss a vecor λ =λ 1,.., λ m such ha 18 and 19 are sasfed, hen he marke s n equlbrum. In eher case, he opmal consumpon s gven by c = I u λ e φ π 20
15 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 78 The only par of he proof ye o deermne s ha f here exss λ =λ 1,.., λ m such ha 18 and 19 are sasfed, hen he resulng marke s n equlbrum. Ths s rue because 1. If he vecor exss, hen we know ha 20 maxmzes uly. 2. Hence, 18 mples ha he commody marke s cleared m =1 c = m e =1 3. From he prevous proof, recall ha porfolo weghs were deermned by he marngale M = [ 0 η u db T u = E π 0 u c u e ] u du From 19 we have n =1 M = 0 whch mples n =1 η = 0. Summng over 17 and hen 16 we fnally fnd m =1 W =0, m =1 θ =0, m =1 θ 0, =0 Equaon 18 s key n he consrucon of a represenave agen. Gven he represenave agen, we shall characerze he parameers of he process. Gven a vecor λ =λ 1,..., λ m, le s defne he funcon I u x, λ = m Iu λ x =1
16 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 79 Ths a connuous, decreasng funcon for x 0,. Defne by U c.; λhenverseofi.; λ, ha s, ha funcon such ha for every x, wehave U c I u x, λ, λ =x 21 Ths funcon s srcly decreasng on 0,. Noce ha he monooncy of he funcon mples I u U c c, λ, λ =c Usng hese defnons, we can rewre 18 as e = I u e φ π, λ We can hen nver hs funcon o oban a formula for he sae-prce densy π = e φ U c e ; λ 22 The form of hs sae prce densy should look famlar. From corollary 1, we have Corollary 2: Under condon A, a fnancal marke s n equlbrum f and only f s sae-prce densy s gven by π n 22 where λ sasfes he sysem of equaons E [ T 0 e φ U c e ; λ I u λ U c e ; λ e d ] = 0 23
17 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 80 In addon, consumpon s gven by c = I u λ U c e ; λ 24 Corollary 2 characerzes an equlbrum, f exss. 3.1 The Represenave Agen Defne he funcon U c; λ = max c 1 0,..,c m 0: m=1 c =c m =1 1 λ u c 25 We hen have Proposon 1: U c; λ s a srcly ncreasng uly funcon sasfyng condon A. In addon du c, λ = U c c, λ dc as defned n 21. The key pon n he proof s o defne ĉ = I u λ U c c; λ 26 and show ha effecvely, hese ĉ sasfy 25. Ths s easy o see. Noce ha 26 mples ha ĉ = λ U c c; λ 27 u c
18 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 81 Hence, consder now any oher c 1,.., c m such ha m =1 c = c. From he src concavy of he uly funcons we have m =1 1 u c m λ =1 1 λ = m 1 =1 = m =1 u ĉ + c ĉ u c ĉ u ĉ + U c c; λ m c λ ĉ =1 1 u ĉ λ So, we have defned a represenave agen who assgn consan weghs λ =λ 1,..., λ m o he varous agens 1 o m. Gven he resul n 23 and 26, he weghs λ 1,...,λ m are chosen so ha he opmal consumpon of he represenave agen equals he aggregae endowmen. We fnally noce he followng homogeney propery of he aggregae represenave uly funcon: For every consan α, wehave U c, αλ = α 1 U c, λ U c c, αλ = α 1 U c c, λ Snce he margnal uly U c c, αλ deermnes he sae prce densy, hs propery s convenen o renormalze he sae prce densy.
19 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 82 Theorem 1: There exss a vecor λ ha sasfes he sysem 23. In addon, f for all cu cc c 1 u c c hs soluon s unque n he sense ha any oher soluon λ mples he exsence of a consan α such ha αu c e, λ =U c e, λ 3.2 Characerzng he Equlbrum We now have all he ngredens o solve back for he parameers of he prce process and he neres rae process. Frs, noce ha we can renormalze he sae prce densy so ha π 0 = U c e 0 ; λ =1 Defne he process ξ = U c e ; λ By Io s lemma we have ξ =1+ U 0 cc e u ; λ de u U 0 ccc e u ; λde u 2 If we assume ha he aggregae endowmen evolves accordng o he Io process de = e μ e, d + e σ e, db
20 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 83 we oban ξ = 1+ 0 U cc e u ; λ e u μ e,u U ccc e u ; λ e 2 u σ e,uσ e,u du + 0 U cc e u ; λ e u σ e,u db u 28 Recall ha from 22, we mus have ha he sae prce densy s π = e φ ξ On he oher hand, we also have ha gven a sysem of prces β,s, each followng a Io s process as descrbed a he begnnng of he eachng noes, we mus have ha he sae prce densy s π = β 1 ξ where ξ =exp 1 2 ν 0 uν udu ν 0 udb u Noce ha snce ξ 0 =1,ξ sasfes he negral equaon ξ =1 0 ξ uν udb u Defne ξ = π e φ = ξ e 0 φ r u du
21 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 84 We hen have ξ =1+ φ r 0 u ξ u d ξ 0 uν udb u 29 By defnon, n equlbrum we mus have π = π or ξ = e φ π = e φ π = ξ Hence, equang he negral equaons 28 and 29 we oban he equales φ r ξ = U cc e ; λ e μ e, U ccc e ; λ e 2 σ e,σ e, ξ ν = U cc e ; λ e σ e, Recall ha by defnon ξ = ξ = U c e ; λ, obanng r ν = φ U cc e ; λ e U c e ; λ μ e, 1 U ccc e ; λ e 2 σ e, σ e, 2 U c e ; λ = U cc e ; λ e U c e ; λ σ e, We can furher defne he relave rsk averson of he represenave agen as γ e ; λ = U cc e ; λ e U c e ; λ
22 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 85 and he relave prudence coeffcen of he represenave agen as q e ; λ = U ccc e ; λ e U cc e ; λ we can rewre he equaons as r ν = φ + γ e ; λ μ e, 1 2 γ e ; λ q e ; λ σ e, σ e, = γ e ; λ σ e, Therefore, we conclude ha 1. The rsk-free rae ncreases lnearly wh he dscoun rae φ and he relave rsk averson coeffcen γ e ; λ, whle decreases wh he relave prudence parameer q e ; λ and wh he varance of he endowmen process σ e, σ e, ; 2. The marke prce of rsk ν ncreases lnearly wh he relave rsk averson coeffcen and he varably of he endowmen process σ e, recall ha ν s a vecor. These resuls mmedaely mply he followng equlbrum concep.
23 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 86 4 The Equy Premum and he Consumpon CAPM From he defnon of ν we have σ ν = μ r 1 d We hen oban μ r 1 d = γ e ; λ σ σ e, Tha s E ds r d = γ e ; λ Cov S ds S, de 30 e Tha s, he expeced excess reurns of asse depends on he relave rsk averson and he covarance beween asse and he aggregae endowmen process. There are wo puzzles here: 1. Equaon 30 mus hold for he marke as a whole. The correlaon beween sock reurns and consumpon growh s anywhere beween.12 and -.05 see Campbell and Cochrane 1999, Table 7 Even wh he opmsc assumpons of correlaon =.12, volaly of reurn =.17 and volaly of consumpon growh =.02, we have expeced excess reurn =.4% even wh γ = 10.
24 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 87 So, γ = 100 would be needed here o oban an equy premum = 4%. Noe ha he rsk free rae may acually end up beng reasonable f γ s hgh enough: For nsance, γ = 100 produces r =0fφ =.02, μ c =.02 and σ e =.02. The reason s ha he precausonary savng move 1/2γ γ +1σ 2 e kcks n. 2. The second puzzle s abou he cross-secon: The consumpon CAPM does no seem o work. Usng he Fama French sze/book-o-marke porfolos as es porfolos, a Fama-MacBeh regresson of reurns on consumpon growh yelds an nsgnfcan coeffcen =.22 and cross-seconal R 2 = 16% see Leau and Ludvgson 2001, Table 3. Recen papers however pon a nose n he consumpon daa and hey show ha f one uses more lags and leads o compue consumpon growh, he resul works ou. Typcal pfal: Noe ha even f a Fama-MacBeh coeffcen urns ou o be posve and sgnfcan and he R 2 s hgh here s sll an ssue of economc sgnfcance: The γ needed o raonalze he resul may be oo hgh, as n he case of he aggregae marke. 4.1 The CAPM From 30 s possble o fnd a bea relaonshp. Defne he 1 d process ψ usng he relaonshp ψ I S σ = e σ e,
25 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 88 We can consder ψ as a self-fnancng sraegy θ 0, θ by choosng θ = ψ and θ 0 o mee he self-fnancng consran. Le S ψ be he value of he porfolo S ψ = ψ S.Wehen have ds ψ S ψ = μ ψ, d + σ ψ, db where σ ψ, = 1 S ψ ψ I S σ = e S ψ σ e, and μ ψ, sasfes he condon 30 μ ψ, r = γ e ; λ σ ψ, σ e, 31 Hence, we fnally have ha for all =1,.., n μ r = γ e ; λ σ σ e, = γ e ; λ Sψ e σ σ ψ, Snce hs holds for he porfolo ψ as well, ha s μ ψ r = γ e ; λ Sψ e σ ψ, σ ψ,
26 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 89 by subsung for he common erm γ e ; λ Sψ e he CAPM bea relaonshp E ds S r d = β E ds ψ S ψ r 1 d we oban where β = σ σ ψ, σ ψ, σ ψ, = cov ds /S,dS ψ /S ψ var ds ψ /S ψ Noe: The condonal CAPM works wh respec o ha asse ha s perfecly correlaed wh he endowmen process and hus he sochasc dscoun facor. Ths need no be he marke porfolo. The exsence of labor ncome, for nsance, generae a wedge beween he sochasc dscoun facor and he marke porfolo: Thus, he CAPM s no supposed o be workng heorecally wh respec o he marke porfolo. Bu even n he case where here s no labor ncome s no obvous ha he marke porfolo s perfecly correlaed wh he endowmen process. We wll do more on hs laer on n he course.
27 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: An Asse Prcng Relaonshp Suppose ha agens sells a fnancal asse whose paymen s s enre endowmen or a fracon of, ha s e. Wha s he far prce of hs sream of paymens? The above resuls mply ha he prce a me of hs clam s S e = 1 [ T E π π u e ] u du 32 [ 1 = U c e E T e φu U c e u e ] u du 33 In fac, f hs was no rue one could fnd a radng sraegy θ 0, θ ha fnances e u and whose value s 32. Ths n urn generaes an arbrage opporuny. Indeed, he value of oal endowmen process s smply S = [ 1 U c e E T e φu ] U c e u e u du 34 We wll use hs prcng equaon ofen.
28 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: A smple example: Log Uly Suppose all he agens have logarhmc uly u c =logc so ha he nverse of he margnal uly s gven by Iu x =x 1 In hs case, he margnal uly of he represenave nvesor s U c c = 1 m λ 1 c =1 We can renormalze he weghs so ha m =1 λ 1 we can assume posve. We hen have U c e 0 =1 = e 0,whch The vecor λ =λ 1,..., λ m has o sasfy he sysem of equaons 23, E [ T 0 e φ U c e ; λ I u λ U c e ; λ e d ] = 0 35 whch becomes E T e φe 0 0 e λ e 0 1 e e d = 0 36
29 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 92 andnurn λ 1 = e 0 E [ T 0 e φ e e d T 0 e φ d ] 37 Hence π = e φ U c e ; λ =e φe 0 e We hen have ha c = I u λ e φ e 0 π = Iu λ = λ 1 e 38 e 0 e Fnally, we have γ e ; λ = e U cc e,λ U c e,λ q e ; λ = e U ccc e,λ U cc e,λ = e e0 /e 2 =1 e 0 /e = e 2e 0 /e 3 e 0 /e 2 =2 Hence, he condon for he marke equlbrum are r ν = φ + μ e, σ e, σ e, = σ e, 4.4 An Alernave Formula for he C-CAPM Before commenng furher, noce also ha an alernave expresson for he 30 can be obaned as follows.
30 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 93 We know ha a an opmum, he followng condon holds see 27 ĉ j = λj U c e; λ 39 u j c Hence, raher han usng he represenave agen uly funcon, we may hnk of usng he margnal uly of agen j defned on he opmal consumpon pah ĉj. Equaon 39 ensures ha he wo approaches are dencal. Defne he process ξ j = 1 u j c ĉ j λ j Defne he sae-prce densy of agen j as π j = β 1 ξ j By gong hrough he same ypo of calculaon, s clear ha everywhere we can subsue he represenave agen uly and endowmen, wh agen uly and consumpon. As a consequence, on hen obans E ds S r d = γ j ĉ j ds Cov S = a j ĉ j Cov ds S ĉ j, dĉj,dĉj
31 Pero Verones Topcs n Dynamc Asse Prcng Sprng 2005 TN#2, page: 94 where a j ĉ j u j cc = u j c ĉ j ĉ j s he coeffcen of absolue rsk averson of agen j. Dvde now boh sdes by a j ĉ j and sum across j =1,..., m. Snce from he marke clearng condon n j=1 ĉ j = e we mus have m Cov ds j=1 S,dĉj = Cov ds S,dĉj we oban E ds r d =Γc Cov ds,de S S The coeffcen Γc = 1 m j=1 a j ĉ j 1 s he coeffcen of absolue rsk averson of he marke self.
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