Modern Dynamic Asset Pricing Models

Transcription

1 Modern Dynamc Asse Prcng Models Lecure Noes 2. Equlbrum wh Complee Markes 1 Pero Verones The Unversy of Chcago Booh School of Busness CEPR, NBER 1 These eachng noes draw heavly on Duffe (1996, Chapers 9 and 1 and Karazas and Shevre (1999, Chaper 3 and 4. They are nended for sudens of Busness 3597 only. Please, do no dsrbue whou my pror consen.

2 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 2 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 pure-exchange economy. Noce n parcular ha here s no producon.

3 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 3 Prmves Le B = ( B 1,..., B d be a d-dmensonal Brownan moon defned on a complee probably space (Ω, F, P and le {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 T E c2 d <. Suppose here are m agens, ndexed by = 1,.., m. Each agen receves an endowmen { e } L+ 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 [ T 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.

4 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 4 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 r udu As n he prevous noes, defne he marke prce of rsk process Assume ha he Novkov s condon s sasfed ( ( 1 T E exp 2 ν = σ 1 (µ r 1 d ν ν d <

5 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 5 Le ξ = exp ( ν db u 1 ν u 2 ν udu 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, 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, θ, S = (β,s. A radng sraegy θ = ( θ, θ fnances a consumpon process c gven ncome e f θ S = θ u d S u + (e u c u du (2 θ T S T = (3

6 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 6 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,θ Λ { ( c, θ, θ L where Λ = + H 2 (S such ha ( θ, θ } fnances c gven e A secury-spo marke equlbrum s a collecon of prce processes (β, S, consumpon processes ( c m ( θ and radng sraeges m such ha gven (β,s, each agen solves (4 and markes =1 =1 clear: m m θ = and c e = =1 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. =1

7 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 7 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, we can use he same echnque used earler o make 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 ( T ξ T = exp ν db u 1 T ν u 2 ν udu Le he dscouned fuure endowmen be denoed by: w = E Q [ T β 1 e d ]

8 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 8 Noce ha he expecaon s under he Q measure. We recall ha for gven nal wealh w, he sac budge consran ha we obaned n TN1 was [ T ] E Q 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 [ T ] [ T ] E Q β 1 c d E Q 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 =. 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 ] [ T ] E π c d E π e d where π s he sae prce densy defned n (1. Noce ha he expecaon s under he orgnal probably measure P. (6

9 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 9 Opmal Consumpon and Porfolo Weghs 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 Iu u c (x = x. Proposon 1: Le he prce process (β,s be 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 ] [ ( T ] E π Iu λ e φ π d = E π e d The relaon wh he resul n TN1 s he followng: 1. The nverse margnal uly funcon: In TN 1 we had u (c,, wh ncluded n he uly funcon. Hence, he relaonshp s Hence, f x = u c (c,, s nverse s u c (c, = e φ u c (c I u (x, = I u ( e φ x Ths explans why we have he erm e φ nsde he nverse uly funcon. (7

10 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 1 Opmal Consumpon and Porfolo Weghs 2. We defned he funcon [ T w (λ = E ] π I u (λ π, d and we mposed w (λ = w. Clearly, equaon (7 s he same condon. We fnally fnd he weghs o suppor he opmal consumpon. Ths s dencal o our fndngs n TN 1, bu s useful o revew hem 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 = θ, β + θ S and he dscouned wealh as Ŵ = β 1 W = θ, + θ S β 1 = θ, + θ Ŝ where we recall ha Ŝ = S β 1 s a marngale under Q.

11 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 11 Opmal Consumpon and Porfolo Weghs Hence dŝ = IŜσ d B where B s a Brownan moon under Q generaed by Grsanov s heorem hrough he formula B = B + From he dynamc budge consran we also have ν u du so ha Ŵ = θ, + θ Ŝ (8 = θ ( u dŝu + e u c u du (9 β 1 u dŵ = β 1 = β 1 ( e c d + θ dŝ (1 ( e c d + θ IŜσ d B (11

12 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 12 Opmal Consumpon and Porfolo Weghs By defnng c, = c e and seng w =, 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: Ŵ = 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 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

13 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 13 Opmal Consumpon and Porfolo Weghs We hen have he followng chan of equales Ŵ = E Q = = ( T β 1 u ( T E ξ T β 1 u ( T E ξ u β 1 u ( c u e u du ( c u eu du = = ξ ( c u eu du ξ = For noaonal convenence, we can defne ( T E ξ T β 1 ( u c u eu du E (ξ T ( T E E u (ξ T β 1 u ξ E ( T π u ( c u e u T JT = ( π u c u eu du ξ ( c u e u du du

14 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 14 Opmal Consumpon and Porfolo Weghs We can rewre he dscouned wealh as Ŵ = 1 ( E J ξ T J 1 ( = M ξ J where we defned he P marngale M as M = E ( J T (12 (13 From he Marngale Represenaon Theorem, here exss a d valued process η ( L 2 d such ha M = M + η u db u (14 where we now se M =. Recall now ha dξ = ξ ν db

15 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 15 Opmal Consumpon and Porfolo Weghs Hence, from Io s Lemma dŵ ( M J = ξ ( 2 M J = + ξ ( 3 M J + ξ ( M J ξ dξ + 1 ( dm ξ dj (dξ 2 1 ( ξ 2 dξ dm dj ν db + 1 ( ( η ξ db π c e d ν ν d + 1 ξ ν η d = Ŵ ν db + 1 η ξ db β 1 +Ŵ ν ν d + 1 ξ ν η d ( c e d Hence, usng agan B = B ν udu we oban dŵ = β 1 ( c e d + 1 ξ ( η + Ŵ ν d B (15

16 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 16 Opmal Consumpon and Porfolo Weghs Comparng (1 wh (15, we hen mus have θ I Ŝ σ = 1 ( η ξ + Ŵ ν = η ξ + Ŵ ν Mulplyng boh sdes by β we fnally oban where θ I Sσ = η π + W ν (16 W = 1 π E ( T ( π u c u e u du (17 Usng he defnon we also fnd he allocaon n bonds θ, W = θ, β + θ S = β 1 ( W θ S

17 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 17 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 = From he resuls n he prevous secon, we hen have he followng m =1 e Corollary 1: In any equlbrum, we mus have m e = =1 I u ( λ e φ π (18 where λ sasfy he sysem of equaons [ T ] ( ( E π I u λ e φ π e d = (19

18 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 18 Equlbrum and he Represenave Agen 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 = Iu λ e φ π (2 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 (2 maxmzes uly. 2. Hence, (18 mples ha he commody marke s cleared m m c = e =1 3. From he prevous proof, recall ha porfolo weghs were deermned by he marngale [ T ] ( M = η udb u = E π u c u e u du =1

19 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 19 From (19 we have n =1 M = whch mples n =1 η =. Summng over (17 and hen (16 we fnally fnd m =1 W =, m =1 θ =, m =1 θ, = 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 m I u (x, λ = Iu (λ x Ths a connuous, decreasng funcon for x (,. Defne by U c (.; λ he nverse of I (.; λ, ha s, ha funcon such ha for every x, we have U c (I u (x, λ,λ = x (21 Ths funcon s srcly decreasng on (,. Noce ha he monooncy of he funcon mples I u (U c (c, λ,λ = c =1

20 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 2 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 The form of hs sae prce densy should look famlar. From corollary 1, we have π = e φ U c (e ; λ (22 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 [ T E e φ U c (e ; λ ( ] Iu (λ U c (e ; λ e d = (23 In addon, consumpon s gven by Corollary 2 characerzes an equlbrum, f exss. c = I u (λ U c (e ; λ (24

21 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 21 The Represenave Agen Defne he funcon U (c; λ = max c 1,..,c m : 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 as defned n (21. du (c, λ dc = U c (c, λ 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.

22 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 22 Ths s easy o see. Noce ha (26 mples ha 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 λ u ( c = = u c (ĉ = λ U c (c; λ (27 m =1 m =1 m =1 1 ( ( u ĉ + ( c ĉ u (ĉ c λ 1 λ u ( ĉ + U c (c; λ 1 λ u ( ĉ m ( c ĉ 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. =1

23 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 23 We fnally noce he followng homogeney propery of he aggregae (represenave uly funcon: For every consan α, we have 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. Theorem 1: There exss a vecor λ ha sasfes he sysem (23. In addon, f for all cu cc (c 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 1

24 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 24 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 = U c (e ; λ = 1 Defne he process ξ = U c (e ; λ By Io s lemma we have ξ = 1 + U cc (e u ; λde u U ccc (e u ; λ (de u 2 If we assume ha he aggregae endowmen evolves accordng o he Io process de = e µ e, d + e σ e, db

25 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 25 Characerzng he Equlbrum we oban ξ = ( U cc (e u ; λe u µ e,u U ccc (e u ; λe 2 u σ e,uσ e,u du 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 ν u ν u 2 du ν u db u

26 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 26 Characerzng he Equlbrum Noce ha snce ξ = 1, ξ sasfes he negral equaon ξ = 1 ξ u ν udb u Defne ξ = π e φ = ξ e (φ r udu We hen have ξ = 1 + (φ r u ξ u d By defnon, n equlbrum we mus have π = π ξ u ν u db u (29 or ξ = e φ π = e φ π = ξ

27 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 27 Characerzng he Equlbrum Hence, equang he negral equaons (28 and (29 we oban he equales (φ r ξ = U cc (e ; λe µ e, U ccc (e ; λe 2 σ e, σ e, ξ ν = U cc (e ; λe σ e, Recall ha by defnon ξ = ξ = U c (e ; λ, obanng r = φ U cc (e ; λe U c (e ; λ µ e, 1 U ccc (e ; λe 2 σ e, σ e, 2 U c (e ; λ ν = U cc (e ; λe 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 and he relave prudence coeffcen of he represenave agen as q (e ; λ = U ccc (e ; λ e U cc (e ; λ

28 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 28 Characerzng he Equlbrum We can rewre he equaons as r = φ + γ (e ; λµ e, 1 2 γ (e ; λq (e ; λ σ e, σ e, ν = γ (e ; λσ 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.

29 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 29 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 [ ] ds E S r d = γ (e ; λ Cov ( ds S, de e (3 Tha s, he expeced excess reurns of asse depends on he relave rsk averson and he covarance beween asse and he aggregae endowmen process.

30 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 3 There are wo puzzles here: 1. Equaon (3 mus hold for he marke as a whole. The correlaon beween sock reurns and consumpon growh s anywhere beween.12 and -.5 (see Campbell and Cochrane (1999, Table 7 Even wh he opmsc assumpons of correlaon =.12, volaly of reurn =.17 and volaly of consumpon growh =.2, we have expeced excess reurn =.4% even wh γ = 1. So, γ = 1 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 = 1 produces r = f φ =.2, µ c =.2 and σ e =.2. 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 crossseconal R 2 = 16% (see Leau and Ludvgson (21, 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.

31 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 31 The CAPM From (3 s possble o fnd a bea relaonshp. Defne he 1 d process ψ usng he relaonshp ψ I S σ = e σ e, We can consder ψ as a self-fnancng sraegy ( θ, θ by choosng θ = ψ and θ self-fnancng consran. o mee he Le S ψ be he value of he porfolo S ψ = ψ S. We hen have where ds ψ S ψ σ ψ, = 1 S ψ = µ ψ, d + σ ψ, db ψ I S σ = e S ψ σ e, and µ ψ, sasfes he condon (3 µ ψ, r = γ (e ; λ σ ψ, σ e, (31

32 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 32 The CAPM Hence, we fnally have ha for all = 1,.., n µ r = γ (e ; λσ 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, by subsung for he common erm γ (e ; λ Sψ e we oban he CAPM bea relaonshp ( ( ds E S r d = β (E ds ψ S ψ r 1 d where β = σ σ ψ, σ ψ, σ ψ, = cov (ds /S, dsψ /Sψ var (ds ψ /Sψ

33 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 33 The CAPM 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.

34 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 34 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 u e π udu [ 1 T ] = U c (e E e φ(u U c (e u e udu (32 (33 In fac, f hs was no rue one could fnd a radng sraegy (θ, θ ha fnances e u value s (32. Ths n urn generaes an arbrage opporuny. Indeed, he value of oal endowmen process s smply [ 1 T ] S = U c (e E e φ(u U c (e u e u du We wll use hs prcng equaon ofen. and whose (34

35 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 35 A smple example: Log Uly Suppose all he agens have logarhmc uly u (c = log (c so ha he nverse of he margnal uly s gven by I u (x = x 1 In hs case, he margnal uly of he represenave nvesor s U c (c = 1 m λ 1 c We can renormalze he weghs so ha m =1 λ 1 = e, whch we can assume posve. We hen have U c (e = 1 The vecor λ = (λ 1,..., λ m has o sasfy he sysem of equaons (23, [ T E e φ U c (e ; λ ( ] Iu (λ U c (e ; λ e d = (35 =1

36 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 36 A smple example: Log Uly whch becomes [ T ( ( λ e ] E e φe e e 1 e d = (36 and n urn λ 1 [ ] T E e φ e e d = e T e φ d (37 Hence We hen have ha c = I u π = e φ U c (e ; λ = e φe e ( λ e φ e π = Iu (λ e e = λ 1 (38 e

37 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 37 A smple example: Log Uly 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 ( e /e 2 = 1 e /e = e 2e /e 3 ( e /e 2 = 2 Hence, he condon for he marke equlbrum are r = φ + µ e, σ e, σ e, ν = σ e,

38 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 38 An Alernave Formula for he C-CAPM Before commenng furher, noce also ha an alernave expresson for he (3 can be obaned as follows. We know ha a an opmum, he followng condon holds (see (27 ( u j c ĉ j = λ j U c (e; λ (39 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 ĉ j. Equaon (39 ensures ha he wo approaches are dencal. Defne he process ξj = 1 ( u j c ĉ j λ j Defne he sae-prce densy of agen j as π j = β 1 ξj

39 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 39 An Alernave Formula for he C-CAPM By gong hrough he same ype 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 [ ] ds ( E r d = γ j ĉ j S = a j (ĉ j ( ds Cov S, dĉj Cov ( ds S ĉ j, dĉ j 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 (ĉ j j and sum across j = 1,..., m. Snce from he marke clearng condon n j=1 ĉj = e we mus have m j=1 ( ds Cov S, dĉ j ( ds = Cov S, dĉ j

40 Pero Verones Modern Dynamc Asse Prcng Models - Equlbrum wh Complee Markes page: 4 An Alernave Formula for he C-CAPM we oban The coeffcen [ ] ds E S ( ds r d = Γ (c Cov S, de Γ (c = 1 ( 1 m j=1 aj ĉ j s he coeffcen of absolue rsk averson of he marke self.