ASSET PRICING IN INTERTEMPORAL CONSUMPTION MODELS OCTOBER 5, 2011 ASSET PRICING APPLICATIONS. The Border of Macro and Finance

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1 ASSET PICING IN INTETEMPOAL CONSUMPTION MODELS OCTOBE 5, 2 The Border o Macro and Finance ASSET PICING APPLICATIONS Lucas-ree model General equilibrium asse pricing Equiy premium puzzle isk-ree rae puzzle Alernaive preerence speciicaions ecursive uiliy (Epsein-Zin) Habi persisence Hyperbolic discouning isk sharing (Consrucing he represenaive consumer) Ocober 5, 2 2

2 The Border o Macro and Finance ASSET PICING APPLICATIONS Lucas-ree model General equilibrium asse pricing Equiy premium puzzle isk-ree rae puzzle Alernaive preerence speciicaions ecursive uiliy (Epsein-Zin) Habi persisence Hyperbolic discouning isk sharing (Consrucing he represenaive consumer) Ocober 5, 2 3 Lucas (978) AN EQUILIBIUM ASSET PICING MODEL Modiy inerpreaion o ineremporal model m ax E β u( c ) { c, a} subjec o c sa d a + sa +,,, 2,... a : end-o-period- holdings o shares o an asse s : period- price o a share d : period- per-share dividend Ocober 5, 2 4 2

3 Lucas (978) AN EQUILIBIUM ASSET PICING MODEL Modiy inerpreaion o ineremporal model m ax E β u( c ) { c, a} subjec o c sa d a + sa +,,, 2,... a : end-o-period- holdings o shares o an asse s : period- price o a share d : period- per-share dividend Equivalenly express budge consrain as c + s ( a a ) d a,,,2,... change in share holdings, i.e., quaniy o savings Supply-side o inancial markes a an equilibrium oucome Do no/canno impose when solving consumer problem! Ocober 5, 2 5 Lucas (978) AN EQUILIBIUM ASSET PICING MODEL FOCs yield asse pricing equaion βλ s E ( d + s ),,, 2,... λ Pricing kernel aka sochasic discoun acor (SDF) Period- asse price Can express in erms o marginal uiliies bu don have o βu'( c ) s E ( d + s ),,, 2,... u'( c ) SDF x Fuure reurn Two componens:. Fuure price o sock 2. Fuure dividend paymen Ocober 5, 2 6 3

4 Lucas (978) AN EQUILIBIUM ASSET PICING MODEL Basic asse pricing equaion βλ s E ( d + s ),,, 2,... λ ecursively subsiue orward β λ β λ s E d + lim E s λ + λ Impose a TVC * * Similar inuiion as lim β u'( c ) a Value o erminal asse holdings (in presen-value uils) mus There: had price o asse, so needed erminal condiion on quaniy Ocober 5, 2 7 Lucas (978) AN EQUILIBIUM ASSET PICING MODEL Basic asse pricing equaion βλ s E ( d + s ),,, 2,... λ ecursively subsiue orward β λ β λ s E d + lim E s λ + λ Impose a TVC ules ou asse price bubbles The basic orm o any asse pricing condiion Ocober 5, 2 8 4

5 Lucas (978) EQUILIBIUM ASSET PICING Basic asse pricing equaion β λ s E d + λ O ( + s ) Lucas (978) model {d },, ollows Markov process Supply-side o inancial markes is a Suppose represenaive consumer βλ s E d,,, 2,... λ wih TVC imposed Implemening represenaive consumer An ininiy o consumers, each indexed by a poin on he uni inerval [,] Each individual is idenical in preerences and endowmens Implies aggregae consumpion demand and asse demand Aggregae consumpion demand One individual s Aggregae Toal measure consumpion x savings o individuals demand demand One individual s savings demand x Toal measure o individuals Ocober 5, 2 9 Lucas (978) EQUILIBIUM ASSET PICING Basic asse pricing equaion β λ s E d + λ O ( + s ) Lucas (978) model {d },, ollows Markov process Supply-side o inancial markes is a Suppose represenaive consumer βλ s E d,,, 2,... λ wih TVC imposed Implemening represenaive consumer An ininiy o consumers, each indexed by a poin on he uni inerval [,] Each individual is idenical in preerences and endowmens Implies aggregae consumpion demand and asse demand Aggregae consumpion demand One individual s Aggregae One individual s consumpion x savings savings demand x demand demand Ocober 5, 2 5

6 Lucas (978) EQUILIBIUM ASSET PICING Basic asse pricing equaion β λ s E d + λ Lucas (978) model {d },, ollows Markov process Supply-side o inancial markes is a Soluion is a consumpion and asse price sequence ha saisies O ( + s ) Sequence o sochasic pricing condiions * * β u'( c ) s E d * + u'( c ) βλ s E d,,, 2,... λ in which TVC ( nobubbles ) is already imposed wih TVC imposed Sequence o budge consrains wih asse-marke clearing condiion a imposed * c d,,, 2,... { c *, * a}, +, 2,... aking as given exogenous law o moion or d Ocober 5, 2 Lucas (978) EQUILIBIUM ASSET PICING Wha i N asses? m ax E β N N N j j j j j j u( c ) { c, a} subjec o c s a d + a j j j + a s,,, 2,... No necessarily Arrow-Debreu asses N no necessarily equal o he number o exogenous saes o naure a j : end-o-period- holdings o shares o asse j s j : period- price o a share o asse j d j : period- per-share dividend rom asse j FOCs yield N asse pricing equaions j βλ j j s E ( d + s ), j, 2,..., N,,, 2,... λ Each has idenical SDF Ocober 5, 2 2 6

7 Lucas (978) EQUILIBIUM ASSET PICING βλ d + s E λ j j j s Demand-side pricing condiion or asse j SDF Gross reurn on one uni o a j Finance applicaions someimes/oen suppose SDF ollows exogenous law o moion May be useul or making progress on undersanding asse pricing i we have he wrong model o (ineremporal) consumpion Ocober 5, 2 3 Lucas (978) EQUILIBIUM ASSET PICING βλ d + s E λ j j j s Demand-side pricing condiion or asse j SDF Gross reurn on one uni o a j Finance applicaions someimes/oen suppose SDF ollows exogenous law o moion May be useul or making progress on undersanding asse pricing i we have he wrong model o (ineremporal) consumpion Consumpion-based asse pricing j j β u'( c ) d + s E j u' ( c) s d + s E u c u c j j β '( ) '( ) j s Growh o marginal uiliy Inerpreaion: on average, in uure saes in which marginal uiliy is low, asse j s reurn mus be high in order o be induced o hold i Ocober 5, 2 4 7

8 The Border o Macro and Finance ASSET PICING APPLICATIONS Lucas-ree model General equilibrium asse pricing Equiy premium puzzle isk-ree rae puzzle Alernaive preerence speciicaions ecursive uiliy (Epsein-Zin) Habi persisence ecursive uiliy (Epsein-Zin) Hyperbolic discouning isk sharing (Consrucing he represenaive consumer) Ocober 5, 2 5 Seminal Finance Applicaion EQUILIBIUM ASSET PICING No-arbirage condiions For any asses a i, a k E i i k k β u'( c ) d + s d + s i k u' ( c ) s s Growh o marginal uiliy Gross reurn on one uni o a i Gross reurn on one uni o a k eurn premium o asse i over asse k Provides oundaion or various empirical ess o marke eiciency or simply he underlying srucure o he model Mehra and Presco (985 JME) Tes how well model does in joinly explaining reurns on aggregae equiy and risk-ree bonds epresenaive agen Only wo asses: a Lucas ree (sock) and risk-ree bond Main resul: equiy premium puzzle Ocober 5, 2 6 8

9 Seminal Finance Applicaion max E uc ( ) subjec o { c, a, b} + + a + sa + c sa b d b,,, 2,... a : end-o-period- holdings o shares o an asse inerpre as sock s : period- price o a share d : period- per-share dividend b : end-o-period- holdings o risk-ree bond inerpre as shor-erm Tbill : period- price o risk-ree bond ha maures in period Asse-pricing condiions βu'( c ) β u'( c ) ( + s ) E u'( c ) s E d u'( c ) β u'( c ) d + s E u'( c ) s No arbirage β u'( c ) + E u'( c ) / s + Ocober 5, 2 7 Seminal Finance Applicaion Can sandard ineremporal consumpion model reasonably joinly explain risk-ree reurns and risky (sock) reurns? β u'( c ) () E (2) u'( c ) β u '( c ) + E u'( c ) s Mehra and Presco (985 JME) Originally posed quesion Answered in he negaive! Equiy premium Ocober 5, 2 8 9

10 Seminal Finance Applicaion Can sandard ineremporal consumpion model reasonably joinly explain risk-ree reurns and risky (sock) reurns? β u'( c ) () E (2) u'( c ) β u '( c ) + E u'( c ) s Mehra and Presco (985 JME) Originally posed quesion Answered in he negaive! Equiy premium An ongoing cenral issue in inance and macroeconomics No a qualiaive shorcoming o basic model A (specacular) quaniaive shorcoming o basic model Good overviews/reviews Kocherlakoa (996 JEL) Mehra and Presco (23, Handbook o Economics and Finance) Cochrane, Chaper 2 Ocober 5, 2 9 Seminal Finance Applicaion Can sandard ineremporal consumpion model reasonably joinly explain risk-ree reurns and risky (sock) reurns? β u'( c ) () E (2) u'( c ) β u '( c ) + E u'( c ) s Equiy premium Sar wih (2) s ( '( ) / '( ),/ ) Cov u c u c E s E[ u ( c + )/ u '( c )] + + Ideniy / s as real reurn on broad sock index (i.e., S&P5) Ideniy / as real reurn on (nominally) risk-ree shor T-bills Expeced reurn on equiy risk-ree reurn + premium or bearing risk Premium depends on covariance o equiy reurn wih marginal uiliy ecall inerpreaion: on average, in uure saes in which marginal uiliy is low, asse j s reurn mus be high in order o be induced o hold i Ocober 5, 2 2

11 Seminal Finance Applicaion s ( '( ) / '( ),/ ) Cov u c u c E s E[ u ( c + )/ u '( c )] + + Expeced reurn on equiy risk-ree reurn + premium or bearing risk Premium depends on covariance o equiy reurn wih marginal uiliy On average, in uure saes in which marginal uiliy is low, asse j s reurn mus be high in order o be induced o hold i Asses ha co-vary posiively wih consumpion over ime (negaively wih uure MU o consumpion) command a high premium because heir reurns are desabilizing i.e., push individual away rom consumpion smoohing across uure risky saes Tes using sandard CA c u( c ) Ocober 5, 2 2 Seminal Finance Applicaion s ( + + ) E Cov ( c / c ),/ s E( c / c) Can es wih variey o aggregae sock reurns (inclusive o dividends) Consruc real risk-ree rae rom nominal shor ineres raes and inlaion Mehra and Presco (985) use daa rom Can apply various sophisicaed economeric ools (GMM, Bayesian, ec) Sample means convey basic problem approximae wih uncondiional E(.) Ocober 5, 2 22

12 Seminal Finance Applicaion s (( / ),/ ) σ Cov c c E s + Ec ( / c) Can es wih variey o aggregae sock reurns (inclusive o dividends) Consruc real risk-ree rae rom nominal shor ineres raes and inlaion Mehra and Presco (985) use daa rom Can apply various sophisicaed economeric ools (GMM, Bayesian, ec) Sample means convey basic problem approximae wih uncondiional E(.) Sample saisics, (annual daa) mean(/ ). mean(/ s ).7 mean(c /c ).8 SD(c /c ).36 SD( s ).274 Cov(c /c, s ).29 Explaining 6 percen equiy premium wih growh raes o aggregae consumpion requires a sand on A σ and hen compuing Cov(.) based on i Ocober 5, 2 23 Seminal Finance Applicaion s (( / ),/ ) σ Cov c c E s + Ec ( / c) Wha are reasonable values o A? Business-cycle models require σ (Econ 7) Microeconomic evidence higher(?) risk aversion Hall (988) evidence indicaes IES. I CA σ (Though Hall disavows ha he is measuring risk aversion Kocherlakoa says argumen is deeply lawed (99, JFin p. 75)) Mehra and Presco asser σ as exreme upper bound on risk aversion Ocober 5,

13 Seminal Finance Applicaion s (( / ),/ ) σ Cov c c E s + Ec ( / c) Wha are reasonable values o A? Business-cycle models require σ (Econ 7) Microeconomic evidence higher(?) risk aversion Hall (988) evidence indicaes IES. I CA σ (Though Hall disavows ha he is measuring risk aversion Kocherlakoa says argumen is deeply lawed (99, JFin p. 75)) Mehra and Presco asser σ as exreme upper bound on risk aversion σ Sandard model explains.3 percen o he 6 percen equiy premium An order o magniude o! Even hough σ already oo high (?) Explaining equiy premium wih sandard model requires σ 5! Ocober 5, 2 25 Seminal Finance Applicaion c () β E (2) c / s (( / ),/ ) σ Cov c c E s + Ec ( / c) σ Sandard model explains.3 percen o he 6 percen equiy premium An order o magniude o! Explaining equiy premium wih sandard model requires σ 5! I accep σ 5 equires acceping virually zero IES (no inconsisen wih Hall 988 ) and generaes anoher anomaly I prior is β , () implies /? Ocober 5,

14 Seminal Finance Applicaion EQUITY PEMIUM AND ISK-FEE ATE PUZZLES c () β E (2) c / s (( / ),/ ) σ Cov c c E s + Ec ( / c) σ Sandard model explains.3 percen o he 6 percen equiy premium An order o magniude o! Explaining equiy premium wih sandard model requires σ 5! I accep σ 5 equires acceping virually zero IES (no inconsisen wih Hall 988 ) and generaes anoher anomaly I prior is β , () implies /. percen higher han empirical risk-ree rae! (Exremely) high σ o ix equiy premium puzzle causes risk-ree rae puzzle! Which can be ose by β.6 exreme impaience! Hyperbolic? Ocober 5, 2 27 Seminal Finance Applicaion EQUITY PEMIUM AND ISK-FEE ATE PUZZLES c () β E (2) c / s (( / ),/ ) σ Cov c c E s + Ec ( / c) Wih CA preerences, resolving boh puzzles requires adoping view ha consumers are EXTEMELY risk averse CA undamenally enangles risk aiudes EXTEMELY impaien and ineremporal aiudes Many assumpions underlying basic model No ransacions coss in inancial markes Everyone paricipaes in all inancial markes i.e., no limied paricipaion A represenaive agen exiss i.e., can ignore heerogeneiy in aggregaing Euler equaions (recalls Aanasio criicism o Hall (978)) Ec Firs (mos naural?) line o aack modiy u(.) o disenangle IES and A Ocober 5,