Mehra and Prescott assert σ = 10 as extreme upper bound on risk aversion

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1 ADDRESSING THE EQUITY PREMIUM AND RISK-FREE RATE PUZZLES OCTOBER, 2 Seminal Finance Applicaion EQUITY PREMIUM PUZZLE s (( + /,/ + σ Cov c c R E s = R R σ + Ec ( + / c Wha are reasonable values o RRA? Business-cycle models require σ (Econ 7 Microeconomic evidence higher(? risk aversion Hall (988 evidence indicaes IES σ. I CRRA 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, 2 2

2 Seminal Finance Applicaion EQUITY PREMIUM AND RISK-FREE RATE PUZZLES σ c + ( β E = (2 c / R s (( + /,/ + σ Cov c c R E s = R R σ + 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 Requires acceping virually zero IES (no inconsisen wih Hall 988 and generaes anoher anomaly I prior is β , ( implies /R. 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, 2 3 Seminal Finance Applicaion EQUITY PREMIUM AND RISK-FREE RATE PUZZLES σ c + ( β E = (2 c / R s (( + /,/ + σ Cov c c R E s = R R σ + Ec ( + / c Wih CRRA preerences, resolving boh puzzles requires adoping view ha consumers are EXTREMELY risk averse CRRA undamenally enangles risk aiudes EXTREMELY 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 RRA Ocober, 2 4 2

3 The Border o Macro and Finance ASSET PRICING APPLICATIONS Lucas-ree model General equilibrium asse pricing Equiy premium puzzle Risk-ree rae puzzle Alernaive preerence speciicaions Recursive uiliy (Epsein-Zin Habi persisence Hyperbolic discouning Risk sharing (Consrucing he represenaive consumer Ocober, 2 5 Equiy Premium Puzzle EXPECTED UTILITY CRRA uiliy σ c u( c = σ IES = and RRA( c = σ σ Ocober, 2 6 3

4 Equiy Premium Puzzle EXPECTED UTILITY a Beginning o planning horizon CRRA uiliy σ c u( c = σ IES = and RRA( c = σ σ For any u(c, sandard liecycle model is in class o expeced uiliy models Economic oucomes during period : income, consumpion, savings Period a Probabiliy q: Realizaion y2 H Probabiliy p: Realizaion y2bar Probabiliy -p-q: Realizaion y2 L a2 Uiliy o risky evens = E(uiliy o he possible risky oucomes von-neumann-morgensern axiomiizaion: INDEPENDENCE AXIOM he mos criical Inuiively: uiliy is addiively separable over he risky oucomes and linear in he probabiliies (Much more deail in Econ 63/64! End o planning horizon Economic oucomes during period 2: sochasic income, sae-coningen consumpion, savings Ocober, 2 7 Equiy Premium Puzzle EXPECTED UTILITY Expeced uiliy Exogenous sae vecor z ollows known Markov process s ma x E β u( c s { cs, as} s= s= { u + β V( a } V( a ; z = max ( c E ; z c, a + Deerminisic case s max β uc ( s { cs, a s } s= s= Addiively separable in he risky oucomes and linear in he probabiliies { uc βv a } V( a max ( ( = c, a + Ocober, 2 8 4

5 Equiy Premium Puzzle Epsein-Zin / Kreps-Poreus uiliy Exogenous sae vecor z ollows known Markov process V( a ; z = max u( c + β c, a ( ; V a z + Curvaure over he risky oucomes and nonlinear in he probabiliies α, α Typical represenaion in lieraure ρ ρ ρ α α ; = m x β ( + c, a V ( a z a c + E V( a ; z Obain more general represenaion by seing V = V ρ α = α ρ Ocober, 2 9 Equiy Premium Puzzle Epsein-Zin / Kreps-Poreus uiliy Exogenous sae vecor z ollows known Markov process V( a ; z = max u( c + β c, a ( ; V a z + Curvaure over he risky oucomes and nonlinear in he probabiliies α, α α = back o expeced uiliy! α require some more condiions on u(. and α I u(., hen V(. (E-Z 989 Theorem 3. α > corresponds o risk aversion [over wha?...] Larger values o α represen greaer degree o risk aversion Ocober, 2 5

6 Equiy Premium Puzzle Epsein-Zin / Kreps-Poreus uiliy Exogenous sae vecor z ollows known Markov process V( a ; z = max u( c + β c, a ( ; V a z + α, α α = back o expeced uiliy! Curvaure over he risky oucomes and nonlinear in he probabiliies (reormulae he recursion i u(. α require some more condiions on u(. and α I u(., hen V(. (E-Z 989 Theorem 3. α > corresponds o risk aversion [over wha?...] Larger values o α represen greaer degree o risk aversion I u(., hen reormulae he recursion as ( β E V( a; z +, and V(. (E-Z 989 Theorem 3. α > corresponds o risk aversion [over wha?...] Larger values o α represen lesser degree o risk aversion Ocober, 2 Equiy Premium Puzzle Epsein-Zin / Kreps-Poreus uiliy Exogenous sae vecor z ollows known Markov process V( a ; z = max u( c + β c, a ( ; V a z + Curvaure over he risky oucomes and nonlinear in he probabiliies α, α α = back o expeced uiliy! α require some more condiions on u(. and α I u(., hen V(. (E-Z 989 Theorem 3. α > corresponds o risk aversion [over wha?...] Larger values o α represen greaer degree o risk aversion Deerminisic case V( a { u βv ( z } ; z = max ( c + a ; c, a + Pursue his case wihou loss o generaliy Idenical o sandard model i no uncerainy! Ocober, 2 2 6

7 Equiy Premium Puzzle Epsein-Zin / Kreps-Poreus uiliy V( a ; z = max u( c + β c, a ( ; V a z + Sequenial represenaion? NONE! Primiive IS he value uncion Reason: sae-non-separable preerences Bu ime-separable preerences α, α Can inerpre preerences as being deined over Curren-period consumpion, c Asses a he end o curren period/beginning o nex period, a a links curren period o sae j, j, nex period hrough budge consrain(s Common inerpreaion o α Curren period j j j j c + a = y + ( + r a c + + a + = y + + ( + r + a Twiss and unwiss he value uncion Sae j nex period Ocober, 2 3 Equiy Premium Puzzle Consumer problem FOCs V( a ; max ( z = u c + λ c, a y + ( + r a a + E V( a; z+ [ ] ( c β α c : u'( c λ = a : Env : V( a ; z = λ( + r Ocober, 2 4 7

8 Equiy Premium Puzzle Consumer problem FOCs V( a ; max ( z = u c + λ c, a y + ( + r a a + E V( a; z+ [ ] ( c β α c : u'( c λ = a : Env : V( a ; z = λ( + r Euler equaion Isolae pricing kernel Ocober, 2 5 Equiy Premium Puzzle Asse pricing represenaion βu ( c V( a ; z = + r + V( a; z+ + + E ( u'( c Pricing kernel aka sochasic discoun acor (SDF α = collapses SDF o sandard IMRS Wih sae-non-separable (i.e., α, kernel depends on IMRS (sandard Value uncion isel!...which in urns depends on wealh How o measure V(.?...i s uils How o measure α? Ocober, 2 6 8

9 Equiy Premium Puzzle Asse pricing represenaion βu ( c V( a; z d s = E u'( c s R V( a; z+ βu ( c V( a ; z + + = E u'( c V( a; z+ R Equiy premium Bonds So does recursive uiliy model help? No clear Disenangling RRA rom IES is useul bu sill require (very high RRA! Lack o empirical guidance on α a shorcoming (Noe: α parameer does no EQUAL RRA requires more general analysis Also problemaic: pricing kernel no invarian o level shis o u(. Ocober, 2 7 Equiy Premium Puzzle How o compue IES? From deerminisic model since IES is a concep disinc rom risk V( a I assume CRRA u(c, IES is { u βv ( z } ; z = max ( c + a ; c, a + IES = σ No dieren rom expeced uiliy model So can have IES (or lower, ollowing Hall (988 and RRA >> simulaneously How o empirically pin down risk aversion parameer in recursive preerences? Work jus beginning now Ocober, 2 8 9

10 The Border o Macro and Finance ASSET PRICING APPLICATIONS Lucas-ree model General equilibrium asse pricing Equiy premium puzzle Risk-ree rae puzzle Alernaive preerence speciicaions Recursive uiliy (Epsein-Zin Habi persisence Hyperbolic discouning Risk sharing (Consrucing he represenaive consumer Ocober, 2 9 Equiy Premium Puzzle Wha i wan o reain expeced uiliy heory? Local risk aversion can be inroduced by supposing lagged consumpion aecs uiliy vc (, c Time-non-separable preerences Sae-separable preerences More generally, could be v(c, c -, c -2,, c -τ As long as resric o inie lags (o preserve Markov propery Illusrae wih one lag: v(c, c - Ocober, 2 2

11 Equiy Premium Puzzle Wha i wan o reain expeced uiliy heory? Local risk aversion can be inroduced by supposing lagged consumpion aecs uiliy vc (, c Time-non-separable preerences Sae-separable preerences More generally, could be v(c, c -, c -2,, c -τ As long as resric o inie lags (o preserve Markov propery Illusrae wih one lag: v(c, c - Two ypical ormulaions Muliplicaive habis Addiive (subracive habis c vc (, c =, γ [,] c γ v( c, c = u( c c, α [,] Pursue his ormulaion Sequenial and recursive ormulaions max E β u( c αc { c, a} = = sae vecor? { u c V a z } V( a, c ; z = max ( +β E (, c ; c, a Ocober, 2 2 Equiy Premium Puzzle max E β u( c αc { c, a} = subjec o c + a = y + ( + r a, =,, 2,... = FOCs c : a : c delivers DISUTILITY in period + because o pre-accumulaed habis/addicions u '( c αβ '( α λ = αc Eu c+ c [ r ] λ + βe λ + (+ + = Euler equaion α αβ '( c αc β ( α αβ E u'( c α u'( c c Eu + = E u' ( c+ c c+ ( + r+ Ocober, 2 22

12 Equiy Premium Puzzle max E β u( c αc { c, a} = subjec o c + a = y + ( + r a, =,, 2,... = FOCs c : a : c delivers DISUTILITY in period + because o pre-accumulaed habis/addicions u '( c αβ '( α λ = αc Eu c+ c [ r ] λ + βe λ + (+ + = Euler equaion α αβ '( c αc β ( α αβ E u'( c α u'( c c Eu + = E u' ( c+ c c+ ( + r+ β( u' ( c+ c β E+ u'( c+ 2 c+ u'( c αc β Eu'( c c = E ( + r + + Pricing kernel aka sochasic discoun acor (SDF Los o leads and lags makes kernel more lexible, in hope(? ha can beer mach asse-pricing acs Ocober, 2 23 Equiy Premium Puzzle { α λ [ y c + r ] + β V a } V( a, c ; z = max u( c c + + a ( a E (, c ; z c, a + FOCs c : a : Env wr a : Env wr c : Euler equaion Ocober,

13 Equiy Premium Puzzle Asse pricing equaions β( '( β E+ u'( c+ 2 c+ '( βeu' ( c c u c c d + s = E u c c + s R = E β ( u c+ c β E+ u'( c+ 2 c+ β Eu '( u'( c c '( c+ c R Equiy premium Bonds Ocober, 2 25 Equiy Premium Puzzle Asse pricing equaions β( '( β E+ u'( c+ 2 c+ '( α αβeu' ( c αc u c c d + s E = u c c + s R = E β ( u c+ c β E+ u'( c+ 2 c+ β Eu '( u'( c c '( c+ c R Equiy premium Bonds Why migh habis help solve equiy premium and risk-ree rae puzzle? Back-o-envelope: compue RRA wih respec o c, given CRRA speciicaion c - c α.8 (empirical esimaes using macro daa 2 2 c u/ c σ c RRA( c = = 5σ u/ c c c Pumps up RRA by acor o 5 Ocober,

14 Equiy Premium Puzzle Asse pricing equaions β( '( β E+ u'( c+ 2 c+ '( βeu' ( c c u c c d + s = E u c c + s R = E β ( u c+ c β E+ u'( c+ 2 c+ β Eu '( u'( c c '( c+ c R Equiy premium Bonds Why migh habis help solve equiy premium and risk-ree rae puzzle? Back-o-envelope: compue RRA wih respec o c, given CRRA speciicaion c - c α.8 (empirical esimaes using macro daa 2 2 c u/ c σ c RRA( c = = 5σ u/ c c c Pumps up RRA by acor o 5 So does habi model help? Yes on risk-ree rae puzzle, less so on equiy premium puzzle, which sill requires implausibly high risk aversion Ocober, 2 27 Equiy Premium Puzzle EQUITY PREMIUM PUZZLE? Oher explanaions o equiy premium and risk-ree rae puzzles? Borrowing consrains Transacions coss o inancial marke rading Limied paricipaion in inancial markes Small probabiliies o disasers / black swans Long-run risk (i.e., changes o he growh rend Heerogeneous individuals Myopic behavior (sor o like hyperbolic A saisical ariac?...! An avalanche o lieraure in macro and inance since Mehra and Presco (985 Overviews Kocherlakoa (996 Journal o Economic Lieraure Mehra and Presco (23, Handbook o Economics o Finance, Chaper 4 Ocober,

15 The Border o Macro and Finance ASSET PRICING APPLICATIONS Lucas-ree model General equilibrium asse pricing Equiy premium puzzle Risk-ree rae puzzle Alernaive preerence speciicaions Recursive uiliy (Epsein-Zin Habi persisence Hyperbolic discouning Risk sharing Aggregaion (Consrucing he represenaive consumer Ocober,

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

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