Asset Pricing with Heterogeneous Consumers

, JPE 1996 Presented by: Rustom Irani, NYU Stern November 16, 2009

Outline Introduction 1 Introduction Motivation Contribution 2 Assumptions Equilibrium 3 Mechanism Empirical Implications of Idiosyncratic Risk 4 5 Empirical Evidence: BCG (JPE, 2002)

Outline Introduction Motivation Contribution 1 Introduction Motivation Contribution 2 Assumptions Equilibrium 3 Mechanism Empirical Implications of Idiosyncratic Risk 4 5 Empirical Evidence: BCG (JPE, 2002)

Motivation Contribution Representative Agent Consumption-Based Asset Pricing Breeden-Lucas consumption-capm model: 1 Complete markets; 2 No trading frictions; 3 Time-additive utility. Aggregate consumption growth only priced factor: Fluctuations too small in data! Euler equation estimation (Hansen & Singleton, 1982): Unreasonably high RRA required to deliver empirical average excess returns; Equity-premium puzzle.

Motivation Contribution What does Market Incompleteness mean? Agents are unable to insure against risks they face: Earnings, health, investments, etc.; Asymmetric information or limited enforcement of contracts... Complete markets require all state-contingent contracts may be exchanged; However, observe very few insurance mechanisms; s we consider have exogenously incomplete markets: markets and institutions as we see them (e.g., stocks & bonds, social security, health insurance, etc.); Don t attempt to micro-found.

Motivation Contribution Market Incompleteness and Asset Pricing Relaxing full consumption insurance promising: Individual consumption growth is much more volatile than aggregate consumption growth; Can this be exploited to get asset pricing right? Key insights: 1 If individual and aggregate consumption risk vary systematically, then individual risk impacts equity premium; 2 Persistence and heteroscedasticity of shocks matters; This paper: Uninsurable earnings risk might matter.

Main Contribution Introduction Motivation Contribution Closed-form solutions in presence of uninsurable earnings risk: X-sectional distribution of consumption growth matters! Misspecification of Euler equation; Implications for risk-return relationship; Particular earnings process constructed s.t. equilibrium pricing kernel depends on x-sectional distn. of consumption; Back-solving (Sims, JBES, 1990); Conversely, a class of models is identified in which market incompleteness is irrelevant: If x-sectional variance of consumption growth is orthogonal to returns then incompleteness irrelevant and Breeden-Lucas CCAPM can be used for asset pricing; Krueger & Lustig (JET, 2009) extends this class.

Outline Introduction Assumptions Equilibrium 1 Introduction Motivation Contribution 2 Assumptions Equilibrium 3 Mechanism Empirical Implications of Idiosyncratic Risk 4 5 Empirical Evidence: BCG (JPE, 2002)

Environment Introduction Assumptions Equilibrium 1 Endowment economy; 2 Continuum of ex-ante identical, infinitely-lived consumers; 3 Finite set of securities available for trade: Market incompleteness arises because cannot insure individual earnings risk using this set of securities!

Market Arrangement Introduction Assumptions Equilibrium At every time t: 1 n securities: Net dividend d jt, ex-dividend price P jt ; Each security in positive net supply; Consumer i has holding θ ijt. 2 T bonds: Default-free discount bond, paying one unit of consumption; Each bond in zero net supply; Consumer i has holding b ijt.

Consumer s Problem Introduction Assumptions Equilibrium V i0 = max {θ it,b it,c it } i,t E 0 [ t=0 ] 1 α ρt Cit e 1 α s.t. C it + θ it P t + b it B T t = I it + θ i,t 1 (P t + d t ) + b i,t 1 B T 1 t I it is consumer i s labor income endowment; Consumption + Savings = Nonfinancial + Financial Income.

Aggregation Introduction Assumptions Equilibrium Add up consumption, dividends and labor income: 1 C t = i I C it; 2 D t = n j=1 d jt; 3 I t = i I I it = C t D t.

Income Process Introduction Assumptions Equilibrium Let M t > 0 be an SDF implied by no-arbitrage; Assume i s labor endowment is as follows: I it = δ it C t D t [ t ( s.t. δ it = exp ηis y s ys 2 /2 )] y t = s=1 ( 2 α 2 + α [ m t + ρ + α c t ] ) 1/2 IID multiplicative unit-root earnings shock η it N(0,1).

Equilibrium Introduction Assumptions Equilibrium An equilibrium is a value function, decision rules for the investor, and pricing functions {V, C, θ, b, P, B } s.t.: 1 Optimality: Given (P, B ), (C, θ, b ) maximizes utility and V is associated value; 2 Market Clearing: j, t i I θ ijt = 1; i I b ijt = 0.

Outline Introduction Mechanism Empirical Implications of Idiosyncratic Risk 1 Introduction Motivation Contribution 2 Assumptions Equilibrium 3 Mechanism Empirical Implications of Idiosyncratic Risk 4 5 Empirical Evidence: BCG (JPE, 2002)

An Equilibrium with Autarky Mechanism Empirical Implications of Idiosyncratic Risk Under the maintained assumptions, if: 1 E [M t ] 0, as t ; 2 M t+1 /M t e ρ (C t+1 /C t ) α ; then there exists an equilibrium with no trade. Given any (I t, P t, B t ) there exists (I it ) consistent with equilibrium concept! No trade means that agent consumes labor earnings each period and does not trade in financial markets: This is unrealistic, but facilitates subsequent analysis; Follows from (unusual) choice of earnings process.

Consumption in Equilibrium Mechanism Empirical Implications of Idiosyncratic Risk In equilibrium, C it = δ it C t, hence: t ln Ct i = ln C t + ɛ i t s.t. ɛ i ( t = η i t y s ys 2 /2 ) s=1 Hence yt+1 2 corresponds to cross-sectional variance of consumption growth (conditional on the aggregate state): ( ) ] C Var [ln xs i t+1 /Ct i Ct+1 i /C t i C t+1, y t+1 = yt+1 2

Stochastic Discount Factor Mechanism Empirical Implications of Idiosyncratic Risk Use equilibrium consumption and EE to extract SDF: 1 = E t [ e ρ ( C i t+1 C i t = 1 = E t [e ρ+ 1 2 α(1+α)y 2 t+1 = M t+1 = e ρ+ 1 2 α(1+α)y 2 t+1 ) α ] R j t+1 ( Ct+1 C t ( ) α Ct+1 C t ) α R j t+1 Notice that M t+1 is a function of the x-sectional distribution! ]

Euler Equation Estimation Mechanism Empirical Implications of Idiosyncratic Risk M t+1 = e ρ+ 1 2 α(1+α)y 2 t+1 ( ) α Ct+1 In the presence of uninsurable idiosyncratic risk (y 2 t+1 0), M t+1 = g( C t+1 C t, y 2 t+1 ); Standard Euler equation (M t+1 = f ( C t+1 C t )) misspecified and estimates of the coefficient of RRA will be biased: C t

Risk-Return Relation Introduction Mechanism Empirical Implications of Idiosyncratic Risk E t [R jt+1 R ft ] = Cov t [R jt+1, M t+1 ] Var t [M t+1 ] } {{ } β jt Var t [M t+1 ] E t [M t+1 ] }{{} λ M t SDF suggests relationship between x-sectional distribution of nonfinancial income risk and asset risk/return: 1 Time-series properties of yt+1 2 will affect market price of risk: In particular, yt 2 in bad times implies counter-cyclical MPR; Counter-cyclical idiosyncratic labor income risk confirmed in STY (JPE, 2004); 2 Returns that covary negatively with yt+1 2 will have higher β and risk premium.

Outline Introduction 1 Introduction Motivation Contribution 2 Assumptions Equilibrium 3 Mechanism Empirical Implications of Idiosyncratic Risk 4 5 Empirical Evidence: BCG (JPE, 2002)

Introduction 1 Asset pricing in an incomplete market endowment economy: Very special example, but results generalize; 2 Modified Euler equation now depends on properties of idiosyncratic risk process: Strong assumptions on income process; No trade in financial markets in equilibrium; Tractability, but at what cost? 3 Asset pricing implications for: Euler equation estimation of preference parameters; Risk/return relationship.

Outline Introduction Empirical Evidence: BCG (JPE, 2002) 1 Introduction Motivation Contribution 2 Assumptions Equilibrium 3 Mechanism Empirical Implications of Idiosyncratic Risk 4 5 Empirical Evidence: BCG (JPE, 2002)

Euler Equation Errors: Set Up Empirical Evidence: BCG (JPE, 2002) Brav, Constantinides & Geczy (JPE, 2002) investigate EE errors using an incomplete markets SDF: 1 CRRA SDF: M t+1 (gt+1 i ) = β(g t+1 i ) γ, where gt+1 i C i t+1 Ct i 2 Assume standard EE holds for every household and asset: ] 1 = E t [β(gt+1) i γ R j t+1 i, j

Euler Equation Errors: Approach Empirical Evidence: BCG (JPE, 2002) Under complete markets, HH s fully insure and equalize their MRS state-by-state: Consumption growth rates are equalized across HHs; CCAPM holds, i.e., M t+1 = β(g t+1 ) γ is a valid SDF; We know this doesn t work! With incomplete markets: We do not have full insurance; HHs do not equate MRS/consumption growth rates; However, if EE holds i, any linear combination of individual SDFs should be valid; BCG test if equally-weighted sum of HH s MRS is valid SDF: M t+1 = βi 1 I i=1 (g i t+1 ) γ

Euler Equation Errors: Approach Empirical Evidence: BCG (JPE, 2002) Interested in EE errors of the form: u t+1 = I 1 (β ) I ( ) (gt+1) i γ R t+1 Rt+1 f i=1 Conduct standard tests of: 1 T T t=1 u t+1 = 0; They find that Euler equation is satisfied for reasonable γ.

Euler Equation Errors: Result 1 Empirical Evidence: BCG (JPE, 2002)

Euler Equation Errors: C&D SDF Empirical Evidence: BCG (JPE, 2002) They also perform the same test for the Constantinides & Duffie (JPE, 1996) SDF: ( Ii=1 c i ) α { M t+1 = β t+1 α(α+1) Ii=1 c t i exp I 1 [ I 2 i=1 log(gt+1 i ) log(g t+1 i )2]} This is equivalent to testing if most of the x-sectional variation of the consumption growth rate is captured by idiosyncratic income shocks that are: 1 Multiplicative; 2 i.i.d. lognormal.

Euler Equation Errors: Result 2 Empirical Evidence: BCG (JPE, 2002) Euler equation errors increase with RRA; However, error is statistically insignificant for RRA> 1; Paper argues that this highlights the importance of the x-sectional skewness of the HH s consumption growth rate, combined with the first two moments, which is a major contribution.

Euler Equation Errors: Final Thoughts Empirical Evidence: BCG (JPE, 2002) Zero Euler equation errors is one dimension along which to test an asset pricing model; There are other dimensions too: 1 Return predictability; 2 Implied wealth-consumption ratio; 3 Persistence of SDF; Can incomplete markets models explain some of these facts?