ASSET PRICING WITH LIMITED RISK SHARING AND HETEROGENOUS AGENTS

Transcription

1 ASSET PRICING WITH LIMITED RISK SHARING AND HETEROGENOUS AGENTS Francisco Gomes and Alexander Michaelides Roine Vestman, New York University November 27, 2007

2 OVERVIEW OF THE PAPER The aim of the paper is to jointly......generate asset pricing moments and household portfolios close to data......while matching some facts concerning consumption and wealth in a production economy Key features of the model Endogenous limited participation Preference heterogeneity Idiosyncratic risk A government that provides bonds Related papers Life-cycle portfolio model: Gomes and Michaelides (2005) Asset pricing with exogenous non-participation: Basak and Cuoco (1998), Guvenen (2005) Concern about matching quantities while producing an equity premium: Krusell and Smith (1997), Favilukis (2007)

3 MODEL Production Y t = Z t K α t L 1 α t Z t = G t U t R K t W K t = α Y t K t δ t = (1 α) Y t L t δ t δ t = δ + s η t Government Ct G + Rt B B t = B t+1 B t + T t }{{}}{{}}{{} Residual Endog. Exog. Taxes: τ K (plus tax on bequests) Plus social security: τ s, replacement rate λ

4 HOUSEHOLDS Preferences { V a = Labor income (1 β)ca 1 1/ψ + β a < a R : Hat i = W t L i a L i a = Paε i i Pa i = exp(f (a))pa 1 i ξi a a r : λp i a R W t Trading technology } 1 ( E a (p a V 1 ρ ) 1 1/ψ 1 1/ψ a+1 ) 1 ρ Indicator for entry in a given time period: I i P Entry cost: FP i aw t X i at = K i at(1 + (1 τ K )R K t ) + B i at(1 + (1 τ K )R B t ) + D at I i P FPi atw t a < a r : D at L i at(1 τ s )W t a a r : D at λp i a r W t

5 EQUILIBRIUM HH problem A life-cycle portfolio problem including some aggregate variables as states such that the HH can forecast next period s factor prices HH specific states: x i at, F i a Aggregate states for forming expectations: k t, U t, η t, P B t Law of motions: k t+1 = Γ K (k t, U t, η t ), P B t+1 = Γ P (k t, U t, η t, P B t ) Equilibrium Same conditions as in a stationary competitive equilibrium Expectations on market prices, or Γ, are verified in equilibrium

6 THE ROLE OF IDIOSYNCRATIC RISK Idiosyncratic risk Stimulates wealth accumulation Without it, the wealth distribution is finite dimensional vector, determined by age and investor type Extreme case: Two infinitely lived agents wealth distr. is captured by a scalar!

7 SOLUTION METHOD (KRUSELL-SMITH) Iterate j=1,2,... until convergence For each (U t, η t )-pair, guess Γ (j) K ln(k t+1 ) + q (j) 10 + q(j) 11 ln(k t) ln(pt+1 B ) + q(j) 20 + q(j) 21 ln(pb t ) and Γ(j) P. That is: Solve HH problem conditional on Γ (j) K and Γ(j) P Simulate households Regress by (U t, η t )-pair to get q (j) Check convergence of Γ (j) K and Γ(j) P (q(j) ), if not then j j + 1 Are R 2 s high enough? If not, adjust the functional forms

8 CALIBRATION Production α = 0.36 σ(u) = 0.01 σ(δ) = 0.16 τ k = 0.2 Households Preferences Type A: ρ A = 1.1, ψ A = 0.1 Type B: ρ B = 5, ψ B = 0.4 ρ A, ψ A set to match participation ψ B, β set to match R f and σ S C, σns C ρ B set to match risk premium Entry cost F = 0.06 (In equilibrium: $2,100)

9 MAIN RESULTS QUANTITIES Participation: 53.1% (51.9%) Equity share: 56.9% (54.8%) σ c S =5.02% (3.6%) σ c NS =2.65% (1.4%) Wealth distr. Model Data P P P

10 MAIN RESULTS ASSET PRICES

11 WHAT FEATURES GENERATE THE EQUITY PREMIUM? (1) KS GM Preferences log utility, stoch β E-Z Firm leverage No Accounted for Aggr shocks Productivity Productivity and depr. One-time entry cost No Yes Tight borrowing constraint Yes Yes Idiosyncratic risk Yes Yes OLG No Yes

12 WHAT FEATURES GENERATE THE EQUITY PREMIUM? (2) Tight borrowing constraint As in Krusell and Smith (1997), a tight borrowing constraint is key E [R R f ] = σ(m) σ(r R f ) corr((r R f ), E [m]) }{{} E [m] Not high enough in K-S }{{} High when borrowing constraint is tight Stochastic depreciation (δ t ) GM need s and σ u to match σ(r R f ) and σ C Idiosyncratic risk A small effect of -0.34% of σ ɛ = σ ξ = 0 Recall Krueger and Lustig (2007)! Non-participation Very modest effect However, in the data, σ S > σns σ(m) Seems like most of the contribution to σ S comes from idiosyncratic risk?

13 THE TENSION BETWEEN PORTFOLIOS AND ASSET PRICES Net supply of bonds Zero net supply and limited participation: stockholders will take a short position Positive net supply gives better portfolio results, but reduces the equity premium - tension b/n portfolio allocations and prices!

14 EXOGENOUS VS. ENDOGENOUS NON-PARTICIPATION Exogenous non-participation Basak and Cuoco (1998), Guvenen (2005), Chien Cole and Lustig (2007) Non-participants will typically be wealthy. They can move prices, also the price of the risky asset Recall that in Guvenen (2005), non-participants are key! Endogenous non-participation Entry cost non-participants are poor. They can not move the price of any asset However, GM need non-participants to match the portfolio facts and the consumption volatility of stockholders

15 EPSTEIN-ZIN PREFERENCES (1) PEOPLE WHO ARE RISK AVERSE IN ATEMPORAL WEALTH GAMBLES ENTER THE STOCK MARKET Two types A: ρ A = 1.1, ψ A = 0.1 B: ρ B = 5, ψ B = 0.4 Participation by type - a little counterintuitive? S NS A 3.7% 46.3% B 49.4% 0.6% Forces that moderate participation according to GM (2005, 2007) Little wealth (e.g. young) implies precautionary savings motive. Stronger motive for type B to enter. Another savings motive is consumption smoothing. Stronger motive for type B to enter. Therefore, type B is much more prone to enter Hence, type B determines the equity premium Due to a high γ B the equity share is (somewhat) modest and the equity premium is high

16 EPSTEIN-ZIN PREFERENCES (2) DOES PREFERENCE FOR THE TIMING OF RESOLUTION OF UNCERTAINTY PLAY A ROLE? θ determines the preference for late or for early resolution of uncertainty θ 1 γ < 1 preference for early resolution 1 1 ψ θ A = 1 90 < 1 θ B = 8 3 > 1 The risk averse type B prefers late resolution of uncertainty!

17 EPSTEIN-ZIN PREFERENCES (3) ALTERNATIVE MODELS Alternative sources of heterogeneity Model 1: ρ A = 3, ψ A = 0.05, β A = 0.7, λ A = 0.8 Model 2: ρ A = 4, ψ A = 0.05, β A = 0.6, λ A = 0.8 Model 1 and Model 2: ρ B = 3, ψ B = 0.6, β B = 0.99, λ B = 0.6 θ B = 3!