Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997

Transcription

1 Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997 Seminar in Asset Pricing Theory Presented by Saki Bigio November / 23

2 Why Heterogeneity? Goals of the Paper Reconciliation of Consumer Based Asset Pricing and Purely Empirical Asset Pricing Theory (CAPM related literature). 2 / 23

3 Why Heterogeneity? Goals of the Paper Reconciliation of Consumer Based Asset Pricing and Purely Empirical Asset Pricing Theory (CAPM related literature). Finding a Consistent description of Joint Distribution of Asset Returns and Consumption both in homoskedastic and heteroskedastic return processes. 2 / 23

4 Why Heterogeneity? Goals of the Paper Reconciliation of Consumer Based Asset Pricing and Purely Empirical Asset Pricing Theory (CAPM related literature). Finding a Consistent description of Joint Distribution of Asset Returns and Consumption both in homoskedastic and heteroskedastic return processes. Finding an analytically tractable representation of the consumer s problem to understand mechanisms underlying asset pricing. 2 / 23

5 Step #1 Log-Linearization of Lifetim Budget Constraint Log-Linearization of Budget Constraint Log-Linearization of Budget Constraint The consumer s Budget Constraint in terms of the m portfolio return: Updating the budgate constraint Dividing both sides by W t : W t+1 = R m,t+1 (W t C t ) (1) C W t = C t + ( t+j ) (2) i=1 i R m,t+j W t+1 W t = R m,t+1 j=1 ( 1 C ) t W t Defining logx = x, and taking logs on both sides we obtain: (3) w t+1 = r m,t+1 + log(1 exp(c t w t )) (4) 3 / 23

6 Step #1 Log-Linearization of Lifetim Budget Constraint Log-Linearization of Budget Constraint Log-Linearization of BC First order taylor expansion of log(1 exp(c t w t ) : ( ( )) co log(1 exp(c t w t )) log 1 exp w o ( exp co + ( ) [(c t w t ) (c o w o )] c 1 exp o w o w o ) and first order taylor expansion yields the following: ( w t+1 r m,t ) ((c t w t )) + k (5) ρ where k is defined in the distributed notes 4 / 23

7 Step #1 Log-Linearization of Lifetim Budget Constraint Log-Linear Consumption Surprise Function Log-Linear Consumption Surprise Function We then can use the trivial Inequality: in lifetime budget : w t+1 = c t+1 + (c t w t ) (c t+1 w t+1 ) (6) c t w t = ρ k (r m,t+i c t+i ) + ρ 1 ρ k (7) i=1 By taking expectations to both sides we obtain: [ ] c t w t = E t ρ k (r m,t+i c t+i ) i=1 + ρ 1 ρ k (8) 5 / 23

8 Step #1 Log-Linearization of Lifetim Budget Constraint Log-Linear Consumption Surprise Function so we can express the unexpected change in consumption as: c t+1 E t [c t+1 ] = w t+1 E t [w t+1 [ ] +E t+1 ρ k (r m,t+i c t+i ) i=1 [ ] E t ρ k (r m,t+i c t+i ) i=1 and regrouping the expectational terms: c t+1 E t [c t+1 ] = [E t+1 E t ] ρ k (r m,t+i ) [E t+1 E t ] c t+i (9) i=0 we use 5 and 6 to get rid of w t+1 E t [w t+1 ]. The goal here is to try to eliminate the terms related to consumption i=1 6 / 23

9 Step #2: Intertemporal Asset Pricing with Constant Variance (Euler Equations) Assumptions Assumptions 1 Log-Normal conditional distribution of Asset Returns and Consumption 2 Epstein-Zin Utility Reminder Log-Normal Distributed Random Variables x t LN(µ, σ 2 ) log (x t ) N(µ, σ 2 ) ) E t [x t ] = exp (µ + σ2 2 VAR [x t ] = ( exp ( σ 2) 1 ) E t [x t ] 2 = ( exp ( σ 2) 1 ) exp ( 2µ + σ 2) Mode [x t ] = exp ( µ σ 2) Median = exp (µ) 7 / 23

10 Step #2: Intertemporal Asset Pricing with Constant Variance (Euler Equations) Assumptions Assumptions Epstein-Zin Utility { U t = (1 β)c 1 γ θ t [ + β (E t U 1 γ t+1 ] 1 )} θ 1 γ θ (10) where θ = 1 γ [ 1 ( 1 σ )] where γ is the coefficient of relative risk-aversion, and σ is the elasticity of intertemporal substitution. 8 / 23

11 Step #2: Intertemporal Asset Pricing with Constant Variance (Euler Equations) Properties of Epstein-Zin: Properties of Epstein-Zin: 1 Properties of Parameters Value of Parameter Value of θ Implication γ 1 θ 0 Irrelevance of Variance Terms σ 1 θ Variance term in EE becomes 0 γ = 1 θ = 1 Standard Time Separable γ = σ = 1 θ = 1 Log Utility 9 / 23

12 Step #2: Intertemporal Asset Pricing with Constant Variance (Euler Equations) Properties of Epstein-Zin: Properties of Epstein-Zin: 1 Properties of Parameters Value of Parameter Value of θ Implication γ 1 θ 0 Irrelevance of Variance Terms σ 1 θ Variance term in EE becomes 0 γ = 1 θ = 1 Standard Time Separable γ = σ = 1 θ = 1 Log Utility 2 If one replaces the Optimal Consumption Policy in terms of Wealth [ ] 1 V t = max U t = (1 β) σ 1 σ C t Wt (11) {c t} W t 9 / 23

13 Step #2: Intertemporal Asset Pricing with Constant Variance (Euler Equations) Properties of Epstein-Zin: 1 The Euler Equation for an asset i s return in terms of a market return r m, t+1 { } 1 Ct 1 = E t [β { } σ 1 θ 1 R i,t+1] C t+1 R m,t+1 (12) 2 The former in terms only of R m,t+1 : { 1 = E t β { Ct C t+1 } 1 σ Rm,t+1 } θ (13) which is a standard euler equation when θ = / 23

14 Step #2: Intertemporal Asset Pricing with Constant Variance (Euler Equations) Second Order Taylor Expansion of Euler Equation Back in our model, we take a 2nd order taylor expansion of this last equation: Reminder 2nd order taylor expansions: In the EE: f (x t ) f (x o ) + f (x o ) (x x o ) + f (x o ) 2! 0 = θ log β θ σ E t c t+1 +θe t r m, t where and reminding that: (x x o ) 2 (14) ( ( ) ) 2 θ V cc + θ 2 V mm 2θ2 σ σ V cm (15) V cc = VAR t ( c t ) = VAR ( c t+1 E t c t+1 ) (16) VAR(Ax + By) = A 2 VAR(x) + B 2 VAR(y) + 2ABCOV (x, y) Possible only when asset returns and consumption are conditionally homoskedastic. Variance and covariance terms will include a time subscript. 11 / 23

15 Step #2: Intertemporal Asset Pricing with Constant Variance (Euler Equations) Second Order Taylor Expansion of Euler Equation By clearing out the change in consumption: where E t c t = µ m + σe t r m,t+1 (17) µ m = σ log β + 1 [ ] θ 2 σ V cc + θσv mm 2θV cm = σ log β + 1 θ 2 σ Var t [ c t+1 σr m,t+1 ] (18) where in the Heteroskedastic version this becomes: µ m,t = σ log β + 1 [ ] θ 2 σ V cc,t + θσv mm,t 2θV cm,t 12 / 23

16 Step #2: Intertemporal Asset Pricing with Constant Variance (Euler Equations) Second Order Taylor Expansion of Euler Equation The log-linear version of the General Euler equation is 0 = θ log β θ σ E t c t+1 + (θ 1) E t r m, t+1 +E t r i, t (19) cross variance related terms (in document) 13 / 23

17 Step #2: Intertemporal Asset Pricing with Constant Variance (Euler Equations) Clearing Out Consumption from Budget Constraint Clearing Out Consumption from Budget Constraint Using this expression one can derive and expression for the Equity Premium E t r i,t+1 r f,t+1 = V ii 2 + θ V ic σ + (1 θ) V im (20) In this set-up the risk-premium is constant over time. In Heteroskedastic Variances need to impose some sort of structure on law of motion µ m,t 14 / 23

18 Step #3: Substituting out Consumption from Lifetime BC In LT Budget Constraint In LT Budget Constraint We now replace the log-linear portfolio euler equation in the intertemporal budget constraint to get: c t w t = (1 σ) E t j=1 ρ j r m,t+j + ρ (k µ m) 1 ρ so it is clear that when σ > 1 income effect dominates the substitution effect. The approximation to the Epstein-Zin Value function is given by: ( ) 1 v t = w t + (c t w t ) = w t + E t ρ j r m,t+j 1 σ that is, the value function is determined by current wealth and a measure of the longrun investment opportunities. j=1 15 / 23

19 Step #3: Substituting out Consumption from Lifetime BC The clearing out the surprise function The clearing out the surprise function Using the Euler equation and plugging it in the consumption surprise equation we obtain: c t+1 E t [c t+1 ] = r m,t+1 E t r m,t+1 + (1 σ) [E t+1 E t ] This equation is important for the following reasons: ρ k (r m,t+i ) 1 Determines conditions on return process process that justify original assumption of joint log-normality. 2 The equation shows consumption smoothing dependent on market information i=0 16 / 23

20 Step #3: Substituting out Consumption from Lifetime BC Digression Heteroskedastic Variances Digression Heteroskedastic Variances Note for Heteroskedastic version of consumption surprise µ m,t may not be cleared out: C t+1 E t [c t+1 ] = Same [E t+1 E t ] ρ k (µ m,t+i ) the new term reflects the influence of changing risk on saving. µ m,t+i is a function of the second moments of consumption i=1 17 / 23

21 Step #3: Substituting out Consumption from Lifetime BC Digression Heteroskedastic Variances Back in Homoskedastic Version Using this equation we directly obtain the following expression: COV t (r i,t+1, c t+1 ) = V i,m + (1 σ) V i,h where ) V i,h = COV t (r i,t+1, [E t+1 E t ] ρ k (r m,t+i ) i=0 18 / 23

22 Step #3: Substituting out Consumption from Lifetime BC Finally Asset Pricing without Consumption Finally Asset Pricing without Consumption Finally, using this terms we arrive to the core equation of the paper: E t r i,t+1 r f,t+1 = V ii 2 + θ σ V im + = V ii 2 + γv im + (γ 1) V ih θ (1 θ) V ih + (1 θ)v im σ 1 Assets can be priced using covariances with returns on (1) invested wealth (2) news regarding future returns 2 The only relevant parameter from the utility function is the Coefficient of Relative Risk Aversion 19 / 23

23 Step #3: Substituting out Consumption from Lifetime BC Finally Asset Pricing without Consumption E t r i,t+1 r f,t+1 = V ii 2 + γv im + (γ 1) V ih 1 Risk Premium (net of Jensen s Inequality Effect): 1 Investment Opportunity Effect: Asset s Covariance with the market portfolio wheighted by γ (risk aversion) 2 Hedging Effect: Assets Covariance with future returns 2 CAPM models only use covariances with return on market portfolio 1 If CRA is γ = 1 2 When V ih, investment opportunities are constant 3 If this is the case one could back out γ to test for reasonable parameterization Mehra-Presccott (1986) 20 / 23

24 Applications and Important Points within the Paper that I skipped Applications and Important Points within the Paper that I skipped Term Structure of Interest Rates E t r i,t+1 r f,t+1 = V ii 2 + γv im + (1 γ) V ih 21 / 23

25 Applications and Important Points within the Paper that I skipped Applications and Important Points within the Paper that I skipped Term Structure of Interest Rates E t r i,t+1 r f,t+1 = V ii 2 + γv im + (1 γ) V ih Heteroskedastic Returns If we define a law of motion for µ m,t µ m,t = µ 0 + ψe t r m,t+1 which hold if V mh,t = V 0 mh + V 1 mh E tr m,t+1 in which case: E t r i,t+1 r f,t+1 = β 1 + β 2 V mm,t which implies that the equity premium is a GARCH process. 21 / 23

26 Applications and Important Points within the Paper that I skipped Applications and Important Points within the Paper that I skipped Term Structure of Interest Rates E t r i,t+1 r f,t+1 = V ii 2 + γv im + (1 γ) V ih Heteroskedastic Returns If we define a law of motion for µ m,t µ m,t = µ 0 + ψe t r m,t+1 which hold if V mh,t = V 0 mh + V 1 mh E tr m,t+1 in which case: E t r i,t+1 r f,t+1 = β 1 + β 2 V mm,t which implies that the equity premium is a GARCH process. Accuracy of the Representation 21 / 23

27 Campbell s Conclusions Campbell s Conclusions 1 Expected excess log return is composed by the 3 elements discussed previously 22 / 23

28 Campbell s Conclusions Campbell s Conclusions 1 Expected excess log return is composed by the 3 elements discussed previously 2 Restricted Heteroskedasticity can be incorporated in the former analysis as discussed 22 / 23

29 Things that I don t have clear Things that I don t have clear General Equilibrium Approach to understand predictability How to link this with a setup similar to Lucas s 78 Asset Pricing in an Exchange Economy 23 / 23

30 Things that I don t have clear Things that I don t have clear General Equilibrium Approach to understand predictability How to link this with a setup similar to Lucas s 78 Asset Pricing in an Exchange Economy More work needs to be done in the Dividend Process (where do this things come from) 23 / 23

31 Things that I don t have clear Things that I don t have clear General Equilibrium Approach to understand predictability How to link this with a setup similar to Lucas s 78 Asset Pricing in an Exchange Economy More work needs to be done in the Dividend Process (where do this things come from) What is the referential portfolio m? 23 / 23