reviewing income and wealth heterogeneity, portfolio...
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Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Reviewing Income and WealthHeterogeneity, Portfolio Choice and
Equilibrium Asset Returnsby P. Krussell and A. Smith, JPE 1997
Seminar in Asset Pricing Theory
Presented by Saki BigioNovember 2007
1 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Why Heterogeneity?
Goals of the Paper
Reconciliation of Consumer Based Asset Pricing and PurelyEmpirical Asset Pricing Theory (CAPM related literature).
Finding a Consistent description of Joint Distribution of AssetReturns and Consumption both in homoskedastic andheteroskedastic return processes.
Finding an analytically tractable representation of the consumer’sproblem to understand mechanisms underlying asset pricing.
2 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Why Heterogeneity?
Goals of the Paper
Reconciliation of Consumer Based Asset Pricing and PurelyEmpirical Asset Pricing Theory (CAPM related literature).
Finding a Consistent description of Joint Distribution of AssetReturns and Consumption both in homoskedastic andheteroskedastic return processes.
Finding an analytically tractable representation of the consumer’sproblem to understand mechanisms underlying asset pricing.
2 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Why Heterogeneity?
Goals of the Paper
Reconciliation of Consumer Based Asset Pricing and PurelyEmpirical Asset Pricing Theory (CAPM related literature).
Finding a Consistent description of Joint Distribution of AssetReturns and Consumption both in homoskedastic andheteroskedastic return processes.
Finding an analytically tractable representation of the consumer’sproblem to understand mechanisms underlying asset pricing.
2 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Step #1 Log-Linearization of Lifetim Budget Constraint
Log-Linearization of Budget Constraint
Log-Linearization of Budget ConstraintThe consumer’s Budget Constraint in terms of the m portfolio return:
Wt+1 = Rm,t+1(Wt − Ct) (1)
Updating the budgate constraint
Wt = Ct +∞∑i=1
Ct+j(i∏
j=1
Rm,t+j
) (2)
Dividing both sides by Wt :
Wt+1
Wt= Rm,t+1
(1− Ct
Wt
)(3)
Defining logX = x , and taking logs on both sides we obtain:
∆wt+1 = rm,t+1 + log(1− exp(ct − wt)) (4)
3 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
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(ct − wt) :
log(1− exp(ct − wt)) ≈ log
(1− exp
(co
wo
))
+− exp
(co
wo
)1− exp
(co
wo
) [(ct − wt)− (co − wo)]
and first order taylor expansion yields the following:
∆wt+1 ≈ rm,t+1 +
(1− 1
ρ
)((ct − wt)) + k (5)
where k is defined in the distributed notes
4 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
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:
∆wt+1 = ∆ct+1 + (ct − wt)− (ct+1 − wt+1) (6)
in lifetime budget :
ct − wt =∞∑i=1
ρk (rm,t+i −∆ct+i ) +ρ
1− ρk (7)
By taking expectations to both sides we obtain:
ct − wt = Et
[ ∞∑i=1
ρk (rm,t+i −∆ct+i )
]+
ρ
1− ρk (8)
5 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Step #1 Log-Linearization of Lifetim Budget Constraint
Log-Linear Consumption Surprise Function
so we can express the unexpected change in consumption as:
ct+1 − Et [ct+1] = wt+1 − Et [wt+1]
+Et+1
[ ∞∑i=1
ρk (rm,t+i −∆ct+i )
]
−Et
[ ∞∑i=1
ρk (rm,t+i −∆ct+i )
]
and regrouping the expectational terms:
ct+1−Et [ct+1] = [Et+1 − Et ]∞∑i=0
ρk (rm,t+i )− [Et+1 − Et ]∞∑i=1
∆ct+i (9)
we use 5 and 6 to get rid of wt+1 − Et [wt+1].The goal here is to try to eliminate the terms related to consumption
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Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Step #2: Intertemporal Asset Pricing with Constant Variance (Euler Equations)
Assumptions
Assumptions
1 Log-Normal conditional distribution of Asset Returns andConsumption
2 Epstein-Zin Utility
Reminder Log-Normal Distributed Random Variables
xt˜LN(µ, σ2) → log (xt) ˜N(µ, σ2)
Et [xt ] = exp
(µ+
σ2
2
)VAR [xt ] =
(exp
(σ2)− 1)Et [xt ]
2 =(exp
(σ2)− 1)exp
(2µ+ σ2
)Mode [xt ] = exp
(µ− σ2
)Median = exp (µ)
7 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Step #2: Intertemporal Asset Pricing with Constant Variance (Euler Equations)
Assumptions
Assumptions
Epstein-Zin Utility
Ut =
{(1− β)C
1−γθ
t + β
(Et
[U1−γ
t+1
] 1θ
)} θ1−γ
(10)
where
θ =1− γ[
1−(
1σ
)]where γ is the coefficient of relative risk-aversion, and σ is the elasticityof intertemporal substitution.
8 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Step #2: Intertemporal Asset Pricing with Constant Variance (Euler Equations)
Properties of Epstein-Zin:
Properties of Epstein-Zin:
1 Properties of ParametersValue of Parameter Value of θ Implicationγ → 1 θ → 0 Irrelevance of Variance Termsσ → 1 θ →∞ Variance term in EE becomes 0γ = 1
σ θ = 1 Standard Time Separableγ = σ = 1 θ = 1 Log Utility
2 If one replaces the Optimal Consumption Policy in terms of Wealth
Vt = max{ct}
Ut =
[(1− β)−σ Ct
Wt
] 11−σ
Wt (11)
9 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Step #2: Intertemporal Asset Pricing with Constant Variance (Euler Equations)
Properties of Epstein-Zin:
Properties of Epstein-Zin:
1 Properties of ParametersValue of Parameter Value of θ Implicationγ → 1 θ → 0 Irrelevance of Variance Termsσ → 1 θ →∞ Variance term in EE becomes 0γ = 1
σ θ = 1 Standard Time Separableγ = σ = 1 θ = 1 Log Utility
2 If one replaces the Optimal Consumption Policy in terms of Wealth
Vt = max{ct}
Ut =
[(1− β)−σ Ct
Wt
] 11−σ
Wt (11)
9 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
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 marketreturn rm,t+1
1 = Et
[β
{Ct
Ct+1
}−1σ{
1
Rm,t+1
}1−θ
Ri,t+1
](12)
2 The former in terms only of Rm,t+1 :
1 = Et
{β{ Ct
Ct+1
}−1σ
Rm,t+1
}θ (13)
which is a standard euler equation when θ = 1.
10 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
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 lastequation:Reminder 2nd order taylor expansions:
f (xt) ≈ f (xo) + f ′ (xo) (x − xo) +f ′′ (xo)
2!(x − xo)
2 (14)
In the EE:
0 = θ log β− θσ
Et∆ct+1+θEtrm,t+1 +1
2
((θ
σ
)2
Vcc + θ2Vmm −2θ2
σVcm
)(15)
whereVcc = VARt(∆ct) = VAR (∆ct+1 − Et∆ct+1) (16)
and reminding that:
VAR(Ax + By) = A2VAR(x) + B2VAR(y) + 2ABCOV (x , y)
Possible only when asset returns and consumption are conditionallyhomoskedastic. Variance and covariance terms will include a timesubscript.
11 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Step #2: Intertemporal Asset Pricing with Constant Variance (Euler Equations)
Second Order Taylor Expansion of Euler Equation
By clearing out the change in consumption:
Et∆ct = µm + σEtrm,t+1 (17)
where
µm = σ log β +1
2
[θ
σVcc + θσVmm − 2θVcm
]= σ log β +
1
2
θ
σVart [∆ct+1 − σrm,t+1] (18)
where in the Heteroskedastic version this becomes:
µm,t = σ log β +1
2
[θ
σVcc,t + θσVmm,t − 2θVcm,t
]
12 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
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
σEt∆ct+1 + (θ − 1) Etrm,t+1 +Etri ,t+1 +... (19)
cross variance related terms (in document)
13 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
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 EquityPremium
Etri,t+1 − rf ,t+1 = −Vii
2+ θ
Vic
σ+ (1− θ) Vim (20)
In this set-up the risk-premium is constant over time. In HeteroskedasticVariances need to impose some sort of structure on law of motion µm,t
14 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
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 theintertemporal budget constraint to get:
ct − wt = (1− σ)Et
∞∑j=1
ρj rm,t+j +ρ (k − µm)
1− ρ
so it is clear that when σ > 1 income effect dominates the substitutioneffect.The approximation to the Epstein-Zin Value function is given by:
vt = wt +
(1
1− σ
)(ct − wt) = wt + Et
∞∑j=1
ρj rm,t+j
that is, the value function is determined by current wealth and a measureof the longrun investment opportunities.
15 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
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 surpriseequation we obtain:
ct+1 − Et [ct+1] = rm,t+1 − Etrm,t+1 + (1− σ) [Et+1 − Et ]∞∑i=0
ρk (rm,t+i )
This equation is important for the following reasons:
1 Determines conditions on return process process that justify originalassumption of joint log-normality.
2 The equation shows consumption smoothing dependent on marketinformation
16 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Step #3: Substituting out Consumption from Lifetime BC
Digression Heteroskedastic Variances
Digression Heteroskedastic Variances
Note for Heteroskedastic version of consumption surprise µm,t may notbe cleared out:
Ct+1 − Et [ct+1] = Same− [Et+1 − Et ]∞∑i=1
ρk (µm,t+i )
the new term reflects the influence of changing risk on saving. µm,t+i is afunction of the second moments of consumption
17 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Step #3: Substituting out Consumption from Lifetime BC
Digression Heteroskedastic Variances
Back in Homoskedastic Version
Using this equation we directly obtain the following expression:
COVt (ri,t+1,∆ct+1) = Vi,m + (1− σ) Vi,h
where
Vi,h = COVt
(ri,t+1, [Et+1 − Et ]
∞∑i=0
ρk (rm,t+i )
)
18 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
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:
Etri,t+1 − rf ,t+1 = −Vii
2+θ
σVim +
θ (1− θ)σ
Vih + (1− θ)Vim
= −Vii
2+ γVim + (γ − 1) Vih
1 Assets can be priced using covariances with returns on (1) investedwealth (2) news regarding future returns
2 The only relevant parameter from the utility function is theCoefficient of Relative Risk Aversion
19 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Step #3: Substituting out Consumption from Lifetime BC
Finally Asset Pricing without Consumption
Etri,t+1 − rf ,t+1 = −Vii
2+ γVim + (γ − 1) Vih
1 Risk Premium (net of Jensen’s Inequality Effect):
1 Investment Opportunity Effect: Asset’s Covariance with themarket 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 = 12 When Vih, investment opportunities are constant3 If this is the case one could back out γ to test for reasonable
parameterizationMehra-Presccott (1986)
20 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Applications and Important Points within the Paper that I skipped
Applications and Important Points within the Paperthat I skipped
Term Structure of Interest Rates
Etri,t+1 − rf ,t+1 = −Vii
2+ γVim + (1− γ) Vih
Heteroskedastic ReturnsIf we define a law of motion for µm,t
µm,t = µ0 + ψEtrm,t+1
which hold if Vmh,t = V 0mh + V 1
mhEtrm,t+1 in which case:
Etri,t+1 − rf ,t+1 = β1 + β2Vmm,t
which implies that the equity premium is a GARCH process.
Accuracy of the Representation
21 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Applications and Important Points within the Paper that I skipped
Applications and Important Points within the Paperthat I skipped
Term Structure of Interest Rates
Etri,t+1 − rf ,t+1 = −Vii
2+ γVim + (1− γ) Vih
Heteroskedastic ReturnsIf we define a law of motion for µm,t
µm,t = µ0 + ψEtrm,t+1
which hold if Vmh,t = V 0mh + V 1
mhEtrm,t+1 in which case:
Etri,t+1 − rf ,t+1 = β1 + β2Vmm,t
which implies that the equity premium is a GARCH process.
Accuracy of the Representation
21 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Applications and Important Points within the Paper that I skipped
Applications and Important Points within the Paperthat I skipped
Term Structure of Interest Rates
Etri,t+1 − rf ,t+1 = −Vii
2+ γVim + (1− γ) Vih
Heteroskedastic ReturnsIf we define a law of motion for µm,t
µm,t = µ0 + ψEtrm,t+1
which hold if Vmh,t = V 0mh + V 1
mhEtrm,t+1 in which case:
Etri,t+1 − rf ,t+1 = β1 + β2Vmm,t
which implies that the equity premium is a GARCH process.
Accuracy of the Representation
21 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Campbell’s Conclusions
Campbell’s Conclusions
1 Expected excess log return is composed by the 3 elements discussedpreviously
2 Restricted Heteroskedasticity can be incorporated in the formeranalysis as discussed
22 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Campbell’s Conclusions
Campbell’s Conclusions
1 Expected excess log return is composed by the 3 elements discussedpreviously
2 Restricted Heteroskedasticity can be incorporated in the formeranalysis as discussed
22 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Things that I don’t have clear
Things that I don’t have clear
General Equilibrium Approach to understand predictabilityHow to link this with a setup similar to Lucas’s 78 ”Asset Pricing inan Exchange Economy”
More work needs to be done in the Dividend Process (where do thisthings come from)
What is the referential portfolio m?
23 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Things that I don’t have clear
Things that I don’t have clear
General Equilibrium Approach to understand predictabilityHow to link this with a setup similar to Lucas’s 78 ”Asset Pricing inan Exchange Economy”
More work needs to be done in the Dividend Process (where do thisthings come from)
What is the referential portfolio m?
23 / 23
Reviewing Income and Wealth Heterogeneity, Portfolio Choice and Equilibrium Asset Returns by P. Krussell and A. Smith, JPE 1997
Things that I don’t have clear
Things that I don’t have clear
General Equilibrium Approach to understand predictabilityHow to link this with a setup similar to Lucas’s 78 ”Asset Pricing inan Exchange Economy”
More work needs to be done in the Dividend Process (where do thisthings come from)
What is the referential portfolio m?
23 / 23