Reviewing Income and Wealth Heterogeneity, Portfolio...

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

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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.

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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)

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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

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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)

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so we can express the unexpected change in consumption as:

ct+1 − Et [ct+1] = wt+1 − Et [wt+1]

+Et+1

[ ∞∑i=1


]

−Et

[ ∞∑i=1


]

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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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)

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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.

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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)

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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.

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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.

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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

]

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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)

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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

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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.

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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

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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

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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 )

)

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Finally Asset Pricing without Consumption


Finally, using this terms we arrive to the core equation of the paper:


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

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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)

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Applications and Important Points within the Paper that I skipped

Applications and Important Points within the Paperthat I skipped

Term Structure of Interest Rates


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

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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

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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?

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