Asset Pricing with Heterogeneous...

IntroductionModel

ResultsConclusionDiscussion

Asset Pricing with Heterogeneous Consumers

Constantinides & Duffie, JPE 1996Presented by: Rustom Irani, NYU Stern

November 16, 2009

Constantinides & Duffie Asset Pricing with Heterogeneous Consumers

IntroductionModel


Outline

1 IntroductionMotivationContribution

2 ModelAssumptionsEquilibrium

3 ResultsMechanismEmpirical Implications of Idiosyncratic Risk

4 Conclusion

5 DiscussionEmpirical Evidence: BCG (JPE, 2002)


IntroductionModel


MotivationContribution

Outline




4 Conclusion



IntroductionModel



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 averageexcess returns;“Equity-premium puzzle.”

...Could relaxing the complete markets assumption help?


IntroductionModel



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 maybe exchanged;

However, observe very few insurance mechanisms;

Models we consider have exogenously incomplete markets:

Model markets and institutions as we see them (e.g., stocks &bonds, social security, health insurance, etc.);Don’t attempt to micro-found.


IntroductionModel



Market Incompleteness and Asset Pricing

Relaxing full consumption insurance promising:

Individual consumption growth is much more volatile thanaggregate 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.


IntroductionModel



Main 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. equilibriumpricing kernel depends on x-sectional distn. of consumption;

“Back-solving” (Sims, JBES, 1990);

Conversely, a class of models is identified in which marketincompleteness is irrelevant:

If x-sectional variance of consumption growth is orthogonal toreturns then incompleteness irrelevant and Breeden-LucasCCAPM can be used for asset pricing;Krueger & Lustig (JET, 2009) extends this class.


IntroductionModel


AssumptionsEquilibrium

Outline




4 Conclusion



IntroductionModel



Environment

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 individualearnings risk using this set of securities!


IntroductionModel



Market Arrangement

At every time t:1 n securities:

Net dividend djt , ex-dividend price Pjt ;Each security in positive net supply;Consumer i has holding θijt .

2 T bonds:

Default-free discount bond, paying one unit of consumption;Each bond in zero net supply;Consumer i has holding bijt .


IntroductionModel



Consumer’s Problem

Vi0 = max{θit ,bit ,Cit}i,t

E0

[ ∞∑t=0

e−ρtC 1−α

it

1− α

]

s.t. Cit + θitPt + bitBTt = Iit + θi ,t−1 (Pt + dt) + bi ,t−1B

T−1t

Iit is consumer i ’s labor income endowment;

Consumption + Savings = Nonfinancial + Financial Income.


IntroductionModel



Aggregation

Add up consumption, dividends and labor income:

1 Ct =∫i∈I Cit ;

2 Dt =∑n

j=1 djt ;

3 It =∫i∈I Iit = Ct − Dt .


IntroductionModel



Income Process

Let Mt > 0 be an SDF implied by no-arbitrage;

Assume i ’s labor endowment is as follows:

Iit = δitCt − Dt

s.t. δit = exp

[t∑

s=1

(ηisys − y2

s /2)]

yt =

(2

α2 + α[∆mt + ρ+ α∆ct ]

)1/2

IID multiplicative unit-root earnings shock ηit ∼ N(0,1).


IntroductionModel



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∈Iθ∗ijt = 1;∫

i∈Ib∗ijt = 0.


IntroductionModel


MechanismEmpirical Implications of Idiosyncratic Risk

Outline




4 Conclusion



IntroductionModel



An Equilibrium with Autarky

Under the maintained assumptions, if:1 E [Mt ]→ 0, as t →∞;2 Mt+1/Mt ≥ e−ρ (Ct+1/Ct)−α ;

then there exists an equilibrium with no trade.

Given any (It ,Pt ,Bt) there exists (Iit) consistent withequilibrium concept!

“No trade” means that agent consumes labor earnings eachperiod and does not trade in financial markets:

This is unrealistic, but facilitates subsequent analysis;Follows from (unusual) choice of earnings process.


IntroductionModel



Consumption in Equilibrium

In equilibrium, Cit = δitCt , hence:

ln C it = ln Ct + εit s.t. εit =

t∑s=1

(ηitys − y2

s /2)

Hence y2t+1 corresponds to cross-sectional variance of

consumption growth (conditional on the aggregate state):

Varxs

[ln

(C i

t+1/Cit

C it+1/C

it

)| Ct+1, yt+1

]= y2

t+1


IntroductionModel



Stochastic Discount Factor

Use equilibrium consumption and EE to extract SDF:

1 = Et

[e−ρ

(C i

t+1

C it

)−αR j

t+1

]

=⇒ 1 = Et

[e−ρ+ 1

2α(1+α)y2

t+1

(Ct+1

Ct

)−αR j

t+1

]

=⇒ Mt+1 = e−ρ+ 12α(1+α)y2

t+1

(Ct+1

Ct

)−αNotice that Mt+1 is a function of the x-sectional distribution!


IntroductionModel



Euler Equation Estimation

Mt+1 = e−ρ+ 12α(1+α)y2

t+1

(Ct+1

Ct

)−αIn the presence of uninsurable idiosyncratic risk (y2

t+1 6= 0),

Mt+1 = g(Ct+1

Ct, y2

t+1);

Standard Euler equation (Mt+1 = f (Ct+1

Ct)) misspecified and

estimates of the coefficient of RRA will be biased:


IntroductionModel



Risk-Return Relation

Et [Rjt+1 − Rft ] =−Covt [Rjt+1,Mt+1]

Vart [Mt+1]︸︷︷︸βjt

· Vart [Mt+1]

Et [Mt+1]︸︷︷︸λM

t

SDF suggests relationship between x-sectional distribution ofnonfinancial income risk and asset risk/return:

1 Time-series properties of y2t+1 will affect market price of risk:

In particular, ↑ y 2t in bad times implies counter-cyclical MPR;

Counter-cyclical idiosyncratic labor income risk confirmed inSTY (JPE, 2004);

2 Returns that covary negatively with y2t+1 will have higher β

and risk premium.


IntroductionModel


Outline




4 Conclusion



IntroductionModel


Conclusion

1 Asset pricing in an incomplete market endowment economy:

Very special example, but results generalize;

2 Modified Euler equation now depends on properties ofidiosyncratic 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.


IntroductionModel


Empirical Evidence: BCG (JPE, 2002)

Outline




4 Conclusion



IntroductionModel



Euler Equation Errors: Set Up

Brav, Constantinides & Geczy (JPE, 2002) investigate EEerrors using an incomplete markets SDF:

1 CRRA SDF: Mt+1(g it+1) = β(g i

t+1)−γ , where g it+1 ≡

C it+1

C it

2 Assume standard EE holds for every household and asset:

1 = Et

[β(g i

t+1)−γR jt+1

]∀ i , j


IntroductionModel



Euler Equation Errors: Approach

Under complete markets, HH’s fully insure and equalize theirMRS state-by-state:

Consumption growth rates are equalized across HHs;CCAPM holds, i.e., Mt+1 = β(gt+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 individualSDFs should be valid;

BCG test if equally-weighted sum of HH’s MRS is valid SDF:

Mt+1 = βI−1∑I

i=1(g it+1)−γ


IntroductionModel



Euler Equation Errors: Approach

Interested in EE errors of the form:

ut+1 = I−1

(β

I∑i=1

(g it+1)−γ

)(Rt+1 − R f

t+1

)Conduct standard tests of: 1

T

∑Tt=1 ut+1 = 0;

They find that Euler equation is satisfied for reasonable γ.


IntroductionModel



Euler Equation Errors: Result 1


IntroductionModel



Euler Equation Errors: C&D SDF

They also perform the same test for the Constantinides &Duffie (JPE, 1996) SDF:

Mt+1 = β

(∑Ii=1 ci

t+1∑Ii=1

cit

)−α

exp{

α(α+1)2

I−1 ∑Ii=1

[log(g i

t+1)− log(g it+1)2

]}

This is equivalent to testing if most of the x-sectionalvariation of the consumption growth rate is captured byidiosyncratic income shocks that are:

1 Multiplicative;2 i.i.d. lognormal.


IntroductionModel



Euler Equation Errors: Result 2

Euler equation errors increase with RRA;

However, error is statistically insignificantfor RRA> 1;

Paper argues that this highlights theimportance of the x-sectional skewnessof the HH’s consumption growth rate,combined with the first two moments,which is a major contribution.


IntroductionModel



Euler Equation Errors: Final Thoughts

Zero Euler equation errors is one dimension along which totest 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?


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