Return Predictability and Stock Market Crashes in a Simple Rational Expectations Model



normal case. Moreover, the general relationship between aggregate relative
risk aversion, the dividend and the price process of the market portfolio is
derived. In Section 2, predictability of excess returns and excess volatility are
shown. Section 3 discusses conditions for stock market crashes. In section 4
an analytic formula for the price of the market portfolio is presented together
with simulations illustrating the previous results. Section 5 concludes.

1 The Economic Setting

Our aim is to analyze a simple model of investors preserving essential plau-
sible properties of a rational expectations equilibrium. Since asset pricing
depends on aggregate relative risk aversion, we first motivate our assump-
tion that aggregate relative risk aversion is declining. Then we analyze the
implications on asset pricing. We consider a pure exchange economy with
a perfect and complete market. All agents have homogeneous and rational
expectations, but different utility functions.

1.1 Investor Heterogeneity and Aggregate Risk Aversion

In this section we argue that declining aggregate relative risk aversion (RRA)
is the normal case. Aggregate RRA is the market’s relative risk aversion as
implied by the stochastic discount factor (pricing kernel) by which claims
to be paid at a given future date are valued in the capital market.
4 In
the case of risk neutrality the stochastic discount factor is constant. With
risk aversion, the stochastic discount factor is declining in some aggregate
variable like wealth or aggregate consumption. The negative elasticity of
the discount factor with respect to this variable defines aggregate RRA.
This variable is given in our model by the dividend of the market portfolio.

Aggregate RRA depends on investors’ RRA. There is little disagreement
that investors display declining absolute risk aversion. But it is controver-
sial whether they display declining RRA. While it is therefore difficult to
justify declining aggregate RRA in a representative investor economy (Ru-
binstein, 1974), we will show that declining aggregate RRA is likely to be
observed if a representative investor does not exist. Consider the following
setup. At each date
τ , aggregate consumption equals aggregate dividend
Dτ . Each agent, indexed by i =1,... ,n, has a time-additive von Neumann-
Morgenstern utility function. She has some initial endowment and trades
in a perfect, complete market. She may consume at each future date. As

4Since in this paper we consider only the characteristics of the market portfolio, we
do not differentiate between the pricing kernel and the asset specific pricing kernel. For a
discussion, see Camara (2003).



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