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

dividend being the only risk factor. The price of the market portfolio at
date t, St , is the present value of all future dividends

∞

exp (-rf (s - t)) DsΦt,sds
t

This price is finite given a sufficiently high risk-free rate and aggregate risk
aversion. exp (-rf (s - t)) Φt,s is the date t-stochastic discount factor for
claims contingent on some state at date s. In an arbitrage-free, complete
market this function is unique. Technically, Φ0_,t is a martingale and Φ_t,s =
Φ0,_s/Φ0,t. Since the dividend is the only risk factor in the market, Φ₀,_t can
be characterized by

dΦ0,t = -η_t^Φ,DσDΦ0,tdWt ,

Φ0,0 ⁼1^,

where aggregate RRA is given by η_t^Φ,D , the negative elasticity of the sto-
chastic discount factor for claims to be paid at date t with respect to the
dividend Dt.⁷ In this setting the asset price St can be characterized by the
following stochastic differential equation

dSt = η_t^Φ,D η_t^S,D σ_D² St - Dt + rfStdt + η_t^S,DσDStdWt
'----------------------------.----------------------------'          "---------*---------'

=μs ( St ) St                   =^ς s ⁽ St ) St

μ_s (S_t) denotes the instantaneous drift which equals the expected instanta-
neous excess return plus the risk-free rate rf .ΣS (St) denotes the instan-
taneous volatility. Both, volatility and drift depend in general on the asset
price St and time t. For simplicity of notation we suppress the time index.
η_t^S,D denotes the elasticity of the asset price St with respect to the dividend
Dt.

2 Predictability of Excess Returns and Excess Vola-
tility

We begin the analysis of return characteristics by looking at the elasticity
of the asset price with respect to the dividend. If this elasticity is equal

⁷ One way to interpret this stochastic discount factor is to assume that at each date
aggregate consumption equals the aggregate dividend. Then the stochastic discount factor
mirrors aggregate marginal utility of consumption. Alternatively, investors may use the
aggregate dividend as an index of welfare with a higher index lowering the stochastic
discount factor.

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