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

the variance of asset returns over finite periods. Hence, declining aggregate
RRA explains the well documented excess volatility.

Proposition 2 (Excess Volatility) Suppose that at each date aggregate
RRA is declining in the dividend and that the dividend is governed by a geo-
metric Brownian motion with constant instantaneous volatility and constant
instantaneous drift. Then the conditional (τ>t= θ) and the unconditional
(τ>t>θ) variance of the total return index exceed the dividend variance,
i.e.

Var(lnVτ - ln Vt | D_θ) >Var(lnDτ - ln Dt | D_θ) .           (6)

Proposition 2 is proved in the appendix.⁹ The conditional variance Var(ln V_τ |D_t)
exceeds the dividend variance Var(ln D_τ∖D_t) because of overreaction. The
same is true of the unconditional variance. Therefore excess volatility is
obtained.

We have seen that asset returns under declining aggregate RRA overreact
compared to constant aggregate RRA. Does the overreaction render asset
returns predictable? First, notice that the instantaneous drift of the total
return index μ_v(S_t) equals the instantaneous drift of stock returns plus the
dividend yield. Hence the instantaneous Sharpe ratio

μv (St) - rf _ ^μs (St) ⁺ D ^{- r}f _ φ ,d

Σ V ( St )   = Σ S ( St )      = ^ηt   '

depends negatively on Dt for declining aggregate RRA, η_t^Φ,D. Therefore the
Sharpe ratio can easily be predicted knowing the current dividend. The pre-
dictability of the Sharpe ratio would directly translate into predictability of
excess returns if the instantaneous return volatility ΣV (St) was non-random.
But changes in volatility might disturb this relationship, the exception being
that the volatility does not increase with the dividend.

Instead of using the fundamental variable, in our model the dividend, for
forecasting excess returns, many forecasts are based on past excess re-
turns. This is successful if excess returns are either positively or nega-
tively autocorrelated. To analyze the serial return dependence, we con-
sider the covariance between the excess return over the time span [t, τ], i.e.
CER_t,_τ ≡ fT ^dV^s — JT rf ds, and the instantaneous expected excess return
at time τ, i.e. μ_v (τ) — rf .10

⁹Proposition 2 assumes declining aggregate RRA. It does not hold in an analogous
manner for increasing aggregate RRA.

¹⁰For a similar analysis see Johnson (2002).

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