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

to 1, then the asset price also follows a geometric Brownian motion since
the dividend is governed by a geometric Brownian motion. If the elasticity
is higher than 1, then the spot price overreacts compared to a geometric
Brownian motion. The following proposition establishes the relationship
between the overreaction and aggregate RRA.

Proposition 1 (Overreaction)Assume that at each date aggregate RRA
is declining in the dividend and that the dividend is governed by a geometric
Brownian motion with constant instantaneous volatility and constant in-
stantaneous drift. Then the elasticity of the asset price with respect to the
dividend is higher than 1.⁸

This proposition is proved in the appendix. To get the intuition for the
overreaction, think about aggregate RRA in terms of RRA of a represen-
tative investor. A representative investor with decreasing RRA requires a
lower excess return for the same risk, the wealthier he is, i.e. the higher the
dividend is. Compared to an investor with constant RRA, her required risk
premium decreases, the wealthier she is. Hence, the price she is willing to
pay for the asset increases with increasing dividend more than under con-
stant RRA. Thus, with declining aggregate RRA an increase in the dividend
induces a decline in the required risk premium which reinforces the purely
fundamental increase of the asset price so that the asset price overreacts
compared to constant aggregate RRA. Similarly, a decline in the dividend
induces an overproportional decline in the asset price.

To draw conclusions about the behavior of excess returns we need to derive
the behavior of the total return index (performance index) Vt. Since the total
return index includes the reinvested dividend payments, its return minus the
risk-free rate is the excess return that we are interested in,

^dVt - rf dt = ^dS^t + D^tdt — rf dt.

Vt       St  St

Note that V_t = α_tS_t with α_t being independent of D_t. Therefore ∂⅛ Dt_t ≡
nV^,D = nS^,D ≡ ∂∂ln DD_t. This implies that Proposition 1 holds equally for the
elasticities η_t^S,D and η_t^V,D . Hence, declining aggregate RRA implies that
the total return index also overreacts. This overreaction translates into
an increase in the instantaneous volatility of returns as the instantaneous
volatility of the total return index is the product ΣV (St) = η_t^V,DσD. This
equals the instantaneous volatility of stock returns ΣS(St) = η_t^S,DσD. The
following proposition establishes that declining aggregate RRA also raises

⁸The corresponding result that asset returns underreact if aggregate RRA is increasing
is shown in the appendix.

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