Proposition 3 (Predictability of asset returns) Suppose that at each
date aggregate RRA is declining in the dividend and that the dividend is gov-
erned by a geometric Brownian motion with constant instantaneous volatility
and constant instantaneous drift. Then, the cumulated excess return and the
instantaneous expected excess return are negatively correlated if the volatility
of excess returns does not increase with the dividend.11
This proposition is proved in the appendix.12 It shows that excess returns are
negatively autocorrelated if aggregate RRA is declining, provided that the
volatility of excess returns does not increase with the dividend. The intuition
for the negative autocorrelation is that if past returns have been strongly
positive, investors are better off implying lower aggregate RRA. Hence, the
required risk premium decreases which lowers future expected excess returns.
This leads to negative autocorrelation. However, Proposition 3 reveals that
autocorrelation might be positive if the volatility of excess returns strongly
increases with the dividend so that the required risk premium increases, too.
This will be illustrated later in our simulations. To sum up, predictability
of asset returns may be caused by declining aggregate RRA.
3StockMarketCrashes
In this section we analyze aggregate RRA in more detail and provide an
explanation for stock market crashes in our simple rational expectations
model. Often a stock market crash like that at the beginning of this decade
is associated with a previous price bubble. Such a bubble is created by a
strong stock price increase which is not driven by a strong improvement in
fundamentals, and a subsequent strong price decline aligning the stock price
again to fundamentals. It is difficult to explain bubbles in a rational expec-
tations framework. Many explanations are based on behavioral departures
from ”rationality” or market imperfections. The explanation of crashes in
this paper relies neither on irrationality nor on market imperfections.
We define a crash as a situation in which a small decline in the fundamentals
triggers a strong decline in the stock price. Conversely, a small improve-
ment in the fundamentals may trigger a strong increase in the stock price.
A bubble that bursts, thus, might be observed if a small improvement in
fundamentals leading to a strong price increase is followed by a small decline
in fundamentals leading to a strong price decline. Such phenomena will be
shown to be fully consistent with a rational expectations equilibrium in a
11The corresponding result for increasing aggregate RRA is shown in the appendix.
12 The conditions established in Proposition 3 are sufficient but not necessary.
10