perfect market. Strongly declining aggregate RRA in a small range of fun-
damentals may be viewed as a shift from a high risk aversion regime to a
low risk aversion regime or vice versa. This regime shift causes the crash.
To illustrate the regime shift, first, consider a market with constant aggre-
gate RRA, η. Then in an infinite horizon model in which the aggregate
dividend follows a geometric Brownian motion, the stock price at date t, St,
is a multiple of the dividend at date t,
∞
St=
s=t
E( Ds∣Dt )
(1 + k )s-t
= Dt
with 1 + k = exp[rf + η σD — μD].
To make things simple, suppose rf equals μD. Then the price dividend ratio
equals [1 — exp(—η σ2D)]-1. Empirical estimates of the dividend volatility
of the market portfolio are around 12.8 percent. Then the price dividend
ratio would be around 61.5 for constant aggregate RRA of 1. Now suppose
that unexpectedly aggregate RRA increases from a constant level of 1 to a
constant level of 10. Then the price dividend ratio would drop to 6.6, i.e.
the price would drop by almost 90 p ercent. Hence the shift from the low to
the high risk aversion regime induces a stock market crash. In the following,
we analyze equilibria with the potential for a stock market crash.
The property required for a crash is that aggregate RRA stays almost con-
stant in the range of low aggregate dividends, then drops sharply with an
increase in dividends and, again, almost stays constant in the upper range.
Even though a precise characterization of the conditions implying these
properties is difficult, we present a condition implying that aggregate RRA
almost stays constant for the low and the high supply range. Hence, in be-
tween, the (negative) slope must be strong so as to move down sufficiently.
Given such a condition, we may observe a stock market crash. The result
proved in the appendix assumes in line with the literature that all investors
have constant RRA, but the level of RRA differs across investors.
Lemma 2 Consider an equilibrium allocation with investors i (i =1,... ,n)
ordered by declining level of constant relative risk aversion. Then the slope
of the aggregate RRA curve relating aggregate RRA to the aggregate dividend
approaches zero for high levels of the aggregate dividend. The same is true
for very low levels if the constant RRA of investor 1 is higher than twice the
constant RRA of every other investor.
This result characterizes situations in which we may observe a locally strong
decline in aggregate RRA. It is driven by the condition that the constant
RRA of the most risk averse investor is much higher than that of the other
11