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



below the current future price and the empirical estimates are subject to
various methodological concerns (see Barone-Adesi et. al. 2004, Bliss and
Panigirtzoglou, 2003 and Hentschel, 2003). Also there is little consensus on
risk aversion of individual investors. Most researchers agree, however, that
investors display declining absolute risk aversion and that their risk aver-
sion levels differ. The first important result in this paper is that under these
conditions aggregate relative risk aversion is likely to decline. Therefore this
paper analyzes asset price processes under declining aggregate relative risk
aversion.

To analyze the effects of declining aggregate relative risk aversion on the
characteristics of the market portfolio return, we consider a model similar
to that of Brennan and Xia (2002) in which the dividend on the market
portfolio is governed by a geometric Brownian motion and the price of the
market portfolio equals the present value of these dividends. The price de-
pends on aggregate relative risk aversion. Since the dividend is exogenously
given, but prices are not, we prefer to define aggregate relative risk aversion
as the negative elasticity of the stochastic discount factor with respect to
the dividend. In an intertemporal model this allows us to characterize risk
preferences independently of endogenous asset prices. If aggregate relative
risk aversion is constant, then the market return is identically and indepen-
dently distributed ruling out return predictability, excess volatility and stock
market crashes. If, however, aggregate relative risk aversion declines with
increasing concurrent dividend, then an increase in the dividend leads to
an overproportional price increase because the risk premium declines. Sim-
ilarly, if the dividend declines, then the stock price declines overproportion-
ally because the risk premium increases. This implies excess volatility and
predictability of market returns. If aggregate relative risk aversion declines
rapidly in some dividend range, then the risk premium declines strongly in
this range so that the price of the market portfolio increases rapidly given
a small increase in dividends. Conversely, a small decline in dividends then
leads to a strong price decline, similar to a crash. If the dividend happens
to first increase and then to decline, then we may observe a stock market
movement which resembles a bubble that bursts. In the language of technical
analysis, the lower bound of this critical dividend range may be interpreted
as the support level and the upper bound as the resistance level. We show
that aggregate relative risk aversion may strongly decline with increasing
dividend if relative risk aversion levels differ strongly across investors.

These important results are new as shown by a brief discussion of the theoret-
ical asset pricing literature. For finite horizon models it is known from Bick
(1990) and Franke et. al. (1999) that if the price of the market portfolio is
governed by a geometric Brownian motion as in the Black and Scholes (1973)
model, then aggregate relative risk aversion is constant. Bick (1990) and He
and Leland (1993) derive characteristics of asset price processes which are



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