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



low dividend levels and low for high dividend levels. For instance, for Spec-
ification 3 the average Sharpe ratio is around 0.18 given an initial dividend
D0 =4. For D0 = 1, we find an astonishingly high average Sharpe ratio
of about 2.65. These fluctuations of the Sharpe ratio may explain why for
certain time periods empirical studies find such high equity premia.

- insert Table 2 here -

More information on the characteristics of the asset price process is provided
in Table 2. This table presents measures of return volatility and of auto-
correlation in returns and return volatility for the benchmark case and for
specifications 1 to 3. The results are given for initial dividends of 1 and 4.
The initial dividend is relevant since it determines the likely dividend paths
underlying the simulation results. Regarding return predictability, the au-
tocorrelation of returns is imp ortant. All displayed return autocorrelations
are negative. Figure 6 reveals that the serial return covariance is slightly
negative everywhere for specification 1, but this is not true for specifications
2 and 3. Here the autocorrelation becomes positive at a dividend level of
about 3.5, reaches a peak at about 3.7 resp. 3.9 and then turns strongly
negative before it moves back close to zero. The intuition for this surprising
result is as follows. When the dividend moves up from, say, 3.8 to 4.2, then
the asset return is strongly positive as it is when the dividend moves further
up in the next period from 4.2 to 4.7 implying positive autocorrelation. But
when it moves further up from 4.7 to 5.2, then the return will be small im-
plying negative autocorrelation. Hence even though Table 2 shows negative
autocorrelations, this only indicates that the local negative autocorrelations
dominate the positive ones in our simulations. These results seem to be in
line with empirical research suggesting short-term momentum and long-term
reversals.15

- insert Figure 6 here -

Finally, Table 2 illustrates excess volatility and volatility clustering. The
volatility of asset returns equals the dividend volatility of 12.8 percent in
the benchmark case. But it is higher for specifications 1 to 3 because of
declining aggregate RRA. For specification 1 and an initial dividend of 1
the return volatility is 16.7 percent on a monthly basis and 16.6 percent on
a 4-year basis. These figures are higher for specifications 2 and 3. They in-
crease dramatically if the initial dividend is 4, i.e. in the center of the crash
potential. The strong price movements in this region produce a predictable,
high volatility. All the autocorrelations in return volatility shown in the last

15 There are different definitions of momentum and reversals. In this paper we define
positive [negative] serial correlation as momentum [reversal].

18



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