The right panel in figure 7.6 graphs the conditional volatility of returns
as a function of the state variable. Since much of the return volatility in our
model is generated by changing risk aversion, the conditional volatility in
any state depends on how sensitive the investor’s risk aversion in that state
is to dividend shocks. Empirically, volatility has been found to be higher
after market crashes than booms, which in our context would mean an up-
ward sloping conditional volatility curve. For much of the range of the state
variable, this is exactly what we find. However, this result is sensitive to
how we make the degree of loss aversion depend on prior outcomes, so we
do not attach too much weight to it.
Table 7.5 presents autocorrelations of log returns and of the price/dividend
ratio. As expected, our model produces negatively autocorrelated returns at
all lags: high prices lower risk aversion and lead to lower returns on average.
These negative autocorrelations imply long horizon mean reversion of the
kind documented by Poterba and Summers (1988) and Fama and French
(1988a). Moreover, the price/dividend ratio is highly autocorrelated in our
model, closely matching its actual behavior.
Since the investor’s risk aversion changes over time, expected returns also
vary, and hence returns are predictable. To demonstrate this, we use our
simulated data to run regressions of cumulative log returns over a j-year
horizon on the lagged dividend/price ratio for j=1, 2, 3, and 4:
(38)where rtis the log return. Table 7.6 presents the slope coefficients βjand
R^2 (j) obtained from our simulated data alongside the empirical values.
Note that our simulated results capture the main features of the empirical
findings, including an R^2 that increases with the return horizon.
rrtt++ 12 ++⋅⋅⋅ +rtjjjttjt+=αβ+ (/)DP+,.PROSPECT THEORY AND ASSET PRICES 257
Figure 7.6. Conditional moments of stock returns. The conditional stock return
mean and volatility are plotted against zt, which measures prior gains and losses: a
low zt, indicates prior gains. The results are for Economy II with b 0 =2 and k=3.
The dotted line in the left-hand panel indicates the level of the risk-free rate.