with j=6, the impulse response peaks at 1.265, while with j=18, the peak
reaches 1.252, neither as high as in the case where j=12. This nonmonoto-
nicity arises because of two competing effects. On the one hand, an increase
in jmeans that there are more generations of momentum traders active in
the market at any one time; hence their cumulative effect should be
stronger, all else equal. On the other hand, the momentum traders ration-
ally recognize the dangers of having a longer horizon—there is a greater
risk that they get caught trading late in the momentum cycle. As a result,
they trade less aggressively, so that φis decreasing in j.
A more clear-cut result appears to emerge when we consider the effect of
jon the time pattern of autocorrelations. As suggested by figure 14.1, the
smaller jis, the faster the autocorrelations begin to turn negative. For ex-
ample, with j=6, the first negative autocorrelation occurs at a lag of 6
months, while with j=18, the first negative autocorrelation occurs at a lag
of 12 months. Thus the intuition from Proposition 2 seems to carry over to
the case of nonzero risk aversion.
In figure 14.2, we examine the effect of changing momentum traders’ risk
tolerance. (This Experiment can equivalently be thought of as varying the rel-
ative proportions of momentum traders and newswatchers.) We set j=z= 12
months, and allow γto vary. As risk tolerance increases, momentum traders
respond more aggressively to past price changes—that is, φincreases. This
A UNIFIED THEORY OF UNDERREACTION 515
Figure 14.2. Cumulative impulse response and momentum traders’ risk tolerance.
The momentum traders’ risk tolerance gamma takes on values of 1/11, 1/7 and 1/3.
Base is the cumulative impulse response without momentum trading. The other pa-
rameter values are set as follows: the information diffusion parameter zis 12, the
momentum traders’ horizon jis 12 and the volatility of news shock is 0.5.