In general, the βs, κs, and φs in Eq. (A.16) have to be determined
numerically as fixed points of equations (A.8) and (A.18) using the
same methodology as described above in section A.3. While solving
for these parameters is computationally difficult, we can character-
ize certain behavior in a covariance stationary equilibrium. Given a
positive one-unit shock that begins to diffuse among newswatchers
at time t, the price underreacts at tfor finite smart money risk toler-
ance, that is,
The price eventually converges to one in a covariance stationary
equilibrium. And in a covariance stationary equilibrium, the price
must also overshoot one. To see this, suppose it does not. Then the
serial correlation in returns would be positive at all horizons. Then
this implies that momentum investors would have φ>0, which by
our previous logic implies that there would be overreaction, thereby
establishing a contradiction.
When the risk tolerance of smart money is infinite, it follows from
the discussion above that without momentum traders prices follow a
random walk. So, the expected return to momentum trading is
zero. Hence, when the risk tolerance of smart money is infinite,
prices following a random walk and no momentum trading is in fact
a covariance-stationary equilibrium. QED
∆P
tzz
=+− <
1
β 1 1.A UNIFIED THEORY OF UNDERREACTION 537