In Proposition 2, we provide sufficient conditions on p 1 and p 2 for prices
to exhibit both underreaction and overreaction, and their form is very sim-
ilar to what we have just obtained. In fact, the argument in Proposition 2 is
essentially the one we have just made, although some effort is required to
make the reasoning rigorous.
Before stating the proposition, we repeat the definitions of overreaction
and underreaction that were presented in section 2. Overreaction can be
thought of as meaning that the expected return following a sufficiently
large number of positive shocks should be lower than the expected return
following the same number of successive negative shocks. In other words,
there exists some number J≥1, such that for all j≥J,
Et(Pt+ 1 −Ptyt=yt− 1 =...=yt−j=y)
−Et(Pt+ 1 −Ptyt=yt− 1 =...=yt−j=−y)<0.
Underreaction means that the expected return following a positive shock
should exceed the expected return following a negative shock. In other words,
Et(Pt+ 1 −Ptyt=+y)−Et(Pt+ 1 −Ptyt=−y)>0.
Proposition 2 below provides sufficient conditions on πL, πH, λ 1 , and λ 2
for these two inequalities to hold.^11
Proposition 2.If the underlying parameters πL, πH, λ 1 , and λ 2 satisfy
p 2 <p 1 < p 2 ,
p 2 ≥0,
then the price function in Proposition 1 exhibits both underreaction
and overreaction to earnings; and are positive constants that
depend on πL, πH, λ 1 , and λ 2 (the full expressions are given in the
Appendix).We now examine the range of values of the fundamental parameters πH,
πL, λ 1 , and λ 2 for which the sufficient conditions for both underreaction and
overreaction are satisfied. Since the conditions in Proposition 2 are some-
what involved, we evaluate them numerically for a large range of values of
the four underlying parameters. Figure 12.1 illustrates one such exercise. We
k^ kk^ kA MODEL OF INVESTOR SENTIMENT 441(^11) For the purposes of Proposition 2, we have made two simplifications in our mathematical
formulation of under- and overreaction. First, we examine the absolute price change Pt+1−Pt
rather than the return. Second, the good news is presumed here to be the event yt=+y, i.e., a
positive change in earnings, rather than better-than-expected earnings. Since the expected
change in earnings Et(yt+ 1 ) always lies between −yand +y, a positive earnings change is in fact
a positive surprise. Therefore, the results are qualitatively the same in the two cases. In the
simulations in section 5.2, we calculate returns in the usual way, and condition on earnings
surprises as well as raw earnings changes.