Thus a crucial insight is that “early” momentum buyers impose a negative
externality on “late” momentum buyers.^4 Ideally, one uses a momentum
strategy because a price increase signals that there is good news about fun-
damentals out there that is not yet fully incorporated into prices. But some-
times, a price increase is the result not of news but just of previous rounds of
momentum trade. Because momentum traders cannot directly condition on
whether or not news has recently arrived, they do not know whether they
are early or late in the cycle. Hence they must live with this externality, and
accept the fact that sometimes they buy when earlier rounds of momentum
trading have pushed prices past long-run equilibrium values.
Although we make two distinct bounded-rationality assumptions, our
model can be said to “unify” underreaction and overreaction in the follow-
ing sense. We begin by modeling a tendency for one group of traders to un-
derreact to private information. We then show that when a second group
of traders tries to exploit this underreaction with a simple arbitrage strat-
egy, they only partially eliminate it and, in so doing, create an excessive mo-
mentum in prices that inevitably culminates in overreaction. Thus, the very
existence of underreaction sows the seeds for overreaction, by making it
profitable for momentum traders to enter the market. Or, said differently,
the unity lies in the fact that our model gets both underreaction and over-
reaction out of just one primitive type of shock: gradually diffusing news
about fundamentals. There are no other exogenous shocks to investor sen-
timent and no liquidity-motivated trades.
In what follows, we develop a simple infinite-horizon model that cap-
tures these ideas. In section 1, we present and solve the basic model, and do
a number of comparative statics experiments. Section 2 contains several ex-
tensions. In section 3, we draw out the model’s empirical implications. Sec-
tion 4 discusses related work, and section 5 concludes.
1 .The Model
A. Price Formation with Newswatchers OnlyAs mentioned above, our model features two classes of traders, newswatch-
ers, and momentum traders. We begin by describing how the model works
when only the newswatchers are present. At every time t, the newswatchers
trade claims on a risky asset. This asset pays a single liquidating dividend at
some later time t. The ultimate value of this liquidating dividend can be
written as: DT=D 0 +∑Tj= 0 εj, where all the ε’s are independently distrib-
uted, mean-zero normal random variables with variance σ^2. Throughout,
A UNIFIED THEORY OF UNDERREACTION 505(^4) As we discuss below, this “momentum externality” is reminiscent of the herding models of
Banerjee (1992), Bikhchadani, Hirshleifer, and Welch (1992), and Scharfstein and Stein (1990).