Eq. (8) allows us to state the following:
Proposition 2:In any covariance-stationary equilibrium, if price changes
are positively correlated at short horizons (e.g., if cov(∆Pt+ 1 , ∆Pt− 1 )
>0), then with risk-neutral momentum traders they are negatively
correlated at a horizon no longer than j+1—that is, it must be that
cov(∆Pt+i, ∆Pt− 1 )<0 for some i≤j.It is useful to explore the differences between Propositions 1 and 2 in
some detail, since at first glance, it might appear that they are somewhat
contradictory. On the one hand, Proposition 1 says that in response to
good news, there is continued upward momentum in prices for at least jpe-
riods, and possibly more (if j<z−1). On the other hand, Proposition 2
suggests that price changes begin to be reversed within j+1 periods, and
quite possibly sooner than that.
The two propositions can be reconciled by noting that the former is a con-
ditionalstatement—that is, it talks about the path of prices from time t
onward, conditional on there having been a news shock at time t. Thus
Proposition 1 implies that if a trader somehow knows for sure that there is a
news shock at time t, he could make a strictly positive expected profit by buy-
ing at this time and holding until time t+j. One might term such a strategy
“buying early in the momentum cycle”—that is, buying immediately on the
heels of news arrival. But of course, such a strategy is not available to the mo-
mentum traders in our model, since they cannot condition directly on the εs.
In contrast, Proposition 2 is an unconditionalstatement about the auto-
covariance of prices. It flows from the requirement that if a trader buys at
time tin response to an unconditional price increase at time t−1, and then
holds until t+j, he makes zero profits on average. This zero-profit require-
ment in turn must hold when momentum traders are risk-neutral, because
the unconditional strategy isavailable to them.
There is a simple reason why an unconditional strategy of buying follow-
ing any price increase does not work as well as the conditional strategy of
buying only following directly observed good news: not all price increases
are news-driven. In particular, a trader who buys based on a price increase
observed at time truns the following risk. It may be “late” in the momen-
tum cycle, in the sense that there has not been any good news for the last
several periods. Say the last good news hit at t−i. If this is the case, the price
increase at time tis just due to a late round of momentum buying. And those
earlier momentum purchases kicked off by the news at t−iwill begin to
be unwound in the very near future (specifically, at t−i+j+1) causing the
trader to experience losses well before the end of his trading horizon.
This discussion highlights the key spillover effect that drives our results. A
momentum trader who is fortunate enough to buy shortly after the arrival
of good news imposes a negative externality on those that follow him. He
512 HONG AND STEIN