where φMand φCnow denote the trading elasticities of the momentum
traders and the contrarians respectively. These elasticities in turn satisfy:
φM=wγcov(Pt+j−Pt, ∆Pt− 1 )/{var(∆P)varM(Pt+j−Pt)} (10)
φC=(1−w)γcov(Pt+j−Pt, ∆Pt− 1 −c)/{var(∆P)varC(Pt+j−Pt)} (11)
Equilibrium now involves a two-dimensional fixed point in (φM, φC) such
that prices are given by equation (9), while at the same time (10) and (11)
are satisfied. Although this is a more complicated problem than before, it is
still straightforward to solve numerically. Of course, this is no longer the end
of the story, since we still need to endogenize w. This can be done by impos-
ing an indifference condition: In an interior solution where 0<w<1, the
utilities of the momentum traders and contrarians must be equalized, so no-
body wants to switch styles. It turns out that the equal-utility condition can
be simply rewritten in terms of either conditional variances or covariances
of prices (see the appendix for a proof ). This gives us:
Proposition 3:In an interior solution with 0<w<1, it must be that:- var(Pt+j−Pt∆Pt− 1 )=var(Pt+j−Pt∆Pt− 1 −c); or equivalently
2.cov((Pt+j−Pt),∆Pt− 1 )=cov((Pt+j−Pt),∆Pt− 1 −c); or equivalently - cov(∆Pt+ 1 , ∆Pt− 1 )+cov(∆Pt+ 2 , ∆Pt− 1 )+...+cov(∆Pt+j,∆Pt− 1 )=
−cov(∆Pt+ 1 ,∆Pt− 1 −c)−cov(∆Pt+ 2 , ∆Pt− 1 −c)−...−cov(∆Pt+j,∆Pt− 1 −c).
The essence of the proposition is that in order for contrarians to be active
in equilibrium (that is, to have w<1) there must be as much profit oppor-
tunity in the contrarian strategy as in the momentum strategy. Loosely
speaking, this amounts to saying that the negative autocorrelations in the
reversal phase must cumulatively be as large in absolute magnitude as the
positive autocorrelations in the initial underreaction phase. Thus adding
the option of a contrarian strategy to the model cannot overturn the basic
result that if there is underreaction in the short run, there must eventually
be overreaction at some later point.
As it turns out, for a large range of parameter values, we can make a much
stronger statement: the contrarian strategy is not used at all, for any choice of
c. Rather, we get a corner solution of w=1, in which all arbitrageurs endoge-
nously choose to use a momentum strategy. This is in fact the outcome for
every set of parameters that appears in figures 14.1–14.3. Thus our previous
numerical solutions are wholly unaffected by adding contrarians to the mix.
In order to get contrarian strategies to be adopted in equilibrium, we
have to crank up the aggregate risk tolerance γto a very high value. This
does two things: first, it drives down the expected profits to the momentum
strategy; and second, it causes the degree of overreaction to increase. Both
of these effects raise the relative appeal of being a contrarian to the point
518 HONG AND STEIN