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2Brennan and Cao (1996) have demonstrated that asymmetric information may lead to

rational momentum trading when information is revealed gradually in the market.

In a noisy capital market with imperfect information, there is room for rational agents

with heterogenous beliefs. In this paper we compare the optimal portfolio strategies of two

types of agents – conformists who believe in the log-normal market consensus and “rebels”

whose alternative beliefs incorporate price predictability. Surprisingly, we find that

momentum trading is optimal for rebels who believe that the market is grossly mispriced.

The empirical work of Lakonishok, Shleifer and Vishny (1994), Jegadeesh and Titman

(1993) and Gatev, Goetzmann and Rouwenhorst (1999) documents profits from momentum

and contrarian strategies that appear to exploit predictable price movements. These profits

raise the question whether such momentum and contrarian strategies are, in fact, the optimal

trading strategies of agents who incorporate price predictability in their alternative beliefs

about the market process. The main contribution of our paper is to enable the empirical

detection of non-conformist beliefs about the possibility of market timing.

Recent papers like DeLong et al. (1990), Barberis et al. (1997) and Hong and Stein

(1999) merely postulate the existence of irrational agents who behave as momentum traders.

However, alternative beliefs need not be irrational – Merton (1980), Summers (1986) and

Poterba and Summers (1989) have argued that it is extremely difficult to distinguish

statistically between randomness and certain forms of price-predictability.

Alternative specifications of the price process were explored in the early work of Merton

(1971) who showed that an agent who believes in an exponential trend always invests more

We employ the Cox-Huang (1989) martingale approach to the terminal wealth portfolio

problem of the agent, to address a collection of issues about the optimal trading strategy

under the alternative beliefs that incorporate price predictability. What will the rebel do?

How well will the rebel do? Can we distinguish the rebel from the typical market

participant?

Our main results show that the rebels act as momentum traders when they believe that

the current market price is far from the “right” price. In contrast, the conformists are

contrarian or momentum traders depending on whether they are more or less risk averse than

the representative agent. Thus, to an outsider, risk-averse rebels whose beliefs are far from

the market’s beliefs may appear as risk-tolerant conformists. The terminal wealth

distribution of rebels whose beliefs are sufficiently different from those of the market are

first-order dominated by certain conformist portfolios. But, in general, the properties of the

terminal wealth distribution under different beliefs suggest that it is difficult to detect wrong

beliefs from limited performance data. Finally, the long-run investment portfolios of both

conformists and rebels need not be heavily biased towards equities. Both the conformist’s

and the rebel’s long-run portfolios include a positive equity component, however equities

need not dominate the portfolio.

Our conformist momentum results generalize and make empirically relevant the

qualitative momentum aspect of the early portfolio insurance work of Leland (1980) and

Brennan and Solanki (1981). Moreover, Leland (1980) considered the optimal portfolio

insurance for agents who have different estimates of the mean holding period return.

Our paper differs from Leland (1980) in two major ways. First, we define alternative

Our main objective is to enable the empirical detection of alternative beliefs in reversion

and the feasibility of market timing. Our dynamic specification of the beliefs in reversion

allows the investment opportunity set to depend on the current market conditions. Hence the

more general dynamic definition of alternative beliefs is necessary in order to capture beliefs

in the possibility of opportunistic market timing. We relate this alternative dynamic

specification to observable trading patterns. In contrast, Leland (1980) emphasizes that his

model does not allow for market timing, while the Brennan and Solanki (1981) model has no

differences in expectations. These models remain within the static log-normal specification,

and there are no perceived changes in the investment opportunity set. On the other hand, we

are able to solve for the more general dynamic beliefs by using the Cox-Huang methodology,

which was not available at the time of the earlier models. Our results explain observed

counter-intuitive momentum trading with deviant beliefs in market reversion to a growth

trend, while remaining within the attractive rational paradigm of utility-maximizing agents.

The wrong beliefs are maintained because of market noise that prevents the timely

unambiguous learning of the proper dynamic specification.

In section I we present the model. Section II analyzes the momentum and contrarian

appearance of the optimal trading for different beliefs about the market dynamics. In section

III we examine the performance of the strategies that are optimal under different beliefs.