In this section, we discuss the implications and generalization of our theoretical analysis. It isworth noting first that, unlike the usual biological population process, wedo not obtain the population dynamic simply by assuming some fitness criterion, e.g., recent profits or expected utility, for the determination of the growth of each type of trader.

Instead, following Blume and Easley (1992), we take the view that there is a natural population dynamic in asset markets that emerges from the process of wealth accumulation. In this process, the endogenously determined growth rate of wealth accumulation governs the relative fitness of each type of trader in the market.

As a result, this dynamic does not depend on individual adaptation as required in the usual learning–imitation process. For example, both DSSW (1990) and Hirshleifer and Luo (2001) assume recent profitability to be the fitness criterion in their imitation processes, whereas Palomino (1996) assumes expected utility as the fitness criterion in his adopted imitation process. Thus, our endogenously determined group wealth accumulation process distinguishes our model from these other models.

The population dynamic in this paper depicts the growth of the wealth of the group, not of individual traders. In fact, the survival of the group of nonrational investors is, to some extent, at the expense of the individual investors. This happens because irrationality (overconfidence or investor sentiment) induces individual traders to trade more aggressively and, as a result, they have a higher expected return as well as a higher variance than rational traders do. This means that individual nonrational investors tend to have a higher probability of going bankrupt than do individual rational investors. In this sense, individual nonrational investors may be subject to the gambler’s ruin problem (Samuelson (1971, 1977)).

In this paper, however, we show that in a large economy where the high variance risk is diversified in the group wealth portfolio, the resulting population dynamic is driven primarily by the expected return differential between the two groups. Therefore, the gambler’s ruin problem at the individual level does not prevent the survival of nonrational investors as a group.

What happens then for a small economy where the high variance risk cannot be diversified away? In this case, there is a nonlinear (concave) relation between return and population growth. Palomino (1996) examines this case and finds that spiteful noise traders can hurt rational investors more than themselves. As a result, even with the high variance risk, the noise traders can still dominate the market if they are moderately over-optimistic and if the fundamental risk is relatively
large.

Remarkably, the condition for the survival of noise traders in a small economy is essentially the same as the condition obtained in our model under a large economy. This invariance result highlights that our main conclusion—moderately nonrational investors can dominate the market, particularly when the fundamental risk is large—is robust to a nonlinear (concave) relation between return and population growth.

Overconfidence acts like a commitment device to aggressive trading in our pairwise contest, but the commitment device effect does not require that overconfident traders move first. In fact, at the beginning of each contest a trader does not know which type of the other trader, rational or nonrational, he or she is going to face.

There are four possible type combinations of the two traders and the probability of each combination is governed by the population distribution at that time. The duopoly model of Kyle and Wang (1997) does not require that overconfident traders move first either. In fact, Kyle and Wang show that if a rational trader moves first, then a moderately overconfident trader will not only outperform the first-move rational trader, but also do better than if he or she were also rational.

On the other hand, if an overconfident trader moves first, then the second mover is better off being overconfident than being rational. This leads to a Nash equilibrium in which both traders are overconfident. This equilibrium is a prisoner’s dilemma in which both traders make less profits than if they both were rational. This Nash equilibrium outcome generalizes the special case of a single overconfident insider
as in Odean (1998).

In our evolutionary model once investors are born to be a certain type (rational or nonrational), they are “programmed” to their type in the evolutionary game. One might argue that overconfident investors should learn over time to change their erroneous belief and eventually converge to the rational belief. Empirical evidence in psychology literature (Kahneman et al. (1982)) shows that people do not update their beliefs rationally. For example, Daniel et al. (1998) consider an updating
rule based on biased self-attribution—a rule by which investors essentially believe “heads I win, tails it’s chance” (Langer and Roth (1975) and Gervais and Odean (2001)). In such a biased learning process, overconfident beliefs need not converge to rational beliefs.

The most interesting empirical implication of our analysis lies in the area of fund management. Under the view of the efficient market hypothesis, all assets are efficiently priced and hence it is optimal to invest passively in the index fund. One can view, therefore, these passive fund managers as the rational traders in the market. On the other hand, fund managers, who are overconfident about their private information or too optimistic about the future prospect of the asset value, tend to disagree with the market efficiency hypothesis and trade actively and aggressively in the market. In other words, these active fund managers tend to manifest

themselves as the nonrational traders in the market. In this context, our analysis of the survival issue provides several new empirical implications for the survival of active fund management. First, although individual active fund managers trade more aggressively and hence die faster than individual passive fund managers, the active fund management style (i.e., the “group”) can still persist in the market, e.g., Fidelity investment group versus Vanguard investment group.

Second, the group of active fund managers survives better in a market with high fundamental risk. This means that active fund management should be more popular in the market where the fundamental value of assets is more difficult to assess. This leads to several testable implications. For example, active fund management should
be more popular in high-risk funds than in low-risk funds. Similarly, active fund management should be more popular in the stock market than in the bond market and more popular in emerging markets than in the U.S. market, etc. Finally, while individual active fund managers may display various degrees of overconfidence or investor sentiment, the surviving active fund managers should exhibit moderate aggressiveness, rather than extreme aggressiveness.

Although in this paper we focus on the survival of nonrational investors in asset markets, our key finding that moderately aggressive, nonrational agents can enhance their survivability seems to have broader applications in other economic settings.

For example, Bernardo andWelch (2000) examine a model of informational cascades and find that overconfident entrepreneurs, who overweigh their private information, can better convey valuable information to the group and hence avoid the bad herding equilibrium. Goel and Thakor (2000) consider a model of leadership and showthat the overconfident manager, who understimates his or her project risk, has a greater chance to be chosen as the leader (CEO) than an otherwise identical rational manager. This is so because the race to CEO is like a winner-take-all game, in which only the extreme positive performance will be awarded.

Prof. F. Albert Wang

Next: Conclusion

Summary: Index