The Evolution of Portfolio Rules in Financial Markets

Emanuela Sciubba - Lucy Cavendish College

The aim of this paper is to test the performance of the standard version of CAPM in a evolutionary framework. We imagine a heterogeneous population of long-lived agents who invest their wealth according to different portfolio rules. In particular, we consider: traders who "believe" in CAPM and use its predictions as a rule of thumb; traders who are endowed with a quadratic utility function and, therefore, display genuine mean-variance behavior; finally, traders who are endowed with a logarithmic utility function and therefore adopt the celebrated "Kelly criterion" for investment. We are interested in the asymptotic behavior of the wealth share of these three types of traders. Following Blume and Easily (JET, 1992) we define notions of dominance, survival and extinction in the market through the limiting value of each trader's wealth share. In particular, the wealth share of a dominating trader is asymptotically bounded away from zero, while extinction occurs when when the wealth share converges almost surely to zero. In a complete securities market with aggregate uncertainty, we show that both CAPM behavior and genuine mean-variance behavior show are evolutionarily unstable. In fact, a state that features strictly positive population shares of traders who either believe in CAPM or are endowed with a quadratic utility function, fails to be Lyapunov stable. Our main results are obtained in a simple setting, where traders have constant and identical savings rates. Under this assumption we are able to prove that logarithmic utility maximizers dominate on any sample path. Furthermore, in the presence of aggregate uncertainty, both those traders who believe in CAPM and those who display a genuine mean-variance behavior are driven to extinction. We then check the robustness of our results allowing for different kinds of heterogeneity among traders. First of all, we allow for more than two types of traders in the market and we prove that, except for special cases, our results are robust in a multipopulation framework. Secondly, we allow for different savings rates among traders. We find that the market selects for the most patient investor. Therefore our results are obviously robust, as far as logarithmic utility maximizers are relatively more patient than CAPM traders. However we prove that this is a sufficient but not necessary condition. Finally, we allow for heterogeneity among CAPM traders in their degree of risk aversion and we prove that our results are robust in this setting.

Scheduled for Session 2.3 Financial Models - III