Heterogeneous Beliefs and the Performances of Optimal Portfolios

The market selection depends on agent's survival index, which is a function of agent's belief and risk preference. When preferences are identical, the survival index of an agent is a decreasing function of his belief accuracy and therefore agent survives if and only if he has the lowest survival index. Following this result, one maybe tempted to think that an agent is expected to perform at least as good as the market if he survives, and he is expected to outperform the market if his belief is more accurate than all other agents' beliefs. We show that the these statements are false in general. In terms of long-run performance, market outperforms those agents who do not have the minimum survival index in the long-run. When multiple agents survive, we show that no agent can outperform the market in the long-run. In terms of the expected performance, all agents are expected to underperform the market even when they all survive in the long-run. When survival indices differ, the fittest agent with the lowest survival index is expected to outperform the market consistently with any given finite investment horizons if and only if his subjective belief is much more accurate than the other agents' beliefs.