Competition Between Infinitely Many Fund Managers and Investor's Welfare

This work analyzes the dynamic competition among an infinite number of managers acting in a financial market with a riskless bond and a risky asset. Each player competes against infinitely many competitors for receiving money flows that depend on her relative performances. We assume that each manager attempts to overperform the industry average performance. We find the closed formula for the optimal policy. We show that when all the agents are identical (homogenous case) the competition induced by the convex incentive affects both the risk aversion of the manager and her optimal policy. The change in the risk aversion and the shift in the risk taking behavior have opposite effects on manager's optimal policy. In the homogenous case the two effects perfectly offset and the optimal policy coincide with the usual Merton policy. We characterize the optimal solution of the problem also in the extended framework allowing for heterogenous groups of managers. In this case the two opposite forces acting on the manager's choice do not balance each other and there is room for the analysis of the change in the risk-taking optimal behavior of managers and in the whole industry as function of the parameters of the utility function of the managers as well as the relative weight of the groups in the population. We study the welfare loss of investors, who let their money being managed by managers, relating to the level of competition in the market.