The Optimal Dynamic Investment Policy for a Fund Manager Compensated with an Incentive Fee

Abstract

We use martingale methods to solve the investment problem of a risk averse fund manager who charges an incentive fee which he cannot hedge in his personal account. An incentive fee is a share in the positive part of the returns on the client’s portfolio net of some benchmark return. The optimal policy is a long-shot; there is always some chance of bankrupting the client, but if the terminal fund value is nonzero, it is in the money by some strictly positive account. We provide explicit expressions for the optimal trading strategy with either the riskless asset or the market portfolio as benchmark and with either constant relative or absolute risk aversion. Rather than trying to maximize volatility, as earlier literature suggests, the manager dynamically adjusts volatility as the assets move in or out of the money. As the manager accumulates profits, he moderates portfolio risk. For example, if the manager has constant relative risk aversion, volatility converges to the Merton constant as fund value grows large. On the other hand, as bankruptcy approaches, portfolio volatility goes to infinity.