A Free Boundary Problem Arising from Hedge Funds Flows

When the value of a hedge fund approaches its running maximum, performance fees are paid and the fund typically experiences positive flows, while large drawdowns routinely lead to negative flows. In a model where investors' flows are a function of the drawdown, we solve in closed form the stochastic optimal control problem of a manager who maximizes the expected value of future fees and anticipates investors' response to the fund’s performance. The problem involves a nonlinear HJB equation which can be linearized though a duality approach, at the cost of replacing it with a free-boundary problem. The verification result relies on an upper bound for the lifetime maximum of a diffusion with negative drift. In contrast to models where outflows are performance-insensitive, a higher drawdown induces the manager to take more risk. Such risk-shifting incentive increases as flows' sensitivity to performance increase, and as managerial ability decreases.