Optimal effort under high-water mark contracts

We mainly develop a model measuring the optimal effort of a risk-neutral hedge fund manager in a continuous time framework. The fund manager chooses the optimal effort to maximize the present value of total fees and reduce liquidation risks, trading off extra return benefits against the cost of the effort. We find that the manager's effort depends on the ratio between the fund's assets under management (AUM) and the high-water mark (HWM), and endogenous fund liquidation has key influence on the dynamics of the effort. Our calibration suggests that when the fund is close to liquidation, the manager exerts greatest effort. The more distant the fund value is from the liquidation boundary, the less effort the manager chooses to make; when the fund value is approaching the HWM, the manager's optimal effort still decreases, but the rate of decline becomes far slower. The optimal effort contributes to both increasing the likelihood of survival for the fund and preserving the fund's going-concern value. A growth of degree of the effort cost, volatility of the AUM, exogenous liquidation probability or endogenous liquidation boundary decreases the optimal effort. We also find empirical evidence that may support our theoretical conclusion.