Portfolio Management with Option Compensation Scheme Under Rank-Dependent Expected Utility

This paper studies the optimal investment problem with an option compensation scheme under rank-dependent expected utilities. Due to the presence of distortion functions and a nonconcave actual utility function, the conventional optimization tools like convex optimization and dynamic programming cannot be applied to this model. To address this challenge, a solution scheme for this nonconvex optimization problem and a procedure for fully solving this problem are proposed and demonstrated. We give explicit forms of optimal policies for a hyperbolic absolute risk-averse (HARA) manager assuming typical types of distortion functions. Numerical analysis illustrates how incentive fee rates and probability distortion influence the optimal investment policies of the fund manager. We find that under a not bad performance, (1) the increase of incentives reduces the asset volatility, (2) an increasing incentive fee rate results in a decreasing probability of bankruptcy, (3) a high risk-seeking degree leads to a high return and a high bankruptcy probability, and (4) a high risk-aversion degree reduces the bankruptcy probability and the asset volatility.