Analysis of the optimal time to withdraw investments from hedge funds with alternative fee structures

Abstract

We study the optimal stopping problem arising from an investor determining the best time to withdraw from a hedge fund with a shared loss fee structure and a positive fee for assets under management—a decision that is critically important to the viability of such products in practice. The optimal solution is characterized as the first exit time of the fund value from a bounded region with upper and lower stopping boundaries. In the infinite horizon case, we present the complete solution to the optimal stopping problem, while in the finite horizon case we derive a pair of coupled integral equations for the stopping bounds and present an asymptotic analysis of the stopping boundaries for small time. The analysis requires new mathematical results extending techniques suitable for options with one exercise boundary to the case of a coupled pair of upper and lower boundaries.