Abstract
In this paper, we consider portfolio optimization under cumulative prospect
theory (CPT). Existing methods for CPT optimization are only available under
particular assumptions that may not hold in practice. We propose the first
numerical method for solving CPT portfolio optimization based on historical
asset return under a general incomplete market setting. Our method is an
alternating direction method of multiplier (ADMM), which introduces an
auxiliary variable to represent the historical return of the associated
portfolio. The main difficulty in our ADMM is that one of its two subproblems
involves optimization with the CPT utility subject to a chain constraint. We
develop two methods to solve this subproblem. The first one is based on the
philosophy of dynamic programming, and the second one is a variant of the
well-known pooling-adjacent-violators algorithm. We further demonstrate the
theoretical convergence of the proposed ADMM method and the two methods for
solving the difficulty subproblem. Our numerical experiments demonstrate the
effectiveness of the proposed method. Based on the proposed method, our
empirical study with real data further demonstrates how the CPT's parameters
influence the investor's investment behavior.