Portfolio Optimization with Cumulative Prospect Theory Utility via Convex Optimization

We consider the problem of choosing a portfolio that maximizes the cumulative prospect theory (CPT) utility on an empirical distribution of asset returns. We show that while CPT utility is not a concave function of the portfolio weights, it can be expressed as a difference of two functions. The first term is the composition of a convex function with concave arguments and the second term a composition of a convex function with convex arguments. This structure allows us to derive a global lower bound, or minorant, on the CPT utility, which we can use in a minorization–maximization (MM) algorithm for maximizing CPT utility. We further show that the problem is amenable to a simple convex–concave (CC) procedure which iteratively maximizes a local approximation. Both of these methods can handle small and medium size problems, and complex (but convex) portfolio constraints. We also describe a simpler method that scales to larger problems, but handles only simple portfolio constraints.