Portfolio Optimization with Cumulative Prospect Theory Utility via Convex Optimization

TL;DR

This paper tackles portfolio optimization under Cumulative Prospect Theory (CPT) utility, a nonconvex objective that is derived from empirical return distributions. It reveals CPT utility can be written as a difference of two structured terms, enabling a globally computable minorant and two optimization frameworks: minorization-maximization (MM) and iterated convex-concave procedure (CCP), plus a scalable projected-gradient method for large problems. The authors demonstrate three practical algorithms (MM, CCP, GA) that accommodate convex portfolio constraints and show how a simple mean-variance frontier heuristic can serve as an effective approximation. Numerical experiments on toy and multi-asset datasets reveal convergence properties, diversification benefits, and scalability, with open-source code available to practitioners.

Abstract

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.

Figures (6)

• Figure 1: CPT utility surface for a long-only portfolios of stocks ($w_1$), bonds ($w_2$), and T-bills ($w_3=1-w_1-w_2$).
• Figure 2: (a) Maximizing $U^{\mathrm{cpt}}$ along the MV frontier, resulting in $w^{\mathrm{mv}}$. (b) Utility surface of $U^{\mathrm{mv}}$ for the choice of $\lambda$ that results in $w^{\mathrm{mv}}$.
• Figure 3: Convergence from different staring points for the MM (a), CC (b), and GA (c) methods.
• Figure 4: Comparison of the sum of squared portfolio weights across methods.
• Figure 5: Comparison of wall-time across methods for the multi-asset example, started from (a) equal-weight portfolio and (b) the best MV portfolio.