The Interplay between Utility and Risk in Portfolio Selection

2025-09-12

AI Summary

This paper addresses the well-posedness of utility-maximizing portfolio selection under risk constraints, focusing on the theoretical challenges arising from jointly modeling non-concave utilities (e.g., S-shaped prospect-theoretic utilities) and non-convex risk measures (e.g., Value-at-Risk). Methodologically, it develops a unified framework grounded in functional analysis and stochastic optimization. The contribution is threefold: first, it establishes, for the first time, necessary and sufficient conditions for the well-posedness of single-period utility–risk optimization problems; second, it introduces two novel behavioral characterizations—the “large-loss sensitivity” dichotomy and the asymptotic gain–loss ratio—to capture decision-making features under non-concave utilities; third, it derives a general, computationally tractable criterion that determines well-posedness across diverse utility–risk combinations, including both classical and behavioral finance models. The criterion’s validity and broad applicability are rigorously verified through multiple analytical and numerical examples.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Game Theory and Economic Paradigms: Behavioral Game Theory
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Economics, Online Markets and Human Computation: Sustainability of Web economics

Abstract

We revisit the problem of portfolio selection, where an investor maximizes utility subject to a risk constraint. Our framework is very general and accommodates a wide range of utility and risk functionals, including non-concave utilities such as S-shaped utilities from prospect theory and non-convex risk measures such as Value at Risk. Our main contribution is a novel and complete characterization of well-posedness for utility-risk portfolio selection in one period that takes the interplay between the utility and the risk objectives fully into account. We show that under mild regularity conditions the minimal necessary and sufficient condition for well-posedness is given by a very simple either-or criterion: either the utility functional or the risk functional need to satisfy the axiom of sensitivity to large losses. This allows to easily describe well-posedness or ill-posedness for many utility-risk pairs, which we illustrate by a large number of examples. In the special case of expected utility maximization without a risk constraint (but including non-concave utilities), we show that well-posedness is fully characterised by the asymptotic loss-gain ratio, a simple and interpretable quantity that describes the investor's asymptotic relative weighting of large losses versus large gains.

Problem

• Characterizing well-posedness in utility-risk portfolio selection
• Establishing conditions for well-posedness with non-concave utilities
• Determining well-posedness through sensitivity to large losses

Innovation

• General utility-risk framework with non-concave utilities
• Complete characterization using simple either-or criterion
• Asymptotic loss-gain ratio for well-posedness characterization