The Interplay between Utility and Risk in Portfolio Selection

ArXiv ID: 2509.10351 “View on arXiv”

Authors: Leonardo Baggiani, Martin Herdegen, Nazem Khan

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.

Keywords: Portfolio Selection, Prospect Theory, Value at Risk (VaR), Utility Maximization, Risk Management

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is mathematically dense, featuring advanced functional analysis and measure-theoretic proofs to characterize well-posedness for general utility-risk pairs, with heavy use of notation and theoretical derivations. However, it is purely theoretical with no empirical data, backtests, or implementation details, focusing solely on abstract existence conditions rather than practical trading strategies.

  flowchart TD
    A["Research Goal:<br>Characterize Well-Posedness of<br>Utility-Risk Portfolio Selection"] --> B["Key Methodology:<br>General Framework with<br>Non-Concave Utilities &<br>Non-Convex Risk Measures"]
    B --> C{"Data/Inputs:<br>Mild Regularity Conditions<br>on Utilities & Risks"}
    C --> D["Computation:<br>Analyze Interplay between<br>Utility and Risk Objectives"]
    D --> E{"Key Finding 1:<br>Sensitivity to Large Losses<br>Is Necessary & Sufficient?"}
    E -- No --> F["Outcome:<br>Ill-Posed Problem"]
    E -- Yes --> G["Key Finding 2:<br>Expected Utility Only:<br>Asymptotic Loss-Gain Ratio"]
    E -- Yes --> H["Outcome:<br>Well-Posed Problem"]