Paper: arXiv 2405.12154

Authors: Martin Herdegen, Nazem Khan, Cosimo Munari

Abstract

Risk and utility functionals are fundamental building blocks in economics and finance. In this paper we investigate under which conditions a risk or utility functional is sensitive to the accumulation of losses in the sense that any sufficiently large multiple of a position that exposes an agent to future losses has positive risk or negative utility. We call this property sensitivity to large losses and provide necessary and sufficient conditions thereof that are easy to check for a very large class of risk and utility functionals. In particular, our results do not rely on convexity and can therefore also be applied to most examples discussed in the recent literature, including (non-convex) star-shaped risk measures or S-shaped utility functions encountered in prospect theory. As expected, Value at Risk generally fails to be sensitive to large losses. More surprisingly, this is also true of Expected Shortfall. By contrast, expected utility functionals as well as (optimized) certainty equivalents are proved to be sensitive to large losses for many standard choices of concave and nonconcave utility functions, including $S$-shaped utility functions. We also show that Value at Risk and Expected Shortfall become sensitive to large losses if they are either properly adjusted or if the property is suitably localized.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a rigorous theoretical exploration of risk and utility functionals, establishing mathematical conditions for sensitivity to large losses. While it demonstrates strong mathematical complexity and clarity, it is entirely theoretical with no empirical backtesting or real-world data validation.

Research Flowchart

  flowchart TD
    A["Research Question: Under what conditions are risk/utility functionals sensitive to large loss accumulation?"] --> B["Mathematical Formulation: Define 'Sensitivity to Large Losses' for general non-convex functionals"]
    B --> C["Analyze Risk & Utility Classes: VaR, Expected Shortfall, Expected Utility, Certainty Equivalents, S-Shaped Utility"]
    C --> D["Derive Necessary & Sufficient Conditions: Evaluate large multiples of loss-exposing positions"]
    D --> E["Key Finding 1: VaR and Expected Shortfall fail sensitivity to large losses"]
    D --> F["Key Finding 2: Expected Utility and Certainty Equivalents pass (including S-shaped functions)"]
    D --> G["Key Finding 3: VaR & Expected Shortfall regain sensitivity via proper adjustments or localization"]