Risk, utility and sensitivity to large losses

TL;DR

This paper characterizes when risk and utility functionals are sensitive to large losses, providing conditions applicable to a broad class of models, including non-convex and non-standard utility functions, with implications for risk measurement.

Contribution

It introduces the concept of sensitivity to large losses and offers necessary and sufficient conditions applicable to various risk and utility functionals, including non-convex cases.

Findings

Value at Risk is generally not sensitive to large losses.

Expected Shortfall also lacks sensitivity to large losses.

Expected utility functionals are often sensitive to large losses, especially with standard utility functions.

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.