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 Expected Shortfall becomes sensitive to large losses if it is either properly adjusted or if the property is suitably localized. The talk is based on joint work with Nazem Khan (Dublin City University) and Cosimo Munari (University of Verona).

Speaker

Martin Herdegen (University of Warwick)