  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.
