Risk measures and comonotonicity: a review

In this paper we examine and summarize properties of several well-known risk measures
that can be used in the framework of setting solvency capital requirements for a risky business.
Special attention is given to the class of (concave) distortion risk measures. We investigate
the relationship between these risk measures and theories of choice under risk. Furthermore we
consider the problem of how to evaluate risk measures for sums of non-independent random
variables. Approximations for such sums, based on the concept of comonotonicity, are proposed.
Several examples are provided to illustrate properties or to prove that certain properties do
not hold. Although the paper contains several new results, it is written as an overview and
pedagogical introduction to the subject of risk measurement. The paper is an extended version
of Dhaene et al.[11].
Keywords Comonotonicity; Distortion; Lognormal; Risk measurer; Theory of choice
under risk.
Mathematics Subject Classification 91B30.