Risk measures and dependencies of risks

In the last few years the properties of risk measures that can be considered as suiting ”best practice” rules in insurance have been studied extensively in the actuarial literature. In Artzner (1999) so-called coherency axioms were proposed to be satisfied for risk measures that are used for providing capital requirements. On the other hand Goovaerts et al. (2003a), (2003b),(2003c) argue that the choice of appropriate set of axioms should depend on the axiomatic ”situation at hand”. In this contribution, we show that so-called concave distortion risk measures are not always consistent with some well-known dependency measures such as Pearson’s r, Spearman’s and Kendall’s  , i.e. higher dependency between random variables does not necessary lead to higher risk measure of corresponding sums. We also test numerically to what extend risk measures are consistent with certain dependency measures and how stable the consistency level is for different one-parametric families of distortion risk measures.