AND

The actual discussions of appropriate risk measures to be used for the calculation of capital requirements in the Solvency II process have concentrated mainly on Value-at-Risk (VaR) and Expected Shortfall (ES). However, only recently the possible influence of dependence structures between the various types of risk or lines of business on such risk measures has drawn more attention [see e.g. WÜTHRICH (2003) or EMBRECHTS, HÖING AND PUCCETTI (2005) for a detailed discussion in connection with VaR]. The purpose of this paper is to investigate in more detail how the total risk distribution depends on different underlying dependence structures (copulas) while keeping the marginal distributions fixed, and how at least approximately such distributions can be calculated explicitly. We give several examples of uncorrelated (but dependent) risks with the same marginals, which show a completely different behavior for the aggregated risk distribution, in particular for the corresponding VaR and ES. Further, the influence of co- and counter-monotonicity of the marginal risks is shown to be totally different in the cases where the expectation of the individual risks is finite or infinite. These observations make it clear that the concept of correlation which is widely used e.g. in geophysical modeling and other professional DFA tools is not an appropriate dependence measure when risk aggregation or reinsurance of combined risks is considered