Multivariate extremes and the aggregation of dependent risks: examples and counter-examples (Q626283)

The authors present several results on properties of risk measures for extreme risks for arbitrary dimensions \(n\geq2\) and dependence structures. The multivariate extreme value theory (MEVT) approach is used which provides a natural framework to discuss diversification of a portfolio under the risk measure Value-at-Risk (VaR). It is shown that the interplay between existence, non-existence of a finite moment, one- or two-sidedness, and symmetry versus asymmetry of the underlying risk distribution functions have to be carefully balanced in order to be able to conclude sub- or super-additivity of quantile based risk measures like VaR. Numerous examples and counter-examples highlight the applicability of the main results.