Worst case portfolio vectors and diversification effects

We consider the problem of identifying the worst case dependence structure of a portfolio X1, . . . ,Xn of d-dimensional risks, which yields the largest risk of the joint portfolio. Based on a recent characterization result of law invariant convex risk measures the worst case portfolio structure is identified as a μ-comonotone risk vector for some worst case scenario measure μ. It turns out that typically there will be a diversification effect even in worst case situations. The only exeptions arise when risks are measured by translated max correlation risk measures. We determine the worst case portfolio structure and the worst case diversification effect in several classes of examples as for example in elliptical, in Euclidean spherical and in Archimedian type distribution classes.