Extreme VaR scenarios in higher dimensions

For a sequence of random variables X ₁, ..., X _n , the dependence scenario yielding the worst possible Value-at-Risk at a given level α for X ₁+...+X _n is known for n=2. In this paper we investigate this problem for higher dimensions. We provide a geometric interpretation highlighting the dependence structures which imply the worst possible scenario. For a portfolio (X ₁,..., X _n ) with given uniform marginals, we give an analytical solution sustaining the main result of Rüschendorf (Adv. Appl. Probab. 14(3):623–632, 1982). In general, our approach allows for numerical computations.