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Under general multivariate regular variation conditions, the extremeValue-at-Risk of a portfolio can be expressed as an integral of a known kernelwith respect to a generally unknown spectral measure supported on the unitsimplex. The estimation of the spectral measure is challenging in practice andvirtually impossible in high dimensions. This motivates the problem studied inthis work, which is to find universal lower and upper bounds of the extremeValue-at-Risk under practically estimable constraints. That is, we study theinfimum and supremum of the extreme Value-at-Risk functional, over the infinitedimensional space of all possible spectral measures that meet a finite set ofconstraints. We focus on extremal coefficient constraints, which are popularand easy to interpret in practice. Our contributions are twofold. Firstly, weshow that optimization problems over an infinite dimensional space of spectralmeasures are in fact dual problems to linear semi-infinite programs (LSIPs) --linear optimization problems in an Euclidean space with an uncountable set oflinear constraints. This allows us to prove that the optimal solutions are infact attained by discrete spectral measures supported on finitely many atoms.Second, in the case of balanced portfolia, we establish further structuralresults for the lower bounds as well as closed form solutions for both thelower- and upper-bounds of extreme Value-at-Risk in the special case of asingle extremal coefficient constraint. The solutions unveil importantconnections to the Tawn-Molchanov max-stable models. The results areillustrated with a real data example.

Distributionally Robust Inference for Extreme Value-at-Risk