Some remarks on value-at-risk optimization

Abstract

We discuss two observations related to value-at-arisk optimization. First we consider a portfolio problem under an infinite number of value-at-risk inequality constraints (modelling first order stochastic dominance). The random data are assumed to be normally distributed. Although this problem is necessarily non-convex, an explicit solution can be derived. Secondly, we provide a (negative) result on quantitative stability of the value-at-risk under variation of the random variable. Although reduced Lipschitz properties (in the sense of calmness) may hold true at continuously distributed random variables under suitable conditions, the result shows that no full Lipschitz property (more generally: Hölder property at any rate) can hold in the neighbourhood of an arbitrary continuously distributed variable. Even worse, this observation holds true with respect to any probability metric weaker than that of total variation.Keywords: value at risk, stochastic dominance, Hölder continuity, quantile functions