Are law-invariant risk functions concave on distributions?
While it is reasonable to assume that convex combinations on the level of random variables lead to a reduction of risk (diversification effect), this is no more true on the level of distributions. In the latter case, taking convex combinations corresponds to adding a risk factor. Hence, whereas asking for convexity of risk functions defined on random variables makes sense, convexity is not a good property to require on risk functions defined on distributions. In this paper we study the interplay...