  It is well known that Expected Shortfall (also called Average Value-at-Risk)
is a convex risk measure, i. e. Expected Shortfall of a convex linear
combination of arbitrary risk positions is not greater than a convex linear
combination with the same weights of Expected Shortfalls of the same risk
positions. In this short paper we prove that Expected Shortfall is a concave
risk measure with respect to probability distributions, i. e. Expected
Shortfall of a finite mixture of arbitrary risk positions is not lower than the
linear combination of Expected Shortfalls of the same risk positions (with the
same weights as in the mixture).
