Adjusted Expected Shortfall

Abstract

We introduce and study the main properties of a class of convex risk measures that refine Expected Shortfall by simultaneously controlling the expected losses associated with different portions of the tail distribution. The corresponding adjusted Expected Shortfalls quantify risk as the minimum amount of capital that has to be raised and injected into a financial position X to ensure that Expected Shortfall ES p (X) does not exceed a pre-specified threshold g(p) for every probability level p ∈ [0, 1]. Through the choice of the benchmark risk profile g one can tailor the risk assessment to the specific application of interest. We devote special attention to the study of risk profiles defined by the Expected Shortfall of a benchmark random loss, in which case our risk measures are intimately linked to second-order stochastic dominance.

Law-invariant functionals that collapse to the mean: Beyond convexity

“…For X, Y ∈ L 0 we say that X and Y are comonotone if for all x, y ∈ R, P(X ≤ x, Y ≤ y) = min{P(X ≤ x), P(Y ≤ y)}. 5 Similarly, we say that X and Y are antimonotone if for all x, y ∈ R, P(X ≤ x, Y ≤ y) = max{P(X ≤ x) + P(Y ≤ y) − 1, 0}. 6 In the proof of the sharp version of the Fréchet-Hoeffding bounds and in the sequel, we will repeatedly use the fact that, by nonatomicity, for all X, Y ∈ L 0 we can always find X ′ ∼ X and Y ′ ∼ Y such that X ′ and Y ′ are comonotone.…”

“…The following representation result from [25] will play a crucial role in our later analysis. In the terminology of [5], it shows that any consistent risk measure on L ∞ can be expressed as a minimum of adjusted Expected Shortfalls.…”