Seven Proofs for the Subadditivity of Expected Shortfall

Subadditivity is the key property which distinguishes the popular risk measures Value-at-Risk and Expected Shortfall (ES). In this paper we offer seven proofs of the subadditivity of ES, some found in the literature and some not. One of the main objectives of this paper is to provide a general guideline for instructors to teach the subadditivity of ES in a course. We discuss the merits and suggest appropriate contexts for each proof. With different proofs, different important properties of ES are revealed, such as its dual representation, optimization properties, continuity, consistency with convex order, and natural estimators.