Subadditivity is the key property which distinguishes the popular risk measures Value-at-Risk andExpected Shortfall (ES). In this paper we offer seven proofs of the subadditivity of ES, some found in the literatureand some not. One of the main objectives of this paper is to provide a general guideline for instructorsto teach the subadditivity of ES in a course. We discuss the merits and suggest appropriate contexts for eachproof.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.
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