An Axiomatic Foundation for the Expected Shortfall

TLDR

The Basel Accords have replaced value‑at‑risk with expected shortfall as the primary market‑risk metric, yet a formal axiomatic basis for ES remains absent. This study introduces four economic axioms—monotonicity, law invariance, prudence, and no reward for concentration—to uniquely define expected shortfall. The authors derive ES by exploiting new concepts such as tail events and risk concentration, demonstrating how these notions underpin the axiomatic characterization. The results establish ES as the first economically grounded, globally dominating regulatory risk measure that rewards diversification and penalizes concentration. The manuscript was accepted by Manel Baucells in the Decision Analysis journal.

Abstract

In the recent Basel Accords, the expected shortfall (ES) replaces the value-at-risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most popular risk measure in financial regulation. Although ES is—in addition to many other nice properties—a coherent risk measure, it does not yet have an axiomatic foundation. In this paper, we put forward four intuitive economic axioms for portfolio risk assessment—monotonicity, law invariance, prudence, and no reward for concentration—that uniquely characterize the family of ES. Therefore, the results developed herein provide the first economic foundation for using ES as a globally dominating regulatory risk measure, currently employed in Basel III/IV. Key to the main results, several novel notions such as tail events and risk concentration naturally arise, and we explore them in detail. As a most important feature, ES rewards portfolio diversification and penalizes risk concentration in a special and intuitive way, not shared by any other risk measure. This paper was accepted by Manel Baucells, decision analysis.