An Axiomatic Foundation for the Expected Shortfall

Management Science, Forthcoming

38 Pages Posted: 22 Jul 2019 Last revised: 17 Mar 2020

See all articles by Ruodu Wang

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science

Ricardas Zitikis

Western University

Date Written: July 19, 2019


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 - which are monotonicity, law invariance, prudence and no reward for concentration - that uniquely characterize the family of ES. The herein developed results, therefore, 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.

Keywords: risk measure, Expected Shortfall, risk concentration, diversification, risk aggregation

JEL Classification: C01, C02, C52, C58, C61, D81, E58, G11, G18, G22, G28

Suggested Citation

Wang, Ruodu and Zitikis, Ricardas, An Axiomatic Foundation for the Expected Shortfall (July 19, 2019). Management Science, Forthcoming, Available at SSRN: or

Ruodu Wang (Contact Author)

University of Waterloo - Department of Statistics and Actuarial Science ( email )

Waterloo, Ontario N2L 3G1

Ricardas Zitikis

Western University ( email )

1151 Richmond Street
Suite 2
London, Ontario N6A 5B8

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