A Theory for Measures of Tail Risk

Mathematics of Operations Research, forthcoming

32 Pages Posted: 22 Sep 2016 Last revised: 31 Jul 2020

See all articles by Fangda Liu

Fangda Liu

Central University of Finance and Economics (CUFE) - China Institute for Actuarial Science

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science

Date Written: October 28, 2019

Abstract

The notion of "tail risk" has been a crucial consideration in modern risk management. To achieve a comprehensive understanding of the tail risk, we carry out an axiomatic study for risk measures which quantify the tail risk, that is, the behavior of a risk beyond a certain quantile. Such risk measures are referred to as tail risk measures in this paper. The two popular classes of regulatory risk measures in banking and insurance, the Value-at-Risk (VaR) and the Expected Shortfall (ES), are prominent, yet elementary, examples of tail risk measures. We establish a connection between a tail risk measure and a corresponding law-invariant risk measure, called its generator, and investigate their joint properties. A tail risk measure inherits many properties from its generator, but not subadditivity or convexity; nevertheless, a tail risk measure is coherent if and only if its generator is coherent. We explore further relevant issues on tail risk measures, such as bounds, distortion risk measures, risk aggregation, elicitability, and dual representations. In particular, there is no elicitable tail convex risk measure rather than the essential supremum, and under a continuity condition, the only elicitable and positively homogeneous monetary tail risk measures are the VaRs.

Keywords: Basel III, tail risk, risk aggregation, elicitability, Value-at-Risk

Suggested Citation

Liu, Fangda and Wang, Ruodu, A Theory for Measures of Tail Risk (October 28, 2019). Mathematics of Operations Research, forthcoming, Available at SSRN: https://ssrn.com/abstract=2841909 or http://dx.doi.org/10.2139/ssrn.2841909

Fangda Liu

Central University of Finance and Economics (CUFE) - China Institute for Actuarial Science ( email )

39 South College Road, Haidian
Beijing, 100081
China

Ruodu Wang (Contact Author)

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

Waterloo, Ontario N2L 3G1
Canada

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