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Quantile-Based Risk Sharing

Forthcoming, Operations Research

40 Pages Posted: 10 Mar 2016 Last revised: 17 Nov 2017

Paul Embrechts

Swiss Federal Institute of Technology Zurich; Swiss Finance Institute

Haiyan Liu

Michigan State University - Department of Mathematics

Ruodu Wang

University of Waterloo - Department of Statistics and Actuarial Science

Date Written: October 24, 2017

Abstract

We address the problem of risk sharing among agents using a two-parameter class of quantile-based risk measures, the so-called Range-Value-at-Risk (RVaR), as their preferences. The family of RVaR includes the Value-at-Risk (VaR) and the Expected Shortfall (ES), the two popular and competing regulatory risk measures, as special cases. We first establish an inequality for RVaR-based risk aggregation, showing that RVaR satisfies a special form of subadditivity. Then, the Pareto-optimal risk sharing problem is solved through explicit construction. To study risk sharing in a competitive market, an Arrow-Debreu equilibrium is established for some simple, yet natural settings. Further, we investigate the problem of model uncertainty in risk sharing, and show that, generally, a robust optimal allocation exists if and only if none of the underlying risk measures is a VaR. Practical implications of our main results for risk management and policy makers are discussed, and several novel advantages of ES over VaR from the perspective of a regulator are thereby revealed.

Keywords: Value-at-Risk, Expected Shortfall, risk sharing, regulatory capital, robustness

JEL Classification: G10, O16

Suggested Citation

Embrechts, Paul and Liu, Haiyan and Wang, Ruodu, Quantile-Based Risk Sharing (October 24, 2017). Forthcoming, Operations Research. Available at SSRN: https://ssrn.com/abstract=2744142

Paul Embrechts

Swiss Federal Institute of Technology Zurich ( email )

ETH-Zentrum
CH-8092 Zurich
Switzerland

Swiss Finance Institute

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4
Switzerland

Haiyan Liu

Michigan State University - Department of Mathematics ( email )

619 Red Cedar Road
East Lansing, MI 48824
United States

Ruodu Wang (Contact Author)

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

Waterloo, Ontario N2L 3G1
Canada

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