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Quantile-Based Risk Sharing

Paul Embrechts¹; Haiyan Liu²; Ruodu Wang³

¹ RiskLab, Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland; and Swiss Finance Institute, 8006 Zurich, Switzerland · ² Department of Mathematics and Department of Statistics and Probability, Michigan State University, East Lansing, Michigan 48824 · ³ Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

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. Furthermore, we investigate the problem of model uncertainty in risk sharing and show that, in general, 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.The e-companion is available at https://doi.org/10.1287/opre.2017.1716 .

DOI

10.1287/opre.2017.1716

Volume

66 (4)

Pages

936-949

Language

en

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Sources

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