Quantile-Based Risk Sharing

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 risk sharing problem is solved through explicit construction. Three relevant issues on optimal allocation are investigated: extra sources of randomness, comonotonicty, and model uncertainty. We 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. In particular, in the context of the calculation of regulatory capital, we provide some general guidelines on how a regulatory risk measure can lead to certain desirable or undesirable properties of risk sharing among firms. Several novel advantages of ES over VaR from the perspective of a regulator are thereby revealed.

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