Serial risk sharing

Introduction

This work introduces a novel risk-sharing rule called serial risk sharing (SRS), inspired by the serial cost-sharing method of Moulin and Shenker (1992, Econometrica 60(5), 1009–1037). Using a serial mechanism, which involves a quantile-based decomposition of initial losses, and a uniform allocation of the resulting components to the relevant agents, we develop an equitable and intuitive risk allocation tailored for heterogeneous losses, serving as a computationally efficient alternative to other risk-sharing rules like conditional mean risk sharing (CMRS) due to the availability of a closed-form formula. In addition to numerical illustrations, the analytical properties of SRS are shown to consolidate its theoretical foundation.

Methods

The quantile-based decomposition of losses is constructed via a novel concept of quantile equivalent loss, defined via the probability levels of agents’ initial losses. By probability and quantile transformations, the distributional properties of the resulting components remain tractable, enabling theoretical analysis of the behavior of SRS. Particularly, when the losses are ordered by first-order stochastic dominance, substantial simplification of the rule is available, further facilitating the analysis of the rule and improving its transparency.

Results

Various theoretical properties of SRS are shown, such as actuarial fairness, full allocation, scale invariance, monotonicity, risk fairness, and convex-order improvement, under suitable assumptions. Additionally, through numerical studies, we demonstrate that SRS exhibits good properties and performs well relative to other risk-sharing rules. The numerical results also reinforce our theoretical findings.

Conclusion

By adapting the serial cost-sharing method put forward by Moulin and Shenker (1992) to the risk-sharing context, we design a novel risk-sharing rule known as serial risk sharing, where losses are decomposed into components based on quantiles. It carries a clear and interpretable structure, with favorable theoretical properties and supportive numerical results.