Optimal risk transfer under quantile-based risk measurers

•Find the optimal risk transfer among multiple insurance players under some quantile-based risk measure criteria.•Develop a new constructive methodology to solve the well-known problem of optimal reinsurance.•Find the set of all optimal solutions for the proposed optimisation problem.

The classical problem of identifying the optimal risk transfer from one insurance company to multiple reinsurance companies is examined under some quantile-based risk measure criteria. We develop a new methodology via a two-stage optimisation procedure which not only allows us to recover some existing results in the literature, but also makes possible the analysis of high-dimensional problems in which the insurance company diversifies its risk with multiple reinsurance counter-parties, where the insurer risk position and the premium charged by the reinsurers are functions of the underlying risk quantile. Closed-form solutions are elaborated for some particular settings, although numerical methods for the second part of our procedure represent viable alternatives for the ease of implementing it in more complex scenarios. Furthermore, we discuss some approaches to obtain more robust results.