Optimal Retention for a Stop-Loss Reinsurance under the VaR and CTE Risk Measures

We propose practical solutions for the determination of optimal retentions in a stop-loss reinsurance. We develop two new optimization criteria for deriving the optimal retentions by, respectively, minimizing the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risks of an insurer. We establish necessary and sufficient conditions for the existence of the optimal retentions for two risk models: individual risk model and collective risk model. The resulting optimal solution of our optimization criterion has several important characteristics: (i) the optimal retention has a very simple analytic form; (ii) the optimal retention depends only on the assumed loss distribution and the reinsurer’s safety loading factor; (iii) the CTE criterion is more applicable than the VaR criterion in the sense that the optimal conditions for the CTE criterion is less restrictive than the corresponding VaR criterion; (iv) if optimal solutions exist, then both VaRand CTE-based optimization criteria yield the same optimal retentions; (v) there exists a threshold risk tolerance level beyond which the insurer optimally should not reinsure her risk. As applications, we extend the results to the individual risk models with dependent risks and use multivariate phase type distributions and multivariate Pareto distributions to illustrate the effect of dependence on the optimal retentions. We also use the compound Poisson distribution and the compound negative binomial distribution to illustrate the optimal retentions in a collective risk model.