It is known that the partial stop-loss contract is an optimal reinsurance form under the VaR risk measure. Assuming that market premiums are set according to the expected value principle with varying loading factors, the optimal reinsurance parameters of this contract are obtained under three alternative single and joint party reinsurance criteria: (i) strong minimum of the total retained loss VaR measure; (ii) weak minimum of the total retained loss VaR measure and maximum of the reinsurer's expected profit; (iii) weak minimum of the total retained loss VaR measure and minimum of the total variance risk measure. New conditions for financing in the mean simultaneously the cedent's and the reinsurer's required VaR economic capital are revealed for situations of pure risk transfer (classical reinsurance) or risk and profit transfer (design of internal reinsurance or reinsurance captive owned by the captive of a corporate firm).

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 condition for the former is less restrictive than the latter; (iv) if optimal solutions exist, then both VaR- and CTE-based optimization criteria yield the same optimal retentions. In terms of applications, we extend the results to the individual risk models with dependent risks and use multivariate phase type distribution, multivariate Pareto distribution and multivariate Bernoulli distribution to illustrate the effect of dependence on 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.