Optimal retention for a stop-loss reinsurance with incomplete information

This paper considers the determination of optimal retention in a stop-loss reinsurance. Assume that we only have incomplete information on a risk X for an insurer, we use an upper bound for the value at risk (VaR) of the total loss of an insurer after stop-loss reinsurance arrangement as a risk measure. The adopted method is a distribution-free approximation which allows to construct the extremal random variables with respect to the stochastic dominance order and the stop-loss order. We derive the optimal retention such that the risk measure used in this paper attains the minimum. We establish the sufficient and necessary conditions for the existence of the nontrivial optimal stop-loss reinsurance. For illustration purpose, some numerical examples are included and compared with the results yielded in Theorem 2.1 of Cai and Tan (2007).