Optimal reinsurance and stop-loss order

During the last two decades, the interest of the actuarial literature in the stochastic orderings has been growing to such a point that they become one of the most important tools to compare the riskiness of different random situations. Our purpose in this note is to derive new results about the optimal reinsurance coverages for the ceding company, when the optimality criterion consists in minimizing the retained risk with respect to the stop-loss order. We so slightly complete the study initiated in Van Heerwaarden, Kaas and Goovaerts [Insurance: Mathematics and Economics 8 (1989) 11–17].