Sensitivity Analysis for Ruin Probabilities: Canonical Risk Model

The surplus process of an insurance portfolio is defined as the wealth obtained by the premium payments minus the reimboursements made at the times of claims. When this process becomes negative (if ever), we say that ruin has occurred. The general setting is the Gambler’s Ruin Problem. In this paper we address the problem of estimating derivatives (sensitivities) of ruin probabilities with respect to the rate of accidents. Estimating probabilities of rare events is a challenging problem, since näıve estimation is not applicable. Solution approaches are very recent, mostly through the use of Importance Sampling techniques. Sensitivity estimation is an even harder problem for these situations. We shall study three methods for estimating ruin probabilities: one via importance sampling (IS), and two others via indirect simulation: the storage process (SP), which restates the problems in terms of a queueing system, and the convolution formula (CF). To estimate the sensitivities, we apply the RPA method to IS, the IPA method to SP and the Score Function method to CF. Simulation methods are compared in temrs of their efficiency, a criterion that appropriately weighs precision and CPU time. As well, we indicate how other criteria such as set-up time and prior formulæ development may actually be problem-dependent.