January 2001, Volume 52, Issue 1, Pages 71 - 81

Theoretical Paper

Sensitivity analysis for ruin probabilities: canonical risk model FJ Vázquez-Abad & P LeQuoc University of Montreal, Canada     Correspondence to: FJ Vacutéquez-Abad, Dr FJ Vázquez-Abad, Department of Computer Science and Operations Research University of Montreal, Canada. E-mail: vazquez@IRO.UMontreal.ca

Keywords

risk;   ruin probabilities;   sensitivity analysis;   rare event simulation;   importance sampling;   estimation efficiency

Abstract

The surplus process of an insurance portfolio is defined as the wealth obtained by the premium payments minus the reimbursements made at the time 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 naïve 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 queuing system, and the convolution formula (CF). To estimate the sensitivities, we apply the Rare Perturbation Analysis (RPA) method to IS, the Infinitesimal Perturbation Analysis (IPA) method to SP and the score function method to CF. Simulation methods are compared in terms 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 formulae development may actually be problem-dependent.

Received November 1998; Accepted June 2000

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