Sharp approximations of ruin probabilities in the discrete time models

Abstract

According to Solvency II directive, each insurance company could determine solvency capital requirements using its own, tailor made, internal model. This highlights the urgency of having fast numerical tools based on practically-oriented mathematical models. From the Solvency II perspective discrete time framework seems to be the most relevant one. In this paper, we propose a number of fast and accurate approximations of ruin probabilities involving some integral operator and examine them along strictly theoretical as well as numerical lines. For a few claim distributions the approximations are shown to be exact. In general, we prove that they converge with an exponential rate to the exact ruin probabilities without any restrictive assumptions on the claim distribution. A fast algorithm to approximate ruin probabilities by a numerical fixed point of the involved integral operator is given. As an application, ruin probabilities for, e.g. normally and Weibull – distributed claims are computed. Comparisons with discrete time counterparts of some continuous time approximation methods are also carried out. Numerical studies show that our approximations are precise both for small and large values of the initial surplus u. In contrast, the empirical De Vylder-type ones strongly depend on the claim distributions and are less precise for small and medium values of u.

Keywords:

• Discrete time risk models
• Ruin probabilities
• Approximations
• Integral operator generated by a risk process
• The theory of fixed points
• Risk management based on internal models
• Solvency II

Acknowledgements

The authors would like to thank Lasse Koskinen for helpful remarks.

Lesław Gajek is the Vice-Chairperson of the Polish Financial Supervision Authority. This article has been performed in a private capacity and the opinions expressed in it should not be attributed to the PFSA.

Notes

¹i.e. the ruin probability considered as a function of the initial surplus u; see Section 2 for details

²comp. Grandell (2000)

³see Asmussen (2000, p. 80)

⁴Grandell (2000, p. 157)

⁵i.e. for some

⁶see e.g. Dickson (2005, pp. 79–81) for technical details