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Ruin theory in insurance and the comparison of methods for approximation of the ruin probability in infinite time horizon
Abstract
The article presents theoretical background of ruin theory and the description of the classical model for the insurer’s surplus. Analytical calculation of the ruin probability are presented for special cases of the single loss distribution (exponential, gamma and mixture of exponential distributions). The main focus is on the analysis of available methods for approximation of the ruin probability in infinite horizon in continuous-time model. The quality of approximation is tested by comparing the approximated ruin probability with the probability determined analytically (if possible) or estimated numerically with use of Pollaczek-Khinchin formula. The approximation errors (in both absolute and relative terms) are shown for selected light-tailed distributions (mixture of exponential distributions, gamma) and heavy-tailed distributions (Pareto, lognormal, Weibull and Burr). The goal of the article is the assessment of the possibilities to use the approximation methods for ruin probability by insurance companies, including areas such as pricing or solvency, especially in the context of Solvency II regime. The analyses performed show that in most cases approximation results are quite satisfying (relative error not exceeding 5%) and the lowest errors are observed for Cramer-Lundberg and de Vylder approximations in case of light-tailed distributions and for Beekman-Bowers and de Vylder approximations in case of heavy-tailed distributions. The approximation quality in general deteriorates in line with the decreasing assumed ruin probability, especially for heavy-tailed distributions.
Keywords:
ruin theory, surplus process, ruin probability, approximation, light-tailed distributions, heavy-tailed distributionsJEL Codes
C13, C46, G22Details
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