A practical solution to the problem of ultimate ruin probability

Abstract

In a heuristic way, comparable to the method used by Bohman (1977) in the derivation of his rule of thumb, we find an approximation to the probability of ruin in an infinite time period, in the classical risk model. The solution is practical because our formula is very simple and because it uses only the three first moments of the distribution function of one claim. The risk reserve Y_t at the instant t depends, among other things, on the expected number λ. of claims in one year, the expected cost α of one claim, the security loading η. We replace the stochastic process Y_t (t⩾0) by a stochastic process Y^′ _t (t⩾0), also interpreted as a risk reserve, in such a way that (i) the distribution function of one claim cost is 1 − e^−x/α′ in the new process, (ii) the new parameters λ′, α′, η′ are fixed in such a way that fd_114_1 Then the probability of ultimate ruin in the initial process is approximated by the probability of ruin in the new process. We verified the quality of the approximation on almost all the cases treated numerically in the actuarial literature. It appeared to be surprisingly good for the practical values of the initial risk reserve u (not too small), except in one single case: less good results, but nevertheless accurate enough for all practical purposes, are obtained when the distribution function of one cost is lognormal. We think that several extensions and modifications of the method developed in this paper are possible. In (i), we took the exponential. distribution because then a closed formula exists for the probability of ruin. Other choices are possible. It is not excluded that the method can be extended to the case of a finite planning horizon. For simplicity, and because it is the most important case, we assumed the claim number process to be a Poisson process. Here also extensions seem possible.