Selected problems involving the probability of ruin for an insurance company

In the actuarial science literature, an insurance company is said to be ruined if, at some time t \u3e 0, the aggregate claims up to time t exceed the sum of the initial surplus and the total premium collected up to time t. The calculation and/or estimation of the probability of ruin is of fundamental importance. In this paper we use computer simulation to estimate the probabilities of ruin over a finite horizon of time when the aggregate claims process is Compound Poisson and the distribution of the claim sizes is: (i) Weibull, and (ii) exponential with a random parameter. The obtained estimates are compared with the case of exponential claim sizes and it is found that, in all but the increasing failure rate Weibull case, the probability of ruin, assuming exponential claims, overestimates the actual probability of ruin. Thus, one should be extremely careful in using the exponential claim formula uncritically