TABLE OF CONTENTS In this chapter we focus again on collective risk models, but now in the long term. We consider the development in time of the capital U( t) of an insurer. This is a stochastic process which increases continuously because of the earned premiums, and decreases stepwise because of the payment of claims. When the capital becomes negative, we say that ruin occurs. Let ?( u) denote the probability that this ever happens, provided that the annual premium and the claims process remain unchanged. This probability is a useful tool for the management since it serves as an indication of the soundness of the insurer's combined premiums and claims process, given the available initial capital u = U(0). A high probability of ruin indicates instability: measures such as reinsurance or raising some premiums should be considered, or the insurer should attract extra working capital.
The probability of ruin enables one to compare portfolios with each other, but we cannot attach any absolute meaning to the probability of ruin, as it doesn't actually represent the probability that the insurer will go bankrupt in the near future. First of all, it might take centuries for ruin to actually happen. Moreover, potential interventions in the process, for instance paying out dividends or raising the premium for risks with an unfavorable claims performance, are ruled out in the determination of the probability of ruin. Furthermore, the effects of inflation on the one hand and the return on...
