Ruin probabilities under capital constraints

In this paper, we generalise the classic compound Poisson risk model, by the introduction of ordered capital levels, to model the solvency of an insurance firm. A breach of the higher capital level, the magnitude of which does not cause further breaches of either the lower level or the so-called intermediate confidence level (of the shareholders), requires a capital injection to restore the surplus to a solvent position. On the other hand, if the confidence level is breached capital injections are no longer a viable method of recapitalisation. Instead, the company can borrow money from a third party, subject to a constant interest rate, which is paid back until the surplus returns to the confidence level and subsequently can be restored to a fully solvent position by a capital injection. If at any point the surplus breaches the lower capital level, the company is considered ‘insolvent’ and is forced to cease trading. For the aforementioned risk model, we derive an explicit expression for the ‘probability of insolvency’ in terms of the ruin quantities of the classical risk model. Under the assumption of exponentially distributed claim sizes, we show that the probability of insolvency is in fact directly proportional to the classical ruin function. It is shown that this result also holds for the asymptotic behaviour of the insolvency probability, with a general claim size distribution. Explicit expressions are also derived for the moment generating function of the accumulated capital injections up to the time of insolvency and finally, in order to better capture the reality, dividend payments to the companies shareholders are considered, along with the capital constraint levels, and explicit expressions for the probability of insolvency, under this modification, are obtained.