On Fair Reinsurance Premiums; Capital Injections in a Perturbed Risk Model
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We consider a risk model in which deficits after ruin are covered by a new type of reinsurance contract that provides capital injections which depend on a chosen level of retention. To allow the insurance company's survival after ruin, the reinsurer would have to inject capital. Assuming that these capital injections are not from the shareholders, but rather obtained through such new reinsurance agreements, the problem here is to determine adequate reinsurance premiums. It seems fair to base the net reinsurance premium on the discounted expected value of any future capital injections. Inspired by the results of Huzak et al. (2004) and Ben Salah (2014) on successive ruin events, we show that an explicit formula for this reinsurance premium exists in a setting where aggregate claims are modeled by a subordinator and a Brownian perturbation. This result is illustrated explicitly for two specific risk models and some numerical examples.
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