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Optimal retention levels, given the joint survival of cedent and reinsurer. (English) Zbl 1143.91029
In the present paper the optimal reinsurance is considered from the point of view of both the interests of the primary insurer (cedent) and reinsurer, as two parties jointly liable for the risk they share. For this purpose two alternative joint risk measures and two optimality criteria for setting the excess of loss retention level are introduced. One such measure is the probability that both cedent and reinsurer survive up to finite time horizon. Then the retention level is said to be optimal if it maximizes the probability of joint survival. The alternative measure is the absolute value of the difference between the probability of survival to a finite moment of the cedent and probability of survival of the reinsurer, given the survival of the cedent up to that moment. The optimal retention level is the value which minimizes this difference. To calculate the optimal retention levels the authors derive explicit expressions for these two criteria, assuming that the originally occurring claims to cedent (claims before reinsurance) have a discrete joint distribution, their arrivals form a Poisson point process and the premium income function is a nonnegative increasing real function. Necessary calculations can be carried out using the corresponding Mathematica modules, developed for this purpose. The quota share contracts are also considered under the same model. It is shown that the probability of joint survival of the cedent and reinsurer coincides with the probability of survival of solely the insurer. A number of examples and numerical comparisons, illustrating the performance of the proposed reinsurance optimality criteria, are presented and discussed.

MSC:
 91B30 Risk theory, insurance (MSC2010) 62P05 Applications of statistics to actuarial sciences and financial mathematics
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