In the presence of reinsurance, an insurer may effectively reduce its (aggregated) loss by partially ceding such a loss to a reinsurer. Stop-loss and quota-share reinsurance contracts are commonly agreed between these two parties. In this paper, we aim to explore a combination of these contracts. The survival functions of the ceded loss and the retained loss are firstly investigated. Optimizing such a reinsurance design is then carried out from the joint perspective of the insurer and the reinsurer. Specifically, we explicitly derive optimal retentions under a criterion of minimizing a convex combination of conditional tail expectations of the insurer’s total loss and the reinsurer’s total loss. In addition, an estimation procedure and more explanations on numerical examples are also presented to find their estimated values.
Based on the settlement period for insurance claim, insurance is divided into 2 types of business which are short-tail business (settlement period <1 year) and long-tail business (settlement period ≥1 year). In long-tail business, it is important for insurance company to have claim reserve in order to settle claims in the future. Claim reserve modelling is done using chain ladder method that is based on the trend of paid claims. Another method that is often used is Bornhuetter-Ferguson which is based on paid claims and also premium. In this paper, Benktander method that combines chain ladder and Bornhuetter-Ferguson using optimal credibility is introduced. Optimal credibility is obtained through minimum mean squared error and minimum variance. Benktander provides moderate reserve compared to chain ladder and Bornhuetter-Ferguson.
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