Abstract
It is well-known that reinsurance can be an effective risk management solution for financial institutions such as the insurance companies. The optimal reinsurance solution depends on a number of factors including the criterion of optimization and the premium principle adopted by the reinsurer. In this paper, we analyze the Value-at-Risk based optimal risk management solution using reinsurance under a class of premium principles that is monotonic and piecewise. The monotonic piecewise premium principles include not only those which preserve stop-loss ordering, but also the piecewise premium principles which are monotonic and constructed by concatenating a series of premium principles. By adopting the monotonic piecewise premium principle, our proposed optimal reinsurance model has a number of advantages. In particular, our model has the flexibility of allowing the reinsurer to use different risk loading factors for a given premium principle or use entirely different premium principles depending on the layers of risk. Our proposed model can also analyze the optimal reinsurance strategy in the context of multiple reinsurers that may use different premium principles (as attributed to the difference in risk attitude and/or imperfect information). Furthermore, by artfully imposing certain constraints on the ceded loss functions, the resulting model can be used to capture the reinsurer’s willingness and/or capacity to accept risk or to control counterparty risk from the perspective of the insurer. Under some technical assumptions, we derive explicitly the optimal form of the reinsurance strategies in all the above cases. In particular, we show that a truncated stop-loss reinsurance treaty or a limited stop-loss reinsurance treaty can be optimal depending on the constraint imposed on the retained and/or ceded loss functions. Some numerical examples are provided to further compare and contrast our proposed models to the existing models.
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Acknowledgments
Cong thanks the funding support from the Waterloo Research institute in Insurance, Securities and Quantitative finance (WatRISQ). Tan acknowledges research funding from the MOE Project of Key Research Institute of Humanities and Social Sciences at Universities(No. 11JJD790004), the Natural Sciences and Engineering Research Council of Canada and the Society of Actuaries Centers of Actuarial Excellence Research Grant.
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Cong, J., Tan, K.S. Optimal VaR-based risk management with reinsurance. Ann Oper Res 237, 177–202 (2016). https://doi.org/10.1007/s10479-014-1584-8
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DOI: https://doi.org/10.1007/s10479-014-1584-8