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Optimal Reciprocal Reinsurance Treaties Under the Joint Survival Probability and the Joint Profitable Probability

Authors

  • Jun Cai,

    1. Jun Cai is at the Department of Statistics and Actuarial Science, University of Waterloo, Canada, and China Institute for Actuarial Science, Central University of Finance and Economics, China. Ying Fang is at the School of Mathematical Sciences, Shandong Normal University, China. Zhi Li is at the Transamerica Reinsurance, the United States. Gordon E. Willmot is at the Department of Statistics and Actuarial Science, University of Waterloo, Canada. The authors can be contacted via e-mail: jcai@uwaterloo.ca, fangying2006nk@yahoo.com.cn, zhi.li@transamerica.com, and gewillmo@uwaterloo.ca, respectively. The authors are grateful to the three anonymous referees and the editor for their careful reading and thoughtful comments. The first and the fourth authors are both thankful for the support from the Natural Sciences and Engineering Research Council of Canada. The fourth author is thankful for support from the Munich Reinsurance Company. The second author gratefully acknowledges the support from the National Natural Science Foundation of China (No. 11126093 and No. 71071088), the National Basic Research Program of China (973 Program, No. 2007CB814905), and the Research Fund for the Doctorial Program of Higher Education of China. The third author thanks the University of Waterloo for support. The original version of the article was presented in 2010 Second World Risk and Insurance Economics Congress at Singapore in July 2010. The authors are also thankful for the comments and suggestions of the participants in the congress on the original version of this article.
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  • Ying Fang,

    1. Jun Cai is at the Department of Statistics and Actuarial Science, University of Waterloo, Canada, and China Institute for Actuarial Science, Central University of Finance and Economics, China. Ying Fang is at the School of Mathematical Sciences, Shandong Normal University, China. Zhi Li is at the Transamerica Reinsurance, the United States. Gordon E. Willmot is at the Department of Statistics and Actuarial Science, University of Waterloo, Canada. The authors can be contacted via e-mail: jcai@uwaterloo.ca, fangying2006nk@yahoo.com.cn, zhi.li@transamerica.com, and gewillmo@uwaterloo.ca, respectively. The authors are grateful to the three anonymous referees and the editor for their careful reading and thoughtful comments. The first and the fourth authors are both thankful for the support from the Natural Sciences and Engineering Research Council of Canada. The fourth author is thankful for support from the Munich Reinsurance Company. The second author gratefully acknowledges the support from the National Natural Science Foundation of China (No. 11126093 and No. 71071088), the National Basic Research Program of China (973 Program, No. 2007CB814905), and the Research Fund for the Doctorial Program of Higher Education of China. The third author thanks the University of Waterloo for support. The original version of the article was presented in 2010 Second World Risk and Insurance Economics Congress at Singapore in July 2010. The authors are also thankful for the comments and suggestions of the participants in the congress on the original version of this article.
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  • Zhi Li,

    1. Jun Cai is at the Department of Statistics and Actuarial Science, University of Waterloo, Canada, and China Institute for Actuarial Science, Central University of Finance and Economics, China. Ying Fang is at the School of Mathematical Sciences, Shandong Normal University, China. Zhi Li is at the Transamerica Reinsurance, the United States. Gordon E. Willmot is at the Department of Statistics and Actuarial Science, University of Waterloo, Canada. The authors can be contacted via e-mail: jcai@uwaterloo.ca, fangying2006nk@yahoo.com.cn, zhi.li@transamerica.com, and gewillmo@uwaterloo.ca, respectively. The authors are grateful to the three anonymous referees and the editor for their careful reading and thoughtful comments. The first and the fourth authors are both thankful for the support from the Natural Sciences and Engineering Research Council of Canada. The fourth author is thankful for support from the Munich Reinsurance Company. The second author gratefully acknowledges the support from the National Natural Science Foundation of China (No. 11126093 and No. 71071088), the National Basic Research Program of China (973 Program, No. 2007CB814905), and the Research Fund for the Doctorial Program of Higher Education of China. The third author thanks the University of Waterloo for support. The original version of the article was presented in 2010 Second World Risk and Insurance Economics Congress at Singapore in July 2010. The authors are also thankful for the comments and suggestions of the participants in the congress on the original version of this article.
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  • Gordon E. Willmot

    1. Jun Cai is at the Department of Statistics and Actuarial Science, University of Waterloo, Canada, and China Institute for Actuarial Science, Central University of Finance and Economics, China. Ying Fang is at the School of Mathematical Sciences, Shandong Normal University, China. Zhi Li is at the Transamerica Reinsurance, the United States. Gordon E. Willmot is at the Department of Statistics and Actuarial Science, University of Waterloo, Canada. The authors can be contacted via e-mail: jcai@uwaterloo.ca, fangying2006nk@yahoo.com.cn, zhi.li@transamerica.com, and gewillmo@uwaterloo.ca, respectively. The authors are grateful to the three anonymous referees and the editor for their careful reading and thoughtful comments. The first and the fourth authors are both thankful for the support from the Natural Sciences and Engineering Research Council of Canada. The fourth author is thankful for support from the Munich Reinsurance Company. The second author gratefully acknowledges the support from the National Natural Science Foundation of China (No. 11126093 and No. 71071088), the National Basic Research Program of China (973 Program, No. 2007CB814905), and the Research Fund for the Doctorial Program of Higher Education of China. The third author thanks the University of Waterloo for support. The original version of the article was presented in 2010 Second World Risk and Insurance Economics Congress at Singapore in July 2010. The authors are also thankful for the comments and suggestions of the participants in the congress on the original version of this article.
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Abstract

A reinsurance treaty involves two parties, an insurer and a reinsurer. The two parties have conflicting interests. Most existing optimal reinsurance treaties only consider the interest of one party. In this article, we consider the interests of both insurers and reinsurers and study the joint survival and profitable probabilities of insurers and reinsurers. We design the optimal reinsurance contracts that maximize the joint survival probability and the joint profitable probability. We first establish sufficient and necessary conditions for the existence of the optimal reinsurance retentions for the quota-share reinsurance and the stop-loss reinsurance under expected value reinsurance premium principle. We then derive sufficient conditions for the existence of the optimal reinsurance treaties in a wide class of reinsurance policies and under a general reinsurance premium principle. These conditions enable one to design optimal reinsurance contracts in different forms and under different premium principles. As applications, we design an optimal reinsurance contract in the form of a quota-share reinsurance under the variance principle and an optimal reinsurance treaty in the form of a limited stop-loss reinsurance under the expected value principle.

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