Core of the Reinsurance Market with Dependent Risks
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Core of the Reinsurance Market with Dependent Risks Jia-Hua Zhang1 · Shu-Cherng Fang2 · Yi-Fan Xu1
Received: 4 December 2016 / Revised: 5 June 2017 / Accepted: 30 June 2017 © Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany 2017
Abstract Baton and Lemaire (Astin Bull 12:57–71, 1981) proved the nonemptiness of the core of a reinsurance market in which the risks of companies are independent. However, cases involving dependent risks have received increasing concerns in modern actuarial science. In this paper, we investigate the nonemptiness of the core of a reinsurance market where the risks of different companies may be dependent. When the exponential utility function is employed, we find an important property on risk premium and show that the core of the market is always nonempty. Keywords Core · Risk premium · Reinsurance market · Exponential utility · Cooperative game Mathematics Subject Classification 91A12 · 91B30
This paper is dedicated to Professor Duan Li in celebration of his 65th Birthday. Jia-Hua Zhang’s research is supported by Fudan University Student Growth Fund Scholarship. Shu-Cherng Fang’s research is supported by US ARO Grant (No. W911NF-15-1-0223). Yi-Fan Xu’s research is supported by the National Natural Science Foundation of China (Nos. 71372113, 71531005) and Join Funding of Fudan University & Taiwan University.
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Yi-Fan Xu [email protected] Jia-Hua Zhang [email protected] Shu-Cherng Fang [email protected]
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School of Management, Fudan University, Shanghai 200433, China
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Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC 27695-7906, USA
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J.-H. Zhang et al.
1 Introduction Traditional insurance theory always assumes independent risks. With the increasing complexity of insurance and reinsurance markets, modern insurance theory raises more concerns on dependent risks [1]. The risks of different companies may actually correlate in many situations. For example, in a car accident, claims of the automobile insurance company and the medical insurance company are clearly dependent [2]. Cooperative game theory provides a natural tool for modeling the reinsurance market [3–5]. To the best of our knowledge, Baton and Lemaire [3] first introduced the collective rationality to a reinsurance market and provided a way to understand reinsurance problem from the viewpoint of cooperative game theory. They found that the core of the market is nonempty when the utility function of each company is of exponential form. This result is important in risk exchange treaties and has often been cited by follow-up researches [6–8]. However, this fundamental result is based on the assumption of independent risks. In the paper, we study a reinsurance market with a finite set of companies, in which the risks involved by different companies may be dependent. Each company faces a risky situation characterized by a random claim amount. Different from Baton and Lemaire [3], these cl
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