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Title: Optimal insurance in the presence of reinsurance
Authors: Zhuang, SC
Boonen, TJ
Tan, KS
Xu, ZQ 
Issue Date: 2017
Source: Scandinavian actuarial journal, 2017, v. , no. 6, p. 535-554
Abstract: This paper studies an optimal insurance and reinsurance design problem among three agents: policyholder, insurer, and reinsurer. We assume that the preferences of the parties are given by distortion risk measures, which are equivalent to dual utilities. By maximizing the dual utility of the insurer and jointly solving the optimal insurance and reinsurance contracts, it is found that a layering insurance is optimal, with every layer being borne by one of the three agents. We also show that reinsurance encourages more insurance, and is welfare improving for the economy. Furthermore, it is optimal for the insurer to charge the maximum acceptable insurance premium to the policyholder. This paper also considers three other variants of the optimal insurance/reinsurance models. The first two variants impose a limit on the reinsurance premium so as to prevent insurer to reinsure all its risk. An optimal solution is still layering insurance, though the insurer will have to retain higher risk. Finally, we study the effect of competition by permitting the policyholder to insure its risk with an insurer, a reinsurer, or both. The competition from the reinsurer dampens the price at which an insurer could charge to the policyholder, although the optimal indemnities remain the same as the baseline model. The reinsurer will however not trade with the policyholder in this optimal solution.
Keywords: Contract design
Distortion risk measures
Premium constraint
Competition constraint
Publisher: Taylor & Francis Scandinavia
Journal: Scandinavian actuarial journal 
ISSN: 0346-1238
EISSN: 1651-2030
DOI: 10.1080/03461238.2016.1184710
Rights: © 2016 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (, which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, and is not altered, transformed, or built upon in any way.
The following publication Zhuang, S. C., Boonen, T. J., Tan, K. S., & Xu, Z. Q. (2017). Optimal insurance in the presence of reinsurance. Scandinavian Actuarial Journal, 2017(6), 535-554 is available at
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