Optimal reinsurance design: a mean-variance approach. (English) Zbl 1414.91175

Summary: In this article, we study an optimal reinsurance model from the perspective of an insurer who has a general mean-variance preference. In order to reduce ex post moral hazard, we assume that both parties in a reinsurance contract are obligated to pay more for a larger realization of loss. We further assume that the reinsurance premium is calculated only based on the mean and variance of the indemnity. This class of premium principles is quite general in the sense that it includes many widely used premium principles such as expected value, mean value, variance, and standard deviation principles. Moreover, to protect the insurer’s profit, a lower bound is imposed on its expected return. We show that any admissible reinsurance policy is dominated by a change-loss reinsurance or a dual change-loss reinsurance, depending upon the coefficient of variation of the ceded loss. Further, the change-loss reinsurance is shown to be optimal if the premium loading increases in the actuarial value of the coverage; while it becomes decreasing, the optimal reinsurance policy is in the form of dual change loss. As a result, the quota-share reinsurance is always optimal for any variance-related reinsurance premium principle. Finally, some numerical examples are applied to illustrate the applicability of the theoretical results.


91B30 Risk theory, insurance (MSC2010)
Full Text: DOI


[1] Arrow, K. J., Uncertainty and the Welfare Economics of Medical Care, American Economic Review, 53, 5, 941-973, (1963)
[2] Balbás, A.; Balbás, B.; Heras, A., Optimal Reinsurance with General Risk Measures, Insurance: Mathematics and Economics, 44, 3, 374-384, (2009) · Zbl 1162.91394
[3] Bernard, C.; Tian, W., Optimal Reinsurance Arrangements under Tail Risk Measures, Journal of Risk and Insurance, 76, 3, 709-725, (2009)
[4] Bernard, C.; He, X.; Yan, J. A.; Zhou, X. Y., Optimal Insurance Design under Rank-Dependent Expected Utility, Mathematical Finance, 25, 1, 154-186, (2015) · Zbl 1314.91134
[5] Borch, K., An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance, Transactions of the 16th International Congress of Actuaries, 597-610, (1960)
[6] Bühlmann, H., Mathematical Methods in Risk Theory, (1970), New York: Springer, New York · Zbl 0209.23302
[7] Cai, J.; Tan, K. S., Optimal Retention for a Stop-Loss Reinsurance under the VaR and CTE Risk Measures, ASTIN Bulletin, 37, 1, 93-112, (2007) · Zbl 1162.91402
[8] Cai, J.; Tan, K. S.; Weng, C.; Zhang, Y., Optimal Reinsurance under VaR and CTE Risk Measures, Insurance: Mathematics and Economics, 43, 1, 185-196, (2008) · Zbl 1140.91417
[9] Carlier, G.; Dana, R.-A., Rearrangement Inequalities in Non-Convex Insurance Models, Journal of Mathematical Economics, 41, 4-5, 483-503, (2005) · Zbl 1106.91038
[10] Cheung, K. C., Optimal Reinsurance Revisited—A Geometric Approach, ASTIN Bulletin, 40, 1, 221-239, (2010) · Zbl 1230.91070
[11] Chi, Y., Optimal Reinsurance under Variance Related Premium Principles, Insurance: Mathematics and Economics, 51, 2, 310-321, (2012) · Zbl 1284.91215
[12] Chi, Y.; Tan, K. S., Optimal Reinsurance under VaR and CVaR Risk Measures: A Simplified Approach, ASTIN Bulletin, 41, 2, 487-509, (2011) · Zbl 1239.91078
[13] Chi, Y.; Tan, K. S., Optimal Reinsurance with General Premium Principles, Insurance: Mathematics and Economics, 52, 2, 180-189, (2013) · Zbl 1284.91216
[14] Chi, Y.; Lin, X. S., Optimal Reinsurance with Limited Ceded Risk: A Stochastic Dominance Approach, ASTIN Bulletin, 44, 1, 103-126, (2014) · Zbl 1290.91082
[15] Cui, W.; Yang, J.; Wu, L., Optimal Reinsurance Minimizing the Distortion Risk Measure under General Reinsurance Premium Principles, Insurance: Mathematics and Economics, 53, 1, 74-85, (2013) · Zbl 1284.91222
[16] Gajek, L.; Zagrodny, D., Optimal Reinsurance under General Risk Measures, Insurance: Mathematics and Economics, 34, 2, 227-240, (2004) · Zbl 1136.91478
[17] Guerra, M.; Centeno, M. D. L., Optimal Reinsurance for Variance Related Premium Calculation Principles, ASTIN Bulletin, 40, 1, 97-121, (2010) · Zbl 1230.91073
[18] Guerra, M.; Centeno, M. D. L., Are Quantile Risk Measures Suitable for Risk-Transfer Decisions?, Insurance: Mathematics and Economics, 50, 3, 446-461, (2012) · Zbl 1262.91093
[19] Huberman, G.; Mayers, D.; Smith, Jr., C. W., Optimal Insurance Policy Indemnity Schedules, Bell Journal of Economics, 14, 2, 415-426, (1983)
[20] Kaluszka, M., Optimal Reinsurance under Mean-Variance Premium Principles, Insurance: Mathematics and Economics, 28, 1, 61-67, (2001) · Zbl 1009.62096
[21] Kaluszka, M., Mean-Variance Optimal Reinsurance Arrangements, Scandinavian Actuarial Journal, 1, 28-41, (2004) · Zbl 1117.62115
[22] Kaluszka, M., Optimal Reinsurance under Convex Principles of Premium Calculation, Insurance: Mathematics and Economics, 36, 3, 375-398, (2005) · Zbl 1120.62092
[23] Kaluszka, M.; Okolewski, A., An Extension of Arrow’s Result on Optimal Reinsurance Contract, Journal of Risk and Insurance, 75, 2, 275-288, (2008)
[24] Kiesel, S.; Rüschendorf, L., On the Optimal Reinsurance Problem, Applicationes Mathematicae, 40, 3, 259-280, (2013) · Zbl 1285.91059
[25] Ohlin, J., On a Class of Measures of Dispersion with Application to Optimal Reinsurance, ASTIN Bulletin, 5, 2, 249-266, (1969)
[26] Raviv, A., The Design of an Optimal Insurance Policy, American Economic Review, 69, 1, 84-96, (1979)
[27] Rüschendorf, L., Mathematical Risk Analysis: Dependence, Risk Bounds, Optimal Allocations and Portfolios, (2013), New York: Springer, New York · Zbl 1266.91001
[28] Shaked, M.; Shanthikumar, J. G., Stochastic Orders, (2007), New York: Springer, New York
[29] Tan, K. S.; Weng, C.; Zhang, Y., VaR and CTE Criteria for Optimal Quota-Share and Stop-Loss Reinsurance, North American Actuarial Journal, 13, 4, 459-482, (2009)
[30] Van Heerwaarden, A. E.; Kaas, R.; Goovaerts, M. J., Optimal Reinsurance in Relation to Ordering of Risks, Insurance: Mathematics and Economics, 8, 1, 11-17, (1989) · Zbl 0683.62060
[31] Young, V. R., Optimal Insurance under Wang’s Premium Principle, Insurance: Mathematics and Economics, 25, 2, 109-122, (1999) · Zbl 1156.62364
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.