×

zbMATH — the first resource for mathematics

Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles. (English) Zbl 1284.91222
Summary: Recently the optimal reinsurance strategy concerning the insurer’s risk attitude and the reinsurance premium principle has been an interesting topic. This paper discusses the optimal reinsurance problem with the insurer’s risk measured by distortion risk measure and the reinsurance premium calculated by a general principle including expected premium principle and Wang’s premium principle as its special cases. Explicit solutions of the optimal reinsurance strategy are obtained under the assumption that both the ceded loss and the retained loss are increasing with the initial loss. We present a new method for discussing the optimal problem. Based on our method, one can explain the optimal reinsurance treaty in the view of a balance between the insurer’s risk measure and the reinsurance premium principle.

MSC:
91B30 Risk theory, insurance (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arrow, K. J., Uncertainty and the welfare economics of medical care, American Economic Review, 53, 941-973, (1963)
[2] Arrow, K. J., Optimal insurance and generalized deductibles, Scandinavian Actuarial Journal, 1-42, (1974) · Zbl 0306.90009
[3] Balbás, A.; Balbás, B.; Heras, A., Optimal reinsurance with general risk measures, Insurance: Mathematics and Economics, 44, 3, 374-384, (2009) · Zbl 1162.91394
[4] Bernard, C.; Tian, W., Optimal reinsurance arrangements under tail risk measures, Journal of Risk and Insurance, 76, 3, 706-725, (2009)
[5] Borch, K., The safety loading of reinsurance premiums, Scandinavian Actuarial Journal, 163-184, (1960) · Zbl 0122.37603
[6] 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
[7] Cai, J.; Tan, K. S.; Weng, C.; Zhang, Y., Optimal reinsurance under var and CTE risk measures, Insurance: Mathematics and Economics, 43, 185-196, (2008) · Zbl 1140.91417
[8] Cheung, K. C., Optimal reinsurance revisited—a geometric approach, Astin Bulletin, 40, 1, 221-239, (2010) · Zbl 1230.91070
[9] 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
[10] Chi, Y.; Tan, K. S., Optimal reinsurance with general premium principles, Insurance: Mathematics and Economics, 52, 180-189, (2012) · Zbl 1284.91216
[11] Dhaene, J.; Denuit, M.; Goovaerts, M. J.; Kaas, R.; Vyncke, D., The concept of comonotonicity in actuarial science and finance: theory, Insurance: Mathematics and Economics, 31, 3-33, (2002) · Zbl 1051.62107
[12] Dhaene, J.; Vanduffel, S.; Goovaerts, M. J.; Kaas, R.; Tang, Q.; Vyncke, D., Risk measures and comonotonicity: a review, Stochastic Models, 22, 573-606, (2006) · Zbl 1159.91403
[13] Froot, K. A., The market for catastrophe risk: a clinical examination, Journal of Financial Economics, 60, 529-571, (2001)
[14] Gajek, L.; Zagrodny, D., Insurer’s optimal reinsurance strategies, Insurance: Mathematics and Economics, 27, 105-112, (2000) · Zbl 0964.62099
[15] Gajek, L.; Zagrodny, D., Optimal reinsurance under general risk measures, Insurance: Mathematics and Economics, 34, 227-240, (2004) · Zbl 1136.91478
[16] Gajek, L.; Zagrodny, D., Reinsurance arrangements maximizing insurer’s survival probability, Journal of Risk and Insurance, 71, 3, 421-435, (2004)
[17] Kaluszka, M., Optimal reinsurance under mean-variance premium principles, Insurance: Mathematics and Economics, 28, 61-67, (2001) · Zbl 1009.62096
[18] Kaluszka, M., Mean-variance optimal reinsurance arrangements, Scandinavian Actuarial Journal, 1, 28-41, (2004) · Zbl 1117.62115
[19] Kaluszka, M., An extension of arrow’s result on optimality of a stop loss contract, Insurance: Mathematics and Economics, 35, 527-536, (2004) · Zbl 1122.91343
[20] Kaluszka, M., Optimal reinsurance under convex principles of premium calculation, Insurance: Mathematics and Economics, 36, 375-398, (2005) · Zbl 1120.62092
[21] Kaluszka, M.; Okolewski, A., An extension of arrow’s result on optimal reinsurance contract, Journal of Risk and Insurance, 75, 2, 275-288, (2008)
[22] Rao, M. M., Measure theoty and integration, (1987), John Wiley New York
[23] Tan, K. S.; Weng, C.; Zhang, Y., Var and CTE criteria for optimal quato-share and stop loss reinsurance, North American Actuarial Journal, 13, 4, 459-482, (2009)
[24] Tan, K. S.; Weng, C.; Zhang, Y., Optimality of general reinsurance contracts under CTE risk measure, Insurance: Mathematics and Economics, 49, 2, 175-187, (2011) · Zbl 1218.91097
[25] Wang, S.; Young, V. R.; Panjer, H. H., Axiomatic characterization of insurance prices, Insurance: Mathematics and Economics, 21, 173-183, (1997) · Zbl 0959.62099
[26] 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.