# zbMATH — the first resource for mathematics

Optimal reinsurance with general premium principles. (English) Zbl 1284.91216
Summary: We study two classes of optimal reinsurance models from the perspective of an insurer by minimizing its total risk exposure under the criteria of value at risk (VaR) and conditional value at risk (CVaR), assuming that the reinsurance premium principles satisfy three basic axioms: distribution invariance, risk loading and stop-loss ordering preserving. The proposed class of premium principles is quite general in the sense that it encompasses eight of the eleven commonly used premium principles listed in [V. R. Young, “Premium principles”, in: J. L. Teugels (ed.) and B. Sundt (ed.), Encyclopedia of actuarial science. Hoboken, NJ: John Wiley & Sons (2004; doi:10.1002/9780470012505.tap027)]. Under the additional assumption that both the insurer and reinsurer are obligated to pay more for larger loss, we show that layer reinsurance is quite robust in the sense that it is always optimal over our assumed risk measures and the prescribed premium principles. We further use the Wang’s and Dutch premium principles to illustrate the applicability of our results by deriving explicitly the optimal parameters of the layer reinsurance. These two premium principles are chosen since in addition to satisfying the above three axioms, they exhibit increasing relative risk loading, a desirable property that is consistent with the market convention on reinsurance pricing.

##### MSC:
 91B30 Risk theory, insurance (MSC2010)
Full Text:
##### References:
 [1] Arrow, K. J., Uncertainty and the welfare economics of medical care, American Economic Review, 53, 5, 941-973, (1963) [2] Artzner, P.; Delbaen, F.; Eber, J. M.; Heath, D., Coherent measures of risk, Mathematical Finance, 9, 3, 203-228, (1999) · Zbl 0980.91042 [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] Balbás, A.; Balbás, B.; Heras, A., Stable solutions for optimal reinsurance problems involving risk measures, European Journal of Operational Research, 214, 3, 796-804, (2011) · Zbl 1219.91064 [5] Bernard, C.; Tian, W., Optimal reinsurance arrangements under tail risk measures, The Journal of Risk and Insurance, 76, 3, 709-725, (2009) [6] Borch, K., 1960. An attempt to determine the optimum amount of stop loss reinsurance. In: Transactions of the 16th International Congress of Actuaries, Vol. I, pp. 597-610. [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] Cheung, K. C., Optimal reinsurance revisited—a geometric approach, Astin Bulletin, 40, 1, 221-239, (2010) · Zbl 1230.91070 [10] 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 [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, 1, 3-33, (2002) · Zbl 1051.62107 [12] Föllmer, H.; Schied, A., Stochastic finance: an introduction in discrete time, (2004), Walter de Gruyter · Zbl 1126.91028 [13] Gajek, L.; Zagrodny, D., Optimal reinsurance under general risk measures, Insurance: Mathematics and Economics, 34, 2, 227-240, (2004) · Zbl 1136.91478 [14] Kaluszka, M., Optimal reinsurance under mean-variance premium principles, Insurance: Mathematics and Economics, 28, 1, 61-67, (2001) · Zbl 1009.62096 [15] Kaluszka, M., Optimal reinsurance under convex principles of premium calculation, Insurance: Mathematics and Economics, 36, 3, 375-398, (2005) · Zbl 1120.62092 [16] Kaluszka, M.; Okolewski, A., An extension of arrow’s result on optimal reinsurance contract, The Journal of Risk and Insurance, 75, 2, 275-288, (2008) [17] Laeven, R. J.A.; Goovaerts, M. J., Premium calculation and insurance pricing, (Encyclopedia of Quantitative Risk Analysis and Assessment, (2008), John Wiley & Sons) [18] Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J., Stochastic processes for insurance and finance, (1999), John Wiley & Sons · Zbl 0940.60005 [19] Shaked, M.; Shanthikumar, J. G., Stochastic orders, (2007), Springer [20] 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) [21] 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 [22] Van Heerwaarden, A. E.; Kaas, R., The Dutch premium principle, Insurance: Mathematics and Economics, 11, 2, 129-133, (1992) · Zbl 0781.62163 [23] Venter, G. G., Premium calculation implications of reinsurance without arbitrage, Astin Bulletin, 21, 2, 223-230, (1991) [24] Wang, S., Insurance pricing and increased limits ratemaking by proportional hazards transforms, Insurance: Mathematics and Economics, 17, 1, 43-54, (1995) · Zbl 0837.62088 [25] Wang, S.; Young, V. R.; Panjer, H., Axiomatic characterization of insurance prices, Insurance: Mathematics and Economics, 21, 2, 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 [27] Young, V. R., Premium principles, (Teugels, J.; Sundt., B., Encyclopedia of Actuarial Science, Vol. 3, (2004), John Wiley& Sons)
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.