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On optimal reinsurance policy with distortion risk measures and premiums. (English) Zbl 1314.91132
Summary: In this paper, we consider the problem of optimal reinsurance design, when the risk is measured by a distortion risk measure and the premium is given by a distortion risk premium. First, we show how the optimal reinsurance design for the ceding company, the reinsurance company and the social planner can be formulated in the same way. Second, by introducing the “marginal indemnification functions”, we characterize the optimal reinsurance contracts. We show that, for an optimal policy, the associated marginal indemnification function only takes the values zero and one. We will see how the roles of the market preferences and premiums and that of the total risk are separated.

MSC:
91B30 Risk theory, insurance (MSC2010)
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[1] Acerbi, C., Spectral measures of risk: a coherent representation of subjective risk aversion, J. Banking Finance, 26, 7, 1505-1518, (2002)
[2] Arrow, K. J., Uncertainty and the welfare economics of medical care, Amer. Econ. Rev., LIII, 5, (1963)
[3] Balbás, A.; Garrido, J.; Mayoral, S., Properties of distortion risk measures, Methodol. Comput. Appl. Probab., 11, 3, 385-399, (2009) · Zbl 1170.91368
[4] Bernard, C.; Tian, W., Optimal reinsurance arrangements under tail risk measures, J. Risk Insurance, 76, 3, 709-725, (2009)
[5] 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. No. 3, pp. 597-610.
[6] Cai, J.; Tan, K. S., Optimal retention for a stop-loss reinsurance under the var and CTE risk measures, Astin Bull., 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 Math. Econom., 43, 1, 185-196, (2008) · Zbl 1140.91417
[8] Cheung, K. C., Optimal reinsurance revisited—a geometric approach, Astin Bull., 40, 1, 221-239, (2010) · Zbl 1230.91070
[9] Cheung, K.; Sung, K.; Yam, S.; Yung, S., Optimal reinsurance under general law-invariant risk measures, Scand. Actuar. J., 2014, 1, 72-91, (2014) · Zbl 1401.91110
[10] Chi, Y.; Tan, K. S., Optimal reinsurance with general premium principles, Insurance Math. Econom., 52, 2, 180-189, (2013) · Zbl 1284.91216
[11] Cont, R.; Deguest, R.; Scandolo, G., Robustness and sensitivity analysis of risk measurement procedures, Quant. Finance, 10, 6, 593-606, (2010) · Zbl 1192.91191
[12] Heimer, C. A., Reactive risk and rational action: managing moral hazard in insurance contracts. vol. 6, (1989), Univ. of California Press
[13] Kaluszka, M., Optimal reinsurance under mean-variance premium principles, Insurance Math. Econom., 28, 1, 61-67, (2001) · Zbl 1009.62096
[14] Kaluszka, M.; Okolewski, A., An extension of arrow’s result on optimal reinsurance contract, J. Risk Insurance, 75, 2, 275-288, (2008)
[15] Kusuoka, S., On law invariant coherent risk measures, (Advances in Mathematical Economics. Vol. 3, Adv. Math. Econ., vol. 3, (2001), Springer Tokyo), 83-95 · Zbl 1010.60030
[16] Rudin, W., Real and complex analysis, (1987), McGraw-Hill, Inc. New York, NY, USA · Zbl 0925.00005
[17] Sereda, E.; Bronshtein, E.; Rachev, S.; Fabozzi, F.; Sun, W.; Stoyanov, S., Distortion risk measures in portfolio optimization, (Guerard, John B., Handbook of Portfolio Construction, (2010), Springer US), 649-673
[18] Wang, S., Insurance pricing and increased limits ratemaking by proportional hazards transforms, Insurance Math. Econom., 17, 1, 43-54, (1995) · Zbl 0837.62088
[19] Wang, S. S.; Young, V. R.; Panjer, H. H., Axiomatic characterization of insurance prices, Insurance Math. Econom., 21, 2, 173-183, (1997) · Zbl 0959.62099
[20] Wu, X.; Zhou, X., A new characterization of distortion premiums via countable additivity for comonotonic risks, Insurance Math. Econom., 38, 2, 324-334, (2006) · Zbl 1132.91019
[21] Young, V. R., Optimal insurance under wang’s premium principle, Insurance Math. Econom., 25, 2, 109-122, (1999) · Zbl 1156.62364
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