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An extension of Arrow’s result on optimality of a stop loss contract. (English) Zbl 1122.91343

Summary: The optimal, from the cedent’s point of view, reinsurance treaties for a fixed reinsurer’s premium are derived. We assume that the cedent wants to minimize a convex risk measure, e.g. the variance, semi-variance, upper partial moment or mean absolute deviation. The reinsurer’s premium is calculated on the basis of the expectation and variance of his/her part of the total risk. The obtained results provide an extension of Arrow’s result on the optimality of a stop loss treaty.

MSC:

91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
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[1] Arrow, K.J., Uncertainty and the welfare economics of medical care, Am. econ. rev., 53, 941-973, (1963)
[2] Arrow, K.J., Essays in the theory of risk bearing, (1971), Markham Publishing Co. Chicago · Zbl 0215.58602
[3] Arrow, K.J., Optimal insurance and generalized deductibles, Scand. actuarial J., 1, 1-42, (1974) · Zbl 0306.90009
[4] Berliner, B., A risk measure alternative to the variance, ASTIN bull., 9, 42-58, (1977)
[5] Borch, K., Optimal insurance arrangements, ASTIN bull., 8, 284-290, (1975)
[6] Borch, K., Economics of insurance, (1990), North-Holland Amsterdam
[7] Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., Nesbitt C.J., 1997. Actuarial Mathematics, second ed. Society of Actuaries, Schaumburg, III. · Zbl 0634.62107
[8] Daykin, C.D.; Pentikäinen, T.; Pesonen, M., Practical risk theory for actuaries, (1994), Chapman & Hall London · Zbl 1140.62345
[9] Deprez, O.; Gerber, H.U., On convex principles of premium calculation, Ins.: mathematics econ., 4, 179-189, (1985) · Zbl 0579.62090
[10] Gajek, L.; Zagrodny, D., Insurer’s optimal reinsurance strategies, Ins.: mathematics econ., 27, 105-112, (2000) · Zbl 0964.62099
[11] Gajek, L.; Zagrodny, D., Optimal reinsurance under general risk measures, Ins.: mathematics econ., 34, 227-240, (2004) · Zbl 1136.91478
[12] Gerber, H.U.; Pafumi, G., Utility functions: from risk theory to finance, N. am. actuarial J., 2, 74-91, (1998) · Zbl 1081.91511
[13] Goovaerts, M.J.; De Vylder, F.; Haezendonck, J., Insurance premiums: theory and applications, (1984), North-Holland Amsterdam · Zbl 0532.62082
[14] Heerwaarden van, A.E.; Kaas, R., The Dutch premium principle, Ins.: mathematics econ., 11, 129-133, (1992) · Zbl 0781.62163
[15] Heerwaarden van, A.E.; Kaas, R.; Goovaerts, M.J., Optimal reinsurance in relation to ordering of risks, Ins.: mathematics econ., 8, 261-267, (1989) · Zbl 0686.62090
[16] Hesselager, O., Some results on optimal reinsurance in terms of the adjustment coefficient, Scand. actuarial J., 2, 80-95, (1990) · Zbl 0728.62100
[17] Kaluszka, M., Optimal reinsurance under Mean-variance premium principles, Ins.: mathematics econ., 28, 61-67, (2001) · Zbl 1009.62096
[18] Kaluszka, M., Mean-variance optimal reinsurance arrangements, Scand. actuarial J., 1, 28-41, (2004) · Zbl 1117.62115
[19] Kaluszka, M., An extension of the gerber-Bühlmann-jewell conditions for optimal risk sharing, ASTIN bull., 34, 27-48, (2004) · Zbl 1098.91073
[20] Markowitz, H.M., Portfolio selection, Efficient diversification of investments, (1959), J. Wiley & Sons New York
[21] Mossin, J., Aspects of rational insurance purchasing, J. polit. economy, 76, 553-568, (1968)
[22] Ohlin, J., On a class of measures of dispersion with application to optimal reinsurance, ASTIN bull., 5, 249-266, (1969)
[23] Pedersen, Ch.S.; Satchell, S.E., An extended family of financial-risk measures, Geneva pap. risk ins. theory, 23, 89-117, (1998)
[24] Raviv, A., The design of an optimal insurance policy, Am. econ. rev., 69, 84-96, (1979)
[25] Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J., Stochastic processes for insurance and finance, (1999), J. Wiley & Sons Chichester · Zbl 0940.60005
[26] Schmitter, H., 2001. Setting optimal reinsurance retentions. Swiss Re Publications.
[27] Smith, V.L., Optimal insurance coverage, J. polit. economy, 76, 68-77, (1968)
[28] Spaeter, S.; Roger, P., The design of optimal insurance contracts: a topological approach, Geneva pap. risk ins. theory, 22, 5-19, (1997)
[29] Young, V.R., Optimal insurance under wang’s premium principle, Ins.: mathematics econ., 25, 109-122, (1999) · Zbl 1156.62364
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