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Stable solutions for optimal reinsurance problems involving risk measures. (English) Zbl 1219.91064
Summary: The optimal reinsurance problem is a classic topic in actuarial mathematics. Recent approaches consider a coherent or expectation bounded risk measure and minimize the global risk of the ceding company under adequate constraints. However, there is no consensus about the risk measure that the insurer must use, since every risk measure presents advantages and shortcomings when compared with others.
This paper deals with a discrete probability space and analyzes the stability of the optimal reinsurance with respect to the risk measure that the insurer uses. We will demonstrate that there is a “stable optimal retention” that will show no sensitivity, insofar as it will solve the optimal reinsurance problem for many risk measures, thus providing a very robust reinsurance plan. This stable optimal retention is a stop-loss contract, and it is easy to compute in practice. A fast linear time algorithm will be given and a numerical example presented.

MSC:
91B30 Risk theory, insurance (MSC2010)
90C90 Applications of mathematical programming
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[1] Annaert, J.; Van Osselaer, S.; Verstraete, B., Performance evaluation of portfolio insurance strategies using stochastic dominance criteria, Journal of banking and finance, 33, 272-280, (2009)
[2] Arrow, K.J., Uncertainty and the welfare of medical care, American economic review, 53, 941-973, (1963)
[3] Artzner, P.; Delbaen, F.; Eber, J.M.; Heath, D., Coherent measures of risk, Mathematical finance, 9, 203-228, (1999) · Zbl 0980.91042
[4] Balbás, A.; Balbás, B.; Heras, A., Optimal reinsurance with general risk measures, Insurance: mathematics and economics, 44, 374-384, (2009) · Zbl 1162.91394
[5] Balbás, A.; Balbás, R.; Garrido, J., Extending pricing rules with general risk functions, European journal of operational research, 201, 23-33, (2010) · Zbl 1177.91143
[6] Benati, S., The optimal portfolio problem with coherent risk measure constraints, European journal of operational research, 150, 572-584, (2003) · Zbl 1033.90060
[7] Bernard, C.; Tian, W., Optimal reinsurance arrangements under tail risk measures, Journal of risk and insurance, 76, 3, 709-725, (2009)
[8] Borch, K., An attempt to determine the optimum amount of stop loss reinsurance, Transactions of the 16th international congress of actuaries I, 597-610, (1960)
[9] Burgert, C.; Rüschendorf, L., Consistent risk measures for portfolio vectors, Insurance: mathematics and economics, 38, 2, 289-297, (2006) · Zbl 1138.91490
[10] 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
[11] 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
[12] Centeno, M.L.; Simoes, O., Optimal reinsurance, Revista de la real academia de ciencias, RACSAM, 103, 2, 387-405, (2009) · Zbl 1181.91090
[13] Cherny, A.S., Weighted V@R and its properties, Finance and stochastics, 10, 367-393, (2006) · Zbl 1101.91023
[14] Deprez, O.; Gerber, U., On convex principles of premium calculation, Insurance: mathematics and economics, 4, 179-189, (1985) · Zbl 0579.62090
[15] Frittelli, M.; Scandolo, G., Risk measures and capital requirements for processes, Mathematical finance, 16, 4, 589-612, (2005) · Zbl 1130.91030
[16] Gajec, L.; Zagrodny, D., Optimal reinsurance under general risk measures, Insurance: mathematics and economics, 34, 227-240, (2004) · Zbl 1136.91478
[17] Goovaerts, M.; Kaas, R.; Dhaene, J.; Tang, Q., A new classes of consistent risk measures, Insurance: mathematics and economics, 34, 505-516, (2004) · Zbl 1188.91087
[18] Kaluszka, M., Optimal reinsurance under convex principles of premium calculation, Insurance: mathematics and economics, 36, 375-398, (2005) · Zbl 1120.62092
[19] Konno, H.; Akishino, K.; Yamamoto, R., Optimization of a long-short portfolio under non-convex transaction costs, Computational optimization and applications, 32, 115-132, (2005) · Zbl 1085.90046
[20] Luenberger, D.G., Optimization by vector spaces methods, (1969), John Wiley & Sons · Zbl 0176.12701
[21] Mansini, R.; Ogryczak, W.; Speranza, M.G., Conditional value at risk and related linear programming models for portfolio optimization, Annals of operations research, 152, 227-256, (2007) · Zbl 1132.91497
[22] Miller, N.; Ruszczynski, A., Risk-adjusted probability measures in portfolio optimization with coherent measures of risk, European journal of operational research, 191, 193-206, (2008) · Zbl 1142.91591
[23] Nakano, Y., Efficient hedging with coherent risk measure, Journal of mathematical analysis and applications, 293, 345-354, (2004) · Zbl 1085.91032
[24] Ogryczak, W.; Ruszczynski, A., From stochastic dominance to Mean risk models: semideviations and risk measures, European journal of operational research, 116, 33-50, (1999) · Zbl 1007.91513
[25] Ogryczak, W.; Ruszczynski, A., Dual stochastic dominance and related Mean risk models, SIAM journal on optimization, 13, 60-78, (2002) · Zbl 1022.91017
[26] Rockafellar, R.T.; Uryasev, S.; Zabarankin, M., Generalized deviations in risk analysis, Finance and stochastics, 10, 51-74, (2006) · Zbl 1150.90006
[27] Samuelson, P.A., The foundations of economic analysis, (1947), Harvard University Press · Zbl 0031.17401
[28] Schied, A., Optimal investments for risk- and ambiguity-averse preferences: A duality approach, Finance and stochastics, 11, 107-129, (2007) · Zbl 1143.91021
[29] Wang, S.S., A class of distortion operators for pricing financial and insurance risks, Journal of risk and insurance, 67, 15-36, (2000)
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