zbMATH — the first resource for mathematics

Marginal indemnification function formulation for optimal reinsurance. (English) Zbl 1348.91196
Summary: In this paper, we propose to combine the Marginal Indemnification Function (MIF) formulation and the Lagrangian dual method to solve optimal reinsurance model with distortion risk measure and distortion reinsurance premium principle. The MIF method exploits the absolute continuity of admissible indemnification functions and formulates optimal reinsurance model into a functional linear programming of determining an optimal measurable function valued over a bounded interval. The MIF method was recently introduced to analyze the reinsurance model but without premium budget constraint. In this paper, a Lagrangian dual method is applied to combine with MIF to solve for optimal reinsurance solutions under premium budget constraint. Compared with the existing literature, the proposed integrated MIF-based Lagrangian dual method provides a more technically convenient and transparent solution to the optimal reinsurance design. To demonstrate the practicality of the proposed method, analytical solution is derived on a particular reinsurance model that involves minimizing Conditional Value at Risk (a special case of distortion function) and with the reinsurance premium being determined by the inverse-S shaped distortion principle.

91B30 Risk theory, insurance (MSC2010)
91B16 Utility theory
62P05 Applications of statistics to actuarial sciences and financial mathematics
Full Text: DOI
[1] Abdellaoui, M., Parameter-free elicitation of utility and probability weighting functions, Manage. Sci., 46, 1497-1512, (2000) · Zbl 1232.91114
[2] Acerbi, C., Spectral measures of risk: a coherent representation of subjective risk aversion, J. Bank. Finance, 26, 1505-1518, (2002)
[3] Arrow, K. J., Uncertainty and the welfare economics of medical care, Amer. Econ. Rev., 53, 941-973, (1963)
[4] Assa, H., On optimal reinsurance policy with distortion risk measures and premiums, Insurance Math. Econom., 61, 70-75, (2015) · Zbl 1314.91132
[5] Balbás, A.; Balbás, B.; Balbás, R.; Heras, A., Optimal reinsurance with general risk measures, Insurance Math. Econom., 44, 374-384, (2009) · Zbl 1162.91394
[6] Balbás, A.; Balbás, B.; Balbás, R.; Heras, A., Optimal reinsurance under risk and uncertainty, Insurance Math. Econom., 60, 61-74, (2015) · Zbl 1308.91075
[7] Borch, K., 1960. An attempt to determine the optimum amount of stop loss reinsurance. In: Transactions of the 16th International Congress of Actuaries.
[8] Cai, J.; Tan, K. S., Optimal retention for a stop-loss reinsurance under the var and CTE risk measures, Astin Bull., 37, 93-112, (2007) · Zbl 1162.91402
[9] Cai, J.; Tan, K. S.; Weng, C.; Zhang, Y., Optimal reinsurance under var and CTE risk measures, Insurance Math. Econom., 43, 185-196, (2008) · Zbl 1140.91417
[10] Cheung, K. C., Optimal reinsurance revisited—a geometric approach, Astin Bull., 40, 221-239, (2010) · Zbl 1230.91070
[11] Cheung, K. C.; Lo, A., Characterizations of optimal reinsurance treaties: a cost-benefit approach, Scand. Actuar. J., (2015), in press
[12] Cheung, K. C.; Sung, K.; Yam, S.; Yung, S., Optimal reinsurance under general law-invariant risk measures, Scand. Actuar. J., 72-91, (2014) · Zbl 1401.91110
[13] Chi, Y.; Tan, K. S., Optimal reinsurance with general premium principles, Insurance Math. Econom., 52, 180-189, (2013) · Zbl 1284.91216
[14] Chi, Y.; Weng, C., Optimal reinsurance subject to vajda condition, Insurance Math. Econom., 53, 170-189, (2013) · Zbl 1284.91217
[15] Cui, W.; Yang, J.; Wu, L., Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles, Insurance Math. Econom., 53, 74-85, (2013) · Zbl 1284.91222
[16] Dhaene, J.; Kukush, A.; Linders, D.; Tang, Q., Remarks on quantiles and distortion risk measures, Eur. Actuar. J., 2, 319-328, (2012) · Zbl 1256.91027
[17] Gonzalez, R.; Wu, G., On the shape of the probability weighting function, Cogn. Psychol., 38, 129-166, (1999)
[18] Ingersoll, J., Non-monotonicity of the tversky-kahneman probability-weighting function: a cautionary note, Eur. Financ. Manage., 14, 385-390, (2008)
[19] Kaluszka, M., Optimal reinsurance under mean-variance premium principles, Insurance Math. Econom., 28, 61-67, (2001) · Zbl 1009.62096
[20] Kaluszka, M.; Krzeszowiec, M., Pricing insurance contracts under cumulative prospect theory, Insurance Math. Econom., 50, 159-166, (2012) · Zbl 1239.91080
[21] Quiggin, J., A theory of anticipated utility, J. Econ. Behav. Organ., 3, 323-343, (1982)
[22] Quiggin, J., Generalized expected utility theory: the rank dependent model, (1992), Springer Science & Business Media · Zbl 0915.90081
[23] Rieger, M. O.; Wang, M., Cumulative prospect theory and the Saint |St. Petersburg paradox, Econom. Theory, 28, 665-679, (2006) · Zbl 1145.91328
[24] Tan, K. S.; Weng, C.; Zhang, Y., Optimality of general reinsurance contracts under CTE risk measure, Insurance Math. Econom., 49, 175-187, (2011) · Zbl 1218.91097
[25] Tversky, A.; Fox, C. R., Weighing risk and uncertainty, Psychol. Rev., 102, 269-283, (1995)
[26] Tversky, A.; Kahneman, D., Advances in prospect theory: cumulative representation of uncertainty, J. Risk Uncertain., 5, 297-323, (1992) · Zbl 0775.90106
[27] Wu, G.; Gonzalez, R., Curvature of the probability weighting function, Manage. Sci., 42, 1676-1690, (1996) · Zbl 0893.90003
[28] Young, V. R., Optimal insurance under wang’s premium principle, Insurance Math. Econom., 25, 109-122, (1999) · Zbl 1156.62364
[29] Zheng, Y. T.; Cui, W., Optimal reinsurance with premium constraint under distortion risk measures, Insurance Math. Econom., 59, 109-120, (2014) · Zbl 1306.91089
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.