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Optimal risk transfer under quantile-based risk measurers. (English) Zbl 1284.91199

Summary: The classical problem of identifying the optimal risk transfer from one insurance company to multiple reinsurance companies is examined under some quantile-based risk measure criteria. We develop a new methodology via a two-stage optimisation procedure which not only allows us to recover some existing results in the literature, but also makes possible the analysis of high-dimensional problems in which the insurance company diversifies its risk with multiple reinsurance counter-parties, where the insurer risk position and the premium charged by the reinsurers are functions of the underlying risk quantile. Closed-form solutions are elaborated for some particular settings, although numerical methods for the second part of our procedure represent viable alternatives for the ease of implementing it in more complex scenarios. Furthermore, we discuss some approaches to obtain more robust results.

MSC:

91B30 Risk theory, insurance (MSC2010)

Software:

robustbase
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References:

[1] Acerbi, C., Spectral measure of risks: a coherent representation of subjective risk aversion, Journal of Banking & Finance, 26, 7, 1505-1518, (2002)
[2] Acerbi, C.; Tasche, D., On the coherence of expected shortfall, Journal of Banking & Finance, 26, 7, 1487-1503, (2002)
[3] Arrow, K. J., Uncertainty and the welfare economics of medical care, American Economic Review, 53, 5, 941-973, (1963)
[4] Bellini, F.; Rosazza Gianin, E., Haezendonck-goovaerts risk measures and Orlicz quantiles, Insurance: Mathematics & Economics, 51, 1, 107-114, (2012) · Zbl 1284.91205
[5] Borch, K., 1960. An attempt to determine the optimum amount of stop loss reinsurance, in: Transactions of the 16th International Congress of Actuaries I, pp. 597-610.
[6] Boudt, K.; Caliskan, D.; Croux, C., Robust explicit estimators for Weibull parameters, Metrika, 73, 2, 187-209, (2011) · Zbl 1206.62028
[7] Brazauskas, V.; Serfling, R., Robust estimation of tail parameters for two-parameter Pareto and exponential models via generalized quantile statistics, Extremes, 3, 3, 231-249, (2000) · Zbl 0979.62016
[8] Brazauskas, V.; Serfling, R., Favorable estimators for Fitting Pareto models: a study using goodness-of-fit measures with actual data, ASTIN Bulletin, 33, 2, 365-381, (2003) · Zbl 1058.62030
[9] Cai, J.; Tan, K. S.; Weng, C.; Zhang, Y., Optimal reinsurance under var and cte risk measures, Insurance: Mathematics & Economics, 43, 1, 185-196, (2008) · Zbl 1140.91417
[10] Centeno, M. L.; Guerra, M., The optimal reinsurance strategy — the individual claim case, Insurance: Mathematics & Economics, 46, 3, 450-460, (2010) · Zbl 1231.91151
[11] Cheung, K. C., Optimal reinsurer revisited — a geometric approach, Astin Bulletin, 40, 1, 221-239, (2010) · Zbl 1230.91070
[12] Chi, Y.; Tan, K. S., Optimal reinsurance under var and CVaR risk measures: a simplified approach, Astin Bulletin, 42, 2, 487-509, (2011) · Zbl 1239.91078
[13] Cont, R.; Deguest, R.; Scandalo, G., Robustness and sensitivity analysis of risk measurement procedures, Quantitative Finance, 10, 6, 593-606, (2010) · Zbl 1192.91191
[14] Croux, C.; Gelper, G.; Mahieu, K., Robust exponential smoothing of multivariate time series, Computational Statistics and Data Analysis, 54, 2999-3006, (2010) · Zbl 1284.62547
[15] Czellar, V.; Karolyi, G. A.; Ronchetti, E., Indirect robust estimation of the short-term interest rate process, Journal of Empirical Finance, 14, 546-563, (2007)
[16] Dell’Aquila, R.; Embrechts, P., Extremes and robustness: a contradiction?, Financial Markets and Portfolio Management, 20, 103-118, (2006)
[17] Denuit, M.; Dhaene, J.; Goovaerts, M. J.; Kaas, R., Actuarial theory for dependent risks: measures, orders and models, (2005), Wiley Chichester
[18] Dhaene, J.; Denuit, M.; Goovaerts, M. J.; Kaas, R.; Vyncke, D., The concept of comonotonicity in actuarial science and finance: theory, Insurance: Mathematics & Economics, 31, 1, 3-33, (2002) · Zbl 1051.62107
[19] Dhaene, J.; Denuit, M.; Goovaerts, M. J.; Kaas, R.; Vyncke, D., The concept of comonotonicity in actuarial science and finance: applications, Insurance: Mathematics & Economics, 31, 2, 133-161, (2002) · Zbl 1037.62107
[20] Dhaene, J.; Kukush, A.; Linders, D.; Qihe, T., Remarks on quantiles and distortion risk measures, European Actuarial Journal, 2, 2, 319-328, (2012) · Zbl 1256.91027
[21] Fernholz, L. T., Reducing the variance by smoothing, Journal of Statistical Planning and Inference, 57, 1, 29-38, (1997) · Zbl 0877.62043
[22] Gather, U., Robust estimation of the mean of the exponential distribution in outlier situations, Communications in Statistics A, 15, 2323-2345, (1986) · Zbl 0603.62041
[23] Gather, U.; Schultze, V., Robust estimation of scale of an exponential distribution, Statistica Neerlandica, 53, 327-341, (1999) · Zbl 1072.62548
[24] Goovaerts, M. J.; Linders, D.; Van Weert, K.; Tank, F., On the interplay between distortion, mean value and haezendonck-goovaerts risk measures, Insurance: Mathematics & Economics, 51, 1, 10-18, (2012) · Zbl 1284.91235
[25] Goovaerts, M. J.; Kaas, R.; Dhaene, J.; Tang, Q., Some new classes of consistent risk measures, Insurance: Mathematics & Economics, 34, 3, 505-516, (2004) · Zbl 1188.91087
[26] Guerra, M.; Centeno, M. L., Optimal reinsurance policy: the adjustment coefficient and the expected utility criteria, Insurance: Mathematics & Economics, 42, 2, 529-539, (2008) · Zbl 1152.91583
[27] Guerra, M.; Centeno, M. L., Optimal reinsurance for variance related premium calculation principles, Astin Bulletin, 40, 1, 97-121, (2010) · Zbl 1230.91073
[28] Haezendonck, J.; Goovaerts, M. J., A new premium calculation principle based on Orlicz norms, Insurance: Mathematics & Economics, 1, 1, 41-53, (1982) · Zbl 0495.62091
[29] Hampel, F. R.; Ronchetti, E. M.; Rousseeuw, P. J.; Stahel, W. A., Robust statistics: the approach based on influence functions, (1986), Wiley New York · Zbl 0593.62027
[30] Huber, P. J., Robust statistics, (1981), Wiley New York · Zbl 0536.62025
[31] Hubert, M.; Vandervieren, E., An adjusted boxplot for skewed distributions, Computational Statistics and Data Analysis, 52, 12, 5186-5201, (2008) · Zbl 1452.62074
[32] Hubert, M.; Dierckx, G.; Vanpaemel, D., Detecting influential data points for the Hill estimator in Pareto-type distributions, Computational Statistics and Data Analysis, 65, 13-28, (2013)
[33] Hürlimann, V., Conditional value-at-risk bounds for compound Poisson risks and a normal approximation, Journal of Applied Mathematics, Issue 3, 141-153, (2003) · Zbl 1012.62110
[34] Jones, M. C.; Marron, J. S.; Sheather, S. J., A brief survey of bandwidth selection for density estimation, Journal of the American Statistical Association, 91, 433, 401-407, (1996) · Zbl 0873.62040
[35] Jones, B. L.; Zitikis, R., Empirical estimation of risk measures and related quantities, North American Actuarial Journal, 7, 4, 44-54, (2003) · Zbl 1084.62537
[36] Jones, B. L.; Zitikis, R., Risk measures, distortion parameters, and their empirical estimation, Insurance: Mathematics & Economics, 41, 2, 279-297, (2007) · Zbl 1193.91065
[37] Kaluszka, M., Optimal reinsurance under mean-variance premium principles, Insurance: Mathematics & Economics, 28, 1, 61-67, (2001) · Zbl 1009.62096
[38] Kaluszka, M., Truncated stop loss as optimal reinsurance agreement in one-period models, Astin Bulletin, 35, 2, 337-349, (2005) · Zbl 1156.62363
[39] 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)
[40] Kou, S. G.; Peng, X.; Heyde, C., External risk measures and basel accords, Mathematics of Operations Research, (2012), (forthcoming) · Zbl 1297.91089
[41] Loisel, S.; Mazza, C.; Rullire, D., Robustness analysis and convergence of empirical finite-time ruin probabilities and estimation risk solvency margin, Insurance: Mathematics & Economics, 42, 2, 746-762, (2008) · Zbl 1152.91594
[42] Ludrovski, M.; Young, V. R., Optimal risk sharing under distorted probabilities, Mathematics and Financial Economics, 2, 2, 87-105, (2009) · Zbl 1255.91182
[43] Marazzi, A.; Ruffieux, C., Implementing M-estimators of the gamma distribution, (Rieder, H., Robust Statistics, Data Analysis, and Computer Intensive Methods, in Honor of Peter Hubers 60th Birthday, Lecture Notes in Statistics, vol. 109, (1996), Springer), 277-297 · Zbl 0846.62031
[44] Marazzi, A.; Ruffieux, C., The truncated mean of an asymmetric distribution, Computational Statistics and Data Analysis, 32, 79-100, (1999)
[45] Marazzi, A.; Barbati, G., Robust parametric means of asymmetric distributions: estimation and testing, Estadistica, 54, 162-163, 47-72, (2003) · Zbl 1034.62016
[46] Maronna, R.; Martin, D.; Yohai, V., Robust statistics — theory and methods, (2006), Wiley New York · Zbl 1094.62040
[47] Rizzo, M. L., New goodness-of-fit tests for Pareto distributions, Astin Bulletin, 39, 2, 691-715, (2009) · Zbl 1178.62051
[48] Serfling, R., Efficient and robust Fitting of lognormal distributions, North American Actuarial Journal, 6, 95-109, (2002) · Zbl 1084.62511
[49] Sheather, S. J.; Jones, M. C., A reliable data-based bandwidth selection method for kernel density estimation, Journal of the Royal Statistical Society series B, 53, 683-690, (1991) · Zbl 0800.62219
[50] Silverman, B. W., Density estimation for statistics and data analysis, (1986), Chapman and Hall New York · Zbl 0617.62042
[51] Tukey, J. W., Exploratory data analysis, (1977), Addison-Wesley Reading, Massachusetts · Zbl 0409.62003
[52] Vandewalle, B.; Beirlant, J.; Christmann, A.; Hubert, M., A robust estimator for the tail index of Pareto-type distributions, Computational Statistics and Data Analysis, 51, 6252-6268, (2007) · Zbl 1445.62102
[53] Van Heerwaarden, A. E.; Kaas, R.; Goovaerts, M. J., Optimal reinsurer in the relation to ordering of risk, Insurance: Mathematics & Economics, 8, 1, 11-17, (1989) · Zbl 0683.62060
[54] Verdonck, T.; Debruyne, M., The influence of individual claims on the chain-ladder estimates: analysis and diagnostic tool, Insurance: Mathematics & Economics, 48, 1, 85-98, (2011) · Zbl 1233.91156
[55] Verlaak, R.; Beirlant, J., Optimal reinsurance programs: an optimal combination of several reinsurance protections on a heterogeneous insurance portfolio, Insurance: Mathematics & Economics, 33, 2, 381-403, (2003) · Zbl 1103.91376
[56] Wang, S.; Young, V. R., Ordering risks: expected utility theory versus yaari’s dual theory of risk, Insurance: Mathematics & Economics, 22, 2, 145-161, (1998) · Zbl 0907.90102
[57] Weissman, I., Estimation of parameters and large quantiles based on the \(k\) largest observations, Journal of the American Statistical Association, 73, 812-815, (1978) · Zbl 0397.62034
[58] Welsch, R.; Zhou, X., Application of robust statistics to asset allocation models, REVSTAT — Statistical Journal, 5, 1, 97-114, (2007) · Zbl 05217609
[59] Young, V. R., Optimal insurance under wang’s premium principle, Insurance: Mathematics & Economics, 25, 2, 109-122, (1999) · Zbl 1156.62364
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