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Conditional tail moments of the exponential family and its related distributions. (English) Zbl 1219.91071

Summary: The risk measure is a central theme in the risk management literature. For good reasons, the conditional tail expectation (CTE) has received much interest in both insurance and finance applications. It provides for a measure of the expected riskiness in the tail of the loss distribution. In this article we derive explicit formulas of the CTE and higher moments for the univariate exponential family class, which extends the natural exponential family, using the canonical representation. In addition we show how to compute the conditional tail expectations of other related distributions using transformation and conditioning. Selected examples are presented for illustration, including the generalized Pareto and generalized hyperbolic distributions. We conclude that the conditional tail expectations of a wide range of loss distributions can be analytically obtained using the methods shown in this article.

MSC:

91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
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[1] Artzner P., Mathematical Finance 9 pp 203– (1999) · Zbl 0980.91042 · doi:10.1111/1467-9965.00068
[2] Barndorff-Nielsen O. E., Levy Processes: Theory and Applications (2001)
[3] Bawa V. S., Journal of Financial Economics 5 (2) pp 189– (1977) · doi:10.1016/0304-405X(77)90017-4
[4] Brown L., Lecture Note-Monograph Series IMS (1986)
[5] Bühlmann H., A Course in Credibility Theory and Its Applications (2005) · Zbl 1108.91001
[6] Cai J., Journal of Applied Probability 42 (3) pp 810– (2005) · Zbl 1079.62022 · doi:10.1239/jap/1127322029
[7] Casella G., Statistical Inference., 2. ed. (2003)
[8] Degen M., ASTIN Bulletin 37 (2) pp 265– (2007) · Zbl 1154.62077 · doi:10.2143/AST.37.2.2024067
[9] Eberlein E., Bernoulli 1 pp 281– (1995) · Zbl 0836.62107 · doi:10.2307/3318481
[10] EmbreChts P., Modelling Extremal Events (1997) · Zbl 0873.62116 · doi:10.1007/978-3-642-33483-2
[11] Furman E., Insurance: Mathematics and Economics 37 (3) pp 635– (2005) · Zbl 1129.91025 · doi:10.1016/j.insmatheco.2005.06.006
[12] Furman E., Astin Bulletin 36 (2) pp 433– (2006) · Zbl 1162.91373 · doi:10.2143/AST.36.2.2017929
[13] Hardy M. R., North American Actuarial Journal 5 (2) pp 41– (2001) · Zbl 1083.62530 · doi:10.1080/10920277.2001.10595984
[14] Jørgenson B., Statistical Propoerties of the Generalized Inverse Gaussian Distribution (1982) · doi:10.1007/978-1-4612-5698-4
[15] Klugman S., Loss Models., 2. ed. (2004) · Zbl 1141.62343
[16] Landsman Z., North American Actuarial Journal 7 (4) pp 55– (2003) · Zbl 1084.62512 · doi:10.1080/10920277.2003.10596118
[17] Landsman Z., Astin Bulletin 35 (1) pp 189– (2005) · Zbl 1099.62122 · doi:10.2143/AST.35.1.583172
[18] Lehmann E. L., Theory of Point Estimation., 2. ed. (1998) · Zbl 0916.62017
[19] Markowitz H., Portfolio Selection (1959)
[20] McNeil A. J., Quantitative Risk Management (2005) · Zbl 1089.91037
[21] Panjer H., Measurement of Risk, Solvency Requirements, and Allocation of Capital within Financial Conglomerates (2002)
[22] Panjer H. H., Insurance Risk Models (1992)
[23] Ramsay C. M., Transactions of Society of Actuaries 45 pp 305– (1993)
[24] Valdez E. A., Belgian Actuarial Bulletin 5 (1) pp 26– (2005)
[25] Vilar-Zanón J. L., Astin Bulletin 37 (2) pp 405– (2007) · Zbl 1154.62023 · doi:10.2143/AST.37.2.2024074
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