zbMATH — the first resource for mathematics

Weighted premium calculation principles. (English) Zbl 1141.91509
Summary: A prominent problem in actuarial science is to define, or describe, premium calculation principles (pcp’s) that satisfy certain properties. A frequently used resolution of the problem is achieved via distorting (e.g., lifting) the decumulative distribution function, and then calculating the expectation with respect to it. This leads to coherent pcp’s. Not every pcp can be arrived at in this way. Hence, in this paper we suggest and investigate a broad class of pcp’s, which we call weighted premiums, that are based on weighted loss distributions. Different weight functions lead to different pcp’s: any constant weight function leads to the net premium, an exponential weight function leads to the Esscher premium, and an indicator function leads to the conditional tail expectation. We investigate properties of weighted premiums such as ordering (and in particular loading), invariance. In addition, we derive explicit formulas for weighted premiums for several important classes of loss distributions, thus facilitating parametric statistical inference. We also provide hints and references on non-parametric statistical inferential tools in the area.

91B30 Risk theory, insurance (MSC2010)
PDF BibTeX Cite
Full Text: DOI
[1] Brazauskas, V., Jones, B.L., Puri, M.L., Zitikis, R., 2007. Estimating conditional tail expectations with actuarial applications in view. Journal of Statistical Planning and Inference (in press-a) · Zbl 1152.62027
[2] Brazauskas, V.; Jones, B.L.; Puri, M.L.; Zitikis, R, Nested \(L\)-statistics and their use in comparing the riskiness of portfolios, Scandinavian actuarial journal, 107, 162-179, (2007) · Zbl 1150.91025
[3] Brazauskas, V., Jones, B.L., Zitikis, R., 2007. Robustification of vector-valued empirical risk measures. Metron (in press-b) · Zbl 1416.62580
[4] Brazauskas, V., Jones, B.L., Zitikis, R., 2007. Robust fitting of claim severity distributions and the method of trimmed moments. North American Actuarial Journal (submitted for publication) · Zbl 1159.62067
[5] Brazauskas, V.; Kaiser, T., Discussion of “empirical estimation of risk measures and related quantities” by B.L. Jones and R. zitikis, North American actuarial journal, 8, July, 114-117, (2004) · Zbl 1085.62504
[6] Castillo, E.; Iglesias, A.; Ruíz-Cobo, R., Functional equations in applied sciences, (2005), Elsevier Amsterdam · Zbl 1071.39022
[7] Denuit, M.; Dhaene, J.; Goovaerts, M.; Kaas, R.; Laeven, R., Risk measurement with the equivalent utility principles, Statistics and decisions, 24, 1-25, (2006) · Zbl 1171.91326
[8] Denneberg, D., Non-additive measure and integral, (1994), Kluwer Dordrecht · Zbl 0826.28002
[9] Dhaene, J.; Vanduffel, S.; Goovaerts, M.J.; Kaas, R.; Tang, Q.; Vyncke, D., Risk measures and comonotonicity: A review, Stochastic models, 22, 573-606, (2006) · Zbl 1159.91403
[10] Furman, E.; Landsman, Z., On some risk-adjusted tail-based premium calculation principles, Journal of actuarial practice, 13, 175-191, (2007) · Zbl 1192.91116
[11] Furman, E.; Landsman, Z., Tail variance premium with applications for elliptical portfolio of risks, ASTIN bulletin, 36, 433-462, (2007) · Zbl 1162.91373
[12] Goovaerts, M.J.; Kaas, R.; Laeven, R.J.A.; Tang, Q., A comonotonic image of independence for additive risk measures, Insurance: mathematics and economics, 35, 581-594, (2004) · Zbl 1122.91341
[13] Goovaerts, M.J., Laeven, R.J.A., 2007. Actuarial risk measures for financial derivative pricing. Insurance: Mathematics and Economics, in press (doi:10.1016/j.insmatheco.2007.04.001) · Zbl 1152.91444
[14] Heilpern, S., A rank-dependent generalization of zero utility principle, Insurance: mathematics and economics, 33, 67-73, (2003) · Zbl 1058.91024
[15] Heilmann, W.R., Decision theoretic foundations of credibility theory, Insurance: mathematics and economics, 8, 77-95, (1989) · Zbl 0687.62087
[16] Helmers, R.; Mangku, I.W.; Zitikis, R., A non-parametric estimator for the doubly-periodic Poisson intensity function, Statistical methodology, 4, 481-492, (2007) · Zbl 1248.62137
[17] Jones, B.L.; Zitikis, R., Empirical estimation of risk measures and related quantities, North American actuarial journal, 7, October, 44-54, (2003) · Zbl 1084.62537
[18] Jones, B.L.; Zitikis, R., Testing for the order of risk measures: an application of \(L\)-statistics in actuarial science, Metron, 63, 193-211, (2005) · Zbl 1416.62587
[19] Jones, B.L.; Puri, M.L.; Zitikis, R., Testing hypotheses about the equality of several risk measure values with applications in insurance, Insurance: mathematics and economics, 38, 253-270, (2006) · Zbl 1088.62126
[20] Kamps, U., On a class of premium principles including the esscher premium, Scandinavian actuarial journal, 1, 75-80, (1998) · Zbl 1031.62505
[21] Kleiber, C.; Kotz, S., Statistical size distributions in economics and actuarial sciences, (2006), Wiley New York
[22] Lehmann, E.L., Some concepts of dependence, Annals of mathematical statistics, 37, 1137-1153, (1966) · Zbl 0146.40601
[23] Lu, Y.; Garrido, J., Doubly periodic non-homogeneous Poisson models for hurricane data, Statistical methodology, 2, 17-35, (2005) · Zbl 1248.86003
[24] Patil, G.P.; Ord, J.K., On size-biased sampling and related form-invariant weighted distributions, Sankhyā, series. B, 38, 48-61, (1976) · Zbl 0414.62015
[25] Patil, G.P.; Rao, C.R., Weighted distributions and size-biased sampling with applications to wildlife populations and human families, Biometrics, 34, 179-189, (1978) · Zbl 0384.62014
[26] Patil, G.P.; Rao, C.R.; Ratnaparkhi, M.V., On discrete weighted distributions and their use in model choice for observed data, Communications in statistics, A: theory and methods, 15, 907-918, (1986) · Zbl 0601.62022
[27] Patil, G.P., Rao, C.R., Zelen, M., 1986b. A computerized bibliography of weighted distributions and related weighted methods for statistical analysis and interpretations of encountered data, observational studies, representativeness issues, and reading inference. Center for Statistical Ecology and Environmental Statistics, Pennsylvania State University, University Park, PA
[28] Rao, C.R., Statistics and truth. putting chance to work, (1997), World Scientific Publishing River Edge, NJ · Zbl 0923.62001
[29] Serfling, R.J., Approximation theorems of mathematical statistics, (1980), Wiley New York · Zbl 0423.60030
[30] Tsanakas, A.; Desli, E., Risk measures and theories of choice, British actuarial journal, 9, 959-991, (2003)
[31] van Heerwaarden, A.E.; Kaas, R.; Goovaerts, M.J., Properties of the esscher premium calculation principle, Insurance: mathematics and economics, 8, 261-267, (1989) · Zbl 0686.62090
[32] Wang, S.S., Insurance pricing and increased limits ratemaking by proportional hazards transforms, Insurance: mathematics and economics, 17, 43-54, (1995) · Zbl 0837.62088
[33] Wang, S.S., Premium calculation by transforming the layer premium density, ASTIN bulletin, 26, 71-92, (1996)
[34] Young, V.R., Premium principles, ()
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.