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Weighted premium calculation principles

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  • Furman, Edward
  • Zitikis, Ricardas
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    Abstract

    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.

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    Bibliographic Info

    Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

    Volume (Year): 42 (2008)
    Issue (Month): 1 (February)
    Pages: 459-465

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    Handle: RePEc:eee:insuma:v:42:y:2008:i:1:p:459-465

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    Web page: http://www.elsevier.com/locate/inca/505554

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    1. Heilpern, S., 2003. "A rank-dependent generalization of zero utility principle," Insurance: Mathematics and Economics, Elsevier, vol. 33(1), pages 67-73, August.
    2. Bruce L. Jones & Ricardas Zitikis, 2005. "Testing for the order of risk measures: an application of L-statistics in actuarial science," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(2), pages 193-211.
    3. Denuit Michel & Dhaene Jan & Goovaerts Marc & Kaas Rob & Laeven Roger, 2006. "Risk measurement with equivalent utility principles," Statistics & Risk Modeling, De Gruyter, vol. 24(1/2006), pages 25, July.
    4. Jones, Bruce L. & Puri, Madan L. & Zitikis, Ricardas, 2006. "Testing hypotheses about the equality of several risk measure values with applications in insurance," Insurance: Mathematics and Economics, Elsevier, vol. 38(2), pages 253-270, April.
    5. Shaun, Wang, 1995. "Insurance pricing and increased limits ratemaking by proportional hazards transforms," Insurance: Mathematics and Economics, Elsevier, vol. 17(1), pages 43-54, August.
    6. Marc J. Goovaerts & Rob Kaas & Roger J.A. Laeven & Qihe Tang, 2004. "A Comonotonic Image of Independence for Additive Risk Measures," Tinbergen Institute Discussion Papers 04-030/4, Tinbergen Institute.
    7. Heilmann, Wolf-Rudiger, 1989. "Decision theoretic foundations of credibility theory," Insurance: Mathematics and Economics, Elsevier, vol. 8(1), pages 77-95, March.
    8. Van Heerwaarden, A. E. & Kaas, R. & Goovaerts, M. J., 1989. "Properties of the Esscher premium calculation principle," Insurance: Mathematics and Economics, Elsevier, vol. 8(4), pages 261-267, December.
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    Cited by:
    1. Kim, Joseph H.T. & Jeon, Yongho, 2013. "Credibility theory based on trimming," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 36-47.
    2. Zaks, Yaniv & Tsanakas, Andreas, 2014. "Optimal capital allocation in a hierarchical corporate structure," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 48-55.
    3. Bolance, Catalina & Guillen, Montserrat & Pelican, Elena & Vernic, Raluca, 2008. "Skewed bivariate models and nonparametric estimation for the CTE risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 386-393, December.
    4. Kaluszka, M. & Laeven, R.J.A. & Okolewski, A., 2012. "A note on weighted premium calculation principles," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 379-381.
    5. Li, Xiaohu & Lin, Jianhua, 2011. "Stochastic orders in time transformed exponential models with applications," Insurance: Mathematics and Economics, Elsevier, vol. 49(1), pages 47-52, July.
    6. Sendov, Hristo S. & Wang, Ying & Zitikis, Ricardas, 2011. "Log-supermodularity of weight functions, ordering weighted losses, and the loading monotonicity of weighted premiums," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 257-264, March.
    7. Li, Hao & Melnikov, Alexander, 2014. "Polynomial extensions of distributions and their applications in actuarial and financial modeling," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 250-260.
    8. Furman, Edward & Landsman, Zinoviy, 2010. "Multivariate Tweedie distributions and some related capital-at-risk analyses," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 351-361, April.
    9. Furman, Edward & Zitikis, Ricardas, 2008. "Weighted risk capital allocations," Insurance: Mathematics and Economics, Elsevier, vol. 43(2), pages 263-269, October.
    10. Choo, Weihao & de Jong, Piet, 2009. "Loss reserving using loss aversion functions," Insurance: Mathematics and Economics, Elsevier, vol. 45(2), pages 271-277, October.
    11. Alexandru V. Asimit & Raluca Vernic & Riċardas Zitikis, 2013. "Evaluating Risk Measures and Capital Allocations Based on Multi-Losses Driven by a Heavy-Tailed Background Risk: The Multivariate Pareto-II Model," Risks, MDPI, Open Access Journal, vol. 1(1), pages 14-33, March.
    12. Sordo, Miguel A., 2009. "Comparing tail variabilities of risks by means of the excess wealth order," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 466-469, December.
    13. Sordo, Miguel A. & Suárez-Llorens, Alfonso, 2011. "Stochastic comparisons of distorted variability measures," Insurance: Mathematics and Economics, Elsevier, vol. 49(1), pages 11-17, July.
    14. Asimit, Alexandru V. & Furman, Edward & Vernic, Raluca, 2010. "On a multivariate Pareto distribution," Insurance: Mathematics and Economics, Elsevier, vol. 46(2), pages 308-316, April.
    15. Galeotti, Marcello & Gürtler, Marc & Winkelvos, Christine, 2009. "Accuracy of premium calculation models for CAT bonds: An empirical analysis," Working Papers IF29V4, Technische Universität Braunschweig, Institute of Finance.

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