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Second-order properties of the Haezendonck–Goovaerts risk measure for extreme risks

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  • Mao, Tiantian
  • Hu, Taizhong
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    Abstract

    The Haezendonck–Goovaerts risk measure is based on the premium calculation principle induced by an Orlicz norm, which is defined via an increasing and convex Young function and a parameter q∈(0,1) representing the confidence level. In this paper, we first reestablish the first-order expansions of the Haezendonck–Goovaerts risk measure for extreme risks with a power Young function in Tang and Yang (2012) in terms of the tail quantile function. Second, we are interested in the calculation of the second-order expansions of the Haezendonck–Goovaerts risk measure as q↑1. We only consider the case in which the risk variable belongs to the max-domain of attraction of an extreme value distribution.

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    Bibliographic Info

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

    Volume (Year): 51 (2012)
    Issue (Month): 2 ()
    Pages: 333-343

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    Handle: RePEc:eee:insuma:v:51:y:2012:i:2:p:333-343

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

    Related research

    Keywords: (Extended) regular variation; Extreme value theory; First-order expansion; Max-domain attraction; Second-order expansion; Second-order regular variation; Young function;

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    References

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    1. Goovaerts, Marc J. & Kaas, Rob & Dhaene, Jan & Tang, Qihe, 2004. "Some new classes of consistent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 34(3), pages 505-516, June.
    2. Tang, Qihe & Yang, Fan, 2012. "On the Haezendonck–Goovaerts risk measure for extreme risks," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 217-227.
    3. Bellini Fabio & Rosazza Gianin Emanuela, 2008. "Optimal portfolios with Haezendonck risk measures," Statistics & Risk Modeling, De Gruyter, vol. 26(2), pages 89-108, March.
    4. Haezendonck, J. & Goovaerts, M., 1982. "A new premium calculation principle based on Orlicz norms," Insurance: Mathematics and Economics, Elsevier, vol. 1(1), pages 41-53, January.
    5. Bellini, Fabio & Rosazza Gianin, Emanuela, 2008. "On Haezendonck risk measures," Journal of Banking & Finance, Elsevier, vol. 32(6), pages 986-994, June.
    6. Goovaerts, Marc & Linders, Daniël & Van Weert, Koen & Tank, Fatih, 2012. "On the interplay between distortion, mean value and Haezendonck–Goovaerts risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 10-18.
    7. Bellini, Fabio & Rosazza Gianin, Emanuela, 2012. "Haezendonck–Goovaerts risk measures and Orlicz quantiles," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 107-114.
    8. Hua, Lei & Joe, Harry, 2011. "Second order regular variation and conditional tail expectation of multiple risks," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 537-546.
    9. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
    10. Nam, Hee Seok & Tang, Qihe & Yang, Fan, 2011. "Characterization of upper comonotonicity via tail convex order," Insurance: Mathematics and Economics, Elsevier, vol. 48(3), pages 368-373, May.
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    Cited by:
    1. Hashorva, Enkelejd & Ling, Chengxiu & Peng, Zuoxiang, 2014. "Second-order tail asymptotics of deflated risks," Insurance: Mathematics and Economics, Elsevier, vol. 56(C), pages 88-101.

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