IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v50y2012i1p217-227.html
   My bibliography  Save this article

On the Haezendonck–Goovaerts risk measure for extreme risks

Author

Listed:
  • Tang, Qihe
  • Yang, Fan

Abstract

In this paper, we are interested in the calculation of the Haezendonck–Goovaerts risk measure, which is defined via a convex Young function and a parameter q∈(0,1) representing the confidence level. We mainly focus on the case in which the risk variable follows a distribution function from a max-domain of attraction. For this case, we restrict the Young function to be a power function and we derive exact asymptotics for the Haezendonck–Goovaerts risk measure as q↑1. As a subsidiary, we also consider the case with an exponentially distributed risk variable and a general Young function, and we obtain an analytical expression for the Haezendonck–Goovaerts risk measure.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:insuma:v:50:y:2012:i:1:p:217-227
    DOI: 10.1016/j.insmatheco.2011.11.007
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668711001296
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2011.11.007?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Krätschmer, Volker & Zähle, Henryk, 2011. "Sensitivity of risk measures with respect to the normal approximation of total claim distributions," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 335-344.
    2. 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.
    3. Davis, Richard & Resnick, Sidney, 1988. "Extremes of moving averages of random variables from the domain of attraction of the double exponential distribution," Stochastic Processes and their Applications, Elsevier, vol. 30(1), pages 41-68, November.
    4. Rob Kaas & Marc Goovaerts & Jan Dhaene & Michel Denuit, 2008. "Modern Actuarial Risk Theory," Springer Books, Springer, edition 2, number 978-3-540-70998-5, September.
    5. Bellini, Fabio & Rosazza Gianin, Emanuela, 2008. "On Haezendonck risk measures," Journal of Banking & Finance, Elsevier, vol. 32(6), pages 986-994, June.
    6. Bellini Fabio & Rosazza Gianin Emanuela, 2008. "Optimal portfolios with Haezendonck risk measures," Statistics & Risk Modeling, De Gruyter, vol. 26(2), pages 89-108, March.
    7. Asimit, Alexandru V. & Furman, Edward & Tang, Qihe & Vernic, Raluca, 2011. "Asymptotics for risk capital allocations based on Conditional Tail Expectation," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 310-324.
    8. 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.
    9. Enkelejd Hashorva & Anthony G. Pakes & Qihe Tang, 2010. "Asymptotics of Random Contractions," Papers 1008.0126, arXiv.org.
    10. Yannick Malevergne & Didier Sornette, 2006. "Extreme Financial Risks : From Dependence to Risk Management," Post-Print hal-02298069, HAL.
    11. 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.
    12. Hashorva, Enkelejd & Pakes, Anthony G. & Tang, Qihe, 2010. "Asymptotics of random contractions," Insurance: Mathematics and Economics, Elsevier, vol. 47(3), pages 405-414, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Niushan Gao & Cosimo Munari & Foivos Xanthos, 2019. "Stability properties of Haezendonck-Goovaerts premium principles," Papers 1909.10735, arXiv.org, revised Aug 2020.
    2. Ling, Chengxiu & Peng, Zuoxiang, 2016. "Tail asymptotics of generalized deflated risks with insurance applications," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 220-231.
    3. Mao, Tiantian & Hu, Taizhong, 2012. "Second-order properties of the Haezendonck–Goovaerts risk measure for extreme risks," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 333-343.
    4. Samuel Drapeau & Mekonnen Tadese, 2019. "Dual Representation of Expectile based Expected Shortfall and Its Properties," Papers 1911.03245, arXiv.org.
    5. Samuel Drapeau & Mekonnen Tadese, 2019. "Relative Bound and Asymptotic Comparison of Expectile with Respect to Expected Shortfall," Papers 1906.09729, arXiv.org, revised Jun 2020.
    6. Xun, Li & Zhou, Yangzhi & Zhou, Yong, 2019. "A generalization of Expected Shortfall based capital allocation," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 193-199.
    7. Tadese, Mekonnen & Drapeau, Samuel, 2020. "Relative bound and asymptotic comparison of expectile with respect to expected shortfall," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 387-399.
    8. Liu, Qing & Peng, Liang & Wang, Xing, 2017. "Haezendonck–Goovaerts risk measure with a heavy tailed loss," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 28-47.
    9. 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.
    10. Ahn, Jae Youn & Shyamalkumar, Nariankadu D., 2014. "Asymptotic theory for the empirical Haezendonck–Goovaerts risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 78-90.
    11. Wei, Li & Yuan, Zhongyi, 2016. "The loss given default of a low-default portfolio with weak contagion," Insurance: Mathematics and Economics, Elsevier, vol. 66(C), pages 113-123.
    12. 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.
    13. Tang, Qihe & Yang, Fan, 2014. "Extreme value analysis of the Haezendonck–Goovaerts risk measure with a general Young function," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 311-320.
    14. Leipus, Remigijus & Paukštys, Saulius & Šiaulys, Jonas, 2021. "Tails of higher-order moments of sums with heavy-tailed increments and application to the Haezendonck–Goovaerts risk measure," Statistics & Probability Letters, Elsevier, vol. 170(C).
    15. Mao, Tiantian & Stupfler, Gilles & Yang, Fan, 2023. "Asymptotic properties of generalized shortfall risk measures for heavy-tailed risks," Insurance: Mathematics and Economics, Elsevier, vol. 111(C), pages 173-192.
    16. Cheung, Ka Chun & Lo, Ambrose, 2013. "General lower bounds on convex functionals of aggregate sums," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 884-896.
    17. Gao, Niushan & Munari, Cosimo & Xanthos, Foivos, 2020. "Stability properties of Haezendonck–Goovaerts premium principles," Insurance: Mathematics and Economics, Elsevier, vol. 94(C), pages 94-99.
    18. Gómez, Fabio & Tang, Qihe & Tong, Zhiwei, 2022. "The gradient allocation principle based on the higher moment risk measure," Journal of Banking & Finance, Elsevier, vol. 143(C).
    19. Wang, Xing & Peng, Liang, 2016. "Inference for intermediate Haezendonck–Goovaerts risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 231-240.
    20. Xun, Li & Jiang, Renqiao & Guo, Jianhua, 2021. "The conditional Haezendonck–Goovaerts risk measure," Statistics & Probability Letters, Elsevier, vol. 169(C).
    21. Asimit, Alexandru V. & Chen, Yiqing, 2015. "Asymptotic results for conditional measures of association of a random sum," Insurance: Mathematics and Economics, Elsevier, vol. 60(C), pages 11-18.
    22. Bellini, Fabio & Rosazza Gianin, Emanuela, 2012. "Haezendonck–Goovaerts risk measures and Orlicz quantiles," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 107-114.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Bellini, Fabio & Rosazza Gianin, Emanuela, 2012. "Haezendonck–Goovaerts risk measures and Orlicz quantiles," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 107-114.
    3. Ahn, Jae Youn & Shyamalkumar, Nariankadu D., 2014. "Asymptotic theory for the empirical Haezendonck–Goovaerts risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 78-90.
    4. Mao, Tiantian & Hu, Taizhong, 2012. "Second-order properties of the Haezendonck–Goovaerts risk measure for extreme risks," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 333-343.
    5. 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.
    6. Yang, Yang & Hashorva, Enkelejd, 2013. "Extremes and products of multivariate AC-product risks," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 312-319.
    7. Cheung, Ka Chun & Lo, Ambrose, 2013. "General lower bounds on convex functionals of aggregate sums," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 884-896.
    8. Tang, Qihe & Yang, Fan, 2014. "Extreme value analysis of the Haezendonck–Goovaerts risk measure with a general Young function," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 311-320.
    9. Zhiping Chen & Qianhui Hu & Ruiyue Lin, 2016. "Performance ratio-based coherent risk measure and its application," Quantitative Finance, Taylor & Francis Journals, vol. 16(5), pages 681-693, May.
    10. Constantinescu, Corina & Hashorva, Enkelejd & Ji, Lanpeng, 2011. "Archimedean copulas in finite and infinite dimensions—with application to ruin problems," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 487-495.
    11. Roy Cerqueti & Raffaella Coppier & Gustavo Piga, 2012. "Corruption, growth and ethnic fractionalization: a theoretical model," Journal of Economics, Springer, vol. 106(2), pages 153-181, June.
    12. Bellini, Fabio & Laeven, Roger J.A. & Rosazza Gianin, Emanuela, 2021. "Dynamic robust Orlicz premia and Haezendonck–Goovaerts risk measures," European Journal of Operational Research, Elsevier, vol. 291(2), pages 438-446.
    13. Bellini Fabio & Rosazza Gianin Emanuela, 2008. "Optimal portfolios with Haezendonck risk measures," Statistics & Risk Modeling, De Gruyter, vol. 26(2), pages 89-108, March.
    14. Wang, Xing & Peng, Liang, 2016. "Inference for intermediate Haezendonck–Goovaerts risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 231-240.
    15. Xun, Li & Zhou, Yangzhi & Zhou, Yong, 2019. "A generalization of Expected Shortfall based capital allocation," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 193-199.
    16. Krätschmer, Volker & Zähle, Henryk, 2011. "Sensitivity of risk measures with respect to the normal approximation of total claim distributions," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 335-344.
    17. Canna, Gabriele & Centrone, Francesca & Rosazza Gianin, Emanuela, 2021. "Haezendonck-Goovaerts capital allocation rules," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 173-185.
    18. Hashorva, Enkelejd, 2015. "Extremes of aggregated Dirichlet risks," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 334-345.
    19. Xun, Li & Jiang, Renqiao & Guo, Jianhua, 2021. "The conditional Haezendonck–Goovaerts risk measure," Statistics & Probability Letters, Elsevier, vol. 169(C).
    20. Guillén, Montserrat & Sarabia, José María & Prieto, Faustino, 2013. "Simple risk measure calculations for sums of positive random variables," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 273-280.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:insuma:v:50:y:2012:i:1:p:217-227. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/505554 .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.