IDEAS home Printed from https://ideas.repec.org/a/spr/metcap/v25y2023i4d10.1007_s11009-023-10066-7.html
   My bibliography  Save this article

Asymptotics of Sum of Heavy-tailed Risks with Copulas

Author

Listed:
  • Fan Yang

    (University of Waterloo)

  • Yi Zhang

    (Zhejiang University)

Abstract

We study the tail asymptotics of the sum of two heavy-tailed random variables. The dependence structure is modeled by copulas with the so-called tail order property. Examples are presented to illustrate the approach. Further for each example we apply the main results to obtain the asymptotic expansions for Value-at-Risk of aggregate risk.

Suggested Citation

  • Fan Yang & Yi Zhang, 2023. "Asymptotics of Sum of Heavy-tailed Risks with Copulas," Methodology and Computing in Applied Probability, Springer, vol. 25(4), pages 1-23, December.
    • Handle: RePEc:spr:metcap:v:25:y:2023:i:4:d:10.1007_s11009-023-10066-7
    DOI: 10.1007/s11009-023-10066-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11009-023-10066-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11009-023-10066-7?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. Rafael Schmidt & Ulrich Stadtmüller, 2006. "Non‐parametric Estimation of Tail Dependence," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 33(2), pages 307-335, June.
    2. Juan-Juan Cai & John H. J. Einmahl & Laurens Haan & Chen Zhou, 2015. "Estimation of the marginal expected shortfall: the mean when a related variable is extreme," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 77(2), pages 417-442, March.
    3. Genest, Christian & Rivest, Louis-Paul, 1989. "A characterization of gumbel's family of extreme value distributions," Statistics & Probability Letters, Elsevier, vol. 8(3), pages 207-211, August.
    4. Abdelaati Daouia & Stéphane Girard & Gilles Stupfler, 2018. "Estimation of tail risk based on extreme expectiles," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 80(2), pages 263-292, March.
    5. Mainik, Georg & Embrechts, Paul, 2013. "Diversification in heavy-tailed portfolios: properties and pitfalls," Annals of Actuarial Science, Cambridge University Press, vol. 7(1), pages 26-45, March.
    6. Barbe, Philippe & Fougères, Anne-Laure & Genest, Christian, 2006. "On the Tail Behavior of Sums of Dependent Risks," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 361-373, November.
    7. Anthony W. Ledford & Jonathan A. Tawn, 1997. "Modelling Dependence within Joint Tail Regions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(2), pages 475-499.
    8. Georg Mainik & Ludger Rüschendorf, 2010. "On optimal portfolio diversification with respect to extreme risks," Finance and Stochastics, Springer, vol. 14(4), pages 593-623, December.
    9. Hua, Lei & Joe, Harry, 2011. "Tail order and intermediate tail dependence of multivariate copulas," Journal of Multivariate Analysis, Elsevier, vol. 102(10), pages 1454-1471, November.
    10. Alink, Stan & Lowe, Matthias & V. Wuthrich, Mario, 2004. "Diversification of aggregate dependent risks," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 77-95, August.
    11. Anthony W. Ledford & Jonathan A. Tawn, 2003. "Diagnostics for dependence within time series extremes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 521-543, May.
    12. 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.
    13. Geluk, J. & de Haan, L. & Resnick, S. & Starica, C., 1997. "Second-order regular variation, convolution and the central limit theorem," Stochastic Processes and their Applications, Elsevier, vol. 69(2), pages 139-159, September.
    14. Hua, Lei & Joe, Harry, 2014. "Strength of tail dependence based on conditional tail expectation," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 143-159.
    15. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    16. Tang, Qihe & Tang, Zhaofeng & Yang, Yang, 2019. "Sharp asymptotics for large portfolio losses under extreme risks," European Journal of Operational Research, Elsevier, vol. 276(2), pages 710-722.
    17. Christian Genest & Johanna Nešlehová & Jean-François Quessy, 2012. "Tests of symmetry for bivariate copulas," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(4), pages 811-834, August.
    18. Mao, Tiantian & Yang, Fan, 2015. "Risk concentration based on Expectiles for extreme risks under FGM copula," Insurance: Mathematics and Economics, Elsevier, vol. 64(C), pages 429-439.
    19. Yanchun Zhao & Tiantian Mao & Fan Yang, 2021. "Estimation of the Haezendonck-Goovaerts risk measure for extreme risks," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2021(7), pages 599-622, August.
    Full references (including those not matched with items on IDEAS)

    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. Fan Yang & Yi Zhang, 2024. "Asymptotics of Sum of Heavy-tailed Risks with Copulas," Papers 2411.09657, arXiv.org.
    2. 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.
    3. Li, Haijun & Wu, Peiling, 2013. "Extremal dependence of copulas: A tail density approach," Journal of Multivariate Analysis, Elsevier, vol. 114(C), pages 99-111.
    4. Ji, Liuyan & Tan, Ken Seng & Yang, Fan, 2021. "Tail dependence and heavy tailedness in extreme risks," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 282-293.
    5. Mainik Georg & Rüschendorf Ludger, 2012. "Ordering of multivariate risk models with respect to extreme portfolio losses," Statistics & Risk Modeling, De Gruyter, vol. 29(1), pages 73-106, March.
    6. Haoyu Chen & Tiantian Mao & Fan Yang, 2024. "Estimation of the Adjusted Standard-deviatile for Extreme Risks," Papers 2411.07203, arXiv.org.
    7. Das Bikramjit & Fasen-Hartmann Vicky, 2019. "Conditional excess risk measures and multivariate regular variation," Statistics & Risk Modeling, De Gruyter, vol. 36(1-4), pages 1-23, December.
    8. Haoyu Chen & Tiantian Mao & Fan Yang, 2024. "Estimation of the adjusted standard‐deviatile for extreme risks," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 51(2), pages 643-671, June.
    9. Xia Han & Liyuan Lin & Hao Wang & Ruodu Wang, 2024. "Diversification quotient based on expectiles," Papers 2411.14646, arXiv.org, revised Nov 2024.
    10. 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.
    11. Cui, Hengxin & Tan, Ken Seng & Yang, Fan & Zhou, Chen, 2022. "Asymptotic analysis of portfolio diversification," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 302-325.
    12. Shaoying Chen & Yang Yang & Zhimin Zhang, 2024. "Asymptotics for credit portfolio losses due to defaults in a multi-sector model," Annals of Operations Research, Springer, vol. 337(1), pages 23-44, June.
    13. Daouia, Abdelaati & Girard, Stéphane & Stupfler, Gilles, 2018. "Tail expectile process and risk assessment," TSE Working Papers 18-944, Toulouse School of Economics (TSE).
    14. Xia Han & Liyuan Lin & Ruodu Wang, 2023. "Diversification quotients based on VaR and ES," Papers 2301.03517, arXiv.org, revised May 2023.
    15. Han, Xia & Lin, Liyuan & Wang, Ruodu, 2023. "Diversification quotients based on VaR and ES," Insurance: Mathematics and Economics, Elsevier, vol. 113(C), pages 185-197.
    16. Daouia, Abdelaati & Girard, Stéphane & Stupfler, Gilles, 2017. "Extreme M-quantiles as risk measures: From L1 to Lp optimization," TSE Working Papers 17-841, Toulouse School of Economics (TSE).
    17. Hua, Lei & Joe, Harry, 2014. "Strength of tail dependence based on conditional tail expectation," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 143-159.
    18. de Haan, Laurens & Neves, Cláudia & Peng, Liang, 2008. "Parametric tail copula estimation and model testing," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1260-1275, July.
    19. 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.
    20. Hua, Lei & Polansky, Alan & Pramanik, Paramahansa, 2019. "Assessing bivariate tail non-exchangeable dependence," Statistics & Probability Letters, Elsevier, vol. 155(C), pages 1-1.

    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:spr:metcap:v:25:y:2023:i:4:d:10.1007_s11009-023-10066-7. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.