IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v50y2012i1p139-149.html
   My bibliography  Save this article

Risk concentration of aggregated dependent risks: The second-order properties

Author

Listed:
  • Tong, Bin
  • Wu, Chongfeng
  • Xu, Weidong

Abstract

Under the current regulatory guidelines for banks and insurance companies, the quantification of diversification benefits due to risk aggregation plays a prominent role. In this paper we establish second-order approximation of risk concentration associated with a random vector X:=(X1,X2,…,Xd) in terms of Value at Risk (VaR) within the methodological framework of second-order regular variation and the theory of Archimedean copula. Moreover, we find that the rate of convergence of the first-order approximation of risk concentration depends on the the interplay between the tail behavior of the marginal loss random variables and their dependence structure. Specifically, we find that the rate of convergence is determined by either the second-order parameter (ρ1) of Archimedean copula generator or the second-order parameter (ρ) of the tail margins, leading to either the so-called dependence dominated case or margin dominated case.

Suggested Citation

  • Tong, Bin & Wu, Chongfeng & Xu, Weidong, 2012. "Risk concentration of aggregated dependent risks: The second-order properties," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 139-149.
    • Handle: RePEc:eee:insuma:v:50:y:2012:i:1:p:139-149
    DOI: 10.1016/j.insmatheco.2011.11.002
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668711001235
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2011.11.002?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. Alink, Stan & Löwe, Matthias & Wüthrich, Mario V., 2005. "Analysis of the Expected Shortfall of Aggregate Dependent Risks," ASTIN Bulletin, Cambridge University Press, vol. 35(1), pages 25-43, May.
    2. Rustam Ibragimov, 2009. "Portfolio diversification and value at risk under thick-tailedness," Quantitative Finance, Taylor & Francis Journals, vol. 9(5), pages 565-580.
    3. Degen, Matthias & Embrechts, Paul & Lambrigger, Dominik D., 2007. "The Quantitative Modeling of Operational Risk: Between G-and-H and EVT," ASTIN Bulletin, Cambridge University Press, vol. 37(2), pages 265-291, November.
    4. Zhou, Chen, 2010. "Dependence structure of risk factors and diversification effects," Insurance: Mathematics and Economics, Elsevier, vol. 46(3), pages 531-540, June.
    5. Jansen, Dennis W & de Vries, Casper G, 1991. "On the Frequency of Large Stock Returns: Putting Booms and Busts into Perspective," The Review of Economics and Statistics, MIT Press, vol. 73(1), pages 18-24, February.
    6. Degen, Matthias & Lambrigger, Dominik D. & Segers, Johan, 2010. "Risk concentration and diversification: Second-order properties," Insurance: Mathematics and Economics, Elsevier, vol. 46(3), pages 541-546, June.
    7. 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.
    8. Ibragimov, Rustam & Walden, Johan, 2008. "Portfolio diversification under local and moderate deviations from power laws," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 594-599, April.
    9. Xavier Gabaix, 2009. "Power Laws in Economics and Finance," Annual Review of Economics, Annual Reviews, vol. 1(1), pages 255-294, May.
    10. Embrechts, Paul & Neslehová, Johanna & Wüthrich, Mario V., 2009. "Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness," Insurance: Mathematics and Economics, Elsevier, vol. 44(2), pages 164-169, April.
    11. Walden, Johan & Ibragimov, Rustam, 2008. "Portfolio Diversification under Local and Moderate Deviations from Power Laws," Scholarly Articles 2640586, Harvard University Department of Economics.
    12. Ibragimov, Rustam & Walden, Johan, 2007. "The limits of diversification when losses may be large," Journal of Banking & Finance, Elsevier, vol. 31(8), pages 2551-2569, August.
    13. Xavier Gabaix & Parameswaran Gopikrishnan & Vasiliki Plerou & H. Eugene Stanley, 2006. "Institutional Investors and Stock Market Volatility," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 121(2), pages 461-504.
    14. Degen, Matthias & Lambrigger, Dominik D. & Segers, Johan, 2010. "Risk concentration and diversification: Second-order properties," LIDAM Reprints ISBA 2010011, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    15. Xavier Gabaix & Parameswaran Gopikrishnan & Vasiliki Plerou & H. Eugene Stanley, 2003. "A theory of power-law distributions in financial market fluctuations," Nature, Nature, vol. 423(6937), pages 267-270, May.
    16. 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.
    17. Chavez-Demoulin, V. & Embrechts, P. & Neslehova, J., 2006. "Quantitative models for operational risk: Extremes, dependence and aggregation," Journal of Banking & Finance, Elsevier, vol. 30(10), pages 2635-2658, October.
    18. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
    19. Ibragimov, Rustam & Walden, Johan, 2007. "The limits of diversification when losses may be large," Scholarly Articles 2624460, Harvard University Department of Economics.
    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. Bikramjit Das & Marie Kratz, 2017. "Diversification benefits under multivariate second order regular variation," Working Papers hal-01520655, HAL.
    2. Mario Fortin & Marcelin Joanis & Philippe Kabore & Luc Savard, 2022. "Determination of Quebec's Quarterly Real GDP and Analysis of the Business Cycle, 1948–1980," Journal of Business Cycle Research, Springer;Centre for International Research on Economic Tendency Surveys (CIRET), vol. 18(3), pages 261-288, November.
    3. Gilles Boevi Koumou & Georges Dionne, 2022. "Coherent Diversification Measures in Portfolio Theory: An Axiomatic Foundation," Risks, MDPI, vol. 10(11), pages 1-19, October.
    4. Das, Bikramjit & Kratz, Marie, 2017. "Diversification benefits under multivariate second order regular variation," ESSEC Working Papers WP1706, ESSEC Research Center, ESSEC Business School.

    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. Ibragimov, Rustam, 2014. "On the robustness of location estimators in models of firm growth under heavy-tailedness," Journal of Econometrics, Elsevier, vol. 181(1), pages 25-33.
    2. Ibragimov, Rustam & Prokhorov, Artem, 2016. "Heavy tails and copulas: Limits of diversification revisited," Economics Letters, Elsevier, vol. 149(C), pages 102-107.
    3. Chollete, Lorán & de la Peña, Victor & Lu, Ching-Chih, 2012. "International diversification: An extreme value approach," Journal of Banking & Finance, Elsevier, vol. 36(3), pages 871-885.
    4. Rustam Ibragimov & Johan Walden, 2011. "Value at risk and efficiency under dependence and heavy-tailedness: models with common shocks," Annals of Finance, Springer, vol. 7(3), pages 285-318, August.
    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. Chollete, Loran & de la Pena , Victor & Lu, Ching-Chih, 2009. "International Diversification: An Extreme Value Approach," UiS Working Papers in Economics and Finance 2009/26, University of Stavanger.
    7. Ibragimov, Rustam & Jaffee, Dwight & Walden, Johan, 2011. "Diversification disasters," Journal of Financial Economics, Elsevier, vol. 99(2), pages 333-348, February.
    8. Moore, Kyle & Sun, Pengfei & de Vries, Casper G. & Zhou, Chen, 2013. "The cross-section of tail risks in stock returns," MPRA Paper 45592, University Library of Munich, Germany.
    9. Ankudinov, Andrei & Ibragimov, Rustam & Lebedev, Oleg, 2017. "Sanctions and the Russian stock market," Research in International Business and Finance, Elsevier, vol. 40(C), pages 150-162.
    10. Rustam Ibragimov & Johan Walden, 2010. "Optimal Bundling Strategies Under Heavy-Tailed Valuations," Management Science, INFORMS, vol. 56(11), pages 1963-1976, November.
    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. Ankudinov, Andrei & Ibragimov, Rustam & Lebedev, Oleg, 2017. "Heavy tails and asymmetry of returns in the Russian stock market," Emerging Markets Review, Elsevier, vol. 32(C), pages 200-219.
    13. Rustam Ibragimov & Marat Ibragimov & Rufat Khamidov, 2010. "Measuring Inequality in CIS Countries: Theory and Empirics," wiiw Balkan Observatory Working Papers 88, The Vienna Institute for International Economic Studies, wiiw.
    14. Chollete, Lorán & de la Peña, Victor & Lu, Ching-Chih, 2011. "International diversification: A copula approach," Journal of Banking & Finance, Elsevier, vol. 35(2), pages 403-417, February.
    15. Kyle Moore & Pengfei Sun & Casper de Vries & Chen Zhou, 2013. "Shape Homogeneity and Scale Heterogeneity of Downside Tail Risk," Working Papers 13-13, Chapman University, Economic Science Institute.
    16. Ibragimov, Rustam & Walden, Johan, 2008. "Portfolio diversification under local and moderate deviations from power laws," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 594-599, April.
    17. Moore, Kyle & Sun, Pengei & de Vries, Casper G. & Zhou, Chen, 2013. "The drivers of downside equity tail risk," MPRA Paper 45591, University Library of Munich, Germany.
    18. Daníelsson, Jón & Jorgensen, Bjørn N. & Samorodnitsky, Gennady & Sarma, Mandira & de Vries, Casper G., 2013. "Fat tails, VaR and subadditivity," Journal of Econometrics, Elsevier, vol. 172(2), pages 283-291.
    19. Ankudinov, Andrei & Ibragimov, Rustam & Lebedev, Oleg, 2017. "Extreme movements of the Russian stock market and their consequences for management and economic modeling," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 45, pages 75-92.
    20. Chen, Zhimin & Ibragimov, Rustam, 2019. "One country, two systems? The heavy-tailedness of Chinese A- and H- share markets," Emerging Markets Review, Elsevier, vol. 38(C), pages 115-141.

    More about this item

    Keywords

    Aggregated risk; Risk concentration; Archimedean copula; Second-order regular variation; Dependence structure;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • G21 - Financial Economics - - Financial Institutions and Services - - - Banks; Other Depository Institutions; Micro Finance Institutions; Mortgages

    Statistics

    Access and download statistics

    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:139-149. 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.