IDEAS home Printed from https://ideas.repec.org/a/eee/insuma/v81y2018icp27-35.html
   My bibliography  Save this article

A multivariate tail covariance measure for elliptical distributions

Author

Listed:
  • Landsman, Zinoviy
  • Makov, Udi
  • Shushi, Tomer

Abstract

This paper introduces a multivariate tail covariance (MTCov) measure, which is a matrix-valued risk measure designed to explore the tail dispersion of multivariate loss distributions. The MTCov is the second multivariate tail conditional moment around the MTCE, the multivariate tail conditional expectation (MTCE) risk measure. Although MTCE was recently introduced in Landsman et al. (2016a), in this paper we essentially generalize it, allowing for quantile levels to obtain the different values corresponded to each risk. The MTCov measure, which is also defined for the set of different quantile levels, allows us to investigate more deeply the tail of multivariate distributions, since it focuses on the variance–covariance dependence structure of a system of dependent risks. As a natural extension, we also introduced the multivariate tail correlation matrix (MTCorr). The properties of this risk measure are explored and its explicit closed-form expression is derived for the elliptical family of distributions. As a special case, we consider the normal, Student-t and Laplace distributions, prevalent in actuarial science and finance. The results are illustrated numerically with data of some stock returns.

Suggested Citation

  • Landsman, Zinoviy & Makov, Udi & Shushi, Tomer, 2018. "A multivariate tail covariance measure for elliptical distributions," Insurance: Mathematics and Economics, Elsevier, vol. 81(C), pages 27-35.
    • Handle: RePEc:eee:insuma:v:81:y:2018:i:c:p:27-35
    DOI: 10.1016/j.insmatheco.2018.04.002
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167668717302561
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.insmatheco.2018.04.002?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. Ignatieva, Katja & Landsman, Zinoviy, 2015. "Estimating the tails of loss severity via conditional risk measures for the family of symmetric generalised hyperbolic distributions," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 172-186.
    2. Furman, Edward & Landsman, Zinoviy, 2006. "Tail Variance Premium with Applications for Elliptical Portfolio of Risks," ASTIN Bulletin, Cambridge University Press, vol. 36(2), pages 433-462, November.
    3. Zinoviy Landsman & Emiliano Valdez, 2003. "Tail Conditional Expectations for Elliptical Distributions," North American Actuarial Journal, Taylor & Francis Journals, vol. 7(4), pages 55-71.
    4. Landsman, Zinoviy & Makov, Udi & Shushi, Tomer, 2016. "Tail conditional moments for elliptical and log-elliptical distributions," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 179-188.
    5. Landsman, Zinoviy & Makov, Udi & Shushi, Tomer, 2016. "Multivariate tail conditional expectation for elliptical distributions," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 216-223.
    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. Yeshunying Wang & Chuancun Yin, 2021. "A New Class of Multivariate Elliptically Contoured Distributions with Inconsistency Property," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1377-1407, December.
    2. Baishuai Zuo & Chuancun Yin & Narayanaswamy Balakrishnan, 2020. "Explicit expressions for joint moments of $n$-dimensional elliptical distributions," Papers 2007.09349, arXiv.org, revised Aug 2020.
    3. Ogasawara, Haruhiko, 2021. "A non-recursive formula for various moments of the multivariate normal distribution with sectional truncation," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    4. Eric C. K. Cheung & Oscar Peralta & Jae-Kyung Woo, 2021. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Papers 2201.11122, arXiv.org.
    5. Chuancun Yin, 2019. "Stochastic ordering of Gini indexes for multivariate elliptical random variables," Papers 1908.01943, arXiv.org, revised Sep 2019.
    6. Baishuai Zuo & Chuancun Yin, 2020. "Conditional tail risk expectations for location-scale mixture of elliptical distributions," Papers 2007.09350, arXiv.org.
    7. Baishuai Zuo & Chuancun Yin, 2022. "Multivariate doubly truncated moments for generalized skew-elliptical distributions with application to multivariate tail conditional risk measures," Papers 2203.00839, arXiv.org.
    8. Ling, Chengxiu, 2019. "Asymptotics of multivariate conditional risk measures for Gaussian risks," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 205-215.
    9. Longobardi, Maria & Pellerey, Franco, 2019. "On the role of dependence in residual lifetimes," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 56-64.
    10. Baishuai Zuo & Chuancun Yin, 2021. "Multivariate tail covariance for generalized skew-elliptical distributions," Papers 2103.05201, arXiv.org.
    11. Baishuai Zuo & Chuancun Yin & Jing Yao, 2023. "Multivariate range Value-at-Risk and covariance risk measures for elliptical and log-elliptical distributions," Papers 2305.09097, arXiv.org.
    12. Roozegar, Roohollah & Balakrishnan, Narayanaswamy & Jamalizadeh, Ahad, 2020. "On moments of doubly truncated multivariate normal mean–variance mixture distributions with application to multivariate tail conditional expectation," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    13. Cheung, Eric C.K. & Peralta, Oscar & Woo, Jae-Kyung, 2022. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 364-389.

    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. Baishuai Zuo & Chuancun Yin & Jing Yao, 2023. "Multivariate range Value-at-Risk and covariance risk measures for elliptical and log-elliptical distributions," Papers 2305.09097, arXiv.org.
    2. Baishuai Zuo & Chuancun Yin, 2022. "Doubly truncated moment risk measures for elliptical distributions," Papers 2203.01091, arXiv.org.
    3. Baishuai Zuo & Chuancun Yin, 2020. "Conditional tail risk expectations for location-scale mixture of elliptical distributions," Papers 2007.09350, arXiv.org.
    4. Shushi, Tomer, 2019. "The Minkowski length of a spherical random vector," Statistics & Probability Letters, Elsevier, vol. 153(C), pages 104-107.
    5. Mohammed, Nawaf & Furman, Edward & Su, Jianxi, 2021. "Can a regulatory risk measure induce profit-maximizing risk capital allocations? The case of conditional tail expectation," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 425-436.
    6. Eini, Esmat Jamshidi & Khaloozadeh, Hamid, 2021. "The tail mean–variance optimal portfolio selection under generalized skew-elliptical distribution," Insurance: Mathematics and Economics, Elsevier, vol. 98(C), pages 44-50.
    7. Huang, Zhenzhen & Wei, Pengyu & Weng, Chengguo, 2024. "Tail mean-variance portfolio selection with estimation risk," Insurance: Mathematics and Economics, Elsevier, vol. 116(C), pages 218-234.
    8. Xiangyu Han & Chuancun Yin, 2022. "Tail Conditional Moments for Location-Scale Mixture of Elliptical Distributions," Mathematics, MDPI, vol. 10(4), pages 1-21, February.
    9. Kim, Joseph H.T. & Kim, So-Yeun, 2019. "Tail risk measures and risk allocation for the class of multivariate normal mean–variance mixture distributions," Insurance: Mathematics and Economics, Elsevier, vol. 86(C), pages 145-157.
    10. Jiang, Chun-Fu & Peng, Hong-Yi & Yang, Yu-Kuan, 2016. "Tail variance of portfolio under generalized Laplace distribution," Applied Mathematics and Computation, Elsevier, vol. 282(C), pages 187-203.
    11. Landsman, Zinoviy & Makov, Udi & Shushi, Tomer, 2016. "Tail conditional moments for elliptical and log-elliptical distributions," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 179-188.
    12. Fouad Marri & Khouzeima Moutanabbir, 2021. "Risk aggregation and capital allocation using a new generalized Archimedean copula," Papers 2103.10989, arXiv.org.
    13. Furman, Edward & Wang, Ruodu & Zitikis, Ričardas, 2017. "Gini-type measures of risk and variability: Gini shortfall, capital allocations, and heavy-tailed risks," Journal of Banking & Finance, Elsevier, vol. 83(C), pages 70-84.
    14. Fouad Marri & Khouzeima Moutanabbir, 2021. "Risk aggregation and capital allocation using a new generalized Archimedean copula," Working Papers hal-03169291, HAL.
    15. Cai, Jun & Wang, Ying & Mao, Tiantian, 2017. "Tail subadditivity of distortion risk measures and multivariate tail distortion risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 75(C), pages 105-116.
    16. Baishuai Zuo & Chuancun Yin, 2021. "Multivariate tail covariance for generalized skew-elliptical distributions," Papers 2103.05201, arXiv.org.
    17. Cheung, Eric C.K. & Peralta, Oscar & Woo, Jae-Kyung, 2022. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 364-389.
    18. Nicole Bauerle & Tomer Shushi, 2019. "Risk Management with Tail Quasi-Linear Means," Papers 1902.06941, arXiv.org, revised Jan 2020.
    19. Ignatieva, Katja & Landsman, Zinoviy, 2021. "A class of generalised hyper-elliptical distributions and their applications in computing conditional tail risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 437-465.
    20. Jaworski, Piotr & Pitera, Marcin, 2017. "A note on conditional covariance matrices for elliptical distributions," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 230-235.

    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:81:y:2018:i:c:p:27-35. 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.