Skip to main content

Vector-Valued Tail Value-at-Risk and Capital Allocation

Methodology and Computing in Applied Probability volume 18, pages 653–674 (2016)Cite this article

Abstract

Enterprise risk management, actuarial science or finance are practice areas in which risk measures are important to evaluate for heterogeneous classes of homogeneous risks. We present new measures: bivariate lower and upper orthant Tail Value-at-Risk. They are based on bivariate lower and upper orthant Value-at-Risk, introduced in Cossette et al. (Insurance: Math Econ 50(2):247–256, 2012). Many properties and applications are derived. Notably, they are shown to be positive homogeneous, invariant under translation and subadditive in distribution. Capital allocation criteria are suggested. Moreover, results on the sum of random pairs are presented, allowing to use a more accurate model for dependent classes of homogeneous risks.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  • Artzner P, Delbaen F, Eber J-M, Heath D (1999) Coherent measures of risk. Math Financ 9(3):203–228

    Article  MathSciNet  MATH  Google Scholar 

  • Balbás A, Balbás R, Jiménez-Guerra P (2012) Vector risk functions. Mediterr J Math 9(3):563–574

  • Barnett V (1976) The ordering of multivariate data. J R Stat Soc Ser A 139(3):318–355

    Article  MathSciNet  Google Scholar 

  • Basel (2003) Trends in risk integration and aggregation, the joint forum, Bank for International Settlements. http://www.bis.org

  • Basel (2004) International convergence of capital measurement and capital standards: a revised framework

  • Bedoui R, Dbabis M (2009) Copulas and bivariate risk measures: an application to hedge funds. 26ème journée internationales d’économie monétaire et financière

  • Belzunce F, Castaño A, Olvera-Cervantes A, Suárez-Llorens A (2007) Quantile curves and dependence structure for bivariate distributions. Comput Stat Data Anal 51(10):5112–5129

    Article  MathSciNet  MATH  Google Scholar 

  • Cossette H, Mailhot M, Marceau M (2012) TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts. Insurance: Math Econ 50(2):247–256

    MathSciNet  MATH  Google Scholar 

  • Cousin A, Di Bernardino E (2013) On multivariate extensions of value-at-risk. J Multivar Anal 119:32–46

    Article  MathSciNet  MATH  Google Scholar 

  • Cousin A, Di Bernardino E (2014) On multivariate extensions of conditional-tail-expectation. Insur Math Econ 55:272–282

    Article  MathSciNet  MATH  Google Scholar 

  • Denuit M, Dhaene J, Goovaerts M, Kaas R (2005) Actuarial theory for dependent risks. Wiley Online Library

  • Di Bernardino E, Fernández-Ponce JM, Palacios Rodríguez F, Rodrígez-Griñolo MR (2015) On multivariate extensions of the conditional Value-at-Risk measure. Insur Math Econ 61:1–16

    Article  MathSciNet  MATH  Google Scholar 

  • Embrechts P, Puccetti G (2006) Bounds for functions of multivariate risks. J Multivar Anal 97(2):526–547

    Article  MathSciNet  MATH  Google Scholar 

  • Fernández-Ponce JM, Suárez-Llorens A (2002) Central regions for bivariate distributions. Austrian J Stat 31:2–3

    Google Scholar 

  • Joe H (1997) Multivariate models and dependence concepts. Chapman and Hall, London

    Book  MATH  Google Scholar 

  • Jouini E, Meddeb M, Touzi N (2004) Vector-valued coherent risk measures. Finance Stochast 8(4):531–552

    Article  MathSciNet  MATH  Google Scholar 

  • Mainik G, Schaanning E (2012) On dependence consistency of CoVaR and some other systemic risk measures, arXiv preprint

  • McNeil AJ, Frey R, Embrechts P (2005) Quantitative risk management. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Müller A, Stoyan D (2002) Comparison methods for stochastic models and risks. Wiley, New York

    MATH  Google Scholar 

  • Ruschendorf L (1982) Random variables with maximum sums. Adv Appl Probab 14(3):623–632

    Article  MathSciNet  MATH  Google Scholar 

  • Shaked M, Shanthikumar JG (2007) Stochastic orders and their applications, 2nd edn. Springer series in Statistics, Springer, New York

    MATH  Google Scholar 

  • Serfling R (2002) Quantile functions for multivariate analysis: approaches and applications. Stat Neerlandica 56(2):214–232

    Article  MathSciNet  MATH  Google Scholar 

  • Sordo MA, Suárez-Llorens A, Bello AJ (2015) Comparison of conditional distributions in portfolios of dependent risks. Insur Math Econ 61:62–69

    Article  MathSciNet  MATH  Google Scholar 

  • Tahar IB, Lépinette E (2012) Vector-valued risk measure processes, Working Paper

  • Tahar IB, Lépinette E (2014) Vector-valued coherent risk measure processes. Int J Theor Appl Finance 17(02)

  • Tasche D (1999) Risk contributions and performance measurement. Report of the Lehrstuhl für mathematische Statistik, TU München

  • Tibiletti L (1993) On a new notion of multidimensional quantile. Metron 51(3–4):77–83

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. École d’Actuariat, Université Laval, Quebec City, Quebec, Canada

    Hélène Cossette & Étienne Marceau

  2. Department of Mathematics and Statistics, Concordia University, Montréal, Quebec, Canada

    Mélina Mailhot

  3. Département de Mathématiques et Informatique, Université du Québec à Trois-Riviàres, Trois-Riviéres, Quebec, Canada

    Mhamed Mesfioui

Authors
  1. Hélène Cossette
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mélina Mailhot
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Étienne Marceau
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mhamed Mesfioui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mélina Mailhot.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cossette, H., Mailhot, M., Marceau, É. et al. Vector-Valued Tail Value-at-Risk and Capital Allocation. Methodol Comput Appl Probab 18, 653–674 (2016). https://doi.org/10.1007/s11009-015-9444-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11009-015-9444-9

Keywords

  • Bivariate Tail Value-at-Risk
  • Multivariate risk measures
  • Capital allocation
  • Copulas
  • Bounds

Mathematics Subject Classification (2010)

  • 62P05
  • 91B30