Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Dependence Modeling

Ed. by Puccetti, Giovanni

1 Issue per year

Covered by:

Open Access
See all formats and pricing
More options …

Quantile of a Mixture with Application to Model Risk Assessment

Carole Bernard / Steven Vanduffel
Published Online: 2015-10-27 | DOI: https://doi.org/10.1515/demo-2015-0012


We provide an explicit expression for the quantile of a mixture of two random variables. The result is useful for finding bounds on the Value-at-Risk of risky portfolios when only partial dependence information is available. This paper complements the work of [4].

Keywords: Model Risk; Rearrangement Algorithm; Mixture

MSC:: 60E05; 60E15


  • [1] Acerbi, C., and D. Tasche (2002). On the coherence of expected shortfall. J. Banking Financ. 26(7), 1487–1503. Google Scholar

  • [2] Bernard, C., L. Rüschendorf, and S. Vanduffel (2015). VaR bounds with a variance constraint. Forthcoming in J. Risk Insurance. Google Scholar

  • [3] Bernard, C., L. Rüschendorf, S. Vanduffel, and J. Yao (2015). How Robust is the Value-at-Risk of Credit Risk Portfolios? Forthcoming in Eur. J. Financ. Google Scholar

  • [4] Bernard, C., and S. Vanduffel (2015). A new approach to assessing model risk in high dimensions, J. Banking Financ. 58, 166–178. Google Scholar

  • [5] Castellacci, G. (2012). A formula for the quantiles of mixtures of distributions with disjoint supports. Available at http:// ssrn.com/abstract=2055022. Google Scholar

  • [6] Embrechts, P., G. Puccetti, and L. Rüschendorf (2013). Model uncertainty and VaR aggregation. J. Banking Financ. 37(8), 2750–2764. Web of ScienceGoogle Scholar

  • [7] Föllmer, H., and A. Schied (2011): Stochastic Finance: an Introduction in Discrete Time. Walter de Gruyter, Berlin. Google Scholar

  • [8] Gaffke, N., and L. Rüschendorf (1981). On a class of extremal problems in statistics. Optimization 12(1), 123–135. Google Scholar

  • [9] Kotz, S., and S. Nadarajah (2004). Multivariate t-distributions and their Applications. Cambridge University Press. Google Scholar

  • [10] Landsman, Z. M., and E. A. Valdez (2003). Tail conditional expectations for elliptical distributions. North Amer. Actuar. J. 7(4), 55–71. Google Scholar

  • [11] McNeil, A. J., R. Frey, and P. Embrechts (2005): Quantitative Risk Management: Concepts, Techniques and Tools: Concepts, Techniques and Tools. Princeton university press. Google Scholar

  • [12] Puccetti, G., and L. Rüschendorf (2013). Sharp bounds for sums of dependent risks. J. Appl. Probab. 50(1), 42–53. CrossrefWeb of ScienceGoogle Scholar

  • [13] Puccetti, G., B. Wang, and R. Wang (2012). Advances in complete mixability. J. Appl. Probab. 49(2), 430–440. CrossrefGoogle Scholar

  • [14] Puccetti, G., B. Wang, and R. Wang (2013). Complete mixability and asymptotic equivalence of worst-possible VaR and ES estimates. Insurance Math. Econom. 53(3), 821–828. Google Scholar

  • [15] Wang, B., and R. Wang (2011). The complete mixability and convex minimization problems with monotone marginal densities. J. Multivariate Anal. 102(10), 1344–1360. Web of ScienceGoogle Scholar

About the article

Received: 2015-06-16

Accepted: 2015-10-14

Published Online: 2015-10-27

Citation Information: Dependence Modeling, Volume 3, Issue 1, ISSN (Online) 2300-2298, DOI: https://doi.org/10.1515/demo-2015-0012.

Export Citation

© 2015 Carole Bernard and Steven Vanduffel. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

Edgars Jakobsons and Steven Vanduffel
Risks, 2015, Volume 3, Number 4, Page 599
Giovanni Puccetti, Ludger Rüschendorf, Daniel Small, and Steven Vanduffel
Scandinavian Actuarial Journal, 2015, Page 1

Comments (0)

Please log in or register to comment.
Log in