Quantile of a Mixture with Application to Model Risk Assessment

Carole Bernard 1  and Steven Vanduffel 2
  • 1 Department of Accounting, Law and Finance at the Grenoble Ecole de Management
  • 2 Department of Economics and Political Sciences at Vrije Universiteit Brussel (VUB)


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].

If the inline PDF is not rendering correctly, you can download the PDF file here.

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

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

  • [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.

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

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

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

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

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

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

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

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

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

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

  • [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.

  • [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.


Journal + Issues

Dependence Modeling aims to provide a medium for exchanging results and ideas in the area of multivariate dependence modeling. Topics include Copula methods, environmental sciences, estimation and goodness-of-fit tests, extreme-value theory, limit laws, mass transportations, measures of association, multivariate distributions and tests, quantitative risk management, risk assessment, risk models, risk measures and stochastic orders and time series.