We derive improved estimates for the model risk of risk portfolios when additional to the marginals some partial dependence information is available.We consider models which are split into k subgroups and consider various classes of dependence information either within the subgroups or between the subgroups. As consequence we obtain improved VaR bounds for the joint portfolio compared to the case with only information on the marginals. Our paper adds to various recent approaches to obtain reliable and usable risk bounds resp. estimates of the model risk by including partial dependence information additional to the information on the marginals. In particular we extend an approach suggested in Bignozzi, Puccetti and Rüschendorf (2015) and in Puccetti, Rüschendorf, Small and Vanduffel (2017), which is based on positive dependence resp. on independence information available for some subgroups.
If the inline PDF is not rendering correctly, you can download the PDF file here.
 Barlow, R. E. and F. Proschan (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.
 Bernard, C., L. Rüschendorf, and S. Vanduffel (2015). Value-at-Risk boundswith variance constraints. J. Risk Insur., to appear. Available at http://dx.doi.org/10.1111/jori.12108.
 Bernard, C., L. Rüschendorf, S. Vanduffel, and R. Wang (2016). Risk bounds for factor models. Finance Stoch., to appear. Available at https://ssrn.com/abstract=2572508.
 Bernard, C., L. Rüschendorf, S. Vanduffel, and J. Yao (2017). How robust is the Value-at-Risk of credit risk portfolios? European J. Financ. 23(6), 507-534.
 Bignozzi, V., G. Puccetti, and L. Rüschendorf (2015). Reducing model risk via positive and negative dependence assumptions. Insurance: Math. Econ. 61, 17-26.
 Block, H. W. and A. R. Sampson (1988). Conditionally ordered distributions. J. Multivariate Anal. 27(1), 91-104.
 Embrechts, P. and G. Puccetti (2006). Bounds for functions of dependent risks. Finance Stoch. 10(3), 341-352.
 Embrechts, P., G. Puccetti, and L. Rüschendorf (2013). Model uncertainty and VaR aggregation. J. Bank. Financ. 37(8), 2750-2764.
 Embrechts, P., B. Wang, and R. Wang (2015). Aggregation-robustness and model uncertainty of regulatory risk measures. Finance Stoch. 19(4), 763-790.
 Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.
 Kelker, D. (1970). Distribution theory of spherical distributions and a location-scale parameter generalization. Sankhy¯a A 32(4), 419-430.
 Mai, J.-F. and M. Scherer (2012). Simulating Copulas. Stochastic Models, Sampling Algorithms and Applications. Imperial College Press, London.
 Müller, A. and M. Scarsini (2001). Stochastic comparison of random vectors with a common copula. Math. Oper. Res. 26(4), 723-740.
 Müller, A. and M. Scarsini (2005). Archimedean copulae and positive dependence. J. Multivariate Anal. 93(2), 434-445.
 Müller, A. and D. Stoyan (2002). Comparison Methods for Stochastic Models and Risks, John Wiley & Sons, Chichester.
 Nelsen, R. B. (2006). An Introduction to Copulas. Second edition. Springer, New York.
 Pan, X., G. Qiu, and T. Hu (2016). Stochastic orderings for elliptical random vectors. J. Multivariate Anal. 148, 83-88.
 Puccetti, G. and L. Rüschendorf (2012a). Bounds for joint portfolios of dependent risks. Stat. Risk Model. 29(2), 107-132.
 Puccetti, G. and L. Rüschendorf (2012b). Computation of sharp bounds on the distribution of a function of dependent risks. J. Comput. Appl. Math. 236(7), 1833-1840.
 Puccetti, G. and L. Rüschendorf (2013). Sharp bounds for sums of dependent risks. J. Appl. Probab. 50(1), 42-53.
 Puccetti, G. and L. Rüschendorf (2014). Asymptotic equivalence of conservative VaR- and ES-based capital charges. J. Risk 16(3), 3-22.
 Puccetti, G., L. Rüschendorf, D. Small, and S. Vanduffel (2017). Reduction of Value-at-Risk bounds via independence and variance information. Scand. Actuar. J. 2017(3), 245-266.
 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.
 Rüschendorf, L. (1981). Characterization of dependence concepts for the normal distribution. Ann. Inst. Statist.Math. 33(1), 347-359.
 Rüschendorf, L. (2004). Comparison of multivariate risks and positive dependence. J. Appl. Probab. 41(2), 391-406.
 Rüschendorf, L. (2017). Risk bounds and partial dependence information. In D. Ferger, W. G. Manteiga, T. Schmidt, and J.-L. Wang (Eds.), From Statistics to Mathematical Finance. Festschrift in Honour of Winfried Stute (2017). Springer, New York.
 Shaked, M. and J. G. Shanthikumar (2007). Stochastic Orders. Springer, New York.
 Tibiletti, L. (2002). As the dependence structure is ffxed, do more risky assets lead to more risky portfolio? Working paper. Available at: http://www.huebnergeneva.org/documents/tibiletti.pdf.
 Wang, B. and R.Wang (2011). The complete mixability and convex minimization problemswith monotonemarginal densities. J. Multivariate Anal. 102(10), 1344-1360.
 Wang, R. (2014). Asymptotic bounds for the distribution of the sum of dependent random variables. J. Appl. Probab. 51(3), 780-798.
 Wei, G. and T. Hu (2002). Supermodular dependence ordering on a class of multivariate copulas. Stat. Probab. Lett. 57(4), 375-385.