Default probabilities and default correlations under stress

We investigate default probabilities and default correlations of Merton-type credit portfolio models in stress scenarios where a common risk factor is truncated. The analysis is performed in the class of elliptical distributions, a family of light-tailed to heavy-tailed distributions encompassing many distributions commonly found in financial modelling. It turns out that the asymptotic limit of default probabilities and default correlations depend on the max-domain of the elliptical distribution's mixing variable. In case the mixing variable is regularly varying, default probabilities are strictly smaller than 1 and default correlations are in (0; 1). Both can be expressed in terms of the Student t-distribution function. In the rapidly varying case, default probabilities are 1 and default correlations are 0. We compare our results to the tail dependence function and discuss implications for credit portfolio modelling. ! Keywords: financial risk management, credit portfolio modelling, stress testing, elliptic distribution, max-domain 
 MSC classification: 60G70, 91G40 ISSN: 1436-9753 ! ! Contact: ! Ludger Overbeck Institute of Mathematics University of Giessen Arndtstrasse 2 35392 Giessen ! Natalie Packham Juniorprofessorin für Quantitative Finance Frankfurt School of Finance & Management Sonnemannstraße 9-11 60314 Frankfurt am Main Phone: +49-69-154008-723 [email protected] Michael Kalkbrener Deutsche Bank AG Taunusanlage 12 60325 Frankfurt am Main The views expressed in this paper are those of the authors and do not necessarily reject the position of Deutsche Bank AG. Default probabilities and default correlations under stress