Credit Risk Modelling and Estimation via Elliptical Copulae

Dependence modelling plays a crucial role within internal credit risk models. The theory of copulae, which describes the dependence structure between a multi-dimensional distribution function and the corresponding marginal distributions, provides useful tools for dependence modelling. The difficulty in employing copulae for internal credit risk models arises from the appropriate choice of a copula function. From the practical point of view the dependence modelling of extremal credit default events turns out to be a desired copula property. This property can be modelled by the so-called tail dependence concept, which describes the amount of dependence in the upperright-quadrant tail or lower-left-quadrant tail of a bivariate distribution. We will give a characterization of tail dependence via a tail dependence coefficient for the class of elliptical copulae. This copula class inherits the multivariate normal, t, logistic, and symmetric general hyperbolic copula. Further we embed the concepts of tail dependence and elliptical copulae into the framework of extreme value theory. Finally we provide a parametric and non-parametric estimator for the tail dependence coefficient. Published in: G. Bohl, G. Nakhaeizadeh, S.T. Rachev, T. Ridder and K.H. Vollmer (eds.), Credit Risk Measurement, Evaluation and Management, Physica-Verlag Heidelberg, 2003, 267-289.