Aim of this paper is to introduce some general theory of copulas, to present some of their useful properties, such as the invariance under strictly increasing transformations and to introduce Archimedean and elliptical copulas. In many areas like for example in modeling credit derivatives or for calculations in actuarial science, the different risk usually can't be assumed to be independent. Thus one needs to consider different risks occurring at the same time affecting each other. One often has a much better idea about the marginal behavior of the individual risks than about their dependence structure. Abe Sklar stated in the central theorem of this work, that copulas enable us to derive a joint multivariate distribution from given one-dimensional marginal distributions, including a postulated dependence structure, i.e. a copula, and vice versa. First, we will present the copula function approach in the bivariate and the multivariate case. We will then have a close look at two representatives, the Archimedean copulas and the elliptical copulas each with two examples. As examples for Archimedean copulas we will introduce the bivariate Clayton copula family and the bivariate Gumbel copula family. As representatives for elliptical copulas we've chosen the d-variate Gauss copula and the d-variate t-copula. Finally, we will talk about survival analysis and some different approaches for measuring default correlation like Kendall's Tau and Spearman's Rho as well as about calibration of copulas based on rank correlations.     «

Aim of this paper is to introduce some general theory of copulas, to present some of their useful properties, such as the invariance under strictly increasing transformations and to introduce Archimedean and elliptical copulas. In many areas like for example in modeling credit derivatives or for calculations in actuarial science, the different risk usually can't be assumed to be independent. Thus one needs to consider different risks occurring at the same time affecting each other. One often has...     »