During the last decade, the dependencies between financial assets have increased due to globalization effects and relaxed market regulation. The standard industrial methodologies like RiskMetrics and CreditMetrics model the dependence structure in the derivatives or in the credit portfolio by assuming multivariate normality of the underlying risk factors. It has been well recognized that many financial assets exhibit a number of features which contradict the normality assumption - namely asymmetry, skewness and heavy tails. Moreover, asset return data suggests also a dependence structure which is quite different from the Gaussian. Recent empirical studies indicate that especially during highly volatile and bear markets the probability for joint extreme events leading to simultaneous losses in a portfolio could be seriously underestimated under the normality assumption. Theoretically, Embrechts et al. show that the traditional dependence measure (the linear correlation coefficient) is not always suited for a proper understanding of the dependency in financial markets. When it comes to measuring the dependence between extreme losses, other measures (e.g. the tail dependence coefficient) are more appropriate. This is particularly important in the credit risk framework, where the risk factors actually enter the model only to introduce a dependence structure in the portfolio. Clearly, appropriate multivariate models suited for extreme events are needed. In this thesis, we consider a portfolio credit risk model in the spirit of CreditMetrics. With respect to the marginal losses, we retain and enhance all features of that model and we incorporate not only the default risk, but also the rating migrations, the credit spread volatility and the recovery risk. The dependence structure in the portfolio is given by a set of underlying risk factors which we model by a general multivariate elliptical distribution. On the one hand, this model retains the standard Gaussian model as a.