The Modelling of Risk of Portfolios with Copulae

Introduction: In the first section, ways of modelling single name credit risks are introduced, in order to give an idea of how equity prices drive possible bond defaults, but more importantly introduce the models that are currently used in practice. Particular emphasis will be placed on Merton ́s model as a means of calculating default probability as it presents the basic idea of deriving the chance of default. Additionally it is the theoretical basis of commercial models like Moody ́s KMV and Credit Metrics. Merton ́s model will be described in detail in the following sections, also addressing the assumption of the Black and Scholes formula, as well as the estimation of the asset value of the firm using observable equity values. The importance of the structural models is that they impose some kind of threshold that has to be passed by the value of the assets of a firm in order for the bond to default. Threshold models are a more general way of implementing structural models and will be discussed in the section thereafter. In the following sections, some basic mathematical properties of copulae will be de- scribed especially paying attention to the way that by defining dependence through copulae one is able to separate the marginal distributions from the joint distribution. The reason for this emphasis is that by doing so desired effects like tail dependence can be implemented in the multivariate model, without having to change the marginal or single name distributions. Moody ́s KMV ́s CreditMetrics which is an industry standard, implicitly uses a Gauß copula in order to model the joint probability of default. Since tail risk is not captured by Gauß copulae, the use of another copula to model the joint default risk will be analysed and compared. As mentioned, the Student-t copula seems to be of special interest here, since although not requiring further dependence measures than the widely used Pearson ́s measure of correlation and a degree of freedom (DoF), it is able to model the tail risk and thus, assign a higher probability to multiple defaults, as it has been shown to be the case in the financial crisis. It remains to be seen if the models that measure tail risk are models that will only perform well in times of crisis. The Gauß copula has without question performed well in economically calm times. So that it may be of interest if and when models such as the Student-t copula will yield the best fit to data. [...]