The multivariate modelling of default risk is a crucial aspect of the pricing of credit derivative products referencing a portfolio of underlying assets, and the evaluation of Value at Risk of such portfolios. This paper proposes a model for the joint dynamic behavior of credit ratings for several firms. Namely, individual credit ratings are modelled by univariate continuous time Markov chain, while their joint dynamics is modelled using copulas. A by-product of the method is the joint laws of the default times of all the firms in the portfolio. The use of copulas allows us to incorporate our knowledge of the modelling of univariate processes, into a multivariate framework. The Normal and Student copulas commonly used in the literature as well as by practitioners do not produce very different estimates of default risk prices. We show that this result is restricted to these two two basic copulas. That is, for any other family of copula, the choice of the copula greatly affects the pricing of default risk.