The multivariate modeling of default risk is a crucial aspect of the pricing of credit derivative products referencing a portfolio of underlying assets and of the evaluation of the value-at-risk of such portfolios. This paper proposes a model for the joint dynamics of credit ratings of several firms. Namely, individual credit ratings are modeled by a univariate continuous time Markov chain, while their joint dynamics are modeled using copulas. A by-product of the method is the joint laws of the default times of all of the firms in the portfolio. The use of copulas allows us to incorporate our knowledge of the modeling 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 basic copulas. That is, for any other family of copula, the choice of the copula greatly affects the pricing of default risk.