Estimating rating transition probabilities with missing data

In this article we provide a rigorous treatment of one of the central statistical issues of credit risk management. Given K- 1 rating categories, the rating of a corporate bond over a certain horizon may either stay the same or change to one of the remaining K - 2 categories; in addition, it is usually the case that the rating of some bonds is withdrawn during the time interval considered in the analysis. When estimating transition probabilities, we have thus to consider a K-th category, called withdrawal, which contains (partially) missing data. We show how maximum likelihood estimation can be performed in this setup; whereas in discrete time our solution gives rigorous support to a solution often used in applications, in continuous time the maximum likelihood estimator of the transition matrix computed by means of the EM algorithm represents a significant improvement over existing methods. Key words: Continuous-time Markov chain, Transition matrix, EM algorithm, Default probability