Estimation in the Continuous Time Mover-Stayer Model with an Application to Bond Ratings Migration

Abstract

The usual tool for modeling bond ratings migration is a discrete, time homogeneous Markov chain. Such model assumes that all bonds are homogeneous with respect to their movement behavior among rating categories and that the movement behavior does not change over time. However, among recognized sources of heterogeneity in ratings migration is age of a bond (time elapsed since issuance). It has been observed that young bonds have a lower propensity to change ratings, and thus to default, than more seasoned bonds. The aim of this paper is to introduce a continuous, time-nonhomogeneuous model for bond ratings migration, which also incorporates a simple form of population heterogeneity. The specific form of heterogeneity postulated bythe proposed model appears to be suitable for modeling the effect of age of a bond on its propensity to change ratings. This model, called a mover-stayer model, is an extension of a time-nonhomogeneous Markov chain. This paper derives the maximum likelihood estimators for the parameters of a continuous time mover-stayer model based on a sample of independent continuously monitored histories of the process, and develops the likelihood ratio test for discriminating between the Markov chain and the mover-stayer model. The methods are illustrated using a sample of rating histories of young corporate issuers. For this sample, the likelihood ratio test rejects a Markov chain in favor of a mover-stayer model. For young bonds with lowest rating the default probabilities predicted by the mover-stayer model are substantially lower than those predicted by the Markov chain.

Keywords: Ratings migration, mover-stayer model, Markov chain, estimation