The bond rating is one of the most important indicators of a corporation's credit quality and therefore its default probability. It was first developed by Moody's in 1914 and by Poor's Corporation in 1922 and it is generally assigned by external agencies to publicly traded debts. Apart from the external ratings by independent rating agencies, there are internal ratings by banks and other financial institutions, Basel Committee on Banking Supervision (2001). External rating data by agencies are available for many years, in contrast to internal ratings. Their short history in most cases does not exceed 5-10 years. Both types of ratings are usually recorded on an ordinal scale and labeled alphabetically or numerically. For the construction of a rating system see Crouhy et al. (2001).
A change in a rating reflects the assessment that the company's credit quality has improved (upgrade) or deteriorated (downgrade). Analyzing these rating migrations including default is one of the preliminaries for credit risk models in order to measure future credit loss. In such models, the matrix of rating transition probabilities, the so called transition matrix, plays a crucial role. It allows to calculate the joint distribution of future ratings for borrowers that compose a portfolio, Gupton et al. (1997). An element of a transition matrix gives the probability that an obligor with a certain initial rating migrates to another rating by the risk horizon. For the econometric analysis of transition data see Lancaster (1990).
In a study by Jarrow et al. (1997) rating transitions were modeled as a time-homogeneous Markov chain, so future rating changes are not affected by the rating history (Markov property). The probability of changing from one rating to another is constant over time (homogeneous), which is assumed solely for simplicity of estimation. Empirical evidence indicates that transition probabilities are time-varying. Nickell et al. (2000) show that different transition matrices are identified across various factors such as the obligor's domicile and industry and the stage of business cycle.
Rating migrations are reviewed from a statistical point of view throughout this chapter using XploRe . The way from the observed data to the estimated one-year transition probabilities is shown and estimates for the standard deviations of the transition rates are given. In further extension, dependent rating migrations are discussed. In particular, the modeling by a threshold normal model is presented.
Time stability of transition matrices is one of the major issues
for credit risk estimation. Therefore, a chi-square test of
homogeneity for the estimated rating transition probabilities is
applied. The test is illustrated by an example and is compared to
a simpler approach using standard errors. Further, assuming time
stability, multi-period rating transitions are discussed. An
estimator for multi-period transition matrices is given and its
distribution is approximated by bootstrapping. Finally, the
change of the composition of a credit portfolio caused by rating
migrations is considered. The expected composition and its
variance is calculated for independent migrations.