The Basel Committee on Banking Supervision, a regulatory body under the Bank of International Settlements, has in its ’New Capital Accord’ proposed a regulatory setup in which banks are allowed to base the capital requirement on their own internal rating systems and to use external rating systems as well. The increased reliance on rating systems as risk measurement devices has further increased the need for focusing on statistical analysis and validation methodology for rating systems. While the formal definitions of ratings by the major agencies do not formally employ a probability, or an interval of probabilities, in their definition of the various categories, any use of the ratings for risk management and capital allocation will have to assign default probabilities to each rating category and to probabilities of transition between non-default categories. There are many statistical issues, of course, in assigning such probabilities. A fundamental problem is the relatively small sample sizes which have to be used for the estimation. Not only are defaults rare in the top categories. Transitions to ’distant’ rating categories are also rare. Most often, transitions involve the transition from a sub-category of a rating class (from, say Baa1 to Baa2 in Moody’s system or from BBB+ to BBB in Standard and Poor’s classification). Hence to observe more transition activity we may choose to include rating modifiers. However, that also greatly increases the number of rating transition probabilities to be estimated and in fact leaves us with even more rare events in our model. So whether or not we include 8 or (say) 18 rating categories leaves us with important estimates which have to be based on few events.
In Lando and Skødeberg (2002), it is shown that using a continuous-time analysis of the rating transition data enables us to meaningfully estimate probabilities of rare transitions, even if the rare transitions are not actually observed in our data set. This is not possible using classical ’multinomial’ techniques, such as those of Carty and Fons (1993) and Carty (1997). In this paper, we show that the continuous-time procedure also allows us to find significantly improved confidence sets for rare events. Our method is based on bootstrapping the generator and we contrast this method with a simple binomial approach and a multinomial approach. Both Nickell, Perraudin, and Varotto (2000) and Höse, Huschens, and Wania (2002) contain estimates of standard deviations and confidence sets, but since they are based on multiomial type estimators, they cannot assign meaningful confidence sets to probabilities of rare events.
The improved understanding of transition probability variability has consequences for a number of issues in credit risk management and in the analysis of credit risk models in general. First, removing the zeros from all events in the matrix of estimated transition probabilities and supplying confidence bands gives us a concrete method of assessing, for example, the proposal put forward by the Basel Committee, of imposing a minimum probability of 0.03% in these cases. As we will see shortly, a proper use of the full information in the data set gives a better impression of the appropriateness of such a limit.
Second, the comparison of actual default probabilities for high rating categories and the default probabilities implied by spreads in corporate bond prices relies on a point estimate of the default probability for a given rating category. While the observed difference between the two quantities is not removed, a proper analysis of confidence sets may still alter the picture significantly.
The usefulness of the method becomes even more transparent when attacking some of the fundamental problems in the analysis of rating transitions, namely the inclusion of business cycle variables and the handling of non-homogeneities, such as industry and country effects, within rating classes. Whenever we try to analyze such problems we do so by introducing more covariates and hence limit the number of events even further for any given value of the covariates. In a related paper, Fledelius, Lando, and Nielsen (2002) consider the use of smoothing techniques to mitigate this problem. In this paper, our focus is on the estimation of transition probabilities without the inclusion of covariates. We consider instead the uncertainty of estimates by comparing a stable period (1995-1999) with a volatile period (1987-1991). There is clear evidence of variations in rating transition probabilities with time. Nickell, Perraudin, and Varotto (2000) demonstrate a dependence on the state of the business cycle, whereas Blume, Lim, and MacKinlay (1998) suggest a change in rating policy as an explanation for the variation.
The outline of the paper is as follows: Section 2 briefly gives a recapitulation of a discrete, multinomial estimation of the transition parameters and a continuous-time method based on the generator. We then describe two ways of obtaining confidence intervals for default events - one based on the discrete method and one based on a bootstrap method. Section 3 describes our data and some of the data cleaning issues one faces when looking at the data set. Section 4 gives our results and section 5 concludes.
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