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.
