3.1. Conceptual considerations

The goal with this section on the statistical background is to introduce the model that will serve as the unifying framework for most of the more technical considerations in the part on validation techniques. We begin with some conceptual considerations.

We look at rating systems in a binary classification framework. In particular, we will show that the binary classification concept is compatible with the idea of having more than two rating grades. For the purpose of this chapter, binary classification is understood in the sense of discriminating between the populations of defaulters and non-defaulters, respectively.

Also for the purpose of this chapter, we assume that the score or rating grade S (based on regression or other methods) assigned to a borrower summarizes the information that is contained in a set of covariates (e.g. accounting ratios). Rating or score variable design, development or implementation ^[3.] is not the topic of this presentation. We want to judge with statistical methods whether rating or scoring systems are appropriate for discrimination between good and bad and are well-calibrated.

With regard to calibration, at the end of the section, we briefly discuss how PDs can be derived from the distributions of the scores in the population of the defaulters and non-defaulters, respectively.

3.2. Basic setting

We assume that with every borrower two random variables are associated. There is a variable S that may take on values across the whole spectrum of real numbers. And there...