5.3 Calibrating the Rating Model

Calibration assigns a default probability estimate to each possible overall score. According to European Directive 2006/48/EC, the rating scale used by the credit institution to group debtors in classes with a reasonably small band of PD values. The directive allows the use of "direct estimates of risk parameters [which] may be seen as the outputs of grades on a continuous rating scale". (Directive 2006/48/EC, Annex VII, Part 4, 4)

There two distinct calibration procedures to be used, depending on the methodology used to build the scoring function.

• Logistic regression: already yields sample-dependent PD estimates, which need to be rescaled to each segment's average PD.
• Statistical and heuristic models (e.g. MDA): calibration assigns PD values to scores; rescaling may be necessary.

 Unless a full data survey is used to generate the data set, external data is necessary to calibrate the rating system. Specifically, for all segments it is necessary to know the a priori PD (the average default rate).


Calibration for logistic regression
Rescaling is done via the Relative Default Frequency, that is, the ration of defaulters to non-defaulters in the sample.

Procedure for rescaling the PD estimates from a logistic regression model:

1. Calculation of the mean PD of the non-defaulted sample.
2. Conversion of mean PD in RDF(sample).
3. Obtaining the mean PD of the portfolio (average default rate per segment).
4. Representation of the model's PD estimates as RDF(unscaled).
5. RDF scaling: RDF(scaled) = RDF(unscaled) * (RDF(portfolio)/RDF(sample))
   
6. Conversion of RDF(scaled) in PD(scaled)

Care must be taken that the default criterion used by the external source (average default rate per segment) be consistent with the default criterion chosen for the rating model (see Basel II definition of default event).

Calibrating for Multiple Discriminant Analysis and other statistical and heuristic models
Procedure:

1. Division of score range into intervals
   
   • Differences between mean PDs of different intervals are sufficiently large.
   • 10 or more classes (min. 7 according to Directive 2006/48/EC)
   • 100 cases per interval to make a reliable estimate of mean PD.
   • Interval widths not necessarily identical.
     
2. RDF(unscaled) is computed for each interval as the ratio of defaulters to non-defaulters.
3. RDF scaling: RDF(scaled) = RDF(unscaled) * (RDF(portfolio)/RDF(sample))
   
4. Conversion of RDF(scaled) in PD(scaled)

A PD value must be assigned to each individual score value. This is done by building an interpolation based on the discrete points representing the classes: average score value within the class vs. default rate of the class. The interpolating function can be an exponential function. Inverting the interpolating function allows the conversion of the PD ranges defining the classes to score ranges.