Rating Models and Validation - Oesterreichische Nationalbank

Chart 16: Indicators in the ÒCrebonÓ Rating System at Bayerische Hypo- und Vereinsbank 25
3.2.2 Regression Models
Like discriminant analysis, regression models serve to model the dependence of
a binary variable on other independent variables. If we apply this general definition
of regression models to credit assessment procedures, the objective is
to use certain creditworthiness characteristics (independent variables) to determine
whether borrowers are classified as solvent or insolvent (dependent binary
variable). The use of nonlinear model functions as well as the maximum likelihood
method to optimize those functions means that regression models also
make it possible to calculate membership probabilities and thus to determine
default probabilities directly from the model function. This characteristic is relevant
in rating model calibration (see section 5.3).
In this context, we distinguish between logit and probit regression models.
The curves of the model functions and their mathematical representation are
shown in chart 17. In this chart, the function denotes the cumulative standard
normal distribution, and the term ( P ) stands for a linear combination of the
factors input into the rating model; this combination can also contain a constant
term. By rescaling the linear term, both model functions can be adjusted to
yield almost identical results. The results of the two model types are therefore
not substantially different.
Due to their relative ease of mathematical representation, logit models are
used more frequently for rating modeling in practice. The general manner in
which regression models work is therefore only discussed here using the logistic
regression model (logit model) as an example.
In (binary) logistic regression, the probability p that a given case is to be
classified as solvent (or insolvent) is calculated using the following formula: 26
p ¼
1
1 þ exp½ ðb0 þ b1 K1 þ b2 K2 þ ::: þ bn KnÞŠ :
25 See EIGERMANN, J., Quantitatives Credit-Rating unter Einbeziehung qualitativer Merkmale, p. 102.
26 In the probit model, the function used is p ¼ ð P Þ, where Nð:Þ stands for standard normal distribution and the following
applies for P : P ¼ b0 þ b1 x1 þ ::: þ bn xn.
Rating Models and Validation
Guidelines on Credit Risk Management 43