W. K. Hardle, R. A. Moro, and D. Schafer
Recursive partitioning, also known as classification and regression trees
(CART) performs classification by orthogonally dividing the data space. At
each step only a division (split) along one of the axes is possible. The axis
is chosen such, that a split along it reduces the variance in each of the sub-
spaces and maximises the variance between them. Entropy based criteria can
also be used. The visible drawback is the orthogonal division itself which im-
poses severe restrictions on the smoothness of the classifying function and
may not adequately capture the correlation structure between the variables.
Orthogonal division means that the separating hyperplane can only consist
of orthogonal segments parallel to the coordinate grid, whereas the boundary
between the classes has a smoothly changing gradient.
The neural network (NN) is a network of linnear classifiers (neurons)
that are connected with one another in a prespecified way. The outputs of
some of the neurons are inputs for others. The performance of a NN greatly
depends on its structure that must be adapted for solving different problems.
The network must be designed manually that requires a substantial experience
from the operator. Moreover, NNs mostly do not povide a global solution but
only a local one. This feature, as well as too much heuristics create many
obstacles on the way of using NNs at the rating departments of banks.
We would like to have a model that is able to select a classifying function
based on very general criteria. The SVM is a statistical technique that in many
applications, such as optical character recognition and medical diagnostics,
showed very good performance. It has a flexible solution and is controlled by
adjusting only few parameters. Its overall good performance and flexibility
make the SVM a suitable candidate [9].
Within a rating methodology each company is described by a set of vari-
ables x, such as financial ratios. Financial ratios, such as debt ratio (leverage)
or interest coverage (earnings before interest and taxes to interest) charac-
terise different sides of company operation. They are constructed on the basis
of balance sheets and income statements. For example, the Bundesbank uses
32 ratios (predictors) computed using the company statements from its cor-
porate bankruptcy data base. The predictors and basic statistics are given in
Table 1. The whole Bundesbank data base covers the period 1987-2005 and
consists of 553500 anonymised statements of solvent and insolvent companies.
Most companies appear in the database several times in different years.
The class y of a company can be either y = -1 (‘successful’) or y = 1
(‘bankrupt’). Initially, an unknown classifier function f : x → y is estimated
on a training set of companies (xi, yi), i = 1, ..., n. The training set represents
the data for companies which are known to have survived or gone bankrupt. In
order to obtain PDs from the estimated scores f , rating practitioners usually
rely on prespecified rating classes (i.e. BBB, C, AA, etc.). A certain range of
scores and PDs belong to each rating class. The ranges are computed on the
basis of historical data. To derive a PD for a newly scored company its score
f is compared with the historical values of f’s for each class. Basing on the