Rating Models and Validation - Oesterreichische Nationalbank

estimates, which means that it is entirely possible to operate under the assumption
of uncorrelated defaults. In any case, however, persistent overestimates of
significance will lead to more frequent recalibration of the rating model, which
can have negative effects on the modelÕs stability over time. It is therefore necessary
to determine at least the approximate extent to which default correlations
influence PD estimates.
Default correlations can be modeled on the basis of the dependence of
default events on common and individual random factors. 104 For correlated
defaults, this model also makes it possible to derive limits for assessing deviations
in the realized default rate from its forecast as significant at certain confidence
levels.
In the approximation formula below, q denotes the confidence level (e.g.
95% or 99.9%), Nc the number of cases observed per rating class, pk the default
rate per rating class, the cumulative standard normal distribution, the probability
density function of the standard normal distribution, and the default
correlation:
(one-sided test): If
pobserved c >Q þ 1
2Nc
0
B
@2Q
1
Q ð1 QÞ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1ð1 qÞ t ffiffiffiffi
1
p
1 p
ð1 2 Þ ð1 qÞ t
pffiffiffiffiffiffiffiffiffiffiffi
ð1 Þ
the default rate in class c is significantly underestimated at the confidence level q.
In this context, t ¼
1 ðp forecast
c Þ; Q ¼ 1
ffiffi
p 1 ðqÞþt
p .
The test above can be used to check for overestimates of the default rate in
individual rating classes as well as the overall sample. 105
The table below (chart 74) compares the results of the calibration test under
the assumption of uncorrelated defaults with the results based on a default correlation
of 0.01. A deviation in the realized default rate from the forecast rate is
considered significant whenever the realized default rate exceeds the upper
limit indicated for the respective confidence level.
ffiffiffiffiffiffi
1
Chart 74: Identification of Significant Deviations in Calibration in the Data Example
(Comparison of Tests for Uncorrelated Cases and Results for a Default Correlation of 0.01)
104 Vasicek One-Factor Model, cf. e.g. TASCHE, D, A traffic lights approach to PD validation.
105 In this context, it is necessary to note that crossing the boundary to uncorrelated defaults ð ! 0Þ is not possible in the approximation
shown. For this reason, the procedure will yield excessively high values in the upper default probability limit for very
low default correlations (< 0.005).
ffiffi
Rating Models and Validation
Guidelines on Credit Risk Management 123
1
C
A