Rating Models and Validation - Oesterreichische Nationalbank

Chart 65: Interpretation of the Bayesian Error Rate as the Lowest Overall Error where p ¼ 50%
Entropy-Based Measures of Discriminatory Power
These measures of discriminatory power assess the information gained by using
the rating model. In this context, information is defined as a value which is
measurable in absolute terms and which equals the level of knowledge about
a future event.
Let us first assume that the average default probability of all cases in the segment
in question is unknown. If we look at an individual case in this scenario
without being able to estimate its credit quality using rating models or other
assumptions, we do not possess any information about the (good or bad) future
default status of the case. The maximum in this scenario is the information an
observer gains by waiting for the future status.
If, however, the average probability of a credit default is known, the information
gained by actually observing the future status of the case is lower in this
scenario due to the previously available information.
These considerations lead to the definition of the Òinformation entropyÓ
value, which is represented as follows for dichotomous events with a probability
of occurrence p for the Ò1Ó event (in this case the credit default):
H0 ¼ fp log 2 ðpÞþð1 pÞ log 2 ð1 pÞg
H0 refers to the absolute information value which is required in order to determine
the future default status, or conversely the information value which is
gained by observing the Òcredit default/no credit defaultÓ event. Thus entropy
can also be interpreted as a measure of uncertainty as to the outcome of an
event.
H0 reaches its maximum value of 1 when p ¼ 50%, that is, when default and
non-default are equally probable. H0 equals zero when p takes the value 0 or 1,
that is, the future default status is already known with certainty in advance.
Conditional entropy is defined with conditional probabilities pð jcÞ instead of
absolute probabilities p; the conditional probabilities are based on condition c.
Rating Models and Validation
Guidelines on Credit Risk Management 113