Remarks on the monotonicity of default probabilities

TL;DR

This paper examines the mathematical relationship between monotonic default probabilities and optimal decision rules in credit scoring, linking ROC curve concavity to score system effectiveness.

Contribution

It establishes that monotonic default probabilities are equivalent to the concavity of the ROC curve, providing a new criterion for evaluating scoring systems.

Findings

Monotonic default probabilities correspond to concave ROC curves.

Optimal decision rules are characterized by the concavity of the ROC curve.

The area under the ROC curve relates to the Information Value metric.

Abstract

The consultative papers for the Basel II Accord require rating systems to provide a ranking of obligors in the sense that the rating categories indicate the creditworthiness in terms of default probabilities. As a consequence, the default probabilities ought to present a monotonous function of the ordered rating categories. This requirement appears quite intuitive. In this paper, however, we show that the intuition can be founded on mathematical facts. We prove that, in the closely related context of a continuous score function, monotonicity of the conditional default probabilities is equivalent to optimality of the corresponding decision rules in the test-theoretic sense. As a consequence, the optimality can be checked by inspection of the ordinal dominance graph (also called Receiver Operating Characteristic curve) of the score function: it obtains if and only if the curve is concave. We conclude the paper by exploring the connection between the area under the ordinal dominance graph and the so-called Information Value which is used by some vendors of scoring systems. Keywords: Conditional default probability, score function, most powerful test, Information Value, Accuracy Ratio.