Bifractal Receiver Operating Characteristic Curves: A Formula for Generating Receiver Operating Characteristic Curves in Credit-Scoring Contexts

Abstract

This paper formulates a mathematical model for generating receiver operating characteristic (ROC) curves without underlying data. Credit scoring practitioners know that the Gini coefficient usually drops if it is only calculated on cases above the cutoff. This fact is not a mathematical necessity, however, as it is theoretically possible to get an ROC curve that keeps the same Gini coefficient no matter how big a share of lowest score cases are excluded from the calculation (a “right-hand” fractal ROC curve). Analogously, a left-hand fractal ROC curve would be a curve that keeps its Gini coefficient constant below any cutoff point. The model proposed here is a linear combination of left- and right-hand ROC curves. A bifractal ROC curve is drawn with just two parameters: one responsible for the shape of the curve and the other responsible for the area under the curve (a Gini coefficient). As is shown in this paper, most real-life credit-scoring ROC curves lie between the two fractal curves. In consequence, the Gini coefficient will be consistently lower when computed only on approved loans.