From the Fermi-Dirac Distribution to PD curves

Abstract

In machine learning applications, and in credit risk modeling in particular, model performance is usually measured by using CAP and ROC curves. The purpose of this paper is to use the statistics of the CAP curve to provide a new method for credit PD curves calibration that are not based on arbitrary choices as the ones that are used in the industry. We map CAP curves to a ball-box problem and we use statistical physics techniques to compute the statistics of the CAP curve from which we derive the shape of PD curves. This approach leads to a new type of shape for PD curves that have not been considered in the literature yet, namely the Fermi-Dirac function which is a two parameter function depending on the target default rate of the portfolio and the target accuracy ratio of the scoring model. We show that this type of PD curve shape is likely to outperform the logistic PD curve that practitioners often use. We suggest that practitioners should stop using logistic PD curves and should adopt the Fermi-Dirac function to improve the accuracy of their credit risk measurement.