From the Fermi–Dirac distribution to PD curves

Purpose: In machine learning applications, and in credit risk modeling in particular, model performance is usually measured by using cumulative accuracy profile (CAP) and receiving operating characteristic 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. Design/methodology/approach: The author maps CAP curves to a ball–box problem and uses statistical physics techniques to compute the statistics of the CAP curve from which the author derives the shape of PD curves. Findings: 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. The author shows that this type of PD curve shape is likely to outperform the logistic PD curve that practitioners often use. Practical implications: This paper has some practical implications for practitioners in banks. The author shows that the logistic function which is widely used, in particular in the field of retail banking, should be replaced by the Fermi–Dirac function. This has an impact on pricing, the granting policy and risk management. Social implications: Measuring credit risk accurately benefits the bank of course and the customers as well. Indeed, granting is based on a fair evaluation of risk, and pricing is done accordingly. Additionally, it provides better tools to supervisors to assess the risk of the bank and the financial system as a whole through the stress testing exercises. Originality/value: The author suggests that practitioners should stop using logistic PD curves and should adopt the Fermi–Dirac function to improve the accuracy of their credit risk measurement.