A Bayesian approach to probability default model calibration: Theoretical and empirical insights on the Jeffreys test

Christophe Hurlin () and Yoann Pull
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Christophe Hurlin: LEO - Laboratoire d'Économie d'Orleans [2022-...] - UO - Université d'Orléans - UT - Université de Tours - NEOLAiA - NEOLAiA European University = Université Européenne NEOLAÏA - UCA - Université Clermont Auvergne
Yoann Pull: LEO - Laboratoire d'Économie d'Orleans [2022-...] - UO - Université d'Orléans - UT - Université de Tours - NEOLAiA - NEOLAiA European University = Université Européenne NEOLAÏA - UCA - Université Clermont Auvergne

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Abstract: The calibration of Probability of Default (PD) models is a cornerstone of model risk management, ensuring alignment between predicted and observed default rates, as required by credit risk regulation and supervisory practices. This paper examines the Jeffreys test as a Bayesian alternative to traditional frequentist methods for the assessment of PD model calibration. Unlike classical binomial tests, which often rely on normal approximations and exhibit poor coverage in low-default or small-sample portfolios, the Jeffreys test constructs credible intervals using a non-informative prior, improving robustness and interpretability. We provide a systematic and regulator-aligned assessment of the Jeffreys test in internal validation settings, with particular emphasis on low- and zero-default segments, small-sample grades, and environments with drifting default risk. Through Monte Carlo simulations, we compare its performance with standard approaches such as the Clopper–Pearson interval and normal approximations, highlighting its superior coverage properties in sparse-data regimes. We further illustrate the approach through two complementary empirical applications: a Low-Default Portfolio constructed from publicly available Standard & Poor's corporate rating histories, and a U.S. mortgage credit portfolio based on the Freddie Mac Single-Family Loan-Level dataset. These applications show how Jeffreys credible intervals and the Jeffreys -value provide transparent diagnostics of calibration and conservatism at the grade or pool level.

Keywords: Credit risk management; Bayesian inference; PD model calibration (search for similar items in EconPapers)
Date: 2026-05
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Published in European Journal of Operational Research, 2026, ⟨10.1016/j.ejor.2026.05.008⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-05674258

DOI: 10.1016/j.ejor.2026.05.008

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