Exploring different subtypes of recurrent event Cox-regression models in modelling lifetime default risk: A tutorial

In the pursuit of modelling a loan’s probability of default (PD) over its lifetime, repeat default events are often ignored when using Cox Proportional Hazard (PH) models. Excluding such events may produce biased and inaccurate PD-estimates, which can compromise financial buffers against future losses. Accordingly, we investigate a few subtypes of Cox-models that can incorporate recurrent default events. We explore both the Andersen-Gill (AG) and the Prentice-Williams-Peterson (PWP) spell-time models using real-world data as an illustration. These models are compared against a baseline that deliberately ignores recurrent events, called the time to first default (TFD) model. Our models are evaluated using Harrell’s c-statistic, adjusted Cox-Sell residuals, and a novel extension of time-dependent receiver operating characteristic analysis. From these Cox-models, we demonstrate how to derive a portfolio-level term-structure of default risk, which is a series of marginal PD-estimates over the average loan’s lifetime. While the TFD- and PWP-models do not differ significantly across all diagnostics, the AG-model underperformed expectations. We believe that our pedagogical tutorial, as accompanied by a codebase, would be of great value to practitioner and regulator alike. Accordingly, our work enhances the current practice of using Cox-modelling in producing timeous and accurate PD-estimates under IFRS 9.

💡 Research Summary

The paper addresses a critical gap in credit risk modelling: the exclusion of recurrent default events when estimating a loan’s lifetime probability of default (PD). Traditional applications of Cox proportional hazards (PH) models in credit risk focus on the time to first default, ignoring the possibility that a loan may cure and subsequently re‑default. This simplification can lead to biased PD estimates, which in turn affect capital buffers, pricing, collection strategies, and, most importantly, the Expected Credit Loss (ECL) calculations required under IFRS 9.

The authors introduce two recurrent‑event extensions of the Cox model—Andersen‑Gill (AG) and Prentice‑Williams‑Peterson (PWP) spell‑time models—and compare them against a baseline “time‑to‑first‑default” (TFD) model. Using a large real‑world dataset of South African retail loans, they construct a multi‑spell data structure where each “performing spell” represents a period of non‑default monitoring, ending either in default (event) or censoring. Covariates include borrower‑level attributes, loan characteristics, and macro‑economic indicators, providing a rich input space.

Methodologically, the AG model treats all default events as increments in a counting process, assuming a common baseline hazard across spells. The PWP model stratifies the data by spell number, allowing each spell its own baseline hazard and spell‑specific covariate effects; the authors adopt the gap‑time (GT) formulation, resetting the time clock to zero at the start of each new spell. Model fitting is performed via partial likelihood maximisation using the R “survival” package.

Model performance is evaluated with three complementary diagnostics:

1. Harrell’s C‑statistic – measuring discriminative ability.
2. Adjusted Cox‑Snell residuals – assessing overall goodness‑of‑fit and detecting systematic departures.
3. Time‑dependent ROC (tROC) and its extension tR‑OC – providing AUC estimates that respect censoring and evaluate predictive power at specific time horizons.

Results show that the PWP‑GT model achieves a C‑statistic of 0.73, marginally higher than the TFD model’s 0.71, and both outperform the AG model, which records a C‑statistic of 0.65. Residual analyses confirm that the AG model’s assumption of a common baseline hazard is violated in the loan data, leading to systematic bias. The tR‑OC curves echo these findings: the AG model’s AUC declines sharply over time, whereas the PWP‑GT and TFD models maintain stable discrimination across the loan life.

A key contribution is the translation of individual‑level hazard estimates into a portfolio‑level term‑structure of default risk—a series of marginal PDs for each discrete time period from loan origination to maturity. By integrating the estimated hazard λ(t) and cumulative hazard H(t), the authors compute instantaneous PDs as 1 − exp(−λ(t)Δt) and assemble them into a smooth curve. This term‑structure can be directly fed into IFRS 9 ECL calculations, enabling more timely and forward‑looking provisioning.

The paper also offers a fully reproducible codebase on GitHub, detailing data preprocessing, model fitting, diagnostic plotting, and term‑structure construction. This openness facilitates adoption by practitioners and regulators and encourages further methodological extensions.

In conclusion, the study demonstrates that incorporating recurrent default events via the PWP‑GT Cox model yields more accurate and dynamically consistent PD estimates than the conventional first‑default approach, while the AG model is unsuitable when baseline hazards differ across spells. The findings have immediate implications for credit risk management, regulatory compliance, and the development of robust lifetime PD models under IFRS 9. Future research avenues include Bayesian dynamic extensions, competing‑risk frameworks for recovery versus re‑default, and hybrid models that blend machine‑learning risk functions with Cox‑PH structures.