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

Arno Botha, Tanja Verster, Bernard Scheepers

TL;DR

The paper investigates dynamic lifetime probabilities of default by comparing three recurrent-event Cox models—TFD (time to first default), AG (calendar-time spells with a common baseline hazard), and PWP (spell-stratified baseline hazards in gap-time). It introduces a resolution-rate diagnostic to assess sampling representativeness and extends time-dependent ROC analysis to handle clustered survival data, enabling estimation of a portfolio-level term-structure of default risk. Using a rich South African mortgage dataset, the study calibrates and benchmarks the models across goodness-of-fit and discriminatory power, deriving portfolio-level default-term estimates and comparing them to Kaplan-Meier baselines. Findings show that, in this dataset, TFD and PWP perform similarly to AG in most diagnostics, with PWP marginally preferred and AG underperforming due to the common-baseline assumption, while recurrence adds limited predictive gain; the work provides practical diagnostics and open code to improve IFRS 9 PD estimation in financial risk management.

Abstract

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.

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

TL;DR

The paper investigates dynamic lifetime probabilities of default by comparing three recurrent-event Cox models—TFD (time to first default), AG (calendar-time spells with a common baseline hazard), and PWP (spell-stratified baseline hazards in gap-time). It introduces a resolution-rate diagnostic to assess sampling representativeness and extends time-dependent ROC analysis to handle clustered survival data, enabling estimation of a portfolio-level term-structure of default risk. Using a rich South African mortgage dataset, the study calibrates and benchmarks the models across goodness-of-fit and discriminatory power, deriving portfolio-level default-term estimates and comparing them to Kaplan-Meier baselines. Findings show that, in this dataset, TFD and PWP perform similarly to AG in most diagnostics, with PWP marginally preferred and AG underperforming due to the common-baseline assumption, while recurrence adds limited predictive gain; the work provides practical diagnostics and open code to improve IFRS 9 PD estimation in financial risk management.

Abstract

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.
Paper Structure (13 sections, 21 equations, 11 figures)

This paper contains 13 sections, 21 equations, 11 figures.

Table of Contents

  1. Introduction and literature review
  2. Different types of recurrent event survival models
  3. Testing sampling representativeness using the resolution rate Lg of type Lg
  4. Term-structure of default risk: Three Cox regression models for recurrent events
  5. Time-dependent ROC-analysis for survival models
  6. A brief review of classical time-dependent ROC-analysis
  7. Dealing with clustered observations: The clustered tROC-extension
  8. Calibrating the recurrent event Cox regression models to mortgage data
  9. Comparing different recurrent event Cox-models across various diagnostics
  10. Conclusion
  11. Appendix
  12. Structuring data according to each recurrent event survival model
  13. A description of selected input variables within each recurrent event Cox-model

Figures (11)

  • Figure 1: Demonstrating the resolution types and recurrence of performing spells over time for a few hypothetical loans.
  • Figure 2: Illustrating the different spell-level data structures respective to each type of recurrent survival model. These data structures are shown for a hypothetical loan that defaulted twice before becoming right-censored. Inspired by ozga2018Additional.
  • Figure 3: Comparing the resolution rates of type $\psi=1$ (Default) over time across the various datasets. The MAE-based AD-measure from \ref{['eq:ResolRate_MAE']} summarises the discrepancies over time for each dataset-pair.
  • Figure 4: Distribution of the maximum number of performance spells experienced per loan, drawn from the full set $\mathcal{D}$.
  • Figure 5: Resolution rate $r_\psi(t')$ of type $\psi=1$ (Default) over reporting time $t'$, calculated per numbered spell and using the cohort-end $t_s$ time scale. Spell numbers beyond three are grouped together simply for graphical fidelity.
  • ...and 6 more figures