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.

翻译：在建立贷款生命周期违约概率模型的过程中，使用Cox比例风险模型时常忽略重复违约事件。排除此类事件可能导致有偏且不准确的违约概率估计，从而削弱抵御未来损失的财务缓冲能力。为此，我们研究了几种能够纳入重复违约事件的Cox模型亚型。我们使用实际数据作为示例，分别探讨了Andersen-Gill模型和Prentice-Williams-Peterson事件间隔时间模型。这些模型与刻意忽略重复事件的基线模型——首次违约时间模型进行了比较。我们采用Harrell's c统计量、调整后的Cox-Sell残差以及时变受试者工作特征分析的新颖扩展方法对模型进行评估。通过这些Cox模型，我们展示了如何推导投资组合层面的违约风险期限结构，即贷款平均生命周期内的一系列边际违约概率估计。虽然首次违约时间模型与Prentice-Williams-Peterson模型在所有诊断指标上均无显著差异，但Andersen-Gill模型的表现未达预期。我们相信，本教程配合代码库将为从业者和监管者提供重要参考。因此，我们的研究改进了当前在IFRS 9准则下运用Cox模型生成及时准确违约概率估计的实践方法。