This paper considers a definition of through-the-cycle (TTC) as independent from an economic state that can result in a time-varying TTC probability of default (PD). A top-down approach is proposed to transform hybrid PDs into TTC PDs through the use of a point-in-time-ness (PITness) parameter as an additional parameter to the Vasicek model, which expresses the dependency of a hybrid PD on a common factor. The proposed framework aims to explain fluctuations in observed default frequency (ODF) and modeled default frequency time series. A novel approach is considered that defines ODF as analogous to an aggregated PIT PD stemming from a perfect foresight model, which is not available to the modeler but can be assumed backward in time for calibration purposes. An elaborate segmentation framework is considered to understand differences in both the Vasicek correlation and the PITness parameter for a portfolio of obligors; this can be applied to both retail and nonretail portfolios.

中文翻译：

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关于违约概率及其与观察到的违约频率和共同因素的关系

本文将贯穿整个周期 (TTC) 的定义视为独立于可能导致随时间变化的 TTC 违约概率 (PD) 的经济状态。提出了一种自上而下的方法，通过使用时间点（PITness）参数作为 Vasicek 模型的附加参数，将混合 PD 转换为 TTC PD，该模型表示混合 PD 对公共 PD 的依赖性因素。所提出的框架旨在解释观察到的默认频率 (ODF) 和建模的默认频率时间序列的波动。考虑了一种新方法，将 ODF 定义为类似于源自完美预见模型的聚合 PIT PD，建模者无法使用该模型，但可以出于校准目的及时假设。一个精心设计的分割框架被认为是理解 Vasicek 相关性和债务人投资组合的 PITness 参数的差异；这可以应用于零售和非零售投资组合。