Calibrating a proportional hazards model with time-correlated covariates: a case study in probability of default modelling for credit risk analysis

Brent Oeyen and Themis Rallis

Quantitative Finance, 2026, vol. 26, issue 2, 273-284

Abstract: Proportional Hazards Models with an exponential duration model encounter a parameter identification problem in case one or more time-dependent covariates are correlated with the time component of the duration formula. Conventional approaches to modelling time-dependent covariates no longer apply, and a new framework is required. A case study is considered from the field of credit risk modelling to propose a framework that solves the parameter identification problem. Specifically, the effect of credit maturities on the default behaviour of a loan portfolio are often not analysed thoroughly, nor is this a regulatory requirement. However, in most portfolios, there can exist a strong negative correlation between the maturity of a loan and the creditworthiness of the customer. This may lead to spurious conclusions when analysing the default behaviour of the portfolio. In this paper, an example – a selected wholesale Low-Default Portfolio (LDP) at ING – of such a correlation is presented and how this impacts the calculation of a Long-Run Average Default Rate (LRADR) estimate, which is used for calibrating regulatory Probability of Default (PD) estimates. A rigorous mathematical framework based on a proportional hazard rate model, where time is correlated with the scale component of the baseline hazard, is introduced to simulate default patterns comparable to the real-world LDP, which also explains how to parametrise such patterns. Finally, several estimation methods for the LRADR estimate given a correlated time component with a credit ranking function are provided and evaluated.

Date: 2026
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DOI: 10.1080/14697688.2025.2601721

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