Lifetime probability of default: Cox model with time-dependent covariates using maximum likelihood

Credit granting institutions need to estimate the probability of default, the chance a customer fails to make repayments as promised (BIS (2006) and IASB (2014)). The Cox model with time-varying covariates (Cox (1972), Crowley and Hu (1977)) is a technique often applied due to its substantial benefits beyond classification approaches (such as logistic regression) whilst achieving similar accuracy (Lessmann et al. (2015), Bellotti and Crook(2009)). However partial likelihood estimation of this model has two shortcomings that remain unaddressed in the literature: (1) the baseline hazard is not estimated, so calculating probabilities requires a further estimation step; and (2) a covariance matrix for both regression parameters and the baseline hazard is not produced. We address these by developing a maximum likelihood method that jointly estimates regression coefficients and the baseline hazard using constrained optimisation that ensures the baseline hazard’s non-negativity. We show in a simulation our technique is more accurate in moderate sized samples and when applied to real home loan data it produces a smoother estimate of the baseline hazard than the Breslow (1972) estimator. Our model could be used to predict life-time probability of default, required under the International Financial Reporting Standard (IFRS) 9 accounting standard.