Forward ordinal probability models for point-in-time probability of default term structure: methodologies and implementations for IFRS 9 expected credit loss estimation and CCAR stress testing

Common ordinal models, including the ordered logit model and the continuation ratio model, are formulated by a common score (ie, a linear combination of given explanatory variables) plus rank-specific intercepts. Sensitivity to the common score is generally not differentiated between rank outcomes. We propose an ordinal model based on forward ordinal probabilities for rank outcomes. In addition to the common score and intercepts, the forward ordinal probabilities are formulated by the rank- and rating-specific sensitivity (for a risk-rated portfolio). This rank-specific sensitivity allows a risk rating to respond to its migrations to default, downgrade, stay and upgrade accordingly. A parameter estimation approach based on maximum likelihood for observing rank-outcome frequencies is proposed. Applications of the proposed model include modeling rating migration probability for point-in-time probability of default term structure for International Financial Reporting Standard 9 expected credit loss estimation and Comprehensive Capital Analysis and Review stress testing. Unlike the rating transition model based on the Merton model, which allows only one sensitivity parameter for all rank outcomes for a rating and uses only systematic risk drivers, the proposed forward ordinal model allows sensitivity to be differentiated between outcomes, and to include entity-specific risk drivers (eg, the downgrade history or credit quality changes for an entity in the previous two quarters can be included). No additional estimation of the asset correlation is required. As an example, the proposed model, benchmarked with the rating transition model based on the Merton model, is used to estimate the probability of default term structure for a commercial portfolio, where for each rating the sensitivities are differentiated between migrations to default, downgrade, stay and upgrade. Our results show that the proposed model is more robust.