Forward Ordinal Probability Models for Point-in-Time Probability of Default Term Structure

Common ordinal models, including the ordered logit model and the continuation ratio model, are structured by a common score (i.e., a linear combination of a list of given explanatory variables) plus rank specific intercepts. Sensitivity with respect to the common score is generally not differentiated between rank outcomes. In this paper, we propose an ordinal model based on forward ordinal probabilities for rank outcomes. The forward ordinal probabilities are structured by, in addition to the common score and intercepts, the rank and rating (for a risk-rated portfolio) specific sensitivity. This rank specific sensitivity allows a risk rating to respond to its migrations to default, downgrade, stay, and upgrade accordingly. An approach for parameter estimation is proposed based on maximum likelihood for observing rank outcome frequencies. Applications of the proposed model include modeling rating migration probability for point-in-time probability of default term structure for IFRS9 expected credit loss estimation and CCAR stress testing. Unlike the rating transition model based on 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 include entity specific risk drivers (e.g., downgrade history or credit quality changes for an entity in last two quarters can be included). No estimation of the asset correlation is required. As an example, the proposed model, benchmarked with the rating transition model based on Merton model, is used to estimate the rating migration probability and probability of default term structure for a commercial portfolio, where for each rating the sensitivity is differentiated between migrations to default, downgrade, stay, and upgrade. Results show that the proposed model is more robust.