Back-Testing Credit Risk Parameters on Low Default Portfolios : A Bayesian Approach with an Application to Sovereign Risk

The validation of the Probabilities of Default estimated by a rating system model is particularly challenging in the context of low default portfolios; for example, Sovereign portfolios tend to exhibit a very low number (often zero) of defaults, so that traditional hypothesis tests are impractical. Motivated by these considerations, and following Dwyer (2007), Tasche (2013), Kruger (2015) and Blümke (2020), we propose a Bayesian approach to back-test credit risk parameters in low default portfolios. In order to assess the calibration of a PD rating system it is necessary to compare the estimated probability of default for each rating class with the observed empirical distribution of defaults. Under a Bayesian approach such comparison is made via the estimation of a posterior probability distribution of the PD given the realized sample default rates and a prior probability distribution. In this work we focus on the validation and back-testing process of an estimated master scale derived from an internal rating system. In particular, using different configurations of prior distributions for the PD, we implement three different procedures: a “hypothesis test”-like approach to validate master scale PD values, which are used to calibrate the prior distribution; a "Bayesian transfer learning"-like approach that merges market data and observed default data to validate the estimated PDs; and a sequential procedure in which, assuming a prior distribution on the PD of the worst rating class and the monotonicity among the classes, the prior distribution of the PD of each class is calibrated on the posterior distribution of the adjacent worse rating class