Bayesian estimation of probabilities of default for low default portfolios

Abstract

The estimation of probabilities of default (PDs) for low default portfolios by means of upper confidence bounds is a well-established procedure in many financial institutions. However, there are often discussions within the institutions or between institutions and supervisors about which confidence level to use for the estimation. The Bayesian estimator for the PD based on the uninformed, uniform prior distribution is an obvious alternative that avoids the choice of a confidence level. It is demonstrated in this paper that in the case of independent default events the upper confidence bounds can be represented as quantiles of a Bayesian posterior distribution based on a prior that is slightly more conservative than the uninformed prior. The paper then describes how to implement the uninformed and conservative Bayesian estimators in the dependent one- and multi-period default data cases and compares their estimates with the upper confidence bound estimates. The comparison leads to a suggestion of a constrained version of the uninformed (neutral) Bayesian estimator as an alternative to the upper confidence bound estimators.