Calibration of rating grades to point-in-time and through-the-cycle levels of probability of default

Mark Rubtsov

Abstract

The paper argues for the need for and importance of the dual calibration of a probability of default (PD) model (ie, calibration to both point-in-time (PIT) and through-the-cycle (TTC) PD levels). It explains why such a calibration should happen at the rating grade level, and how PD estimates can be assigned to a bank’s existing rating grades, ie, without introducing any changes to either the underlying (hybrid) rating function or the rating scale. This is done by first translating the traditional asymptotic single risk factor model underlying the Basel risk-weight formulas from its original obligor level to the rating grade level, and then determining the model’s parameters by using a maximum likelihood estimation. Standard deviations of the estimates obtained then provide a link to the regulatory margin of conservatism. Rating migration adds a new dimension to the model and leads to procyclical deviations of TTC PDs per grade from their long-term average counterparts. A link between rating migration and calibration levels gives a new definition to the rating function’s degree of PIT-ness, which becomes a key parameter responsible for the TTC stability at the portfolio level. The closed-form solutions obtained are flexible with respect to different patterns of rating migration and explicitly account for a prediction error in the business cycle model. The proposed technique is illustrated on a sample of corporate customers.