Estimation Error in IRB Models: An Analytic Solution to the Margin of Conservatism

The internal ratings-based approach (IRB) to capital requirements for credit risk is a set of models that allow banks assessing exposures’ credit risk, and calculate the minimum regulatory capital they need to hold under the Basel framework. IRB models develop on a sound theoretical basis, and apply statistical techniques on historical data to estimate the probability of default (PD), loss given default (LGD) and exposure at default (EAD) of each exposure. These estimates are subject to uncertainty and errors and these errors can lead to underestimation of the capital needed to cover the credit risk. To account for estimation risk, the Basel framework requires that IRB models incorporate a margin of conservatism (MoC), that can be implemented by introducing an adjustment factor reflecting the magnitude of the estimation error. However, to date, an agreed solution shared by banks and regulators/supervisors has not yet been defined. In this paper, considering the theoretical approach suggested by Gourieroux and Zakoian (2013), we conform their approach to the Asymptotic Single Risk Factor (ASRF) model that is the baseline for the derivation of the credit risk measures under the IRB approach, based on Merton (1974), Vasicek (2002) and Gordy (2003). This theoretical approach allows us to set up an analytical correction to the IRB supervisory formula accounting for the estimation error related to the estimation of the PD parameters. We also prove that it is possible to define an analytical correction that accounts for the uncertainty in estimating the asset correlation. The proposed correction is tested with Monte Carlo simulations. The proposed approach in this paper does not require introducing additional elements in the ASRF model (e.g. prior distributions or other parameters), and the implemented correction ensures that the probability of observing an exception (i.e. a default rate higher than the estimated quantile of the default rate distribution) is equal to the desired confidence level. An empirical application of our approach is presented on real data drawn from the Statistical Database of the Bank of Italy.