Analytical adjustment of the IRB for Supervisory Formula for the estimation error

The Internal-Ratings Based (IRB) approach is aimed at providing a measure of the maximum loss that a credit portfolio could generate over a year and with a given confidence level. As most of the standard risk measures, such as the Value-at-Risk (VaR), or the Expected Shortfall (ES), also the IRB measure depends on some parameters that must be estimated. The usual plug-in approach, consisting in substituting the theoretical formulas the parameters with their estimates, does not consider the effect of the additional uncertainty generated by the estimation error. In this paper, we develop an analytical correction to the IRB formula that enables us to correct for the parameters uncertainty, using the theoretical setting developed by Gourieroux and Zakoïan (2013) and, to our knowledge, this is the first application of that approach to credit risk. This approach provides an approximated correction that depends on the variance-covariance matrix of the estimated parameters and does not require specific assumptions on their prior distribution, avoiding also computationally intensive Monte Carlo simulations. We show the validity of our correction on simulated data and show that our results are consistent with Tarashev (2010) who adopts Bayesian methods. We argue that our approach is more flexible and suited to be extended to the estimation of other parameters of the IRB formula. The opinions are of the authors and do not necessarily coincide with those of their Institutions.