Estimation Error and Uncertainty in IRB Supervisory Formula: An Analytical Approach

Banks and financial institutions assess credit risk to determine the likelihood of borrower default. Accurate assessment of credit risk is essential to set aside appropriate capital to safeguard against potential losses. Regulatory capital requirements ensure banks maintain adequate financial resilience, reducing systemic risk. Supervisory Formula Method (SFM), under the Internal Ratings-Based (IRB) approach defined by Basel II and Basel III, typically quantifies credit risk for securitized exposures. This formula calculates regulatory capital requirements based on the underlying asset pool's risk parameters, but its accuracy depends on the precision of underlying estimates. Precise measurement enhances financial stability by preventing under- or over-estimation of risk while Inaccurate risk measurement can lead to severe financial and regulatory consequences. IRB methodologies primarily focus on calculating the maximum potential loss that could arise from credit exposures within a portfolio over a year and a given confidence level. This enables institutions to understand and prepare for extreme risk scenarios. Commonly within the IRB framework, Risk Metrics include: Value-at-Risk (VaR), Expected Shortfall (ES). The IRB approach's accuracy significantly depends on robust parameter estimation: reliable estimation of these parameters is fundamental for achieving precise risk evaluations and regulatory compliance. The usual plug-in approach, consisting to substitute the theoretical formulas with the parameters with their estimates, does not consider the effect of the additional uncertainty generated by the estimation error. In many standard derivations and presentations of risk measures like the Value-at-Risk or the Expected Shortfall, it is assumed that all the model’s parameters are known. In practical cases, the parameters must be estimated and this introduces an additional source of uncertainty that is usually not accounted for. If parameters are to be estimated, then also the VaR includes an additional source of randomness that must be taken into account. As long as estimates are uncertain, the estimation risk must be considered in the model. The Prudential Regulators have formally raised the issue of errors stemming from the internal model estimation process in the context of credit risk, calling for margins of conservatism (MoC) to cover possible underestimation in capital requirements Notwithstanding this requirement, to date, a solution shared by banks and regulators/supervisors has not yet been found. The level of capital requirement generated by the IRB approach crucially depends on asset correlation, a parameter that enters the regulatory risk weight formula and it is determined by the Regulators. Several studies have estimated the asset correlations and found that the empirical values are materially different from the regulatory calibration included in the Basel framework. (Casellina et al., The calibration of the IRB supervisory formula – EBA Staff Paper N. 17 – 09/2023). 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.