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

In this paper we explore the possibility to adapt the methodology developed by Gourieroux (2012) where it is derived, under quite general conditions, a Value at Risk measure that is adjusted for estimation errors. We show that it is possible to obtain an analytic correction of the Supervisory Formula that enables to correct at the same time for the estimation error of the PD and the asset correlation. Besides, providing a solution to the problem of the MoC, we think our approach opens to the possibility of letting banks to estimate the asset correlation by themselves. Indeed, through extensive Montecarlo simulations, we prove that it would be enough to plugin in our revised formula the maximum likelihood estimation of the PD and the asset correlation and the associated matrix of variance and covariance of the estimators, to obtain an estimate of the unconditional PD percentile which is guaranteed not to be exceeded with a probability equal to the desired confidence level. Furthermore, with the aim to preserve the current expression of the Supervisory Formula, we suggest exploiting our approach to correct the confidence level actually introduced in the Supervisory Formula so to obtain the desired confidence level. We believe that the merit of this paper is to provide an approach that opens up the possibility of allowing banks to estimate, in addition to the PD, also the asset correlation parameter, thus overcoming the need for the regulator to set the same values for this parameter for all banks.