A review of the fundamentals of the Fundamental Review of the Trading Book: standard foreign exchange rules are highly asymmetric with respect to reporting currencies

Abstract

We examine the mathematical formalism underlying the Basel Committee on Banking Supervision’s “Revised standardized approach of the Fundamental Review of the Trading Book”. One of its goals is to provide a simple, uniform methodology for market risk that applies to all banks independently of their internal models. The formalism exhibits subtle nonlinearities that can have deep implications and deserve careful analysis. We demonstrate that the capital charge for the linear part of the foreign exchange book can differ substantially, depending on the reporting currency. Indeed, we prove that the linear charge is the same for any reporting currency if and only if the correlation between risk-weighted positions in the currencies is 50%, and the intra-bucket correlation is 100%. We show that the capital charge for the nonlinear part of the book fails to be invariant regardless of the correlation values. We demonstrate (in the most recent version of the rules) up to 58% and 300% anomalies for the linear and nonlinear charges, respectively, depending on the book. We also show the rather surprising result that, for a given reporting currency, the capital charges are only functionally dependent on the original underlying rates of the portfolio and are not necessarily dependent on the rates referencing the reporting currency. The anomaly is therefore due to the noninvariance of the aggregation mechanism under a change of reporting currency. We solve the problem by offering a proposal that carefully restores the invariance, using the same regulatory framework and without changing the underlying conservatism. Our analysis also has the practical application of facilitating the transformations of the computations in any reporting currency, possibly circumventing the need for system changes in some cases.