The profit-and-loss attribution test

In this paper, we analyze the failure probabilities of the profit-and-loss attribution (PLA) test as defined in the final market risk standard published in January 2016 by the Basel Committee on Banking Supervision. We calculate theoretical failure probabilities under the assumption that both the hypothetical and unexplained profit-and-loss (P&L) for an individual instrument are normally distributed random variables with zero mean and a prescribed ratio of their respective variances. In addition, we assume that the hypothetical P&Ls across different instruments in the same trading desk have a constant correlation, as do the unexplained P&Ls. We present results for the probabilities of failing the PLA test within different horizons and the steady-state proportion of desks that a bank might expect to maintain accreditation in order to use the internal model approach, assuming a minimum period of delay associated with the reaccreditation process subsequent to a desk failing the PLA test. Our analysis explains why the PLA test is likely to have a high failure probability, making it difficult to pass over a sustained period.