Pitfalls in Backtesting Historical Simulation VaR Models

Historical Simulation (HS) and its variant, the Filtered Historical Simulation (FHS), are the most widely used Value-at-Risk forecast methods at commercial banks. These forecast methods are traditionally evaluated by means of the unconditional backtest. This paper formally shows that the unconditional backtest is always inconsistent for backtesting HS and FHS models, with a power function that can be even smaller than the nominal level in large samples. Our ndings have fundamental implications in the determination of market risk capital requirements, and also explain Monte Carlo and empirical ndings in previous studies. We also propose a data-driven weighted backtest with good power properties to evaluate HS and FHS forecasts. Finally, our theoretical ndings are conrmed in a Monte Carlo simulation study and an empirical application with three U.S. stocks. The empirical application shows that multiplication factors computed under the current regulatory framework are downward biased, as they inherit the inconsistency of the unconditional backtest.