Multilevel and Tail Risk Management

ABSTRACT We introduce backtesting methods to assess Value-at-Risk (VaR) and expected shortfall (ES) that require no more than desktop VaR violations as inputs. Maintaining an integrated perspective, our methodology relies on multiple testing combine evidence the frequency dynamic evolution of violations, capture information a single threshold can provide about magnitude violations. Contributions include formal finite sample analysis joint distribution multi-threshold limiting results unify discrete continuous definitions cumulative across thresholds. Simulation studies demonstrate power advantages proposed tests, particularly with small samples when underlying models are unavailable assessors. Results also reinforce usefulness CaViaR approaches not just for but ES back-tests. Empirically, we data by Bloomberg exchange traded funds. find tail risk is adequately reflected via wide spectrum available measures. useful prescriptions empirical practice and, generally, recent arguments in favor combined tests forecasts management.