Multilevel and Tail Risk Management[Backtesting Expected Shortfall]

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 VaR perspective, our methodology relies on multiple testing to combine evidence on the frequency and dynamic evolution of violations, and to capture more information than a single threshold can provide about the magnitude of violations. Contributions include a formal finite sample analysis of the joint distribution of multi-threshold violations, and limiting results that unify discrete and continuous definitions of cumulative violations across thresholds. Simulation studies demonstrate the power advantages of the proposed tests, particularly with small samples and when underlying models are unavailable to assessors. Results also reinforce the usefulness of CaViaR approaches not just for VaR but also as ES back-tests. Empirically, we assess desktop data by Bloomberg on exchange traded funds. We find that tail risk is not adequately reflected via a wide spectrum of models and available measures. Results provide useful prescriptions for empirical practice and, more generally, reinforce the recent arguments in favor of combined tests and forecasts in tail risk management.