Backtesting Expected Shortfall via Multi-Quantile Regression

In this article, we propose a new approach to backtest Expected Shortfall (ES) exploiting the definition of ES as a function of Value-at-Risk (VaR). Our methodology examines jointly the quality of VaRs along the tail distribution of the risk model, and encompasses the Basel Committee recommendation of verifying quantiles at risk levels 97.5%, and 99%. We introduce four easy-to-use backtests which regress the ex-post losses on the VaR forecasts in a multi quantile regression model, and test the resulting parameter estimates. Monte-Carlo simulations show that our tests are powerful to detect various model misspecifications. We apply our backtests on S&P500 returns over the period 2007-2012. Our tests clearly identify misleading ES forecasts in this period of financial turmoil. Empirical results also show that the detection abilities are higher when the evaluation procedure involves more than two quantiles, which should accordingly be taken into account in the current regulatory guidelines.

Couperier O. and Leymarie J. (2021) Backtesting Expected Shortfall via Multi-Quantile Regression. Working Paper.