E-backtesting

In the recent Basel Accords, the Expected Shortfall (ES) replaces the Value-at-Risk (VaR) as the standard risk measure for market risk in the banking sector, making it the most important risk measure in financial regulation. One of the most challenging tasks in risk modeling practice is to backtest ES forecasts provided by financial institutions. To design a model-free backtesting procedure for ES, we make use of the recently developed techniques of e-values and e-processes. Backtest e-statistics are introduced to formulate e-processes for risk measure forecasts, and unique forms of backtest e-statistics for VaR and ES are characterized using recent results on identification functions. For a given backtest e-statistic, a few criteria for optimally constructing the e-processes are studied. The proposed method can be naturally applied to many other risk measures and statistical quantities. We conduct extensive simulation studies and data analysis to illustrate the advantages of the model-free backtesting method, and compare it with the ones in the literature. This paper was accepted by Agostino Capponi, finance. Funding: R. Wang acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada [Grants RGPIN-2024-03728 and CRC-2022-00141]. J. Ziegel acknowledges financial support from the Swiss National Science Foundation. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.01659 .