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. Model-free e-statistics are introduced to formulate e-processes for risk measure forecasts, and unique forms of model-free e-statistics for VaR and ES are characterized using recent results on identification functions. For a given model-free e-statistic, optimal ways of 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.

翻译：在最新的巴塞尔协议中，预期亏损（ES）取代风险价值（VaR）成为银行业市场风险的标准风险度量指标，使其成为金融监管中最重要的风险度量。风险建模实践中最具挑战性的任务之一是对金融机构提供的ES预测进行回测。为设计一种无模型的ES回测程序，我们利用了近年来发展的e值与e过程技术。引入无模型e统计量以构建风险度量预测的e过程，并利用关于识别函数的最新结果刻画了VaR和ES的无模型e统计量的唯一形式。针对给定的无模型e统计量，研究了e过程的最优构建方式。所提方法可自然推广至众多其他风险度量与统计量。我们通过大量模拟研究与数据分析，展示了该无模型回测方法的优势，并与文献中现有方法进行了比较。