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.

翻译：在近期的《巴塞尔协议》中，期望损失取代风险价值成为银行业市场风险的标准度量指标，使其成为金融监管中最重要的风险度量工具。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期望损失预测进行回测。为了设计一种无模型的期望损失回测方法，我们利用了最近发展的e值与e过程技术。通过引入回测e统计量来构建风险度量预测的e过程，并基于识别函数的最新研究成果，刻画了风险价值与期望损失回测e统计量的独特形式。针对给定的回测e统计量，我们研究了优化构建e过程的若干准则。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，阐明了这种无模型回测方法的优势，并与文献中的现有方法进行了比较。