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：基于e值的回测方法 在最近的巴塞尔协议中，预期亏损（Expected Shortfall，ES）替代了价值风险（Value-at-Risk，VaR）成为银行业市场风险的标准风险度量，因此成为金融监管中最为重要的风险度量。在风险建模实践中, 回测金融机构提供的ES预测是最具挑战性的任务之一。为了设计一种基于模型的自由回测ES程序，我们利用了最近发展的e值和e进程技术。我们引入模型自由的e统计量，以形式化风险度量预测的e进程，并使用近期对于鉴别函数的结果来表征VaR和ES的模型自由的独特形式的e统计量。我们对于给定的模型自由e统计量，研究了构建e进程的最优方法。提出的方法自然可以应用于许多其他的风险度量和统计量。我们开展了广泛的仿真研究和数据分析，以说明模型自由的回测方法的优点，并将其与文献中的方法进行比较。