Extending Backtests of Value-at-Risk and Expected Shortfall Forecasts

风险预测是金融风险管理的核心任务之一。两个最流行的量化风险的指标是风险价值和预期缺口。鉴于最近的2007- 2008年金融危机，对这些风险指标进行充分预测的作用怎么强调都不过分。与任何其他预测一样，有许多模型能够发布这些预测。因此，有必要从统计上评估这些模型中哪一个是适当的。为此，已经开发了相当多的回测。从统计学的角度来看，这些测试往往显示出严重的尺寸失真。这意味着正确的预测模型不会以预先指定的概率（例如）5%被拒绝，而是以（例如）50%的概率被拒绝。这些扭曲可能有两个原因。首先，由于风险预测只关注分布的尾部，因此回测中有意义的观测值数量很少。因此，统计学中通常使用的渐近参数不能依赖。其次，除了极少数例外，回测忽略了模型参数必须在风险价值和预期短缺预测之前估计的事实。该项目的前两个目标是应对回测的这两个挑战。关于大小失真的第一个原因，我们打算发展一个理论，允许处理的情况下，很少有可用的观察。第二，我们打算改进现有的稀缺测试的用户友好性。到目前为止，这些测试需要对从业者进行详细的个案分析。我们也将探讨这两个问题是否也可以同时处理，以发展一种可以普遍应用的方法。这意味着在测试之前必须获得所有数据。该项目的最终目标是建立风险预测监测程序，以处理新的观测结果。这允许快速可靠地检测预测模型故障。这种程序不仅对金融公司有意义，而且对监督这些金融公司的管理机构也有意义。

One of the key tasks in financial risk management is to forecast risk. Two of the most popular measures that quantify risk are the Value-at-Risk and the Expected Shortfall. In light of the recent financial crisis of 2007-8, the role of adequate forecasts of these risk measures cannot be overstated. As with any other forecast, there is a multitude of models capable of issuing these predictions. Consequently, there is a need to statistically assess which of these models is adequate. To this end, quite a few backtests have been developed. From a statistical point of view, these tests often display serious size distortions. This means that a correct forecasting model is not rejected with the pre-specified probability of (say) 5%, but with (say) 50% probability. These distortions can have two causes. First, as risk forecasts only concern the tails of the distribution, the number of meaningful observations in backtests is small. Thus, asymptotic arguments typically used in statistics cannot be relied upon. Second, with very few exceptions, backtests ignore the fact that model parameters have to be estimated prior to Value-at-Risk and Expected Shortfall forecasting.The first two aims of the project are to deal with these two challenges for backtests. Regarding the first reason for the size distortions, we intend to develop a theory that allows to deal with the situation of very few useable observations. For the second, we intend to improve the user-friendliness of the scarce available tests. So far, these tests require an involved case-by-case analysis of the practitioner. We will also investigate whether the two problem can also be tackled at once to develop a method that can be applied universally.The backtests in the literature are all of a one-shot type. This means that all data must be available before testing. The final aim of the project is to construct monitoring procedures for risk forecasts that can deal with new incoming observations. This allows to detect forecasting model failure quickly and reliably. Such a procedure may not only be interesting for financial firms, but also for regulatory bodies monitoring these financial firms.