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

金融风险管理的核心任务之一是对风险进行预测。量化风险的两个最流行的指标是风险价值和预期缺口。鉴于最近发生的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为过。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合适的。为此，已经开发了相当多的回测。从统计学的角度来看，这些测试往往表现出严重的大小扭曲。这意味着一个正确的预测模型不会以预先指定的5%的概率被拒绝，而是以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，在回溯测试中有意义的观察数量很少。因此，统计学中通常使用的渐近论点是不可靠的。其次，除了极少数例外，回溯测试忽略了这样一个事实，即模型参数必须在风险值和预期缺口预测之前进行估计。项目的前两个目标是应对回溯测试的这两个挑战。关于尺寸扭曲的第一个原因，我们打算发展一种理论，允许处理可用观测非常少的情况。对于第二个，我们打算改善现有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复杂的个案分析。我们还将研究这两个问题是否也可以同时解决，以开发一种具有普遍适用性的方法。文献中的回测都是一次性的。这意味着所有数据在测试前都必须可用。该项目的最终目标是建立风险预测的监测程序，以处理新的即将到来的观测。这允许快速可靠地检测预测模型故障。这样的程序可能不仅对金融公司感兴趣，而且对监督这些金融公司的监管机构也是如此。

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.