Measure 5

 What is most important in the context of risk management is not whether a large number of run-of-the-mill movements were predicted well at the expense of a few large movements - but exactly the contrary. In fact, we can well afford a general over-estimate or under-estimate of risk with regard to small movements in exchange for better performance in the tail. Measure 5 presents the mean log-likelihood of all movements beyond a certain percentile level rather than all movements exceeding a certain confidence level. As discussed in section  the results of model 5 in figures 5u and 5m clearly show that tail emphasized optimization used in model 5 achieves decreased sensitivity of forecasting performance to the size of the event as desired. In addition, figures 5u and 5m are the only figures presented where the X and Y axis are not both model dependent. The X axis depends only on the data and the corresponding mean log-likelihoods refer to the same set of events - making comparison of models meaningful.