Measure 4

 Measure 4 looks at the mean log-likelihood of all exceedent events beyond the confidence level plotted on the X axis. We present figures 4u and 4m only to illustrate the difference in relation to measure 5 which we strongly recommend as a solid performance measure. While the mean log-likelihood against confidence level does help in establishing the degree of exceedence - it fails as a valid comparative measure between models. This is because the exceedent events referred to by a specific choice on the X axis do not correspond to the same set of events for different models. Differing sets of exceedent events at the same confidence level imply differing theoretical maximums for the mean log-likelihood of exceedence - making direct comparison fallacious. (In addition we note that figure 4m has no expression of mean log-likelihood for model 5 at the $99\%$ level because both contributing portfolios do not contribute an exceedent event. This again highlights the problem of passing legislation based entirely on evaluations at the $99\%$ level.)