Measure 3

 In spite of the fact that measure 3 is what we refer to as a digital measure (like measures 1 and 2) - it brings brings out an important point which is missed by measures 1 and 2. This is the clustering property of exceedences. This measure determines if the frequency of consecutive exceedences is higher than expected at a given confidence level. It is expected that models 1 and 2 which make no attempt to capture the autocorrelation of volatility should behave the worst in terms of this measure. It is therefore surprising to find that model 3 fares worse than model 1 with respect to this measure. (The consecutive loss exceedence ratio of model 2 in the univariate context at the $99\%$ confidence level is 11.01 - not plotted in figure 3u for clarity of presentation.)

We have already pointed out in section  the serious caveat with regard to low number of exceedence events when focusing on the $99\%$ confidence levels. We again emphasize that while it is legitimate that any regulatory authority should be looking at these high levels for the evaluation of risk on a day to day basis once a model is accepted - the criteria of accepting this model should be based on performance of the model over an entire range of extreme levels starting from say the $75\%$ level. Figure 3m provides an example of this problem. In figure 3m we have only plotted data when there is at least one exceedence event beyond a certain confidence level (contributed by at least one of the two portfolios over which the measurement is averaged). Thus at the $99\%$level the absence of any entry in figure 3m indicates that none of the models admit a single event of consecutive exceedences over a 1000 day out-of-sample period. It is only when viewing the approach to the $99\%$ level that we can pick out the relative performance of the models and the consistent superiority of model 5.