Model 1

In terms of the measures presented it turns out that historical simulation is not as bad as one would expect for such a simplistic model - which makes no attempt to capture the dynamics of the market and which is based entirely on the series x^_(P) without referring to the characteristics of the individual risk factors x^_(k). It has reasonable results in terms of the digital measures but has very bad quality in terms of the degree of exceedence as reported in figures 4 and 5. We believe that by introducing smoothing in the conditional distribution determined from the 250 day empirical distribution - with the constraint that the probability density must monotonically decrease away from the center - the performance of this model can be considerably improved. If in addition the smoothed 250 day empirical distribution based on X_t (expression ) could be modulated so as to confirm to the variance of a GARCH(1,1) process (optimized with tail emphasis) then the resultant model may be extremely powerful. The caveat of such a scheme however is the complexity introduced in the portfolio function if this methodology is employed in the usual way - where the stochastic process predicts each series x^_(k) rather than directly predicting the portfolio series x^_(P). For now, we leave this proposal of such a hybrid approach - mixing historical simulation and stochastic methods - as food for thought for future investigations.