Conclusions

• Model judgement is dramatically improved by visualization of model performance over an adequately large range of high confidence levels.
• A sophisticated measurement strategy should include the following in addition to the conventional measures based on exceedence counts:
  • Measures (or variants thereof) sensitive to degree of exceedence (such as measures 4 and 5).
  • Measures discriminating model behaviour for large and small movements (such as measure 5).
  • Measures which assess model behaviour with respect to market dynamics (such as measure 3).
• The comparative performance of models in the univariate and multivariate contexts provide a systematic approach to model building which allows discrimination between the quality of the diagonal and off-diagonal elements of the covariance matrix.
• The method of tail-emphasized optimization brings striking improvement in model performance - particularly for the large movements to which VaR methodologies are most vulnerable.
• The usage of implied covariances in models 4 and 5 to estimate $\Sigma_t$ has shown remarkable stability and performance.
• The issues of underestimated VaR due to rank defects or near-singularity of $\Sigma_t$, and stochastic errors in estimation of $\Sigma_t$ - particularly in conjunction with frequent portfolio adjustments - are non trivial.
• Underestimated VaR may be improved by analyzing VaR as a function of portfolio space in the neighbourhood of the candidate portfolio.