An Analysis of Performance Measures using Copulae

We have carried out a detailed comparison of the statistical properties and the relationships between a set of five performance measures using 14 UK based Investment Trusts over a sample period ranging from 1980 to 2001. Our results suggest very clearly that there is almost no difference between Jensen’s Alpha, the Treynor-Mazuy (TM) measure and the Positive Period Weighting(PPW) measure over our sample period and amongst our set of Investment Trusts. This would seem to indicate that there is no timing ability within these fund managers. The Sharpe Ratio clearly provides different signals regarding performance than the other measures and is the only absolute measure in the set of measures we have considered. While simple correlation analysis suggests that there is a high degree of dependence between most of the measures we have shown that there is a lack of significant concordance between the Sharpe Ratio and all the other measures. This indicates the inadequacy of correlation analysis with non-gaussian data. We have also shown that the Sharpe Ratio exhibits negative left tail area dependence with respect to Jensen’s Alpha, TM and PPW but is independent in the left tail from the Higher Moment measure of Hwang and Satchell, that is when poor performance is indicated. Jensen’s Alpha, TM and the HM measure do not seem to show any significant asymptotic left tail dependency. All the measures appear to be asymptotically independent in their upper tail when good performance is indicated. These results are further refined by non-asymptotic quantile regression results which indicate finite sample dependency the HM measure and Jensen’s Alpha throughout the body of their conditional distribution and in the left tail but not the upper tail.