An analysis of performance measures using copulae 195
where
1
[·] is the inverse of the standard normal cumulative distribution
function with correlation coefficient r
x
. This posterior distribution then pro-
vides all the information needed to construct the aggregated performance
measure and to conduct inference on it in any decision framework. The ques-
tion now turns on how to extract the aggregate performance measure from
this distribution and this depends on the specification of the decision maker’s
utility or loss function. We could simply use the mean or the median of this
distribution but as emphasized by Christoffersen and Diebold (1997), what
function of the non-Gaussian posterior distribution serves as the optimal esti-
mator of the underlying performance measure will depend critically on the
asymmetric loss function the fund manager is almost certainly going to hold.
In particular we expect that fund managers would be substantially more loss
averse than for an equivalent profit on the up side. This issue is much more
complex than can be developed here but see Hwang and Salmon (2002) for
an extended discussion of performance measure aggregation.
7.6 CONCLUSIONS
We have carried out a fairly 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 TreynorMazuy (TM) measure and the Positive Period
Weighting (PPW) measure over our sample period and among our set of 14
investment trusts. This would seem to indicate that there is no timing ability
within these fund managers. The Sharpe ratio clearly provides different sig-
nals 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 concor-
dance 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 perfor-
mance 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 regres-
sion results which indicate finite sample dependency of the HM measure and

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