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Tests of the Correlation Between Portfolio Performance Measures

Journal 35: Zicklin-Capco Institute Paper Series in Applied Finance

Chris Adcock, Nelson Areal, Manuel Armada, Maria Ceu Cortez, Benilde Oliveira, Florinda Silva

This paper reports an investigation into measures of portfolio performance. The Sharpe ratio is the natural performance measure when asset returns come from any elliptically symmetric distribution, regardless of the investor utility function and subject only to regularity conditions. Under such distributions, the measures of portfolio performance which are in common use are monotonic functions of the Sharpe ratio. It is shown that for large sample sizes the correlation between measures of performance which are functions of the Sharpe ratio is asymptotically equal to unity.

The correct specification for tests of the correlation between portfolio performance measures is therefore the null hypothesis ρ = 1. A multivariate test of the correlations between several measures of performance is presented. This may be used in either a multivariate or bivariate setting. The paper presents a detailed example based on a number of FTSE indices. Performance measures are computed both parametrically using the normal distribution and using sample estimates. The new test does not lead to the rejection of the null hypothesis that all correlations are equal to unity. This suggests that despite the evidence of non-normality in returns there seems to be little gained in abandoning the Sharpe ratio.

The choice of measures of risk and performance are critical tasks in the area of portfolio performance evaluation. The traditional theoretical basis of modern portfolio theory is the mean-variance model of Markowitz (1952). In the context of portfolio performance, the mean-variance model and the implied assumption of normally distributed returns lead directly to the use of the Sharpe ratio as the performance measure of choice. In practice, though, asset returns are frequently not normally distributed. Furthermore, as Markowitz himself pointed out, variance, or more precisely volatility, is not necessarily a good measure of risk. Responding to these features, numerous different risk measures have been developed. In particular, recent years have seen the development of risk measures which are specifically concerned with the tails of the distribution of asset returns, or which are intrinsically asymmetric; the so-called downside risk measures. Naturally, the non-normality of returns, the perceived shortcomings in the use of volatility, and the advent of new measures all raise the question of whether the risk-adjusted performance of investment portfolios is sensitive to the method of evaluation used. There are numerous papers in the recent finance literature that present empirical studies of portfolio performance measurement.

Many of these studies provide evidence which suggests that the choice of a particular performance measure has little or no effect on the rankings of a set of investment portfolios. This finding, if true, offers the opportunity for some simplifications in the task of portfolio evaluation and fund manager selection. Pfingsten et al. (2004) compared rank correlations for several risk measures on the basis of an investment bank’s 1999 trading book. They concluded that different measures of performance resulted in largely identical rankings. Eling and Schuhmacher (2006) conducted a study of hedge fund indices data from 1994 to 2003 and drew similar conclusions. The findings from this study were extended significantly in Eling and Schuhmacher (2007) using data from individual hedge funds. The results of this second study were the same as the study of hedge fund indices, in that identical or very similar rank orderings across hedge funds were found. These authors concluded that despite significant deviations of hedge fund returns from a normal distribution, the first two moments seemed to describe the distributions sufficiently well, at least from the perspective of performance measurement. Similar results were also obtained by Eling (208) for a large sample of different types of investment funds.

In contrast to these findings, Ornelas et al. (2010) and Zakamouline (2011) argue that the choice of the performance measure does influence the evaluation of investment funds. In their studies these authors test the equality of ranks based on several performance measures and identify significant differences for some of the performance measures studied. The differences in the empirical findings reported by the papers cited in the preceding paragraph make it clear that the analysis of portfolio performance is not a trivial task. In addition, Zakamouline (2011) points out that many performance measures are monotonic functions of the Sharpe ratio when they are computed using a formula based on the normal distribution. In such cases, ranking portfolios using different measures will produce no additional insights, as the ranks will be identical. Furthermore, under the assumption of normality of portfolio returns, if performance measures are computed using sample quantities, for example a sample semi-variance, then appeal to standard asymptotic theory indicates that rankings will be similar, even if not identical. Notwithstanding the debate about rankings, it remains clear that sample correlations and rank correlations between pairs of performance measures are invariably numerically very different from zero. Not surprisingly, the standard tests of correlation then also lead invariably to the rejection of the hypothesis that the correlation between a pair of performance measures is zero.


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