Dec
8

## Why doesn’t the choice of performance measure matter?

Choosing an appropriate performance measure is important for fund investors, nevertheless, many researchers find empirically that the choice of measures does not matter because those measures generate identical rank ordering, even though the distribution of fund returns is non-normal. In this paper we certify their findings by proving the monotonicity of several widely used performance measures when the distribution is a location-scale family. The mutual fund monthly return data from 1997 to 2015, together with simulation results, collaborate with our proof.

An adequate risk-adjusted return performance measure to select investment funds is crucial for financial analysts and investors. Sharpe ratio has become a standard measure by adjusting the return of a fund by its standard deviation (Sharpe, 1966), nevertheless, practitioners often question this measure mainly for its invalidity if the distribution of fund returns is beyond normal (Kao, 2002; Amin and Kat, 2003; Gregoriou and Gueyie, 2003, Cavenaile, et al, 2011, Di Cesare, et al, 2014). Several new measures have been proposed and investigated to overcome this limitation of the Sharpe ratio, however, Eling (2008)

finds choosing a performance measure is not critical to mutual fund evaluation, Eling and Schuhmacher (2007) compare the Sharpe ratio with 12 other measures for hedge funds and conclude that the Sharpe ratio and other measures generate virtually identical rank ordering, despite the significant deviations from normal distribution. Similar evaluation includes Eling and Faust (2010) on funds in emerging markets, Auer and Schuhmacher (2013) on hedge funds, and Auer (2015) on commodity investments.

This paper proves that several widely used performance measures are monotonic if the distribution of asset returns is a LS family, a family of univariate probability distributions parametrized by a location and a non-negative scale parameters that is commonly applied in finance (Levy and Duchin, 2004). Our proof certifies the empirical findings in other studies on the indifference of choosing a performance measure when valuing a fund. We show that those measures generate virtually the same rank ordering using monthly mutual fund return data from 1997 to 2005 and Monte-Carlo simulations. Therefore this paper contributes to both the academia and industry by clarifying the phenomenon.

For example, the below figure plots the correlation and confidence intervals based on 2000 simulations for each sample size. For simplicity, we show the results for the Sharpe (ρ1), the Sharpe-Omega (ρ2) and the Sortino ratio (ρ3) only. Consistent with the previous finding, the rank correlation among these performance measures is roughly equal, and is approaching one with the increase of sample size.

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An adequate risk-adjusted return performance measure to select investment funds is crucial for financial analysts and investors. Sharpe ratio has become a standard measure by adjusting the return of a fund by its standard deviation (Sharpe, 1966), nevertheless, practitioners often question this measure mainly for its invalidity if the distribution of fund returns is beyond normal (Kao, 2002; Amin and Kat, 2003; Gregoriou and Gueyie, 2003, Cavenaile, et al, 2011, Di Cesare, et al, 2014). Several new measures have been proposed and investigated to overcome this limitation of the Sharpe ratio, however, Eling (2008)

finds choosing a performance measure is not critical to mutual fund evaluation, Eling and Schuhmacher (2007) compare the Sharpe ratio with 12 other measures for hedge funds and conclude that the Sharpe ratio and other measures generate virtually identical rank ordering, despite the significant deviations from normal distribution. Similar evaluation includes Eling and Faust (2010) on funds in emerging markets, Auer and Schuhmacher (2013) on hedge funds, and Auer (2015) on commodity investments.

This paper proves that several widely used performance measures are monotonic if the distribution of asset returns is a LS family, a family of univariate probability distributions parametrized by a location and a non-negative scale parameters that is commonly applied in finance (Levy and Duchin, 2004). Our proof certifies the empirical findings in other studies on the indifference of choosing a performance measure when valuing a fund. We show that those measures generate virtually the same rank ordering using monthly mutual fund return data from 1997 to 2005 and Monte-Carlo simulations. Therefore this paper contributes to both the academia and industry by clarifying the phenomenon.

For example, the below figure plots the correlation and confidence intervals based on 2000 simulations for each sample size. For simplicity, we show the results for the Sharpe (ρ1), the Sharpe-Omega (ρ2) and the Sortino ratio (ρ3) only. Consistent with the previous finding, the rank correlation among these performance measures is roughly equal, and is approaching one with the increase of sample size.

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