Abstract

The Sharpe ratio from Sharp (1966) is a popularly used performance measure to compare funds or to create fund rankings. However, the Sharpe ratio has some limitations and well-known shortcomings that have been extensively discussed in the literature.

Dowd (1999) highlighted the importance of dealing with net rather than gross portfolio exposures and suggested implementing a value at risk (VaR) approach to risk-return analysis.

Dowd (2000) pointed out the limiting assumption that returns of the assets used for the calculation of the Sharpe ratio are normally distributed.

Harding (2002) criticized the definition of risk in the construction of the Sharpe ratio. He argued that the return of an asset is a definite and meaningful quantity, whilst risk is not.

Harding (2002) considered the problem that large positive returns are penalized in the calculation of the Sharpe ratio. The removal of the highest returns from a data series could actually increase the Sharpe ratio by reducing the “risk” - a case of “reductio ad absurdum”.

Goetzman, Ingersoll, Spiegel and Welch (2002) showed that the Sharpe ratio may be manipulated with option-like strategies.

Lo (2002) argued that the constituents of the Sharpe ratio are unknown quantities that must be estimated statistically and are, therefore, subject to estimation error.

Cerny (2003) found that the Sharpe ratio is closely related to quadratic utility and argued that there is a one-to-one relationship between the maximum quadratic utility attainable in the market and the Sharpe ratio.

The two major generally prevailing criticism are (1) that mutual funds with negative returns cannot be compared meaningful using the Sharpe ratio and (2) when the return distributions deviate from normality, it may lead to unreasonable results.

Therefore, we employ two performance measures, which refined the Sharpe ratio, the Israelsen’s Sharpe ratio and the new R-Sharpe ratio, to evaluate the performances of funds with negative returns. Then, this study examines which performance measurement approaches are efficient for non-normality on the distribution of asset returns. In addition, we engage modern performance measures to calculate downside risks.

We collect daily returns of 26 equity funds from September 2021 to September 2022. This funds represents the peer group “Equity fund of funds global - dynamic” of the GELD-Magazin, No. 11/2022.

The study carried out, that the choice of the performance measure has an impact on the ranking results of the used peer group and thus on the assessment.

1. Introduction

The modern portfolio theory of Markowitz based on a mean-variance model indicates that the investors always choose the optimal risky portfolio with the highest Sharpe ratio within a feasible set. According Van and Duong (2021), the Sharpe ratio is a natural definition of a performance measure. Also, the mean-variance model assumes that the investor has negative exponential utility and that the risky asset returns are normally distributed.

Preceding literature illustrates that the Sharpe ratio (SR) generates biases in performance evaluation (1) if returns are negative and (2) if returns distribution deviates from normal distribution because SR is derived under the mean-variance model with the strict assumption of either quadratic preferences or customarily distributed returns. When returns are negative or the return distributions deviate from normality, it may lead to unreasonable results. Therefore, this study examines which performance measurement approaches are efficient for negative returns and non-normality on the distribution of asset returns.

We collect daily returns of 26 equity funds from September 2021 to September 2022. Then, we employ the Sharpe ratio and two performance measures, which refined the Sharpe ratio, the Israelsens’s Sharpe ratio, and the R-Sharpe ratio, to evaluate the performances of equity funds in the negative market phases 2022. In addition, we engage three modern performance measures (Modified MVAR Sharpe ratio, Lower Partial Moments and Omega ratio) to calculate downside risks. The downside risk of an investment is the maximum loss that can occur owing to the uncertainty present in the realized return (Dowd, 2005). In a typical distribution chart, investors with a long-term position are worried about the returns that fall to the right side of the distribution and investors with a short-term position focus on the left side of the chart. Thus, investors need performance measures they can provide information on the downside risk potential as well as being able to capture the potential returns that fall on the right side of the distribution chart. All three estimators consider the first four moments of distribution in computing performance measures.

The detail of this chapter is organized as follows. Section 2 reviews the two methodologies of the refined Sharpe ratios and present three approaches for calculating nonparametric estimations and downside risks. The data basis and research methodology is described in section 3. Section 4 presents the empirical results of the study. The final section is the conclusion.

2. Approaches to measuring performance

2. 1 Approaches to measuring performance with negative returns

Several alternative refinements of the original Sharpe ratio have been developed to rank funds in periods of negative excess returns. We find that modified versions of the Sharpe ratio generally result in more consistent performance rankings as the original Sharpe ratio.

2.1.1 Performance measures based on volatility

The normal distribution is based on the assumption that all returns are symmetrically distributed around their mean. It is a theoretical model and is fully defined by the parameters "expected value" and "variance". The normal distribution is a good approximation for the long-term returns of the global bond and stock market.

2.1.2 Israelsen's modified Sharpe ratio

Many studies have proposed adjustments to the traditional Sharpe Ratio to overcome two major shortcomings, namely the inability to account for negative returns and higher moments. For example, Israelsen's (2005) study suggests adding an exponent to the standard deviation (risk denominator) to improve the Sharpe ratio estimate when excess returns (rp − rf) are negative.

Israelsen's modified Sharpe ratio (ImSRp) is calculated using historical returns as follows:

2.1.3 R-Sharpe ratio

The R-Sharpe ratio method from Roland Rupprechter is based on the idea of adjusting risk premiums in such a way that portfolios or funds with negative returns can also be correctly ranked according to their expected return/variance ratio. The R-Sharpe ratio indicates the risk premium adjusted by delta(s) per unit of total risk assumed. Here, delta(s) = delta (rpw;1) = s2 - s1, where rpw is the arithmetic mean of the returns of the worst performance portfolio in the peer group. The R-Sharpe ratio (RSRp) is calculated using historical returns as follows:

The general rule is that the higher the R-Sharpe ratio, the better the return-to-risk performance of a particular fund. The use of this method is reasonable if funds produce negative return-to-risk ratios. The R-Sharpe ratio is quite simple to calculate and give logically comprehensible results. Cross-asset class comparisons are possible if the funds to be compared have similar distribution characteristics.

2.2 Downside performance measure

If the returns are not normally distributed, for example if funds use options for leverage or hedging, the risks associated with fat tails or left-skewed distributions are significantly underestimated and the performance is presented too positively Favre/Singer (2002). The use of derivatives into a portfolio can cause a positive or negative skewness in the return distribution and at the same time have an impact on the kurtosis of the return distribution. A change in the skewness and kurtosis of the return distribution in turn has the effect that the probability of achieving a certain return changes and the probability assumptions of the normal distribution no longer fully apply.

This is the reason why Favre and Galeano (2002) include skewness and kurtosis in the calculation by deriving the modified Value at Risk based on the Cornish-Fisher development.

Furthermore, according to Chen, He & Zhang (2011), the Omega Ratio is another useful ratio in the context of non-normally distributed returns and downside risks.

2.2.1 Modified (MVaR) Sharpe ratio

According to Gregoriou & Gueyie (2003), the modified Sharpe ratio considers the excess return of an investment to the ratio of a modified Value at Risks, whereby this ratio extends the normal Sharpe ratio by the factors skewness and kurtosis.

According to Gregoriou et al. (2003), the formula can be defined mathematically as follows:

2.2.2 Lower Partial Moments

The lower partial moments (LPMs) measure risk by negative deviations of the stock returns in relation to a minimal acceptable return, or return threshold τ. As LPMs consider only negative deviations of returns from τ, they seem to be a more appropriate measure of risk than the standard deviation, which considers both negative and positive deviations from the expected return. The advantage of the LPM measures is that no prerequisites with regard to the form of distribution have to be fulfilled. The LPM measures can be applied to any distribution of returns.

The literature proposes a minimum return τ of 0 percent as nominal capital preservation, the inflation rate as real capital preservation, the risk-free interest rate as achieving the minimum opportunity cost, the expected value of the return on a given asset as ensuring the expected increase in wealth, and a market index as comparing performance with the market.

Given a sample in the size of T observations, the following distribution-free estimator of the nth-order LPM measure is obtained:

About the parameter n, LPM with the order 0 < n < 1 can express ’risk seeking’, for n = 1 ’risk neutrality’, and n > 1 ’risk aversion’ behaviour for the investor. Thus, the higher n the more risk averse an investor. The LPMs of order 0 and 1 are, respectively, the shortfall probability and the expected shortfall.

2.2.3 Omega ratio

In an enlightening research, Keating and Shadwick (2002b) and Keating and Shadwick (2002a) introduce the Omega ratio, and claimed that this universal performance measure, designed to redress the information impoverishment of traditional mean- variance analysis would address these concern. They emphasize that the Omega metric has the great ad- vantage over traditional measures to encapsulate all information about risk and return as it depends on the full return distribution, as well as to avoid looking at a specific level as this measure is provided for all level, hence entitling each investor to look at his/her risk appetite level.

Strictly speaking, the omega ratio is defined as the probability-weighted ratio of gains over losses at a given level of expected return. In financial words, this ratio determines the quality of the investment bet relative to the return threshold.

Omega (Ω is calculated by adjusting the excess return with the LPM measures of order 1, 2 or 3.

According to Bacon (2008), the Omega Sharpe ratio is simply Ω - 1, which should produce identical performance rankings as the original Omega (Ω) ratio, which can be represented as follows Keating & Shadwick (2002):

3. Data and Research Methodology

We calculate steady daily returns based on the daily redemption values of the equity mutual funds.

For the performance analysis, we use the period from September 2021 to September 2022. The choice of the one-year interval is justified by the existence of negative excess returns as a result of the negative stock market phase in 2022.

Another argument for this time horizon is the possibility of comparing the risk-adjusted and downside risk results after applying different performance measures with the peer group of GELD-Magazin, who is ranked by returns.

For the threshold return we use 10%, for the risk free rate 0.50%. Previous analyses show that the choice of interest rate has little or no impact on the results. Although level differences may arise due to the choice of interest rate, they do not exert any influence on the ranking, Roßbach (1991).

We collect daily returns of 26 equity funds from September 2021 to September 2022. This funds represents the peer group “Equity fund of funds global - dynamic” of the GELD-Magazin, No. 11/2022.

4. Empirical Results

4. 1 Results of approaches to measuring performance with negative returns

4.1.1 Ranking based on returns

Table 1 reports the first two moments of the peer group. As we can see from the table, all funds exhibit negative returns. The fund with the lowest volatility (12.74%) has a higher return compared to the fund with the highest volatility (22.28%).

4.1.2 Rankings by application the Sharpe ratio

Table 2 lists the ranking based on the Sharpe ratio. As result of the negative stock market performance 2022 all funds have a negative Sharpe ratio. In such cases it is often considered paradoxical that a fund with greater standard deviation and weaker performance may nonetheless have a higher (less negative) excess return Sharpe ratio and thus be considered to have been "better". We saw this effect also in our study. The table 2 illustrate, that the fund in position 5 has despite his higher volatility and lower return the better SR compared with the fund in position 1. Thus, we suggest that, for performance evaluation, the SR in phases of negative returns should not be used.

4.1.3 Ranking if the Israelsen’s Sharpe ratio is used

Israelsen (2005) added an exponent to the standard deviation (risk denominator), to improve the Sharpe ratio. Table 3 reports the rankings based on the Israelsen’s Sharpe ratio (ImSR). As we can see, the ImSR produce a more consistent ranking then the SR. But looking at figure 1, we found that the results do not exactly reflects the distributions of the top four funds. Thus, we suggest that, the ImSR should be used with caution for performance evaluation.

4.1.4 Ranking based on the R-Sharpe ratio

The solution from Roland Rupprechter is to add a constant difference value (delta rpw; 1) to the numerator of the Sharpe ratio (excess return). The higher excess return divided by the standard deviation results in (1) positive numbers and (2) leaves the risk to return ratio compared to the SR unchanged. Using this modification results in an intuitive ranking. This is confirmed by the results of figure 1. The fund with the narrowest distribution leads the ranking. The other three funds to be seen in the graph also comply with the ranking in table 4. Thus, we can conclude that the R-Sharpe ratio produce, with the view of the Markowitz mean-variance model, correct risk-adjusted rankings if all or some returns of the peer group funds are negative.

4.2 Results of downside performance measure

4.2.1 Modified (MVaR) Sharpe ratio

For calculating the MVaR, we used white noise error terms created from a GARCHNIG model. Adjusting the excess return with the modified Value at Risk results in the performance measure modified MVAR Sharpe ratio (MSR). Our analysis shows that the range of MSR results was with -1.94% to -7.60% unexpectedly wide. The mean value was 4.27%.

The biggest surprise was that a fund with volatility in the first quintile had with -7.22% the second worst result. With using the MSR we could see significant risk of loss that investors might sustain. Therefore, the variance result may be misleading because some other funds may induce similar big losses than this fund, but variance cannot detect them.

4.2.2 Lower Partial Moments (LPM)

Before we started the LPM analysis, we tested the fund returns with the Jarque-Bera (JB) Test of their distributions. The JB test is a statistical test that uses skewness and kurtosis in the data to test whether there is a normal distribution and was proposed by Jarque and Bera (1980).

Normality is one of the assumptions for many statistical tests, like the t test or F test. A normal distribution has a skew of zero (i.e.it’s perfectly symmetrical around the mean) and a kurtosis of three; kurtosis tells how much data is in the tails and gives an idea about how “peaked” the distribution is. It’s not necessary to know the mean or the standard deviation for the data in order to run the test.

At a significance level of 5%, the hypothesis of normal distribution is rejected if JB > 6. This value was exceeded by 7 of 26 funds. It therefore makes sense to carry out a more sophisticated investigation with the Lower Partial Moments.

As table 5 shows, there were major shifts in the ranking when considering the lower partial moments. The jump of one fund up 12 places was the largest forward change. His probability of losses lies at 52%. When losses occur, they are on average 0.374%. This fund benefited from its above-average right skewed distribution compared to the peer group. Hence, use of the Lower partial moments becomes significant whenever the data does not conform to the assumptions of the Sharpe ratio weighted.

4.2.3 Omega ratio

As presented in Keating and Shadwick (2002b) and Keating and Shadwick (2002a), for an asset whose return r has a cumulative probability distribution function F and θ is the target return threshold defining what is considered a gain versus a loss, the Omega ratio is defined as:

When θ is set to zero the gain-loss-ratio by Bernardo and Ledoit arises as a special case Bernardo and Ledoit (2000). The selling point of the omega ratio compared to Sharpe ratio and other traditional risk ratio is that at first sight, it seems to depend on the entire return distribution through the cumulative probability distribution function F as well as not rely on any particular moments in terms of value and even existence, making it intellectually very attractive, so Benhamou, Guez, Paris (2020). Graphically, for a θ value of 2,00 percent and the cumulative distribution given by figure 2, the Ω(θ) ratio is defined as the ratio of the red area over the green area.

The omega ratio can help to understand how well an investment strategy mitigates tail risk. Advisors will want to compare the omega ratios of a few funds to properly assess this. An example of this from the peer group: the fund with the lowest volatility (blue line) in table 1 achieved a better loss limitation than the fund in position 26 (red line) over the one-year period under review. However, there is a cost to this: The price of protecting on the downside is to give up some of the returns on the upside. Figure 3 illustrates this trade-off.

Conclusions

Although the Sharpe ratio is a popular performance measurement, it has various limitations. Sharpe ratios can be negative. This is the situation if the performance of mutual funds has underperformed the risk-free rate. If investors pick the one with the highest Sharpe ratio, they choosing the fund with the greatest volatility, rather than the least, since a negative number divided by a large number is greater than that negative number divided by a small number. Therefore, we used two performance measures, which refined the Sharpe ratio, the Israelsen’s Sharpe ratio (ImSR) and the R-Sharpe ratio (RSR), to evaluate the performances of funds with negative returns. Our empirical findings indicate that the ImSR and RSR produce a more similar ranking than the Sharpe ratio. Due to his unchanged risk to return ratio compared to the SR, we can conclude that the RSR produce - with the view of the Markowitz mean-variance model - correct risk-adjusted rankings if all or some returns of the peer group funds are negative. We also examined three modern performance measures (Modified MVAR Sharpe ratio, Lower Partial Moments and Omega ratio) to calculate downside risks. The results show that down side risks exist which would not detect from the variance-model.

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