A finely measured performance - beyond the Sharpe ratio...

Edhec's Walter Gehin continues Edhec's examination of hedge fund performance measurement techniques by analysing absolute performance metrics that are not based on the Sharpe ration and which, furthermore, attempt to take skewness and kurtosis into account...

In this article, the third of a series of articles on hedge fund performance measurement, we discuss absolute performance measures not based on the Sharpe ratio.

Several new performance measures are not based on the Sharpe ratio. They are innovative in that they attempt to take skewness and kurtosis into account.

Stutzer index

The Stutzer index was introduced by Stutzer (2000)1. It is based on the behavioural hypothesis that investors aim to minimise the probability that the excess returns over a given threshold will be negative over a long time horizon.

When the portfolio has a positive expected excess return, this probability will decay to zero at an exponential decay rate as the time horizon increases.

It is equal to the maximum decay rate to zero of the expected excess return: the higher the Stutzer index, the longer the time horizon and the better the hedge fund. The Stutzer index downgrades the ranking of funds whose skewness is strongly negative and whose kurtosis is strongly positive, while it upgrades the ranking of funds whose skewness is near zero and whose kurtosis is not strongly positive.

Bacmann and Scholz (2003)2 compare the rankings of 44 hedge fund indices with the Stutzer index and the Sharpe ratio.

The database used, provided by CSFB/Tremont, HFR and Stark, covers the period of January 1994 to February 2003. Four indices are drawn from the traditional universe (MSCI World Index, Russell 2000, S&P 500 and the Salomon World Government Bond Index). Fifteen indices are normally distributed according to the Jarque-Bera statistic at the 5% significance level.

In comparison with the Sharpe ratio, 37 funds have the same ranking according to the Stutzer index. However, if we consider the higher moments for the indices whose rank improves, the negative skewness turns positive in the case of the Stutzer index.

The positive kurtosis decreases from 7.22 to 3.69. For the indices whose rank deteriorates, the negative skewness significantly increases from -0.82 to -2.95. The positive kurtosis increases strongly from 7.22 to 19.17.

In contrast to the previous results, ranks are similar when the authors only consider the traditional indices, whatever the performance measure.

This confirms that higher moments are the source of the mismatch between the Sharpe ratio and the Stutzer index.

Omega

The Omega measure was introduced by Keating and Shadwick (2002)3. It incorporates all the moments of the return distribution, including skewness and kurtosis.

Moreover, in contrast to the Sharpe ratio, ranking is always possible, whatever the threshold. It requires no assumptions on the return distribution or on the utility function of the investor.

Omega is expressed as the ratio of the gain with respect to the threshold and the loss with respect to the same threshold:



where L is the required return threshold, a and b are the return intervals and F(x) is the cumulative distribution of returns below threshold L.

At a defined level of threshold, the higher the Omega, the better.

Gupta, Kazemi and Schneeweis (2003)4 give an intuitive expression of Omega:



where C(L) is essentially the price of a European call option written on the investment and P(L) is essentially the price of a European put option written on the investment.

De Souza and Gokcan (2004)5 provide the Omega formula in a discrete case:



where R+ (R -) is the return above (below) a threshold L.

Bacmann and Scholz (2003) compare the rankings of 44 hedge fund indices with the Omega and the Sharpe ratio.

In comparison with the Sharpe ratio, 36 funds have the same ranking according to the Omega, but if we consider the higher moments for the indices whose rank improves, the negative skewness decreases from -0.75 to -0.45.

The positive kurtosis decreases from 7.18 to 4.09.

For the indices whose rank deteriorates, the negative skewness significantly increases from -0.75 to -2.60.

The positive kurtosis increases strongly from 7.18 to 16.85 in the case of the Omega.

Ranks are similar when only the traditional indices are considered. This confirms that the Sharpe ratio tends to underestimate or overestimate the performance results in the context of hedge funds.

Sharpe-Omega6

Presented by Gupta, Kazemi and Schneeweis (2003), the Sharpe-Omega has identical features to the Omega, whilst keeping the same risk approach as the Sharpe ratio. It is introduced in the following way:



This indicator has the particular quality of being proportional to (1-omega). Consequently, it provides strictly the same rankings as the Omega. Through numerical examples in the case of changes in the distribution of an investment's return, the authors show that the Sharpe-Omega is most sensitive to the mean and the variance, and is less impacted by skewness and kurtosis.

Using monthly data from January 1994 to May 2003, Gupta et al. estimate the Omega and Sharpe-Omega for the S&P 500 index, the CSFB convertible arbitrage index and the CSFB equity market neutral index.

For different levels of threshold, the two indicators give the same rankings for the three indices.

Sharpe-Omega is successively calculated by successively modifying only the mean and the threshold (while standard deviation = 5%, skewness = 0, kurtosis = 3), only the standard deviation and the threshold (while mean = 1%, skewness = 0, kurtosis = 3), only the skewness and the threshold (while mean = 1%, standard deviation = 5%, kurtosis = 3), and only the kurtosis and the threshold (while mean = 1%, standard deviation = 5%, skewness = 0).

It appears that changes in mean and standard deviation have the most pronounced impact on the Sharpe-Omega, confirming Keating and Shadwick's (2002) conclusions on Omega.

AIRAP

Sharma (2004)7 introduces a risk-adjusted performance measure dedicated to hedge funds.

It is known as the Alternative Investments Risk Adjusted Performance (AIRAP).

AIRAP is constructed on the basis of the expected utility theory. The selected form of utility is a Constant Relative Risk Aversion (CRRA). AIRAP is formulated as follows:

- when c (Arrow-Pratt coefficient) is different from 1 and greater than or equal to 0:



where:



and pi is the frequency of percentage returns.

- when c is equal to 1:



Sharma recommends an Arrow-Pratt coefficient (represented by c) from 1 to 10. Because a geometric mean is used to measure the average performance, c = 1 corresponds to risk neutrality (in this case the risk premium is nil)8.

Cases with c comprised between 0 and 1 assume that rational investors accept the risk of insolvency, but according to the author this is implausible.

Adopting a cautious view, the author assumes c = 4. This corresponds to a case where investors accept a risk of a maximum loss of 20.7% of their wealth.

An approach that only involves using the ratio of gross and net assets is inadequate for taking into account the impact of leverage on the performance of hedge funds, because of the presence of derivatives. This justifies a risk-based approach. AIRAP captures the impact of leverage through a credit for the higher mean and a penalty for the higher volatility as a function of the CRRA parameter.

The optimal leverage, which maximises AIRAP for a range of CRRA, can be defined by standard optimisation techniques.

According to Sharma, AIRAP presents several advantages. It takes leverage and investor preferences into account.

Unlike traditional risk-adjusted performance measures, AIRAP penalises negative skewness and positive kurtosis. Moreover, it is scale invariant and can be used for non-directional strategies, unlike the Treynor ratio.

Data covers the period of January 1997 to December 2001. At the index level, the data is provided by EACM. At the individual fund level, the data is provided by Hedge Fund Research (HFR).

Rank reversals between Sharpe and AIRAP and between Jensen's alpha and AIRAP are presented with 19 different levels of Constant Relative Risk Aversion for the Hedge Fund Research universe.

The percentage of Sharpe ratio rank reversals is between 99% and 100%, while the percentage of Jensen's alpha rank reversals is between 98% and 100%. The Spearman rank correlation confirms the lack of correlation between standard measures and the AIRAP. At the intra-strategy level, even if the rank reversal is somewhat lower, it also indicates discrepancies between the Sharpe ratio and AIRAP.

Kappa

Kappa, introduced by Kaplan and Knowles (2004)9, is presented as a generalised downside risk-adjusted performance measure.

"Generalised" means that this indicator can become any risk-adjusted return measure, through a single parameter.



where  is the expected periodic return, is the investor's minimum acceptable or threshold periodic return and LPM is the lower partial moment.

It becomes apparent that the Sortino ratio is equal to K2 and Omega to K1+1. And n is strictly greater than 0.

Kappa can be calculated in two ways: it can use discrete return data or a parameter-based calculation. A discrete calculation gives robust results, but it is a strict requirement.

A parameter-based calculation involves deriving a continuous return distribution from the values of the first four moments, that is, mean, standard deviation, skewness and kurtosis.

Kaplan and Knowles test Kappa on a database provided by HFR, that covers the period from January 1990 to February 2003. They focus on 11 hedge fund indices. Firstly, for each hedge fund strategy, Kappa is calculated with n being equal to 1 or 2, with a successive threshold of 0% or 1%. It is stated that the difference between the results obtained through the two methods (discrete or parameter-based) increases when the threshold decreases.

In such cases, Kappa has to be handled cautiously. Secondly, the rankings obtained through the two methods are compared successively with n = 1, 2 and 3, and with a threshold of -1%, -0.5% and 0%. In terms of ranking, the parameter-based method provides similar results to the discrete method. The parameter n has the greatest impact on the ranking: only two strategies (emerging markets and event-driven) have the same ranking regardless of what n is, for a threshold of 0%.

With n being equal to 1, 2 or 3, an inverse relationship between the threshold and the value of Kappa appears. The steepness of the Kappa curve decreases when the parameter n increases.

Considering the sensitivity of Kappa to skewness, when the threshold is above (below) the mean return, it is insensitive (sensitive). When Kappa is sensitive, it is a negative function of n.