Performance measures are important tools for management decisions. They induce a total (or partial) order of investment opportunities so that agents can reduce their decisions regarding these investments to a simple comparison of these coefficients. Such decisions are, for instance, concerned with ranking investment opportunities or evaluating money managers, such as fund managers.
The Sharpe ratio (introduced as and also called reward-to-variability ratio), proposed by Sharpe (1966, 1994), is one of the most prominent performance measures. It is the ratio of the mean over the standard deviation of the expected excess return of an investment opportunity.
It thus corrects the expected return by taking into account a specific type of risk taken by the investor. Its justification requires some restrictions on either the distributions of the returns or the investor’s preferences. The typical distributional assumption is that all returns under consideration belong to the same location-scale family, see Meyer (1987) and Schuhmacher and Eling (2011) for a recent application of this argument. The most common example is the normal distribution. Moreover, the Sharpe ratio is adequate if investors only care about
the mean and variance of an investment. However, it is well known that financial returns very often exhibit non-normal characteristics, such as (negative) skewness and excess kurtosis, which differ between assets (cf. Agarwal and Naik, 2004; Malkiel and Saha, 2005). This rules out the case that return distributions belong to the same location-scale family. Furthermore, empirical and experimental studies show that it is unlikely that investors do not care about these higher order moments (cf. Golec and Tamarkin, 1998; Harvey and Siddique, 2000). A prominent field of application for the Sharpe ratio is fund-ranking. Since the Sharpe ratio ignores differences in higher order moments the question arises as to what extent the ranking changes if one accounts for non-normality.
This has led to the development of various performance measures which take into account these stylized facts of financial returns. Most of them either replace the mean with a different reward measure or they substitute the standard deviation with a different measure of the (relevant) risk taken by the investor, or both. However, most of these measures are proposed in a rather ad hoc way and their economic foundation is rather vague. For an overview we refer to Cogneau and Hübner (2009a,b), Eling and Schuhmacher (2007) and Farinelli et al. (2008) and the references therein.
In this chapter we propose a new performance measure that, in contrast to the Sharpe ratio, meets the natural requirement that it is strictly monotone with respect to stochastic dominance and can account not only for the mean and variance but also for higher moments.
The performance measure is obtained by dividing the mean of an investment opportunity by its economic index of riskiness proposed by Aumann and Serrano (2008) (AS index henceforth).
As opposed to the risk measures mentioned above, the AS index is mainly derived from a choice theoretic axiom, namely the axiom of duality. It requires an index to reflect the following natural notion of less risky: given that an investment is accepted by some agent, less risk averse agents acceptless risky investments. As such it is an economically motivated axiom and Aumann and Serrano (2008) therefore termed their indexeconomic index of riskiness. To emphasize that our performance measure is based on such an economic risk measure we refer to it as the economic performance measure (EPM). If investment returns are normally distributed, the EPM and the
Sharpe ratio produce the same ranking of these investments. The EPM, thus, generalizes the Sharpe ratio with respect to non-normal distributions.
Moreover, we extend the continuity result of Aumann and Serrano (2008) and show that if the distribution of the returns converges to the normal distribution, the EPM converges to two times the squared Sharpe ratio. Thus, the EPM also asymptotically induces the same ranking as the Sharpe ratio. This is especially appealing in connection with the aggregational Gaussianity property of financial returns. This property states that for decreasing sampling frequency, e.g.
going from daily to monthly and down to yearly returns, the return distribution approximates the normal distribution, see e.g. Cont (2001) and Rydberg (2000). While the Sharpe ratio is appropriate for low frequency returns, the new performance measure is appropriate for both low and high frequency returns, with no disadvantages compared to the Sharpe ratio in the former case.
We propose a parametric and a non-parametric moment estimator for the EPM. For para-metric estimation we assume that returns follow a normal inverse Gaussian (NIG) distribution proposed by Barndorff-Nielsen (1997). As the NIG distribution is analytically tractable and has several attractive properties it is widely used in financial applications. It allows to model skew-ness and semi-heavy tails. We derive a closed form expression for the EPM of NIG-distributed random variables (e.g. excess returns) in terms of the first four moments. This makes explicit the dependence on skewness and kurtosis and provides a moment estimator for the EPM that is virtually as easy to compute as the Sharpe ratio. For non-parametric estimation the crucial idea is to use a moment condition that corresponds to the defining equation of the AS index.
Results on asymptotic normality can readily be inferred from the literature on the method of moments. In a simulation study we address the issue of estimation uncertainty. Given that higher moments are important to investors, our results suggest that even for data sets with a limited number of observations, rankings based on the EPM are superior to Sharpe ratio rankings.
We apply our two estimators to rank mutual funds and hedge funds via the EPM and compare the results with a Sharpe ratio ranking. Imposing the parametric assumption of
NIG-distributed returns yields a ranking of the funds that is very similar to the one implied by the non-parametric estimation of the EPM, which indicates that the NIG-distribution is a reasonable choice. While the distributions of the excess returns from the mutual funds are close to Gaussian, the distributions of the hedge funds returns show pronounced skewness and excess kurtosis. As a consequence, the ranking of the mutual funds is very similar under the Sharpe ratio and the EPM. For the hedge funds, however, the two measures yield different rankings. In particular, if a fund’s return distribution has relatively low (high) skewness and/ or relatively high (low) excess kurtosis, the fund is typically ranked lower (higher) by the EPM than by the Sharpe ratio.
The remainder of this chapter is structured as follows. The next section introduces the eco-nomic performance measure. We derive properties of this new measure and discuss its relation to the Sharpe ratio and other performance measures. In Section 3.3 we suggest estimators for the EPM and conduct a Monte Carlo experiment. Section 3.4 provides an empirical illustra-tion using mutual funds and hedge funds return data. Secillustra-tion 3.5 concludes. The Appendix contains supplementary calculations.