پایه و اساس تصمیم گیری تئوری برای اندازه گیری عملکرد پاداش به خطر

In this paper we prove that partial-moments-based performance measures (e.g., Omega, Kappa, upside-potential ratio, Sortino–Satchell ratio, Farinelli–Tibiletti ratio), value-at-risk-based performance measures (e.g., VaR ratio, CVaR ratio, Rachev ratio, generalized Rachev ratio), and other admissible performance measures are a strictly increasing function in the Sharpe ratio. The theoretical basis of this result is the location and scale property and two other plausible and mild conditions. Our result provides a decision-theoretic foundation for all these frequently used performance measures. Moreover, it might explain the empirical finding that all these measures typically lead to very similar rankings.

The most popular reward-to-risk performance measure is the Sharpe ratio (see, e.g., Alexander and Baptista, 2010, Darolles and Gourieroux, 2010, Ding et al., 2009, Eling and Faust, 2010, Szakmary et al., 2010 and Serban, 2010). The least restrictive sufficient condition for expected utility to imply Sharpe ratio rankings is the location and scale (LS) property (see Sinn, 1983 and Meyer, 1987). This property requires that the random returns from the investment funds in the choice set differ from one another only by location and scale parameters. Schuhmacher and Eling (2011) argue that the LS property is also sufficient for expected utility to imply drawdown-based performance measure rankings. Hence, the same conditions that provide an expected utility foundation for the Sharpe ratio also provide a foundation for drawdown-based performance measures. Their result shows that drawdown-based performance measures will lead to the same ranking as the Sharpe ratio if the random returns satisfy the LS property. Thus the question arises as to whether the LS property is sufficient to ensure consistency between expected utility and other performance measures that differ from the Sharpe ratio by the risk and reward measure employed. To answer this question, we argue that any admissible risk measure should satisfy two conditions: first, it should satisfy positive homogeneity, which is an important axiom in most axiomatic systems (see Kijima and Ohnishi, 1993, Pedersen and Satchell, 1998, Artzner et al., 1999 and Rockafellar et al., 2006); second, adding a positive constant to an investment fund’s random excess rate of return should not increase the investment fund’s risk. This condition contains the mutually incompatible axioms of translation invariance (see Artzner et al., 1999) and shift invariance (see Kijima and Ohnishi, 1993, Pedersen and Satchell, 1998 and Rockafellar et al., 2006) as special cases. Similarly, any admissible reward measure should also satisfy two conditions: positive homogeneity and that adding a positive constant to an investment fund’s random excess rate of return does increase the investment fund’s reward. The main result is that under the LS property, any admissible performance measure is a strictly increasing function in the Sharpe ratio. An admissible performance measure is using admissible risk and reward measures. This finding has two important implications. First, it provides a decision-theoretic foundation for lower-partial-moments, value-at-risk, and other admissible performance measures that differ from the Sharpe ratio by the risk and reward measure employed. Second, since the normal, the extreme value, and many other distributions commonly used in finance satisfy the LS property (see Schuhmacher and Eling, 2011), the finding may explain the empirical observation that rank correlations between the Sharpe ratio and alternative performance measures are extremely high (see Eling and Schuhmacher, 2007 and Eling et al., 2010). This paper is structured as follows. In Section 2 we present our main result. In Section 3 we demonstrate the main result using well-known risk and reward measures. A numerical illustration is presented in Section 4. We conclude in Section 5.

The main result is that under the location and scale property, any admissible performance measure is a strictly increasing function in the Sharpe ratio. This means that for these distributions, performance ranking will be the same regardless of whether it is conducted via an admissible performance measure or via the Sharpe ratio. This finding has an important implication with regard to the use of admissible performance measures: using admissible performance measures is theoretically justified under the same conditions as is the Sharpe ratio. In other words, the same conditions that provide a decision-theoretic foundation for the Sharpe ratio also provide a decision-theoretic foundation for admissible performance measures. Schuhmacher and Eling (2011) show that the beta, extreme value, gamma, logistic, normal, Student’s t, uniform, Weibull, and normal inverse Gaussian (NIG) satisfy the LS property. Hence, for many distributions commonly used in finance, admissible performance measures are strictly increasing functions in the Sharpe ratio. To avoid creating misunderstanding with respect to the LS property, we emphasize four aspects. First, a set of investment funds satisfies the LS property only if their returns follow the same LS distribution. For example, if fund A’s return follows a normal distribution and fund B’s return follows a logistic distribution, this set of investment funds does not satisfy the LS property. Second, distributions with three or four parameters, like the gamma or NIG distribution, satisfy the LS property only if the shape parameters are fixed. For example, if the returns of two funds follow the NIG distribution with different shape parameters, this set of the investment funds does not satisfy the LS property. Third, LS distributions justify use of the Sharpe ratio in selecting the best investment funds, i.e., no combinations of investment funds are allowed. LS distributions do not justify using the Sharpe ratio in constructing the best portfolio of investment funds. This follows from the fact that combinations of investment funds with the risk-free asset satisfy the LS property, while combinations of two investment funds in general do not (for details, see Meyer and Rasche, 1992). When combinations of investment funds are allowed, a stronger condition on the distributions is necessary, e.g., elliptical or q-radial distributions (see Chen et al., 2011). Fourth, if a set of investment funds satisfies the LS property, the investment funds will exhibit identical skewness and kurtosis; this is a critical aspect of the LS property. However, Schuhmacher (2012) shows that results similar to those set forth in this paper hold for the generalized LS property as defined by Meyer and Rasche (1992). The advantage of the generalized LS property is that it does not imply identical skewness and kurtosis for the investment funds. Finally, note that a number of conditions have been identified that imply consistency between the expected utility and the mean-standard deviation approach. Well known among these are quadratic utility, normality, and elliptical symmetry; almost unknown are the LS property and its generalization, the two conditions of concern in this paper.