Nanyang Business School Forum on Risk Management and Insurance

# Higher-order Omega: A Performance Index with a Decision-Theoretic Foundation

**Tags:** almost stochastic dominance, acceptance dominance, performance index, optimal hedge ratio, EGRIE

**More from:** Hongwei Bi, Rachel J Huang, Larry Y Tzeng, Wei Zhu

**Editor’s Note:**Posted by Rachel J Huang, Professor at Department of Finance, National Central University, Taiwan; Hongwei Bi, Professor at School of Insurance and Economics, University of International Business and Economics, China; Larry Y Tzeng, Professor at Department of Finance, National Taiwan University, Taiwan; Wei Zhu, Professor at School of Insurance and Economics, University of International Business and Economics, China. This paper was presented at the 45th Annual Seminar of EGRIE, Nuremberg, Germany on September 17-19, 2018.

An important question in modern decision analysis is how to choose a good performance measure to properly evaluate decisions or risky prospects. One of the well-known performance indices is the Sharpe ratio proposed by Sharpe (1966). The ratio works well when the evaluated prospects follow normal distributions or the decision makers have mean-variance preferences. However, normality is difficult to observe in reality. Moreover, the literature indicates that decision makers do exhibit preferences on higher moments of their investments, such as skewness and kurtosis. Thus, to take higher moments of the distribution and decision makers’ preferences into consideration, the blossoming literature has adopted a utility-based approach to provide and support different performance measures.

In this paper, we contribute to this line of the literature by establishing a new performance index referred to as the *N*th-order Omega. The index includes Omega as proposed by Keating and Shadwick (2002) as a special case where *N* = 1. The *N*th-order Omega does not have any assumption regarding the distribution and takes into account the higher moments of the evaluated gambles. Moreover, and more importantly, the index is established by adopting an approach that is free of a utility functional form. Instead of assuming a utility functional form of the decision makers, we only require partial information on preferences, e.g., risk aversion, prudence, temperance, etc. Since our requirement regarding preferences is less demanding, the theoretical support for our index broadens the applications of the index.

The *N*th-order Omega is rooted in the concept of acceptance dominance proposed by Hart (2011). Acceptance dominance is defined by stating that a distribution *F* dominates another distribution *G* for a specific set of decision makers if *G* is always rejected, as long as *F* is rejected. In other words, if *F* dominates *G* in terms of acceptance dominance, it means that *F* is accepted more, and thus rejected less, than *G*, i.e., *F* performs better than *G* for this set of decision makers.

To build the decision-theoretic foundation of the *N*th-order Omega, we modify Hart’s (2011) acceptance dominance in two respects. First, we compare gambles with a target constant payoff *L* rather than compare gambles through rejection as in Hart (2011). Doing so can broaden the applications since, in practice, a decision/gamble is usually compared with a target payoff to evaluate the performance (Bernardo and Ledoit, 2000; Keating and Shadwick, 2002). In addition, if *L* is set to zero, it then reduces to the case where gambles are compared through rejection. Second, for the set of decision makers, we exclude some agents with pathological preferences as in the literature on almost stochastic dominance which is firstly proposed by Leshno and Levy (2002). When Hart (2011) defined acceptance dominance, he focused on all risk-averse decision makers. As pointed out by Leshno and Levy (2002) and Tzeng et al. (2013), it is difficult to reach a common consensus on comparing gambles because some risk-averse decision makers have pathological preferences. Almost stochastic dominance overcomes this problem by excluding these pathological preferences. Along this direction, we follow almost *N*th-degree stochastic dominance (ANSD) as proposed by Tzeng et al. (2013) to focus on most economically relevant decision makers without pathological preferences and establish a new acceptance dominance.

Specifically, we define the new order referred to as “almost *N*th-order acceptance dominance”. It says that for a given target payoff *L*, a distribution *F* dominates a distribution *G* in terms of almost *N*th-order acceptance dominance, so that if *F* is dominated by *L* for the decision makers defined by ANSD, then *G* is also dominated by *L* for the same set of decision makers. Note that this order is still incomplete. To obtain a complete order, we further require this order to be substantive, i.e., it should hold for all *εN*, which is a preference parameter in ANSD. We show that for a given target *L*, a distribution *F* dominates a distribution *G* in terms of almost *N*th-order acceptance dominance for all *εN* if and only if the *N*th-order Omega of *F* is not less than that of *G*. Based on the equivalence, we clarify the decision-theoretic foundation of the index.

We show that the *N*th-order Omega is monotonic with respect to Nth-degree stochastic dominance (NSD) in the sense that if all decision makers with mixed risk aversion up to degree *N*, defined as in Caballé and Pomansky (1996), prefer one distribution to another, then our index will show that the preferred distribution has a better performance. Therefore, the *N*th-order Omega provides a nice extension of NSD by completing the partial ordering of NSD. Furthermore, this important property provides a theoretical guideline for the choice of the order. For example, for insatiable, risk-averse and prudent decision makers, a 3rd-order Omega would be a suitable performance index to evaluate gambles. For insatiable, risk-averse, prudent and temperate decision makers, a 4th-order Omega should be considered. Several other properties of the Nth-order Omega are further examined. Moreover, to facilitate its application, an equivalent representation of the *N*th-order Omega is derived.

Finally, to demonstrate the applicability of the index, we show how to apply Omega, the 2nd-order Omega and the 3rd-order Omega in deriving the optimal hedge ratio. We first provide an analytical solution of the optimal hedge ratio that maximizes Omega under the assumption that spot and futures returns follow a joint normal distribution. We then find numerical solutions for the optimal hedge ratios that respectively maximize Omega, the 2nd-order Omega and the 3rd-order Omega. The comparative statics of the optimal hedge ratios with respect to a change in the threshold L is further examined.

The complete paper is available at:

https://www.vwrm.rw.fau.de/egrie-2018/conference-programme/.

Dear Larry (and research team): I didn’t get a chance to tell you after your presentation of this paper at EGRIE how much I like the work. It is a very good advance on the Omega measure.

Great work!

Mike