4.5.4. Omega Ratio

The Omega ratio introduced by Keating and Shadwick (2002) is one of the most important performance measures by using the probability-weighted ratio of gains and losses for any threshold return target. Guo et al. (2017b) developed the properties to study the relationships between the Omega ratio and (a) the first-order SD; (b) the second-order SD for risk-averters; and (c) the second-order SD for risk-seekers. Chow et al. (2019b) further established the necessary conditions between the Omega ratio and SD for both risk-averters and risk-seekers, and demonstrated that the Omega ratio outperforms the Sharpe ratio in many cases.

#### 4.5.5. Economic Performance Measure

Homm and Pigorsch (2012) developed the economic performance measure (EPM), which is related to Behavioral Finance, because it is related to SD, which, in turn, is related Behavioral Finance. Thus, EPM is related to Behavioral Finance. To make the EPM become useful in the comparison of different assets, Niu et al. (2018) developed the theory of construction confidence intervals for EPMs, including one-sample and two-sample confidence intervals, and derived the asymptotic distributions for one-sample confidence interval and for two-sample confidence interval for samples that are independent. The testing approach developed by Niu et al. (2018) is robust for many dependent cases.

#### 4.5.6. Other Risk Measures and Performance Measures

Because variance gives the same weight in measuring downside risk as well as upside profit for any prospect, it is not a good measure to capture the downside risk. To circumvent the limitation, several risk measures and performance measures have been proposed, including Value-at-Risk (VaR, Jorion 2000; Guo et al. 2019b), conditional-VaR (C-VaR, Rockafellar and Uryasev 2000; Guo et al. 2019b), Kappa ratio (Kaplan and Knowles 2004), Farinelli and Tibiletti (FT, Farinelli and Tibiletti 2008), economic performance measure (Homm and Pigorsch 2012), and others. In addition, Ma and Wong (2010) proved that VaR is related to first-order SD and C-VaR is related to second-order SD. Niu et al. (2017) proved that the Kappa ratio is related to SD, and Guo et al. (2019a) proved that the F–T ratio is related to SD for both risk-averse and risk-seeking investors under some conditions.

#### 4.5.7. Applications of Risk Measures and Performance Measures

There are many applications of using risk measures and performance measures to test for market efficiency, and check whether there is any anomaly in the market. Examples include Sharpe (1966), Keating and Shadwick (2002), Kaplan and Knowles (2004), Broll et al. (2006, 2011, 2015), Wong et al. (2008, 2018a), Leung and Wong (2008a), Fong et al. (2008), Abid et al. (2009, 2013, 2014), Qiao et al. (2010, 2012, 2013), Lean et al. (2010a, 2010b, 2013, 2015), Homm and Pigorsch (2012), Chan et al. (2012, 2019a, 2019b), Bai et al. (2013), Qiao and Wong (2015), Hoang et al. (2015a, 2015b, 2018, 2019), Tsang et al. (2016), Guo et al. (2017b, 2019a), Mroua et al. (2017), Niu et al. (2017), Bouri et al. (2018), and Chow et al. (2019b), among others.

#### 4.5.8. Indi fference Curves

The indi fference curve, which was first developed by Tobin (1958), is one of the important areas in Behavioral Finance, because it reveals the behavior of both risk-averters and risk-seekers in the mean and variance diagram. Tobin (1958) proved that the indi fference curve is increasingly convex for risk-averters, decreasingly convex for risk-seekers, averse (seeking) investors, and is horizontal for risk-neutral investors when assets follow the normal distribution. Meyer (1987) extended the theory by relaxing the assumption of normality and including the location-scale (LS) family to the theory of indi fference curve.

Wong (2006, 2007) extended the theory to include the general conditions that were presented in Meyer (1987)Meyer Wong and Ma (2008) further extended the theory by introducing some general non-expected utility functions and the LS family with general n random seed sources, and established some important properties of the theory of indi fference curves. To date, the literature only discusses indi fference curves for risk-averters and risk-seekers. Broll et al. (2010) extended the theory by examining the behavior of indi fference curves for investors with S-shaped utility functions.

#### *4.6. Two-Moment Decision Models and Dynamic Models with Background Risk*

The two-moment decision model is related to Behavioral Finance because it can be used to measure the behaviors of both risk-averters and risk-seekers. Many works have been done in this area. For instance, Alghalith et al. (2017) showed that the change of the price of expected energy will a ffect the demand for both energy and non-risky inputs, but the uncertain energy price only a ffects uncertain energy price but not the demands for the non-risky inputs for any risk-averse firm.

Alghalith et al. (2017) showed that the variance of energy price a ffects the demands of both non-risky inputs and energy decrease when the variance is vulnerable, but does not a ffect the demands of the non-risky inputs when there is only uncertain in energy price for any risk-averse firm. Guo et al. (2018a) contributed to the MV model with multiple additive risks by establishing some properties on the marginal rate of substitution between mean and variance. They also illustrated the properties by using the MV model with multiple additive risks to study banks' risk-taking behaviors.

In addition, Alghalith et al. (2016) extended the theory of the stochastic factor model with an additive background risk and the dynamic model with either additive or multiplicative background risks by including a general utility function in the models in which the risks are correlated with the factors in the models.
