*4.4. Stochastic Dominance*

Stochastic dominance (SD) is one of the most important areas in Behavioral Finance, because SD can be used to compare the performance of di fferent assets, which is equivalent is the preferences of investors with di fferent utilities. We discuss some important SD papers in this section. Readers may refer to Levy (2015), Sriboonchitta et al. (2009), and the references therein for more information.

#### 4.4.1. Stochastic Dominance for Risk-Averters and Risk-Seekers

SD is one of the most important selection rules for both risk-averters and risk-seekers. Quirk and Saposnik (1962), Fishburn (1964, 1974), Hadar and Russell (1969, 1971), Hanoch and Levy (1969), Whitmore (1970), Rothschild and Stiglitz (1970, 1971), Tesfatsion (1976), Meyer (1977), and others developed the SD rules for risk-averters. On the other hand, Hammond (1974), Meyer (1977), Hershey and Schoemaker (1980), Stoyan (1983), Myagkov and Plott (1997), Wong and Li (1999), Li and Wong (1999), Anderson (2004), Post and Levy (2005), Wong (2006, 2007), Levy (2015), Guo and Wong (2016), and others developed the SD rules for risk-seekers.

#### 4.4.2. Stochastic Dominance for Investors with (Reverse) S-Shaped Utility Functions

Friedman and Savage (1948) observed that many individuals buy insurance, as well as lottery tickets, and it is well-known that utility with strictly concavity or strictly convexity cannot explain this phenomenon. To solve this problem, academics developed S-shaped and reverse S-shaped utility functions, while Levy and Levy (2002, 2004) and others developed the SD theory for investors with S-shaped and reverse S-shaped utility functions. We call the SD theory for individual with S-shaped

prospect SD (PSD) and the SD theory for individual with reverse S-shaped utility function Markowitz SD (MSD).

Wong and Chan (2008) extended the PSD and MSD theory to the first three orders. They developed several important properties for MSD and PSD. For example, they showed that the dominance of assets in terms of PSD (MSD) is equivalent to the expected utility preference of the assets for investors with (reverse) S-shaped utility function.

#### 4.4.3. Almost Stochastic Dominance

The theory of almost stochastic dominance (ASD) was developed by Leshno and Levy (2002, LL) to measure a preference for "most" decision makers but not all decision makers. However, Tzeng et al. (2013, THS) found that expected-utility maximization does not hold in the second-degree ASD (ASSD) defined by LL. Thus, they suggested to use another definition of the ASSD definition that possesses the property. Nonetheless, Guo et al. (2013) proved that, though LL's ASSD does not satisfy the expected-utility maximization, it possesses the hierarchy property, whereas though THS's ASSD possesses the expected-utility maximization, it does not possess the hierarchy property.

Guo et al. (2014) extended the ASD theory by developing the necessary conditions for the ASD rules. In addition, they established several important properties for ASD. Guo et al. (2016) further extended the theory of ASD theory by including the theory of ASD for risk-seekers. In addition, they established some relationships between ASD for both risk-averters and risk-seekers. Tsetlin et al. (2015) established the theory of generalized ASD (GASD), and Guo et al. (2016) compared ASD and GASD and pointed out their advantages and disadvantages.

#### 4.4.4. Stochastic Dominance Tests and Applications

There are many stochastic dominance tests that can be used in Behavioral Finance. The commonly used SD tests include Davidson and Duclos (2000), Linton et al. (2005), Linton et al. (2010), Bai et al. (2011b, 2015), and Ng et al. (2017). We note that Lean et al. (2008) conducted simulations to compare the performance of different SD tests and found that the SD test developed by Davidson and Duclos (2000) has decent size and power.

The SD tests developed by Bai et al. (2011b, 2015) are improvements of the SD test developed by Davidson and Duclos (2000). Thus, the SD tests developed by Bai et al. (2011b, 2015) also have decent size and power. Ng et al. (2017) conducted simulations and found that their proposed SD test also had decent size and power. Chan et al. (2019a) developed a third-order SD test.

There are many studies that have applied stochastic dominance tests to test for market efficiency and check whether there is any anomaly in the market (for example, Fong et al. (2005, 2008), Wong et al. (2006, 2008, 2018a), Lean et al. (2007, 2010a, 2010b, 2013, 2015), Gasbarro et al. (2007, 2012), Wong (2007), Chiang et al. (2008), Abid et al. (2009, 2013, 2014), Qiao et al. (2010, 2012, 2013), Chan et al. (2012), Qiao and Wong (2015), Hoang et al. (2015a, 2015b, 2018, 2019), Tsang et al. (2016), Mroua et al. (2017), Bouri et al. (2018), and Valenzuela et al. (2019), among others).

#### *4.5. Risk Measures and Performance Measures*

The theory of risk measures and performance measures is one of the most important theories in Behavioral Finance because it can be used to measure the preferences of investors from different types of utilities. We discuss some different risk measures in this section.

#### 4.5.1. Mean Variance Rules

Markowitz (1952b) and Tobin (1958) first proposed the mean–variance (MV) selection rules for risk-averters, while Wong (2007) extended the theory by introducing the MV selection rules for risk-seekers and established the relationship between SD and MV rules for both risk-averters and risk-seekers. Chan et al. (2019b) further extended the theory by introducing the moment rules for both risk-averters and risk-seekers, and establishing the relationship between SD and the moment rules.
