**3. Consistency Results**

We will use the term "theorem" to state new results obtained in this paper and "proposition" to state some well-known results. Some academics may believe that the SSD is consistent with the Omega ratio because they assert the following:

$$\text{if}\quad X \succeq\_{SSD} Y,\quad \text{then}\quad \Omega\_X(\eta) \succeq \Omega\_Y(\eta)\quad \text{for any}\quad \eta \in \mathbb{R},\tag{10}$$

where <sup>Ω</sup>*X*(*η*) is the Omega ratio for *X* defined in (7) or (8). The above assertion is in Darsinos and Satchell (2004) and others. We first establish the following property to state that the argumen<sup>t</sup> in (10) may not be correct:

**Property 1.** *SSD alone is not sufficient to imply* <sup>Ω</sup>*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*) *for any η.*

Property 1 implies that the assertion made by Darsinos and Satchell (2004) and others may not be always correct. We construct the following example to support the argumen<sup>t</sup> stated in Property 1.

**Example 1.** *Consider two prospects X and Y having the following distributions:*

$$X = 10 \quad \text{with } prob. \ 1, \text{ and } \ Y = \begin{cases} \ 1 & \text{with } prob. \ 2/3 \\\ \ 11 & \text{with } prob. \ 1/3 \end{cases}. \tag{11}$$

*Then, we get μX* = 10 *and μY* = 13/3 *and obtain the following:*

$$F\_X^{(2)}(\eta) = \begin{cases} 0 & \text{if } \eta < 10 \\ \eta - 10 & \text{if } \eta \ge 10 \end{cases}, \quad F\_Y^{(2)}(\eta) = \begin{cases} 0 & \text{if } \eta < 1 \\ 2(\eta - 1)/3 & \text{if } 1 \le \eta < 11 \\ \eta - 13/3 & \text{if } \eta \ge 11 \end{cases},$$

$$F\_X^{(2)R}(\eta) = \begin{cases} 10 - \eta & \text{if } \eta < 10 \\ 0 & \text{if } \eta \ge 10 \end{cases}, \quad F\_Y^{(2)R}(\eta) = \begin{cases} 13/3 - \eta & \text{if } \eta < 1 \\ (11 - \eta)/3 & \text{if } 1 \le \eta < 11 \\ 0 & \text{if } \eta \ge 11 \end{cases}.$$

*It follows that F*(2) *X* (*η*) ≤ *F*(2) *Y* (*η*), *for all η* ∈ *R. That is, X SSD Y. However, for any* 10 ≤ *η* < 11*, we have F*(2)*<sup>R</sup> X* (*η*) ≡ 0 < *F*(2)*<sup>R</sup> Y* (*η*)*. Recalling the definition of* <sup>Ω</sup>*X*(*η*)*, we can conclude that* <sup>Ω</sup>*X*(*η*) ≡ 0 < <sup>Ω</sup>*Y*(*η*) *for any* 10 ≤ *η* < 11*, and thus, the statement "*Ω*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*) *for any η" does not hold.*

To complement Property 1, we establish the following property:

**Property 2.** *SRSD alone is not sufficient to imply* <sup>Ω</sup>*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*) *for any η.*

We construct the following example to support the argumen<sup>t</sup> stated in Property 2.

**Example 2.** *Consider two prospects X and Y as follows:* 

$$X = \begin{cases} \begin{array}{c} 2 \quad \text{with } prob. \ 1/2 \\ 8 \quad \text{with } prob. \ 1/2 \end{array} \quad \text{and} \ \mathcal{Y} = \begin{cases} \begin{array}{c} 3 \quad \text{with } prob. \ 2/3 \\ 6 \quad \text{with } prob. \ 1/3 \end{array} . \end{cases} \tag{12}$$

.

*We have μX* = 5 *and μY* = 4 *and obtain the following:*

$$\begin{aligned} F\_X^{(2)}(\eta) &= \begin{cases} 0 & \text{if } \eta < 2 \\ (\eta - 2)/2 & \text{if } 2 \le \eta < 8, \end{cases} & F\_Y^{(2)}(\eta) = \begin{cases} 0 & \text{if } \eta < 3 \\ 2(\eta - 3)/3 & \text{if } 3 \le \eta < 6, \\ \eta - 4 & \text{if } \eta \ge 6 \end{cases} \\\ F\_X^{(2)R}(\eta) &= \begin{cases} 5 - \eta & \text{if } \eta < 2 \\ 4 - \eta/2 & \text{if } 2 \le \eta < 8, \end{cases} & F\_Y^{(2)R}(\eta) = \begin{cases} 4 - \eta & \text{if } \eta < 3 \\ 2 - \eta/3 & \text{if } 3 \le \eta < 6. \\ 0 & \text{if } \eta \ge 6 \end{cases} \end{aligned}$$

*It follows that F*(2)*<sup>R</sup> X* (*η*) ≥ *F*(2)*<sup>R</sup> Y* (*η*), *for all η* ∈ *R. This concludes that X SRSD Y. However, for η* = 3.3*, we can get:*

$$\Omega\_X(\eta) = \frac{F\_X^{(2)R}(\eta)}{F\_X^{(2)}(\eta)} = \frac{4 - \eta/2}{(\eta - 2)/2} = \frac{8 - \eta}{\eta - 2} = 3.615.$$

$$\Omega\_Y(\eta) = \frac{F\_Y^{(2)R}(\eta)}{F\_Y^{(2)}(\eta)} = \frac{2 - \eta/3}{2(\eta - 3)/3} = \frac{6 - \eta}{2\eta - 6} = 4.5.$$

*That is,* <sup>Ω</sup>*X*(*η*) < <sup>Ω</sup>*Y*(*η*)*. In fact, for any* 3 < *η* < 7 − √<sup>13</sup>*, we have* <sup>Ω</sup>*X*(*η*) < <sup>Ω</sup>*Y*(*η*)*, and thus, the statement "*Ω*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*) *for any η" does not hold.*

Properties 1 and 2 tell us that SSD and SRSD alone are not sufficient to imply <sup>Ω</sup>*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*) for any *η*. Then, one may ask: what is the relationship between <sup>Ω</sup>*X*(*η*) and <sup>Ω</sup>*Y*(*η*) when there is SSD or SRSD? Guo et al. (2016) and Balder and Schweizer (2017) provide an answer. In this paper, we restate their result to extend the work by Darsinos and Satchell (2004) and others by first deriving the relationship between SSD (for risk averters) and the Omega ratio:

**Proposition 1.** *For any two returns X and Y with means μX and μY and Omega ratios* <sup>Ω</sup>*X*(*η*) *and* <sup>Ω</sup>*Y*(*η*)*, respectively, if X SSD Y, then* <sup>Ω</sup>*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*) *for any η* ≤ *μX.*

Now, it is clear that Proposition 1 extends the results of Darsinos and Satchell (2004) by restricting the range of the return threshold. We note that Balder and Schweizer (2017) obtain a similar result of Proposition 1. However, we have independently derived Proposition 1 and reported the results in Guo et al. (2016). Moreover, our proof is different from Balder and Schweizer (2017).

In addition, we also study the relationship of second-order risk-seeking stochastic dominance and the corresponding Omega ratios. A dual result as stated in Theorem 1 is obtained. Finally, the relationship between first-order stochastic dominance and the Omega ratios is established in Corollary 2. Some simple examples (Examples 1 and 2) are presented to show that SSD or SRSD alone are not sufficient to imply <sup>Ω</sup>*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*) for any *η*.

Here, we provide a short proof<sup>2</sup> as follows: although it is true that if *X SSD Y*, then *μX* − *η* ≥ *μY* − *η* for any *η*. However, the sign of *μX* − *η* and *μY* − *η* can be negative. To be precise, for *η* > *μX* ≥ *μY*, 0 > *μX* − *η* ≥ *μY* − *η*. In this situation, we can get:

$$\frac{\mu\_X - \eta}{F\_X^{(2)}(\eta)} \le \frac{\mu\_X - \eta}{F\_Y^{(2)}(\eta)}.$$

Furthermore, we note that:

$$\frac{\mu\_Y - \eta}{F\_Y^{(2)}(\eta)} = \frac{\mu\_Y - \mu\_X}{F\_Y^{(2)}(\eta)} + \frac{\mu\_X - \eta}{F\_Y^{(2)}(\eta)} \le \frac{\mu\_X - \eta}{F\_Y^{(2)}(\eta)} \cdot \frac{\eta}{\mu\_X}$$

Consequently, we cannot determine the sign of *μX*−*η F*(2) *X* (*η*) − *μY*−*η F*(2) *Y* (*η*). Thus, we cannot determine the sign of <sup>Ω</sup>*X*(*η*) − <sup>Ω</sup>*Y*(*η*). However, for any *η* ≤ *μ<sup>X</sup>*, we can have *μX* − *η* ≥ 0, and thus, we have:

$$\frac{\mu\_X - \eta}{F\_X^{(2)}(\eta)} \ge \frac{\mu\_X - \eta}{F\_Y^{(2)}(\eta)} \ge \frac{\mu\_Y - \eta}{F\_Y^{(2)}(\eta)}.$$

This implies that <sup>Ω</sup>*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*), and thus, the assertion of Proposition 1 holds.

In the proof of Proposition 1, one could conclude that if *X SSD Y*, then <sup>Ω</sup>*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*) for any *η* ≤ *μ<sup>X</sup>*. However, for *η* > *μ<sup>X</sup>*, we cannot determine which one is larger if we are using SSD. However, one could consider employing the SD (RSD) theory for risk seeking (refer to Equation (6)) in the study. By doing so, we establish the following theorem to state the relationship between the SRSD and Omega ratio:

**Theorem 1.** *For any two returns X and Y with means μX and μY and Omega ratios* <sup>Ω</sup>*X*(*η*) *and* <sup>Ω</sup>*Y*(*η*)*, respectively, if X SRSD Y, then* <sup>Ω</sup>*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*) *for any η* ≥ *μY.*

Here, we give a short proof as follows: assume that *X SRSD Y*. This is equivalent to *F*(2)*<sup>R</sup> X* (*η*) = ∞*η* (1 − *FX*(*ξ*))*dξ* ≥ *F*(2)*<sup>R</sup> Y* (*η*). Recall that *F*(2)*<sup>R</sup> X* (*η*) = *F*(2) *X* (*η*) − (*η* − *μX*) ≥ 0. This yields the following equation:

$$\frac{1}{\Omega\_X(\eta)} = \frac{F\_X^{(2)}(\eta)}{F\_X^{(2)R}(\eta)} = \frac{F\_X^{(2)R}(\eta) + (\eta - \mu\_X)}{F\_X^{(2)R}(\eta)} = 1 + \frac{\eta - \mu\_X}{F\_X^{(2)R}(\eta)}.$$

<sup>2</sup> We note that our proof is different from that of Balder and Schweizer (2017).

Further, we note that *X SRSD Y* implies *μX* ≥ *μY*. Thus, for *η* ≥ *μY*, we obtain:

$$\frac{\eta - \mu\_X}{F\_X^{(2)R}(\eta)} = \frac{\eta - \mu\_Y}{F\_X^{(2)R}(\eta)} + \frac{\mu\_Y - \mu\_X}{F\_X^{(2)R}(\eta)} \le \frac{\eta - \mu\_Y}{F\_X^{(2)R}(\eta)} \le \frac{\eta - \mu\_Y}{F\_Y^{(2)R}(\eta)}.$$

In other words, we can ge<sup>t</sup> <sup>Ω</sup>*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*) for any *η* ≥ *μY*, and thus, the assertion of Theorem 1 holds.

We note that Darsinos and Satchell (2004) assert that SSD is consistent with the Omega ratio; that is, the relationship in Equation (10) holds. However, we find that the consistency of SSD and the Omega ratio holds only when we restrict the range of return threshold, as stated in our Proposition 1 and Theorem 1. From Proposition 1 and Theorem 1, one could then derive the following theorem to state the relationship between the FSD and Omega ratio:

**Theorem 2.** *If the SSD and SRSD hold, then the Omega ratio dominance also holds. In particular, this is the case when the FSD holds.*

We give a short proof as follows: if *X FSD Y*, by using the hierarchy property (Levy (1992, 1998, 2015); Sriboonchitta et al. (2009)), we obtain both *X SSD Y* and *X SRSD Y*. From Proposition 1 and Theorem 1, we have <sup>Ω</sup>*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*) for any *η* ≤ *μX* and *η* ≥ *μY*. Since *μX* ≥ *μY*, we have <sup>Ω</sup>*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*) for any *η* ∈ *R*, and thus, the assertion of Theorem 2 holds.

#### **4. Testing Market Efficiency, Arbitrage Opportunity and Anomaly**

In this section, we will discuss how to apply the theory developed in this paper to examine whether the market is efficient, whether there is any arbitrage opportunity in the market and whether there is any anomaly in the market. To do so, we consider the following four pairs of hypotheses:

$$H\_0^{SSD}: \quad X \succ\_{SSD} Y \quad \text{versus} \quad H\_1^{SSD}: X \succ\_{SSD} Y \tag{13}$$

$$H\_0^{SRSD}: X \not\succ\_{SRSD} Y \quad \text{versus} \quad H\_1^{SRSD}: X \succ\_{SRSD} Y \tag{14}$$

$$H\_0^{FSD}: X \not\succ\_{FSD} Y \quad \text{versus} \quad H\_1^{FSD}: X \succ\_{FSD} Y \tag{15}$$

$$H\_0^{\text{CD}}: X \not\succ\_{\text{OD}} Y \quad \text{versus} \quad H\_1^{\text{OD}}: X \succ\_{\text{OD}} Y \tag{16}$$

To test whether there is any SSD in two assets as stated in (13), we can apply Proposition 1 to test whether <sup>Ω</sup>*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*) for any *η* ≤ *μ<sup>X</sup>*. If this is true, then we could have *X SSD Y*. Similarly, to test whether there is any SRSD in two assets as stated in (14), we can apply Theorem 1 to test whether <sup>Ω</sup>*X*(*η*) ≥ <sup>Ω</sup>*Y*(*η*) for any *η* ≥ *μY*. If this is true, then we could have *X SRSD Y*. Last, to test whether there is any FSD in two assets as stated in (15), we can apply Theorem 2 and Definition 1 to test whether *X OD Y*. If this is true, then we could have *X FSD Y*. Readers may ask: why should we test *HSSD* 1 in (13), *HSRSD* 1 in (14), *HFSD* 1 in (15), and *<sup>H</sup>OD*1 in (16)? The answer is that we want to test whether there is any arbitrage opportunity in the market, whether there is any anomaly and whether the market is efficient. We first discuss testing arbitrage opportunity and anomaly, and, thereafter, discuss testing market efficiency and investor rationality in the next subsections.

#### *4.1. Arbitrage Opportunity and Anomaly*

It is well known from the market efficiency hypothesis that if one can ge<sup>t</sup> an abnormal return, then the market is considered inefficient, and there could exist arbitrage opportunity and anomaly. Thus, in order to test arbitrage opportunity and anomaly, one can apply Theorem 2 and Definition 1 to test *<sup>H</sup>OD*1 in (16) and check whether *X OD Y*. If *X OD Y*, then applying Theorem 2, we can conclude that *X FSD Y* could be true. Jarrow (1986) and Falk and Levy (1989) have claimed that if FSD exists, under certain conditions, arbitrage opportunities also exist, and investors will increase not only their expected utilities, but also their wealth if they shift from holding the dominated asset to the dominant one. One may consider it a financial anomaly.

However, Wong et al. (2008) have shown that if FSD exists statistically, arbitrage opportunities may not exist, but investors can increase their expected utilities, as well as their expected wealth, but not their wealth if they shift from holding the dominated asset to the dominant one. In this paper, we call this situation "expected arbitrage opportunity" or "arbitrage opportunity in expectation"; this means that if *X OD Y* appears many times and if investors could buy *X* and short sell *Y* each time, then on average, they could not only increase their expected utility, but also increase their expected wealth. In this situation, one may believe that there could be arbitrage opportunity and anomaly.

Falk and Levy (1989), Bernard and Seyhun (1997) and Larsen and Resnick (1999) comment that if there exists first-order dominance of a particular asset over another, but the dominance does not last for a long period, market efficiency and market rationality cannot be rejected. In general, the first-order dominance should not last for a long period of time because if the market is rational and efficient, then market forces will adjust the market so that there is no FSD. For example, if Property A dominates Property B at the FSD, then all investors would buy Property A and sell Property B. This will continue driving up the price of Property A relative to Property B, until the market price of Property A relative to Property B is high enough to make the marginal investor indifferent between Properties A and B. In this situation, we conclude that the market is still efficient and that investors are still rational. In the traditional theory of market efficiency, if one is able to earn an abnormal return for a considerable length of time, the market is considered inefficient. If new information is either quickly made public or anticipated, the opportunity to use the new information to earn an abnormal return is of very limited value. On the other hand, if the first-order dominance can hold for a long time and all investors can increase their expected wealth by switching their asset choice, we claim that the market is inefficient and that investors are irrational. However, sometimes FSD could still be held for a long period if investors do not realize such dominance exists or there are some reasons for the investors to buy the dominated asset. For example, investors could prefer to buy a bigger property for their status, even if the price is too high. If the FSD relationship among some assets still exists over a long period of time, then we could have arbitrage opportunity and anomaly, that market is inefficient and that investors are not rational.

#### *4.2. Market Efficiency and Rationality*

In last section, if *HOD* 1 in (16) such that *X OD Y* is not rejected over a long period of time, then we conclude that there could be arbitrage opportunity and anomaly, that the market is inefficient, and that investors are not rational. Nonetheless, if *HOD* 1 in (16) is rejected, should we conclude that the market is efficient and that investors are rational? Here, we would like to recommend academics and practitioners to further examine the higher order SD, say, for example, the second-order SD, before they conclude that the market is efficient.

Falk and Levy (1989) have argued that, given two assets, X and Y, if by switching from X to Y (or by selling X short and holding Y long), an investor can increase expected utility, the market is inefficient. SSD does not imply any arbitrage opportunity, but it does imply the preference of one asset over another by risk-averse investors. For example, if we apply Proposition 1 to test whether <sup>Ω</sup>*A*(*η*) ≥ <sup>Ω</sup>*B*(*η*) for any *η* ≤ *μA* and find that it is true, then we could have *A SSD B*, and thus, Property A dominates Property B by SSD. In this situation, one would not make an expected profit by switching from Property B to Property A, but switching would allow risk-averse investors to increase their expected utility. In this situation, could we conclude that the property market is not efficient?

We sugges<sup>t</sup> that this claim could be made if one believes that the market only contains risk-averse investors. However, it is well known that the market could have other types of investors (see, for example, Friedman and Savage (1948), Markowitz (1952), Thaler and Johnson (1990), Broll et al. (2010) and Egozcue et al. (2011) for more discussion). Under the assumption that the market could contain more than one type of investor, such as risk averters, as well as risk seekers, in this situations, academics could apply Theorem 1 to test whether <sup>Ω</sup>*B*(*η*) ≥ <sup>Ω</sup>*A*(*η*) for any *η* ≥ *μA*. If this is true, then we could have *B SRSD A*, and thus, Property A dominates Property B by SSD and Property B dominates Property A by SRSD. If this is the case, then risk averters could prefer to buy Property A, while risk seekers prefer to invest in Property B. Then, equilibrium could be reached in the sense that both Properties A and B can be sold well, and there is no upward or downward pressure on the price of both Properties A and B, while both risk averters and risk seekers could ge<sup>t</sup> what they want. Under these conditions, Qiao et al. (2012) argue that the market is still efficient and investors are still rational. On the other hand, if Property A dominates Property B by both SSD and SRSD, then one could conclude that the market is inefficient. However, if Property A dominates Property B by both SSD and SRSD, then Property A dominates Property B by FSD. We have discussed this case in the above.
