**1. Introduction**

It is well known that the standard deviation is not a good measure of risk because it penalizes upside deviation, as well as downside deviation. Additionally, it is also poor at measuring risk with asymmetric payoff profiles. The poor performance of the standard deviation will lead to poor performance of the Sharpe ratio, which establishes a relationship between the ratio of return versus volatility (Kapsos et al. (2014); Guastaroba et al. (2016)). A number of studies developed some theories that propose to circumvent the limitations. For example, Homm and Pigorsch (2012) develop an economic performance measure based on Aumann and Serrano's (2008) index of riskiness. They prove that the proposed economic performance measure is consistent with firstand second-order stochastic dominance (SD). Keating and Shadwick (2002) propose to use the Omega ratio, the probability weighted ratio of gains versus losses to a prospect or the ratio of upside returns (good) relative to downside returns (bad), to replace the Sharpe ratio to measure the risk return performance of a prospect. Thus, the Omega ratio considers all moments, while the Sharpe ratio considers only the first two moments of the return distribution in the construction. According to Caporin et al. (2016), Bellini et al. (2017) and the references provided therein, the Omega ratios are strongly related to expectiles, which are a type of inverse of the Omega ratio and present interesting properties as risk measures. Guastaroba et al. (2016) discuss the advantages of using the Omega ratio further. Thus, the Omega ratio has been commonly used by academics and practitioners as noted by Kapsos et al. (2014) and the references therein.

It is well known that the SD theory can be used 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 (Sriboonchitta et al. (2009); Levy (2015)), and thus, academics are interested in checking whether there is any relationship between any risk measure with SD. The work from Darsinos and Satchell (2004) and others can be used to establish the relationship between the second-order SD (SSD) and the Omega ratio. By using two counterexamples, we first demonstrate that SSD and/or second-order risk-seeking SD (SRSD) alone for any two prospects is not sufficient to imply Omega ratio dominance (OD) and that the Omega ratio of one asset is always greater than that of the other one. We then extend the work of Darsinos and Satchell (2004) and others by proving that the preference of SSD (for risk averters) implies the preference of the corresponding Omega ratios are selected only when the return threshold is less than the mean of the higher return asset. On the other hand, the preference of SRSD (for risk seekers) implies the preference of the corresponding Omega ratios only when the return threshold is larger than the mean of the smaller return asset. Lastly, we develop the relationship between the first-order SD (FSD) and the Omega ratio in such a way that the preference of FSD for any investor with increasing utility functions does imply the preference of the corresponding Omega ratios for any return threshold.

Qiao and Wong (2015) apply SD tests to examine the relationship between property size and property investment in the Hong Kong real estate market. They do not find any FSD relationship in their study. Tsang et al. (2016) extend their work to reexamine the relationship between property size and property investment in the same market. They sugges<sup>t</sup> to analyze both rental and total yields and find the FSD relationship of rental yield in adjacent pairings of different housing classes in Hong Kong. Based on their analysis on both rental and total yields, they conclude that investing in a smaller house is better than a bigger house. We note that analyzing both rental and total yields is not sufficient to draw such a conclusion. To circumvent the limitation, we extend their work by applying the Omega ratio to examine the relationship between property size and property investment in the Hong Kong real estate market. In addition to analyzing the rental yield, we recommend analyzing the price yields of different houses. We find that a smaller house dominates a bigger house in terms of rental yield, and there is no dominance between smaller and bigger houses in price yield. Our findings lead us to conclude that regardless of whether the buyers are risk averse or risk seeking, they will not only achieve higher expected utility, but also obtain higher expected wealth when buying smaller properties. This implies that the Hong Kong real estate market is not efficient, and there are expected arbitrage opportunities and anomalies in the Hong Kong real estate market. Our findings are useful for real estate investors in their investment decision making and useful to policy makers in real estate for their policy making to make the real estate market become efficient.

The rest of this paper is organized as follows: Section 2 presents the formal definitions of the SD rules and Omega ratios. We then show our main results about the consistency of Omega ratios with respect to the SD in Section 3. In Section 4, we 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. An illustration of the Hong Kong housing market is included in Section 5. The final section offers our conclusion.

#### **2. Definitions of Stochastic Dominance and Omega Ratios**

We first define cumulative distribution functions (CDFs) for *X* and *Y*:

$$F\_Z^{(1)}(\eta) = F\_Z(\eta) = P(Z \le \eta) \text{, for } \ Z = X, Y \text{ .} \tag{1}$$

We define the second-order integral of *Z*, *F*(2) *Z* ,

$$F\_Z^{(2)}(\eta) = \int\_{-\infty}^{\eta} F\_Z^{(1)}(\xi) d\xi \quad \text{for} \quad Z = X, Y; \tag{2}$$

and define the second-order reverse integral, *F*(2)*<sup>R</sup> Z*, of *Z*

$$F\_Z^{(2)R}(\eta) = \int\_{\eta}^{\infty} (1 - F\_Z^{(1)}(\xi)) d\xi \quad \text{for} \quad Z = X, Y. \tag{3}$$

If *Z* is the return, then *F*(1) *Z* (*η*) is the CDF of the return up to *η* and *F*(2) *Z* (*η*) is the second-order integral of *Z* up to *η*, that is the probability of the CDF of the return up to *η*, and *F*(2)*<sup>R</sup> Z* (*η*) is the second-order reverse integral of *Z* up to *η*, that is the reverse integration of the reverse CDF of the return up to *η*. We call *F*(*i*) *Z* the *i*-th-order integral of *Z*, which will be used to define the SD theory for risk averters (see, for example, Quirk and Saposnik (1962)). On the other hand, we call *F*(*i*)*<sup>R</sup> Z* the *i*-th-order reversed integral, which will be used to define the SD theory for risk seekers (see, for example, Hammond (1974)). Risk averters typically have a preference for assets with a lower probability of loss, while risk seekers have a preference for assets with a higher probability of gain. When choosing between two assets *X* or *Y*, risk averters will compare their corresponding *i*-th order SD integrals *F*(*i*) *X* and *F*(*i*)*<sup>R</sup> Y* and choose *X* if *F*(*i*) *X* is smaller, since it has a lower probability of loss. On the other hand, risk seekers will compare their corresponding *i*-th order RSD integrals *F*(*i*)*<sup>R</sup> X* and *F*(*i*)*<sup>R</sup> Y* and choose *X* if *F*(*i*)*<sup>R</sup> X*is larger since it has a higher probability of gain.

Following the definition of stochastic dominance (Hanoch and Levy (1969)), prospect *X* first-order stochastically dominates prospect *Y*:

$$\text{if and only if } \ F\_X^{(1)}(\eta) \le F\_Y^{(1)}(\eta) \quad \text{for any } \eta \in \mathbb{R}\_\prime \tag{4}$$

which is denoted by *X FSD Y*; prospect *X* second-order stochastically dominates prospect *Y*:

$$\text{if and only if } \ F\_X^{(2)}(\eta) \le F\_Y^{(2)}(\eta) \quad \text{for any } \eta \in \mathbb{R}\_\star \tag{5}$$

which is denoted by *X SSD Y*. Here, FSD and SSD denote first- and second-order stochastic dominance, respectively.

Next, we follow Levy (2015) to define risk-seeking stochastic dominance (RSD)<sup>1</sup> for risk seekers. Prospect *X* stochastically dominates prospect *Y* in the sense of second-order risk seeking:

$$\text{if and only if } \ F\_X^{(2)R}(\eta) \ge F\_Y^{(2)R}(\eta) \quad \text{for any } \eta \in \mathbb{R}, \tag{6}$$

which is denoted by *X SRSD Y*. Here, SRSD denotes second-order RSD.

<sup>1</sup> Levy (2015) denotes it as RSSD, while we denote it as RSD.

Quirk and Saposnik (1962), Hanoch and Levy (1969), Levy (2015) and Guo and Wong (2016) have studied various properties of stochastic dominance (for risk averters), while Hammond (1974), Meyer (1977), Stoyan and Daley (1983), Li and Wong (1999), Wong and Li (1999), Wong (2007), Levy (2015) and Guo and Wong (2016) have developed additional properties of risk-seeking stochastic dominance for risk seekers. One important property for SD is that SSD and SRSD are equivalent to the expected-utility maximization for (second-order) risk-averse and risk-seeking investors, respectively, while FSD is equivalent to the expected-utility/wealth maximization for any investor with increasing utility functions.

We turn to define <sup>Ω</sup>*X*(*η*) as follows:

$$
\Omega\_X(\eta) = \frac{\int\_{\eta}^{\infty} (1 - F\_X(\xi)) d\xi}{\int\_{-\infty}^{\eta} F\_X(\xi) d\xi}. \tag{7}
$$

Here, *η* is called the return threshold. For any investor, returns below (above) her/his return threshold are considered as losses (gains). Thus, the Omega ratio is the probability weighted ratio of gains to losses relative to a return threshold.

According to Darsinos and Satchell (2004), we can also rewrite <sup>Ω</sup>*X*(*η*) as follows:

$$\Omega\_X(\eta) = \frac{F\_X^{(2)R}(\eta)}{F\_X^{(2)}(\eta)} = \frac{F\_X^{(2)}(\eta) - (\eta - \mu\_X)}{F\_X^{(2)}(\eta)} = 1 + \frac{\mu\_X - \eta}{F\_X^{(2)}(\eta)}.\tag{8}$$

We state the following Omega ratio dominance (OD) rule by using the Omega ratio:

**Definition 1.** *For any two prospects X and Y with Omega ratios,* <sup>Ω</sup>*X*(*η*) *and* <sup>Ω</sup>*Y*(*η*)*, respectively, X is said to dominate Y by the Omega ratio or X is said to Omega ratio dominate Y, denote by:*

$$X \succeq\_{\text{OD}} Y \quad \text{if} \quad \Omega\_X(\eta) \succeq \Omega\_Y(\eta) \text{ for any } \eta \in \mathbb{R}. \tag{9}$$
