Stochastic dominance and Omega ratio: Measures to examine market efficiency, arbitrage opportunity, and anomaly

Guo, Xu; Jiang, Xuejun; Wong, Wing-Keung Article Stochastic dominance and Omega ratio: Measures to examine market efficiency, arbitrage opportunity, and anomaly Economies Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Guo, Xu; Jiang, Xuejun; Wong, Wing-Keung (2017) : Stochastic dominance and Omega ratio: Measures to examine market efficiency, arbitrage opportunity, and anomaly, Economies, ISSN 2227-7099, MDPI, Basel, Vol. 5, Iss. 4, pp. 1-16, https://doi.org/10.3390/economies5040038 This Version is available at: https://hdl.handle.net/10419/197040 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ economies Article Stochastic Dominance and Omega Ratio: Measures to Examine Market Efficiency, Arbitrage Opportunity, and Anomaly Xu Guo 1, Xuejun Jiang 2and Wing-Keung Wong 3,4,5,6,* 1School of Statistics, Beijing Normal University, Beijing 100875, China; [email protected] 2 Department of Mathematics, South University of Science and Technology of China, Shenzhen 518055, China; [email protected] 3Department of Finance and Big Data Research Center, Asia University, Taichung 41354, Taiwan 4Department of Economics and Finance, Hang Seng Management College, Hong Kong, China 5Department of Economics, Lingnan University, Hong Kong, China 6Department of Finance, College of Management, Asia University, 500, Lioufeng Rd., Wufeng, Taichung 41354, Taiwan *Correspondence: [email protected] Academic Editor: Ralf Fendel Received: 23 August 2017; Accepted: 2 October 2017; Published: 19 October 2017 Abstract: Both stochastic dominance and Omegaratio 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. In this paper, we first study the relationship between stochastic dominance and the Omega ratio. We find that second-order stochastic dominance (SD) and/or second-order risk-seeking SD (RSD) alone for any two prospects is not sufficient to imply Omega ratio dominance insofar that the Omega ratio of one asset is always greater than that of the other one. We extend the theory of risk measures by proving that the preference of second-order SD implies the preference of the corresponding Omega ratios only when the return threshold is less than the mean of the higher return asset. On the other hand, the preference of the second-order RSD implies the preference of the corresponding Omega ratios only when the return threshold is larger than the mean of the smaller return asset. Nonetheless, first-order SD does imply Omega ratio dominance. Thereafter, we apply the theory developed in this paper to examine the relationship between property size and property investment in the Hong Kong real estate market. We conclude 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 investors and policy makers in real estate. Keywords: stochastic dominance; Omega ratio; risk averters; risk seekers; utility maximization; market efficiency; anomaly JEL Classification: C0, D81, G10 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 Economies 2017,5, 38; doi:10.3390/economies5040038 www.mdpi.com/journal/economies Economies 2017,5, 38 2 of 16 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 suggest 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 2presents 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 Economies 2017,5, 38 3 of 16 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 Xand Y: F(1) Z(η) = FZ(η) = P(Z≤η), for Z=X,Y. (1) We define the second-order integral of Z,F(2) Z, F(2) Z(η) = Zη −∞F(1) Z(ξ)dξfor Z=X,Y; (2) and define the second-order reverse integral, F(2)R Z, of Z F(2)R Z(η) = Z∞ η(1−F(1) Z(ξ))dξfor Z=X,Y. (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)R 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)R 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)R 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)R X and F(i)R Y and choose Xif F(i)R Xis 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: if and only if F(1) X(η)≤F(1) Y(η)for any η∈R, (4) which is denoted by XFSD Y; prospect Xsecond-order stochastically dominates prospect Y: if and only if F(2) X(η)≤F(2) Y(η)for any η∈R, (5) which is denoted by XSSD Y . Here, FSD and SSD denote firstand second-order stochastic dominance, respectively. Next, we follow Levy (2015) to define risk-seeking stochastic dominance (RSD) 1 for risk seekers. Prospect Xstochastically dominates prospect Yin the sense of second-order risk seeking: if and only if F(2)R X(η)≥F(2)R Y(η)for any η∈R, (6) which is denoted by XSRSD Y. Here, SRSD denotes second-order RSD. 1Levy (2015) denotes it as RSSD, while we denote it as RSD. Economies 2017,5, 38 4 of 16 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 ΩX(η)as follows: ΩX(η) = R∞ η(1−FX(ξ))dξ Rη −∞FX(ξ)dξ. (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 ΩX(η)as follows: ΩX(η) = F(2)R X(η) F(2) X(η)=F(2) X(η)−(η−µX) F(2) X(η)=1+µX−η F(2) X(η). (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, ΩX(η) and ΩY(η) , respectively, X is said to dominate Y by the Omega ratio or X is said to Omega ratio dominate Y, denote by: XOD Y if ΩX(η)≥ΩY(η)for any η∈R. (9) 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: if XSSD Y, then ΩX(η)≥ΩY(η)for any η∈R, (10) where ΩX(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 argument in (10) may not be correct: Property 1. SSD alone is not sufficient to imply ΩX(η)≥ΩY(η)for any η. Property 1implies that the assertion made by Darsinos and Satchell (2004) and others may not be always correct. We construct the following example to support the argument stated in Property 1. Example 1. Consider two prospects X and Y having the following distributions: X=10 with prob. 1, and Y =(1with prob. 2/3 11 with prob. 1/3 . (11) Economies 2017,5, 38 5 of 16 Then, we get µX=10 and µY=13/3 and obtain the following: F(2) X(η) = (0 if η<10 η−10 if η≥10 ,F(2) Y(η) =      0 if η<1 2(η−1)/3 if 1 ≤η<11 η−13/3 if η≥11 , F(2)R X(η) = (10 −ηif η<10 0 if η≥10 ,F(2)R Y(η) =      13/3 −ηif η<1 (11 −η)/3 if 1 ≤η<11 0 if η≥11 . 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)R X(η)≡ 0 <F(2)R Y(η) . Recalling the definition of ΩX(η) , we can conclude that ΩX(η)≡ 0 <ΩY(η) for any 10 ≤η<11, and thus, the statement “ΩX(η)≥ΩY(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 ΩX(η)≥ΩY(η)for any η. We construct the following example to support the argument stated in Property 2. Example 2. Consider two prospects X and Y as follows: X=(2with prob. 1/2 8with prob. 1/2 and Y =(3with prob. 2/3 6with prob. 1/3 . (12) We have µX=5and µY=4and obtain the following: F(2) X(η) =      0 if η<2 (η−2)/2 if 2 ≤η<8, η−5 if η≥8 F(2) Y(η) =      0 if η<3 2(η−3)/3 if 3 ≤η<6, η−4 if η≥6 F(2)R X(η) =      5−ηif η<2 4−η/2 if 2 ≤η<8, 0 if η≥8 F(2)R Y(η) =      4−ηif η<3 2−η/3 if 3 ≤η<6. 0 if η≥6 It follows that F(2)R X(η)≥F(2)R Y(η) ,for all η∈R . This concludes that XSRSD Y . However, for η= 3.3, we can get: ΩX(η) = F(2)R X(η) F(2) X(η)=4−η/2 (η−2)/2 =8−η η−2=3.615. ΩY(η) = F(2)R Y(η) F(2) Y(η)=2−η/3 2(η−3)/3 =6−η 2η−6=4.5. That is, ΩX(η)<ΩY(η) . In fact, for any 3 <η< 7 −√13 , we have ΩX(η)<ΩY(η) , and thus, the statement “ΩX(η)≥ΩY(η)for any η” does not hold. Properties 1and 2tell us that SSD and SRSD alone are not sufficient to imply ΩX(η)≥ΩY(η) for any η . Then, one may ask: what is the relationship between ΩX(η) and ΩY(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: Economies 2017,5, 38 6 of 16 Proposition 1. For any two returns X and Y with means µX and µY and Omega ratios ΩX(η) and ΩY(η) , respectively, if X SSD Y, then ΩX(η)≥ΩY(η)for any η≤µX. Now, it is clear that Proposition 1extends 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 1and 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 1is obtained. Finally, the relationship between first-order stochastic dominance and the Omega ratios is established in Corollary 2. Some simple examples (Examples 1and 2) are presented to show that SSD or SRSD alone are not sufficient to imply ΩX(η)≥ΩY(η)for any η. Here, we provide a short proof 2 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: µX−η F(2) X(η)≤µX−η F(2) Y(η). Furthermore, we note that: µY−η F(2) Y(η)=µY−µX F(2) Y(η)+µX−η F(2) Y(η)≤µX−η F(2) Y(η). 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 ΩX(η)−ΩY(η). However, for any η≤µX, we can have µX−η≥0, and thus, we have: µX−η F(2) X(η)≥µX−η F(2) Y(η)≥µY−η F(2) Y(η). This implies that ΩX(η)≥ΩY(η), and thus, the assertion of Proposition 1holds. In the proof of Proposition 1, one could conclude that if XSSD Y , then ΩX(η)≥ΩY(η) for any η≤µX . However, for η>µX , 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 ΩX(η) and ΩY(η) , respectively, if X SRSD Y, then ΩX(η)≥ΩY(η)for any η≥µY. Here, we give a short proof as follows: assume that XSRSD Y . This is equivalent to F(2)R X(η) = R∞ η( 1 −FX(ξ))dξ≥F(2)R Y(η) . Recall that F(2)R X(η) = F(2) X(η)−(η−µX)≥ 0. This yields the following equation: 1 ΩX(η)=F(2) X(η) F(2)R X(η)=F(2)R X(η) + (η−µX) F(2)R X(η)=1+η−µX F(2)R X(η). 2We note that our proof is different from that of Balder and Schweizer (2017). Economies 2017,5, 38 7 of 16 Further, we note that XSRSD Yimplies µX≥µY. Thus, for η≥µY, we obtain: η−µX F(2)R X(η)=η−µY F(2)R X(η)+µY−µX F(2)R X(η)≤η−µY F(2)R X(η)≤η−µY F(2)R Y(η). In other words, we can get ΩX(η)≥ΩY(η) for any η≥µY , and thus, the assertion of Theorem 1holds. 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 1and 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 1and Theorem 1, we have ΩX(η)≥ΩY(η) for any η≤µX and η≥µY . Since µX≥µY, we have ΩX(η)≥ΩY(η)for any η∈R, and thus, the assertion of Theorem 2holds. 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: HSSD 0:X6SSD Yversus HSSD 1:XSSD Y(13) HSRSD 0:X6SRSD Yversus HSRSD 1:XSRSD Y(14) HFSD 0:X6FSD Yversus HFSD 1:XFSD Y(15) HOD 0:X6OD Yversus HOD 1:XOD Y(16) To test whether there is any SSD in two assets as stated in (13), we can apply Proposition 1to test whether ΩX(η)≥ΩY(η) for any η≤µX . 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 1to test whether ΩX(η)≥ΩY(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 2and Definition 1to 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 HOD 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 get 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 2and Definition 1to test HOD 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 Economies 2017,5, 38 8 of 16 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 1to test whether ΩA(η)≥ΩB(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 suggest 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, Economies 2017,5, 38 15 of 16 Flavin, Marjorie, and Shinobu Nakagawa. 2008. A model of housing in the presence of adjustment costs: A structural interpretation of habit persistence. American Economic Review 98: 474–95. Friedman, Milton, and L. J. Savage. 1948. The Utility Analysis of Choices Involving Risk. Journal of Political Economy 56: 279–304. Jarrow, Robert. 1986. The relationship between arbitrage and first order stochastic dominance. Journal of Finance 41: 915–21. Kallberg, Jarl G., Crocker H. Liu, and D. Wylie Greig. 1996. The role of real estate in the portfolio allocation process. Real Estate Economics 24: 359–77. Keating, Con, and William F. Shadwick. 2002. A universal performance measure. Journal of Performance Measurement 6: 59–84. Kapsos, Michalis, Nicos Christofides, and Berç Rustem. 2014. Worst-case robust Omega ratio. European Journal of Operational Research 234: 499–507. Larsen, Glen A., and Bruce G. Resnick. 1999. A performance comparison between cross-sectional stochastic dominance and traditional event study methodologies. Review of Quantitative Finance and Accounting 12: 103–12. Levy, Haim. 1992. Stochastic dominance and expected utility: Survey and analysis. Management Science 38: 555–93. Levy, Haim. 1998. Stochastic Dominance: Investment Decision Making under Uncertainty. Boston: Kluwer. Levy, Haim. 2015. Stochastic Dominance: Investment Decision Making Under Uncertainty, 3rd ed. New York: Springer. Li, Chi-Kwong, and Wing-Keung Wong. 1999. Extension of Stochastic Dominance Theory to Random Variables. RAIRO-Operations Research 33: 509–24. Markowitz, Harry. 1952. The Utility of Wealth. Journal of Political Economy 60: 151–6. Meyer, Jack. 1977. Second Degree Stochastic Dominance with Respect to a Function. International Economic Review 18: 476–87. Niu, Cuizhen, Wing-Keung Wong, and Lixing Zhu. 2016. First Stochastic Dominance and Risk Measurement. MPRA Paper 75027, University Library of Munich, Munich, Germany. Niu, Cuizhen, Wing-Keung Wong, and Qunfang Xu. 2017. Kappa Ratios and (Higher-Order) Stochastic Dominance. Risk Management 19: 245–53. Qiao, Zhuo, Ephraim Clark, and Wing-Keung Wong. 2012. Investors’ Preference towards Risk: Evidence from the Taiwan Stock and Stock Index Futures Markets. Accounting & Finance 54: 251–74. Qiao, Zhuo, and Wing-Keung Wong. 2015. Which is a better investment choice in the Hong Kong residential property market: A big or small property? Applied Economics 47: 1670–85. Quirk, James P., and Rubin Saposnik. 1962. Admissibility and measurable utility functions. Review of Economic Studies 29: 140–46. Sriboonchitta, Songsak, Wing-Keung Wong, Sompong Dhompongsa, and Hung T. Nguyen. 2009. Stochastic Dominance and Applications to Finance, Risk and Economics. Boca Raton: Chapman and Hall/CRC. Stoyan, Dietrich, and Daryl J. Daley. 1983. Comparison Methods for Queues and Other Stochastic Models. New York: John Wiley & Sons. Thaler, Richard H., and Eric J. Johnson. 1990. Gambling with the House Money and Trying to Break Even: The Effects of Prior Outcomes on Risky Choice. Management Science 36: 643–60. Tsang, Chun-Kei, Wing-Keung Wong, and Ira Horowitz. 2016. Arbitrage opportunities, efficiency, and the role of risk preferences in the Hong Kong property market. Studies in Economics and Finance 33: 735–54. Wong, Wing-Keung. 2006. Stochastic Dominance Theory for Location-Scale Family. Advances in Decision Sciences 2006: 1–10. doi:10.1155/JAMDS/2006/82049. Wong, Wing-Keung. 2007. Stochastic dominance and mean-variance measures of profit and loss for business planning and investment. European Journal of Operational Research 182: 829–43. Wong, Wing-Keung, and Chi-Kwong Li. 1999. A note on convex stochastic dominance. Economics Letters 62: 293–300. Economies 2017,5, 38 16 of 16 Wong, Wing-Keung, Kok Fai Phoon, and Hooi Hooi Lean. 2008. Stochastic dominance analysis of Asian hedge funds. Pacific-Basin Finance Journal 16: 204–23. Ziering, Barry, and Willard McIntosh. 2000. Property size and risk: Why bigger is not always better. Journal of Property Management 5: 62–70. c 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).