**5. Illustration**

Investment in property is important in both consumption and investment decisions (Henderson and Ioannides (1987)). Ziering and McIntosh (2000) argue that housing size is important in determining the risk and return of housing and conclude that the largest class of housing provides investors with the highest return and the greatest volatility. However, Flavin and Nakagawa (2008) document that investing in larger houses does not reduce risk, while Kallberg et al. (1996) show that smaller property offers impactful diversification benefits for investment portfolios with high return aspirations. On the other hand, Cannon et al. (2006) explain housing returns by volatility, price level and stock-market risk, and Ghent and Owyang (2010) investigate supply and demand to explain movements in the housing market.

The housing market in Hong Kong plays a very important role in the Hong Kong economy (Haila (2000)), and Hong Kong is one of the most expensive housing markets in the world in terms of both prices and rents (Tsang et al. (2016)). 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 and find the FSD relationship in rental yield in any adjacent pairing of the five well-defined housing classes in Hong Kong. In empirical studies, very few studies could discover the existence of any FSD relationship, and it is very important to obtain the FSD relationship (if there is any) because this information is very helpful to investors. For example, the findings from Tsang et al. (2016) imply that by shifting investing from the largest class of housing to the smallest class of housing, investors could obtain higher expected utility, as well as higher expected wealth from rental income.

In this paper, 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. We recommend that analysts apply the Omega ratio to examine whether there is any FSD relationship between any pair of variables being studied because it is easier to obtain the Omega ratio. The Omega ratio could serve as a complementary tool for the FSD test, and thus, we recommend that analysts use both the Omega ratio and FSD test in their analysis. The existence of dominance from both the Omega ratio and FSD test could assert the existence of the FSD relationship between the variables being examined. In addition, our illustration could also serve our purpose to demonstrate whether the theory developed in this paper holds true.

In order to readdress the issue studied by Tsang et al. (2016), we first use the same rental yield data used in Tsang et al. (2016) to compare monthly property-market rental yields in private domestic units of five different housing classes from January 1999–December 2013 in Hong Kong. The data are obtained from the Rating and Valuation Department of the Hong Kong SAR. The monthly rental yields for each class are calculated by dividing the average rent within the class by the average sale price for houses in the class for that month. Private domestic units are defined as independent dwellings with separate cooking facilities and bathrooms (and/or lavatories). They are sub-divided into five classes

by reference to floor area: Class A salable area less than 40 m2; Class B salable area of 40–69.9 m2; Class C salable area of 70–99.9 m2; Class D salable area of 100–159.9 m2; and Class E salable area of 160 m<sup>2</sup> or above.

To analyze the rental yield and to illustrate Theorem 2, we set *A* = rental yield of Class A and *E* = rental yield of Class E and present the summary statistics of the rental yields for Classes *A* and *E* in Table 1.


**Table 1.** Summary statistics for *X* and *Y*.

Note: *A* = the rental yield of Class A, *E* = the rental yield of Class E, and std = standard deviation. *t*- and F-tests report the *p*-values of the tests.

We first test the following hypotheses:

$$\begin{array}{ll} H\_0^\mu: & \mu\_A = \mu\_E \quad \text{versus} \quad H\_1^\mu: \mu\_A > \mu\_E \end{array} \tag{17}$$

for rental yield. The result of the *t*-test in Table 1 concludes that the mean rental yield of *A* is significantly higher than that of *E*. Thereafter, we test the following hypotheses:

$$H\_0^\sigma: \quad \sigma\_A = \sigma\_\mathcal{E} \quad \text{versus} \quad H\_1^\sigma: \sigma\_A < \sigma\_\mathcal{E} \tag{18}$$

for rental yield. The result of the *F*-test in Table 1 does not reject that the variances of the rental yields of both *A* and *E* are the same. Applying the mean-variance rule for risk averters Markowitz (1952) that *A* is better than *E* if *μA* ≥ *μ<sup>E</sup>*, *σA* ≤ *σE* and there is at least one strictly inequality, we conclude that risk averters prefer Property *A* to Property *E* based on rental yield. On the other hand, if we apply the mean-variance rule for risk seekers (Wong (2006, 2007); Guo et al. (2017)) that *A* is better than *E* provided that *μA* ≥ *μ<sup>E</sup>*, *σA* ≥ *σE* and there is at least one strict inequality, we conclude that risk seekers prefer Property *A*to Property *E* based on rental yield under the condition that *A* and *E* belong to the same location-scale family or the same linear combination of location-scale families Wong (2006, 2007). Nonetheless, this conclusion cannot imply the existence of the first-order SD relationship between Properties *A* and *E* based on rental yield if *A* and *E* do not belong to the same location-scale family or the same linear combination of location-scale families. To circumvent the limitation, this paper recommends that academics and practitioners use the Omega ratio rule as discussed in this paper. Thus, we turn to applying the Omega ratio rule to analyze whether there is any first-order SD relationship between Properties *A* and *E* based on rental yield.

We note that for the existence of the Omega ratio, we need *Z* < *η* with *Z* = *A*, *E*. To satisfy this condition, we choose *η* > max(min(*A*), min(*E*)). In addition, the term (*η* − *Z*)+ should not be too small. If not, the Omega ratios will be very large. Thus, in this illustration, we set *η* ≥ max(min(*A*), min(*E*)) + 0.5%. Furthermore, for *η* ≥ max(max(*A*), max(*E*)), we have (*Z* − *η*)+ ≡ 0. Thus, we set the upper-bound for *η* as max(max(*A*), max(*E*)). We exhibit the plot in Figure 1.

From the figure, it is clear that <sup>Ω</sup>*A*(*η*) ≥ <sup>Ω</sup>*E*(*η*) for any *η* ∈ *R*. We skip displaying plots of other pairs of variables because all the plots draw the same conclusion. We find that Class A dominates Classes B, C, D and E, Class B dominates Classes C, D and E, Class C dominates Classes D and E and Class D dominates Class E, by using the Omega ratio rule. We summarize the results of the Omega ratio dominance in Table 2. The results in the table are read based on row versus column. For example, the cell in Row A and Column B tells us that Class A dominates Class B by the Omega ratio and is denoted by OD, while the cell in Row B and Column A means that Class B does not dominate Class A by the Omega ratio, as denoted by ND.

**Figure 1.** The plots of Omega ratios of rental yields of Class A and Class E. The dotted and solid line represent the results of Class A and Class E, respectively.


**Table 2.** Pairwise comparison between rental yields.

OD is Omega ratio dominance defined in Definition 1. ND means no Omega ratio dominance.

To check whether a smaller house (any house in the group with the smaller size) is better than a bigger house (any house in the group with the bigger size), only comparing their rental yields is not good enough. Tsang et al. (2016) sugges<sup>t</sup> analyzing both rental and total yields. 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. We explain the reasons as follows: Tsang et al. (2016) find that (a) the smaller house dominates the bigger house in terms of rental yield, and (c) there is no dominance between the smaller and bigger houses in total yield where total yield = rental yield + price yield. Under (a) and (c), it is possible that (b') the bigger house dominates the smaller house in terms of the price yield, and thus, under (a), (b') and (c), we cannot conclude that the smaller house is a better investment than the bigger house. To circumvent the limitation, in addition to analyzing the rental yield, we recommend analyzing the price yield as follows: We set *A* = price yield of Class A and *E* = price yield of Class E and present the summary statistics of the price yields for Classes *A* and *E* in Table 3.

**Table 3.** Summary statistics of the price yield for Classes *A* and *E*.


Note: *A* = the price yield of Class A and *E* = the price yield of Class E. *t*- and F-tests report the *p*-values of the tests.

We first test the null hypothesis *Hμ*0 that *μA* = *μE* versus the alternative hypothesis *Hμ*1 that *μA* > *μE* as shown in (17) for the price yield. The result of the *t*-test in Table 3 does not reject that the mean price yields for *A* and *E* are the same. Thereafter, we test the null hypothesis *<sup>H</sup><sup>σ</sup>*0 that *σA* = *σE* versus the alternative hypothesis *<sup>H</sup><sup>σ</sup>*1 that *σA* < *σE* as shown in (18) for the price yield. The result of the *F*-test in Table 3 concludes that the variance of the price yield of *A* is significantly smaller that that of *E*. Thus, applying the mean-variance rules, we can conclude that risk averters prefer to invest in *A* rather than *E*, but risk seekers are indifferent between *A* and *E*. Nonetheless, this conclusion cannot imply any first-order SD relationship between Properties *A* and *E* based on the price yield. In this paper, we recommend that academics and practitioners use the Omega ratio rule as discussed in this paper.

Continuing with our analysis in the rental yield, we find that when *η* ≥ 0.0054, <sup>Ω</sup>*A*(*η*) is smaller than <sup>Ω</sup>*E*(*η*), while when *η* < 0.0054, <sup>Ω</sup>*A*(*η*) is larger. Thus, there is no OD relationship between *A* and *E*. To illustrate our results empirically, we set *η* ∈ [0.0054, max(max(*A*), max(*E*))]. The related results are exhibited in Figure 2. From this figure, it is clear that the Omega ratio of Class E is larger than that of Class A. For the Omega ratio dominance, different from the analysis for the rent yields, there is no dominance relationship between A and E in terms of the price yield by using the Omega ratio, and thus, we conclude that there is no FSD relationship between A and E in terms of the price yield.

**Figure 2.** The plots of Omega ratios of the price yield of Class A and Class E. The dotted and solid line represent the results of Class A and Class E, respectively.

Tsang et al. (2016) find that Class A SSD dominates Class E in terms of total yield. We conduct the Omega ratio test analysis for this issue. Our findings are consistent with Tsang et al. (2016). Since using both rental yield and price yield could draw the conclusion that investing in the smaller house is better than the bigger house, we skip reporting the OD results for the total yield.

Recall that Tsang et al. (2016) have shown that (a) the smaller house dominates the bigger house in terms of rental yield and (c) there is no dominance between smaller and bigger houses in total yield. Under (a) and (c), it is possible that (b') the bigger house dominates the smaller house in terms of the price yield.

In this paper, we find that (a) the smaller house dominates the bigger house in terms of rental yield, and (b) there is no dominance between smaller and bigger houses in price yield. We note that total yield = rental yield + price yield. Findings (a) and (b) can ge<sup>t</sup> either (c) that the smaller house dominates the bigger house in terms of total yield or (c') there is no dominance between smaller and bigger houses in terms of the total yield. No matter under (a), (b) and (c) or under (a), (b) and (c') (actually, we find (c') in our paper), we 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 and policy makers in real estate for their policy making to make the real estate market become efficient.

Last, we note that though our paper finds that there exists "expected arbitrage opportunity" in the Hong Kong real estate market, however, it is very difficult, if not impossible, to short sell a property in Hong Kong. Thus, it is not easy to explore this "expected arbitrage opportunity". Nonetheless, if an investor would like to buy a big house to stay in Hong Kong and sell it a couple years later, then, she/he may consider buying a few smaller houses with the same amount of funds in total, rent out all the smaller houses she/he bought, rent a bigger house for her/him to stay and sell all the properties she/he bought as her/his plan a couple of years later. In this way, she/he will ge<sup>t</sup> positive net rental income each month (since the rental yield of the smaller house OD dominates that of the bigger house), while the price yield has no difference when she/he sells the big house or the small houses (since there is OD dominance between smaller and bigger houses in terms of price yield). Thus, when she/he sells all her/his properties, she/he still gets net profit by the rental rental if she/he chooses to buy small houses.
