Premium-Discount Patterns in Exchange-Traded Funds (ETFs): Evidence from the Tracker Fund of Hong Kong (TraHK)

Transcription

1 Premium-Discount Patterns in Exchange-Trae Funs (ETFs): Evience from the Tracker Fun of Hong Kong (TraHK) Karen, H.Y. Wong Department of Accounting, Finance an Law, The Open University of Hong Kong, Hong Kong. Fax: Tel.: Goron, Y.N. Tang Department of Finance an Decision Science, Hong Kong Baptist University, Hong Kong Fax: Tel.:

2 Premium-Discount Patterns in Exchange-Trae Funs (ETFs): Evience from the Tracker Fun of Hong Kong (TraHK) Abstract Tracker Fun of Hong Kong (TraHK), the first Exchange-trae fun (ETF) in Asia, is esigne to trace the Hang Seng Inex (HSI). However, the TraHK oes not track the HSI successfully at the market opens. Results also show that the TraHK tracks the HSI more efficiently with a higher number of transactions. Compare positive an negative premiums, the positive premiums, between the HSI an TraHK, have higher mean values than those of negative premiums. For intraweek perioicity, a U-shape pattern forms. Key wors: Exchange-trae funs; HSI; intraay; intraweek 2

3 Premium Patterns in Exchange-Trae Funs (ETFs): A Case of Tracker Fun of Hong Kong (TraHK) Karen H. Y. Wong a an Goron Y. N. Tang b, a Department of Accounting, Finance an Law, The Open University of Hong Kong, Kowloon, Hong Kong b Department of Finance an Decision Sciences, Hong Kong Baptist University, Kowloon, Hong Kong Introuction After the first launche in 1990, exchange-trae funs (ETFs) has evelope rapily aroun the worl. In November 1999, the Tracker Fun of Hong Kong (TraHK), the first ETFs in Asia, was liste in Hong Kong Stock Exchange. The characteristic of the TraHK is its unique benchmark, the HSI. It is expecte that investors will be able to buy or sell their units on the Stock Exchange at a price that is close to the Net Asset Value (NAV) per unit but this cannot be guarantee. 1 Although the NAV of the TraHK is ajuste base on the upate HSI, the TraHK price may not be equal to the actual NAV of the TraHK. There is a potential arbitrage between the ETF shares an the unerlying assets. Consequently, numbers stuies fin the potential arbitrage via examine a tracking performance of ETFs. [See for example, Gastineau (2002). Jares an Lavin (2004). Lin, Chan, an Hsu (2005). Engle an Sarkar (2006).] In this paper, the tracking performance is efine as the premium, rather than the tracking error. The tracking error is commonly use by inex funs, but the term is inappropriate in this case. The tracking error is the stanar eviation of the ifference between returns on the inex fun an its benchmark. Premium, however, is the ifference between the Corresponing author. Tel.: ; fax: ; khywong@ouhk.eu.hk 1 Risk Associate with TraHK. See 3

4 price of the TraHK an the level of the HSI. As the TraHK is not an inex fun an hence stanar eviation is not use in the calculation, the premium therefore represents the tracking performance of the TraHK, an thus seems more suitable parameter to use than the tracking error. As the cash market changes rapily with time, the premium will vary at ifferent time perios as well. During traing hours, the HSI is upate every 15 secons. The NAV of the TraHK is upate once the HSI is reporte. The time lag shoul be consiere in this situation. In aition, investors show their expectation of the TraHK by buying or selling. In Taiwan market, Lin, Chan an Hsu confirme that Taiwan Top 50 Tracker Fun is pricing efficient in the Taiwan stock market. Will TraHK be a goo inicator to replicate the performance of the HSI? The purpose of this paper seeks to have a eeper unerstaning in the tracking characteristics of TraHK, that is, the ifference between premiums from ifferent time perios. Intraweek an intraay patterns in the premium are also investigate. Extensive stuies have summarize the intraweek an intraay patterns of both cash an futures markets. The existence of the ay-of-the-week effect in cash markets has been foun (Cross, 1973; French, 1980; Simrlock an Starks, 1984 an Harris, 1986). The ayof-the-week effect was also been prove in futures market (Dyl an Maberly, 1986; Junkus, 1986; Keim an Smirlock, 1987). However, Gibbons an Hess (1981) extene the stuy of S&P 500 aily returns from the perio of 1962 to the perio of They confirme negative Monay returns, but cannot fin any variance seasonality. Also, using the traing an nontraing returns on S&P 500 futures inex uring perios, Cornell (19852) coul not fin any ay-of-the-week effect. For intraay patterns, A U- shape pattern has been foun in many markets (Woo, McInish an Or, 1985; Harris, 4

5 1986; Amihu an Menelson, 1987; Joran et al., 1988; Ekman, 1990, 1992; Brock an Kleion, 1992; Ferguson, Mann, an Schneck, 1993; Lee an Linn, 1994). Besies, Smirlock an Starks (1986) foun the weeken effect from hourly returns of the Dow Jones Inustrial Average (DJIA). The effect has shifte from characterizing active traing on Monay to characterizing the nontraing weeken. Hence, the relation between transaction an premium was examine in this stuy. The paper is organize into five sections. Section Chapter 1I escribes the ata use an the calculation of the premium. Section Chapter 1II introuces statistical analysis, nonparametric, normality an correlation tests. Section Chapter 1III presents empirical evience an iscusses results, an Section Chapter 1IV provies concluing remarks. I. Data A. Sources The HSI compile by HSI Services Limite, is the benchmark of the Hong Kong stock market. The inex composition consists of 33 liste companies. These companies form aroun 70% of the total value of the market. The inex was compute after 31 July 1964, but was not operative until November Nowaays, the inex is recore at 15- secon intervals. The transaction ata of the TraHK is obtaine from the Hong Kong Stock Exchange (HKEx). Data consist of ate, time, price an number of shares trae in every transaction. In this stuy, we examine 164,253 transactions of the TraHK. With numerous transactions, a thin traing problem an the corresponing iscounts an premiums wien effect woul be ignore (Barney, 2000). 5

6 B. Perio In this stuy, we consier two sets of ata (1) the HSI an (2) the price of the TraHK. The stuy perio starts from 12 November to 30 November The entire perio covers more than a year, totaling 267 traing ays. With four traing hours in a ay, we ivie it into minute intervals. Data are also separate into five weekays. We can then test whether there are any ay-of-the-week an/or intraay effects. There are two missing pieces of ata. One is from 24 December 1999, as the afternoon traing session was cancelle for Christmas Eve. The other piece of missing ata arose because the computer system of the Hong Kong Stock Exchange broke own on 12 June 2000, causing the ay s traing to be elaye by an hour. Transactions starte at 11:00 then. C. The calculation of premiums The calculation of the premium is unertaken step by step. The premium is firstly calculate in each of a pair of transactions using the matche recore inex of the 15- secon interval. The premium, PM, is use to stuy the mispricing between the HSI an ajuste price, AP, that is, the price of the TraHK * 1000 (AP is use because each unit of the TraHK represents 1/1000 of the HSI). The premium of ajuste price j in interval i on ay is efine as below: ( AP HSI ) i, j i, j PM i, j = 100% i = 1, 2, (1) HSI i, j 2 The TraHK of Hong Kong was first issue on 12 November The issue price was HK$

7 We then carrie out an in-epth stuy, in which the ata were ivie into three groups: premium, positive premium an negative premium. We not only stuie the existence of any trens in these groups, but also examine whether there was any relationship ue to the sign of the premium. II. Methoology As the HSI operates on a 15-secon basis, the transaction price of the TraHK is matche with the inex every 15 secons. Thus, a pair of HSI an TraHK prices is reporte every 15 secons. We then group the available ata into minute classes for every weekay. In each 15-minute interval within same time perio an on the same weekay, we calculate a mean, a stanar eviation an a range of premium values, an count the number of transactions of the TraHK. Within a traing ay, there is a total of four hours traing in the morning (10:00 12:30) an afternoon (14:30 16:00) sessions. Hence, we have approximately 4, minute intervals (16 time intervals in a ay multiplie by 267 traing ays). In each interval, we have four measures on premium (mean, stanar eviation, range an number of transactions) calculate base on the transaction ata matche at 15-secons. Finally, we apply the Chi-Square statistic an the normality test to examine our hypotheses. The Chi-Square statistic is use to test if the istribution of a variable has the same location parameter across ifferent groups. In this stuy, we use the Kruskal-Wallis Test to compare the intraay an/or intraweek populations. Because the Kruskal-Wallis Test only etermines whether a ifference exists, we further test the normality of 7

8 populations using other metho. The correlations of the ata are also examine. We measure the extent to which premiums are relate base on ifferent weekay an 15- minute time intervals. We also examine whether the variances of premiums are associate with the numbers of transactions in each time perio. We further classify our ata into two atasets: (i) positive premium an (ii) negative premium. As mentione earlier, we efine the premium as the ajuste price of the TraHK minus the HSI. Thus, a positive premium is the surplus an a negative premium is the iscount. If the premium is in surplus, investors can buy the HSI to exchange the TraHK units an vice versa. In aition, the classification of the premium can ignore the skewness of the istribution of the premium. In orer to examine any ifference in the tests results concerning the nature of the premium, we repeat above tests in both atasets. A. Mean The average value of the premium in each interval is calculate. We first calculate the mean value of the premium in each interval within the same ay. Then, the average mean of the premium in each interval is measure. PM i 1 = N i N i j = 1 PM i, j for i =1,2,, 16. (2)\ = 1, 2,, 5. N i = Number of transactions recore for interval i on ay. 8

9 PM 1 = N i i = 1 PM i for i = 1,2,, 16. (3) = 1, 2,, 5. for i = Number of ays over which transaction is recore in interval i. B. Stanar Deviation Following Schwert (1989), 3 the stanar eviation for each interval in each ay is calculate as below: STD = i N i i= 1 2 ( PM i, j PM ) i N i for i = 1, 2,, 16. (4) = 1, 2,, 5. From 267 traing ays in the sample perio, the variability is measure in each ata pool of the same interval. We first obtain one average premium from the same 15-minute time interval in a single ay. The average premium is then compare with others in the same interval an we calculate the stanar eviation for that interval. The stanar eviation of every interval of every ay is available, allowing stuies of the intraay an intraweek patterns of premium volatility. 3 Schwert (1989) uses this metho to estimate the monthly variance of the S&P 500 portfolio as the sum of the square aily return eviations from the average aily return in the month. 9

10 C. Range Within the same 15-minute interval of a ay, the range of premiums is efine as the ifference between the maximum an minimum premiums. Thus, every ay uring the stuy perio provies ranges from every 15-minute interval. Base on all transactions in the same 15-minute interval, the average of the range of premium is calculate per 15- minute interval an per weekay. Similarly to the stanar eviation, the mean range is a measure of volatility. The mean range gives a measure of variability by consiering the overall tracking of premiums uring a certain interval. D. Number of transactions Similarly, we fin the number of transactions in every interval in the same ay. The average number of transactions in the same interval is thereby obtaine. For the overall perio, there are a total of 80 ifferent intervals for number of transaction within a week. All number of transactions shows the level of activity of the TraHK in each interval of a ay. E. Testing Statistics Three test measures, Chi-square statistic, normality test an correlation analysis, are consiere in this stuy. Thus, a series of hypotheses are firstly set up. Two hypotheses are examine to fin the existence of ifferences across the week an the significance of ifferences. Day-of-the-week Effects Hypothesis H (a) : The statistical measure, a, is the same across all five weekays for interval t. 10

11 for a = mean, stanar eviation, range an number of transactions t = 1, 2, 3,, 16 Intraay Effects Similarly, another hypothesis is teste to measure ifferences between the various intervals. Hypothesis H (b) : The statistical measure, a, is the same across all 16 intervals for ay. for a = mean, stanar eviation, range an number of transactions = 1, 2, 3,, 5 F. Correlation Correlation tests are applie to premiums an to the number of transaction stuies. Both stuies are ivie into three groups: 1) All premiums 2) positive premiums an 3) negative premiums. The correlation test is a measure of the egree of linear relationship between two variables. The correlation coefficient is in the range of plus one (+1) an minus one (-1). The strength of the relationship is measure after taking the absolute value of the correlation coefficient. 11

12 We first apply correlation tests to premiums in both the intraay an the intraweek stuies. In this regar, correlation tests stuy relationships between premiums in various intraay time intervals, an also examine the relationship between premiums in various intraweek time perios. A number of stuies have examine the relationship between volatility an volume. For our stuy, has the relationship between variances an number of transactions been establishe? From the results of escriptive statistics in former sections, we examine the correlation relationship between variances an number of transactions in every 15-minute interval. Furthermore, correlation results may be ue to particular 15-minute time perios. Each time interval may have an influence on the others. Thus, the correlation test is also use to look for intraay an/or intraweek patterns. III. Empirical Results A. Positive an Negative Premiums The mean of the premiums measures the average benchmark performance of the TraHK. Table 1 an Figure 1 clearly show that high premium always exists on Thursays. Moreover, all premiums are non-normally istribute across a week an also across a ay. Compare with the result of the mean, the range of the premiums (see Table 2 an Figure 2) unergoes many changes. Due to the formation of a W-shape pattern uring the course of a week, high values of ranges are recore on Monays, Friays an even on Wenesays, but not Thursays. All large ifferences are recore in the first 15-minute interval of the morning session (10:15). Being an alternative measure of volatility, the results of stanar eviation are similar to those of the range. Large values of stanar eviation (see Table 3 an Figure 3) always occur in the first 15-minute interval of the 12

13 morning session (10:15). Moreover, large values of stanar eviation occur mostly on Monays, after the weeken break. In aition to Monay, Tuesay is the secon weekay on which a large fluctuation of premiums is recore. There is also one 15- minute interval (16:00) in which large fluctuations are note. Before the en of the week, there are three 15-minute intervals (10:30, 10:45 an 12:15), which show large fluctuations, on Friays. Consiering the absolute value of premium, the highest stanar eviation of a ay exists uring the first 15-minute interval of the session. Two patterns are foun in the view of number of transactions (see Table 4 an Figure 4). A W-shape pattern of ay-of-theweek is forme. High numbers of transactions are recore on both Monays an Friays. These results woul be expecte, as more transactions are mae either at the beginning of the week or at the en of the week. A W-shape pattern also exists for a single ay, as transactions are active at the morning opening an afternoon opening an closing of the markets. B. Positive Premiums Table 5 an Figure 5 show that the mean value of positive premiums; the values are almost at the same level for all weekays. From Table 6, Table 7, Figure 6 an Figure 7, the range an stanar eviation reach a maximum uring the first 15-minute interval (10:15) of the ay on every weekay, except Friays. It is interesting that many transactions are complete uring the first 15-minute interval (10:15) an at the last 15- minute interval (16:00) of a ay (see Table 8 an Figure 8). 13

14 C. Negative Premiums Table 9 an Figure 9 give us a simple intraay picture of the mean of the negative premiums. An L-shape pattern is shown in Figure 10. Table 10 shows that negative premiums are marke a large range at the beginning of a ay (10:15). From the results of calculating stanar eviations (see Table 11 an Figure 11), we know that fluctuations are active at the beginning of the ay ue to the fact that all high mean values fall uring the first 15-minute interval (10:15) of a ay. At the same time, negative premiums always show high transactions uring the first 15-minute intervals of the morning (10:15) an the afternoon (14:45) sessions. In aition, negative premiums always occur either on Monays or Friays (Shown in Table 12 an Figure 12). D. Intraay Pattern Consiering the absolute values of the premiums, the highest stanar eviation of a ay exists uring the first 15-minute interval of the morning session (10:15) (see Table 3 an Figure 3). There is no oubt that the TraHK oes not benchmark the HSI closely at the market opens. From the results of number of transactions with positive an negative premiums (see Table 8 an Table 12), we fin that positive premiums are mainly recore uring three 15-minute intervals: 1) the first interval of the morning session (10:15) on Monays, Tuesays an Thursays; 2) the first interval of the afternoon session (14:45) on Friays; an 3) the last interval of the afternoon session (16:00) on Wenesays an Friays. Similarly, negative premiums also exist uring the same three 15-minute intervals but on ifferent weekays: 1) the first interval of the morning session (10:15) on Monays an 14

15 Friays; 2) the first interval of the afternoon session (14:45) on Wenesays an Thursays; an 3) the last interval of the afternoon session (16:00) on Tuesays. Overall, the average means of the positive premium are higher than those of the negative premiums. Almost all average means of the positive premiums (from Table 5) are greater than 0.5, which are even greater than the average means of the original premiums (from Table 1) ataset, while the average means of the negative premium (see Table 9) are range from -0.2 to It is interesting to note that all of the statistical intraay results are significantly nonnormality at the 1% level. The results show a non-normal istribution. Without consiering tables of mean, all of the results are significantly ifferent to those in the atasets of both the positive an the original premiums. E. Day-of-the-week Pattern Thursay can be consiere as the peak ay of the week in terms of the average mean of the premiums from Table 1 an Figure 1. Eleven out of minute intervals contain the highest average means. However, patterns of average means of positive an negative premiums are ifferent from those of the overall premiums. For positive premiums (see Table 5), although there are 8 out of minute intervals in which the highest average means can be foun on Thursays, there are also five 15-minute intervals which contain the highest average means on Tuesays. However, 10 out of minute intervals recor the highest mean values for the negative premiums on Wenesays. 15

16 Comparing atasets of three kins of premium, there are no same ay-of-the-week patterns in regar to the range an stanar eviation. From the results of the premiums in Table 2, large values for range are recore on Monays an Friays. Similarly, high values of stanar eviation occur on Monays an Friays, while more fluctuation also exists on Tuesays (see Table 3 an Figure 3). For positive premiums, large ranges exist on Monays, Wenesays an Friays (see Table 6 an Figure 6 ). From the results of stanar eviations, shown in Table 7, high values for stanar eviation are evenly istribute over weekays. As there are larger values for ranges on Wenesays, more 15- minute intervals are recore with high values in stanar eviation. For number of transactions, three atasets are recore, showing almost the same ay-ofthe-week patterns as each other. From Table 4 an Figure 4, we know that more transactions are complete on both Monays an Friays than at other times uring the week. Furthermore, more transactions are mae on Friays than on Monays. Thus, the results of number of transactions, with regar to both positive (see Table 8) an negative premium (see Table 12) premiums are consiere here. Most of the number of transactions takes place on Monays an Friays. The results seen on Wenesays are interesting, in terms of both positive an negative premiums. Traing with positive premiums is active on Wenesays, while there are fewer transactions with negative premiums. Table 4 oes not show that fewer transactions are mae on Wenesays. Hence, the prices of the TraHK will always be higher than the HSI on Wenesays. From the number of transactions, in terms of both positive an negative premiums, we know that both atasets show W-shape patterns. In fact, more transactions gave rise to negative premiums. 16

17 Although the statistical results show no significant ifferences, all non-normality tests results are significant ifference. Nearly 90% of the results are non-normally istribute. However, regarless of positive or negative premiums, only about 10% of the number of transactions falls within a normal istribution. F. Intraay an Day-of-the-week Correlation of Premiums In Table 13, Table 14 an Table 15, it can be seen that in all of the intraay tests, all of the ata are positively correlate within a single ay. This statement is strongly supporte as all relate results are significant. Moreover, the positive premiums, similarly to the original premiums, prouce high correlation coefficients ( ) whereas the negative premiums have correlation coefficients between 0.3 an 0.8. For the overall premiums (Table 13), the value of the correlation epens on the time interval. Closer time intervals are recore as having higher correlation coefficients. However, this effect is not seen with either positive or negative premiums. However, negative correlations exist in our intraweek stuy. It can be seen from the original premiums (see Table 16), that there are negative correlations between Thursays an other weekays. Moreover, the positive premiums show negative correlations for two pairs of workays: Monays an Friays, an Tuesays an Friays (see Table 17). From Table 18, we iscover that negative premiums give us a ifferent picture. All ata show highly positive correlations. Furthermore, 70% of the ata have a p-value of less than

18 G. Correlation between Variances an Number of Transactions of Premiums Table 19, Table 20 an Table 21 show the correlations between variances an number of transactions of original premiums, positive premiums an negative premiums within a single ay an across a week. From the results of Table 19, there are 51 out of minute intervals showing negative correlations, emonstrating that variances an number of transactions are inversely relate. Almost all correlations are non-significant. There are only five 15-minute intervals with significant correlation coefficients: 12:30 on Tuesays, 15:30 on Thursays, 10:15, 10:30 an 15:15 on Friays. Ignoring their positive or negative nature, these correlation coefficients all have high values, from to This characteristic can also be foun in the positive an negative premiums. For the positive premiums, there are only four 15-minute intervals with positive significant correlations (11:30 on Monays, 11:15 an 15:30 on Thursays, an 15:15 on Friays). These intervals also have high correlation coefficients, from to Positive significant correlations are also foun in the negative premiums. These correlation coefficients have high values, from to There are also nine 15-minute intervals with significant correlations. These results are consistent with those of Jones et al. (1994). They fin that volatility is positively relate to the number of transactions. IV. Conclusion Using more than one year s worth of ata, this paper stuies the intraay an the ay-ofthe-week patterns of the benchmark of the TraHK. The Fun tracks the HSI. Thus, the price of the Fun shoul theoretically show the same trens as the HSI. After performing a number of tests, we may ascertain the efficiency of the TraHK. 18

19 The intraay stuy shows that the traing of the TraHK is active uring several time intervals, such as the first 15-minute interval (10:00 10:15) in the morning, the first 15- minute interval (14:30 14:45) in the afternoon an the last 15-minute interval (15:45 16:00) in the afternoon. It is important that the TraHK unsuccessfully tracks the HSI at the beginning of the ay. We believe that potential profit occurs when wier eviations happen when the market opens. Jares an Lavin (2004) ha prove that eviation creates ETF returns. Base on the operation, the TraHK will only benchmark the HSI once the HSI has opene an the upate historical ata for the HSI are available. The TraHK will not track the HSI successfully unless the HSI opens. It is also worth mentioning that positive premiums have higher mean values than those of negative premiums. Investors show their expectation of the market by investing in the TraHK. Referring to the results of the intraay correlation, the general tren for a single ay can be easily estimate as positive correlations are forme. During the course of a week, a U-shape pattern forms. As with other stocks, the TraHK is popular on Monays an Friays. It is interesting that the Fun tracks the HSI more efficiently with a higher number of transactions. However, tracking becomes more ifficult as the week progresses. Hence, high average means exist on Thursays. The phenomenon is significant with negative premiums. Thursays an other weekays have negative correlations. Similarly, we fin that negative correlations occur on two other pairs of weekays: (i) Monays an Friays, an (ii) Tuesays an Friays. Furthermore, none of the statistics show significant ifferences using the Chi-Square tests even for the 19

20 non-normally istribute ata. We also foun that there is little correlation between the variances an the number of transactions. 20

21 Table 1 Intraay an Intraweek Patterns of the 15-minute Interval Mean Premium 21 Time Mon Tue We Thur Fri Total CHISQ P>CHISO W:Normal P<W 10: < : < : < : < : < : < : < : < : < : < : < : < : < : < : < : <.0001 Total <.0001 CHISQ P>CHISO W:Normal P<W <.0001 <.0001 <.0001 <.0001 <.0001 <.0100 Notes: The stuy covers across a week. The traing ay is ivie into 16 ivisions lasting 15 minutes each. For the moel N i 1 PM = PM i i, j Ni j= 1 for i =1,2,, 80. N i = Number of transactions recore for interval i on ay. 1 = N i for i = 1,2,, 80. PM PM i i = 1 for i = Number of ays that transaction is recore in interval i.

22 Table 2 Intraay an Intraweek Patterns of the 15-minute Interval Range Premium 22 Time Mon Tue We Thur Fri Total CHISQ P>CHISO W:Normal P<W 10: < : < : < : < : < : < : < : < : < : < : < : < : < : < : < : <.0001 Total <.0001 CHISQ P>CHISO <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 W:Normal P<W <.0001 <.0001 <.0001 <.0001 <.0001 <.0100 Notes: The stuy covers across a week. The traing ay is ivie into 16 ivisions lasting 15 minutes each.

23 Table 3 Intraay an Intraweek Patterns of the 15-minute Interval Stanar Deviation Premium 23 Time Mon Tue We Thur Fri Total CHISQ P>CHISO W:Normal P<W 10: < : < : < : < : < : < : < : < : < : < : < : < : < : < : < : <.0001 Total <.0001 CHISQ P>CHISO <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 W:Normal P<W <.0001 <.0001 <.0001 <.0001 <.0001 <.0100 Notes: The stuy covers across a week. The traing ay is ivie into 16 ivisions lasting 15 minutes each. For the moel STD i = N i i = 1 2 ( PM i, j PM ) i N i for i = 1, 2,, 80.

24 Table 4 Intraay an Intraweek Patterns of the 15-minute Interval Number of Transactions 24 Time Mon Tue We Thur Fri Total CHISQ P>CHISO W:Normal P<W 10: < : < : < : < : < : < : < : < : < : < : < : < : < : < : < : <.0001 Total <.0001 CHISQ P>CHISO <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 W:Normal P<W <.0001 <.0001 <.0001 <.0001 <.0001 <.0100 Notes: The stuy covers across a week. The traing ay is ivie into 16 ivisions lasting 15 minutes each.

25 Table 5 Intraay an Intraweek Patterns of the Average of Positive Mean Premium 25 Time Mon Tue We Thur Fri Total CHISQ P>CHISO W:Normal P<W 10: < : < : < : < : < : < : < : < : < : < : < : < : < : < : < : <.0001 Total <.0001 CHISQ P>CHISO W:Normal P<W <.0001 <.0001 <.0001 <.0001 <.0001 <.0100 Notes: The stuy covers across a week. The traing ay is ivie into 16 ivisions lasting 15 minutes each. For the moel N i 1 PM = PM for i =1,2,, 80. i i, j Ni j= 1 N i = Number of transactions recore for interval i on ay. 1 = N i for i = 1,2,, 80. PM PM i i = 1 for i = Number of ays that transaction is recore in interval i.

26 Table 6 Intraay an Intraweek Patterns of the Range of Positive Mean Premium 26 Time Mon Tue We Thur Fri Total CHISQ P>CHISO W:Normal P<W 10: < : < : < : < : < : < : < : < : < : < : < : < : < : < : < : <.0001 Total <.0001 CHISQ P>CHISO <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 W:Normal P<W <.0001 <.0001 <.0001 <.0001 <.0001 <.0100 Notes: The stuy covers across a week. The traing ay is ivie into 16 ivisions lasting 15 minutes each.

27 Table 7 Intraay an Intraweek Patterns of the Stanar Deviation of Positive Mean Premium 27 Time Mon Tue We Thur Fri Total CHISQ P>CHISO W:Normal P<W 10: < : < : < : < : < : < : < : < : < : < : < : < : < : < : < : <.0001 Total <.0001 CHISQ P>CHISO < <.0001 <.0001 <.0001 <.0001 W:Normal P<W <.0001 <.0001 <.0001 <.0001 <.0001 <.0100 Notes: The stuy covers across a week. The traing ay is ivie into 16 ivisions lasting 15 minutes each. For the moel STD i = N i i = 1 2 ( PM i, j PM ) i N i for i = 1, 2,, 80.

28 Table 8 Intraay an Intraweek Patterns of the Number of Transactions of Positive Mean Premium 28 Time Mon Tue We Thur Fri Total CHISQ P>CHISO W:Normal P<W 10: < : < : < : < : < : < : < : < : < : < : < : < : < : < : < : <.0001 Total <.0001 CHISQ P>CHISO <.0001 <.0001 <.0001 <.0001 <.0001 <.0001 W:Normal P<W <.0001 <.0001 <.0001 <.0001 <.0001 <.0100 Notes: The stuy covers across a week. The traing ay is ivie into 16 ivisions lasting 15 minutes each.

29 Table 9 Intraay an Intraweek Patterns of the Average of Negative Mean Premium 29 Time Mon Tue We Thur Fri Total CHISQ P>CHISO W:Normal P<W 10: < : < : < : < : < : < : < : < : < : < : < : < : < : < : < : <.0001 Total <.0001 CHISQ P>CHISO <.0001 W:Normal P<W <.0001 <.0001 <.0001 <.0001 <.0001 <.0100 Notes: The stuy covers across a week. The traing ay is ivie into 16 ivisions lasting 15 minutes each. For the moel N i 1 PM = PM for i =1,2,, 80. i i, j Ni j= 1 N i = Number of transactions recore for interval i on ay. 1 = N i for i = 1,2,, 80. PM PM i i = 1 for i = Number of ays that transaction is recore in interval i.

30 Table 10 Intraay an Intraweek Patterns of the Range of Negative Mean Premium 30 Time Mon Tue We Thur Fri Total CHISQ P>CHISO W:Normal P<W 10: < : < : < : < : < : < : < : < : < : < : < : < : < : < : < : <.0001 Total <.0001 CHISQ P>CHISO <.0001 W:Normal P<W <.0001 <.0001 <.0001 <.0001 <.0001 <.0100 Notes: The stuy covers across a week. The traing ay is ivie into 16 ivisions lasting 15 minutes each.

31 Table 11 Intraay an Intraweek Patterns of the Stanar Deviation of Negative Mean Premium 31 Time Mon Tue We Thur Fri Total CHISQ P>CHISQ W:Normal P<W 10: < : < : < : < : < : < : < : < : < : < : < : < : < : < : < : <.0001 Total <.0001 CHISQ P>CHISO <.0001 W:Normal P<W <.0001 <.0001 <.0001 <.0001 <.0001 <.0100 Notes: The stuy covers across a week. The traing ay is ivie into 16 ivisions lasting 15 minutes each. For the moel STD i = N i i = 1 2 ( PM i, j PM ) i N i for i = 1, 2,, 80.

32 Table 12 Intraay an Intraweek Patterns of the Number of Transactions of Negative Mean Premium 32 Time Mon Tue We Thur Fri Total CHISQ P>CHISO W:Normal P<W 10: < : < : < : < : < : < : < : < : < : < : < : < : < : < : < : <.0001 Total <.0001 CHISQ P>CHISO <.0001 W:Normal P<W <.0001 <.0001 <.0001 <.0001 <.0001 <.0100 Notes: The stuy covers across a week. The traing ay is ivie into 16 ivisions lasting 15 minutes each.

33 Table 13 Intraay Correlation of Premium Time Interval 10:15 10:30 10:45 11:00 11:15 11:30 11:45 12:00 12:15 12:30 14:45 15:00 15:15 15:30 15:45 16:00 10: : : : : : : : : : : : : : : : Notes: The stuy covers across a traing ay which is ivie into 16 ivisions lasting 15 minutes each. Pearson Correlation Coefficients Prob > r uner HO: Rho = 0 (All are.0001)

34 Table 14 Intraay Correlation of Positive Premium Time Interval 10:15 10:30 10:45 11:00 11:15 11:30 11:45 12:00 12:15 12:30 14:45 15:00 15:15 15:30 15:45 16:00 10: : : : : : : : : : : : : : : : Notes: The stuy covers across a traing ay which is ivie into 16 ivisions lasting 15 minutes each. Pearson Correlation Coefficients Prob > r uner HO: Rho = 0(All are <.0001)