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The tick/volatility ratio as a determinant ofthe compass rose patternChun I Lee a , Ike Mathur b & Kimberly C. Gleason ca Department of Finance/CIS , Loyola Marymount University , USAb Department of Finance , Southern Illinois University , Carbondale, USAc Department of Finance , Florida Atlantic University , USAPublished online: 16 Aug 2006.

To cite this article: Chun I Lee , Ike Mathur & Kimberly C. Gleason (2005) The tick/volatility ratioas a determinant of the compass rose pattern, The European Journal of Finance, 11:2, 93-109, DOI:10.1080/1351847032000137438

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The European Journal of FinanceVol. 11, No. 2, 93–109, April 2005

The Tick/Volatility Ratio as a Determinant ofthe Compass Rose Pattern

CHUN I LEE∗, IKE MATHUR∗∗ & KIMBERLY C. GLEASON†∗Department of Finance/CIS, Loyola Marymount University, USA, ∗∗Department of Finance,Southern Illinois University, Carbondale, USA, †Department of Finance, Florida Atlantic University, USA

ABSTRACT This study provides evidence that low frequency data masks certain returns phenomena in theforeign exchange (forex) market. It is shown that the compass rose pattern is entirely absent in daily returnsin the spot and futures forex markets. In contrast, the intraday returns, especially those for holding periods ofless than an hour, clearly exhibit the pattern. Monte Carlo investigation of the tick/volatility ratio providesconvincing evidence that the pattern appears only if the tick/volatility ratio is above some threshold level.Since intraday returns have a ratio above the threshold value, they exhibit the pattern. On the other hand, theabsence of the pattern in daily returns is due to the fact that the spot and futures currency returns examinedhave a ratio much smaller than the threshold value. Overall, the evidence is consistent with the hypothesisthat the tick/volatility ratio is a determinant of the compass rose pattern. The economic implications of thispattern are discussed.

1. Introduction

Many researchers view the foreign exchange (forex) market as the most efficient of the financialmarkets. Yet, tales of predictability and nonlinearity abound regarding the exchange rate series(Chappell and Eldridge, 1997), and the foreign exchange (forex) market is a notorious vehiclefor technical trading schemes (Neely and Weller, 2003). The use of higher frequency data mayuncover important information regarding return generating processes. Indeed, a large body ofevidence has surfaced regarding the price patterns evident in the time series of returns in variousmarkets. Deviations from certain conditions (e.g. conditions of no cointegration by Elyasianiand Kocagil, 2001 and serial independence by Kramer and Runde, 1997) in the data have beenidentified in studies using high frequency data that are not detected at lower frequencies, raisingimportant questions about market efficiency and asset pricing.

In particular, the foreign exchange series yields interesting intraday phenomena, such asnegative autocorrelation (Goodhart, 1988), leptokurtosis (Caporole et al., 1998), and interdayseasonality and heteroscedasticity (Muller et al., 1990). Harvey and Huang (1991) identify sev-eral features of the forex market that may foster time series characteristics different from thoseobserved in equity markets: large size and volume, overlapping business hours, electronic trading,more liquidity, lower transactions costs, and a demand for foreign currency to settle transactions.

Correspondence Address: Ike Mathur, Department of Finance, Southern Illinois University, Carbondale, IL 62901-4626,USA. Email: Imathur@cba.siu.edu

1351-847X Print/1466-4364 Online/05/020093–17 © 2005 Taylor & Francis Group LtdDOI: 10.1080/1351847032000137438

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Almeida et al. (1998) conclude that the use of low frequency data may mask patterns that areonly discernable at higher frequencies. Goodhart and Figliuoli (1991) find that predictabilityfrom past data is very high when frequencies at the one-minute level are used. In any event, thefailure to adequately analyze price behavior has implications for the presence of predictability(Brooks, 1997), as well as the economic implications of these patterns.

Of particular interest is the compass rose pattern. The compass rose was named by Crack andLedoit (1996) and is the pattern that may be observed in a scatterplot under certain conditionswhen the returns at time t are plotted against the lagged returns at time t−1. It is a structure thattakes the form of evenly spaced rays radiating from the origin with the thickest rays pointing inthe major directions of the compass. Crack and Ledoit (1996) document its existence in the stockmarket, and postulate that the pattern is caused by discrete jumps in prices. They state that thereare three necessary and sufficient conditions that the price series of a stock must satisfy for thepattern to appear. First, the price level has to be large relative to the price change from the last tothe current period. This condition reflects the intensity or volatility of price movements over theshort term. Second, the price changes occur in small discrete jumps. This condition requires thatthe asset is liquid and trades under orderly market conditions. Third, the price of the asset mustvary over a relatively wide range. This condition reflects the volatility or intensity of the asset’sprice movements over the long term.

If one of the conditions, that prices vary over a wide range, is not satisfied, while the othertwo conditions hold, then the resulting pattern is a grid structure, rather than a compass structure.Szpiro (1998) investigates these conditions further and shows that in addition to these two types ofmicrostructure patterns (compass rose and grid), there exists a nanostructure that becomes visibleonly when one drops the conditions that tick size and the number of ticks in price changes is smallrelative to the price level. If the definition of the compass rose pattern is extended to include all threetypes of patterns, then based on Szpiro’s argument, the only necessary and sufficient conditionfor the existence of the pattern is the requirement that price changes take place in discrete jumps.The progression from randomness to the pattern as the discreteness in price changes increases,demonstrated in the Monte Carlo experiment by Kramer and Runde (1997), supports the notionthat discreteness indeed is a determinant of the pattern.

The empirical evidence clearly demonstrates the existence of the pattern in the forex and equitymarkets. However, empirical evidence from futures markets by Lee et al. (1999) (LGM) doesnot support the pervasiveness of the pattern in non-equity markets, and, contrary to Crack andLedoit’s claim, effective tick size fails to explain the appearance/absence of the pattern. Theimplication of non-pervasiveness of the compass rose raises the question of whether the patternis microstructure-driven.

A casual examination of the instruments examined in LGM reveals that futures contracts oncurrencies do not universally exhibit the pattern. Yet Szpiro (1998) finds definitive evidenceof the compass rose pattern in the forex market, using transaction data. Thus, it is of interest todetermine the source of these differential findings. In particular, the difference in results may existbecause the instruments LGM examine are futures contracts, while those examined in Szpiro arespot forex rates, or, because differential holding periods-daily versus intraday-are examined inthese two studies. Additionally, price jumps in the forex market are usually very small, 0.0001(either in terms of $/unit of foreign currency or foreign currency/$) for both daily and intradayforex rates, compared to the jumps of 1/32 evidenced in equity markets. With this much smallermagnitude in price jumps, one would think that the discreteness, or lack of it, in price jumps, asSzpiro demonstrates, should manifest itself in both intraday and daily series in the forex market.Therefore, the absence of the pattern in daily series, in contrast to its presence in intraday series,

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suggests that the discrete jumps in prices may not be the sole determinant of the pattern. Thisopens the door for an investigation of other factors explaining the enigmatic manifestation of thepattern pattern across various time series related to the length of the holding period over whichreturns are generated. The volatility, as measured by the standard deviation of returns, may alsobe a determinant of the pattern.

The purpose of this paper is to investigate these issues using forex rates and price series ofvarious holding periods in spot and futures markets. The forex market provides the opportunityto further question the ubiquity of the compass rose pattern across markets, to better understandthe underlying dynamics of exchange rates, and to investigate the characteristics the newly tradedEuro. The results show that while the compass rose pattern fails to manifest itself in the daily seriesof exchange rates, both spot and futures, all intraday series strongly demonstrate the existence ofthe pattern. Monte Carlo simulation results show that this difference in results is related to the ratioof the tick size and the volatility of returns (hereafter, the tick/volatility ratio). The pattern existswhen the ratio is greater than some threshold value. On the other hand, when the ratio is smallerthan the threshold values, which is the case for daily series, no pattern emerges. This evidencetherefore demonstrates that in addition to discreteness in price changes, the tick/volatility ratiois also a determinant of the pattern.

2. Data and Methodology

2.1 Data

Daily price series of the nearby contracts of 34 forex futures contracts traded on seven exchangesin five countries and 52 daily spot forex rates were obtained from the CRB-Bridge Database. Thespot and forex rates included 50 direct rates against the US dollar and 17 cross-rates not involvingthe US dollar. In addition, intraday series of various lengths (1 minute, 5 minute, and hourly)of four forex rates ( Euro, British Pound, Swiss Franc, and Japanese Yen) were obtained fromDow Jones Telerate Services. The time period of these intraday series varied depending on theavailability of data in Telerate on the day they were obtained. The inclusion of the Euro, whichdebuted on 1 January 1999, provided an opportunity to examine whether it, like the Pound, SwissFranc, and Yen, also exhibited the pattern. For futures contracts, only nearby contracts were usedand next to nearby contracts replaced the nearby contracts at the beginning of the month when thenearby contracts reached the expiration month. Therefore, there is no confounding aggregationof data on contracts of different maturities. A table listing these contracts/series along with theexchanges where they were traded (in the case of futures contracts) is available from the authors.

2.2 Methodology

Depending on the period – either daily or intraday – considered, return of period t , Rt , is calculatedas the change of closing price of period t , Pt − Pt−1, over the closing price of period Pt−1, i.e.

Rt = (Pt − Pt−1)/Pt−1 (1)

Equivalently, considering that prices move only in discrete jumps (h), the ratio of next to currentperiod returns can be expressed as

Rt+1/Rt = [(Pt+1 − Pt)/Pt ]/[(Pt − Pt−1)/Pt−1] (2)

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which can be simplified to nt+1h/nth = nt+1/nt , where nt is the number of ticks that the pricechange represents.

This use of ‘returns’ in the context of futures contracts warrants further explanation. Whilethe initial futures price is never really invested per se, as is the initial price of a stock, it is stillappropriate to use it in the analysis of price movements in futures markets. The invested moneyat time t − 1 is not the price of the futures contract; rather it is the initial margin, which is muchless. As a result of the leverage effect, the actual return on the invested money is a multiple ofthe measure of return. Therefore, the actual returns on invested money can be viewed as a lineartransformation of the returns used in the study. Hence, it is appropriate to examine returns definedas such in order to understand the characteristics of the actual returns.

After computing returns, a scatterplot is generated of Rt+1 against Rt . In the case that returnsremain relatively stable overtime, the plot will indicate that returns tend to cluster on lines radiatingfrom the origin, with the thickest rays emanating in the major directions (north, south, east, andwest), resembling a compass. Under the condition identified by Crack and Ledoit (1996) that theprice exhibits a relatively large variance over the time period examined, the data points create theappearance of the compass rose. Finally, Monte Carlo simulations are conducted to examine theinfluence of the tick/volatility ratio on the emergence of the pattern.

Figure 1A. Plots for Spot British Pound/US Dollar Daily Returns

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3. Results

3.1 Daily and Intraday Returns in Spot and Futures Forex Markets

Figure 1 reports the return patterns for the British pound and the British pound futures contracttraded on the International Monetary Market, using lower frequency (daily) data. Figure 1A and 1Bshow that there is no compass rose pattern in these daily series. Results for other currencies, bothspot and futures, are the same and not reported to conserve space.

Figure 1B. Plots for Futures British Pound/US Dollar Daily Returns

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Figure 2A. Plots for Spot British Pound/US Dollar One Minute Returns

However, the pattern appears when the data frequency is increased. Results for intraday seriesare reported in Fig. 2. Figure 2A–2C shows that spot intraday returns on British pound do exhibit agrid pattern. That the pattern is a grid, rather than the compass rose pattern shown in Szpiro (1998),may be due to the fact that only 4318 observations are used in these figures, compared with thenearly 1.5 million observations that are available in Szpiro. Another explanation for this gridpattern may be because 1 and 5 minute price changes during the sample period of about a week,14/1/1999 to 22/1/1999, experienced a relatively smaller range. When one extends the holdingperiod to an hour, the pattern, still a grid, starts to get blurred, which again may be due to a muchsmaller number of observations, 401, used in the graph. Returns for Japanese Yen and SwissFrance also show the similar grid pattern for 1 and 5 minute returns, and a blurred grid patternfor hourly returns.

The availability of the Euro offers an excellent opportunity to examine the pattern for the newlytraded currency. Figure 3A–3C report the pattern for the 1 and 5 minute and hourly returns onthe Euro. The plot indicates the presence of the pattern in the Euro and reveals that the hourlyreturns have a clearer grid pattern than those for the three currencies. Again, this clearer picture

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Figure 2B. Plots for Spot British Pound/US Dollar Five Minute Returns

may be due to the fact that more observations, 959, are available for the hourly Euro returns. Forthe 1 and 5 minute returns the pattern for Euro is similar to those for individual currencies.

3.2 Monte Carlo Simulation

To investigate if and how the tick/volatility ratio determines the presence of pattern, the followingMonte Carlo experiment is conducted. Six sets of simulated lognormal return series are generated.The mean and standard deviation of these six sets vary, ranging from 0 to 0.0005 for mean,and 0.0002 to 0.006 for standard deviation. These means and standard deviations are chosen tocorrespond to the actual observed values in intraday and daily returns in the forex market. Withineach set, there are six separate series, each corresponding to a tick/volatility ratio of 1/100, 1/50,1/10, 1/5, 1/4, and 10/1.

One conclusion that can immediately be drawn from the results reported in Section 3.1 is thatdespite the small discreteness of price jumps in both spot and futures forex markets, the compassrose pattern is entirely missing in the daily series of returns in these markets. In contrast, one

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Figure 2C. Plots for Spot British Pound/US Dollar One Hour Returns

can easily observe the pattern in intraday series. As discussed in the Introduction, factors otherthan just discrete jumps in prices are apparently responsible for the pattern. The immediate logicalreasoning following this observation suggests that the factor must be something inherently relatedto the difference between the daily and intraday holding periods. Since both daily and intradayseries in the forex market have similar magnitudes in terms of price changes, which is usually0.0001, it is not likely that the tick size is a candidate since it only relates to the magnitude ofthe price changes. The most likely candidate is volatility, or the range of fluctuation of the pricechanges. This logic leads one to examine the issue of volatility of price changes relative to pricechanges. The factor further investigated in this paper is the tick/volatility ratio. Analysis of thedata shows that this ratio decreases with increases in the holding period for the four currenciesexamined.

Monte Carlo simulations are conducted to investigate how the tick/volatility ratio determineswhether the pattern manifests itself. Out of the six different sets of simulated returns, results forone set are reported in Fig. 4. Results for the other sets are similar and not reported to conservespace.

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Figure 3A. Plots for Spot Euro/US Dollar One Minute Returns

Simulation results with mean 0.0005 and standard deviation 0.006 are shown in Fig. 4. Thiscombination of mean and standard deviation is chosen for reporting because it is closer to thoseof the daily returns in the forex market. Figure 4A shows the graph for the returns when thetick/volatility ratio is 1/10. No pattern is noticed for this case.

However, as the frequency of the data increases (i.e. the tick/volatility ratio increases) thepattern is brought into focus. In Fig. 4B, when the ratio equals 1/5, the pattern starts to emerge,notably for the main directions of the compass. The pattern becomes more visible in Fig. 4C, whenthe tick/volatility ratio is 1/4. The pattern is very evident when the ratio is 10/1 (Fig. 4D). In thelast case, the large discreteness of the prices causes the returns to be reduced to just a few possible

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Figure 3B. Plots for Spot Euro/US Dollar Five Minute Returns

values, but the compass rose pattern still manifests itself in the NE, NW, SW, and SE directions.To summarize, the results from Fig. 4 indicate that for returns with larger standard deviations, asin the case of daily series, the threshold value for the emergence of the pattern is much higher. Inview of the observed standard deviations and the tick/volatility ratio of daily returns, the absenceof the pattern can be attributed to the fact that these returns have a tick/volatility ratio smallerthan the threshold values that separate the regimes of pattern and no-pattern.

We conduct another set of simulations with mean zero and standard deviation 0.0002. Thismean/standard deviation combination is chosen because it is very close to the actual observedintraday mean/standard deviation values. The results (not reported here) show that when thetick/volatility ratio is 1/100, the pattern fails to emerge. However, when the ratio is 1/50, agrid pattern emerges. The pattern becomes clearer as the ratio increases to 1/10, and 1/4. Theresults suggest that the compass rose pattern can easily emerge in return series that have verysmall standard deviations, e.g. 0.0002. Under these circumstances, the threshold value for thetick/volatility ratio for the pattern to start to emerge is as small as 1/50. Since 1 and 5 minutes

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Figure 3C. Plots for Spot Euro/US Dollar One Hour Returns

and hourly returns have small volatility, they easily satisfy the requirement that their tick/volatilityratios exceed the threshold value, hence the existence of the pattern in these intraday series.

4. Conclusions and Implications for Future Research

The results reported in Section 3 show that the compass rose pattern is entirely absent inthe dailyseries of returns in the spot and futures forex markets. However, the compass rose pattern or gridpattern are observed with higher frequency data, especially those for holding periods of less thanan hour.

As explained in the Introduction and Section 3, a factor other than the discreteness of pricechanges related to the length of the holding period may be responsible for the difference in resultsbetween Spziro (1998) and LGM. The Monte Carlo investigation of the tick/volatility ratio inSection 3.2 provides convincing evidence that the pattern appears only if the tick/volatility ratiois above some threshold level. For the case that the standard deviation is 0.0002, the thresholdlevel is very small. Hence, we see that the pattern emerges when the ratio is 1/50. With a greater

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Figure 4A. Monte Carlo Simulation Results. Mean = 0.0005; Standard Deviation = 0.006; Tick/VolatilityRatio = 1/10

value for the standard deviation, such as 0.006, the threshold is 1/4. Since intraday returns fallinto the former category of a smaller standard deviation, while daily returns fall into the latter, thesimulation results explain why the pattern exists in intraday series but not in daily series. Overall,the evidence is consistent with the hypothesis that tick/volatility is a determinant of the compassrose pattern.

The results have important economic implications. First, although we document the compassrose pattern in two dimensions, it also can exist in higher dimensions. Crack and Ledoit (1996)show a three-dimensional pattern in their Fig. 9. This pattern can be reasonably well expected

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Tick/Volatility Ratio as a Determinant of the Compass Rose Pattern 105

Figure 4B. Monte Carlo Simulation Results. Mean = 0.0005; Standard Deviation = 0.006; Tick/VolatilityRatio = 1/5

to exist in n dimensions also. Some nonlinear n-dimensional procedures such as the BDS testfor chaos (Brock et al., 1991) test for structural patterns. Exceeding certain tick/volatility ratioswould generate the compass rose, and would bias the results towards finding chaos when none ispresent. Some evidence that further clarifies these issues is provided by Wang and Wang (2002),who extend the present work on the tick size and volatility to the angular distribution and thestrength estimations for the compass rose. Further research is still warranted on this issue.

Second, conditional heteroscedasticity models often utilize squared contemporaneous andlagged returns, similar to the scatterplots for the compass rose. We already know that discreteness

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Figure 4C. Monte Carlo Simulation Results. Mean = 0.0005; Standard Deviation = 0.006; Tick/VolatilityRatio = 1/4

introduces estimation biases (Harris, 1990). It is possible that the compass rose, i.e. the combina-tion of tick size and volatility, also introduces biases in conditionally heteroscedastic estimates,which suggests that conditional volatility forecasting using high frequency data, including, butlimited to the foreign exchange market, may be impacted. Recent research by Amilon (2003) indi-cates that corrections to GARCH models for problems caused by discreteness must be made. Ourresults indicate that it is possible to use the tick/volatility ratio to determine when discreteness isrelevant and must be accounted with corrections to a host of parametric tests applied to asset pric-ing models, which assume continuity. As Amilon (2003) points out, discreteness is not necessarily

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Figure 4D. Monte Carlo Simulation Results. Mean = 0.0005; Standard Deviation = 0.006; Tick/VolatilityRatio = 10/1

problematic when returns and volatility are forecasted for indexes or for high-priced securities,but may be when returns and volatility are forecasted for low-priced stocks or for individualfinancial instruments. These securities would meet the tick/volatility ratio, where discretenesswould impact empirical tests, and corrections to empirical tests to account for discreteness shouldbe made.

A further implication of the detection of the compass rose pattern is that it may enable tradersand analysts to improve out of sample forecasts for those time series where the pattern is evident.As an illustration, consider an analyst who fits an ARMA-GARCH model (Bollerslev et al., 1992)

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on say n returns and then forecasts the (n + 1)th out-of-sample return. Refer to the nth and (+1)hreturns as the 2-tuple (x, y). Suppose the compass rose has a 2-tuple (x ′, y ′) such that the squaredEuclidean distance between the two 2-tuples is the smallest for all 2-tuples on the compass roseand (x, y). Then, the y ′ return from the compass rose could be used as the forecast rather than they forecast from the ARMA-GARCH model. Given the numerous forecasting models utilized, itis possible that the information from the compass rose may help in improving the forecasts fromARMA-GARCH and similar models.

Finally, some markets have moved towards decimalization. For example, the New York StockExchange phased in decimalization on 28August 2000. Decimalization has resulted in smaller ticksizes and lower volatility. In the compass rose, the emergence of the pattern depends on both thetick size and the volatility. Thus, decimalization may influence the continued presence or absenceof the compass rose, indicating that market microstructure patterns that may be influenced by thecompass rose need continued further analysis.

Acknowledgements

The authors would like to thank Richard McDonald of the Chicago Mercantile Exchange, ananonymous referee and an associate editor of this journal for their helpful comments on earlierdrafts of the paper that have helped improve its quality.

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