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The timing of opening trades and pricing errors.

After demonstrating that a zero investment trading strategy that buys stocks with overnight returns below the market average and sells stocks with overnight returns above the market average earns more than 1% monthly profit, I demonstrate that this profit is greater for stocks that start trading more quickly than for other stocks. These results control for trading costs. The resulting pricing errors are a material portion of stock price volatility and suggest that a quick response to overnight information adds non-information-based stock volatility to stock prices.

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Opening prices are known to be more volatile than closing prices after controlling for the amount of information arrivals (Amihud and Mendelson, 1987; Stoll and Whaley, 1990; Stoll, 2000). This suggests a certain degree of inefficiency in opening prices which is yet to be fully understood. I consider the possibility that this behavior may be related to traders overacting to overnight information. In particular, traders who trade very quickly after the markets open are driving prices away from fundamental values by not waiting to fully incorporate all available information.

The speed of the reaction to information is often regarded as an indicator of stock market efficiency. If stock prices react slowly to information, there is an opportunity to profit from public information, thus violating semi-strong form market efficiency, (1) However, it is equally inefficient if the rapid reactions are not accurate reflections of the information given. French and Roll (1986) and Kim and Verrechia (1994) note that stock price volatility increases when investors cannot accurately estimate the true value of information. According Chan (1993), the estimation is more difficult when stocks react faster than others as investors update their estimates of value by learning from the price reactions of similar stocks. (2) With limited price data from other firms to consult, investors may incorrectly estimate the true value of the stock.

Although this idea may be an important challenge to the trend equating the speed of reaction and market efficiency, there has been little supporting empirical evidence regarding the relationship between pricing error and the speed of reaction. I explore this question by examining opening trades from September 1997 to January 2001 when the NASDAQ had a continuous trading structure at the market open instead of the opening call auction process that has been used for other trading periods. Within this period, I can observe the exact sequence of opening trades after a long period of non-trading hours in which overnight information had been accumulated. I use data from this period to examine how accurately investors interpret overnight information and how the time of the opening trade of a stock is related to its pricing error.

I find two notable patterns in the NASDAQ opening prices. First, opening prices exhibit economically significant pricing errors. The pricing errors are estimated using a simple strategy of buying stocks whose overnight return (close to opening trade) is lower than the equal-weighted market return and selling stocks whose overnight return is higher than the market return. This strategy enables me to estimate the amount of mean reversion of a stock opening price after controlling for the overall market movement. I find that this strategy yields more than 1% return per month, after controlling for bid-ask spread and scarce trading issues. More importantly, the stocks that traded sooner after the market opening exhibit greater mean reversion. The difference in mean reversion between the fastest reaction groups and the slowest reaction groups is 0.54% per day. I find that the significant correlation between market-adjusted mean reversion and the speed of opening exists even after controlling for standard liquidity variables such as firm size, bid-ask spread, and volume. This pattern indicates that earlier opening increases significant errors in stock prices.

This paper adds to the existing evidence related to trading and differences of opinion. Chan (1993) and Kim and Verrechia (1994) model a process

whereby stock returns overshoot or underreact after news due to differences of opinion, and stock price volatility increases as a result. In a similar context, Banerjee and Kremer (2010) argue that trading activity can be positively correlated with stock price volatility as trading activity is an indicator of differences of opinion among investors. These differences of opinion models argue that a large part of stock price volatility may result from differences in interpreting the given information implying that investors' views affect stock price volatility. The models also indicate some irrational behaviors, such as a stock price bubble, are possible especially when investors are interpreting information in a biased way. There is empirical evidence (mostly indirect) to support the predictions of those models, but as Banerjee and Kremer (2010) acknowledge, factors, such as market friction, may also explain the anomalies.

Similarly, mean reversion in stock prices has been explained by trading friction. Lo and MacKinlay (1990) find that the returns on small stocks follow the returns of small stocks. Chordia and Swaminathan (2000) and Gervais, Kaniel, and Mingelgrin (2001) confirm that the returns of low volume stocks follow the returns of high volume stocks. These papers argue that small stocks and low volume stocks receive common information later than other stocks and demonstrate mean reversion as a result. This paper's finding that faster reaction increases mean reversion contradicts this explanation.

My findings also address the larger volatility of opening prices documented by Amihud and Mendelson (1987), Stoll and Whaley (1990), and Stoll (2000). This paper's results indicate that abnormally high volatility in opening prices may exist because traders do not have sufficient prior trading data from other stocks at market open. As a result, their interpretations of overnight information contain more noise. In a similar context, Cao, Ghysels, and Hathaway (2000) find that pre-opening quotes can reduce the errors in opening prices.

The rest of paper is organized as follows. Section I discusses hypotheses and empirical methods. Section II measures the overall amount of errors in opening prices. Section III tests the correlation between reaction speed and error in prices. Section IV examines the association between error and stock price volatility. Section V provides my conclusions.

I. Hypothesis and Method

The main hypothesis of this paper is that pricing error increases with the speed of trading. This hypothesis is developed from the difference of opinion literature. I use the assumptions of French and Roll (1986) and Kim and Verrechia (1994) that investors make imperfect estimations of the true value of given information. Further, I follow Chan (1993) in that information is common across the stock market and investors consult the prices of other stocks to update their estimations of the information given. Investors' estimates would be less accurate if investors do not have many prices to consult. As a result, a trade that is made before other stocks are traded would include more errors.

The model of Chan (1993) assumes that the equal-weighted market-wide return is an unbiased estimation of information (since it is the cross-sectional mean). Stock prices that moved above (below) the market return should have a subsequent return below (above) the market return to correct their errors. (3) 1 measure whether stock prices revert to market return after an initial reaction (trade) to the arrival of information and if the degree of reversion is positively related to the speed of the reaction. I define mean reversion as stock prices moving toward the market-wide return after deviating from it. Figure 1 illustrates the mean reversion.

Strong assumptions of this model are that information is market-wide and has the same effect on all stock prices. A test based on these assumptions ignores individual differences in information sets and their effects. If these individual differences are distributed without bias, however, one would be able to find the predicted pattern from market-wide data.

Another obstacle to testing this hypothesis is that the initial stock price reaction to an information event is difficult to identify. For example, one could use prices after an earnings announcement by GM to observe price reactions to the announcement. However, with cross-consulting, a researcher cannot tell whether a change in the price of Ford stock is a reaction to GM's earnings announcement or to an hour earlier news piece regarding increasing gasoline prices. To address this issue, I use NASDAQ market opening prices from September 1997 to January 2001. During this period, the NASDAQ did not use a regular call auction process at the open. Instead, it used a continuous trading system, so that the first transaction after 9:30 a.m. was the opening price. These opening prices are the (non-complied) first reactions to overnight information. I can observe when these reactions occurred from the time stamp on the transactions. I use the opening prices to study the correlation between the timing of the initial trade and pricing errors.

Opening price data come from the Financial Markets Research Center (FMRC) at Vanderbilt University. The FMRC maintains a daily market microstructure database that is constructed from NYSE Trade and Quote (TAQ) data. The database includes all stocks in the TAQ except those with daily prices below $3. The database includes daily variables compiled from TAQ, such as the time of the opening trade, the average bid-ask spread, and the dollar volume. The FMRC data set provides the opening price, the closing price, and the noon price. The opening price is the first traded price after the official market opening at 9:30 a.m. EST. The closing price is the last traded price before the official market closing at 4:00 p.m. EST. The noon price is the traded price closest to 12:00 p.m. EST. If a stock does not have more than ten trades during a given day, I drop that day's observation. This filter ensures that I use stocks with considerable trading activity and reduces the problem of stale prices. Additionally, I control for stock splits and stock dividends by deleting the returns in the window [-1, +1] of the event date.

I measure the overnight return as the return between the Day t-1 closing price and the Day t opening price. The morning return is the return between today's opening price and today's noon price. Figure 2 is a schematic of the types of returns I use.

The pricing error of opening prices is measured from the mean reversion of the overnight return. If we assume that reversal occurs in the first several hours after the market opens, it will be captured in the return from open to noon. This assumption is based on Masulis and Shivakumar (2002) who document that it takes approximately one to three hours for NASDAQ prices to fully reflect the value of seasoned equity offering announcements.

II. Mean Reversion of Opening Prices

The goals of the series of tests performed in this section are to measure the overall degree of mean reversion in overnight prices. First, I measure the frequency of the cross- sectional mean reversion in overnight returns. The following equation measures whether a stock's overnight return tends to converge with the market-wide return during the morning trading hours:

[r.sup.i.sub.morning] = [alpha] + [lambda] x ([r.sup.i.sub.overnight] - [[bar.r].sub.overnight]) + [[gamma].sub.1] x [[bar.].sub.morning]. (1)

* [r.sup.i.sub.morning] is a stock's return between today's first price and the noon price.

* [[bar.r].sub.morning] is the market average morning return.

* [r.sup.i.sub.overnight] is a stock's return between yesterday's closing

price and today's opening price.

* [[bar.r].sub.overnight] is the market average overnight return.

To calculate the market-wide average overnight return of each day, I use all of the stocks in my sample that had their first transaction within one minute after the market opens. This selection is made to minimize the overlap between the period of market average calculation and the period of mean reversion estimation. If there is a cross-sectional convergence, the sign of coefficient ,k should be negative. I estimate Equation (1) for each stock using all available time-series observations, except in the cases in which an opening price has a time stamp later than 10:00 a.m. I then count the stocks with negative [lambda].

Panel A of Table I reports that 88% of the stocks have negative [lambda] indicating that there is a greater tendency for stock returns to converge with the market-wide average. A simple sign test confirms that this pattern did not occur by chance. Panel B indicates the [lambda] coefficients that are significant at the 5% level. Again, 65% of the coefficients are negative and significant, while only 2% of the coefficients are positive and significant. Stocks that had lower overnight returns than the market average have a strong tendency to have higher morning returns. Stocks with higher overnight returns have a tendency toward lower morning returns. This convergence implies that stock returns continuously overreact or underreact to overnight returns and then converge back to the market average.

Next, to estimate the degree of mean reversion, I develop a feasible zero investment strategy that uses the reversion pattern. The amount of profit acquired from the trading strategy will be a measure of the mean reversion and the pricing error in opening prices. To make the trading strategy feasible, the market-wide average overnight return of a given day is calculated from the stocks that had opening transactions in the first minute after the market opening on that day. This setting allows traders to get an idea about the market average overnight return before constructing their positions. I assume a trader starts buying or selling the stocks that had their first transactions three minutes after the market opens. (4) Therefore, a trader can calculate the market-wide average in the first minute, digest the prior price data until two minutes from open, and then purchase the stocks that moved below the estimated market-wide average and sell the stocks that moved above the average price after a minute of trading. One can do programmed trading based on this logic, and such automated trading procedures will take less time to execute than this paper's setting. The trader puts equal weight in two (buy/sell) portfolios and the stocks in each portfolio also have equal weights. The trader expands their position until 10:00 a.m., holds the position for a few hours, and then clears the position at the noon trading price. The return is measured by comparing the price change between the opening price and the noon price (a simple arithmetic return). By design, the return of the trading strategy is the average open-to-noon returns that moved in the opposite direction of (mean reversion) their overnight returns.

Table II reports the result of trading. Profit is measured at a daily level and averaged during a year. Panel A reports raw profit, while Panel B provides the profit from mid- quote returns. In Panel C, I assume that the investor only trades those stocks that deviated more than their previous day's bid-ask spread from the market average to avoid trading on bid-ask bounce. Additionally, I subtract the stock's daily bid-ask spread divided by the stock price from each stock's return in Panel C. Thus, Panels B and C demonstrate profits after controlling for the bid- ask spread in two different ways.

Panel A reports that trading earns a daily profit in a range from 0.7% to 1.2%. Panels B and C also determine that zero investment trading strategy demonstrates significant profit after controlling for the bid-ask spread. The trading strategy indicates a daily profit of approximately 0.3% in Panel B and 0.4% in Panel C. All of the numbers are significantly different from zero at the 1% level. By comparing raw returns to mid-quote and spread controlled returns, one can see that the bid-ask spread is not creating the mean reversion pattern. The profit easily sums to a 1% monthly profit assuming a 20-day trading period per month. This result indicates that stocks go through a significant pattern of overshooting (or underreacting) and then converging every day. Stock price volatility would be increased due to this pattern. Note that Jegadeesh (1990) finds a 2.49% monthly return from short-term mean reversion, but his data period had much larger bid-ask spreads, which may offset the reversion effects. Table II also finds that the degree of mean reversion is not a one-time phenomenon in the earlier years. Rather, the bid-ask spread controlled results in Panels B and C are stable over time. This pattern indicates that the mean reversion occurred consistently throughout the sample period.

III. Speed of Reaction and Degree of Mean Reversion

After establishing that there is a significant amount of mean reversion in opening prices, I test the main hypothesis. That is, the relationship between reaction speed and the size of the mean reversion. The reaction speed to overnight information can be calculated by the time between the official market opening (9:30 a.m.) and the actual time when a stock first traded (opening delay). I rank the opening delay into deciles. The degree of mean reversion is measured by the profit of the zero investment strategy described in the previous section. A trader waits one minute to calculate the market-wide average and then starts trading from minute three. Note that I do not use those stocks that had their first transaction after 10:00 a.m. or stocks that had fewer than ten trades during that day. I also control for the bid-ask spread by adopting the approach in Table II, Panel C. The investor only trades stocks that deviated more than its previous day's bid-ask spread from the market average. Then, a stock's daily bid-ask spread is subtracted from each stock's return. I find qualitatively similar results by using mid-quote returns, which are another way of controlling bid-ask spread. As such, the tables based on mid- quote returns are not reported.

Table III determines that the mean reversion is larger when stocks open faster than others. The zero investment strategy earns a 0.74% spread controlled return from the stocks that open the fastest, but its return drops to 0.20% for those stocks that open the slowest. (5) The difference is 0.54% daily and it is statistically significant at the 1% level. There is almost a monotonic correlation between the speed of the trade and the degree of the mean reversion.

I expand my analysis by first classifying by firm size and then classifying by the opening delay. This type of classification is also used by Chordia and Swaminathan (2000) and Gervais et al. (2001). The logic of these classifications is that size is correlated with a stock's response time to information (Lo and MacKinlay, 1990), and size is related to the amount of information available (Brennan and Hughes, 1991). Therefore, controlling for size would demonstrate a clearer effect of opening delay. I classify stocks into size quartiles using the previous month's market value. Then, in each size quartile, I classify by the opening delay quartile, following the same method as in Table III. This process yields 4 x 4 = 16 clusters, and I calculate the spread controlled profit of the trading strategy for each cluster.

Table IV demonstrates that after controlling for size, the profit of the investment strategy is still greater when a stock reacts faster to overnight information. For all size quartiles, I observe that profit is the largest in the fastest opening stocks, while profit is smallest in the slowest opening stocks. This result confirms that the convergence pattern is more consistent with the difference of opinion models. Also, I find that the relationship between reaction time and profit size is not strongest in the smallest size quartile. The difference is largest for Size Quartile 3 (0.82%), which is the next largest quartile. The smallest size quartile reports a 0.33% difference and the largest size quartile indicates a 0.14% difference. I find that all of the differences are statistically significant at the 10% level. These results confirm that the convergence pattern is not restricted to small stocks that may be infrequently traded or that have abnormally high bid-ask spreads.

I also report multi-factor regression results on the correlation between speed and mean reversion. The purpose of this analysis is to see whether the relationship still holds after controlling for other known factors of mean reversion. Lo and MacKinlay (1990) and Brennan and Hughes (1991) find that firm size is a factor of mean reversion, while Chordia and Swaminathan (2000) and Gervais et al. (2001) determine that stock volume is a factor of mean reversion. Those papers concur that large firm stocks with higher volume react faster to information, and argue those stock prices are, on average, more accurate. This view is in opposition to the implication from this paper's hypothesis. It would be worthwhile to test whether this paper's results still hold after considering all of those factors together.

Let the mean reversion of a stock be the size of the open-to-noon return in the opposite direction of the overnight return. If the open-to-noon return is in the same direction as the overnight return, it would be a drift rather than a mean reversion and it would be recorded as a negative mean reversion. Thus, I introduce an indicator variable that is positive (negative) one if the sign of open-to-noon return is the opposite of (same as) the overnight return. I multiply this indicator variable and the absolute value of the open-to-noon return to measure the size of an opposite direction movement.

mean reversion = I x |Open - to - noon return], (2)

where I = 1 if the open-to-noon return has the opposite sign of the overnight return and I = -1 if the open-to-noon return has the same sign as the overnight return

The basic regression equation is:

Mean [reversion.sub.i,t] =a + b1 x [Open_time.sub.i,t] + b2 x [Firm_size.sub.i,t] + b3 x [Dollar_volume.sub.i,t] + [[epsilon].sub.i,t]. (3)

Open time is the time stamp when the first trade occurred. To measure firm size, I calculate the daily market value of a stock as the product of the daily closing prices and the number of shares outstanding. The daily market value is averaged by every calendar month to reduce the effect of daily volatility in closing prices. The firm size variable in the equation is an average market value of a firm during a month in which the trading occurred. Dollar volume is the product of the daily volume and the daily average price. As a control variable, I divide the bid-ask spread by the stock price and include it in the regression equation. I also insert a dummy variable for the days that had trading halts. This is to see if extreme volatility in a few trading days is driving my result. I use ordinary least square (OLS) estimation with heteroskedasticity-robust errors.

Table V confirms that the correlation between reaction speed and mean reversion is highly significant after size and volume control. The negative coefficient on the open time variable indicates that a stock trading late has less mean reversion. The coefficient on reaction speed is much more significant (t-stat -11.61) when compared to size (t-stat 4.21) or volume (t-stat -4.80). The regression results indicate that the mean reversion of fast trading stocks is a different phenomenon when compared to size or volume related mean reversions. (6)

IV. Non-Information-Related Volatility and Convergence

In this section, I test the relationship between mean reversion and non- information-based volatility. The quality of overnight information varies by stock and if it is the quality that determines the degree of mean reversion, I should see relatively little correlation between noninformation-based volatility and the degree of mean reversion. I borrow a clever measure of non-information-based volatility from the literature regarding opening price volatility. Amihud and Mendelson (1987) and Stoll and Whaley (1990) measure non-information-based volatility in opening prices. Their method compares the stock returns from two consecutive opening prices with the stock returns from two consecutive closing prices. Let the opening return be the return between two consecutive opening prices and let the closing return be the return between two consecutive closing prices. Figure 3 illustrates two returns.

Then, monthly standard deviations of opening returns and closing returns are calculated. The opening return standard deviation divided by the closing return standard deviation is dubbed the "opening friction" in Stoll (2000). The opening friction measure is a non- information-related volatility as the two standard deviation measures share the same 24-hour amount of information. By utilizing the ratio, the information contents cancel out. Amihud and Mendelson (1987) and Stoll and Whaley (1990) find that the ratio is approximately 1.2 indicating that the opening return standard deviation is 20% higher than the closing return standard deviation after controlling for information. Although the existence of this phenomenon is documented, there is not much evidence regarding what is causing this higher volatility of the opening return. This paper may yield some clues about the phenomenon as this higher volatility is not surprising in the difference of opinion framework. One may assume that the opening price is the first reaction to overnight information, and the differences of opinion about that information would be the largest after long non-trading hours. The closing price has fewer differences of opinion as there has been continuous trading before the price is set. Therefore, I examine whether the time of the opening trade is also associated with the variations in the amount of opening friction.

I check the correlation between the size of convergence and the opening friction measure, and I classify the sample into opening friction deciles and report the spread controlled profit of the zero investment strategy.

Table VI reports that the spread controlled profit increases with the size of opening friction, a measure of non-information-related volatility. In the largest opening friction deciles, profit is 0.88%, while profit is 0.03% in the smallest opening friction deciles. Thus, the convergence documented in this paper is strongly related to non-information-related volatility. The strength of the correlation indicates that the convergence pattern is a significant source of stock price volatility. (7) The results of this paper also indicate that the opening friction puzzle is related to differences of opinion about given information. Opening prices have more volatility as investors have less opportunity to learn from other prices and update their estimations using overnight information.

V. Summary and Conclusion

This paper documents a significant daily mean reversion pattern in stock returns. Stocks that moved higher than the market average at opening have more negative returns afterward, while stocks that moved lower than the market average at opening have more positive returns. An investor can create a feasible zero investment strategy that can yield a 1% return per month after controlling for the bid-ask spread and scarce trading. The size of the convergence is actually stronger for the stocks that react faster to overnight information. I further demonstrate that this convergence is strongly correlated with non-information based volatility. Together, these results imply that there is a high degree of differences of opinion in overnight returns, and the differences of opinion and resulting volatility are both increasing with the speed of trading.

One of this paper's contributions is to empirically determine that the speed of reaction can increase errors in stock prices. Differences of opinion models point out that fast trades are made without sufficient price information from other stocks. As a result, traders have more differences of opinion regarding existing information. The difficulty in interpreting this information leads to higher pricing errors in stock prices. I find that this interpretation problem generates a strong mean reversion pattern, and that pattern is more prominent for those stocks that are trading more rapidly. My mean reversion pattern is different from other documented mean reversion patterns such as Lo and MacKinlay (1990), Chordia and Swaminathan (2000), and Gervais et al. (2001) in that faster reaction generates stronger mean reversions.

Another contribution of this paper is to find new empirical evidence regarding differences of opinion models. The results in this paper are consistent with the differences of opinion models in Chan (1993), Kim and Verrechia (1994), and Banerjee and Kremer (2010). Also, this paper explains why opening friction exists in the stock market. The opening friction, named by Stoll (2000), is the abnormally high volatility of opening prices.

Taken together, my results suggest that one contributor to volatility in stock prices is the difficulty in converting qualitative information into a quantitative price.

References

Amihud, Y. and H. Mendelson, 1987, "Trading Mechanisms and Stock Returns: An Empirical Investigation," Journal of Finance 42, 533-553.

Banerjee, B. and I. Kremer, 2010, "Disagreement and Learning: Dynamic Patterns of Trade," Journal of Finance 65, 1269-1302.

Brennan, M. and R Hughes, 1991, "Stock Prices and the Supply of Information," Journal of Finance 46, 1665-1691.

Cao, C., E. Ghysels, and E Hathaway, 2000, "Price Discovery Without Trading: Evidence from the NASDAQ Preopening," Journal of Finance 55, 1339-1365.

Chan, K., 1993, "Imperfect Information and Cross-Autocorrelation Among Stock Prices," Journal of Finance 48, 1211-1230.

Chordia, T. and B. Swaminathan, 2000, "Trading Volume and Cross-Autocorrelations in Stock Returns," Journal of Finance 55,913-935.

Evans, M. and R. Lyons, 2008, "How is Macro News Transmitted to Exchange Rates?" Journal of Financial Economics 88, 26-50.

French, K. and R. Roll, 1986, "Stock Return Variances: The Arrival of Information and the Reaction of Traders," Journal of Financial Economics 17, 5-26.

Gervais, S., R. Kaniel, and D. Mingelgrin, 2001, "The High Volume Return Premium," Journal of Finance 56, 877-919.

Jegadeesh, N., 1990, "Evident of Predictable Behavior of Security Returns," Journal of Finance 45, 881-898.

Kim, O. and R. Verrecchia, 1994, "Market Liquidity and Volume Around Earnings Announcements," Journal of Accounting and Economics 17, 41-67.

Lo, A. and C. MacKinlay, 1990, "When are Contrarian Profits Due to Stock Market Overreaction?" Review of Financial Studies 3, 175-205.

Masulis, R. and L. Shivakumar, 2002, "Does Market Structure Affect the Immediacy of Stock Price Response to News?" Journal of Financial and Quantitative Analysis 37, 617-648.

Stoll, H., 2000, "Friction," Journal of Finance 55, 1479-1514.

Stoll, H. and R. Whaley, 1990, "Stock Market Structure and Volatility," Review of Financial Studies 3, 37-71.

This project began as part of my Ph.D. dissertation at the Owen Graduate School of Management, Vanderbilt University. I wish to especially thank my dissertation chair, Hans Stoll, and other dissertation committee members for their excellent guidance and generous support. I acknowledge the data support from the Financial Markets Research Center (FMRC) of Vanderbilt University. I am grateful to Marc Lipson (Editor) and an anonymous referee for their suggestions and comments. I would also like to express my gratitude to Richard Smith for his comments.

(1) Lo and MacKinlay (1990), Chordia and Swaminathan (2000), and Gervais, Kaniel, and Mingelgrin (2001) find that some stocks react slower than others.

(2) French and Roll (1986) find the volatility arising from differences in interpreting given information is a minor part of the overall stock price volatility. Perhaps this was true in relatively thin markets in the earlier days, but as bid-ask spreads have become smaller and the order execution process has become faster, the differences of opinion may now have a significant effect on stock volatility. Recently, Evans and Lyons (2008) determine that differences in the interpretation of macroeconomic information can account for one-third of foreign exchange market volatility.

(3) Most investors cannot reach the market-wide return with their initial trades because the market-wide return may be calculated only after a sufficient number of individual reactions have occurred.

(4) A stock must have a transaction in the first 30 minutes of the market opening in order to be included in this analysis.

(5) I acquire qualitatively similar or stronger results by using raw returns or mid-quote returns.

(6) I find that the mean reversion is somewhat stronger in large stocks, contrary to the earlier findings that prices of large stocks are more accurate. This result may be due to the fact that overall market liquidity is much higher in recent years and small stocks do not necessarily suffer from the scarce trading problem as before. Also, Model 4, which is without bid-ask spread control, demonstrates negative coefficients on size.

(7) Note that because I am using all of the stocks in the TAQ data, it is unlikely that peculiar overnight information on a handful of stocks will affect the whole result. Differences in information would cancel out if the convergence is at the market-wide level.

Sukwon Thomas Kim *

* Sukwon Thomas Kim is an Assistant Professor of Finance in the A. Gary Anderson Graduate School of Management at the University of California in Riverside, CA.

Table I. Mean Reversion Frequency

I run the following regression for each stock every year and report
the sign of lambda.

Model: [r.sup.I.sub.morning] = [alpha] + [lambda] x
([r.sub.i.sub.overnight] - [[bar.r].sub.overnight]) + [[gamma].sub.l]
x [[bar.r].sub.morning.

* [r.sup.i.sub.morning] is a stock's return between today's
first price and noon price.

* [[bar.r].sup.i.sub.morning] is the market average of morning
returns.

* [r.sup.i.sub.overnight] is a stock's return between previous
day's closing price and today's opening price.

* [[bar.r].sup.i.sub.morning] is the market average of
overnight returns.

The average overnight return is calculated from the overnight returns
of the stocks that opened in the first minute after the market
opening. The sample is comprised of NASDAQ stocks from September 1997
to January 2001. Panel A reports all negative and positive
coefficients. The sign test indicates the probability of acquiring the
observed number of signs if there is a 50-50 chance to have either a
positive or negative sign. Panel B provides the coefficients
significant at the 5% level.

Panel A. Number of Negative and Positive lambda Signs

                                       Cumulative
                 Frequency   Percent   Frequency

Negative Signs     5,917     87.72%      5,917
Positive Signs      828      12.28%      6,745

Panel B. Number of lambda Signs Significant at the 5% Level

                                       Cumulative
                 Frequency   Percent   Frequency

Negative Signs     4,376     64.88%      4,376
Positive Signs      167       2.48%      4,543
Insignificant      2,202     32.64%      6,745

Panel A. Number of Negative and Positive lambda Signs

                 Cumulative    Sign Test
                  Percent     Significance

Negative Signs     87.72%         < 1%
Positive Signs    100.00%         < 1%

Panel B. Number of lambda Signs Significant at the 5% Level

                 Cumulative
                  Percent

Negative Signs     64.88%
Positive Signs     67.36%
Insignificant     100.00%

Table II. A Trading Strategy Using a Convergence Pattern

First, an investor calculates an average of overnight returns using
those stocks opened from the market opening until 9:31 a.m. Then,
among those stocks that opened later than two minutes after the
market opening, the investor purchases those stocks that open below
the average and sells stocks that open above the average. The
investor puts equal weight in buying and selling to achieve a zero
investment. They accumulate the position until 10:00 a.m. and
liquidate it at the noon trading price. The following table reports
the average daily profit from this trading strategy. The sample
includes NASDAQ stocks from September 1997 to January 2001.

                   Daily Return from          Daily Return from
                   Stock that Opened          Stock that Opened
Year            Above Market Average (A)   Below Market Average (B)

Panel A. Raw Profit By Calendar Year

1997 (from               -0.67%                     0.51%
  September)
1998                     -0.88%                     0.37%
1999                     -0.64%                     0.24%
2000                     -0.83%                     0.09%
2001 (January            -0.19%                     0.46%
  only)

Panel B. Mid-Quote Profit By Calendar Year

1997 (From               -0.03%                     0.21%
  September)
1998                     -0.33%                     0.03%
1999                     -0.19%                     0.05%
2000                     -0.59%                     0.01%
2001 (January             0.02%                     0.27%
  only)

Panel C. Profit After Subtracting Bid-Ask Spread

1997 (from               -0.09%                     0.29%
  September)
1998                     -0.24%                     0.26%
1999                     -0.07%                     0.22%
2000                     -0.37%                     0.17%
2001 (January            -0.09%                     0.34%
  only)

                  Daily Profit
                    from the
Year            Position (B - A)

Panel A. Raw Profit By Calendar Year

1997 (from           1.18%
  September)
1998                 1.25%
1999                 0.88%
2000                 0.92%
2001 (January        0.65%
  only)

Panel B. Mid-Quote Profit By Calendar Year

1997 (From           0.24%
  September)
1998                 0.36%
1999                 0.24%
2000                 0.60%
2001 (January        0.25%
  only)

Panel C. Profit After Subtracting Bid-Ask Spread

1997 (from           0.37%
  September)
1998                 0.50%
1999                 0.29%
2000                 0.54%
2001 (January        0.43%
  only)

Table III. Mean Reversion and Reaction Speed

I employ NASDAQ stocks from September 1997 to January 2001. I
measure a stock's opening delay as the difference between the first
transaction time and 9:30 a.m. when the market officially opens. I
do not use stocks whose first transaction occurred after 10:00 a.m.
or stocks that had fewer than ten trades per day. I rank the
opening delay of stocks into deciles and report the profit of the
trading strategy. I assume the investor only trades a stock whose
overnight return deviated more than the previous day's bid-ask
spread from the market-wide return. I also subtract daily bid-ask
spreads from every return. The last row compares the returns of the
fastest decile and slowest decile. The standard error of the
difference can be found in parentheses.

Deciles of         Daily Return from          Daily Return from
Opening            Stock that Opened          Stock that Opened
Delay           Above Market Average (A)   Below Market Average (B)

1 (Fastest)              -0.43%                     0.31%
2                        -0.22%                     0.35%
3                        -0.30%                     0.27%
4                        -0.40%                     0.30%
5                        -0.21%                     0.25%
6                        -0.26%                     0.16%
7                        -0.26%                     0.24%
8                        -0.13%                     0.16%
9                        -0.17%                     0.18%
10 (Slowest)             -0.05%                     0.15%
Difference              -0.38% *                   0.16% *
  Between               (0.10%)                    (0.05%)
  Fastest and
  Slowest

Deciles of        Daily Profit
Opening             from the
Delay           Position (B - A)

1 (Fastest)          0.74%
2                    0.57%
3                    0.57%
4                    0.70%
5                    0.46%
6                    0.42%
7                    0.50%
8                    0.29%
9                    0.35%
10 (Slowest)         0.20%
Difference          0.54% *
  Between           (0.11%)
  Fastest and
  Slowest

* Significant at the 0.10 level.

Table IV. Mean Reversion and Reaction Speed--Size Stratified Result

My sample is comprised of NASDAQ stocks from September 1997
to January 2001. I measure a stock's opening delay as the
difference between the first transaction time and 9:30 a.m. when
the market officially opens. I do not use stocks whose first
transaction occurred after 10:00 a.m. or stocks that had fewer than
ten trades per day. First, I rank a stock's market value into
quartiles. Then, within each size quartile, I rank by opening delay
quartiles. I report the profit of the trading strategy for each
cluster. I assume the investor only trades a stock whose overnight
return deviated more than a previous day's bid-ask spread from the
market-wide return. I also subtract daily bid-ask spreads from
every return.

                              Daily Return from
Quartiles of    Quartiles     Stock that Opened
Market          of Opening      Above Market
Value             Delay          Average (A)

1 (Smallest)   1 (Fastest)         -0.06%
1 (Smallest)   2                   -0.04%
1 (Smallest)   3                   -0.03%
1 (Smallest)   4 (Slowest)          0.03%

2              1 (Fastest)         -0.43%
2              2                   -0.27%
2              3                   -0.25%
2              4 (Slowest)         -0.16%

3              1 (Fastest)         -0.46%
3              2                   -0.39%
3              3                   -0.23%
3              4 (Slowest)          0.20%

4 (Largest)    1 (Fastest)         -0.34%
4 (Largest)    2                   -0.32%
4 (Largest)    3                   -0.24%
4 (Largest)    4 (Slowest)         -0.27%

               Daily Return from    Daily Profit
Quartiles of   Stock that Opened      From the
Market            Below Market        Position
Value             Average (B)          (B-A)

1 (Smallest)         0.47%              0.53%

1 (Smallest)         0.31%              0.35%
1 (Smallest)         0.32%              0.35%
1 (Smallest)         0.22%              0.20%

2                    0.37%              0.80%
2                    0.27%              0.54%
2                    0.14%              0.39%
2                    0.18%              0.34%

3                    0.26%              0.72%
3                    0.22%              0.61%
3                    0.20%              0.43%
3                    0.10%             -0.10%

4 (Largest)          0.24%              0.58%
4 (Largest)          0.19%              0.51%
4 (Largest)          0.18%              0.42%
4 (Largest)          0.17%              0.44%

Table V. Determinants of Mean Reversion - A Regression Approach

Mean reversion = I x Open - to - noon return, (2)

where I = 1 if the open-to-noon return has the opposite sign
of overnight return and I = -I if the open-to
noon return has the same sign of overnight return.

[Mean reversion.sub.i,t] = a + b 1 x [Open_time.sub.i,t] + b2 x
[Firm_size.sub.i,t] + b3 x [Dollar_volume.sub.i,t] +
[[epsilon].sub.i,t]. (3)

I use a multivariate regression model to test the correlation between
the speed of trading and the degree of mean reversion. I control for
known determinants of mean reversion such as firm size (market value),
stock dollar volume, and bid-ask spread. The mean reversion of
overnight return is defined as the subsequent open-to-noon return
in the opposite direction of the overnight return. t-values are
in parentheses.

Variables                         Model 1     Model 2

Time of the opening trade         -1.64 *     -1.70 *
                                 (-11.61)    (-11.86)

Firm size (market value)           0.74 *      0.75 *
                                   (4.21)      (4.24)

Dollar volume                     -0.25 *     -0.24 *
                                  (-4.80)     (-4.79)

Bid-ask spread to stock price      0.23 *      0.24 *
                                  (12.17)     (12.53)

Trading halt dummy                  -0.48       -0.50
                                  (-1.11)     (-1.17)

Year fixed effects                  Yes         No

Observations                      388,227     388,227

Adjusted [R.sup.2]                  0.39%       0.37%

Variables                         Model 3     Model 4

Time of the opening trade         -1.71 *     -1.56 *
                                 (-11.91)    (-10.67)

Firm size (market value)           0.78 *     -0.68 *
                                   (4.32)    (-3.12)

Dollar volume                     -0.25 *     -0.29 *
                                  (-4.96)     (-5.05)

Bid-ask spread to stock price      0.24 *
                                  (12.52)

Trading halt dummy

Year fixed effects                  No          No

Observations                      388,227     390,113

Adjusted [R.sup.2]                  0.37%       0.13%

* Significant at the 0.10 level.

Table VI. Correlation Between Mean Reversion and
Non-Information-Related Volatility

I define opening return as the return between two consecutive
opening prices and closing return as the return between two
consecutive closing prices. I use the monthly standard deviation of
opening returns and closing returns. The opening return standard
deviation divided by the closing return standard deviation is the
"opening friction." I report the spread controlled profit of the
trading strategy by opening friction deciles. The last row compares
the returns of smallest decile and largest decile. The standard
error of the difference is in parentheses.

Deciles of              Daily Return from      Daily Return from
Opening                 Stock that Opened      Stock that Opened
Friction                Above Market A(A)   Below Market Average (B)

1 (Smallest)                  0.23%                  0.26%
2                             0.20%                  0.11%
3                             0.00%                  0.13%
4                             0.00%                  0.16%
5                            -0.13%                  0.24%
6                            -0.27%                  0.26%
7                            -0.47%                  0.26%
8                            -0.52%                  0.29%
9                            -0.54%                  0.30%
10 (Largest)                 -0.62%                  0.27%
Difference Between           -0.85% *                -0.01%
  Fastest and Slowest        (0.11%)                (0.07%)

Deciles of                Daily Profit
Opening                     from the
Friction                Position (B-A)

1 (Smallest)                 0.03%
2                           -0.09%
3                            0.13%
4                            0.16%
5                            0.37%
6                            0.53%
7                            0.73%
8                            0.81
9                            0.84%
10 (Largest)                 0.89%
Difference Between           0.86% *
  Fastest and Slowest       (0.12%)

* Significant at the 0.10 level.
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Author:Kim, Sukwon Thomas
Publication:Financial Management
Article Type:Statistical data
Geographic Code:1USA
Date:Sep 22, 2013
Words:7252
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