Tick Size and Limit Order Execution: An Examination of Stock Splits
By Tom M. Arnold Department of Finance Kelly School of Business University of Indiana 10th and Fee lane Bloomington, IN 47405-1701 and Marc L. Lipson Department of Banking and Finance Terry College of Business University of Georgia Athens, GA 30602-6253
October 1997
We would like to thank James Angel, Shane Corwin, Jason Greene, and Jeff Netter for helpful comments. All remaining errors are the joint property of the authors. Contact Marc L. Lipson by mail or as follows: email at
[email protected]; FAX at (706) 542-9434; phone at (706) 5423644. Please do not quote without permission.
Tick Size and Limit Order Execution: An Examination of Stock Splits
Abstract We study the effects of tick size on limit order executions by examining the proportion of limit orders executed on NYSE/AMEX listed stocks around stock splits. Like recent changes which allow the major U.S. exchanges to quote prices on sixteenths, stock splits alter the pricing grid available to market makers. We explore whether this change affects market liquidity through its impact on the willingness of traders to submit limit orders. We find that both total trading volume and limit order executions increase subsequent to a stock split. More importantly, the proportion of limit orders increases significantly and substantially. For example, the proportion of share volume identified as limit orders increases by about 45 percent following two-for-one stock splits. The percentage increase in limit order activity is also increasing in trade size - the proportion more than doubles for orders over 5,000 shares - and suggests that sophisticated traders are more sensitive to differences in pricing structures. As expected, the changes are more pronounced for larger stock splits. Our results are consistent across a range of methodologies – we examine the time-series of mean and median proportions, changes in individual firm proportions, pairs of trading days matched on trading activity and price changes, and the results of a logistic regression analysis.
Tick Size and Limit Order Execution: An Examination of Stock Splits 1. Introduction Competition between equity markets has grown fierce as the world’s markets compete for ever-growing trading volume and a wave of new equity issues. This competition is nowhere more apparent than in the United States where sweeping new regulations and exchange initiated reforms are rapidly transforming the markets. With the recent reductions in minimum trading increments (tick size) by all three major equity markets in the United States, the effect of tick size on trading behavior and market quality has become an area of particular interest.1 The simplest argument in favor of reduced tick sizes is that finer pricing grids allow competing market makers to set the lowest possible spreads. Relaxing the pricing constraint, particularly for those securities trading near the minimum allowable spread, then reduces trading costs and improves liquidity.2 Offsetting this advantage are a number of less easily articulated costs. For example, where unrestrained price competition leads to spreads which compensate market makers for variable costs, a coarse pricing grid generates revenue to cover fixed investments and therefore encourages more extensive market making. This increase in
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Barclay, Christie, Harris, Kandel and Schultz (1997) provide a detailed study of Securities and Exchange Commission initiated market reforms targeted toward Nasdaq trading. Weaver (1996), Bacidore (1997), and Harris (1997) discuss issues related to changes in relative tick sizes, often referred to by the term “decimalization” though decimalization is only one possible change in tick sizes. 2 Peake (1995) provides arguments along these lines while Harris (1994a) evaluates the effect of grid size of pricing. Chordia and Subrahmanyam (1995) also suggest that smaller price grids will reduce payment for order flow, though Battalio and Holden (1996) argue that this may not be the case.
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participation by market makers improves market liquidity.3 Coarse pricing grids also facilitate negotiation by limiting the number of alternatives considered.4 Another important advantage to coarse pricing grids is that they help enforce time priority in a limit order book (Harris (1991, 1994b) and Seppi (1997)). With a small tick size, a floor trader can strategically jump ahead of standing limit orders by marginally improving on the limit order’s price. Limit orders are therefore more likely to execute when conditions are unfavorable (e.g. when the order is more likely to execute against an informed trade) and traders will be less willing to submit these orders.5 Since limit orders provide liquidity to the market (as opposed to market orders, which consume liquidity), market liquidity is reduced.6 In this paper we examine whether changes in relative tick sizes influences the decision of traders to submit limit orders. Specifically, we examine changes around stock splits in the proportion of trading activity identified as limit orders. The principal advantage of studying changes around stock splits is that we have numerous uncorrelated events.7 The principal
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Grossman and Miller (1988) formalize this argument while Harris (1991) provides empirical evidence of a relationship between tick sizes and market maker profits. Ball, Torous, and Tschoegl (1985), Harris (1991) and Grossman, Miller, Cone, Fischel and Ross (1997) all examine the propensity of market makers to quote on a price grid coarser than what is available. 4 Grossman, Miller, Cone, Fischel and Ross (1997) provide a detailed discussion of this argument and examine endogenously generated price grids in a number of markets. 5 Glosten (1994) presents a model which describes this trade-off. Handa and Schwartz (1996) explore these issues and provide empirical evidence suggesting limit orders are optimal for participants with well-balanced portfolios. However, Handa and Schwartz (1996) do not examine actual limit orders, but measure the execution prices of various execution strategies implemented against actual price data. 6 Given strict price and time priority, Harris (1994b) predicts that a reduction in relative tick size will narrow spread but reduce depth. On the other hand, Greene (1995) presents evidence that limit orders improve spreads and also absorb adverse selection costs. 7 The principal alternative would be to look at changes in limit order execution around the time that exchanges altered their minimum tick sizes. The difficulty with those studies is that trading strategies may be highly correlated across securities and the results may be driven by factors unrelated to the market change. Though they do not examine limit orders, per se, Ahn, Cao and Choe (1996), Weaver (1996), and Bacidore (1997) find negligible changes in market liquidity around market changes in mandated tick sizes. On the other hand, Barclay, Christie, Harris, Kandel and Schultz (1997) examine market quality around the January 20, 1997 implementation of SEC rules requiring Nasdaq market makers to execute or display customer limit orders and found a significant decline in trading costs.
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disadvantage is that stock splits are associated with changes in trading volume and volatility.8 We control for these effects in a number of ways: we examine the time-series of mean and median executed limit orders as a proportion of trading activity, changes in individual firm proportions, and pairs of trading days matched on trading activity and price changes. We also employ a logistic regression analysis. To identify limit orders, we use the methodology described in Greene (1995). We find that both total trading volume and limit order executions increase in our sample subsequent to stock splits. More importantly, the proportion of limit orders increases significantly and substantially. For example, the proportion of share volume identified as limit orders increases by about 45 percent following two-for-one stock splits. The percentage change in executions is also increasing in order size; the proportion increases by about 30 percent for order less than 1,000 shares and more than doubles for orders over 5,000 shares. This suggests that sophisticated traders are more sensitive to differences in pricing structures. As expected, the changes are more pronounced for larger stock splits. Thus, our results demonstrate that limit order activity increases with larger relative tick sizes. Interestingly, the recent reductions in tick sizes by the major U.S. exchanges may have little permanent impact on market activity. As Harris (1994a) and Anshuman and Kalay (1993) point out, a firm ultimately controls the relative tick size by choosing a trading range for its share price.9 If firms believed that finer tick sizes would reduce trading costs, they would allow their
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Copeland (1979) and Lamoureux and Poon (1987) find a decrease in split-adjusted volume whereas Lakonishok and Lev (1987) find no significant changes and Desai, Nimalendran and Venkataraman (1996) find that volume increases. These differences are most likely due to differences in the timing of samples – our sample is most similar to Desai, Nimalendran and Venkataraman (1996) and we also find an increase in some measures of volume. Copeland (1979), Ohlson and Penman (1985), Conroy, Harris and Benet (1990), Koski (1997), and Desai, Nimalendran and Venkataraman (1996) all document increases in volatility. 9 It is also possible that dealers will effectively quote on a grid that is coarser than allowed. This may happen to facilitate market making, as suggested by Grossman, Miller, Cone, Fischel and Ross (1997) or to raise profits as in
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shares to trade at high prices. Our results provide empirical evidence that finer tick sizes reduce the willingness of traders to provide market liquidity through limit orders. This countervailing effect may provide a justification for the observation by Lakonishok and Lev (1987) that stock splits return prices to a “normal” trading range. Similarly, Angel (1997) shows that relative tick sizes have been constant over long periods of time and across different exchange structures. Some empirical support linking relative tick size and limit order activity can be inferred from the work of Hasbrouck (1992). Hasbrouck (1992) examined the cross-sectional relative frequency of limit order executions using the TORQ data set and found that limit order activity is decreasing in share prices (and therefore increasing in relative tick sizes). By examining changes in the time-series of tick sizes, we can control for firm characteristics and directly gauge the impact of tick size on trading activity. A number of studies have examined the impact of stock splits on market quality. Most have been motivate by the positive announcement effect associated with stock splits and the link between stock value and liquidity suggested by Amihud and Mendelson (1986) and Brennan and Subrahmanyam (1994). Results of empirical studies have been mixed: Merton (1987), Lamoureux and Poon (1987), Brennan and Hughes (1991), and Maloney and Mulherin (1992) present evidence that the pool of investors increases after a stock split while Conroy, Harris and Benet (1990), Koski (1995), Desai, Nimalendran, and Venkataraman (1996) find that relative spreads and volatility increase. Our results suggest that the changes in the trading environment are more subtle and varied than might be captured in simple measures of trading activity and trading costs.
Dutta and Madhavan (1997) and Kandel and Marx (1997). Evidence of such quoting practices is provided by Christie and Schultz (1994).
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The rest of this paper is organized as follows. Section 2 discusses our sample and presents summary statistics. Section 3 presents our results. Section 4 concludes.
2. Sample and Summary Statistics We begin with all the stock splits by NYSE or AMEX firms listed in the Center for Research in Security Prices (CRSP) data set during 1993-1996 that had 3-for-2 (1.5-for-one), 2for-one or greater than 2-for-one stock splits. We then use the NYSE Trade and Quote (TAQ) data set to obtain trading data for each day in our examination period which extends from 60 days prior to the first day of trading after the split through sixty days of trading after the split. We exclude the time period extending ten days before split is executed to ten days after the split is executed to eliminate any unusual trading activity or return behavior associated with the split date.10 A trading day is included if there is at least one trade and one quote on that day (though not necessarily one limit order). This sample, which is used for matched trading-day pairs and regression analysis, contains 27,841 trading-days from 302 splits. For our remaining tests, we also require that each firm included in our analysis have valid daily trading observations for every trading day in the examination period as well as at least one buy and one sell on each trading day. 11 This left 15,300 trading days from 153 stock spits; 46 splits that were 1.5-for-one, 96 that were 2-for-one, and 13 that were greater than 2-for-one.
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The standard deviation of returns on the first five days of the excluded window range from 1.34% to 2.7% with an average of 1.77% and rises to 12.47% on the first day of trading after the split. Similarly, the number of trades (share volume) ranges from 188 (263,400) to 232 (371,510) with an average of 205 (297,564) and rises to 484 (647,710) on the first day of trading after the split. These figures were calculated from the sample of splits with valid trading figures for every day in the twenty day exclusion period. These temporary effects appear to be eliminated by the seventh day of trading after the stock split. Our empirical results are essentially unchanged if we include the excluded twenty days of trading or if we exclude a forty day trading period also centered on the split date. 11 We use the tick tick-test to categorize trades as buys or sells. We also adjust the timing or quotes to reflect differences in the reporting speed of quotes and trades. Finally, we group trades which execute within 5 seconds at
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The TAQ data do not identify what proportion of any executed order is executed against a limit order. However, the NYSE and AMEX require specialists to indicate the depth at posted quotes and this depth reflects limit orders as well as the specialist’s own willingness to trade. Following the algorithm presented in Greene (1995), we use changes in posted depth to measure limit order executions. Though not a perfect measure of limit order executions, tests by Greene (1995) show that it is relatively accurate. Any noise introduced by this algorithm is likely to bias against finding significant changes. Finally, since the TAQ data report depth only up to 99,999 shares, we cannot identify limit orders of 100,000 shares or greater. The Greene (1995) algorithm is described in detail in the Appendix. Summary statistics are presented in Table 1. As expected, average prices fall for our sample of stock splits. Consistent with the work of Conroy, Harris and Benet (1990), we find that dollar quoted spreads (ask prices less bid prices) and effective spreads (twice the difference between execution prices and the midpoint of the prevailing quotes) decrease while relative spreads increase. The median volatility of returns from the midpoint of the first quote of the day to the midpoint of the last quote of the day (daytime volatility) increases whereas there is no significant change in means. We use quotes, rather than transaction prices, to eliminate transient effects from bid-ask error. Trading activity measures are calculated from daily totals for each trading day in our sample. There is little empirical evidence on limit order strategies and little theoretical distinction between purchasing and selling a security. However, to better control for the link between price movements and limit order activity and to document any differences if they exist, we examine buy and sell trading activity separately in addition to total trading activity.
the same price and volume into a single trade. These procedures and their justifications are described in Lee and
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If we examine all executed orders, the number of trades increases while (split adjusted) share volume declines. There is evidence of a decrease in (split adjusted) dollar volume only for market buys. The change in the number of trades is substantial. For all trades, the median increases from 86 to 108 - an increase of about twenty-five percent. Taken together, these changes suggest that the nature of trading behavior is altered after the stock split. Such changes may result from the new pricing grid or possibly from changes in the trading clientele of the firm. Given the different behavior of share volume and number of trades, we examine proportions based on both of these activity measures. We do not present our analysis of proportions based on dollar volume since prices do not change substantially within each trading day and our results are therefore virtually identical to those for share volume. Furthermore, differences between mean and median values for some variables suggest that the trading statistics are not always normally distributed. For this reason, we use both parametric and nonparametric tests. Our results for executed limit orders are quite different from the results for all executed orders. We find significant and substantial increases in most of our measures of trading activity. For example, the median number of trades executed each day increase from 10 to 18 and the median daily share volume increases from 4,700 shares to 6,600 shares. Note that for limit orders, the activity of sells is listed before buys while the reverse was true for all executed orders. This was done to facilitate comparisons since market buys (sells) are executed against limit order sells (buys).
Ready (1991).
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3. Empirical Results This section presents our formal tests of limit order activity. The first tests employ standard event study methodology. To control for any correlation within firms, we then examine changes in limit order activity for each firm. To control for characteristics of the trading environment, we add tests of matched pairs of trading days where pairs are matched on the number of trades, share volume, dollar volume and daytime price change. Finally, we conduct a firm-by-firm logistic regression analysis using a wide range of controls. 3.1
Initial Tests Our first analysis looks at the time-series of mean and median limit order activity. We
consider three stock split categories (1.5-for-one, 2-for-one and greater than 2-for-one) as well as two measures of limit order activity (the proportion of limit orders based on both the number of trades and one based on share volume). The measure based on the number of trades compares the number of trades against which limit orders were executed to the total number of executed trades.12 The measure based on volume compares the number of shares executed as limit orders to the number of shares executed for all orders. When we partition by sells and buys, the proportions are calculated from market orders of the same type. 13 The results are shown in Table 2. In every case we find a significant increase in limit order activity. For example, the proportion of limit orders by number of trades increases from 15.55 percent to 17.13 percent for 1.5-for-one stock splits (a 10 percent increase), from 14.17
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Recall that only the active side of the market is reported in trade and quote data. For example, the portion of a trade satisfied by the specialist and the portion satisfied by the limit order book are not reported. Thus, a number of limit orders may be executed against a single market order and we must therefore examine a ratio based on the number of executed orders containing limit orders rather than the actual number of limit orders.
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percent to 19.20 percent for 2-for-one stock splits (a 35 percent increase), and from 8.51 percent to 18.38 percent for greater than 2-for-one stock splits (a 215 percent increase). Interestingly, the magnitude of change is increasing in the degree of the stock split and the results after the stock split are not substantially different across the three samples. These results suggest that all the stock splits lead to post-split trading behaviors which are comparable and that the change is driven by changes in the relative grid size and not a result of other changes in the trading environment. Our results are consistent with Angel (1997), who argues that firms split their stocks to generate an optimal relative grid size. 3.2
Firm-Matched Changes While independence across firms is likely to hold since there is little calendar time
clustering of stock splits, measures are likely to be correlated across time within firms if there are any unmodeled firm characteristics. Table 3 presents a test of changes in limit order proportions in which we calculate the average proportion for a given firm in the pre-split period and subtract it from the average for the firm in the post-split period. We present our analysis for 2-for-one splits only, the results for 1.5-for-one splits are weaker and the results for greater than 2-for-one splits are stronger. To benchmark changes in the levels, we list the mean proportion before the stock split and to facilitate discussion we calculate the change as a percentage of the pre-split mean.14 The pre-split averages are, of course, the same as in Table 2.
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Our results are robust to definitions of these measures. If we consider only trades under 100,000 (since we cannot identify limit orders above this size) or calculate limit buys and sells relative to all trading rather than matching buys and sells, our results are essentially unchanged. 14 We also calculated the average (across firms) of percentage increase rather than the percentage increase in the firm averages. The results are qualitatively identical but more pronounced. However, large individual firm percentage changes can be driven by small pre-split values.
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We also include a breakdown by trade size since we would expect traders submitting larger orders are more sophisticated and therefore more sensitive to changes in the grid size. We partition based on the number of shares traded and distinguish between trades less than or equal to 1,000 shares (small), trades greater than 1,000 shares and less than 5,000 shares (medium), and trades greater than 5,000 shares (large). Specifically, we obtain the number or share volume of limit orders executed in each category and calculate this activity as a proportion of the total activity in each category. Our results on mean values are statistically significant and substantial in every case using standard event study methodology. By number of trades, the proportion for all limit orders increases by about 35%. More importantly, we find that the percentage increase in limit order activity is steadily increasing in the size of the trade. For example, the increase in limit selling activity measured by the number of trades is about 31 percent for small trades and rises to about 125 percent for large trades. These increases suggest that traders are very sensitive to the grid size and adjust their trading strategies accordingly. The median changes are close in value and qualitatively similar to the mean changes. Tests of median changes are conducted as follows. The median value across all firms is calculated for each event-time trading-day and a Wilcoxon signed rank test is used to compare the pre-split and post-split samples. All results are significant. Analysis of the time-series standard error of the median values, not reported, yields similar conclusions. The percentage changes, also not reported, are quantitatively and qualitatively similar. 3.3
Matched Pairs of Trading Days As a further control, we begin with our full sample of stock splits (we no longer require a
trade on every day in our study period) and construct matched pairs of days based on trading 10
activity characteristics. We match each trading day in the pre-split trading period with a trading day in the post-split period as long as the matching characteristic is within 1% for trading activity measures and 10% for daytime price changes. Trading days are selected without replacement. We match on the number of trades in the day, the share volume in the day, the dollar volume in the day, and the daytime price change (based on opening to closing quotes). To conserve space, we present only the results for 1.5-for-one and 2-for-one stock splits. The results for splits greater than 2-for-1 are qualitatively identical to the 2-for-one results. The results are shown in Table 4. The statistical significance of our tests is almost eliminated for the 1.5-for-one sample, but still strong for the 2-for-one sample. This suggests that there are important changes in the trading environment that affect average limit order activity. In addition, we confirm that if any effects are actually a product of the split itself, they should be more pronounced for larger stock splits. In general, the magnitudes of the changes are comparable to those in Table 3 and are comparable across all of our control groups. These results confirm our principal findings. 3.4
Regression Analysis As a final test, we conduct a logistic regression analysis of the proportion of trades each
day that contain limit orders. In our tests, each trading day can be thought of as a series of trials (trades) influenced by conditions associated with that day (our independent variables) and some trials result in positive outcomes (limit orders). We conduct our regression for each firm in our sample and then present tests of the cross-sectional distribution of parameters. This approach provides controls both for firm level characteristics and also for variables affecting the trading environment. The only statistical assumption underlying the tests is stock splits events are
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independent across firms. We also conduct our analysis separately for 1.5-for-one and 2-for-one stock splits as well as distinguishing between limit sells and limit buys. Because of the scant empirical and theoretical work in this area, we have little guidance in choosing our control variables or in predicting the sign of coefficients. We have chosen a set of variables for which at least some argument for inclusion might be made. The principal test is essentially whether the proportion of limit orders executed increases after the stock split and not whether any of the other variables is statistically important. Our independent variables are a constant, an indicator that the day is from the post-split period (POST), the squared residual return on a trading day from the mean return in the pre-split or post-split time period as appropriate (DEV), the one period lag of DEV (LDEV), the splitadjusted trading price (PRICE), the volume of shares traded conditioned on buys and sells (VOL, BVOL, SVOL), the absolute value of the daytime return during the day (ABSRET), and the product of ABSRET and a dummy variable indicating that the stock price moved up during the day (UP) or down during the day (DN). The variable POST is the variable of interest in our study. DEV captures the possibility that market participants may condition trading activities on unusual stock price behavior. Similarly, since a limit order book may be exhausted one day by execution of orders in the prior day and because unusual deviations one day may affect behavior the next, we include the oneday lag of DEV. The variable PRICE may capture any effect of grid size not attributable to the change in the grid size over our study period. VOL, BVOL, and SVOL control for the effect of total trading activity on limit orders, limit buys, and limit sells, respectively. While the use of proportions already controls for any linear relation between the number of trades and the number of limit orders, the volume measure will
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be able to capture any non-linearity. The absolute price change captures the relationship between movements in price and the limit order book. Specifically, we expect that when prices move, more of the limit order book will be executed. For this reason, when we look at limit sells (buys) we interact ABSRET with the dummy variable UP (DN) so that we only have movements in a direction that is likely to lead to limit executions of the type considered. Our results are shown in Table 5. Consistent with our hypothesis, the variable POST is significant and positive in all but one case. The coefficients in a logistic regression, like any non-linear regression model, do not reflect the marginal effects of a change in the independent variable. Furthermore, the effect of an independent variable varies with the levels of all independent variables. To gauge the impact of POST on our regression, we calculate the change in the probability of a limit order trade as POST varies between zero and one, with the remaining explanatory variables evaluated at their mean values. This is listed as POST EFFECT in Table 5. For the 1.5-for-one stock split, POST EFFECT is about 1 percent and for the 2-for-one split it is about 5%. These values are consistent with the values generated in our previous tests. As for the remaining explanatory variables, we find that unusual price movements seem to lead to lower limit order executions while price levels themselves are unimportant. Only for limit sells associated with 1.5-for-one stock splits do we have strong evidence that total volume matters. Consistent with the fact that larger price movements lead to more limit order executions, UP*ABSRET and DN*ABSRET are positive and significant. As might be expected, ABSRET itself is not statistically significant when we examine all orders because when prices move up (down), we find more executions of limit sells (buys) but less executions of limit buys (sells). Only when we disentangle these effects can the influence of price movements on limit order executions be identified.
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While the test of the POST variable is the principal concern of this analysis, we also present statistical evidence that our specification describes the data. First, we present the average Chi-Squared test statistic (equal to –2 times the log of the likelihood function evaluated at the maximum likelihood estimates) for all the firm level regressions. This measure of fit is distributed Chi-Squared under the null hypothesis that all explanatory variables in the model are zero. It is significant in every case. We also list the percentage of firm level regressions for which the chi-squared test was not significant at the 10% level. This is less than 10% in every case.
4. Conclusion We examine the executions of limit orders around stock splits in order to determine whether changes in relative tick sizes lead to changes in limit order activity. Consistent with the work of Harris (1994b), who predicts a greater use of limit orders when the relative tick size is larger, we find an increase in the use of limit orders following stock splits. This increase is larger and more often significant for 2-for-one or greater stock splits. We also find that the effect of stock splits is more pronounced for large trades than smaller trades. Our principal results are also robust to numerous tests, including tests that control for firm specific characteristics, trading day characteristics, and both firm and trading day characteristics. While our results, particularly how they differ across split magnitudes, strongly suggest that the relative grid size influences the execution of limit orders, stock splits may change the trading environment in some manner unrelated to relative tick sizes which we cannot measure and therefore control for. Never the less, our results demonstrate and important link between trading strategies and stock splits. Furthermore, since limit orders provide liquidity
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rather than consume it, our results provides some evidence favoring larger tick sizes and lends additional empirical support for optimal trading ranges.
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Appendix The algorithm used to infer limit orders was developed and tested by Greene (1997). It is based on the assumption that market markers display the demand in the limit order book through quoted depths. This display of interest is mandated (though not necessarily enforced) by the NYSE/AMEX. Consider a limit order to buy 2,000 shares at the quoted ask price. If the market marker is willing to purchase 1,000 shares, the market maker will post the depth of the market as 3,000 shares. Now suppose a trade occurs for 1,000 shares at the ask and the market maker executes this against the limit order. The market maker’s revised depth will be 2,000 shares. Thus, the change in the ask depth indicates a potential limit order execution. The elements used to infer whether a transaction at time t is a limit order are the following: n We obtain the following variables: St and Pt the size and price of the trade at time t, respectively Bt and At the bid and ask prevailing at time t, respectively Bt+1 and At+1 the bid and ask of the quote immediately after the trade at t D(B)t and D(A)t the quoted depth in the bid and ask of prevailing quote at time t D(B)t+1 and D(A)t+1 the quoted depth in the bid and ask of prevailing quote at time t + 1 n We assign a buy or sell indicator to each trade based, denoted Qt where Qt =1 for a market buy and Qt = -1 for a market sell, based on the trade price relative to the midpoint of the prevailing quotes. Using these elements we calculate the extent of a limit order execution, denoted Lt, and whether that limit order is a market sell or buy, denoted Tt , at a time t as follows: (1) Determine the change in depth, denoted ∆Dt , which is a potential limit order: If Qt =1 then if At > At+1 then ∆Dt = 0 if At = At+1 then ∆Dt = Max[D(A)t - D(A)t+1, 0 ] if At < At+1 then ∆Dt = D(A)t If Qt =-1 then if Bt > Bt+1 then ∆Dt = 0 if Bt = Bt+1 then ∆Dt = Max[D(B)t - D(B)t+1, 0 ] if Bt < Bt+1 then ∆Dt = D(B)t (2) Determine the inferred limit order, denoted Lt: If Pt = At or Pt = Bt , then Lt = Min (∆Dt , St), else Lt = 0 A few points should clarify the workings of this algorithm. Only decreases in depth are inferred to be limit orders, when the quoted prices move away from the midpoint, the quoted depth prior to the trade is assumed to be the limit order amount, and if quotes move into the spread, no limit order is assume to have been executed. Finally, a limit order can not exceed the amount of the transaction. For a discussion on the accuracy and details of implementation the reader is referred to the original presentation in Greene (1995).
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Dutta, Prajit K., and Ananth Madhavan, 1996, Competition and collusion in dealer markets, Journal of Finance 52, 245-276. Glosten, Lawrence, 1994, Is the electronic open limit order book inevitable? Journal of Finance 49, 1127-1161. Greene, Jason T., 1995, The impact of limit order executions on trading costs in NYSE stocks. An empirical examination, Indiana University working paper. Grossman, Sanford J., Merton H. Miller, Daniel R. Fischel, Kenneth R. Cone, and David J. Ross, 1997, Clustering and competition in dealer markets, Journal of Law and Economics 40, 23-60. Grossman, Sanford J. and Merton H. Miller, 1988, Liquidity and market structure, Journal of Finance 43, 617-663. Handa, Puneet and Robert A. Schwartz, 1996, Limit order trading, Journal of Finance 51, 1835-1861. Harris, Lawrence, 1991, Stock price clustering and discreteness, Review of Financial Studies 4, 389415. Harris, Lawrence, 1994a, Minimum price variations, discrete bid-ask spreads and quotation sizes, Review of Financial Studies 7, 149-178. Harris, Lawrence, 1994b, Optimal dynamic order submission strategies in some stylized trading problems, University of Southern California working paper. Harris, Lawrence, 1997, Decimalization: A review of the arguments and evidence, working paper. Hasbrouck, Joel, 1992, using the TORQ database, NYSE working paper. Kandel, Eugene, and Leslie M. Marx, 1997, Preferencing and payment for order flow on Nasdaq, University of Rochester working paper. Koski, Jennifer Lynch, 1995, Measurement effects and the variance of returns after stock splits and stock dividends, University of Washington working paper. Lakonishok, Josef, and Baruch Lev, 1987, Stock splits and stock dividends: Why, who and when, Journal of Finance 42, 1347-70. Lamoureux, Christopher G. and Percy Poon, 1987, The market reaction to stock splits, Journal of Finance 42, 1347-1370. Lee, Charles M. C., and Mark J. Ready, 1991, Inferring trade direction from intraday data, Journal of Finance 46, 733-746.
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Maloney, Michael T. and J. Harold Mulherin, 1992, The effect of splitting on the ex: A microstucture reconciliation, Financial Management 21, 44-59. Merton, Robert, 1987, A simple model of capital market equilibrium with incomplete information, Journal of Finance 42, 483-510. Ohlson, James A. and Stephen H. Penman, 1985, Volatility increases subsequent to stock splits: An empirical aberration, Journal of Financial Economics 14, 251-266. Peake, Junius W., 1995, Brother can you spare a dime? Let’s decimalize U.S. equity markets, in Robert A. Schwartz, ed.: Global Equity markets: Technological, Competitive and Regulatory Challenges (Irwin Professional, Chicago). Seppi, Duane J., 1997, Liquidity provision with limit orders and a strategic specialist, Review of Financial Studies 10, 103-150. Weaver, Daniel G., 1996, Decimalization and market quality, Marquette University working paper.
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Table 1 Summary Statistics This table presents summary statistics for the sample of stock splits that had at least one buy and one sell during the 100-day examination period. Trading activity measures are daily values. Both mean and median values are given with indicated significance from univariate t tests (means) and Wilcoxon signed rank tests (medians). Daytime variance is the square of the difference between the daytime return (the return from the midpoint of the first quote of the day to the midpoint of the last quote of the day) and the mean daytime return during the pre-split or post-split examination period (as appropriate) for each firm. Share volume and dollar volume are adjusted to a pre-split basis. Mean Before
Median After
Before
After
Panel A: Trading Environment Average Price ($) Dollar Quoted Spread ($) Relative Quoted Spread (%) Dollar Effective Spread ($) Relative Effective Spread (%) Daytime Volatility (%)
64.55 0.395 0.721 0.192 0.362 0.078
32.63*** 0.303*** 1.063*** 0.158*** 0.556*** 0.042
52.62 0.375 0.586 0.156 0.302 0.006
29.68*** 0.250*** 0.961*** 0.142*** 0.502*** 0.011***
Panel B: Daily Trade Activity – All Executed Orders Number of Trades
All Trades Market Buys Market Sells
171.11 93.83 72.28
233.11*** 132.26*** 100.85***
86.00 46.00 39.00
108.00*** 59.00*** 49.00***
Share Volume
All Trades Market Buys Market Sells
237,902 127,935 109,936
219,186** 112,579*** 106,601
106,500 53,900 45,500
92,125*** 46,000*** 42,650**
Dollar Volume ($million)
All Trades Market Buys Market Sells
17.178 9.264 7.913
16.644 8.648** 7.996
6.014 3.308 2.557
5.621 2.766*** 2.554
Panel C: Daily Trade Activity – Executed Limit Orders Number of Trades
All Trades Limit Sells Limit Buys
26.67 15.16 11.51
48.98*** 29.09*** 19.88***
10 5 4
18*** 9*** 7***
Share Volume
All Trades Limit Sells Limit Buys
21,024 12,016 9,008
27,649*** 15,527*** 12,122***
4,700 2,300 1,700
6,600*** 3,300*** 2,600***
Dollar Volume ($million)
All Trades Limit Sells Limit Buys
1.430 0.815 0.614
1.967*** 1.111*** 0.856***
*** Denotes significance at the 1% level ** Denotes significance at the 5% level * Denotes significance at the 10% level
20
0.246 0.123 0.088
0.376*** 0.189*** 0.145***
Table 2 Proportion of Limit Order Executions This table presents the mean and median proportion of executed limit orders associated with 1.5-for-one, 2-for-one, and greater than 2-for-one stock splits. Proportions are calculated using both the number of trades and the share/dollar volume of trades, and are given for all trades, limit order buys (market sells) and limit order sells (market buys). Both the mean and median values are given. Tests of significance for the mean values are standard event studies; the mean value subsequent to the event (stock split) is evaluated relative to the standard error of the time series of daily mean values prior to the event. Test of significance for median values are comparisons of the pre-split daily median firm values with the post-split daily median firm values using a Wilcoxon signed rank test.
Mean Before
Median After
Before
After
1.5-for-one stock splits: 46 firms Number
All Limit Sell Limit Buy
15.55 15.89 14.81
17.13*** 17.95*** 16.74***
11.72 11.11 10.34
14.28*** 13.81*** 13.33**
Volume
All Limit Sell Limit Buy
9.85 10.73 9.46
11.33*** 12.37** 10.89***
5.28 4.75 3.99
6.67*** 6.66*** 5.47***
2-for-one stock splits: 94 firms Number
All Limit Sell Limit Buy
14.17 14.52 13.58
19.20*** 19.69*** 18.38***
12.58 12.50 11.56
18.16*** 18.18*** 16.77***
Volume
All Limit Sell Limit Buy
8.28 9.04 7.97
12.06*** 13.36*** 11.33***
5.15 5.19 4.34
8.72*** 9.01*** 7.32***
Greater than 2-for-one splits: 13 firms Number
All Limit Sell Limit Buy
8.51 8.92 8.10
18.38*** 19.67*** 16.86***
7.05 6.80 6.25
18.36*** 19.24*** 16.12***
Volume
All Limit Sell Limit Buy
5.44 6.07 5.08
13.70*** 15.25*** 12.52***
3.15 3.16 2.48
11.18*** 12.21*** 9.79***
*** Denotes significance at the 1% level ** Denotes significance at the 5% level
21
Table 3 Test of Changes in Limit Order Execution Using Changes In Firm Averages for 2-for-One Splits This table analyzes changes in the proportion of limit order executions by examining the changes in limit order executions for individual firms. The mean value of firm average proportions before the split is presented as a basis for evaluating the magnitude of the change. The mean and mean change as a percent of the mean value before the split are given as well as median change. Analysis is conducted on all trades and separately on limit order buys, limit order sells, and three size categories within buys and sells: less than 1,000 shares (small), between 1,000 and 5,000 shares (medium), and greater than 5,000 shares (large). Proportions are given relative to the total quantity in each size category and are calculated both from the number of trades and share volume. Tests of significance test whether changes are different from zero and are univariate t tests for means and Wilcoxon signed rank tests for medians. There are 96 observations (firms). Mean Before
Change
Percentage Change
Median Change
By Number of Trades All Limit Order Trades
14.17
5.03***
35.49
4.75***
Limit Sells
All Sells Small Sells Medium Sells Large Sells
14.52 16.49 9.77 4.87
5.11*** 5.09*** 5.96*** 6.07***
35.07 30.86 61.00 124.64
4.89*** 5.11*** 5.07*** 4.04***
Limit Buys
All Buys Small Buys Medium Buys Large Buys
13.58 15.38 8.91 3.15
4.79*** 5.42*** 5.27*** 4.94***
35.27 35.24 59.14 156.82
5.13*** 5.77*** 5.33*** 2.70***
8.28
3.78***
45.65
3.29***
By Volume of Shares All Limit Order Trades Limit Sells
All Sells Small Sells Medium Sells Large Sells
9.04 15.25 9.77 4.26
4.31*** 4.88*** 6.39*** 5.47***
47.67 32.00 65.40 128.40
4.03*** 4.76*** 5.15*** 3.40***
Limit Buys
All Buys Small Buys Medium Buys Large Buys
7.97 14.09 9.21 2.64
3.35*** 5.30*** 5.37*** 4.13***
42.03 37.61 58.30 156.43
3.00*** 5.53*** 5.23*** 2.20***
*** Denotes significance at the 1% level
22
Table 4 Test of Changes in Limit Order Execution Using Matched Pairs of Trading Days This table analyzes changes in the proportion of limit order executions by examining matched pairs of trading days in which trading days are matched as follows. For each firm, a trading day prior to the stock split is matched to a trading day after the stock split based on daily number of trades, daily share volume, or the change in quoted prices during the day. A match is retained only if the post-stock-split day’s matching variable is within 1 percent of the pre-stock-split variable for trading activity measures and within 10 percent for price changes. Matches are selected without replacement in the post-stock-split trading period. Since the number of matches will vary from test to test, the number of observations is given along with the measured change in proportions. Values are mean changes in proportions express as percent. 1.5-for-one Change in Number of Proportion Observations
2-for-one Change in Number of Proportion Observations
Number of Trades Matched (Proportions based on number of trades)
All Trades Limit Sells Limit Buys
1.315 1.123 0.781
2410*** 2276** 2258
4.375 3.956 4.380
2592*** 2511*** 2484**
0.455 0.601 1.201
1301 1259 1254**
3.177 3.865 2.801
2004*** 1947*** 1968***
0.462 0.885 -0.199
1190 1161 1158
1.314 2.186 0.585
1415*** 1401*** 1397
0.201 -0.071 -0.045
1106 1100 1057
4.626 4.571 4.442
1712*** 1706*** 1656***
-0.285 -0.032 0.641
1106 1100 1057
3.553 4.209 3.050
1712*** 1706*** 1656***
Share Volume Matched (Proportions based on shares/dollars)
All Trades Limit Sells Limit Buys
Dollar Volume Matched (Proportions based on shares/dollars)
All Trades Limit Sells Limit Buys
Price Change Matched (Proportions based on number of trades)
All Trades Limit Sells Limit Buys
Price Change Matched (Proportions based on shares/dollars)
All Trades Limit Sells Limit Buys *** Denotes significance at the 1% level ** Denotes significance at the 5% level
23
Table 5 Logistic Regression Analysis of Limit Order Execution Results of logistic regressions of the daily proportion of executed orders that contain limit orders. Regressions are executed for each firm and the table below provides summary statistics on the cross-sectional distribution of parameter estimates. Mean values are given with medians in parentheses below. Test statistics are t tests for means and Wilcoxon signed rank tests for medians. Results are presented separately for 1.5-for-one and 2-for-one stock splits. The independent variables are: a dummy variable equal to one on days subsequent to the stock split (POST), the squared excess daytime return (relative to the mean daytime return for the sample period) (DEV), one day lagged DEV (LDEV), split adjusted opening quote (PRICE), volume in millions of shares calculated for all trades, limit buys, limit sells (VOL, BVOL, SVOL, respectively), absolute value of daytime return (ABSRET), and dummy variables for price increases or decreases during the day (UP, DN). The measure called “POST EFFECT” is the average marginal change in the probability of a limit order when POST is one rather than zero with all other variables are evaluated at their mean values. We present the average across regressions of a Chi-Squared test of goodness of fit under the null that all explanatory variables are zero. We also report the proportion of Chi-Squared p values for each firm’s regression which not significant at the 10% level.
Constant POST
All Trades
1.5-for-one Limit Sells
-1.229** (-1.141)**
-1.554*** (-1.177)***
0.139* (0.147)**
0.108 (0.031)*
DEV
-113.54 (-33.53)
LDEV
-38.42 (-1.35)
-69.80 (-8.84)
PRICE
-0.051 (-0.005)
-0.017 (-0.009)
VOL (×106)
0.475 (-0.041)
BVOL (×106)
-219.25*** (-93.08)***
1.562* (0.115)* -205.42*** (-109.19)***
Limit Buys
-2.253*** (-2.057)***
-2.326*** (-1.840)***
-2.304*** (-2.144)***
0.412*** (0.428)***
0.391*** (0.374)***
0.141*** (0.426)***
-143.28 (14.44)
-221.65*** (-99.68)***
-250.40*** (-96.74)*
-20.61 (-26.03)**
-76.61*** (-28.94)**
-101.85*** (-34.15)***
-62.35** (-4.98)*
-0.022* (0.004)
0.007 0.002
0.009 0.002
0.005 0.002
8.616*** (0.276)***
-0.004 (-0.014) -1.062 (-0.043)
0.307 (-1.616)
UP*ABSRET
-0.346 (-0.016)** -1.145 (-2.780) 6.001*** (5.020)***
5.661*** (3.704)***
DN*ABSRET POST EFFECT Chi-Squared Percent p < 0.10
-1.241** (-1.563)***
2-for-one Limit Sells
All Trades
-0.158 (-0.059)*
SVOL (×106) ABSRET
Limit Buys
5.149* (4.176)* 1.22 304*** 3.50
0.60 140*** 6.77
1.77 186*** 8.74
*** Denotes significance at the 1% level ** Denotes significance at the 5% level
* Denotes significance at the 10% level
24
4.926*** (2.401)*** 5.31 662*** 2.67
5.37 392*** 4.46
4.94 292*** 1.78
