Trade-Size and Price Clustering: The Case of
Short Sales
Benjamin M. Blau
Department of Economics and Finance
Huntsman School of Business
Utah State University
[email protected]
Bonnie F. Van Ness
Department of Finance
School of Business
University of Mississippi
[email protected]
Robert A. Van Ness
Department of Finance
School of Business
University of Mississippi
[email protected]
First version: November 15th, 2007
Current version: August 17th, 2009
The authors would like to thank Kathleen Fuller, Andriy Shkilko, Mark Van Boening and
seminar participants at the University of Mississippi, Mississippi State University, and
the 2009 Eastern Finance Association for their comments.
Trade-Size and Price Clustering: The Case of
Short Sales
Abstract:
This paper compares the clustering of short sales and regular (non-short) trades on round
prices and round trade sizes. Our findings show that short sellers prefer round sizes and
round prices less than regular (non-short) traders. Further, we find evidence suggesting
that size and price rounding decisions are not correlated for short sales on the NYSE and
are negatively correlated for our Nasdaq sample. Combined, the results in this study
support the idea that short sellers are less concerned with negotiation costs, cognitive
processing costs, and the costs of revealing information to the market and more
concerned with exploiting their private information.
I. Introduction
After showing that traders prefer round trade sizes and round prices, Alexander
and Peterson (2007) contend that three hypotheses best explain the observed trade-size
and price clustering. First, the behavioral hypothesis posits that traders think in round
numbers and therefore choose round sizes and prices to minimize cognitive processing
costs (Wyckoff, 1963; Niederhoffer and Osborne 1966; and Ikenberry and Weston 2003).
Second, clustering is consistent with the negotiations hypothesis, which suggests that
buyers and sellers of assets attempt to minimize the costs of continuing negotiations by
settling on round prices and round sizes (Ball, Torous, and Tschoegl, 1985 and Harris,
1991). Third, the stealth trading hypothesis argues that informed traders will attempt to
disguise their information by breaking up larger trades into medium-sized trades – not too
large to reveal their information to other sophisticated traders – and not too small because
of transactions costs (Kyle, 1985; Foster and Vishwanathan, 1990; and Barclay and
Warner, 1993). Results in Alexander and Peterson (2007) are consistent with the idea
that sophisticated traders may attempt to disguise their trades using round sizes as they
show that the price impact of round trades is greater than the price impact of unrounded
trades, particularly for the medium trade-size category.
In a separate stream of literature, Diamond and Verrecchia (1987) present theory
showing that short sellers are more informed that other traders. Empirical research
confirms this prediction as short sellers are shown to consistently predict negative returns
(Dechow et al., 2001; Desai et al., 2002; Diether, Lee, and Werner, 2009; Senchack and
Starks, 1993; Aitken et al., 1998; Desai et al., 2002; and Christophe, Ferri, and Angel,
2004). Recently, Boehmer, Jones, and Zhang (2008) show that larger short sales contain
1
more information than smaller short sales indicating that short sellers do not stealth trade,
which is used to partially explain clustering on round trade sizes. Further, Diether, Lee,
and Werner (2009) show that short sellers target stocks that become temporarily
overvalued and are successful in predicting price reversals. Boehmer and Wu (2008)
show that daily short activity adds to the informational efficiency in stock prices by
shorting overvalued stocks and thus reducing pricing errors. If informed short sellers do
not wish to disguise their trades and their tradable information depends on temporary
pricing inefficiencies, then short sellers are likely to be less concerned with cognitive
processing costs and negotiations costs and more concerned with exploiting their
information. This assertion motivates the need to compare the preferences of short
sellers’ and non-short traders’ for round trade sizes and round prices. This paper
provides this comparison. We hypothesize that short sales will cluster less on round
trades sizes and round prices than regular (non-short) trades.
Results in this paper show that the percentage of short sales that execute at round
sizes is significantly less than the percentage of non-short trades that execute at round
sizes, particularly for the larger trade-size category, which likely contains more informed
short sales (Boehmer, Jones, and Zhang, 2008). These results are robust to a variety tests
similar to those reported in Alexander and Peterson (2007) and substantiate our claim that
short sellers prefer round trade sizes less than non-short traders.
In unreported results, we examine the intraday return predictability of short
sellers. First, we find consistency with the notion that larger short sales contain the most
information about future stock price movements (Boehmer, Jones, and Zhang, 2008) as
larger short sales are followed by the most negative intraday returns. After separating
2
short sales into rounded and unrounded trades, respectively, we find that large unrounded
short sales are followed by greater negative returns than large rounded short sales. This
result rejects the idea that short sellers attempt to disguise their private information by
using round trade sizes.
In additional tests, we find that the percentage of short sales that execute at
multiples of $0.05, $0.10, $0.25, $0.50, and $1.00 is significantly lower than the
percentage of non-short trades that execute at these round price fractions, thus supporting
the idea that short sellers also prefer round prices less than non-short traders. Less
preference for round prices indicates that short sellers are not as concerned about
processing costs and the costs of further negotiations.
To provide more robust tests of the behavioral and negotiations hypotheses, we
follow Alexander and Peterson (2007) and examine the simultaneous occurrence of round
short sales executing at round prices. Using a bivariate probit method, we model price
rounding and trade-size rounding decisions simultaneously. After controlling for factors
that affect price and trade-size rounding, we find no statistical correlation between the
likelihood of a round short sale executing at a round price for NYSE stocks. The
correlation is significantly negative for NASDAQ stocks. This result makes the case for
the negotiations and behavioral hypotheses less compelling when examining short selling
data.
Combined, this study shows short sellers are less concerned about costs that arise
from continuing negotiations, cognitive processing costs, and the costs of revealing their
private information to other sophisticated market participants. Instead, short sellers are
likely more concerned with exploiting their private information.
3
While our results, which are obtained using short-sale transactions data, differ
from the findings in Alexander and Peterson (2007) who examine all trades, our study
indirectly provide robustness for their findings. Because short-sale volume makes up
nearly 25 percent of trade-volume on the NYSE and 31 percent of trade-volume on
NASDAQ (Diether, Lee, and Werner, 2009), the conclusions in Alexander and Peterson
would be even stronger when excluding short sales.
The rest of this paper follows. Section II describes the data. Section III develops
empirical predictions and presents the tests of our predictions. Section IV concludes.
II. Data Description
Short-sale transactions data that is available in response to Regulation SHO
(January 2005) and Trades and Quotes (TAQ) data are used in order to distinguish
between short sales and non-short trades. Following Alexander and Peterson (2007), we
randomly choose 200 NYSE-listed stocks and 200 Nasdaq-listed stocks from a universe
of NYSE and Nasdaq-listed stocks that trade every day of the third quarter in 2006 (July
1, 2006 to September 30, 2006) and have a price greater than $5.1 We obtain daily
returns, prices, and market capitalization from the Center for Research on Security Prices
(CRSP). Using the daily returns, we calculate return volatility, which is defined as the
standard deviation of returns from day t-10 to day t. Following Diether, Lee, and Werner
(2009), we also use a measure of price volatility that is obtained by dividing the
difference between the daily high price and the daily low price by the daily high price.
1
Each stock that trades every day of the third quarter 2006 and has price greater than $5 is assigned a
randomly generated number, from which, we choose 200 NYSE and Nasdaq stocks with the lowest random
numbers. The choice of random stocks follows the sample selection of Alexander and Peterson (2007).
4
When matching the short-sale transaction data to TAQ, there are occasions when
short sales only make up part of the trade. We exclude these trades in attempt to isolate
the difference between unique short sales and unique non-short trades.2
Table 1 reports statistics that describe our random sample. Panel A reports the
average daily price, return, market cap (in $ thousands), return volatility, price volatility,
non-short, and short-sale volume for the NYSE sample while Panel B presents the
statistics that describe the Nasdaq sample. The average NYSE-listed stock has a daily
price of $32.72 and daily return of 0.0002. Panel B shows that the average stock in the
Nasdaq sample has a price of $22.64, a return of 0.0004, and higher returns and price
volatility than NYSE-listed stocks. When comparing the short volume to the non-short
volume, it appears that short volume relative to non-short volume is higher for the
Nasdaq sample than for the NYSE sample, which is consistent with the findings of
Diether, Lee, and Werner (2009).
III. Empirical Tests and Results
In this section, we develop empirical predictions using relevant theoretical and
past empirical literature. We then test our hypotheses. First, we compare the degree of
clustering on round trade sizes for short sales and non-short trades. Second, we examine
the clustering of prices on round fractions for both types of trades. Third, we turn to the
simultaneous occurrence of rounded sizes and prices (both short and non-short).
IV.A Rounded Trade Sizes
2
We also exclude trades that execute on regional exchanges so that trades for stocks listed on the two
primary exchanges are included.
5
Wyckoff (1963) and Niederhoffer and Osborne (1966) argue that individuals are
behaviorally motivated by round numbers, therefore, buying and selling will occur in
round numbers. Ikenberry and Weston (2003) contend that traders will prefer round
numbers in attempt to minimize cognitive processing costs. Alexander and Peterson
(2007) argue that a trader’s preference for round trade sizes is consistent with, what they
term, the behavioral hypothesis.
Observing clustering on round sizes may also be consistent with the idea that
traders settle on round numbers in order to mitigate further negotiations costs (Ball,
Torous, and Tschoegl, 1985, and Harris, 1991). Denoted the negotiations hypothesis,
Alexander and Peterson (2007) argue that clustering on round sizes is also consistent with
this idea.
Alexander and Peterson (2007) assert that round sizes may be used by informed
traders in order to disguise their information. This argument parallels the stealth trading
hypothesis presented in Barclay and Warner (1993).3 Stealth trading is defined as the
occurrence of informed traders breaking up larger trades into medium-sized trades in
order to disguise their information. Medium trades are optimal because large trades can
reveal the informed investors’ private information to other sophisticated market
participants and small trades are subject to higher overall transaction costs. In the
3
Theory suggests that informed traders spread their trades over time (Kyle, 1985, Foster and
Vishwanathan, 1990, and Admati and Pfleiderer, 1988). In their seminal study on stealth trading, Barclay
and Warner (1993) document that approximately 90% of the cumulative price change is made up from
medium-sized trades for their sample of stocks that are tender offer targets in the 1980s. Chakravarty
(2001) argues that, if institutional traders are more informed than individual traders (Lo and MacKinlay,
1990; Meulbroek, 1992; Cornell and Sirri, 1992; Chakravarty and McConnell, 1997; and Koski and
Scruggs, 1998), then institutions are more likely to engage in stealth trading strategies. Consistent with his
conjecture, Chakravarty finds that approximately 80% of the cumulative price change is made up from
medium-sized, institutional trades while virtually none of the cumulative price change is made up from
medium-sized individual trades.
6
framework of round trade-size clustering, Alexander and Peterson argue that round sizes
may act as an additional disguise for the information contained in certain trades.
The recent consensus in the literature is that short sales contain information about
future stock prices (Diamond and Verrecchia, 1987; Aitken et al., 1998; Christophe,
Ferri, and Angel, 2004; Boehmer, Jones, and Zhang, 2008). Further, evidence indicates
that short sellers are contrarian traders (Diether, Lee, and Werner, 2009) and add to the
informational efficiency in prices by reducing pricing errors when stocks become
temporarily overvalued (Boehmer and Wu, 2008). We hypothesize that because short
sellers are informed about the true value of stocks and their information about overvalued
stocks is based on differences between the trading price and the true price, short sellers
will likely be less concerned about costs that arise from additional negotiations, cognitive
processing, and costs of revealing information to the market. Instead, short sellers are
likely more concerned with exploiting their information. Therefore, trade-size clustering
on round sizes should occur less frequently for short sales than for non-short sales.
An important caveat should be made as the data used in the analysis consists of
the trade size at execution as opposed to the order size of the trade. Because of the lack
of availability of order data, we use trade data, similar to those used in Alexander and
Peterson (2007). Alexander and Peterson do use order data for a portion of their analysis
and find that the results for the trade data and order data during the same time period are
qualitatively similar and therefore argue that using trade data to extend their time period
is justified.
7
Because of this argument, we follow previous research and use trade data as an
approximation of trader’s order preferences when examining clustering.4
Figures 1.A through 1.D show the clustering of trades on certain sizes for both
non-short trades and short sales of NYSE and Nasdaq-listed stocks. In figures 1.A and
1.B, trades on the NYSE tend to cluster on multiples of 1,000 shares and multiples of
5,000 shares. Also, the figures show some evidence of clustering on multiples of 500
shares. When comparing the two figures, it appears that non-short trades have a slightly
more distinct clustering pattern than short sales.
Figures 1.C and 1.D show the clustering of trades on certain sizes for the Nasdaq
sample. Clustering on round sizes appears to be more prevalent in the Nasdaq sample
than in the NYSE sample, which is consistent with findings in Alexander and Peterson
(2007).5
In order to test for a difference in the trade-size choice of short sellers and nonshort traders, we calculate the percentage of trades that execute on multiples of 500
shares, 1,000 shares, and 5,000 shares. Using standard t-tests, we determine whether
there is a significant difference between trade-size rounding of short sales and non-short
trades.
Table 2 reports the percentage of non-short trades that occur on multiples of 500,
1,000, and 5,000 shares for the two samples. Panel A presents the results for the NYSE
sample while Panel B report the results for the Nasdaq sample. In Panel A, we see that
4
The empirical predictions are based on the comparison of the non-short trades and short sales. If there is a
systematic bias between trade and order data, it will probably exist for both short sales and non-short
trades.
5
While the figures appear to show more clustering on Nasdaq, we do not know if there is more/less
clustering on Nasdaq than the NYSE as do not directly compare the two exchanges, and that is outside the
scope of this study (the NYSE and Nasdaq samples are not matched).
8
the percentage of non-short trades that occur at round sizes is significantly higher for
multiples of 500 shares, 1,000 shares, and 5,000 shares. Indeed, clustering on round
multiples of 500 shares tends to occur 10.5 percent ([5.8146 - 5.2062]/ 5.8146) more
often for non-short trades than for short sales while clustering on round multiples of
1,000 shares is 15.3 percent higher for non-short trades. Results in column (3) are
pervasive as the percentage of non-short trades that execute on round multiples of 5,000
shares (0.1577 percent) is nearly three times larger than the percentage of short sales that
execute on similar round multiples (0.0569 percent). Results for the Nasdaq sample differ
somewhat. In Panel B we observe that large Nasdaq trades (non-short) are more likely to
cluster on round sizes than large Nasdaq short sales (column three).
The evidence from Table 2 supports the notion that short sales cluster less on
round sizes than non-short trades, particularly for large trades. This result is consistent
with our hypothesis that information-motivated short sellers are less concerned with costs
associated with foregoing rounded trades and is reconcilable with current literature.
Boehmer, Jones, and Zhang (2008) contend that informed short sellers do not stealth
trade and instead execute larger trade sizes. If short sellers’ informativeness decreases
the likelihood of clustering on round sizes, then large short sales, which are found to
contain the most information, should drive the observation of less overall clustering.
Findings in Table 2 are consistent with this argument.
We next follow Alexander and Peterson (2007) by attempting to determine
whether traders’ prefer round trades to unrounded trades by examining the percentage of
trades that are multiples of 500 shares relative to the proportion of trades that are plus 100
shares or minus 100 shares from the multiples. This method, which is used in Alexander
9
and Peterson (2007), analyzes trades that are 400, 500, 600, 900, 1,000, 1,100, 1,400,
1,500, 1,600 shares; etc.. If traders are indifferent between round sizes and unrounded
sizes, then the percentage of trades at multiples of 500 shares will be approximately 33
percent. If traders prefer round sizes, the percentage of trades at multiples of 500 shares
will be significantly greater than 33 percent. Similar intuition applies when examining
the percentage of trades that occur at multiples of 1,000 shares relative to trades that
occur at plus or minus 500 shares from the multiple. Hence, we include trades of 500,
1,000, 1,500, 2,000, 2,500, 3,000 shares, etc. If traders prefer multiples of 1,000 shares,
then the percentage of rounded trades will be significantly greater than 33%.
A binomial z-statistic tests the following hypothesis,
1

H 0 : Z  3

H : Z  1
 A
3
where Z is the percentage of trades (both short and non-short) that are multiples of 500
and 1,000 shares, respectively. Since research finds that non-short traders break up larger
trades into smaller trades in order to disguise their information to other market
participants (Barclay and Warner, 1993; Chakravarty, 2001; and Alexander and Peterson,
2007) but short selling research finds that informed short sellers prefer larger trade sizes
(Boehmer, Jones, and Zhang, 2008), the differences between the percentage of trades at
round sizes should be a function of the size of the trade. Since our hypothesis suggests
that informed short sellers are less concerned with negotiations costs, cognitive
processing costs, and the costs of revealing their information to market, large short sales,
which contain the most information, are less likely to be rounded than large non-short
trades.
10
We expect the difference between trade-size rounding for non-short trades and
short sales to be increasing across trade sizes.6 Trades are separated into three categories
similar to Boehmer, Jones, and Zhang (2008). Small trades are trades less than 2,000
shares, medium-sized trades are trades between 2,000 and 5,000 shares while large trades
are trades greater than or equal to 5,000 shares.7 Table 3 Panel A reports the percentages
of small non-short trades and small short sales that are rounded along with a p-value that
is obtained from the binomial z-test. First, we find that non-short traders of small sizes
prefer round multiples of 500 shares to unrounded sizes across each trade-size category
as the percentage of round trades is significantly greater than 33%. Small non-short
traders do not prefer multiples of 1,000 shares for the small trade-size category as the
percentage of rounded non-short trades is significantly less than 33 percent, suggesting
that small non-short traders prefer trades of 500 shares or 1,500 shares to trades of 1,000
shares. The rounding preference is similar between samples. Second, we find that short
sellers of small sizes may also prefer round sizes to unrounded sizes as results for short
sales are quite similar to the results for non-short trades. Because our objective in this
section is to compare the trade-size preferences of non-short traders and short sellers, we
examine the difference between the percentages of rounding for non-short trades and the
percentages of rounding for short sales. Interestingly, the difference between non-short
percentages and short sale percentages in Panel A shows that short sellers of small sizes
prefer rounded trades more than non-short traders as evidenced by 500 share multiples on
6
For this reason, we do not examine the percentage of trades at rounded multiples of 5,000 shares because
these multiples are all categorized as large trades and our hypothesis argues that the rounding decision is
based on these trade-size categories.
7
Boehmer, Jones, and Zhang (2008) separates short sales into small trade sizes (< 500), medium-small
trade sizes (500 to 1,999), medium-large trade sizes (2,000 to 4,999), and large trade sizes (5,000). The
small size group is extended to 2,000 shares in order to test if smaller short sales are executed on a multiple
of 500 shares.
11
the NYSE and 1,000 share multiples for the Nasdaq sample. A similar examination of
Panel B shows that medium-sized short sales and medium-sized non-short trades are
similarly rounded. The only exception is for multiples of 1,000 shares as the NYSE short
sales cluster less on round sizes while Nasdaq short sales tend to cluster more on round
sizes.
Comparing the preferences of round multiples in Panel C reveals that nearly
systematically, clustering on round multiples is much lower for short sales than for nonshort trades as the reported differences are positive in each column (the difference in [3]
is insignificant). These results support the notion that large short sales are motivated by
information and informed short sellers are less concerned with negotiations costs and
cognitive processing costs. Short sellers of large sizes may be more concerned with the
information contained in the transaction, not necessarily the terms of the transaction.
Additional tests of the hypothesis that clustering depends on trade sizes are
provided in a regression framework. Similar to Alexander and Peterson (2007), the
following equation is estimated:
ln perci = δ0 + δ1D500_Si + δ2D500_Mi + δ3D500_Li + δ4D1000_Si + δ5D1000_Mi + δ6D1000_Li +
δ7D5000i + εi (1)
The dependent variable is the natural log of the percentage of trades that occur at size i.
The independent variables are seven dummy variables that capture whether or not size i is
a round trade size while accounting for the size of the trade. D500_S is equal to one if size
i is 500, 1,000, or 1,500 shares. D500_M is equal to one if size i is 2,000, 2,500, 3,000,
3,500, 4,000, or 4,500 shares. Likewise, D500_L is equal to one if size i is equal to
multiples of 500 greater than or equal to 5,000 shares. The other dummy variables are
12
similarly defined. All trade sizes that are multiples of 100 shares and less than or equal to
15,000 shares are included in the analysis. Equation (1) is estimated for non-short trades
and for short sales and a comparison of the estimated parameters is discussed. If
informed short sellers prefer round sizes less than non-short traders, then the estimates for
δ3, the coefficient for trades rounded to 500 shares in the large size category, and δ6, the
coefficient for trades rounded to 1,000 shares in the large size category, will be
significantly larger for non-short trades than for short sales.
The results estimating equation (1) using robust White (1980) standard errors are
reported in Table 4. Panel A reports the regression results. Our first observation in the
table is that clustering on round sizes is not robust to different trade sizes. The estimates
for 1,000 share dummy variables (δ1000_S, δ1000_M, and δ1000_L) are not significantly
different from zero. When comparing the estimates for δ500_S, δ500_M, δ500_L, and δ5000 in
columns (1) and (2), we see that each of the estimates in column (1) is larger than in
column (2).8 Again, the estimates for for δ500_S, δ500_M, δ500_L, and δ5000 in column (3) are
larger than the same estimates in column (4).9 There is no evidence of clustering on large
multiples of 500 shares for NYSE or Nasdaq short sales while non-short trades appear to
cluster on large multiples of 500 shares. This result is also consistent with the notion that
short sellers of large sizes are not concerned with the terms of the transaction. The
results of Table 4 provide some support for the notion that non-short traders prefer round
sizes more than informed short sellers.
8
F-statistics comparing the estimates are 1.38 (p-value = 0.240) for δ500_S, 2.92 (p-value = 0.088) for
δ500_M, 44.53 (p-value = 0.000) for δ500_L, and 1.00 (p-value = 0.318) for δ5000.
9
F-statistics comparing the estimates are 0.39 (p-value = 0.531) for δ500_S, 0.51 (p-value = 0.476) for
δ500_M, 1.45 (p-value = 0.229) for δ500_L, and 1.26 (p-value = 0.263) for δ5000.
13
Thus far, we find evidence consistent with our hypothesis that short sales cluster
less on round sizes than non-short trades because informed short sellers are less likely to
be concerned with cognitive processing costs, negotiations costs, and the costs of
revealing information to the market. This evidence appears to be driven by larger short
sales, which likely contain the most information (Boehmer, Jones, and Zhang, 2008).
Because our hypothesis is based on the informativeness of different sized short sales, we
examine the ability of short sales to predict negative intraday returns.
In unreported results, we examine 30-minute and 60-minute returns after rounded
and unrounded short sales.10 We also partition these tests by different trade sizes. For
NYSE short sales we find that 60-minute returns following large short sales are nearly
twice as negative as the 60-minute returns that follow small short sales. Consistent with
Boehmer, Jones, and Zhang (2008), we find that returns are monotonically decreasing
across increasing trade sizes, indicating that larger short sales contain more information
than smaller short sales. We find similar results for Nasdaq short sales although returns
following larger short sales are only slightly more negative than returns that follow small
short sales. Both Boehmer, Jones, and Zhang (2008) and Diether, Lee, and Werner
(2009b) suggest that stealth trading is more likely to occur in Nasdaq stocks than in
NYSE stocks because of less restrictive price-tests, which may partially explain our
results.
In general, our untabulated results show that unrounded short sales, particularly in
the larger trade-size category, are better at predicting negative intraday returns than
rounded short sales. These results are consistent with the idea that short sellers do not
attempt to disguise the information in their trades by using round trade sizes.
10
Results are available upon request.
14
IV.B Rounded Prices
We now examine the preferences of short sellers and non-short traders for round
prices. Alexander and Peterson (2007) argue that clustering on round prices may exist
for the some of the same reasons that trade-size rounding exists, namely the behavioral
and negotiations hypotheses.
Figures 2.A to 2.D show the frequency of non-short trades and short sales at $0.01
increments for both the NYSE and Nasdaq samples. Similar to previous research, the
figures show substantial clustering on round price fractions. Whole dollars appear to be
more popular than half-dollars, which are more popular than quarters. Further, tenths
(multiples of dimes) appear to be more popular than twentieths (multiples of nickels). In
figures 2.A and 2.B, it appears that price clustering on the NYSE occurs more frequently
for non-short trades than for short sales as the price increments between nickels are
smoother for non-short trades than for short sales. In figures 2.C and 2.D, it is more
difficult to determine if price clustering on Nasdaq differs between non-short trades and
short sales.
Similar to our initial analysis of trade-size rounding, we calculate the percentages
of non-short trades and short sales that execute at round prices. We compare the
percentages between non-short trades and short sales to determine if non-short trades
execute at rounded prices more than short sales. Round prices are defined as multiples of
$1.00, $0.50, $0.25, $0.10, and $0.05. Table 5 reports the results of the analysis. Panel
A reports price rounding for both non-short trades and short sales as well as the
difference between non-short and short trades for both the NYSE and the Nasdaq sample.
15
The NYSE results show that non-short trades execute at round prices significantly more
than short sales. This result is consistent with the hypothesis that short sellers, who are
more likely to be informed than non-short traders, are less concerned with negotiation
and cognitive processing costs.
Panels B through D show price rounding across the different trade sizes, where
the different trade sizes are defined as before. The results hold for each trade size in the
NYSE sample. However, the results are only significant for Nasdaq stocks in the small
trade-size category although there is some evidence in support of our hypothesis in
medium-sized trades.11 This observation may be due to the lack of large Nasdaq short
sales.
IV.C Simultaneous Price and Trade-Size Rounding
Following Alexander and Peterson (2007), we examine instances when both
round short sales and round non-short trades execute at round price fractions. Alexander
and Peterson argue that simultaneous trade-size rounding and price rounding provide a
more robust test of the negotiation and behavioral hypotheses. Therefore, trades that are
multiples of 500 shares and trades that are plus and minus 100 shares from the multiple
are examined. The cross-sectional mean of the percentage of round trades that occur at
round prices is estimated. We expect that informed short sellers will be less concerned
with negotiations and cognitive processing costs, which will be manifested in fewer
round short sales at round prices relative to non-short trades.
11
Figures 2.C and 2.D provide possible explanations. The results in Table 5, panels C and D may be a
result of sparse short-selling activity in the larger trade-size categories for the Nasdaq sample.
16
Table 6 presents the results of the t-tests for both the NYSE and the Nasdaq
samples. As previously, multiples of $1.00, $0.50, $0.25, $0.10, and $0.05 are shown.
Panel A reports the results for trades (both non-short and short) of all sizes. Consistent
with our expectations, round short sales occur less frequently at round prices and trade
sizes than non-short trades. The results from these t-tests further strengthen earlier
findings, which suggest that short sellers are less concerned with negotiations and
cognitive processing costs.
In Panel A, the analysis is performed across all trade sizes, while panel B is small,
panel C is medium, and panel D is large trade sizes. If larger short sales contain more
information (Boehmer, Jones, and Zhang, 2008) than smaller short sales, the difference
between the percentage of round trades at round prices will increase across trade sizes
because informed short sellers are less likely to be concerned with the terms of the
transaction. Short sellers prefer round trade sizes at round prices significantly less than
non-short traders for the NYSE sample. Further, the difference is increasing across
increasing trade sizes for the NYSE sample.12
A pooled bivariate probit regression provides additional tests of the simultaneous
occurrence of round sizes and round prices. Alexander and Peterson (2007) use a
bivariate probit model and show that, after controlling for other factors that influence the
rounding decision, there is a positive correlation between the likelihood that a trade is
rounded in size and the likelihood that a trade is rounded in price. Therefore, the
following equations are estimated simultaneously.
round size1 = β0 + β1 returnj,i,t-30 + β2 shortsizej,i,d + β3 volumej,i,d + β4 sizej,i,d +
12
While short sellers of Nasdaq stocks prefer round trade sizes at round prices less than non-short traders in
the small trade-size category, we do not find significance in larger trade sizes, which may, again, be
explained by fewer instances of short-selling activity in larger trade-size categories.
17
β5 r_voltj,i,d + β6 p_voltj,i,d + ε1
(2)
round price2 = β0 + β1 returnj,i,t-30 + β2 shortsizej,i,d + β3 volumej,i,d + β4 sizej,i,d +
β5 r_voltj,i,d + β6 p_voltj,i,d + ε2
Rho = Cov(ε1, ε2| X1, X2)
(3)
(4)
where the dependent variable in equation (2) is binary and equal to one if short sale i is a
multiple of 500 shares. Following Alexander and Peterson (2007), the dependent
variable in equation (3) is equal to one if short sale i occurs on a round price, which is
defined as a multiple of $0.05 or 1/20th of a dollar. The past 30 minute return for trade i
of stock j is included as a control. Other control variables include the size of the short
sale, the daily volume, size (market cap), return volatility, and price volatility (defined
before). Rho, in equation (4), is the estimated covariance of the error terms in equations
(2) and (3) conditioned on the control variables used in the analysis. Alexander and
Peterson find a positive estimate for Rho suggesting that, after controlling for other
factors that influence the rounding decision, traders prefer round sizes and round prices
simultaneously. If private information about the true value of stocks motivates short
sellers to trade quickly and be less concerned with the transaction terms, then we expect
the estimate for rho to be non-positive.
Table 7 reports the results from the estimating equations (2) and (3) using only
short sales. Consistent with the argument that information in short sales depends on past
intraday returns, the estimate for β1 is negative suggesting that after periods of high
returns, short sellers do not prefer round sizes or round prices. Size rounding is
positively related to the size of the short sale. Size and price rounding occur more on
18
high volume days. Return and price volatility are positively (negatively) related to the
likelihood of trade-size (price) rounding for both samples. In general, price and size
rounding occur more in lower cap stocks, which is consistent with the notion that smaller
stocks have less of a following by analysts and the costs of negotiating for equilibrium
prices are high, therefore, buyers and sellers settle on round sizes and round prices.
The estimate of primary interest is rho, the conditional covariance of the error
terms for the two equations. We observe non-positive estimates for both the NYSE and
Nasdaq samples (the estimate for rho is virtually zero in the NYSE sample and negative
in the Nasdaq sample), which suggest that, after controlling for other factors that
influence rounding decisions, short sellers are indifferent to round sizes at round prices.
These results differ from Alexander and Peterson (2007), whose findings are consistent
with negotiations and behavioral hypotheses, that traders prefer round sizes at round
prices. The results in our paper show that short sellers are, at least indifferent, suggesting
that they are less concerned with negotiation and cognitive processing costs. Our results
infer that short sellers’ trade-size choices are motivated differently than the trade-size
choices of non-short traders. Short sellers, who are contrarian in past returns and add to
the informational efficiency in prices, are arguably less concerned with costs of
negotiations, cognitive processing costs, and the costs of revealing their information to
the market. Since Boehmer and Wu (2008) finds that short sales reduce pricing errors as
informed traders are able to execute short sales for stocks that become overvalued, short
sellers likely view the timing of and the information contained in the transaction as more
important than the terms of the transaction.
19
IV. Conclusion
Alexander and Peterson (2007) show that trades tend to cluster on round sizes and
argue that this observed clustering is explained primarily by three hypotheses. The
behavioral hypothesis suggests that traders think in round number and therefore prefer
round trade sizes and round prices (Wyckoff, 1963; Niederhoffer and Osborne, 1966; and
Ikenberry and Weston 2003). The negotiations hypothesis argues that traders will settle
on round numbers to avoid the additional costs of continuing negotiations (Ball, Torous,
and Tschoegl, 1985; Grossman et al., 1997; and Harris, 1991). The stealth trading
hypothesis contends that informed traders are motivated to break up their larger trades
into smaller trades to avoid revealing their information to other sophisticated market
participants (Kyle, 1985; Admati and Pfleiderer, 1988; and Barclay and Warner, 1993).
Alexander and Peterson argue and find some evidence that using round sizes may also
disguise the information of informed traders.
Following Alexander and Peterson (2007), this study examines the clustering of
trades on round sizes and at round prices, but separates trades into two distinct categories,
short sales and regular (non-short) trades. Our purpose in doing so is motivated by two
streams of research. First, several papers document that short sellers are informed about
the true value of stocks (Diamond and Verricchia, 1987; Senchack and Starks, 1993;
Aitken et al., 1998; Desai et al., 2002; and Christophe, Ferri, and Angel, 2004) and that
larger short sales contain more information about future firm performance than smaller
short sales (Boehmer, Jones, and Zhang, 2008). The latter result indicates that short
sellers do not stealth trade, which, in the framework of trade-size clustering, suggests that
short sellers have less motivation to prefer round sizes. Second, Diether, Lee, and
20
Werner (2009) show that short sellers are contrarian in contemporaneous and past daily
returns and are able to accurately predict price reversals while Boehmer and Wu (2008)
show that daily short selling reduces pricing errors and add to the informational
efficiency of stock prices. Combined, these studies indicate that short sellers are
informed about the true value of stocks and tend to target stocks that become temporarily
overvalued. If the profitability of short selling depends on information about temporary
overvaluation, then short sellers are likely not willing to settle on round prices nor are
they likely to be overly concerned about cognitive processing costs. For these reasons,
we hypothesize that short sales will cluster less on both round trade sizes and round
prices.
Consistent with our contention, we first find that short sales generally cluster less
on round trade sizes than non-short trades. This result is driven by larger short sales,
which contain the most information (Boehmer, Jones, and Zhang, 2008) suggesting that
short sellers are less concerned with minimizing negotiations costs and cognitive
processing costs, and are more concerned with exploiting the information in their trades.
Second, we show that short sales cluster less on round prices than non-short trades. Price
rounding for short sales is driven by smaller cap stocks on days with high volume
indicating that less known stocks during high levels of uncertainty motivate short sellers
to prefer round prices. Third, using short sales transactions data, we find that price
rounding and trade-size rounding is independent for the NYSE sample and negatively
correlated for the Nasdaq sample. Alexander and Peterson (2007) argue that the
simultaneous occurrence of price rounding and trade-size rounding is a robust test of the
negotiations and behavioral hypothesis. Observing a non-positive correlation between
21
these two types of rounding indicates that short sellers are not concerned with the costs of
negotiating and the cognitive processing costs. In unreported results, we find that large
short sales predict larger negative returns than smaller short sales. Further we observe
that large unrounded short sales predict negative returns better than large rounded short
sales indicating that informed short sellers do not use round sizes to disguise the
information in their trades. Combined, results from this study suggest that short sellers
are more concerned with exploiting their private information.
Further, our results provide additional robustness to the findings of Alexander and
Peterson (2007). While short sales make up nearly 24 percent of trading volume on the
NYSE and 31 percent on Nasdaq (Diether, Lee, and Werner, 2009), our findings suggest
that the results in Alexander and Peterson would be stronger in magnitude than first
documented if short sales were excluded.
22
References:
Admati A. and P. Pfleiderer, 1988, “A Theory of Intraday Patterns: Volume and Price
Variability”. Review of Financial Studies, 1, 3-40.
Aitken, M., A. Frino, M. McCorry, and P. Swan, 1998, “Short Sales are Almost
Instantaneously Bad News: Evidence from the Australian Stock Exchange”.
Journal of Finance, 53, 2205-2223.
Alexander, G.J. and M.A. Peterson, 2007, “An Analysis of Trade-Size Clustering and its
Relation to Stealth Trading”. Journal of Financial Economics, 84, 435-471.
Ball, C.A., W.A. Torous, and A.E. Tschoegl, 1985, “The Degree of Price resolution: The
Case of the Gold Market”. Journal of Futures Markets 5, 29-43.
Barclay, M.J. and J.B. Warner, 1993, “Stealth Trading and Volatility: Which Trades
Move Prices?”. Journal of Financial Economics, 34, 281-306.
Bessembinder, H., 2003, “Trade Execution Costs and Market Quality after
Decimalization”. Journal of Financial and Quantitative Analysis, 38, pg. 747777.
Boehmer E., C.M. Jones, and X. Zhang, 2008, “Which Shorts are Informed?” Journal of
Finance 63, 491-527.
Boehmer, E., and J. Wu, 2008, “Short Selling and the Informational Efficiency of
Prices”. Working paper, Texas A&M University.
Chakravarty, S. and J.J. McConnell, 1997, “An Analysis of Prices, Bid/Ask Spreads and
Bid/Ask Depths Surrounding Ivan Boesky’s Illegal Trading in Carnation’s
Stock”. Financial Management, 26, 18-34.
Chakravarty, S., 2001, “Stealth Trading: Which Trader’s Trades Move Prices?”. Journal
of Financial Economics, 61, 289-307.
Christophe, S.; M. Ferri; and J. Angel, 2004, “Short selling Prior to Earnings
Announcements”. Journal of Finance, 59, 1845-1875.
Cornell, B. and E. Sirri, 1992, “The Reaction of Investors and Stock Prices to Insider
Trading”. Journal of Finance, 47, 1031-1059.
Dechow, P., A. Hutton, L. Meulbroek, and R. Sloan, 2001, Short-sellers, fundamental
analysis, and stock returns, Journal of Financial Economics 61, 77-106.
23
Desai, H., K. Ramesh, S. Thiagarajan, and B. Balachandran, (2002). “An Investigation
of the Information Role of Short Interest in the NASDAQ Market”. Journal of
Finance, 52, 2263-2287.
Diamond, D. and R. Verrecchia, 1987, “Constraints on Short Selling and Asset Price
Adjustment to Private Information”. Journal of Financial Economics, 18, 277312.
Diether, K., K.-H. Lee, and I. Werner, 2009, “Short-sale strategies and return
predictability”. Review of Financial Studies 22, 575-607.
Diether, K., K.-H. Lee, and I. Werner, 2009b, “It’s SHO Time, Short-Sale Price Tests and
Market Quality”. Journal of Finance 64, 37-63.
Foster, F.D., and S. Viswanathan, 1993, “A Theory of the Interday Variations in Volumn,
Variance, and Trading Costs in Securities Markets”. Review of Financial Studies
4, 593-624.
Grossman, S., M. Miller, K. Cone, D. Fischel, and D. Ross, 1997. “Clustering and
Competition in Asset Markets”. Journal of Law and Economics, 40, 23-60.
Harris, L., 1991, “Stock Price Clustering and Discreteness”. Review of Financial
Studies, 4, 389-415.
Ikenberry, D., and J.P. Weston, 2003, “Clustering in US Stock Prices after
Decimalization”. Working paper, Rice University.
Koski, J.L. and J.T. Scruggs, 1998, “Who Trades on the Ex-Dividend Day? Evidence
from the NYSE Audit File Data”. Financial Management, 27, 58-72.
Kyle, A.S., 1985, “Continuous Auctions and Insider Trading”. Econometrica, 53, 13151335.
Lo, A. W. and A.C. MacKinlay, 1990, “When are Contrarian Profits Due to Stock Market
Overreaction?”. Review of Financial Studies, 3, 175-206.
Meulbroek, L., 1992, “An Empirical Analysis of Illegal Insider Trading”. Journal of
Finance, 47, 1661-1699.
Moulton, P., 2005, “You Can’t Always Get What You Want: Trade-size Clustering and
Quantity Choice in Liquidity”. Journal of Financial Economics, 78, 89-119.
Niederhoffer, V., and M.F.M. Osbourne, 1966, “Market Making and Reveral on the
Stock Exchange”. Journal of the American Statistical Association, 61, 897-916.
24
Senchack, A. J., and L.T. Starks, 1993, “Short-sale Restrictions and Market Reaction to
Short-Interest Announcements”. Journal of Financial and Quantitative Analysis
28, 177-194.
Wyckoff, P., 1963, Psychology of Stock Market Timing. Prentice-Hall, Englewood
Cliffs, NJ.
25
Table 1
Descriptive Statistics
The table presents statistics that describe the sample of stocks used in the analysis. The sample includes 400 (200 from the NYSE and 200 from Nasdaq)
randomly selected ordinary common stocks (share code 10 or 11), that trade every day of third quarter 2006, and have a price greater than $5. CRSP daily
ending prices, returns, and market capitalization are included as well as two measures of volatility. Return volatility is defined as the standard deviation of daily
returns from day t-10 to day t, where day t is the current trading day. Price volatility is defined and the daily high price less the daily low price divided by the
daily high price. After matching the Reg SHO data to the TAQ data, we obtain which trades where short sales and which trades were non-short (or regular)
trades. Panel A reports the statistics for the NYSE sample while panel B reports the results for the Nasdaq sample.
Panel A. NYSE Sample
Non Short Sale
Short Sale
Price
Return
Market Cap
Return Volatility
Price Volatility
Volume
Volume
Mean
32.72
0.0002
6,795,153.62
0.0188
0.0255
374,198.53
60,516.33
St. Dev
19.42
0.0021
15,126,176.55
0.0070
0.0084
691,041.64
84,143.38
Min
5.90
-0.0073
78,866.98
0.0041
0.0074
8,385.48
485.48
Max
116.23
0.0070
103,663,861.00
0.0397
0.0494
7,431,688.79
916,209.23
N
200
200
200
200
200
200
200
Panel B. Nasdaq Stocks
Mean
22.64
0.0004
940,748.70
0.0244
0.0335
55,365.07
33,546.90
St. Dev
17.81
0.0025
2,105,532.66
0.0088
0.0106
70,344.67
39,977.04
Min
5.21
-00093
22,573.51
0.0024
0.0036
1,656.10
423.41
Max
202.05
0.0080
24,454,986.02
0.0579
0.0690
492,575.33
252,104.70
N
200
200
200
200
200
200
200
26
Table 2
Trade Size Rounding
The table reports the percentage of short sales that are executed on round trade sizes. Panel A reports the
raw percentages of non-short trades and short sales that are rounded at multiples of 500 shares, 1,000
shares, and 5,000 shares with their differences for all stocks in our sample for the NYSE sample. Similarly,
Panel B reports the results for the Nasdaq sample. We report t-statistics in parentheses.
Panel A. NYSE Sample
% 500 Shares
% 1,000 Shares
% 5,000 Shares
[1]
[2]
[3]
Non-short Trades
5.8146%
1.8281%
0.1577%
Short Sales
5.2062
1.5485
0.0569
Difference
0.6084***
(0.000)
0.2796***
(0.000)
0.1007***
(0.000)
Non-short Trades
1.8994%
0.6316
0.0704
Short Sales
1.9804
0.8143
0.0537
Difference
-0.0810
(0.374)
-0.1830**
(0.011)
0.0168**
(0.019)
Panel B. Nasdaq Stocks
*,**,*** Statistically significant at the 0.10, 0.05, and 0.01 levels
27
Table 3
Rounding Preference
The table reports the percentage of short sales and non-short trades that are executed on round trade sizes. The
table shows the percentage of non-short trades and short sales that are rounded on multiples of 500 shares
compared with non-short trades and short sales that are either 100 shares less than or greater than the multiple
of 500 shares. Similarly, the percentage of non-short trades and short sales that are of multiples of 1,000 shares
relative to non-short trades and short sales that are either 500 shares greater than or less than the multiple of
1,000 shares are reported. A binomial Z-statistic tests whether rounded short sales are preferred to the
alternative. That is, *,**, and *** indicate that the percentage of short sales of 500 shares, for instance, is
significantly greater than the percentage of short sales that are 400 shares and the percentage of short sales that
are 600 shares. Panel A presents the results for small trades (less than 2,000 shares), while Panel B and Panel C
report results for medium trades (between 2,000 and 5,000 shares) and large trades (greater than 5,000 shares),
respectively. The results are reported separately for the NYSE sample and the Nasdaq sample. In each panel,
we also calculate the difference between the percentages for non-short trades and short sales with a
corresponding p-value.
Panel A. Small Trade-size Category
NYSE Stocks
Nasdaq Stocks
% Rounded 500
% Rounded
% Rounded 500
% Rounded
Shares
1,000 Shares
Shares
1,000 Shares
[1]
[2]
[3]
[4]
Non-Shorts
Short Sales
Difference
36.1818***
(0.000)
36.6854***
(0.000)
-0.5036**
(0.032)
24.0305***
(0.000)
24.1373***
(0.000)
-0.1070
(0.732)
50.4196***
(0.000)
50.1735***
(0.000)
0.2461
(0.742)
26.7109***
(0.000)
29.8209***
(0.000)
-3.1130***
(0.000)
46.1503***
(0.000)
43.8325***
(0.000)
2.3180**
(0.028)
73.4371***
(0.000)
72.4355***
(0.000)
1.0016
(0.783)
48.6437***
(0.000)
53.3305***
(0.000)
-4.6870*
(0.052)
72.8373***
(0.000)
63.1714***
(0.000)
9.6660***
(0.000)
82.0838***
(0.000)
81.2988***
(0.000)
0.7850
(0.480)
87.8021***
(0.000)
83.8388***
(0.000)
3.9630***
(0.005)
Panel B. Medium Trade-size Category
Non-Shorts
Short Sales
Difference
46.8599***
(0.000)
45.2111***
(0.000)
1.6488
(0.107)
Panel C. Large Trade-size Category
Non-Shorts
Short Sales
Difference
57.6111***
(0.000)
51.7165***
(0.000)
5.8946***
(0.003)
28
Table 4
Regression Results
The table reports the results from estimating the following equation.
ln perci = δ0 + δ1D500_Si + δ2D500_Mi + δ3D500_Li + δ4D1000_Si + δ5D1000_Mi + δ6D1000_Li + δ7D5000i + εi
The dependent variable is the natural log of the percentage of trades that occur at size i. We include as
independent variables seven dummy variables that capture whether or not size i is a rounded trade size while
accounting for the size of the trade. D500_S is equal to unity if size i is 500, 1,000, or 1,500 shares. D 500_M is
equal to one if size i is 2,000, 2,500, 3,000, 3,500, 4,000, or 4,500 shares. Likewise, D 500_L is equal to unity if
size i is equal to multiples of 500 greater than or equal to 5,000 shares. The other dummy variables are
similarly defined. We estimate the difference between the estimated coefficients from the non-short trades
regression and the short sales regression and report a t-statistic testing whether the difference is significant. We
include all trade sizes that are multiples of 100 shares and less than 15,000 shares. The p-values calculated
from robust White (1980) standard errors are reported in parentheses.
NYSE sample
Nasdaq sample
Non-short trades
Short sales
Non-short trades
Short sales
[1]
[2]
[3]
[4]
intercept
-9.7794***
-8.1821***
-9.5942***
-9.1610***
(0.000)
(0.000)
(0.000)
(0.000)
δ500_S
10.0856***
8.2941***
8.3519***
7.7524***
(0.000)
(0.000)
(0.000)
(0.000)
δ500_M
7.1101***
4.9822***
4.9039***
4.3475***
(0.000)
(0.000)
(0.000)
(0.000)
δ500_L
1.9448***
-0.0197
0.5631*
0.2496
(0.000)
(0.958)
(0.031)
(0.354)
δ1000_S
0.1375
0.3037
0.5382
0.6092
(0.958)
(0.911)
(0.745)
(0.706)
δ1000_M
0.8492
0.8194
1.0930
1.1528
(0.629)
(0.652)
(0.322)
(0.284)
δ1000_L
0.4351
0.0218
0.1103
0.0909
(0.318)
(0.967)
(0.757)
(0.813)
δ5000
2.0853***
1.3846*
2.6216***
2.1063***
(0.003)
(0.073)
(0.000)
(0.000)
Adj R2
0.1098
0.0811
*,**,*** Statistically significant at the 0.10, 0.05, and 0.01 levels
29
0.0146
0.0205
Table 5
Price Rounding
The table reports the raw percentage of trades (both short and non-short) that occur on multiples of $1.00, $0.50, $0.25, $0.10, and $0.05. Panel A reports the
results for all sizes while panels B through D report the results for small, medium, and large sizes, respectively. We calculate the difference between the
percentage of non-short trades and short sales that occur at the pricing increment and use t-test to determine if the difference is statistically significant. We
report p-values in parentheses.
Panel A. All Sizes
NYSE Stocks
Nasdaq Stocks
$1.00
$0.50
$0.25
$0.10
$0.05
$1.00
$0.50
$0.25
$0.10
$0.05
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
Non-shorts
2.04%
3.71%
6.29%
14.27%
26.16%
2.97%
5.11%
8.24%
16.35%
29.10%
Shorts
1.57
Difference
0.47***
(0.000)
Panel B. Small Sizes
Non-shorts
1.93
Shorts
1.54
Difference
0.39***
(0.000)
Panel C. Medium Sizes
Non-shorts
0.069
Shorts
0.028
Difference
0.042***
(0.000)
Panel D. Large Sizes
Non-shorts
0.045
Shorts
Difference
0.010
2.94
5.24
12.43
23.36
2.44
4.18
7.03
14.66
26.42
0.77***
(0.000)
1.05***
(0.000)
1.84***
(0.000)
2.80***
(0.000)
0.53***
(0.000)
0.93***
(0.000)
1.21***
(0.000)
1.69***
(0.000)
2.68***
(0.000)
3.51
5.99
13.67
25.16
2.95
5.06
8.16
16.21
28.85
2.87
5.12
12.17
22.91
2.40
4.11
6.94
14.47
26.11
0.64***
(0.000)
0.87***
(0.000)
1.50***
(0.000)
2.25***
(0.000)
0.54***
(0.000)
0.94***
(0.000)
1.22***
(0.000)
1.73***
(0.000)
2.74***
(0.000)
0.125
0.194
0.387
0.659
0.025
0.038
0.060
0.104
0.176
0.054
0.091
0.191
0.342
0.027
0.045
0.064
0.000
0.000
0.071***
(0.000)
0.103***
(0.000)
0.196***
(0.000)
0.317***
(0.000)
-0.002
(0.727)
-0.007
(0.272)
-0.005
(0.542)
0.104***
(0.000)
0.176***
(0.000)
0.073
0.108
0.209
0.340
0.009
0.015
0.026
0.041
0.074
0.018
0.031
0.064
0.108
0.011
0.017
0.025
0.048
0.077
0.232***
(0.000)
-0.003
(0.184)
-0.002
(0.460)
0.001
(0.834)
-0.007
(0.159)
-0.003
(0.682)
0.035*** 0.055***
0.078***
0.145***
(0.000)
(0.000)
(0.000)
(0.000)
*,**,*** Statistically significant at the 0.10, 0.05, and 0.01 levels
30
Table 6
Trade-Size and Price Rounding
The table reports the percentage of short sales and non-short trades that are executed on round trade sizes at round prices. We include trades that are multiples of
500 shares compared with short sales that are either 100 shares less than or greater than the multiple of 500 shares and calculate the percentage of rounded trades
that occur at the round price increments. Panel A reports the results for all sizes while panels B through D report the results for small, medium, and large sizes,
respectively. We report p-values from testing the difference between the mean percentages for non-shorts and shorts in parentheses.
Panel A. All Sizes
NYSE Stocks
Nasdaq Stocks
$1.00
$0.50
$0.25
$0.10
$0.05
$1.00
$0.50
$0.25
$0.10
$0.05
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
Non-shorts
1.184
2.057
3.262
6.994
12.202
2.967
4.796
7.248
12.991
21.367
Shorts
0.826
Difference
0.359***
(0.000)
Panel B. Small Sizes
Non-shorts
2.683
Shorts
1.912
Difference
0.771***
(0.000)
Panel C. Medium Sizes
Non-shorts
5.145
Shorts
3.539
Difference
1.476***
(0.001)
Panel D. Large Sizes
Non-shorts
8.989
Shorts
Difference
6.023
1.529
2.546
5.572
10.114
2.463
4.037
6.153
11.714
19.836
0.528***
(0.000)
0.715***
(0.000)
1.423***
(0.000)
2.088***
(0.000)
0.441**
(0.036)
0.625**
(0.018)
0.973***
(0.005)
1.184***
(0.008)
1.481***
(0.007)
4.713
7.650
16.716
29.744
5.233
8.333
12.497
22.757
37.672
3.603
6.222
13.858
25.724
4.011
6.664
10.374
20.624
35.034
1.110***
(0.000)
1.428***
(0.000)
2.857***
(0.000)
4.020***
(0.000)
1.091***
(0.000)
1.428***
(0.000)
1.871***
(0.000)
1.966***
(0.001)
2.514***
(0.000)
8.524
12.744
24.975
41.614
5.929
9.701
15.513
26.302
44.851
6.719
9.901
19.554
33.761
6.167
9.744
14.753
28.167
45.515
1.635***
(0.003)
2.659***
(0.000)
5.254***
(0.000)
7.636***
(0.000)
-0.182
(0.849)
-0.136
(0.905)
0.732
(0.596)
-2.311
(0.208)
-1.139
(0.556)
14.004
19.491
35.385
56.532
6.052
10.563
15.837
27.466
44.947
9.862
14.112
27.008
43.001
7.020
9.956
15.110
28.288
47.974
11.959***
(0.000)
-0.729
(0.617)
0.713
(0.689)
1.102
(0.589)
-0.120
(0.960)
-2.231
(0.934)
2.891***
3.720***
5.051***
7.075***
(0.007)
(0.006)
(0.001)
(0.000)
*,**,*** Statistically significant at the 0.10, 0.05, and 0.01 levels
31
Table 7
Pooled Bivariate Probit Estimates
The table presents the results of from estimating the following equations simultaneously.
round size1 = β0 + β1returnj,i,t-30 + β2shortsizej,i,d + β3volumej,i,d + β4sizej,i,d + β5r_voltj,i,d +
β6p_voltj,i,d + ε1
round price2 = β0 + β1returnj,i,t-30 + β2shortsizej,i,d + β3volumej,i,d + β4sizej,i,d + β5r_voltj,i,d +
β6p_voltj,i,d + ε2
The dependent variables are binary variables that capture whether the short sale was rounded in size (multiple
of 500 shares) or rounded in price (multiple of $0.05). The independent variables are the lagged 30 minute
return, the short sale size, the daily volume, the size (market cap), the daily return volatility, and the daily price
volatility. To test for a correlation between the size rounding decision and the price rounding decision, the
bivariate probit model estimates rho, which is defined as the following.
Rho = Cov(ε1, ε2| X1, X2)
which is defined as the estimated covariance of the error terms from the two objective functions conditioned on
the matrices of control variables used in the estimation. We perform estimate the model for the NYSE sample
and the Nasdaq sample. The estimated parameters are reported along with the corresponding p-value (in
parentheses).
NYSE Stocks
Nasdaq Stocks
Trade-size
Trade-size
Rounding
Price Rounding
Rounding
Price Rounding
[1]
[2]
[3]
[4]
intercept
rett-30
short size
volume
cap
r_volt
p_volt
rho
-2.2900***
(0.000)
-0.6277
(0.149)
0.2742***
(0.000)
0.0254***
(0.000)
-0.0124***
(0.000)
0.3309
(0.158)
0.8394***
(0.000)
-0.7163***
(0.000)
-7.0195***
(0.000)
0.0514***
(0.000)
0.1809***
(0.000)
-0.1751***
(0.000)
-3.4251***
(0.000)
-10.2956***
(0.000)
1.3E-6
(0.996)
-5.1542**
(0.000)
-0.5725
(0.305)
0.7984**
(0.000)
0.0028
(0.606)
-0.0121*
(0.047)
2.6868**
(0.000)
2.3561**
(0.000)
0.7450***
(0.000)
-0.7270
(0.224)
-0.0174**
(0.021)
0.1512***
(0.000)
-0.2179***
(0.000)
-4.3166***
(0.000)
-8.5658***
(0.000)
-0.0444**
(0.000)
*,**,*** Statistically significant at the 0.10, 0.05, and 0.01 levels
32
NYSE Non-Short Trade-Size Clustering
10,000,000
1,000,000
Log Frequency
100,000
10,000
1,000
100
10
1
0
10
20
30
40
50
60
70
80
90
100
Trade Size (00s)
Figure 1.A. The figure shows the logarithmic frequency of the number of non-short trades
for each trade size on the NYSE.
NYSE Short-Sale Size Clustering
10,000,000
1,000,000
Log Frequency
100,000
10,000
1,000
100
10
1
0
10
20
30
40
50
60
70
80
90
100
Trade Size (00s)
Figure 1.B. The figure shows the logarithmic frequency of the number of short sales for each
trade size on the NYSE.
Nasdaq Non-Short Trade-Size Clustering
10,000,000
1,000,000
Log Frequency
100,000
10,000
1,000
100
10
1
0
10
20
30
40
50
60
70
80
90
100
Trade Size (00s)
Figure 1.C. The figure shows the logarithmic frequency of the number of non-short trades for
each trade size on Nasdaq.
Nasdaq Short-Sale Size Clustering
10,000,000
1,000,000
Log Frequency
100,000
10,000
1,000
100
10
1
0
10
20
30
40
50
60
70
80
90
100
Trade Size (00s)
Figure 1.D. The figure shows the logarithmic frequency of the number of short sales for each
trade size on Nasdaq.
33
NYSE Non-Short Trade Price Clustering
180,000
160,000
140,000
Frequency
120,000
100,000
80,000
60,000
40,000
20,000
0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
Ce nts
Figure 2.A. The figure shows the frequency of non-short trades at each cent increment for
NYSE stocks.
NYSE Short-Sale Price Clustering
35,000
30,000
Frequency
25,000
20,000
15,000
10,000
5,000
0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
Ce nts
Figure 2.B. The figure shows the frequency of short sales at each cent increment for NYSE
stocks.
Nasdaq Non-Short Trade Price Clustering
80,000
70,000
Frequency
60,000
50,000
40,000
30,000
20,000
10,000
0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
Ce nts
Figure 2.C. The figure shows the frequency of non-short trades at each cent increment for
Nasdaq stocks.
Nasdaq Short Sale Price Clustering
45,000
40,000
35,000
Frequency
30,000
25,000
20,000
15,000
10,000
5,000
0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
Ce nts
Figure 2.D. The figure shows the frequency of short sales at each cent increment for Nasdaq
stocks.
34