The Effect of Round Number Bias in
U.S. and Chinese Stock Markets
Tiansheng Guo 14
Princeton University
Advisor: Professor Wei Xiong
Assistant Instructor: Michael Sockin
April 24, 2013
This paper represents my own work in accordance with University regulations.
Abstract
This paper explores round number bias in a set of U.S. and Chinese large-cap and small-
cap stocks over a recent time period. It aims to distinguish manifested bias from inherent bias:
due to differences in market conditions, the degree of observed bias does not necessarily reflect
inherent bias of investors. The paper finds that U.S. stock prices manifest a lot more clustering
around round numbers than Chinese stocks, but after taking into account liquidity and price
levels, the degree of bias is about the same. In the U.S., small-caps exhibit more bias than large-
caps, but in China, it is the opposite. For Chinese stocks, we find no evidence for excess next day
returns around round numbers, but for U.S. stocks, there is negative excess returns, except for
U.S. large stocks, which has positive excess return if its previous-day closing price ends with
both decimals being round.
I. Introduction
The exploitation of round number bias is ubiquitous in retail and grocery markets
(grocery retailing). Prices are most often set just slightly less than a round number ($9.99, $9.95),
exploiting the irrational way in which our minds convert numerical symbols to analog
magnitudes for decision-making: prices just below a round number will be perceived to be a lot
smaller than the round number price due to the change in the leftmost digit (Thomas and
Morwitz, 2005). Because this slight drop in price is perceived by the mind to be proportionally
more, price is perceived to be lower than the value of a product, causing a discontinuity around
round number prices.
These round number biases extend beyond real assets into financial assets. Aggarwal and
Lucey (2005) presented evidence of barriers in gold prices due to round number bias, with
2 of 38
important effects on the conditional mean and variance. Johnson, Johnson and Shanthikumar
(2008) found significant differences in returns following previous-day closing prices around
round numbers in U.S. stock markets. In China, retail investors dominate the securities market,
and we expect round number bias to be more pronounced. These studies suggest investors biases
for round numbers are a source of irrationality and affect the price levels, which may result in
excess returns.
In this paper, we explore round number bias by analyzing price clustering around round
numbers and excess returns conditional on previous-day round numbers, for U.S. and China
during the time period 2001-2011. We compare the degree of bias between U.S. and Chinese
large-cap and small-cap stocks, which few previous studies have done, especially after the
decimalization of U.S. stock market in 2001. In order to make the comparison valid, we use a
methodological process for choosing stocks so that the U.S. and Chinese stock data are
comparable. We also control for varying amounts of liquidity and price levels in the different
data sets that may affect observed bias.
We expect that there will be little or insignificant effect from round number bias in the
U.S. stock market, due to the greater presence of more rational hedge funds and institutions, but
expect that in the Chinese stock market, where individual investors dominate, round number bias
should be greater. The results of this paper is interesting both practically and theoretically: a
significant finding for an uneven distribution of price levels (e.g. prices end in round numbers
more often) would challenge the price equals value sense of market efficiency because there is
no reason that value should end in certain digits more often than others; even if the effect of
3 of 38
round number bias on returns is too small to present arbitrage opportunities, the findings can still
help the precisely time high-frequency trades.
II.Literature Review
Round number bias is an innate human cognitive bias, and is present in prices and other
metrics. Pope and Simonsohn (2010) found that in baseball, the proportion of batters who hit .
300 (1.4%) was four times greater than those who hit .299 (0.38%), and there were also more
instance of .301 than of .298. They also found that students who take the SAT are much more
likely to retake it if they score just below a round number, even when there were no round
number bias on the side of the admissions officers. Individuals were willing to exert extra effort
to perform just above rather than below such numbers, hoping that the change in the left-most
digit will be seen as a much greater improvement than the marginal effort that is exerted.
The innate cognitive bias for certain numbers is also reflected in how individuals view
prices. Thomas and Morwitz (2005) found that nine-ending prices affect perception when the
leftmost digit changes, and that these effects are not limited to certain types of prices or products.
In financial markets, if the same preferences hold for certain numbers, we should see certain
price levels appear in trades more frequently than numbers that have no preferences, and perhaps
even excess returns.
Johnson, Johnson and Shanthikumar (2008) found that investors trade differently when
closing prices are just below a round number versus just above; when prices were just below a
round number, there were more selling, and if just above, more buying. They also found that
4 of 38
returns following closing prices just above a round number are significantly higher than returns
following prices just below.
Sonnemans (2003) examined the Dutch stock market during 1990-2001 and found that
for individual Dutch stocks, price levels cluster around round numbers, and round numbers act as
price barriers. Furthermore, it presents an interesting natural experiment: after January 1, 1999,
stock prices started to be listed in euros, converted from guilders (2.20371 guilders = 1 euro)
while guilders remained the currency of daily life. Immediately after this conversion and
numerical changes in prices, clustering in round guilder prices disappeared while price clustering
in round euro prices formed.
For Shanghai and Shenzhen stock exchanges, Brown and Mitchell (2004) used daily
opening, high, low, and closing prices, to analyze the final digit of the prices, and found
extremely clear clustering. On the SSE, the prices of A-shares traded were more than twice as
likely to end in 8 as in 4 for the period 1994-2002, but the effect has dissipated a little over time.
They also found much weaker preference for 8 for the corresponding B-shares on both
exchanges.
There are several explanations for round number bias in price levels of stocks. First,
security analysts tend to round forecasts, especially when there is much uncertainty, so traders
who read the forecasts will have expectations that are clustered around round numbers
(Herrmann, Thomas 2005). Second, when well-publicized stocks surpass significant price price
levels, there is more media coverage that would drive up the sentiment; even for indices that are
arbitrarily scaled (and does not say much about fundamentals), Donaldson and Kim (1993) found
support and resistance levels in round numbers in DJIA, but did not find these biases in less
5 of 38
popular indices. Third, traders may set target prices (aspiration level) for their stock, usually at
round numbers, so stop and limit orders may be clustered around round numbers.
Because of these encouraging findings from past studies, we analyze how the effect of the
bias differs in two drastically different countries, U.S. and China, using most recent data, which
few previous studies have done. It is interesting to compare these two countries; although round
number bias is caused by an innate cognitive flaw that is present in societies using arabic
numerals, U.S. and China have very different set of investors, laws, financial systems, culture,
and wealth distribution, which can all influence the degree of round number bias present in their
respective stock markets. If round number bias does manifest differently in U.S. and China,
futher study can be conducted on isolating which characteristic differences of U.S. and China
makes their markets more susceptible to this apparent irrationality.
This paper will run tests for price clustering and abnormal returns for daily closing prices
of 18 U.S. large-cap, 18 U.S. small-cap, 18 Chinese large-cap, and 18 Chinese small-cap stocks.
Another innovation of this paper is that it takes into account the possibility of liquidity and price
level as possible confounding variables to our findings in round number bias, so that the bias
manifested by investors does not equal their inherent bias. We will perform the same analysis
(price clustering, next day returns) on these numbers, and then adjust for liquidity and price
levels.
Because our data sets are more recent than those in previous studies, we expect to find
less evidence for round number bias in China, with the possibility of even excess returns around
round numbers, and expect to find even smaller effects of round number bias in U.S. stock data,
assuming U.S. investors are more sophisticated. The findings of this paper can illustrate how
6 of 38
round number bias has persisted through the most recent decade, in which there was a boom in
trading volume and investor sophistication. It can also show how large-caps and small-caps
manifest bias differently within U.S. and China.
III. Data
To analyze price clustering around round numbers and next day returns conditional on
round number prices, we will study daily closing prices and daily returns with cash dividend
reinvested, of a set of 36 U.S. stocks traded on NYSE and 36 Chinese stocks traded on the SSE
(A shares only), for the decade 6/1/2001 to 5/31/2011, which are all found on Wharton Research
Data Services. The starting date of 6/1/2001 is after the complete decimalization of the U.S.
stock market. The data sets exclude financials, are chosen randomly, and encompass a variety of
industries.
Among the 36 U.S. and 36 Chinese stocks, half are large-cap stocks and half are small-
cap stocks. The 18 U.S. large cap stocks are drawn from the 50 largest U.S. stocks, and the 18
Chinese large cap stocks are drawn from the 50 largest on the SSE. The 18 U.S. small cap stocks
are drawn from the market cap range 500M-800M, and the 18 Chinese small cap stocks are
drawn from stocks in the SSE SmallCap Index (China Securities Index).
The following Figure 1 lists the stocks used in the four data sets, Chinese large-cap,
Chinese small-cap, U.S. large-cap, and U.S. small-cap:
7 of 38
Figure 1: List of all stocks used
The number of firm-day observations over the 2001-2011 decade are as follows: 32,167
for Chinese Large Cap; 40,004 for Chinese Small Cap; 53,918 for US Large Cap; 40,647 for US
Small Cap stocks. A complete data set ideally contains 10*252*18 = 45360 firm-day
observations over the 10 years for 18 stocks, but there were some missing and extra data that
have negligible impact on our analysis. All closing prices that are 1.00 or below were deleted to
prevent cases where the leading digits are also the ending digits, to avoid complications with
Benfords Law, which states that leading digits in naturally occurring data is not uniform. Stocks
go through mergers and acquisitions and become listed under another ticker, yielding extra data,
or as with small stocks and Chinese stocks, data for earlier time periods were not available
because those companies were not publicly traded as early as 2001. Missing or extra data has
little impact as long as all observations belong in the correct category (US Large, Chinese small,
etc.).
The reason for using price levels as opposed to other measures such as P/E, P/B is that
8 of 38
prices levels are the final numbers seen when executing trades, although P/E or P/B may have
just as much evidence for numerical pricing biases, since they are especially susceptible to
security analysts rounding of forecasts or investors targets. In any case, many studies have
looked at just price levels and found robust results. Aggarwal and Lucey (2005) and Sonnemans
(2003) both used daily, unadjusted closing prices and found significant results in price clustering.
Johnson, Johnson and Shanthikumar (2008) used previous day closing prices that are just above
or below a round number to examine returns. For Shanghai, Brown and Mitchell (2004) used
daily opening, high, low, and closing prices, analyzing the final digit of the prices to observe
clustering, also with significant results. We use daily closing prices because they attract more
investor attention than a random price level during the day, and can linger in the minds of
investors after a day of trading, capturing much of the behavioral biases.
The reasons for drawing data from U.S. or China and large cap or small cap, are that
there is plentiful data from the two countries, and the financial markets of these two countries are
so different in terms of listed companies, investors, and regulations, that many extensions and
further studies can be done based on this finding; we expect different sets of investors to be
trading in large cap and small cap stocks, and different number of analysts covering the stocks,
so we expect the magnitude of round number biases to differ across market caps and countries.
For Chinese stocks, we draw from A shares listed on the SSE because it has a large market
capitalization and is not open to foreign investors, unlike the HKSE.
We choose the period June 2001 to June 2011 because the NYSE reported prices in
fractions (1/16) before 2001. The benefit of this decade is that we see a rise out of the dot-com
bubble and another rise and fall in prices from the Great Recession, which would allow a larger
9 of 38
range of price levels and potential for certain prices to cluster. This decade is interesting to
analyze because the advent of online trading allows many more unsophisticated traders to
participate in the stock market, but at the same time, institutional investors become more
sophisticated.
IV. Methodology
The paper will use a two-part analysis. The first part will analyze U.S. and Chinese stock
data for price clustering around round numbers. The second part will analyze next day returns
conditioning on round number closing price. Round number will be defined as prices ending in
one round decimal ($XY.Z0) or two round decimals ($XY.00).
Price clustering is defined as prices levels at which proportionally more trades occur, and
abnormal next day returns as a significant regression coefficient on a variable measuring round
number. If there were no price clustering, then the decimals of stock prices should be
distributed uniformly from .00 to .99. If there were no abnormal returns, then a previous day
closing price that ended in a round number would have no significant explanatory power in the
next day returns.
The price clustering analysis will be graphically presented in a frequency chart, tallying
the occurrences of round number closing price, categorized by country (U.S. vs. China) and size
(large cap vs. small cap), followed by a linear regression (with binary dependent variable). The
next day returns analysis will be conducted with linear regressions, as opposed to probit, for
easier interpretation of the coefficients. It uses ifone as a binary variable for the last decimal
10 of 38
being a round number, iftwo for both decimals, China for Chinese firms, and big for large
cap stocks. The two binary variables ifone and iftwo will be interacted with different
combinations of the other variables. An example regression is shown below:
reti = 0 + 1ifonei + 2iftwoi + 3ifonei China + 4iftwoi China + 5ifonei big + 6iftwoi big
This paper makes a distinction between manifested and inherent bias. Due do differences
in market conditions (liquidity, price levels) across China and U.S., the observed round number
bias may be an amplified measure of investors inherent bias. A second-round analysis takes this
into account and includes measures of liquidity and price levels to take out their effects from
price clustering and next day returns. Due to inaccessibility of order-book data, we use volume
as a crude measure of liquidity that may not be valid when comparing China and U.S., but can be
used to compare within those countries.
V. Price Clustering
First, we analyze manifested bias through simple price clustering analysis. Then, we
control for liquidity and price levels as amplifiers of bias, to observe inherent bias.
Results- Simple Price Clustering
11 of 38
The following graphs tally daily closing prices by last ending-decimal only, compared to
a line of average representing the expected number of observations assuming a uniform
distribution of price levels.
Figure 2
12 of 38
Figure 3
Figure 4
13 of 38
Figure 5
In all four data sets, there is a robust and persistent clustering around prices of the form
WX.Y0 and WX.Y5. Clustering is much stronger in U.S. data sets than in Chinese data sets, and
slightly stronger in small cap stocks than in large cap stocks. For U.S. data sets, clustering is
especially pronounced in prices that end in 5s, or WX.Y5, much more so than Chinese data
sets.
Next, we zoom in the same data sets by tallying closing prices by the last two ending-
decimals, compared to a line representing expected frequency given a uniform distribution.
14 of 38
Figure 6
Figure 7
15 of 38
Figure 8
Figure 9
16 of 38
The findings of two decimals analysis support that of one decimal: round number
bias in U.S. data is manifested much more than in Chinese data, and for Chinese data, bias in
small cap stocks is much more than in large cap stocks. Most of the prices ending in a 0 as the
last decimal have another 0 or a 5 as the decimal before it, so that much of the occurrences of
WX.Y0 are accounted for by WX.00 and WX.50. In the U.S., round number bias is so
strong that prices ending in .00 occured twice as often as in a uniform distribution. Prices
ending in .X0 (.10, .20, .30 etc.), and especially .50 all occurred more than the uniform
distribution in both U.S. and China, and additionally in U.S. only, all prices ending in .X5
occurred more than uniform.
Note that Chinese investors prefered prices ending in .X0 and not .X5, while U.S.
investors strongly prefered both. Additionally, in both U.S. large and small caps, .25 and .75
had the greatest occurrences of all prices ending in .X5, and are the only two price endings that
are greater than their round-number counterparts, .20 and .70. This preference for quarter
prices in the U.S. and not China can be explained by the pervasive use of the Quarter coin as a
currency, which is a foreign concept to the Chinese. Frequent use of the Quarter among the
U.S. population strengthens their familiarity and affinity for the .25 values. Another explanation
for the clustering around quarter values is the lingering effects of the time period prior to
decimalization of stock prices, which occurred right before our sample period, so U.S. investors
are used to trading in 2/8, 12/16.
It is also interesting to see that Chinese data, especially for small caps, had a preference
for .99 and .98 and .88 that is not seen at all in U.S. data. In Chinese small cap data in
particular, .88 had more occurrences than any other non-round number (except .99 and .98).
17 of 38
This can be attributed to 8 as a lucky number in the Chinese culture, with 88 being even
luckier; however, its unlucky counterpart 4 did not show any difference from the average
(investors did not avoid trading around that number).
The following linear regressions quantifies probabilities of seeing at least the last decimal
as a round number (ifone), and seeing both decimals as round numbers (iftwo).
Figure 10
The table below summarizes regression results. For example, in Chinese Big stocks, there is a
0.11195 probability of seeing the last decimal as round, and 0.01576 of seeing both as round.
Figure 11
18 of 38
Discussion- Simple Price Clustering
The manifested bias in the data is statistically significant, and we can rank the strength of
bias (from weak to strong): Chinese large cap, Chinese small cap, U.S. large cap, and U.S. small
cap. The result of this initial survey is not surprising, and the significantly more clustering seen
in U.S. data does not prove that U.S. investors are inherently more biased than Chinese investors.
The probability of seeing a trasaction on a round number is tightly tied to the bid-ask spread: if
the bid-ask spread is wide, it has a greater chance of including a round number, giving the same
investor more chances of choosing a round price in the neighborhood of prices. Also, pure
frequency of seeing a round number does not accurately measure degree of bias. If the price level
of a share is higher, a one-cent difference in price is a smaller fraction of total value traded, so
that a biased trader is penalized less for his round number bias. Therefore, greater clustering in
U.S. data sets may be explained by 1) high bid-ask spread, due to low liquidity, and 2) high
nominal price per share, meaning U.S. investors incur less cost for being biased. This means that
manifested bias does not translate to inherent bias of investors. The next section shows price
clustering analysis adjusting for liquidity and price levels, to analyze inherent bias of investors.
Results- Price Clustering Adjusted for Liquidity and Price Levels
First, we show that liquidity and price level effects can confound our price clustering
analysis. More liquidity should mean less round number bias, while higher price levels should
allow for more bias. If data sets with lower liquidity and higher price levels happen to have
19 of 38
higher level of round number bias, then bias can actually be driven by liquidity and price level
effects, and not inherent round number bias of investors. If there are confounding effects, we
need to adjust for liquidity and price level.
Due to inaccessibility of bid-ask spread on Chinese data, we use volume as a proxy. The
following table summarizes the average volume per firm-day, measured in millions of shares:
Figure 12
We see that Chinese stocks and large-cap stocks trade a lot more, so liquidity can confound price
clustering (resulting in less clustering in Chinese data).
For U.S. data sets, we have bid-ask spread data. The table below shows the probabilities
of observing round number closing prices for U.S. data over the years:
Figure 13
20 of 38
Probabilities of seeing round number prices fell consistently and significantly over the decade, to
converge with the uniform distribution. This decrease is associated with the decrease in bid-ask
fraction, calculated by closing bid-ask spread divided by closing price, suggesting that liquidity
and a narrower bid-ask window possibly reduced investors manifested bias.
The following table shows that price levels are also different across the four data sets:
We see that U.S. stocks and large-cap stocks trade at higher price levels, which is also consistent
with the direction of round number bias; U.S. investors are penalized less as a percentage of
their investment for a one-cent error. Therefore, price levels could also have amplified or
dampened investors inherent bias.
To adjust for liquidity and price level effects, we 1) add volume into the regression, and
2) use frequency weighted by price level. The weighted frequency variables weightedifone and
weightediftwo are calculated using the following:
weightedifone = 100* ifonePricelevel weightediftwo = 100*iftwo
Pricelevel
The weighted variables reflect degree of bias more accurately: for smaller price levels, weighted
Figure 14
21 of 38
variables are greater, representing the higher percentage costs that investors incur for being
biased by one-cent.
The result of linear regressions after controlling for differences in liquidity and price
levels are presented below:
The following table summarizes regression (2) from above, and presents a measure of
inherent bias in each of the four data sets:
Figure 15
Figure 16
22 of 38
The effects can be seen as a rescaled measure of degree of bias, with higher meaning
more bias, though it is hard to interpret. The interpretation is as follows: given the same volume,
being a U.S. small-cap stock (weightedifone = 1.86229, weightediftwo = 0.26531) on average
increases the chance of seeing the last one (or two) decimals as round, as a one-cent fraction of
their closing price, by 1.86229% (or 0.26531%). For example, (holding constant volume), a U.S.
small stock trading around $23 has an increased 10.60% chance of seeing a first decimal round
than if it were a U.S. big stock:
weightedifone = 100* ifonePricelevel , (1.86229 1.40144) =100* ifone
23 , ifone = .10600
But if it were trading around $4, the difference would be 1.8434% for a small stock over a big
stock.
We also notice that volume has a negative coefficient as expected, since it reduces the
amount of bias through a tighter bid-ask spread. An increase in a million shares per firm-day on
average reduces its weightedifone and weightediftwo by 0.01464% and 0.00125%. This is a
substantial impact given that average volume of the four data sets vary widely (Figure 12).
After controlling for liquidity and price levels, Figure 16 shows that the ranking of degree
of bias has changed: (from weakest to strongest) U.S. large-cap, Chinese small-cap, Chinese
large-cap, and U.S. small-cap. The result suggest that in the U.S., small-cap stocks exhibit more
bias than large-caps, but in China, it is the reverse. This apparent contradiction is explored in the
later Discussion section.
23 of 38
Because volume may not have an equal impact across U.S. and China, for the U.S., we
can use bid-ask spread data, which directly measures the window of prices surrounding a
possible round number. The variable usbidaskfrac and its powers are calculated as:
usbidaskfrac = closingask closingbidclosingprice, usbidaskfrac2 = usbidaskfrac( )2
Regression (2) in Figure 17 takes into account that usbidaskfrac may have a non-linear
effect on degree of round number bias. It also includes interaction variables that accounts for the
possibility that bid-ask window may not have an equal impact on U.S. large-caps and small-caps.
Figure 17
24 of 38
The results here support the previous results that were found using volume as a proxy for
liquidity. The negative coefficient on big means that U.S. big stocks exhibit less bias holding
constant price level and bid-ask ratio. Note that the coefficients on usbidaskfrac is positive as
expected, so that higher spread induces more bias. However, the coefficients on the interaction
term bigxusbidask is negative, so that higher bid-ask spread induces bias for small stocks more
so than for big stocks, possibly due to the already narrow spread in big stocks. All this is
consistent for prices that end in two round decimals or just one.
The coefficients can be interpreted similarly as before. For example, a U.S. small-cap
trading at around $23 with a bid-ask fraction of 0.01 has an 0.0349 lower probability of
observing the last decimal as round, than if it were a large-cap:
usweightedifone = .14755 big .40671 big usbidaskfrac +0.04613 big bidaskfrac2 .00087 big usbidaskfrac3= 0.1516
weightedifone = 100* ifonePricelevel , ifone = .0349
Overall, U.S small-cap investors seem to be inherently more biased toward round numbers.
Discussion- Price Clustering Adjusted for Liquidity and Price Levels
It seems contradictory that in the U.S., smaller stocks exhibit more bias, while in China,
smaller stocks exhibit less bias. This finding can be explained by the fact that investors of large-
caps and small-caps are different in U.S. and China, in characteristics and motives. Kumar
(2009) shows that in U.S., individual investors with lower income and less education tend to
25 of 38
gamble in small and local stocks, giving small-cap stocks more speculative qualities and more
room for bias. Also small-cap stocks are more likely to sell-out or buy-in completely; their
investors are are more likely to take a new position or exit entirely, while turnover in large-caps
are driven by existing holders who are merely trading around their positions (Cevik, Thomson
Reuters). U.S. large-caps have more analyst coverage (Bhushan, 1989) and more information
available than small-caps, with prices adjusting faster to new information (Hong, Lim, Stein
2000), reducing round number bias. On the other hand, Hong, Jiang, Zhao (2012) find that in
China, small local stocks are traded more by richer, more educated households in developed
areas for status reasons (Keeping up with the Wangs). These investors may actually be more
sophisticated than investors who trade large-caps, resulting in less bias in Chinese small-caps.
After accounting for liquidity and price level effects, it is surprising to see that overall,
U.S. data would still be similarly biased as Chinese data, even when there should be more noise
trading in China. It is very possible that because of different market conditions and laws around
trading, volume in U.S. has different impact than volume in China, and that volume may not be a
good control for liquidity effect in round number bias (see Discussion- Abnormal Returns).
The most important explanation, however, is probably the selection of the time period. As we
saw in Figure 13, most of clustering in U.S. occurred earlier in the decade, and decreased
dramatically over the years, with the final few years exhibiting less bias than in Chinese data.
This can be due to the narrowing bid-ask spread, or due to investors slowly adjusting to the
recent decimal system for trading stocks, which never affected Chinese investors. Further studies
can be done with bid-ask spread data for this data set, even using future data to avoid the
lingering effects of decimalization.
26 of 38
VI. Abnormal Returns
Like price clustering, abnormal returns based on round numbers is complicated due to the
obvious positive correlation between bid-ask spread and probability of trading on a round
number: given that investors gravitate toward round number prices, having a larger bid-ask
window (more round numbers to choose from) will allow for more biases. For Chinese data, we
use volumeCHN (measured in millions of shares) as a measure of liquidity due to the
inaccessibility of bid-ask spread, and use its powers, volumeCHN2 and volumeCHN3 as
before to take into account nonlinearity. Because daily rate of return is small, we scale up to
percentage return, ret = 100*RET , and then take its next day lagged returns. Again, we use
weighted frequency, which is frequency of seeing one or two round decimals weighted by the
inverse of their closing price. Variables weightedifone and weightediftwo are meant to capture
degree of bias net of price levels, so that the greater the variables, the more serious the biases.
27 of 38
The variables weightedifoneCHN and weightediftwoCHN, are not statistically
significant in any of the regressions, and has little explanatory power on next day returns.
Volume surprisingly has a positive effect on next day returns, and does not seem to be capturing
liquidity premium (see later Discussion).
For U.S. data sets, we use bid-ask fraction instead of volume, with next-day returns in
percentages:
Figure 18
28 of 38
Regression (2) in the above Figure 19 shows that in the U.S., there is causal and statistic
significance for degree of bias (weightedifone, etc.) on next day returns. For small-caps, more
bias (in both one and two decimals) means lower next day returns, with two-decimals having
even more effect. For large-caps, more bias in one-decimal similarly means lower returns.
However, for large-caps, the effect of having both decimals as round is surprisingly large and
positive, strong enough to overwhelm the usual negative effect from round number bias,
generating higher next day returns.
Due to weighing of the variables, coefficients may be hard to interpret. For example,
holding constant bid-ask fraction, a stock trading at $23.40 (only last decimal as round) is
expected to have a -0.03487% lower next day return than if it were not round.
Figure 19
29 of 38
weightedifoneUS = 100* ifone23.40 , retUS = 0.00816 weightedifoneUS , retUS = 0.03487%
For a small-cap stock trading at $80.00 (both decimals round):
retUS = 0.00816 weightedifoneUS .01467 weightediftwoUS = -.02854%
But if it were a large-cap stock:
retUS = .00816 weightedifoneUS .01467 weightediftwoUS +.03445 weightediftwoUS big = .01452%
We also observe that next day returns are increasing in bid-ask spread fraction, so that our
bid-ask measure have captured liquidity premium. This was the opposite when regressing
Chinese returns (Figure 18) using volume as a liquidity measure, where more volume resulted in
higher next day returns (see Discussion).
Discussion- Abnormal Returns
In China, round number bias seemed to have no explanatory power in next day returns in
our regression. This could be due to using volume, which may not be a good control for liquidity.
Mei, Scheinkman, and Xiong (2009) find that trading volume of Chinese shares was not mainly a
result of liquidity. In our regression, volume had positive and significant explanatory power on
next day returns, which failed to take into account liquidity effect in our data. Our findings on
volume is also inconsistent with previous literature. Naughton, Truong, and Veeraraghavan
(2007) found no strong link between volume and returns, and Lee and Rui (2000) found that
30 of 38
trading volume does not Granger-cause stock market returns on any of Chinas four stock
exchanges. This analysis can be repeated in the future by someone with access to data on
Chinese bid-ask spread as a measure of liquidity.
In the U.S., we saw negative excess returns for round numbers, except for large-cap
stocks ending in two round decimals, for which it was positive. Negative returns in U.S. small-
caps is supported by past literature. Wang (2011) finds psychological bias toward round numbers,
and finds positive return for prices ending in $X.01, and negative return for prices just below. It
is also supported by Johnson, Johnson, and Shanthikumar (2008), who find returns following
closing prices just above a round number are significantly higher than returns following prices
just below. The following figure from JJS (2008) shows midpoint-based excess returns by last
digit of previous-day closing price:
Figure 20
31 of 38
The excess return around 0 in JJS (2008) above is consistent with U.S. small-caps and large-
caps in our data, both in direction and magnitude.
The higher return in large-caps can be explained by disproportionate amount of media
attention that the big stocks attract when surpassing an important barrier, usually a round
number, driving up sentiment. Donaldson and Kim (1993) found support and resistance levels in
round numbers in DJIA, which is only an index that is arbitrarily scaled, and round numbers do
not say much about fundamentals. They also find that there were no support and resistance levels
in less popular indices. Future studies can look into this by taking more lagged returns- for
example, next day returns may be higher, but excess returns two days or a week later may be
negative.
VII. Conclusion
Because many previous studies have found positive results but with different data sets
and older time periods, we expected to find similar robustness in clustering in newer data, but
was uncertain whether the effect would be weaker or greater. The increase in sophistication and
narrowing of bid-ask spread should give investors less chances to manifest round number bias,
but may be countered by increase in noise trader participation.
Indeed, price clustering effect was significant and robust, across China and U.S., large
and small caps. However, seeing that U.S. data clustered significantly more than Chinese data
questions whether U.S. investors are inherently more biased. After observing each year
32 of 38
individually in the 2001-2011 data, we saw that round number clustering in the U.S. has
decreased substantially as the bid-ask spread has narrowed, to match that of the Chinese. After
controlling for liquidity and price level effects that have amplified bias for U.S. data, we see that
the degree of round number bias is similar for U.S. and China. However, a contradictory finding
is that there is more round number clustering for small-caps in the U.S., but large-caps in China.
This suggest that small-cap traders in China may be more sophisticated than large-cap traders,
but small-cap traders in U.S. may be more speculative than large-cap traders.
As for excess returns, our findings were inconclusive for Chinese stocks, but for U.S.
stocks, findings were consistent with past literature. Generally, small-cap and large-cap stocks
showed negative next-day excess return around round numbers, with the exception of large-caps
ending in two round decimals, which was positive. This can justify short-term momentum
strategies for U.S. large-caps when they hit significant barriers. The positive excess return can be
explained by the disproportionate amount of media attention it receives and the resulting
sentiment.
The findings of this paper opens up interesting topics for future research. We have only
looked at excess returns for numbers ending in 0s, and future studies can expand the definition
of round number to include $X.50 or $X.25, and even X.88 for China, which showed
clustering in our analysis. It would be more interesting to extend past the decimal point, for
prices in $X00.00, or X88.88 for China. At the same time, analysis can be done with leading
digits to see which attracts more bias. Given that clustering in U.S. has decreased dramatically
after the decimalization of stock markets, it would be interesting to see whether it is due to
33 of 38
increased sophistication of institutional traders, or due to decreased bid-ask spread due to
increased liquidity, or due to steady adjustment to the new decimal system.
34 of 38
Works Cited
Aggarwal, R., and B. Lucey. "Psychological Barriers in Gold Prices?" Review of Financial
Economics 16.2 (2007): 217-30. Print.
Baker, Malcolm, Xin Pan, and Jeffrey Wurgler. "The Psychology of Pricing in Mergers and
Acquisitions." (2009): n. pag. Print.
Bhushan, Ravi. "Firm Characteristics and Analyst following." Journal of Accounting and
Economics 11.2-3 (1989): 255-74. Print.
Bourghelle, David, and Alexis Cellier. "Limit Order Clustering and Price Barriers on Financial
Markets: Empirical Evidence from Euronext." (2007): n. pag. Print.
Brown, P., and J. Mitchell. "Culture and Stock Price Clustering: Evidence from The Peoples'
Republic of China." Pacific-Basin Finance Journal 16.1-2 (2008): 95-120. Print.
Cevik, Arzu. "Understanding Investor Behavior: Trends in Buying & Selling Large-Cap Stocks
& the Implications for Small-Cap Stocks." Thomson Reuters Corporate Solutions. N.p.,
24 Jan. 2013. Web.
China. China Securities Regulatory Commission. CHINAS SECURITIES AND FUTURES
MARKETS. N.p.: n.p., n.d. Web.
Fernald, John, and John H. Rogers. "Puzzles in the Chinese Stock Market." Review of Economics
and Statistics 84.3 (2002): 416-32. Print.
Gromb, Denis, and Dimitri Vayanos. "Limits of Arbitrage." Annual Review of Financial
Economics 2.1 (2010): 251-75. Print.
35 of 38
Gu, G.-F., W. Chen, and W.-X. Zhou. "Quantifying Bid-ask Spreads in the Chinese Stock Market
Using Limit-order Book Data." The European Physical Journal B 57.1 (2007): 81-87.
Print.
Herrmann, Don, and Wayne B. Thomas. "Rounding of Analyst Forecasts." The Accounting
Review 80.3 (2005): 805-23. Print.
Herrmann, Roland, and Anke Moeser. "Do Psychological Prices Contribute to Price Rigidity?
Evidence from German Scanner Data on Food Brands." Agribusiness 22.1 (2006): 51-67.
Print.
Hong, Harrison, Terence Lim, and Jeremy C. Stein. "Bad News Travels Slowly: Size, Analyst
Coverage, and the Profitability of Momentum Strategies." The Journal of Finance 55.1
(2000): 265-95. Print.
Hong, Harrison, Wenxi Jiang, and Bin Zhao. "Trading for Status." (2012): n. pag. Web.
Johnson, Edward, Nicole B. Johnson, and Devin Shanthikumar. "Round Numbers and Security
Returns." (2008): n. pag. Print.
Kang, Joseph. "Contrarian and Momentum Strategies in China Stock Market: 1993-2000."
Pacific-Basin Finance Journal 10.3 (2002): 243-65. Print.
Klumpp, Joni M., B. Wade Brorsen, and Kim B. Anderson. "Producers Preferences for round
Number Prices." Agricultural Finance Review 67.2 (2007): 377-85. Print.
Kumar, Alok. "Who Gambles in the Stock Market?" The Journal of Finance 64.4 (2009):
1889-933. Print.
36 of 38
Lee, Cheng F., and Oliver M. Rui. "Does Trading Volume Contain Information to Predict Stock
Returns? Evidence from China's Stock Markets." Review of Quantitative Finance and
Accounting 14.4 (2000): 341-60. Print.
Mei, Jianping, Jose Scheinkman, and Wei Xiong. "Speculative Trading and Stock Prices:
Evidence from Chinese A-B Share Premia." Annals of Economics and Finance (2009): n.
pag. Web.
Naughton, Tony, Cameron Truong, and Madhu Veeraraghavan. "Momentum Strategies and Stock
Returns: Chinese Evidence." Pacific-Basin Finance Journal 16.4 (2008): 476-92. Print.
Osborne, M. F. M. "Periodic Structure in the Brownian Motion of Stock Prices." Operations
Research 10.3 (1962): 345-79. Print.
Pope, Devin, and Uri Simonsohn. "Round Numbers as Goals : Evidence From Baseball, SAT
Takers, and the Lab." (2011): n. pag. Print.
Pope, Devin, and Uri Simonsohn. "Round Numbers as Goals : Evidence From Baseball, SAT
Takers, and the Lab." Psychological Science (2010): n. pag. Web.
Shea, Christopher. "The Power of Round Numbers." Wall Street Journal 15 Nov. 2012: n. pag.
Print.
Sonnemans, J. "Price Clustering and Natural Resistance Points in the Dutch Stock Market: A
Natural Experiment." European Economic Review 50.8 (2006): 1937-950. Print.
Thomas, Manoj, and Vicki Morwitz. "Penny Wise and Pound Foolish: The Left-Digit Effect in
Price Cognition." Journal of Consumer Research 32.1 (2005): 54-64. Print.
37 of 38
Wang, Amanda Ling Qian. Investor Psychological Bias towards Number Preferences in Stock
Price Endings: Rationality Vs Irrationality. Diss. Massey University, 2011. N.p.: n.p.,
n.d. Print.
Weld, William C., Roni Michaely, Richard H. Thaler, and Shlomo Benartzi. "The Nominal Share
Price Puzzle." Journal of Economic Perspectives 23.2 (2009): 121-42. Print.
Wyss, Rico Von. Measuring and Predicting Liquidity in the Stock Market. Diss. Diss. Nr. 2899
Wirtschaftswiss. St. Gallen, 2003., 2004. N.p.: n.p., n.d. Print.
Xu, Xiaoming, Vikash Ramiah, and Sinclair Davidson. "Noise Trading, Underreaction,
Overreaction and Information Pricing Error Contaminate the Chinese Stock
Market." (n.d.): n. pag. Print.
38 of 38