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

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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

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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

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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

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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

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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:

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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

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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

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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

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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

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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

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Figure 3

Figure 4

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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.

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Figure 6

Figure 7

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Figure 8

Figure 9

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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).

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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

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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

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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

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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

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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

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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.

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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

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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

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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.

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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.

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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

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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

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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

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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

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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

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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

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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.

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