Penny Wise, Dollar Foolish: The Left-Digit Effect in Security Trading*

Transcription

1 Penny Wise, Dollar Foolish: The Left-Digit Effect in Security Trading* Utpal Bhattacharya Indiana University Craig W. Holden** Indiana University Stacey Jacobsen Indiana University February 2010 Abstract The left-digit effect is defined as when a change in the left-most digit of a price (e.g., 7 to 6 when $7.00 drops to $6.99) dramatically affects the perception of the magnitude. Using a random sample of more than 100 million stock transactions, we find excess buying by liquidity demanders when the price starts above an integer and then drops below the integer. Conversely, we find excess selling by liquidity demanders when the price starts below an integer and then rises to the integer or above it. This is true under three buy-sell ratio measures, in multivariate regressions with various controls, and in multiple robustness checks. We consider the left-digit effect and two other possible explanations that are not mutually exclusive. We test which of the three explanations predominates. We find that liquidity demanders who buy when the price falls below an integer or who sell when the price rises to an integer earn lower 24- hour returns than other benchmark liquidity demanders and, in aggregate, lose $350 million per year. This finding plus two other findings suggest that the left-digit effect predominates over the two other explanations. JEL classification: C15, G12, G20. Keywords: Left-digit effect, nine-ending prices, trading strategies. * We thank Bob Jennings, Sreeni Kamma, Shanker Krishnan, Brian Wolfe, and seminar participants at Indiana University, the Investment Industry Regulatory Organization of Canada, and McMaster University. ** Corresponding author. Address: Kelley School of Business, Indiana University, 1309 E. Tenth St., Bloomington, IN Tel.: ; fax: ; address: cholden@ indiana.edu

2 Penny Wise, Dollar Foolish: The Left-Digit Effect in Security Trading Abstract The left-digit effect is defined as when a change in the left-most digit of a price (e.g., 7 to 6 when $7.00 drops to $6.99) dramatically affects the perception of the magnitude. Using a random sample of more than 100 million stock transactions, we find excess buying by liquidity demanders when the price starts above an integer and then drops below the integer. Conversely, we find excess selling by liquidity demanders when the price starts below an integer and then rises to the integer or above it. This is true under three buy-sell ratio measures, in multivariate regressions with various controls, and in multiple robustness checks. We consider the left-digit effect and two other possible explanations that are not mutually exclusive. We test which of the three explanations predominates. We find that liquidity demanders who buy when the price falls below an integer or who sell when the price rises to an integer earn lower 24- hour returns than other benchmark liquidity demanders and, in aggregate, lose $350 million per year. This finding plus two other findings suggest that the left-digit effect predominates over the two other explanations.

3 PENNY WISE, DOLLAR FOOLISH: THE LEFT-DIGIT EFFECT IN SECURITY TRADING 1. Introduction What is the difference when a price drops from $7.00 to $6.99? Computation yields a one cent decline. But a quick approximation based only on the left-most digit suggests a one dollar drop! The leftdigit effect is defined as when a change in the left-most digit of a price dramatically affects the perception of the magnitude. In other words, when assessing the drop from $7.00 to $6.99, some people anchor on the left-most digit changing from 7 to 6, and believe it is a $1 drop. Brenner and Brenner (1982) theorize that humans economize on their limited memory in storing the price of thousands of goods. They note that the economic value of remembering the first digit is much greater than the economic value of remembering the second digit, which is much greater than the economic value of remembering the third digit, and so on. People may not bother to round $6.99 up to $7.00, because this involves an extra costly mental operation. The left-digit effect is, therefore, a manifestation of bounded rationality in the sense of Simon (1957), who postulated that humans experience limits in processing information. In this paper, we test whether the left-digit effect exists in security trading. We ask whether a change in the price s left-digit affects security trading. Specifically, is there excess buying by liquidity demanders when the price starts above an integer and then drops below the integer? Conversely, is there excess selling by liquidity demanders when the price starts below an integer and then rises to the integer or above it? We choose all trades of 100 randomly selected firms each year from 2001 to 2006, which is the decimal pricing era. This gives us a sample of 137 million trades. Following Huang and Stoll (1997), trades above the bid-ask midpoint are classified as liquidity demander buys, trades below the midpoint are classified as liquidity demander sells, and trades equal to the midpoint are discarded. 1 1 Discarding midpoint trades avoids any contamination from misclassifying midpoint trades. Lee and Ready (1991) only claim a 75% success rate in classifying midpoint trades, which is equivalent to a 25% error rate. Lee and Radhakrishna (2000) empirically verify the 75% success rate / 25% error rate of the Lee and Ready algorithm. 1

4 We first do the fall below integer analysis. The fall below integer sample includes all transactions after the ask price drops to penetrate an integer from above and the digits after the decimal point remain in the.90 to.99 region (e.g., the ask falls from $7.02 to $6.97). The fall below nickel benchmark includes all transactions after the ask price drops to penetrate a nickel-ending threshold N from above and the digits after the decimal point remain in the N-.01 to N-.10 region (e.g., the ask falls from $7.17 to $7.12). The idea is to control for falling price scenarios that do not trigger a left-digit price change. We choose the nickel thresholds N =.15,.25,.35,.45,.55,.65,.75 and.85 to avoid overlapping with the.90 to.99 region. We find that the difference in median (mean) buy-sell ratio of liquidity demanders for the fall below integer sample minus the fall below nickel benchmark is significantly positive. This is true across three alternative measures of the buy-sell ratio: (1) the number of buy trades vs. sell trades, (2) the number of shares bought vs. shares sold, and (3) the dollar value bought vs. the dollar value sold. Next, we check this result in multivariate regressions that include trade size dummies, price level dummies, firm size dummies, institutional ownership dummies, share volume dummies, penny-ending dummies (e.g., X.X0 X.X9), exchange dummies, and year dummies. We find that the difference in the coefficients between the sample and the benchmark is significantly positive in these regressions under all three of our buy/sell ratio measures. We then do the rise to integer analysis and the rise above integer analysis. The rise to integer sample includes all transactions after the bid price rises to hit an integer from below and remains there. The rise to nickel benchmark for this sample includes all transactions after the bid price rises to hit a nickel ending price N from below and remains there, where N is defined as above. The rise above integer sample includes all transactions after the bid price rises to penetrate an integer from below and the digits after the decimal point remain in the.01 to.10 range. The rise above nickel benchmark for this sample includes all transactions after the bid price rises to penetrate a nickel ending price N from below and the digits after the decimal point remain in the range N+.01 and N+.10. N is defined as above. We find that the difference in the median (mean) buy-sell ratio of liquidity demanders for both the rise to integer and rise above integer samples minus their corresponding benchmarks is significantly negative. 2

5 This is true across all three definitions of the buy-sell ratio. Our results are confirmed in multivariate regressions that are run in the same manner as in the previous fall below integer analysis. To check the robustness of our results, we break out our sample by price level categories, by institutional holding categories, and by share volume categories. We find that our results hold across all subsamples with rare exceptions. Are there alternative explanations for these results? Could buy/sell imbalances be driven by technical trading strategies triggered by crossing integer thresholds? Specifically, we consider strategies which buy when the price breaks through an integer ceiling (the resistance level) and which sell when the price breaks through an integer floor (the support level). These predictions go in exactly the opposite direction as our evidence and, therefore, we reject this explanation. Could buy/sell imbalances be driven by value trading strategies triggered by hitting or crossing integer thresholds? For example, suppose that an investor engages in fundamental analysis and determines that a particular stock is worth $40. If the stock price starts to decline and no new information causes the investor to change his valuation, then the stock will be considered a buy if the stock hits $40 or goes below it. Theoretically, a buy trade could be triggered by hitting or crossing any feasible price threshold (e.g., $32.73), but in practical terms investors may have a preference for integer thresholds (e.g., $32.00). With integer thresholds, this alternative explanation would predict excess buying when prices fall to or below the integer, and symmetrically, excess selling when prices rise to or above the integer. Could buy/sell imbalances be due to stock price clustering on round prices? Harris (1991) shows that clustering is highest on integers, second highest on half-dollars, etc. 2 But the evidence from the clustering literature is non-directional, in the sense that analyzing trade and quote frequency by price point does not distinguishing between the buys vs. sells of liquidity demanders. We consider the combination of clustering and undercutting. Undercutting is when a new limit sell (buy) is submitted at a penny lower (higher) than the existing ask (bid). For example, suppose that the current ask is $7.00. A 2 See also Osborne (1962), Neiderhoffer (1965, 1966), Christie and Schultz (1994), Kavajecz (1999), Chakravarty, Harris, and Wood (2001), and Simaan, Weaver, and Whitcomb (2003), Kavajecz and Odders-White (2004), and Ahn, Cai, and Cheung (2005). 3

6 new limit sell undercuts at $6.99 and sets the new ask price. Sometime later, a market buy hits the new ask price. So a buy trade is recorded below the integer. Conversely, if the current bid is $7.00 and a new limit buy undercuts at $7.01, then a market sell will lead to a sell trade above the integer. Thus, if bid and ask quotes are frequently on integers, then undercutting will frequently lead to buying below the integer and selling above the integer. Note that this hypothesis, unlike the left-digit effect, does not predict excess selling when prices rise to an integer. It is important to mention that the three possible explanations for our finding left-digit effect, value trading with integer triggers, and clustering on integers plus undercutting are not mutually exclusive hypotheses. All three could be happening at the same time. We are interested in determining which of these three explanations is more prevalent than the other two. We already have some evidence. As we discussed before, excess buying (selling) when prices fall below (rise above) an integer was an implication of all three explanations, but excess selling when prices rise to an integer was not an implication of the clustering on integers plus undercutting hypothesis. Since we find that sells outnumber buys after bid prices have risen to hit an integer, it is evidence against the support for clustering on integers plus undercutting hypothesis. To obtain more evidence, we consider the marketing literature. This literature finds that nineending prices (i.e., $6.99) are very common and profitable in retailing. Nine-ending prices are found to be heavily dominant based on surveys of retailers pricing practices (Schindler and Kirby, 1997) and based on UPC retail scanning data (Stiving and Winer, 1997). Nine-ending prices are found to significantly increase retailers profits (Anderson and Simester, 2003; Blattberg and Neslin, 1990; Monroe, 2003; and Stiving and Winer. 1997). In other words, aggressive retailers earn more profits by exploiting the leftdigit effect. 3 3 A key limitation of the marketing literature is that in retailing, there are only customer buys, but no customer sells. In the stock market, we observe both liquidity demander buys and liquidity demander sells. Thus, we are able to examine whether there is excess selling when prices rise to or above the integer, as well as excess buying when prices fall below the integer. 4

7 Returning to securities markets, we hypothesize that liquidity demanders who are motivated to trade by the left-digit effect are not so much exploited by others as they tend to hurt themselves through their own price impact. Specifically, the price impact of excess buying tends to push the price (and the midpoint) up, whereas the price impact of excess selling tends to push the price (and the midpoint) down. Bounded rational traders are excited to trade by the left-digit price change and proceed to do so despite the price impact. Thus, liquidity demanders who succumb to the left-digit effect should make lower returns than other benchmark liquidity demanders. Alternatively, if liquidity demanders follow positivealpha value trading strategies triggered by hitting or crossing integer thresholds, then they will earn higher returns than other benchmark liquidity demanders. Finally, if buy/sell imbalances are the result of neutral integer clustering and undercutting by liquidity providers, then liquidity demanders will earn the same returns as other benchmark liquidity demanders. We compute both the trade price returns and the midpoint returns that result from buy trades when prices fall below the integer and closing the position 24 hours later. Similarly, we compute both the trade price returns and the midpoint returns that result from (short) sell trades when prices rise to or above the integer and closing the position 24 hours later. We also compute these 24 hour returns around nickelending prices, and these serve as our benchmark returns. Compared to these benchmark returns, we find that buying when prices fall below the integer yields lower 24-hour, trade price returns and lower 24- hour, midpoint returns. We find that selling when prices rise to the integer yields lower 24-hour, trade price returns and lower 24-hour, midpoint returns. These findings indicate that the left-digit effect is the prevalent influence on returns. In summary, though all three explanations may co-exist, the left-digit effect seems to have the prevailing influence on returns. To determine the economic significance of the left-digit effect, we make a rough estimate of the wealth transfer implied by these lower 24-hour returns. We find that liquidity demanders who succumb to the left-digit effect make $350 million less per year than other benchmark liquidity demanders. The left-digit effect hypothesis makes two further predictions that the other two hypotheses do not make. First, the effect of a first left-digit change should be higher than the effect of a second-left digit 5

8 change. In other words, the change from $20.00 to $19.99 should have a higher effect than the change from $21.00 to $ This is because, if the human brain focuses only on the first left digit, the former is a change of $10, whereas the latter is a change of $1. We compare first left-digit changes in two digit integers (i.e., $10, $20 $90) with second left-digit changes in two digit integers (i.e., $11, $12, $19, $21, $22,, $99). We find that first left-digit changes indeed yield more pronounced buy-sell imbalances than second-left digit changes. Second, the effect of the first left-digit change should be higher around two-digit integers than around one-digit integers. In other words, the change from $20.00 to $19.99 should have a higher effect than the change from $9.00 to $8.99. This is because, if the human brain focuses only on the first left digit, the former is a change of $10, whereas the latter is a change of $1. We compare left-digit changes around two-digit integers (i.e., $20, $30 $90) with left-digit changes around one-digit integers (i.e., $2, $3 $10). We find that left-digit changes around these two-digit prices indeed yield more pronounced buy-sell imbalances than left-digit changes around one-digit prices. These two findings also support the left-digit effect. Recent papers by Bagnoli, Park, and Watts (2006) and Johnson, Johnson and Shanthikumar (2007) are the closest to our paper. Using a sample of end-of-day prices, they find that if the end-of-day price is just below an integer (just above an integer), the overnight or next day return is lower (higher). There are three differences with our paper. First, these two papers look at overnight or next day returns starting from closing prices only, whereas we analyze all transactions throughout the day from a highfrequency intraday data set. Second, unlike them, we identify buys and sells of liquidity demanders. Third, and most important, since we can identify buys and sells of liquidity demanders, we can directly test the left-digit hypothesis. We find evidence in favor of this hypothesis. Johnson, Johnson and Shanthikumar (2007) test a number of different hypotheses that can explain their findings the left-digit effect is not one of their hypotheses and they come to no definite conclusions. Bagnoli, Park, and Watts (2006) observe returns only and then infer next day buying/selling behavior from the returns. Specifically, they observe that closing prices ending in 9 (1) yield negative (positive) overnight returns. They infer that closing prices ending in 9 (1) predict future net selling (buying) the following day. Hence, they conclude 6

9 that 0-ending round numbers represent a psychological barrier or hurdle that is difficult to break through. We examine direct evidence of buys and sells, rather than inferring future buying and selling patterns, and the direct evidence supports the left-digit effect. The paper is organized as follows. Section 2 develops hypotheses and the research design. Section 3 describes the decimal era data and our methodology. Section 4 provides descriptive statistics and documents buy-sell imbalances when the left-digit changes. Section 5 tests three alternative explanations of the buy-sell imbalances. Section 6 concludes. The appendix extends our analysis to include the $1/8 th and $1/16 th tick size eras. 2. Hypotheses and Research Design The null hypothesis that we test is that investors are prone to the left-digit effect when a price change causes the left-digit to change. 4 First, if ask prices fall below an integer, the left-digit effect predicts that liquidity demanders are motivated to buy. Second, if bid prices rise to hit an integer or rise above an integer, the left-digit effect predicts that liquidity demanders are motivated to sell. This gives us our first hypothesis. H1: Buys of liquidity demanders should outnumber their sells after ask prices have fallen below an integer, their sells should outnumber their buys after bid prices have risen to hit an integer, and their sells should outnumber their buys after bid prices have risen above an integer. Although the bounded-rationality explanation (the left-digit effect) predicts the buy-sell imbalances of hypothesis H1, two other rational explanations (value trading with integer triggers or clustering on integers plus undercutting) may predict some of these imbalances as well. Next we turn to determining which of these three explanations predominates. 4 A related literature has explored cognitive foundations based on Simon (1957). Thomas and Morwitz (2005), in a series of five experiments, provide a cognitive account of when and why the left-digit effect manifests itself. They explain the effect of a left digit change on price magnitude perceptions seems to be a consequence of the way the human mind converts numerical symbols to analog magnitudes on an internal mental scale. Since this symbol to analog conversion is an automatic process, the left digit effect seems to be occurring automatically, that is, without consumers awareness encoding the magnitude of a multi-digit number begins even before we finish reading all the digits Since we read numbers from left to right, while evaluating 3.99, the magnitude encoding process starts as soon as our eyes encounter the digit 3. Consequently, the encoded magnitude of $3.99 gets anchored on the left most digit (i.e., $3) and becomes significantly lower than the encoded magnitude of $

10 We use three different approaches to distinguish between the various explanations. First, as we discussed before, excess buying (selling) when prices fall below (rise above) an integer was an implication of all three explanations, but excess selling when prices rise to an integer was not an implication of the clustering around integers plus undercutting hypothesis. So, in H1, if we find that sells outnumber buys after bid prices have risen to hit an integer, it would be evidence against the clustering around integers plus undercutting hypothesis. Second, we consider returns. If liquidity demanders are motivated to trade by the left-digit effect, then the price impact of their excess buying will tend to push the price (and the midpoint) up and the price impact of their excess selling will tend tends to push the price (and the midpoint) down. Thus, they will earn lower returns than other benchmark liquidity demanders. So, if the left-digit effect predominates, then: H2A: Buying by liquidity demanders when ask prices fall below an integer and selling by liquidity demanders when bid prices rise to an integer or rise above an integer should earn lower 24 hour returns than analogous benchmark returns around non-integers. If value trading with integer triggers or if clustering on integers plus undercutting predominates, then the opposite should happen: H2B: Buying by liquidity demanders when ask prices fall below an integer and selling by liquidity demanders when bid prices rise to an integer or rise above an integer should earn same or higher 24 hour returns than analogous benchmark returns around non-integers. Third, we devise two other tests that are unique to the left-digit effect explanation. In this decimal era, as we move from a price of $1 to $ in dollar increments, the first left-digit changes around the integers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, and 90. The second left-digit changes around other two-digit integers (11, 12,, 19, 21, 22,, 99). If the left-digit effect exists, a first leftdigit change should be more dramatic than a second left-digit change. In other words, the change from $20.00 to $19.99 should have a higher effect than the change from $21.00 to $ This is because, if 8

11 the human brain focuses only on the first left digit, the former is a change of $10, whereas the latter is a change of $1. This gives us our next test: H3: When ask prices fall below an integer, buys by liquidity demanders should outnumber their sells more around first left-digit changes in two-digit integers (20, 30, 40, 50, 60, 70, 80, and 90) than around second left-digit changes in two-digit integers (21, 22,, 29, 31, 32,, 99). In addition, when bid prices rise to hit an integer or rise above an integer, their sells should outnumber their buys more around first left-digit changes in two-digit integers than around second left-digit changes in two-digit integers. The next test is to check whether the effect of the first left-digit change is higher around two-digit integers than around one-digit integers. In other words, the change from $20.00 to $19.99 should have a higher effect than the change from $9.00 to $8.99. This is because, if the human brain focuses only on the first left digit, the former is a change of $10, whereas the latter is a change of $1. This gives us our final test: H4: When ask prices fall below an integer, buys by liquidity demanders should outnumber their sells more around two-digit integers (20, 30, 40, 50, 60, 70, 80, and 90) than around one-digit integers (1, 2, 3, 4, 5, 6, 7, 8, 9, 10). In addition, when bid prices rise to hit an integer or rise above an integer, sells should outnumber buys more around two-digit integers than around one-digit integers. 3. Data and Methodology The intraday data that we use comes from the New York Stock Exchange (NYSE) Trade And Quote (TAQ) dataset from 2001 to Because of the massive computation involved in our analysis, we select a random sample of traded stocks. Following the methodology of Hasbrouck (2009), a selected stock must meet five criteria to be eligible: (1) it has to be a common stock, (2) it has to be present on the first and last TAQ master file for the year, (3) it has to have the NYSE, American Stock Exchange (AMEX), or National Association of Securities Dealers Automated Quotations (NASDAQ) as the primary listing exchange, (4) it should not change primary exchange, ticker symbol or its CUSIP code over the year, and (5) it has to be listed in the Center for Research in Security Prices (CRSP) database. 9

12 We start with eligible firms in 2001, put them into 5 quintiles based on price, and then randomly select 20 firms from each quintile. We next roll forward to If firms that were selected in 2001 are eligible in 2002, then they remain in the sample; otherwise, they are replaced by new firms that are randomly selected from all eligible 2002 firms. This process is repeated year-by-year through This means that every year we have all trade and quote data of a random sample of 100 traded stocks. The body of our paper analyzes 137,335,376 trades from the decimal era. 5 The appendix extends our analysis to include 7,347,675 trades during the $1/8 th tick size era and 15,992,073 trades during the $1/16 th tick size era. 6 We then use the following screens on the trade and quote data. Only quotes/trades during normal market hours (between 9:30AM and 4:00PM) are considered. Cases in which the bid or ask price or bid or ask size equaled 0 are deleted. In addition, cases in which the bid price was greater than the ask price, or the ask price was twice as big as the bid price are deleted. We delete all prices equal to or greater than $100 and less than $2. The quote condition has to be normal, which excludes cases in which trading has been halted. We calculate the National Best Bid and Offer (NBBO) across all 9 exchanges and across all market makers for any given second. Each trade is then matched to the NBBO in the prior second, as recommended in Henker and Wang (2006). The market capitalization and the share volume of each stock are obtained from CRSP. From CDA Spectrum, we obtain the Institutional Ownership (IO) data of each firm. The fall below integer sample is constructed as follows. Keep it in the data set if (i) the previous best ask is 1 integer higher than the current best ask, (ii) the digits after the decimal point of the previous best ask are in [.00,.10] and (iii) the digits after the decimal point of the current best ask are in [.90,.99]. If all three conditions are met, then collect all trades that occur while the ask quote remains in [.90,.99]. An example of the fall below integer sample is all trades after the ask quote falls from $10.01 to $ The decimal era begins 1/29/01 for NYSE and AMEX and begins 4/2/01 for NASDAQ. Our dataset ends 12/31/06. 6 The TAQ data starts 1/4/93. The $1/8 tick size era ends 6/23/97 for NYSE, 5/6/97 for AMEX, and 6/1/97 for NASDAQ. The $1/16 tick size era ends 1/28/01 for NYSE and AMEX, and 3/31/01 for NASDAQ. 10

13 The fall below nickel benchmark is constructed as follows. Keep it in the benchmark if (i) the previous best ask is the same integer as the current best ask, (ii) the digits after the decimal point of the previous best ask are in [N+.00, N+0.10] and (iii) the digits after the decimal point of the current best ask are in [N-0.10,.N-0.01], where N is one of the nickel thresholds N =.15,.25,.35,.45,.55,.65,.75, and.85. If all three conditions are met, then collect all trades that occur while the ask quote remains in [N-0.10, N- 0.01]. An example for the nickel threshold N =.15 is all trades after the best ask falls from $10.16 to $ The rise to integer sample and its rise to nickel benchmark and the rise above integer sample and its rise above nickel benchmark are constructed in an analogous manner. 4. The Buy-Sell Imbalances of Liquidity Demanders A. Descriptive Statistics To get a feel for buy-sell imbalances, we start with some descriptive statistics. For each.xx price point, we aggregate all buys and all sells for each firm in each year (e.g., trades at $1.99, $2.99, $3.99, etc. are aggregated at the.99 price point). The buy-sell ratio is then computed for each firm-year. This ratio is computed in three different ways: number of buys / number of sells, shares bought / shares sold, and dollars bought / dollars sold. The median of these three ratios over all firm-years is then computed for each price point from.00 to.99. Figure 1A shows the median number of buys /number of sells by.xx price point, Figure 1B shows the median shares bought /shares sold by.xx price point, and Figure 1C shows the median dollars bought /dollars sold by.xx price point. Interestingly, all three figures show that the highest ratio of buys to sells by liquidity demanders occurs at trade prices ending in.99, and the lowest ratio of buys to sells by liquidity demanders occurs at trade prices ending in.01. At trade prices ending in.99, their buys exceed their sells by 67% if measured by number of trades, by 65% if measured by number of shares, and by 62% if measured by dollar value. At trade prices ending in.01, their buys are lower than their sells by about 17% if measured by number of trades, by about 16% if measured by number of shares, and by 15% if measured by dollar value. 11

14 Further investigation of Figures 1A, 1B, and 1C, reveals a regular pattern every ten cents. Figure 1D explores this further by showing the median buy/sell ratios of liquidity demanders by penny-ending price points:.x0,.x1,,.x9. Interestingly, these buy-sell ratios by penny-ending price points are nearly identical for all three buy-sell ratio measures. Specifically, we notice that the highest buy-sell ratios are at prices ending in.x9 and the lowest buy-sell ratios are at prices ending in.x1. Similarly, the second highest ratios are at prices ending in.x4 and the second lowest ratios are at prices ending in.x6. This is consistent with clustering on dimes (.X0) that is undercut by limit sells at.x9 to yield excess buying at.x9, and undercut by limit buys at.x1 to yield excess selling at.x1. Further, this is also consistent with clustering on nickels (.X5) that is undercut by limit sells at.x4 to yield excess buying at.x4, and undercut by limit buys at.x6 to yield excess selling at.x6. All of the descriptive statistics in Figures 1A 1D suggest that buying and selling by liquidity demanders is far from being uniformly distributed. But they all share a basic limitation; they are based on static prices. The patterns shown are independent of the price path to get to a particular price point. However, the left-digit effect is fundamentally about what happens when the left-digit changes. To test the left digit effect, we focus on price paths where the left-digit changes. B. Buy-Sell Imbalance Tests The left-digit effect predicts that changes in prices at different price levels affect security trading differently. Specifically, an investor prone to the left-digit effect, is motivated to buy when an ask quote falls below an integer. Similarly, an investor prone to the left-digit effect, is motivated to sell when a bid quote rises to hit an integer or rises above an integer. Table I gives us the results of test H1 the test which checks whether buys of liquidity demanders outnumber their sells after ask prices have fallen below an integer, and their sells outnumber their buys after bid prices have risen to hit an integer, and their sells outnumber their buys after bid prices have risen above an integer. Panel A gives the difference in median (mean) buy-sell ratio of liquidity demanders between the fall below integer sample and the fall below nickel benchmark. The columns give the results for the 12

15 three buy-sell ratio measures: number of buys /number of sells, shares bought /shares sold, and dollars bought /dollars sold. All six differences (mean and median X three buy-sell measures) are large positive values and are statistically significant at the 1% level. This is strong evidence in favor of the left-digit effect. Panel B gives the difference in median (mean) buy-sell ratio of liquidity demanders between the rise to integer sample and the rise to nickel benchmark. All six differences are large negative values and are statistically significant at the 1% level. This is strong evidence in favor of the left-digit effect. Panel C gives the difference in median (mean) buy-sell ratio of liquidity demanders between the rise above integer sample and the rise above nickel benchmark. All six differences are negative values and three of the differences are statistically significant at the 1% level. This is strong evidence in favor of the left-digit effect. It should be noted that the results of Panels A and C could be explained by the other two hypotheses value trading with integer triggers, and clustering on integers plus undercutting as well, but the results of Panel B cannot be explained by the clustering on integers plus undercutting hypothesis. Table II brings together the results of Table I in a multivariate setting. Column 1 provides the results from a logistic regression in which the dependent variable is 1 for a buy trade by a liquidity demander or 0 for a sell trade by a liquidity demander. Similarly, column 2 provides the regression results from an OLS regression in which the dependent variable is +shares bought for a buy trade by a liquidity demander or -shares sold for a sell trade by a liquidity demander, and column 3 provides the results from an OLS regression in which the dependent variable is +dollars bought for a buy trade by a liquidity demander or -dollars sold for a sell trade by a liquidity demander. The table reports the difference in regression coefficients between three indicator variables that select trades in the fall below integer, rise to integer, and rise above integer samples and the corresponding indicator variables that select trades in the fall below nickel, rise to nickel, and rise above nickel benchmarks. Although the coefficients are not shown, each regression includes the following controls: trade size dummies, price 13

16 level dummies, firm size dummies, institutional ownership and share volume dummies, penny-ending dummies (e.g., X.X0 X.X9), and exchange and year dummies. Table II confirms the left-digit results found Table 1. All three differences in coefficients for fall below integer fall below nickel are positive and significantly greater than zero at the 1% level. All three differences in coefficients for rise to integer rise to nickel are negative and significantly less than zero at the 1% level. All three differences in coefficients for rise above integer rise above nickel are negative and significantly less than zero at the 1% level. Overall, Table II tells us that the results of Table I are robust to controlling for firm-specific, trade-specific, exchange-specific, year-specific effects. Moreover, note again the results for the rise to integer rise to nickel can only be explained by the left-digit effect. In Tables III through V, we examine the robustness of the left-digit effect by various market, firm, and trade characteristics. Table III repeats the analysis of the difference in median buy-sell ratios carried out in Table I for price level quintiles. We find that the left-digit effect is very consistent across all price level quintiles. In unreported results, we computed the difference in mean buy-sell ratios and obtained the same qualitative results. Tables IV and V repeat the analysis by institutional ownership terciles and by share volume terciles, respectively. In both tables, we see that the left-digit effect exists across all classifications with rare exceptions. Again, unreported results show the same qualitative results for the difference in mean buy-sell ratios. Thus, we conclude that the left-digit effect is quite robust. 5. Tests of Alternative Explanations What explains the buy-sell imbalances documented in the prior section? In this section, we test three alternative explanations: (1) the left-digit effect, (2) value trading with integer triggers, and (3) clustering on integers plus undercutting. These explanations are not mutually exclusive. All three could partially explain the buy-sell imbalances. However, the three explanations have very different implications for 24 hour returns. A. Relative Return Tests 14

17 First we test hypotheses H2A and H2B whether the 24 hour returns from buying when the price drops below the integer and selling when the price rises to the integer or rises above the integer are lower or not lower than the benchmark 24 hour returns around non-integers. The left-digit explanation predicts lower returns, whereas value trading with integer triggers or clustering on integers plus undercutting predicts returns that are not lower. We compute 24-hour returns in two different ways. First, we compute 24-hour, trade price returns. For every buy when the price drops below the integer, we compute the return to buying at the actual trade price and then selling at the bid price 24 hours later to close the position. 7 Similarly, for every sell when the price rises to or above the integer, we compute the return to (short) selling at the actual trade price and then buying at the ask price 24 hours later to close the position. Second, we compute 24-hour, midpoint returns. For every observation of a buy when the price drops below the integer, we compute the return to buying at the contemporaneous quote midpoint price and then selling at the quote midpoint price 24 hours later to close the position. For every observation of a sell when the price rises to or above the integer, we compute the return to (short) selling at the contemporaneous quote midpoint price and then buying at the quote midpoint price 24 hours later to close the position. The same methods are used to compute 24-hour, trade price returns and 24-hour, midpoint returns for the non-integer benchmarks. These non-integer benchmarks are the fall below nickel buys, rise to nickel sells, rise above nickel sells, and all remaining buys and sells. Table VI reports the results of multivariate regressions. In Panel A, the dependent variable is the 24-hour, trade price return. In Panel B, the dependent variable is 24-hour, midpoint return. The first three rows of both panels report relative 24-hour returns, defined as the difference in regression coefficients between three indicator variables for the fall below integer buys, rise to integer sells, and rise above integer sells samples and the corresponding indicator variables for the fall below nickel buys, rise to 7 For example, if there is a buy at 11:00.00 am on day t, then a return is computed from buying at the trade price to selling at the bid price at 11:00.00 am on day t hour returns are slightly cleaner than returns until the end of the day, because they avoid the end-of-the-day pricing anomaly documented in Harris (1989). 15

18 nickel sells, and rise above nickel sells benchmarks. 8 Each regression includes controls for price level, firm size, institutional ownership, share volume, penny-ending (e.g., X.X0 X.X9), exchange, and year. The first column is for the full sample and adds controls for trade size. The next three columns break out the sample by trade size based on the Lee and Radhakrishna (2000) methodology. 9 Lee and Radhakrishna (2000) show that what they classify as small trades are a good proxy for individual trades and what they classify as large trades are a good proxy for institutional trades. In Panel A, the relative 24-hour, trade price return for Fall Below Integer Buys (FBIB) is % and is significantly negative with a p-value less than For Rise To Integer Sales (RTIS), the relative return is % and is significantly negative with a p-value of For Rise Above Integer Sales (RAIS), the relative return is not significant at the 1% level. In Panel B, the relative 24-hour, midpoint return to for Fall Below Integer Buys is % and is significantly negative with a p-value less than For Rise To Integer Sales, the relative return is % and is significantly negative with a p-value less than For Rise Above Integer Sales, the relative return is insignificant. Overall for the full sample, Fall Below Integer Buys yield relative 24-hour, trade price and midpoint returns that are significantly negative. This strongly supports hypothesis H2A that the left-digit effect predominates in these samples, and is evidence against hypothesis H2B that the other two reasons predominate. The Rise To Integer Sells yield relative 24-hour, trade price and midpoint returns that are significantly negative. This again strongly supports hypothesis H2A that the left-digit effect predominates in these samples, and is evidence against hypothesis H2B that the other two reasons predominate. The Rise Above Integer Sells yield relative 24-hour, trade price and midpoint returns that are insignificant. 8 Note that negative relative returns do not imply that arbitrage profits can be made net of transaction costs. Rather, they suggest that liquidity demanders who succumb to the left-digit effect earn lower returns on these trades compared to other benchmark liquidity demanders. 9 Firms are first classified as small, medium or large based on the Fama-French market capitalization terciles for each year (see Small (medium, large) firms having trade sizes of less than 2,500 (2,500, 5,000) are small trades. Small (medium, large) firms having trade sizes of less than 20,000 (50,000, 100,000) but more than 2,500 (2,500, 5,000) are medium trades. Small (medium, large) firms having trade sizes larger than 20,000 (50,000, 100,000) are large trades. 16

19 This supports hypothesis H2B that value trading with integer triggers or clustering on integers plus undercutting predominates in this sample. Again, all three phenomena may co-exist, but the left-digit effect predominates for both Fall Below Integer Buys and Rise To Integer Sells, whereas the other two explanations predominate for Rise Above Integer Sells. Columns 2, 3, and 4 report separate regressions for subsamples of small trades, medium trades, and large trades, respectively. Most of the relative returns are significantly negative at the 1% level. Specifically, there are eighteen relative returns for three trade sizes (small, medium, and large) X three samples (FBIB, RTIS, and RAIS) X two types of 24-hour returns (trade price and midpoint). Thirteen of these eighteen relative returns are significantly negative at the 1% level. Also, there is an important skew towards small trades. Five out of six relative returns for small trades (individuals) are significantly negative versus two out of six relative returns for large trades (institutional) are significantly negative. Further, the magnitudes of the negative relative returns are much more negative for small trades. Specifically, the relative returns for Small Trades of Fall Below Integer Buys are much more negative than for medium and large trades (-13 basis points for trade price returns and -13 basis points for midpoint returns for small trades versus larger than -5 basis points for any category of medium and large trades). Thus, small traders are more susceptible to the left-digit effect than institutional traders. In order to determine the economic significance of the left-digit effect, we make a very rough estimate of the wealth transfer implied by these negative relative 24-hour returns. Basically, we see how often and how large the three types of trades (FBIB, RTIS, and RAIS) are, and use that information to determine the aggregate size of the wealth transfer / year. We compute as follows Wealth transfer / year = [(Relative return on FBIB) * (Average dollar value of FBIB) * (# of FBIB) + (Relative return on RTIS) * (Average dollar value of RTIS) * (# of RTIS) + (Relative return on RAIS) * (Average dollar value of RAIS) * (# of RAIS)] * [Size of the TAQ dataset during as multiple of our sample size] / (6 years) = [( %) * ($17,500 / trade) * (4,627,983 FBIB trades in our sample) 17

20 + (-0.008%) * ($17,266 / trade) * (2,071,799 RTIS trades in our sample) + (0.010%) * ($16,777 / trade) * (485,169 RAIS trades in our sample)] * [(3,721 eligible firms / year on average) / (100 firms / year in our sample)] / (6 years) = -$305.6 million / year. We are assuming that our random sample of 100 firms is representative of all firms. The -$305.6 million /year figure is based on relative 24-hour trade price returns. Repeating the calculation using relative 24-hour midpoint returns, we obtain -$399.1 million / year. Averaging the two figures we obtain a wealth transfer estimate of approximately -$350 million / year. Clearly this is a sizable amount of money. And this is a somewhat conservative estimate of the wealth transfer / year as we are ignoring ineligible firms, such as those firms that changed their listing exchange, ticker symbol or CUSIP code. B. Other Tests of the Left-Digit Hypothesis Table VII tests both hypotheses H3 and H4. Hypothesis H3 looks at the difference between first left-digit changes and second left-digit changes. To be precise, it checks whether, after ask prices fall below an integer, buys by liquidity demanders outnumber their sells more around first left-digit changes in two-digit integers (20, 30, 40, 50, 60, 70, 80, and 90) than around second left-digit changes in two-digit integers (11, 12,, 19, 21, 22,, 99). Similarly, H3 checks whether their sells outnumber their buys more around first left-digit changes in two-digit integers than around second left-digit changes in twodigit integers after bid prices rise to hit an integer or rise above an integer. In Panel A, we look at the Fall Below Integer vs the Fall Below Nickel. In the first column, the difference in median buy-sell ratios for a first left-digit change in two digit integers is 18%. It is 16% for a second left-digit change in two-digit integers. For all three buy-sell conventions, the difference is always larger for a first left-digit change than a second left-digit change. In Panel B, we look at the Rise To Integer vs the Rise To Nickel. In Panel C, we look at the Rise Above Integer vs the Rise Above Nickel. In Panels B and C and across the three buy-sell measures, the first left-digit change is a larger 18

21 negative value or is equal to the second left-digit change, except for one case. On balance, we find support for hypothesis H3 that the first left-digit effect is stronger than the second left-digit effect. Hypothesis H4 looks at whether the effect of the first left-digit change is higher around two digit integers than around one digit integers. To be precise, it checks whether, after ask prices fall below an integer, buys by liquidity demanders outnumber their sells more around the two-digit integers (20, 30, 40, 50, 60, 70, 80, and 90) than the one-digit integers (1, 2, 3, 4, 5, 6, 7, 8, 9, 10). Similarly, H4 checks whether their sells outnumber their buys more around the two-digit integers than the one-digit integers after bid prices rise to hit or rise above an integer. In Table VII Panel A, we find that the first left-digit changes around two-digit integers are larger than first left-digit changes around one-digit integers for all three buy-sell conventions. In Panels B and C, we find that the first left-digit changes around two-digit integers are larger negative values than first left-digit changes around one-digit integers in four out of six cases. On balance, we find support for hypothesis H4 that the first left-digit change is higher around two digit integers than around one digit integers. Table VIII tests hypotheses H3 in a multivariate setting. Controls are added for: trade size, price level, firm size, institutional ownership, share volume, penny-ending (e.g., X.X0 X.X9), exchange, and year. The table reports the difference in coefficients [e.g., (Fall Below Integer) X (First Left-Digit Changes) vs. (Fall Below Integer) X (Second Left-Digit Changes)]. We find that the difference in coefficients always has the predicted sign and is statistically significant at the 1% level in 8 out of 9 tests. This provides strong support for hypotheses H3 even after controlling for other influences. Table IX tests hypotheses H4 in a multivariate setting. Controls are added for trade size, price level, firm size, institutional ownership, share volume, penny-ending (e.g., X.X0 X.X9), exchange, and year. The table reports difference in difference results [e.g., (Fall Below Integer) X (First Left-Digit Changes in Two Digit Integers >= 20) less (Fall Below Nickel) X (Nickel Thresholds > 20) vs. (Fall Below Integer) X (First Left-Digit Changes in One-Digit Integers <= 10) less (Fall Below Nickel) X (Nickel Thresholds < 10)]. We find that the first left-digit change is stronger around two-digit integers than around one-digit integers for two situations Fall Below Integer and Rise To Integer. For the third 19