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Attention bias and 52-Week High Price Momentum

Abstract

In this paper, we propose a simple modification of 52-week high price momentum

strategy. We show that the 52-week high momentum profits based on stocks with a

recent 52-week high price and slow-accumulation closing price are significantly higher

than the stocks with a distant 52-week high price and rapid-accumulation closing price.

The Fama-French three-factor alpha of our modified 52-week high momentum strategy

is larger than twice of original George and Hwang (2004) momentum strategy.

JEL: G11, G12, G14

Keywords: 52-week, attention, recency, anchor, momentum

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1. Introduction

Investors’ cognitive bias has been widely documented to play an important role in

determining asset prices, for example, Barber and Odean (2008), Hou, Peng, and Xiong

(2009), Hirshleifer, Lim, and Teoh (2012), and Li and Yu (2012). We explore the

hypothesis that behavioral anchoring bias interacts with attention bias.

George and Hwang (2004) indicate that investors use the 52-week high as an

anchor when assessing the up-trend stock price. The anchoring bias leads to investors

to underreact to good news about the stocks whose prices are near their 52-week high

prices. For example, when good news in the prior year pushes a stock’s closing price

near a new 52-week high, traders are reluctant to bid the price of the stock higher even

if the information warrants it. As a result, they underreact to good news. When the

information eventually prevails, and the price goes up, momentum occurs. They

confirm that a momentum strategy based on the nearness of current closing price to its

52-week high price can earn significantly positive abnormal returns. The momentum is

denoted as GH momentum.

In this paper, first, according to Bhootra and Hur (2013), we construct a recency

ratio (RR) to measure the distance to the past 52-week high price. Bhootra and Hur

(2013) show that anchoring bias is stronger for stocks with recent 52-week high price

than stocks with the distant 52-week high price. The stocks that attain the 52-week high

price in the recent past significantly outperform the stocks that attain the 52-week price

in the distant past. Since investors pay too much attention to the anchor (anchoring bias),

investors underreact to the positive (negative) news about the stocks whose prices are

near (far from) their 52-week high price. The timing of 52-week high price (i.e., the

anchor) affects the profitability of GH momentum.

Second, following Da, Gurun, and Warachka (2014), we create an information

discreteness (ID) measure to proxy for investors’ attention. Specifically, we use the

percentage of positive daily returns relative negative daily returns to estimate

information discreteness (i.e., accumulation of formation-period return or closing price)

that captures the relative frequency of small signals. Da, Gurun, and Warachka (2014)

document that investors are inattentive to information arriving continuously in small

amounts, which is denoted as the frog in the pan (FIP) hypothesis. Specifically, a series

of frequent gradual changes attracts less attention than infrequent dramatic changes.

The profits of Jegadeesh and Titman (1993) price momentum for stocks with

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continuous information, are higher than stocks with discrete information. Because of

investors’ under (or limited) attention to the dynamic changes of price, the investors

tend to underreact to continuous information. We argue that the accumulation of closing

price affects the profitability of GH momentum.

GH measure is the ratio of closing price to the 52-week high price. Recency ratio

affects GH measure through the timing of 52-week high price, i.e., the denominator of

GH measure, while the information discreteness influences GH measure by the

accumulation of closing price, i.e., numerator the of GH measure. Moreover, recency

ratio is positively associated with the level of investors’ attention and is negatively

associated with the level of investors’ underreaction, while in contrast, information

discreteness is negatively associated with the level of investors’ attention and is

positively associated with the level of investors’ underreaction. Combining these two

attention-bias related measures, we hypothesize that stocks whose 52-week high price

occur in the recent past and their closing prices have gradually moved up will have a

higher magnitude of underreaction than stocks whose 52-week high price occur in the

distant past and their closing prices have dramatically moved up. That is, if both the

recency hypothesis and frog in the pan hypothesis hold, then RR and ID should enhance

the GH momentum.

The results confirm our conjectures. First, the GH momentum is the stronger

in stocks with continuous price accumulation than stocks with discrete price

accumulation. The alpha (Fama-French three-factor model) of GH momentum in

continuous information is 1.89% with t-statistics of 12.32, and the alpha of GH

momentum in stocks with discrete information is 1.64% with t-statistics of 13.10.

The alpha of spread in GH momentum between continuous information and

discrete information is 0.25% with t-statistics of 3.31. Further, the GH momentum

is the stronger in the recent 52-week high group than in the distant 52-week high

group. The alpha of GH momentum in stocks with recent 52-week high is 1.72%

with t-statistics of 13.94, and the alpha of GH momentum in distant 52-week high

is 1.51% with t-statistics of 12.20. The alpha of spread in GH momentum between

recent and distant 52-week high groups is 0.21% with t-statistics of 1.69. Consistent

with our hypothesis that the GH momentum strategy is higher for stocks with a recent

52-week high price and having the closing price gradually changed than for stocks with

a distant 52-week high price and having the closing price suddenly changed.

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Moreover, we test whether one subsumes or dominates the other. Specifically, we

explore the interaction between rerecncy ratio and information discreteness on GH

momentum by independently forming triple sorting portfolios. Interestingly, the results

indicate that after controlling for recency ratio, the information discreteness can

improve GH momentum only for stocks with moderate recency ratio. On the other hand,

after controlling for information discreteness, the recency ratio can improve GH

momentum only for stocks with moderate information discreteness. The limited

attention hypothesis provides the following explanation.1 If the RR is relatively high,

i.e., the timing of 52-week high price is very close to the current time, since investors

pay too much attention to 52-week high price, the relative importance of the how a

closing price is formed will be neglected. As a result, investors’ attention to how the

closing price is formed (quick or slow) might be indifferent. On the other hand, when

the RR is relatively low, i.e., the timing of 52-week high price is far from the current

time, investors pay too little attention to the stock. As a result, the accumulation process

of closing price will also be neglected and induce indifferent attention toward how the

closing price is formed (quick or slow). Similarly, when the ID is relatively high, i.e.,

the closing price changes quickly, investors are significantly attracted by the sudden

changes of closing price. As a result, too much attention to the current closing price will

mitigate the relative importance of the timing of 52-week high price. Moreover, when

the ID is relatively low, i.e., the closing price changes slowly, investors pay little

attention to the stocks whenever the timing of 52-week high price is recent or distant.

We modify the GH measure by summing the GH, RR, and -ID to simultaneously

incorporating recency ratio and information discreteness into account, and document

that the modified GH ratio (MGH) greatly improves the performance of original GH

momentum. The alpha of MGH is 2.01% and statistically significant with t-statistic

of 14.51. Particularly, the alpha of MGH momentum is larger than twice of GH

momentum strategy.

Our paper is related to two strands of literature: The literature on 52-week high

price momentum and the literature on investor behavior bias. The paper contributes to

the literature in threefold. First, no study introduces and compares different types of

1 Many studies show that Investor’s attention is a limited cognitive resource which can prevent them

from immediately processing all available information (Hirshleifer and Teoh 2003; Sims 2003; Peng and

Xiong 2006).

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limited attention bias at the same time. Second, we investigate how the anchoring bias

and attention bias interacts with each other. Third, we show that combing two types of

attention biases can significantly enhance the profit of a 52-week high price momentum

strategy.

2. Literature reviews

2.1. The anchoring hypothesis

Tversky and Kahneman (1974) suggest that individuals who use one of the most

common heuristics (anchoring and adjustment) to make estimates, start from an initial

value (the anchor) and adjust this upwards or downwards to account for the

information they have available. George and Hwang (2004) have documented that

investors tend to use 52-week high price as an important reference point in

decision making. They argue that the profitability of 52-week high strategy arises

because investors are reluctant to bid up the price of stocks trading near their 52-

week high price beyond the 52-week high, even if positive information warrants

a higher valuation. On the other hand, investors are unwilling to bid down the price

of stocks in response to negative information, when these stocks are trading at

prices far below the 52-week peak price. As the true information is eventually

realized, the subsequent prices are corrected. The price momentum occurs.

2.2. The attention-bias hypothesis

Much psychological literature establishes that there are limits to the central cognitive-

processing capacity of the human brain. In the finance area, many participants,

particularly individual investors, can devote only limited attention to their portfolios. For

example, Peng and Xiong (2006) show that the limited attention of investors cause

investors to process more market-wide information than firm-specific information.

Similarly, Li and Yuan (2012) indicate that market-wide attention-grabbing events cause

investors to pay increased attention to their portfolios, thereby increasing trading activity

and, in turn, influencing stock prices.

Bhootra and Hur (2013) show that anchoring bias appears to be negatively related

to the distance of the anchor. They argue that the investors’ reluctance to bid up the

price of a stock in response to positive information would be particularly strong

when the stock has recently traded at an elevated price. If investors put more weight

on recent information due to the recency bias, then the straightforward implication

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is that their tendency to underreact would be significantly stronger when the 52-

week high occurred recently. On the other hand, if the 52-week high occurred early

during the year, the investors’ response to information is likely to be relatively more

complete, resulting in weaker underreaction. As a result, investors’ over attention to

the anchor will cause a serious anchoring bias, i.e., underreaction.

On the other hand, Da, Gurun, and Warachka (2014) show that a series of gradual

changes attracts less attention than sudden dramatic changes. They document a frog-in-

the-pan (FIP) hypothesis that originates from investors’ under attention. This hypothesis

predicts that investors are less attentive to information arriving continuously in small

amounts than to information with the same cumulative stock price implications that

arrives in large amounts at discrete time points. The FIP hypothesis predicts that ID has

a conditional relationship with momentum.

3. Portfolio results

The data of the sample period is from January 1965 to December 2017. The relevant

firm-level accounting data are drawn from COMPUSTAT. Stocks listed on the NYSE,

the AMEX and the NASDAQ with ordinary common equity (security type 10 or 11

from Center for Research in Security Prices (CRSP)) are included. The firms must have

valid monthly returns over the past 12 months. We exclude stocks with price less than

$5 at the end of the portfolio formation month to prevent from illiquid trading. Further,

we also exclude stocks with the smallest NYSE market capitalization decile. The Fama

and French three factors are downloaded from the Kenneth French’s website.2

According to George and Hwang (2004), at the end of each month t, we use the

price ratio of the closing price to the previous 12-month high price as the sorting index

to sort stocks into deciles. We denote this as GH momentum. The price adjusted for

stock splits and dividends using CRSP price adjustment factor. The winner (loser)

portfolio defined as stocks with the highest 10% (lowest) GH ratio.

GHt = Closing Pricet

52−week high price (1)

For the comparison purpose, we also measure JT (Jegadeesh and Titman, 1993)

momentum profits. JT portfolios are constructed as follows. For each month t, we sort

2 Please refer to the Kenneth French’s website:

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

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all stocks into deciles by their cumulative raw returns during the previous 12 months.

We skip one month between portfolio formation and holding period to avoid the effects

of bid-ask bounce, price pressure, and any lagged reaction.

Following Da, Gurun, and Warachka (2014), we calculate information

discreteness (ID) to proxy for the speed of stock prices. The ID is determined by the

sign of daily returns and ignores the magnitude by equally weighting each observed

return. The ID is denoted as follows.

ID = sgn(PRET) × [%neg −%pos ], (2)

where PRET is the cumulative return during the formation period. sgn(PRET) is

denoted as the sign of PRET. sgn(PRET) = 1 if PRET > 0 and sgn(PRET) = -1 if PRET

< 0. %neg and %pos are the percentage of days during the formation period with

positive and negative returns.

The higher ID implies discrete information and lower ID implies continuous

information. For example, for winner stocks, their PRET is high. A high percentage of

positive returns (%pos > %neg) means that PRET is constructed by many small

movements of positive returns. From Eq (2), a high percentage of positive return for

winner stock suggests a low value in ID implying the continuous information. In the

extreme case, if a series of daily returns are all positive, then ID will be a minimum

value of -1. On the other hand, if only a few positive returns contribute to PRET of

winner stock, then ID is closing to +1 and the information is discrete.

Following Bhootra and Hur (2013), at the end of each month, we estimate the

recency ratio, RR, for each stock as follows:

RR = 1 - number of days since 52−week high price

364 (3)

The recency ratio is adversely related to the number of days since the 52-week

high price. In extreme case, the number of days since the 52-week high price is 0 if the

closing price at time t is the 52-week high price and the RR ratio equal to 1. At the end

of each month, the stocks with the highest 10% RR are classified as winner portfolio,

and the lowest 10% RR of the firms are allocated as loser portfolio.

The portfolios are held for six overlapping months. The portfolio returns are

equally weighted. We construct a zero-investment portfolio by buying the top ten

percentile winner and selling the bottom ten percentile loser stocks. The overlapping

holding period implies time diversification. That is, for each month t, one-sixth of

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stocks need to be replaced with new winners and losers.

We begin our analysis by examining the results of portfolios formed on JT,

GH, and RR measures. Table 1 of Panel A reports the average monthly returns

over the 6-month holding period. Consistent with GH, we find evidence of strong

momentum based on the nearness of the current price to 52-week high price. The

GH loser portfolio earns a return of 0.62% per month, while the winner portfolio

has a return of 1.35% per month. The spread of 0.73% between loser and winner

portfolio returns is statistically significant with a t-statistic of 3.31. The

corresponding Fama and French (1993) alpha is 1.03% and also statistically

significant with a t-statistic of 5.03.

We find evidence of strong momentum based on RR measure as well. The RR

loser portfolio earns a return of 0.76% per month, while the RR winner portfolio

return is 1.36% per month. The corresponding difference of 0.60% per month (t-

statistic = 4.69) and the alpha of 0.78% (t-statistic = 6.32) are statistically

significant at the 1% level. The significant JT momentum profits are also

documented. The JT loser portfolio earns a return of 0.53% per month, while the

JT winner portfolio return is 1.40% per month. The corresponding difference of

0.87% per month (t-statistic = 4.09) and the alpha of 1.14% (t-statistic = 5.58) are

statistically significant at the 1% level.

As indicated by prior research, the loser stocks tend to experience positive

return in January, and therefore, the zero-cost winner minus loser portfolio tends to

earn much higher return after excluding the January returns. To examine the impact

of January seasonality, we report the returns to JT, GH, and RR based momentum

strategies separately for January and non-January months in Panels B and C of

Table 1. The results confirm that the loser stocks earn higher positive returns in

January, and thus induce negative momentum profits in January. The non-January

momentum returns are particularly pronounced, especially for excluding large

positive January returns of losers; the winner portfolio returns are similar with or

without exclusion of January returns. The JT strategy generates a monthly return of

1.07% with an alpha of 1.31%; GH strategy earns a monthly return of 0.97% with

an alpha of 1.23%, and the RR strategy generates a monthly return of 0.75% with

an alpha of 0.90%.

[Table 1 Here]

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We investigate the profitability of a strategy that uses information of 52-week

high price as well as the information discreteness. The stocks are independently

sorted into five by five portfolios based on information discreteness (ID) and

52-week high price (GH). The results of these tests are reported in Table 2. For

each of the five ID portfolios (continuous to discrete information), we report the

raw and risk-adjusted returns to GH winner and loser portfolios and the difference

in returns between winner and loser portfolios. The results show that the profits of

GH momentum are the stronger in low information discreteness (continuous) group

of than in high information discreteness (discrete) group. For example, the raw

returns (alpha) of GH momentum profits in low information discreteness are 1.60%

(1.89%) with t-statistics of 9.26 (12.32) and the raw returns (alpha) of GH

momentum profits in high information discreteness are 1.38% (1.64%) with t-

statistics of 8.87 (13.10). The raw (alpha) spread of the momentum profits between

low (continuous) and high (Discrete) ID groups is 0.22% (0.25%) with t-statistics

of 2.81 (3.31). In sum, we show that the profitability of GH momentum strategy

can be significantly enhanced by conditioning on the information discreteness.

[Table 2 Here]

We explore the profitability of a strategy that uses information of 52-week

high price as well as the recency ratio. The stocks are independently sorted into

five by five portfolios based on recency ratio (RR) and 52-week high price

(GH). The results of these tests are reported in Table 3. For each of the five RR

portfolios (Distant to Recent information). Consistent with Bhootra and Hur (2013),

the results show that the profits of GH momentum are the stronger in the recent

group than in the distant group. For example, the raw returns (alpha) of GH

momentum profits in recent RR are 1.45% (1.72%) with t-statistics of 9.82 (13.94)

and the raw returns (alpha) of GH momentum profits in distant RR are 1.21%

(1.51%) with t-statistics of 8.16 (12.20). The raw (alpha) spread of the momentum

profits between recent RR and distant RR groups is 0.24% (0.21%) with t-statistics

of 1.92 (1.69). In sum, we show that the profitability of GH momentum strategy

can be significantly enhanced by conditioning on the recency of 52-week high price.

[Table 3 Here]

In addition to separately investigating the impact of recency ratio and information

discreteness on GH momentum, we explore the interaction between recency ratio and

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information discreteness on GH momentum by forming independent triple sorting

portfolios in Table 4. Several interesting findings of Table 4 are summarized as follows.

First, after controlling for recency ratio, the information discreteness can improve GH

momentum only for stocks with moderate recency ratio. After controlling for

information discreteness, the recency ratio can improve GH momentum only for stocks

with moderate information discreteness. For example, the alpha of GH momentum

profit for stocks with low ID (high RR) of 0.84% (0.92%) is significantly higher than

alpha of GH momentum profit for stocks with high ID (low RR) of 73% (73%).

However, the improvement only for stocks with moderate recency ratio (information

discreteness), that is, the Middle RR (Middle ID).

The above results are aligned with the limited attention hypothesis that investor’s

attention is a limited cognitive resource which can prevent them from immediately

processing all available information (Hirshleifer and Teoh 2003; Sims 2003; Peng and

Xiong 2006). Recency ratio affects GH ratio through the timing of 52-week high price,

i.e., the denominator of GH ratio, while the information discreteness influences GH

ratio by the formation of closing price, i.e., numerator the of GH ratio. If the RR is

relatively high, i.e., the timing of 52-week high price is very close to the current time,

investors will commit a serious recency bias. The serious recency bias (too much

attention to 52-week high price) mitigates the relative importance of the formation of

closing price. That is, when the timing of 52-week high price is close to the current time,

the investors’ attention to how the closing price is formed (quick or slow) might be

indifferent. On the other hand, when the RR is relatively low, i.e., the timing of 52-

week high price is far from the current time, investors commit less recency bias. In

contrast, the investor might have a distant bias. The distant bias also mitigates the

relative importance of the formation of closing price because investors pay too little

attention to the stock. As a result, when the timing of 52-week high price is far from the

current time, the investors’ attention to how the closing price is formed (quick or slow)

is also indifferent. The analysis of the influence of ID on RR is in a similar manner.

When the ID is relatively high, i.e., the closing price changes quickly, investors are

significantly attracted by the sudden changes of closing price. As a result, too much

attention to the current closing price will mitigate the relative importance of the timing

of 52-week high price. Moreover, when the ID is relatively low, i.e., the closing price

changes slowly, investors will pay very little attention to the stocks whenever the timing

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of 52-week high price is recent or distant. Furthermore, we document that the GH

momentum is greatly improved by incorporating recency ratio and information

discreteness into account.

Second, when we look at extreme cases, the GH momentum performance is greatly

improved by incorporating the recency ratio and information discreteness into account.

Specifically, consistent with our hypothesis that the profitability of 52-week high

momentum strategy is higher for stocks with a recent 52-week high price and having

the closing price gradually changed than for stocks with a distant 52-week high price

and having the closing price suddenly changed. For instance, the alpha of GH

momentum profit for stocks with low ID and high RR of 0.97% is significantly higher

than that for stocks with high ID and low RR of 83%. The monthly alpha between these

two extreme portfolios is 0.14% with t-statistics of 1.93, around 1.68% a year.

[Table 4 Here]

The anchor of the 52-week high price is associated with investors’ underreaction.

We have shown that RR is positively related to the magnitude of underreaction and, ID

is negatively correlated with the magnitude of underreaction. As a result, we construct

a new modified GH measure which is defined as:

MGH = GH + RR – ID, (4)

where GH, RR, and ID are defined the same as previously mentioned. The measure

MGH simultaneously incorporates anchoring bias and two limited attention bias

measures into account. The performance of decile portfolios based on MGH is shown

in Table 5.

We find evidence of strong momentum based on the MGH. The loser portfolio

earns a return of 0.25% per month, while the winner portfolio has a return of 1.99%

per month. The spread of 1.74% in loser and winner portfolio returns is statistically

significant with a t-statistic of 11.03. The corresponding Fama and French (1993)

alpha is 2.01% and also statistically significant with a t-statistic of 14.51.

Particularly, the Fama-French three-factor alpha of MGH momentum is larger than twice of

original GH momentum strategy. The performance of January and non-January months

are shown in Panels B and C of Table 5. Similarly, the results confirm that the loser

stocks earn higher positive returns in January, and thus induce negative momentum

profits in January. The non-January momentum returns are particularly pronounced,

especially for excluding large positive January returns of losers; the winner

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portfolio returns are similar with or without exclusion of January returns. The

strategy generates a monthly return of 1.96% with a Fama–French alpha of 2.19%.

[Table 5 Here]

4. Cross-sectional regression results

To compare the profitability among different momentum strategy, we estimate the

contributions of the various portfolios formed in month t – j to the month t return by

estimating the following cross-sectional regression:

Rit=b0.jt+b1.jtRi,t−1+b2.jtSIZEi,t−1+b3.jtJTHi,t−j+b4.jtJTLi,t−j+b5.jtRRHi,t−j+b6.jtRRLi,t−j+b7.jtGHHi,t−j+

b8.jtGHLi,t−j+b8.jtMGHHi,t−j+b9.jtMGHLi,t−j+eit, (5)

where Rit is the return of stock i in month t; SIZEi,t−1 is the natural logarithm of stock

i's market capitalization at the end of previous month; JTHi,t−j equals one if stock i’s

past performance over the 12-month period (t–j–12, t–j) is in the top 30% when

measured by JT’s performance criterion, and is zero otherwise; JTLi,t−j equals one if

stock i’s past performance over the period (t–j–12, t–j) is in the bottom 30% when

measured by JT’s performance criterion, and is zero otherwise. The rest of variables are

defined similarly.

Table 6 indicates that the presence of short-term return reversals: the

coefficients on past month returns are significantly negative. The negative

relationship between firm size and f u tu re returns is also documented. As for

momentum strategy, we find significant JT, GH, RR, and MGH momentum. The

returns on GH’s winner minus loser strategy is 0.22% per month (t-statistic =

11.13). When the January month is excluded from the sample, the momentum

profits of these three strategies are enhanced.

[Table 6 Here]

We examine the role of recency ratio in profitability of momentum by modifying

Fama and MacBeth cross-sectional regression of George and Hwang (2004) and

Bhootra and Hur (2013) as follows.

Rit=b0.jt+b1.jtRi,t−1+b2.jtSIZEi,t−1+b3.jtGHHi,t−j+b4.jtGHLi,t−j+b5.jtRRHi,t−j+b6.jtRRLi,t−j+b7.jtRRHi,t−

j*GHHi,t−j+b8.jtRRHi,t−j*GHLi,t−j+b9.jtRRLi,t−j*GHHi,t−j+b10.jtRRLi,t−j*GHLi,t−j+eit, (6)

where RR is the recency ration defined in Equation (3). RRH (RRL) is a dummy

variable that equals 1 for 30% of the stocks with the highest (lowest) recency ratio at

the end of month t - j, and is 0 otherwise. Table 7 shows that the GH momentum strategy

is more significant in stocks with high RR than stocks with low RR. For example, the

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interactive term GHH*RRH is significantly positive at 1% level of significance, the

term GHL*RRH is insignificantly negative. The difference between in coefficients on

GHH*RRH and GHL*RRH is 0.15% with t-statistics of 5.02. The difference between

coefficients GHH*RRL and GHL*RRL is 0.04% with t-statistics of 1.96. We conduct

a difference-in-difference test, i.e., the difference of (GHH*RRH - GHL*RRH) -

(GHH*RRL - GHL*RRL) of 0.10% with t-statistics of 4.31, confirming that GH

momentum profits are significantly higher in stocks with high RR. The non-January

results also indicate the same conclusion.

[Table 7 Here]

We also examine whether information discreteness affects the profitability of 52-

week momentum strategy. The ID is defined by Equation (2). IDH (IDL) is a dummy

variable that equals 1 for 30% of the stocks with the highest (lowest) ID ratio at the end

of month t - j, and is 0 otherwise.

Rit=b0.jt+b1.jtRi,t−1+b2.jtSIZEi,t−1+b3.jtGHHi,t−j+b4.jtGHLi,t−j+b5.jtIDHi,t−j+b6.jtIDLi,t−j+b7.jtIDHi,t−j*

GHHi,t−j+b8.jtIDHi,t−j*GHLi,t−j+b9.jtIDLi,t−j*GHHi,t−j+b10.jtIDLi,t−j*GHLi,t−j+eit, (7)

where ID is the information discreteness defined in Equation (2). IDH (IDL) is a

dummy variable that equals 1 for 30% of the stocks with the highest (lowest)

information discreteness at the end of month t - j, and is 0 otherwise.

Table 8 shows that the GH momentum strategy is more significant in stocks with

low ID than stocks with high ID. For example, the difference in coefficients on

GHH*IDL and GHL*IDL is 0.15% with t-statistics of 5.12. The difference in

coefficients GHH*IDH and GHL*IDH is 0.05% with t-statistics of 1.62. A difference-

in-difference test, i.e., difference between (GHH*IDL - GHL*IDL) - (GHH*IDH -

GHL*IDH) of 0.10% with t-statistics of 2.64, confirming the evidence that the

performance GH momentum is also driven by ID. The non-January results also indicate

the same conclusion.

[Table 8 Here]

We explore the interaction between recency ratio (RR) and information

discreteness (ID) on GH momentum. The joint test of two explanations are adopted by

the following equation.

Rit=b0.jt+b1.jtRi,t−1+b2.jtSIZEi,t−1+b3.jtGHHi,t−j+b4.jtGHLi,t−j+b5.jtRRHi,t−j+b6.jtRRLi,t−j+b7.jtIDHi,t−j

+b8.jtIDLi,t−j+b9.jtIDHi,t−j*RRHi,t−j*GHHi,t−j+b10.jtIDHi,t−j*RRHi,t−j*GHLi,t−j+b11.jtIDHi,t−j*RRLi,

t−j*GHHi,t−j+b12.jtIDHi,t−j*RRLi,t−j*GHLi,t−j+b12.jtIDLi,t−j*RRHi,t−j*GHHi,t−j+b14.jtIDLi,t−j*RRHi,

t−j*GHLi,t−j+b15.jtIDLi,t−j*RRLi,t−j*GHHi,t−j+ b16.jtIDLi,t−j*RRLi,t−j* GHLi,t−j +eit,

(8)

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The results are shown in Table 9. Since there are too many coefficients in Eq. (8),

we only report the interaction coefficients, and the rest of the unreported coefficients

are quite similar to previous results. First, after controlling for information discreteness,

the recency ratio (RR) can enhance 52-week high momentum profits only for stocks

with high ID group. The result is consistent with average raw returns in Panel A of Table

4.3 For example, the coefficient of GHIDHRR is 0.14% with t-statistics of 2.77 and

coefficient of GHIDLRR is 0.05% with t-statistics of 0.97. On the other hand, after

controlling for recency ratio, the information discreteness add a very limited

contribution to the profitability of 52-week high price momentum. Specifically, the

coefficient of GHIDRRH is -0.03% with t-statistics of -0.39 and coefficient of

GHIDRRL is 0.06% with t-statistics of 1.38. The results are consistent with our

conjecture that because of the investors’ limited attention, the ID and RR are

particularly stronger in stocks with moderate attention. We compare 52-week high

momentum profits for stocks with distant 52-week high anchor and continuous closing

price formation to stocks with recent 52-week high anchor and discrete closing price

formation. The average difference-in-difference between GH momentum in low ID and

high RR group and GH momentum in high ID and low RR group is 0.11% with t-

statistics of 2.27, suggesting that adding information of RR and ID significantly

enhances the GH momentum.

[Table 9 Here]

5. Conclusion

This paper explores the roles of two attention-bias related variables, i.e., distance of the

anchor and information discreteness, in explaining anchoring bias of 52-week high

price momentum strategy. The anchoring bias, i.e., investors viewing the 52-week high

price as a reference point, leads to investors to underreact to good news about the stocks

whose prices are near their 52-week high price (George and Hwang 2004). We show

that both attention-bias measures, i.e., a distance of the anchor and information

discreteness, simultaneously affect the profitability of 52-week high strategies.

Specifically, the 52-week high momentum profits based on stocks with recent 52-week

high price and slow-accumulation closing price (or low information discreteness) is

3 The results of Panel B of Table 4 suggest that RR (ID) can affect alpha of GH momentum only in

stocks with moderate ID (RR). However, we use raw return to estimate coefficients in Table 9.

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significantly higher than the stocks with distant 52-week high price and rapid-

accumulation closing price (high information discreteness). The Fama-French three

factor alpha of a modified 52-week high momentum strategy incorporating recency and

information discreteness is about twice as large for original GH momentum strategy.

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References

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overreaction in asset markets. Journal of Finance 54, 2143–2184.

Hong, H., Lim, T., Stein, J.C., 2000. Bad news travels slowly: size, analyst coverage,

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implications for stock market efficiency. Journal of Finance 48, 65–91.

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and Biases. Cambridge University Press, New York.

Tversky, A., Kahneman, D., 1974. Judgement under uncertainty: heuristics and biases.

Science 185, 1124–1131.

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Table 1. Monthly performance of momentum strategy The sample period is from January 1965 to December 2017. We exclude the stocks with price

less than $5 and stocks below NYSE minimum decile market capitalization. The stocks are

sorted into decile portfolios based on JT, GH, and RR measures. Momentum is defined as a

zero-cost portfolio that long-buy top winner portfolio and short-sell loser portfolio. The JT is

denoted as the cumulative prior 12-month raw return excluding the most recent month. GH is

the ratio of current price to 52-week high price. Recency ratio (RR), is defined as 1 – number of

days since 52-week high price/364. The monthly average raw returns and Fama and French

(1993)’s alpha of each momentum portfolio during all periods are provided Panels A and B.

Panel C provides the Fama and French (1993)’s alpha excluding January. t-statistics are in

parentheses. The monthly raw returns and Fama and French (1993)’s alpha of each momentum

portfolio are provided. t-statistics are in parentheses.

JT GH RR

Loser Winner WML Loser Winner WML Loser Winner WML

Panel A: All periods

Raw returns 0.533 1.402 0.869 0.618 1.351 0.733 0.760 1.364 0.604 (1.78) (4.76) (4.09) (1.92) (6.32) (3.31) (3.40) (6.40) (4.69)

FF3 alpha -0.813 0.326 1.139 -0.670 0.360 1.030 -0.437 0.344 0.781 (-6.09) (3.12) (5.58) (-4.68) (4.47) (5.03) (-5.32) (5.57) (6.32)

Panel B: January

Raw returns 3.602 2.229 -1.373 3.336 1.456 -1.881 2.499 1.545 -0.954 (2.82) (2.15) (-1.72) (2.27) (1.98) (-1.58) (2.81) (1.96) (-2.15)

FF3 alpha 0.240 -0.133 -0.373 -0.056 -0.331 -0.275 -0.008 -0.430 -0.422 (0.38) (-0.29) (-0.42) (-0.07) (-0.95) (-0.25) (-0.03) (-1.68) (-0.85)

Panel C: Excluding January

Raw returns 0.254 1.327 1.072 0.371 1.341 0.970 0.602 1.347 0.745 (0.84) (4.32) (4.91) (1.15) (6.00) (4.54) (2.63) (6.09) (5.60)

FF3 alpha -0.543 0.771 1.310 -0.384 0.848 1.229 -0.084 0.822 0.902 (-4.12) (7.04) (6.35) (-2.84) (10.42) (6.26) (-0.98) (12.85) (7.18)

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Table 2 Independent double sorts on information discreteness and 52-

week high price momentum The sample period is from January 1965 to December 2017. We exclude the stocks with price

less than $5 and stocks below NYSE minimum decile market capitalization. The stocks are

independently sorted into 5 by 5 portfolios based on information discreteness (ID) and 52-week

high price (GH). ID is determined by the sign of daily returns and ignores the magnitude by equally

weighting each observed return. ID = sgn(PRET) × [%neg −%pos ] where PRET is the cumulative

return during the formation period. sgn(PRET) is denoted as the sign of PRET. sgn(PRET) = 1 if PRET

> 0 and sgn(PRET) = -1 if PRET < 0. %neg and %pos are the percentage of days during the formation

period with positive and negative returns. GH is the ratio of current price to 52-week high price.

CMD denotes that the momentum profit of Continuous portfolio minus the momentum profit

of Discrete portfolio. The monthly average raw returns and Fama and French (1993)’s alpha of

each momentum portfolio during all periods are provided in Panels A and B. Panel C provides

the Fama and French (1993)’s alpha excluding January. t-statistics are in parentheses.

Panel A: Raw returns

Continuous 2 3 4 Discrete

Loser 0.227 0.419 0.454 0.502 0.363

(0.76) (1.44) (1.59) (1.76) (1.25)

Winner 1.824 1.803 1.823 1.781 1.743

(9.07) (9.20) (9.32) (9.28) (9.03)

CMD

WML 1.597 1.384 1.369 1.279 1.380 0.217

(9.26) (8.48) (8.76) (8.30) (8.87) (2.81) Panel B: FF3 alpha

Loser -1.080 -0.893 -0.861 -0.787 -0.901

(-8.98) (-8.05) (-8.58) (-8.00) (-9.40)

Winner 0.807 0.782 0.811 0.764 0.733

(13.91) (15.00) (15.36) (14.57) (13.24)

CMD

WML 1.887 1.675 1.672 1.552 1.635 0.253

(12.32) (11.76) (12.46) (12.04) (13.10) (3.31) Panel C: Exclude January FF3 alpha

Loser -1.199 -1.017 -0.958 -0.868 -0.972

(-10.76) (-9.67) (-9.78) (-9.28) (-10.55)

Winner 0.859 0.833 0.870 0.807 0.772

(14.43) (15.70) (16.12) (15.01) (13.59)

CMD

WML 2.059 1.850 1.828 1.675 1.744 0.314

(14.28) (13.61) (13.95) (13.56) (14.27) (4.18)

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Table 3 Independent double sorts on recency ratio and 52-week high price

momentum The sample period is from January 1965 to December 2017. We exclude the stocks with price

less than $5 and stocks below NYSE minimum decile market capitalization. The stocks are

independently sorted into 5 by 5 portfolios based on recency ratio (RR) and 52-week high price

(GH). Recency ratio (RR), is defined as 1 – number of days since 52-week high price/364. GH is the

ratio of current price to 52-week high price. RMD denotes that the momentum profit of Recent

portfolio minus the momentum profit of Distant portfolio. The monthly average raw returns

and Fama and French (1993)’s alpha of each momentum portfolio during all periods are

provided in Panels A and B. Panel C provides the Fama and French (1993)’s alpha excluding

January. t-statistics are in parentheses.

Panel A: Raw returns

Distant 2 3 4 Recent

Loser 0.356 0.406 0.502 0.443 0.524

(1.24) (1.39) (1.69) (1.49) (1.76)

Winner 1.571 1.613 1.748 1.891 1.977

(8.19) (8.30) (8.98) (9.41) (9.45)

RMD

WML 1.214 1.207 1.246 1.448 1.453 0.238 (8.16) (7.73) (7.65) (9.48) (9.82) (1.92) Panel B: FF3 alpha

Loser -1.002 -0.900 -0.777 -0.806 -0.713

(-8.20) (-7.55) (-6.57) (-8.20) (-6.24)

Winner 0.509 0.554 0.711 0.889 1.011

(7.32) (9.09) (11.94) (15.36) (13.35)

RMD

WML 1.510 1.454 1.488 1.695 1.724 0.214 (12.20) (11.17) (11.17) (13.80) (13.94) (1.69) Panel C: Exclude January FF3 alpha

Loser -1.122 -1.009 -0.877 -0.855 -0.649

(-9.41) (-8.86) (-7.80) (-8.80) (-5.50)

Winner 0.523 0.576 0.751 0.936 1.094

(7.17) (9.07) (12.31) (15.72) (14.47)

RMD

WML 1.646 1.584 1.629 1.792 1.743 0.098 (13.88) (12.69) (12.71) (14.73) (13.84) (0.78)

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Table 4 Independent triple sorts on recency ratio, information

discreteness, and 52-week high price The sample period is from January 1965 to December 2017. We exclude the stocks with price

less than $5 and stocks below NYSE minimum decile market capitalization. The stocks are

independently sorted into 3 by 3 by3 portfolios based on recency ratio (RR), information

discreteness (ID), and and 52-week high price (GH). Recency ratio (RR), is defined as 1 – number of

days since 52-week high price/364. ID is determined by the sign of daily returns and ignores the

magnitude by equally weighting each observed return. ID = sgn(PRET) × [%neg −%pos ] where

PRET is the cumulative return during the formation period. sgn(PRET) is denoted as the sign of PRET.

sgn(PRET) = 1 if PRET > 0 and sgn(PRET) = -1 if PRET < 0. %neg and %pos are the percentage of

days during the formation period with positive and negative returns. GH is the ratio of current price

to 52-week high price. The monthly average raw returns and Fama and French (1993)’s alpha

of long winner and short loser portfolio during all periods are provided in Panels A and B. t-

statistics are in parentheses.

Panel A: Raw returns

Low RR Middle RR High RR RRHML

Low ID 1.066 1.045 1.161 0.095

(8.70) (8.24) (9.77) (1.15)

Middle ID 0.902 0.916 1.092 0.190

(8.11) (7.81) (9.28) (2.45)

High ID 0.989 0.951 1.141 0.152

(9.25) (7.94) (9.57) (1.98)

High RR & Low ID

- Low RR & high ID

IDLMH 0.076 0.094 0.020 0.171

(1.47) (1.71) (0.30) (2.13)

Panel B: FF3 alpha Low RR Middle RR High RR RRHML

Low ID 0.901 0.842 0.970 0.069

(8.65) (7.56) (9.43) (0.82)

Middle ID 0.728 0.720 0.922 0.193

(7.69) (7.15) (9.07) (2.44)

High ID 0.828 0.727 0.943 0.115

(9.29) (7.19) (9.20) (1.49)

High RR & Low ID

- Low RR & high ID

IDLMH 0.073 0.115 0.026 0.142

(1.39) (2.07) (0.39) (1.93)

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Table 5. Monthly performance of modified 52-week high price momentum

strategy The sample period is from January 1965 to December 2017. We exclude the stocks with price

less than $5 and stocks below NYSE minimum decile market capitalization. The stocks are

sorted into decile portfolios based on adjusted 52-week high price measure. Momentum is defined

as a zero-cost portfolio that long-buy top winner portfolio and short-sell loser portfolio. The

adjusted 52-week high price measure is defined as: MGH = GH + RR – ID, where GH denotes the ratio

of current price to 52-week high price, RR (Recency ratio), is defined as 1 – number of days since

52-week high price/364, and ID (information discreteness) is defined as sgn(PRET) × [%neg

−%pos ] where PRET is the cumulative return during the formation period. sgn(PRET) is denoted as

the sign of PRET. sgn(PRET) = 1 if PRET > 0 and sgn(PRET) = -1 if PRET < 0. %neg and %pos are the

percentage of days during the formation period with positive and negative returns. The monthly

average raw returns and Fama and French (1993)’s alpha of each momentum portfolio during

all periods are provided Panels A and B. Panel C provides the Fama and French (1993)’s alpha

excluding January. t-statistics are in parentheses.

Loser 2 3 4 5 6 7 8 9 Winner WML

Panel A: All periods

Raw returns 0.247 0.650 0.806 0.893 0.981 1.153 1.245 1.376 1.567 1.988 1.741

(0.88) (2.65) (3.51) (4.09) (4.67) (5.64) (6.24) (6.90) (7.96) (10.00) (11.03)

FF3 alpha -1.054 -0.582 -0.392 -0.259 -0.146 0.055 0.165 0.309 0.542 0.992 2.045

(-10.11) (-7.60) (-6.91) (-5.67) (-3.44) (1.37) (4.13) (7.58) (11.22) (16.70) (14.51)

Panel B: January

Raw returns 3.136 2.931 2.533 2.191 1.980 1.956 1.817 1.913 1.995 2.474 -0.662

(2.60) (2.73) (2.61) (2.38) (2.29) (2.31) (2.21) (2.24) (2.54) (3.07) (-0.87)

FF3 alpha -0.261 -0.246 -0.415 -0.557 -0.568 -0.467 -0.452 -0.374 -0.021 0.526 0.787

(-0.49) (-0.59) (-1.43) (-2.08) (-2.68) (-2.54) (-2.54) (-2.10) (-0.11) (2.21) (1.11)

Panel C: Excluding January

Raw returns -0.015 0.443 0.649 0.775 0.890 1.080 1.193 1.327 1.529 1.944 1.959

(-0.05) (1.78) (2.78) (3.49) (4.14) (5.16) (5.83) (6.52) (7.54) (9.52) (12.65)

FF3 alpha -1.152 -0.640 -0.411 -0.250 -0.121 0.092 0.217 0.370 0.602 1.040 2.192

(-11.30) (-8.66) (-7.40) (-5.74) (-2.92) (2.30) (5.45) (9.08) (12.30) (17.03) (15.81)

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Table 6 Fama-MacBeth cross-sectional regression For each month from January 1965 to December 2017, the following cross-sectional regression is

estimated. Rit=b0.jt+b1.jtRi,t−1+b2.jtSIZEi,t−1+b3.jtJTHi,t−j+b4.jtJTLi,t−j+b5.jtRRHi,t−j+b6.jtRRLi,t−j+b7.jtGHHi,t−j+

b8.jtGHLi,t−j +b8.jtMGHHi,t−j+b9.jtMGHLi,t−j++eit, Average parameter values are the time series averages

from the cross-sectional estimates of stock variables, and t-statistics are the time-series averages divided

by the time-series standard errors. Ri,t-1 and Sizei;t are the return and log (market capitalization) of

stock i in month t-1. JTH (JTL) is a dummy variable that equals 1 for 30% of the stocks with

largest (smallest) 12-month cumulative returns at the end of month t- j, and is 0 otherwise, RRH

(RRL) is a dummy variable that equals 1 if the stock belongs to top (bottom) 30% of the stocks

based on their recency ratio at the end of month t - j, and is 0 otherwise, GHH (GHL) is a

dummy variables that equals 1 for 30% of the stocks with largest (smallest) GH measure at the

end of month t - j, and is 0 otherwise, and MGHH (MGHL) is a dummy variables that equals 1

for 30% of the stocks with largest (smallest) MGH measure at the end of month t - j, and is 0

otherwise

Panel A: All periods Panel B: January excluded

Model1 Model2 Model3 Model4 Model1 Model2 Model3 Model4

1 Intercept 2.168 3.030 2.183 2.995 1.669 2.526 1.594 2.481

(4.31) (5.77) (3.82) (5.48) (3.33) (4.80) (2.78) (4.50)

2 Rt-1 -0.025 -0.028 -0.024 -0.025 -0.019 -0.023 -0.019 -0.020

(-6.66) (-7.03) (-6.08) (-6.18) (-5.00) (-5.51) (-4.54) (-4.70)

3 MV -0.074 -0.147 -0.086 -0.146 -0.036 -0.109 -0.047 -0.108

(-2.39) (-4.65) (-2.57) (-4.50) (-1.18) (-3.46) (-1.39) (-3.33)

4 JTH 0.030 0.030

(1.27) (1.18)

5 JTL -0.074 -0.100

(-3.53) (-4.66)

6 GHH 0.107 0.105

(12.91) (12.29)

7 GHL -0.117 -0.140

(-6.01) (-7.00)

8 RRH 0.070 0.090

(3.32) (4.17)

9 RRL -0.044 -0.055

(-2.61) (-3.16)

10 MGHH 0.108 0.109

(11.99) (11.91)

11 MGHL -0.109 -0.130

(-6.86) (-7.98)

12=4-5 JT 0.104 0.129

(3.63) (4.36)

13=6-7 GH 0.224 0.245

(11.13) (11.97)

14=8-9 RR 0.114 0.145

(4.37) (5.38)

15=10-11 MGH 0.218 0.238

(11.48) (12.35)

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Table 7 Fama-MacBeth cross-sectional regression: Momentum profits

conditional on recency ratio For each month from January 1965 to December 2017, the following cross-sectional regression is

estimated. Rit=b0.jt+b1.jtRi,t−1+b2.jtSIZEi,t−1+b3.jtGHHi,t−j+b4.jtGHLi,t−j+b5.jtRRHi,t−j+b6.jtRRLi,t−j+b7.jtRRHi,t−j

*GHHi,t−j+b8.jtRRHi,t−j*GHLi,t−j+b9.jtRRLi,t−j*GHHi,t−j+b10.jtRRLi,t−j*GHLi,t−j+eit, Average parameter

values are the time series averages from the cross-sectional estimates of stock variables, and t-statistics

are the time-series averages divided by the time-series standard errors. Ri,t-1 and Sizei;t are the return

and log (market capitalization) of stock i in month t-1. JTH (JTL) is a dummy variable that

equals 1 for 30% of the stocks with largest (smallest) 12-month cumulative returns at the end

of month t- j, and is 0 otherwise, RRH (RRL) is a dummy variable that equals 1 if the stock

belongs to top (bottom) 30% of the stocks based on their recency ratio at the end of month t - j,

and is 0 otherwise, and GHH (GHL) is a dummy variables that equals 1 for 30% of the stocks

with largest (smallest) GH measure at the end of month t - j, and is 0 otherwise.

All No Jan.

1 Intercept 3.278 2.787

(6.30) (5.31)

2 Rt-1 -0.031 -0.026

(-8.34) (-6.75)

3 MV -0.157 -0.121

(-5.00) (-3.84)

4 GHH 0.059 0.054

(5.17) (4.53)

5 GHL -0.108 -0.130

(-4.48) (-5.31)

6 RRH -0.051 -0.037

(-2.50) (-1.80)

7 RRL -0.015 -0.026

(-0.87) (-1.52)

8 GHH*RRH 0.136 0.136

(6.53) (6.25)

9 GHH*RRL 0.030 0.035

(1.72) (1.96)

10 GHL*RRH -0.009 0.007

(-0.31) (0.23)

11 GHL*RRL -0.012 -0.008

(-0.56) (-0.35)

12 = 9-11 GHRRL 0.042 0.043

(1.96) (1.96)

13 = 8-10 GHRRH 0.146 0.129

(5.02) (4.35)

14 = 13-12 GHRR 0.104 0.086

(4.31) (3.51)

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Table 8 Fama-MacBeth cross-sectional regression: Momentum profits

conditional on information discreteness For each month from January 1965 to December 2017, the following cross-sectional regression is

estimated. Rit=b0.jt+b1.jtRi,t−1+b2.jtSIZEi,t−1+b3.jtGHHi,t−j+b4.jtGHLi,t−j+b5.jtIDHi,t−j+b6.jtIDLi,t−j+b7.jtIDHi,t−j*

GHHi,t−j+b8.jtIDHi,t−j*GHLi,t−j+b9.jtIDLi,t−j*GHHi,t−j+b10.jtIDLi,t−j*GHLi,t−j+eit, Average parameter values

are the time series averages from the cross-sectional estimates of stock variables, and t-statistics are the

time-series averages divided by the time-series standard errors. Ri,t-1 and Sizei;t are the return and log

(market capitalization) of stock i in month t-1. JTH (JTL) is a dummy variable that equals 1 for

30% of the stocks with largest (smallest) 12-month cumulative returns at the end of month t- j,

and is 0 otherwise, IDH (IDL) is a dummy variable that equals 1 if the stock belongs to top

(bottom) 30% of the stocks based on their information discreteness at the end of month t - j,

and is 0 otherwise, and GHH (GHL) is a dummy variables that equals 1 for 30% of the stocks

with largest (smallest) GH measure at the end of month t - j, and is 0 otherwise.

All No Jan.

1 Intercept 3.105 2.568

(6.00) (4.96)

2 Rt-1 -0.029 -0.024

(-7.50) (-5.96)

3 MV -0.151 -0.114

(-4.86) (-3.67)

4 GHH 0.099 0.106

(6.97) (7.19)

5 GHL -0.063 -0.078

(-2.78) (-3.30)

6 IDH -0.010 -0.000

(-0.56) (-0.02)

7 IDL 0.000 0.008

(0.00) (0.43)

8 GHH*IDH -0.009 -0.025

(-0.40) (-1.09)

9 GHH*IDL 0.039 0.028

(1.85) (1.28)

10 GHL*IDH -0.056 -0.054

(-2.08) (-1.97)

11 GHL*IDL -0.109 -0.125

(-3.91) (-4.38)

12 = 9-11 GHIDL 0.148 0.154

(5.12) (5.02)

13 = 8-10 GHIDH 0.047 0.029

(1.62) (0.99)

14 = 12-13 GHID 0.101 0.124

(2.64) (3.14)

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Table 9 Fama-MacBeth cross-sectional regression: Momentum profits

conditional on information discreteness and recency ratio For each month from January 1965 to December 2017, the following cross-sectional regression is

estimated.Rit=b0.jt+b1.jtRi,t−1+b2.jtSIZEi,t−1+b3.jtGHHi,t−j+b4.jtGHLi,t−j+b5.jtRRHi,t−j+b6.jtRRLi,t−j+b7.jtIDHi,t−j

+b8.jtIDLi,t−j+b9.jtIDHi,t−j*RRHi,t−j*GHHi,t−j+b10.jtIDHi,t−j*RRHi,t−j*GHLi,t−j+b11.jtIDHi,t−j*RRLi,t−j*GHHi,t−j

+b12.jtIDHi,t−j*RRLi,t−j*GHLi,t−j+b13.jtIDLi,t−j*RRHi,t−j*GHHi,t−j+b14.jtIDLi,t−j*RRHi,t−j*GHLi,t−j+b15.jtIDLi,t

−j*RRLi,t−j*GHHi,t−j+b16.jtIDLi,t−j *RRLi,t−j* GHLi,t−j +eit. Average parameter values are the time series

averages from the cross-sectional estimates of stock variables, and t-statistics are the time-series averages

divided by the time-series standard errors. Ri,t-1 and Sizei;t are the return and log (market

capitalization) of stock i in month t-1. JTH (JTL) is a dummy variable that equals 1 for 30% of

the stocks with largest (smallest) 12-month cumulative returns at the end of month t- j, and is 0

otherwise, RRH (RRL) is a dummy variable that equals 1 if the stock belongs to top (bottom)

30% of the stocks based on their recency ratio at the end of month t - j, and is 0 otherwise, and

GHH (GHL) is a dummy variables that equals 1 for 30% of the stocks with largest (smallest)

GH measure at the end of month t - j, and is 0 otherwise.

Coefficients All No Jan.

1 GHH*IDH*RRH 0.061 0.042

(2.33) (1.54)

2 GHL*IDH*RRH -0.094 -0.089

(-2.15) (-1.94)

3 GHH*IDL*RRH 0.144 0.146

(6.07) (5.84)

4 GHL*IDL*RRH 0.014 0.033

(0.33) (0.75)

5 GHH*IDH*RRL -0.006 -0.003

(-0.25) (-0.13)

6 GHL*IDH*RRL -0.022 -0.010

(-0.74) (-0.33)

7 GHH*IDL*RRL 0.001 -0.005

(0.04) (-0.17)

8 GHL*IDL*RRL -0.079 -0.090

(-2.92) (-3.23)

9 = 1-2 GHIDHRRH 0.155 0.131

(3.23) (2.59)

10 = 3-4 GHIDLRRH 0.130 0.113

(2.92) (2.44)

11= 5-6 GHIDHRRL 0.016 0.007

(0.50) (0.21)

12 = 7-8 GHIDLRRL 0.080 0.085

(2.32) (2.37)

13 = 9-11 GHIDHRR 0.139 0.124

(2.77) (2.36)

14 = 10-12 GHIDLRR 0.050 0.028

(0.97) (0.52)

15 = 10-9 GHIDRRH -0.025 -0.018

(-0.39) (-0.26)

16 = 12-11 GHIDRRL 0.064 0.078

(1.38) (1.65)

17 = 10-11 GHIDRR 0.114 0.106

(2.27) (2.05)

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