r. david mcleantax loss selling assumes that investors sell their worst performing (or best...

Momentum Meets Reversals*(Job Market Paper)

R. David McLean

First Draft: November 1, 2004This Draft: January 9, 2005

Abstract

This paper studies momentum and long-term reversals concurrently. Reversals are not sensitive to the business cycle, but have a robust, positive, and significant relation with the costs and risks associated with arbitrage. Reversals could therefore be the result of mispricings that persist in the presence of sophisticated investors, who are motivated to correct mispricings, but are deterred from doing so by arbitrage costs. Momentum profits do not have a positive relation with arbitrage costs, so it is unlikely that such frictions can account for its persistence. A theory that integrates momentum and reversals needs to account for these important differences.

* Finance Department, Fulton Hall 330, Carroll School of Management, Boston College, Chestnut Hill, MA 02467. Contact: [email protected]. I am grateful to David Chapman, Tarun Chordia, Karl Diether (discussant), Richard Evans, Wayne Ferson, Cliff Holderness, Bing Liang, Jeffrey Pontiff, and Phil Strahan for helpful comments. This paper was presented at the Inquire Europe Autumn Symposium in Vienna, the 2005 FMA Doctoral Consortium, the 2005 FMA Special PhD Student Sessions, and the Boston College Brown Bag Workshop. I would also like to thank the Evelyn Brust Research and Education Foundation for financial support.

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1. IntroductionThere is a large body of empirical evidence showing that the cross-section of

stock returns can be predicted with past returns. For example, Jegadeesh and

Titman (1993, 2001) document a momentum effect over three to twelve month

horizons; winners continue to be winners and losers continue to be losers.

Rouwenhorst (1998, 1999) and Chui, Titman, and Wei (2000) find that

momentum portfolios are profitable in various developed and emerging

international markets. Over longer horizons, the predictability seems to be in the

other direction. DeBondt and Thaler (1985, 1987) and Chopra, Lakonishok, and

Ritter (1992) document long-term reversals over two to five year horizons; losers

become winners and winners become losers.1

What causes momentum and reversals is still is a matter of debate. Over the

previous decade there have been numerous papers that attempt to explain

momentum but far fewer that attempt to explain reversals. This difference may

in large part be due to the Fama and French (1996) finding that the Fama-French

three-factor model explains reversals but not momentum. One important

contribution of this paper is to show that once the seasonality of the reversal

portfolio is taken into account the Fama-French three-factor model no longer

explains reversals.

Previous studies have shown that reversals occur predominantly in January (see

Debondt and Thaler (1985 and 1987), Zarowin (1990), Ball, Kothari, and Shanken

(1995), and Moskowitz and Grinblatt (2004)). In the 78-year sample used in this

paper, reversals only occur in January. In my sample, the reversal portfolio’s

Fama-French three factor alpha is positive and significant in January (2.10, t-

statistic = 2.01), yet negative and significant outside of January (-.25, t-statistic = -

1 Jegadeesh (1990) and Lehmann (1990) report price reversals at monthly and weekly intervals, however I do not study these effects in this paper.

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1.73).2 In the full sample I, like Fama and French, find that the Fama and French

three-factor alpha is equal to zero, but this result is the consequence of

combining the positive alpha in January with the negative alphas of the non-

January months, during which there are no reversals to begin with.

If the Fama-French three factor model cannot explain momentum and reversals

then what can? The main goal of this paper is to test whether the persistence of

momentum and reversals profits is related to holding costs associated with

trading in each of the portfolios. In textbooks arbitrage is costless, and

arbitrageurs immediately eliminate any mispricing. However, DeLong et al.

(1990) and Shleifer and Vishny (1990, 1997) reason that real world arbitrage is

costly and that a mispricing will only be arbitraged away when arbitrage benefits

exceed arbitrage costs.3 If momentum and reversals both represent profits from

mispricing, then these papers predict that the magnitude of both effects be

should related to arbitrage costs.

How does one identify arbitrage costs empirically? Pontiff (1996) identifies two

types of arbitrage costs: transaction costs and holding costs. Transactions costs

are incurred when positions are opened or closed, while holding costs are

incurred every period that the position is open. Pontiff (2005) reviews all of the

existing studies in which both transactions costs and holding costs have been

related to anomalies. He concludes that in each of these studies holding costs, in

particular unhedgeable or idiosyncratic volatility, is the most important arbitrage

cost. In some of these studies idiosyncratic volatility renders transaction costs

2 I find that the momentum portfolio exhibits the opposite seasonality; its’ Fama-French three factor alpha is negative and significant in January, while positive and significant outside of January. 3 Pure or textbook arbitrage requires no capital and entails no risk. Shleifer and Vishny (1990) use the word arbitrage to describe: “trading based on knowledge that the price of an asset is different from its fundamental value.”

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insignificant in multivariate tests.4 Therefore, an important and unanswered

question is whether holding costs, especially idiosyncratic risk, can help to

explain the persistence of the momentum and reversal anomalies.

Pontiff (1996) and Shleifer and Vishny (1997) first identified idiosyncratic

variance (risk) as an important holding cost to arbitrageurs. The argument is as

follows: The larger a position that an arbitrageur takes in a given security, the

less diversified the arbitrageur becomes, and the more the arbitrageur exposes

her portfolio to the idiosyncratic risk of that given security (clearly the systematic

risk can be hedged, however the idiosyncratic risk cannot). Therefore, for a

given alpha, arbitrageurs will allocate a relatively small portfolio weight to a

high idiosyncratic risk stock; reducing the influence that arbitrage has on the

stock’s price (see Pontiff 2005 for a proof and illustrative example). In this paper,

idiosyncratic risk is the standard deviation of the residual that is generated by

regressing a stock’s excess return on the market’s excess return.

Another holding cost proxy used in this study is the percentage of shares held by

institutions (institutional holdings). Stocks with low institutional holdings are

more difficult to borrow and when borrowed have a higher risk of recall. When

a stock is shorted the short-seller does not have access to the proceeds of the

short sale, but receives a rate of interest known as the rebate rate on the proceeds.

D’Avolio (2001) shows that low institutional holdings predict low and even

negative rebate rates; so institutional holdings are a proxy for short sale holding

costs. Nagel (2005) studies a variety of return anomalies (not momentum or

reversals) and finds that underperformance is more pronounced among stocks

4 The anomalies reviewed by Pontiff (2005) include closed-end fund mispricings (Pontiff, 1996), Standard and Poor’s depository receipts (Ackert and Tian, 2000), index inclusion (Wurgler and Zhuravskaya, 2002), the book-to-market effect (Ali, Hwang, and Trombley, 2003), SEO underperformance (Pontiff and Schill, 2004), post-earnings announcement drift (Mendehall, 2004), the accrual anomaly (Mashruwala, Rajgopal, and Shevlin, 2005), and short interest (Duan, Hu, and McLean 2005).

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with low institutional ownership. Nagel interprets his results as evidence that

short sale costs have limited arbitrage.5

The results in this paper with respect to reversals and holding costs can be

summarized as follows. I find that reversals have a strong, positive, and

significant relation to arbitrage holding costs. Reversal profits are more than

seven times greater in high versus low idiosyncratic risk stocks, and more than

nine times greater in stocks with low institutional holdings than in stocks with

high institutional holdings. These results imply that reversals could be the result

of mispricings, which are shielded by the risks and costs associated with

arbitrage. I also find that reversals are not sensitive to the business cycle, and

even have higher returns in recessions, although the difference between

expansion and recession returns is not statistically significant. 6

There is some evidence in other studies that reversals are related to transaction

costs. Studies by Fama and French (1988), Zarowin (1990), and Chopra,

Lakonishok, and Ritter (1992) show that reversals are greatest in small stocks,

while Ball, Kothari, and Shanken (1995) show that reversals mainly occur in slow

priced stocks. I find that reversals are monotonically related to transaction cost

proxies (price and size), in that reversals are greatest in small and low-priced

stocks and then decrease as size and price increase. However in multivariate

tests, I show that idiosyncratic risk has the strongest affect on the reversal

portfolio, rendering price and size insignificant. This is consistent with the

conclusion of Pontiff (2005) that idiosyncratic risk is the single most important

cost faced by arbitrageurs.

5 Geczy, Musto, and Reed (2002) find that after controlling for short sale constraints and costs a short only, equally-weighted momentum portfolio is still profitable.6 Chordia and Shivakumar (2002) find that momentum is positive in expansions and negative in recessions. Griffin, Ji and Martin (2003) do not find this relation in international markets.

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These results may also help to explain the existence of long-term reversals.

Previous studies have argued that tax loss selling (due to the documented

seasonality) and/or investor overreaction are the root causes of long-term

reversals.7 The results in this paper accommodate both of these explanations.

Tax loss selling assumes that investors sell their worst performing (or best

performing if one is a short-seller) securities at the end of the year, which

depresses prices, and then buy the same securities back at the beginning of the

year, causing the reversal. However tax-loss selling is in itself an insufficient

explanation, for it does not explain why other investors do not buy the securities

that the tax selling investors are selling at year-end. I show that reversals have a

monotonic relation with arbitrage costs, so the results in this paper suggest that

arbitrage risk may deter rational investors from buying a sufficient amount of the

tax-loss selling securities, thus allowing prices to depress at year end, and then

reverse in January when the tax-loss sellers buy their shares back. To the extent

that reversals may be caused by overreaction, arbitrage costs again explain why

the effect is not arbitraged away.

In find that momentum bears a much different relation to arbitrage holding costs

than does reversals. There is no statistical difference in momentum profits

between high and low idiosyncratic risk stocks. In regression tests idiosyncratic

risk is shown to have an insignificant relation with momentum profits.

Momentum is weakest in stocks with the least institutional holdings, which is the

opposite of what a costly arbitrage explanation would predict. Therefore

momentum is not related to arbitrage holding costs.

Several studies have examined the relation between momentum and transaction

costs. If the persistence of momentum were due to transaction costs that 7 Debondt and Thaler (1985, 1987), Barberis, Shleifer and Vishny (1998), Daniel, Hirshleifer, and Subrahmanyam (1998), and Hong and Stein (1999) attribute reversals to overreaction. Grinblatt and Moskowitz (2004) suggest that reversals are in large part caused by tax-loss selling.

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discouraged arbitrageurs from correcting mispricings, then we would expect

momentum profits to be greatest in the stocks with the highest transaction costs.

However this is not what the literature has shown. For example Hong, Lim, and

Stein (2002) find that momentum is negative in stocks in the lowest NYSE size

decile, while Jegadeesh and Titman (2001) find that momentum is negative in

stocks with prices less than $5. Lee and Swaminathan (2002) find that

momentum is most profitable in high volume stocks and least profitable in low

volume stocks. All three of these papers suggest that momentum is weakest

when transaction costs are highest.

Korajcyk and Sadka (2004) and Lesmond, Schill, and Zhou (2004) estimate

transaction costs and both of these studies find that momentum profits do not

exceed transaction costs in equally weighted portfolios. However an optimizing

arbitrageur would not trade an equally weighted momentum portfolio,

especially given the evidence that momentum is weakest in stocks with the

highest transaction costs. Consistent with this notion, Korajcyk and Sadka (2004)

find that momentum profits do exceed transaction costs in both value and

liquidity-weighted portfolios. Lesmond, Schill, and Zhou (2004) find that the

most profitable momentum stocks tend to have high transaction costs, however

they also show that momentum profits are not monotonically related to

transaction costs.

Consistent with these other studies, I find that momentum is also not

monotonically related to size or price, but is monotonically related to volume (as

in Lee and Swaminathan, 2000), and is greatest in high volume stocks, which is

the opposite of what a costly arbitrage story would predict. Multivariate

regression tests confirm that momentum has an either neutral or negative

relation to both transaction and holding cost proxies.

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A mispricing explanation for momentum is tenuous, as momentum does not

have a monotonic relation with arbitrage costs. If momentum is the result of

mispricings, then there must be an unidentified deterrent to arbitrage that allows

the anomaly to persist. A behavioral theorist could argue that arbitrageurs

simply choose to avoid trading in the stocks that drive this anomaly, or perhaps

even trade on momentum, making the anomaly worse (see Delong et al., 1990b).

However one then needs to explain why arbitrageurs apparently correct

mispricings related to reversals and the other anomalies reviewed by Pontiff

(2005), but not momentum.

So is momentum caused by risk? If it is, then very small and very low-priced

stocks would need to be negatively correlated with this unknown source of risk,

for in these stocks momentum is negative. Trading volume would also have to

be negatively correlated with this source of risk, as trading volume is negatively

correlated with momentum. This source of risk was also absent prior to 1940, as

momentum did not exist prior to 1940 (Jegadeesh and Titman, 1993), and this

source of risk has increased since 1990, or at least the covariance of individual

stocks with this source of risk has increased since 1990, as momentum has

strengthened since 1990 (see Jegadeesh and Titman, 2001).

Finally the results in this study may also be useful for understanding how

momentum and reversals may be related to one another. Jegadeesh and Titman

(1993, 2001) show that the profits of momentum portfolios reverse and become

negative over long horizons; this could be evidence that momentum and the

long-term reversals first documented in Debondt and Thaler (1985) are related.8

8 The reversal is significant in the 1965-1981 subperiod, but insignificant in the 1982-1998 subperiod. Lee and Swaminathan (2000) find that reversal of momentum portfolios is related to volume. Griffin, Ji and Martin (2003) document a momentum-reversal pattern in international markets. Cooper, Gutierrez, and Hammed (2004) find that momentum is related to the state of the market, but reversal of the momentum portfolio is not.

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A number of theoretical papers have even predicted that long-term reversals are

caused by near-term momentum.9 However, Moskowitz and Grinblatt (2004)

note that linking the two anomalies is questionable, as momentum does not

exhibit the same seasonality as reversals. Conrad and Kaul (1998) also show that

momentum and reversals have each been strongest at different points in time. In

this study I show that reversals do not share momentum’s sensitivity to the

business cycle. I explore the cross-sectional differences between the two effects

and conclude that each effect is greatest in different types of stocks. A theory

that wishes to integrate these two anomalies would need to account for the

differences documented in this paper.

To further study how momentum and reversals might be related I form a

combined momentum-reversal strategy, which conditions on both medium and

long-horizon past returns. The resulting portfolio is long momentum winners

that are also reversal losers, and short momentum losers that are also reversal

winners. I decompose the returns of this combined portfolio into three parts, a

pure momentum part, a pure reversal part, and an interactive part that arises

from simultaneous membership in both portfolios. An original finding of this

paper is that there is an interactive effect; it explains about 10% of the returns of

the combined portfolio. Momentum accounts for about 60% of the combined

portfolio’s returns, while reversals account for about 30%.

The fact that there are separate and significant momentum and reversal

components within the combined portfolio again reinforces the notion that

momentum and reversal are in large part separate effects. Over long horizons,

9 See Barberis, Shleifer and Vishny (1998), Daniel, Hirshleifer, and Subrahmanyam (1998), and Hong and Stein (1999). Referring to these three papers Hirshleifer (2001) writes “In all these models the misperceptions that drive momentum are also the drivers of long-term reversals. These models imply that if there is some market segmentation, then those assets with the largest momentum effects should also have the largest reversal effects…”

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the momentum component of this portfolio reverses, while the reversal

component remains positive. This suggests that the long-horizon reversals of

momentum documented by Jegadeesh and Titman (2001) and the long-term

reversal of Debondt and Thaler (1985) can happen simultaneously in the same

stock, which again suggests that momentum and reversals are largely separate

effects. However there is an interaction, and this would imply that at some

level, or at least among some stocks, the two effects might be related.

The rest of this paper is organized as follows. Section 2 discuses some of the

related literature. Section 3 describes the data and some initial results. Section 4

is the costly arbitrage analysis. Section 5 explores the roles of different economic

states and January on the effects and Section 6 concludes.

2. Related LiteratureIn a contemporaneous paper Arena, Haggard, and Yan (2005) find that

momentum (losers only) is stronger in stocks with high idiosyncratic risk, which

is not what I find. Differences in sample construction explain why Arena et al.’s

findings are different than mine.10 Arena et al. drop all stocks with prices under

$5 and market caps in the lowest NYSE decile. These are stocks for which

arbitrage is costly and idiosyncratic risk is very high; a costly arbitrage study that

excludes such stocks may overlook important empirical findings. Arena et al. are

discarding a number of stocks that would presumable be in their high

idiosyncratic risk portfolio and make it perform worse, as momentum performs

poorly among the smallest and stocks and stocks with prices under $5 (see Hong,

Lim, and Stein, 2002 and Jegadeesh and Titman, 2001). Furthermore, Arena et al.

only use data post 1965, and I find that their results do not hold in the 1940-1965

period for which data are available and momentum is profitable.

10 If I follow Arena et al’s sample construction methods then I get similar results to theirs.

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Ali and Trombley (2005) argue that momentum is caused by short sale

constraints in stocks returns. They show that the worst performing momentum

losers are stocks with low institutional ownership. However, similar to Arena et

al., Ali and Trombley begin their study by discarding all stocks with market

values below the second NYSE/AMEX market value decile, which are precisely

the stocks for which institutional ownership is low and short sale constraints are

most likely to bind. These are also the stocks for which momentum is either

weak or negative (see Hong, Lim, and Stein, 2002).

3. Data and Preliminary ResultsIn this section I assign stocks to portfolios based on past returns. I use monthly

data from CRSP; my sample period in this Section is from January 1940 through

December 2003. The sample begins in 1940 due to the finding of Jegadeesh and

Titman (1993) that momentum returns were negative prior to 1940. However the

results do not qualitatively change if the entire CRSP dataset is used, in which

case the analysis would begin in 1930. To be in my sample, each stock had to

have at least five years of data prior to a portfolio formation date. I first perform

monthly sorts of all of the stocks in my sample based on past returns from t-6 to

t-1. This was the preferred portfolio formation horizon of Jegadeesh and Titman

(1993, 2001) [JT hereafter]. I start the momentum horizon at t-1 months to avoid

the microstructure issues described by Jegadeesh (1990). Each stock is placed

into one of five momentum portfolios based on its ranking, i.e. stocks with the

highest past returns (winners) are placed in portfolio 5 and those with the lowest

past returns (losers) are place in portfolio 1. To create the reversal portfolios I

repeat the described process, only I measure past returns from t-60 to t-6. The

sorting and therefore the two portfolios are created independently. I begin the

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long horizon measurement at t-6 to avoid overlapping with the momentum

horizon.11

A difference between this paper and JT is that JT use 10 portfolios, and calculate

their momentum profits as P10-P1. In this paper I only construct five portfolios,

so my momentum profits should appear a bit weaker than JT’s. The point in this

paper is not to document the existence of momentum, but rather to study the

momentum affect across subsamples of stocks. Using larger portfolios allows me

to have stronger signal to noise properties in my tests; the portfolios remain large

enough so that they are diversified and therefore idiosyncratic volatilities

typically do not affect the standard errors. However, in earlier periods (before

1935, which I examine in Section 5) the cross-sorted portfolios contain as few as

10 stocks, but average about 50 stocks.

3.1. Returns of the Different Past Return PortfoliosPanel A of Table 1 displays the returns of three past return portfolios. The

returns are in percent and are monthly averages. The standard errors for the t-

statistics are calculated using the method of Newey and West (1987). Each

month the returns for each portfolio are calculated, the reported numbers are the

average of the monthly return time-series. The Momentum profits are calculated

by subtracting the equally weighted returns of quintile 1 (losers) from quintile 5

(winners). Reversal profits are calculated by subtracting the equally weighted

buy and hold the returns of quintile 5 (winners) from quintile 1 (losers). The

combined portfolio is long stocks that are both momentum winners and reversal

losers and short stocks that are both momentum losers and reversal winners.

In Table 1, we observe positive momentum profits of .81% (t-statistic = 7.79) and

.32% (t-statistic = 3.63) over 6 and 12-month holding periods. This result implies

11 A portfolio that chooses t-3 to t-1 (for momentum) and from t-36 to t-3 (for reversals) produces similar results.

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that momentum profits may begin to reverse after month 6, consistent with

Lewellen (2002), who shows that the returns of the momentum portfolio become

negative in month 7. Panel A shows that the momentum portfolio reverses and

becomes negative in Years 2 through 5. The momentum portfolio produces

negative returns of –.18% (t-statistic = -31.99), –.16% (t-statistic = -1.33), –.11% (t-

statistic = -1.28), and –.12% (t-statistic = -1.44) in Years 2, 3, 4, and 5.

Table 1 also reveals strong evidence of reversals. We observe positive reversal

profits of .47% (t-statistic = 2.98) and 86% (t-statistic = 5.04) over 6 and 12-month

holding periods. The profits of this portfolio are persistent out to year five, in

which the return is .25% (t-statistic = 2.53). One should note that the reversal in

Year one (47%) is more than double the initial reversal of the momentum

portfolio in Year 2 (-.18%), so the reversal of the momentum portfolio could at

most explain only a small fraction of long-term reversals.

The combined portfolio in Table 1 has very high returns of 1.41% (t-statistic =

8.23) and 1.33% (t-statistic = 7.70) over 6 and 12-month holding periods. The

returns remain positive out through year 5, in which the return is .11% (t-statistic

= .95). The combined portfolio outperforms momentum by .62% (t-statistic =

3.80) and 1.01% (t-statistic = 5.93) over 6 and 12-month horizons. The combined

portfolio continues to beat momentum out through year 5. Like those of the

reversal portfolio, the returns of the momentum-reversal portfolio dissipate in a

somewhat linear fashion as the horizon lengthens. The combined portfolio

outperforms reversals by .94% (t-statistic = 8.65) and .47% (t-statistic = 4.72) over

6 and 12-month horizons. However in Year 2 the combined portfolio

underperforms reversals and does so out through year 5. It seems that stocks in

the combined undergo a lager portion of their reversal in the first year, and

therefore the portfolio has lower returns in the later years.

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3.2. Decomposing the Combined Portfolio’s ProfitIn this Section I decompose the profits of the combined portfolio. To estimate the

different components I perform monthly Fama-Macbeth cross-sectional

regressions using the following regression model

rt+K,i = aK + b1 Momentumt,i + b2Reversalt,i + b3Combinedt,i + et+K,i (1)

Where the subscript i refers to stock i. rt+K,i is the future return for stock i.

Momentum = 1 if the stock is a momentum winner, -1 if the stock is a momentum

loser and 0 otherwise. Reversal = 1 if the stock is a reversal loser, -1 if the stock is

a reversal winner and 0 otherwise. Combined = 1 if both Momentum and Reversal

also =1, -1 if both Momentum and Reversal = -1 and 0 otherwise. Winners and

losers are defined as in the previous section; winners are in the high past return

quintiles and losers are in the low past return quintiles. When a stock is in the

combined portfolio, the Momentum, Reversal and Combined dummies will all be

on and will all be of the same sign, 1 if it is a buy and –1 if it is sell. I report the

time series average of the coefficients from each cross-section.

In Table 2, at the 6-month and 1 year horizons the Combined coefficient is positive

and significant.12 This implies that there are positive expected returns in excess

of the expected returns from the pure momentum and reversal portfolios. This

also implies that at some level these two effects may be related, as the Combined

coefficient can be viewed as a type of interaction term. The combined coefficient

is .07% (t-statistic = 1.81) and .11% (t-statistic = 2.66) over 6 and 12-month

horizons. Note that the portfolio is a long-short portfolio, so when Combined=–1,

the expected return from the sell side is -.07% and -.11% over the 6 and 12-month 12 The expected return of the long portion of the Combined portfolio is a+b1 +b2 +b3. The expected return of the short portion of the Combined portfolio is –[ a -b1 -b2 -b3]. Therefore, the expected return of the Combined portfolio, which is equally long and short is 2 * [b1 +b2 +b3]. The expected return of the Momentum stratgey is 2 * b2 and the expected return of the Reversal portfolio is 2 * b3.

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horizons. So according to the 12-month return regression coefficient the total

monthly returns that arise form the interaction are about .22% per month, or

about 2.64% a year, and this is in addition to the expected returns from the pure

momentum and reversal portfolios.

The expected return of the Combined portfolio is two times the sum of the

Momentum, Reversal and Combined coefficients. This amount is displayed in the

Long-Short total row. The regression implies that the expected return of the

combined portfolio is about 1.33% per month, in the first 6 months, which is very

close to the 1.41% per month that was estimated in Table 1. The returns of this

portfolio are then decomposed into the Momentum, Reversal and Combined

components. The Combined coefficient contributes 10% and 16% over the 6 and

12-month horizons. In year 3, the Combined coefficient’s contribution becomes

negative. Note that overall the Combined portfolio still has positive profits,

which are displayed in the Long-Short Total row, however after Year 3, the

Portfolio’s profits are completely driven by the Reversal component, the

Momentum and Combined components make negative contributions.

Despite the significance of the combined coefficient, the results in this Section

imply that momentum and long-term reversals are independent effects, as the

coefficients for both portfolios are statistically significant over the first 6 and 12-

month periods. In year 2 the momentum coefficient becomes negative (-.07%, t-

statistic = -1.74), while the reversals coefficient remains positive and significant

(.30%, t-statistic = 3.71). These results clearly show that the reversal of the

momentum portfolio documented by Jegadeesh and Titman (1993, 2001) is

distinct from the long-term reversal first documented by Debondt and Thaler

(1985), as both reversals can happen at the same time within the same portfolio of

stocks.

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4. The Role of Arbitrage Costs in the Relative Strength Portfolios’ Profits In this Section I explore what role arbitrage costs play in the profits of the

momentum, reversal, and combined portfolios.

4.1 Momentum and Reversal Portfolios Cross-Sorted on Costly Arbitrage ProxiesThis Section examines whether the profitability of the momentum and reversal

portfolios varies with measures of arbitrage costs. The momentum and reversal

portfolios are created as they were in the previous Section. Idiosyncratic risk

(IR), institutional holdings (IH) are the holding cost proxies, while size (MV),

price (PRC), and trading volume (TV) are the trading cost measures. Each month

all of the stocks in the sample are sorted independently on each of the costly

arbitrage proxies. Each stock is then placed into one of three portfolios for each

of the arbitrage cost proxies based on its ranking for that proxy. Trading volume

data are available beginning in 1963 and institutional holdings data are available

on a quarterly basis beginning 1980. The other measures are available for the

entire sample.

The first panel of Table 3 cross-sorts the momentum winners and losers into

three different idiosyncratic risk (IR) portfolios. The difference between the high

and low IR momentum portfolios is only .10% per month (t-statistic = 1.00). The

difference between the high and low IR reversal portfolios is much stronger,

averaging .71% per month (t-statistic = 5.16). 28% of the momentum portfolio’s

profits occur in high IR stocks, while 73% of the reversal portfolio’s profits occur

in high IR stocks. These results are consistent with a costly arbitrage explanation

for the persistence of reversals, but not for momentum.

The second panel cross-sorts the momentum winners and losers into three

different institutional holdings (IH) portfolios. The difference between the high

and low institutional holdings momentum portfolios is .49% per month (t-statistic

= 2.49). This is the opposite of what a costly arbitrage story would predict. The

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difference between the high and low IH reversal portfolios is large, averaging –

1.20% per month (t-statistic = -3.35). 18% of the momentum portfolio’s profits

occur in low IH stocks, while 73% of the reversal portfolio’s profits occur in low

IH stocks. These results are again consistent with a costly arbitrage explanation

for the persistence of reversals, but not momentum.

The third panel cross-sorts the momentum winners and losers into three

different market value (MV) portfolios. The difference between the large and

small stock momentum portfolios is only .31% per month (t-statistic = 3.45). This

is the opposite of what a costly arbitrage story would predict. However

momentum appears to be strongest among mid-cap stocks, as the returns of that

portfolio average .90% per month, (t-statistic = 9.00), higher than both the large

and small stock portfolios. The difference between the large and small stock

reversal portfolios is striking, it averages -.74% per month (t-statistic = -6.02).

This result is consistent with the numerous studies that have found that reversals

are stronger in small stocks. 27% of the momentum portfolio’s profits occur in

low MV stocks, while 70% of the reversal portfolio’s profits occur in low MV

stocks. These results are once again consistent with a costly arbitrage

explanation for the persistence of reversals, but not momentum.

The fourth panel cross-sorts the momentum winners and losers into three

different price (PRC) portfolios. The difference between the high and low price

momentum portfolios is only .32% per month (t-statistic = 3.89). This is again the

opposite of what a costly arbitrage story would predict. Further momentum

appears to be equally strong among the medium and high priced stocks and

among the low priced stocks. The difference between the high and low price

reversal portfolios is large, averaging -.96% per month (t-statistic = -6.78). 33% of

the momentum portfolio’s profits occur in low PRC stocks, while 84% of the

reversal portfolio’s profits occur in low PRC stocks. These results are again

- 17 -

consistent with a costly arbitrage explanation for the persistence of reversals, but

not momentum.

The fifth panel cross-sorts the momentum winners and losers into three different

trading volume (TV) portfolios. The difference between the high and low

volume momentum portfolios is .54% per month (t-statistic = 4.31). In fact, 51%

of the momentum portfolio’s profits occur in high TV stocks. This result is

consistent with the findings of Lee and Swaminathan (2000). This is the opposite

of what a costly arbitrage story would predict. The difference between the high

and low volume reversal portfolios is large, averaging -.10% per month (t-statistic

= -.61). These results are not consistent with a costly arbitrage explanation for

the persistence of either reversals or momentum.

Figure 1 plots the differences described in this Section. The patterns in Figure 1

reinforce the notion that a costly arbitrage story can explain the persistence of

reversals, but not momentum. Figure 2 plots the percentage of each portfolio

occurring in high arbitrage cost group. One can clearly see that reversals

predominantly arise in stocks that are costly to arbitrage, but momentum does

not.

4.2 Combined Portfolios Cross-Sorted on Costly Arbitrage Measures Table 5 is the same as Table 4, only in Table 5 the combined portfolio is cross-

sorted on each arbitrage cost measures, rather than the momentum and reversal

portfolios. Recall that the combined portfolio is long stocks that are both

momentum winners and reversal losers and short stocks that are both

momentum losers and reversal winners. In the previous Section it was shown

that reversals seemed to be stronger in high arbitrage cost stocks, but momentum

is not. Therefore it is unclear as to what type of relation the combined portfolio

will have to arbitrage cost proxies.

- 18 -

The left hand side of the first panel cross-sorts the combined portfolio into three

different idiosyncratic risk (IR) portfolios. The difference between the high and

low IR stock combined portfolios is .82% per month (t-statistic = 3.77). 69% of the

combined portfolio’s profits occur in high IR stocks, these results are what a

costly arbitrage story would predict. The right hand side of the second panel

cross-sorts the combined portfolio into three different institutional holdings (IH)

portfolios. The difference between the high and low IH combined portfolios is

only -.73% per month (t-statistic = -1.98). This is what a costly arbitrage story

would predict. As with reversals, the evidence is that arbitrage holding costs can

help to explain the persistence of the combined portfolio.

The left hand side of the second panel cross-sorts the combined portfolio into

three different market value (MV) portfolios. The difference between the large

and small stock combined portfolios is -.40% per month (t-statistic = -2.29). The

right hand side of the second panel cross-sorts the combined portfolio into three

different PRC portfolios. The difference between the high and low price stock

combined portfolios is only -.60% per month (t-statistic = -3.03). This is what a

costly arbitrage story would predict. The third and final panel cross-sorts the

combined portfolio into three different TV portfolios. The difference between the

high and low volume combined portfolios is .02% per month (t-statistic = .09).

4.3 Multivariate Regression AnalysesTable 6 explores the costly arbitrage hypothesis in a multivariate setting. I create

portfolios for momentum, reversals and the combined portfolio. MOM=1 if the

stocks is a momentum winner, -1 if the stocks is a momentum loser, and 0

otherwise. REV=-1 if the stocks is a reversal winner, 1 if the stocks is a reversal

loser, and 0 otherwise. COMB= if both MOM and REV=1, -1 if both MOM and

REV=-1, and 0 otherwise.

- 19 -

I again consider five different costly arbitrage measures idiosyncratic risk (IR),

size (MV), price (PRC), institutional holdings (IH), and trading volume (TV).

Each stock is also placed into one of three costly arbitrage portfolios for each

measure. The costly arbitrage portfolios in this section are constructed so that

portfolio 3 contains the stocks with the highest arbitrage costs; e.g. low PRC, low

MV, low TV, and low IH, but high IR stocks. So the portfolio rankings are done

on IH-1, MV-1, PRC-1, TV-1, and IR. Each stock is then assigned a value of 1, 2, or 3

for each characteristic depending on which portfolio it gets placed in for that

characteristic. The cross-sorted portfolios therefore take on values of –3, -2, -1, 0,

1, 2, or 3. For example a low price momentum winner would take on a value of

3, while a low price momentum loser would take on a value of –3. If the costly

arbitrage hypothesis is correct, then we expect to see positive and significant

interactions.

In all of the regressions the dependent variable is the average monthly return in

the six-months subsequent to portfolio formation. In Regression 1 the only

variable is MOM. Its coefficient value is .29% (t-statistic = 2.96). This means

when MOM=1 the expected momentum premium is .29%, when MOM=-1 the

expected momentum premium is -.29%, and the total momentum profit from a

long-short portfolio is .58%.

In Regression 2 the interaction portfolios and characteristic portfolios are

included. There are three interactions that are significant, but incorrectly signed.

MOM*IR (coefficient = -.16, t-statistic = -2.77), MOM*MV-1 (coefficient = -.10, t-

statistic = -2.17), and MOM*VOL-1 (coefficient = -13, t-statistic = -2.12) all imply

that momentum is stronger when arbitrage costs are lower. MOM*PRC-1

(coefficient = .13, t-statistic = 1.18) is the only correctly signed, albeit insignificant

coefficient. The MOM coefficient is now much larger (coefficient = 1.03, t-statistic

- 20 -

= 4.79), as compared to its value in Regression 1, implying that the interactions

and control variables actually dampen the momentum profits.

In Regression 3 the only variable is REV. Its coefficient value is .40% (t-statistic =

2.29). This means that the total reversal profit from a long-short portfolio is .80%.

In Regression 4 the interaction portfolios and characteristic portfolios are

included. There is one interaction that is correctly signed. REV*IR (coefficient =

.35, t-statistic = 5.20) implies that reversals are stronger when arbitrage costs are

higher. This result is consistent with the conclusions of Pontiff (2005) that

idiosyncratic risk is the greatest holding cost faced by arbitrageurs. In this table

it has rendered the other cost proxies insignificant. The REV coefficient is now

negative (coefficient = -.95, t-statistic = .3.77), implying that the interactions and

control variables fully explain reversal profits.

In Regression 5 the only variable is COMB. Its coefficient value is .77% (t-statistic

= 5.73). This means that the total combined profit from a long-short portfolio is

1.54%. In Regression 6 the interaction portfolios and characteristic portfolios are

included. There are two interactions that are significant and correctly signed.

COMB*IR (coefficient = .20, t-statistic = 2.20), and COMB*PRC-1 (coefficient = .22,

t-statistic = 1.83) both imply that combined portfolio’s returns are stronger when

arbitrage costs are higher. The COMB coefficient is now much insignificant

(coefficient = 0.01, t-statistic = .02), implying that the interactions and control

variables fully explain the combined portfolio’s profits.

The results in the Section support the univariate results. Momentum does not

seem to be a function of arbitrage costs. However both the reversal and the

combined portfolio’s returns covary strongly with arbitrage costs.

- 21 -

5. Relative Strength Portfolios’ Profits at Different Points in TimeIn this Section I compare the momentum, reversal, and combined effects in

expansions and contractions and in January versus non-January months. I also

slightly expand my sample to 1930-2003, as the early part of the 1930’s was

contraction years. The returns are also now monthly, meaning only the first

month’s returns after portfolio formation are studied. This is done mainly so the

performance measurement in this Section can be related to the results in Fama

and French (1996).

5.1 Expansions versus ContractionsTable 6 compares momentum, reversals, and combined the portfolio in

expansions and contractions. Chordia and Shivakumar (2002) find that

momentum is only profitable in expansions and is negative in recessions. I get

the same result; momentum profits are .39 (t-statistic = 6.84) in expansion months

and -.57 (t-statistic = -.18) in recession months. However the same is not true for

reversals, reversal profits are .67 (t-statistic = 6.84) in expansion months and 1.14

(statistic = 1.60) in recession months. Although reversal profits are on average

greater in recessions, the difference between reversal profits in expansion and

recession month is not statistically significant (t-statistic = -.76). The fact that the

profits of the reversal portfolio do not vary with the business cycle is consistent

wit the notion that reversals may be caused by mispricings. As in the previous

sections, the evidence in this Section is that there are important differences

between momentum and reversals.

The combined portfolio has higher returns in expansions (mean = 1.19 t-statistic =

8.53) than in contractions (mean = .60 t-statistic = 2.12), although the difference

(mean =.59 t-statistic = .99) is not statistically significant. However the Sharpe

ratio of the combined portfolio is 3.16 in expansions versus .68 in contractions,

and the CAPM and Fama-French alphas are only signifanct in expansion as well,

so this portfolio does exhibit some business cycle risk.

- 22 -

5.2 January versus Non-JanuaryPrevious studies have shown that reversals are for the most part profitable only

in the month of January (Moskowitz and Grinblatt, 2004), while momentum is

only profitable outside of January (Jegadeesh and Titman, 1993, 2001). Like these

studies I find that momentum is negative in January and positive outside of

January, while reversals are profitable in January, but not statistically different

from zero outside of January. Like reversals the combined effect is also strongest

in January (difference = 1.90, t-statistic = 2.92), but is still active outside of

January (mean = .90, t-statistic = 3.93).

Fama and French (1996) claim that their three-factor model explained reversals.

Table 7 shows that the reversal’s alpha from the Fama-French three factor model

is positive and significant in January (alpha = 2.10, t-statistic = 2.01), and negative

and significant outside of January (alpha = -.25, t-statistic = -1.73). Therefore, the

Fama and French’s finding is the result of combining the negative alpha months

(during which there are nor reversals) with the positive alpha month.13

5. ConclusionThis paper documents several important empirical facts regarding momentum

and long-term reversals. First, once the seasonality of long-term reversals is

taken into account, the Fama-French three-factor model no longer explains

reversals. Reversals do not fluctuate with the business cycle, however reversals

are monotonically related to idiosyncratic risk and other arbitrage costs. The

profits of the reversal portfolio increase as these costs increase, which suggests

that reversals may be the result of mispricings, which are not arbitraged away

due to the costs associated with arbitrage activity. Like the numerous other

anomalies reviewed by Pontiff (2005), idiosyncratic risk dominates the other cost

proxies in multivariate tests.

13 In the entire sample I find that the alpha from a regression of the reversal portfolio on the Fama-French three factor model is .00 (t-statistic = -0.02)

- 23 -

Momentum profits on the other hand do not appear to be related to arbitrage

costs. Korajcyk and Sadka (2004) have already shown that an optimizing

arbitrageur can generate robust profits in momentum portfolios if trading costs

are taken into account. I find no difference in returns between high and low

idiosyncratic risk momentum portfolios. Therefore, momentum is either some

type of risk, or for some yet unidentified reason arbitrageurs avoid trading in

momentum stocks altogether, which might explain why the magnitude of the

anomaly is not related to its arbitrage costs.

Finally, momentum and reversals have important cross-sectional differences.

Reversals tend to appear in small, low-priced stocks, which are avoided by

institutions. This result implies that individuals cause reversals. Momentum on

the other hand is very weak in stocks that are avoided by institutions and

appears in predominantly high volume, medium-sized stocks. Gutierrez and

Pirinsky (2005) contend that momentum is actually caused by institutions; this

hypothesis might help to explain why momentum does so poorly in small stocks

and stocks that are avoided by institutions. A theory that wishes to integrate

these two anomalies would need to account for the differences documented in

this paper.

- 24 -

References

Ackert, L.F., and Y.S. Tian, 2000, Arbitrage and valuation in the market for Standard and Poor’s depository receipts, Financial Management, 71-88

Arena, M., K. Haggard, and Y.S. Xuemin, 2005, Price momentum and idiosyncratic volatility, University of Missouri - Columbia Working Paper

Ashiq, A. and M. Trombley, 2005, Short-selling costs and momentum in stocks returns, Journal of Business Finance and Accounting, forthcoming

Ball, R., S.P. Kothari, and J. Shanken, 1995, Problems in measuring portfolio performance: An application to contrarian investment strategies, Journal of Financial Economics 38, 79-107

Barberis, N., A. Shleifer, and R. Vishny, 1998, A model of investor sentiment, Journal of Financial Economics 49, 307-343

Barberis, N., A. Shleifer, 2003, Style investing, Journal of Financial Economics 68, 161-199

Debondt, W., and D. Thaler, 1985, Does the Stock Market Overreact?, Journal of Finance 55, 793-805

Debondt, W., and D. Thaler, 1987, Further Evidence on Overreaction and Stock Market Seasonality?, Journal of Finance 57, 557-581

DeLong, B., A. Shleifer, L. Summers, and R. Waldman, 1990, Positive feedback investment portfolios and destabilizing rational speculation, Journal of Finance 45, 379-395

Duan Y., Hu G., and R.D. McLean, 2005, Idiosyncratic risk, short-sellers, and stock returns, Boston College Working Paper

Chopra, N., J. Lakonishok, and J. Ritter, 1992, Measuring abnormal performance: Do stocks overreact?, Journal of Financial Economics 31, 235-268

Chordia, T., and L. Shivakumar, 2002, Momentum, business cycle, and time varying expected returns, Journal of Finance, 57, No.2

Chui, A., S. Titman and K.C.J. Wei, 2000, Momentum, ownership structure, and financial crises: An analysis of Asian stock markets, working paper, University of Texas at Austin.

- 25 -

Conrad, J. G. Kaul, 1998, An anatomy of trading strategies, Review of Financial Studies, No. 3, 489-519

Cooper, M., R. C. Gutierrez Jr., and A. Hameed, 2004, Journal of Finance, No.3 1345-1365

Daniel, K., D. Hirshleifer, and A. Sunbrhamanyam, 1988, Investor psychology and security market under and overreactions, Journal of Finance, 53, pp 1839-1886

Daniel, K., and S. Titman, 1999, Market efficiency in an irrational world, Financial Analyst Journal, 55, 28-40

D’Avolio, G., 2002, The market for borrowing stock, Journal of Financial Economics66, 271-306

DeBondt, W. and R. Thaler, 1985, Does the stock market overreact? Journal of Finance 40, 28-40

DeBondt, W., and R. Thaler, 1987, Further evidence of investor overreaction and stock market seasonality, Journal of Finance 42, 557–581.

Delong, B., Shleifer, A., Summers, W., and R. Waldman, 1990, Noise trader risk in financial markets, Journal of Political Economy 98, 703-738

De Long, J. B., A. Shleifer, L. H. Summers, and R. J. Waldmann, 1990, Positive feedback investment portfolios and destabilizing rational speculation, Journal of Finance 45, 379–395

Duan, Y., G. Hu, and R. D. McLean, 2005, Idiosyncratic risk, short-sellers, and stocks returns, Boston College Working Paper

Fama, E.F. and K.R. French, 1988, Permanent and Temporary Components of Stocks Prices, Journal of Political Economy, Vol.6, No.2, 246-273

Fama, E.F. and K.R. French, 1992, The cross-section of expected stocks returns, Journal of Finance 47, 427-265

Fama, E.F., and K.R. French, 1996, Multifactor explanations of asset pricing anomalies, Journal of Finance 51, 55−84.

Friedman, M., 1953, The case for flexible exchange rates, Essays in Positive Economics (University of Chicago Press), 157–203.

- 26 -

George, T. and C. Hwang, 2004, The 52-week high and momentum investing, Journal of Finance, 59, 2145-2176

Griffin, J., S. Ji, and S. Martin, 2003, Momentum investing and business cycle risk: Evidence from pole to pole, Journal of Finance Forthcoming

Grinblatt, M., and T. Moskowitz, 2003, Predicitng stock price movements from past returns: the role of consistency in tax-loss selling, Journal of Financial Economics 71, 541-579

D. Hirshleifer, 2001, Investor Psychology and Asset Pricing, Journal of Finance 56,1533-1597

Hong, H., and J. Stein, 1999, A unified theory of underreaction, momentum trading and overreaction in asset markets, Journal of Finance 45, 265-295

Jegadeesh, Narasimhan, 1990, Evidence of predictable behavior of security returns, Journal of Finance 45, 881-898

Jegadeesh, N. and S. Titman, 1993, Returns to buying winners and selling losers: Implications for stock market efficiency, Journal of Finance 48, 65-91

Jegadeesh, N. and S. Titman, 2001, Profitability of momentum portfolios: An evaluation of alternative portfolios, Journal of Finance 56, 699-720

Liu, X. L., J. B. Warner, and L. Zhang, 2005, Momentum Profits and Macroeconomic Risk, University of Rochester and NBER Working Paper

Lee C., and B. Swaminathan, 2000, Price momentum and trading volume, Journal of Finance 55, 1217-1269

Mashruwala, C., Rajgopal, S., and T. Shevlin, 2005, Why is the accrual anomaly not arbitraged away? The role of idiosyncratic risk and transaction costs, Forthcoming Journal of Accounting and Economics

Mendenhall, R., 2004, Arbitrage risk and post-earnings announcement drift, Journal of Business 77, 875-894

Pontiff, J., 1996, Costly Arbitrage: Evidence from Closed-End Funds, Quarterly Journal of Economics 111, 1135-1151

- 27 -

Pontiff, J., 2005, Costly arbitrage and the Myth of idiosyncratic risk, Boston College Working Paper

Pontiff, J. and M. J. Schill, 2004, Long-Run Seasoned Equity Offering Returns: Data Snooping, Model Misspecification, Or Mispricing? A Costly Arbitrage Approach,” Boston College Working Paper

Rouwenhorst, G. 1998, International momentum portfolios, Journal of Finance 53, 267-284

Rouwenhorst, G. 1999, Local return factors and turnover in emerging stock martkets, Journal of Finance 54, 1439-1464

Shleifer, A., 1986, Do demand curves for stocks slope down?, Journal of Finance41, 579-590

Shleifer, A. and R. Vishny, 1990, Equilibrium Short Horizons of Investors and Firms, American Economic Review 80, 148–153.

Shleifer, A. and R. Vishny, 1997, The limits of arbitrage, Journal of Finance, LII (1), 35–55

Wurgler, J. and E. Zhuravskaya, 2002, Does arbitrage flatten demand curves for stocks?, Journal of Business 75, 583-60

P. Zarowin, 1990, Size, Seasonality, and Stock Market Overreaction, Journal of Financial and Quantitative Analysis 25, 113-125

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Table 1: Returns of the Momentum, Reversal, and Combined PortfoliosThis Table displays the equal-weighted average monthly returns of three different past return portfolios. At the beginning of each month between January 1940 and December 2003 all of the stocks in the CRSP database are ranked on their past returns from t-1 to t-6 and from t-6 to t-60. The two rankings are performed independently. Each stock is placed into one of five different momentum quintiles based on its t-1 to t-6 ranking. The momentum portfolio returns are calculated by mimicking a portfolio that buys the high past return quintile and sells the low past return quintile. Each stock is also placed into one of five different reversal quintiles based on its t-6 to t-60 ranking. The reversal portfolio returns are calculated by mimicking a portfolio that buys the low past return quintile and sells the high past return quintile. The combined portfolio buys stocks that are in both momentum and reversal buys and sells stocks that are both momentum and reversal sells. The future returns are equally weighted, and are calculated over the different horizons listed across the first row. The t-statistics are reported in parentheses, with standard errors corrected for autocorrelation by the method of Newey and West (1987).

Table 1: Past Return Strategies 1940 to 2003

Strategy t to t+6Return

Year 1 Return

Year 2 Return

Year 3 Return

Year 4 Return

Year 5 Return

Combined 1.41 1.33 0.28 0.23 0.21 0.11(8.23) (7.70) (1.92) (1.66) (1.85) (0.95)

Momentum 0.81 0.32 -0.18 -0.16 -0.11 -0.12(7.79) (3.63) (-1.99) (-1.33) (-1.28) (-1.44)

Reversal 0.47 0.86 0.58 0.48 0.39 0.25(2.98) (5.04) (3.33) (3.29) (3.58) (2.53)

Comb-Mom 0.60 1.01 0.47 0.39 0.32 0.23(3.80) (5.53) (2.67) (2.61) (2.65) (2.02)

Comb-Rev 0.94 0.47 -0.29 -0.25 -0.18 -0.15(8.65) (4.72) (-2.82) (-1.86) (-2.22) (-1.63)

- 29 -

Table 2: Regression DecompositionThis Table reports the time-series average of slope coefficients estimated from monthly Fama-Macbeth cross-sectional regressions. The regression model is

rt+K,i = aK + b1 Momentumt,i + b2Reversalt,i + b3Combinedt,i + et+K,i

Where the subscript i refers to stock i. rt+K,i is the future return for stock i. Momentum = 1 if the stock is a momentum winner, -1 if the stock is a momentum loser and 0 otherwise. Reversal = 1 if the stock is a reversal loser, -1 if the stock is a reversal winner and 0 otherwise. Combined = 1 if both Momentum and Reversal also =1, -1 if both Momentum and Reversal = -1 and 0 otherwise. Winners are in the high past return quintiles; losers are in the low past return quintiles. The past return quintiles were constructed on monthly basis. The Long-Short total is the expected returns of the combined portfolio, which is equal to two times the sum of the Combined, Momentum, and Reversal coefficients. The t-statistics are reported in parentheses, with standard errors corrected for autocorrelation by the method of Newey and West (1987). The sample period is from January 1940 to December 2003.

Table 2: Past Return Strategies Decomposition: 1940 to 2003

Coefficients t to t+6Return

Year 1 Return

Year 2 Return

Year 3 Return

Year 4 Return

Year 5 Return

Intercept 1.32 1.45 1.33 1.56 1.28 1.24(7.55) (8.90) (7.99) (8.03) (8.13) (8.28)

Combined 0.07 0.11 -0.07 -0.06 -0.06 -0.03(1.81) (2.66) (-1.92) (-1.29) (-1.77) (-0.87)

Momentum 0.40 0.15 -0.07 -0.05 -0.03 -0.04(8.76) (3.71) (-1.74) (-1.07) (-1.05) (-1.17)

Reversal 0.20 0.41 0.30 0.23 0.20 0.13(3.31) (5.28) (3.71) (3.45) (3.92) (2.96)

Observations 762 756 744 732 720 708

Long-Short Portfolio 1.33 1.35 0.31 0.25 0.24 0.10

Combined Component 10% 16% -47% -45% -51% -65%Momentum Component 60% 22% -45% -38% -22% -85%Reversal Component 30% 62% 193% 183% 173% 250%

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Table 3: Momentum and Reversal Portfolios Cross-Sorted on Arbitrage CostsThis Table displays the equal-weighted average monthly returns of different cross-sorted momentum and reversal portfolios. At the beginning of each month between December 1940 and December 2003 all of the stocks in the CRSP database are ranked independently on past returns from t-1 to t-6 (momentum) and from t-6 to t-60 (reversals). Each stock is placed into one of five different momentum quintiles based on its t-1 to t-6 ranking. The momentum portfolio returns are calculated by mimicking a portfolio that buys the high past return quintile and sells the low past return quintile. Each stock is also placed into one of five different reversal quintiles based on its t-6 to t-60 ranking. The reversal portfolio returns are calculated by mimicking a portfolio that buys the low past return quintile and sells the high past return quintile. The winners and losers of each portfolio are placed into one of three idiosyncratic risk (IR), market value (MV), price (PRC), trading volume (TV), and institutional holdings (IH) portfolios. The rankings for the characteristics are done independently of the momentum and reversal rankings. TV data are available beginning in 1963 and INST data are available on a quarterly basis beginning in 1980. The returns reported are the average monthly returns over the six-month period subsequent to portfolio formation. The returns are in percents. T-statistics are calculated using the method of Newey and West (1987).

Table 3: Cross-Sorted Momentum and Reversal Portfolios (1940-2003)

Idiosyncratic Risk Idiosyncratic Risk

Momentum Low Med High H-L t-statistic Reversal Low Med High H-L t-statistic% of Portfolio 35% 37% 28% % of Portfolio 24% 36% 40%% of Profit 30% 42% 28% % of Profit 6% 21% 73%

Loser 0.85 0.88 1.00 0.15 (0.86) Loser 1.36 1.49 1.72 0.36 (1.91)

Winner 1.52 1.78 1.77 0.25 (1.56) Winner 1.25 1.22 0.90 -0.35 (-2.11)

W-L 0.67 0.90 0.77 0.10 L-W 0.11 0.26 0.82 0.71

t-statistic (7.49) (10.55) (7.50) (1.00) t-statistic (0.96) (2.16) (5.01) (5.16)

Institutional Holdings (1980-2003) Institutional Holdings (1980-2003)


Loser 1.47 0.59 0.88 -0.59 (-2.41) Loser 2.30 1.55 1.38 -0.92 (-2.79)

Winner 1.79 1.79 1.69 -0.10 (-0.61) Winner 0.96 1.11 1.24 0.28 (1.95)

W-L 0.32 1.20 0.81 0.49 L-W 1.34 0.44 0.14 -1.20

t-statistic (1.37) (4.18) (4.10) (2.49) t-statistic (3.01) (1.80) (0.47) (-3.35)

- 31 -

Table 3 Continued

Market Value Market Value


Loser 1.51 0.75 0.72 -0.79 (-5.56) Loser 2.11 1.42 1.31 -0.80 (-5.34)

Winner 1.94 1.65 1.46 -0.48 (-3.91) Winner 1.13 1.10 1.07 -0.06 (-0.59)

W-L 0.43 0.90 0.74 0.31 L-W 0.98 0.32 0.24 -0.74


Price (1940-2003) Price (1940-2003)


Loser 1.38 0.83 0.69 -0.69 (-4.45) Loser 1.89 1.37 1.12 -0.77 (-4.62)

Winner 1.90 1.67 1.52 -0.37 (-2.60) Winner 0.94 1.06 1.12 0.19 (1.62)

W-L 0.52 0.84 0.83 0.32 L-W 0.96 0.31 0.00 -0.96


Trading Volume (1963-2003) Trading Volume (1963-2003)


Loser 1.18 0.76 0.42 -0.76 (-4.95) Loser 1.73 1.42 1.24 -0.49 (-2.63)

Winner 1.82 1.76 1.60 -0.22 (-1.44) Winner 1.27 1.11 0.87 -0.39 (-2.54)

W-L 0.64 0.99 1.18 0.54 L-W 0.46 0.30 0.36 -0.10


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Table 4: Cross-Sorted Combined PortfoliosThis Table displays the equal-weighted average monthly returns of different cross-sorted momentum and reversal portfolios. At the beginning of each month between December 1940 and December 2003 all of the stocks in the CRSP database are ranked independently on their past returns from t-1 to t-6 (momentum) and from t-6 to t-60 (reversals). The combined portfolio buys stocks that are in both momentum and reversal buys and sells stocks that are both momentum and reversal sells. The buys and sells of the portfolio are placed into one of three idiosyncratic risk (IR), market value (MV), price (PRC), trading volume (TV), and institutional holdings (IH) portfolios. The rankings for the autocorrelations and other characteristics are done independently of the momentum and reversal rankings. TV data are available beginning in 1963 and INST data are available on a quarterly basis beginning in 1980. The returns reported are the average monthly returns over the six-month period subsequent to portfolio formation. The returns are in percents. T-statistics are calculated using the method of Newey and West (1987).

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Table 4: Combined Portfolios (1940-2003)

Idiosyncratic Risk Institutional Holdings (1980-2003)

Combined Low Med High H-L t-statistic Combined Low Med High H-L t-statistic% of Portfolio 10% 32% 58% % of Portfolio 52% 31% 17%% of Profit 6% 25% 69% % of Profit 67% 38% 13%

Loser 0.81 0.96 0.45 -0.36 (-1.67) Loser 0.35 0.44 0.70 0.34 (1.93)

Winner 1.64 2.03 2.10 0.46 (1.94) Winner 2.11 2.12 1.73 -0.38 (-1.12)

W-L 0.83 1.07 1.65 0.82 W-L 1.76 1.69 1.03 -0.73


Price (1940-2003) Trading Volume (1963-2003)

Combined Low Med High H-L t-statistic Combined Low Med High H-L t-statistic% of Portfolio 52% 31% 17% % of Portfolio 21% 29% 50%% of Profit 57% 24% 11% % of Profit 21% 27% 52%

Loser 0.49 0.64 0.54 0.05 (0.32) Loser 0.60 0.48 0.04 -0.57 (-3.61)

Winner 2.01 1.69 1.46 -0.55 (-2.86) Winner 2.20 1.96 1.65 -0.55 (-3.01)

W-L 1.52 1.05 0.92 -0.60 W-L 1.60 1.47 1.61 0.02

t-statistic (10.68) (7.74) (5.48) (-3.03) t-statistic (9.35) (7.62) (6.69) (0.09)

Market Value

Combined Low Med High H-L t-statistic% of Portfolio 46% 33% 22%% of Profit 51% 31% 18%

Loser 0.63 0.49 0.52 -0.11 (-0.84)

Winner 2.15 1.75 1.63 -0.51 (-2.71)

W-L 1.52 1.26 1.12 -0.40

t-statistic (9.19) (8.43) (6.90) (-2.29)

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Table 5: Multivariate Regressions AnalysesMOM = 1 if the stock is a momentum winner, -1 if the stock is a momentum loser and 0 otherwise. REV = 1 if the stock is a reversal loser, -1 if the stock is a reversal winner and 0 otherwise. COMB= 1 if both MOM and REV also =1, -1 if both MOM and REV = -1 and 0 otherwise. Winners are in the high past return quintiles; losers are in the low past return quintiles. The past return quintiles were constructed on monthly basis. The stocks are also placed into one of three idiosyncratic risk (IR), market value (MV), price (PRC), institutional holdings (IH), and trading volume (TV) portfolios. These characteristic portfolios in this section are constructed so that portfolio 3 contains the stocks with the highest arbitrage costs; e.g. low PRC, low MV, low TV, and low IH, but high IR stocks. So the portfolio the rankings are done on IH-1, MV-1, PRC-1, and TV-1. The sample is from 1980-2003 and consists of quarterly data.

Table 5: Multivariate Regression Analyses

Variable Reg 1 Reg 2 Variable Reg 3 Reg 4 Variable Reg 5 Reg 6

U 1.41 0.49 U 1.43 0.86 U 1.41 0.68(6.39) (1.54) (6.49) (3.12) (6.39) (2.14)

MOM 0.29 1.03 REV 0.40 -0.90 COMB 0.77 0.01(2.96) (4.79) (2.29) (-3.88) (5.73) (0.02)

MOM*IR -0.16 REV*IR 0.35 COMB*IR 0.20(-2.77) (5.20) (2.20)

MOM*INST-1 -0.05 REV*INST-1 0.11 COMB*INST-1 0.02(-0.83) (1.51) (0.20)

MOM*MV-1 -0.10 REV*MV-1 0.11 COMB*MV-1 -0.15(-2.17) (1.05) (-1.36)

MOM*PRC-1 0.13 REV*PRC-1 -0.01 COMB*PRC-1 0.22(1.18) (-0.17) (1.83)

MOM*VOL-1 -0.13 REV*VOL-1 0.05 COMBVOL-1 -0.03(-2.12) (0.69) (-0.32)

IR 0.06 IR 0.06 IR 0.08(0.47) (0.48) (0.58)

INST-1 -0.10 INST-1 -0.07 INST-1 -0.09(-1.57) (-1.18) (-1.40)

MV-1 0.30 MV-1 0.28 MV-1 0.29(5.82) (2.28) (5.54)

PRC-1 0.02 PRC-1 -0.18 PRC-1 -0.09(0.19) (-2.03) (-0.86)

VOL-1 0.20 VOL-1 0.17 VOL-1 0.19(2.44) (2.20) (2.31)

N 93 93 N 93 93 N 93 93

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Table 6: Returns in Different Economic StatesThis Table reports the average one-month returns of the momentum, reversal, and combined portfolios in expansions and recessions. The sample period is from 1930-2003 and includes all stocks on the CRSP tape with at least 60-months of history.

Table 6. Returns in Different Economic States (1930-'03)Expansion Combined Momentum Reversal

Mean 1.19 0.39 0.67t-statistic (8.53) (6.84) (2.70)Sharpe Ratio 3.16 1.06 1.53

CAPM Alpha 1.24 0.64 0.39

t-statistic (5.34) (2.88) (1.61)

FF Alpha 1.02 0.87 -0.07

t-statistic (4.78) (4.17) (-0.45)

Observations 715 715 715

Contraction Combined Momentum Reversal

Mean 0.60 -0.57 1.14t-statistic (2.12) (-0.18) (1.60)Sharpe Ratio 0.68 -0.78 1.44

CAPM Alpha 0.66 -0.46 1.08

t-statistic (0.94) (-0.88) (1.62)

FF Alpha 0.21 -0.08 0.34

t-statistic (0.33) (-0.18) (0.87)


Means: E-C 0.59 0.96 -0.47

t-statistic (0.99) (1.66) (-0.76)

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Table 7: Returns In and Out of JanuaryThis Table reports the average one-month returns of the momentum, reversal and combined portoflioss in January and in non-January months. The sample period is from 1930-2003 and includes all stocks on the CRSP tape with at least 60-months of history.

Table 7. Returns in January vs. Non-January Months (1930-'03)January Combined Momentum Reversal

Mean 2.83 -6.03 8.27t-statistic (3.00) (-5.85) (7.26)Sharpe Ratio 7.53 -16.42 19.00

CAPM Alpha 2.96 -5.22 0.39

t-statistic (2.97) (-5.06) (1.61)

FF Alpha 0.10 -0.22 2.10

t-statistic (0.09) (-1.91) (2.01)


Non-January Combined Momentum Reversal

Mean 0.93 0.79 0.06t-statistic (3.93) (3.70) (0.29)Sharpe Ratio 7.53 -16.42 19.00

CAPM Alpha 1.04 1.04 1.08

t-statistic (4.47) (5.32) (1.62)

FF Alpha 1.01 1.11 -0.25

t-statistic (4.69) (6.07) (-1.73)


Means: J-NJ 1.90 -6.82 8.21

t-statistic (2.92) (-8.45) (10.03)

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Figure 1: Profits of the Cross-Sorted Momentum and Reversal Portfolios

Momentum, Reverals, and Idiosyncratic Risk

0.00

0.20

0.40

0.60

0.80

1.00

Low Med High

MomentumReversals

Momentum, Reversals, and Market Value

0.000.200.400.600.801.001.20

Low Med High

MomentumReversal

Momentum, Reversals, and Price

0.000.200.400.600.801.001.20

Low Med High

MomentumReversal

Momentum, Reversals, and Trading Volume

0.000.200.400.600.801.001.201.40

Low Med High

MomentumReversal

Momentum, Reversals, and Institutional Holdings

0.000.200.400.600.801.001.201.401.60

Low Med High

MomentumReversal

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Figure 2: Percentage of Portfolios’ Profits in High Arbitrage Cost Stocks

Percentage of Profits in High Arbitrage Cost Stocks

0%

10%20%

30%40%

50%60%

70%80%

90%

High IR Low INST Low MV Low PRC Low TV

MomentumReversals

r. david mcleantax loss selling assumes that investors sell their worst performing (or best...

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