Momentum Meets Reversals* (Job Market Paper)

Transcription

1 Momentum Meets Reversals* (Job Market Paper) R. David McLean First Draft: November 1, 2004 This 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 Contact: McLeanRA@BC.edu. 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.

2 1. Introduction There 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

3 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

4 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)

5 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

6 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

7 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

8 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 subperiod, but insignificant in the 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

9 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 - 8 -

10 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 Literature In 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 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

11 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 Results In 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 The sample begins in 1940 due to the finding of Jegadeesh and Titman (1993) that momentum returns were negative prior to However the results do not qualitatively change if the entire CRSP dataset is used, in which case the analysis would begin in 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

12 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 Returns of the Different Past Return Portfolios Panel 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

13 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 = ),.16% (t-statistic = -1.33),.11% (tstatistic = -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

14 3.2. Decomposing the Combined Portfolio s Profit In 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+b 1 +b 2 +b 3. The expected return of the short portion of the Combined portfolio is [ a -b 1 -b 2 -b 3]. Therefore, the expected return of the Combined portfolio, which is equally long and short is 2 * [b 1 +b 2 +b 3]. The expected return of the Momentum stratgey is 2 * b 2 and the expected return of the Reversal portfolio is 2 * b

15 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

16 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 Proxies This 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 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

17 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

18 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 crosssorted 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

19 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 Analyses Table 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

20 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

21 = 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

22 5. Relative Strength Portfolios Profits at Different Points in Time In 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 , 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 Contractions Table 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

23 5.2 January versus Non-January Previous 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 Conclusion This 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)

24 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

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