Decomposing Momentum Spread

Transcription

1 Decomposing Momentum Spread James Tengyu Guo February 20, 2017 Abstract We find the momentum spread (the difference of the past returns between winners and losers) is negatively predicting momentum returns in the long-run, not in the following month. We further decompose the momentum spread into old and new, and find that the old momentum spread is the one driving the predictability. We propose a new measure - momentum spread diff, argue that it is a purer measure to identify the destabilizing mechanism of excessive arbitrage, and find supportive evidence consistent with four hypotheses. Momentum spread diff predicts an 8.8% reversal of momentum strategy in the second year after adjusting Fama-French three factors, which is the highest among measures using momentum spreads. Finally, we show possible cross-predictability of other strategies using momentum spread diff. JEL-Classification: G12, G14 Keywords: Momentum, Overreaction, Destabilizing Mechanism of Excess Arbitrage. This project originates from my second-year Research Paper for the Ph.D. program in Finance program at London School of Economics. I show my greatest gratitude to Dong Lou for his consistent supervision, support and help. London School of Economics. t.guo3@lse.ac.uk i

2 Contents 1 Introduction 1 2 Related Literature 2 3 Data and Factor/ Spread Construction Predictability of Momentum Spreads Decomposing Momentum Spreads Momentum Age A Modified Model based on Hong and Stein (1999) Momentum Spread Diff Cross Predictability and Factor Timing Strategy 13 6 Conclusion 14

3 1 Introduction In financial markets, long-short strategies are widely used by hedge funds, especially the ones who focus on equity market and pursue market neutral returns, to take advantage of many stylized facts coming from either different risk loadings, or inefficiencies due to some behavioral reasons or various frictions. Asset managers are making money by trading on these factors/ strategies, and in the meantime, taking some risks which some other investors (usually the counter-parties) do not want to, achieving better risksharing or making the financial market more efficient by correcting mispricing. The most important and implemented ones include size, value, momentum, idiosyncratic volatility, betting against beta and etc. All these strategies are implemented by a portfolio construction with their respective characteristic measures. Trading/ arbitraging behavior on these strategies will also change the measures and their spreads (the difference of the average characteristic measures between long-leg and the short-leg) of them. For example, for size strategy, it is longing the small cap stocks and shorting the large cap stocks, so trading on size will make the size spread smaller, and the same logic works for value strategy. However, for momentum strategy, arbitrageurs are buying the winners and selling the losers, which will produce a price impact and broaden the momentum spread. If the arbitrageurs cannot know how much capital is trading on this momentum type unanchored strategy by Stein (2009), they may create a bubble of excessive arbitrage, which may destabilize the market subsequently. Some research has worked on this issue: Lou and Polk (2012) use the correlation between stock returns and Huang (2015) use momentum gap 1 to proxy arbitrage activity, and they both find that higher momentum activity will follow by a negative return on momentum. This paper pushes Huang (2015) s result further, and finds that momentum spread is negatively predicting momentum returns in the long-run, but not in the following month. At the beginning of each month, among momentum stocks, we classify them into subgroups based on their momentum age, which we define as the number of consecutive months that a stock has been identified as momentum stocks up to that point of time, both for winners and losers. Through this kind of classification, we also find evidence that the negative predictability is coming from the old momentum spread, but not the new one. One caveat of momentum spread is that, it is also measuring the market dispersion before the momentum formation, which is regarded as another market state variable in the literature. To get a purer proxy for the excessive arbitrage, we propose a new 1 In Huang (2015), he defines the momentum gap as the difference between the 75th and 25th percentiles of the distribution of cumulative stock returns from month t 12 to t 2, which is slightly different from the momentum spread. 1

4 measure, momentum spread diff, which is the difference between old momentum spread and new momentum spread, and argue it also negatively predicts the momentum returns in the long-run. We find the data supports the implication: after adjusting Fama-French three factors, momentum spread diff has the highest predicting power about the longrun reversal of momentum strategy. We argue that this kind of Diff-in-Diff type of measure, filters out other drivers in the momentum spreads, and helps to identify the destabilizing mechanism of arbitrage activity - excessive arbitrage beyond fundamental. Finally, we present some results of using momentum spread diff to predict other factor returns. The logic follows that if momentum bubble busts, we would like to expect investors to allocate capital to other strategies. The rest of this paper is organized as following: we first review the literature; then we discuss the data and the factor/ spread construction; in section 4, we decompose the momentum spreads to propose our new measure momentum spread diff, and provide evidence supporting our results; section 5 checks cross-predictability and section 6 concludes. 2 Related Literature Arbitrageurs can make the market more liquid and efficient, but they can also cause instability by taking high leverages and overcrowding as Stein (2009) pointed out. Jacobs and Levy (2014) document some stylized fact of smart beta investing, and also point out that overcrowding may lead to overvaluation and factor crash. Among all strategies, momentum is most prone to this excess arbitrage argued by Lou and Polk (2012). They infer arbitrage activity from return correlations and propose a modified time-series momentum strategy. Some other research support the idea of decomposing momentum into different groups: Chen, Kadan, and Kose (2009) find that a two-way sorting based on long-term and recent performance can accommodate the two effects by distinguishing between fresh and stale winners and losers, which is consistent with the story that investors mistakenly respond to shocks to firm fundamentals as if they are going to continue in the long run. Novy-Marx (2012) finds that momentum is primarily driven by the stocks performances in the twelve to seven months prior to portfolio formation. Daniel, Klos, and Rottke (2016) find a Betting Against Winners strategy that goes short the overpriced winners and long other winners generates a Sharpe-ratio of 1.08, due to the disagreement and short-sale constraints. The closest research is Huang (2015) s momentum gap, and in which he finds that momentum gap is negatively predicting momentum returns in the next month. We consider our result different and more robust than his for three reasons: first, his result of predicting next month momentum return is not robust in our sample, and as we argue 2

5 later, the main reason for this could be the imperfect timing capability of the peak of momentum bubble; second, the momentum is not a linear strategy, trading the decile momentum stocks shows much higher return than trading quintile momentum stocks 2, so by construction, momentum arbitrage will have a larger effect on the momentum spreads between top decile and bottom decile; third, apart from arbitrage activity, there are many other drivers which are driving momentum spreads/ gaps, so without purging out these drivers, it is difficult to pin down the destabilizing mechanism of excessive arbitrage. However, our measure, the momentum spread diff, fits this purpose. 3 Data and Factor/ Spread Construction All firm level data is from WRDS, by combining CRSP for stock information and Compustat for accounting information. Fama-French factors are from Kenneth French website 3. We measure the benchmark momentum spread as the difference of the weighted lagged 12-month cumulative return (skipping the most recent month) between the top and bottom decile stocks, and the weight is the same weight weighting the returns. By construction, the benchmark momentum spread at any point of time is positive: Spread mom = Ret t 12,t 2,W inners Ret t 12,t 2,Losers. 3.1 Predictability of Momentum Spreads (Insert Table 1 and Table 2) In Huang (2015), he defines the Momentum Gap as the difference between the 75th and 25th percentiles of the distribution of cumulative stocks returns. Table 1 replicates his finding: in column MOM and MOM ex, the raw and adjusted momentum returns in month t are reported, and clearly, there is a downward trend as the momentum gap increases. However, the magnitude of this relationship is not as big as in Huang s sample period. Data spans from 1965 to 2010, excluding We also check the difference between the group 1 and group 5, and find the negative difference are insignificant for both the raw returns (with a t-stat -1.41) and Fama-French adjusted returns (with a 2 This can be checked by simply comparing the average of UMD and MOM factors on French s website. 3 We show our sincere gratitude to Ken French for supplying the Fama-French factors. 4 Since there is a huge crash of momentum strategy in 2009 (see Daniel and Moskowitz (2013)), so we excluding the months in 2008 to make sure the second year reversal predictability is not driven by this extreme case. Our results are even stronger after taking 2008 into account. 3

6 t-stat -1.53). Latter columns report the subsequent cumulative returns of the same momentum portfolio without rebalancing. We can see that the negative predictability of momentum gap continues up to two years after the formation. Table 2 uses momentum spread to sort all months. We can see that in the short-run, there is no consistent pattern across groups. However, we find very strong predictive power for momentum strategy using momentum spread in the long-run, that is after six months of the momentum formation to the end of year three. The magnitude of momentum reversal in group 5 is much larger than the counterparty in Table 1, with the cumulative momentum returns from month 7 to month 26, -16.1% 5, more negative than -12.0% in Table 1, and also significant in different time spans. The main reason we use the momentum spread, instead of momentum gap, is that, as mentioned before, momentum is not a linear strategy, trading the extreme decile momentum stocks shows much higher profit than trading quintile momentum stocks. If this gap/spread type of measure is truly capturing momentum arbitrage activity as Huang (2015) argues in his paper, we would expect the momentum arbitrage will push the extreme measures wider. In another word, the measures using extreme stocks (in the top or bottom decile) should contain more information, proxy the momentum arbitrage activity better and hence predict the momentum reversal more strongly, than the measures using the non-extreme stocks (at the 75th or 25th cut-off). However, as shown in his paper, if he is using another measuring, the difference between the 90th and 10th percentiles of the distribution of cumulative stocks returns, his results are less robust. Another advantage of using momentum spread is that, it takes into account more available information in the part return difference among the momentum stocks, by taking the weighted average of the past returns, instead of using the cut-off. 4 Decomposing Momentum Spreads The main problem of using momentum spread is that there are too many drivers/ mechanisms which are driving the time variation of these spreads. Since the momentum spreads are the difference of the past cumulative returns between the winners and the lowers, it is by construction measuring the return dispersion in the market. There are several researches arguing market dispersion per se is a (proxy of) state variable, see Stivers and Sun (2010), and Loungani, Rush, and Tave (1990) and Fortin and Araar (1997) find its relationship to real economy, such as labor. So without identifying the determinants of the underlying drivers of momentum spreads and purging way the irrelevant ones, we are quite parsimonious to interpret the negative predictability as 5 4.2% 8.2% 3.7% = 16.1% and 2.3% 9.1% 0.6% = 12.0% for the latter number. 4

7 evidence for excessive arbitrage channel. Our contribution is that, by borrowing the Diff-in-Diff type of idea, we propose a new measure, momentum spread diff, to purge away other drivers in the momentum spreads and leave a purer proxy of excessive momentum arbitrage activity. We now use the past cumulative returns in the last 6 months, t 6 to t 1, instead of t 12 to t 2, to calculate the momentum spread. The main reason we do this is that, since momentum ranking is based on the t 12 to t 2 returns and both of old and new need to meet the momentum criteria (cut-off) first, the correlation between old spreads and new spreads calculated from the same ranking period returns will be too high. By using a time horizon (t 6 to t 1) from the ranking horizon, the correlation between old and new decreases, which helps us to study their differences. This modification is also consistent with the result in Novy-Marx (2012): since he finds that momentum return with the one-year ranking period, is mainly positively driven by the first half of the year, one would expect the second half of the year is negatively driving the momentum return, in a relative sense. This natuarally links to the destabilizing arbitrage. 4.1 Momentum Age Spread mom = Ret t 6,t 1,W inners Ret t 6,t 1,Losers. At the end of every month, after selecting the momentum stocks by their past performance, we classify them into subgroups based on their momentum age. We define momentum age as the number of months that a stock has been consecutively identified as a momentum stock in the past few months, both for winners and losers: new (1), old (2-6), ancient (7 and more months). For example, at the end of September, Microsoft shows up in the momentum portfolio as a winner (in the top decile), and it is not a winner in the last August, we assign age one to it. A month later, at the end of October, Microsoft again stays as a winner, then its momentum age at this point of time is two. Table 3 reports the relative composition of momentum stocks in terms of their age. Stocks with momentum age larger than seven months consist of less than 40 percent of the momentum portfolio, which highlights the high turnover of momentum strategy. (Insert Table 3) The idea we want to elaborate here, is that if a stock has been in the momentum portfolios for longer horizon, it will be pushed by the momentum traders even higher or lower depending on being a winner or a loser. If a stock is old enough, we conjecture 5

8 it has been pushed very high by momentum arbitrageurs, and a stronger and sooner reversal will follow. To show this, we form momentum portfolios based on their ages: MOM age = Ret W inners age Ret Losers age. (1) Figure 1: Cumulative momentum returns for three momentum portfolios with different momentum age (no balancing after formation) up to 36 months. In Figure 1, we follow Jegadeesh and Titman (2001) and plot the cumulative returns of momentum portfolios 6, but for different ages. Consistent with our prediction, the figure shows that the older the momentum portfolio is, the sooner and the stronger it will revert. For the fresh momentum stocks (with age 1, 2 or 3 months), the reversal effect is hardly there. This result is consistent Chen, Kadan, and Kose (2009) s finding, which says the fresh momentum stocks (similar to our new momentum stocks) will perform better than the others. With this evidence in mind: if momentum stocks with different age are behaving differently after the formation, we would expect the momentum stocks (and their spreads) with different ages are also carrying different information at the formation time. That is the point where we start to treat the old momentum spread and new momentum spread differently. 6 These portfolios are equal-weighted, and the ranking is based on past returns from t-12 to t-2. Small caps with market cap in the NYSE bottom decile and stocks with price less than $5 are excluded. 6

9 4.2 A Modified Model based on Hong and Stein (1999) In the classical paper by Hong and Stein (1999), A unified theory of underreaction, momentum trading, and overreaction in asset markets, they build an equilibrium model with bounded rationality. In there model, there are two types of investors, Newswatchers and Momentum traders, both risk averse with CARA utility. Their rationality is bounded in a way: for news-watchers, they only process fundamental information; and for momentum traders, they only trade on the past price change. In a simplified version of their model, the price follows a process: P t = P t 1 + I t + X t X t 12, (2) where I t is the change in the fundamental information and X t is the price impact from momentum trading activity, which follows X t = φ P t 1, (3) φ the equilibrium demand elasticity, and 12 is the holding period: after 12 periods, momentum traders will close their position, which brings in an opposite price impact with a lag of 12, X t 12. In their original model, they assume the size of momentum traders is constant over time, which we believe is not true in real world. The arbitrage capital has a huge timevariation in practice, and meanwhile, it is unknown to arbitrageurs. Stein (2009) argue that without knowing the exact amount of arbitrage activity, if all arbitrageurs are just optimizing their own leverage without considering the externality, financial markets will suffer possible destabilization. We modify Hong and Stein (1999) s model by introducing shocks to momentum arbitrage capital γ t, which could come from either crowding or leverage, and most importantly, is not observed by other momentum traders 7. The new momentum trade follows: X t = γ t φ P t 1, (4) and in their original model, γ t = 1. With this modification, there is an amplification effect in the momentum positive feedback: dx t dγ t 1 γt = γ t φ d P t 1 dγ t 1 = γ t φ(φ P t 2 ). (5) Momentum positive feedback captures that momentum trading in this period will generate a price impact, and this will attract more momentum trading next period in 7 So the momentum trader will follow their original trading strategy. 7

10 the same direction, since momentum trading is simply based on the past change in prices. In Equation 5, the amplification effect captures, if there is a large momentum shock in the last period, the positive feedback will be amplified. Figure 2: Hong and Stein (1999) Modified - early arbitrage shock Figure 2 and Figure 3 show impulse response function of prices if there is a fundamental shock at t = 13. We take the benchmark parameterization in Hong and Stein (1999), with the z = 12 measuring the (linear) rate of information flow and higher values of z implying slower information diffusion, and j = 12 the momentum traders horizon, and the volatility of the new shocks 0.5, and the demand elasticity φ = determined in equilibrium with the momentum traders risk tolerance γ = 1/3. The blue solid line is the bench mark case when there are only news-watchers, and the red dashed line is the result of overreaction in Hong and Stein (1999) 8. The black dotted lines are the results of our modification by introducing excessive momentum arbitrage shocks. The difference between Figure 2 and Figure 3 is that, in Figure 2, the arbitrage shock shows up around t = 6, and in Figure 3 the arbitrage shock shows up around t = 12. Both figures clearly show that with excessive momentum arbitrage capital, the long-term reversal of momentum will be larger, and possibly postponed. There are three main hypotheses from the model, and later we will present empirical evidence consistent with them: 8 These two lines are replicated identically with respect to the ones in the Hong and Stein (1999), see lines marked Base and gamma=1/3 their figure 2, and also numbers in their table AII. 8

11 Figure 3: Hong and Stein (1999) Modified - late arbitrage shock Hypothesis 1: With the increase the momentum age, the momentum characteristic spread, the difference of the past cumulative returns between the winners and losers, will be larger. In the figures, the spread is the distance between the line and the x axis, and we can see it is increasing with time before t = 13. This effect is also extant in Hong and Stein (1999) s original model. Hypothesis 2: For momentum stocks with the same age, more momentum arbitrage capital will drive up the momentum spread, resulting stronger reversal in the long-run. Hypothesis 3: Because of the positive feedback effect, the price impact coming from momentum trading will increase with the momentum age, relative to the fundamental part. Old momentum spreads will carry more information about the momentum arbitrage activity than new momentum spreads, which, in turn, carry more fundamental information. Older spreads will predict the long-term reversal of momentum more strongly. This effect is stronger with momentum arbitrage shocks due to the amplification effect. (Insert Table 4 and Table 5) In Table 4, we find evidence consistent with the Hypothesis 1: as the momentum age increase, the benchmark momentum spreads increase monotonically, across groups. 9

12 In Table 5, we find old spread is larger than new spread (88.4% > 88.0% > 71.3%), consistent with the Hypothesis 1 9. Within each momentum age, the larger the spread, the stronger the reversal will be, consistent with the Hypothesis 2. If we focus on the prediction for year 2 and year 3 (mom same and mom same 25 36), the predicting power for momentum reversal is increasing with momentum age, 8.7% > 13.1% > 13.5% and 3.8% > 5.4% > 7.6%, which is consistent with the Hypothesis 3. The ancient momentum spreads have slight lower predictive power, which can be justified since the information in these stocks is complicated. Another difference here is that, to compute the momentum spreads, we are no longer using the cumulative returns from t 12 and t 2, but the ones from t 6 and t 1. There are two reasons we make this change: first, if we are using the same cumulative return with the momentum characteristic from t 12 and t 2, we will end up in very high correlation between the old and new spreads, since both of them need to meet the very strong condition of being momentum stocks first; second, as Novy-Marx (2012) pointed out, the momentum phenomenon is mainly driven by cumulative returns from the first half of the one-year ranking period (stronger than the whole ranking period), and considering the whole ranking period is positively generating momentum, we expect the second half contains negative information about momentum. We also run a conditional regression regression, to compare the predictive power of the old and new momentum spreads controlling the other, as in Equation 6. We first sort all months into five groups m, based on their new (old) momentum spreads, and then within each group, we sort months into five subgroups n, based on their old (new) momentum spreads. And then we demean the old (new) momentum spreads, the second sorting variable, by subtracting their means in each group from the first-step sorting. The idea of doing this is to eliminating the comovement of these two spreads, considering they are the same contemporaneous measures just for different stock pools. Cumu Ret m,n = a + b (Spread m,n Spread m ) + ɛ m,n. (6) Results of these conditional regressions are shown in Table 6. We can see that after controlling the old mom spread, the new momentum spread shows little predictive power for long-term reversal of momentum strategy, and it is not the case the other way around. (Insert Table 6) 9 We are using the t 6 to t 1 spreads now. As for the spreads among the ancient momentum stocks, they are smaller than the old ones, which seems to be inconsistent with the model prediction. However, we argue that actually fits the model quite well: since Table 5 are using the cumulative returns in the past six months to calculate the spreads, instead of the ranking period t-12 to t-2, the ancient ones are by construction gaining more returns in the first part of the ranking year and lower returns in the second part of the ranking year. 10

13 4.3 Momentum Spread Diff With the results in Table 6, can we conclude that the old momentum spread is the best predictive indicator in this kind? We say no. The same problem is still haunting: too many drivers/ mechanisms are driving the spread. We propose a new measure based on momentum spreads - momentum spread diff - the difference between old momentum spreads and new spreads. Here is an example: Spread Old = 0.2 Fundamental Momentum Arbitrage + Other Drivers; Spread New = 0.8 Fundamental Momentum Arbitrage + Other Drivers. Take the difference between the two, we get: Mom Spread Diff = Spread Old Spread New = 0.6 Fundamental Momentum Arbitrage + o(other Drivers). (7) For illustration, we take arbitrary numbers 0.8 and 0.2 to capture the old and new spreads containing different weights of information from fundamental and momentum arbitrage. By taking the difference, we till cannot disentangle the fundamental part or the price impact part from momentum arbitrage activity. However, we argue that this measure serves as a purer and stronger indicator to identify the destabilizing mechanism of excessive arbitrage for two reasons: First, this measure does not suffer the too-many-driver problem: since both spreads are measuring the cumulative returns from t 6 to t 1, and the only difference is that the stocks being measured are different - old and new, so if there are other drivers apart from the fundamental news and momentum arbitrage activity, these drivers are likely to affect both spreads in a similar dimension and magnitude. Taking the difference between the old and new, exactly purges out the other drivers (at least their firstorder effect), and leaves the momentum arbitrage (with a positive sign) excessive to the fundamental (with a negative sign) and the smaller higher-order effect of other drivers (captured by o(other Drivers) in Equation 7). Second and more importantly, as we argued before, old momentum spreads contain more information about arbitrage activity and the new ones contain more about fundamental information, in a relatively sense. By taking the difference between the old and new spreads, the information about fundamental will be negative and the information about momentum arbitrage will be positive. By construction, this is a direct measure of excessive momentum arbitrage - the momentum arbitrage activity beyond the fundamental information - resulting in the destabilizing mechanism of excessive 11

14 momentum arbitrage: since both negative fundamental component and positive arbitrage component are predicting a bad performance of momentum in future, leading to another hypothesis below. Hypothesis 4: Momentum spread diff will negatively predict momentum performance in the long-run. (Insert Table 8 and Table 9) Table 8 and Table 9 show the predicting power of momentum spread diff. Table 8 define the old momentum stocks with momentum ages two and three months, and Table 8 for two to six months. We can see there is no predictability in the shortrun, which is not surprising if we consider momentum spread diff is measuring the momentum arbitrage activity: in group 1, few people are trading on momentum, so it will not be profitable; and in group 5, many people are trading on momentum, pushing the winners up and losers down, but it is difficult to time the peak of the momentum bubble. However, in the long-run (year 2 and year 3), the reversal shows up, consistent with the Hypothesis 4. (Insert Table 10) Table 10 shows the predicting power of momentum spread diff for different sample periods. We find the reversal effect is stronger and also happening sooner after the momentum is discovered by academics. Possibly there is more momentum arbitraging in this subsample periods, with another fact that the size of institutional investors grow rapidly in recent twenty years. However, the year two reversal magnitude predicted is not as large as using the Old Spread alone (8.5% < 13.5%). We argue this does not undermine our result. We adjust the contemporaneous Fama-French three factors for our raw returns since we want to distinguish novel predictability effects. The adjusted returns are defined as the sum of α t t+τ and the fitted value of ɛ t t+τ in the full-sample regression, as in Equation 8. MOM t t+τ = α t t+τ +β MKT MKT RF t t+τ +β SMB SMB t t+τ +β HML HML t t+τ +ɛ t t+τ. (8) (Insert Table 11 ) Table 11 shows that, after controlling the Fama French three factors, Momentum Spread Diff has the highest predicting power for the long-term reversal of momentum strategy, year 2 and year 3, within the momentum spread family, apart from the value 12

15 spread. This is also a strong evidence that after purging out both other drivers of momentum spreads and other factor components in the subsequent momentum returns, the momentum spread diff, which we argue captures the excessive momentum arbitrage most precisely, does have a strong negative predictive power for the long-run reversal of momentum strategy. 5 Cross Predictability and Factor Timing Strategy Another piece of information from Table 11 is that, after controlling Fama-French factors, the predictability of long-run reversal decreases for all measures, for momentum spread diff 10.9% 8.8%, original momentum spread with ranking period from t 12 to t % 6.8%, and value spread 20.4% 15.2%. Put differently, these measures/ spreads can predict other factors as well. (Insert Table 12) In Table 12, we notice that spreads of size, value, momentum and beta are predicting momentum and size strategies with opposite signs, comparing to predicting value, ivol and beta strategies, for most of the cases. For the value, ivol and beta strategy, they are all longing the relatively low risk and high value stocks with high book-to-market ratio, low idiosyncratic volatility and small beta, and shorting the counterparts. These strategies are consistent with the idea of contrarian investing, while momentum is more aggressive by chasing the winners. There is also anecdotal evidence suggesting that hedge funds, especially high frequency trading funds, prefer to hold stocks with high ivol and large beta, for intraday trading for both directions. Combo t = 1 3 (HML t + V OL t + BAB t ). (9) We simply consider a equal weighted combo factor for value, ivol and beta strategy, as in Equation 9. We use a moving average (MA) strategy to time the factor returns: comparing the Mom Spread Diff with its MA in the last 12 months. Mom Spread Diff is the part orthogonal to the cumulative market returns and the market volatility in the last 36 months, which have been found been able to predict momentum returns by Cooper, Gutierrez, and Hameed (2004) and Wang and Xu (2015) respectively. (Insert Table 13 and Table 14) Table 13 and Table 14 show that with high spread diff (proxy for high momentum arbitrage), combo strategy performs bad in the short-run, and good in the long-run. It is the same story in the momentum, in the short-run, arbitrageurs are still pushing 13

16 momentum, without knowing exact time the bubble will bust. However, in the longrun, momentum will perform badly and combo will perform well. It is possible that when arbitrageurs find momentum is not profiting anymore, they decide to move their capital to pursue other strategies. This substitute effect (without identification) is stronger in the recent 20 years, possibly with the increasing market share of professional institutional investors, allocating capital from time to time. 6 Conclusion Different from Huang (2015), we find the momentum spread is negatively predicting momentum returns in the long-run, but not in the following month. We argue that excessive momentum arbitrage will drive up the spread, and will lead to lower momentum returns, but this reversal will not show up instantly, but rather in the long-run (mostly after six months). Based on a modified version of Hong and Stein (1999) s model, we decompose the momentum spread into old and new ones based on the momentum age of the stocks, and find that the old momentum spread is the one driving the predictability. We push the model further and propose a new measure - momentum spread diff, and find empirical evidence consistent with the model prediction. We argue momentum spread diff is a much purer measure to capture the excessive momentum arbitrage activity. After adjusting Fama-French factors, our new measure has the highest predictability for long-run reversal of momentum strategy among the momentum spread family, which helps to pin down the destabilizing mechanism of excessive arbitrage. We also find possible cross-predictability using momentum spread diff, and argue that this could be due to the time-varying allocation of capital by the arbitrageurs. 14

17 References Chen, L., O. Kadan, and E. Kose (2009): Fresh Momentum,. Cooper, M. J., R. C. Gutierrez, and A. Hameed (2004): Market states and momentum, The Journal of Finance, 59(3), Daniel, K. D., A. Klos, and S. Rottke (2016): Betting Against Winners, Social Science Electronic Publishing. Daniel, K. D., and T. J. Moskowitz (2013): Momentum crashes, Swiss Finance Institute Research Paper, (13-61), Fortin, M., and A. Araar (1997): Sectoral shifts, stock market dispersion and unemployment in Canada, Applied Economics, 29(6), Hong, H., and J. C. Stein (1999): A unified theory of underreaction, momentum trading, and overreaction in asset markets, The Journal of Finance, 54(6), Huang, S. (2015): The momentum gap and return predictability, in WFA 2015 Seattle Meetings Paper. Jacobs, B. I., and K. N. Levy (2014): Smart Beta versus Smart Alpha, Journal of Portfolio Management, 40(4), 4 7. Jegadeesh, N., and S. Titman (2001): Profitability of Momentum Strategies: An Evaluation of Alternative Explanations, Journal of Finance, 56(2), Lou, D., and C. Polk (2012): Comomentum: Inferring Arbitrage Activity from Return Correlations, Social Science Electronic Publishing. Loungani, P., M. Rush, and W. Tave (1990): Stock market dispersion and unemployment, Journal of Monetary Economics, 25(3), Novy-Marx, R. (2012): Is momentum really momentum?, Journal of Financial Economics, 103(3), Pastor, L., and R. F. Stambaugh (2003): Liquidity Risk and Expected Stock Returns, Journal of Political Economy, 111(3). Stein, J. C. (2009): Presidential address: Sophisticated investors and market efficiency, The Journal of Finance, 64(4), Stivers, C., and L. Sun (2010): Cross-Sectional Return Dispersion and Time Variation in Value and Momentum Premiums (Digest Summary), Journal of Financial and Quantitative Analysis, 45(4), Wang, K. Q., and J. Xu (2015): Market volatility and momentum, Journal of Empirical Finance, 30,

18 Table 1: Forecasting Momentum Returns with Momentum Gap. This table reports the average returns of the same momentum portfolio without any rebalancing after formation, for the first month (mom ex means Fama-French factors adjusted), 1-3, 4-6, 7-12, and months. All months are sorted into five groups based on their Momentum Gap realized in the last month. 5-1 means the difference between the average returns in the group with largest spreads and the group with the smallest. Rank No. Obs. Mom Gap MOM MOM ex mom same 1 3 mom same 4 6 mom same 7 12 mom same mom same % 1.7% 2.4% 4.2% 3.3% 3.4% 3.1% 1.3% (3.98) (6.40) (5.70) (4.04) (3.76) (2.46) (0.81) % 0.8% 1.6% 3.4% 3.2% 2.1% 0.2% -0.6% (1.28) (2.76) (3.70) (3.64) (2.19) (0.14) (-0.33) % 1.8% 2.5% 4.0% 2.3% 1.5% -3.3% -0.1% (3.35) (5.19) (3.88) (3.02) (1.51) (-1.82) (-0.07) % 0.4% 0.8% 0.5% 1.1% 1.8% -4.8% -4.6% (0.38) (0.76) (0.38) (0.88) (1.33) (-2.36) (-2.50) % 0.4% 0.8% 0.0% -1.0% -2.3% -9.1% -0.6% (0.42) (0.82) (-0.00) (-0.79) (-1.21) (-3.86) (-0.33) 16

19 Table 2: Forecasting Momentum Returns with Momentum Spread. This table reports the average returns of the same momentum portfolio without any rebalancing after formation, for the first month (mom ex means Fama-French factors adjusted), 1-3, 4-6, 7-12, and months. All months are sorted into five groups based on their Momentum Spread realized in the last month. 5-1 means the difference between the average returns in the group with largest spreads and the group with the smallest. Rank No. Obs. Mom Spread MOM MOM ex mom same 1 3 mom same 4 6 mom same 7 12 mom same mom same % 0.9% 1.3% 1.4% 0.9% 3.5% 3.2% 0.9% (1.14) (1.89) (1.13) (0.93) (3.93) (2.64) (0.44) % 0.8% 1.2% 3.8% 3.8% 3.2% -2.0% -0.8% (0.95) (1.74) (3.98) (4.11) (3.52) (-1.20) (-0.41) % 1.6% 2.3% 3.7% 3.7% 1.7% -3.4% 1.6% (3.26) (4.49) (5.06) (4.44) (1.79) (-2.05) (0.94) % 1.4% 1.9% 3.2% 1.4% 2.2% -3.5% -2.6% (2.65) (3.30) (3.13) (1.51) (1.71) (-1.57) (-1.56) % 0.4% 1.3% 0.1% -0.9% -4.2% -8.2% -3.7% (0.47) (1.24) (0.10) (-0.65) (-2.19) (-3.40) (-2.15) 17

20 Table 3: Percentage of Momentum Age in Momentum Stocks. Since we are using NYSE breakpoints to sort the momentum stocks, the whole sample winners will be slightly more than the whole sample losers. Momentum Age All 1 2&3 4&5&6 7&7+ Losers 45.4% 11.0% 11.6% 9.7% 13.2% Winners 54.6% 11.7% 13.0% 11.5% 18.5% Table 4: Percentage of Momentum Age in Momentum Stocks. This table reports the benchmark momentum spreads for different momentum ages. Each column, all months are classified into five groups according to their benchmark momentum spreads with different momentum age. Benchmark Spreads Momentum Age Groups 1 to 5 1 2&3 4&5&6 7&7+ Group % 80.9% 93.4% 102.7% Group % 95.9% 113.0% 126.8% Group % 108.4% 127.8% 146.7% Group % 124.0% 151.4% 172.8% Group % 174.6% 218.7% 256.0% 18

21 Table 5: Forecasting Momentum Returns with Mom Spreads for Different Ages. This table reports the average returns of the same momentum portfolio without any rebalancing after formation, for 1-3, 4-6, 7-12, and months. All months are sorted into five groups based on their momentum spreads for different ages, realized in the last month. 5-1 means the difference between the average returns in the group with largest spreads and the group with the smallest. Momentum Age Rank No. Obs. Mom Spread Diff mom same 1 3 mom same 4 6 mom same 7 12 mom same mom same % 3.6% 3.9% 4.3% 0.4% 1.0% (4.38) (4.49) (4.35) (0.34) (0.63) % -0.2% -1.2% -3.8% -8.2% -2.9% (-0.14) (-0.89) (-1.96) (-3.32) (-1.67) % -5.0% -8.1% -8.7% -3.8% (-2.43) (-3.20) (-3.74) (-3.09) (-1.67) 2& % 4.3% 3.6% 4.1% 0.6% 2.4% (4.59) (4.06) (4.30) (0.42) (1.64) % 1.5% -1.4% -4.3% -12.5% -3.0% (1.03) (-0.98) (-2.29) (-4.79) (-1.87) % -5.0% -8.4% -13.1% -5.4% (-1.61) (-3.02) (-3.97) (-4.46) (-2.49) 4&5& % 2.5% 1.8% 3.9% 0.7% 1.8% (2.40) (2.26) (4.21) (0.49) (1.28) % 1.6% -0.5% -1.4% -12.8% -5.8% (1.09) (-0.33) (-0.66) (-4.84) (-3.11) % -2.3% -5.2% -13.5% -7.6% (-0.50) (-1.42) (-2.31) (-4.54) (-3.25) 7& % 0.4% 1.0% 2.4% 0.5% 1.4% (0.39) (1.08) (2.55) (0.38) (1.02) % 1.3% 0.5% -1.6% -9.5% -5.0% (0.99) (0.43) (-0.86) (-4.04) (-2.91) % -0.5% -4.0% -10.0% -6.5% (0.55) (-0.30) (-1.91) (-3.67) (-2.90) 19

22 Table 6: Regressing Momentum Returns on New/ Old Mom Spread. This table reports the coefficient of regressing cumulative returns of momentum in 7-12, and months, on demeaned spreads. 5% statistical significance is indicated in bold. Dependent Variable First Control Independent Variable mom same 7 12 mom same mom same Mom Spread New Mom Spread Old Mom Spread Old Mom Spread New (-2.91) (-5.10) (-2.28) (-2.18) (-0.93) (-0.19) Table 7: Spreads and Diffs correlation with contemporaneous market factors. This table reports the time-series correlation coefficients between variables. MKT Liquidity is the aggregate liquidity in Pastor and Stambaugh (2003). TEDrate is the difference between the interbank rate and the risk-free rate. NAV LS and g NAV LS is the (log) net asset value of Long-Short strategy hedge funds and its growth rate. MKT Liquidity TEDrate NAV LS g NAV LS Old Mom Spread New Mom Spread Mom Spread Diff (Old - New) Table 8: Forecasting Momentum Returns with Mom Spread Diff (2&3-1). This table reports the average returns of the same momentum portfolio without any rebalancing after formation, for 1-3, 4-6, 7-12, and months. All months are sorted into five groups based on their momentum spread diff realized in the last month. 5-1 means the difference between the average returns in the group with largest measure and the group with the smallest. The old momentum stocks with momentum ages two and three months, and new ones for just one month. Rank No. Obs. Mom Spread Diff mom same 1 3 mom same 4 6 mom same 7 12 mom same mom same % 1.7% -0.1% 1.4% 0.4% -0.1% (1.43) (-0.15) (1.29) (0.28) (-0.04) % 2.1% 2.9% 3.6% -0.9% 1.0% (2.57) (3.41) (3.26) (-0.57) (0.51) % 4.6% 3.1% 0.8% -2.6% -1.3% (3.93) (3.32) (0.83) (-1.39) (-0.76) % 3.8% 2.5% 2.6% -2.6% -1.4% (3.51) (2.55) (2.29) (-1.40) (-0.74) % 0.0% 0.5% -1.9% -8.1% -2.8% (0.04) (0.42) (-1.06) (-3.31) (-1.74) % 0.7% -3.3% -8.5% -2.8% (-0.95) (0.42) (-1.58) (-2.97) (-1.18) 20

23 Table 9: Forecasting Momentum Returns with Mom Spread Diff (2&3&4&5&6-1). This table reports the average returns of the same momentum portfolio without any rebalancing after formation, for 1-3, 4-6, 7-12, and months. All months are sorted into five groups based on their momentum spread diff realized in the last month. 5-1 means the difference between the average returns in the group with largest measure and the group with the smallest. The old momentum stocks with momentum ages between two and six months, and new ones for just one month. Rank No. Obs. Mom Spread Diff mom same 1 3 mom same 4 6 mom same 7 12 mom same mom same % 1.3% -0.5% 1.7% -0.4% 0.4% (1.11) (-0.47) (1.66) (-0.25) (0.23) % 2.9% 2.9% 3.5% 0.3% 1.5% (3.56) (3.51) (3.62) (0.16) (0.76) % 2.5% 1.6% 1.3% 0.8% 2.1% (2.15) (2.26) (1.24) (0.48) (1.17) % 4.1% 4.0% 2.1% -3.2% -4.6% (4.08) (3.61) (1.71) (-1.56) (-2.44) % 1.5% 0.8% -2.1% -11.3% -4.0% (1.06) (0.62) (-1.13) (-4.77) (-2.41) % 1.3% -3.8% -10.9% -4.4% (0.12) (0.78) (-1.80) (-3.85) (-1.89) Table 10: Forecasting Momentum Returns with Mom Spread Diff (2&3&4&5&6-1) for different subsamples. This table reports the average returns of the same momentum portfolio without any rebalancing after formation, for 1-3, 4-6, 7-12, and months. All months are sorted into five groups based on their momentum spread diff realized in the last month. 5-1 means the difference between the average returns in the group with largest measure and the group with the smallest. Sample Period Rank No. Obs. Mom Spread Diff mom same 1 3 mom same 4 6 mom same 7 12 mom same mom same % 2.6% 1.2% 1.9% -0.3% 0.8% (1.76) (1.00) (1.48) (-0.14) (0.37) % 2.4% 1.8% 4.7% -8.5% -5.3% (1.57) (1.29) (2.82) (-3.32) (-2.06) % 0.6% 2.8% -8.2% -6.2% (-0.06) (0.33) (1.32) (-2.52) (-1.81) % -0.4% -2.3% 1.6% -1.8% -0.4% (-0.23) (-1.38) (0.92) (-0.74) (-0.15) % 0.0% -0.9% -8.6% -15.5% -5.2% (-0.01) (-0.39) (-2.49) (-3.88) (-1.85) % 1.4% -10.2% -13.7% -4.8% (0.13) (0.52) (-2.64) (-2.90) (-1.31) 21

24 Table 11: Forecasting table after controlling FF factors. This table reports the average returns of the same momentum portfolio without any rebalancing after formation, for 1-3, 4-6, 7-12, and months. All months are sorted into five groups based on their momentum spread diff realized in the last month. 5-1 means the difference between the average returns in the group with largest measure and the group with the smallest. Predictor Rank mom same 1 3 mom same 4 6 mom same 7 12 mom same mom same Mom Spread Diff Mom Spread Old Mom Spread New Mom spread 2-12 Value spread 5 2.0% 1.7% 0.3% -8.2% -0.9% (1.59) (1.54) (0.20) (-3.78) (-0.55) % 1.0% -3.1% -8.8% -3.6% (-0.30) (0.70) (-1.71) (-3.32) (-1.66) 5 1.1% 0.2% -1.4% -7.2% 1.2% (0.90) (0.13) (-0.82) (-3.02) (0.71) % -3.8% -6.1% -7.8% -3.5% (-1.80) (-2.45) (-3.20) (-2.85) (-1.61) 5 0.7% 0.0% 0.2% -4.2% 0.4% (0.60) (0.04) (0.15) (-1.91) (0.22) % -4.2% -5.0% -5.4% -2.2% (-2.30) (-3.05) (-2.66) (-2.13) (-0.98) 5 1.8% 1.0% -0.7% -4.1% -0.3% (1.38) (0.91) (-0.47) (-2.10) (-0.15) % -1.0% -5.3% -6.8% -3.5% (-0.35) (-0.69) (-2.93) (-2.96) (-1.43) 5 3.5% 1.0% -0.5% -11.3% 1.4% (2.51) (0.78) (-0.27) (-5.09) (0.83) % -4.6% -6.8% -15.2% -2.7% (-1.72) (-2.88) (-3.40) (-5.79) (-1.05) 22

25 Table 12: One year cumulative factor returns regressing on characteristic spreads. This table reports the regression coefficients of cumulative factor returns at month t on spreads at month t-1. At end of each month, all stocks are sorted into different quintiles according to their respective characteristic measures independently, except for momentum with deciles. The factor returns is the zero-cost portfolio longing the top quintile (decile) and shorting the bottom quintile (decile) for size, value, momentum and ivol, and BAB factor is formed following?. Stocks with price less than 5$ are excluded from our sample, together with the ones in the NYSE bottom size decile, to minimize market microstructure bias to our results. The t-stat in the parentheses, is adjusted for serial-dependence with 6 lags using? standard error. Characteristics Spreads (t-1) Dep. Var. SIZE VALUE MOM IVOL BETA MKTRFt-1 SMB SMB (4.76) (-1.60) (1.25) (6.69) (1.82) (-1.58) HML HML (-0.65) (3.08) (2.02) (2.48) (-0.54) (-0.28) MOM MOM (1.17) (-1.82) (-1.12) (-1.25) (3.04) (1.42) VOL VOL (-2.92) (3.20) (1.86) (-3.50) (-2.41) (1.54) BAB BAB (-1.70) (1.60) (1.56) (-1.65) (-0.55) (1.17) MKTRF MKTRF (0.81) (-2.08) (-2.09) (-0.07) (-0.61) (-0.02) 23

26 Table 13: Combo Factor Returns. This table reports the average returns of the combo strategy, for 1-3, 4-6, 7-12, and months. All months are sorted into two groups based on their momentum spread diff realized in the last month, and its relative value comparing to its MA in the last 12 month <MA12 >MA12 Variable Mean t Value Variable Mean t Value MA cumu % 3.22 MA cumu % 1.64 MA cumu % 3.73 MA cumu % 1.2 MA cumu % 3.99 MA cumu % 3.17 MA cumu % 5.09 MA cumu % 5.18 MA cumu % 3.6 MA cumu % 7.46 Table 14: Combo Factor Returns in This table reports the average returns of the combo strategy, for 1-3, 4-6, 7-12, and months. All months are sorted into two groups based on their momentum spread diff realized in the last month, and its relative value comparing to its MA in the last 12 month <MA12 >MA12 Variable Mean t Value Variable Mean t Value MA cumu % 1.24 MA cumu % -0.4 MA cumu % 1.65 MA cumu % -0.6 MA cumu % 0.24 MA cumu % 1.15 MA cumu % 0.01 MA cumu % 2.37 MA cumu % 0.69 MA cumu %