The bottom-up beta of momentum

The bottom-up beta of momentum Pedro Barroso First version: September 2012 This version: November 2014 Abstract A direct measure of the cyclicality of momentum at a given point in time, its bottom-up beta with respect to the market, forecasts both the returns and the risk of the strategy. Challenging a potential risk-based explanation, a highly cyclical momentum portfolio forecasts both higher risk and lower returns for the strategy. The results show robustness out-of-sample (OOS) and controlling for other variables. One predictive regression of monthly momentum returns on its bottom-up beta produces an OOS R-square of 2.41%. This contrasts with the usual negative OOS R-squares of similar predictive regressions for the market excess return. I thank Alan Gregory, Miguel Ferreira, Pedro Santa-Clara, and Ronald Masulis for their suggestions. University of New South Wales, School of Banking and Finance. E-mail: p.barroso@unsw.edu.au 1

1 Introduction Unconditionally, the Fama-French factors do not explain the risk or the returns of momentum (Fama and French (1996)). But Grundy and Martin (2001) show this is because momentum portfolios have time-varying systematic risk and this is not captured in unconditional regressions. The beta of the winners-minus-losers portfolio should depend by construction on the previous returns of the market. For instance, in late 1999, and after good returns in the stock market, the winners were mostly high beta stocks while the laggards were low beta stocks. Hence the momentum portfolio, short on previous losers and long on previous winners, should have a high beta by design. By contrast, at the end of 2008, in an extreme bear market, previous losers should be typically stocks with high betas such as financial firms at the time, while the group of winner stocks would have low betas. Thus the momentum portfolio would have a negative beta by construction. I compute the bottom-up beta of momentum using monthly returns from January 1950 to December 2012 for all stocks in the Center of Research for Security Prices (CRSP). The bottom-up betas change quite substantially over time, ranging from a minimum of -1.71 to a maximum of 2.09 and are positively related to previous returns in the market. Bottom-up betas are much better at explaining the risk of momentum than one unconditional regression. The bottom-up betas with respect to the Fama-French factors explain 39.59 percent of the out-of-sample variation in momentum returns, nearly 17 times more than one unconditional regression. They also compare favorably to other methods of estimation of the conditional beta. On the other hand, it is known that it is not persistence in returns of the Fama- French factors that explains momentum profits (Grundy and Martin (2001), Blitz, Huij, and Martens (2011), Chaves (2012)). As such, large loadings of momentum on some other factors should constitute a warning against the attractiveness of investing in the strategy. The expected return of momentum diminishes significantly with the absolute loading of the strategy on the market. An increase of one standard deviation in the absolute value of the market beta leads to a 1.18 percentage points decrease in momentum s 2

monthly expected return. One regression of momentum returns using this predictive variable holds an out-of-sample (OOS) R-square of 2.41%, a very high value for a monthly frequency. This contrasts with the predictability of the market, where similar regressions typically have negative R-squares (Goyal and Welch (2008)). In fact, the predictive models that show more robustness for the market achieve OOS R-squares of about 1% for the same frequency, so the predictability of momentum returns is more than double that of the market. The absolute loading of momentum on the market also forecasts its risk. One predictive regression of the realized variance of momentum on the absolute value of its beta holds an OOS R-square of 8.25 percentage points. The results of predictive regressions for risk and returns are robust controlling for other variables found relevant in the literature. In particular, the absolute loading on the market is not subsumed by the market being in a bear state (Cooper, Gutierrez, and Hameed (2004)) and the recent volatility of the market (Wang and Xu (2011); and Tang and Xu (2012)). One possible use of time-varying betas is to hedge the systematic risk of momentum (Grundy and Martin (2001); Daniel and Moskowitz (2011); and Martens and Oord (2014)). I examine the usefulness of bottom-up betas to hedge the risk of momentum in real time and find that it does not avoid its large drawdowns. The hedged strategy still has a high excess kurtosis and a pronounced left-skew, so the crash risk is not removed by hedging. Daniel and Moskowitz (2011) estimate the beta of momentum with a regression of daily returns and they also find momentum investors could not use time-varying betas to avoid the crashes in real time. This contrasts with the benefits of using the timevarying volatility of momentum to manage its risk (Barroso and Santa-Clara (2014)). 2 The time-varying beta of momentum The momentum portfolio changes its composition as new stocks join the group of previous losers or the group of previous winners. This changing composition induces time variation in the beta of the portfolio with respect to the market. I compare three methods of estimation of these time-varying betas: i) a bottom-up approach; ii) estimating beta as a linear function of factor returns in the formation period; iii) a high-frequency 3

beta estimated from daily returns. The bottom-up beta of the momentum portfolio is the weighted average of individual betas in the portfolio: N t β BU,t = w i,tˆβi,t (1) i=1 where N t is the number of stocks in the portfolio at time t, w i,t is the weight of stock i in the WML portfolio and ˆβ i,t is the beta of the individual stock estimated from past monthly returns. This beta relies only on past information, known before time t, so its forecasts are out-of-sample (OOS) by construction. 1 Figure 1 shows the bottom-up beta of the momentum strategy and compares it with the unconditional beta, obtained from running a regression of the WML on the market with the full sample. The unconditional beta is -0.27. So on average the losers portfolio has a higher beta than the winners portfolio. But this unconditional beta masks substantial time-variation in the composition of the WML portfolio. The bottom-up beta ranges from -1.71 to 2.09. Therefore, the momentum strategy is at times highly exposed to the overall stock market, while at other times it is actually negatively related with the market. Grundy and Martin (2001) show that the time-varying systematic risk of momentum is due to the return of the overall market during the formation period. After bear markets, winners tend to be low-beta stocks while losers are high-beta stocks. By shorting losers to go long winners, the WML portfolio will have by construction a negative beta. The opposite happens after bull markets. Figure 2 illustrates this point, confirming the results of Grundy and Martin (2001) and Korajczyk and Sadka (2004). The bottom-up beta, obtained from the individual stock level, is approximately a linear function of market returns during the formation period. Following Grundy and Martin (2001), I estimate this beta running an OLS regression: r wml,t = α + β 0 rmrf t + β 1 rmrf t rmrf t 2,t 12 (2) where rmrf t 2,t 12 is the cumulative excess return of the market during the formation period of the momentum portfolio. Then the time-varying linear beta is: 1 I describe with more detail the method of estimation of the bottom-up betas in the annex. 4

β L,t = β 0 + β 1 rmrf t 2,t 12 (3) Another method to estimate the time-varying beta of momentum is to use the daily returns of the WML portfolio and those of the market. Following Daniel and Moskowitz (2011) we regress at the end of each month the daily returns of momentum on the market in the previous 126 sessions ( 6 months). 2 As the bottom-up beta, the onemonth lagged high-frequency beta (β HF,t 1 ) produces an OOS forecast of momentum s exposure to the market. Table 1 presents descriptive statistics for different estimates of beta. The unconditional beta is the estimate from on OLS regression, which is constant. The first two rows show the in-sample results for β unc and β L using returns from 1964:02 to 2010:12. 3 The third and fourth rows present the OOS results for the same variables. Here I use the period from 1950:01 to 1964:01 to obtain initial estimates of the betas, producing an OOS forecast for the following month. Then I re-iterate the procedure every month till the end of the sample using an expanding window of monthly observations. The resulting OOS period is from 1964:02 to 2010:12. The in-sample (out-of-sample) linear beta varies between a minimum of -1.74 (- 1.60) and a maximum of 1.61 (1.46). The high-frequency beta varies even more from a minimum of -1.94 to a maximum of 2.16. So all estimates show there is substantial time-variation in the market exposure of the momentum strategy. A relevant test is whether time-varying betas explain the risk and returns of momentum. For each estimation method I obtain the hedged momentum return as: z t = r wml,t β t rmrf t (4) where β t is the conditional beta at time t. Table 2 shows that time-varying risk does not explain the excess returns of the momentum strategy. The mean excess return of the market-hedged momentum strategy ranges from 1.19 percent per month to 1.47 percent per month. All the t-statistics 2 They estimate the beta using 10 lags of daily returns to correct for stale quotes. This correction does not improve results in my sample period, so I only repport results from regressions with no lags. 3 To facilitate the comparison, the same sample period is examined for all methods. The daily returns of the Fama-French factors is only available starting in 1963:07. This restricts the comparable sample period to start in 1964:02. 5

exceed four, so they are highly significant. This confirms the results of Grundy and Martin (2001) and Blitz, Huij, and Martens (2011). Grundy and Martin (2001) show that time-varying risk does not explain the alpha of momentum. Conversely, Blitz, Huij, and Martens (2011) and Chaves (2012) show that momentum profits come from persistence in returns at the individual stock level, rather than in the factors themselves. As a result, hedging market risk has little effect on the alpha of momentum. Momentum has more beta risk in good times (in bull markets) than in bad times (in bear markets). The opposite pattern should hold for time-varying beta to explain momentum average excess returns. But taking time-varying betas into consideration enhances substantially the understanding of momentum s risk. The R-squared (1 var(zt) var(r wml,t )) improves OOS from just 1.93 percent for the unconditional model to values ranging from 15.78 percent to 24.81 percent using the conditional models. 4 This is 8 to nearly 13 times more than the unconditional model. The bottom-up beta performs particularly well OOS, with the highest R-square among those considered. 3 Exposure to the Fama-French factors One unconditional OLS regression of monthly momentum returns on the Fama-French factors from 1964:02 to 2010:12 holds (t-statistics in parenthesis): r W ML,t = 1.71 0.34 r RMRF,t 0.04 r SMB,t 0.47 r HML,t (5.70) ( 4.91) ( 0.42) ( 4.51) The regression has an adjusted R-square of just 5.63 percent and a significant positive alpha of 1.71 percent, confirming the result in Fama and French (1996) that their factors do not explain the risk and returns of momentum. Still, this improves substantially the fit of the CAPM, which has an adjusted R-square of just 2.54 percent for the same period. This is mainly because momentum is significantly and negatively related to value. Yet, as for the exposure to the market, these estimates mask substantial time-variation in risk. 4 Note that the OOS r-squared can assume negative values. Goyal and Welch (2008) show this is often the case with predictive regressions. 6

Figure 3 shows the exposure of momentum to the market (RMRF), value (HML), and size (SMB) factors. Just as for the market, exposure to value and size varies a lot. 5 Table 3 shows the descriptive statistics of the bottom-up betas. The exposure of the momentum portfolio to size and value varies even more than the exposure to the market. For the HML factor, the beta ranges from -3.06 to 2.61, while for the market it ranges only from -1.62 to 1.58. The standard deviation of the betas with respect to size and value are, respectively, 0.82 and 0.83. This is more than the standard deviation of β rmrf. Table 4 shows the average excess returns of the hedged portfolio with respect to the Fama-French factors, its t-statistics and R-square. As for the CAPM, the conditional models do not explain the mean excess returns of momentum which have always a positive mean with significant t-statistics ranging between 3.57 (for the bottom-up beta) and 6.48 (for the beta linear with past returns). However, the conditional models improve considerably the understanding of the systematic risk of momentum. In sample, the R-square of the unconditional model is 6.13 percent, which is reduced to only 2.37 percent in the out-of-sample (OOS) test. The high-frequency beta, used in Daniel and Moskowitz (2011), produces an OOS R-square of 29 percent. In spite of this large improvement, the high frequency approach underperforms other measures of systematic risk. The linear model has an OOS R-square of 37.72 percent. Even more so, bottom-up betas explain 39.59 percent of the systematic risk of momentum OOS. This is nearly 17 times more than the unconditional model. As for the CAPM, the bottom-up beta is the best conditional model to explain the systematic risk of momentum. 4 Predicting the risk and return of momentum At times, the momentum portfolio is well characterized by a directional exposure on the persistence of returns for some factor. The bottom-up betas constitute a direct and real-time measure of this. But the source of momentum returns is more likely to be persistence in returns at the stock level, not in factor returns. As such, a larger beta (in absolute value) should forecast a lower return for the momentum portfolio. 5 In unreported results, I find that the betas w.r.t. to each factor are positively related. Yet most of their variation is idiosyncratic (more than 90 percent). They do not move in tandem with each other. 7

Table 6 shows the results of variations of a predictive regression for momentum returns from 1964:02 to 2010:12: r W ML,t = α + β 1 ˆβ RMRF,t 1 + β 2 ˆβ SMB,t 1 + β 3 ˆβ HML,t 1 + β 4 Bear t 1 +β 5 RV rmrf,t 1 where the first three regressors are the absolute values of the (lagged) bottom-up betas of momentum w.r.t. the Fama-French factors. The fourth regressor is an indicator variable that assumes the value of 1 if the cumulative return on the stock market has been negative in the previous three years, and zero otherwise. The fifth regressor is the (lagged) realized volatility of the stock market, which is computed from the daily returns of the market in the previous month. The first three models in table 6 show, one by one, the significance of each bottom-up beta. The bottom-up betas of the size and value factors are not significant, but the beta with respect to the market, with a t-statistic of -3.97, is significant by any conventional level. This shows that an extreme positive, or negative, loading on the market factor predicts a lower momentum return. One standard deviation increase in the absolute value of the market-beta of momentum is associated with a reduction in the monthahead expected return of 1.18 percentage points. So this is an economically significant result. The last column shows the out-of-sample (OOS) R-square of the regressions. A positive value indicates that the regression outperformed the historical mean when forecasting the momentum premium. For the size and value betas, the negative OOS R-square show that the regressions performed worse out-of-sample than the historical mean, what should be expected since the regressors lack significance even in-sample. But the regression using the beta of momentum w.r.t. the market shows a high and positive OOS R-square of 2.41 percentage points. This contrasts with the usual negative OOS R-squares of regressions predicting the equity premium, even when using in-sample significant variables (Goyal and Welch (2008)). In model four, I test the three betas simultaneously. The beta w.r.t. the market forecasts returns, but not the other betas. The last two regressors are controls for other variables that have been found to predict the return of momentum. Cooper, Gutierrez, and Hameed (2004) show that momentum performs badly after bear markets. Both Wang and Xu (2011) and Tang and 8

Xu (2012) show that momentum has lower returns in volatile markets. Testing all these simultaneously shows that the market-beta of momentum is significant controlling for the other variables. This regression is not robust OOS as using too many regressors increases the chances for in-sample overfitting, even if some predictors are indeed significant. Daniel and Moskowtiz (2011) find that momentum crashes are due to the option-like payoffs of the losers portfolios in extreme bear markets. As the bottom-up market beta of momentum tends to be very negative in those situations, it is pertinent to examine if my results are not driven by negative betas and momentum crashes alone. To test for this, the following OLS regression distinguishes between positive and negative bottom-up betas. It holds (t-statistics in parenthesis): r W ML,t = 2.56 3.46 ˆβ+ RMRF,t 1 +4.70 ˆβ RMRF,t 1 (6.05) ( 2.71) (3.75) where ˆβ + RMRF,t 1 is the value of the beta of momentum w.r.t. the market if positive and ˆβ RMRF,t 1 the same if negative. The results are stronger for the left tail of the bottom-up beta. A marginal increase in the negative beta reduces the expected return of momentum more than in a positive beta. But the predictability is quite significant in the right tail as well. A high positive beta predicts lower returns for momentum, with a t-statistic of -2.71, the variable is significant at the 1% level. So the result is not driven by the negative betas, momentum crashes and option-like payoffs alone. 6 As there is a large momentum crash in the sample, I also test for non-linearities in this relation running a step-wise regression: r W ML,t = 2.64 5.18 ˆβ m,t 1 3.17 ˆβ m +,t 1 (6.19) ( 4.00) ( 2.55) where ˆβ m,t 1 is the absolute value of the bottom-up beta w.r.t. the market if this value is below the median and ˆβ m +,t 1 the same if it is above the median. I find the results are not totally driven by observations with particularly high absolute values of beta w.r.t the market. In fact, there is a slightly stronger association for smaller absolute values of the predictor variable. Another important issue is whether the bottom-up betas of momentum help predict its risk. Table 7 presents variations of the predictive regression: 6 In fact, in unreported results, I find there is no supportive evidence that the market-beta of momentum forecasts market reversals, which have been interpreted as the cause of momentum crashes. 9

r W ML,t = α + β 1 ˆβ RMRF,t 1 + β 2 ˆβ SMB,t 1 + β 3 ˆβ HML,t 1 + β 4 Bear t 1 +β 5 RV rmrf,t 1 + β 6 RV wml,t 1 where the dependent variable is the realized variance of momentum, at a monthly frequency. The regressors are the same as above for predicting returns, except for the last one which is the lagged value of momentum s realized variance. Barroso and Santa- Clara (2014) show the risk of momentum is highly predictable using its own lagged value as a predictor. As such it is important to control for this variable. The first three regressions show that the betas w.r.t. the market and value factors are significant, but the beta of the size factor is not. The regressions that use the value and market betas are robust, with an OOS R-square as high as 8.25 percentage points for the market. The fourth regression includes all three variables and confirms this result. The fifth regression includes the two control variables (the lagged risk of the market and the state of the market). The bottom-up betas of the market and value factors remain significant. The last model includes the lagged realized variance of momentum. This variable with a t-statistic of 21.88 is highly significant and it increases the OOS R-square from 22.77 percentage points to 53.84 percentage points. The significance of the bottom-up beta of momentum w.r.t. the market remains significant but the beta w.r.t. the value factor does not. In general, the predictability of momentum s risk is much higher than the predictability of its returns. The most relevant predictor is the lagged value of momentum s risk. Still, the bottom-up beta of momentum w.r.t the market is significant in every regression considered and robust OOS. So a highly cyclical (or counter-cyclical) momentum portfolio predicts a higher risk going forward. 5 Systematic risk and momentum crashes Grundy and Martin (2001) find that hedging the time-varying risk exposures of momentum produces stable returns. Daniel and Moskowitz (2011) show that this relies on using ex post information and that hedging in real time with time-varying betas does not avoid the momentum crashes. However, their method of estimating the time varying risk is with top-down regressions of daily data the high-frequency beta. I find this is a less satisfactory approach to capture the time-varying systematic risk of momentum 10

(although it still clearly outperforms an unconditional model). Bottom-up betas provide a superior method to estimate the time-varying exposure of momentum to other factors. This leads to the question of whether hedging this timevarying risk with a more suitable method could avoid the large drawdowns of momentum. Table 5 shows the performance of hedged portfolios using bottom-up betas. Hedging market risk or the Fama-French factor exposures reduces the excess kurtosis and leftskewness of returns, without a clear effect on the Sharpe ratio (it improves using the CAPM but decreases using the Fama-French factors). The reduction in crash risk is modest though. The hedged strategies have an excess kurtosis exceeding 5 and a leftskew almost as pronounced as the WML strategy. Figure 4 shows the cumulative return of the WML and its hedged versions during the last two years of our sample. This period includes the major momentum crash since the great depression March-April 2009. The WML factor accumulated a return of -74.87 percent in the period. The hedged portfolios performed slightly better. The CAPMhedged portfolio accumulated a negative return of -66.21 percent and the Fama-French portfolio -63.66 percent. So hedging with time-varying betas would have a relatively minor impact during the momentum crash. The crash is still there after hedging. This confirms the result of Daniel and Moskowitz (2011) that hedging the time-varying betas does not avoid the crashes in real time. Other practical considerations should also limit the potential usefulness of the idea of hedging the risk of momentum. While there are future contracts that could be easily used to hedge the market-risk of momentum, there are no such obvious analogues for the size and value factors. Besides, the hedge ratios exhibit substantial time variation and very high values at particular times. Anderson, Bianchi, and Goldberg (2012) show that time-varying leverage can seriously dampen the results of a strategy after transaction costs. So this should be another source of concern for a hedged momentum portfolio. But to all of these considerations, one can add a very simple fact: the performance of hedged momentum, even before transaction costs and other practical implementation issues are taken into account, is not impressive. 11

6 Conclusion When the previous returns of a factor are high, the momentum portfolio rotates from low-beta stocks to high-beta stocks on that factor. This changes the betas of momentum over time. Conditional betas capture the systematic risk of momentum much better than an unconditional model. Using the Fama-French factors, the out-of-sample R-square increases from just 2.37 percent for the unconditional model to as much as 39.59 percent using conditional betas. The bottom-up betas perform particularly well in capturing time variation in systematic risk. They achieve the best results comparing to the linear and the high-frequency beta, both with the CAPM and the Fama-French factors. Using this method to hedge the time-varying betas of momentum does not avoid its crashes though or translate into any significant improvement. Conventional momentum is a less appealing strategy whenever it relies excessively on a (positive or negative) loading in the market factor. This forecasts both lower returns and higher risk for the strategy. This happens because the stocks in both legs of the portfolio are most likely selected on the basis of their high exposure to the market in the formation window. The market may have performed very well, or very badly, in that period but that says very little about its performance going forward. On the other hand, it does convey important information about the future performance of momentum. 12

Annex To estimate the bottom-up betas, I use data from CRSP (Center for Research in Security Prices) with monthly returns for all stocks listed in the NYSE, AMEX or NASDAQ from January 1950 to December 2010. Following the standard practice, the momentum portfolios are sorted according to accumulated returns in the formation period which is from month t-12 to month t 2. The stocks are classified into deciles using as cutoff points the universe of all firms listed on the NYSE. This way there is an equal number of firms listed in the NYSE in each decile. This is to prevent the possibility of very small firms dominating either the long or short leg of the portfolio. In order to be considered in the portfolio a firm s stock must have a valid return in month t 2, a valid price in month t 13, and information on the market capitalization of the firm in the previous month. We take into consideration the delisting return of a stock whenever it is available. Individual stocks are value-weighted within each decile. The return of the winner-minus-losers (WML) is simply the return of the top decile portfolio, sorted on previous momentum, minus the return of the bottom decile portfolio. I estimate the beta of each individual stock running an OLS regression of its monthly excess return on the excess return of the market from t 61 to t 2, the end of the formation period. I require at least 24 valid returns in that period to estimate the beta. The market return is the value-weighted return of all stocks in the CRSP universe, as obtained from Kenneth French s online data library. 13

References [1] Anderson, R., S. Bianchi, and L. Goldberg (2012). Will My Risk Parity Strategy Outperform?, Financial Analysts Journal, 68(6), pp. 75-93. [2] Barroso, P. and P. Santa-Clara (2014). Momentum has its moments, forthcoming in the Journal of Financial Economics. [3] Blitz, D., J. Huij, and M. Martens (2011). Residual momentum, Journal of Empirical Finance, Vol 18, 3. pp. 506-521. [4] Chaves, D. (2012). Eureka! A momentum strategy that also works in Japan, working paper. [5] Cooper, M., R. Gutierrez Jr., and A. Hameed (2004). Market states and momentum, The Journal of Finance, 59 (3), pp. 1345-1366. [6] Daniel, K., and T. Moskowitz (2011). Momentum crashes, Working paper. [7] Fama, E. and K. French (1992). The cross-section of expected stock returns, The Journal of Finance, vol. 47, no. 2, pp. 427-465. [8] Fama, E. and K. French (1996). Multifactor explanations of asset pricing anomalies, The Journal of Finance, vol. 51, no. 1, pp. 55-84. [9] Goyal, A., and I. Welch (2008). A comprehensive look at the empirical performance of equity premium prediction, Review of Financial Studies 21, pp. 1455-1508. [10] Grundy, B., and J. S. Martin (2001). Understanding the nature of the risks and the source of the rewards to momentum investing, Review of Financial Studies, 14, pp. 29-78. [11] Jegadeesh, N., and S. Titman (1993). Returns to buying winners and selling losers: Implications for stock market effi ciency, The Journal of Finance, vol. 48, no. 1, pp. 65-91. [12] Martens, M., and A. Oord (2014). Hedging the time-varying risk exposures of momentum returns, Journal of Empirical Finance, vol. 28, pp. 78-89. [13] Tang, F., and J. Mu (2012). Market Volatility and Momentum, working paper. 14

[14] Wang, K., and W. Xu (2011). Market Volatility and Momentum, working paper. 15

Tables and Figures Market beta of WML portfolio 2 1.5 1 Beta of WML portfolio 0.5 0 0.5 1 β bu 1.5 β unc 60 65 70 75 80 85 90 95 00 05 10 Time Figure 1. Bottom-up and unconditional beta of the WML portfolio. The bottom-up beta of the WML portfolio is obtained from the previous 5 years of monthly returns of individual stocks. Returns from 1955:03 to 2010:12. 16

2.5 Market beta of WML and return in the formation period 2 1.5 Beta of WML portfolio 1 0.5 0 0.5 1 1.5 β t 2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Gross return on market portfolio from t 12 to t 2 β t = 2.76 + 2.55 rmrf t 2,t 12 Figure 2. The bottom-up beta of the WML portfolio w.r.t. the market and the previous one-year return on the market portfolio. All returns from 1955:03 to 2010:12. Beta Mean Max Min STD(B) β unc (IS) -0.26 - - - β L (IS) -0.12 1.61-1.74 0.52 β unc -0.08 0.07-0.24 0.07 β L -0.06 1.46-1.60 0.38 β BU -0.05 2.09-1.71 0.57 β HF 0.15 2.16-1.94 0.62 Table 1. Descriptive statistics of different betas of momentum. The first is the beta from one unconditional regression of the returns of the WML portfolio on the market. The second is a conditional regression where the beta depends linearly on the previous one-year return of the market. The third and fourth rows present the results for these same regressions but out-of-sample. The fifth row shows the results for the bottom-up beta of momentum w.r.t. the market. The final row shows the results for the highfrequency beta estimated from daily returns of momentum in the previous 6 months. The bottom-up and high-frequency betas are out-of-sample. All betas are from 1964:02 to 2010:12. The final column shows the standard deviation of the beta estimate. 17

Beta z t z t /σ zt R 2 β unc (IS) 1.47 4.95 2.71 β L (IS) 1.43 5.36 22.14 β unc 1.40 4.70 1.93 β L 1.37 5.11 20.39 β BU 1.34 5.15 24.81 β HF 1.19 4.32 15.78 Table 2. Performance of time-varying betas in explaining the excess returns and risk of the momentum strategy. The first column shows the average return of a hedged momentum strategy using the estimated betas as hedge ratios. The second column shows the t-statistic of the mean return of the hedged momentum strategy. The final column shows the r-squared of the model for the risk of momentum. The first two rows present results for in-sample betas and the others for out-of-sample betas. All hedged returns and betas are from 1964:02 to 2010:12. 1 Load on Rm Rf Beta 0 1 1960 1970 1980 1990 2000 2010 Beta 2 1 0 1 Load on SMB 1960 1970 1980 1990 2000 2010 2 Load on HML Beta 0 2 1960 1970 1980 1990 2000 2010 Figure 3. The loading of the WML on the FF factors using bottom-up betas. The bottom-up betas are estimated from the betas of individual stocks in the momentum portfolio. The sample period is from 1955:02 to 2010:02. 18

Beta Mean Max Min STD(B) Rm-Rf -0.03 1.58-1.62 0.40 SMB -0.03 2.25-1.91 0.82 HML 0.07 2.61-3.06 0.83 Table 3. Descriptive statistics of the conditional betas of the WML portfolio (estimated from the bottom-up) for the Fama-French factors. The bottom-up betas are estimated from the betas of individual stocks in the momentum portfolio. All betas are from 1955:03 to 2010:12. Beta z t z t /σ zt R 2 β unc (IS) 1.71 5.85 6.13 β L (IS) 1.48 6.48 42.73 β unc 1.65 5.56 2.37 β L 1.39 5.85 37.72 β BU 0.84 3.57 39.59 β HF 1.29 5.1 29.00 Table 4. Performance of time-varying betas with respect to the Fama-French factors in explaining excess returns and risk of the momentum strategy. The first column shows the mean return of a hedged momentum strategy, using the given model to estimate the beta of momentum each moment in time. The second column shows the t-statistic of the mean and the third the r-square of the model. The first two rows present results for in-sample betas and the others for out-of-sample betas. All hedged returns and betas are from 1964:02 to 2010:12. Max Min Mean STD KURT SKEW Sharpe WML 25.54-45.53 16.14 24.65 7.89-1.52 0.65 Market hedged 23.41-38.05 15.98 21.33 5.77-1.22 0.75 FF hedged 24.07-37.12 9.87 19.09 6.81-1.17 0.52 Table 5. Performance of the hedged portfolios. The hedged portfolios consist in the long-short momentum portfolio complemented with a loading on the risk factor chosen to offset the time-varying beta of momentum each point in time. In each case the betas are estimated bottom-up. The results are thus OOS. Kurt stands for excess kurtosis. The mean return, the standard deviation and the Sharpe ratio are all annualized. Returns from 1964:02 to 2010:12. 19

1.2 1.1 WML CAPM hedged FF hedged 1 0.9 cumulative return 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0 5 10 15 20 25 Figure 4. Momentum crash. The cumulative returns of the WML portfolio (blue), the momentum portfolio hedged for market risk (green) and for the Fama-French factors (red). Returns from 2008:12 to 2010:12. 20

Model Const RMRF SMB HML Bear RV rmrf adj-r 2 OOS R 2 1 2.55-4.10 - - - - 2.56 2.41 [6.03] [-3.97] - - - - - - 2 1.80 - -0.65 - - - 0.00-0.15 [3.33] - [-0.99] - - - - - 3 1.23 - - 0.20 - - -0.16-1.02 [2.59] - - [0.36] - - - - 4 2.68-4.26-0.61 0.50 - - 2.54 1.30 [3.73] [-4.07] [-0.94] [0.89] - - - - 5 2.76-3.79-0.66 0.59-0.24-82.84 2.43-0.01 [3.81] [-3.28] [-1.01] [1.04] [-0.32] [-1.13] - - Table 6. Predictive regressions of the return of the WML portfolio on the absolute value of the bottom up-betas of momentum with respect to the Fama-French factors. In model 5, we include the bear market indicator function and the lagged one-month realized variance of the market. The bear market indicator assumes the value of 1 if the cumulative return of the market for the previous 3 years is negative. The returns are from 1964:02 to 2010:12. The coeffi cients are multiplied by 100 for readability. Its t-statistics are in squares brackets. The R-squares are in percentage points. The last column shows the out-of-sample r-squared, this is computed using an expanding window of observations after an initial in-sample period of 240 months. Model Const RMRF SMB HML Bear RV rmrf RV wml adj-r 2 OOS R 2 1 0.04 1.19 - - - - - 10.32 8.25 [0.67] [8.10] - - - - - - - 2 0.50 - -0.16 - - - - 0.31 0.32 [6.20] - [-1.66] - - - - - - 3 0.08 - - 0.46 - - - 5.49 5.23 [1.16] - - [5.80] - - - - - 4-0.10 1.08-0.10 0.36 - - - 13.80 11.44 [-0.95] [7.41] [-1.15] [4.66] - - - - - 5-0.14 0.80-0.09 0.31-0.16 88.31-25.40 22.77 [-1.47] [5.33] [-1.00] [4.17] [-1.62] [9.29] - - - 6 0.06 0.27-0.05 0.01-0.08 4.78 73.76 59.85 53.84 [0.84] [2.44] [-0.82] [0.22] [-1.08] [0.60] [21.88] - - Table 7. Predictive regressions of the monthly realized variance of the WML portfolio on the absolute value of the bottom-up betas of momentum with respect to the Fama-French factors. In model 5, we include the bear market indicator function, the lagged one-month realized variance of the market and the lagged one-month realized variance of momentum itself. The bear market indicator assumes the value of 1 if the cumulative return of the market for the previous 3 years is negative. The forecasted realized variances are from 1964:02 to 2010:12. The coeffi cients are multiplied by 100 for readability, its t-statistics are in squares brackets. The R-squares are in percentage points. The returns are from 1964:02 to 2010:12. The coeffi cients are multiplied by 100 for readability. Its t-statistics are in squares brackets. The R-squares are in percentage points. The last column shows the out-of-sample r-squared, this is computed using an expanding window of observations after an initial in-sample period of 240 months. 21