Dynamic factor exposure of momentum investing
Evidence from the S&P500
MSc BA Finance Thesis
Christof G.W. Tenkink
University of Groningen, Faculty of Economics and Business
October 2012
Supervisor: Dr. A. Plantinga
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Dynamic factor exposure of momentum investing
Evidence from the S&P500
Abstract
The view that momentum investing yields significant returns has been well accepted. However the source of the returns and the interpretation of the evidence are widely debated. In recent decades the risk of this investing strategy has been examined intensively. Most academics however make use of static models to track the risk exposure. This might result in less accurate estimations of the risk-adjusted profitability of momentum investing. This paper provides evidence that momentum investing has to deal with dynamic factor exposure during investment periods. Factor loadings change over time and seem to be dependent on factor return realizations and portfolio return realizations.
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1. Introduction
Momentum strategies have been quite successful over the decades, see for example Rouwenhorst (1998) and Jegadeesh and Titman (2001). The source of the returns and the interpretation of the evidence, however, are still widely debated. In recent decades the risk of this investment strategy has been examined intensively. Well-known studies by Rouwenhorst (1998) and Jegadeesh Titman (2001) make use of static models to track the risk exposure of momentum portfolios. However, if the factor risk exposure of a portfolio changes over time, this may result in less accurate estimations of the risk-adjusted profitability of momentum investing. To track the exposure to common risk factors the most obvious model to use is the three-factor model which has outperformed the CAPM in repeated tests of Fama & French. They found pervasive evidence that CAPM is unable to explain return on size and Price-to-Book sorted portfolios. In this paper I focus on the Fama-French three factor model to adjust the moment profits for common risk factors. I will reveal the impact of dynamic risk factors on the profitability of momentum investing.
The existence of momentum is a market anomaly which classical finance struggles to explain. Evidence of momentum strategies earning significant profits constitutes a rejection of the efficient markets hypothesis, which is a cornerstone of modern finance. According to the efficient market hypothesis, stock returns should be uncorrelated over time. Abnormal stock returns are assumed to be solely caused by the arrival of new information. As long as differences in stock returns can be explained by differences in systematic risk, it is in line with efficient market theory. Therefore momentum profits can only be considered to be an anomaly if they cannot be explained by systematic risk factors. Conrad and Kaul (1998) were the first to adopt a risk-based view on momentum profits which would be in line with efficient market theory. They state that cross-sectional dispersion in expected returns can explain momentum. When return variations are due to common non-diversifiable components they are in line with market efficiency. When they are due to idiosyncratic components they are diversifiable and it is an arbitrage opportunity. However Moskowitz and Grinblatt (1999) show that the effect of cross-sectional dispersion is not strong enough to completely explain momentum profits.
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dissipate after the investment period, a finding difficult to reconcile with standard notions of priced financial risk.
Because of the inability to explain the anomaly, academics have been motivated to attribute the appearance of momentum to cognitive biases of investors, which belong in the field of behavioral finance. Biases in the behavior of investors can lead to imperfect markets. Several empirical studies however argue that momentum profits do not contradict market efficiency. Some academics have argued that the profits are just the result of data mining. This criticism became increasingly unlikely with the passage of time as momentum has been prevalent over decades and on stock markets all over the world except Japan, which I will illustrate in the literature section. The bad performance of the strategy in the last decade (2000-2010) reinforces the doubts regarding the existence of real momentum profits.
Dynamic factor betas
The inability to explain momentum profits could indicate market inefficiency but it could also be due to a miss-specified asset pricing model. This problem motivates researchers to experiment with alternative asset pricing models to provide a risk-based explanation of momentum profits. One potential problem of traditional asset pricing models like the Three-Factor model is the fact that their factor betas are assumed to be stable over time, which does not seem to be the case in practice. Momentum investing most likely has to deal with dynamic factor betas over time. Because factor risk is associated with uncertainty, this will have an impact on the compositions of winner portfolios as well. If for example small-cap stocks have outperformed large-caps in the formation period, they will be more present in the winner portfolio than in the loser portfolio. If this is the case we can attribute part of the momentum effect to the small-cap effect. However, if small-caps happen to underperform in the subsequent ranking period, the loading of the winner portfolio on the SMB factor will likewise decrease. So the factor exposure of momentum portfolios will differ due to different factor realizations in their ranking period and the accompanying factor loadings of a portfolio. This feature should be incorporated in asset pricing tests.
Because factor returns show normally a positive autocorrelation over time, there will probably be a certain degree of co-variation between factor loadings of momentum portfolios and the contemporaneous factor premiums. This will probably have an inhibitory effect on the momentum profits. These kinds of factor dynamics are well recognized by the recent momentum literature.
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building will have a smoothing effect on this time-variation of individual betas, but only up to a certain degree. With respect to momentum investing the biggest changes can be expected in the extreme decile portfolios based on past returns (the winner and the loser portfolio). Because the stocks in these portfolios exhibit sharp price moves, there may probably occur substantial changes in financial leverage. As is well-known, leverage has a huge impact on stock risk (Black, 1976).
This leverage will be subject to changes when momentum investing works out as it should. Decreasing leverage means -ceteris paribus- that the market beta will decline. In that case winner stock prices continue to increase and they will eventually become growth (high Price-to-Book ratio) as well as large-cap stocks. When losers continue to s they will -ceteris paribus- eventually become value stocks. So when momentum investing is successful we may expect to eventually invest long in growth/large-cap stocks and short in value/small-growth/large-cap stocks. This will have a decreasing effect on the riskiness of the portfolio during the investment period. The static Fama-French model however, does not account for this changing risk exposure. In this paper I will focus on the widely used static Fama-French model to adjust for risk. My aim is to show that this static model does not give an accurate picture of the risk implications of momentum investing.
This leaves us with the following research question: Is momentum investing subject to dynamic factor exposure between and during the investment periods?
When momentum investing is indeed subject to changes in risk factor loadings, the risk-adjusted returns of momentum investing might be misjudged. As stated above, when stock price momentum is present, the risk of the momentum portfolio is expected to decrease. Because the loadings on the Fama-French factors are only measured once, the decrease in risk is not incorporated in the calculation of the risk-adjusted returns. Misspecified beta dynamics can negatively affect asset-pricing tests as well as portfolio composition choices. If momentum portfolios are formed based on past stock-specific returns we need to know a stock’s loading on the Fama-French factors during the ranking period. If these loadings change within a period they are not well estimated using a static model and we will not be able to determine the part of the return that is stock-specific. This will weaken the implementation of the investment strategy because only the stock-specific part is diversifiable and therefore valuable for the investor.
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available for private investors. Thereby it prevents the momentum profits to be exaggerated because of a too strong presence of small caps which normally demand higher risk compensation and show generally higher momentum profits. By doing this I can check whether the riskiness of momentum portfolios changes from the ranking- to the investment period. This may result in a better way to adjust momentum returns for factor risk and it could lead to a better approach to select stocks for a momentum investing strategy.
In asset pricing literature, the common practice to find risk-adjusted returns is to run a time-series regression of portfolio returns on the risk factor returns. However, while running the regression over the full sample period, one implicitly assumes that the factor betas of these portfolios are constant over time. For individual stock or some characteristic based portfolio this might be true, however, for momentum portfolios this seems very implausible. Because momentum portfolios take on different compositions of stocks over time in response to changing factor premiums, the factor loadings will exhibit drastic changes as well. Lewellen and Nagel (2005) show a simple test of the conditional CAPM using direct estimates of conditional alphas and betas from short-window regressions. Therefore the need to specify conditioning information is not present anymore.
Many academics followed an approach of conditioning factor betas on macro-economic state variables which has several difficulties. Most important is the puzzle to identify the right state variables. By shortening the estimation period for the CAPM or Fama-French regressions this problem is circumvented. There is no need to choose the right conditioning variables anymore, because the short OLS regressions provide direct estimates of assets conditional alphas and betas. A problem that comes up with this approach is the fact that non-synchronous price movements can have a substantial impact on betas that are estimated over short intervals. Lo and MacKinlay (1990) show that especially small stocks tend to react with a delay to common news. Therefore a daily or weekly beta will miss much of the small-stock covariance with market returns. This problem is mitigated because the tests in this paper focus on value-weighted portfolios of the S&P 500 index.
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2. Literature Review
The first evidence of the momentum effect was found by Jegadeesh and Titman (1993) with their momentum study in the US. Their sample contains NYSE and AMEX stocks in the period from January 1965 to December 1985. Their strategy of buying past winner stocks and shorting past loser stocks resulted in a 12% compounded average excess return per year. The portfolios were constructed by ranking the stocks on their past six month return and with a holding period of six months as well. To be sure that momentum was not limited to US markets more research was needed. Rouwenhorst (1998) found evidence for momentum in all the twelve European countries within his sample. He included 2190 firms in the period from 1978 to 1995. His results were in line with the results of Jegadeesh and Titman with similar average monthly returns around 1% per month.
Many researchers focus on common risk factors to explain momentum. The Capital Asset Pricing Model (CAPM) uses only one variable, the market beta, to describe the returns of a portfolio or stock with the returns of the market as a whole. In contrast, the Fama–French model uses three variables. Fama and French (1992) came with this extension because they noted that small-caps tended to perform better than big-caps and value stocks outperformed growth stocks. They added these two factors to the CAPM to reflect a portfolio's exposure to these two classes:
( ) (1)
Here R is the portfolio's rate of return, Rf is the risk-free return rate, and Rm is the return of the whole stock market. The model says that the expected return on a portfolio in excess of the risk-free rate [E( )] is explained by the co-variation of its return to three factors: the market excess return ( ) the difference between the return of small-stocks and big-cap stocks (SMB); and the difference between the return of low price-to-book stocks and high price-to-book stocks (HML).
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Carhart (1997) included momentum as a factor to the Fama-French model. His aim was not to provide risk interpretations, but only to explain returns. He reports that momentum profits unexplained by common factors and transaction cost were strongly concentrated in strong underperformance by the bottom decile. His four-factor model outperformed the CAPM as well as the Fama-French model in the tests. He concluded that these findings are after all consistent with market efficiency. Because he does not provide us with risk interpretations of the momentum factor I do not include this factor in my regression equation. Furthermore, this makes it possible for me to compare the alphas of the momentum strategies using different regression methods. By using the Carhart model, the part of the alphas caused by eventual momentum would be captured away by the loading on the momentum factor.
As mentioned in the introduction, Conrad and Kaul (1998) state that momentum strategies are only profitable because of cross-sectional variation in mean returns and hence just are a compensation for changing risk with respect to certain stocks. They conclude that momentum profits are not due to time series patterns in stock returns and therefore no prove of market inefficiency. Tai (2003) suggests a conclusion in line with Conrad and Kaul. According to his research, momentum returns are not really an anomaly and therefore not without risk. He argues that the abnormal returns are a compensation for higher exposure to common risk factors of these stocks.
Debondt and Thaler (1985) report that their winner quintile firms have a market value that is on average almost twice the market value of the loser quintile firms. Despite this observation they neglect to perform a statistical test for the equality of size between the groups. Zarowin (1990) however, reports that the fact that losers tend to be smaller sized firms than winners is the cause that losers outperform winners over a 3-year investment horizon. What makes this finding even stronger is the fact that in periods when winners are smaller than losers, winners outperform losers. When controlling for size, they found little evidence of any return discrepancy, except for the month January. Moskowitz and Grinblatt (1999) found the contrasting result that although the momentum profits decline after adjusting for size, book-to-market equity and individual stock momentum, they still remain significant. According to them much of the momentum effects are explained by industry influences. They found evidence that industry momentum strategies are more profitable than individual stock momentum strategies in terms of raw returns. This seems plausible because industry components may be more difficult to diversify which makes them more risky.
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that the difference of momentum between small and big stocks is not so big, when the smallest decile is disregarded.
Jegadeesh and Titman (2001) report that winners and losers tend to have a smaller than average market value in the sample. This is not a surprise, because small stocks are more volatile, which makes them a more likely candidate to be in an extreme return sorted portfolio. Furthermore they indicate that the market betas are almost equal for winner and loser portfolios. However, the losers load somewhat more on the small-caps and value stocks than the winners. It can be concluded that loser portfolios are exposed to slightly more risk. The Fama-French alpha they report is 1.36 percent which is larger than the raw return of 1.23 percent. The difference arises because the shorted losers are more sensitive to the Fama-French risk factors.
Fama and French (1996) conclude that the profitability of a momentum strategy cannot be explained by its unconditional factor exposure. This might be true, however, Chan (1988) and Jagannathan and Wang (1996) argue that when a momentum portfolio loads heavily on a factor, it may appear to earn abnormal returns, because exposure to that factor requires a high return. Such a possibility could explain momentum for certain periods, in the case that the factors themselves exhibit positive momentum.
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Since beta is in part a measure of relative financial and operational leverage, the betas of winner/loser stocks should decrease/increase between the formation and investment periods. Empirical evidence dating back at least to Fama and French (1997) shows that time-invariant regression techniques yield risk loadings that are imprecisely estimated because true betas experience substantial variation through time. Ang and Chen (2005) try to deal with this problem by taking mean reverting first-order auto-regressions to model variation in the betas of value premiums. They do this in a highly parameterized latent variable model that also includes assumptions like the variation over time of the expected excess market return and its volatility. This method may be somewhat over-specified which may cause the model to fail if one of their assumptions is not solid.
For that reason, it seems more reliable to model beta dynamics by running short-window (rolling sample) OLS regressions. Fama and French (2006) examine four alternative specifications to estimate beta. They use slope dummies to allow for periodic changes in beta. They used respectively a constant beta for the whole sample period, a single break in beta on a specific date, beta changes every 5 years, and annual changes of beta at the portfolio formation point. They regarded the R² of the regressions as a guideline to judge which of the specifications is best, adjusted for the degrees of freedom. The shortening of the estimation intervals resulted in an increased or unchanged R² for every portfolio. They inferred that picking up more variation in the true beta more than compensated for the loss in degrees of freedom. They infer that it is best to estimate beta on a yearly base when portfolios are rebalanced. Because their portfolio compositions used to change every year by rebalancing this result seems plausible. Grundy and Martin (2001) in contrast, report that the conclusions of their research are not influenced by the length of the period they use to estimate the factor betas.
In this paper I will carry out short-window regressions following the approach of Fama & French to strive for a better estimation technique of factor betas. Their results indicate a possibility to improve the risk adjustment technique of raw momentum profits. Because these portfolios are rebalanced every six months the betas estimations should be allowed to change more often.
3. Methodology
To find an answer for the research question I first set up the momentum portfolios following the approach of Jegadeesh and Titman (2001). I construct weekly portfolios by ranking stocks based on their past six-month cumulative returns except for the most recent month to avoid contaminating momentum profits with the very short-term return reversal reported by Lehmann (1990) and Lo and MacKinlay (1990).
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portfolios are held for the same period of months as the ranking period and are rebalanced in January and July of each year. Within the portfolios the stocks are value-weighted which is not in line with the approach of Jegadeesh and Titman (2001). I choose to do this because setting up value-weighted portfolios is in the spirit of minimizing variance because return variances and size are negatively related. Furthermore it is more realistic in terms of real-life investment opportunities.
The raw momentum profit is calculated as follows: Momentum profit (recent winners – recent losers) = Logarithmic return P1- Logarithmic return P10
This measure does not take into account the riskiness of the portfolio, because it is just the raw return. To adjust for risk, the raw returns are regressed on the weekly Fama-French factor returns to measure the covariance with these risk factors. To measure the factor loadings in the usual way I collected the weekly HML and SMB factor returns from the Fama-French website. Furthermore, I collected the excess market returns from the Fama-French website.
The resulting regression-coefficients proxy for the exposure of the momentum portfolios to these risk factors. It makes sense to run regressions during the ranking periods and the investment periods and to compare them with each other to see if the betas change over time. The factor model considered is the three-factor Fama- French model:
The regression is of the form: (2) The excess market return: ( - ) is the value-weighted return on all NYSE, AMEX, and NASDAQ stocks (from CRSP) minus the one-month Treasury bill rate (from Ibbotson Associates). SMB is the weekly return on the Fama French small-cap portfolio minus the big-cap portfolio. HML is the weekly return on the low Price-to-Book portfolio minus the high Price-to-Book portfolio.
First of all I will run a single regression over the whole period of 1991-2011. This will show if momentum investing was profitable during this period. Furthermore it provides the average risk loadings for this period. Because I do not believe that the risk loadings are stable I will run regressions over shorter intervals.
The question if factor loadings are dynamic can be translated into the following hypothesis: : The Fama-French factor loadings of momentum portfolios are stable over time.
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months). I present this in the data section. When there are clear developments in these measured risk loadings this indicates changing riskiness of the portfolios that is not captured by the static Fama-French model.
To test the stability of the factor betas during the sample period I run a Chow test to test for structural breaks of the parameters. The unrestricted regression to test for structural breaks is given by the following equation:
(3) Where = 1 during and zero otherwise. In other words, takes the value one for observations in the first subsample and zero for observations in the second subsample. The Chow test is a standard F-test of the joint restriction: : .
If the Chow tests show no rejections I cannot reject the null hypothesis and conclude that the factor exposure of momentum portfolios is not dynamic. In that case we could make use of the static three-factor model. When the null hypothesis is rejected by the Chow test it makes sense to adjust to dynamic risk factors when calculating momentum profits. This will help to improve performance measurement of portfolio managers on shorter horizons, for example. When calculating risk-adjusted profits over a recent period it does not make sense to use risk loadings that are based on long-term averages.
With dynamic factor betas it makes sense to make use of short-term beta estimations for the periods surrounding the breakpoint. To examine if beta estimations are more accurate over shorter horizons I allow betas to change every five years and even every six months. Therefore I will run the regressions for winner-, loser- and momentum portfolios for the periods 1992-1996, 1997-2001, 2002-2006 and 2007-2011. Because in the paper of Fama French (2006) the estimation period of one year provided the best results, I will even try to estimate betas for all the 40 ranking- and investment periods. Afterwards we can check whether the R² of the model with short-window beta estimations is higher than the model with the longer-window beta estimations. This indicates how well the different models fit the data.
The main objective however was to examine whether possible changes of the risk loadings depend on the sign of momentum portfolio return realizations. I expect this relation because increasing stock prices drive stocks in the direction of growth stocks and large-cap stocks. To illustrate how this might work, consider the following line of reasoning:
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probably decrease. Furthermore I expect the loading on the value premium (HML) to decrease. Because an increase in equity for individual stocks will ceteris paribus result in less financial leverage, the market beta should decrease as well. When equity decreases in value the story goes vice versa and the factor loadings are expected to increase. The loser portfolio stocks’ risk loadings are expected to go in the opposite direction, but because the strategy takes a short position in this portfolio the momentum portfolio is affected in the same way as by the long position in the winner portfolio. To test the possible change in risk loadings, I will run separate regressions for the periods where the portfolio returns were positive and for the periods where the portfolio returns were negative. Then I can see whether the average betas for these separate regressions statistically differ from each other. If this is the case we can conclude that there is a relationship between risk factor loadings and past momentum returns. A Chow test can determine again, whether the factor loadings are statistically different between positive and negative portfolio changes.
4. Data Collection
The dataset contains weekly total return series of the S&P 500 stocks between 1991 and 2011. Dividends are reinvested in the index. 27 banks/financial institutions have been excluded from the dataset because of their different capital structure. 48 stocks have not been found in the database of Thomson Datastream. Stocks with less than 24 months of consecutive return data or market values are excluded. The data is almost free of survivorship bias, since I included all the stocks that have been delisted from the S&P 500 during the 20 year period, provided that they were available on Datastream. The total number of included stocks is 847 (355 dead stocks). The sample size of active stocks per year ranges between 399 and 474. In January 1991 each ranking portfolio decile contained 34 stocks. By the final ranking period each decile contained 47 listed firms.
The Fama-French three factor-model factor returns and the description below have been downloaded from the Fama-French website. The excess market return: ( – ) is the value-weighted return on all NYSE, AMEX, and NASDAQ stocks (from CRSP) minus the one-month Treasury bill rate (from Ibbotson Associates). SMB is the weekly return on the Fama French small-cap portfolio minus the big-cap portfolio. HML is the weekly return on the low to-Book portfolio minus the high Price-to-Book portfolio. All factors are on a weekly basis for the period between 1992 and 2011.
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To be included in the tests, a firm must have stock prices for December of year t -1 and June of t. I do not include firms until they have appeared on Thomson Datastream for 1 year.
Table 1. Summary statistics of momentum portfolio returns Portfolio mean return standard
deviation variance maximum return minimum return P1 0.15% 3.4% 0.12% 16% -27% P2 0.12% 3.0% 0.09% 12% -18% P3 0.07% 2.9% 0.09% 12% -21% P4 0.15% 2.6% 0.07% 17% -19% P5 0.18% 2.9% 0.08% 13% -29% P6 0.20% 2.7% 0.08% 22% -17% P7 0.15% 3.0% 0.09% 19% -23% P8 0.14% 3.0% 0.09% 14% -19% P9 0.17% 3.1% 0.09% 18% -21% P10 0.06% 3.6% 0.13% 26% -25%
Summary statistics of the logarithmic raw returns of momentum portfolios consisting of S&P 500 stocks during the period 1992-2011.
Table 1 shows the raw returns of the momentum portfolios. This means that they are not yet adjusted for factor risk. We see that the standard deviation of the raw returns is highest in the extreme return portfolios which is in line with findings of Jegadeesh and Titman (1993, 2001).
Table 2. Portfolio characteristics Summary statistics Market value
Ranking period Market value Investment period PtB ratio Ranking period PtB ratio Investment period P1 15 17 4.1 4.1 P2 18 19 3.8 3.7 P3 19 19 3.6 3.5 P4 17 18 3.3 3.3 P5 18 19 3.3 3.3 P6 19 19 3.3 3.3 P7 18 19 3.3 3.2 P8 17 17 3.2 3.2 P9 15 16 3.1 3.1 P10 11 12 2.9 3.0
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In the existing literature betas are tracked by regressing the portfolio returns on the factor returns. Table 2 shows us the loadings on the HML- and SMB factor directly via the average market values and the market-weighted Price-to-Book ratios of the momentum portfolios during the formation period and the investment period. This is a more direct measure. From table 2 we can conclude that the winner stocks (P1) have certainly higher market values than the loser stocks (P10). The same is true regarding the price-to-book ratios. An explanation could be that winner/loser stocks are the stocks with the biggest increase/decrease in price over the recent period which is reflected in market values as well as the price-to-book ratios.
In comparison with the other portfolios we can conclude that the winner (P1) and loser (P10) portfolios start with a relatively low market value at the start of the ranking period which reflects relatively high loadings on small stocks. Small stocks are more volatile so it is not that surprising that these stocks show up in the extreme return performance portfolios. Surprisingly there seems to be no clear pattern in the development of market values and Price-to-Book ratios from the ranking period to the investment period. This might be due to the fact that I show averages over the whole sample period. Increases of the measures in successful momentum periods may be offset by decreases in unsuccessful periods.
To test for first order autocorrelation in the residuals of the portfolio returns I perform a Durban-Watson test. The Durbin-Durban-Watson statistics are insignificant for every single portfolio as is shown in table 3. Therefore the null hypothesis that the errors are independent of one another cannot be rejected. This is one of the conditions that must be satisfied to use OLS regressions.
Table 3. Test for Autocorrelation in residuals Durbin-Watson statistic P1 2.14 P2 2.13 P3 1.96 P4 2.10 P5 2.02 P6 2.14 P7 2.13 P8 2.02 P9 2.08 P10 2.05
17 5. Results
Table 4 shows the regression results with risk adjustment by the static Fama French-model. I do not allow betas to change during this period. Jegadeesh and Titman (1993) for example let their betas change once during the period of 1965 until 1998. So this first table is in line with this relative static way of modeling.
Table 4. Factor loadings during investment period
Weekly alpha β Market β SMB β HML R²
P1 (Winners) 0.0372% 1.10 -0.02 -0.32 0.60 (0.54) (37.96)** (-1.62) (-5.15)** P2 -0.0007% 1.00 -0.02 -0.04 0.69 (-0.14) (48.07)** (-0.53) (1.15) P3 -0,05% 1.01 -0.18** 0.05 0.66 (-0.91) (45.13)** (-4.93)** (2.21)* P4 0.0289% 0.93 -0.13** 0.19 0.71 (0.65) (50.55)** (-3.87)** (6.18)** P5 0.0437% 0.97 -0.06 0.27 0.64 (0.40) (42.51)** (-1.48) (6.37)** P6 0.0661% 0.95 -0.06 0.35 0.67 (0.64) (45.53)** (-2.34)* (9.98)** P7 0.0015% 1.00 0.00 0.28 0.68 (0.03) (45.96)** (-0.48) (6.06)** P8 0.0070% 0.99 -0.15 0.26 0.64 (0.12) (43.28)** (-4.08)** (6.64)** P9 0.0469% 1.02 -0.09 0.18 0.66 (0.34) (44.76)** (-2.11)* (4.55)** P10 (Losers) -0.1027% 1.14 0.02 0.42 0.60 (-1.44) (39.38)** (0.01) (8.20)** WML 0,14% -0.04 -0.04 -0.74 0.28 (1.27) (-2.06)* (-1.07) (-8.89)**
The weekly raw portfolio returns are regressed on the weekly Fama-French factor returns. This table shows the risk-adjusted returns as well as the portfolio loadings on these risk factors. The corresponding t-statistics are listed between parentheses. The ** and * denote significance on the 1% and 5% level respectively. The average risk-free rate has been subtracted from the portfolio return to calculate alpha. WML denotes the momentum portfolio: winners minus losers.
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As can be seen from the table, none of the portfolios earns significant momentum profits. The annualized alpha for the winner portfolio (P1) is 1.6%. Because the loser portfolio (P10) has a negative annualized alpha of -5.1% the momentum portfolio (WML) earns a positive annualized risk-adjusted profit of 6.7%. However the alpha of the momentum portfolio is not significant either. One possible explanation might be the fact that I did not hedge the January returns which have usually a negative impact on the momentum profits (see for example Jegadeesh and Titman, 2001). But it is questionable if this extension would have caused the profits to become significant.
From the table I conclude that extreme return portfolios P1 and P10 have the highest market betas. Risky stocks with higher market betas have more volatile returns so their presence in these portfolios is to be expected. This is in line with Jegadeesh and Titman (2001). The SMB betas are mostly negative which might be due to the fact that the S&P 500 consists of large-cap stocks. Only P1 and P2 have negative loadings on the HML factor. Because these have been the best performing portfolios over the past six months they are more likely to be growth stocks because of their risen market value. The factor loadings in this table are estimated once for the whole sample period. The regression results show significant relationships between the portfolio returns and especially the market return and the HML-return, but I do not expect the betas to be stable during the sample period.
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Table 5. Factor loadings with betas allowed to change every 5 year
1992-1996
variable alpha ß Market ß SMB ß HML R²
P1 -0.0013 1.41 0.09 -0.30 0.55 (-0.99) (12.11)** (0.61) (-2.02)* P10 0.0000 1.04 0.37 -0.01 0.55 (-0.01) (13.08)** (3.69)** (-0.05) WML -0.0019 0.38 -0.31 -0.28 0.16 (-1.05) (2.4)* (-1.54) (-1.40) 1997-2001
variable alpha ß Market ß SMB ß HML R²
P1 0.0014 1.03 -0.28 -0.31 0.56 (0.76) (11.15)** (-2.48)* (-2.09)* P10 -0.0025 0.96 -0.27 0.27 0.67 (-1.32) (9.86)** (-2.24)* (1.72)* WML 0.004 0.07 -0.56 -0.58 0.34 (1.51) (0.09) (-1.87) (-3.16)** 2002-2006
variable alpha ß Market ß SMB ß HML R²
P1 -0.0010 1.09 0.28 0.13 0.69 (-0.90) (17.52)** (2.34)* (1.01) P10 0.0002 1.11 0.17 -0.26 0.66 (0.18) (16.15)** (1.56) (-1.79)* WML -0,0012 -0.02 0.11 -0.13 0.23 (-1.03) (-0.35) (0.65) (0.93)** 2007-2011
variable alpha ß Market ß SMB ß HML R²
P1 0.0009 1.01 -0.00 -0.26 0.80 (0.90) (27.83)** (-0.06) (-3.60)* P10 0.0005 1.18 0.01 0.56 0.82 (0.34) (24.41)** (0.07) (5.83)** WML 0,0004 -0.16 -0.01 -0.83 0.26 (0.38) (-2.58)* (-0.09) (-6.37)**
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** and * denote significance on the 1% and 5% level respectively The respective Risk-free rates have been subtracted to calculate alpha
From table 5 we see that the factor betas differ between the four periods of five years. The market beta of the momentum portfolio is positive during the first decade (1992-2001) but negative for the second decade (2002-2011). The Chow tests to test for beta stability showed rejections at the 5% significance level between all the four periods. This shows clearly that we should allow betas to change in order to adjust for dynamic factor betas.
Regarding the other factors I note that the HML beta takes on significant positive as well as significant negative values for certain periods. This is a clear evidence of dynamic behavior. The table shows that during the last three periods of time the sign of the HML loadings of the winner- and loser portfolio differ from each other every time. This is probably caused by changing factor returns. Grundy and Martin (2001) found evidence that winner portfolios load positively on the factors that provided good returns in the past period. Loser portfolios load negatively on this factors. Besides the fact that the winner- and loser portfolio have opposite signs on their HML betas, these signs also change every of the last three periods. This shows clearly that factor betas exhibit dynamic behavior, which is in line with the evidence of Grundy and Martin (2001). With respect to the SMB factor betas I am somewhat more cautious because the factor betas are mostly not significant, which prevents me from drawing conclusions about their stability.
Table 5 makes clear that the regression in table 4 where betas are not allowed to change misses and neglects the development of changing factor betas during the period 1991-2011. Therefore we cannot accurately determine what the risk-adjusted momentum profit actually was over sub-periods of time. The R² does not improve really by shortening the estimation period, although the last period (2006-2011) in table 5 shows values of R² above 80% which are the highest values so far.
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Table 6. Average alpha and factor loadings during investment periods
Chance of significance
P1 (winners) mean stdev t-stat p 5% p 1%
Alpha -0.003 0.007 -0.02 8% 3%
ß Market 1.197 0.254 2.03* 93% 88%
ß SMB -0.058 0.339 -0.07 10% 5%
ß HML -0.323 0.434 -0.39 18 10%
Chance of significance
P10 (losers) mean stdev t-stat p 5% p 1%
Alpha -0.001 0.003 0.05 3% 0%
ß Market 1.124 0.221 3.86** 83 % 75%
ß SMB -0.104 0.348 0.30 15 % 8%
ß HML -0.143 0.485 0.12 25% 18 %
This table shows the average Alpha and the average of the factor-betas of momentum portfolios consisting of S&P 500 stocks during 40 investment periods of six months over the period 1992-2011. The * and ** denote significance on the 5% and the 1% respectively
The chance of significance denotes the percentage of periods wherein the 40 separate beta estimates where significant on the 5% level and the 1% level respectively.
Table 6 shows again the factor loadings, but now these are based on 40 separate regressions for the investment periods. Only the market betas are significant using these short estimation periods of six months. Because the factor loadings for the SMB and HML premium are mostly not significant I cannot draw conclusions from them. Apparently the six-month periods are too short to estimate all the three factor betas using my approach. For periods where all the beta coefficients are positive I choose to run a Chow-test to test for structural breakpoints between ranking and investment periods. Several Chow-tests were significant at the 5% level, which is evidence that at least in some periods the factor loadings change from the ranking period to the investment period.
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I only observe a significant decrease in market beta after a positive return of the momentum portfolio, which is in line with theory. Less leverage results –ceteris paribus- in less exposure to market risk, (see Black, 1976)
Table 7. Change in factor loadings from ranking to investment period
P1 mean Positive return Negative return
ß Market -0.21 -0.20 0.12 (-0.36) (-0.29) (0.72) ß SMB -0.24 -0.29 0.09 (-0.22) (-0.24) (0.20) ß HML -0.16 -0.19 0.00 (-0.16) (-0.23) (0.01)
P10 mean Positive return Negative return
ß Market 0.18 -0.21 0.10 (0.47) (-1.24) (0.76) ß SMB -0.03 0.11 0.16 (-0.03) (0.23) (0.16) ß HML -0.18 -0.46 0.11 (-0.21) (-0.61) (0.12)
WML mean Positive return Negative return
ß Market -0.37 -0.42 -0.11 (-0.53) (-1.79)* (-0.77) ß SMB -0.22 -0.15 0.42 (0.17) (-0.13) (0.39) ß HML -0.34 -0.43 0.20 (-0.23) (-0.30) (0.13)
The weekly raw portfolio returns are regressed on the weekly Fama-French factor returns. This table shows the risk-adjusted returns as well as the portfolio loadings on these risk factors. The corresponding t-statistics are listed between parentheses. The regression equation is as follows:
The mean change is the average of the 40 differences in factor loadings between the ranking period and the investment period: ß ranking period – ß investment period. The 2nd and 3rd column show the change in beta after respectively a positive- and negative portfolio return.
* denotes significance at the 10% level
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momentum portfolio is significant at the 10% level. Market beta is expected to decrease when equity rises (due to the positive return) so this is line with theory.
After all the approach to estimate betas by only using the 26 weekly observations during the investment periods seems too short to draw conclusions about factor beta stability. Therefore I run regressions by putting all the positive- as well as the negative return periods together. The extra data points may improve the beta estimations. The results are shown in table 8.
Table 8. Separate regressions for positive- and negative portfolio returns.
Postive portfolio return periods Negative portfolio return periods Variable Market ß SMB ß HML ß R² Market ß SMB ß HML ß R²
P1 1.12 -0.1 -0.48 0.74 1.09 0.1 -0.04 0.46 (41,9)** (-1.95)* (-9.55)** (16.5)** (0.80) (-0.36) P10 1.24 0.20 0.36 0.72 1.00 0.19 0.23 0.50 (30.07)** (2.38)* (4.86)** (14.33)** (1.90)* (1.86)* P1-P10 -0.12 -0.14 -0.56 0.06 0.02 -0.04 -0.19 0.10 (-2,39)* (-1.02) (-5.49)** (0.26) (-1.01) (-1.23)*
This table shows the regression of weekly raw portfolio returns on the weekly Fama-French factor returns. The first column shows the estimated factor betas for the periods where the respective portfolio returns were positive. The second column shows the estimated factor betas for periods where the respective portfolio returns were negative. The corresponding t-statistics are listed between parentheses. The regression equation is as follows:
The chow test of no breakpoints for one of the equation regressors was significant with a p-value of 0,000. ** and * denote significance at the 1% and 5% levels, respectively.
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Since the conventional methods to measure the riskiness of momentum portfolios make use of static betas, this dynamic risk pattern is not incorporated in the models. Therefore the risk-adjusted profits in periods of return continuation are underestimated and the profits in periods of return reversal are underestimated. On average this flaw on both sides will be canceled out over the whole period, but for specific periods the momentum profits will be incorrectly estimated. In applications like performance measurement this might be very misleading. This finding proves the usefulness of shorter estimation periods to estimate factor loadings.
6. Conclusion
To find out whether momentum portfolios based on past six-month returns have dynamic risk loadings I made use of short time OLS regressions. Using this approach I found that no significant momentum profits were present over the period 1991-2011 for stocks of the S&P500 index. This finding is evidence in line with market efficiency. The chow test for structural breakpoints between periods of five years showed clear rejections of the null hypothesis that all portfolio risk factor loadings are stable over time. I report clear changing factor betas between the four periods of five years. The latter periods show also the highest values of R² of all the regressions. This emphasizes the usefulness of shortening the estimation period of factor betas. Fama and French (2006) report that their shortest estimation period of one year provides the highest R², indicating the model with the highest explaining power. Unfortunately, in my shortest period regressions of six months with weekly return data the SMB- and HML-betas were mostly not significant. Therefore these tests do not allow me to draw conclusions regarding the SMB- and HML risk loadings. Regarding the market beta I report the expected decrease after a positive return of the momentum portfolio.
The fact that the risk loadings vary over time makes clear that the static Fama French three-factor model is not the best way to adjust momentum portfolio returns for risk. Two developments lead to changes in factor loadings. The fact that factor returns change over time ensures that ‘past winner’ portfolios load differently per period depending on which style of investing (big-/smallcap or value/growth) provides the highest risk compensation. This was found by Grundy and Martin (2001). In this paper I show that portfolio factor loadings change as a consequence of increasing/decreasing equity values. From the results it becomes evident that betas move in different directions depending on the sign of the momentum portfolio returns. This is in line with theory. Risk declines when equity increases, ceteris paribus, as was found already by Fischer Black (1976). This reveals that factor loadings even change during investment periods.
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the return for bearing factor risk is not influencing the stock selection any more. Following their approach the relevance of changing factor risk compensations fades away.
The fact that I found no significant momentum profits in the sample period points in the direction of market efficiency. None of the regression methods showed any evidence of significant momentum profits. By shortening the estimation period I know that there were no significant momentum profits in any of the four periods of five years. The strategy of buying past winners and selling past losers does not seem to work anymore these days.
The results in this paper indicate that after controlling for risk the profits of the strategy will be higher in periods where momentum works out as it should, but lower in periods where it does not work. Because every momentum portfolio -that is held for a long enough period- will have years with positive profits and years with negative profits, the static estimation method will have a blurring impact on the evaluation of momentum profits. Therefore risk loadings of momentum portfolios should definitely be allowed to change when evaluating the profitability of this even for the layman appealing investment strategy.
In future research it might be interesting to model the changing betas in a more efficient way. The optimal estimation period should not be too short as the insignificant factor betas in the six-month regressions suggest. However, the estimation period of factor loadings should definitely not be too long, because the results in this paper made clear that this leads to inaccurate risk adjustments of momentum profits.
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