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2 Journal of Banking & Finance 35 (2011) Contents lists available at ScienceDirect Journal of Banking & Finance journal homepage: Risk adjustment and momentum sources Jun Wang a, Yangru Wu b,c, a Quantitative Analytics, Standard and Poor s, New York, NY 10041, USA b Rutgers Business School Newark and New Brunswick, Rutgers University, USA c Chinese Academy of Finance and Development, Central University of Finance and Economics, Beijing, China article info abstract Article history: Received 6 October 2009 Accepted 10 October 2010 Available online 20 October 2010 JEL classification: G10 G11 G12 We show that the conventional procedure of risk adjustment by running full-sample time-series Fama French three-factor regressions is not appropriate for momentum portfolios because the procedure fails to allow for the systematic dynamics of momentum portfolio factor loadings. We propose a simple procedure to adjust risks associated with the Fama French three factors for momentum portfolios. Using our proposed method, the Fama French three factors can explain approximately 40% of momentum profits generated by individual stocks and nearly all of momentum returns from style portfolios. Ó 2010 Elsevier B.V. All rights reserved. Keywords: Momentum Risk adjustment Market efficiency 1. Introduction The profitability of momentum strategies is well documented since the work of Jegadeesh and Titman (1993). Buying the bestperforming stocks and shorting the worst-performing ones during the past 3 12 months and holding the zero-cost portfolio for the subsequent 3 12 months can earn significant profits both in the US and international equity markets (Chan et al., 1996, 2000; Rouwenhorst, 1998; Balvers and Wu, 2006; Griffin et al., 2003). The profitability of such trading strategies is robust to sub-sample periods (Jegadeesh and Titman, 2001; Grundy and Martin, 2001). The issue under heated debate is, however, the sources of momentum. The dominant view is that momentum profits cannot be explained by popular asset pricing models, such as the capital asset pricing model (CAPM) or the Fama and French (1993) three-factor model (Fama and French, 1996; Grundy and Martin, 2001). Therefore, the stock price momentum is widely regarded as the most persistent asset pricing anomaly that poses a big challenge to the long-established efficient markets hypothesis and motivates researchers to explore behavioral explanations (Barberis et al., 1998; Daniel et al., 1998; Hong and Stein, 1999; Han and Grinblatt, Corresponding author at: Rutgers Business School Newark and New Brunswick, Rutgers University, USA. Tel.: ; fax: addresses: jun_wang@standardandpoors.com (J. Wang), yangruwu@ andromeda.rutgers.edu (Y. Wu). 2005). Other authors, nevertheless, present evidence that momentum profits are rewards for assuming additional systematic risks and thus have nothing to do with market inefficiency, providing empirical support for theoretical models that associate momentum returns with fundamental risks (Conrad and Kaul, 1998; Berk et al., 1999; Harvey and Siddique, 2000; Chordia and Shivakumar, 2002, 2006; Johnson, 2002; Lewellen and Shanken, 2002; Avramov et al., 2007; Liu et al., 2008; Sagi and Seasholes, 2007). Perhaps the most powerful evidence provided by the non-riskbased view is the inability of traditional asset pricing models to account for the momentum profitability. Adjusting momentum returns by either the CAPM or the Fama French three-factor model does not reduce the returns; instead it strengthens the raw returns in most cases. However, as Fama (1970) puts it, any test of market efficiency involves the joint hypothesis problem. The test must assume an equilibrium asset pricing model that defines normal asset returns and the rejection of the null hypothesis may be due to either market inefficiency or misspecification of the assumed equilibrium model. The joint hypothesis problem motivates some researchers to experiment with alternative asset pricing specifications used for risk adjustment of momentum returns. For example, Ahn et al. (2003) use the stochastic discount factor estimated nonparametrically from a set of industry portfolios to account for the risks associated with momentum trading strategies. Wang (2003) constructs a nonparametric pricing kernel that represents a flexible form of the Fama French three-factor model and uses the model to adjust momentum returns. Yao (2002) adopts a dynamic principal /$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi: /j.jbankfin

3 1428 J. Wang, Y. Wu / Journal of Banking & Finance 35 (2011) component method to extract latent factors from a cross-section of stock returns to account for the momentum profitability. Harvey and Siddique (2000) demonstrate that adding the conditional skewness to the Fama French three-factor model helps explain momentum. These authors find that momentum strategies no longer earn significant abnormal returns if risks are adjusted by their alternative models, suggesting that momentum profits are a compensation for assuming systematic risks. 1 This article is a new effort to unravel momentum sources in the direction of risk adjustment of momentum profits. Unlike the aforementioned recent studies, we do not pursue a new equilibrium model to adjust for risks; instead we focus on the most widely used linear Fama French three-factor model. Numerous studies establish the connection of the Fama French factors with the real fundamental risk exposures (Fama and French, 1995; Liew and Vassalou (2000); Brennan et al., 2004), so the use of this model can mitigate the potential data-mining or overfitting problems for the nonparametric or principal component techniques. We show that it is flawed to use the full-sample unconditional time-series regression of momentum portfolio returns on either excess market return or on Fama French three factors to find the risk-adjusted momentum returns because the procedure ignores the dynamic nature of the factor loadings of momentum portfolios. Specifically, if we consider the Fama French three-factor model as an appropriate equilibrium model for both individual stocks and portfolios, the winners should load much more heavily on the three factors than the losers when the factors earn positive premia on average during the ranking periods of momentum strategies and the reverse will be true when the factor premia are negative. If the factor premia are positively autocorrelated over ranking and holding periods, as is typically the case in reality, there should be some degree of positive covariation between the factor loadings of the winner-loser momentum portfolio and the contemporaneous factor premia. The conventional unconditional risk adjustment ignores the dynamic relationship between momentum portfolio factor loadings and factor premia by implicitly assuming that the factor betas are constant over time and consequently underestimating the contribution of the common risk factors to momentum profits. We propose a simple approach to allow for the dynamic nature of momentum portfolio betas by adjusting common risk factors at the individual stock level. Using our procedure, the risk-adjusted momentum returns are reduced uniformly and substantially for a variety of momentum strategies, albeit they remain statistically significant in most cases. The fact that the Fama French three factors cannot fully explain momentum profits under the proposed approach may suggest that both risk factors and behavioral factors play a role in the generation of momentum effect, but it could also be a result of the inadequacy of this model as an equilibrium model for the stocks that underlie momentum portfolios. For the latter possibility, momentum profits could be better accounted for if a more adequate model could be identified. We attempt to distinguish these two conjectures by comparing the difference in Fama French three-factor-adjusted momentum returns between individual-stock-based momentum strategies and some portfolio-based momentum strategies, or the so-called style momentum explored by Barberis and Shleifer (2003). Contrary to the individual stock momentum, the profits from most style momentum strategies become both statistically and economically insignificant after they are adjusted for risks at the individual component portfolio level. In related work, Wang (2002) implements Fama French threefactor adjustment for style momentum returns by allowing for the dynamic nature of momentum portfolio betas (he calls beta rotation ) and finds that style momentum returns can be explained away by a properly designed adjustment scheme. Although both this article and Wang s paper aim at correcting for the same flaw committed in the previous literature, we focus on individual stock momentum and so our conclusion is more general. In addition, we make an effort to establish the link between the ability of an equilibrium model to capture momentum returns and its ability to explain the returns of the momentum-underlying stocks or portfolios. The remainder of this paper is organized as follows. Section 2 describes the data. Section 3 proposes a simple risk adjustment procedure that allows for the systematic dynamics of factor loadings for momentum portfolios. This section also presents riskadjusted profitability of individual stock momentum strategies. Section 4 presents risk-adjusted returns for style momentum strategies using our risk adjustment procedure and Section 5 concludes. 2. The data This paper uses two sets of data. The first dataset includes the monthly returns for all the stocks listed in NYSE and AMEX over the period from January 1965 to December 2002, obtained from the CRSP monthly tape. We only consider all domestic primary stocks (CRSP share codes 10 and 11) and so closed-end funds, Real Estate Investment Trusts (REITs), trusts, American Depository Receipts (ADRs), and foreign stocks are excluded from the analysis. This dataset is comparable to the data used in many related studies, such as Fama and French (1996), Jegadeesh and Titman (1993, 2001), and Ahn et al. (2003). A total of 6640 stocks are used in the study. The number of stocks used in our momentum strategies for each month ranges from a minimum of 1820 to a maximum of The summary statistics for this dataset are provided in Panel AofTable 1, in which we also report the summary statistics for NYSE and AMEX equal-weighted and value-weighted market indices and the Fama French three factors. 2 The annualized mean returns for individual stocks, the equal-weighted index, and the value-weighted index are 12%, 14%, and 10.8%, respectively, during the sample period. The average first-order autocorrelation in individual stock monthly returns is negative, while the first-order autocorrelations in equal-weighted and value-weighted market indices are both positive, a result consistent with the findings of Campbell et al. (1997). We also find that the first-order autocorrelations for the Fama French three factor-mimicking portfolios are all positive and autocorrelations in the short and intermediate horizons (1 12 months) are mostly positive for the SMB portfolio and the HML portfolio, a finding that will be used in the following section. We run a time-series regression of the monthly returns for each stock in our sample on the contemporaneous Fama French three factors and the results show that on average only about 25% of crosssectional variations in monthly stock returns can be explained by the Fama French three-factor model. The other set of data we use in this article is the monthly returns for some characteristic-based stock portfolios over the period from January 1965 to December These portfolios include 30 industry portfolios based on four-digit SIC code, 10 size portfolios sorted by market capitalization of the universe of NYSE, AMEX, and NASDAQ stocks, and 25 size-b/m portfolios double-sorted by the market capitalization quintile and the book-to-market ratio quintile. The datasets are obtained from Kenneth French s website and the detailed descriptions about the construction of these 1 Studies that strive to explain momentum based on time-varying risk premia and conditioning information also include Gu and Huang (2010), Guo (2006), and Wu (2002), among others. 2 The NYSE and AMEX market indices are extracted from CRSP monthly tape and the data for Fama-French three factors are obtained from Kenneth French s website. We thank Kenneth French for generously making these data available to the public.

4 J. Wang, Y. Wu / Journal of Banking & Finance 35 (2011) Table 1 Summary statistics. Individual stocks Equal-weighted index Value-weighted index Excess market return SMB HML Panel A: Individual stocks, market indices, and Fama French three factors Mean return Standard deviation q q q q q q q q q q q q b m b SMB b HML Adjusted R Mean return Standard deviation b m b SMB b HML Adj. R 2 q 1 q 2 q 3 q 6 q 12 Panel B: Average statistics for 30 industry portfolios, 10 size portfolios, and 25 size-b/m portfolios Industry Size Size-B/M This table reports summary statistics for monthly return data of individual stocks, equal-weighted and value-weighted NYSE/AMEX market indices, Fama French three factors (Excess market returns, SMB and HML), 30 industry portfolios, 10 size portfolios, 10 B/M portfolios and 25 size-b/m portfolios. Panel A presents the average annualized mean returns in proportion, standard deviations, and 1 12 lagged autocorrelations (q 1 to q 12 ) for 6640 NYSE and AMEX-listed stocks, equal-weighted and value-weighted NYSE/AMEX market indices and Fama French three factor-mimicking portfolios. Also reported are the beta estimates and adjusted R 2 for the regression of stock returns on Fama French three factors. Panel B reports the average annualized mean returns in proportion, standard deviations, sample autocorrelations at lag 1, 2, 3, 6 and 12, Fama French three factor loadings and the associated adjusted R 2 for 30 industry portfolios, 10 size portfolios, and 25 size-b/m portfolios. The details for construction of these portfolios are provided in Kenneth French s website. The sample period is from January 1965 to December portfolios are also provided on that website. The summary statistics for these portfolio datasets are presented in Panel B of Table 1. We report annualized mean returns, standard deviations of portfolio returns, and some short-term or intermediate-term autocorrelations for each set of portfolios. We also present the average Fama French three factor loadings for each set of portfolio, along with the adjusted R 2 of the Fama French three-factor model. We observe that the Fama French three-factor model has good explanatory power for these portfolios, especially for those portfolios constructed on the basis of size and/or book-to-market ratio. The average adjusted R 2 for industry portfolios, size portfolios, and size-b/m portfolios are 65%, 96%, and 91%, respectively, an expected result that is consistent with a series of papers by Fama and French (1992, 1993, 1995, 1996). 3. Individual stock momentum and risk adjustment 3.1. Raw momentum returns and conventional risk adjustment 3 We also implement the strategies by skipping one month between sorting and holding periods and obtain similar results. We implement a variety of momentum strategies based on the combinations of ranking period L and holding period H. At each month t, we rank stocks contained in the CRSP monthly tape throughout the past L months by their cumulative returns during the past L months. We then buy $1 worth of the equal-weighted portfolio that consists of decile stocks with the highest cumulative returns (the winner) and sell short $1 worth of the equal-weighted portfolio consisting of the decile stocks with the lowest cumulative returns (the loser) and hold this zero-cost arbitrage portfolio for the next H months. We then examine the average returns during the H months for the winner, the loser, and the winner-loser arbitrage portfolio. 3 For a complete view of momentum profits, we select all the combinations of L = 3, 6, 9, 12 months and H =3, 6, 9, 12 months. To enhance the power of tests, we follow Jegadeesh and Titman (1993) by examining momentum portfolios with overlapping holding months. Specifically, for a momentum strategy with H holding periods, at any given month t, the trading strategy will hold a series of zero-cost portfolios, including the portfolio selected in the current month t and the H 1 portfolios that were selected in the past H 1 months, thus each portfolio accounting for the weight of 1/H. This means that we only revise at most 1/H of the stocks in any given month and carry over the rest from the preceding month. We show the results of momentum returns in Panel A of Table 2. We compute the t-ratios in this table as well as in subsequent tables using Newey and West (1987) heteroscedasticity and autocorrelation consistent standard errors with (H 1) lags. Our results confirm the findings of Jegadeesh and Titman (1993). All the momentum strategies, except the 3-month ranking/3-month holding strategy, earn statistically and economically significant profits. The most successful strategy is the 12-month ranking/3-month holding strategy, which generates an annual mean return of 14.2% with a t-ratio of 4.300, which is significant at the 1% level. 4 To understand whether momentum profits are a compensation for risks or a result of irrationalities, two asset pricing models are widely used for the purpose of risk adjustment: the CAPM and the Fama French three-factor model. Previous studies find that 4 Lo and MacKinlay (1990) demonstrate that momentum profit can be decomposed into the autocovariances of the factor premia, the cross-sectional autocovariances of the individual stocks, and the cross-sectional variance of the mean returns of the individual stocks. Such a decomposition will provide useful insight about the sources of momentum. However, in order to properly compute the full cross-sectional autocovariance matrices of individual stocks, all stocks must have no missing observations for the full sample. While our data set contains 6640 stocks, only 317 stocks have continuous time-series observations for the full sample. Choosing only those 317 stocks for the analysis will subject our study to severe survivorship bias. We therefore do not pursue such a decomposition.

5 1430 J. Wang, Y. Wu / Journal of Banking & Finance 35 (2011) Table 2 Profitability of individual stock momentum strategies. H =3 H =6 H =9 H =12 Mean return t-ratio Mean return t-ratio Mean return t-ratio Mean return t-ratio Panel A: Raw momentum returns L = 3 Winner Loser Winner-loser L = 6 Winner Loser Winner-loser L = 9 Winner Loser Winner-loser L = 12 Winner Loser Winner-loser Panel B: Fama French three-factor-adjusted momentum returns L = 3 Winner Loser Winner-loser L = 6 Winner Loser Winner-loser L = 9 Winner Loser Winner-loser L = 12 Winner Loser Winner-loser This table reports the annualized mean returns in proportion and t-ratios for 16 momentum strategies, distinguished by the combination of ranking period L and holding period H. As in Jegadeesh and Titman (1993), momentum portfolios are formed by purchasing the 10% (decile) stocks with the highest cumulative returns over the past L months and selling short the 10% stocks with the lowest cumulative returns over the past L months. Portfolios are then held for the subsequent H months. The data cover the period from January 1965 through December The numbers in bold highlight the momentum returns that are significant at the 5% level or better. The t-ratios in this table as well as in subsequent tables are computed using Newey and West (1987) heteroscedasticity and autocorrelation consistent standard errors with (H 1) lags. Panel A reports raw momentum return. Panel B reports the results (alphas) from full-sample time-series regressions of excess momentum returns on Fama French three factors: R e it ¼ a i þ b im ðr mt R ft Þþb is R SMB;t þ b iv R HML;t þ e i;t : neither of these two models has any explanatory power for momentum returns (see, e.g., Fama and French, 1996; Jegadeesh and Titman, 1993). When the time-series of momentum returns are regressed on either the contemporaneous excess market returns or the Fama French three factors, the resulting regression intercepts (alphas) are quite significant and in most cases even larger than the raw returns. We will report results of risk-adjusted returns using the Fama French three-factor model to adjust for risk throughout the paper. The results on CAPM risk-adjusted returns are similar and are not reported. Specifically, risk-adjusted returns (alphas) are generated from the Fama French three-factor model as follows: R e it ¼ a i þ b im ðr mt R ft Þþb is R SMB;t þ b iv R HML;t þ e i;t ð1þ where R e it is the excess return for the winner or the loser portfolio, or the returns for the zero-cost winner-loser portfolio, R mt is the return for the value-weighted CRSP market index, R ft is the 1-month T-bill rate, R SMB,t is the return on the zero-cost portfolio that buys largecapitalization stocks and sells small-capitalization stocks, and R HML,t is the return on the zero-cost portfolio that buys high book-to-market stocks and sells low book-to-market stocks. The regression slopes b im, b is, and b iv are corresponding loadings on market factor, size factor, and value factor, respectively. Results reported in Panel BofTable 2 show that momentum returns look even more abnormal in the framework of the Fama French three-factor model because the risk-adjusted returns are uniformly larger and more significant than the raw returns. Not surprisingly, this piece of evidence has always been cited in favor of non-risk-based explanation of momentum phenomenon. Our position in this article is that the results in Panel B of Table 2 should be interpreted with great caution. We will allocate the following several sections to clarify our arguments and provide evidence in favor of our claim that common risk factors, here the Fama French three factors, do have relevance to momentum returns Dynamic factor loadings of momentum portfolios In the literature, the common practice to find risk-adjusted returns is to run a time-series regression of stock/portfolio returns on the risk factor premia. A key assumption implicit in this sort of full-sample time-series regression is the constancy of the factor loadings (betas) over time for the stocks/portfolios under consideration. This may be true for individual stocks and some characteristic-based portfolios, but for momentum portfolios, this assumption is generally violated. The main reason is that the stock compositions of momentum portfolios will change considerably over time in response to movements in factor premia, leading to drastic changes in factor loadings. To see why it is so in the framework of the Fama French three-factor model, we first assume that returns for individual stocks are generated by the Fama French three risk factors. The literature has established the connection of these three factors with the fundamental risk exposures in the real economy (Fama and French, 1995; Liew and Vassalou (2000); Brennan et al., 2004). The use of the Fama French model could therefore mitigate the potential data-mining or overfitting problems for the nonparametric or principal component approaches. Since the winning and the losing decile portfolios are constructed on the basis of the cumulative returns of the universe of individual stocks during the ranking period, the compositions of these two decile extreme portfolios will depend on the average risk premia associated with the Fama French three factors during the ranking period. If the risk premium on one factor, say, the size factor, is positive during this period, the winner portfolio will generally consist of stocks

6 J. Wang, Y. Wu / Journal of Banking & Finance 35 (2011) with high loadings on the size factor and the loser portfolio will mainly consist of stocks with low size betas. Consequently, the winner has a high size beta and the loser has a low size beta. If, on the other hand, the size premium is negative during the ranking period, stocks with low size betas will mostly be picked for the winner and stocks with high size betas will be picked for the loser. As a result, the winner tends to have a low size beta and the loser tends to have a high size beta. The market factor and value factor have the same dynamic pattern. We can make two important points from the above illustration. First, the Fama French three factor loadings for momentum portfolios (the winner, the loser, and the winner-loser zero-cost portfolio) are time-varying, calling into question the conventional procedure of risk adjustment by running full-sample time-series regressions. Second, the pattern of time-variation for betas is not arbitrary, instead there is a systematic relationship between beta variations and the changes in risk premia on common factors. Specifically, the winner s/loser s beta on a factor will covary positively/ negatively with the ranking-period risk premium of this factor and consequently the beta for the zero-cost winner-loser portfolio (the winner s beta minus the loser s beta) should covary positively with the ranking-period risk premium on the corresponding factor. The systematic time-variation of factor loadings of momentum portfolios has important implications for risk adjustment of momentum profits and therefore for the attribution of momentum sources. Intuitively, if there exists some degree of positive autocorrelation for the factor premia (this is true for our sample period, see Panel A of Table 1), then the positive correlation of factor loadings of winner-loser momentum portfolio with the ranking-period factor premia will lead to the positive covariation of factor loadings with the contemporaneous risk premia, neglecting this will lead to the underestimation of the contribution of risk factors to momentum profits. We discuss this point more formally as follows. Again, we assume that stock returns are generated by the Fama French three factors, then the excess returns for the winner and the loser can be expressed as follows: R Wt ¼ a Wt þ b Wm;t ðr mt R ft Þþb Ws;t R SMB;t þ b Wv;t R HML;t þ e Wt ð2þ R Lt ¼ a Lt þ b Lm;t ðr mt R ft Þþb Ls;t R SMB;t þ b Lv;t R HML;t þ e Lt ð3þ The b on each of the three factors is the average of betas of the individual stocks that are included in the winner or loser at each point in time, hence the subscript of t to allow for the time-variation of the portfolios. The momentum profits are thus obtained by taking the difference between Eqs. (2) and (3), namely: R W L;t ¼ R Wt R Lt ¼ða Wt a Lt Þþðb Wm;t b Lm;t ÞðR mt R ft Þ þðb Ws;t b Ls;t ÞR SMB;t þðb Wv;t b Lv;t ÞR HML;t þðe Wt e Lt Þ ð4þ In conventional unconditional regressions, factor loadings are assumed to be constant over time. With this assumption and by taking unconditional expectations on both sides of (4), we get the average momentum profits over time as: EðR W L;t Þ¼Eða Wt a Lt ÞþEðb Wm;t b Lm;t ÞEðR mt R ft Þ þ Eðb Ws;t b Ls;t ÞEðR SMB;t ÞþEðb Wv;t b Lv;t ÞEðR HML;t Þ þ Eðe Wt e Lt Þ ð5þ where E is the expectation operator. The first term on the righthand side of (5), E(a Wt a Lt ), is simply the three-factor-adjusted returns reported in Panel B of Table 2 of this article and in numerous previous papers. Since the factor loadings are typically negative, the risk-adjusted returns will be higher than the average raw momentum returns, E(R W L,t ). This evidence is often cited as against the risk-based story of momentum effect. However, as we discussed above, factor loadings of the winner-loser portfolio should be positively correlated with the corresponding factor premium. In light of this, when taking expectations on both sides of Eq. (4) and using the identity E(X Y) = cov(x, Y) + E(X)E(Y), we should have the following: EðR W L;t Þ¼Eða Wt a Lt ÞþEðb Wm;t b Lm;t ÞEðR mt R ft Þ þ cov½ðb Wm;t b Lm;t Þ; ðr mt R ft ÞŠ þ Eðb Ws;t b Ls;t ÞEðR SMB;t Þþcov½ðb Ws;t b Ls;t Þ; R SMB;t Š þ Eðb Wv;t b Lv;t ÞEðR HML;t Þþcov½ðb Wv;t b Lv;t Þ; R HML;t Š þ Eðe Wt e Lt Þ ð6þ Comparing Eq. (6) with Eq. (5), we find that there are three additional covariance terms in (6) when we take into consideration the covariation of momentum portfolios factor loadings with the corresponding factor premium. Since the covariances are typically positive, it is conceivable that the excess returns obtained from conventional unconditional time-series regressions (the first term on the right-hand side of Eq. (5)) are biased upward and the contributions of common risk factors to momentum profits (including the three covariance terms) are correspondingly biased downward. A natural question to ask is: how important are the three extra covariance terms for risk adjustments? We provide results for the most profitable 12-month ranking/3-month holding period momentum portfolio. We estimate the betas for the zero-cost momentum portfolio at each time t as the average of betas for the stocks that are included in the portfolio at time t. We estimate betas for individual stocks by running 36-month rolling regressions following Grundy and Martin (2001). To check for robustness, we also estimate the betas by running 60-month rolling regressions. The results are similar and are not reported. For each month t, we run regressions of monthly excess returns for each stock during the past 36 months on the contemporaneous Fama French three factors: R i;s R fs ¼ a i þ b im ðr ms R fs Þþb is R SMB;s þ b iv R HML;s þ e i;s ; s ¼ t 36; t 35;...; t 1: ð7þ Although our argument for systematically time-varying betas for momentum portfolios is valid even in the case of the unconditional Fama French three-factor model for individual stocks, we favor the conditional version of the model and hence rolling regressions to find the component stocks beta estimates. There is a large body of evidence to support the time-variation of betas for individual stocks (see, e.g., Campbell and Vuolteenaho, 2004; Ferson and Harvey, 1999; Harvey, 1989). On the other hand, betas for the extreme decile portfolios based on past returns (the winner and the loser) are more likely to be subject to changes over time: the sharp moves in price for the stocks included in these portfolios will lead to drastic changes in financial leverage for the issuing firms and leverage is well-known to have a big impact on risks (Black, 1976). We find that, for our 12-month ranking/3-month holding winner-loser momentum portfolio, the covariance of market factor loading with the monthly market factor premium over the period from January 1965 to December 2002 is and the corresponding correlation is 1.79%. The covariances of loadings on the size factor and the value factor with the contemporaneous size and value factor premia are and , respectively, and the corresponding correlations are 8.09% and 9.74%. These three covariance terms combined contribute an approximate 5.8% annual return, which amounts to 41% of the raw return of 14.2%. However, this is not the full story. The three product terms in Eq. (6), the products of expected betas and the corresponding expected factor premia, may also make contributions to the raw momentum returns. In the next sections, we will examine risk adjustments of momentum profits by explicitly allowing for the systematic dynamics of momentum portfolio betas.

7 1432 J. Wang, Y. Wu / Journal of Banking & Finance 35 (2011) A simple procedure for risk adjustment of momentum returns We propose a simple procedure for risk adjustment of momentum returns that circumvents the difficulty of beta dynamics modeling. Our method is based on Eq. (4), where betas for the winners and the losers are computed as the averages of betas of stocks that are included in the corresponding portfolios and the three factor betas of an individual stock are estimated using 36-month rolling regressions of excess stock returns on the Fama French three factors, as in Eq. (7). For example, the market factor loading at time t can be found as: ^b ðw LÞm;t ¼ ^b Wm;t ^b Lm;t ¼ 1 N Wt XNWt i¼1 ^b im;t 1 N Lt XNLt i¼1 ^b im;t where N Wt is the number of stocks included in the winner at time t and N Lt is the number of stocks included in the loser at time t. The size and value factor loadings can be found in the same way. The risk-adjusted momentum return for month t, R adj W L;t, is then: R adj W L;t ¼ R W L;t ^b ðw LÞm;t ðr mt R ft Þ ^b ðw LÞs;t R SMB;t ^b ðw LÞv;t R HML;t The mean risk-adjusted momentum return during the full sample period is simply the time-series average of R adj W L;t. It can be easily shown that our method is equivalent to adjusting risks at the level of individual stocks that are components of the momentum portfolio. Specifically, at each month t, we run 36-month rolling regression of monthly excess returns for each stock that is included in the momentum portfolio on the contemporaneous Fama French three factors. We then use the beta estimates for each stock to find the risk-adjusted return for that stock in month t (sum of the alpha and the residual): R adj i;t ¼ R i;t R ft ^b im ðr mt R ft Þ ^b is R SMB;t ^b iv R HML;t ð10þ The risk-adjusted momentum profits for month t, R adj W L;t, is then the average of R adj i;t across all the winning stocks minus the average of R adj i;t across all the losing stocks and the risk-adjusted momentum returns during the full sample period is simply the time-series average of R adj W L;t. To distinguish our method from other risk adjustment procedures, we term it as risk adjustment at the individual stock level since it explicitly considers the individual stock composition of the momentum portfolios. ð8þ ð9þ The results of risk adjustment at the individual stock level are reported in Table 3. Comparing the results with the raw returns in Panel A of Table 2, we find that risk-adjusted momentum returns are reduced uniformly for all 16 strategies and on average the Fama French three-factor model explains about 34% of the raw momentum returns across the 16 strategies. Most of the risk-adjusted returns, however, remain significant at the 5% level (13 out of 16 cases, compared with 15 out of 16 cases for raw returns), although all the t-ratios are substantially lower. It is interesting to compare our results with several recent related studies. For example, Ahn et al. (2003) report that their nonparametric conditional benchmark can explain approximately 63% of the raw returns. Yao (2002) shows that about 32% of the raw returns can be explained away by 3 dynamic factors and 90% by 6 dynamic factors using the dynamic principal component method. Both studies use a similar sample period to ours. However, these authors use flexible asset pricing benchmarks which allow them more degrees of freedom to match the data and they find that only more than three factors can have superior explanatory power. We use the original linear Fama French three-factor model, which is economically well justified. A more striking and relevant comparison can be made with the findings of Grundy and Martin (2001). They report a 0.78% monthly raw return for the 6-month/1-month momentum strategy over the time period from 1966 to 1995 for the NYSE and AMEX-listed stocks. They also use the Fama French three-factor rolling regressions to find risk-adjusted returns, but different from our procedure, they run the regressions for the winner-loser portfolios directly. Using this method, they find that the risk-adjusted momentum return is either higher than or very close to the raw return, depending on the rolling windows used. Although they also recognize the systematic dynamics of betas for momentum portfolios, they nevertheless claim that, from their findings, recognizing the dynamic factor exposure of a momentum strategy tends to magnify, rather than explain away, its profitability. Our findings and conclusions are different from theirs, conceivably due to the fact that their method implicitly assumes that momentum portfolio factor loadings are constant over the rolling windows. It is unlikely that factor premia change little over 36 months or 60 months and therefore it is hard to justify the constant beta assumption for momentum portfolios. We compare the results from different risk adjustment procedures using our dataset for 6-month ranking/1- month holding strategy with 1-month gap between ranking and holding periods, the exact strategy considered by Grundy and Martin (2001). The annualized raw momentum return is 11.1%, the Table 3 Risk-adjusted profitability of stock momentum strategies: risk adjustment at individual stock level. H =3 H =6 H =9 H =12 Mean return t-ratio Mean return t-ratio Mean return t-ratio Mean return t-ratio L = 3 Winner Loser Winner-loser L = 6 Winner Loser Winner-loser L = 9 Winner Loser Winner-loser L = 12 Winner Loser Winner-loser This table presents the annualized risk-adjusted returns in proportion and t-ratios for 16 momentum strategies based on individual stocks. Risk adjustments are made at the level of individual stocks that are components of the winner-loser portfolios using 36-month rolling regressions. The risk-adjusted momentum returns are computed as the average Fama French three-factor-adjusted returns of the stocks included in the winners minus the average Fama French three-factor-adjusted returns of the stocks included in the losers. The sample period covers January 1965 through December The numbers in bold highlight the risk-adjusted momentum returns that are significant at the 5% level or better.

8 J. Wang, Y. Wu / Journal of Banking & Finance 35 (2011) Table 4 Profitability of momentum strategies formed by sorting past risk-adjusted returns. H =3 H =6 H =9 H =12 Mean return t-ratio Mean return t-ratio Mean return t-ratio Mean return t-ratio L = 3 Winner Loser Winner-loser L = 6 Winner Loser Winner-loser L = 9 Winner Loser Winner-loser L = 12 Winner Loser Winner-loser This table reports the annualized mean returns in proportion and t-ratios for 16 momentum strategies, distinguished by the combination of ranking period L and holding period H. In each month, we run rolling regression of each stock s excess return on the three Fama French factors using prior 36-month data and compute the risk-adjusted return (the alpha). Momentum portfolios are formed by purchasing the 10% (decile) stocks with the highest cumulative risk-adjusted returns over the past L months and selling short the 10% stocks with the lowest cumulative risk-adjusted returns over the past L months. Portfolios are then held for the subsequent H months. The data cover the period from January 1965 through December The numbers in bold highlight the momentum returns that are significant at the 5% level or better. risk-adjusted return using the conventional full-sample Fama French three factor regression is 14.9%, the risk-adjusted momentum return using the Grundy Martin procedure is 14.6%, while the risk-adjusted return with risk adjustment made at the individual stock level is 9.0%. The results demonstrate that the risk-adjusted momentum returns should reduce under the appropriate risk adjustment scheme Momentum portfolio formed by sorting risk-adjusted returns In this subsection, we check for the robustness of our procedure for risk adjustment from another angle. We investigate the importance of the firm-specific component of stock returns in driving momentum by conducting the following test. In each month, we run the rolling regression of each stock s excess return on the three Fama French factors using prior 36-month data and compute the risk-adjusted return (the alpha). Momentum portfolios are formed by purchasing the 10% (decile) stocks with the highest cumulative risk-adjusted returns over the past L months and selling short the 10% stocks with the lowest cumulative risk-adjusted returns over the same period. Portfolios are then held for the subsequent H months. Table 4 reports the profitability of these strategies. Compared to the results of Panel A of Table 2 where momentum portfolios are formed by sorting prior raw returns of stocks, we find that the excess returns of momentum portfolios formed by sorting prior risk-adjusted stock returns are uniformly lower and less significant. In particular, the excess returns for the three combinations, (L =3,H = 6), (L =6,H = 3), and (L = 12, H = 12), which are significant at the 5% level or better in Panel A of Table 2, now become statistically insignificant in Table 4. The average excess return across all 12 momentum portfolios formed by sorting prior raw stock returns is 9.00% (see Panel A of Table 2). In contrast, the average excess return across the 12 momentum portfolios formed by sorting prior risk-adjusted stock returns is 5.23%, a 42% reduction (see Table 4). This figure compares well with the corresponding average risk-adjusted return of 12 momentum strategies using our risk adjustment procedure, which is 6.12% (see Table 3). 4. Style momentum and risk adjustment In this section, we use some characteristic-based portfolios that are extensively used in prior literature to implement momentum strategies, including 30 industry portfolios, 10 portfolios sorted by market capitalization, and 25 portfolios double-sorted by size and book-to-market ratio. The procedure for implementing style momentum strategies is the same as that used for individual stock momentum strategies, except that we use portfolio data here instead of individual stock data. For industry portfolios and size-b/ M portfolios, we buy the 3 best-performing industry portfolios/ size-b/m portfolios and sell short the 3 worst-performing industry portfolios/size-b/m portfolios for the past L months and hold the zero-cost momentum portfolio for the next H months. For size portfolios, the single best-performing portfolio is bought and the single worst-performing portfolio is shorted to construct the zero-cost momentum portfolio. In either case, the winner and the loser both consist of about 10% of the portfolios. The results for style momentum profits are reported in Panel A of Table 5. For brevity, we only present momentum returns for 6-month ranking period strategies. The results confirm the existence of significant style momentum, as predicted by Barberis and Shleifer (2003) and documented by Moskowitz and Grinblatt (1999), Lewellen (2002), and Chen and DeBondt (2004). We proceed to explore the potential sources of style momentum. As a first try, we find risk-adjusted momentum returns (alphas) by running conventional full-sample regressions of monthly excess momentum returns on the contemporaneous Fama French three factors, as in Eq. (1), for the three sets of portfolios, respectively. The alpha estimates are reported in Panel B of Table 5. We find that nearly all the risk-adjusted returns obtained in this way are larger and more significant than the corresponding raw returns for all the three sets of portfolios. The findings highlight the failure of the conventional unconditional Fama French three-factor model to account for the style momentum returns, but just as in the case of individual stock momentum, this does not mean common risk factors are not relevant to momentum profits because the potential dynamic factor loadings for momentum portfolios render the use of the unconditional Fama French three-factor model problematic. To adjust for the influence of dynamic betas for style momentum portfolios, we implement risk adjustments at the component style portfolio level. Just as in the case of individual stock momentum, for each set of the portfolios, at each month t, we first run 36- month rolling regressions of the individual portfolio excess returns on the contemporaneous Fama French three factors and then use the estimated betas to adjust for returns of the individual portfolios that are included in the momentum portfolio in that month. The results are reported in Panel C of Table 5. We find that, after the beta dynamics are allowed for, the risk-adjusted industry momentum returns are uniformly reduced, compared with the raw returns reported in Panel A, but they are still statistically sig-

9 1434 J. Wang, Y. Wu / Journal of Banking & Finance 35 (2011) Table 5 Profitability of style momentum strategies. H =3 H =6 H =9 H =12 Mean return t-ratio Mean return t-ratio Mean return t-ratio Mean return t-ratio Panel A: Profitability of style momentum strategies Industry Max3-Min Size Max1-Min Size-B/M Max3-Min Panel B: Three-factor-adjusted momentum returns using the conventional time-series regressions Industry Max3-Min Size Max1-Min Size-B/M Max3-Min Panel C: Three-factor-adjusted momentum returns with risk adjustment at the component portfolios level Industry Max3-Min Size Max1-Min Size-B/M Max3-Min This table presents raw and risk-adjusted momentum returns (annualized and in proportion) based on 30 industry portfolios, 10 size portfolios and 25 size-b/m portfolios. Only the strategies with 6-month ranking period are reported. Panel A reports raw momentum returns for the three sets of portfolios. The three best-performing and three worst-performing portfolios are selected to construct the momentum portfolios for industry and size-b/m portfolios, and the best-performing portfolio and the worstperforming portfolio are selected for size momentum strategies. Panel B reports risk-adjusted momentum returns using the conventional Fama French three-factor timeseries regressions. Panel C reports risk-adjusted momentum returns using Fama French three-factor model as the benchmark and risk adjustments are made on the component individual portfolio level using 36-month rolling regressions. The sample period covers January 1965 through December The numbers in bold highlight the (risk-adjusted) momentum returns that are significant at the 5% level or better. nificant. The result reconfirms the findings made by Moskowitz and Grinblatt (1999), who find that industry momentum profits decline but remain significant after adjusting for size, book-tomarket equity, and individual stock momentum. Different from both individual stock momentum and industry momentum, however, results on the risk-adjusted momentum returns are quite impressive for the size portfolios and size-b/m portfolios. The three-factor-adjusted returns for size momentum strategies all become insignificant and three out of four even turn negative. Momentum profits for size-b/m portfolios all decline substantially and half of the strategies become insignificant. Our results are consistent with those of Lewellen (2002) and Wang (2002), who also find that the Fama French three-factor model has good explanatory power for momentum profits produced by portfolios sorted by size and/or book-to-market ratio. One reason why the Fama French three-factor model has a higher power in explaining style momentum than in explaining individual stock momentum can be because the portfolio betas which are now constituent parts of the style momentum strategy are estimated more precisely than the betas of individual securities. To summarize, Table 5, along with Table 3, highlights the sharp contrast between individual stock momentum or industry momentum on one side and the size or size-b/m portfolio momentum on the other side. After allowing for the dynamic pattern of factor loadings for momentum portfolios, the Fama French three-factor model does a good job in explaining momentum returns for some style portfolios, especially for the portfolios sorted by size and/or value features of stocks. However, the model leaves a substantial portion of momentum profits unexplained for individual stocks or industry portfolios. 5. Conclusions There is considerable evidence of momentum in stock returns and some portfolio returns. The leading view in the literature is that momentum profits are driven by firm-specific component of stock returns. This view is typically corroborated by results from conventional risk adjustments using the CAPM or the Fama French three factor time-series regressions, which indicate that the winning portfolios are no riskier than the losing portfolios and that the risk-adjusted momentum returns remain significant and are even higher than the raw returns. This piece of evidence is often cited as suggesting the non-risk-based explanations of momentum effect. We call into question the dominant view on the momentum sources by first pointing out a flaw in the conventional full-sample time-series regressions commonly used for risk adjustments of momentum returns. We find that the Fama French three factor loadings for the winner-loser momentum portfolios are time-varying, or more precisely, they are positively correlated with the contemporaneous corresponding factor premia. This regularity challenges the use of conventional unconditional Fama French three factor time-series regressions in risk adjustment of momentum portfolios in that the model is only valid on the basis of constant factor loadings. To correct for this flaw, we propose an alternative risk adjustment procedure, which is still based on the linear Fama French three-factor model but risk adjustments are made at the level of the individual stocks that are the component assets of the momentum portfolios. We find that risk-adjusted momentum returns obtained this way decline substantially compared with the raw returns, although most of them still remain statistically significant. We report that, on average, about 34% of the raw momentum returns can be accounted for by the Fama French three-factor model. We also examine the sources associated with the profitability of style momentum strategies. We find that, using our risk adjustment procedure, style momentum profits for the size and bookto-market sorted portfolios can be nearly fully explained away by the Fama French three-factor model, while a large portion of industry momentum profits is left unexplained by the model. Acknowledgements We would like to thank Ivan Brick, Hui Guo, Iftekhar Hasan, Dongcheol Kim, Sterling Yan, Feng Zhao, the editor, the anonymous referees and seminar participants at Rutgers University and at the FMA meeting for helpful comments. We thank Henry Wu for editorial assistance. Yangru Wu would also like to thank Rutgers Business School and the Whitcomb Center for Financial Services for financial support. Part of this work was completed while Wu visited the Central University of Finance and Economics. All errors and omissions remain our own. References Ahn, D., Conrad, J., Dittmar, R., Risk adjustment and trading strategies. Review of Financial Studies 16,