Style Rotation, Momentum, and Multifactor Analysis

Kevin Q. Wang¤

March, 2003

¤Joseph L. Rotman School of Management, University of Toronto. Comments are welcome!E-mail: kwang@rotman.utoronto.ca. I would like to thank Raymond Kan for helpful commentsand Eric Kirzner for providing information on ETFs. I would also like to thank the Social Sciencesand Humanities Research Council of Canada and the Connaught Fund at the University of Torontofor research support.

Style Rotation, Momentum, and Multifactor Analysis

Abstract

Investment style rotation has been an issue of long-lasting interests to practitioners. Con-

sistent with the literature, we ¯nd that a style momentum and a logit-based style rotation

strategies generate high returns. Surprisingly, the Fama-French three factor model appears to

fail completely at explaining the pro¯tability of the strategies, although the model captures

much of the variation in the underlying style returns. This paper provides an explanation

about why high risk-adjusted payo®s can be obtained for dynamic strategies even if the un-

derlying assets are perfectly priced by a factor model. For the style rotation strategies, we

show that neither pricing errors of the three factor model nor cross-style di®erences in aver-

age returns are the primary cause of the pro¯ts. The dynamic strategies induce signi¯cant

multifactor beta rotation. It is the covariances between the rotating betas and the factors

that are the most important source of payo®s to the style rotation strategies.

JEL Classi¯cation: G11; G14

Keywords: Style rotation; Momentum strategies; Risk adjustment; Beta rotation; Return

decomposition; Multifactor analysis; Equity style management

1. Introduction

The notion of equity styles has been around for decades. An equity style is simply an equity

class, a portfolio of stocks that share a common characteristic (e.g., small-cap stocks). A large

body of both academic and industry research has been devoted to style investing. In recent

years, average return di®erences between styles, such as the di®erence between growth and

value stocks, have become the focus of many investigations. For example, Rosenberg, Reid,

and Lanstein (1985), Fama and French (1992), Lakonishok, Shleifer, and Vishny (1994),

and Roll (1997), among many others, have examined the long-term relative performances

between growth, value, small-cap, and large-cap stocks. Meanwhile, the potential success

of style rotation strategies has also attracted numerous studies (e.g., Beinstein (1995), Fan

(1995), Sorensen and Lazzara (1995), Kao and Shumaker (1999), Levis and Liodakis (1999),

and Asness et al. (2000)). These studies conclude that various dynamic style strategies are

pro¯table and suggest that relative performances between equity styles are time-varying and

predictable. In addition to the attempts to explore investment strategies, the concept of

styles has also been utilized in the evaluation of managed portfolios. Most notably, Sharpe

(1992) proposes an asset class factor model for performance attribution of mutual funds.

Daniel et al. (1997), Fung and Hsieh (1997), and Ibbotson and Kaplan (2000) have extended

Sharpe's style analysis in several ways.

In this article, we provide a multifactor analysis of style momentum. Style momentum is

a combination of style rotation and momentum strategies. Speci¯cally, we consider a set of

size and book-to-market sorted portfolios that represent well-known investment styles, and

rank the style portfolios in each month according to their returns over the previous month.

A style momentum strategy buys the winner style and short-sells the loser style. This style

strategy generates signi¯cant pro¯ts. Over the period from 1960 to 2001, the average return

of the winner is, on an annualized basis, more than 16 percent higher than that of the loser.

This return di®erence is signi¯cantly larger than the di®erence between the average returns

of any two style portfolios. More surprisingly, conventional risk adjustment using the Fama

and French (1993) three factor model appears to strengthen, rather than explain, the style

momentum pro¯ts, although the model does capture much of the variation in the returns of

the underlying style portfolios.1 The Fama-French three factor regressions do not provide

any evidence that the strategy of buying the winner is any riskier than that of buying the

loser. According to the regression intercepts, the risk-adjusted return di®erence between the

1The conventional risk adjustment procedure is to run an ordinary least squares time series regressionof the strategy's excess return on the common risk factors and then take the regression intercept as therisk-adjusted return (i.e., Jensen's alpha).

1

winner and loser strategies is even larger than the raw return di®erence.

The puzzling performance of the style momentum strategy is consistent with several

explanations. First, since the Fama-French three factor model is imperfect in pricing the

style portfolios, the style momentum pro¯ts may arise from pricing errors of the model. On

one hand, the pricing errors may result from investor irrationality. For example, Barberis,

Shleifer, and Vishny (1998), Daniel, Hirshleifer, and Subrahmanyam (1998), and Hong and

Stein (1999) propose theories based on investor cognitive biases that can generate momentum

and other anomalies. On the other hand, a rational story can be built on Conrad and

Kaul's (1998) argument that the cross-sectional dispersion in average returns may explain

momentum. That is, the style momentum pro¯ts (raw returns) may be generated by the

cross-style di®erences in average returns, but the Fama-French model fails to accurately

capture the cross-section of the average returns. As a result of this mispricing, the three

factor model does not explain the performance of style momentum.

Second, the style momentum pro¯ts may be due to cycles or non-stationarity in style

returns. Barberis and Shleifer (2001) recently provided an interesting theory that irrational

trend-chasing investors can generate cyclical investment styles. Their model can generate

strong pro¯ts for style-level momentum. However, even to generate style non-stationarity, it

is not necessary to assume investor irrationality. For example, it su±ces that style returns

are generated by a risk-based model with non-stationary time-varying parameters.2 The

pricing errors explanation and the non-stationarity explanation are obviously not mutually

exclusive. Both are general enough to be consistent with either investor irrationality or

market e±ciency. Finally, a closely-related alternative explanation is that the style momen-

tum pro¯ts may be due to the time-varying risk of the style portfolios. In this story, the

style betas with respect to the Fama-French three factors change signi¯cantly over time,

although both the style betas and the style returns are strictly stationary. In other words, a

time-varying beta version of the three factor model may explain the style momentum.3

This article o®ers a di®erent explanation. As the ¯rst step, three examples are provided

to illustrate the e®ects of pricing errors. Three sets of excess returns are constructed from

the standard Fama-French three factor regressions. The ¯rst set is obtained by removing the

regression intercepts. These returns are perfectly correlated with the actual style returns,

but the cross-section of the average returns is perfectly captured by the three factor model.

The second set of excess returns is obtained by removing both the regression intercepts and

2Roll (1997) has considered such a non-stationary factor model.3Stationary factor-based risk models with time-varying factor loadings (or factor betas) are popular in

the recent literature of asset pricing. For example, Ferson and Harvey (1999) have studied a time-varyingbeta version of the Fama-French three factor model.

2

the regression residuals. These returns are perfectly captured by the constant beta version

of the Fama-French model, such that the time series regressions for these returns produce

intercepts that are all equal to zero and R2's that are all equal to one. To generate the third

set of returns, the sample means of the factors are ¯rst subtracted from the factors; then the

construction is identical to that for the second set, so that there are no pricing errors with

respect to the three factor model. In this case, there is no cross-style di®erence in average

returns.4 Using each of these three sets of returns, we replicate the style momentum strategy

to obtain the strategy's returns and run the conventional risk adjustment regressions.

The results are striking. In all three cases, returns on the style momentum strategy

are high, and so are the risk-adjusted returns. For example, for the second set of returns

described above, the winner's monthly average return is 1.27 percent higher than the loser's,

while the risk-adjusted di®erence is 1.33 percent per month.5 The results imply that neither

pricing errors of the three factor model nor cross-sectional di®erences in average returns are

the primary cause of style momentum. In addition, since the constant beta version of the

three factor model is able to generate high momentum returns in these cases, the results

suggest that a simple explanation of style momentum may exist, making complex theories

based on either time-varying betas or non-stationarity in style returns unnecessary. Finally,

the results demonstrate that the commonly used time series regression procedure for risk

adjustment is problematic in evaluating style momentum.

The key to reconciling these ¯ndings is that the style momentum strategy induces sig-

ni¯cant multifactor beta rotation. Intuitively, as the styles take turns to be the winner and

the loser over time, the factor betas of the winner and the loser rotate between the style

betas. More importantly, the rotating betas may be correlated with the factors. When the

book-to-market factor is high, for example, it is more likely that the winner (loser) will be a

style with high (low) book-to-market factor beta. If this factor is autocorrelated, the rotating

beta of the momentum strategy may thus be correlated with the future value of the factor.

Indeed, we ¯nd that the three factor betas of the style momentum strategy rotate drasti-

cally over time and that the rotating betas are correlated with the corresponding factors.

To illustrate this point analytically, we provide an example of a relative strength strategy,

which shows that momentum pro¯ts can arise in a constant beta model even if there is no

cross-style di®erence in average returns. We prove that as long as the correlation between

the rotating beta and the factor is non-zero, the risk-adjusted return of the relative strength

strategy, obtained by the conventional regression approach, is generally non-zero.

4By design, the average excess return for each style is zero in this case.5These numbers are only slightly lower than those for the momentum strategy based on actual returns.

3

We propose a simple approach to multifactor risk adjustment of style momentum. The

method takes beta rotation into account and tests whether the average conditional alpha of

the style momentum strategy is zero. We implement the method with the standard three

factor regressions for the individual styles and use a bootstrap procedure to incorporate

estimation noise associated with the regressions. The test results indicate that the average

conditional alpha of the buying-winner-selling-loser strategy is statistically di®erent from

zero; however, it is rather small (0.23%), approximately only 20 percent of the raw payo®

(1.37%) to the strategy. Next, we propose a decomposition method to analyze the sources

of pro¯ts to style momentum. Given a multifactor asset pricing model, the decomposition

method divides a dynamic strategy's average return into four components. Two of the

components are contributed by errors in pricing the style returns: the regression intercepts

and the regression residuals. The remaining two components are contributed by common risk

factors: the products of average beta values and factor risk premiums and the covariances

between the rotating betas and the factors. Our results show that it is the covariances

between the rotating betas and the factors that are the single most important source of

payo®s to style momentum.

Momentum-based trading strategies, ¯rst documented by Jegadeesh and Titman (1993),

have attracted considerable attention. Using data from 1965 to 1989, Jegadeesh and Titman

¯nd that stocks with high returns over the past three to twelve months continue to outper-

form stocks with low past returns over the same period. The pro¯tability of momentum

strategies that are constructed by buying past winners and short-selling past losers appears

to be surprisingly robust. For example, Rouwenhorst (1998) shows that there also exist

signi¯cant pro¯ts to individual stock momentum strategies in twelve European countries.

Chan, Jegadeesh, and Lakonishok (1996) ¯nd that although short-term return continuation

is somewhat related to under-reaction to earnings information, stock price momentum is

not subsumed by momentum in earnings. Recently, Jegadeesh and Titman (2001) provide

evidence that momentum pro¯ts continued in the 1990s, suggesting that their initial ¯ndings

are not a result of data mining.

In addition to the robustness of momentum pro¯ts, the in°uential ¯nding of Fama and

French (1996) has fueled a fast growing literature on the anomaly. Fama and French ¯nd

that momentum is the only CAPM-related anomaly unexplained by the Fama-French three

factor model. According to the three factor regression results, the winner portfolio is not

riskier than the loser portfolio, suggesting that the winner's average return should not exceed

that of the loser. In other words, instead of explaining momentum, the three factor model

4

strengthens it.6 The risk adjustment result is so puzzling that Fama (1998) describes the

momentum e®ect as an anomaly that is \above suspicion."

Researchers are divided on the issue of how to explain the risk-adjusted pro¯tability

of momentum strategies. Many have been tempted to conjecture that momentum pro¯ts

result from market ine±ciency. Barberis, Shleifer, and Vishny (1998), Daniel, Hirshleifer,

and Subrahmanyam (1998), and Hong and Stein (1999) propose behavioral theories that

attribute momentum to investor cognitive biases. Hong, Lim, and Stein (2000), Lee and

Swaminathan (2000), and Jegadeesh and Titman (2001) provide evidence consistent with

the behavioral models. In contrast, several authors have provided e±cient-markets-based

explanations of the momentum anomaly. Conrad and Kaul (1998) show that cross-sectional

dispersion in average stock returns can generate momentum. Berk, Green, and Naik (1999)

provide a theoretical model where the cross-sectional dispersion in risk and expected returns

generates momentum pro¯ts. Chordia and Shivakumar (2002) conduct an empirical study

and conclude that momentum pro¯ts can be explained by a set of lagged macroeconomic

variables. Johnson (2002) shows that a single-¯rm model with a standard pricing kernel can

generate momentum when expected dividend growth rates vary over time.

None of these papers, however, explains the puzzle of why the Fama-French three factor

model, so successful with numerous other anomalies, completely fails to capture momentum.

In this regard, our results are intriguing. While style-level momentum is as pro¯table as

individual stock momentum, conventional risk adjustment using the Fama-French model

completely fails to explain the pro¯tability of style momentum. This is even more puzzling

than in the case of individual stock momentum, since it is well-known that the three factor

model does a very good job of capturing returns on the size and book-to-market portfolios.

Our results show that the failure of the Fama-French model to explain style momentum

is primarily due to a °aw in the conventional risk adjustment method. The conventional

method ignores multifactor beta rotation of style momentum. Once this is adjusted for, the

three factor model can account for most of the style momentum pro¯ts.

Style rotation includes a rich array of dynamic style-based strategies. Style momentum

is only a special case. With the rapid development of exchange-traded funds (ETFs), style

rotation strategies have become an attractive alternative to dynamic strategies based on

individual stocks.7 For instance, to short-sell three hundred loser stocks in an individual

6See Haugen and Baker (1996) and Jegadeesh and Titman (2001) for similar results.7For example, Atkinson and Green (2001), Wiandt and McClatchy (2002), and Mazzilli et al. (2002)

explain in detail why ETFs are regarded as one of the most exciting new investment vehicles. In particular,the authors point out that ETFs provide a convenient and cost-e®ective tool to implement style rotationstrategies.

5

stock momentum strategy is far more intimidating than to short-sell an ETF that represents

a loser style. In this article, we contribute a new risk adjustment approach and a new

decomposition method, both of which are simple to implement and are applicable to general

style rotation strategies. To further demonstrate the methodological contribution to the

growing literature on style rotation, we analyze a style timing strategy built on a three factor

logit model. This strategy uses the logit model to predict relative style performance and

constructs a rotation procedure accordingly. We ¯nd that rotation based on the predictions

from the three factor logit model generates signi¯cant pro¯ts. Again, due to ignorance of

beta rotation, the conventional risk adjustment method fails to explain the pro¯ts. Our

proposed approach gives rise to a completely di®erent conclusion.

Finally, it is noteworthy to contrast our multifactor analysis of style momentum with

several recent empirical studies that recognize the time-varying factor exposure of momentum

strategies. In the case of individual stock momentum, Grundy and Martin (2001) emphasize

that momentum strategies induce time-varying factor betas. However, we ¯nd that when

applied to style momentum, their proposed regression approach does not e®ectively account

for the e®ects of beta rotation. Lewellen (2002) and Wang (2002) have studied momentum

based on size and book-to-market portfolios. Lewellen's focus is on cross-serial covariances

between returns and implications of the negative average of the autocovariances that he

¯nds.8 Wang focuses on a new asset pricing test constructed with a nonparametric pricing

kernel that represents a °exible form of the Fama-French three factor model. While both

Lewellen and Wang recognize the time-varying beta feature of style momentum, neither of

them discusses the issue of cross-style beta rotation in any detail.

The article is organized as follows. Section 2 presents the style portfolios, data, returns

of style momentum, and risk adjustment results using the conventional regression method.

Section 3 describes the three cases on e®ects of pricing errors of the Fama-French three factor

model. It also includes the example of the relative strength strategy, empirical results on

multifactor beta rotation, and a look at the Grundy-Martin regressions. Section 4 proposes

the risk adjustment approach and the return decomposition method that incorporate beta

rotation. It includes empirical results from the application of these methods to style momen-

tum. Section 4 also presents results for strategies based on di®erent formation and holding

periods. Section 5 describes the style rotation strategy built on the three factor logit model

and reports empirical results. The article concludes in Section 6.

8Lewellen has considered portfolios formed by sorting the residuals of the Fama-French three factorregressions. However, it is unclear whether these portfolios can be used to infer risk-adjusted pro¯ts tomomentum strategies, because the portfolio weights based on the residuals are di®erent from those based onthe style returns.

6

2. Pro¯tability of Style Momentum

2.1. Style Portfolios

We examine a momentum investment strategy that rotates among nine style portfolios.

Speci¯cally, in any given month, the strategy selects a winner and a loser on the basis of

returns over the previous month. The winner (loser) is the style portfolio that has the highest

(lowest) return over the previous month among the nine style portfolios. We focus on the

momentum strategy that buys the winner style of last month and short-sells the loser style

of last month. We also look at the strategy that buys only the winner and the strategy that

buys only the loser.

The nine portfolios are selected from Fama and French's (1993) twenty-¯ve value-weighted

size and book-to-market (BE/ME) portfolios that are double-sorted by ¯ve size quintiles and

¯ve book-to-market quintiles. The nine size-BE/ME portfolios are chosen to represent nine

di®erent investment styles:

Small-Cap Growth Small-Cap Neutral Small-Cap Value(SZ1-BM1) (SZ1-BM3) (SZ1-BM5)

Mid-Cap Growth Mid-Cap Neutral Mid-Cap Value(SZ3-BM1) (SZ3-BM3) (SZ3-BM5)

Large-Cap Growth Large-Cap Neutral Large-Cap Value(SZ5-BM1) (SZ5-BM3) (SZ5-BM5)

where (SZ1, SZ3, SZ5) and (BM1, BM3, BM5) are three out of the ¯ve size quintiles and

three out of the ¯ve book-to-market quintiles, respectively. See Fama and French (1993) for

details on the portfolio construction.

The investment styles represented by these portfolios are well-known in practice. As

discussed by Mazzilli et al. (2002), for example, three S&P/BARRA indexes: S&P 500,

Mid-Cap 400, and Small-Cap 600, which represent di®erent size styles, are sorted by book-

to-market ratios to create additional six style indexes: S&P 500 Growth, S&P 500 Value,

Mid-Cap 400 Growth, Mid-Cap 400 Value, Small-Cap 600 Growth, and Small-Cap 600 Value.

7

The S&P/BARRA style indexes are widely accepted among practitioners. Exchange-traded

funds (ETFs) are available on all of these style indexes. Another example is the recent

study of Levis and Liodakis (1999) on pro¯tability of style rotation strategies in the United

Kingdom. Following the procedure of Fama and French (1993), the authors construct nine

style indexes that are stock portfolios double-sorted by size and BE/ME.

Summary statistics of the nine size-BE/ME portfolios and the Fama and French (1993)

three factors are reported in Table 1. The data consist of monthly observations for the sample

period from January 1960 to December 2001.9 The nine style portfolios exhibit signi¯cant

dispersion in average returns, ranging from an average monthly excess return of 0.28 percent

for the small-growth style to 1.06 percent for the large-value style. Table 1 also reports the

regressions of the style portfolios' excess returns on the three factors of Fama and French.

The three factors capture strong common variation in the returns of the style portfolios. The

R2 is high for each of the regressions, ranging from 0.81 to 0.95. The three factor betas exhibit

patterns that are well known.10 The market factor betas (bi, i = 1; ¢ ¢ ¢ ; n) are relatively °at,varying between 0.93 and 1.10. In contrast, the size factor betas (si, i = 1; ¢ ¢ ¢ ; n) and thebook-to-market factor betas (hi, i = 1; ¢ ¢ ¢ ; n) are more disperse, with cross-style di®erencesclearly related to size and BE/ME, respectively. Finally, although most of the regression

intercepts (®i, i = 1; ¢ ¢ ¢ ; n) seem small, three of them have t-statistics above 2 in absolute

value, and they are not small relative to the average excess returns on the styles.

2.2. Returns of Style Momentum

Momentum in the style portfolios seems to be as strong as that in individual stocks. For

the 1960-2001 period, as reported in Table 2, the strategy of buying the winner style earned

an average monthly excess return of 1:21 percent. In contrast, the strategy of buying the

loser style has an average monthly excess return of ¡0:16 percent. Thus, on average, thewinner strategy outperforms the loser strategy by 1.37 percent per month, or 16.44 percent

per year. This means that style momentum is as pro¯table as individual stock momentum in

terms of raw pro¯ts. For example, Jegadeesh and Titman (1993) ¯nd a similar magnitude of

di®erences between the returns to their portfolios of winners and losers that are constructed

from individual stocks. Next, to check robustness, we divide the sample into the 1960-1980

9We choose the sample period since most studies on momentum and the Fama-French model use samplesfrom the early 1960s. As a robustness check, we have veri¯ed that the results for the 1932-2001 period areindeed similar. Data on some of the size-BE/ME portfolios before 1932 are missing. We conjecture thatearlier data on the portfolios may be much less reliable. All the data are provided on French's web site.10In this article, we refer to the regression slope on each factor as the beta with respect to that factor.

8

and 1981-2001 periods. In both periods, the winner strategy signi¯cantly outperforms the

loser strategy. The t-statistics reveal that the average return di®erence between the winner

and the loser is statistically signi¯cant in all of the three time periods.

Using the conventional risk adjustment method (described in footnote 1), the Fama-

French three factor model strengthens style momentum. We compute the risk-adjusted

returns of style momentum by applying the standard regression procedure to the Fama-

French three factor model. The results are reported in Table 2. According to the t-statistics,

the risk-adjusted return (or the regression intercept) is statistically signi¯cant in every case.

For the 1960-2001 period, the winner strategy has a risk-adjusted return of 0:64 percent,

while the loser's risk-adjusted return is ¡0:83 percent. The market and size factor betasof the winner are lower than those of the loser, while the book-to-market betas are similar.

The model predicts an average excess return of 0.57 percent for the winner strategy, lower

than the average excess return of 0.67 percent predicted for the loser.11 In other words,

according to this commonly applied procedure, the winner strategy is not riskier than the

loser strategy. Therefore, the risk-adjusted return di®erence (1.47 percent) between the

winner and the loser strategies is even higher than the raw return di®erence (1.37 percent).

The results from the other two periods lead to the same conclusion.

These results are quite puzzling. First, the winner-loser return di®erence is much larger

than the di®erences between average returns of the styles. Intuitively, this makes it di±cult to

imagine that the style momentum phenomenon is generated by the cross-sectional dispersion

in the average style returns. Second, the risk-adjusted pro¯ts to style momentum are higher

than the raw pro¯ts, even though the Fama-French model does a good job of capturing

the variation in the style returns. Consistent with Fama and French (1996), Jegadeesh

and Titman (2001), and Grundy and Martin (2001), this result suggests that none of the

momentum pro¯ts can be attributed to compensation for risk. Finally, factor models have

long shaped the way ¯nancial economists and practitioners think about risk. To be priced

by the Fama-French model, the momentum payo® (or the winner-loser return di®erence)

should covary with the common factors and at least one of the multifactor betas should be

signi¯cantly positive. According to the three factor regressions, however, the winner-loser

return di®erence over time does not signi¯cantly covary with the size and book-to-market

factors. The market beta is the only statistically signi¯cant beta, but it is negative.

11For an investment strategy, the average excess return predicted by a factor model is the di®erencebetween the sample average excess return and the alpha.

9

3. Multifactor Risk Adjustment

3.1. Have We Got It Wrong?

Intuitively, it is tempting to conjecture that pricing errors of the Fama-French three factor

model are the main cause of the high risk-adjusted payo® to style momentum. To examine

this possibility, we start with the three factor regression equations for style portfolios:

rit = ®i + biRMRFt + siSMBt + hiHMLt + "it; (1)

where rit is the excess return on the i-th style portfolio, for i = 1; ¢ ¢ ¢ ; n; RMRFt, SMBt, andHMLt are time-t values of the market factor (excess return on the market), the size factor,

and the book-to-market factor of Fama and French, respectively.

If the pricing errors of the three factor model are responsible for style momentum, the

risk-adjusted return must come from either the average return errors (®i's) or the regression

residuals ("it's) or both. In general, the expected return of a dynamic style-based strategy

will be entirely attributed to the three factors if

®i = 0 and Et¡1("it) = 0; (2)

for i = 1; ¢ ¢ ¢ ; n, where Et¡1 denotes the expectation conditional on information availableup to time t¡ 1. If the conditions in (2) are satis¯ed, the three factor model can perfectlyexplain the expected return on any style rotation strategy, including style momentum. To

illustrate, suppose that a dynamic strategy has portfolio weights wit¡1, for i = 1; ¢ ¢ ¢ ; n. By(1), a portion of the excess return on the strategy is due to

Pni=1wit¡1®i and

Pni=1wit¡1"it.

The conditions in (2) guarantee thatPn

i=1wit¡1®i = 0 and E(Pn

i=1wit¡1"it) = 0.

We design three experiments to check what happens if we \remove" the pricing errors.

The results, based on standard time series regressions of (1), are presented in three cases in

Table 3. In the ¯rst case, the excess returns on the style portfolios are determined by the

¯rst of the following equations (i.e., equation (3)):

rit = biRMRFt + siSMBt + hiHMLt + "it; (3)

rit = biRMRFt + siSMBt + hiHMLt; (4)

rit = bi(RMRFt ¡ RMRF) + si(SMBt ¡ SMB) + hi(HMLt ¡HML); (5)

where bi, si, and hi are the regression slope estimates, and "it is the ¯tted regression residual.

In this case, the excess return rit is perfectly correlated with the actual excess return rit, but

10

the three factor regression for rit has a zero intercept. Apparently, this is designed to check

the e®ects of removing the alphas (i.e., ®i = 0). The second case goes one step further. The

excess returns on the style portfolios are determined by (4), where the residuals ("it's) are

also removed. This is a case in which both conditions in (2) hold. By construction, the three

factor model perfectly explains these returns, such that the three factor regression intercepts

are all equal to zero and the R2's are all equal to one. In the last case, we adjust for the

cross-sectional dispersion in average returns of the style portfolios. The excess returns are

determined by (5), where each of the three factors is replaced by the deviation from its mean.

Thus, the average style returns are all equal to the riskfree rate.

Using each of the three sets of returns, we replicate momentum strategies, measure mo-

mentum returns, and obtain risk-adjusted returns by applying the conventional three factor

time series regressions. The results are impressive, as both the raw and risk-adjusted payo®s

to the strategy of buying the winner style and selling the loser style are high. The average

return di®erence between the winner and the loser strategies is statistically signi¯cant in

each case, with an estimated value of 1.27 percent in the ¯rst two cases and 1.22 percent in

the last case. The more striking ¯nding is that the risk-adjusted return di®erence between

the winner and the loser is not only statistically signi¯cant but also very close to the average

return di®erence in each of the three cases. In other words, the conventional risk adjustment

method gives rise to the conclusion that using the three factor model, virtually none of the

pro¯ts to style momentum can be attributed to compensation for risk, even when the model

captures perfectly the returns on the underlying style portfolios! The results clearly indicate

that the commonly-used risk adjustment approach is problematic in the evaluation of style

momentum.

3.2. An Example

Why does the conventional risk adjustment method fail? In particular, why does it fail in the

second and third cases described above, even if the three factor model explains 100 percent

of the variation in the style returns and completely captures the cross-section of the average

style returns?

We present a simple example to illustrate why the conventional approach is problematic.

We use a relative strength strategy of Lo and MacKinlay (1990) that has the following

portfolio weights:

wit¡1 =1

n(rit¡1 ¡ ¹rt¡1); (6)

for i = 1; ¢ ¢ ¢ ; n, where rit¡1 is the excess return of the style i and ¹rt¡1 is the excess return

11

on the equal-weighted portfolio. The relative strength strategy, though not identical to the

buying-winner-selling-loser strategy of Jegadeesh and Titman (1993), is technically conve-

nient since the weights are linear in the returns. For this reason, it has appeared in numerous

articles on momentum. To our knowledge, however, no one has analyzed the risk adjustment

regression method with the relative strength strategy.

To ease exposition, all of the styles' excess returns are assumed to be perfectly captured

by a constant single beta model

rit = ¯ift;

where ¯i is the beta of style i, for i = 1; ¢ ¢ ¢ ; n, and ft is a factor that is normally distributedwith mean ¹ and variance ¾2. Assume that the factor ft is autocorrelated and let ½ denote

the ¯rst autocorrelation.12

The excess return on the relative strength strategy is

rpt = ¯pt¡1ft;

where rpt =Pn

i=1wit¡1rit and ¯pt¡1 =Pn

i=1wit¡1¯i. It should be noted that the beta ¯pt¡1of the strategy is not constant over time, even though the excess return rit is determined by

the constant beta model.

In the conventional risk adjustment procedure, one runs an ordinary least squares regres-

sion of rpt on ft and then takes the regression intercept as the risk-adjusted return of the

relative strength strategy. The regression estimates of the intercept (Jensen's alpha) and the

slope converge to ap and bp that minimize the expected value of the squared error:

Ejrpt ¡ ap ¡ bpftj2: (7)

In Appendix A, we show that

ap = cov(¯pt¡1; ft)µ1¡ ¹

2

¾2

¶; (8)

bp =1

n

nXi=1

(¯i ¡ ¹)2(1 + ½)¹; (9)

12This is consistent with perfectly e±cient markets, as autocorrelation in a common risk factor may arisefrom time-variation in the factor risk premium (e.g., Fama and French (1989) and Ferson and Harvey (1991)).In practice, however, autocorrelation in the return of a factor mimicking portfolio may be contaminated byproxy errors (i.e., Roll's critique) and non-synchronous trading (e.g., Lo and MacKinlay (1990)). For theFama-French model, it is di±cult to disentangle empirically among the causes, as autocorrelation in thefactors is fairly low.

12

and

cov(¯pt¡1; ft) =1

n

nXi=1

(¯i ¡ ¹)2¾2½: (10)

These expressions shed light on the conventional risk adjustment method. In particular,

equation (8) shows that the covariance between the strategy's beta and the factor contributes

to a non-zero regression intercept.13 Let's consider the intuition behind why the strategy's

beta may be correlated with the factor. When ft¡1 is positive (negative), the winner is thestyle that has the highest (lowest) beta, while the loser is the one that has the lowest (highest)

beta.14 Therefore, the strategy's beta ¯pt¡1 is changing over time, and correlated with ft¡1.If ft is autocorrelated, or equivalently, if ft¡1 is correlated with the conditional risk premiumEt¡1(ft), the beta of the dynamic strategy may be correlated with ft. Equation (10) showsthat as long as the factor ft is autocorrelated and there is cross-sectional dispersion in betas,

the covariance between the strategy's beta ¯pt¡1 and the factor ft will di®er from zero. Thisexample illustrates that the conventional risk adjustment method is °awed. Even though

the factor model completely captures the n style returns, the risk-adjusted return or the

regression intercept of the relative strength strategy is generally non-zero.

In this example of the relative strength strategy, as equation (9) shows, the regression

factor loading bp is determined by the cross-sectional variance of betas and the unconditional

risk premium ¹. The model's predicted return on the strategy, bpE(ft), is always non-

negative. As a result, the model can still explain a portion of the expected return on the

relative strength strategy unless ¹ = 0. Thus, in this example, the strategy's expected return

is always higher than ap (when ¹ 6= 0 and ½ 6= ¡1), although the di®erence can be small.The strategy's expected return is identical to the intercept ap when ¹ = 0.

3.3. Multifactor Beta Rotation

The constant beta version of the Fama-French three factor model is widely applied. We

focus on this simple version to demonstrate multifactor beta rotation associated with style

momentum. Assuming that the three factor regression equation (1) and the conditions in

(2) hold, the excess return on any dynamic strategy can be expressed as

rpt = ®pt¡1 + bpt¡1RMRFt + spt¡1SMBt + hpt¡1HMLt + "pt;

13Typically, ¹2

¾2 6= 1. For example, as one can verify by Table 1, ¹2 is small relative to ¾2 for any of theFama-French three factors.14If there is an error term in this factor model, the intuition is still valid. When ft¡1 is positive and very

high, the winner tends to be a style with a high beta, while the loser tends to be a style with a low beta.

13

where rpt =Pn

i=1wit¡1rit, ®pt¡1 =Pn

i=1wit¡1®i = 0, "pt =Pn

i=1wit¡1"it, and in particular,the three factor betas of the strategy are

bpt¡1 =nXi=1

wit¡1bi; (11)

spt¡1 =nXi=1

wit¡1si; (12)

hpt¡1 =nXi=1

wit¡1hi: (13)

Our construction of momentum portfolios follows that of Jegadeesh and Titman (1993), such

that the portfolio weights are di®erent from those of the relative strength strategy. In this

subsection, we consider both the winner strategy and the loser strategy. The winner strategy

has the following weights

wit¡1 =

8<: 1 if rit¡1 = max1·j·n rjt¡1,

0 otherwise,

for i = 1; ¢ ¢ ¢ ; n. In a similar manner, the weights for the strategy of buying the loser styleare de¯ned by wit¡1 = 1 if rit¡1 = min1·j·n rjt¡1 and wit¡1 = 0 otherwise, for i = 1; ¢ ¢ ¢ ; n.A dynamic strategy may induce signi¯cant beta rotation over time. For example, consider

the winner strategy de¯ned above. Let the style betas be estimated by the standard time

series regressions.15 If the small-growth style (SZ1-BM1) is the winner for a particular month,

for example, the size and book-to-market factor betas of the winner strategy for that month

are 1:42 and ¡0:26, respectively. However, if the large-value style (SZ5-BM5) turns out tobe the winner next month, the winner's size and book-to-market factor betas become ¡0:08and 0:86, respectively! Hence, even if the style returns are generated by the constant beta

model, the return of a dynamic strategy is determined by a conditional model, such that the

betas of the strategy may vary signi¯cantly over time.

Figures 1, 2, and 3 plot the three factor betas over time for the winner and loser strategies.

As di®erent styles take turns being the winner and the loser over time, the three factor betas

of the strategies rotate among the betas of the nine styles. The ¯gures vividly show that the

dynamic strategies are associated with signi¯cant beta rotation. Table 4 presents descriptive

statistics of the rotating betas. The winner and the loser have the same range (maximum

and minimum values) for each of the three betas. The average betas are not very di®erent

15See Table 1 for the three factor beta estimates for the style portfolios.

14

between the winner and the loser. The size and book-to-market factor betas of both the

winner and the loser have similar standard deviations. The autocorrelation coe±cients of all

the betas are weak: they indicate that the betas jump drastically over time with little serial

dependence (as illustrated by the ¯gures). Most importantly, all the covariances between

the rotating betas and the corresponding factors are positive for the winner strategy, and all

are negative for the loser strategy.

Multifactor beta rotation can give rise to signi¯cant momentum pro¯ts. We revisit the

third case of Section 3.1 (or case (iii) of Table 3) to highlight this point. In this case, the

three factors have zero mean by construction, and hence, the average raw momentum return

is identical to the risk-adjusted return obtained by the conventional method. By equation

(5), the average return on a momentum strategy (or any dynamic strategy) can be expressed

as:

1

T

TXt=1

rpt =1

T

TXt=1

bpt¡1(RMRFt ¡ RMRF) + 1

T

TXt=1

spt¡1(SMBt ¡ SMB)

+1

T

TXt=1

hpt¡1(HMLt ¡HML)

=1

T

TXt=1

(bpt¡1 ¡ ¹bp)(RMRFt ¡ RMRF) + 1

T

TXt=1

(spt¡1 ¡ ¹sp)(SMBt ¡ SMB)

+1

T

TXt=1

(hpt¡1 ¡ ¹hp)(HMLt ¡ HML)

= cov(bpt¡1;RMRFt) + cov(spt¡1; SMBt) + cov(hpt¡1;HMLt):

This clearly shows that the momentum payo® is the sum of the covariances between the

rotating betas and the factors. Neither pricing errors of the three factor model nor cross-

sectional di®erences in average style returns exist in this case. The covariances are the only

source of the high payo® to the buying-winner-selling-loser strategy.

3.4. Grundy-Martin Regressions

Grundy and Martin (2001) were the ¯rst to emphasize the time-varying beta feature of

momentum strategies. They have convincingly shown that momentum strategies based on

individual stocks are associated with time-varying factor exposure in accordance with the

performance of common risk factors during the formation period. Grundy and Martin have

considered a simple regression method to adjust for this dynamic exposure. The regression

15

approach allows each of the three Fama-French factor betas to take three possible values,

depending on the factor's return over the formation interval. Applying it to our context, the

regression equation takes the following form:

rpt = ®p + bdownD1;downt¡1 RMRFt + b°atD

1;°att¡1 RMRFt + bupD

1;upt¡1 RMRFt

+sdownD2;downt¡1 SMBt + s°atD

2;°att¡1 SMBt + supD

2;upt¡1 SMBt

+hdownD3;downt¡1 HMLt + h°atD

3;°att¡1 HMLt + hupD

3;upt¡1 HMLt + "pt; (14)

where each of the dummy variables is de¯ned by whether the factor's return over the forma-

tion month is at least 1 standard deviation below its mean, within 1 standard deviation of

its mean, or at least 1 standard deviation above its mean, respectively. For example, for the

market factor,

D1;±t¡1 =

8<: 1 if RMRFt¡1 is of type ±,

0 otherwise,

where RMRFt¡1 is of type \down," \°at," or \up," if RMRFt¡1 is at least 1 standarddeviation below its mean, within 1 standard deviation from its mean, or at least 1 standard

deviation above its mean, respectively. The dummy variables D2;±t¡1 and D

3;±t¡1 are de¯ned in

a similar way for the SMB and HML factors.

We run the Grundy-Martin regressions for four sets of returns. The ¯rst set is the actual

returns on the nine style portfolios. The other three sets of returns correspond to the three

cases of Section 3.1, given by (3), (4), and (5), respectively. The results are presented in

Table 5. For brevity, we only report the regression intercepts, which are the risk-adjusted

returns according to the regression method.

The intercepts of the Grundy-Martin regressions are di®erent from those of the conven-

tional three factor regressions reported in Table 2 and Table 3. However, the intercepts for

the buying-winner-selling-loser strategy (W¡L) remain quite large and all are statisticallysigni¯cant according to the t-statistics. The intercept of the Grundy-Martin regression for

the actual returns is 1.06 percent, compared to the raw payo® of 1.37 percent. For the

remaining three sets of style returns constructed in Section 3.1, the intercepts are slightly

above one percent per month in the ¯rst and second sets, but lower in the last case. Recall

that we removed the pricing errors of the Fama-French model for the style portfolios in the

three cases. Speci¯cally, in the second and third cases (i.e., (ii) and (iii) of Table 5), the style

returns are entirely captured by the three factors, so that the momentum pro¯ts should be

completely attributed to the three common factors. Therefore, the results of Table 5 show

that the Grundy-Martin regression method is so imprecise that it is inadequate to adjust for

the risk associated with the style momentum strategy.

16

4. Sources of Pro¯ts to Style Momentum

4.1. Risk Adjustment for Style Momentum

We propose a simple risk adjustment approach to incorporate the e®ects of beta rotation.

This method can be applied to any style rotation strategy.

Suppose that excess returns on the style portfolios have the following factor structure

rit = ®i + ¯ift + "it; (15)

where ¯i is a 1£ k vector of betas, for i = 1; ¢ ¢ ¢ ; n, and ft is a k £ 1 vector of common riskfactors. The coe±cients in (15) are assumed to be constant over time.16 For example, the

widely applied constant beta version of the Fama-French three factor model presented in (1)

is a special case of (15). As pointed out in Section 3.1, pro¯ts to any dynamic style strategy

can be completely attributed to the factors if both the intercepts and the conditional mean

of the regression errors are equal to zero.

To be precise, suppose that the portfolio weights of a dynamic strategy are wit¡1 fori = 1; ¢ ¢ ¢ ; n. Then, if ®i = 0 and Et¡1("it) = 0, the excess return of the strategy is

rpt = ¯pt¡1ft + "pt;

where rpt =Pn

i=1wit¡1rit, ¯pt¡1 =Pn

i=1wit¡1¯i, and "pt =Pn

i=1wit¡1"it such that

Et¡1("pt) = 0:

In other words, the strategy's return follows a conditional factor model, where the beta

vector ¯pt¡1 is time-varying because the weights wit¡1 change over time.

The part of the strategy's return that is not attributable to the factors, which we call

the adjusted return, is

ARt ´ rpt ¡ ¯pt¡1ft: (16)

A natural and testable implication for the dynamic strategy is

E(ARt) = 0;

16One can view (15) as a special case of the following conditional structure

rit = ®it¡1 + ¯it¡1ft + "it;

where if the factor model holds, ®it¡1 = 0 and Et¡1("it) = 0. Our set-up in (15) is the constant beta versionof this conditional model.

17

since

E(ARt) = E[Et¡1(ARt)] = E[Et¡1("pt)] = 0:

Intuitively, to test whether E(ARt) = 0 is to test whether the average conditional alpha

of the dynamic strategy is zero. It should be noted that our approach is di®erent from

any method that examines portfolios formed by sorting regression intercepts or residuals.

For example, Lewellen (2002) has considered portfolios that are formed by sorting the three

factor regression residuals. These portfolios have di®erent weights from the portfolios that

are created by sorting style returns, and thus it is unclear whether they can serve as the

basis for drawing inferences about the average alpha of the momentum strategy.

As the winner and the loser rotate between the styles, the volatility of the adjusted return

ARt de¯ned by (16) also rotates, because the standard deviations of the regression residuals

"it can di®er signi¯cantly across styles. For example, Table 1 shows that the standard

deviations of the three factor regression residuals for the nine style portfolios range from

1.24 to 2.51. A simple method to adjust for the heteroscedasticity is to standardize the

adjusted return17

SARt ´rpt ¡ ¯pt¡1ft

¾pt¡1; (17)

where ¾pt¡1 is the time-(t ¡ 1) conditional standard deviation of "pt. The test based onSARt may be statistically more e±cient, but the test based on ARt is economically easier

to interpret. We implement both of them in empirical tests. In each case, we start with the

three factor regressions reported in Table 1. Using the estimates of the three factor betas

and the residual standard deviations, we obtain ARt and SARt for the winner (W), the loser

(L), and the buying-winner-selling-loser (W¡L) strategies.For both time series ARt and SARt, we compute the standard t-statistics to draw infer-

ences about their sample means. While they may be informative, these t-statistics do not

incorporate estimation errors from the three factor regressions for the style portfolios. To

cope with this issue, we implement a bootstrap procedure to obtain p-values that incorporate

the estimation errors. Appendix B provides a description of the bootstrap method.

Our approach e®ectively incorporates multifactor beta rotation. Table 6 shows that our

tests produce results that are drastically di®erent from those of the common OLS regression

procedure reported in Table 2. For example, for the sample period from January 1960 to

December 2001, the winner strategy has an impressive average excess return of 1.23 percent.

Without taking beta rotation into account, the three factor regression gives us an alpha of

0.64 percent (with t-statistic = 4:12, see Table 2) as the risk-adjusted return. In contrast,

17This is a common practice in the event study literature.

18

after adjusting for beta rotation, the average conditional alpha of the winner strategy is only

0.02 percent (t-statistic = 0:27)! For the buying-winner-selling-loser (W¡L) strategy, theOLS regression produces an alpha of 1.47 percent, higher than the average return di®erence

of 1.37 percent between the winner and the loser. Again, in contrast, the average of the

adjusted return is only 0.23 percent (t-statistic = 1:86), though it is marginally signi¯cant

at the 5% level according to the bootstrap test. As shown in Table 6, the results for the

other two periods, 1960-1980 and 1981-2001, are similar.

4.2. Components of Momentum Returns

In addition to the inference approach of section 4.1, we propose a return decomposition

method for dynamic trading strategies and apply it to analyze the sources of pro¯ts to style

momentum.

Given a factor model de¯ned by (15), we can divide style returns into four parts

rit = ®i + ¯i¹f + ¯i(ft ¡ ¹f) + "it; (18)

where ¹f is the sample average of the factor ft. The ¯rst two parts, ®i and ¯i¹f , sum up to

the average return of the style. ®i is Jensen's alpha and ¯i ¹f is the average return predicted

by the factor model. The last two parts, ¯i(ft ¡ ¹f) and "it, represent time-variation in the

style return. ¯i(ft¡ ¹f) is the time-variation of the return captured by the factor model and

"it is the unexplained regression error.

Consequently, we can break down the return of any dynamic style strategy into four

components

rpt = ®pt¡1 + ¯pt¡1 ¹f + ¯pt¡1(ft ¡ ¹f) + "pt; (19)

where rpt is the excess return on the strategy, such that rpt =Pn

i=1wit¡1rit, ®pt¡1 =Pni=1wit¡1®i, ¯pt¡1 =

Pni=1wit¡1¯i, and "pt =

Pni=1wit¡1"it. By the decomposition, there

are four sources or four components of the average return on the style strategy:

¹rp = ¹®p + ¹p¹f + ¯pt¡1(ft ¡ ¹f) + ¹"p; (20)

where the ¯rst and last components, ¹®p and ¹"p, are due to pricing errors of the factor

model (15). That is, ¹®p comes from non-zero intercepts, ®i's, of (15), while a non-zero ¹"p

can be generated if Et¡1("it) 6= 0. The other two components are related to the factors.

The component ¹p¹f is the product of average betas and risk premiums, while the term

¯pt¡1(ft ¡ ¹f) is the sum of the covariances between the rotating betas and the factors.

19

This method of return decomposition is interesting for several reasons. First, it is di®erent

from the approach that examines portfolios formed by sorting components of the returns

(e.g., a portfolio formed by sorting regression intercepts or residuals). In general, a portfolio

based on a component has di®erent weights than a portfolio based on the total return.

Moreover, the sum of returns on all the portfolios sorted by components is not equal to the

return on the momentum strategy that ranks winners and losers by total returns. Second,

the decomposition highlights the e®ects of beta rotation. It shows that as the momentum

strategy's betas rotate between style betas over time, the average return of the strategy is

attributable not only to the products of average betas and risk premiums, ¹p¹f , but also to

the covariances between the betas and the factors, ¯pt¡1(ft ¡ ¹f). Third, the method permits

existence of pricing errors of the factor model and provides a useful way to examine the

e®ects of pricing errors. This is important since ¯nancial economists are still unsettled on

the matter of appropriate factor pricing models.18

We apply the method to analyze the sources of pro¯ts to style momentum. The results,

reported in Table 7, show that the sum of the covariances between the rotating betas and

the factors, the component ¯pt¡1(ft ¡ ¹f), is the most important source of the payo® to the

strategy of buying the winner and selling the loser. For the sample period from January

1960 to December 2001, the \W¡L" strategy produces an average return di®erence of 1.37percent between the winner and the loser. Our decomposition shows that the covariances

between the betas and the factors contribute 1.10 percent, which is approximately 80 percent

of the raw return di®erence. By the bootstrap test, this component is statistically signi¯cant

at the 0.1% level (t-statistic = 4:76). The standard deviation column also shows that much

of variation in the style momentum return is due to variation of the term ¯pt¡1(ft ¡ ¹f).

Consistent with the results of Table 6, the last component ¹"p is the second largest, with an

estimated value of 0.20 percent. The t-statistic for the component ¹"p is below 2, but the

bootstrap test indicates that it is signi¯cant at the 5% level.

4.3. Formation and Holding Periods

So far our focus has been on the style momentum strategy that has a formation period of

1 month and a holding period of 1 month. We now take a look at the e®ects of di®erent

formation and holding periods.

18In this article, we focus on the Fama-French three factor model. However, the model is not withoutcontroversy. For example, see MacKinlay (1995), Ferson and Harvey (1999), and Daniel, Hirshleifer, andSubrahmanyam (2001).

20

Suppose that a strategy ranks the styles by their returns over the previous L months.

Let Ri¿ (L) be the gross return on style i over an L-month interval from month ¿ ¡L+ 1 tomonth ¿ . The weights of this ranking strategy are de¯ned as

wi¿ (L) =

8>>>><>>>>:1 if Ri¿ (L) = max1·j·nRj¿ (L);

¡1 if Ri¿ (L) = min1·j·nRj¿ (L);

0 otherwise.

Following Jegadeesh and Titman (1993), the momentum trading strategy with an L-month

formation period and an H-month holding period is a combination of the past H ranking

strategies. Speci¯cally, the weights of this (L-month, H-month) strategy are

wit¡1 =1

H

t¡1X¿=t¡H

wi¿ (L):

Now the method of (20) can be applied to decompose the strategy's return.

We present the results in the three panels of Table 8. Panel A presents the cases with a

1-month holding period and a formation period ranging from three months to sixty months.

Panel B includes the cases with a 1-month formation period and a holding period of various

lengths. In Panel C, both the formation and holding periods range from three months to

sixty months. For brevity, we only report the cases in which L = H in Panel C, and we

focus on the buying-winner-selling-loser (W¡L) strategy in all the cases of Table 8.Panel A of Table 8 shows that the sum of the covariances between the betas and the factors

(i.e., the component ¯pt¡1(ft ¡ ¹ft)) is the most important source of the momentum return

when the formation period is less than or equal to 12 months. More than half of the average

momentum return is attributed to this component. For longer formation periods, except for

the 24-month case, this is the only component that is signi¯cant at the 5% level according

to the bootstrap test. Interestingly, the ¯rst two components, ¹®p and ¹p¹f , increase with

the length of the formation interval. This pattern is intuitive, since for a longer formation

interval, it is more likely that the winner (loser) is the style portfolio that has high (low)

average return. Consistent with this intuition, contribution to the momentum payo® from

the variation of the style return (i.e., the two components ¯pt¡1(ft ¡ ¹ft) and ¹"p) becomes

less important as the length of the formation period increases.

For e®ects of the holding period, Panel B shows that the sum of covariances between the

betas and the factors is the largest component of the average momentum pro¯t in all the

cases, though it decreases with the length of the holding interval. In addition, in most cases,

21

this is the most statistically signi¯cant component according to the bootstrap test. The

pattern that the covariances between the betas and the factors weaken as the holding period

increases is also intuitive. With a long holding period, by construction, the momentum

portfolio consists of many winners and losers from the distant past. If the factors do not

possess long memory, the betas of the winners and the losers from the distant past may have

little to do with the current values of the factors.

Finally, the results of Panel C are consistent with what we observe from the ¯rst two

panels. When both the formation and holding periods get longer, the sum of the covariances

between the betas and the factors decreases. This component of the covariances remains

the largest until L = H = 24 months. For the last three cases in Panel C, the component

becomes much smaller or even negative. Nonetheless, for the last three cases, the momentum

pro¯ts and all the components are statistically insigni¯cant at the 10% level according to

the bootstrap test. Consistent with Panel A, we see that the ¯rst two components (¹®p and¹p¹f) become more important as the formation and the holding periods increase.19

4.4. Discussion

The ability of a factor pricing model to explain dynamic portfolio strategies is not equivalent

to the ability to explain the cross-sectional di®erences in average returns of the basic assets.

Our analysis suggests that even when a factor model can price the cross-section of the average

style returns perfectly, it may fail to explain the momentum strategy because momentum

pro¯ts may be (largely or entirely) due to some latent missing factors.

For example, suppose that the styles' excess returns are determined by

rit = ¯ift + "it;

for i = 1; ¢ ¢ ¢ ; n, and that an unidenti¯ed factor gt is embedded in the error term"it = bigt + eit;

where gt and eit are such that E(gt) = 0, Et¡1(eit) = 0, gt is independent from ft, and eit is

independent from both gt and ft.

19Similar to the industry momentum e®ect documented by Moskowitz and Grinblatt (1999), we ¯nd thatthe (1-month, 1-month) strategy has the highest payo®. Given the existence of many di®erent combinationsof formation and holding periods, it is natural to question whether style momentum pro¯ts are a statisticalartifact. Applying White's (2000) reality check method, we conducted a test that incorporates the e®ectsof searching over 81 strategies (by combining the nine formation periods and the nine holding periods). We¯nd that the momentum pro¯ts are still highly statistically signi¯cant. In a di®erent context, Daniel andTitman (1999) implemented a test based on a di®erent bootstrap method.

22

This model can explain the cross-section of the average style returns perfectly since

E(rit) = ¯iE(ft)

for i = 1; ¢ ¢ ¢ ; n. However, the model may fail to account for pro¯ts to a dynamic strategy.To illustrate this point, let wit¡1 be the weights of a dynamic strategy, for i = 1; ¢ ¢ ¢ ; n. Thenthe excess return of the strategy is

rpt = ¯pt¡1ft + "pt;

where rpt =Pn

i=1wit¡1rit, ¯pt¡1 =Pn

i=1wit¡1¯i, and "pt =Pn

i=1wit¡1"it. Thus, even ifbeta rotation (associated with the identi¯ed factors in the vector ft) is correctly taken into

account, the residual "pt can still contribute to a non-zero payo®. The contribution of the

residual "pt is from the covariance between the missing factor gt and its rotating beta bpt¡1:

E("pt) = E(bpt¡1gt) = cov(bpt¡1; gt);

where bpt¡1 =Pn

i=1wit¡1bi.

This is an interesting point, given that the tests in the literature on factor models are

typically focused on whether the models can capture the cross-section of average returns.

Our analysis shows that even a perfect pass of a model in tests that are targeted on regression

intercepts does not guarantee that the model can explain dynamic strategies.

The performance of a factor model also depends on functional form assumptions for the

betas. In this article, we focus on the constant beta version of the Fama-French three factor

model for style portfolios. In general, the tests can be based on time-varying beta versions

that extend (15). For example, one may consider specifying ¯it¡1 = zt¡1Bi, where zt is a1£ l vector of conditioning variables and Bi is a l£ k parameter matrix. The potential gainof such an extension is that the conditional betas for styles may provide an additional source

of correlations between the momentum strategy's betas and the factors. In an extreme case,

even if there is no cross-style dispersion in the betas, a style strategy's beta may still be

correlated with the factors if the conditioning variables zt¡1 are correlated with the factorsft. However, this potential gain is over-shadowed by the misspeci¯cation issue of conditional

betas stressed by Ghysels (1998). As Ghysels shows, misspeci¯ed conditional beta models

often underperform the constant beta models. Another reason to focus on the constant beta

case is that it is the simplest version of the Fama-French model, it is widely applied, and

it performs quite well for the size and BE/ME portfolios. To the best of our knowledge,

a well-speci¯ed conditional beta version of the three factor model that leads to signi¯cant

improvement over the constant beta version is not yet available.

23

5. Rotation on Predicted Relative Performance

5.1. A Three-Factor Logit Approach

Style rotation includes a rich array of dynamic strategies. Style momentum is only a special

case among all style-based strategies. In general, one may ¯rst build a model to predict the

relevant style spreads over time and then construct a style rotation strategy that adjusts

portfolio weights according to the prediction of relative style performance. For example,

Beinstein (1995), Fan (1995), Sorensen and Lazzara (1995), Kao and Shumaker (1999), Levis

and Liodakis (1999), and Asness et al. (2000) investigate models that forecast di®erences

between returns on growth and value strategies according to measures of aggregate economic

and ¯nancial conditions. These studies focus on variables such as the earnings yield on S&P

500, the slope of the yield curve, corporate credit spreads, corporate pro¯ts, spreads in

valuation multiples, expected earnings growth spreads, and other macroeconomic measures.

A three factor logit approach to style rotation is considered in this section. We use a logit

model based on the Fama-French three factors to predict relative style performance. To the

best of our knowledge, the three factors have never been used as predictors in the existing

literature on style rotation.20 Our purpose here is not to promote a particular style timing

strategy; rather, we aim to illustrate that the issue and the solution that we have considered

for style momentum are of general importance to the growing literature on studies of style

rotation.

The style timing strategy is constructed as follows. Let Smallt be the sum of time-t

returns on the styles SZ1-BM1, SZ1-BM3, and SZ1-BM5. Let Larget be the sum of time-t

returns on the styles SZ5-BM1, SZ5-BM3, and SZ5-BM5. Moreover, let Growtht be the sum

of time-t returns on the styles SZ1-BM1, SZ3-BM1, and SZ5-BM1, and let Valuet be the sum

of time-t returns on the styles SZ1-BM5, SZ3-BM5, and SZ5-BM5. The same logit model,

with di®erent parameter values, is applied to predict the signs of the size spread and the

value spread:

p1t = Prob(Smallt+1 > Larget+1jxt) =exp(a1xt)

1 + exp(a1xt); (21)

p2t = Prob(Valuet+1 > Growtht+1jxt) = exp(a2xt)

1 + exp(a2xt); (22)

20Levis and Liodakis (1999) utilized a logit model to construct style rotation strategies. They did notconsider the Fama-French three factors for prediction of relative performance. Furthermore, none of thestudies cited above has used the Fama-French model for risk-adjustment of style timing strategies.

24

where xt = (1 RMRFt SMBt HMLt)0, and a1 and a2 are two 1£ 4 parameter vectors. The

model is estimated by the maximum likelihood method. See Maddala (1983) for details

about the logit model and the estimation method.

The rotation strategy is based on rolling-window estimates. In any given month t, the

logit model parameters are estimated over a ¯ve-year window from month t ¡ 60 to montht ¡ 1. The conditional probability estimates p1t and p2t are obtained from the parameter

estimates. Three dynamic portfolios are constructed with the estimated logit probabilities.

The ¯rst purchases the predicted winner (PW). This strategy selects SZ1 if p1t > 0:55, SZ3

if 0:45 · p1t · 0:55, and SZ5 if p1t < 0:45. At the same time, the strategy selects BM5

if p2t > 0:55, BM3 if 0:45 · p2t · 0:55, and BM1 if p2t < 0:45. The combination of the

predicted winners in the size and BE/ME quintiles de¯nes the PW portfolio. The second

portfolio is to buy the predicted loser (PL), which is the opposite of the PW portfolio. That

is, the combination of the predicted size quintile loser and predicted BE/ME quintile loser

gives the PL strategy. Finally, the third portfolio is the one that buys the PW portfolio and

short-sells the PL portfolio (PW¡PL).

5.2. Empirical Results

Panel A of Table 9 presents the maximum likelihood estimates of the logit model for the

full sample period from January 1960 to December 2001. The estimates show that both the

lagged market factor (RMRF) and the lagged size factor (SMB) are statistically signi¯cant

predictors of the relative performance between small-cap and large-cap stocks. The positive

coe±cients suggest that small-cap stocks tend to perform better during the following month

when the small-cap stocks and the market are doing relatively well in the current month.

The lagged HML factor and the intercept are statistically insigni¯cant, although neither is

trivial in terms of the estimate magnitudes and the t-statistics. In contrast, the lagged HML

factor and the intercept are statistically signi¯cant in the logit regression for the relative

performance between value and growth stocks. This means that the lagged value spread has

the power to predict the sign of the value spread. The style rotation strategy is implemented

on the ¯ve-year rolling window estimates, which change slowly over time. The averages of

these estimates are similar to the full sample estimates reported in Panel A. For brevity,

these rolling window estimates are not reported.

The strategies built on the probabilities of relative performance are fairly di®erent from

the style momentum strategies. We ¯nd that the predicted winner (PW) strategy selects the

previous month's winner style in less than 25 percent of the months throughout the sample

25

period. Similarly, the predicted loser (PL) portfolio is di®erent from the previous month's

loser style in more than 75 percent of the months in the sample. The return on the strategy

of buying PW and short-selling PL, or the di®erence rPW¡rPL, has a correlation coe±cient of0.41 with the return di®erence rW¡rL. Thus, the buying-PW-selling-PL strategy (PW¡PL)is quite di®erent from the buying-winner-selling-loser strategy (W¡L) that was consideredin the previous sections.

As reported in Panel B, the PW strategy generates high raw returns, with an average

excess return of 0.98 percent per month. The PL strategy produces an average excess return

of 0.11 percent per month. The conventional risk adjustment method does not explain the

di®erence of 0.87 percent, which is more than 10 percent per year. The PW strategy has an

alpha of 0.35 percent from the three factor OLS regression. This implies that the predicted

average excess return for the PW portfolio is 0.63 percent. The PL strategy has an alpha

of ¡0:50 percent, so that the regression predicts an average excess return of 0.61 percent forthe PL portfolio. The three factor regression cannot explain the return di®erence between

PW and PL. The regression intercept for the buying-PW-selling-PL strategy is 0.85 percent,

nearly identical to the raw return di®erence. All of the three factor loadings are statistically

insigni¯cant in the regression for the PW¡PL strategy, which has a low R2 of 2 percent.Panel C of Table 9 shows that the conventional risk adjustment procedure fails due to

the ignorance of beta rotation. The average returns for both PW and PL portfolios are

mainly due to two components, the sample means of ¯pt¡1 ¹f and ¯pt¡1ft, where ft = ft ¡ ¹f .

For the di®erence between PW and PL, however, its average is largely attributed to the

mean of the component ¯pt¡1ft. Panel C indicates that approximately 70% of the average

return di®erence between the predicted winner and loser is from the covariances between

the rotating betas and the factors. Much of the variation in the return di®erence between

PW and PL is due to the component ¯pt¡1ft. In contrast to the results of Table 7 for stylemomentum, the second largest component of the payo® to the PW¡PL strategy is ¹p ¹f (i.e.,the products of average betas and factor risk premiums). In this case, the pricing errors of

the Fama-French model, whether the regression intercepts or the residuals, contribute little

to the pro¯tability of the timing strategy.

In sum, built on the three factor logit model to predict relative style performance, the

style rotation strategy that buys the predicted winner and short-sells the predicted loser

generates impressive returns. Using the conventional risk adjustment method, the Fama-

French model fails to explain the average return on the strategy. In contrast, once beta

rotation is appropriately taken into account, the average return of this timing strategy is

largely captured by the Fama-French three factor model.

26

6. Conclusions

Style momentum appears to be an interesting alternative to individual stock momentum.

With the exploding growth of ETFs, it is much easier to deal with a small number of style

ETFs than thousands of individual stocks as in the momentum strategies of Jegadeesh and

Titman (1993). Style momentum is as pro¯table as individual stock momentum in terms of

raw payo®s. Style momentum also replicates the puzzling results of Fama and French (1996)

for the risk-adjusted pro¯tability of individual stock momentum. According to conventional

risk adjustment using the three factor time series regressions, the strategy of buying the

winner style is not riskier than that of buying the loser style. Consistent with the ¯ndings of

Fama and French (1996) and Jegadeesh and Titman (2001) for momentum strategies based

on individual stocks, this result indicates that none of the pro¯ts to the buying-winner-

selling-loser style strategy can be attributed to compensation for risk.

In this article, we have shown that the high risk-adjusted payo® to style momentum is

an illusion created by a °aw in the conventional risk adjustment method. The key to our

explanation is that the style momentum strategy induces multifactor beta rotation. In a

multifactor asset pricing model, the average return of a dynamic strategy with beta rotation

is determined not only by products of average beta values and factor risk premiums but

also by covariances between betas and factors. It is the covariances between the rotating

betas and the common risk factors that the conventional method has missed. The covariance

estimates indicate that for any of the three Fama-French factors, the factor beta of the winner

tends to be higher (lower) than that of the loser when the factor is expected to be high (low)

next month. This is why we ¯nd that the strategy of buying the winner is indeed riskier than

that of buying the loser, even though the average values of both strategies' betas contribute

little to explaining the winner-loser return di®erential.

Evaluation of style rotation strategies is an important topic in tactical asset allocation

and equity style management. This article demonstrates that beta rotation plays a signi¯cant

role in risk adjustment for dynamic style strategies. To emphasize the e®ects of cross-style

beta rotation, we focus on the simplest constant beta version of the Fama-French model for

the style returns. Our results show that rotation among constant style betas is su±cient

to account for most of the pro¯ts to the style momentum strategy (and the timing strategy

based on the logit model). A step that may further improve the explanatory power is to

apply the three factor model with time-varying betas for the underlying style portfolios. If

the style betas are time varying, the betas of the style momentum strategy will vary over

time, so that the strategy's betas can be correlated with the risk factors even without any

27

cross-style dispersion in the betas. However, the potential gain from a model with time-

varying style betas is matched by the thorny speci¯cation problem that Ghysels (1998) has

highlighted. This is an extension that remains to be explored.

A tantalizing future project is to pursue an extension to individual stock momentum. To

analyze dynamic strategies that rank stocks on the basis of returns, beta rotation can be

important, especially if there is a multifactor asset pricing model that can capture much of

the variation in individual stock returns. Although the risk adjustment approach and the

decomposition method that we propose in this article are clearly applicable, there are some

challenging issues that have to be cautiously dealt with, in order to e®ectively analyze the

sources of pro¯ts to individual stock momentum.

To account for beta rotation among stocks, a di±cult issue is that a large number of

stocks may have multifactor betas that are highly non-stationary. The nature of beta non-

stationarity may also be heterogeneous across di®erent stocks. The beta non-stationarity

generates a host of di±cult econometric problems that do not appear tractable. Another

challenging issue is that it is di±cult to precisely estimate the betas of (several thousands

of) individual stocks, even if all the betas are strictly stationary. With stationary betas, it

is still unclear how to resolve the issue of misspeci¯cation for beta dynamics. One simple

approach that avoids any functional form speci¯cation is to run constant beta rolling-window

regressions. The length of the rolling-windows can be kept short (e.g., 36 or 60 months) in

practice to generate enough variation in the beta estimates. However, there is no convincing

reason that this is an e®ective way to capture time-variation in the betas.21

The ability of an asset pricing model to explain a dynamic portfolio strategy depends on

how well the model can capture returns of the underlying basic assets. On the other hand, to

incorporate the e®ects of multifactor beta rotation, we need to e®ectively capture betas of the

basic assets. Given the uncertainty regarding performance of the Fama-French three factor

model at the individual stock level and the challenges mentioned above, there is no basis

to make any strong claims about sources of individual stock momentum. Whatever one's

conjecture, whether multifactor beta rotation can account for individual stock momentum

is certainly an intriguing question that warrants further research.

21It is logically inconsistent to assume that betas take on di®erent values for di®erent but overlappingestimation windows. In addition, ¯nite sample estimation biases or noises can seriously impact results,especially for short estimation windows (e.g., Jegadeesh and Titman (2002)).

28

Appendix

A. The Relative Strength Strategy

In this appendix, we prove the risk adjustment regression results (8), (9), and (10) for the

relative strength strategy. The expressions (8) and (9) for ap and bp follow from an extension

of Stein's lemma. Let x, y, and z be jointly normally distributed. Then,

cov(x; yz) = cov(x; y)E(z) + cov(x; z)E(y):

This result can be easily established with Stein's lemma. We use it without proof.22

Since the portfolio weights de¯ned in (6) are linear in the excess returns, the beta ¯pt¡1is normally distributed, given the normality of the common factor. Thus,

cov(¯pt¡1ft; ft) = cov(¯pt¡1; ft)E(ft) + E(¯pt¡1)var(ft):

On the other hand, the solution to (7) is

ap = E(rpt)¡ bpE(ft)bp =

cov(rpt; ft)

var(ft):

Therefore,

bp =cov(rpt; ft)

var(ft)=cov(¯pt¡1; ft)E(ft) + E(¯pt¡1)var(ft)

var(ft)

= E(¯pt¡1) + ¹cov(¯pt¡1; ft)var(ft)

:

The intercept estimate converges to

ap = E(rpt)¡ bpE(ft) = E(¯pt¡1ft)¡ E(¯pt¡1)E(ft)¡ ¹2cov(¯pt¡1; ft)var(ft)

= cov(¯pt¡1; ft)µ1¡ ¹

2

¾2

¶:

22The proof is available upon request. Stein's lemma states that if x is normally distributed, and g is asmooth function such that Ejg0(x)j exists, then

cov[g(x); x] = var(x)E[g0(x)]:

29

Finally,

cov(¯pt¡1; ft) = cov

ÃnXi=1

wit¡1¯i; ft

!= cov

Ã1

n

nXi=1

(¯i ¡ ¹)¯ift¡1; ft!

=1

n

nXi=1

(¯i ¡ ¹)¯icov(ft¡1; ft) =1

n

nXi=1

(¯i ¡ ¹)2cov(ft¡1; ft);

and similarly

E(¯pt¡1) = E

ÃnXi=1

wit¡1¯i

!= E

Ã1

n

nXi=1

(¯i ¡ ¹)¯ift¡1!=1

n

nXi=1

(¯i ¡ ¹)2¹:

Putting everything together, we have established results (8), (9), and (10).

B. The Bootstrap Test

The t-statistics reported in Tables 6 through 8 as well as Panel C of Table 9 do not take into

account estimation errors associated with the Fama-French three factor beta estimates and

the regression intercepts. We perform a bootstrap test in each case that incorporates the

estimation noise. The test is built on the stationary bootstrap of Politis and Romano (1994)

that uses overlapping blocks with lengths that are sampled randomly from the geometric

distribution. The advantage of random block lengths is that the resulting bootstrap data

series is stationary. Details of the stationary bootstrap test procedure are described below.

First, for t = 1; ¢ ¢ ¢ ; T , de¯ne xt as the vector that includes all the relevant variablesxt ´ (r1t ¢ ¢ ¢ rnt f1t ¢ ¢ ¢ fkt):

Let STAT denote a t-statistic (for a sample mean) that is computed by the standard proce-

dure. We resample the data to obtain B time series fxbjtg; t = 1; ¢ ¢ ¢ ; T , for j = 1; ¢ ¢ ¢ ; B. Foreach resampled time series, we compute a t-statistic, denoted by STATbj, which is a resam-

pled version of STAT. We then use the following statistics to compute the p-value associated

with the t-statistic: pT (STATbj ¡ STAT)

for j = 1; ¢ ¢ ¢ ; B. We compare pT (STAT) to the quantiles of pT (STATbj¡STAT) to obtainthe p-value. The bootstrap p-value may be de¯ned as the probability in favor of the null

hypothesis that the mean is equal to zero, i.e.,

Prob[pT (STATbj ¡ STAT) >

pT (STAT)]:

30

By this de¯nition, a p-value that is close to zero indicates statistically signi¯cant evidence

that the mean is positive.

For t = 1; ¢ ¢ ¢ ; T , and j = 1; ¢ ¢ ¢ ; B, xbjt is de¯ned by

xbjt = x´jt ;

where ´jt is a random index chosen according to the stationary bootstrap algorithm of

Politis and Romano. To implement this method, we choose a smoothing parameter q and

then proceed in three steps as follows:

1. For t = 1, draw ´j1 as a random variable, uniformly distributed over f1; ¢ ¢ ¢ ; Tg,independently of other variables.

2. Increase t by 1. If t > T , stop. Otherwise, draw a standard uniform random variable

u, independently of other variables.

² If u < q, draw ´jt as a random variable, uniformly distributed over f1; ¢ ¢ ¢ ; Tg,independently of other variables.

² If u ¸ q, set ´jt = ´jt¡1 + 1; if ´jt > T , set ´jt = 1.

3. Repeat the second step.

Theoretically, the parameter q should change with the sample size (i.e., q ´ qT ), such

that 0 < qT · 1, qT ! 0, and TqT !1, as T !1. Following Sullivan, Timmermann, andWhite (1999), we set q = 0:1, which corresponds to an average block length of 10. We have

tried various values of q in empirical tests and ¯nd that the test results are not sensitive to

the choice of q.

31

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35

Table 1

Descriptive Statistics

This table presents summary statistics for the Fama-French three factors and the nine style portfo-lios. Panel A reports the sample means, standard deviations, ¯rst autocorrelation coe±cients, and thecontemporaneous correlation matrix for the three factors. Panel B presents sample means and stan-dard deviations for excess returns on the nine style portfolios, which are selected from the twenty-¯vevalue-weighted size and book-to-market sorted portfolios of Fama and French. The nine portfolios areSZ1-BM1, SZ1-BM3, SZ1-BM5, SZ3-BM1, SZ3-BM3, SZ3-BM5, SZ5-BM1, SZ5-BM3, and SZ5-BM5,where (SZ1, SZ3, SZ5) and (BM1, BM3, BM5) are three out of the ¯ve size quintiles and three out ofthe ¯ve book-to-market quintiles, respectively. See Fama and French (1993) for details on the portfolioconstruction. Panel B also reports the ordinary least squares regressions for the styles:

rit = ®i + biRMRFt + siSMBt + hiHMLt + "it

where rit is the excess return on the i-th style portfolio, for i = 1; ¢ ¢ ¢ ; n; RMRFt, SMBt, and HMLt aretime-t values of the market factor (excess return on the market), the size factor, and the book-to-marketfactor of Fama and French, respectively. The t-statistics are reported in parentheses. The sample periodis from January 1960 to December 2001.

Panel A: Fama-French Three Factors

sample standard ¯rstaverage deviation autocorrelation correlation matrix

RMRF 0:47 4:44 0:06 1:00 0:30 ¡0:42SMB 0:16 3:20 0:08 0:30 1:00 ¡0:30HML 0:43 2:93 0:13 ¡0:42 ¡0:30 1:00

36

Table 1

Descriptive Statistics (Continued)

Panel B: Style Portfolios

Three factor regressionaverage s.d. ® b s h R2 ¾"

SZ1-BM1 0:28 8:13 ¡0:33 1:04 1:42 ¡0:26 0:91 2:51(¡2:86) (36:74) (37:96) (¡6:06)

SZ1-BM3 0:79 6:05 0:03 0:93 1:12 0:32 0:94 1:45(0:48) (56:60) (52:05) (12:70)

SZ1-BM5 1:06 5:88 0:12 0:98 1:09 0:70 0:95 1:36(1:99) (64:04) (53:91) (29:93)

SZ3-BM1 0:39 6:83 ¡0:06 1:09 0:73 ¡0:42 0:95 1:55(¡0:83) (62:68) (31:74) (¡16:07)

SZ3-BM3 0:65 4:94 ¡0:12 1:02 0:44 0:50 0:90 1:59(¡1:60) (56:97) (18:61) (18:33)

SZ3-BM5 0:92 5:29 ¡0:04 1:10 0:54 0:83 0:90 1:64(¡0:54) (59:40) (21:99) (29:51)

SZ5-BM1 0:43 4:84 0:17 0:97 ¡0:26 ¡0:38 0:93 1:24(3:08) (69:61) (¡13:89) (¡17:79)

SZ5-BM3 0:56 4:31 0:02 0:98 ¡0:24 0:27 0:84 1:71(0:29) (50:68) (¡9:52) (9:18)

SZ5-BM5 0:63 4:68 ¡0:21 1:05 ¡0:08 0:86 0:81 2:03(¡2:28) (45:70) (¡2:75) (24:72)

37

Table 2

Returns of Style Momentum

This table presents statistics for three strategies. The winner strategy buys the winner style of theprevious month, while the loser strategy buys the loser style of the previous month. The third strategyis the buying-winner-selling-loser strategy; its pro¯tability is measured by the di®erence between returnson the winner and the loser. The table reports the sample means and standard deviations for the excessreturns on the winner and the loser, as well as the di®erence between the two (i.e., W¡L). It also reportsthe three factor OLS regressions for the three strategies (just like the regressions for the style portfoliosin Table 1). The t-statistics are reported in parentheses.

Three factor regressionaverage s.d. ® b s h R2

01/1960-12/2001:Winner 1:21 5:70 0:64 0:90 0:51 0:12 0:65

(4:75) (4:12) (23:49) (10:07) (2:01)Loser ¡0:16 6:55 ¡0:83 1:10 0:58 0:10 0:73

(¡0:54) (¡5:30) (28:59) (11:34) (1:72)W¡L 1:37 5:99 1:47 ¡0:20 ¡0:07 0:02 0:03

(5:12) (5:42) (¡3:04) (¡0:78) (0:16)

01/1960-12/1980:Winner 1:32 5:57 0:76 0:96 0:42 0:24 0:75

(3:78) (4:22) (20:48) (6:36) (3:14)Loser ¡0:30 6:47 ¡0:90 1:07 0:64 0:04 0:79

(¡0:74) (¡4:77) (21:72) (9:20) (0:50)W¡L 1:63 5:01 1:66 ¡0:11 ¡0:22 0:20 0:05

(5:15) (5:29) (¡1:30) (¡1:88) (1:50)

01/1981-12/2001:Winner 1:09 5:85 0:54 0:85 0:51 0:02 0:57

(2:95) (2:14) (13:01) (6:17) (0:22)Loser 0:01 6:64 ¡0:77 1:13 0:56 0:14 0:67

(0:01) (¡3:04) (17:37) (6:87) (1:41)W¡L 1:08 6:84 1:31 ¡0:28 ¡0:05 ¡0:12 0:03

(2:52) (2:94) (¡2:42) (¡0:37) (¡0:67)

38

Table 3

E®ects of the Pricing Errors

This table reports results for three cases on the pricing errors of the Fama-French model for the sampleperiod from January 1960 to December 2001. In cases (i), (ii), and (iii), the excess returns on the styleportfolios are assumed to be determined by equations (3), (4), and (5), respectively. In each case, basedon these assumed style returns, the winner strategy, the loser strategy, as well as the buying-winner-selling-loser strategy are constructed. The following results are then obtained by repeating the sameprocedure as in Table 2.

Three factor regressionaverage s.d. ® b s h R2

Case (i):Winner 1:25 5:73 0:68 0:90 0:51 0:11 0:65

(4:90) (4:38) (23:45) (10:12) (1:95)Loser ¡0:02 6:45 ¡0:66 1:09 0:49 0:06 0:72

(¡0:08) (¡4:19) (28:22) (9:63) (1:04)W¡L 1:27 5:95 1:34 ¡0:19 0:02 0:05 0:02

(4:80) (4:96) (¡2:90) (0:23) (0:52)

Case (ii):Winner 1:28 5:58 0:67 0:89 0:63 0:18 0:72

(5:15) (4:96) (26:69) (14:45) (3:51)Loser 0:01 6:11 ¡0:66 1:12 0:43 0:13 0:77

(0:02) (¡4:93) (33:75) (9:79) (2:65)W¡L 1:27 5:84 1:33 ¡0:23 0:21 0:04 0:03

(4:90) (5:06) (¡3:52) (2:42) (0:45)

Case (iii):Winner 0:63 5:54 0:62 0:88 0:59 0:10 0:72

(2:54) (4:68) (26:19) (13:49) (1:96)Loser ¡0:59 6:17 ¡0:61 1:13 0:46 0:20 0:77

(¡2:16) (¡4:59) (33:58) (10:37) (3:92)W¡L 1:22 5:89 1:22 ¡0:26 0:13 ¡0:10 0:03

(4:66) (4:74) (¡3:92) (1:54) (¡1:02)

39

Table 4

Beta Rotation of Style Momentum

This table presents descriptive statistics for the three betas, bpt¡1, spt¡1, and hpt¡1, of the winnerand the loser strategies. The betas are obtained in two steps. First, the three factor OLS regressionestimates are obtained for all of the nine style portfolios. The three betas are then given by (11),(12), and (13), respectively. For the correlation corr(¯jpt¡1; f

jt ), the beta ¯

jpt¡1 represents bpt¡1, spt¡1,

and hpt¡1, for j = 1; 2; 3, while the factor fjt represents RMRFt, SMBt, and HMLt, for j = 1; 2; 3,respectively. The sample period is from January 1960 to December 2001.

Winner LoserRMRF SMB HML RMRF SMB HMLbpt¡1 spt¡1 hpt¡1 bpt¡1 spt¡1 hpt¡1

average 1:02 0:45 0:24 1:02 0:52 0:13s.d. 0:05 0:62 0:51 0:05 0:68 0:51maximum 1:10 1:42 0:86 1:10 1:42 0:86minimum 0:93 ¡0:26 ¡0:42 0:93 ¡0:26 ¡0:42¯rst autocorr. 0:06 0:16 0:15 0:04 0:24 0:04

corr(¯jpt¡1; fjt ) 0:04 0:15 0:14 ¡0:10 ¡0:18 ¡0:11

(0:79) (3:33) (3:23) (¡2:30) (¡4:14) (¡2:55)

corr(¯jpt¡1; fjt¡1) 0:14 0:68 0:60 ¡0:21 ¡0:64 ¡0:59

(3:07) (20:94) (16:87) (¡4:88) (¡18:61) (¡16:24)

40

Table 5

Intercepts of the Grundy-Martin Regressions

This table presents the Grundy-Martin regressions for the three strategies: Winner, Loser, and W¡L(buying-winner-selling-loser). In these regressions, each of the three Fama-French factor betas is allowedto take on three values, depending on the value of the factor. See equation (14) for details. Four setsof style returns are used. The ¯rst set is the historical returns reported in Table 1 (denoted \ActualReturns"). The other three sets are identical to those used in the three cases of Table 3. The regressionintercepts and the associated t-statistics are reported. The sample period is from January 1960 toDecember 2001.

Winner Loser W ¡ Lintercept t-stat intercept t-stat intercept t-stat

Actual Returns 0:20 1:07 ¡0:86 ¡4:53 1:06 3:22

Case (i) 0:25 1:36 ¡0:82 ¡4:26 1:07 3:25

Case (ii) 0:44 2:71 ¡0:57 ¡3:52 1:01 3:19

Case (iii) 0:39 2:45 ¡0:43 ¡2:65 0:82 2:61

41

Table 6

Risk Adjustment for Style Momentum

Three time series are obtained for each of the three strategies, i.e., Winner, Loser, and W¡L (buying-winner-selling-loser) strategies. For the Winner strategy, for example, the three time series are theexcess return (rW), the adjusted return (ARW) de¯ned in (16), and the standardized adjusted return(SARW) de¯ned in (17). The table reports the mean, the standard deviation, and the t-statistic of themean for each variable. A bootstrap test is conducted in each case. The superscript of the t-statisticis assigned to be 0, 1, 2, or 3. The superscript of 0 indicates that the bootstrap test does not rejecta zero mean at the 5% signi¯cance level. The superscripts of 1, 2, and 3 indicate that the bootstraptest rejects at the 5%, 1%, and 0.1% level, respectively. Details of the bootstrap test are provided inAppendix B.

01/1960 ¡ 12/2001 01/1960 ¡ 12/1980 01/1981 ¡ 12/2001mean s.d. t-stat mean s.d. t-stat mean s.d. t-stat

rW 1:21 5:70 4:75(3) 1:32 5:57 3:77(3) 1:09 5:85 2:95(3)

ARW 0:02 1:86 0:27(0) 0:11 1:74 1:00(0) ¡0:04 1:94 ¡0:32(0)SARW 0:03 1:02 0:56(0) 0:08 0:98 1:22(1) 0:00 1:06 ¡0:02(0)

rL ¡0:16 6:55 ¡0:54(0) ¡0:30 6:47 ¡0:74(0) 0:01 6:64 0:01(0)

ARL ¡0:21 2:07 ¡2:26(1) ¡0:10 1:84 ¡0:86(0) ¡0:31 2:21 ¡2:25(0)SARL ¡0:08 1:08 ¡1:62(0) ¡0:06 1:03 ¡0:89(0) ¡0:11 1:13 ¡1:54(0)

rW ¡ rL 1:37 5:99 5:11(3) 1:63 5:01 5:14(3) 1:08 6:84 2:51(2)

ARW¡L 0:23 2:79 1:86(1) 0:21 2:53 1:32(0) 0:27 2:95 1:48(0)

SARW¡L 0:10 1:47 1:58(1) 0:13 1:40 1:51(1) 0:11 1:51 1:13(0)

42

Table 7

Sources of Pro¯ts to Style Momentum

By the three factor regressions, each style return is divided into four components: rit = ®i+¯i¹f+¯ift+

"it, for i = 1; ¢ ¢ ¢ ; n, where ft = ft ¡ ¹f . Accordingly, the return on a strategy has four components:rpt = ®pt¡1 + ¯pt¡1 ¹f + ¯pt¡1ft + "pt, where ®pt¡1 =

Pni=1wit¡1®i, ¯pt¡1 =

Pni=1wit¡1¯i, "pt =Pn

i=1wit¡1"it, and wit¡1, for i = 1; ¢ ¢ ¢ ; n, are the portfolio weights of the strategy. For returns on eachof the three strategies (i.e., the Winner, the Loser, and the W¡L strategies) and the four components,the table reports the means, the standard deviations, and the t-statistics of the sample means. Abootstrap test is conducted in each case. The superscript of the t-statistic is assigned to be 0, 1, 2, or 3.The superscript of 0 indicates that the bootstrap test does not reject a zero mean at the 5% signi¯cancelevel. The superscripts of 1, 2, and 3 indicate that the bootstrap test rejects at the 5%, 1%, and 0.1%level, respectively. Details of the bootstrap test are provided in Appendix B.

Winner Loser W ¡ Lmean s.d. t-stat mean s.d. t-stat mean s.d. t-stat

01/1960-12/2001

rpt 1:21 5:70 4:75(3) ¡0:16 6:55 ¡0:54(0) 1:37 5:99 5:11(3)

®pt¡1 ¡0:06 0:16 ¡8:57(2) ¡0:09 0:18 ¡11:28(3) 0:03 0:25 2:51(0)

¯pt¡1 ¹f 0:66 0:23 64:92(1) 0:62 0:22 64:49(1) 0:04 0:37 2:29(0)

¯pt¡1ft 0:53 5:42 2:18(3) ¡0:57 6:16 ¡2:08(3) 1:10 5:18 4:76(3)

"pt 0:08 1:86 1:02(0) ¡0:12 2:05 ¡1:29(0) 0:20 2:77 1:64(1)

01/1960-12/1980

rpt 1:32 5:57 3:77(3) ¡0:30 6:47 ¡0:74(0) 1:63 5:01 5:14(3)

®pt¡1 ¡0:01 0:15 ¡0:56(0) ¡0:02 0:15 ¡2:26(0) 0:02 0:22 1:18(0)

¯pt¡1 ¹f 0:58 0:28 33:24(1) 0:53 0:27 31:35(1) 0:05 0:47 1:60(0)

¯pt¡1ft 0:64 5:26 1:92(3) ¡0:73 6:18 ¡1:87(3) 1:37 4:07 5:33(3)

"pt 0:12 1:73 1:05(1) ¡0:08 1:86 ¡0:67(0) 0:19 2:52 1:22(0)

01/1981-12/2001

rpt 1:09 5:85 2:95(3) 0:01 6:64 0:01(0) 1:08 6:84 2:51(2)

®pt¡1 ¡0:08 0:27 ¡4:39(2) ¡0:19 0:35 ¡8:78(3) 0:12 0:45 4:12(3)

¯pt¡1 ¹f 0:75 0:28 41:65(0) 0:67 0:28 37:72(0) 0:08 0:48 2:73(2)

¯pt¡1ft 0:38 5:59 1:08(2) ¡0:34 6:16 ¡0:89(1) 0:72 6:13 1:87(1)

"pt 0:04 1:96 0:29(0) ¡0:12 2:15 ¡0:90(0) 0:16 2:91 0:86(0)

43

Table 8

Formation and Holding Periods

This table extends Table 7 for di®erent formation and/or holding periods. Table 8 is focused on thebuying-winner-selling-loser strategy (i.e., W¡L) for the sample period from January 1960 to December2001. Panel A reports the results for di®erent formation periods with the holding period of one month.Panel B reports the results for di®erent holding periods while the formation period has the ¯xed lengthof one month. Panel C reports the results for the case where the lengths of both the formation andholding periods equal to each other but range from 3 months to 60 months.

Panel A: Di®erent Formation Periods

mean t-stat mean t-stat mean t-stat mean t-stat

3-month 6-month 9-month 12-month

rpt 0:92 3:24(2) 0:59 2:08(1) 0:90 3:11(3) 1:20 4:10(3)

®pt¡1 0:05 4:18(1) 0:07 6:36(1) 0:08 6:46(0) 0:09 7:46(0)

¯pt¡1 ¹f 0:07 4:28(1) 0:08 4:95(0) 0:10 5:67(0) 0:12 6:96(0)

¯pt¡1ft 0:59 2:31(2) 0:41 1:69(1) 0:53 2:09(2) 0:78 3:06(3)

"pt 0:22 1:91(2) 0:02 0:15(0) 0:20 1:71(2) 0:22 1:92(3)

24-month 36-month 48-month 60-month

rpt 0:65 2:59(2) 0:86 3:43(3) 0:74 2:92(3) 0:71 2:86(2)

®pt¡1 0:15 11:95(1) 0:18 13:44(0) 0:19 15:48(0) 0:21 17:77(0)

¯pt¡1 ¹f 0:14 8:42(0) 0:15 8:60(0) 0:20 12:57(0) 0:25 15:23(0)

¯pt¡1ft 0:20 0:89(0) 0:45 2:02(3) 0:27 1:27(1) 0:19 0:94(1)

"pt 0:16 1:44(2) 0:10 0:79(0) 0:07 0:59(0) 0:05 0:41(0)

44

Table 8

Formation and Holding Periods (Continued)

Panel B: Di®erent Holding Periods

mean t-stat mean t-stat mean t-stat mean t-stat

3-month 6-month 9-month 12-month

rpt 0:58 2:85(2) 0:39 2:68(2) 0:35 2:76(2) 0:38 3:49(3)

®pt¡1 0:03 4:24(1) 0:03 5:55(0) 0:03 6:48(0) 0:03 7:43(0)

¯pt¡1 ¹f 0:04 3:72(0) 0:04 5:30(0) 0:04 6:38(0) 0:04 7:24(0)

¯pt¡1ft 0:41 2:33(1) 0:30 2:38(2) 0:23 2:06(2) 0:27 2:72(3)

"pt 0:10 1:29(0) 0:03 0:46(0) 0:06 1:25(1) 0:05 1:33(1)

24-month 36-month 48-month 60-month

rpt 0:20 2:95(3) 0:18 3:37(3) 0:17 3:37(3) 0:14 2:94(3)

®pt¡1 0:03 10:42(0) 0:03 12:80(0) 0:03 14:37(0) 0:03 15:73(0)

¯pt¡1 ¹f 0:04 9:10(0) 0:04 11:32(0) 0:04 13:41(0) 0:04 15:16(0)

¯pt¡1ft 0:12 1:97(2) 0:10 2:06(3) 0:08 1:78(3) 0:05 1:09(1)

"pt 0:02 0:57(0) 0:02 0:65(0) 0:02 0:91(1) 0:02 1:01(1)

45

Table 8

Formation and Holding Periods (Continued)

Panel C: Changing Both Formation and Holding Periods

mean t-stat mean t-stat mean t-stat mean t-stat

3m-3m 6m-6m 9m-9m 12m-12m

rpt 0:43 1:79(1) 0:46 1:90(1) 0:53 2:32(2) 0:53 2:57(2)

®pt¡1 0:05 5:31(0) 0:07 8:01(1) 0:07 8:54(0) 0:09 9:77(0)

¯pt¡1 ¹f 0:07 5:49(1) 0:08 6:53(0) 0:10 7:36(0) 0:11 8:69(0)

¯pt¡1ft 0:24 1:11(0) 0:20 0:92(0) 0:31 1:48(2) 0:26 1:41(1)

"pt 0:07 0:76(0) 0:11 1:23(1) 0:05 0:66(0) 0:07 0:85(0)

24m-24m 36m-36m 48m-48m 60m-60m

rpt 0:56 2:76(2) 0:39 1:78(0) 0:28 1:34(0) 0:23 1:13(0)

®pt¡1 0:16 16:33(0) 0:18 18:02(0) 0:19 20:25(0) 0:20 23:48(0)

¯pt¡1 ¹f 0:14 10:94(0) 0:15 12:40(0) 0:21 20:67(0) 0:27 26:37(0)

¯pt¡1ft 0:22 1:14(1) 0:02 0:12(0) ¡0:16 ¡0:85(0) ¡0:25 ¡1:40(0)"pt 0:05 0:61(0) 0:04 0:40(0) 0:04 0:38(0) 0:01 0:14(0)

46

Table 9

Style Rotation on A Three Factor Logit

Panel A reports the maximum likelihood estimates of the logit model parameters de¯ned in (21) and(22). The t-statistics are provided in parentheses. Panel B and Panel C are for the three strategiesconstructed from the estimated logit probabilities over ¯ve-year rolling windows. PW is the strategythat buys the predicted winner, and PL is the strategy that buys the predicted loser. The thirdstrategy, denoted PW¡PL, buys the predicted winner and short-sells the predicted loser. In terms ofthe construction, Panel B is identical to Table 2, and Panel C is identical to Table 7, with the logit-model-based strategies replacing the momentum strategies. See Table 2 and Table 7 for details. Thesample period is from January 1960 to December 2001.

Panel A: Estimates of the Logit Model

Intercept RMRF SMB HML

Small-Cap vs. Large-Cap ¡0:14 0:10 0:09 0:07(¡1:45) (3:92) (2:86) (1:79)

Value vs. Growth 0:21 0:03 0:00 0:08(2:22) (1:35) (¡0:15) (2:33)

47

Table 9

Style Rotation on A Three Factor Logit (Continued)

Panel B: Returns of the Strategy

Three factor regressionaverage s.d. ® b s h R2

PW 0:98 5:70 0:35 0:96 0:46 0:26 0:68(3:88) (2:25) (25:11) (9:34) (4:61)

PL 0:11 6:15 ¡0:50 1:01 0:56 0:11 0:74(0:40) (¡3:27) (27:05) (11:69) (1:91)

PW¡PL 0:87 5:78 0:85 ¡0:05 ¡0:10 0:16 0:02(3:40) (3:05) (¡0:73) (¡1:17) (1:54)

Panel C: Sources of Pro¯ts to the Strategy

PW PL PW ¡ PLmean s.d. t-stat mean s.d. t-stat mean s.d. t-stat

rpt 0:98 5:70 3:64(3) 0:11 6:15 0:38(0) 0:87 5:78 3:19(3)

®pt¡1 ¡0:04 0:18 ¡4:91(1) ¡0:07 0:23 ¡6:11(1) 0:03 0:13 4:12(0)

¯pt¡1 ¹f 0:75 0:23 67:65(1) 0:59 0:26 48:23(1) 0:16 0:47 7:29(0)

¯pt¡1ft 0:26 5:44 1:01(2) ¡0:34 6:13 ¡1:18(1) 0:61 5:43 2:35(3)

"pt 0:01 1:73 0:16(0) ¡0:07 1:81 ¡0:76(0) 0:08 2:29 0:72(0)

48

1965 1970 1975 1980 1985 1990 1995 20000.940.960.98

11.021.041.061.08

Win

ner R

MR

F be

ta b

Panel A.

1965 1970 1975 1980 1985 1990 1995 20000.940.960.98

11.021.041.061.08

Lose

r RM

RF

beta

b

Panel B.

Figure 1. Plots of the market factor betas of the winner and the loser

The market factor beta for the winner is bpt¡1 =Pni=1wit¡1bi where wit¡1 = 1 if rit¡1 = max1·j·n rjt¡1

and wit¡1 = 0 otherwise. The weights for the loser strategy are such that wit¡1 = 1 if rit¡1 = min1·j·n rjt¡1and wit¡1 = 0 otherwise. The market factor betas for the style portfolios (bi, i = 1; ¢ ¢ ¢ ; n) are estimated bythe three factor time series regressions. The regression estimates are reported in Table 1.

49

1965 1970 1975 1980 1985 1990 1995 2000

0

0.5

1W

inne

r SM

B be

ta s

Panel A.

1965 1970 1975 1980 1985 1990 1995 2000

0

0.5

1

Lose

r SM

B be

ta s

Panel B.

Figure 2. Plots of the size factor betas of the winner and the loser

The size factor beta for the winner is spt¡1 =Pni=1wit¡1si where wit¡1 = 1 if rit¡1 = max1·j·n rjt¡1 and

wit¡1 = 0 otherwise. The weights for the loser strategy are such that wit¡1 = 1 if rit¡1 = min1·j·n rjt¡1and wit¡1 = 0 otherwise. The size factor betas for the style portfolios (si, i = 1; ¢ ¢ ¢ ; n) are estimated by thethree factor time series regressions. The regression estimates are reported in Table 1.

50

1965 1970 1975 1980 1985 1990 1995 2000-0.4

-0.2

0

0.2

0.4

0.6

0.8

Win

ner H

ML

beta

h

Panel A.

1965 1970 1975 1980 1985 1990 1995 2000-0.4

-0.2

0

0.2

0.4

0.6

0.8

Lose

r HM

L be

ta h

Panel B.

Figure 3. Plots of the book-to-market factor betas of the winner and the loser

The book-to-market factor beta for the winner is hpt¡1 =Pni=1wit¡1hi where wit¡1 = 1 if rit¡1 =

max1·j·n rjt¡1 and wit¡1 = 0 otherwise. The weights for the loser strategy are such that wit¡1 = 1 if

rit¡1 = min1·j·n rjt¡1 and wit¡1 = 0 otherwise. The book-to-market factor betas for the style portfolios

(hi, i = 1; ¢ ¢ ¢ ; n) are estimated by the three factor time series regressions. The regression estimates arereported in Table 1.

51