Momentum is not Momentum Robert Novy-Marx | University of Chicago and NBER This Draft: June, 2008 Abstract Momentum is completely explained by firms’ returns twelve to seven months prior to portfolio formation, not by a tendency of rising and falling stocks to keep rising and falling. This result holds when returns are value or equal weighted, across size quin- tiles, and across sample periods. There is also considerably more momentum in large stocks than is commonly recognized. Attenuation bias introduced by including recent returns in the typical momentum strategy sorting criteria is particularly acute for large stocks, biasing large cap momentum strategies’ returns toward zero. A strategy that buys winners and sells losers, defined as the upper and lower quintiles of cumulative returns over the first half of the year preceding portfolio formation, generates average returns of over twelve percent per year among the largest, most liquid stocks. Keywords: Momentum, factor models. JEL Classification: G12. I would like to thank Gene Fama, Andrea Frazzini, Toby Moskowitz, and Milena Novy-Marx. Financial support from the Center for the Research in Securities Prices at the University of Chicago Graduate School of Business is gratefully acknowledged. | University of Chicago Graduate School of Business, 5807 S Woodlawn Avenue, Chicago, IL 60637. Email: [email protected].
Transcript
Momentum is not Momentum�
Robert Novy-Marx|
University of Chicago and NBER
This Draft: June, 2008
Abstract
Momentum is completely explained by firms’ returns twelve to seven months prior
to portfolio formation, not by a tendency of rising and falling stocks to keep rising and
falling. This result holds when returns are value or equal weighted, across size quin-
tiles, and across sample periods. There is also considerably more momentum in large
stocks than is commonly recognized. Attenuation bias introduced by including recent
returns in the typical momentum strategy sorting criteria is particularly acute for large
stocks, biasing large cap momentum strategies’ returns toward zero. A strategy that
buys winners and sells losers, defined as the upper and lower quintiles of cumulative
returns over the first half of the year preceding portfolio formation, generates average
returns of over twelve percent per year among the largest, most liquid stocks.
Keywords: Momentum, factor models.
JEL Classification: G12.
�I would like to thank Gene Fama, Andrea Frazzini, Toby Moskowitz, and Milena Novy-Marx. Financial
support from the Center for the Research in Securities Prices at the University of Chicago Graduate Schoolof Business is gratefully acknowledged.
|University of Chicago Graduate School of Business, 5807 S Woodlawn Avenue, Chicago, IL 60637.
1 It is common to skip one month (or one week) prior to portfolio formation, to remove the large short-horizon reversals associated with bid-ask bounce, which cannot be exploited by any implementable trad-ing strategy. Considerable attention has also focused on the length of the holding period after portfolioformation. For example, Jegadeesh and Titman (1993) consider the “J=K-strategies,” which form portfo-lios based on stock performance over the previous J months, and hold the portfolios for K months, whereJ, K 2 f3, 6, 9, 12g.
2 Quintiles defined by NYSE breaks.
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2.2 Results
Figure 1 shows the average one month returns (left), volatilities (center), and Sharpe ratios
(right), for various momentum strategies. The strategies are all winners-minus-losers quin-
tile portfolios. They differ based on the sorting criteria, i.e., what constitutes “winners” and
“losers.” Value-weighted results appear as blue (dark) bars, while equal-weighted results
are depicted in green (light).
The top panel shows results for strategies based on portfolios sorted on cumulative re-
turns from twelve to “lag” months (inclusive) prior to portfolio formation, i.e., MOM12,lag.
The second set of bars (lag = 2) in each figure in the top panel corresponds to the “stan-
dard” momentum strategies, which form portfolios on the basis of performance over the
preceding year, excluding the month immediately prior to portfolio formation.
The top, left hand figure shows a pronounced hump shape in the value-weighted return
spread. Sorting based on the previous year’s returns, and sorting based on returns in the
single month beginning twelve months prior to portfolio formation, both yield high-low
spreads of about 60 basis points per month. The strategy based on returns from only the
first three months of the year prior to portfolio formation, MOMvw12,10, produces almost the
same spread, and a higher Sharpe ratio, than the momentum strategy based on the standard
criterion of past performance over the first eleven months of the year prior to portfolio
formation, MOMvw12,2 (spreads of 80.2 basis points per month vs. 81.8 basis points per
month and annual Sharpe ratios of 0.67 vs. 0.53). The hump peaks at about 113 basis
points per month for MOMvw12,6. The strategy that only uses returns from the first half
of the year prior to portfolio formation, MOMvw12,7, generates 107 basis points per month.
A similar, though less pronounced pattern is observed with the equal-weighted strategies.
The spreads are relatively flat on strategies that ignore recent returns out to as much as six
months prior to portfolio formation. After that, skipping more months prior to portfolio
formation begins to significantly reduce the strategies’ return spreads.
The top, middle figure shows a strong, monotonic decreasing relation between months
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1 2 3 4 5 6 7 8 9 101112−0.5
0
0.5
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MOM(12,lag)A
ve
rag
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etu
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lag1 2 3 4 5 6 7 8 9 101112
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lag1 2 3 4 5 6 7 8 9 101112
−0.5
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Sh
arp
e R
atio
(a
nn
ua
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lag
1 2 3 4 5 6 7 8 9 101112131415−0.5
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rcent/
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lag1 2 3 4 5 6 7 8 9 101112131415
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MOM(lag,lag)
Sharp
e R
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lag1 2 3 4 5 6 7 8 9 101112131415
−0.5
0
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MOM(lag+5,lag)
Sharp
e R
atio (
annual)
lag
Figure 1: Returns, Standard Deviations, and Sharpe Ratios to Momentum StrategiesThe figures show average monthly returns, monthly standard deviations, and annual Sharpe ratios to winner-minus-looser quintile portfolios. The top panel shows MOM12,lag, where portfolios are formed based onreturns over the period starting twelve months prior to portfolio formation and ending lag months prior toportfolio formation (inclusive). The middle panel shows MOMlag,lag, where portfolios are formed basedon returns lag months prior to portfolio formation. The bottom panel shows MOMlagC5,lag, based on thereturns over a six month window beginning lag+5 months prior to portfolio formation and ending lag monthsprior to portfolio formation. Blue (dark) bars show value-weighted results, while green (light) bars showequal-weighted results. The sample period covers January 1974 through January 2007.
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skipped prior to portfolio formation and the volatility of the strategies. This relation is es-
pecially pronounced with the equal-weighted strategies. The observed pattern is consistent
with mean-reverting stochastic volatility. The portfolio selection criteria, which select for
stocks that have experienced large movements, bias both the winner and loser portfolios
toward stocks that have high volatility, and consequently toward stocks that have volatili-
ties higher than their own long-run average. A longer interval prior to portfolio formation
provides time for these volatilities to revert downward.
The top, right figure shows the strategies’ Sharpe ratios. The pattern here is largely
inherited from the return spreads, “tilted” counter-clockwise by the downward sloping
volatility profile. The value-weighted Sharpe ratio profile is strongly hump-shaped, peak-
ing at MOMvw12,6. The equal-weighted Sharpe ratio profile is initially upward-sloping over
the first half of the year, with strategies that skip more months prior to portfolio formation
having higher Sharpe ratios, though the profile is relatively flat beyond six months.
The middle panel shows strategies based on portfolios sorted on a single month’s re-
turns, i.e., MOMlag,lag. The general trend is upward sloping spreads, for both value and
equal-weighted strategies, which fall off a cliff after twelve months. The profile is largely
consistent with the results of Jegadeesh (1990), and the return-response profile studied in
Heston and Sadka (2008) in their study of seasonality in the cross-section of returns. The
abrupt drop-off after one year is one of the most striking features of the figure, and poses a
significant hurdle for stories, either rational or behavioral, that attempt to explain momen-
tum.
The bottom panel shows strategies based on portfolios sorted on a six month return
window, up to and including the month “lag” months prior to portfolio formation, i.e.,
MOMlagC5,lag. The hump-shaped pattern in return spreads and Sharpe ratios is pronounced,
for both value and equal-weighted strategies. The volatility profile shows that the portfolios
“settle down” after seven or eight months.
All the patterns look the same in the “early” and “late” periods (1974-1991 and 1991-
2007), though the spreads and Sharpe ratios are slightly higher in the first half of the sample,
7
as momentum was both less profitable and riskier in the second half of the sample (and in
particular, from 2000 on).
3 Intermediate Horizon versus Recent Past Performance
Figure 1 suggests that momentum derives largely from a firm’s past performance at interme-
diate horizons, not its recent performance. This section shows this formally, both paramet-
rically, using Fama-MacBeth (1973) regressions, and non-parametrically, using portfolios
constructed by double-sorting on firms’ past performances at the two horizons.
3.1 Fama-MacBeth Regression Results
Table 1 reports results of Fama-MacBeth regressions of returns on intermediate horizon and
recent past performance, r12,7 and r6,2. The first four specifications show equal-weighted
results, while the next four show value-weighted results.3
Specifications (1), (2) and (3) show that past performance at all three horizons (twelve to
two months, twelve to seven months, and six to two months) predict returns in univariate re-
gressions, but that the correlation is strongest with intermediate horizon past performance.
Specification (4) shows that recent past performance provides little incremental informa-
tion beyond that contained in intermediate horizon past performance. Moreover, including
both past performances as explanatory variables increases the significance of the coefficient
on intermediate horizon past performance relative to its univariate estimate, while reducing
the significance of the coefficient on recent past performance.4
3 The value-weighted results employ weighted least squares in each cross-sectional regression, wherethe weights are firms’ market capitalizations at the beginning of the period. That is, while each periodthe equal-weighted results minimize the residual cross-sectional variance (i.e., minimize
Pj �2
tj ), the value-weighted results minimize the residual cross-sectional variance weighted by firms’ market capitalizations(i.e., minimize
Pj me(t�1)j �2
tj ).4 Table 1 reports results using all of the data. Winsorizing the data at the one and 99 percent levels
strengthens all of the results, increasing both the magnitudes and the significances of all of the coefficientestimates. These improvements are only marginal for the value-weighted results, but significant for theequal-weighted results, where outlying observations on small firms weaken the correlations.
The table reports the results of Fama-MacBeth regressions of firm returns on past perfor-mance. Past performance is measured at horizons of twelve to two months (r12,2), twelve toseven months (r12,7), and six to two months (r6,2). The equal-weighted results minimize the
residual variance in each cross-section (i.e., minimizeP
j �2tj ), while the value-weighted results
minimize the residual variances weighted by firms’ beginning of period market capitalizations(i.e., minimize
Pj me(t�1)j �2
tj ).
All of these results hold in both the early and late subsamples of the data, January
1974 to July 1990 and August 1990 to January 2007, respectively. Subsample results
are provided in appendix A.1. Regressions that control for size and book-to-market also
yield qualitatively identical results.5 These results strongly suggest that momentum derives
mainly from intermediate horizon past performance.
3.2 Double sorts on “early” and “late” returns
Another way to identify, non-parametrically, whether intermediate horizon or recent past
performance drives momentum is to look at the returns to portfolios formed by double
5 In these regressions, the independent variable is a stock’s return minus the value-weighted return toa benchmark portfolio consisting of stocks with similar size and industry-adjusted book-to-market. Thesebenchmark portfolios are constructed using the first two sorts of Daniel et. al. (1997). First stocks are sortedinto quintile portfolios on the basis of market capitalization, using NYSE breaks. Each of these portfoliosis then sorted into quintiles on the basis of industry-adjusted book-to-market (book-to-market divided byindustry book-to-market, where the industry definitions are the Fama-French 49), again using NYSE breaks.
All the results in this paper are robust to controlling for size and book-to-market using this procedure.
9
sorting on these two variables. I use an unconditional double sorting procedure, which
does not produce any “thin cells” because return auto-correlations are low and the two
sorts consequently relatively uncorrelated.6
Table 2 reports the average monthly returns to these double sorted portfolios. The
results are striking. Firms that had high returns from twelve to seven months in the past
have high expected returns, even if they are in the bottom recent return quintile. Firms
that had low returns from twelve to seven months in the past have low expected returns,
even if they are in the top recent return quintile. Conversely, recent winners perform poorly
on average if they had poor intermediate horizon past performance, while recent losers
perform well on average if they had strong intermediate horizon past performance.
Sorting on intermediate horizon past performance also generates significantly greater
variation in expected returns than sorting on recent past performance. On a value-weighted
basis, the return spreads between winner and loser portfolios based on intermediate horizon
past performance are nearly an order of magnitude higher than those based on recent past
performance. On an equal-weighted basis, the return spreads based on intermediate horizon
past performance are roughly twice as large as those based on recent past performance.
Again, these patterns hold in both the early and late subsamples of the data (January
1974 to July 1990 and August 1990 to January 2007, respectively). Subsample results are
again provided in appendix A.1.
6 The corners are slightly thick by number of firms, because the early return winners and losers portfoliosare both biased towards volatile stocks, which are more likely to end up in the late return winners and losersportfolios. However, no cell averages fewer that 160 firms (2.98% of the total). The corners are slightly thinby market capitalization (except for the winners-winners corner), because extreme returns are more commonamong small stocks.
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TABLE 2
AVERAGE MONTHLY EXCESS RETURNS
(BASIS POINTS PER MONTH) TO PORTFOLIOS
DOUBLE SORTED ON r12,7 AND r6,2
Panel A: value-weighted portfolio returns
r6,2 quintile
(All) (L) (2) (3) (4) (W) (W-L)
(All) 58.1 51.0 67.8 60.7 51.1 67.9 16.9
(L) 0.3 -29.9 19.2 20.5 -4.1 5.7 35.7
(2) 42.7 42.1 77.4 54.8 24.6 54.9 12.8
(3) 59.7 64.1 84.3 50.0 58.3 74.1 10.0
(4) 72.7 46.7 71.3 82.2 86.9 78.4 31.7
(W) 107.3 91.2 97.1 117.7 106.4 127.6 36.4r 12
,7q
uin
tile
(W-L) 107.3 121.1 77.9 97.2 110.5 112.9
Panel B: equal-weighted portfolio returns
r6,2 quintile
(All) (L) (2) (3) (4) (W) (W-L)
(All) 94.8 70.2 93.0 97.3 93.1 116.1 45.9
(L) 53.9 52.7 44.5 60.4 49.2 63.6 10.9
(2) 93.7 80.2 93.0 99.2 86.8 119.7 39.5
(3) 102.9 78.4 109.4 98.7 104.2 131.9 53.5
(4) 114.3 83.9 112.5 113.3 123.5 144.0 60.1
(W) 140.2 105.5 150.7 147.9 143.1 172.5 67.0r 12
,7q
uin
tile
(W-L) 86.3 52.8 106.2 87.9 93.9 108.9
Average monthly excess returns (basis points per month) to portfolios double sorted onr12,7 and r6,2 (stock’s cumulative return over the period twelve to seven months (inclu-sive) prior to portfolio formation, and a stock’s cumulative return over the period six to twomonths (inclusive) prior to portfolio formation, respectively). The sample period coversJanuary 1974 to January 2007.
11
Table 3 reports the results of time-series regressions of the returns to the winners-minus-
losers spread portfolios, in each direction, on UMD (value-weighted and equal-weighted,
respectively).7
The left hand side of Table 3 shows that, on a value-weighted basis, recent winners ac-
tually fail to significantly outperform recent losers across intermediate horizon return quin-
tiles. The root-mean-squared return spread is 27.8 basis points per month, and a Gibbons-
Ross-Shanken (GRS) test fails to reject that the true expected return spreads are jointly zero
(F5,392 D 0.66, for a p-value of 65.6%). These small return spreads are observed despite
the fact that the strategies have large UMD loadings. They consequently have significant
negative alphas relative to UMD. The root-mean-squared pricing error relative ot UMD is
45.5 basis points per month, and a (GRS) test strongly rejects that the true pricing errors
are jointly different from zero (F5,391 D 3.32, for a p-value of 0.6%).
At the same time, intermediate horizon winners significantly outperform intermediate
horizon losers across recent return quintiles. The root-mean-squared return spread is 107.0
basis points per month, and a (GRS) test emphatically rejects that the true expected return
spreads are jointly zero (F5,392 D 8.14, for a p-value of 0.000%). These strategies have
smaller UMD loadings, and consequently generate significant positive alphas relative to
UMD. The root-mean-squared pricing error relative to UMD is 56.2 basis points per month,
and a (GRS) test strongly rejects that the true pricing errors are jointly different from zero
(F5,391 D 5.23, for a p-value of 0.000%).
The right hand side of Table 3 shows the same basic patterns for the equal-weighted
strategies. The sorts on intermediate horizon past performance generate larger return spreads
than the sorts on recent past performance, though a GRS test does reject the irrelevance of
recent performance.8 Despite their lower returns, the recent horizon winner-minus-loser
7 I do not include the Fama-French factors in these regressions, both because the momentum strategies’unconditional loadings on these factors are generally small and insignificant, and because momentum portfo-lios’ loadings on these factors vary over time. Results that control for size and book-to-market by adjustingstocks’ returns using the benchmark portfolios described in footnote 5 yield qualitatively identical results.
8 The root-mean-squared return spread on the high-minus-low portfolios sorted on recent past perfor-mance is 50.3 basis points per month, and 92.2 basis points per month for the portfolios sorted on intermedi-ate horizon past performance. In both cases GRS tests reject that the spreads are jointly zero (F5,392 D 5.32
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TABLE 3
CONDITIONAL WINNER-MINUS-LOSER
RETURN SPREADS AND UMD LOADINGS TO
PORTFOLIOS DOUBLE SORTED ON r12,7 AND r6,2
value-weighted results equal-weighted results
E[re ] ˛ UMD E[re ] ˛ UMD
Panel A: MOM6,2, conditioned on r12,7
low 35.7 -33.6 0.890 10.9 -47.3 0.919
[1.22] [-1.46] [16.20] [0.39] [-2.18] [18.28]
2 12.8 -57.6 0.905 39.5 -06.9 0.772
[0.49] [-3.07] [20.24] [1.69] [-0.39] [18.86]
3 10.0 -61.6 0.919 53.5 -0.9 0.815
[0.39] [-3.46] [21.68] [2.35] [-0.05] [21.80]
4 31.7 -33.0 0.831 60.1 7.5 0.768
[1.20] [-1.62] [17.13] [2.70] [0.46] [20.09]
high 36.4 -32.0 0.879 67.0 18.9 0.739
[1.33] [-1.53] [17.64] [3.12] [1.18] [20.01]
Panel B: MOM12,7, conditioned on r6,2
low 121.1 71.6 0.636 52.8 8.2 0.607
[4.83] [3.29] [12.29] [2.62] [0.49] [15.69]
2 77.9 24.2 0.690 106.2 69.5 0.473
[3.21] [1.21] [14.53] [6.55] [5.07] [14.90]
3 97.2 37.0 0.733 87.9 55.0 0.533
[3.72] [1.76] [15.42] [5.25] [4.27] [17.89]
4 110.5 58.1 0.673 93.9 57.6 0.508
[4.89] [3.21] [15.60] [5.71] [4.46] [16.98]
high 121.9 73.2 0.625 108.9 74.4 0.427
6-2
retu
rnq
uin
tile
(r6
,2)
12
-7re
turn
qu
inti
le(r
12
,7)
[5.28] [3.77] [13.52] [6.80] [5.52] [13.72]
The table reports the average monthly returns to momentum strategies basedon recent past performance (winner-minus-loser quintiles) constructed withinintermediate horizon past performance quintiles (Panel A), and the averagemonthly returns to momentum strategies based on intermediate horizon pastperformance constructed within recent past performance quintiles (Panel B).The table also reports results of time-series regressions of these strategies’ re-turns on UMD. The sample period covers January 1974 to January 2007.
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portfolios have much larger UMD loadings, and consequently have small, generally in-
significant alphas relative to UMD, which are negative on average. The intermediate hori-
zon winner-minus-loser portfolios have large, significant alphas relative to UMD.9
4 Information in Intermediate-Horizon Past Performance
The previous section shows that momentum derives primarily from intermediate horizon,
not recent, past performance. I now show that what information is contained in recent past
performance regarding expected returns is largely redundant to the “older” information
contained in intermediate horizon past performance.
This is done by analyzing the relations between the returns to the standard momentum
strategy based on returns twelve to two months prior to portfolio formation, MOM12,2, and
the strategies that use only the “early” and “late” information used to construct the standard
strategy, MOM12,7 and MOM6,2, respectively.
4.1 Strategies’ return characteristics
Table 4 shows the basic properties of these strategies. Results are presented for both value
and equal-weighted strategies. The first panel considers MOM12,2, and shows the standard
results. MOMvw12,2 and MOMew
12,2 generate significant return spreads of 82 and 93 basis
points per month, respectively. The three Fama-French factors explain essentially none of
the strategies’ return variation, but the strategies are well priced by UMD.
The second panel considers MOM12,7. The results are similar, except that the strategies
have a significant positive alpha relative to UMD. MOMvw12,7 and MOMew
12,7 generate highly
significant returns of 107 and 87 basis points per month, respectively. The three Fama-
and F5,392 D 13.57, respectively, both for p-values of 0.000%).9 The root-mean-squared pricing errors relative to UMD for the spread portfolios sorted on recent and
intermediate past performance are 28.7 and 57.0 basis points per month, respectively. In both cases GRStests reject that the pricing errors are jointly zero (F5,391 D 3.70, for a p-value of 0.3%, and F5,391 D 10.39,for a p-value of 0.000%, respectively).
Momentum strategy average returns and factor loadings, January 1974 to Jan-uary 2007. MOMn,m denotes the winner-minus-loser portfolio (quintiles), wherewinners and losers are based on cumulative returns from n to m months (inclusive)prior to portfolio formation. The super-scripts signify whether portfolio returns arevalue-weighted (vw) or equal-weighted (ew).
15
French factors explain little of the strategies’ return variation. Their alphas relative to
UMD, both significant, are 46 and 36 basis points per month, respectively.
The third panel considers MOM6,2. In contrast to the 12-2 and 12-7 strategies, MOMvw6,2
and MOMew6,2 do not generate significant return spreads. They do load heavily on UMD,
however, even more so than the 12-7 strategies. These strategies consequently have nega-
tive alphas relative to UMD, of -55.5 and -21.8 basis points per month, respectively. Like
the 12-2 and 12-7 strategies, they are essentially orthogonal to the Fama-French factors.
4.2 Spanning tests
Table 5 presents results from spanning tests conducted using the standard momentum
strategies, MOM12,2, and the momentum strategies based only on the “early” and “late”
information employed by the standard strategy, MOM12,7 and MOM6,2.
These spanning tests regress a “test” strategy’s returns on the returns to one or more
“explanatory” strategies. The intercept’s test-statistic is the information ratio of the test
strategy benchmarked to the mimicking portfolio constructed from the explanatory strate-
gies. An insignificant intercept suggests an investor could achieve statistically identical
expected returns, while exposing herself to less volatility, trading only in the explanatory
strategies. A statistically significant intercept suggests the test strategy improves the invest-
ment opportunity set, and consequently contributes significant information.10
The table shows both value-weighted (specifications 1-4) and equal-weighted (specifi-
cations 5-8) results. Both these sets of results support the hypothesis that the information
in recent past performance is largely redundant to that in intermediate horizon past perfor-
mance. Specification (1) shows that the returns to the 12-7 strategies are large, and more
10 For example, the three Fama-French factors span long-run reversals, because the abnormal returns to“contrarian” strategies, which buy long-term losers and sell-long term winners, are insignificant relative tothe Fama-French factors. A strategy that only takes positions in the Fama-French factors can therefore gen-erate statistically identical expected returns, without exposing an investor to the residual variance from theregression of the contrarian strategies returns on the Fama-French factors. Conversely, momentum is “out-side” the span of the Fama-French factors, because no combination of the Fama-French factors “explains” theabnormal returns to momentum, in the sense that it generates a large, significant three-factor alpha. Addingmomentum to the Fama-French factors consequently significantly improves the investment opportunity set.
Time-series regressions employing the returns to momentum strategies constructed using past perfor-mance over different horizons. MOMn,m is the returns to the winner-minus-loser portfolio (quintiles),where winners and losers are based on cumulative returns from n to m months (inclusive) prior to portfo-lio formation. The sample period covers January 1974 through January 2007.
17
significant than the returns to the 12-2 strategies, while the returns to the 6-2 strategies are
insignificant. Specifications (2) and (3) show that the 12-7 strategies explain the returns to
both the 12-2 strategies and the 6-2 strategies, while the converse are false. After control-
ling for MOM12,2, the “unexplained” returns to MOM12,7 are 54.9 and 26.9 basis points
per month (value-weighted and equal-weighted), with t-statistics of almost four and almost
three. MOM6,2, explains little of the abnormal returns to either MOM12,2, or MOM12,7.
After controlling for MOM6,2, the “unexplained” returns to MOM12,2 are 65.3 and 50.1
basis points per month (value and equal-weighted, respectively), and the “unexplained” re-
turns to MOM12,7 are 101.7 and 64.8 basis points per month (value and equal-weighted,
respectively), all with t-stats above four.
Once again, qualitatively identical results hold in both the early and late subsamples of
the data (January 1974 to July 1990 and August 1990 to January 2007), and are provided
in appendix A.1.
5 Alternative Momentum Factor
UMD fails to accurately price either the 12-7 and 6-2 strategies (Table 4), but the 12-7
strategy accurately prices both the 12-2 and 6-2 strategies (Table 5). This suggests that a
“UMD-like” factor constructed using criteria based on intermediate horizon past returns
should perform better than canonical UMD in pricing tests.
I consequently construct “UMD12,7” along the lines of UMD, using the corner port-
folios from the intersection of a tertile sort on past returns with the “big/small” sort on
market equity. The procedure differs only insofar as “winners” and “losers” are determined
by performance twelve to seven months, not twelve to two months, prior to portfolio for-
mation. UMD12,7 has the same average returns as UMD over the sample (80 basis points
per month), but is significantly less volatile (monthly standard deviations of 3.0 and 4.2
percent, respectively), resulting in a significantly higher Sharpe ratio (0.92, as opposed to
0.65). The monthly returns to the two factors are 81.6% correlated over the sample. Trail-
18
Jan 76 Jan 81 Jan 86 Jan 91 Jan 96 Jan 01 Jan 06−4
−3
−2
−1
0
1
2
3
4
5
Date
traili
ng o
ne y
ea
r a
ve
rag
e m
on
thly
re
turn
s (
%)
UMD
UMD(12,7)
Figure 2: Trailing one year average monthly returns to UMD and UMD12,7
UMD12,7 is constructed employing the same methodology used to construct UMD, exceptthat winners and losers are defined using past performance over the period 12 to seven months
in the past. That is, UMD12,7 is 12(large winners�large losersCsmall winners�small losers),
where “large” (“small”) denotes firms above (below) the median market capitalization ofNYSE firms, respectively, and “winners” (“losers”) are firms that performed better (worse)than 70 percent of NYSE firms over the first half of the preceding year.
ing one-year average monthly returns to UMD and UMD12,7 are shown in figure 2, in blue
(dark) and green (light), respectively.
Table 6 shows that UMD12,7 does indeed perform better pricing portfolios sorted on
r12,7 and r6,2. The first panel shows that UMD12,7 prices UMD, but the converse is false.
The second panel demonstrates that both factors price the 12-2 strategies. The third and
fourth panels show that UMD12,7 prices the 12-7 and 6-2 strategies, while UMD does not.
19
TABLE 6
REGRESSIONS OF MOMENTUM STRATEGY RETURNS
ON UMD AND UMD12,7
dependent factor loadings
variable const. UMD UMD12,7 adj.-R2 (%)
UMD -11.7 1.129 66.9[-0.95] [28.28]
UMD12,7 33.2 0.593[3.77]
MOMvw12,2 -13.4 1.223 90.8
[-1.64] [62.58]
-29.3 1.339 62.4[-1.73] [25.67]
MOMew12,2 16.0 0.992 69.9
[1.17] [30.31]
7.5 1.080 43.4[0.39] [17.47]
MOMvw12,7 45.9 0.790 55.1
[3.05] [22.05]
0.2 1.351 84.8[0.03] [47.04]
MOMew12,7 36.3 0.650 54.8
[2.91] [21.92]
9.3 0.978 65.1[0.84] [27.22]
MOMvw6,2 -52.5 0.920 62.0
[-3.48] [25.46]
-30.1 0.616 14.5[-1.30] [8.24]
MOMew6,2 -22.0 0.868 56.3
[-1.36] [22.61]
-8.6 0.682 18.1[-0.38] [9.42]
Momentum strategy factor loadings, over the sample period Jan-uary 1974 to January 2007. UMD comes from Ken French’s web-site. UMD12,7 is constructed using the methodology used to constructUMD, but the “winner” and “loser” tertiles are based on returns twelveto seven months prior to portfolio formation. MOMn,m is the winner-minus-loser portfolio (quintiles), where winners and losers are basedon cumulative returns from n to m months (inclusive) prior to portfo-lio formation. The super-scripts signify whether portfolio returns arevalue-weighted (vw) or equal-weighted (ew).
20
UMD12,7 also performs better pricing the 25 portfolios double sorted on r12,7 and r6,2,
employed in Tables 2 and 3. The root-mean-squared pricing errors of the value-weighted
returns to the 25 portfolios, relative to the three Fama-French factors plus UMD, is 30.1
basis points per month, and a GRS test rejects the hypothesis that the pricing errors are
jointly zero at the 0.000% level (F25,368 D 2.86). Relative to the Fama-French factors and
UMD12,7 the root-mean-squared pricing error falls to 21.2 basis points per month. A GRS
test again rejects the hypothesis that the pricing errors are jointly zero, but less emphatically
(F25,368 D 2.28, for a p-value of 0.1%).
Improvements are smaller using equal-weighted returns. Using UMD12,7 instead of
UMD, the root-mean-squared pricing error falls from 27.7 to 22.9 basis points per month,
but in both cases GRS tests reject that the pricing errors are jointly zero at the 0.000% level
(F25,368 D 4.26 and F25,368 D 3.27, respectively).11
6 Relation to Other Known Results
This section places my main results, that intermediate horizon past performance, not recent
past performance, drives momentum, in the context of the literature. It also demonstrates
the robustness of the results.12 In particular, this section shows that:
1. the results hold after controlling for the twelve month effect identified by Jegadeesh
(1990) and recently studied in greater detail by Heston and Sadka (2008);
2. the results hold across size quintiles, though recent past performance is a marginally
significant contributor to momentum in the smallest stocks;
11 UMD12,7 also performs better than UMD pricing the 25 portfolios double sorted on r12,2 and size,which is somewhat surprising given that UMD is constructed using r12,2. The root-mean-squared pricingerror on the 25 portfolios is 22.9 basis points per month relative to FF3+UMD, but falls to 21.3 basis pointsper month relative to relative to FF3+UMD12,7. The GRS F-statistic on the pricing errors is also lower usingUMD12,7.
12 As noted earlier, the main results hold in both the early and late subsamples of the data, and are robust tocontrolling for size and value effects by adjusting returns using size and book-to-market matched benchmarkportfolios.
21
3. momentum is driven primarily by intra-industry effects, and to a lesser extent by
industry momentum, though what little short horizon momentum exists is driven
by intra-industry lead-lag effects like those analyzed in Hou (2007), facts that are
consistent with both Moskowitz and Grinblatt’s (1999) result that industries largely
explain momentum, and Asness, Porter and Stevens’ (2000) seemingly contradictory
result that momentum is stronger within industries;
4. intermediate horizon momentum cannot be explained by capital gains overhang or
disposition effects, though capital gains overhang does explain the poor January per-
formance of intermediate horizon momentum strategies, as it does the poor January
performance of momentum strategies more generally; and
5. the results are essentially unrelated to the consistency of performance result of Grin-
blatt and Moskowitz (2004), which appear to be driven by a non-linearity in the
relation between past performance and expected returns, and not by anything related
to consistency.
6.1 Twelve Month Effect
Performance twelve months prior to portfolio formation is particularly important for pre-
dicting expected returns (Jegadeesh (1990), Heston and Sadka (2008)), especially for small
stocks. While this twelve month effect contributes to the profitability of momentum strate-
gies, it cannot explain the difference in the performance of strategies based on intermediate
horizon and recent past performance.
Panel A of Table 7 reports the results of Fama-MacBeth regressions of returns on past
performance over the month twelve prior, and the periods 11 to seven months and six to
two months prior. Intermediate horizon past performance, even excluding month twelve
prior, contains significant predictive power for forecasting returns. The coefficient on r11,7
is highly significant, especially in the value-weighted regression. The coefficients on r12,12
are also highly significant, especially in the equal-weighted regressions, consistent with the
22
TABLE 7
TESTING THE ROLE OF TWELVE MONTH EFFECT
Panel A: slope coefficients (�102) and [test-statistics] from
Fama-MacBeth regressions of the form rtj D ˇ̌̌ 0xtj C �tj
The table reports Fama-MacBeth regressions of firms’ returns on their prior performance in the monthtwelve prior (r12,12), and over the periods eleven to seven months prior (r11,7) and six to two months prior(r6,2). The table also reports the results of time-series regressions of the standard 12-2 momentum strategy’sreturns (MOM12,2) on the returns to momentum strategies based on performance in the month twelve prior(MOM12,12), and over the period eleven to seven months prior (MOM11,7). The sample covers January1974 through January 2007.
results of Jegadeesh (1990) and Heston and Sadka (2008).13 The information in r11,7 is,
however, distinct from that in r12,12. The coefficient on r11,7 is not significantly affected
by the inclusion of r12,12 as an explanatory variable.14
Panel B of Table 7 shows that MOM11,7 alone explains the abnormal returns to the
standard 12-2 strategy, while MOM12,12 does not. MOM11,7 prices MOM12,2, in the sense
13 While the coefficients on r12,12 are larger than those on r11,7, there is more than twice the variation inr11,7 as there is in r12,12, and the two variables consequently contribute similarly to explaining the variationin returns.
14 The coefficients on r11,7 is 0.548, with a test statistic of 2.91, in an equal-weighted regression thatexcludes r12,12. In a value-weighted regression the coefficient is 1.317, with a test statistic of 4.49.
23
that the information ratio of the standard 12-2 strategy is insignificant relative to the strategy
constructed on the basis of past performance over the period 11 to seven months prior.
MOM11,7 prices MOM12,2 alone or in conjunction with MOM12,12, value-weighting or
equal-weighting the strategy returns. In contrast, MOM12,12 does not price MOM12,2;
MOM12,2 has a large information ratio relative to MOM12,12.
MOM11,7 and MOM12,12 also have large information ratios with respect to each other,
reflecting the distinct nature of the information in r11,7 and r12,12. On a value-weighted ba-
sis, MOM11,7 generates 77.8 basis points per month relative to MOM12,12, with a test statis-
tic of 3.64, while MOM12,12 generates 45.8 basis points per month relative to MOM11,7,
with a test statistic of 2.36. On an equal-weighted basis, MOM11,7 generates 49.5 basis
points per month relative to MOM12,12, with a test statistic of 2.58, while MOM12,12 gen-
erates 63.4 basis points per month relative to MOM11,7, with a test statistic of 4.84. That
is, both r11,7 and r12,12 matter, though intermediate horizon past performance matters more
for large stocks, while the twelve month effect is more important for small stocks.
Double sorting on r12,12 and r11,7 produces results consistent with those presented in
Table 7. Results of these double sorts, which produce large, significant return spreads
in both directions, are provided in Appendix A.2 (Table 19). These return spreads are
significantly larger in the r11,7 direction when value-weighted, but slightly larger in the
r12,12 direction when equal-weighted, reflecting the importance of past performance at both
horizons, and the particular importance of the twelve month effect among small stocks.
6.2 Size Effects
The value-weighted return spread between portfolios of “winners” and “losers” increases
when the most recent six months of returns is ignored in the portfolio selection procedure
(Table 4). It is also well known that standard 12-2 momentum is concentrated in small
stocks. These facts suggest that robustness tests should include an investigation of the
effects of firm size.
24
This section consequently discusses results of both parametric and non-parametric tests,
which control for size, of the relation between past performance and expected returns. The
parametric tests consist of Fama-MacBeth regressions, performed within size quintiles,
of returns on past performance. The non-parametric tests investigate the performance of
momentum strategies constructed within size quintiles. Time-series average characteristics
of the size portfolios, which were constructed using NYSE breaks, are given in Table 8.
average capitalization ($106) 45 227 513 1,341 10,569
Market equity quintile break points based on NYSE stocks only. The sample coversJanuary 1974 through January 2007.
Panel A of Table 9 reports slope coefficients from Fama-MacBeth regressions, per-
formed within size quintiles, of returns on past performance at both recent and interme-
diate horizons. Because the value-weighted and equal-weighted results are similar after
controlling for size, only value-weighted results are presented. In these regressions, the
coefficient on intermediate horizon past performance is large and highly significant across
size quintiles. In contrast, the coefficient on recent past performance is only significant in
the smallest two size quintiles.15 That is, it appears that while intermediate horizon mo-
15 One striking difference between the value and equal-weighted results is the poor equal-weighted perfor-mance of the micro cap strategies relative to their value weighted performance. Within the small quintile thecoefficient on r6,2 in equal-weighted Fama-MacBeth regressions is only 0.191 (�10�2), less than one-fifththat of the value-weighted result reported in Table 9, and insignificant. This results because the smallest 10%of stocks exhibit no momentum. The absence of momentum among the smallest stocks significantly reducesthe slope coefficient on past performance in equal-weighted Fama-MacBeth regressions, despite the fact thatthese stocks represent, on average, only 0.06% of the market by capitalization. Interestingly, the slope co-efficient on intermediate horizon past performance for micro-cap stocks remains large and significant. Inequal-weighted Fama-MacBeth regressions the coefficient on r12,7 for the smallest stocks is 0.567 (�10�2),with a test statistic of 3.59.
25
TABLE 9
FAMA-MACBETH REGRESSION RESULTS AND MOMENTUM
STRATEGIES’ PERFORMANCE, BY SIZE QUINTILE
Panel A: slope coefficients (�102) and [test-statistics] from
Fama-MacBeth regressions of the form rtj D ˇ̌̌ 0xtj C �tj
independent size quintile
variables (small) (2) (3) (4) (big)
r12,7 0.894 0.812 1.258 1.307 1.595
[6.16] [4.91] [5.50] [5.12] [4.58]
r6,2 1.006 1.004 0.503 0.295 -0.115
[4.04] [3.94] [1.77] [0.88] [-0.27]
Panel B: strategy excess returns (basis points per month)
MOMi12,7 120.2 112.2 108.9 91.1 103.1
[7.30] [6.60] [5.43] [4.25] [4.14]
MOMi6,2 123.8 108.4 58.4 40.5 -8.6
[5.53] [4.86] [2.40] [1.56] [-0.32]
Panel C: ˛s from MOMi12,7 D ˛ C ˇ MOMi
6,2 C �t
MOMi12,7 69.0 74.7 92.1 75.3 105.4
[4.88] [4.78] [4.86] [3.96] [4.41]
Panel D: ˛s from MOMi6,2 D ˛ C ˇ MOMi
12,7 C �t
MOMi6,2 31.8 41.8 12.1 -11.3 -41.0
[1.61] [1.99] [0.51] [-0.48] [-1.55]
The table reports results of Fama-MacBeth regressions of firms’ returns on theirrecent and intermediate horizon past performances, conducted within size quin-tiles. The table also reports the average returns to momentum strategies based onrecent and intermediate horizon past performance, constructed within size quin-
tiles (MOMin,m, where winners and losers are based on cumulative returns from
n to m months (inclusive) prior to portfolio formation, and the portfolios containstocks only from size quintile i), and results of spanning tests employing htesestrategies’ returns.
mentum affects the whole market, whatever short run momentum exists is driven solely by
the bottom 6% of the market. Moreover, the significance of the coefficient on r12,7 exceeds
that on r6,2 even among the smallest stocks.
The non-parametric tests reinforce the results of the Fama-MacBeth regressions. Panel
26
B shows the average monthly returns to momentum strategies that control for firm size.
These strategies consist of winner-minus-loser quintile portfolios constructed within size
quintiles, and are denoted by MOMin,m, where winners and losers are based on cumulative
returns from n to m months (inclusive) prior to portfolio formation and the super-script i
identifies the size quintile. Again, because the equal and value weighted results are quite
similar after controlling for size, only value-weighted results are presented.
Panel B demonstrates that sorting on intermediate past performance generates large,
highly significant return spreads across size quintiles. In contrast, sorting on recent past
performance only generates large, highly significant return spreads in the smallest two size
quintiles, while generating a marginally significant return spread in the third quintile, and
insignificant spreads in the largest two quintiles.16 In fact, for the largest stocks information
regarding recent past performance is worse than useless. Skipping the six months prior to
portfolio formation in the performance criterion yields a significant increase in the returns
to large cap momentum strategies: while the large cap 12-7 strategy generates a highly
significant 103 basis points per month, the large cap 12-2 strategy generates barely half
these returns, an insignificant 56.1 basis points per month.
Panels C and D show that even though small-cap, short horizon momentum strategies
are profitable, small-cap momentum strategy are still more profitable when based on in-
termediate horizon past performance. Panels C and D present results of spanning tests
of momentum strategies based on intermediate and recent past performance, constructed
within size quintiles. Panel C shows that across size quintiles the information ratios of the
strategies based on intermediate past performance are large relative to the strategies based
on recent past performance. Panel D demonstrates that the converse is false. While some
short horizon momentum exists in the bottom 6% of the market, it appears that recent past
16 Again, the absence of momentum among the smallest stocks significantly reduces the equal-weightedmicro cap strategy’s performance relative to its value-weighted performance. Within the small quintile theequal-weighted winner-minus-loser spreads is much smaller, and less significant, than its value-weightedcounterpart. As in the Fama-MacBeth regressions, the performance reduction is more severe for recent pastperformance. The average return to the equal-weighted micro-cap momentum strategy based on recent pastperformance (r6,2) is 49.2 basis points per month, with a test-statistic of 2.11, while that based on intermediatehorizon past performance (r12,7) is 77.1 basis points per month, with a test-statistic of 4.45.
27
performance provides little information regarding future performance beyond that already
present in intermediate horizon past performance.
6.3 Industry Effects
The previous section demonstrates that while momentum is driven by intermediate horizon
past performance, some short horizon momentum exists among the smallest stocks. This
section shows that these two effects derive from different sources. Intermediate horizon
momentum is driven by intra-industry effects, and to a lesser extent by industry momentum.
Short horizon momentum, to the extent it exists, is driven by small stocks following their
industries, i.e., by intra-industry lead-lag effects like those analyzed by Hou (2007).
In order to study the role industry plays in the profitability of momentum strategies, I
employ variables that measure past industry performance and past performance relative to
industry performance. For each stock r indusn,m denotes the cumulative return to the stock’s
industry over prior months n to m (inclusive), and r reln,m denotes the cumulative returns of
the stock in excess of its industry’s return over the same period.17
Panel A of Table 10 reports results of Fama-MacBeth regressions of stocks’ returns on
past performance relative to industry, and past industry performance, measured at interme-
diate and short horizons. Two things about the results stand out.
First, intermediate horizon past performance is more important than recent past perfor-
mance, both within and across industries. The magnitudes and significances of the coeffi-
cients on intermediate horizon past performance exceed those on recent past performance,
in and across industries, in both the value and equal weighted regressions.
Second, performance relative to industry matters more for large stocks, while industry
performance matters more for small stocks. The magnitudes and significances of the coef-
ficients on the intra-industry variables are greater in the value-weighted regressions, which
put more weight on large stocks. The magnitudes and significances of the coefficients on
the industry variables are greater in the equal-weighted regressions, which put relatively
17 I employ the Fama-French 49 industries, and value-weight industry returns.
and [test-statistics] from Fama-MacBeth regressions
of the form rtj D ˇ̌̌ 0xtj C �tj
independent variables
r rel12,7 r rel
6,2 r indus12,7 r indus
6,2
equal-weighted 0.601 0.193 2.725 2.492
results [4.09] [0.83] [6.64] [4.13]
value-weighted 1.170 0.549 2.531 -0.019
results [5.73] [2.18] [4.07] [-0.03]
Panel B: non-parametric results– intra-industry
and industry momentum strategy excess returns
(basis points per month) and alphas relative to UMD
value-weighted equal-weighted
portfolio returns portfolio returns
strategy E[re ] ˛ UMD E[re ] ˛ UMD
MOMrel12,7 81.9 42.6 0.505 61.0 22.3 0.497
[5.30] [3.67] [18.31] [3.94] [1.83] [17.75]
MOMrel6,2 38.1 -12.7 0.652 26.5 -26.2 0.677
[2.14] [-1.08] [23.28] [1.30] [-1.75] [18.95]
MOMindus12,7 53.7 3.1 0.649 76.8 43.0 0.433
[2.45] [0.18] [15.47] [4.61] [3.01] [12.72]
MOMindus6,2 7.2 -50.0 0.734 73.0 24.3 0.626
[0.31] [-2.75] [16.94] [3.52] [1.48] [15.97]
The table reports results of Fam-MacBeth regressions of firms’ returns on their indus-tries’ past performance, and their past performances relative to their industries, at bothshort and intermediate horizons. r indus
n,m denotes the cumulative returns to a firm’s industry,
and r reln,m a stock’s cumulative return in excess of its industry’s return, over the months
n to m (inclusive) in the past. The table also reports the expected returns to momentumstrategies based on past industry performance, and strategies based on past performancesrelative to industry, at both recent and intermediate horizons, as well as results of time-
series regressions of these strategies’ returns on UMD. MOMindusn,m and MOMrel
n,m denotethe winner-minus-loser quintile portfolios, where winners and losers are based on cumu-lative industry returns and returns in excess of industry returns, respectively, from n to mmonths (inclusive) prior to portfolio formation. The sample covers January 1974 throughJanuary 2007.
29
more weight on small stocks. The most striking difference between the equal and value
weighted results is the coefficient on recent industry performance. While the coefficient
on recent industry performance in the equal-weighted regressions is highly significant (the
test-statistic is 4.13), the coefficient in the value-weighted regressions is completely in-
significant. That is, it appears as if recent industry performance has no predictive power
regarding the returns of large stocks, but significant power predicting the returns of small
stocks.
These results are largely consistent with both the seemingly contradictory results of
Moskowitz and Grinblatt (1999), which finds that industry momentum strategies are more
profitable than momentum strategies that control for industry, and those of Asness, Porter
and Stevens (2000), which finds that industry-relative momentum strategies significantly
outperform those based on industry performance. Moskowitz and Grinblatt consider equal-
weighted strategies based on six months of past returns, the precise combination for which
industry performance matters more than industry-relative performance. Asness, Porter and
Stevens find their strongest results using value-weighted strategies, driven by large stocks
for which past industry-relative performance is more important than past industry perfor-
mance at both intermediate horizons and short horizons.
Analysis of the performance to portfolios sorted on the basis of prior returns supports
the results of the Fama-MacBeth regressions. Panel B of Table 10 reports the average
monthly returns to momentum strategies based on the four measures of past performance
employed in the parametric tests of Panel A. The table also reports results of time series
regressions of these strategies’ returns on UMD.
These non-parametric test results tell the same story as the parametric tests. Whether
value-weighted or equal-weighted, intra-industry momentum strategies and industry mo-
mentum strategies are both more profitable when based on intermediate-horizon past per-
formance than when based on recent past performance. Value-weighted momentum strate-
gies are more profitable when formed on the basis of industry-relative performance, while
equal weighted strategies are more profitable when formed on the basis of industry per-
30
formance. Inspection of the returns to the strategies formed on the basis of recent past
performance suggests that what short horizon momentum exists is largely driven by small
stocks following industry performance. Inspection of the returns to the strategies formed
on the basis of intermediate horizon past performance suggests that momentum, as a whole,
is primarily driven by intra-industry effects at longer horizons.
In assessing the relative importance of intra-industry versus industry variation in past
performance for predicting future returns, it is also useful to consider the performance of
UMD-like factors. As in section 5, these factors are constructed using the same method-
ology used to construct UMD, except here “winners” and losers” are defined using either
the performance of a stock relative to its industry or the performance of its industry, and
denoted UMDrel and UMDindus, respectively. Factors are constructed on the basis of past
performance over the periods 12 to seven months and six to two months prior to portfolio
construction.
The first four panels of Table 11 show the excess returns to these four factors and span-
ning tests of these strategies and UMD. The first panel demonstrates that the intermediate
horizon strategy based on industry-relative performance has a large information ratio rela-
tive to UMD. The information ratio of UMDrel12,7 relative to UMD exceeds that of UMD rel-
ative to the three Fama-French factors over the same period.18 At the same time, UMDrel12,7
prices UMD well. The information ratio of UMD relative to UMDrel12,7 is insignificant. In
contrast, the next three panels show that UMD prices the intermediate horizon industry
momentum strategy, and both the short horizon intra-industry and the short horizon indus-
try strategies, but has a significant information ratio relative to each of these strategies. It
looks, roughly, like intra-industry variation in past performance at intermediate horizons
primarily drives momentum.
The last three panels of Table 11 look directly at the relations between these four alter-
native momentum factors, and yield results consistent with the first four panels. UMDrel12,7
18 The test-statistic on the intercept of a time-series regression of UMD on the three Fama-French factorsover the same period is 4.73.
31
TABLE 11
SPANNING TESTS EMPLOYING
ALTERNATIVE MOMENTUM FACTORS AND UMD:
TIME-SERIES REGRESSIONS OF THE FORM
yt D ˛ C ˇxt C �t
dependent independent
variable (y) variable (x) E[re ] ˛ ˇ adj-R2
UMDrel12,7 UMD 64.1 34.5 0.380 55.8%
[6.07] [4.83] [22.73]
UMD UMDrel12,7 -16.3 1.471
[-1.13]
UMDrel6,2 UMD 33.2 -10.0 0.544 69.5%
[2.42] [-1.30] [30.31]
UMD UMDrel6,2 36.9 1.280
[3.15]
UMDindus12,7 UMD 49.3 14.6 0.446 39.9%
[3.38] [1.27] [16.25]
UMD UMDindus12,7
33.5 0.899
[2.06]
UMDindus6,2 UMD 26.3 -18.3 0.563 44.0%
[1.48] [-1.35] [17.79]
UMD UMDindus6,2 58.7 0.784
[3.72]
UMDrel12,7
UMDrel6,2
50.6 0.389 24.7%
[5.49] [11.43]
UMDrel6,2 UMDrel
12,7 -6.5 0.638
[-0.53]
UMDrel12,7 UMDindus
12,7 46.4 0.358 24.4%
[4.99] [11.35]
UMDindus12,7
UMDrel12,7
5.3 0.687
[0.40]
UMDrel12,7 UMDindus
6,2 61.0 0.114 3.6%
[5.87] [3.96]
UMDindus6,2 UMDrel
12,7 5.6 0.334
[0.30]
Time-series regressions using the returns to UMD, and UMD-like factors con-structed on the basis of intermediate horizon and recent past performance relativeto industry performance (UMDrel
12,7 and UMDrel6,2, respectively), and intermediate
horizon and recent past industry performance (UMDindus12,7 and UMDindus
6,2 , respec-
tively). The sample covers January 1974 through January 2007.
32
has a large significant positive alpha relative to each of UMDrel6,2, UMDindus
12,7 and UMDindus6,2 .19
At the same time, UMDrel12,7 prices each of the other factors well; none of the other factors
has a significant alpha relative to UMDrel12,7.
6.4 Capital Gains Overhang, Disposition Effects, and Seasonality
Frazzini (2006) considers the role that capital gains (or losses) experienced by holders of
a stock play, through the disposition effect, in the speed that new information about that
stock is impounded into prices. He consequently constructs a measure of the capital gains
overhang for each stock. This measure is defined as the percentage deviation of the ag-
gregate cost basis from the current price, where the aggregate cost basis is the holding
weighted average purchase price of the stock by mutual funds in the Thompson Financial
CDA/Spectrum Mutual Funds database. Note that this measure is at least somewhat pos-
itively correlated, mechanically, with past returns. While Frazzini (2006) focuses on the
interaction of his measure of capital gains overhang and post-announcement price drift, the
measure itself may be used to predict returns.
The “gains-minus-losses” (GML) strategy consists of buying high gain stocks (top quin-
tile) and selling high loss stocks (bottom quintile). I only consider the equal-weighted
strategy, as the effect is concentrated in small stocks. Equal-weighted GML generates 86.2
basis points per month, with a test-statistic of 3.44, over the Frazzini (2006) sample period
from April 1980 through August 2002.
Disposition effects associated with unrealized gains and losses do not explain the per-
formance of the 12-7 strategy, and cannot, contrary to the results of Grinblatt and Han
(2005), even explain the performance of the 12-2 strategy. Table 12 shows results of span-
ning tests on the capital gains overhang strategy, GML, and the three equal-weighted mo-
mentum strategies, MOMew12,2, MOMew
12,7 and MOMew6,2. The first two panels demonstrate
that both the 12-2 and 12-7 strategies price GML, while the converse are false. The third
19 UMDrel12,7 also has a large information ratio relative to the three factors jointly. Its alpha relative to the
other three factors is 38.2 basis points per month, with a test-statistic of 4.77.
33
TABLE 12
SPANNING TESTS OF MOMENTUM AND
CAPITAL GAINS OVERHANG STRATEGIES:
TIME-SERIES REGRESSIONS OF THE FORM
yt D ˛ C ˇxt C �t
dependent independent
variable (y) variable (x) ˛ ˇ adj.-R2
GML MOMew12,2 -15.3 0.741 78.3
[-1.27] [31.15]
MOMew12,2 GNL 45.8 1.059
[3.21]
GML MOMew12,7 -4.9 0.806 51.0
[-0.26] [16.73]
MOMew12,7 GNL 58.3 0.635
[3.66]
GML MOMew6,2 46.3 0.750 44.4
[2.45] [14.67]
MOMew6,2 GNL 2.0 0.595
[0.12]
Time-series regressions using the equal-weighted returns to mo-mentum strategies based on past performance at different horizons(MOMew
n,m), and the equal-weighted returns to the capital “gains-minus-losses” strategy (GML). The sample period covers April 1980through August 2002, employed in Frazzini (2006).
panel shows that GML prices the 6-2 strategy, while the converse is false. It appears that the
12-7 strategy contains all of the pertinent pricing information in the 12-2 strategy, which
in turn contains all of the pertinent pricing information in the gains-minus-losses strategy,
which in turn contains all of the pertinent pricing information in the 6-2 strategy.
6.4.1 Disposition Effect and Seasonality
While a trading strategy based on a univariate sort on the capital gains overhang fails to
explain momentum, controlling for capital gains overhang when constructing momentum
idence for the tax effects story for standard momentum strategies’ poor January perfor-
mance.
Table 13 presents results obtained by double sorting on past returns and capital gains
overhang. Caution must be taken because the two sorts are correlated, yielding relatively
thin high-low and low-high corners (in particular, the losers with large capital gains over-
hang portfolio averages only 27 firms).
Even so, Table 13 argues strongly that the capital gain overhang’s power to predict
returns in univariate sorts derives completely from its correlation with past returns. The
left half of the table shows that the capital gains overhang has no power predicating returns
after controlling for differences in past returns.
TABLE 13
EQUAL-WEIGHTED CONDITIONAL SPREADS AND UMD LOADINGS
FROM DOUBLE SORTS ON r12,2 AND CAPITAL GAINS OVERHANG
GML, conditioned MOMew12,2, conditioned
on r12,2 on capital gains
E[re ] ˛ UMD E[re ] ˛ UMD
low -18.4 -51.3 0.339 low 173.8 108.1 0.679
[-0.65] [-1.85] [5.22] [6.72] [5.55] [14.89]
2 -15.8 -43.6 0.295 2 144.2 79.3 0.671
[-0.81] [-2.34] [6.56] [6.72] [5.55] [14.89]
3 -15.6 -38.7 0.239 3 146.6 94.7 0.536
[-0.77] [-1.95] [5.15] [6.97] [5.83] [14.10]
4 -32.3 -59.7 0.283 4 178.4 118.7 0.617
[-1.18] [-2.20] [4.46] [6.24] [4.87] [10.80]
high 01.5 -37.8 0.406 high 193.7 121.6 0.745
pas
tre
turn
(r1
2,2
)q
uin
tile
[0.05] [-1.44] [6.62]
cap
ital
gai
nover
han
gq
uin
tile
[5.99] [4.56] [11.92]
Time-series regressions using the returns to momentum strategies constructed withincapital gains quintiles, and the returns to the capital gains-minus-losses strategy con-structed within past performance quintiles. GML denotes the equal-weighted gains-minus-losses portfolio (quintiles). MOMew
12,2 denotes the equal-weighted winner-minus-loserportfolio (quintiles), based on cumulative returns from twelve to two months (inclusive)prior to portfolio formation. The sample period covers April 1980 to August 2002, em-ployed in Frazzini (2006).
35
Controlling for capital gains overhang does, however, improve momentum’s perfor-
mance. This result is similar in spirit to the main finding of Frazzini (2006), which finds
that controlling for capital gains overhang improves the performance of earnings momen-
tum strategies. The right half of Table 13 shows that within capital gains quintiles the
equal-weighted 12-2 momentum strategies generate 144 to 194 basis points per month,
with test-statistics ranging from 5.99 to 6.72. These compare favorably to the uncondi-
tional equal-weighted 12-2 strategy, which generates 137 basis points per month, with a
test-statistic of 4.57, over the Frazzini (2006) sample period April 1980 through August
2002.
Figure 3, which plots the average returns to the ten long/short strategies shown in Ta-
ble 13 by calendar month, provides compelling evidence for the tax effects story for mo-
mentum’s poor January performance. The momentum portfolios have positive expected
January returns after controlling for capital-gains overhang. The gains-minus-losses port-
folios, after controlling for past returns, are only distinguished by their terrible January
performance.
Because capital gains overhang drives momentum’s poor January performance, we
should expect to see the 12-7 strategies outperform the 12-2 and 6-2 strategies in January.
Returns from prior months twelve through seven are mechanically less correlated with past
one year returns, and consequently with unrealized capital gains and losses, than are returns
from prior months twelve through two. Returns from prior months twelve through seven
are slightly more correlated with past one year returns than are returns from prior months
six through two, but these returns from farther back are less correlated with the aggregate
capital gains overhang, because more of the gains or losses associated with returns in the
more distant past have been realized in the course of six months of normal trading. It is
thus reasonable to ask how much of the 12-7 strategy’s outperformance of the 12-2 and 6-2
strategies is due to differences in their average January returns.
36
J F M A M J J A S O N D
0
2
4
6MOM(12,2)
gain
s =
low
J F M A M J J A S O N D
0
2
4
6
gain
s =
2
J F M A M J J A S O N D
0
2
4
6
gain
s =
3
J F M A M J J A S O N D
0
2
4
6
ga
ins =
4
J F M A M J J A S O N D
0
2
4
6
gain
s =
hig
h
J F M A M J J A S O N D
−6
−4
−2
0
2
4GML
past re
turn
s =
low
J F M A M J J A S O N D
−6
−4
−2
0
2
4
past re
turn
s =
2
J F M A M J J A S O N D
−6
−4
−2
0
2
4past re
turn
s =
3
J F M A M J J A S O N D
−6
−4
−2
0
2
4
past
retu
rns =
4
J F M A M J J A S O N D
−6
−4
−2
0
2
4
past
retu
rns =
hig
h
Figure 3: Conditional Average Returns to MOMew12,2 and GML, by Calendar Month
The figures show the average monthly returns (percent), by calendar month, to 1) momentum strate-gies constructed within capital gain overhang quintiles (left), and 2) capital gain/loss strategiesconstructed within past return quintiles (right).
37
Figure 4, which depicts average returns by month for the 12-2, 12-7 and 6-2 strategies,
confirms these predictions. The average January returns to the 12-7 strategies, 0.45 and
-2.40 percent value and equal-weighted, respectively, exceed those to the 12-2 strategies
(-1.38 and -4.72 percent) and the 6-2 strategies (-1.62 and -5.32 percent).
J F M A M J J A S O N D
−2
−1
0
1
2
3
4MOM(12,2)
J F M A M J J A S O N D
−2
−1
0
1
2
3
4MOM(12,7)
J F M A M J J A S O N D
−2
−1
0
1
2
3
4MOM(6,2)
Figure 4: Returns to Momentum Strategies, by MonthThe figures show the average returns, by calendar month, to the three momentum strategiesMOM12,2, MOM12,7 , MOM6,2. Value-weighted returns are shown in blue (dark), while equal-weighted returns are shown in green (light). The January returns to MOMew
12,2 and MOMew6,2 are
-4.72% and -5.32%, respectively.
This difference in January returns is insufficient, however, to explain the superior per-
formance of the 12-7 strategies, as is readily apparent in spanning tests on the 12-7 and
12-2 strategies that omit January returns. Results of these tests are provided in Table 14.
The first panel shows that even after omitting January returns MOMvw12,7 still completely
explains the abnormal returns to MOMvw12,2, while the converse is false.20 The equal-
weighted results, presented in the second panel, are equivocal. After omitting January
returns MOMew12,7 “explains” the abnormal returns to MOMew
12,2, while the converse is false,
but the significance of both these results is marginal.
20 Omitting January does weaken the result slightly, with the test-statistic on the alpha of the value-weighted 12-7 strategy relative to the value-weighted 12-2 strategy falling from 3.89 to 3.12.
38
TABLE 14
SPANNING TESTS OF MOM12,2 AND MOM12,7
OMITTING JANUARY RETURNS
dependent independent
variable (y) variable (x) ˛ ˇ adj.-R2
MOMvw12,2 MOMvw
12,7 0.4 0.897 60.4
[0.03] [23.53]
MOMvw12,7 MOMvw
12,2 44.5 0.674
[3.12]
MOMew12,2 MOMew
12,7 22.6 1.052 69.7
[1.88] [28.90]
MOMew12,7 MOMew
12,2 20.5 0.663
[2.16]
Time-series regressions using the returns, excluding January re-turns, to momentum strategies formed on the basis of prior year, or in-termediate horizon, past performance. MOMn,m denotes the winner-minus-loser portfolio (quintiles), where winners and losers are basedon cumulative returns from n to m months (inclusive) prior to portfo-lio formation. The super-script denotes whether the strategies’ returnsare value-weighted (vw) or equal-weighted (ew). The sample coversJanuary 1974 through January 2007.
6.5 Consistency of Performance
Grinblatt and Moskowitz (2004) argues that high past returns achieved with a series of
steady positive months generates larger expected returns than the same level of past returns
achieved through a few extraordinary months. They find that “consistent winners,” defined
as stocks that had positive returns in at least eight of the eleven months from t - 12 to t -
2 (inclusive), significantly outperform other stocks, even after controlling for the level of
past performance.21
It is possible that this result is driven by the importance of intermediate horizon past
performance. “Consistent winners” are unlikely to have performed poorly over the crucial
21 Watkins (2003) also considers the role of “return consistency” on future stock performance. He showsthat firms that had positive (negative) returns every month for six straight months significantly outperform(underperform) the market over the following six months.
39
period spanning the first half of the previous year. A “consistent winner,” as defined by
Grinblatt and Moskowitz (2004), is guaranteed to have had at least two winning months
over this period, and is likely to have had at least four winning months over that time. It
is therefore reasonable to ask if the power of the “consistent winners” indicator derives
from its ability to help distinguish stocks that performed well at intermediate horizons
from stocks that have performed equally well over the past year due to exceptional recent
performance.
A series of Fama-MacBeth regressions tests whether the fact that intermediate horizon
past performance, and not recent past performance, has power predicting returns is related
to the consistency of performance result. These tests regress stocks’ returns on past per-
formance, measured at different horizons, and an indicator as to whether the stocks were
consistent winners, IC W . Following Grinblatt and Moskowitz (2004), this variable takes
the value one, for each stock and in each month, if the stock had positive returns in eight of
the first eleven months of the preceding year, and zero otherwise.
This consistent winner indicator conflates two effects. Stocks that have “won” in eight
out of eleven months tend to be both “big winners” and consistent performers, i.e., stocks
in the upper tail of the past performance distribution, and also stocks that have realized
low return volatility over the same period. In an attempt to disentangle these effects, in
some specifications both a “big winners” indicator and realized volatility are included as
explanatory variables. The big winners indicator, IBW , takes the value one, for each stock
and in each month, if the stock was in the top quintile of performers over the first eleven
months of the previous year, and zero otherwise. Realized volatility, �12,2, is the annualized
standard deviation of monthly returns over the same period.22
Specification (1) of Table 15 shows that the consistency of performance result of Grin-
blatt and Moskowitz (2004) does not explain the disparity in power between intermediate
22 The slope coefficients from a Fama-MacBeth regression of the consistent winners indicator, I C W , ontothe big winners indicator and realized volatility, I BW and �12,2, are 0.293 and -0.198 respectively, with test
statistics of 58.8 and -31.6. The time-series average of the cross-sectional variation in I C W explained byI BW and �12,2 is 15.9%.
40
horizon and recent past performance for predicting returns. In both Panel A, which includes
prior year performance (r12,2), and Panel B, which includes both intermediate horizon and
recent past performance (r12,7 and r6,2), the slope on the “consistent winners” indicator
is positive and significant. Panel B also reveals, however, that intermediate horizon past
performance proves a strong predictor of expected returns, while recent past performance
has essentially no predictive power, even after controlling for whether stocks are consistent
winners.
Specification (2) suggests a strong non-linearity in the relation between past perfor-
mance and expected returns.23 It shows that top past performers generate significantly
higher expected returns, even after controlling for the level of past performance. This spec-
ification also fails to find a significant role for consistency of performance, as measured by
realized volatility.
Specification (3) suggests that the significance of the consistent winners variable derives
from this non-linearity in the relation between past performance and expected returns, and
not from anything having to do with “winning” consistently. Including both the consistent
winners and big winners indicators as regressors dramatically reduces the slope coeffi-
cient on the consistent winners indicator, to the point that it is no longer significant, while
leaving the slope coefficient on the big winners indicator, and its significance, essentially
unchanged.
Specifications (4) to (6) repeat the tests of specifications (1) to (3) on a value weighted
basis, and yield qualitatively identical results.
Appendix A.2 shows that even less support exists for the thesis that “winner consis-
tency” matters, after controlling for past performance, if past performance is measured by
relative rank in the cross section.
23 Further evidence of this non-linearity in the relation between past performance and expected returns isprovided in the Appendix A.2. Tests performed there suggests a stronger linear relation between the relativeranking of past performance and expected returns than between the level of past performance and expectedreturns.
The table reports the results of Fama-MacBeth regressions of firms’ returns onpast performance. Past performance is measured at horizons of twelve to two months(r12,2), twelve to seven months (r12,7), and six to two months (r6,2). The “consistent
winners” indicator, I C W , takes the value one if a stock had positive returns in eight ofthe first eleven months of the preceding year, and zero otherwise. The “big winners”indicator, I BW , takes the value one if a stock was in the upper quintile of past perfor-mance in the first eleven months of the preceding year, and zero otherwise. Realizedvolatility, �12,2, is the annualized standard deviation of monthly returns over firsteleven months of the preceding year. The equal-weighted results minimize the resid-ual variance in each cross-section (i.e., minimize
Pj �2
tj ), while the value-weighted
results minimize the residual variances weighted by firms’ market capitalizations (i.e.,minimize
Pj me(t�1)j �2
tj ). The sample period covers January 1974 through January
2007.
42
7 Conclusion
“Momentum” does not accurately describe the returns to buying winners and selling losers.
On average recent winners under-perform the market if they had poor intermediate horizon
past performance, while recent losers out-perform the market if they had strong intermedi-
ate horizon performance. This fact is inconsistent with the standard notion of momentum,
that “rising stocks tend to keep rising, while falling stocks tend to keep falling.”
These findings pose significant difficulties for models that purport to explain momen-
tum. Popular behavioral explanations predicated on biases in the way that investors inter-
pret information generate positive short-lag autocorrelations in prices. According to these
explanations, security prices underreact to news, which is incorporated slowly into prices,
yielding price momentum. Popular rational explanations predicated on positive correla-
tions between past performance and risk exposure generate similar predictions of short-lag
autocorrelations in prices.
Such short-lag autocorrelations are inconsistent with the data. Rather, the observed
term structure of momentum information exhibits 1) significant information in past per-
formance at horizons of twelve to seven months, 2) recent returns that are largely irrele-
vant after controlling for performance at intermediate horizons, and 3) an abrupt drop-off
at twelve months, beyond which there is no return predictability after controlling for the
Fama-French factors. Explanations consistent with the observed term structure of momen-
tum are not readily apparent, and provide a significant challenge for future research.
Understanding that intermediate horizon past performance drives momentum facilitates
the design of more profitable trading strategies. Including information regarding largely
irrelevant recent past performance in the portfolio selection criteria introduces an errors-
in-variables problem, which reduces performance. Ignoring recent performance when se-
lecting stocks significantly improves momentum strategy Sharpe ratios. Performance is
particularly enhanced in liquid, large cap stocks, which exhibit far more “momentum” than
is commonly recognized.
43
A Appendix
A.1 Subsample results
My primary result, that intermediate horizon past performance, not recent past performance, drives
“momentum,” is equally strong in both halves of the data. I demonstrate this by repeating the main
tests from section 3 in the early and late sub-samples, January 1974 to July 1990 and August 1990
to January 2007, respectively.
Table 16 shows the results of Fama-MacBeth regressions. Both subsamples exhibit the same
pattern observed in Table 1: larger, more significant slope coefficients on r12,7 than on r12,2 , and
smaller, less significant coefficients on r6,2 . In panel A, which reports the equal-weighted results,
all of the coefficients and their test-statistics are smaller in the late sample than in the early sample.
This is consistent with momentum strategies’ relatively poor recent performance. Surprisingly, the
value-weighted results, reported in Panel B, are generally stronger in the late sample.
Table 17 shows non-parametrically that expected returns correlate more strongly with interme-
diate horizon past performance than with recent past performance. The Table reports the average
returns to portfolios formed by double sorting on recent and intermediate horizon past performance.
Both subsamples exhibit the same pattern observed in Table 2. Panel A shows that in both subsam-
ples intermediate horizon winners on average significantly outperform the market, even if they were
recent losers, while intermediate horizon losers on average significantly underperform the market,
even if they were recent winners. Panel B shows similar results for the portfolios’ equal-weighted
returns. The return spreads between winner and loser portfolios based on intermediate horizon past
performance exceed those based on recent past performance by almost an order of magnitude in
Panel A (value-weighted results), and by a factor of two in Panel B (equal-weighted results).
Table 18 shows that the primary results from the spanning tests of Table 5 hold in both the early
and late sub-samples. Panel A shows that the 12-7 strategies price the 12-2 strategies, while Panel
B shows that the converse is false.24 Panel C demonstrates that the 6-2 strategies fail to price
the 12-2 strategies.
24 The positive alpha of MOMew12,7 with respect to MOMew
12,2 is insignificant in the late sample.
44
TABLE 16
FAMA-MACBETH REGRESSIONS RESULTS BY SUB-SAMPLES
rtj D ˇ̌̌ 0xtj C �tj
slope coefficients (�102) and [test-statistics]
under alternative specifications
early sample late sample
independent January 1974 - July 1990 August 1990 - January 2007
variables (1) (2) (3) (4) (1) (2) (3) (4)
Panel A: equal-weighted results
r12,2 0.617 0.352
[2.88] [1.41]
r12,7 0.867 0.835 0.555 0.533
[3.96] [4.07] [2.26] [2.25]
r6,2 0.290 0.251 0.313 0.278
[0.92] [0.82] [0.72] [0.67]
Panel B: value-weighted results
r12,2 0.826 0.688
[2.98] [2.20]
r12,7 1.543 1.470 1.328 1.298
[3.96] [4.05] [3.40] [3.55]
r6,2 0.352 0.293 0.403 0.259
[0.83] [0.72] [0.75] [0.50]
The table reports the results of Fama-MacBeth regressions of firm returns on past performance,like those presented in Table 1, in the early and late subsamples of our data. Past performance ismeasured at horizons of twelve to two months (r12,2), twelve to seven months (r12,7), and six totwo months (r6,2). The equal-weighted results minimize the residual variance in each cross-section
(i.e., minimizeP
j �2tj ), while the value-weighted results minimize the residual variances weighted
by firms’ beginning of period market capitalizations (i.e., minimizeP
j me(t�1)j �2tj ). The early
subsample covers January 1974 through July 1990, while the late subsample covers August 1990through January 2007.
45
TABLE 17
AVERAGE MONTHLY EXCESS RETURNS
(BASIS POINTS PER MONTH) TO PORTFOLIOS
DOUBLE SORTED ON r6,2 AND r12,7
BY SUB-SAMPLES
early sample late sample
January 1974 - July 1990 August 1974 - January 2007
Average monthly excess returns (basis points per month) to portfolios double sorted on r12,7 and r6,2
(stock’s cumulative return over the period twelve to seven months (inclusive) prior to portfolio formation,and a stock’s cumulative return over the period six to two months (inclusive) prior to portfolio formation,respectively), like those presented in Table 2, in the early and late subsamples of our data. The earlysubsample covers January 1974 through July 1990, while the late subsample covers August 1990 throughJanuary 2007.
46
TABLE 18
SPANNING TESTS ON MOM12,2, MOM12,7 AND MOM6,2:
TIME-SERIES REGRESSIONS OF THE FORM
MOM t D ˇ̌̌ 0xt C �t
regression intercepts and [t-stats]
value-weighted equal-weighted
independent returns returns
variable (early) (late) (early) (late)
Panel A: MOM12,2 as dependent variable
constant -3.3 -32.3 -6.3 -4.0
[-0.16] [-1.15] [-0.42] [-0.20]
MOM12,7 0.892 0.965 1.042 1.243
[18.16] [16.75] [21.43] [26.91]
adj-R2(%) 62.4 58.7 69.8 78.6
Panel B: MOM12,7 as dependent variable
constant 44.9 61.1 32.8 19.3
[2.54] [2.78] [2.79] [1.37]
MOM12,2 0.702 0.610 0.672 0.633
[18.16] [16.75] [21.43] [26.91]
adj-R2(%) 62.4 58.7 69.8 78.6
Panel C: MOM12,2 as dependent variable
constant 79.5 50.7 61.7 38.8
[3.96] [1.99] [5.04] [2.65]
MOM6,2 0.898 0.862 0.912 0.962
[17.33] [19.01] [26.31] [38.11]
adj-R2(%) 60.1 64.7 77.7 88.1
Time-series regressions using momentum strategies based on differentsorting periods. MOMn,m is returns to the winner-minus-loser portfo-lio (quintiles), where winners and losers are based on cumulative returnsfrom n to m months (inclusive) prior to portfolio formation, and the super-script denotes whether the strategy’s returns is value or equal weighted.The early subsample covers January 1974 through July 1990, while thelate subsample covers August 1990 through January 2007.
47
A.2 Additional Tests
TABLE 19
AVERAGE MONTHLY EXCESS RETURNS
(BASIS POINTS PER MONTH) TO PORTFOLIOS
DOUBLE SORTED ON r12,12 AND r11,7
Panel A: value-weighted results
r11,7 quintile
(All) (L) (2) (3) (4) (W) (W-L)
(All) 58.1 14.7 41.8 55.5 68.3 102.7 88.0
(L) 35.4 -24.7 22.9 31.1 49.7 69.1 93.8
(2) 43.8 17.3 23.3 45.7 73.6 79.5 62.2
(3) 56.9 10.6 59.6 50.3 68.6 105.3 94.8
(4) 67.2 46.6 55.5 73.9 75.1 101.9 55.3
(W) 98.4 63.6 64.8 74.8 93.4 158.2 94.6r 12
,12
qu
inti
le
(W-L) 59.5 88.3 41.9 43.7 43.7 89.1
Panel B: equal-weighted results
r11,7 quintile
(All) (L) (2) (3) (4) (W) (W-L)
(All) 94.8 63.1 88.2 98.0 111.6 126.6 63.5
(L) 64.5 23.5 53.6 75.5 93.2 100.5 76.9
(2) 91.1 51.5 83.1 93.5 1.06.5 132.1 80.6
(3) 100.6 61.4 103.3 101.1 1.17.0 129.5 68.1
(4) 113.2 92.9 105.7 113.3 1.23.6 140.7 47.8
(H) 133.8 117.2 122.2 129.6 1.49.4 164.9 47.8r 12
,12
qu
inti
le
(W-L) 69.3 93.8 68.6 54.1 56.2 64.5
Average monthly excess returns (basis points per month) to portfolios double sorted onr12,12 and r11,7, a stock’s return twelve months previously and a stock’s cumulative returnover the period eleven to seven months (inclusive) prior to portfolio formation, respectively.The sample period covers January 1974 to January 2007.
The table reports the results of Fama-MacBeth regressions of firms’ returns on past perfor-mance and past performance ranking. Past performance is measured at horizons of twelve totwo months (r12,2), twelve to seven months (r12,7), and six to two months (r6,2). The past per-formance rankings (CDFr12,2
, CDFr12,7and CDFr6,2
) are the fraction of firms that had lowercumulative returns over the corresponding test periods. The equal-weighted results minimizethe residual variance in each cross-section (i.e., minimize
Pj �2
tj ), while the value-weighted re-
sults minimize the residual variances weighted by firms’ market capitalizations (i.e., minimizePj me(t�1)j �2
The table reports the results of Fama-MacBeth regressions of firms’ returns onpast performance. Past performance is measured at horizons of twelve to two months(r12,2), twelve to seven months (r12,7), and six to two months (r6,2). The “consistent
winners” indicator, I C W , takes the value one if a stock had positive returns in eight ofthe first eleven months of the preceding year, and zero otherwise. The “big winners”
indicator, I BW , takes the value one if a stock was in the upper quintile of past perfor-mance in the first eleven months of the preceding year, and zero otherwise. Realizedvolatility, �12,2, is the annualized standard deviation of monthly returns over firsteleven months of the preceding year. The equal-weighted results minimize the resid-ual variance in each cross-section (i.e., minimize
Pj �2
tj ), while the value-weighted
results minimize the residual variances weighted by firms’ market capitalizations (i.e.,minimize
Pj me(t�1)j �2
tj ). The sample period covers January 1974 through January
2007.
50
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