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Electronic copy available at: http://ssrn.com/abstract=1663266
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Momentum – Reversal Strategy
Hsin-Yi Yu
Assistant Professor, Department of Finance, National University of Kaohsiung
700, Kaohsiung University Rd., Nanzih District 81148, Kaohsiung, Taiwan
Tel: +886-7-5919709; Fax: +886-7-5919329; Email: [email protected]
Li-Wen Chen
Assistant Professor, Department of Finance, National Chung Cheng University
168 University Road, Minhsiung Township, Chiayi County 62102, Taiwan
Tel: +886-5-2720411 ext. 24213; Fax: +886-5-2720818; Email: [email protected]
__________
This data and ideas of this paper derive from the authors’ study at Business School, University of Edinburgh. We are
sincerely grateful for the valuable comments from Akindynos-Nikolaos Baltas, Jason Wei, Ned Hawley, Russ Abbott,
and the seminar participants at the 18th SFM Annual Meeting, the workshop in Kyushu University, National Chung
Cheng University, National TsingHua University, and National DongHua University. All errors are the sole property
of the author.
Electronic copy available at: http://ssrn.com/abstract=1663266
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Momentum – Reversal Strategy
Abstract
Various theories have been presented to explain momentum and reversals in stock returns. Based
on the model of Hong and Stein (1999), this paper creates a hybrid strategy to avoid the losses
from the reversal phase. The risk-adjusted returns of the new strategy are significantly higher
than those of the traditional momentum strategy. Moreover, the risk-adjusted returns of the new
strategy cannot be fully explained by Carhart’s four-factor model. Such a finding is robust in
different time periods and size quintiles. Overall, this paper exploits the interaction between
heterogeneous investors and generates distinctive applications.
Keywords: Momentum; Reversal; Overreaction; Underreaction; Timing Ability
JEL code: G11, G12, G14
Electronic copy available at: http://ssrn.com/abstract=1663266
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I. Introduction
Can a strategy be created that avoids the process of momentum toward reversal and obtains
higher risk-adjusted returns? The phenomenon of price momentum has been documented in
many studies, for example Jegadeesh and Titman (hereafter JT) (1993, 2001) and Chan,
Jegadeesh, and Lakonishok (1996). Previous studies have used a host of different theories to
explain the phenomenon and find that over the intermediate term – three to twelve months −
return continuation or momentum is observed. Moreover, intermediate-term momentum and
long-term reversals are sequential components of the process by which the market absorbs news.
This paper constructs a hybrid strategy which not only buys winners and sells losers, but also
avoids the reversal phases of stocks caused by investor overreaction to enhance the risk-adjusted
returns. We find that the risk-adjusted returns of the hybrid strategy are significantly higher than
those of the momentum strategy and cannot be explained by Carhart’s four-factor model.
The profitability of momentum strategy is one of the most well known CAPM-related anomalies
unexplained by the Fama-French three-factor model (Fama and French, 1993). The
cross-sectional return predictability appears to be prevalent in different markets (Rouwenhorst,
1998; Doukas and McKnight, 2005) and different asset classes (Asness, Moskowitz, and
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Pedersen, 2009). It also exists between and within industries (Moskowitz and Grinblatt, 1999;
Hameed, Huang and Mian, 2010). The highly debated explanations for price momentum are
rationalized along two major finance categories: the theory of market frictions (Barberis, Shleifer,
and Vishny, 1998; Hong and Stein, 1999), and the behavioral theory of market inefficiency
(Daniel, Hirshleifer, and Subrahmanyam, 1998). Both theories attempt to explain the
co-existence of intermediate-term momentum and long-term reversal in stock returns as the
result of systematic violations of rational behavior by investors (George and Hwang, 2004). The
theory of Daniel et al. (1998) is based on investor overconfidence and variations in confidence
arising from biased self-attribution. Momentum occurs because traders overreact to private
information, and confirming public information triggers further overreaction. Long-term
reversals occur as the overreaction is corrected in the long run.
On the other hand, Hong and Stein’s (1999) model emphasizes the interaction between
heterogeneous investors rather than behavioral biases. Two kinds of investors – newswatchers
and momentum traders – observe different pieces of private information at different points in
time. Momentum occurs because newswatchers who only observe fundamentals are slow to
revise their previous impression when new information arrives. Such underreaction gives a
chance to momentum traders, who only condition on past returns, to arbitrage away any profit
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left behind by the newswatchers. Such a price increase attracts more momentum traders and
culminates in overreaction. Reversals can be observed by correcting such overreaction. In both
theories, intermediate-term momentum and long-term reversals are sequential components of
how the market digests news. Following the Hong-Stein model, Hong, Lim, and Stein (2000)
provide empirical evidence to support the notion that information diffuses gradually in the
market.
If reversals do occur, the profits gained by chasing the past winners and short selling the past
losers would inevitably be reduced when the phenomenon of reversals starts. Specifically, if a
stock in the winner group is in the overreaction phase caused by investors, it is not the best buy,
because its impending reversal would reduce the profit. Similarly, in the loser group, a stock
experiencing overreaction is also not a best sell, because the coming reversal would lower the
profit of short selling. Thus, if we can discover which stocks are more likely to experience the
overreaction caused by investors, we can avoid trading these stocks, which are in the reversal
phase, and enhance the risk-adjusted returns. Based on this logic, a better strategy is to buy
winners and sell losers that are less likely to experience investor overreaction in the near future.
The problem is how to know whether a stock in the winner (loser) group is likely to experience
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overreaction and approach reversal in the near future. Different from the behavioral viewpoint1,
Hong and Stein (1999) argue that momentum traders who only condition on the past returns
would flock to stocks which experience a price jump brought about by newswatchers. That is, a
momentum trader’s order at time t is an increasing function of the return from time t-2 to t-1.
This kind of momentum trading creates a further price jump and attracts more momentum
traders2. Finally, the initial reaction of prices in the direction of fundamentals is accelerated and
generates overreaction. If this is true, one can observe the trend of past returns to identify which
stocks are likely to set off more momentum trading and then have their prices be pushed above
the long-run equilibrium in an accelerative way.
This paper develops two methods to analyze the trend of past returns. We first divide stocks into
the winner and loser groups by using a similar methodology to that of JT (1993). Subsequently,
1 Along the behavioral avenue, overreaction is due to the departure from the classical assumptions of strict
rationality on the part of investors. However, it is not easy to determine where to start, because there is a huge
number of such departures that investors may entertain. Hence, it is difficult to perceive when irrational behavior
starts through a regular and unitary method.
2 The Hong-Stein model classifies momentum traders into early traders and late traders. “Early” momentum buyers
would impose a negative externality on “late” momentum buyers. Sometimes a price increase is the result not of
news but just of previous rounds of momentum trade. Because momentum traders cannot directly condition on
whether or not news has recently arrived, they do not know whether they are early or late in the cycle.
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we further classify stocks in the winner and loser groups into two groups by observing the trend
of past returns. The first method is the Comparison Method: this focuses on the comparison
between the past J-month and the past 12-month geometric average rate of return (GARR),
where J is less than 12. Given that momentum traders only condition on past returns, the
increasing pattern of past returns of the winner stocks would induce more momentum trading and
then push the prices acceleratively above the fundamentals. Thus, stocks in this group are more
likely to amplify the momentum effect, but ultimately lead to reversals. Following this thinking,
if a stock’s GARR of the past J months exceeds the past 12-month GARR in the winner group,
the stock price is more likely to experience an accelerative increase. Hence, we can expect that
this stock would have a relatively high probability of reversals. Conversely, if the GARR of the
past J months is below the past 12-month GARR for a stock, the stock price is less likely to
increase acceleratively; thus, it is difficult to determine when overreaction would be triggered.
Similarly, in the loser group, if the GARR in the past J months for a stock is lower than the past
12-month GARR, the decreasing pattern of the past 12-month returns would attract more
momentum traders and push the price above the fundamentals in an accelerative way. Thus, it is
more likely for this stock to face reversals in the near future, whereas, if the GARR of a stock in
the past J months exceeds the past 12-month GARR, the probability of having reversals in the
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near future is relatively low.
We further generalize the comparison method as the Convex-Concave Method. This method uses
the GARRs of the past 1 to 12 months to identify the convex or concave relationship between the
past 12-month GARRs (Y axis) and time (X axis). The convex-concave method is used to
characterize the criteria of the comparison method. The convex (concave) relationship is similar
to the criteria that the GARR of the past J months is higher (lower) than the past 12-month
GARR, where J is less than 12. To be specific, observing a convex (concave) relationship
implies that more (less) momentum trading is set off, because momentum traders simply follow
past returns. Thus, a convex (concave) relationship suggests that the probability of having
reversals in the near future is high (low) in the winner group but low (high) in the loser group.
Unlike the comparison method, the convex-concave method is sure to provide a consistent
classification for one stock. For example, it is possible that a stock has a higher past 5-month but
lower past 6-month GARR than its past 12-month GARR. In this case, the comparison method
cannot properly classify such a stock if both past 5-month and 6-month GARRs are considered.
However, the convex-concave method considers all the GARRs of the past 1 to 12 months
simultaneously without this issue.
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The sample covers all common stocks listed in NYSE, AMEX, and NASDAQ from the CRSP
monthly file. The sample period is from January 1965 to December 2009. Momentum portfolios
are constructed using a similar methodology to that of JT (1993).
In addition to the traditional momentum strategy, four extra strategies are constructed. Among
the four strategies, the momentum-reversal strategy (MRS), which buys stocks with a lower
probability of reversals in the winner group and sells stocks which have lower probability of
reversals in the loser group, achieves the highest risk-adjusted returns. The risk-adjusted return of
MRS is significantly higher than that of the traditional momentum strategy documented in JT
(1993) and cannot be explained fully by the Fama-French three-factor and Carhart’s four-factor
models (Carhart, 1997). This suggests that MRS generates additional profit which is not included
in the conventional systematic risk factors. All these findings are not sensitive to firm size and
time.
As MRS tries to avoid the phenomenon of reversals, the above finding inspires a question: Is it
possible that the abnormal returns are derived from timing ability, especially switching between
the momentum and contrarian strategies? An investor with momentum timing ability is able to
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correctly predict which strategy – momentum or contrarian – will outperform the other at a
subsequent time and then select the correct strategy or increase the use of the correct strategy.
Therefore, one may expect that the risk-adjusted returns of MRS are caused by the ability to
switch to the contrarian strategy when the efficacy of the momentum strategy declines.
The empirical results indicate that the risk-adjusted returns obtained by using MRS are not
attributable to market, size, and B/M timing abilities. Meanwhile, the coefficients of momentum
timing ability are negatively significant. This finding reveals that from the viewpoint of
momentum timers, a person who implements MRS would wrongly exclude a winner (loser)
stock and deviate from the momentum strategy. However, an investor who implements MRS can
only infer that reversal is approaching when he observes the convex-shaped past returns. He
cannot predict the exact time of the reversals. Therefore, from a momentum timer’s perspective,
a person implementing MRS may sometimes wrongly exclude a winner (loser) stock from the
winner (loser) portfolio. The reason for excluding these stocks from a portfolio is that they are
more likely to experience reversals in the near future.
In conclusion, this paper designs a hybrid strategy – momentum-reversal strategy (MRS) – to
exploit the underreaction from newswatchers and the overreaction from momentum traders. It is
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a practical application of the Hong-Stein model. To do so, we begin by analyzing the trend of
past returns. From the trend, we can observe whether the phenomenon of reversals is more likely
to occur in the near future. The tests show that this hybrid strategy can earn significantly higher
risk-adjusted returns than the traditional momentum strategy formed by JT (1993). Further, the
returns of the hybrid strategy cannot be rationalized with Carhart’s four-factor model.
The remainder of this paper is organized as follows. Section 2 describes the data and the
methodology. Section 3 presents the results of MRS by using both the comparison and the
convex-concave methods. Section 4 tests whether the profit of MRS is conditional on time and
firm size, along with a discussion about the robustness checks. Section 5 investigates whether the
risk-adjusted returns earned by MRS can be explained by Carhart’s four-factor model or specific
timing ability. The conclusions are set out in Section 6.
II. Data and Methodology
A. Data and the Momentum Strategy
All common stocks (share codes 10 and 11) listed in the New York, American Stock Exchanges
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and NASDAQ from the Center for Research in Security Prices (CRSP) monthly file are used in
this study. The sample period is from January 1965 to December 2009. In summary, the sample
comprises a total of 22,421 stocks3.
To capture the momentum anomaly, I adopt the similar methodology in JT (1993) to calculate
monthly returns of the momentum strategies. In each month t, all stocks are sorted on their
returns for the past 12 months from t-1 to t-124. Different from JT (1993), I place less emphasis
on the tails of the performance distribution. I sort the sample into only three parts based on past
12-month performance: P1, which includes the worst-performing 30 percent; P2, which includes
the middle 40 percent; and P3, which includes the best-performing 30 percent. The highest group
is called the “winners” (W), and the lowest group is the “losers” (L). This is similar to the
measure used by Moskowitz and Grinblatt (1999), Rouwenhorst (1998) and Hong and Stein
(2000).
3 I also use the sample which includes all stocks listed in the New York, American Stock Exchanges and NASDAQ
to do analysis. All empirical results remain unchanged. The results are more significant by using all stocks. The
number of all stocks in total is 26,827.
4 Some may argue that a calendar month should be skipped between the evaluation and holding periods to avoid
artificial return predictability caused by microstructure issues (e.g., bid-ask bounces). Therefore, I also repeat all
analyses in this paper by sorting all stocks on their returns for the past 11 months from t-2 to t-12. All major findings
are unchanged.
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The Fama-French three-factor model is used to measure the risk-adjusted returns of momentum.
(1) tptHMLpt
SMBpt
RMRFpptp eHMLSMBRMRFr ,, ++++= βββα
where tpr , is the monthly return on a portfolio in excess of the one-month T-bill return; pα is
the risk-adjusted return of portfolio i, and tRMRF is the return on the market portfolio in excess
of the risk-free rate. tSMB and tHML are value-weighted, zero-investment, factor-mimicking
portfolios for size and book-to-market equity in stock returns respectively. These factor data are
collected from the website of Kenneth R. French.
B. The Comparison Method
Previous studies usually neglect the trend of past returns. This implies unrealistically that the
change of past returns does not bring new information to investors. Hence, this method analyzes
the trend of past returns to determine which stock is more likely to induce overreaction and
approaches reversals. First, I classify stocks into the winner (W) and loser (L) groups based on
the similar methodology of JT (1993). In the winner group, stocks are further classified into two
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categories: the return-increasing winner (IW); and return-decreasing winner (DW), by using the
geometric average rate of return (GARR) in the following.
(2) ∏=
−++−=12
1
12
1
,, )1(1k
ktiGARR
ti RR
where tiR , is the raw return5 of stock i at time t.
If the past J-month GARR exceeds the past 12-month GARR, this stock would be classified as
the return-increasing winner (IW) group. Otherwise, it is classified as the return-decreasing
winner (DW) group. GARR is used to unify the unit of return of differing period lengths. The
number of months (J) ranges from 1 to 11. Based on the Hong-Stein model, a price jump caused
by newswatchers would attract momentum traders and then initiate a further price increase. Such
a cycle would push prices past the fundamental values acceleratively. As stocks in the IW group
have a better recent J-month GARR compared to the past 12-month returns, such a price jump is
5 Since the hybrid strategy is based on the irrational activity of momentum traders (i.e., overreaction), it is
reasonable to assume that those trend chasers are not sophisticated enough to calculate excess returns or abnormal
returns. Therefore, raw returns, rather than excess returns or abnormal returns, are used to compute GARR.
Nevertheless, excess returns are also used to repeat all analysis in this paper and obtain similar results. For brevity,
only the results using raw returns are report.
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more likely for these stocks to attract momentum traders and face reversals in the near future.
Conversely, the stocks in the DW group that have worse recent J-month returns than the past
12-month returns are less tempting to the momentum buyers and thereby less likely to face
reversals in the near future.
Similarly, stocks of the loser group are divided into two categories: the geometric average rate of
return (GARR) exceeding that of the past 12-month GARR is classified as the return-increasing
loser (IL); otherwise it is classified as the return-decreasing loser (DL). The number of months (J)
ranges from 1 to 11. Since stocks in both IL and DL groups have been categorized to the loser
group (L), when the recent J-month return of a stock is even worse, it means that this stock can
attract more momentum traders who only condition on past returns to short sell the stock. Hence,
the stocks in the DL group possess a higher possibility of reversals. Contrarily, the stocks in the
IL group are less attractive for the momentum traders because their returns are improved. Thus,
unlike the stocks in the IW group, the stocks in the IL group have a lower possibility of
triggering overreaction and subsequent reversals.
C. The Convex-Concave Method
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In some special cases, the comparison method may lead to an ambiguous outcome. This method
treats each month independently. For example, if the past 5-month geometric average rate of
return (GARR) of a stock in the winner group exceeds its past 11-month, the stock can be
classified into the IW group. However, it is possible that the past 6-month GARR of the same
stock is below the past 11-month GARR. In this situation, the same stock would be categorized
to the DW group. To conclude, the comparison method does not consider all GARRs of the past
1 to 11 months concurrently and may not be able to provide consistent stock classification.
Given this, the convex-concave method is developed. First, the stocks are divided into the winner
(W) and loser (L) groups based on the similar methodology of JT (1993). For each stock in the
winner and loser groups, the multiple regression estimated by ordinary least squares in Equation
(3) is used to determine whether the possibility of reversals of a stock is high.
(3) 2, ttR ti γβα ++=
where tiR , is the monthly raw return of stock i at time t. t is an ordinal variable, which is equal
to 1, 2, 3… or 12 for the indication of the past 12-, 11-, 10-… or 1-month respectively. At the
beginning of each month, the returns from all the previous 12 months are used to conduct the
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regression and determine the sign of coefficients6. If the coefficient of γ is positive (negative),
the return of the stock is a convex (concave) function of time.
In the winner group, the convex-shaped returns are more likely to attract momentum buyers and
approach reversals. On the contrary, the concave-shaped returns suggest that the growing speed
of returns decreases as time goes by. These stocks are less likely to kindle trend chasers’ interests
and have a lower probability to face reversals in the near future. Therefore, in the winner group,
the stocks are classified with positiveγ into the return-increasing winner (IW) group, while the
stocks with negativeγ are placed into the return-decreasing winner (DW) group.
In the loser group, the convex-shaped stock returns (IL group) represent return improvement.
This pattern is more likely to reduce momentum traders’ interests and lessen the overreaction.
Therefore, these stocks are less likely to have reversals in the near future. In contrast, the
concave-shaped stock returns are more likely to confront the reversal in the near future.
Accordingly, in the loser group, stocks with positiveγ are classified into the return-increasing
6 The past raw returns in previous 24 and 36 months are also used to determine the convexity and concavity. All
major results remain unchanged. To compare the findings in this paper with those in previous studies, the results of
using the past 12-month returns are presented.
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loser (IL) group, otherwise into the return-decreasing loser (DL) group.
D. The Trading Strategies
It is reasonable to avoid stocks whose return patterns are more likely to induce overreaction and
thereby trigger reversal. By doing so, we should be able to enhance risk-adjusted returns.
Following this line of thinking, the traditional momentum strategy and four alternative trading
strategies are constructed in the following.
Strategy 1: Buy winners (W) and sell losers (L).
Strategy 2: Buy winners (W) and sell stocks in the return-increasing loser (IL).
Strategy 3: Buy stocks in the return-decreasing winner (DW) and sell losers (L).
Strategy 4: Buy stocks in the return-decreasing winner group (DW) and sell stocks in the
return-increasing loser group (IL).
Strategy 5: Buy stocks in the return-increasing winner group (IW) and sell stocks in the
return-decreasing loser (DL).
The first strategy is the traditional momentum strategy documented in JT (1993). Strategy 2
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avoids the reversals of the stocks in the loser group and Strategy 3 avoids the reversals of the
stocks in the winner group. Strategy 4 is the momentum-reversal strategy (MRS), which avoids
the reversals of the stocks in both winner and loser groups. This means that the buying profit of
Strategy 4 is not reduced by the reversals of stocks in the winner group and the short-selling
profit would be enhanced by excluding the stocks that are more likely to have reversals in the
near future. In contrast to Strategy 4, Strategy 5 buys and sells stocks possessing higher
probability of having reversals. We expect that Strategy 4 would generate the highest
risk-adjusted returns, even higher than the traditional momentum profit, while Strategy 5 earns
the lowest risk-adjusted returns, even lower than the traditional momentum profit.
At the beginning of each month, these five equally-weighted portfolios are constructed and held
for one month. Portfolios are then reformed according to the same criteria at the beginning of the
following month.
E. Timing Ability
It is easy to consider MRS (Strategy 4) as a combination of the momentum strategy and timing
ability. Four timing models are adopted to explore whether the profit of MRS is due to timing
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activities. These four timing are introduced below.
Treynor and Mazuy (1966), and Henriksson and Merton (1981) demonstrate two models of
measuring market timing ability based on the CAPM-based model.
TM – Treynor and Mazuy (1966) model:
(4) tititiiti RMRFRMRFr ,2
,1, εγβα +⋅+⋅+=
HM – Henriksson and Merton (1981) model:
(5) tititiiti RMRFRMRFr ,*
,1, εγβα +⋅+⋅+=
ttt RMRFRMRFIRMRF ⋅>= }0{*
where tir , is the month t excess return of the mutual fund i (net return minus T-bill return); iα
is the risk-adjusted return that cannot be explained by the model; tRMRF is month t excess
return on a value-weighted aggregate market proxy portfolio. }{conditionI is an indicator
function that equals one if the condition is true, and zero if otherwise.
Volkman (1999), Bollen and Busse (2001), and Chen, Adams, and Taffler (2011) apply the two
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models to Carhart’s four-factor model to create the factor timing models. This means that the
models of Treynor and Mazuy (1966) and Henriksson and Merton (1981) can be applied, not
only to the excess market return tRMRF , i.e. market timing ability, but also to other factors, i.e.
tSMB , tHML , and tMOM . Applying the two models to tSMB measures the size timing
ability; the ability to choose between small and big capitalization companies. Similarly, tHML
captures the book-to-market timing ability; the ability to choose between value and growth stock,
while tMOM reveals the momentum-strategy timing ability; the ability to choose between
momentum and contrarian strategies (Chen et al., 2011). Like fund managers, these four timing
abilities, i.e. market timing, size timing, book-to-market timing, and momentum-strategy timing,
may be possessed by investors who earn risk-adjusted returns. All four timing measures are
included to demonstrate the kind of timing ability held by MRS. The two timing ability models
are as follows:
CTM –Treynor and Mazuy (1966) and Carhart (1997) factor timing model:
(6) tititititi
titititiiti
MOMHMLSMBRMRF
MOMpHMLhSMBsRMRFr
,2
,42
,32
,22
,1
,
εγγγγ
βα
+⋅+⋅+⋅+⋅
+⋅+⋅+⋅+⋅+=
CHM –Henriksson and Merton (1981) and Carhart (1997) factor timing model:
22
(7) tititititi
titititiiti
MOMHMLSMBRMRF
MOMpHMLhSMBsRMRFr
,*
,4*
,3*
,2*
,1
,
εγγγγ
βα
+⋅+⋅+⋅+⋅
+⋅+⋅+⋅+⋅+=
ttt
ttt
ttt
ttt
MOMMOMIMOM
HMLHMLIHML
SMBSMBISMB
RMRFRMRFIRMRF
⋅>=
⋅>=
⋅>=
⋅>=
}0{
}0{
}0{
}0{
*
*
*
*
where tir , , iα and tRMRF use the same calculations in equations (4) and (5); tSMB , tHML ,
and tMOM are returns on value-weighted, zero-investment factor-mimicking portfolios for size,
book-to-market equity, and one-year momentum in stock returns respectively. }{conditionI is
an indicator function that equals one if the condition is true, and zero if otherwise.
{ }iiii ,4,3,2,1 ,,, γγγγ are measures of market timing, size timing, book-to-market timing and
momentum-strategy timing respectively.
III. The Empirical Evidence on the Momentum-Reversal Profits
In this section, two methods – the comparison method and the convex-concave method – are
used to classify stocks. The risk-adjusted returns of the traditional momentum strategy are
confirmed first. Further, the two methods are used to test whether MRS can earn higher
risk-adjusted returns than the traditional momentum strategy.
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A. Momentum Profits and Reversal
Many studies document a significant momentum profit (e.g. JT, 1993, 2001; George and Hwang,
2004) by using the Fama-French three-factor model. First, I repeated the methodology of JT
(1993) in order to confirm whether momentum profits continued until recently, suggesting that
the original results were not a product of data snooping bias.
Table 1 reports risk-adjusted returns of winner (W), loser (L) and momentum (W-L) portfolios
using the similar methodology of JT (1993) under the Fama-French three factor model. It can be
observed that the momentum profit is significant at the 5 percent significance level when stocks
are sorted into terciles based on the past 12-month returns. The risk-adjusted return is 54.40 basis
points per month, which is approximately 6.73 percent annually.
[INSERT TABLE 1 HERE]
B. Momentum-Reversal Profits: The Comparison Method
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Is it possible for us to exploit both the underreaction of newswatchers and the overreaction of
momentum traders to achieve higher risk-adjusted returns? This section uses the comparison
method to classify stocks and observes whether MRS can outperform the traditional momentum
strategy.
Table 2 reports the performance of winner (W), loser (L) and four other portfolios defined by the
comparison method: winners with a lower possibility of reversals (DW); winners with a higher
possibility of reversals (IW); losers with a lower possibility of reversals (IL); and losers with a
higher possibility of reversals (DL). The comparison method compares the past J-month
geometric average rate of return (GARR) with the past 12-month GARR to determine the
possibility of facing reversals in the near future. The first row in Table 2 indicates the number of
month(s) used to measure GARR, i.e. the value of J. For example, when J is equal to 6, we
compare the past 6-month GARR with the past 12-month GARR. If the past 6-month GARR
exceeds the past 12-month GARR, the stock in the winner group would be classified into the IW
group. The Fama-French three-factor model is adopted to measure the risk-adjusted returns.
[INSERT TABLE 2 HERE]
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It is obvious that the risk-adjusted returns of IL portfolios are the lowest, lower even than those
of the loser portfolio (L). Interestingly, the risk-adjusted returns of the DL portfolios are positive.
In other words, if we keep purchasing the stocks whose past J-month GARRs are below their
own past 12-month GARRs in the loser group each month, we can finally earn positive
risk-adjusted returns. It is clear that short-selling those stocks in the DL group is not a good idea,
despite them being in the loser group (L).
On the other hand, the trend of past returns in the winner group is not as important as that in the
loser group for the momentum traders. We can observe that the differences in the risk-adjusted
returns between the DW, IW, and W portfolios are not as pronounced as those between the IL,
DL, and L portfolios. This finding implies that the decreasing pattern of returns for the loser
stocks would set off momentum trading on a larger scale, while the increasing pattern of returns
for the winner stocks would trigger momentum trading on a smaller scale.
Table 3 presents the risk-adjusted returns of the traditional momentum strategy and four new
trading strategies. Strategy 1 is the traditional momentum strategy documented by JT (1993).
Strategies 2 and 3 are designed to avoid the reversals triggered by the stocks in the loser and
winner groups respectively. MRS (Strategy 4) is proposed to avoid the reversals of the stocks in
26
both the winner and loser groups, whereas Strategy 5 simply buys and sells stocks with a higher
possibility of reversals. The first row measures the number of month(s) (J) used to measure
GARR. The first column lists five trading strategies. JT (1993) investigate extensively the impact
of the January effect on momentum portfolio returns; therefore, I also calculate the risk-adjusted
returns of all trading strategies under the situation where the month of January is included or
excluded.
[INSERT TABLE 3 HERE]
Two features stand out. First and foremost, once we avoid the stocks with high possibility of
reversals, the risk-adjusted returns are boosted. Strategies 2 to 4, which avoid the reversals of
different stock groups, earn higher risk-adjusted returns than the traditional momentum strategy
(Strategy 1). In Panel A, the traditional momentum profit is 54.40 basis points per month,
approximately 6.73 percent per year. The risk-adjusted returns of Strategy 4 are usually the
highest among the five strategies. The peak of Strategy 4 is 195.47 basis points per month, i.e.,
approximately 26.15 percent annually. These findings support the arguments in the Hong-Stein
model and the inference that avoiding the reversals can produce higher risk-adjusted returns.
27
Secondly, comparing the numbers in Panels A and B, the portfolio returns without January are
much larger for all five strategies. In Panel B, the traditional momentum profit is 105.32 basis
points per month. Implementing Strategy 4 can earn 226.24 basis points per month at most, a
striking 30.80 percent per year. This corroborates the finding of JT (1993), who argue that there
is a seasonal effect in the performance of the portfolios, and that the returns of the portfolios are
lowest in January. In addition, in Panel A of Table 3, Strategy 5 achieves the least profit among
the five strategies (-67.81 basis points per month). This finding is consistent with the expectation
that buying and selling stocks with a higher possibility of reversals achieves the least profit
among the five strategies.
Note that once we short sell stocks in the IL group, the risk-adjusted returns are enhanced. The
level of enhancement is much higher than buying stocks in the DW group. The asymmetric
reaction supports the argument above that momentum traders may have different attitudes toward
winner stocks with an increasing pattern of returns and loser stocks with a decreasing pattern of
returns.
C. Momentum-Reversal Profits: The Convex-Concave Method
28
Table 4 reports the risk-adjusted returns of winner (W), loser (L) and four other portfolios
defined by the convex-concave method. We can observe that the risk-adjusted return of the DW
portfolio is the highest, the W portfolio is in the middle, and the IW portfolio has the lowest
return. Compared to the risk-adjusted returns of the winner portfolio (W), those of IW and DW
portfolios diverge in different directions. In the loser group, it can be also noted that the DL
portfolio experiences positive risk-adjusted returns, while the IL portfolio produces lower
negative risk-adjusted returns than the loser group (L) under the Fama-French three-factor model.
Thus, both the comparison method and the convex-concave method demonstrate similar
classification characteristics.
[INSERT TABLE 4 HERE]
Five long-short trading strategies have been constructed for the convex-concave method. The
empirical results are similar to those of the comparison method. Table 5 presents the
risk-adjusted returns of the traditional momentum strategy and four trading strategies which use
the convex-concave method to classify stocks. Apparently, Strategy 4 generates the highest
risk-adjusted return and Strategy 5 produces the lowest return under the Fama-French
three-factor model. The highest risk-adjusted return is 189.61 basis points per month (25.28
29
percent annually) earned by Strategy 4 when January is excluded. Conversely, Strategy 5
produces the lowest risk-adjusted return, regardless of whether January is included or excluded.
The expected value of the difference between the returns of Strategy 4 and the returns of Strategy
1 is significantly non-zero (Z value = 4.702, p-value = 0.000). These findings are consistent with
those using the comparison method, as discussed in the previous section.
[INSERT TABLE 5 HERE]
IV. Further Evidence of the Momentum-Reversal Strategy
In this section, several tests are conducted to confirm the robustness of the findings. First, the
holding period of the MRS portfolio is extended to explore whether we can continue to profit
from MRS in the long run. Furthermore, similar to JT (2001), we delete stocks whose market
capitalizations are in the smallest NYSE/AMEX/NASDAQ decile to repeat previous analysis.
We also sort stocks into quintiles and deciles to observe whether the conclusion remains
unchanged. Finally, we also test whether the previous results are robust in terms of time and firm
size. Both the comparison and the convex-concave methods are used in the analysis. Since the
results are similar, only the results for the convex-concave method are presented.
30
A. Holding Period
The duration of the risk-adjusted returns earned by one trading strategy is of interest to academia.
DeBondt and Thaler (1985) find long-term reversals over a period of three to five years.
Lehmann (1990) and Jegadeesh (1990) also document the phenomenon of reversals, but for
horizons of one week and one month respectively. Over the intermediate term − three to twelve
months − momentum is observed instead (JT, 1993). If MRS (Strategy 4) can efficiently avoid
long-term reversals, we should be able to earn risk-adjusted returns by implementing MRS when
the holding period is extended to three or five years.
I classify stocks into different groups by using the convex-concave method. The holding periods
vary from 3 to 60 months. Since the results are similar when using the comparison method, for
brevity I only report the results using the past 12 months as the evaluation period under the
convex-concave method.
To increase the power of the tests, the strategies examined include portfolios with overlapping
holding periods. Similar to the J-month/K-month strategy of JT (1993) (hereafter the rebalance
31
strategy), a strategy that selects stocks on the basis of returns over the past 12 months and holds
them for K months is constructed as follows: At the beginning of each month t, the stocks are
ranked according to their returns for the past 12 months. Based on these rankings and the
convex-concave methods, the IW, DW, IL, DL portfolios are formed. Since I focus on Strategy 4,
every month, the strategy takes a long position in the DW portfolio and a short position in the IL
portfolio, held for K months. I close the position initiated in month t-K in both the DW and IL
portfolios, and take a new position of the two portfolios using the screening criteria of month t.
Therefore, in each month, we revise 1/K of the stocks in the long and short portfolios, and carry
over the rest from the previous month. That is, portfolios are rebalanced monthly to maintain
equal weights. In addition to the rebalanced portfolios, a series of buy and hold portfolios are
also examined.
[INSERT TABLE 6 HERE]
Table 6 presents the persistence of the risk-adjusted returns for Strategy 4. Although the
significance of the risk-adjusted returns gradually decreases, the returns never become
significantly negative. Collectively, the results are supportive of the notion that there is no
reversal for Strategy 4 over a period of three to five years. The finding is robust when the
32
January returns are removed. Strategy 4 can continue to create significant risk-adjusted returns
by holding the portfolio for over three years.
B. Delete Outliers
Conrad and Kaul (1993) point out that much of the evidence of long horizon mean reversion in
DeBondt and Thaler (1985) is due to the inclusion of low-priced stocks. Like JT (2001), I
exclude all stocks priced under $5 at the beginning of the holding period and all stocks with
market capitalizations that would place them in the smallest NYSE/AMEX/NASDAQ decile.
The test results indicate that the deletion of low-priced and illiquid stocks have little effect on all
the results presented above.
C. Quintiles and Deciles
We place less emphasis on the tails of the performance distribution, so stocks are grouped into
terciles to obtain the winner and loser stocks in the above analysis. Some may argue that this
classification is different from that of JT (1993), who sort stocks into deciles. Therefore, we also
rank stocks into quintiles and deciles to repeat all the analyses above. The results demonstrate
33
that the sorting criterion does not materially alter the major conclusion: Strategies 4 and 5 still
generate the highest and lowest returns among the five trading strategies respectively.
D. Do Momentum-Reversal Strategy Profits Change Over Time?
To investigate whether the findings are conditional on time, the performance of the trading
strategies in nine non-overlapping sub-periods is examined. Each sub-period lasts 5 years. Table
7 presents the risk-adjusted returns of the traditional momentum and four other trading strategies
conditional on time by using the convex-concave method to classify stocks.
[INSERT TABLE 7 HERE]
The evidence indicates that the risk-adjusted returns of the traditional momentum strategy are
significantly positive from Jan. 1980 to Dec. 1989, but insignificant during the other sub-periods.
The profit of Strategy 1 achieves the peak, 105.21 basis points per month (13.38 percent annually)
during January 1980 to December 1984. Although the momentum profits are limited in some
time periods, the traditional momentum strategy still can earn significant risk-adjusted returns
when the whole period is used as the sample as documented in Table 1. Strategies 2 to 4 also
34
achieve their peaks from January 1980 to December 1984. The highest risk-adjusted returns can
be observed in Strategy 4 during January 1980 to December 1984 as 223.63 basis points per
month in Panel B, which is nearly 30.40 percent annually. Collectively, the profits of all trading
strategies are conditional upon time.
Meanwhile, the argument that MRS (Strategy 4) is better than the traditional momentum strategy
(Strategy 1) is not dependent on time. The risk-adjusted returns of Strategy 4 are always higher
than the traditional momentum profits, whereas those of Strategy 5 are always lower than the
traditional momentum profits in every sub-period. These empirical results can be reviewed more
clearly in Figure 1, which plots the risk-adjusted returns of five strategies in different sub-periods
under the convex-concave method. It is found that the line of Strategy 4, illustrated in Figure 1,
is always higher than that of Strategy 1, and the line of Strategy 5 is always located at the bottom
in different sub-periods.
[INSERT FIGURE 1 HERE]
E. Size Partition
35
This section examines whether the central findings are conditional on firm size. The common
view is that small firms have more volatile returns and are, therefore, more likely to limit
investor ability to arbitrage the momentum phenomenon for profit. For example, Merton (1987),
and Grossman and Miller (1988) argue that market making or arbitrage capacity may be less in
small-capitalization stocks. Conversely, Hong et al. (2000) indicate that small firms have more
limited investor participation. Thinner market-making capacity can lead to more pronounced
supply shock induced reversals, thereby eroding the momentum risk-adjusted returns. The focus
of this section is to investigate whether the risk-adjusted returns of MRS are conditional on firm
size.
The stocks are sorted to quintiles along the size dimension. The size partition is based on market
capitalization at the beginning of each month, with the rank of 1 being the smallest and the rank
of 5 being the largest. Table 8 presents the risk-adjusted returns of five trading strategies under
the convex-concave method and Figure 2 illustrates the results.
[INSERT TABLE 8 HERE]
As can be seen, there is an inverted U-shape in Figure 2. Except for Strategy 4, the risk-adjusted
36
returns of the other four strategies are negative for the very smallest stocks. The result can be
interpreted as suggestive evidence in favor of the argument that supply shocks induced by
thinner market-making capacity could lead to a lower momentum profit in small stocks (Hong et
al., 2000). By the second size quintile, the risk-adjusted returns are positive for five strategies.
The profit of Strategy 4 reaches a peak in the second size quintile. The profit is a striking 238
basis points per month (t-statistic=8.76), which is almost twice the value for the sample as a
whole. After the second size quintile, the profit of Strategy 4 declines monotonically. The shape
of Figure 2 is similar to that of Figure 1 in Hong et al. (2000).
[INSERT FIGURE 2 HERE]
However, even though the risk-adjusted returns of five strategies are not size-neutral, the
argument that the risk-adjusted returns of Strategy 4 are higher than those of the traditional
momentum strategy for each size quintile is supported. In Figure 2, it is clear that the
risk-adjusted returns earned by Strategy 4, which avoids the reversals in both the winner and
loser groups, are always higher than those earned by Strategy 1 among firms in different size
quintiles. In contrast, the risk-adjusted returns of Strategy 5, which trades stocks with a higher
possibility of reversals, are always the lowest under the convex-concave method.
37
V. Four-factor Model and Timing Ability
Carhart (1997) constructs the four-factor model by using Fama and French’s (1993) three-factor
model plus an additional factor capturing JT’s (1993) one-year momentum anomaly. If MRS can
efficiently avoid the losses caused by the phenomenon of reversals, it can be expected that the
risk-adjusted returns earned by MRS cannot be explained by Carhart’s four factors. Moreover, we
also investigate whether the risk-adjusted return of MRS is attributable to timing activity.
A. Carhart’s Four-factor Model
If a trading strategy simply reflects the dynamics of returns in accordance to the systematic
factors’ impact, then we should observe an insignificant intercept in a factor model. Following
this line of thinking, the extra profits earned by Strategy 4 should not be explained fully by
Carhart’s four-factor model, because the risk-adjusted return of Strategy 4 does not simply come
from momentum. The momentum factor here is computed as the equally weighted average of
firms with the highest 30 percent 12-month returns lagged one month minus the equal-weight
average of firms with the lowest 30 percent 12-month returns lagged one month. It can be seen
38
that this factor is established by simply implementing the traditional momentum strategy, which
means that the losses caused by reversals are involved therein.
Table 9 reports Carhart’s alphas of the traditional momentum and four other trading strategies.
The convex-concave method is employed to classify stocks. As anticipated, there remain
significant risk-adjusted returns for Strategy 4 based on Carhart’s four-factor model. It can be
confirmed that under the use of Carhart’s four-factor model, Strategy 4 still produces the highest
risk-adjusted returns among the five strategies. The monthly risk-adjusted return of Strategy 4 is
38.92 basis points (approximately 4.88 percent annually), and such a return does not become
insignificant after using Carhart’s four-factor model.
[INSERT TABLE 9 HERE]
A point which merits discussion is that once part of the returns of the trading strategies is
explained by momentum exposure, the strategies which do not avoid the stocks in the DL group
would earn significantly negative risk-adjusted returns. In Table 9, we can observe that as long as
stocks in the DL group are not excluded from the portfolios, i.e. Strategies 3 and 5, negative
risk-adjusted returns would be obtained under Carhart’s four-factor model. However, Strategy 2
39
still possesses positive risk-adjusted returns without excluding the stocks in the IW group from
the portfolios.
The results corroborate previous studies. Hong et al. (2000) show that the momentum effect
comes from losers, as opposed to winners, because investors are more prone to underreact to bad
news than to good news7 . Meanwhile, the Hong-Stein model also indicates that only
newswatchers care about fundamentals and react to news. Taken together, we can expect that the
underreaction to bad news to come mainly from newswatchers rather than momentum traders. If
momentum traders know the facts, they would be more willing to flock to the stocks having bad
news and decreasing returns, as opposed to the stocks having good news, because they can
exploit this underreaction to a larger extent. Given this, it is more likely for stocks with bad news
and decreasing returns to culminate in overreaction and induce reversals. Therefore, excluding
stocks in the DL group can avoid reversals more efficiently than excluding stocks in the IW
group. Thus, Strategy 2 still generates positive risk-adjusted returns, but Strategies 3 and 5 do
not.
7 Hong et al. (2000) demonstrate that when firms are sitting on good news, managers probably have every incentive
to push this news out to investors as fast as possible. In contrast, when there is bad news, managers would announce
and report the news to investors sluggishly.
40
B. Timing Ability of the Momentum-Reversal Strategy
This section includes four different timing models to test whether the observed risk-adjusted
returns of MRS can be attributed largely to the timing activities, especially momentum timing
ability. Momentum timing is the ability to recognize which strategy, momentum or contrarian,
would generate higher returns in the near future. Since MRS attempts to predict when the
reversal is approaching and sell the winners and buy the losers, this behavior looks like
momentum timing – switching between investing winners and losers. It is reasonable to speculate
that the risk-adjusted returns of MRS are due to momentum timing ability. However, there are
two concerns. First, reversal is a gradual phenomenon. Investors who implement MRS cannot tell
the exact time of reversals. They can only be aware that the reversal is approaching through
observing the trend of past returns. Secondly, once an investor reduces the weight of using the
momentum strategy, it is not necessary for him to immediately shift to the contrarian strategy.
Therefore, from the viewpoint of momentum timers, it is very likely for the MRS users to sell
(buy) winners (losers), as the momentum strategy is still profitable.
[INSERT TABLE 10 HERE]
41
Table 10 shows the results. For brevity, only the results of the convex-concave method are
reported, although both methods are examined and obtain similar results. In Panels A and B, we
can observe that Strategy 4 possesses market timing ability under the TM and HM models.
However, the adjusted R2 in both models is very low.
In Panels C and D, the CTM and CHM models are introduced. First, we find that the adjusted R2
increases obviously. However, the coefficients of market timing ability become insignificant.
Meanwhile, the coefficient of momentum timing is significantly negative. That is, the investors
implementing MRS would be regarded as having negative momentum timing ability from the
viewpoint of momentum timers. A negative coefficient of momentum timing ability reflects that
an investor wrongly determines which strategy – momentum or contrarian – is profitable. In
other words, a person with negative momentum timing ability would reduce the use of the
momentum or contrarian strategy when that strategy is still profitable. For MRS implementers,
the reason for excluding some stocks is that those stocks possess a relatively high possibility of
reversals in the near future. However, they can only observe that the reversals are approaching
but cannot know when the reversals would occur exactly. As long as the excluded stocks in the
winner (loser) groups do not immediately swap to the loser (winner) group and produce profit for
42
the opposite strategy, these MRS users would be viewed as having negative momentum timing
ability.
VI. Conclusion
Various theories have been proposed to explain momentum in stock returns. Along the theoretical
avenue, Hong and Stein (1999) put forward a unified theory to explain that momentum comes
from gradual information diffusion and the interaction between heterogeneous investors. Given
this model, this paper designs a hybrid strategy – the momentum-reversal strategy (MRS) – to
find out which stocks are more likely to attract momentum traders and then induce reversals by
observing the trend of past returns.
The empirical results demonstrate that the risk-adjusted returns of MRS are significantly higher
than those of the traditional momentum strategy documented by JT (1993). Moreover, the
risk-adjusted returns earned by MRS cannot be explained fully by the Fama-French three-factor
and Carhart’s four-factor models. Such a finding is robust in different time periods and size
quintiles. In brief, if we can efficiently exploit the profits caused by the interaction between
different types of investors, the anomaly due to return autocorrelations can be further
43
strengthened.
Our findings reveal that the traditional momentum strategy does not fully extract information
from past returns. MRS provides a way to extract information and generate higher returns. Future
research can explore patterns and regularities for other anomalies to refine them toward a higher
return. Overall, we hope that this hybrid strategy provides some applicable principles which can
be applied to all known anomalies.
44
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48
TABLE 1
The Risk-Adjusted Momentum Returns under the Fama-French Three-Factor Model
Winner (W) Loser (L) Winner-Loser
(W-L)
Risk-adjusted returns
41.85
(4.972)
[0.000]
-12.55
(-0.622)
[0.534]
54.40
(2.222)
[0.027]
This table reports the average monthly portfolio returns from January 1965 to December 2009
for the momentum investing strategy under the Fama-French three-factor model. The sample
includes all common stocks on CRSP. We do not exclude NASDAQ stocks. Portfolios are
formed based on past 12-month returns respectively. All portfolios are held for 1 month. The
winner (loser) portfolio is the equally weighted portfolio of the top (bottom) 30% of stocks
ranked by past 12-month returns. The t-statistics are in parentheses and the p values are in
brackets.
49
TABLE 2
The Risk-Adjusted Returns of Portfolios: The Comparison Method
1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10 months
W
41.85
(4.972)
[0.000]
IW
17.05
(1.699)
[0.090]
29.94
(2.884)
[0.004]
39.65
(3.748)
[0.000]
38.10
(3.659)
[0.000]
41.11
(3.913)
[0.000]
41.47
(4.016)
[0.000]
43.32
(4.133)
[0.000]
40.05
(3.854)
[0.000]
43.74
(4.173)
[0.000]
35.91
(3.537)
[0.000]
DW
66.96
(6.802)
[0.000]
52.46
(5.441)
[0.000]
41.14
(4.400)
[0.000]
42.54
(4.456)
[0.000]
43.97
(4.604)
[0.000]
41.17
(4.405)
[0.000]
43.39
(4.707)
[0.000]
40.04
(4.426)
[0.000]
36.44
(4.112)
[0.000]
44.54
(5.063)
[0.000]
L
-12.55
(-0.622)
[0.534]
IL
-128.51
(-6.929)
[0.000]
-133.56
(-7.066)
[0.000]
-112.96
(-6.236)
[0.000]
-101.61
(-5.736)
[0.000]
-95.08
(-5.415)
[0.000]
-89.28
(-5.210)
[0.000]
-78.68
(-4.464)
[0.000]
-74.02
(-4.328)
[0.000]
-65.88
(-4.051)
[0.000]
-60.48
(-3.589)
[0.000]
DL 84.87 84.77 69.38 56.01 51.01 37.60 29.79 29.16 28.26 29.74
50
(3.617)
[0.000]
(3.636)
[0.000]
(2.949)
[0.003]
(2.422)
[0.016]
(2.188)
[0.029]
(1.647)
[0.100]
(1.303)
[0.193]
(1.282)
[0.200]
(1.243)
[0.215]
(1.348)
[0.178]
This table reports the average monthly portfolio returns from January 1965 to December 2009 for six portfolios under the Fama-French
three-factor model. The sample includes all stocks on CRSP. All portfolios are held for 1 month. The winner (W) portfolio is the equally
weighted portfolio of the top 30% of stocks ranked by past 12-month returns. Conversely, the loser (L) portfolio is the equally weighted portfolio
of the bottom 30% of stocks. In the winner group, we further divided the stocks into two categories. If the past J-month geometric average rate
of return (GARR) exceeds the past 12-month GARR, this stock would be classified into the top increasing (IW) category, otherwise the top
decreasing (DW) category. Similarly, in the loser group, if the past J-month return is better than the past 12-month return, this stock would be
classified into the bottom increasing (IL) category, otherwise the bottom decreasing (DL) category. The first row indicates the value of J. The
t-statistics are in parentheses and the p values are in brackets.
51
TABLE 3
The Risk-Adjusted Returns of the Momentum Strategy and the Strategies Avoiding Reversals: The Comparison Method
Panel A: January included
1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10
months
Strategy 1
(W – L)
54.40
(2.222)
[0.027]
Strategy 2
(W – IL)
170.36
(7.591)
[0.000]
175.21
(7.787)
[0.000]
154.80
(7.125)
[0.000]
143.46
(6.687)
[0.000]
136.93
(6.406)
[0.000]
131.13
(6.268)
[0.000]
120.53
(5.633)
[0.000]
115.87
(5.476)
[0.000]
107.73
(5.296)
[0.000]
102.33
(4.896)
[0.000]
Strategy 3
(DW – L)
79.51
(3.373)
[0.001]
65.01
(2.761)
[0.006]
53.69
(2.300)
[0.022]
55.09
(2.367)
[0.018]
56.52
(2.430)
[0.015]
53.71
(2.310)
[0.021]
55.94
(2.402)
[0.017]
52.59
(2.267)
[0.024]
48.98
(2.103)
[0.036]
56.88
(2.439)
[0.015]
Strategy 4
(DW – IL)
195.47
(8.818)
[0.000]
185.84
(8.351)
[0.000]
154.10
(7.240)
[0.000]
144.15
(6.856)
[0.000]
139.05
(6.646)
[0.000]
130.45
(6.402)
[0.000]
122.07
(5.868)
[0.000]
114.06
(5.554)
[0.000]
102.32
(5.163)
[0.000]
104.95
(5.187)
[0.000]
Strategy 5
(IW – DL)
-67.81
(-2.345)
-54.83
(-1.898)
-29.73
(-1.021)
-17.91
(-0.619)
-9.91
(-0.340)
3.87
(0.134)
13.64
(0.470)
10.87
(0.378)
15.57
(0.542)
6.12
(0.219)
52
[0.019] [0.058] [0.308] [0.536] [0.734] [0.894] [0.638] [0.706] [0.588] [0.827]
Panel B: January excluded
1 month 2 months 3 months 4 months 5 months 6 months 7 months 8 months 9 months 10
months
Strategy 1
(W – L)
105.32
(5.010)
[0.000]
Strategy 2
(W – IL)
207.55
(9.771)
[0.000]
211.84
(9.723)
[0.000]
191.79
(9.223)
[0.000]
179.55
(8.719)
[0.000]
174.73
(8.758)
[0.000]
170.40
(8.863)
[0.000]
162.10
(8.529)
[0.000]
155.04
(8.083)
[0.000]
147.62
(7.959)
[0.000]
141.86
(7.502)
[0.000]
Strategy 3
(DW – L)
124.01
(5.898)
[0.000]
110.31
(5.328)
[0.000]
100.29
(4.939)
[0.000]
100.80
(4.956)
[0.000]
102.34
(5.008)
[0.000]
100.57
(4.978)
[0.000]
102.97
(5.078)
[0.000]
99.72
(4.932)
[0.000]
95.74
(4.745)
[0.000]
102.55
(5.047)
[0.000]
Strategy 4
(DW – IL)
226.24
(10.446)
[0.000]
216.85
(9.868)
[0.000]
186.76
(9.019)
[0.000]
175.03
(8.502)
[0.000]
171.75
(8.574)
[0.000]
165.66
(8.686)
[0.000]
159.76
(8.439)
[0.000]
149.45
(7.839)
[0.000]
138.04
(7.515)
[0.000]
139.21
(7.449)
[0.000]
Strategy 5
(IW – DL)
-1.90
(-0.079)
[0.937]
8.89
(0.360)
[0.719]
32.46
(1.290)
[0.198]
46.18
(1.877)
[0.061]
52.35
(2.085)
[0.038]
64.14
(2.595)
[0.010]
74.31
(2.991)
[0.003]
71.97
(2.943)
[0.003]
76.38
(3.106)
[0.002]
67.65
(2.850)
[0.005]
53
The monthly risk-adjusted returns of five trading strategies from January 1965 to December 2009 under the Fama-French three-factor model are
reported in this table. The sample includes all common stocks on CRSP. All portfolios are held for 1 month. We compare the past J-month
geometric average rate of return (GARR) with the past 12-month GARR to classify stocks in the winner (W) and loser (L) groups into four
groups: IW, DW, IL and DL. The first row indicates the values of J. Five trading strategies are constructed. Strategy 1 is the traditional
momentum strategy, which focuses on buying winners (W) and selling losers (L). Strategies 2, 3 and 4 respectively avoid reversals of different
groups. Strategy 2 is to buy winners (W) and sell bottom increasing (IL). Strategy 3 is to buy top decreasing (DW) and sell the losers (L).
Strategy 4 (MRS) is to buy top decreasing (DW) and sell bottom increasing (IL). Strategy 5 is to buy top increasing (IW) and sell bottom
decreasing (DL). The t-statistics are in parentheses and the p values are in brackets.
54
TABLE 4
The Risk-Adjusted Returns of Portfolios: The Convex-Concave Method
Fama-French three-factor model
W
41.85
(4.972)
[0.000]
IW (convex)
26.23
(2.719)
[0.007]
DW (concave)
52.78
(5.308)
[0.000]
L
-12.55
(-0.622)
[0.534]
IL (convex)
-86.51
(-4.128)
[0.000]
DL (concave)
45.45
(2.134)
[0.033]
This table reports the average monthly portfolio returns from January 1965 to
December 2009 for six portfolios under the Fama-French three-factor model. The
sample includes all common stocks on CRSP. All portfolios are held for 1 month. The
winner (W) portfolio is the equally weighted portfolio of the top 30% of stocks ranked
by past 12-month returns. Conversely, the loser (L) portfolio is the equally weighted
portfolio of the bottom 30% of stocks. We use the regression to further classify stocks
in the winner and loser groups: 2, ttR ti γβα ++= , where
tiR , is the monthly raw return
of stock i at time t. t is equals to 1, 2, 3… or 12 for the indication of the past 12-, 11-,
10-… or 1-month respectively. In the winner group, we classify the stocks with
positiveγ into the top increasing (IW) group and the stocks with negativeγ into the
55
top decreasing (DW) group. In the loser group, stocks with positiveγ are classified
into the bottom increasing (IL) group, otherwise into the bottom decreasing (DL)
group. The t-statistics are in parentheses and the p values are in brackets.
56
TABLE 5
The Risk-Adjusted Returns of the Momentum Strategy and the Strategies Avoiding
Reversals: The Convex-Concave Method
Fama-French three-factor model
January included January excluded
Strategy 1
(W – L)
54.40
(2.222)
[0.027]
105.32
(5.010)
[0.000]
Strategy 2
(W – IL)
128.36
(5.195)
[0.000]
178.40
(8.087)
[0.000]
Strategy 3
(DW – L)
65.33
(2.593)
[0.010]
116.53
(5.387)
[0.000]
Strategy 4
(DW – IL)
139.29
(5.411)
[0.000]
189.61
(8.226)
[0.000]
Strategy 5
(IW – DL)
-19.21
(-0.744)
[0.457]
32.49
(1.475)
[0.141]
This table reports the monthly risk-adjusted returns of five trading strategies from
January 1965 to December 2009 under the Fama-French three-factor model. The
sample includes all common stocks on CRSP. All portfolios are held for 1 month. We
classify stocks in the winner (W) and loser (L) groups into four groups – IW, DW, IL
and DL – by using the convex-concave method. Five trading strategies are constructed.
Strategy 1 is the traditional momentum strategy, which focuses on buying winners (W)
and selling losers (L). Strategies 2, 3, and 4 respectively avoid reversals of different
groups. Strategy 2 is to buy winners (W) and sell bottom increasing (IL). Strategy 3 is
to buy top decreasing (DW) and sell the losers (L). Strategy 4 (MRS) is to buy top
decreasing (DW) and sell bottom increasing (IL). Strategy 5 is to buy top increasing
(IW) and sell bottom decreasing (DL). The t-statistics are in parentheses and the p
values are in brackets.
57
TABLE 6
The Risk-Adjusted Returns of the Momentum-Reversal Strategy with Longer Holding
Periods
Holding
period
(months)
3 6 9 12 24 36 48 60
Panel A: Rebalanced strategy
January
included
130.10
(5.689)
[0.000]
110.58
(5.700)
[0.000]
85.04
(4.930)
[0.000]
59.97
(3.838)
[0.000]
27.17
(2.288)
[0.023]
19.08
(1.915)
[0.056]
15.51
(1.809)
[0.071]
10.99
(1.417)
[0.157]
January
excluded
172.71
(8.251)
[0.000]
148.27
(8.209)
[0.000]
120.36
(7.438)
[0.000]
93.25
(6.380)
[0.000]
52.86
(4.894)
[0.000]
40.05
(4.359)
[0.000]
32.98
(4.131)
[0.000]
26.24
(3.622)
[0.000]
Panel B: Buy and hold strategy
January
included
129.78
(5.666)
[0.000]
106.49
(5.520)
[0.000]
79.53
(4.707)
[0.000]
55.20
(3.672)
[0.000]
24.92
(2.204)
[0.028]
16.47
(1.660)
[0.097]
13.24
(1.540)
[0.124]
9.98
(1.306)
[0.192]
January
excluded
169.37
(7.815)
[0.000]
140.73
(7.626)
[0.000]
111.10
(6.836)
[0.000]
84.83
(5.881)
[0.000]
48.55
(4.613)
[0.000]
36.15
(3.873)
[0.000]
29.92
(3.683)
[0.000]
24.51
(3.386)
[0.001]
This table reports the monthly risk-adjusted returns of Strategy 4 from January 1965
to December 2009 under the Fama-French three-factor model. The sample includes all
common stocks on CRSP. Firms with the highest 30 percent past 12-month returns are
grouped to the winner portfolio (W) and firms with the lowest 30 percent past
12-month returns are grouped to the loser group (L). Firms in the winner (W) and
loser (L) groups are further classified to four groups – IW, DW, IL and DL – by using
the convex-concave method. Strategy 4 (MRS) is to buy top decreasing (DW) and sell
bottom increasing (IL). The risk-adjusted returns in Panel A are calculated by a series
of portfolios that were rebalanced monthly to maintain equal weights and those in
Panel B are calculated by a series of buy and hold portfolios. The t-statistics are in
parentheses and the p values are in brackets.
58
TABLE 7
The Risk-Adjusted Returns of the Momentum Strategy and the Strategies Avoiding Reversals: Sub-periods
Time Period
Jan. 1965 –
Dec. 1969
Jan. 1970 –
Dec. 1974
Jan. 1975 –
Dec. 1979
Jan. 1980 –
Dec. 1984
Jan. 1985 –
Dec. 1989
Jan. 1990 –
Dec. 1994
Jan. 1995 –
Dec. 1999
Jan. 2000 –
Dec. 2004
Jan. 2005 –
Dec.2009
Strategy 1
(W – L)
51.38
(0.818)
[0.418]
-40.78
(-0.971)
[0.336]
-14.98
(-0.254)
[0.801]
105.21
(2.172)
[0.034]
82.53
(1.941)
[0.057]
32.11
(0.760)
[0.450]
72.99
(1.143)
[0.258]
-113.36
(-0.786)
[0.435]
-40.90
(-0.535)
[0.594]
Strategy 2
(W – IL)
95.52
(1.482)
[0.146]
81.71
(1.761)
[0.084]
14.92
(0.243)
[0.809]
197.51
(3.929)
[0.000]
158.61
(3.369)
[0.001]
108.92
(2.373)
[0.021]
154.22
(2.563)
[0.013]
-74.11
(-0.525)
[0.601]
30.59
(0.389)
[0.699]
Strategy 3
(DW – L)
54.90
(0.766)
[0.448]
-23.76
(-0.505)
[0.616]
24.01
(0.366)
[0.716]
131.33
(2.724)
[0.009]
94.57
(2.183)
[0.033]
21.11
(0.482)
[0.632]
90.78
(1.524)
[0.133]
-104.24
(-0.698)
[0.488]
-54.29
(-0.691)
[0.492]
Strategy 4
(DW – IL)
99.03
(1.350)
[0.184]
98.73
(1.916)
[0.060]
53.91
(0.789)
[0.433]
223.63
(4.380)
[0.000]
170.65
(3.517)
[0.001]
97.92
(2.076)
[0.043]
172.01
(2.986)
[0.004]
-64.99
(-0.437)
[0.664]
17.20
(0.210)
[0.834]
Strategy 5
(IW – DL)
0.54
(0.009)
-115.44
(-2.376)
-101.14
(-1.568)
-16.77
(-0.330)
0.38
(0.008)
-24.95
(-0.550)
-21.46
(-0.270)
-127.64
(-0.846)
-78.96
(-0.955)
59
[0.993] [0.021] [0.123] [0.742] [0.994] [0.585] [0.788] [0.401] [0.344]
This table presents the average monthly risk-adjusted returns under the Fama-French three-factor model for five trading strategies in
non-overlapping sub-periods. The first row indicates the limit of a sub-period. The sample includes all common stocks on CRSP. All portfolios
are held for 1 month. Firms in the winner (W) and loser (L) groups are further classified to four groups – IW, DW, IL and DL – by using the
convex-concave method. Five trading strategies are constructed. Strategy 1 is the traditional momentum strategy, which focuses on buying
winners (W) and selling losers (L). Strategies 2, 3, and 4 respectively avoid reversals of different groups. Strategy 2 is to buy winners (W) and
sell bottom increasing (IL). Strategy 3 is to buy top decreasing (DW) and sell the losers (L). Strategy 4 is to buy top decreasing (DW) and sell
bottom increasing (IL). Strategy 5 is to buy top increasing (IW) and sell bottom decreasing (DL). The t-statistics are in parentheses and the p
values are in brackets.
60
TABLE 8
The Risk-Adjusted Returns of the Momentum Strategy and the Strategies Avoiding
Reversals: Size Partition
Firm Size
1 (small) 2 3 4 5 (large)
Strategy 1
(W – L)
-157.32
(-4.853)
[0.000]
127.41
(5.101)
[0.000]
134.16
(5.718)
[0.000]
94.79
(4.343)
[0.000]
66.80
(3.225)
[0.001]
Strategy 2
(W – IL)
-6.47
(-0.195)
[0.846]
223.73
(8.451)
[0.000]
176.77
(7.251)
[0.000]
125.42
(5.591)
[0.000]
85.33
(4.103)
[0.000]
Strategy 3
(DW – L)
-132.81
(-3.915)
[0.000]
141.46
(5.582)
[0.000]
128.82
(5.300)
[0.000]
105.53
(4.501)
[0.000]
86.30
(3.916)
[0.000]
Strategy 4
(DW – IL)
19.21
(0.552)
[0.581]
238.00
(8.760)
[0.000]
171.51
(6.686)
[0.000]
136.23
(5.565)
[0.000]
104.61
(4.634)
[0.000]
Strategy 5
(IW – DL)
-295.49
(-8.024)
[0.000]
43.02
(1.600)
[0.110]
99.19
(3.924)
[0.000]
63.16
(2.776)
[0.006]
37.40
(1.705)
[0.089]
This table reports the average monthly risk-adjusted returns of the five trading
strategies under different firm size quintiles. Stocks are sorted independently into
quintiles, ranking by market capitalization: 1 being the smallest and 5 being the
largest. The numbers in this table are the risk-adjusted returns under the Fama-French
three-factor model. The sample includes all common stocks on CRSP. All portfolios
are held for 1 month. The convex-concave method is adopted to categorize stocks into
four groups – IW, DW, IL, and DL. Five trading strategies are constructed. Strategy 1
is the traditional momentum strategy, which focuses on buying winners (W) and
selling losers (L). Strategies 2, 3, and 4 respectively avoid reversals of different
groups. Strategy 2 is to buy winners (W) and sell bottom increasing (IL). Strategy 3 is
to buy top decreasing (DW) and sell the losers (L). Strategy 4 is to buy top decreasing
(DW) and sell bottom increasing (IL). Strategy 5 is to buy top increasing (IW) and
sell bottom decreasing (DL). The t-statistics are in parentheses and the p values are in
61
brackets.
62
TABLE 9
The Risk-Adjusted Returns of the Momentum Strategy and the New Strategies
Avoiding Reversals under Carhart’s Four-Factor Model
Fama-French three-factor model Carhart’s four-factor model
Strategy 1
(W – L)
54.40
(2.222)
[0.027]
-46.34
(-3.440)
[0.001]
Strategy 2
(W – IL)
128.36
(5.195)
[0.000]
30.16
(2.053)
[0.041]
Strategy 3
(DW – L)
65.33
(2.593)
[0.010]
-37.58
(-2.661)
[0.008]
Strategy 4
(DW – IL)
139.29
(5.411)
[0.000]
38.92
(2.455)
[0.014]
Strategy 5
(IW – DL)
-19.21
(-0.744)
[0.457]
-116.49
(-6.937)
[0.000]
This table reports the monthly risk-adjusted returns of five trading strategies from
January 1965 to December 2009 under Carhart’s four-factor model. Carhart’s
four-factor model is: tpt
MOMpt
HMLpt
SMBpt
RMRFpptp eMOMHMLSMBRMRFr ,, +++++= ββββα , where
tpr , is the monthly return on a portfolio of funds in excess of the one-month T-bill
return; pα is the risk-adjusted return of portfolio i,
tRMRF is the return on the
market portfolio in excess of the risk-free rate, and tSMB ,
tHML and tMOM are
value-weighted, zero-investment, factor-mimicking portfolios for size,
book-to-market equity, and one-year momentum in stock returns respectively. All
common stocks on CRSP are included. All portfolios are held for 1 month. We
classify stocks in the winner (W) and loser (L) groups into four groups – IW, DW, IL
and DL – by using the convex-concave method. Five trading strategies are
constructed. Strategy 1 is the traditional momentum strategy, which focuses on
buying winners (W) and selling losers (L). Strategies 2, 3, and 4 respectively avoid
reversals of different groups. Strategy 2 is to buy winners (W) and sell bottom
increasing (IL). Strategy 3 is to buy top decreasing (DW) and sell the losers (L).
Strategy 4 is to buy top decreasing (DW) and sell bottom increasing (IL). Strategy 5
63
is to buy top increasing (IW) and sell bottom decreasing (DL). The t-statistics are in
parentheses and the p values are in brackets.
64
TABLE 10
Timing Ability of the Momentum-Reversal Strategy
intercept RMRF timing SMB timing HML timing MOM
timing Adj-R2
Panel A: Treynor and Mazuy (1966) model (TM)
158.22
(5.430)
[0.000]
-1.73
(-2.745)
[0.006]
0.045
Panel B: Henriksson and Merton (1981) model (HM)
203.45
(5.026)
[0.000]
-0.46
(-2.662)
[0.008]
0.044
Panel C: Treynor and Mazuy (1966) and Carhart (1997) factor timing model
(CTM)
62.71
(3.316)
[0.001]
0.15
(0.360)
[0.719]
-0.08
(-0.127)
[0.899]
-1.08
(-1.101)
[0.272]
-0.75
(-2.846)
[0.005]
0.668
Panel D: Henriksson and Merton (1981) and Carhart (1997) factor timing model
(CHM)
93.31
(3.252)
[0.001]
0.15
(1.273)
[0.203]
-0.10
(-0.664)
[0.507]
-0.18
(-1.040)
[0.299]
-0.32
(-3.003)
[0.003]
0.670
This table reports the results of the timing abilities of MRS (Strategy 4), which
avoids the reversal of winner and loser by using the convex-concave method. All
common stocks on CRSP are included. Panel A refers to the traditional Treynor and
Mazuy (1966) model (TM), and Panel B refers to the traditional Henriksson and
Merton (1981) model (HM). Panel C refers to the Treynor and Mazuy (1966) and
Carhart (1997) factor timing model (CTM), and Panel D refers to the Henriksson
and Merton (1981) and Carhart (1997) factor timing model (CHM). The CTM model
contains Carhart’s (1997) four factors – market excess return (RMRF), Fama and
French’s (1993) factor-mimicking portfolios for size (SMB) and book-to-market
equity (HML), and Carhart’s (1997) factor-mimicking portfolio for one-year return
momentum (MOM) – and the squares of Carhart’s four factors. The t-statistics are in
parentheses and the p values are in brackets.
65
-150
-100
-50
0
50
100
150
200
250
Jan. 1965–Dec.1969
Jan. 1970–Dec.1974
Jan. 1975–Dec.1979
Jan. 1980–Dec.1984
Jan. 1985–Dec.1989
Jan. 1990–Dec.1994
Jan. 1995–Dec.1999
Jan. 2000–Dec.2004
Jan. 2005–Dec.2009
Time
Ris
k-a
dju
ste
d R
etu
rn (
ba
sis
po
int)
Strategy 1
Strategy 2
Strategy 3
Strategy 4
Strategy 5
FIGURE 1 The Profits of Five Trading Strategies in Sub-periods. The profits of five strategies are plotted against different sub-periods. Each
sub-period lasts for five years. The convex-concave method is adopted to categorize stocks into four groups.
66
-400
-300
-200
-100
0
100
200
300
1 2 3 4 5
Firm Size
Ris
k-a
dju
ste
d R
etu
rn (
ba
sis
po
int)
Strategy 1
Strategy 2
Strategy 3
Strategy 4
Strategy 5
FIGURE 2 The Profits of Five Trading Strategies Conditional on Firm Size. The
profits of five strategies are plotted against firm size quintiles, 1 (smallest) to 5
(largest). The convex-concave method is adopted to categorize stocks into four
groups.