Discussion Paper No. 38

Time Series Residual
Momentum
Hongwei Chuang
March, 2015

Data Science and Service Research
Discussion Paper

Center for Data Science and Service Research
Graduate School of Economic and Management
Tohoku University27-1 Kawauchi, Aobaku
Sendai 980-8576, JAPANTime Series Residual Momentum
Hongwei Chuanga,∗a Graduate School of Economics and Management, Tohoku University27-1 Kawauchi, Aoba Ward, Sendai 980-8576, Japan

Abstract
The momentum strategy as described in the seminal work of Jegadeesh and Titman (1993) leads to stream of studies on theoretical work of momentum effectand empirical analysis in different financial assets and markets of several countries.
However, the characteristics displayed by the momentum strategy have often beingargued that was associated with risk exposures to pricing factors and time-varying
risk. Especially, the investment portfolio’s performance endanger profound drawdown risk when the market rebounds after financial crisis. In this study, I proposed
the time series residual momentum strategy to mitigate the magnitude of losses. If
the stock positively (negatively) deviates from predicted intrinsic values during ashort-term period, it is denoted as over-valuation (under-valuation). Through investigating the U.S. stock market, the empirical results show the proposed strategy can
not only achieve significant improvement for the conventional momentum strategy,
but also can substantially reduce the drastic losses from financial crises.
Key words: Momentum; Fama-French Factors; Asset Pricing; Financial Crisis
JEL classification: C3; G2

1. Introduction
As Isaac Newton said “What goes up must come down.” Everything inearth should obey the law of gravity. Therefore, it is natural to assume thelaw of gravity can also be applied in financial market. However, academic
studies have found out that stocks which perform better than the other stocks
in recent past will continue for a certain period. The phenomenon is called“momentum” effect. Past studies also have confirmed that this effect has been
observed not only in the stock market but also in commodity and currency
markets. The profit cannot be explained away by saying that high-performancestocks are more risky, or by arguing that the trading cost would eat up all the
profit, instead they are considered potential profitable.
This anomaly also drives a juggernaut through one of the tenets in financial theory, which is the efficient market hypothesis. In the weak-form efficient
market hypothesis, the past price movements should provide no guide to fu1

ture price changes. Investors should have no logical reasons to prefer the pastwinners to the past losers. Therefore, we are going to revisit the conventional
momentum strategy in the U.S. stock market by using the residual analysis over a longer period from January 1930 to December 2010. Our proposed
strategy can control the short-term behavior and intermediate continuation ofa stock.
The intermediate-horizon return continuations are first reported by Jegadeesh and Titman (1993). They propose the momentum strategy, whichbuys the past winning stocks over the previous three to twelve months andsells the past losing stocks over the same period exhibits substantial profit.
Although these results have been widely accepted, researchers are still in debate over the sources of the profit and the interpretation of the evidence. The
momentum effect is not only remarkably persistent in stock market but also incommodity and currency markets. Okunev and White (2003) find momentum
effect in currencies. Erb and Harvey (2006) find momentum effect in commodities. Moskowitz, Ooi and Pedersen (2012) find momentum effect in exchange traded futures contracts. Moreover, the momentum effect is worldwide.
Rouwenhorst (1998) finds evidence of momentum effect in developed stock
markets, and Rouwenhorst (1999) documents momentum effect in emerging
markets.
Although the momentum strategy tends to work well on the winners andthe losers over the past three-to-twelve months, the momentum effect disappears for longer periods in three or five year. Just as trees do not grow to
the sky, share prices do not rise forever. Many investigators have argued themomentum strategies to display characteristics that are often associated withfactors of price risk. Chordia and Shivakumar (2002) reveal the profits of momentum strategies to exhibit strong variation across the business cycle. In
their study, they show the conventional momentum strategy earns 14.70% annualized return during expansions and loses 8.70% during recessions over theperiod from January 1930 to December 2009. Cooper, Gutierrez and Hameed(2004) further examine the variation of average returns to the U.S. equity momentum strategies. They find in “UP” states, which are defined by the lagged
three-year return of the market, the historical mean of returns of a equallyweighted momentum strategy is 0.93% per month. In “DOWN” states, thehistorical mean of returns of a equally-weighted momentum strategy is -0.37%
per month. It means those losers often experience strong gains after the market
collapses. In such circumstances, implementation of the momentum strategy
will result in persistent strings of negative returns. It leads to a momentum
crash.
In this paper, we propose a more effective strategy, time series residual
momentum, which ranks all eligible stocks based on their total returns andresidual returns in the recent past months. The advantage of the time series2momentum strategy is not only to reduce the time-varying exposures but also
to eliminate the “value risk,” which occurs when a stock is overvalued. Therefore, this time series residual momentum strategy performs better than theconventional momentum strategy and the the residual momentum strategy.
Most important of all, it can avoid the huge loss of the conventional momentum strategy after 2000.
To sum, in past decades, financial academics and practitioners have recognized that the momentum strategy can generate significant profit and thatthe momentum effect exists in many financial markets. But, fewer people notice the collapse of the momentum is profound especially after the financial
crisis. In this study, we adjust the conventional momentum strategy in the
U.S. stock market by using the residual analysis over the period from January 1930 to December 2010. The modification we proposed can take care of
the short-term behavior and the intermediate continuation of a stock. The
empirical results show the modification can largely improve the conventional
momentum strategy.
2. Momentum Crash
We first examine the conventional momentum strategy over a longer period in the U.S. stock market from January 1930 to December 2010. Mostof time, the market appears to underreact to the public information andresults in consistent price momentum. However, in extreme conditions, thepast losers of the momentum strategy often comprise a very high premium.
When the market conditions are going better, those past losers will experiencestrong gains. This leads to a “momentum crash.” It is because the returns of
the conventional momentum strategies are highly-skewed. Therefore, investors
who implement the conventional momentum strategy will experience stringsof negative returns, especially after the market collapses.
Data and Portfolio Formation
The data for the study is from CRSP monthly files. Our data consists ofall domestic, primary stocks listed on the New York Stock Exchange (NYSE),
American Stock Exchange (AMEX), and Nasdaq (NASDAQ) stock markets
during the period from January 1929 to December 2010. We utilize only the
returns on common shares with a 10 or 11 CRSP share-code. Close-end funds,
Real Estate Investment Trust (REITs), unit trusts, American Depository Receipts (ADRs), and foreign stocks are excluded from the analysis. We also
exclude stocks with price below $1 during the formation periods to reduce themicrostructure effect associated with low-price stocks.
Following most of the literature, we rank the stocks based on their past
twelve-month returns excluding the most recent month. Then we assign thesestocks into 10 equally weighted portfolios at the end day of each formation
3

Fig. 1. Momentum Portfolio Formation

month. Each portfolio is held for one month following the formation month.
The reason why we focus on their past twelve-month returns excluding themost recent month is that this momentum definition is currently most broadly
used and readily available through the PR1YR factor of Carhart (1997). The
momentum strategy is typically to disentangle the intermediate horizon momentum effect from the short reversal effect documented by Jegadeesh (1993)
and Lehmann (1990).
Figure 1 illustrates time line of the portfolio formation process for the
momentum returns in May 2009. The ranking period returns are the cumulative returns from the close of the last trading day in April 2008 to the closeof the last trading day in March 2009. We sort all firms that meet the datarequirements into 10-decile portfolios which are labeled P1 to P10 according
to their cumulative returns over the ranking period. The 10% of firms withthe highest ranking period returns go into portfolio P10, the “[W]inner”-decile
portfolio, and those with the lowest ranking period returns go into portfolio
P1, the “[L]oser”-decile portfolio. The return on a zero investment “WinnerMinus-Loser” (WML) portfolio is the difference of the return on the Winner
and the return on the Loser portfolio in each period.
The monthly returns of the decile portfolios are based on the equallyweighted returns. Decile membership does not change in a month, except forthe case of delisting. We also consider the overlapping portfolios approach,which is a strategy that holds a series of portfolios selected in the current
month and the previous month. The market return we used is Dow Jones
Industry Average (DJIA) index downloaded from Yahoo! Finance. 1 The riskfree rate and the cross-sectional Fama-French three factors are downloaded
from the data library of Kenneth R. French. 2
Momentum Portfolio Performance
Table 1 presents the moments of the momentum decile portfolios fromJanuary 1930 to December 2010. For all portfolios, Mean, Std, Skew and Kurt
1

We use DJIA instead of S&P 500 because the first trading day of S&P 500 is Jan
3, 1950.
2 http://mba.tuck.dartmouth.edu/pages/faculty/ken.french4denote the full-period realized mean, stand deviation, skewness and kurtosisof the monthly returns to the portfolios. The average return of the winner
portfolio is 1.63% per month and is 0.69% of the loser portfolio. A self-financing
strategy that buys the top 10%, Winner, and sells the bottom 10%, Loser,
produces profits of 0.94% per month. The result is consistent with the existing
literature, which states that there is a strong momentum premium over the
past 80-year period.
Table 1
Twelve-Month Momentum Portfolio Characteristics, January 1930-December 2010
Portfolio

Mean

Std

Skew

Kurt

P1 (Past Loser)

0.69%

0.10672.386617.9023P2

0.82%

0.08191.624114.4532P3

1.01%

0.07812.176020.5349P4

1.06%

0.07191.555815.7259P5

1.15%

0.06901.167813.2865P6

1.20%

0.06630.946512.3870P7

1.28%

0.06450.41847.8186P8

1.41%

0.0676

0.67139.8730P9

1.54%

0.07180.33666.9300P10 (Past Winner)

1.63%

0.08701.029413.4764

WML

0.94%

0.0701-2.7992

22.7026

The skewness of the past loser portfolio is 2.3866, and the skewness ofthe past winner portfolio is 1.0294. The skewness of the past loser portfolio is
more than twice of the skewness of the past winner portfolio. One should notethat the past loser portfolios are considerably positively-skewed than the pastwinner portfolios. Therefore, the skewness of WML portfolio is -2.7992, whichis largely negative-skewed. Another thing we should note is that the kurtosisof WML portfolio is 22.7026, which is also considerably large than usual.
The skewness of returns on momentum portfolios is not surprising. For
example, Wall Street calls a high momentum stock that suddenly collapses
as a “torpedo.” A torpedoed stock is a stock hit by new negative information
after a long period of positive information. As the stock price has continuously
increased in the past, investors may expect that the company’s business is ingood conditions. Hence, the upward trend in earnings will more likely continue. However, it remains possible that the trend of increasing earnings may
suddenly reverse. When it occurs, the positive effect on the stock’s earnings
may also be reversed, ex. some stocks in the technology sector, where new innovations make existing technologies obsolete. Those fast-growing companies
5Fig. 2. Cumulative Monthly Returns of WML, DJIA and Risk-free, January
1930-December 2010

which depend on existing technologies might be affected. The growth of thesefirms suddenly stop.
Figure 2 presents the cumulative monthly returns 3 for investing $1 initially in (1) the risk-free asset; (2) the DJIA index; and (3) the zero investment
portfolio, WML portfolio, respectively, over the period from January 1930 to
December 2010. On the right side of the plot, we present the final dollar values
for each of the three portfolios. We can see the final value of WML is $448.73,
DJIA is $46.59, and the risk-free asset is $17.78. The momentum strategy does
earn a significant return.
Momentum Crashes
Good things don’t last long. The momentum strategy appears to loose
their profitability in the recent years. Again, we plot the cumulative monthly
returns for investments in the risk-free asset, the DJIA index, and the WML
portfolio from January 2000 to December 2010 as shown in Figure 3. We
can find the WML portfolio loses almost 70% at the end of 2010. In fact,
it lost 1.46% per annum over the period from January 2000 to December
2010. The large losses of the WML portfolio occured in the first half of 2009,3

The cumulative return on an implementable strategy is an investment at time 0
and fully reinvested at each time point. During the period, there is no cash put inor
QTtaken out. R(t, T ) denotes the cumulative return between time t to T , R(t, T ) =s=t+1 (1 + Rs ).

6

Fig. 3. Cumulative Monthly Returns of WML, DJIA and Riskfree, January 2000-December 2010

especially in March, April, and May. The losses were 18.22%, 39.52%, and14.29% respectively. The rough under-performance appears when the market
rebounds from the bottom. We argue that the large negative skewness of theWML portfolio can cause the momentum crash.
Table 2
Top 10 Worst-Month WML Portfolio Returns
Rank

Month

Past Loser

Past Winner

WML

DJIA

1

1932/0777.76%

15.58%-62.18%25.79%

2

2001/0178.79%

17.74%-61.05%

0.92%

3

1932/08

100.08%43.18%-56.91%35.76%

4

1939/0981.67%

24.78%-56.89%11.72%

5

2009/04

48.12%

8.61 %-39.52%

7.35%

6

2002/1137.68%

8.26 %-29.42%

5.94%

7

1932/01

29.66%

5.14 %-24.53%-1.73%

8

1975/0144.12%

20.02%-24.10%14.19%

9

1938/06

38.84%

16.52%-22.32%24.26%

10

1974/0124.72%

4.84 %-19.88%0.55%

Table 2 presents the top ten worst-month returns of the WML strategy over the periods from January 1930 to December 2010. The table also
7

gives the contemporaneous monthly returns of DJIA. These returns often occur when the market has dramatic downturns, and during a month where
the market gradually rises. For example, during the two-month period from
July to August in 1932, the DJIA increased 61.55%, but the return of theWML portfolio decreased 119.09%. As we can also see from the Table 2, during the two-month period, returns of the Past Loser deciles are 77.76% and100.08% outperform the returns of the Past Winner deciles which are 15.58%
and 43.18% respectively. Thus, the strong momentum reversals can be characterized as a momentum crash. The momentum crash happens in the short sideof the portfolio, the past losers portfolios, which are crashing up rather thandown. We should also notice that these momentum crashes are also clusteredas shown in Figure 4. The strong performance of the past losers occurs overthe span of multiple months.
Robustness Check
For robustness check, we also plot the cumulative monthly WML portfolio
returns over the period from January 1930 to December 2010 by considering
different formation periods, i.e., we rank the stocks based on the past j-month
return, where j=3, 6, and 9. The result is shown as Figure 5. We can see thecumulative monthly WML return based on the past 3-month return is $2.79.
The cumulative monthly WML return basing on the past 6-month return is$7.11. And the cumulative monthly WML return basing on the past 9-month
return is $17.11. The best performance of cumulative monthly WML returnis based on past 12-month return, which reaches $448.73. Thus, we focus on
the 12-month formation period as our momentum strategy analysis hereafter.
Table 3 presents the moments of the momentum decile portfolios fromJanuary 1930 to December 2010 based on the past 3-month return. The skewness of the past loser portfolios is 1.5887, and the skewness of the past winner portfolios is 0.3401. The skewness of WML portfolios is -2.8086, which is
negative. The kurtosis of WML portfolios is 22.8985. Table 4 presents the moments of the momentum decile portfolios from January 1930 to December 2010
based on the past 6-month return. The skewness of the past loser portfolios
is 2.0914, and the skewness of the past winner portfolios is 0.0396. The skewness of WML portfolios is -2.6249, which is negative. The kurtosis of WML
portfolios is 19.7919. Table 5 presents the moments of the momentum decile
portfolios from January 1930 to December 2010 based on the past 9-month
return. The skewness of the past loser portfolios is 2.1495, and the skewnessof the past winner portfolios is 0.8497. The skewness of WML portfolios is-3.1013, which is negative. The kurtosis of WML portfolios is 22.4271.
Further, we plot the cumulative monthly WML portfolio returns over theperiod from January 2000 to December 2010 by considering different formation
periods as shown in Figure 6. We can see the cumulative monthly WML return8Fig. 4. Momentum Portfolio Performance, January 1930-December 20109

Table 3
Characteristics of 3-month Momentum Portfolios, January 1930-December 2010
Portfolio

Mean

Std

Skew

Kurt

P1 (Past Loser)

0.91%

0.1013

1.5887

10.4791P2

1.09%

0.08711.894115.8793P3

1.23%

0.08072.118219.4230P4

1.19%

0.07191.216713.2133P5

1.15%

0.06710.61088.6659P6

1.17%

0.06640.782311.5043P7

1.15%

0.06440.38497.1533P8

1.16%

0.06640.3782

7.8832P9

1.16%

0.07000.25456.3014

P10 (Past Winner)

1.24%

0.08050.3401

6.3945WML

0.33%

0.0613-2.8086

22.8985

Table 4Characteristics of 6-month Momentum Portfolios, January 1930-December 2010
Portfolio

Mean

Std

Skew

Kurt

P1 (Past Loser)

0.95%

0.10732.091413.6075

P2

1.02%

0.08782.045416.3411P3

1.06%

0.0794

1.769215.2567P4

1.16%

0.0761

2.191220.9869P5

1.16%

0.0692

1.392814.3128P6

1.18%

0.06801.082013.3095P7

1.17%

0.0641

0.36247.9755P8

1.18%

0.0644-0.10765.2800P9

1.35%

0.0689-0.0565

5.1462

P10 (Past Winner)

1.44%

0.08110.03964.4203WML

0.49%

0.0691-2.6249

19.7919

based on the past 3-month return is $0.51. The cumulative monthly WML
return based on the past 6-month return is $0.47. And the cumulative monthly
WML return based on the past 9-month returns is $0.2. It shows that the
momentum crash won’t eliminate no matter how many months of the past
j-month return we consider, where j=3, 6, 9, and 12.
10

Fig. 5. j-Month Momentum Portfolios Performance, January 1930-December 201011

Table 5
Characteristics of 9-month Momentum Portfolios, January 1930-December 2010
Portfolio

Mean

Std

Skew

Kurt

P1 (Past Loser)

0.82%

0.1066

2.149515.7713P2

0.97%

0.08702.279119.8423P3

1.02%

0.07801.648915.1072P4

1.07%

0.0713

1.255012.4999P5

1.09%

0.0686

1.1951

12.7131P6

1.19%

0.06651.037412.1164P7

1.28%

0.06680.8437

10.9467P8

1.28%

0.06730.4746

10.1607P9

1.44%

0.0717

0.48688.7271

P10 (Past Winner)

1.48%

0.08520.8497

11.3598

WML

0.66%

0.0736-3.101322.4271

3. Time Series Residual Momentum Strategy
Many investigators have argued that the characteristics displayed by the
momentum strategy are often associated with factors of price risk. Chordia andShivakumar (2002) show that the profits of momentum strategies to exhibit
strong variation across the business cycle. They report that the conventional
momentum strategy earns 14.70% annualized return during expansions andloses 8.70% during recessions over the period from January 1930 to December
2009.
In addition, Blitz, Huij and Martens (2011) also argue that the conventional momentum strategies exhibit the time-varying exposures to FamaFrench three factors. To counter this drawback, they propose the “residual
momentum strategy” to reduce the time-varying exposures by ranking all
stocks based on its own residual returns instead of the total returns. Blitz,
Huij and Martens (2011) show that the residual momentum strategy can earn
profits that are about twice as large as the conventional momentum strategy
with more consistency over time.
The literature on momentum strategies focus on the relative performanceof securities in cross-sectional perspective. Moskowitz, Ooi and Pedersen (2012)
propose the “time series momentum strategy” in equity index, currency, commodity, and bond futures. They only focus on a security’s past returns. They
found that the past twelve-month returns of a security to show strong positive
influence on its future return with persistent effect for about one year, followedby partially-reversed role over longer horizons. This strategy can also deliver12

Fig. 6. Cumulative Monthly Returns of j-Month Momentum Portfolios, January
2000-December 2010

substantial abnormal returns with little exposure to the standard asset pricing
factor to perform well during extreme market condition.
The advantage of cross-sectional residual momentum is to reduce the time13

varying exposures. On the other hand, the advantage of the time series momentum is that the security’s past returns provide strong positive predictability to
its future returns. Hence, it is natural to combine the cross-sectional residual
momentum strategy and time series momentum strategy into the “time series
residual momentum strategy.” Our empirical evidence shows the strategy can
avoid the huge loss in recent years.
Cross-Sectional Residual Momentum
The residual momentum strategy proposed by Blitz, Huij and Martens(2011) is also known as a “cross-sectional” residual momentum strategy, since
their approach focused on the cross-sectional relative performance in residual
returns of securities. Their formula is shown as follows. In each month t, theresidual returns are estimated by using Fama-French three factors model, thatis,(1)
rti = αi + β1i MKTt + β2i SMBt + β3i HMLt + itiwhere rt is the excess return of stock i in month t. MKTt , SMBt , and HMLt
are the excess returns of factor-mimicking portfolios for the market, size and,value in month t. αi , β1i , β2i and β3i are parameters to be estimated. And it is
the residual return.
Only the stocks with complete return history of over 36-month rolling
regression window were included. The purpose of using 36-month rolling windows is to ensure having a sufficient number of return observations to obtainaccurate estimates for stock exposures to the market, size, and value. In orderto obtain the residual return, the regression over 36-month rolling windows,
was run, i.e., over the period from t − 36 to t − 1 to get the estimated parameters α̂i , β̂1i , β̂2i and β̂3i of each stock i. The estimated residual return of stocki in month t is
eit = rti − (β̂1i MKTt + β̂2i SMBt + β̂3i HMLt ),(2)

where the estimated α̂i is not included, since the intercept of regression served
as a general control for the misspecification in the model for expected stock
returns.
At each formation month t, they rank all eligible stocks into 10-decile
portfolios according to the estimated residual returns from past twelve months,
standardized by the standard deviation over the same period of time. The cumulative monthly returns of the residual momentum strategy and total return
momentum strategy are plotted by Blitz, Huij and Martens (2011) in Figure7.
As a result, the highest value of the residual momentum strategy overthe period from January 1930 to December 2010 was about $10,000, larger
than those of total return momentum. This suggested that the performance of
14

Fig. 7. Cumulative monthly returns plot by Blitz, Huij and Martens (2011), January
1930-December 2010

the residual momentum strategy is more consistent in compared to the conventional momentum strategy. In addition, this also exposed how poorly theconventional momentum strategy has performed during “The Great Depression” that started from 1929 and lasted until the late 1930s or early 1940s.
Time Series Momentum
The literature on momentum strategies focus on the relative performanceof securities in cross-sectional perspective. Moskowitz, Ooi and Pedersen (2012)
propose the “time series momentum strategy” in equity index, currency, commodity, and bond futures, with focus on past returns of a security. In theirstudy, they first examine the time series predictability of future returns acrossdifferent time horizons. In other words, they regress the excess return rt in themonth t on its lagged h months return, rt−h . Both of the returns are scaled22by their ex-ante volatilities, σt−1
and σt−h−1
,
rt−h
rt
= α + βh 2+ t ,2σt−1
σt−h−1(3)where the ex-ante volatility is annualized as follows:
σt2 = 261∞
X

(1 − δ)δ i (rt−1−i − r¯t )2 .

(4)

i=0

261 is the factor which scales the variance to be annual, r¯t is the exponential
weighted average return, and the weight (1 − δ)δ i add up to 1. The parameterδ is chosen so that the center of mass of the weight is 60 days. The way toadjust futures returns by their ex-ante volatilities is similar to use the Generalized Least Squares instead of Ordinary Least Squares. They find a positive15

Fig. 8. Time Series Predictability by Moskowitz, Ooi and Pedersen (2010)t-statistic for the first twelve months, indicating the significant continuationand trend in its return.
Figure 8 shows that t-statistics are positive in twelve months. This indicates the significant return continuation or trends. In their study, the moststrong time series momentum effect are on the properties of the past twelvemonth time series momentum and one month holding period. The time series
momentum strategy is defined to take a long position if the past twelve-month
cumulative excess return is positive and a short position if the past twelvemonth cumulative excess return is negative. The position is scaled by the
ex-ante annualized volatility that is 0.60%. Therefore, the return of the timeseries momentum strategy for security i from time t to t + 1 is as follow:isign(rt−12,t)

0.06% ir,
σti t,t+1(5)i
is the return of security i from
where σti is defined as same as (4) and rt,t+1
time t to t + 1.

Time Series Residual Momentum
Previous literatures have generally attributed the momentum effect tofirm-specific returns, arguing that investors have either under-reacted or belatedly overreacted to the firm-specific returns. Recently, Lewellen (2002) studied the momentum effect in focusing on the role of industry, size, and B/Mfactors. Lewellen (2002) argued that it is impossible to explain the significant component of momentum through firm-specific returns with behavioral
16

model. Instead, that the excess covariance, not underreaction, can explain the
momentum effect in the portfolios.
It is well known that the momentum effect is a cross-sectional result andit is not the same as positive autocorrelation. As Lo and MacKinlay (1990)
showed in 1990, the possibility of momentum effect been caused by the autocorrelation in returns, the lead-lag relation among stocks (cross-serial correlation) or the cross-sectional dispersion in unconditional means. Later, Lewellen(2002) has also proposed a simpler strategy based on Lo and Mackinlay’s
theory to consider holding assets in proportion to the market-adjust returns,
shown as followed.
The proportion weight of asset i in month t is
wti =

1 i
m
(r − rt−1
),
N t−1

(6)

m
i
is the corresponding returnis the asset’s return in month t − 1, rt−1
where rt−1
on the equally-weighted index, and N is the total number of stocks. Assumingthat returns have unconditional mean µ ≡ E[rt ] and autocovariance matrixΩ ≡ E[(rt−1 − µ)(rt − µ)0 ]. Therefore, the portfolio return in month t equals

πt =

X
i

wti rti =1 X im(r − rt−1)rti ,
N i t−1(7)

and the expected profit is
1 X m i
1 X i iE[ (rt−1 rt )] − E[ (rt−1
rt )]
N
N
i
i
1 X
=
(ρi + µ2i ) − (ρm + µ2m ),N i

E[πt ] =(8)

where ρi and ρm are the autocovariances of asset i and the equally-weighted
index m. (8) shows that the profit of the portfolio depends on the magnitudeof asset autocovariances relative to market’s auto-covariance. (8) can also be
written into a matrix notation. Let the average autocovariance equal tr(Ω)/Nand the autocovariance of the market portfolio equal 10 Ω1/N 2 . tr(·) denotes
the sum of the diagonals and 10 is a vector of ones. Therefore, (4.8) equals
11
tr(Ω) − 2 10 Ω1 + σµ2
N
N
N −1
1
=
tr(Ω) − 2 [10 Ω1 − tr(Ω)] + σµ2 ,2NNE[πt ] =(9)where σµ2 is the cross-sectional variance of unconditional expected returns [collecting the µ2i and µ2m terms in (8)].
The decompositions of Lewellen (2002) suggest the profit of momentum17

Table 6
Categories of Momentum Strategy

Momentum

Residual Momentum

Cross Section

Jegadeesh and Titman (1993), ...

Blitz, Huij and Martens (2011)

Time Series

Moskowitz, Ooi and Pedersen (2012)

Our study

to rise in three ways as followed. First, stocks are positively autocorrelated,
so firms currently with high returns are expected to have high returns in
the future. Second, since the stock returns are negatively correlated to thelagged returns of other stocks, high future returns can be predicted by theother stocks’ poor performances. Third, stocks simply have high unconditional
expected returns relative to the other stocks. Overall, the empirical finding of
Lewellen (2002) has concluded the lead-lag relations to play an important role
among stocks.
One of the advantages in cross-sectional residual momentum is that it
consists of the ability to reduce the time-varying exposure effect of the conventional momentum strategies. This time series momentum shows that the
future returns of a stock can be forecast by its past returns. In addition,in the decomposition of Lewellen (2002), it is natural to combine the crosssectional residual momentum strategy with time series momentum strategy
into the “time series residual momentum strategy.” Summary of the different
categories in current momentum strategy is shown in Table 6.
Methodology
Without loss of generality, we assume that there are n securities with Ti to
be the length of security i. To implement the time series residual momentum
strategy, the first step is to estimate the time series residual returns of eachsecurity. In following with Blitz, Huij and Martens (2011), we ran the following
regression through the same equation as equation (1) in month t, for eachsecurity i, where the residual returns eit is obtained by equation (2).
We propose a measure of value risk, denoting qti (j) for security i based on
its own past j-month time series residual return in the month t. The qti (j) is18

Fig. 9. Q-Q plot

defined as:

Pt−j+1 i
eti
qt (j) = Φ( qt),j × σ̂t2(10)

where σ̂t2 is the variance of eit over the same period, j is the strength of value
risk over the past j months ( j = 3, 6, 9 or 12), and Φ is the cumulative
function of a standard Normal distribution.
qti (j) can be viewed as a proxy measure of the present value relative to its
long-term equilibrium level. qti (j) is used to describe whether or not a stock’s
recent j -month return has been overvalued. Through this formula, we can form
the portfolios based on independently sort all eligible stocks by their own past
12-month accumulative monthly return and j -month value-risk measure, qti (j)at each formation month t.
In argument regarding to whether or not

Pt−j+1 i√t

et

j×σ̂t2

follows a Standard
Pt−j+1 i

Normal distribution can be confirmed by a Q-Q plot of √t

19et

j×σ̂t2for a stockwith its permno number 10006, as shown in Figure 9. The Q-Q plot wasconstructed using different j -month, j=3, 6, 9 and 12. In j=3, the sample
quantiles were approvimate to the Standard Normal distribution, confirmingthat it is not inappropriate to use Φ as the transformation.
In order to form the portfolios, the stocks were first assigned into one of 10
portfolios based on their cumulative returns over the previous twelve months.
The top 10% of firms with the highest ranking period returns were filed into
portfolio P10, the “[W]inner” decile portfolio. On the other hand, the bottom
10% of firms with the lowest ranking period returns were filed into portfolio
P1, the “[L]oser” decile portfolio. Besides, the stocks are also independently
assigned into one of the 3 portfolios with the categories of [U]nder, [M]iddle
and [O]ver, based on the value-risk measures over the same period of time.
In combination of the two separately arranged ranking methods, up to 30
value-risk momentum portfolios were obtained. “WU-LO” was defined as themonthly portfolio return of the “[W]inner that is [U]nder-valued minus the[L]oser that is [O]ver-valued.” A return on a zero investment of a WU-LO
portfolio is the difference between the return in the undervalued winner andovervalued loser portfolio in each time period. In this study, only monthly
return over next one month was focused based on a equally-weighted averageof the portfolio returns.
Empirical results
The results are shown in Table 7, where the standard deviations are in theparentheses. As seen from the table, when j = 3, the monthly average returnof the WU-LO portfolio is 2.44% that is equal to 33.57% annual return. For j= 6, the monthly average return of the WU-LO portfolio is 1.67% that is equal
to 22.06% annual return. For j = 9, the monthly average return of the WU-LO
portfolio is 1.74% that is equal to 23.01% annual return. Overall, the returnsof these three WU-LO portfolios outperformed the conventional momentum
strategy. We further use one-tailed t-test to examine the difference between
the returns of WU-LO portfolio and WML. For j = 3, the t-test statistic of
WU-LO portfolio to WML is 3.4675 with p-value < 0.01. For j = 6, the t-test
statistic of WU-LO portfolio to WML is 1.4003 with p-value < 0.1. For j = 9
and j = 12, the t-test statistics are not significant at 5% significant level.
Initially, our study focused on the consideration of short-termed contrarian and the continuation in intermediate momentum of each stock. As a result,the improvement in conventional momentum strategy with value risk was observed, mainly due to the undervaluing of a firm at its high momentum. At thiscircumstance, the stock price will tend to rise instead of fall in the following
month. Thus, our strategy was aimed at controlling the short-term behaviorof a stock and the persistent intermediate momentum.
20

21

Figure 10 plot the monthly portfolio return of the time series momentum
strategy (j = 3) over the period from July 1967 to December 2010. As a result, the time series momentum strategy was shown to perform better thanthe conventional momentum strategy as shown in Figure 3.4. In addition, the
WU-LO portfolio returns were observed to be more stable than the returns ofconventional momentum portfolio, though we also recognized the imperfectionof our strategy with incapability of avoiding a huge loss of -37.86% in April
2009. Further plotting on the cumulative monthly return of conventional momentum strategy and time series residual momentum strategy over the time
period from July 1967 to December 2010 was also shown in Figure 11. As aresult, the cumulative monthly return of WU-LO is $116,926.2; whereas the
cumulative monthly return of WML was shown to be $69.09. In considering
the performance after 2000, the cumulative monthly return was also plotted forconventional momentum strategy and time series residual momentum strategy
over the time period from January 2000 to December 2010, as shown in Figure
12. The WML was shown to lose about 70% in December 2010 compared to
its initial investment in January 2000. As for the value using our strategy,217.10% was observed in December 2010 comparing to its initial investment
in January 2000.
4. Conclusions and Future Researches
The momentum effect has been widely accepted in financial market. But,
fewer people notice the collapse of the momentum is profound especially after
financial crisis. There are “momentum crashes.” Just as trees do not growto the sky, likewise, share prices do not rise forever. The momentum crashis because that conventional momentum strategy exhibits substantial timevarying exposures to the market.
In this study, we revisit the conventional momentum strategy in the U.S.
stock market by the residual analysis and the price-risk adjustment over a long
period of time, from January 1930 to December 2010. Such time-varying exposures of the conventional momentum strategy can be reduced by the residual
analysis and the price-risk adjustment. The idea was when a firm is undervalued (under-priced) at its high momentum; the stock price tends to rise
instead of fall in the following month. Conversely, when a firm is overvalued(over-priced) at its low momentum, the stock price tends to fall instead ofrise. The strategies that we proposed can control the short-term behavior ofa stock and the intermediate continuation.
Overall, our empirical results showed that through these two strategies,
significant improvement was observed in compared to the conventional momentum strategy, suggesting it to be a better way for portfolios managementin future studies. Even more, this new concept of residual analysis can also be
further applied to other financial markets as commodities and currencies.
22Fig. 10. Time Series Momentum Portfolio Performance, July 1967-December 201023

Fig. 11. Cumulative Monthly Returns of WML, WU-LO, July 1967-December 201024

Fig. 12. Cumulative Monthly Returns of WML, WU-LO, January 2000-December
2010

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