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Time series momentum

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Journal of Financial Economics

Journal of Financial Economics 104 (2012) 228–250
0304-40

doi:10.1

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journal homepage: www.elsevier.com/locate/jfec
Time series momentum$
Tobias J. Moskowitz a,n, Yao Hua Ooi b, Lasse Heje Pedersen b,c

a University of Chicago Booth School of Business and NBER, United States
b AQR Capital Management, United States
c New York University, Copenhagen Business School, NBER, CEPR, United States
a r t i c l e i n f o

Article history:

Received 16 August 2010

Received in revised form

11 July 2011

Accepted 12 August 2011
Available online 11 December 2011

JEL classification:

G12

G13

G15

F37

Keywords:

Asset pricing

Trading volume

Futures pricing

International financial markets

Market efficiency
5X & 2011 Elsevier B V. .

016/j.jfineco.2011.11.003

thank Cliff Asness, Nick Barberis, Gene F

Hentschel, Brian Hurst, Andrew Karolyi, John L

hard Thaler, Adrien Verdelhan, Robert Vishny

rgler, and seminar participants at NYU a

s in Denver, CO for useful suggestions and d

nd Haibo Lu for excellent research assistance

iative on Global Markets at the University

of Business and CRSP for financial support.

esponding author.

ail address:

oskowitz@chicagobooth.edu (T.J. Moskowitz

Open access under CC B
a b s t r a c t

We document significant ‘‘time series momentum’’ in equity index, currency, commod-

ity, and bond futures for each of the 58 liquid instruments we consider. We find

persistence in returns for one to 12 months that partially reverses over longer horizons,

consistent with sentiment theories of initial under-reaction and delayed over-reaction.

A diversified portfolio of time series momentum strategies across all asset classes

delivers substantial abnormal returns with little exposure to standard asset pricing

factors and performs best during extreme markets. Examining the trading activities of

speculators and hedgers, we find that speculators profit from time series momentum at

the expense of hedgers.

& 2011 Elsevier B.V. Open access under CC BY-NC-ND license.
1. Introduction: a trending walk down Wall Street

We document an asset pricing anomaly we term ‘‘time
series momentum,’’ which is remarkably consistent across
very different asset classes and markets. Specifically, we
find strong positive predictability from a security’s own
past returns for almost five dozen diverse futures and
ama, John Heaton,

iew, Matt Richard-

, Robert Whitelaw,

nd the 2011 AFA

iscussions, and Ari

. Moskowitz thanks

of Chicago Booth

).

Y-NC-ND license.
forward contracts that include country equity indexes,
currencies, commodities, and sovereign bonds over more
than 25 years of data. We find that the past 12-month
excess return of each instrument is a positive predictor of
its future return. This time series momentum or ‘‘trend’’
effect persists for about a year and then partially reverses
over longer horizons. These findings are robust across a
number of subsamples, look-back periods, and holding
periods. We find that 12-month time series momentum
profits are positive, not just on average across these assets,
but for every asset contract we examine (58 in total).

Time series momentum is related to, but different
from, the phenomenon known as ‘‘momentum’’ in the
finance literature, which is primarily cross-sectional in
nature. The momentum literature focuses on the relative
performance of securities in the cross-section, finding that
securities that recently outperformed their peers over the
past three to 12 months continue to outperform their

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T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250 229
peers on average over the next month.1 Rather than focus
on the relative returns of securities in the cross-section,
time series momentum focuses purely on a security’s own
past return.

We argue that time series momentum directly
matches the predictions of many prominent behavioral
and rational asset pricing theories. Barberis, Shleifer, and
Vishny (1998), Daniel, Hirshleifer, and Subrahmanyam
(1998), and Hong and Stein (1999) all focus on a single
risky asset, therefore having direct implications for time
series, rather than cross-sectional, predictability. Like-
wise, rational theories of momentum (Berk, Green, and
Naik, 1999; Johnson, 2002; Ahn, Conrad, and Dittmar,
2003; Liu and Zhang, 2008; Sagi and Seasholes, 2007) also
pertain to a single risky asset.

Our finding of positive time series momentum that
partially reverse over the long-term may be consistent
with initial under-reaction and delayed over-reaction,
which theories of sentiment suggest can produce these
return patterns.2 However, our results also pose several
challenges to these theories. First, we find that the
correlations of time series momentum strategies across
asset classes are larger than the correlations of the asset
classes themselves. This suggests a stronger common
component to time series momentum across different
assets than is present among the assets themselves. Such
a correlation structure is not addressed by existing beha-
vioral models. Second, very different types of investors in
different asset markets are producing the same patterns
at the same time. Third, we fail to find a link between
time series momentum and measures of investor senti-
ment used in the literature (Baker and Wurgler, 2006; Qiu
and Welch, 2006).

To understand the relationship between time series
and cross-sectional momentum, their underlying drivers,
and relation to theory, we decompose the returns to a
1 Cross-sectional momentum has been documented in US equities

(Jegadeesh and Titman, 1993; Asness, 1994), other equity markets

(Rouwenhorst, 1998), industries (Moskowitz and Grinblatt, 1999),

equity indexes (Asness, Liew, and Stevens, 1997; Bhojraj and

Swaminathan, 2006), currencies (Shleifer and Summers, 1990), com-

modities (Erb and Harvey, 2006; Gorton, Hayashi, and Rouwenhorst,

2008), and global bond futures (Asness, Moskowitz, and Pedersen, 2010).

Garleanu and Pedersen (2009) show how to trade optimally on momen-

tum and reversal in light of transaction costs, and DeMiguel, Nogales,

and Uppal (2010) show how to construct an optimal portfolio based on

stocks’ serial dependence and find outperformance out-of-sample. Our

study is related to but different from Asness, Moskowitz, and Pedersen

(2010) who study cross-sectional momentum and value strategies

across several asset classes including individual stocks. We complement

their study by examining time series momentum and its relation to

cross-sectional momentum and hedging pressure in some of the same

asset classes.
2 Under-reaction can result from the slow diffusion of news (Hong

and Stein, 1999), conservativeness and anchoring biases (Barberis,

Shleifer, and Vishny, 1998; Edwards, 1968), or the disposition effect to

sell winners too early and hold on to losers too long (Shefrin and

Statman, 1985; Frazzini, 2006). Over-reaction can be caused by positive

feedback trading (De Long, Shleifer, Summers, and Waldmann, 1990;

Hong and Stein, 1999), over-confidence and self-attribution confirma-

tion biases (Daniel, Hirshleifer, and Subrahmanyam, 1998), the repre-

sentativeness heuristic (Barberis, Shleifer, and Vishny, 1998; Tversky

and Kahneman, 1974), herding (Bikhchandani, Hirshleifer, and Welch,

1992), or general sentiment (Baker and Wurgler, 2006, 2007).
time series and cross-sectional momentum strategy fol-
lowing the framework of Lo and Mackinlay (1990) and
Lewellen (2002). This decomposition allows us to identify
the properties of returns that contribute to these patterns,
and what features are common and unique to the two
strategies. We find that positive auto-covariance in
futures contracts’ returns drives most of the time series
and cross-sectional momentum effects we find in the
data. The contribution of the other two return
components—serial cross-correlations and variation in
mean returns—is small. In fact, negative serial cross-
correlations (i.e., lead-lag effects across securities), which
affect cross-sectional momentum, are negligible and of
the ‘‘wrong’’ sign among our instruments to explain time
series momentum. Our finding that time series and cross-
sectional momentum profits arise due to auto-covar-
iances is consistent with the theories mentioned above.3

In addition, we find that time series momentum captures
the returns associated with individual stock (cross-sec-
tional) momentum, most notably Fama and French’s UMD
factor, despite time series momentum being constructed
from a completely different set of securities. This finding
indicates strong correlation structure between time series
momentum and cross-sectional momentum even when
applied to different assets and suggests that our time
series momentum portfolio captures individual stock
momentum.

To better understand what might be driving time
series momentum, we examine the trading activity of
speculators and hedgers around these return patterns
using weekly position data from the Commodity Futures
Trading Commission (CFTC). We find that speculators
trade with time series momentum, being positioned, on
average, to take advantage of the positive trend in returns
for the first 12 months and reducing their positions when
the trend begins to reverse. Consequently, speculators
appear to be profiting from time series momentum at the
expense of hedgers. Using a vector auto-regression (VAR),
we confirm that speculators trade in the same direction as
a return shock and reduce their positions as the shock
dissipates, whereas hedgers take the opposite side of
these trades.

Finally, we decompose time series momentum into the
component coming from spot price predictability versus
the ‘‘roll yield’’ stemming from the shape of the futures
curve. While spot price changes are mostly driven by
information shocks, the roll yield can be driven by
liquidity and price pressure effects in futures markets
that affect the return to holding futures without necessa-
rily changing the spot price. Hence, this decomposition
may be a way to distinguish the effects of information
dissemination from hedging pressure. We find that both
of these effects contribute to time series momentum, but
3 However, this result differs from Lewellen’s (2002) finding for

equity portfolio returns that temporal lead-lag effects, rather than

auto-covariances, appear to be the most significant contributor to

cross-sectional momentum. Chen and Hong (2002) provide a different

interpretation and decomposition of the Lewellen (2002) portfolios that

is consistent with auto-covariance being the primary driver of stock

momentum.

4 We also confirm the time series momentum returns are robust

among more illiquid instruments such as illiquid commodities (feeder

cattle, Kansas wheat, lumber, orange juice, rubber, tin), emerging market

currencies and equities, and more illiquid fixed income futures (not

reported).

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250230
only spot price changes are associated with long-term
reversals, consistent with the idea that investors may be
over-reacting to information in the spot market but that
hedging pressure is more long-lived and not affected by
over-reaction.

Our finding of time series momentum in virtually
every instrument we examine seems to challenge the
‘‘random walk’’ hypothesis, which in its most basic form
implies that knowing whether a price went up or down in
the past should not be informative about whether it will
go up or down in the future. While rejection of the
random walk hypothesis does not necessarily imply a
rejection of a more sophisticated notion of market effi-
ciency with time-varying risk premiums, we further show
that a diversified portfolio of time series momentum
across all assets is remarkably stable and robust, yielding
a Sharpe ratio greater than one on an annual basis, or
roughly 2.5 times the Sharpe ratio for the equity market
portfolio, with little correlation to passive benchmarks in
each asset class or a host of standard asset pricing factors.
The abnormal returns to time series momentum also do
not appear to be compensation for crash risk or tail
events. Rather, the return to time series momentum tends
to be largest when the stock market’s returns are most
extreme—performing best when the market experiences
large up and down moves. Hence, time series momentum
may be a hedge for extreme events, making its large
return premium even more puzzling from a risk-based
perspective. The robustness of time series momentum for
very different asset classes and markets suggest that our
results are not likely spurious, and the relatively short
duration of the predictability (less than a year) and the
magnitude of the return premium associated with time
series momentum present significant challenges to the
random walk hypothesis and perhaps also to the efficient
market hypothesis, though we cannot rule out the exis-
tence of a rational theory that can explain these findings.

Our study relates to the literature on return autocorrela-
tion and variance ratios that also finds deviations from the
random walk hypothesis (Fama and French, 1988; Lo and
Mackinlay, 1988; Poterba and Summers, 1988). While this
literature is largely focused on US and global equities,
Cutler, Poterba, and Summers (1991) study a variety of
assets including housing and collectibles. The literature
finds positive return autocorrelations at daily, weekly, and
monthly horizons and negative autocorrelations at annual
and multi-year frequencies. We complement this literature
in several ways. The studies of autocorrelation examine, by
definition, return predictability where the length of the
‘‘look-back period’’ is the same as the ‘‘holding period’’ over
which returns are predicted. This restriction masks signifi-
cant predictability that is uncovered once look-back periods
are allowed to differ from predicted or holding periods. In
particular, our result that the past 12 months of returns
strongly predicts returns over the next one month is missed
by looking at one-year autocorrelations. While return con-
tinuation can also be detected implicitly from variance
ratios, we complement the literature by explicitly docu-
menting the extent of return continuation and by construct-
ing a time series momentum factor that can help explain
existing asset pricing phenomena, such as cross-sectional
momentum premiums and hedge fund macro and managed
futures returns. Also, a significant component of the higher
frequency findings in equities is contaminated by market
microstructure effects such as stale prices (Richardson,
1993; Ahn, Boudoukh, Richardson, and Whitelaw, 2002).
Focusing on liquid futures instead of individual stocks and
looking at lower frequency data mitigates many of these
issues. Finally, unique to this literature, we link time series
predictability to the dynamics of hedger and speculator
positions and decompose returns into price changes and roll
yields.

Our paper is also related to the literature on hedging
pressure in commodity futures (Keynes, 1923; Fama and
French, 1987; Bessembinder, 1992; de Roon, Nijman, and
Veld, 2000). We complement this literature by showing
how hedger and speculator positions relate to past futures
returns (and not just in commodities), finding that
speculators’ positions load positively on time series
momentum, while hedger positions load negatively on
it. Also, we consider the relative return predictability of
positions, past price changes, and past roll yields. Gorton,
Hayashi, and Rouwenhorst (2008) also link commodity
momentum and speculator positions to the commodities’
inventories.

The rest of the paper is organized as follows. Section 2
describes our data on futures returns and the positioning
of hedgers and speculators. Section 3 documents time
series momentum at horizons less than a year and
reversals beyond that. Section 4 defines a time series
momentum factor, studying its relation to other known
return factors, its performance during extreme markets,
and correlations within and across asset classes. Section 5
examines the relation between time series and cross-
sectional momentum, showing how time series momen-
tum is a central driver of cross-sectional momentum as
well as macro and managed futures hedge fund returns.
Section 6 studies the evolution of time series momentum
and its relation to investor speculative and hedging
positions. Section 7 concludes.

2. Data and preliminaries

We describe briefly the various data sources we use in
our analysis.

2.1. Futures returns

Our data consist of futures prices for 24 commodities,
12 cross-currency pairs (from nine underlying currencies),
nine developed equity indexes, and 13 developed govern-
ment bond futures, from January 1965 through December
2009. These instruments are among the most liquid
futures contracts in the world.4 We focus on the most
liquid instruments to avoid returns being contaminated
by illiquidity or stale price issues and to match more

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250 231
closely an implementable strategy at a significant trade
size. Appendix A provides details on each instrument and
their data sources, which are mainly Datastream, Bloom-
berg, and various exchanges.

We construct a return series for each instrument as
follows. Each day, we compute the daily excess return of
the most liquid futures contract (typically the nearest or
next nearest-to-delivery contract), and then compound
the daily returns to a cumulative return index from which
we can compute returns at any horizon. For the equity
indexes, our return series are almost perfectly correlated
with the corresponding returns of the underlying cash
indexes in excess of the Treasury bill rate.5

As a robustness test, we also use the ‘‘far’’ futures
contract (the next maturity after the most liquid one). For
the commodity futures, time series momentum profits are
in fact slightly stronger for the far contract, and, for the
financial futures, time series momentum returns hardly
change if we use far futures.

Table 1 presents summary statistics of the excess
returns on our futures contracts. The first column reports
when the time series of returns for each asset starts, and
the next two columns report the time series mean
(arithmetic) and standard deviation (annualized) of
each contract by asset class: commodities, equity indexes,
bonds, and currencies. As Table 1 highlights, there is
significant variation in sample mean returns across
the different contracts. Equity index, bonds, and curren-
cies yield predominantly positive excess returns, while
various commodity contracts yield positive, zero, and
even negative excess average returns over the sample
period. Only the equity and bond futures exhibit statisti-
cally significant and consistent positive excess average
returns.

More striking are the differences in volatilities across
the contracts. Not surprisingly, commodities and equities
have much larger volatilities than bond futures or cur-
rency forward contracts. But, even among commodities,
there is substantial cross-sectional variation in volatilities.
Making comparisons across instruments with vastly dif-
ferent volatilities or combining various instruments into a
diversified portfolio when they have wide-ranging vola-
tilities is challenging. For example, the volatility of natural
gas futures is about 50 times larger than that of 2-year US
bond futures. We discuss below how we deal with this
issue in our analysis.
2.2. Positions of traders

We also use data on the positions of speculators and
hedgers from the Commodity Futures Trading Commission
(CFTC) as detailed in Appendix A. The CFTC requires all large
traders to identify themselves as commercial or non-com-
mercial which we, and the previous literature (e.g.,
Bessembinder, 1992; de Roon, Nijman, and Veld, 2000), refer
to as hedgers and speculators, respectively. For each futures
5 Bessembinder (1992) and de Roon, Nijman, and Veld (2000) compute

returns on futures contracts similarly and also find that futures returns are

highly correlated with spot returns on the same underlying asset.
contract, the long and short open interest held by these
traders on Tuesday are reported on a weekly basis.6

Using the positions of speculators and hedgers as
defined by the CFTC, we define the Net speculator position
for each asset as follows:

Net speculator position

¼
Speculator long positions�Speculator short positions

Open interest
:

This signed measure shows whether speculators are
net long or short in aggregate, and scales their net
position by the open interest or total number of contracts
outstanding in that futures market. Since speculators and
hedgers approximately add up to zero (except for a small
difference denoted ‘‘non-reported’’ due to measurement
issues of very small traders), we focus our attention on
speculators. Of course, this means that net hedger posi-
tions constitute the opposite side (i.e., the negative of Net
speculator position).

The CFTC positions data do not cover all of the futures
contracts we have returns for and consider in our analysis.
Most commodity and foreign exchange contracts are
covered, but only the US instruments among the stock
and bond futures contracts are covered. The third and
fourth columns of Table 1 report summary statistics on
the sample of futures contracts with Net speculator
positions in each contract over time. Speculators are net
long, on average, and hence hedgers are net short, for
most of the contracts, a result consistent with
Bessembinder (1992) and de Roon, Nijman, and Veld
(2000) for a smaller set of contracts over a shorter time
period. All but two of the commodities (natural gas and
cotton) have net long speculator positions over the
sample period, with silver exhibiting the largest average
net long speculator position. This is consistent with
Keynes’ (1923) conjecture that producers of commodities
are the primary hedgers in markets and are on the short
side of these contracts as a result. For the other asset
classes, other than the S&P 500, the 30-year US Treasury
bond, and the $US/Japanese and $US/Swiss exchange
rates, speculators exhibit net long positions, on average.
Table 1 also highlights that there is substantial variation
over time in Net speculator positions per contract and
across contracts. Not surprisingly, the standard deviation
of Net speculator positions is positively related to the
volatility of the futures contract itself.

2.3. Asset pricing benchmarks

We evaluate the returns of our strategies relative to
standard asset pricing benchmarks, namely the MSCI
World equity index, Barclay’s Aggregate Bond Index, S&P
GSCI Index, all of which we obtain from Datastream, the
long-short factors SMB, HML, and UMD from Ken French’s
Web site, and the long-short value and cross-sectional
6 While commercial traders likely predominantly include hedgers,

some may also be speculating, which introduces some noise into the

analysis in terms of our classification of speculative and hedging trades.

However, the potential attenuation bias associated with such misclassi-

fication may only weaken our results.

Lexis Zhang

Table 1
Summary statistics on futures contracts.Reported are the annualized mean return and volatility (standard deviation) of the futures contracts in our

sample from January 1965 to December 2009 as well as the mean and standard deviation of the Net speculator long positions in each contract as a

percentage of open interest, covered and defined by the CFTC data, which are available over the period January 1986 to December 2009. For a detailed

description of our sample of futures contracts, see Appendix A.

Data start date Annualized mean Annualized volatility Average net speculator

long positions

Std. dev. net speculator

long positions

Commodity futures

ALUMINUM Jan-79 0.97% 23.50%

BRENTOIL Apr-89 13.87% 32.51%

CATTLE Jan-65 4.52% 17.14% 8.1% 9.6%

COCOA Jan-65 5.61% 32.38% 4.9% 14.0%

COFFEE Mar-74 5.72% 38.62% 7.5% 13.6%

COPPER Jan-77 8.90% 27.39%

CORN Jan-65 �3.19% 24.37% 7.1% 11.0%

COTTON Aug-67 1.41% 24.35% �0.1% 19.4%

CRUDE Mar-83 11.61% 34.72% 1.0% 5.9%

GASOIL Oct-84 11.95% 33.18%

GOLD Dec-69 5.36% 21.37% 6.7% 23.0%

HEATOIL Dec-78 9.79% 33.78% 2.4% 6.4%

HOGS Feb-66 3.39% 26.01% 5.1% 14.5%

NATGAS Apr-90 �9.74% 53.30% �1.6% 8.9%

NICKEL Jan-93 12.69% 35.76%

PLATINUM Jan-92 13.15% 20.95%

SILVER Jan-65 3.17% 31.11% 20.6% 14.3%

SOYBEANS Jan-65 5.57% 27.26% 8.2% 12.8%

SOYMEAL Sep-83 6.14% 24.59% 6.7% 11.2%

SOYOIL Oct-90 1.07% 25.39% 5.7% 12.8%

SUGAR Jan-65 4.44% 42.87% 10.0% 14.2%

UNLEADED Dec-84 15.92% 37.36% 7.8% 9.6%

WHEAT Jan-65 �1.84% 25.11% 4.3% 12.1%

ZINC Jan-91 1.98% 24.76%

Equity index futures

ASX SPI 200 (AUS) Jan-77 7.25% 18.33%

DAX (GER) Jan-75 6.33% 20.41%

IBEX 35 (ESP) Jan-80 9.37% 21.84%

CAC 40 10 (FR) Jan-75 6.73% 20.87%

FTSE/MIB (IT) Jun-78 6.13% 24.59%

TOPIX (JP) Jul-76 2.29% 18.66%

AEX (NL) Jan-75 7.72% 19.18%

FTSE 100 (UK) Jan-75 6.97% 17.77%

S&P 500 (US) Jan-65 3.47% 15.45% �4.6% 5.4%

Bond futures

3-year AUS Jan-92 1.34% 2.57%

10-year AUS Dec-85 3.83% 8.53%

2-year EURO Mar-97 1.02% 1.53%

5-year EURO Jan-93 2.56% 3.22%

10-year EURO Dec-79 2.40% 5.74%

30-year EURO Dec-98 4.71% 11.70%

10-year CAN Dec-84 4.04% 7.36%

10-year JP Dec-81 3.66% 5.40%

10-year UK Dec-79 3.00% 9.12%

2-year US Apr-96 1.65% 1.86% 1.9% 11.3%

5-year US Jan-90 3.17% 4.25% 3.0% 9.2%

10-year US Dec-79 3.80% 9.30% 0.4% 8.0%

30-year US Jan-90 9.50% 18.56% �1.4% 6.2%

Currency forwards

AUD/USD Mar-72 1.85% 10.86% 12.4% 28.8%

EUR/USD Sep-71 1.57% 11.21% 12.1% 18.7%

CAD/USD Mar-72 0.60% 6.29% 4.7% 24.1%

JPY/USD Sep-71 1.35% 11.66% �6.0% 23.8%

NOK/USD Feb-78 1.37% 10.56%

NZD/USD Feb-78 2.31% 12.01% 38.8% 33.8%

SEK/USD Feb-78 �0.05% 11.06%

CHF/USD Sep-71 1.34% 12.33% �5.2% 26.8%

GBP/USD Sep-71 1.39% 10.32% 2.7% 25.4%

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250232

Lexis Zhang

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250 233
momentum factors across asset classes from Asness,
Moskowitz, and Pedersen (2010).

2.4. Ex ante volatility estimate

Since volatility varies dramatically across our assets
(illustrated in Table 1), we scale the returns by their
volatilities in order to make meaningful comparisons
across assets. We estimate each instrument’s ex ante
volatility st at each point in time using an extremely
simple model: the exponentially weighted lagged squared
daily returns (i.e., similar to a simple univariate GARCH
model). Specifically, the ex ante annualized variance st

2

for each instrument is calculated as follows:

s2t ¼ 261
X1
i ¼ 0

ð1�dÞdiðrt�1�i�rtÞ
2 , ð1Þ

where the scalar 261 scales the variance to be annual, the
weights ð1�dÞdi add up to one, and rt is the exponentially
weighted average return computed similarly. The para-
meter d is chosen so that the center of mass of the
weights is

P1
i ¼ 0ð1�dÞd

i
i¼ d=ð1�dÞ ¼ 60 days. The volati-

lity model is the same for all assets at all times. While all
of the results in the paper are robust to more sophisti-
cated volatility models, we chose this model due to its
simplicity and lack of look-ahead bias in the volatility
estimate. To ensure no look-ahead bias contaminates our
results, we use the volatility estimates at time t�1
applied to time-t returns throughout the analysis.

3. Time series momentum: Regression analysis and
trading strategies

We start by examining the time series predictability of
futures returns across different time horizons.

3.1. Regression analysis: Predicting price continuation and

reversal

We regress the excess return rst for instrument s in
month t on its return lagged h months, where both
returns are scaled by their ex ante volatilities sst�1
(defined above in Section 2.4):

rst=s
s
t�1 ¼ aþbhr

s
t�h=s

s
t�h�1þe

s
t : ð2Þ

Given the vast differences in volatilities (as shown in
Table 1), we divide all returns by their volatility to put
them on the same scale. This is similar to using General-
ized Least Squares instead of Ordinary Least Squares
(OLS).7 Stacking all futures contracts and dates, we run a
pooled panel regression and compute t-statistics that
account for group-wise clustering by time (at the monthly
level). The regressions are run using lags of h¼1, 2, y, 60
months.

Panel A of Fig. 1 plots the t-statistics from the pooled
regressions by month lag h. The positive t-statistics for the
first 12 months indicate significant return continuation or
7 The regression results are qualitatively similar if we run OLS

without adjusting for each security’s volatility.
trends. The negative signs for the longer horizons indicate
reversals, the most significant of which occur in the year
immediately following the positive trend.

Another way to look at time series predictability is to
simply focus only on the sign of the past excess return.
This even simpler way of looking at time series momen-
tum underlies the trading strategies we consider in the
next section. In a regression setting, this strategy can be
captured using the following specification:

rst=s
s
t�1 ¼ aþbhsignðr

s
t�hÞþe

s
t : ð3Þ

We again make the left-hand side of the regression
independent of volatility (the right-hand side is too since
sign is either þ1 or �1), so that the parameter estimates
are comparable across instruments. We report the t-
statistics from a pooled regression with standard errors
clustered by time (i.e., month) in Panel B of Fig. 1.

The results are similar across the two regression specifi-
cations: strong return continuation for the first year and
weaker reversals for the next 4 years. In both cases, the data
exhibit a clear pattern, with all of the most recent 12-month
lag returns positive (and nine statistically significant) and the
majority of the remaining lags negative. Repeating the panel
regressions for each asset class separately, we obtain the
same patterns: one to 12-month positive time series momen-
tum followed by smaller reversals over the next 4 years as
seen in Panel C of Fig. 1.

3.2. Time series momentum trading strategies

We next investigate the profitability of a number of
trading strategies based on time series momentum. We
vary both the number of months we lag returns to define
the signal used to form the portfolio (the ‘‘look-back
period’’) and the number of months we hold each portfo-
lio after it has been formed (the ‘‘holding period’’).

For each instrument s and month t, we consider
whether the excess return over the past k months is
positive or negative and go long the contract if positive
and short if negative, holding the position for h months.
We set the position size to be inversely proportional to
the instrument’s ex ante volatility, 1=sst�1, each month.
Sizing each position in each strategy to have constant
ex ante volatility is helpful for two reasons. First, it makes
it easier to aggregate strategies across instruments with
very different volatility levels. Second, it is helpful econ-
ometrically to have a time series with relatively stable
volatility so that the strategy is not dominated by a few
volatile periods.

For each trading strategy (k,h), we derive a single time
series of monthly returns even if the holding period h is
more than one month. Hence, we do not have overlapping
observations. We derive this single time series of returns
following the methodology used by Jegadeesh and Titman
(1993): The return at time t represents the average return
across all portfolios at that time, namely the return on the
portfolio that was constructed last month, the month
before that (and still held if the holding period h is greater
than two), and so on for all currently ‘‘active’’ portfolios.

Specifically, for each instrument, we compute the
time-t return based on the sign of the past return from

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-5
-4
-3
-2
-1
0
1
2
3
4
5
6

Month lag

t-statistic by month, all asset classes

-3
-2
-1
0
1
2
3
4
5
6
7

Month lag

t-statistic by month, all asset classes

-3

-2

-1

0

1

2

3

4

5

t-
S

ta
tis

tic
t-

S
ta

tis
tic

t-
S

ta
tis

tic

t-
S

ta
tis

tic
t-

S
ta

tis
tic

t-
S

ta
tis

tic

Month lag

Commodity futures

-4

-3

-2

-1

0

1

2

3

Month lag

Equity index futures

-3

-2

-1

0

1

2

3

4

Month lag

Government bond futures

-4

-3

-2

-1

0

1

2

3

4

5

Month lag

Currencies

Fig. 1. Time series predictability across all asset classes. We regress the monthly excess return of each contract on its own lagged excess return over various
horizons. Panel A uses the size of the lagged excess return as a predictor, where returns are scaled by their ex ante volatility to make them comparable across

assets, Panel B uses the sign of the lagged excess return as a predictor, where the dependent variable is scaled by its ex ante volatility to make the regression

coefficients comparable across different assets, and Panel C reports the results of the sign regression by asset class. Reported are the pooled regression

estimates across all instruments with t-statistics computed using standard errors that are clustered by time (month). Sample period is January 1985 to

December 2009. (A) Panel A: rst=s
s
t�1 ¼ aþbhr

s
t�h
=ss

t�h�1
þest ; (B) Panel B: r

s
t=s

s
t�1 ¼ aþbhsignðr

s
t�h
Þþest ; (C) Panel C: Results by asset class.

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250234
time t�k�1 to t�1. We then compute the time-t
return based on the sign of the past return from t�k�2
to t�2, and so on until we compute the time-t return
based on the final past return that is still being used from
t�k�h to t�h. For each (k,h), we get a single time series
of monthly returns by computing the average return of all
of these h currently ‘‘active’’ portfolios (i.e., the portfolio
that was just bought and those that were bought in the
past and are still held). We average the returns across all
instruments (or all instruments within an asset class), to
obtain our time series momentum strategy returns,
r

TSMOMðk,hÞ
t .

To evaluate the abnormal performance of these
strategies, we compute their alphas from the following

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T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250 235
regression:

r
TSMOMðk,hÞ
t ¼ aþb1MKTtþb2BONDtþb3GSCItþsSMBt

þhHMLtþmUMDtþet , ð4Þ
Table 2
t-statistics of the alphas of time series momentum strategies with different loo

Reported are the t-statistics of the alphas (intercepts) from time series regr

look-back and holding periods on the following factor portfolios: MSCI World

SMB, and UMD Fama and French factors from Ken French’s Web site. Panel A re

for equity index futures, Panel D for bond futures, and Panel E for currency for

1 3 6

Panel A: All assets

Lookback period (months) 1 4.34 4.68 3.8

3 5.35 4.42 3.5

6 5.03 4.54 4.9

9 6.06 6.13 5.7

12 6.61 5.60 4.4

24 3.95 3.19 2.4

36 2.70 2.20 1.4

48 1.84 1.55 1.1

Panel B: Commodity futures

Lookback period (months) 1 2.44 2.89 2.8

3 4.54 3.79 3.2

6 3.86 3.53 3.3

9 3.77 4.05 3.8

12 4.66 4.08 2.6

24 2.83 2.15 1.2

36 1.28 0.74 0.0

48 1.19 1.17 1.0

Panel C: Equity index futures

Lookback period (months) 1 1.05 2.36 2.8

3 1.48 2.23 2.2

6 3.50 3.18 3.4

9 4.21 3.94 3.7

12 3.77 3.55 3.0

24 2.04 2.22 1.9

36 1.86 1.66 1.2

48 0.81 0.84 0.5

Panel D: Bond futures

Lookback period (months) 1 3.31 2.66 1.8

3 2.45 1.52 1.1

6 2.16 2.04 2.1

9 2.93 2.61 2.6

12 3.53 2.82 2.5

24 1.87 1.55 1.6

36 1.97 1.83 1.7

48 2.21 1.80 1.5

Panel E: Currency forwards

Lookback period (months) 1 3.16 3.20 1.4

3 3.90 2.75 1.5

6 2.59 1.86 2.3

9 3.40 3.16 2.6

12 3.41 2.40 1.6

24 1.78 0.99 0.5

36 0.73 0.42 �0.0

48 �0.55 �1.05 �1.4
where we control for passive exposures to the three major
asset classes—the stock market MKT, proxied by the excess
return on the MSCI World Index, the bond market BOND,
proxied by the Barclays Aggregate Bond Index, the
k-back and holding periods.

essions of the returns of time series momentum strategies over various

Index, Lehman Brothers/Barclays Bond Index, S&P GSCI Index, and HML,

ports results for all asset classes, Panel B for commodity futures, Panel C

wards.

Holding period (months)

9 12 24 36 48

3 4.29 5.12 3.02 2.74 1.90

4 4.73 4.50 2.60 1.97 1.52

3 5.32 4.43 2.79 1.89 1.42

8 5.07 4.10 2.57 1.45 1.19

4 3.69 2.85 1.68 0.66 0.46

4 1.95 1.50 0.20 �0.09 �0.33

4 0.96 0.62 0.28 0.07 0.20

6 1.00 0.86 0.38 0.46 0.74

1 2.16 3.26 1.81 1.56 1.94

0 3.12 3.29 1.51 1.28 1.62

4 3.43 2.74 1.59 1.25 1.48

9 3.06 2.31 1.27 0.71 1.04

4 1.85 1.46 0.58 0.14 0.57

4 0.58 0.18 �0.60 �0.33 �0.14

7 �0.25 �0.34 �0.03 0.34 0.65

4 1.01 0.92 0.75 1.16 1.29

9 3.08 3.24 2.28 1.93 1.28

1 2.81 2.78 2.00 1.57 1.14

9 3.52 3.03 2.08 1.36 0.88

9 3.30 2.64 1.96 1.21 0.75

3 2.58 2.02 1.57 0.78 0.33

6 1.70 1.49 0.87 0.43 0.13

6 0.90 0.66 0.34 0.02 0.08

8 0.44 0.36 0.12 0.01 0.23

4 2.65 2.88 1.76 1.60 1.40

0 1.99 1.80 1.27 1.05 1.00

8 2.53 2.24 1.71 1.36 1.37

8 2.55 2.43 1.83 1.17 1.40

7 2.42 2.18 1.47 1.12 0.96

2 1.66 1.58 1.01 0.90 0.64

0 1.62 1.73 1.13 0.75 0.91

3 1.43 1.26 0.72 0.73 1.22

6 2.43 2.77 1.22 0.83 �0.42

4 3.05 2.55 1.02 0.10 �0.84

2 2.82 2.08 0.62 �0.16 �1.14

5 2.35 1.72 0.20 �0.38 �1.17

5 1.25 0.71 �0.29 �1.01 �1.67

3 0.27 �0.05 �1.15 �1.88 �2.27

4 �0.42 �0.96 �1.67 �2.04 �2.42

1 �1.62 �1.79 �2.02 �2.34 �2.32

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T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250236
commodity market GSCI, proxied by the S&P GSCI Index—as
well as the standard Fama-French stock market factors SMB,
HML, and UMD for the size, value, and (cross-sectional)
momentum premiums. For the evaluation of time series
momentum strategies, we rely on the sample starting in
1985 to ensure that a comprehensive set of instruments have
data (see Table 1) and that the markets had significant
liquidity. We obtain similar (and generally more significant)
results if older data are included going back to 1965, but
given the more limited breadth and liquidity of the instru-
ments during this time, we report results post-1985.

Table 2 shows the t-statistics of the estimated alphas
for each asset class and across all assets. The existence
and significance of time series momentum is robust
across horizons and asset classes, particularly when the
look-back and holding periods are 12 months or less. In
addition, we confirm that the time series momentum
results are almost identical if we use the cash indexes
for the stock index futures. The other asset classes do not
have cash indexes.

4. Time series momentum factor

For a more in-depth analysis of time series momen-
tum, we focus our attention on a single time series
momentum strategy. Following the convention used in
the cross-sectional momentum literature (and based on
the results from Fig. 1 and Table 2), we focus on the
properties of the 12-month time series momentum strat-
egy with a 1-month holding period (e.g., k¼12 and h¼1),
which we refer to simply as TSMOM.

4.1. TSMOM by security and the diversified TSMOM factor

We start by looking at each instrument and asset
separately and then pool all the assets together in a
diversified TSMOM portfolio. We size each position (long
or short) so that it has an ex ante annualized volatility of
40%. That is, the position size is chosen to be 40%/st�1,
where st�1 is the estimate of the ex ante volatility of the
contract as described above. The choice of 40% is incon-
sequential, but it makes it easier to intuitively compare
our portfolios to others in the literature. The 40% annual
volatility is chosen because it is similar to the risk of an
average individual stock, and when we average the return
across all securities (equal-weighted) to form the portfo-
lio of securities which represent our TSMOM factor, it has
an annualized volatility of 12% per year over the sample
period 1985–2009, which is roughly the level of volatility
exhibited by other factors such as those of Fama and
French (1993) and Asness, Moskowitz, and Pedersen
(2010).8 The TSMOM return for any instrument s at time
t is therefore:

rTSMOM,s
t,tþ1

¼ signðrst�12,tÞ
40%

sst
rst,tþ1: ð5Þ
8 Also, this portfolio construction implies a use of margin capital of

about 5–20%, which is well within what is feasible to implement in a

real-world portfolio.
We compute this return for each instrument and each
available month from January 1985 to December 2009.
The top of Fig. 2 plots the annualized Sharpe ratios of
these strategies for each futures contract. As the figure
shows, every single futures contract exhibits positive
predictability from past one-year returns. All 58 futures
contracts exhibit positive time series momentum returns
and 52 are statistically different from zero at the 5%
significance level.

If we regress the TSMOM strategy for each security on the
strategy of always being long (i.e., replacing ‘‘sign’’ with a 1 in
Eq. (5)), then we get a positive alpha in 90% of the cases (of
which 26% are statistically significant; none of the few
negative ones are significant). Thus, a time series momentum
strategy provides additional returns over and above a passive
long position for most instruments.

The overall return of the strategy that diversifies across
all the St securities that are available at time t is

rTSMOMt,tþ1 ¼
1

St

XSt
s ¼ 1

signðrst�12,tÞ
40%

sst
rst,tþ1:

We analyze the risk and return of this factor in detail
next. We also consider TSMOM strategies by asset class
constructed analogously.

4.2. Alpha and loadings on risk factors

Table 3 examines the risk-adjusted performance of a
diversified TSMOM strategy and its factor exposures.
Panel A of Table 3 regresses the excess return of the
TSMOM strategy on the returns of the MSCI World stock
market index and the standard Fama-French factors SMB,
HML, and UMD, representing the size, value, and cross-
sectional momentum premium among individual stocks.
The first row reports monthly time series regression
results and the second row uses quarterly non-overlap-
ping returns (to account for any non-synchronous trading
effects across markets). In both cases, TSMOM delivers a
large and significant alpha or intercept with respect to
these factors of about 1.58% per month or 4.75% per
quarter. The TSMOM strategy does not exhibit significant
betas on the market, SMB, or HML but loads significantly
positively on UMD, the cross-sectional momentum factor.
We explore the connection between cross-sectional and
time series momentum more fully in the next section, but
given the large and significant alpha, it appears that time
series momentum is not fully explained by cross-sectional
momentum in individual stocks.

Panel B of Table 3 repeats the regressions using the
Asness, Moskowitz, and Pedersen (2010) value and
momentum ‘‘everywhere’’ factors (i.e., factors diversified
across asset classes) in place of the Fama and French
factors. Asness, Moskowitz, and Pedersen (2010) form
long-short portfolios of value and momentum across
individual equities from four international markets, stock
index futures, bond futures, currencies, and commodities.
Similar to the Fama and French factors, these are cross-
sectional factors. Once again, we find no significant
loading on the market index or the value everywhere
factor, but significant loading on the cross-sectional

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0.0

0.2

0.4

0.6

0.8

1.0

1.2

G
ro

ss
s

ha
rp

e
ra

tio

Sharpe ratio of 12-month trend strategy

Commodities Currencies Equities Fixed Income

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

Z
in

c

A
E

X

Ill
iq

ui
di

ty
(

N
or

m
al

iz
ed

r
an

k
ba

se
d

on
d

ai
ly

tr
ad

in
g

vo
lu

m
e)

Commodities Currencies Equities Fixed Income

Illiquidity of futures contracts
Correlation (Sharperatio, Illiquidity) = -0.16

Fig. 2. Sharpe ratio of 12-month time series momentum by instrument. Reported are the annualized gross Sharpe ratio of the 12-month time series
momentum or trend strategy for each futures contract/instrument. For each instrument in every month, the trend strategy goes long (short) the contract

if the excess return over the past 12 months of being long the instrument is positive (negative), and scales the size of the bet to be inversely proportional

to the ex ante volatility of the instrument to maintain constant volatility over the entire sample period from January 1985 to December 2009. The second

figure plots a normalized value of the illiquidity of each futures contract measured by ranking contracts within each asset class by their daily trading

volume (from highest to lowest) and reporting the standard normalized rank for each contract within each asset class. Positive (negative) values imply

the contract is more (less) illiquid than the median contract for that asset class.

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250 237
momentum everywhere factor. However, the returns to
TSMOM are not fully captured by the cross-sectional
everywhere factor—the alpha is still an impressive
1.09% per month with a t-stat of 5.40 or 2.93% per quarter
with a t-stat of 4.12.

4.3. Performance over time and in extreme markets

Fig. 3 plots the cumulative excess return to the
diversified time series momentum strategy over time
(on a log scale). For comparison, we also plot the cumu-
lative excess returns of a diversified passive long position
in all instruments, with an equal amount of risk in each
instrument. (Since each instrument is scaled by the same
constant volatility, both portfolios have the same ex ante
volatility except for differences in correlations among
time series momentum strategies and passive long stra-
tegies.) As Fig. 3 shows, the performance over time of the
diversified time series momentum strategy provides a
relatively steady stream of positive returns that outper-
forms a diversified portfolio of passive long positions in
all futures contracts (at the same ex ante volatility).

We can also compute the return of the time series
momentum factor from 1966 to 1985, despite the limited
number of instruments with available data. Over this
earlier sample, time series momentum has a statistically

Table 3
Performance of the diversified time series momentum strategy.

Panel A reports results from time series regressions of monthly and non-overlapping quarterly returns on the diversified time series momentum

strategy that takes an equal-weighted average of the time series momentum strategies across all futures contracts in all asset classes, on the returns of

the MSCI World Index and the Fama and French factors SMB, HML, and UMD, representing the size, value, and cross-sectional momentum premiums in

US stocks. Panel B reports results using the Asness, Moskowitz, and Pedersen (2010) value and momentum ‘‘everywhere‘‘ factors instead of the Fama and

French factors, which capture the premiums to value and cross-sectional momentum globally across asset classes. Panel C reports results from

regressions of the time series momentum returns on the market (MSCI World Index), volatility (VIX), funding liquidity (TED spread), and sentiment

variables from Baker and Wurgler (2006, 2007), as well as their extremes.

Panel A: Fama and French factors

MSCI

World

SMB HML UMD Intercept R2

Monthly
Coefficient 0.09 �0.05 �0.01 0.28 1.58% 14%

(t-Stat) (1.89) (�0.84) (�0.21) (6.78) (7.99)

Coefficient 0.07 �0.18 0.01 0.32 4.75% 23%

Quarterly (t-Stat) (1.00) (�1.44) (0.11) (4.44) (7.73)

Panel B: Asness, Moskowitz, and Pedersen (2010) factors

MSCI

World

VAL Everywhere MOM

Everywhere

Intercept R2

Monthly
Coefficient 0.11 0.14 0.66 1.09% 30%

(t-Stat) (2.67) (2.02) (9.74) (5.40)

Coefficient 0.12 0.26 0.71 2.93% 34%

Quarterly (t-Stat) (1.81) (2.45) (6.47) (4.12)

Panel C: Market, volatility, liquidity, and sentiment extremes

MSCI

World

MSCI World

squared

TED spread TED spread top

20%

VIX VIX top 20%

Quarterly
Coefficient �0.01 1.99

(t-Stat) (�0.17) (3.88)

Quarterly
Coefficient �0.001 �0.008

(t-Stat) (�0.06) (�0.29)

Quarterly
Coefficient 0.001 �0.003

(t-Stat) (0.92) (�0.10)

Sentiment
Sentiment top

20%

Sentiment bottom

20%

Change in

sentiment

Change in sentiment top

20%

Change in sentiment

bottom 20%

Quarterly
Coefficient 0.03 �0.01 �0.01

(t-Stat) (0.73) (�0.27) (�0.12)

Quarterly
Coefficient �0.01 0.02 0.01

(t-Stat) (�1.08) (1.25) (0.66)

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250238
significant return and an annualized Sharpe ratio of 1.1,
providing strong out-of-sample evidence of time series
momentum.9

Fig. 3 highlights that time series momentum profits
are large in October, November, and December of 2008,
which was at the height of the Global Financial Crisis
when commodity and equity prices dropped sharply,
bond prices rose, and currency rates moved dramatically.
Leading into this period, time series momentum suffers
losses in the third quarter of 2008, where the associated
price moves caused the TSMOM strategy to be short in
many contracts, setting up large profits that were earned
in the fourth quarter of 2008 as markets in all these asset
classes fell further. Fig. 3 also shows that TSMOM suffers
sharp losses when the crisis ends in March, April, and May
9 We thank the referee for asking for this out-of-sample study of old

data.
of 2009. The ending of a crisis constitutes a sharp trend
reversal that generates losses on a trend following strat-
egy such as TSMOM.

More generally, Fig. 4 plots the TSMOM returns against
the S&P 500 returns. The returns to TSMOM are largest
during the biggest up and down market movements. To
test the statistical significance of this finding, the first row
of Panel C of Table 3 reports coefficients from a regression
of TSMOM returns on the market index return and
squared market index return. While the beta on the
market itself is insignificant, the coefficient on the market
return squared is significantly positive, indicating that
TSMOM delivers its highest profits during the most
extreme market episodes. TSMOM, therefore, has payoffs
similar to an option straddle on the market. Fung and
Hsieh (2001) discuss why trend following has straddle-
like payoffs and apply this insight to describe the perfor-
mance of hedge funds. Our TSMOM strategy generates
this payoff structure because it tends to go long when the

$100

$1,000

$10,000

$100,000

19
85

19
87

19
89

19
91

19
93

19
95

19
97

19
99

20
01

20
03

20
05

20
07

20
09

G
ro

w
th

o
f $

10
0

(lo
g

sc
al

e)

Date

Time Series Momentum Passive Long

Fig. 3. Cumulative excess return of time series momentum and diversified passive long strategy, January 1985 to December 2009. Plotted are the
cumulative excess returns of the diversified TSMOM portfolio and a diversified portfolio of the possible long position in every futures contract we study.

The TSMOM portfolio is defined in Eq. (5) and across all futures contracts summed. Sample period is January 1985 to December 2009.

-15%

-10%

-5%

0%

5%

10%

15%

20%

25%

30%

-25%

T
im

e-
se

rie
s

m
om

en
tu

m
r

et
ur

ns

S&P 500 returns
25%15%5%-5%-15%

Fig. 4. The time series momentum smile. The non-overlapping quarterly returns on the diversified (equally weighted across all contracts) 12-month time
series momentum or trend strategy are plotted against the contemporaneous returns on the S&P 500.

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250 239
market has a major upswing and short when the market
crashes.

These results suggest that the positive average TSMOM
returns are not likely to be compensation for crash risk.
Historically, TSMOM does well during ‘‘crashes’’ because
crises often happen when the economy goes from normal
to bad (making TSMOM go short risky assets), and then
from bad to worse (leading to TSMOM profits), with the
recent financial crisis of 2008 being a prime example.

4.4. Liquidity and sentiment

We test whether TSMOM returns might be driven or
exaggerated by illiquidity. We first test whether TSMOM
performs better for more illiquid assets in the cross-section,
and then we test whether the performance of the diversified
TSMOM factor depends on liquidity indicators in the time
series. For the former, we measure the illiquidity of each
futures contract using the daily dollar trading volume
obtained from Reuters and broker feeds. We do not have
historical time series of daily volume on these contracts, but
use a snapshot of their daily volume in June 2010 to
examine cross-sectional differences in liquidity across the
assets. Since assets are vastly different across many dimen-
sions, we first rank each contract within an asset class by
their daily trading volume (from highest to lowest) and
compute the standard normalized rank of each contract by
demeaning each rank and dividing by its standard deviation,
i.e., (rank-mean(rank))/std(rank). Positive (negative) values
imply a contract is more (less) illiquid than the median
contract for that asset class. As shown in the bottom of
Figure 2, we find little relation between the magnitude of
the Sharpe ratio of TSMOM for a particular contract and its
illiquidity, as proxied by daily dollar trading volume. The
correlation between illiquidity and Sharpe ratio of a time
series momentum strategy by contract is �0.16 averaged
across all contracts, suggesting that, if anything, more liquid
contracts exhibit greater time series momentum profits.

We next consider how TSMOM returns co-vary in
aggregate with the time series of liquidity. The second row
of Panel C of Table 3 reports results using the Treasury
Eurodollar (TED) spread, a proxy for funding liquidity as
suggested by Brunnermeier and Pedersen (2009), Asness,
Moskowitz, and Pedersen (2010), and Garleanu and
Pedersen (2011), and the top 20% most extreme realizations
of the TED spread to capture the most illiquid funding
environments. As the table shows, there is no significant
relation between the TED spread and TSMOM returns,
suggesting little relationship with funding liquidity. The
third row of Panel C of Table 3 repeats the analysis using

Lexis Zhang

Lexis Zhang

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250240
the VIX index to capture the level of market volatility and
the most extreme market volatility environments, which
also seem to correspond with illiquid episodes. There is no
significant relationship between TSMOM profitability and
market volatility either.

At the bottom of Panel C of Table 3, we also examine
the relationship between TSMOM returns and the senti-
ment index measures used by Baker and Wurgler (2006,
2007). We examine both the level of sentiment and its
monthly changes (first differences) and examine the top
and bottom extremes (20%) of these variables. As the
regressions indicate, we find no significant relationship
between TSMOM profitability and sentiment measures,
even at the extremes.

4.5. Correlation structure

Table 4 examines the correlation structure of the time
series momentum strategies and compares them to the
correlation structure of passive long positions in the
contracts. The first row of Panel A of Table 4 reports the
average pair-wise correlation of time series momentum
returns among contracts within the same asset class. The
correlations are positive within each asset class, ranging
from 0.37 to 0.38 for equities and fixed income futures to
0.10 and 0.07 for commodities and currencies. Part of this
correlation structure reflects the comovement of the
returns to simply being passive long (or short) in each
instrument at the same time. The second row of Panel A of
Table 4 reports the average pair-wise correlation of
passive long positions within each asset class and, except
Table 4
Correlations of time series momentum strategy returns within and

across asset classes.

Panel A reports within each asset class the average pair-wise correla-

tion of each instruments’ 12-month time series momentum strategy

returns, as well as a passive long position in each instrument. Panel B

reports the correlation of time series momentum strategies and passive

long positions across asset classes, where an equal-weighted average of

the instruments within each asset class is first formed and then the

correlations between the equal-weighted strategies across asset classes

are calculated. Correlations are calculated from monthly returns over the

period January 1985–December 2009.

Panel A: Average pair-wise correlation within asset class

Commodities Equities Fixed

income

Currencies

TSMOM strategies 0.07 0.37 0.38 0.10

Passive long

positions

0.19 0.60 0.63 �0.04

Panel B: Average correlation across asset classes

Correlations of TSMOM strategies

Commodities 1

Equities 0.20 1

Fixed income 0.07 0.21 1

Currencies 0.13 0.20 0.05 1

Correlations of passive long positions

Commodities 1

Equities 0.17 1

Fixed income �0.12 �0.03 1

Currencies �0.12 �0.20 0.02 1
for currencies, passive long strategies exhibit higher
correlations than time series momentum strategies
within an asset class.

Panel B of Table 4 shows the average correlation of
time series momentum strategies across asset classes.
Here, we first compute the return of a diversified portfolio
of time series momentum strategies within each asset
class and then estimate the correlation of returns for
TSMOM portfolios across asset classes. All of the correla-
tions are positive, ranging from 0.05 to 0.21. For compar-
ison, the table also shows the correlations across asset
classes of diversified passive long positions. For every
asset class comparison, the correlation of time series
momentum strategies across asset classes is larger than
the corresponding correlation of passive long strategies,
many of which are negative.

Summing up the results from both panels, time series
momentum strategies are positively correlated within an
asset class, but less so than passive long strategies. However,
across asset classes, time series momentum strategies
exhibit positive correlation with each other, while passive
long strategies exhibit zero or negative correlation across
asset classes. This last result suggests that there is a
common component affecting time series momentum stra-
tegies across asset classes simultaneously that is not present
in the underlying assets themselves, similar to the findings
of Asness, Moskowitz, and Pedersen (2010) who find
common structure among cross-sectional momentum stra-
tegies across different asset classes.

5. Time series vs. cross-sectional momentum

Our previous results show a significant relationship
between time series momentum and cross-sectional
momentum. In this section, we explore that relationship
further and determine how much overlap and difference
exist between our time series momentum strategies and
the cross-sectional momentum strategies commonly used
in the literature.

5.1. Time series momentum regressed on cross-sectional

momentum

Panel A of Table 5 provides further evidence on the
relationship between time series momentum (TSMOM)
and cross-sectional momentum (XSMOM) by regressing
the returns to our time series momentum strategies—
diversified across all instruments and within each asset
class—on the returns of cross-sectional momentum stra-
tegies applied to the same assets. Specifically, following
Asness, Moskowitz, and Pedersen (2010), we apply a
cross-sectional momentum strategy based on the relative
ranking of each asset’s past 12-month returns and form
portfolios that go long or short the assets in proportion to
their ranks relative to the median rank.10
10 Asness, Moskowitz, and Pedersen (2010) exclude the most recent

month when computing 12-month cross-sectional momentum. For

consistency, we follow that convention here, but our results do not

depend on whether the most recent month is excluded or not.

Lexis Zhang

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250 241
The first row of Panel A of Table 5 reports results
from the TSMOM strategy diversified across all assets
regressed on the XSMOM strategy that is diversified
across those same assets. As before, time series momen-
tum and cross-sectional momentum are significantly
related, with the beta of TSMOM on XSMOM equal to
0.66 with a t-statistic of 15.17 and R-square of 44%.
However, as the intercept indicates, TSMOM is not fully
captured by XSMOM, exhibiting a positive and significant
alpha of 76 basis points per month with a t-statistic of
5.90. So, TSMOM and XSMOM are related, but are not
the same.

The second row of Panel A of Table 5 repeats the
regression using XSMOM strategies for each asset class,
including individual stocks. TSMOM is related to XSMOM
across all of the different asset classes, including indivi-
dual equity momentum, which is not even included in the
TSMOM strategy, and this is after controlling for exposure
to XSMOM from the other four asset classes. TSMOM still
exhibits a significant alpha, however, and is therefore not
fully captured by cross-sectional momentum strategies in
these asset classes.

Repeating these regressions using the TSMOM returns
for each asset class separately, we find a consistent
pattern. TSMOM is related to XSMOM within each asset
class, with R-squares ranging from 56% in currencies (FX)
to 14% in fixed income, but TSMOM is not captured by
XSMOM. The alphas of TSMOM remain significantly
positive for every asset class. We also see some interest-
ing cross-asset relationships among TSMOM and XSMOM.
For instance, not only is TSMOM for commodities corre-
lated with XSMOM for commodities, but also with
XSMOM for currencies. Likewise, TSMOM among equity
index futures is not only correlated with XSMOM among
those equity indexes but also with XSMOM among indi-
vidual stocks. And, TSMOM for fixed income is correlated
with XSMOM for fixed income and XSMOM for equity
indexes. These results indicate significant correlation
structure in time series and cross-sectional momentum
across different asset classes, consistent with our earlier
results and those of Asness, Moskowitz, and Pedersen
(2010).
11 This relation between mean returns is exact if annual returns are

computed by summing over monthly returns. If returns are com-

pounded, this relation is approximate, but an exact relation is straight-

forward to derive, e.g., using the separate means of monthly and annual

returns.
5.2. A simple, formal decomposition

We can more formally write down the relationship
between time series (TSMOM) and cross-sectional
(XSMOM) momentum. Following Lo and Mackinlay
(1990) and Lewellen (2002), we can describe a simple
cross-sectional and time series momentum strategy on
the same assets as follows. For cross-sectional momen-
tum, we let the portfolio weight of instrument i be wXS,it ¼
ð1=NÞðrit�12,t�r

EW
t�12,tÞ, that is, the past 12-month excess

return over the equal-weighted average return,
rEWt�12,t ¼ ð1=NÞ

PN
i ¼ 1 r

i
t�12,t . The return to the portfolio is

therefore

rXSt,tþ1 ¼
XN
i ¼ 1

wXS,it r
i
t,tþ1:
Next, assuming that the monthly expected return is11

mi ¼ Eðrit,tþ1Þ ¼ Eðr
i
t�12,tÞ=12 and letting m ¼ ½m

1,. . .,mN �0,
Rt,s ¼ ½r

1
t,s,. . .,r

N
t,s�
0, and O¼ E½ðRt�12,t�12mÞðRt,tþ1�mÞ0�, the

expected return to cross-sectional momentum (XSMOM)
can be decomposed as

E rXSt,tþ1
� �

¼
trðOÞ

N
�

10O1
N2
þ12s2m

¼
N�1

N2
trðOÞ�

1

N2
10O1�trðOÞ
� �

þ12s2m , ð6Þ

where tr is the trace of a matrix, 1 is an (N�1) vector of
ones, and s2m is the cross-sectional variance of the mean
monthly returns mi.

Eq. (6) shows that cross-sectional momentum profits can
be decomposed into an auto-covariance component
between lagged 1-year returns and future 1-month returns
(the diagonal elements of O captured by the first term), a
cross-covariance component capturing the temporal leads
and lags across stocks (the off-diagonal elements of O
captured by the second term), and the cross-sectional
variation in unconditional mean returns (the third term).
As emphasized by Lewellen (2002), cross-sectional momen-
tum profits need not be generated by positive autocorrela-
tion in returns (i.e., time series predictability). If cross-serial
covariances across stocks are negative, implying that high
past returns of an asset predict lower future returns of other
assets, this, too, can lead to momentum profits. Likewise,
large cross-sectional variation in mean returns can also lead
to momentum profits since, on average, assets with the
highest mean returns will have the highest realized returns.

The returns to time series momentum can be decom-
posed similarly if we let the portfolio weights be wTS,it ¼
ð1=NÞrit�12,t . Then the expected return is

EðrTSt,tþ1Þ ¼ E
XN
i ¼ 1

wTS,it r
i
t,tþ1

!
¼

trðOÞ
N
þ12

m0m
N
: ð7Þ

As these equations highlight, time series momentum
profits are primarily driven by time series predictability
(i.e., positive auto-covariance in returns) if the average
squared mean returns of the assets is small. Comparing
Eqs. (6) and (7), we see that time series momentum
profits can be decomposed into the auto-covariance term
that also underlies cross-sectional momentum (plus the
average squared mean excess return). The equations thus
provide a link between time series and cross-sectional
momentum profitability, which we can measure in the
data to determine how related these two phenomena are.

Panel B of Table 5 computes each of the components of
the diversified 12-month cross-sectional and time series
momentum strategies across all assets and within each
asset class according to the equations above. We report
the three components of the cross-sectional momentum
strategy: ‘‘Auto’’ refers to the auto-covariance or time
series momentum component, ‘‘Cross’’ refers to the cross-
serial covariance or lead-lag component, and ‘‘Mean’’

Table 5
Time series momentum vs. cross-sectional momentum.

Panel A reports results from regressions of the 12-month time series momentum strategies by asset class (TSMOM) on 12-month cross-sectional momentum strategies (XSMOM) of Asness, Moskowitz, and

Pedersen (2010). Panel B reports results from the decomposition of cross-sectional momentum and time series momentum strategies according to Section 4.2, where Auto is the component of profits coming

from the auto-covariance of returns, Cross is the component coming from cross-serial correlations or lead-lag effects across the asset returns, Mean is the component coming from cross-sectional variation in

unconditional mean returns, and Mean squared is the component coming from squared mean returns. Panel C reports results from regressions of several XSMOM strategies in different asset classes, the Fama-

French momentum, value, and size factors, and two hedge fund indexes obtained from Dow Jones/Credit Suisse on our benchmark TSMOM factor.

Panel A: Regression of TSMOM on XSMOM

Independent variables

XSMOM ALL XSMOM COM XSMOM EQ XSMOM FI XSMOM FX XSMOM US stocks Intercept R2

Dependent variable TSMOM ALL 0.66 0.76% 44%

(15.17) (5.90)

TSMOM ALL 0.31 0.20 0.17 0.37 0.12 0.73% 46%

(7.09) (4.25) (3.84) (8.11) (2.66) (5.74)

TSMOM COM 0.65 0.57% 42%

(14.61) (4.43)

TSMOM COM 0.62 0.05 0.02 0.14 0.05 0.51% 45%

(13.84) (1.01) (0.50) (3.08) (1.06) (3.96)

TSMOM EQ 0.39 0.47% 15%

(7.32) (3.00)

TSMOM EQ 0.07 0.28 0.04 0.06 0.24 0.43% 22%

(1.29) (5.07) (0.67) (1.11) (4.26) (2.79)

TSMOM FI 0.37 0.59% 14%

(6.83) (3.77)

TSMOM FI �0.03 0.18 0.34 0.01 0.03 0.50% 17%

(�0.62) (3.05) (6.19) (0.20) (0.48) (3.15)

TSMOM FX 0.75 0.42% 56%

(19.52) (3.75)

TSMOM FX 0.04 0.00 �0.01 0.75 �0.01 0.40% 56%

(1.07) (�0.04) (�0.17) (18.89) (�0.24) (3.49)

T
.J.

M
o

sk
o

w
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e
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a
l.

/
Jo

u
rn

a
l

o
f

F
in

a
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cia
l

E
co

n
o

m
ics

1
0

4
(2

0
1

2
)

2
2

8
–

2
5

0
2

4
2

Panel B: Decomposition of TSMOM and XSMOM

XSMOM decomposition TSMOM decomposition

Auto Cross Mean Total Auto Mean squared Total

ALL 0.53% �0.03% 0.12% 0.61% 0.54% 0.29% 0.83%

COM 0.41% �0.13% 0.11% 0.39% 0.43% 0.17% 0.59%

EQ 0.74% �0.62% 0.02% 0.14% 0.83% 0.17% 1.00%

FI 0.32% �0.10% 0.05% 0.27% 0.35% 0.70% 1.05%

FX 0.71% �0.55% 0.02% 0.18% 0.80% 0.17% 0.96%

Panel C: What factors does TSMOM explain?

Independent variable

TSMOM ALL Intercept R2

Dependent variable XSMOM ALL 0.66 �0.16% 44%

(15.17) (�1.17)

XSMOM COM 0.65 �0.09% 42%

(14.61) (�0.66)

XSMOM EQ 0.39 0.29% 15%

(7.32) (1.86)

XSMOM FI 0.37 �0.14% 14%

(6.83) (�0.87)

XSMOM FX 0.75 �0.19% 56%

(19.52) (�1.71)

UMD 0.49 �0.28% 13%

(6.56) (�0.93)

HML �0.07 0.43% 1%

(�1.46) (2.08)

SMB �0.01 0.10% 0%

(�0.26) (0.49)

DJCS MF 0.55 �0.30% 33%

(9.60) (�1.37)

DJCS MACRO 0.32 0.52% 14%

(5.64) (2.38)

T
.J.

M
o

sk
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w
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l.

/
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rn

a
l

o
f

F
in

a
n

cia
l

E
co

n
o

m
ics

1
0

4
(2

0
1

2
)

2
2

8
–

2
5

0
2

4
3

12 Lequeux and Acar (1998) also show that a simple timing trade

tracks the performance of currency hedge funds.

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250244
refers to the contribution from unconditional mean
returns, as well as their sum (‘‘Total’’). We also report
the two components to TSMOM: the Auto and mean-
squared return components, as well as their sum.

As Panel B of Table 5 shows, time series and cross-
sectional momentum are related but different. The auto-
covariance component contributes just about all of the
cross-sectional momentum strategy profits across all assets.
The cross-serial or lead-lag component contributes nega-
tively to XSMOM and the cross-sectional variation in means
has a small positive contribution to cross-sectional momen-
tum profits. The contribution of these components to cross-
sectional momentum strategies is also fairly stable across
asset classes, with the dominant component consistently
being the auto-covariance or time series piece.

The decomposition of time series momentum shows
that the main component is the auto-covariance of
returns. Squared mean excess returns are a much smaller
component of TSMOM profits, except for fixed income.
Since the cross-sectional correlation of lead-lag effects
among assets contributes negatively to XSMOM, it is not
surprising that TSMOM, which does not depend on the
cross–serial correlations across assets, produces higher
profits than XSMOM.

We also regress the returns of time series momentum
as defined in Eq. (7) (which is linear as opposed to using
the sign of the past return as before) on the returns to
cross-sectional momentum as defined in Eq. (6). We find
that time series momentum has a significant alpha to
cross-sectional momentum, consistent with our earlier
results in Panel A of Table 5 that use a slightly different
specification. We investigate next whether the reverse is
true. Does TSMOM explain XSMOM?

5.3. Does TSMOM explain cross-sectional momentum and

other factors?

Panel C of Table 5 uses TSMOM as a right-hand-side
variable, testing its ability to explain other factors. We first
examine XSMOM to see if TSMOM can capture the returns
to cross-sectional momentum across all asset classes as well
as within each asset class. As the first five rows of Panel C of
Table 5 show, TSMOM is able to fully explain cross-sectional
momentum across all assets as well as within each asset
class for commodities, equity indexes, bonds, and curren-
cies. The intercepts or alphas of XSMOM are statistically no
different from zero, suggesting TSMOM captures the return
premiums of XSMOM in these markets. The only positive
alpha is for XSMOM in equity indexes, which has a marginal
1.86 t-statistic. We also regress the Fama-French cross-
sectional momentum factor for individual US equities,
UMD, on our TSMOM portfolio. The UMD factor is created
from individual US equities and hence has no overlap with
any of the assets used to comprise our TSMOM factor.
Nevertheless, TSMOM is able to capture the return premium
to UMD, which has a positive 0.49 loading on TSMOM and
an insignificant �0.28 alpha (t-stat¼�0.93). We also
examine the other Fama-French factors HML, the value
factor, and SMB, the size factor. HML loads negatively on
TSMOM, so TSMOM naturally cannot explain the value
effect, and SMB has a loading close to zero.
Finally, we also examine the returns to two popular
hedge fund indexes that trade globally across many
assets: the ‘‘Managed Futures’’ hedge fund index and
‘‘Global Macro’’ hedge fund index obtained from Dow
Jones/Credit Suisse from 1994 to 2009. As the last two
rows of Panel C of Table 5 show, both hedge fund indexes
load significantly on our TSMOM factor and in the case of
the Managed Futures index, the TSMOM factor captures
its average return entirely. Hence, TSMOM is a simple
implementable factor that captures the performance
metric of Fung and Hsieh (2001), which they show
explains hedge fund returns as well.12

The strong performance of TSMOM and its ability to
explain some of the prominent factors in asset pricing,
namely, cross-sectional momentum as well as some
hedge fund strategy returns, suggests that TSMOM is a
significant feature of asset price behavior. Future research
may well consider what other asset pricing phenomena
might be related to time series momentum.

6. Who trades on trends: Speculators or hedgers?

To consider who trades on time series momentum,
Fig. 5 shows the Net speculator position broken down by
the sign of the past 12-month return for each instrument
with available CFTC data. Specifically, for each futures
contract, Fig. 5 plots the average Net speculator position
in, respectively, the subsample where the past 12-month
return on the contract is positive (‘‘Positive TSMOM’’ ) and
negative (‘‘Negative TSMOM’’) de-meaned using the aver-
age Net speculator position for each instrument. The
figure illustrates that speculators are, on average, posi-
tioned to benefit from trends, whereas hedgers, by defini-
tion, have the opposite positions. Speculators have longer-
than-average positions following positive past 12-month
returns, and smaller-than-average positions following
negative returns, on average. Said differently, speculators
have larger positions in an instrument following positive
returns than following negative returns. This trend-fol-
lowing pattern of speculative positions is found for every
contract except the S&P 500 futures, where Net speculator
positions have opposite signs, though are close to zero.
Since we also know that time series momentum is
associated with positive abnormal returns, these results
indicate that speculators profit, on average, from these
position changes at the expense of hedgers.

6.1. The evolution of TSMOM

We next consider the dynamics of these trading posi-
tions over time. Our previous results suggest that time
series momentum lasts about a year and then starts to
reverse. We investigate these return patterns in more
depth and attempt to link them to the evolution of trading
positions.

Examining the evolution of TSMOM and trading pat-
terns may help distinguish theories for momentum. For

-25%

-20%

-15%

-10%

-5%

0%

5%

10%

15%

20%

D
e-

m
ea

ne
d

ne
t s

pe
cu

la
to

r
po

si
tio

n

Positive TSMOM Negative TSMOM

Fig. 5. Net speculator positions. For each futures contract, the figure plots the average de-meaned Net speculator position in, respectively, the subsample
where the past 12-month returns on the contract are positive (‘‘Positive TSMOM’’ ) and negative (‘‘Negative TSMOM’’ ). The figure illustrates that

speculators are on average positioned to benefit from trends, whereas hedgers, by definition, have the opposite positions.

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250 245
example, if initial under-reaction to news is the cause of
TSMOM, then this part of the trend should not reverse,
whereas the part of a trend that is driven by over-reaction
should reverse as prices eventually gravitate back toward
fundamentals.

To consider the evolution of TSMOM, we perform an
event study as follows. For each month and instrument,
we first identify whether the previous 12-month excess
returns are positive or negative. For all the time–
instrument pairs with positive 12-month past returns,
we compute the average return from 12 months prior to
the ‘‘event date’’ (portfolio formation date) to 36 months
after. We do the same for the time–instrument pairs with
negative past 12-month returns. We then standardize the
returns to have a zero mean across time and across
the two groups (for ease of comparison), and compute
the cumulative returns of the subsequent months follow-
ing positive past-year returns (‘‘Positive TSMOM’’) and
negative past-year returns (‘‘Negative TSMOM’’), respec-
tively, where we normalize the cumulative returns to be
one at the event date.

Panel A of Fig. 6 shows the cumulative returns condi-
tional on positive and negative time series momentum.
The returns to the left of the event date are, of course,
positive and negative by construction. To the right
of the event date, we see that the positive pre-formation
returns continue upward after the portfolio formation for
about a year, consistent with a time series momentum
effect, and then partially reverse thereafter. This is
consistent with both initial under-reaction and delayed
over-reaction as predicted by sentiment theories such as
Daniel, Hirshleifer, and Subrahmanyam (1998) and
Barberis, Shleifer, and Vishny (1998). While the reversal
after a year suggests over-reaction, the fact that only
part of the post-formation upward trend is reversed
suggests that under-reaction appears to be part of the
story as well. Similarly, the negative pre-formation trend
is continued for a year until it partially reverses as
well.

Panel B of Fig. 6 shows the evolution of Net speculator
positions that coincide with the positive and negative
time series momentum returns. Specifically, for each
instrument and month, we compute the average Net
speculator position for each month from 12 months prior
to the event (portfolio formation date) to 36 months after
portfolio formation for both positive and negative trends.
We see that for positive TSMOM, speculators increase
their positions steadily from months �12 to 0, building
up to the formation date. Likewise, speculators decrease
their positions steadily from the negative TSMOM event
date. These patterns are not by construction since we split
the sample based on returns and not based on Net
speculator positions. After the event date, speculators’
positions begin to mean-revert towards their (positive)
average levels, and plateau at about a year (and maybe
slightly longer for negative TSMOM), which is when the
trend in returns starts to reverse.

The patterns in Fig. 6 indicate that while speculator
positions are consistent with trading on TSMOM, they do
not appear to keep piling into the trade with a lag. In fact,
speculators appear to reduce their trend chasing up to the
point where positive returns from following TSMOM
disappear. Conversely, hedgers, who are on the other side
of these trades, appear to be increasing their positions
steadily in the direction of the trend. This suggests that if
over-reaction is caused by such trading, it would have to
come from hedgers, not speculators. While the direction
of causality between returns and trading positions is

0.85

0.9

0.95

1

1.05

1.1

1.15

C
um

ul
at

iv
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re
tu

rn
s

Event time (months)

Positive TSMOM Negative TSMOM

-0.04
-0.02

0
0.02
0.04
0.06
0.08

0.1
0.12

-12

N
et

s
pe

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to
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po
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tio
n

Event time (months)

Positive TSMOM Negative TSMOM

-6 0 6 12 18 24 30 36

-12 -6 0 6 12 18 24 30 36

Fig. 6. Event study of time series momentum. For each month and instrument, we identify whether the previous 12-month returns are positive or
negative and compute the average return from 12 months prior to the ‘‘event date’’ (portfolio formation date) to 36 months after following positive past-

year returns (‘‘Positive TSMOM’’) and negative past-year returns (‘‘Negative TSMOM’’). We standardize the returns to have a zero mean across time and

across the two groups (for ease of comparison), and compute the cumulative returns of the subsequent months, where we normalize the cumulative

returns to be 1 at the event date. Panel B plots the Net speculator position as defined by the CFTC conditional on positive and negative past returns.

(A) Panel A: Cumulative returns in event time; (B) Panel B: Net speculator positions in event time.

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250246
indeterminate, these results also suggest that trading
positions of speculators and hedgers are closely linked
to the profitability of time series momentum, where
speculators appear to be profiting from trends and rever-
sals at the expense of hedgers.

6.2. Joint dynamics of returns and trading positions

For a more formal analysis of trading patterns and
returns, we study the joint dynamics of time series
momentum returns and the change in Net speculator
positions using a vector autoregressive (VAR) model. We
estimate a monthly bivariate VAR with 24 months of lags
of returns and changes in Net speculator positions and
plot the impulse response of returns and Net speculator
positions from a return shock. We need to include more
than 12 months of lags to capture delayed reversal, but
our results are robust to choosing other lag lengths
between 12 and 24.

We perform a Cholesky decomposition of the
variance–covariance matrix of residuals with the return
first, and consider a one-standard-deviation shock to the
returns of the contract. (The return response to an initial
return shock is qualitatively the same regardless of the
ordering of the Cholesky decomposition because of a
limited feedback with positions.) The response to this
impulse is plotted in Fig. 7, both in terms of the effect on
the cumulative return to the contract and the cumulative
changes in Net speculator positions. As the figure shows,
returns continue to rise for about a year and then partially
reverse thereafter following the return shock. Net spec-
ulator positions increase contemporaneously with the
return shock and then mean-revert to zero at about
a year. These results are consistent with our pre-
vious findings and confirm that speculative positions
match the return patterns of time series momentum.
Speculators seem to profit from TSMOM for about a year
and then revert to their average positions at the same
time the TSMOM effect ends; all at the expense of
hedgers.

The patterns indicate that speculators profit from
time series momentum, while hedgers pay for it. One
explanation might be that speculators earn a premium
through time series momentum for providing liquidity
to hedgers. We explore this possibility further by exam-
ining the predictability of returns using trading positions
as well as different components of futures returns.
Specifically, we investigate whether changes in the under-
lying spot price or the shape of the futures curve
(e.g., ‘‘roll yield’’) are driving the time series predict-
ability and how each of these lines up with trading
positions.

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

-0.7%

0.0%

0.7%

1.3%

2.0%

2.6%

3.3%

4.0%

0

N
et

S
pe

cu
la

to
r

P
os

iti
on

C
um

ul
at

iv
e

R
et

ur
n

Time Relative to Shock (Months)

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

-0.7%

0.0%

0.7%

1.3%

2.0%

2.7%

3.3%

4.0%

N
et

s
pe

cu
la

to
r

po
si

tio
n

C
um

ul
at

iv
e

re
tu

rn

Time relative to shock (months)

Cumulative return (left axis)

Net speculator position (right axis)

Cumulative return (left axis)

Net speculator position (right axis)

Cumulative return (left axis)

Net speculator position (right axis)

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

25.0%

30.0%

-0.7%

0.0%

0.7%

1.3%

2.0%

2.7%

3.3%

4.0%

N
et

s
pe

cu
la

to
r

po
si

tio
n

C
um

ul
at

iv
e

re
tu

rn

Time relative to shock (months)

363330272421181512963

0 363330272421181512963

0 363330272421181512963

Fig. 7. Impulse response from a shock to returns. Plotted are the cumulative returns and speculators’ net positions in response to a one standard deviation
shock to total returns on the futures contract (Panel A), returns on the spot asset (Panel B) and returns to rolling the contract (Panel C). The impulse response is

based on an estimated vector autoregressive model using monthly returns with 24 lags of returns and Net speculator positions that assumes coefficients are

the same across all contracts, with a Cholesky decomposition of the shock. (A) Panel A: Futures returns; (B) Panel B: Spot returns; (C) Panel C: Roll returns.

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250 247
6.3. Predictability of positions, price changes, and roll yield

We decompose the past return of each futures contract
into the change in the price of the underlying spot asset
and the return that is related to the shape of the futures
curve, called the ‘‘roll return’’ or ‘‘roll yield.’’

We define the underlying spot price changes in excess
of the risk-free rate as

Price changet�12,t ¼
Pricet�Pricet�12

Pricet�12
�r

f
t�12,t

,

where prices are measured as the nearest-to-expiration
futures price and r

f
t�12,t

is the risk-free interest rate over
the 12-month period. We then define the roll return by
the following decomposition:

Futures returnt�12,t ¼ Price changet�12,tþRoll returnt�12,t :

In financial futures with little storage costs or conve-
nience yield, the roll return is close to zero, but, in
commodity markets, the roll return can be substantial.
The futures return is calculated from the nearest-to-
expiration or next-to-nearest expiration contract, whose
maturity date may not be in the same month as the spot
return calculation, which is based only on the nearest-to-
expiration contract.

Table 6
Time series predictors of returns: Spot prices, roll returns, and positions.

Reported are results from regressions of the monthly futures return on the previous 12 months’ futures return (‘‘Full TSMOM’’), previous 12 months’

change in spot price (‘‘Spot price MOM’’), past 12-month roll return (‘‘Roll MOM’’), and the 12-month change and average level in speculators’ aggregate

net (i.e., long minus short) positions as a percent of open interest (‘‘Net speculator position’’). Also reported are interactions between the change in Net

speculator positions and the spot and roll returns over the previous 12 months.

Full

TSMOM

Spot price

MOM

Roll

MOM

Chg net speculator

position

Net speculator

position

Spot MOM�Chg net

spec pos

Roll MOM�Chg net

spec pos

Intercept R2

Coefficient 0.019 0.09% 0.6%

t-Stat (3.57) (1.31)

Coefficient 0.014 0.12% 0.3%

t-Stat (2.29) (1.71)

Coefficient 0.024 0.08% 0.3%

t-Stat (3.22) (1.08)

Coefficient 0.007 0.12% 0.2%

t-Stat (2.67) (1.63)

Coefficient 0.007 0.08% 0.2%

t-Stat (2.33) (1.12)

Coefficient 0.017 0.004 0.09% 0.7%

t-Stat (3.13) (1.65) (1.33)

Coefficient 0.018 0.002 0.08% 0.6%

t-Stat (3.31) (0.76) (1.14)

Coefficient 0.017 0.030 0.08% 0.6%

t-Stat (2.74) (3.90) (1.03)

Coefficient 0.014 0.030 0.005 0.07% 0.8%

t-Stat (2.12) (3.94) (1.89) (0.99)

Coefficient 0.014 0.030 0.005 0.023 0.015 0.06% 0.8%

t-Stat (2.17) (3.94) (1.78) (1.38) (0.49) (0.77)

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250248
Our conjecture is that hedgers’ price pressure affects
mostly the roll returns, whereas information diffusion
affects mostly spot price changes. To see why, recall first
Keynes’ (1923) basic idea that hedging pressure must affect
required returns to give speculators an incentive to provide
liquidity by taking the other side of the trade. Since hedging
takes place in futures markets, hedging pressure would
affect futures prices and thus lead to a roll yield as each
futures contract expires at the spot price. When hedgers,
such as commodity producers, are shorting the futures, this
leads to positive roll return and what Keynes called ‘‘normal
backwardation.’’ On the other hand, information diffusion
(which is the driver of several of the behavioral theories),
would simply affect price changes.

Panel B of Fig. 7 plots the impulse response of spot
price changes and Net speculator positions by repeating
the VAR we ran above, replacing the total futures returns
with the spot price changes only. The impulse response of
spot returns and Net speculator positions matches those
for total returns: trends exist for about a year and then
reverse and Net speculative positions mirror that pattern.
This is consistent with initial under-reaction and delayed
over-reaction being due to information diffusion rather
than hedging pressure.

Panel C of Fig. 7 plots the impulse response from
replacing total returns with the roll return in the VAR.
Here, the picture looks quite different. A shock to roll
returns is associated with a continued upward trend to
roll returns and a small effect on Net speculator positions.
This is consistent with hedgers having stable positions in
the same direction for extended periods of time and being
willing to give up roll returns to enjoy hedging benefits.
Speculators who take the other side, profit from momen-
tum as a premium for providing liquidity to hedgers.
Finally, Table 6 revisits the return predictability
regressions we started with, focusing on 12-month return
predictability, but examines the predictive power of the
spot versus roll return, as well as their interaction with
speculative trading positions. We regress the return of
each futures contract on the past 12-month return of each
contract, the spot price change of each contract, the roll
return of each contract, and the change and level of Net
speculator positions. The first five rows of Table 6 report
the univariate regression results for each of these vari-
ables, which are all significant positive predictors of
futures returns.

In multivariate regressions, however, the change in
Net speculator positions drops slightly and becomes
insignificant, indicating that controlling for past returns
reduces some of the predictive power of speculative
positions. This is consistent with the idea that roll return
and speculator positions both capture hedging pressure,
though measured differently and neither being a perfect
measure for hedging pressure. Spot price changes and roll
returns have almost the same predictive regression coef-
ficient in the multivariate regression, hence, their joint
predictive power (as measured by the R-square) is the
same as the univariate predictability of their sum,
which is the total futures return. Finally, the last row of
Table 6 includes interaction terms between the spot and
roll returns and Net speculator positions. While all the
coefficients are positive, indicating that when changes in
Net speculator positions move in the same direction as
returns, there is stronger positive predictability of future
returns, the results are not statistically significant.

The results in Table 6 indicate that time series momen-
tum is not purely driven by one component of futures
returns. Both the spot return change and roll yield provide

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250 249
predictive power for futures returns. In addition, as
the VAR results show, there is an interesting dynamic
between time series momentum and Net speculator and
hedging positions. Speculators seem to ride the trend for
about a year, eventually reducing their positions and
taking the opposite side before it reverses. In the process,
they earn positive excess returns at the expense of
hedgers, who may be willing to compensate speculators
for liquidity provision in order to maintain their hedge.
7. Conclusion

We find a significant time series momentum effect
that is remarkably consistent across the nearly five dozen
futures contracts and several major asset classes we study
over the last 25 years. The time series momentum effect is
distinct from cross-sectional momentum, though the two
are related. Decomposing both time series and cross-
sectional momentum profits, we find that the dominant
force to both strategies is significant positive auto-covar-
iance between a security’s excess return next month and
it’s lagged 1-year return. This evidence is consistent with
initial under-reaction stories, but may also be consistent
with delayed over-reaction theories of sentiment as the
time series momentum effect partially reverses after
one year.

Time series momentum exhibits strong and consistent
performance across many diverse asset classes, has small
loadings on standard risk factors, and performs well in
extreme periods, all of which present a challenge to the
random walk hypothesis and to standard rational pricing
models. The evidence also presents a challenge to current
behavioral theories since the markets we study vary
widely in terms of the type of investors, yet the pattern
of returns remains remarkably consistent across these
markets and is highly correlated across very different
asset classes. Indeed, correlation among time series
momentum returns is stronger than the correlation of
passive long positions across the same asset classes,
implying the existence of a common component to time
series momentum that is not present in the underlying
assets themselves.

Finally, the link between time series momentum
returns and the positions of speculators and hedgers
indicates that speculators profit from time series momen-
tum at the expense of hedgers. This evidence is consistent
with speculators earning a premium via time series
momentum for providing liquidity to hedgers. Decompos-
ing futures returns into the effect of price changes, which
captures information diffusion, and the roll return, which
captures how hedging pressure affects the shape of the
futures curve, we find that shocks to both price changes
and roll returns are associated with time series momen-
tum profits. However, only shocks to price changes
partially reverse, consistent with behavioral theories of
delayed over-reaction to information, and not hedging
pressure.

Time series momentum represents one of the most
direct tests of the random walk hypothesis and a number
of prominent behavioral and rational asset pricing
theories. Our findings present new evidence and chal-
lenges for those theories and for future research.
Appendix A. Data sources

A.1. Equity indexes

The universe of equity index futures consists of the
following nine developed equity markets: SPI 200 (Aus-
tralia), CAC 40 (France), DAX (Germany), FTSE/MIB (Italy),
TOPIX (Japan), AEX (Netherlands), IBEX 35 (Spain), FTSE
100 (UK), and S&P 500 (U.S). Futures returns are obtained
from Datastream. We use MSCI country-level index
returns prior to the availability of futures returns.
A.2. Bond indexes

The universe of bond index futures consists of the
following 13 developed bond markets: Australia 3-year
Bond, Australia 10-year Bond, Euro Schatz, Euro Bobl,
Euro Bund, Euro Buxl, Canada 10-year Bond, Japan 10-
year Bond (TSE), Long Gilt, US 2-year Note, US 5-year
Note, US 10-year Note, and US Long Bond. Futures returns
are obtained from Datastream. We use JP Morgan coun-
try-level bond index returns prior to the availability of
futures returns. We scale daily returns to a constant
duration of 2 years for 2- and 3-year bond futures, 4
years for 5-year bond futures, 7 years for 10-year bond
futures, and 20 years for 30-year bond futures.
A.3. Currencies

The universe of currency forwards covers the following
ten exchange rates: Australia, Canada, Germany spliced
with the Euro, Japan, New Zealand, Norway, Sweden,
Switzerland, UK, and US We use spot and forward interest
rates from Citigroup to calculate currency returns going
back to 1989 for all the currencies except for CAD and
NZD, which go back to 1992 and 1996, respectively. Prior
to that, we use spot exchange rates from Datastream and
Interbank Offered Rate (IBOR) short rates from Bloomberg
to calculate returns.
A.4. Commodities

We cover 24 different commodity futures. Our data on
Aluminum, Copper, Nickel, Zinc are from London Metal
Exchange (LME), Brent Crude, Gas Oil, Cotton, Coffee,
Cocoa, Sugar are from Intercontinental Exchange (ICE),
Live Cattle, Lean Hogs are from Chicago Mercantile
Exchange (CME), Corn, Soybeans, Soy Meal, Soy Oil,
Wheat are from Chicago Board of Trade (CBOT), WTI
Crude, RBOB Gasoline spliced with Unleaded Gasoline,
Heating Oil, Natural Gas are from New York Mercantile
Exchange (NYMEX), Gold, Silver are from New York
Commodities Exchange (COMEX), and Platinum from
Tokyo Commodity Exchange (TOCOM).

T.J. Moskowitz et al. / Journal of Financial Economics 104 (2012) 228–250250
A.5. Position of traders data

We obtain speculator net length and open interest data
from the CFTC Commitments of Traders Report Web site
for the following futures: Corn, Soybeans, Soy Meal, Soy
Oil, Wheat traded on Chicago Board of Trade (CBOT), Live
Cattle, Lean Hogs, Australian Dollar, Canadian Dollar,
Swiss Franc, British Pound, Japanese Yen, Euro FX, New
Zealand Dollar, S&P 500 traded on Chicago Mercantile
Exchange (CME), Cotton, Coffee, Cocoa, Sugar traded on
Intercontinental Exchange (ICE), WTI Crude, RBOB Gaso-
line spliced with Unleaded Gasoline, Heating Oil, Natural
Gas traded on New York Mercantile Exchange (NYMEX),
and Gold, Silver traded on New York Commodities
Exchange (COMEX). The data cover the period January
1986 to December 2009.

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Time series momentum
Introduction: a trending walk down Wall Street
Data and preliminaries
2.1. Futures returns
Positions of traders
Asset pricing benchmarks
Ex ante volatility estimate

Time series momentum: Regression analysis and trading strategies
Regression analysis: Predicting price continuation and reversal
3.2. Time series momentum trading strategies

Time series momentum factor
4.1. TSMOM by security and the diversified TSMOM factor
4.2. Alpha and loadings on risk factors
4.3. Performance over time and in extreme markets
4.4. Liquidity and sentiment
4.5. Correlation structure

Time series vs. cross-sectional momentum
5.1. Time series momentum regressed on cross-sectional momentum
5.2. A simple, formal decomposition
5.3. Does TSMOM explain cross-sectional momentum and other factors?

Who trades on trends: Speculators or hedgers?
6.1. The evolution of TSMOM
6.2. Joint dynamics of returns and trading positions
6.3. Predictability of positions, price changes, and roll yield

Conclusion
Data sources
A.1. Equity indexes
A.2. Bond indexes
A.3. Currencies
A.4. Commodities
A.5. Position of traders data

References