Time-Varying Expected Momentum Profits
Dongcheol Kim *
Tai-Yong Roh †
Byoung-Kyu Min ‡
Suk-Joon Byun §
This draft: August 2012
Abstract:
This paper examines the time variations of expected momentum profits using a two-state Markov
switching model with time-varying transition probabilities to evaluate the empirical relevance of
recent rational theories of momentum profits. We find that in the expansion state the expected
returns of winner stocks are more affected by aggregate economic conditions than those of loser
stocks, while in the recession state the expected returns of loser stocks are more affected than those
of winner stocks. Consequently, expected momentum profits display strong procyclical variations.
We argue that the observed momentum profits are the realization of such expected returns and can
be interpreted as the procyclicality premium. We provide a plausible explanation for time-varying
momentum profits through the differential effect of leverage and growth options across business
cycles.
JEL classification: G12; G14
Keywords: Momentum; Time-varying expected returns; Markov switching regression model;
Business cycle; Procyclicality; Growth options
*
Korea University Business School, Anam-dong, Seongbuk-gu, Seoul 136-701, Korea. Phone:
+82-2-3290-2606, Fax: +82-2-922-7220; E-mail: [email protected]
†
Graduate School of Finance, KAIST (Korea Advanced Institute of Science and Technology)
Business School, Hoegiro, Dongdaemoon-gu, Seoul, 130-722, Korea. Phone: +82-2-958-3968
E-mail: [email protected]
‡
Corresponding author. Institute of Financial Analysis, University of Neuchâtel, Pierre-à-Mazel 7,
2000 Neuchâtel, Switzerland. Phone: +41-32-718-15-74, E-mail: [email protected]
§
Graduate School of Finance, KAIST (Korea Advanced Institute of Science and Technology)
Business School, Hoegiro, Dongdaemoon-gu, Seoul, 130-722, Korea. Phone: +82-2-958-3968
E-mail: [email protected]
Time-Varying Expected Momentum Profits
Abstract
This paper examines the time variations of expected momentum profits using a two-state Markov
switching model with time-varying transition probabilities to evaluate the empirical relevance of
recent rational theories of momentum profits. We find that in the expansion state the expected
returns of winner stocks are more affected by aggregate economic conditions than those of loser
stocks, while in the recession state the expected returns of loser stocks are more affected than those
of winner stocks. Consequently, expected momentum profits display strong procyclical variations.
We argue that the observed momentum profits are the realization of such expected returns and can
be interpreted as the procyclicality premium. We provide a plausible explanation for time-varying
momentum profits through the differential effect of leverage and growth options across business
cycles.
JEL classification: G12; G14
Keywords: Momentum, Time-varying expected returns, Markov switching regression model,
Business cycle, Procyclicality, Growth options
1. Introduction
The cross-sectional difference in average stock returns across their recent past performance has
become one of the most controversial issues in academia as well as industry since the pioneering
work of Jegadeesh and Titman (1993). A simple momentum strategy buying recent winners and
selling recent losers generates both statistically and economically significant profits. There are two
explanations for the sources of these momentum profits in the literature: One is that momentum
profits result from investors’ irrational underreaction to firm-specific information (e.g., Barberis,
Shleifer, and Vishny, 1998; Daniel, Hirshleifer, and Subrahmanyam,1998; Hong and Stein, 1999;
Jiang, Lee, and Zhang, 2005; Zhang, 2006; Chui, Titman, and Wei, 2010). Another is a rational riskbased explanation stating that momentum profits are realizations of risk premiums because winner
stocks are riskier than loser stocks (e.g., Conrad and Kaul, 1998; Berk, Green, and Naik, 1999;
Johnson, 2002; Ahn, Conrad, and Dittmar, 2003; Bansal, Dittmar, and Lundblad, 2005; Sagi and
Seasholes, 2007; Liu and Zhang, 2008).
In contrast to the extensive aforementioned literature on the cross-sectional aspects of
momentum, the intertemporal aspects of momentum profits have received much less attention.
Studies of the intertemporal aspects of momentum profits focus on procyclical time variations in
momentum profits. Johnson (2002) and Sagi and Seasholes (2007) provide the theoretical insight
that momentum profits are likely to be procyclical. According to Johnson (2002), winner stocks
have higher exposure to growth rate risk than loser stocks. Since expected growth rates tend to be
high in expansions and growth rate risk is accordingly high, expected returns on momentum
portfolios should be higher in expansions than in recessions. In a similar vein, the model of Sagi
and Seasholes (2007) suggests that winner stocks tend to have more valuable growth options in
3
expansions than in recessions and such firms are riskier and associated with higher expected returns
in expansions, since growth options are riskier than assets in place.
There is also empirical evidence of the procyclicality of momentum profits. Chordia and
Shivakumar (2002) show that profits of momentum strategies can be explained by a set of lagged
macroeconomic variables that are related to business cycles and payoffs to momentum strategies
disappear after stock returns are adjusted for their predictability based on these macroeconomic
variables. These authors also find that momentum trading delivers reliably positive profits only
during expansionary periods but negative, statistically insignificant profits during recessions. Their
findings uncover procyclical time variations in momentum profits. Cooper, Gutierrez, and Hameed
(2004) also find that momentum profits depend on the state of the market in a procyclical way.
Average momentum profits are positive following periods of up markets but negative following
periods of down markets. However, these authors interpret these results as consistent with the
overreaction models of Daniel, Hirshleifer, and Subrahmanyam (1998) and Hong and Stein (1999). 5
In our view, a possible reason behind the discrepancy in the above authors’ different
interpretations is that the above two studies do not link time-series and cross-sectional properties of
expected returns. For example, the empirical specification used by Chordia and Shivakumar (2002),
regressing momentum payoffs on the lagged macroeconomic variables, does not impose a
covariance between momentum portfolio returns and the pricing kernel. Thus, we cannot
5
Cooper, Gutierrez, and Hameed (2004) report that a multifactor macroeconomic model of returns, as used
by Chordia and Shivakumar (2002), does not explain momentum profits after controlling for market frictions.
Additionally, these authors report that the macroeconomic model cannot forecast the time-series of out-ofsample momentum profits, whereas the lagged return of the market can. Hence, they suggest that the lagged
return of the market is the type of conditioning information that is relevant in predicting the profitability of
the momentum.
4
discriminate whether winners are riskier than losers or vice versa from their results. 6 Cooper,
Gutierrez, and Hameed (2004) also find that asymmetries conditional on the state of the market
complement the evidence of asymmetries in factor sensitivities, volatility, correlations, and
expected returns and thus argue that asset pricing models, both rational and behavioral, need to
incorporate (or predict) such regime switches. Stivers and Sun (2010) show that time variation in
momentum profits can be tied to variation in the market’s cross-sectional return dispersion. They
regard this return dispersion as a leading counter-cyclical state variable based according to the
theory of Gomes, Kogan, and Zhang (2003) and Zhang (2005). These authors find that the recent
cross-sectional return dispersion is negatively related to the subsequent momentum profits and thus
suggest that momentum profits are procyclical. 7
This paper aims to combine the time-series and cross-sectional implications of the
profitability of momentum trading. As Fama (1991, p. 1610) states, “In the end, I think we can hope
for a coherent story that relates the cross-section properties of expected returns to the variation of
expected returns through time.” This paper seeks to provide empirical evidence for such a coherent
story for momentum. To do so, we adopt the two-state Markov switching regression framework
with time-varying transition probabilities by following Perez-Quiros and Timmermann (2000) and
Gulen, Xing, and Zhang (2011). This flexible econometric model allows us to combine the crosssectional evidence on past stock returns with the time-series evidence on the evolution in
conditional returns and to describe asymmetries in the response of momentum profits to aggregate
economic conditions across the state of the economy by incorporating regime switches.
6
Chordia and Shivakumar (2002) admit to this weakness in their approach: “We do not impose crosssectional asset pricing constraints in this study. Proponents of the behavioral theories may argue that, to be
rational, the payoff to momentum strategies must covary with risk factors” (p. 988).
7
Stivers and Sun (2010) also show that the recent cross-sectional return dispersion is shown to be positively
related to the value premium and thus suggest that the value premium is countercyclical.
5
We also examine a differential response in expected returns to shocks to aggregate
economic conditions between winner and loser stocks across the state of economy and the
procyclicality of momentum profits. By employing a similar approach, Perez-Quiros and
Timmermann (2000) examine whether a differential response exists in expected returns to shocks to
aggregate economic conditions between small and large firms. Gulen, Xing, and Zhang (2011) also
examine a differential response in expected returns between value and stock firms and find strong
counter-cyclicality of the value premium. Our paper is not the first to examine the procyclicality of
momentum profits. Chordia and Shivakumar (2002), Cooper, Gutierrez, and Hameed (2004), and
Stivers and Sun (2010) already documented procyclical variations in momentum profits. Unlike the
previous literature, however, this paper show that the risks of winner and loser stocks are
asymmetrical across business cycles and time-variation in riskiness is a driving force for timevariation in momentum profits. In particular, we provide a plausible explanation for why winners
are riskier than losers in expansions but losers are riskier than winners in recessions.
We document two main findings. First, in the recession state, loser stocks tend to have
greater loadings on the conditioning macroeconomic variables than winner stocks, while in the
expansion state winner stocks tend to have greater loadings on these variables than loser stocks. In
other words, in recessions loser (winner) stocks are most (least) strongly affected, while in
expansions winner (loser) stocks are most (least) strongly affected. This indicates that returns on
momentum portfolios react asymmetrically to aggregate economic conditions in recession and
expansion states. Second, the asymmetries in winner and loser stocks’ risk across the states of the
economy lead to strong procyclical time-variations in the expected momentum profits. The expected
momentum profit estimated from the Markov switching regression model tends to be positive and
spike upward just before entering a recession (i.e., the peak of the business cycle), while it becomes
negative during recessions, reaching a maximal negative value at the end of a recession. The above
6
two findings are robust to using alternative instrumental variables in modeling state transition
probabilities and to using industry momentum portfolios as an alternative set of test assets. We also
examine the economic significance of out-of-sample predictability of the model by setting up a
simple stylized trading rule based on the prediction. The results show that the economic
significance of out-of-sample predictability is particularly significant when this trading rule is
applied to loser stocks and during a recession state.
The first main finding above implies that the riskiness of winner and loser stocks is
different across business cycles and, consequently, momentum profits are time-varying. We provide
a plausible explanation for time-varying momentum profits through the differential effect of
leverage and growth options across business cycles. During expansions, growth options have a
higher effect and leverage has a lower effect, and winner stocks tend to have greater growth options
and lower leverage. As a result, winner stocks are riskier in expansions. On the contrary, during
recessions, growth options have a lower effect and leverage has a higher effect, and loser stocks
tend to have lower growth options and higher leverage. Thus, loser stocks are riskier in recessions.
We argue that leverage and growth options are the underlying driving forces for the different
riskiness of winner and loser stocks and for time-varying momentum profits.
The remainder of this paper proceeds as follows. Section 2 discusses the sources of timevarying momentum profits. Section 3 presents a method to estimate the two-state Markov switching
regression model with time-varying transition probabilities. Section 4describes the data and the
empirical results for the model fitted to momentum portfolios. Section 5 provides a plausible
explanation for the observed time-varying momentum profits. Section 6 sets forth a summary and
conclusions.
7
2. Sources of Time-Varying Expected Momentum Profits
In his theoretical model, Johnson (2002) argues that stock prices area convex function of expected
growth, meaning that growth rate risk increases with growth rates and thus, stock price changes (or
stock returns) should be more sensitive to changes in expected growth when the expected growth is
higher. If exposure to this risk carries a positive premium, expected returns rise with growth rates.
Other things being equal, firms with large recent positive price moves (winners) are more likely to
have had positive growth rate shocks than firms with large recent negative price moves (losers).
Hence, a momentum sort will tend to sort firms by recent growth rate changes and sorting by
growth rate changes will also tend to sort firms according to growth rate levels and hence by endof-period expected returns. In other words, recent winners (losers) will tend to have both higher
(lower) growth rate changes in the recent past and higher (lower) subsequent expected returns.
Motivated by Johnson’s (2002) theoretical work, Liu and Zhang (2008) examine why the risk
exposure of winners on the growth rate of industrial production differs from those of losers.
Assuming that the growth rate of industrial production is a common factor summarizing firm-level
changes of expected growth, these authors document that winners have temporarily higher average
future growth rates than losers. More importantly, they find that the expected growth risk as defined
by Johnson (2002) is priced and the increases with expected growth. 8
In their theoretical model, Sagi and Seasholes (2007) show that a firm’s revenues, costs,
and growth options combine to explain momentum profits and they exercise their theoretical
insights to show that momentum strategies using firms with high revenue growth volatility and
8
Liu and Zhang (2008) also find that in many specifications this macroeconomic risk factor explains more
than half of momentum profits and conclude that risk plays an important role in driving momentum profits.
However, some papers report different results. For example, Grundy and Martin (2001) and Avramov and
Chordia (2006) report that controlling for time-varying exposures to common risk factors does not affect
momentum profits. Griffin, Ji, and Martin (2003) show that the model of Chen, Roll, and Ross (1986) does
not provide any evidence that macroeconomic risk variables can explain momentum.
8
valuable growth options outperform traditional momentum strategies. Their model suggests that
firms with valuable growth options exhibit higher autocorrelation than firms without such growth
options, because firms that performed well in the recent past are better poised to exploit their
growth options. Since growth options are riskier than assets in place, such firms are riskier and are
thus associated with higher expected returns. Winner stocks that have good recent performance are
likely to have riskier growth options than loser stocks that have bad recent performance.
Subsequently, winner stocks should earn higher expected returns than loser stocks. Importantly, the
Sagi and Seasholes (2007) model implies that momentum profits should be procyclical: “During up
markets, firms tend to move closer to exercising their growth options, which tends to increase return
autocorrelations. During down markets, firms tend to move closer to financial distress, which tends
to decrease return autocorrelations” (p. 391).
The above theoretical models suggest that momentum profits are procyclical. The expected
growth rates mentioned by Johnson (2002) are high in expansions and growth rate risk is
accordingly high. Since trading strategies based on momentum tend to have high exposure to this
risk, their expected returns should be higher in expansions than in recessions. In a similar vein,
procyclical stocks tend to have greater growth rate risk and more valuable growth options in
expansions than in recessions and thus such firms are riskier and associated with higher expected
returns in expansions. According to Johnson (2002) and Sagi and Seasholes (2007), recent winner
stocks are likely to have greater growth rate risk and riskier growth options and should earn higher
expected returns than recent loser stocks. Therefore, observed momentum profits (or returns on
winner-minus-loser portfolios, hereafter WML) are realizations of such expected returns and can be
interpreted as the procyclicality premium.
9
3. An Econometric Model of Time-Varying Expected Returns
Based on Sagi and Seasholes’ (2007) theoretical model, we argue that momentum profits are
procyclical because of the extent of exercising growth options across business cycles. To
empirically examine the procyclical behavior of momentum profits, the Markov switching
regression framework is appropriate since it can accommodate the time-varying behavior of
momentum profits across business cycles and business cycles can be regarded as states. In this
regard, we employ the Perez-Quiros and Timmermann (2000) Markov switching regression
framework with time-varying transition probabilities based on Hamilton (1989) and Gray (1996).
Let 𝑟𝑡 be the return of a test asset in excess of the riskless return at time 𝑡 and let 𝑿𝑡−1 be a
vector of conditioning variables available up to time 𝑡 − 1 used to predict 𝑟𝑡 . The Markov
switching specification takes all parameters (the intercept term, slope coefficients, and volatility of
excess returns) as a function of a single, latent state variable, 𝑆𝑡 . Specifically,
𝑟𝑡 = 𝛽0,𝑆𝑡 + 𝜷′𝑆𝑡 𝑿𝑡−1 + 𝜀𝑡 ,
𝜀𝑡 ~𝑁�0, 𝜎𝑆2𝑡 �,
(1)
where 𝑁(0, 𝜎𝑆2𝑡 ) denotes a normal distribution with mean zero and variance 𝜎𝑆2𝑡 . In a two-state
Markov switching specification, 𝑆𝑡 = 1 or 𝑆𝑡 = 2, meaning that the parameters to be estimated
are either 𝜃1 = �𝛽0,1 , 𝜷1′ , 𝜎12 � or 𝜃2 = �𝛽0,2 , 𝜷′2 , 𝜎22 �.
Since the above Markov switching model allows the risk and expected return to vary (or
transit) across two states, it is necessary to specify how the underlying states evolve through time.
We assume that the state transition probabilities follow a first-order Markov chain as follows:
𝑝𝑡 = 𝑃𝑟𝑜𝑏(𝑆𝑡 = 1| 𝑆𝑡−1 = 1, 𝒚𝑡−1 ) = 𝑝(𝒚𝑡−1 ),
1 − 𝑝𝑡 = 𝑃𝑟𝑜𝑏(𝑆𝑡 = 2| 𝑆𝑡−1 = 1, 𝒚𝑡−1 ) = 1 − 𝑝(𝒚𝑡−1 ),
𝑞𝑡 = 𝑃𝑟𝑜𝑏(𝑆𝑡 = 2| 𝑆𝑡−1 = 2, 𝒚𝑡−1 ) = 𝑞(𝒚𝑡−1 ),
10
(2)
(3)
(4)
1 − 𝑞𝑡 = 𝑃𝑟𝑜𝑏(𝑆𝑡 = 1| 𝑆𝑡−1 = 2, 𝒚𝑡−1 ) = 1 − 𝑞(𝒚𝑡−1 ),
(5)
where 𝒚𝑡−1 is a vector of variables publicly available at time 𝑡 − 1 and affects the state transition
probabilities between times 𝑡 − 1 and 𝑡 . Although the standard formulation of the Markov
switching model assumes the state transition probabilities to be constant, it would be more
reasonable to assume that the probability of staying in a state depends on prior conditioning
information, 𝒚𝑡−1 , and thus is time-varying, since investors are likely to possess information about
the state transition probabilities superior to that implied by the model with constant transition
probabilities. The literature shows that the economic leading indicator (Filardo, 1994; Perez-Quiros,
2000), interest rates (Gray, 1996; Gulen, Xing, and Zhang, 2011), or the duration of the time spent
in a given state (Durland and McCurdy, 1994; Mahue and McCurdy, 2000) is used as prior
condition information.
We estimate the above two-state Markov switching model using maximum likelihood
methods. 9 Let 𝜽 = (𝜃1 , 𝜃2 ) denote the vector of parameters to be estimated in the likelihood
function. The probability density function of the return, conditional on being state 𝑗, is Gaussian
defined as
𝑓(𝑟𝑡 | 𝛺𝑡−1 , 𝑆𝑡 = 𝑗; 𝜽) =
1
�2𝜋 𝜎𝑗2
𝑒𝑥𝑝 �−
2
�𝑟𝑡 − 𝛽0,𝑆𝑡 − 𝜷′𝑆𝑡 𝑿𝑡−1 �
2𝜎𝑗2
�
(6)
for 𝑗 = 1, 2. The information set 𝛺𝑡−1 contains 𝑿𝑡−1 , 𝑟𝑡−1 , 𝒚𝑡−1 , and lagged values of these
variables. Then, the log-likelihood function is
𝑇
𝐿(𝑟𝑡 | 𝛺𝑡−1 ; 𝜽) = � 𝑙𝑜𝑔[𝜙(𝑟𝑡 |𝛺𝑡−1 ; 𝜽)] ,
𝑡=1
9
(7)
Another estimation approach is a Bayesian approach based on numerical Bayesian methods such as the
Gibbs sampler and Markov Chain Monte Carlo methods (Kim and Nelson, 1999).
11
where the density function 𝜙(𝑟𝑡 |𝛺𝑡−1 ; 𝜃) is simply obtained by summing the probability-weighted
state probabilities across the two states. It is defined as
2
𝜙(𝑟𝑡 |𝛺𝑡−1 ; 𝜽) = � 𝑓(𝑟𝑡 | 𝛺𝑡−1 , 𝑆𝑡 = 𝑗; 𝜽) 𝑃𝑟𝑜𝑏(𝑆𝑡 = 𝑗|𝛺𝑡−1 ; 𝜽),
(8)
𝑗=1
where 𝑃𝑟𝑜𝑏(𝑆𝑡 = 𝑗|𝛺𝑡−1 ; 𝜽) is the conditional probability of being in state 𝑗 at time 𝑡 given
information at time 𝑡 − 1. The conditional state probabilities can be obtained from the standard
probability theorem:
2
𝑃𝑟𝑜𝑏(𝑆𝑡 = 𝑖|𝛺𝑡−1 ; 𝜽) = � 𝑃𝑟𝑜𝑏(𝑆𝑡 = 𝑖|𝑆𝑡−1 = 𝑗, 𝛺𝑡−1 ; 𝜽) 𝑃𝑟𝑜𝑏(𝑆𝑡−1 = 𝑗|𝛺𝑡−1 ; 𝜽).
𝑗=1
(9)
By Bayes’ rule, the conditional state probabilities can be written as
𝑃𝑟𝑜𝑏(𝑆𝑡−1 = 𝑗|𝛺𝑡−1 ; 𝜃)
=
𝑓(𝑟𝑡−1 |𝑆𝑡−1 =
2
∑𝑗=1 𝑓(𝑟𝑡−1 |𝑆𝑡−1
𝑗, X𝑡−1 , y𝑡−1 , 𝛺𝑡−2 ; 𝜽) 𝑃𝑟𝑜𝑏(𝑆𝑡−1 = 𝑗|X𝑡−1 , y𝑡−1 , 𝛺𝑡−2 ; 𝜽)
. (10)
= 𝑗, X𝑡−1 , y𝑡−1 , 𝛺𝑡−2 ; 𝜽) 𝑃𝑟𝑜𝑏(𝑆𝑡−1 = 𝑗|X𝑡−1 , y𝑡−1 , 𝛺𝑡−2 ; 𝜽)
The conditional state probabilities 𝑃𝑟𝑜𝑏(𝑆𝑡 = 𝑖|𝛺𝑡−1 ; 𝜽) are driven by iterating recursively
equations (9) and (10) and the parameter estimates of the likelihood function are obtained (Gray,
1996).Variations in the state probabilities are evidence that the conditional expected return is timevarying.
4. Empirical Results
4.1. Data and Model Specification
We use monthly excess returns (raw returns minus the one-month Treasury bill rate) on the
momentum decile portfolios as test assets. 10 Momentum portfolios are constructed in accordance
10
The one-month Treasury bill rates are obtained
(http://mba.tuck.dartmouth.edu/pages/faculty/ken.french).
12
from
Kenneth
French’s
website
with Jegadeesh and Titman (1993) by sorting all stocks every month into one of 10 decile portfolios
based on the past six-month returns and holding the deciles for the subsequent six months. We skip
one month between the end of the portfolio formation period and the beginning of the holding
period to avoid potential microstructure biases. All stocks in a given portfolio have equal weight.
Portfolio 1 is the past loser, while Portfolio 10 is the past winner. Gulen, Xing, and Zhang (2011)
examine the time-varying behavior of the expected value premium and show that the expected value
premium displays strong countercyclical variations, while we show that the expected momentum
profits display strong procyclical variations. To compare the opposite time-varying behaviors of
these two stock return regularities, we select the sample period from January 1954 to December
2007, as in Gulen, Xing, and Zhang (2011).
Table 1 presents the mean, standard deviation, skewness, and kurtosis of monthly excess
returns on the 10 decile momentum portfolios. The mean excess returns monotonically increase
from 0.352% per month for the past loser portfolio (Portfolio 1) to 1.306% per month for the past
winner portfolio (Portfolio 10). The mean return on the WML is quite significant: 0.954% per
month. A distinct pattern is found in skewness, which almost monotonically decreases from 1.359
for the loser portfolio to -0.644 for the winner portfolio. Portfolios 1 through 4 are positivelyskewed, while Portfolios 5 through 10 are negatively-skewed. These results indicate that past (shortterm) winners are preferred to past losers in the mean-variance framework, but this may not
necessarily be true when considering the third moment, since positively-skewed portfolios should
be preferred to negatively-skewed portfolios. This is consistent with the Arrow–Pratt notion of risk
aversion.
To show that momentum returns are asymmetrically affected by macroeconomic variables
across states (or business cycles), we model the excess returns of each of the momentum portfolios
as a function of an intercept term and lagged values of the one-month Treasury bill rate, the default
13
spread, the inflation rate, and the dividend yield. These variables are commonly used in the
literature on the predictability of stock returns. As in Perez-Quiros and Timmermann (2000) and
Gulen, Xing, and Zhang (2011), we use the one-month Treasury bill rate (TB) as a state variable
proxying for investors’ expectations of future economic activity. According to Fama (1981), an
unobserved negative shock to real economic activity induces a higher Treasury bill rate through an
increase in the current and expected future inflation rate. He argues that a negative correlation
between stock returns and inflation is not a causal relation but is proxying for a positive relation
between stock returns and real activity. Thus, the Treasury bill rate, which is an indicator of the
short-term interest rate, tends to have a negative relation with stock returns (e.g., Fama and Schwert,
1977; Fama, 1981, Campbell, 1987; Glosten, Jagannathan, and Runkle, 1993). Berk, Green, and
Naik (1999) present a theoretical model predicting that changes in interest rates will affect expected
stock returns differently across firms and providing a direct link between cross-sectional dispersions
of expected stock returns and interest rates. Interest rates should be a true cause of ex post stock
returns, because an increase (decrease) in the real interest rate induces a reduction (increase) in
stock values.
The default spread (DEF) is defined as the difference between yields on Baa-rated
corporate bonds and 10-year Treasury bonds from the Federal Reserve Economic Data at the
Federal Reserve Bank of St. Louis and is included to capture the effect of default premiums. Fama
and French (1989) show that the major movements in DEF seem to be related to long-term business
cycle conditions and the default spread forecasts high returns when business conditions are
persistently weak and low returns when conditions are strong. Indeed, the default spread is one of
the most frequently used conditioning variables in predicting stock returns (e.g., Keim and
Stambaugh, 1986; Fama and French, 1988; Kandel and Stambaugh, 1990; Jagannathan and Wang,
1996; Chordia and Shivakumar, 2002).
14
The inflation rate (INFL) is defined as the one-month log difference in the Consumer Price
Index (seasonally-adjusted) from the U.S. Bureau of Labor Statistics. Both economic theory and
traditional idea imply that stock returns and inflation should be positively correlated, since equities
are "hedges" against inflation because they represent claims to real assets. However, the United
States and other industrialized countries exhibit a significant negative correlation between inflation
and real stock returns in the post-war periods (e.g., Fama and Schwert, 1977; Fama, 1981; Geske
and Roll, 1983; Danthine and Donaldson, 1986; Stulz, 1986; Kaul,1987, 1990;Marshall, 1992;
Boudoukh, Richardson, and Whitelaw,1994; Bakshi and Chen, 1996). 11 This negative correlation
between inflation and real stock returns is often termed the stock return–inflation puzzle. Many
authors have tried to resolve this puzzle (e.g., Fama, 1981; Marshall, 1992; Geske and Roll, 1983;
Kaul, 1987). In particular, Kaul (1987) tries to explain this puzzle by hypothesizing that the relation
between stock returns and inflation is caused by the equilibrium process in the monetary sector.
More importantly, these relations vary over time in a systematic manner, depending on the
influence of money demand and supply factors. Post-war evidence from the United States, Canada,
the United Kingdom, and Germany indicates that the negative stock return–inflation relations are
caused by money demand and counter-cyclical money supply effects. On the other hand, procyclical movements in money, inflation, and stock prices during the 1930s lead to relations that are
either positive or insignificant.
The dividend yield (DIV) is defined as the sum of dividend payments accruing to the
Center for Research in Security Prices (CRSP) value-weighted market portfolio over the previous
12 months divided by the contemporaneous level of the index at the end of the month. The standard
valuation model indicates that stock prices are low relative to dividends when discount rates and
11
In contrast to existing evidence of a negative relation at short horizons, Boudoukh and Richardson (1993)
find evidence to suggest that long-horizon nominal stock returns are positively related to both ex ante and ex
post long-term inflation.
15
expected returns are high and vice versa. Thus, the dividend yield (usually measured by the ratio of
dividends to price) varies with expected returns. Thus, the dividend yield proxies for time-variation
in the unobservable risk premium. There is ample empirical evidence that the dividend yield
predicts future stock returns (e.g., Keim and Stambaugh, 1986; Campbell and Shiller, 1988; Fama
and French, 1988; Kandel and Stambaugh, 1990). 12
To capture the movements of momentum portfolio returns, we specify equation (1) by
including the above-mentioned return predictable variables in the following conditional mean
equation:
𝑟𝑖𝑡 = 𝛽𝑖0,𝑆𝑡 + 𝛽𝑖1,𝑆𝑡 𝑇𝐵𝑡−1 + 𝛽𝑖2,𝑆𝑡 𝐷𝐸𝐹𝑡−1 + 𝛽𝑖3,𝑆𝑡 𝐼𝑁𝐹𝐿𝑡−1 + 𝛽𝑖4,𝑆𝑡 𝐷𝐼𝑉𝑡−1 + 𝜀𝑖𝑡 ,
(11)
where 𝑟𝑖𝑡 is the monthly excess return for the 𝑖 𝑡ℎ decile momentum portfolio at time 𝑡, 𝜀𝑖𝑡 is the
2
normally distributed random error term with mean zero and variance 𝜎𝑖,𝑆
, and 𝑆𝑡 = {1, 2}. The
𝑡
2
, is allowed to
regressors are lagged by one month. The conditional variance of excess returns, 𝜎𝑖,𝑆
𝑡
depend only on the state of economy:
2
𝑙𝑛�𝜎𝑖,𝑆
� = 𝜆𝑖,𝑆𝑡 .
𝑡
(12)
We do not include autoregressive conditional heteroskedasticity (ARCH) effects in the conditional
volatility equation. Table 1 shows the first-order autocorrelations of the squared raw excess returns
and the squared residuals from the conditional mean equation of (11) in each of the 10 decile
momentum portfolios. Only one squared raw excess returns and three squared residuals out of 10
portfolios have significant first-order autocorrelation coefficient estimates at the five percent level.
These results indicate that ARCH effects are less important in the conditional volatility in our
framework.
12
Ang and Bekaert (2007) report that the dividend yield does not univariately predict excess returns, but the
predictive ability of the dividend yield is considerably enhanced, at short horizons, in a bivariate regression
with the short rate.
16
Following Gray (1996) and Gulen, Xing, and Zhang (2011), we model the time-varying
state transition probabilities to be dependent on the level of short interest rates, Treasury bill rates,
as follows:
𝑖
= 1, 𝒚𝑡−1 � = 𝛷�𝜋0𝑖 + 𝜋1𝑖 𝑇𝐵𝑡−1 �,
𝑝𝑡𝑖 = 𝑃𝑟𝑜𝑏�𝑆𝑡𝑖 = 1| 𝑆𝑡−1
𝑖
1 − 𝑝𝑡𝑖 = 𝑃𝑟𝑜𝑏�𝑆𝑡𝑖 = 2| 𝑆𝑡−1
= 1, 𝒚𝑡−1 �,
𝑖
= 2, 𝒚𝑡−1 � = 𝛷�𝜋0𝑖 + 𝜋2𝑖 𝑇𝐵𝑡−1 �,
𝑞𝑡𝑖 = 𝑃𝑟𝑜𝑏�𝑆𝑡𝑖 = 2| 𝑆𝑡−1
𝑖
1 − 𝑞𝑡𝑖 = 𝑃𝑟𝑜𝑏�𝑆𝑡𝑖 = 1| 𝑆𝑡−1
= 2, 𝒚𝑡−1 �,
(13)
(14)
(15)
(16)
where 𝑇𝐵𝑡−1 is the one-month–lagged Treasury rate, and 𝛷(∙) is the cumulative probability
density function of a standard normal variable. For robustness checks, we also use two alternative
instruments in the state transition probability equations instead of the Treasury bill rate: the
Composite Leading Indicator and the industrial production growth rate. However, the results are
qualitatively similar, as reported in Section 4.5.
4.2. Estimation Results
4.2.1. Identifying the States
Table 2 reports the estimation results of the parameters in equations (11) through (16). The constant
parameter estimate in the conditional mean equation in state 1 (𝛽𝑖0,1 ) is much lower than that in
state 2 (𝛽𝑖0,2 ) in all momentum portfolios. The constant term in state 1 monotonically increases
across the portfolios from the loser to the winner portfolios and is more precisely estimated. Eight
out of 10 constant terms in state 1 are significantly estimated at the 1% level and all 10 constant
term estimates are negative. In contrast, there is no particular pattern in the constant term in state 2
and any of the 10 constant terms are not significantly estimated. The conditional standard deviation
17
estimate in state 1 (𝜎𝑖,1 ) is greater than that in state 2 (𝜎𝑖,2 ) in all portfolios. All conditional
volatilities are highly significantly estimated. Schwert (1989) and Hamilton and Lin (1996) find that
return volatilities are higher in recession periods than in expansion periods. Their findings are
verified with historical National Bureau of Economic Research (NBER) business cycle dates.
Table 1 reports that average returns in the recession periods are lower in all portfolios than
those in the expansion periods, while standard deviations in the recession periods are much higher
in all portfolios than those in the expansion periods. The overall average returns of all portfolios are
0.686% and 1.000% in recession and expansion periods, respectively, while the standard deviations
are 7.524% and 4.728% in recession and expansion periods, respectively. In sum, state 1 is
associated with low average returns and high conditional volatilities, while state 2 is associated with
high average returns and low conditional volatilities. These results indicate that state 1 is the
recession state and state 2 is the expansion state.
To further identify the states, Figure 1 plots the time-series of the state transition
probabilities of being in state 1 (Panel A) and state 2 (Panel B) at time 𝑡 conditional on the
information set at time 𝑡 − 1, 𝑃𝑟𝑜𝑏�𝑆𝑡𝑖 = 1| 𝛺𝑡−1 ; 𝜽� and 𝑃𝑟𝑜𝑏�𝑆𝑡𝑖 = 2| 𝛺𝑡−1 ; 𝜽�, for the winner
portfolio. As shown in Panels A and B of Figure 1, the state transition probabilities of being in state
1 for the winner portfolio tend to increase and are relatively high during the NBER recession
periods (shaded areas), while the state transition probabilities of being in state 2 tend to decrease
and are low during the NBER recession periods and high during the NBER expansion periods.
Figure 1 also plots the time-series of the probability of being in state 1 (Panel C) and state 2 (Panel
D) for the loser portfolio. A similar but clearer pattern is found for the loser portfolio. The state
transition probabilities of being in state 1 spike more clearly during the recession periods and are
lower during the expansion periods. The pattern of the state transition probabilities of being in state
18
2 appears obviously opposite. In sum, the results suggest that state 1 with low average returns and
high volatilities can be regarded as the recession state and state 2 with high average returns and low
volatilities can be regarded as the expansion state.
4.2.2. Estimation of the Conditional Mean Equations
Table 2 reports the estimation results of the conditional mean equation (11) for each of the 10
momentum portfolios. We also estimate the same Markov switching regression model for each of
10 decile book-to-market portfolios. 13 We report only the difference in the coefficient estimates
between the value (highest book-to-market) portfolio and the growth (lowest book-to-market)
portfolio (HML) in the last column of Table 2 to compare the results of WML with those of HML.
The coefficient estimates on the one-month Treasury bill rate are mostly significant and all
negative for the 10 momentum portfolios in both states 1 and 2 (𝛽𝑖1,1 and 𝛽𝑖1,2 ), which means that
when the short-term interest rate increases in the previous month, the returns of all momentum
portfolios decrease in the current month. There is no monotonic pattern in these coefficients across
the portfolios. Moving from the loser to the winner portfolio, the coefficients have a U-shape in
state 1 (-0.444 for Portfolio 1, -0.932 for Portfolio 5, -0.508 for Portfolio 10), but an inverse Ushape in state 2 (-0.410 for Portfolio 1, -0.097 for Portfolio 5, -0.341 for Portfolio 10). In other
words, the two extreme portfolios (i.e., the loser and winner portfolios) have a similar sensitivity of
returns to changes in short-term interest rates, regardless of the expansion or recession states.
However, the middle momentum portfolios have a different sensitivity across the states. In the
recession state (state 1), the Treasury bill rate has a greater negative impact on the middle
13
Returns on the 10 decile book-to-market portfolios were obtained from Kenneth French’s website at
Dartmouth College (http://mba.tuck.dartmouth.edu/pages/faculty/ken.french).
19
momentum portfolio returns than on the two-end momentum portfolio returns; in the expansion
state (state 2), however, it has a smaller negative impact on the middle momentum portfolio returns.
The coefficient estimates on the default spread in both states (𝛽𝑖2,1 and 𝛽𝑖2,2 ) are all
positive and mostly statistically significant at the 5% level. They exhibit a systematic variation
across the portfolios in both states. In the recession state, moving from the loser portfolio to the
winner portfolio, the coefficient on DEF decreases largely monotonically from 6.091(with𝑡-statistic
of 3.44) to 0.975(with 𝑡-statistic of 1.29). The difference in the coefficients between the winner and
loser portfolios is marginally statistically significant: -5.116 (with 𝑡-statistic of -1.78). In the
expansion state, however, the coefficient increases monotonically from 0.454 (with 𝑡-statistic of
0.92) to 1.779 (with 𝑡-statistic of 2.88). The difference in the coefficients between the winner and
loser portfolios is 1.325 (with 𝑡-statistic of 1.33).The negative value of WML in the recession state
and the positive value of WML in the expansion state indicate that loser stocks are more affected by
the credit condition of the market in the recession state than are winner stocks, but the reverse
occurs in the expansion state. This implies that momentum profits are procyclical.
These above findings are exactly the opposite to those of Gulen, Xing, and Zhang (2011),
who find the counter-cyclicality of the value premium (measured by HML). We find that moving
from the growth portfolio (with low book-to-market) to the value portfolio (with high book-tomarket), the coefficient on DEF increases monotonically in the recession state, while it decreases
monotonically in the expansion state. This pattern is exactly the opposite of that in the momentum
portfolios. Table 2 reports that the differences in the coefficient estimates on DEF between the
value portfolio and the growth portfolio (HML) are 1.151 and -0.032 in the recession and expansion
states, respectively. These results indicate that value stocks are more affected by the credit condition
in the market than growth stocks in the recession state, but the reverse occurs in the expansion state;
20
that is, the value premium is counter-cyclical, which is consistent with Gulen, Xing, and Zhang
(2011).
As in the previous literature, the coefficient estimates on the inflation rate (𝛽𝑖3,1 and 𝛽𝑖3,2 )
are negative for all momentum portfolios, which indicates that high inflation in the previous period
negatively affects current stock returns. The results for the inflation rate also indicate the
procyclicality of the momentum profits. Moving from the loser to the winner portfolios, in the
recession state, the coefficient decreases largely monotonically in magnitude from -6.985(with 𝑡-
statistic of -1.29) to -0.348 (with 𝑡-statistic of -0.17), although all coefficient estimates are
statistically insignificant. The difference in the coefficients between the winner and loser portfolios
is 6.637 (with 𝑡-statistic of 0.68). In the expansion state, however, the coefficient increases largely
monotonically in magnitude from -0.832(with 𝑡-statistic of -0.70) to -4.599 (with 𝑡-statistic of -
4.04). Seven out of the 10 coefficient estimates are statistically significant at the 5% level. The
difference in the coefficients between the winner and loser portfolios is -3.766 (with 𝑡-statistic of -
1.77). These results indicate that loser stocks are more negatively affected by inflation in the
recession state, while winner stocks are more negatively affected by inflation in the expansion state.
The coefficient estimates on the dividend yields (𝛽𝑖4,1 and 𝛽𝑖4,2 ) are all positive and
mostly statistically significant. In the recession state, the coefficients for the loser and winner
portfolios are 4.173 (with 𝑡-statistic of 2.83) and 2.233(with 𝑡-statistic of 3.00), respectively. In the
expansion state, the coefficients for the loser and winner portfolios are 0.701 (with 𝑡-statistic of
2.26) and 0.501(with 𝑡-statistic of 1.55), respectively. We find hardly any particular pattern in the
coefficients on the dividend yield. We simply observe that the pattern in the coefficient on WML is
the opposite of that on HML.
21
Overall, returns on the momentum portfolios react asymmetrically to the macroeconomic
conditioning variables across recession and expansion states. To confirm the differential responses
of momentum returns to aggregate economic conditions in the recession and expansion states, it is
necessary to test whether the coefficients on the four conditioning variables (the Treasury bill rate,
the default spread, the inflation rate, and the dividend yield) are equal across states. We employ a
likelihood ratio test for the null hypotheses: 𝛽𝑖𝑘,𝑆𝑡 =1 = 𝛽𝑖𝑘,𝑆𝑡 =2 for 𝑘 = 1, 2, 3, 4 for each of the
10 momentum portfolios. Table 3 reports the unrestricted and restricted log-likelihood values and
𝑝-values from a standard 𝜒 2 distribution for the 10 momentum portfolios. Seven out of the 10
portfolios have a 𝑝-value less than 1% and all portfolios have 𝑝-values less than 5%, indicating
that the null hypothesis is strongly rejected. It is statistically confirmed, therefore, that the
conditional mean equation is state-dependent and the responses of momentum profits to the
conditioning variables are asymmetric across states.
4.3 A Bivariate Joint Model for Loser and Winner Stocks’ Expected Returns
4.3.1. Model Specification
So far the Markov switching regression models for excess returns have been estimated separately
(i.e., univariately) for each of the 10 momentum portfolios. That is, the condition that the recession
and expansion states occur simultaneously for all test portfolios is not imposed in the estimation.
The joint framework allows us to impose a common state process that drives all excess return series.
Since there are difficulties in estimating a multivariate joint model when the excess returns of all 10
portfolios and the loser and winner portfolios are our main target portfolios, we consider a bivariate
framework that simultaneously estimates the conditional mean equations for both loser and winner
portfolios. This bivariate framework can model the time-varying momentum profit and test its
22
procyclical variations. As in Perez-Quiros and Timmermann (2000) and Gulen, Xing, and Zhang
(2011), the bivariate Markov switching regression model is as follows. Let 𝒓𝑡 = (𝑟𝑡𝐿 , 𝑟𝑡𝑊 )′ be a
(2 × 1) vector consisting of excess returns on the loser and winner portfolios, 𝑟𝑡𝐿 and 𝑟𝑡𝑊 ,
respectively. Then, the joint conditional mean equation is specified as follows:
𝒓𝑡 = 𝜷0,𝑆𝑡 + 𝜷1,𝑆𝑡 TB𝑡−1 + 𝜷2,𝑆𝑡 DEF𝑡−1 + 𝜷3,𝑆𝑡 TERM𝑡−1 + 𝜷4,𝑆𝑡 DIV𝑡−1 + 𝜺𝑡 ,
′
(17)
𝑊
𝐿
where 𝜷𝑘,𝑆𝑡 is a (2 × 1) coefficient vector with elements �𝛽𝑘,𝑆
, 𝛽𝑘,𝑆
� for 𝑘 = 1, 2, 3, 4, and
𝑡
𝑡
𝜺𝑡 is a (2 × 1) vector of normal residuals with mean zero and covariance matrix ∑𝑆𝑡 , 𝑆𝑡 = {1, 2}.
Here ∑𝑆𝑡 is a positive semidefinite (2 × 2) covariance matrix of the residuals from the loser and
winner portfolios’ excess returns in state 𝑆𝑡 . For estimation convenience, we assume the form of the
conditional covariance matrix as follows:
𝑙𝑛�∑𝑖𝑖,𝑆𝑡 � = 𝜆𝑖,𝑆𝑡
𝑓𝑜𝑟 𝑖 = 𝑗
�
1/2
1/2
∑𝑖𝑗,𝑆𝑡 = 𝜌𝑆𝑡 (∑𝑖𝑖,𝑆𝑡 ) (∑𝑗𝑗,𝑆𝑡 )
𝑓𝑜𝑟 𝑖 ≠ 𝑗
(18)
In other words, we assume that the diagonal elements of this variance-covariance matrix, ∑𝑖𝑖,𝑆𝑡 ,
depend only on the state of economy, as in the univariate case of equation (11).The off-diagonal
elements, ∑𝑖𝑗,𝑆𝑡 , assume a state-dependent correlation between the residuals, denoted 𝜌𝑆𝑡 .We also
do not include ARCH effects in the conditional volatility equation. The transition probabilities from
the univariate model are maintained:
𝑝𝑡 = 𝑃(𝑆𝑡 = 1|𝑆𝑡−1 = 1) = Φ(𝜋0 + 𝜋1 TB𝑡−1 ),
𝑞𝑡 = 𝑃(𝑆𝑡 = 2|𝑆𝑡−1 = 2) = Φ(𝜋0 + 𝜋2 TB𝑡−1 ).
4.3.2. Estimation Results
23
(19)
(20)
Table 4 presents the estimation results of the bivariate Markov switching regression model. The
coefficient estimates on the conditioning variables for both loser and winner portfolios are
qualitatively similar to those from the univariate model specification in Table 2, implying that
imposing a common state process changes little. In particular, the asymmetries in the coefficients on
the macroeconomic variables in the conditional mean equation are very similar to those found in the
univariate model specification. Table 4 also reports testing results for the proposition that the
asymmetry in the coefficients observed for the loser portfolio across recession and expansion states
equals the asymmetry for the winner portfolio. For each set of the coefficients, we test the null
hypothesis that
𝑊
𝑊
𝐿
𝐿
− 𝛽𝑘,2
� = �𝛽𝑘,1
− 𝛽𝑘,2
� 𝑓𝑜𝑟 𝑘 = 1, 2, 3, 4.
�𝛽𝑘,1
(21)
None of the nulls of identical asymmetries across states for loser and winner stocks is rejected at
standard significance levels for the slope coefficients on the four macroeconomic variables in the
conditional mean equation. These results are in contrast with those reported by Perez-Quiros and
Timmermann (2000) for size portfolios and with those of Gulen, Xing, and Zhang (2011) for bookto-market portfolios. These authors report that the null hypotheses of identical asymmetries for
small and large stocks and for growth and value stocks are strongly rejected using the conditioning
variables similar to ours. As in the univariate model specification, states 1 and 2 are identified as the
recession state with low average returns and high volatility and as the expansion state with high
average returns and low volatility, respectively.
Figure 2 plots the expected excess returns obtained from the univariate and bivariate
Markov switching models for the winner portfolio (Panel A), the loser portfolio (Panel B), and the
WML (Panel C). The solid line denotes the expected excess returns obtained from the univariate
Markov switching model and the dashed lines denote the expected excess returns obtained from the
24
bivariate model. The shaded areas indicate NBER recession periods. All panels show that the
expected excess returns of the loser and winner portfolios and the expected momentum profit
(WML) display time-variations across the states of the economy.
Panels A and B of Figure 2 show that the series obtained from the univariate and joint
bivariate models for the loser and winner portfolios are approximately similar. The expected returns
of both loser and winner portfolios tend to increase during recession periods but decrease during
expansion periods. However, the loser portfolio tends to display this pattern more strongly than the
winner portfolio. As a result, as seen in Panel C, the expected momentum profit (WML) tends to
decrease sharply and have a negative value during recessions but to increase and have a positive
value just after recessions and during expansions. It tends to be lower during recessions than during
expansions. For example, during the whole sample period, the expected momentum profit is 1.1%
per month during expansions (i.e., low volatility states), while it is -0.8% per month during
recessions (i.e., high volatility states) and the difference in expected momentum profits between the
two states is highly significant (with t-statistic of 18.67). Although there is a little discrepancy in the
expected excess returns of WML obtained from the univariate and bivariate models, this pattern is
overall very similar. This procyclical behavior of the expected momentum profit is the opposite of
the counter-cyclical behavior of the value premium shown by Gulen, Xing, and Zhang (2011).
These authors illustrate that the value premium increases sharply during the later stages of
recessions but decreases just after recessions and that it tends to be higher during recessions than
during expansions.
To further investigate the opposite behavior of the momentum profit and the value
premium across business cycles, we examine the correlation coefficients between the expected
excess returns of WML and the growth rates of procyclical macroeconomic variables such as the
gross domestic product (GDP) and industrial production. They are positive: 0.12 and 0.13 for the
25
GDP and industrial production, respectively. On the other hand, the correlation coefficients between
the expected excess returns of HML and these two macroeconomic variables are negative: -0.21 and
-0.21, respectively. 14
4.4. Trading Rules Based on Out-of-Sample Predictions
To avoid potential problems from over-fitting a complex nonlinear model with a large number of
parameters being estimated as in this paper, it is necessary to examine out-of-sample predictability
of the model. Since the conditional mean equation (11) uses one-month–lagged predictive
macroeconomic variables, it can be used to predict the current month’s returns using conditioning
information available up to the previous month. We follow Perez-Quiros and Timmermann (2000)
and Gulen, Xing, and Zhang (2011) to do a recursive out-of-sample prediction of excess returns for
the loser and winner portfolios. Specifically, we first start to estimate our bivariate Markov
switching regression model by using observations from January 1954 to December 1976 and predict
the return for the next month (January 1977) based on the estimated parameters and the values of
the conditioning information on the most recent month (December 1976). 15 In this way, we reestimate the nonlinear model by adding one new month to the previous estimation window (starting
from January 1954) to ensure that we have enough in-sample observations to precisely estimate the
model and compute the predicted returns for each of the loser and winner portfolios. Consequently,
we obtain the predicted returns from January 1977 to December 2007.
Figure 3 plots the predicted excess returns obtained from the bivariate Markov switching
regression model for the loser portfolio (Panel A), the winner portfolio (Panel B), and the WML
14
Since the GDP is of quarterly frequency, the monthly excess returns are transformed into quarterly returns
by compounding monthly returns over each quarter.
15
The initial sample period from 1954 to 1976 follows from Perez-Quiros and Timmermann (2000) and
Gulen, Xing, and Zhang (2011).
26
portfolio (Panel C). For comparison, the out-of-sample predicted excess returns (with the solid line)
are overlaid with the in-sample predicted excess returns (with the dotted line). The in-sample
predictions are obtained from the one-time estimation of the same bivariate Markov switching
regression model using the whole-period observations from January 1954 to December 2007. The
out-of-sample predictions are highly correlated with the in-sample predictions. Their correlation
coefficients are 0.73 for the winner portfolio, 0.81 for the loser portfolio, and 0.80 for the winnerminus-loser portfolio. For the portfolios, the out-of-sample and in-sample predictions have similar
average returns and standard deviations. However, the out-of-sample predictions have slightly lower
average excess returns but slightly higher standard deviation than the in-sample predictions. For the
winner portfolio, the average returns are 0.93% and 1.16% for the out-of-sample and in-sample
predictions, respectively, while the standard deviations are 1.90% and 1.30%, respectively. For the
loser portfolio, the average returns are 0.14% and 0.18% for the out-of-sample and in-sample
predictions, respectively, while the standard deviations are 2.83% and 2.18%, respectively. For the
winner-minus-loser portfolio, the average returns are 0.79% and 0.98% for the out-of-sample and
in-sample predictions, respectively, while the standard deviations are 1.10% and 0.98%,
respectively.
The economic significance of the out-of-sample prediction can be measured by the
performance of a simple stylized trading rule based on the prediction. We follow the trading rule of
Perez-Quiros and Timmermann (2000), under which, if the predicted excess return is positive, we
take a long position in a given momentum portfolio under consideration (the loser or winner
portfolio) and otherwise switch the position into the one-month Treasury bill. Table 5 presents the
average returns, standard deviations, and Sharpe ratios over the whole period (Panel A) and the
NBER recession states (Panel B) and NBER expansion states (Panel C) for such switching
portfolios when the trading rule is based on the loser and winner portfolios, respectively. This table
27
also presents these return and risk characteristics for Treasury bills and the buy-and-hold strategy
that reinvests all funds in the portfolio under consideration. Table 5 shows that the economic
significance of out-of-sample predictability is particularly significant for the switching portfolio
based on the loser portfolio and during the recession state.
In the whole period (Panel A of Table 5), the switching portfolio based on the loser
portfolio outperforms the buy-and-hold strategy of the loser portfolio in terms of risk-return
characteristics (higher average return of 16.35% versus 9.10%, lower standard deviation of 20.58%
versus 28.29%, and thus a higher Sharpe ratio of 0.511 versus 0.115). However, this outperformance
of the switching portfolio over the buy-and-hold strategy is not apparently observed as much for the
winner portfolio as for the loser portfolio. In the recession states (Panel B of Table 5), the
outperformance of the switching portfolio over the buy-and-hold strategy is conspicuous for both
loser and winner portfolios. Panel B of Table 5 shows that the switching portfolio outperforms the
buy-and-hold strategy applied to both portfolios. For the loser portfolio, the average return, standard
deviation, and Sharpe ratio of the switching portfolio are 38.31%, 30.88%, and 0.962, respectively,
and the corresponding statistics of the buy-and-hold strategy are 5.85%, 40.42%, and -0.068,
respectively. For the winner portfolio, the switching portfolio similarly outperforms the buy-andhold strategy. In the expansion states (Panel C), however, the switching portfolio outperforms the
buy-and-hold strategy only for the loser portfolio. These two trading strategies perform similarly for
the winner portfolio.
4.5. Robustness Tests
Following Gray (1996), we use the one-month Treasury bill rate as the instrument in modeling the
state transition probabilities. Since the estimation results of the conditional mean equation may be
28
sensitive across states to the choice of the instrument variable in the state transition probabilities, it
would be necessary to conduct robustness tests by using alternative instruments instead of the
Treasury bill rate. We choose two alternative instruments in the state transition probabilities: One is
the two-month–lagged value of the year-on-year log difference in the Composite Leading Indicator
(𝛥𝐶𝐿𝐼𝑡−2 ), by following Perez-Quiros and Timmermann (2000), and the other is the one-month-
lagged monthly growth rate of industrial production (𝑀𝑃𝑡−1 ), defined as 𝑀𝑃𝑡 = 𝑙𝑛 𝐼𝑃𝑡 − 𝑙𝑛 𝐼𝑃𝑡−1 ,
where 𝐼𝑃𝑡 is the index level of industrial production at month 𝑡. 16
As in Table 2, Tables 6 and 7 present the estimation results of the univariate Markov
switching model for the 10 decile momentum portfolios when the instrumental variables in
modeling state transition probabilities are 𝛥𝐶𝐿𝐼𝑡−2 and 𝑀𝑃𝑡−1 , respectively. The overall results
from using these new instrumental variables are similar to those from Table 2, which uses the
Treasury bill rate as the instrumental variable. That is, the inferences from Table 2 are robust to
changes in the specification of the state transition probabilities. The null hypotheses of identical
asymmetries across states for loser and winner stocks are also not rejected at standard significance
levels in the slope coefficients on the four conditioning variables (not reported in the tables). 17
To further examine the procyclicality of momentum profits, we also estimate the univariate
Markov switching regression model as in Table 2 for 10 decile industry momentum portfolios. We
construct industry momentum portfolios by sorting 49 industry portfolios every month into one of
10 decile industry momentum portfolios based on the past six-month returns and holding the deciles
for six subsequent months. 18 We also skip one month between the end of the portfolio formation
period and the beginning of the holding period to avoid potential microstructure biases. All industry
16
17
18
The industrial production indexes are obtained from the Federal Reserve Bank of St. Louis.
The results are available upon request.
The returns on the 49 industry portfolios are obtained from French’s
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.
29
website
at
momentum portfolios in a given decile have equal weights. Table 8 presents the estimation results,
which are qualitatively similar to those in Table 2 obtained from using the momentum portfolios
with individual stocks. In particular, the signs and magnitudes of the coefficient estimates on the
conditioning macroeconomic variables for the winner-minus-loser industry momentum portfolio
(WML (industries)) are similar to those for the winner-minus-loser portfolio from individual stocks
(WML (stocks)) in Table 2.
5. A Plausible Explanation for Time-Varying Momentum Profits
We have shown that during the expansion state winner stocks are riskier than loser stocks, while
during the recession state loser stocks are riskier than winner stocks. Consequently, the expected
momentum profits display strong procyclical variations. We now examine the potential driving
sources of time-variations in expected momentum profits.
Other things being equal, firms with large recent positive price moves (winners) are more
likely to decrease their (financial) leverage than firms with large recent negative price moves
(losers). Hence, a momentum sort will tend to sort firms by recent leverage changes. Since higher
leverage implies higher systematic risk (Mandelker and Rhee, 1984), losers are riskier than winners;
hence momentum trading should have lower expected returns. With the presence of growth options,
however, winner stocks become riskier than loser stocks, as discussed in Section 2. Winner stocks
that have had recent good performance are more likely to increase the value of growth options than
loser stocks that have had recent bad performance. Since growth options are riskier than assets in
place, winners are riskier than losers and hence momentum trading should have higher expected
returns. Therefore, the riskiness and expected return of momentum portfolios result from the
relative importance of the leverage and growth options effect. During expansions, when growth
30
options have a higher effect and leverage has a lower effect, winners are riskier than losers.
Likewise, during recessions when growth options have a lower effect and leverage has a higher
effect, losers are riskier than winners.
To provide a plausible explanation for the time-varying momentum profits observed in the
previous section, it is necessary to show that the degree of growth options and leverage differ across
momentum portfolios and that macro-level leverage and growth options covary with the business
cycle. According to the above arguments, we expect winner stocks to have higher growth options
and lower leverage than loser stocks and aggregate leverage to be lower during expansions than
recessions, while aggregate growth options are expected to be higher during expansions than
recessions.
5.1. Momentum, Leverage, and Growth Options
This section examines how leverage and growth options differ across momentum portfolios. We use
the asset-to-equity and debt-to-equity ratios as proxies for leverage and the market-to-book equity
and market-to-book asset ratios as proxies for growth options. 19 To compute the asset-to-equity ratio
of a portfolio, we cross-sectionally aggregate the book values of assets and the market values of
equity of all firms included in the portfolio and compute the ratio of the aggregate book value of
assets to the aggregate market value of assets every month, when momentum portfolios are
rebalanced. Likewise, we compute the other ratios of the portfolio.
19
The asset-to-equity ratio is defined as the ratio of the book value of assets (Compustat annual item AT) to
the market value of equity. The debt-to-equity ratio is defined as the ratio of total assets minus book equity
(Compustat annual item CEQ) to market equity, following Bhandari (1988). Following Sagi and Seasholes
(2007), the market-to-book equity is defined as the ratio of market equity to book equity, and the market-tobook asset is defined as the ratio of the sum of book debt and market equity to the book value of assets, as in
Goyal, Lehn, and Racic (2002).
31
Table 9 presents the time-series averages of the asset-to-equity, debt-to-equity, market-tobook equity, and market-to-book asset ratios over the whole period from 1963 to 2007. Moving
from the loser portfolio to the winner portfolio, we observe a nearly monotonically decreasing
relation between past stock returns and the measures of leverage. The asset-to-equity ratio decreases
from 2.216 for loser stocks to 1.164 for winner stocks. We also observe a similar pattern in the debtto-equity ratio. In contrast, we observe an opposite pattern in the variables proxying for growth
options. The market-to-book equity ratio monotonically increases across momentum portfolios from
1.429 (the loser portfolio) to 2.681 (the winner portfolio). The market-to-book asset ratio also
monotonically increases across portfolios from 1.183 (the loser portfolio) to 1.716 (the winner
portfolio). The differences in the values of all four ratios between the loser and winner portfolios are
statistically significant at the 1% level.
To shed further light on the role of leverage and growth options in sorting momentum
portfolios, we examine how leverage and growth options evolve before and after portfolio
formation. To do this, we take the values of the four ratios proxying for leverage and growth options
over the period from 𝑡 − 36 months to 𝑡 + 36 months, where 𝑡 is the portfolio formation month
and varies from January 1966 to December 2004, and compute the averages over the period
(t − 36, t + 36). Figure 4 illustrates the values of the four proxy ratios of the loser and winner
portfolios over the period (t − 36, t + 36). It shows that the winner portfolio has lower values of the
leverage proxy variables (asset-to-equity and debt-to-equity ratios) and greater values of the growth
option proxy variables (market-to-book equity and market-to-book asset ratios) than the loser
portfolio does over the portfolio formation period (six months before portfolio formation). In fact,
the spread in the value of each proxy variable between the winner and loser portfolios sharply
32
increases over the portfolio formation period and peaks at the portfolio formation month (month 0).
The spread begins to decrease after the portfolio formation month but remains positive.
Overall, the results in Table 9 and Figure 4 show that sorting firms on past stock returns is
related to sorting firms on leverage and growth options.
5.2. Leverage and Growth Options across Business Cycles
To provide a plausible explanation for time-varying momentum profits over business cycles, it is
necessary to show that (macro-level) leverage and growth options covary with business cycles,
since leverage and growth options are implicit driving forces in sorting momentum portfolios.
Figure 5 plots the aggregate values of the two proxy variables for leverage (the asset-toequity ratio in Panel A and the debt-to-equity ratio in Panel B) and two other proxy variables for
growth options (the market-to-book ratio in Panel C and the market-to-book ratio in Panel D) along
with the NBER contraction period over 1963–2007. The aggregate leverage exhibits strong
countercyclical variation. The two leverage proxy variables (in Panels A and B) sharply increase
during recessions and tend to decrease during expansions. On the contrary, the aggregate growth
options exhibit strong procyclical variation. The two growth option proxy variables (in Panels C and
D) sharply decrease during recessions and tend to increase during expansions. Table 10 shows that
the averages of the aggregate leverage variables are higher during recessions than during
expansions (2.671 versus 2.036 for the asset-to-equity ratio and 0.685 versus 0.566 for the debt-toequity ratio), while the averages of the aggregate growth option variables are higher during
expansions than during recessions (2.104 versus 1.527 for the market-to-book equity ratio and 1.299
versus 1.139 for the market-to-book asset ratio). The differences in the averages between
expansions and recessions are all statistically significant at the 1% level.
33
The results in Figure 4 and Table 10 indicate that winner stocks are riskier during
expansions, since these stocks tend to have greater growth options and lower leverage during
expansions when growth options have a higher effect and leverage has a lower effect. Conversely,
loser stocks are riskier during recessions, since these stocks tend to have lower growth options and
greater leverage during recessions when growth options have a lower effect and leverage has a
higher effect.
6. Conclusion
We examine the procyclicality of momentum profits using the two-state Markov switching
regression framework of Perez-Quiros and Timmermann (2000) and find that momentum profits
display strong procyclical variation. Our results show that in the recession state loser stocks tend to
have greater loadings on the conditioning macroeconomic variables than winner stocks, while in the
expansion state winner stocks tend to have greater loadings on those variables than loser stocks. In
other words, in recessions loser (winner) stocks are most (least) strongly affected by aggregate
economic conditions, whereas in expansions winner (loser) stocks are most (least) strongly affected.
This indicates that returns on momentum portfolios react asymmetrically to the aggregate economic
conditions in recession and expansion states. This asymmetry across recession and expansion states
for loser stocks is identical to the asymmetry for winner stocks. This identical asymmetry for
winner and loser stocks is contrasted with the results reported by Perez-Quiros and Timmermann
(2000) for size portfolios and by Gulen, Xing, and Zhang (2011) for book-to-market portfolios.
Using conditioning variables similar to ours, these authors report that identical asymmetries for
small and large stocks and for growth and value stocks are strongly rejected.
34
To further confirm the procyclicality of momentum profits, we plot the momentum profit
estimated from the Markov switching regression model with NBER recession dates. The
momentum profit (or winner-minus-loser) tends to sharply decrease and have a negative value
during recessions but to increase and have a positive value just after recessions and during
expansions. It is higher in expansion periods and lower in recession periods. This procyclical timevarying behavior of the expected momentum profit is the opposite of the counter-cyclical behavior
of the value premium shown by Gulen, Xing, and Zhang (2011). The above results are robust to
using alternative instrumental variables in modeling state transition probabilities and to using
industry momentum portfolios as an alternative set of test assets.
We also examine the economic significance of out-of-sample predictability of the model
by setting up a simple stylized trading rule based on the prediction. Under this trading rule, if the
predicted excess return is positive, we take a long position in the loser or winner portfolio and
otherwise we switch the position into the one-month Treasury bill. The results show that the
economic significance of out-of-sample predictability is particularly significant for the switching
portfolio based on the loser portfolio and during the recession state.
The overall results indicate that the expected returns of winner stocks co-move more with
aggregate economic variables in expansion states than those of loser stocks and the expected
momentum profits display procyclical time-variations. The possible reason that winner stocks do
well in expansions is that they tend to have higher exposure to growth rate risk and more valuable
growth options in expansions than in recessions and thus should have higher expected returns in
expansions. We argue, therefore, that momentum profits are the realizations of such expected
returns and can be interpreted as the procyclicality premium.
35
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Table 1 Moments of Monthly Excess Returns for Ten Decile Momentum Portfolios
This table reports the mean, standard deviation, skewness, and kurtosis of excess returns (in percent) on
the momentum portfolios which are constructed in accordance with Jegadeesh and Titman (1993). That
is, all stocks are sorted every month into one of ten decile portfolios based on past six-month returns,
and held for six months. Excess returns are calculated as the difference between monthly stock returns
and the one-month Treasury bill rate. The data for the one-month Treasury bill rate are from Kenneth
French’s Web site. ‘WML’ indicates ‘Winner’ portfolio minus ‘Loser’ portfolio. 𝜌1 �𝑟𝑖𝑡2 � is the first-
order autocorrelation of the squared raw excess returns, and 𝜌1 �𝜀𝑖𝑡2 � he 1st-order autocorrelation of the
squared residuals from the conditional mean equation (11) in the Markov switching regression model.
Numbers in brackets indicate 𝑝-values of the first-order autocorrelations. The sample period is from
January 1954 to December 2007.
Momentum
portfolio
Mean
Standard
deviation
Skewness
Kurtosis
Loser
2
3
4
5
6
7
8
9
Winner
0.352
0.581
0.758
0.785
0.853
0.892
0.949
0.998
1.094
1.306
8.178
6.094
5.472
5.115
4.880
4.776
4.788
4.902
5.170
6.050
1.359
0.560
0.327
0.164
-0.116
-0.260
-0.420
-0.573
-0.713
-0.644
7.241
4.712
5.300
5.498
5.123
5.107
5.349
4.948
4.518
3.223
WML
0.954
5.226
-2.864
19.313
40
ρ1 �𝑟𝑖𝑡2 �
0.091 [0.021]
0.065 [0.101]
0.069 [0.078]
0.058 [0.138]
0.049 [0.218]
0.035 [0.379]
0.036 [0.367]
0.028 [0.471]
0.034 [0.393]
0.048 [0.225]
ρ1 �ε2𝑖𝑡 �
0.062 [0.117]
0.067 [0.090]
0.091 [0.021]
0.095 [0.016]
0.083 [0.035]
0.075 [0.056]
0.069 [0.080]
0.066 [0.095]
0.060 [0.129]
0.050 [0.204]
Table 2 Parameter Estimates for the Univariate Markov Switching Model of Excess Returns on Ten Decile Momentum
Portfolios
The following univariate two-state Markov switching model is estimated for excess returns on each momentum decile portfolio i:
𝑟𝑖𝑡 = 𝛽𝑖0,𝑆𝑡 + 𝛽𝑖1,𝑆𝑡 𝑇𝐵𝑡−1 + 𝛽𝑖2,𝑆𝑡 𝐷𝐸𝐹𝑡−1 + 𝛽𝑖3,𝑆𝑡 𝐼𝑁𝐹𝐿𝑡−1 + 𝛽𝑖4,𝑆𝑡 𝐷𝐼𝑉𝑡−1 + 𝜀𝑖𝑡
2
𝜀𝑖𝑡 ~𝑁(0, 𝜎𝑖,𝑆
), 𝑆𝑡 = {1, 2}
𝑡
𝑖
𝑖
𝑖
𝑖
𝑝𝑡 = 𝑃�𝑆𝑡 = 1�𝑆𝑡−1
= 1� = 𝛷�𝜋0𝑖 + 𝜋1𝑖 𝑇𝐵𝑡−1 �; 1 − 𝑝𝑡𝑖 = 𝑃�𝑆𝑡𝑖 = 2�𝑆𝑡−1
= 1�
𝑖
𝑖
𝑖
𝑖
𝑖
𝑖
𝑖
𝑖
𝑞𝑡 = 𝑃�𝑆𝑡 = 2�𝑆𝑡−1 = 2� = 𝛷�𝜋0 + 𝜋2 𝑇𝐵𝑡−1 �; 1 − 𝑞𝑡 = 𝑃�𝑆𝑡 = 1�𝑆𝑡−1
= 2�,
where 𝑟𝑖𝑡 is the monthly excess return for a given decile portfolio and 𝑆𝑡𝑖 is the regime indicator. TB is the one-month Treasury bill rate from the
Federal Reserve Economic Data (FRED) at Federal Reserve Bank of St. Louis, DEF is the spread between Moody's seasoned Baa-rated corporate bond
yield and 10-year treasury bond yield from the FRED, INFL is the 1-month log-difference in the CPI index (seasonally adjusted) from the BLS (Bureau
of Labor Statistics), and DIV is the sum of dividend payments accruing to the CRSP value-weighted market portfolio over the previous 12 months
divided by the contemporaneous level of the index. Numbers in parentheses are t-statistics (parameter estimates divided by standard errors). ‘WML’
indicates ‘Winner’ portfolio minus ‘Loser’ portfolio, and ‘HML’ is the highest book-to-market decile portfolio minus the lowest book-to-market decile
portfolio. The sample period is from January 1954 to December 2007.
Parameters
Momentum portfolio (𝑖)
Loser
2
Parameters in the conditional mean equation:
-0.189
-0.136
Constant, state 1 (𝛽𝑖0,1 )
(-2.95)
(-3.62)
-0.005
-0.002
Constant, state 2 (𝛽𝑖0,2 )
(-0.38)
(-0.16)
-0.444
-0.432
TB, state 1 (𝛽𝑖1,1 )
(-0.74)
(-1.16)
-0.410
-0.288
TB, state 2 (𝛽𝑖1,2 )
(-3.10)
(-2.61)
3
-0.121
(-3.49)
0.002
(0.11)
-0.382
(-0.99)
-0.243
(-1.61)
4
-0.098
(-3.81)
0.005
(0.52)
-0.440
(-1.61)
-0.263
(-2.92)
5
-0.086
(-3.84)
-0.012
(-1.16)
-0.932
(-3.43)
-0.097
(-0.99)
41
6
-0.085
(-3.82)
-0.010
(-1.00)
-0.907
(-3.41)
-0.107
(-1.12)
7
-0.079
(-3.55)
-0.009
(-0.78)
-0.868
(-3.26)
-0.081
(-0.81)
8
-0.083
(-0.61)
-0.007
(-0.12)
-0.816
(-0.59)
-0.167
(-0.23)
9
-0.081
(-3.09)
-0.006
(-0.62)
-0.698
(-2.65)
-0.224
(-2.20)
Winner
-0.047
(-1.85)
-0.003
(-0.16)
-0.508
(-1.62)
-0.341
(-2.46)
WML
HML
0.142
(1.45)
0.002
(0.08)
-0.064
(-0.06)
0.069
(0.27)
-0.079
(-1.14)
0.042
(0.78)
-0.119
(-0.25)
-0.398
(-3.09)
Table 2 (Continued)
DEF, state 1 (𝛽𝑖2,1 )
6.091
(3.44)
0.454
(0.92)
-6.985
(-1.29)
-0.832
(-0.70)
4.173
(2.83)
0.701
(2.26)
3.959
(3.91)
0.557
(1.22)
-3.224
(-1.24)
-0.586
(-0.57)
3.012
(3.55)
0.551
(2.06)
3.363
(3.95)
0.607
(1.51)
-2.361
(-1.11)
-0.941
(-0.57)
2.751
(3.71)
0.435
(1.14)
2.487
(3.76)
0.795
(2.18)
-1.733
(-1.01)
-1.781
(-2.13)
2.710
(3.35)
0.290
(1.43)
1.463
(2.44)
1.136
(3.21)
-0.931
(-0.65)
-2.035
(-2.47)
3.647
(4.16)
0.533
(2.51)
1.330
(2.21)
1.059
(2.92)
-0.802
(-0.57)
-2.196
(-2.73)
3.636
(4.16)
0.545
(2.62)
1.171
(1.85)
1.009
(2.61)
-0.699
(-0.49)
-2.314
(-2.85)
3.472
(3.90)
0.526
(2.33)
0.877
(0.43)
1.096
(1.07)
-0.927
(-0.13)
-2.441
(-3.02)
3.725
(1.88)
0.533
(0.77)
0.709
(1.05)
1.277
(3.48)
-1.298
(-0.68)
-3.052
(-3.50)
3.645
(3.90)
0.523
(2.66)
0.975
(1.29)
1.779
(2.88)
-0.348
(-0.17)
-4.599
(-4.04)
2.233
(3.00)
0.501
(1.55)
-5.116
(-1.78)
1.325
(1.33)
6.637
(0.68)
-3.766
(-1.77)
-1.939
(-0.83)
-0.200
(-0.33)
1.151
(0.61)
-0.032
(-0.05)
-0.036
(-0.01)
-1.090
(-0.85)
1.929
(1.40)
-0.945
(-2.35)
2.073
(6.32)
-22.619
(-2.73)
-12.214
(-1.66)
1.946
(6.21)
-10.831
(-1.66)
-7.351
(-1.25)
1.997
(5.27)
-9.203
(-1.12)
-7.518
(-1.08)
1.983
(6.44)
-6.835
(-1.39)
-7.402
(-1.39)
2.040
(6.97)
-4.220
(-0.92)
-5.409
(-1.07)
2.040
(6.92)
-4.098
(-0.87)
-5.103
(-0.98)
2.000
(6.31)
-3.514
(-0.70)
-4.674
(-0.84)
1.945
(1.04)
-3.839
(-0.12)
-2.349
(-0.04)
1.781
(4.91)
-0.215
(-0.03)
1.811
(0.21)
1.972
(4.85)
-7.298
(-1.10)
-8.433
(-1.20)
-0.102
0.410
15.322
-19.386
3.781
-6.337
0.130
0.087
0.075
0.067
0.062
0.061
0.061
0.064
0.067
0.077
-0.053
0.012
(12.82)
(14.29)
(12.53)
(15.73)
(16.57)
(16.53)
(16.40)
(3.08)
(14.73)
(14.03)
0.048
0.036
0.033
0.029
0.029
0.029
0.030
0.032
0.034
0.040
-0.008
0.012
(19.48)
(20.59)
(17.51)
(15.04)
(21.27)
(21.06)
(20.38)
(3.17)
(19.10)
(15.52)
DEF, state 2 (𝛽𝑖2,2 )
INFL, state 1 (𝛽𝑖3,1 )
INFL, state 2 (𝛽𝑖3,2 )
DIV, state 1 (𝛽𝑖4,1 )
DIV, state 2 (𝛽𝑖4,2 )
Transition probability parameters:
Constant (𝜋0𝑖 )
TB, state 1 (𝜋1𝑖 )
TB, state 2 (𝜋2𝑖 )
Standard deviations:
σ, state 1 (σ𝑖,1 )
σ, state 2 (σ𝑖,2 )
42
Table 3 Tests for Equality of the Slope Coefficients Across States in the Markov Switching Model
The following univariate Markov switching model is estimated for excess returns on each momentum decile portfolio i:
𝑟𝑖𝑡 = 𝛽𝑖0,𝑆𝑡 + 𝛽𝑖1,𝑆𝑡 𝑇𝐵𝑡−1 + 𝛽𝑖2,𝑆𝑡 𝐷𝐸𝐹𝑡−1 + 𝛽𝑖3,𝑆𝑡 𝐼𝑁𝐹𝐿𝑡−1 + 𝛽𝑖4,𝑆𝑡 𝐷𝐼𝑉𝑡−1 + 𝜀𝑖𝑡
2
),
𝜀𝑖𝑡 ~𝑁(0, 𝜎𝑖,𝑆
𝑡
𝑝𝑡𝑖
𝑞𝑡𝑖
=
=
𝑃�𝑆𝑡𝑖
𝑃�𝑆𝑡𝑖
=
=
𝑆𝑡 = {1, 2}
𝑖
1�𝑆𝑡−1
𝑖
2�𝑆𝑡−1
𝑖
= 1� = 𝛷�𝜋0𝑖 + 𝜋1𝑖 𝑇𝐵𝑡−1 �; 1 − 𝑝𝑡𝑖 = 𝑃�𝑆𝑡𝑖 = 2�𝑆𝑡−1
= 1�
𝑖
= 2� = 𝛷�𝜋0𝑖 + 𝜋2𝑖 𝑇𝐵𝑡−1 �; 1 − 𝑞𝑡𝑖 = 𝑃�𝑆𝑡𝑖 = 1�𝑆𝑡−1
= 2�,
where 𝑟𝑖𝑡 is the monthly excess return for a given decile portfolio and 𝑆𝑡𝑖 is the regime indicator. TB is the one-month Treasury bill rate from the
Federal Reserve Economic Data (FRED) at Federal Reserve Bank of St. Louis, DEF is the spread between Moody's seasoned Baa-rated corporate bond
yield and 10-year treasury bond yield from the FRED, INFL is the 1-month log-difference in the CPI index (seasonally adjusted) from the BLS (Bureau
of Labor Statistics), and DIV is the sum of dividend payments accruing to the CRSP value-weighted market portfolio over the previous 12 months
divided by the contemporaneous level of the index. Numbers in parentheses are t-statistics (parameter estimates divided by standard errors). The
likelihood ratio tests are conducted on the null hypothesis that the coefficient are equal across states, i.e., 𝛽𝑖𝑘,𝑆𝑡=1 = 𝛽𝑖𝑘,𝑆𝑡=2 for 𝑘 = 1, 2, 3, 4, for
momentum portfolio i. The sample period is from January 1954 to December 2007.
Loser
2
3
4
5
Unrestricted log likelihood value
814
992
1059
1102
1129
Restricted log likelihood with βik,St=1 = βik,St=2 for 𝑘 = 1, 2, 3, 4
808
985
1054
1096
1121
0.01
0.01
0.03
0.02
0.00
6
7
8
9
Winner
Unrestricted log likelihood value
1138
1132
1110
1069
952
Restricted log likelihood with βik,St=1 = βik,St=2 for 𝑘 = 1, 2, 3, 4
1130
1124
1102
1060
946
0.00
0.00
0.00
0.00
0.02
Momentum portfolio (𝑖)
p-value
p-value
43
Table 4 Estimation Results of the Bivariate Markov Switching Model for Excess
Returns to the Loser and Winner Portfolios
The bivariate Markov switching regression model for the loser and winner portfolios’ excess returns is
specified as follows:
𝒓𝑡 = 𝜷0,𝑆𝑡 + 𝜷1,𝑆𝑡 TB𝑡−1 + 𝜷2,𝑆𝑡 DEF𝑡−1 + 𝜷3,𝑆𝑡 TERM𝑡−1 + 𝜷4,𝑆𝑡 DIV𝑡−1 + 𝜺𝑡 ,
where 𝒓𝑡 = (𝑟𝑡𝐿 , 𝑟𝑡𝑊 )′ be a (2 × 1) vector consisting of excess returns on the loser and winner portfolios,
𝑊 ′
𝐿
𝑟𝑡𝐿 and 𝑟𝑡𝑊 , respectively, 𝜷𝑘,𝑆𝑡 is a (2 × 1) coefficient vector with elements �𝛽𝑘,𝑆
, 𝛽𝑘,𝑆
� for 𝑘 = 1, 2, 3, 4,
𝑡
𝑡
and 𝜺𝑡 ~𝑁(0, ∑𝑆𝑡 ), 𝑆𝑡 = {1, 2}. The conditional variance-covariance matrix, ∑𝑆𝑡 , has the following form:
ln (∑𝑖𝑖,𝑆𝑡 ) = 𝜆𝑆𝑖 𝑡 ,for 𝑖 = 𝑗 and ∑𝑖𝑗,𝑆𝑡 = 𝜌𝑆𝑡 (∑𝑖𝑖,𝑆𝑡 )1/2 (∑𝑗𝑗,𝑆𝑡 )1/2 for 𝑖 ≠ 𝑗. The transition probabilities are
defined as
𝑝𝑡 = 𝑃(𝑆𝑡 = 1|𝑆𝑡−1 = 1) = Φ(𝜋0 + 𝜋1 TB𝑡−1 ) and 𝑞𝑡 = 𝑃(𝑆𝑡 = 2|𝑆𝑡−1 = 2) = Φ(𝜋0 + 𝜋2 TB𝑡−1 ),
where Φ(∙) is the cumulative density function of a standard normal variable. TB is the one-month Treasury
bill rate from the Federal Reserve Economic Data (FRED) at Federal Reserve Bank of St. Louis, DEF is the
spread between Moody's seasoned Baa-rated corporate bond yield and 10-year treasury bond yield from the
FRED, INFL is the 1-month log-difference in the CPI index (seasonally adjusted) from the BLS (Bureau of
Labor Statistics), and DIV is the sum of dividend payments accruing to the CRSP value-weighted market
portfolio over the previous 12 months divided by the contemporaneous level of the index. Numbers in
parentheses are t-statistics (parameter estimates divided by standard errors). The sample period is from
January 1954 to December 2007.
Loser portfolio
(L)
Winner portfolio
(W)
Mean parameters:
Constant, state 1 (β0,1 )
Constant, state 2 (β0,2 )
-0.124 (-1.68)
-0.018 (-1.39)
-0.063 (-1.28)
-0.006 (-0.50)
TB, state 1 (β1,1 )
TB, state 2 (β1,2 )
-0.638 (-0.89)
-0.440 (-3.38)
-0.426 (-0.91)
-0.237 (-2.01)
4.744 ( 2.16)
0.727 ( 1.57)
2.603 ( 1.76)
0.590 ( 1.29)
INFL, state 1 (β3,1 )
INFL, state 2 (β3,2 )
-9.545 (-1.29)
-1.033 (-0.97)
-6.192 (-1.24)
-0.996 (-1.01)
4.394 ( 2.55)
0.911 ( 2.89)
2.426 ( 2.29)
0.771 ( 2.69)
DEF, state 1 (β2,1 )
DEF, state 2 (β2,2 )
DIV, state 1 (β4,1 )
DIV, state 2 (β4,2 )
Standard deviation parameters:
0.140 (12.36)
σ, state 1
0.052 (23.11)
σ, state 2
0.091
0.048
44
(12.70)
(24.03)
Tests for identical asymmetries
W
Constant: βL0,1 − βL0,2 = βW
0,1 − β0,2
Log likelihood value
p-value
W
W
L
L
TB: β1,1
− β1,2
= β1,1
− β1,2
Log likelihood value
p-value
W
DE βL2,1 − βL2,2 = βW
2,1 − β2,2 F:
Log likelihood value
p-value
W
INFL: βL3,1 − βL3,2 = βW
3,1 − β3,2
Log likelihood value
p-value
W
DIV: βL4,1 − βL4,2 = βW
4,1 − β4,2
Log likelihood value
p-value
2116
(0.35)
2116
(0.99)
2116
(0.19)
2116
(0.46)
2115
(0.12)
Table 4 (Continued)
Parameters common to both loser and winner portfolios
Correlation parameters:
ρ, state 1
ρ, state 2
Transition probability parameters:
Constant
TB, state 1
TB, state 2
Unconstrained log likelihood
0.710
0.850
(15.14)
(57.48)
1.905
-28.609
-11.171
(7.06)
(-4.48)
(-2.24)
2116
45
TB: π1 = π2
Log likelihood value
p-value
2107
(0.00)
Table 5 Trading Results Based on the Out-of-Sample Prediction from the Bivariate
Markov Switching Regression Model
The buy-and-hold strategy reinvests all funds in a given momentum under consideration (the loser or winner)
portfolio. The switching portfolios take a long position in the momentum portfolio if the excess return
recursively predicted from the bivariate Markov switching regression model is positive, otherwise the position
switches into the one-month Treasury bill. Average returns and standard deviations are annualized. The
sample period is from January 1977 to December 2007.
Loser portfolio
Treasury bill
Buy-and-hold
Winner portfolio
Switching
portfolio
Buy-and-hold
Switching
portfolio
Panel A: The whole period (January 1977 to December 2007)
Average return
Std dev of return
Sharpe ratio
5.83
0.26
9.10
28.29
0.115
16.35
20.58
0.511
22.04
20.71
0.783
21.52
13.94
1.126
20.16
26.62
0.435
33.35
18.67
1.326
22.28
19.86
0.846
19.98
13.17
1.102
Panel B: Recession states
Average return
Std dev of return
Sharpe ratio
8.59
0.34
5.85
40.42
-0.068
38.31
30.88
0.962
Panel C: Expansion states
Average return
Std dev of return
Sharpe ratio
5.47
0.23
9.53
26.38
0.154
46
13.48
18.74
0.428
Table 6 Parameter Estimates for the Univariate Markov Switching Model of Excess
Returns on Ten Decile Momentum:
△CLI As an Alternative Instrument in Modeling State Transition Probabilities
The following univariate Markov switching model was estimated for excess returns on each momentum
decile portfolio i:
𝑟𝑖𝑡 = 𝛽𝑖0,𝑆𝑡 + 𝛽𝑖1,𝑆𝑡 𝑇𝐵𝑡−1 + 𝛽𝑖2,𝑆𝑡 𝐷𝐸𝐹𝑡−1 + 𝛽𝑖3,𝑆𝑡 𝐼𝑁𝐹𝐿𝑡−1 + 𝛽𝑖4,𝑆𝑡 𝐷𝐼𝑉𝑡−1 + 𝜀𝑖𝑡
2
𝜀𝑖𝑡 ~𝑁(0, 𝜎𝑖,𝑆
), 𝑆𝑡 = {1, 2}
𝑡
𝑖
𝑖
𝑖
𝑖
𝑝𝑡 = 𝑃�𝑆𝑡 = 1�𝑆𝑡−1
= 1� = 𝛷�𝜋0𝑖 + 𝜋1𝑖 𝛥𝐶𝐿𝐼𝑡−2 �; 1 − 𝑝𝑡𝑖 = 𝑃�𝑆𝑡𝑖 = 2�𝑆𝑡−1
= 1�
𝑖
𝑖
𝑖
𝑖
𝑖
𝑖
𝑖
𝑖
𝑞𝑡 = 𝑃�𝑆𝑡 = 2�𝑆𝑡−1 = 2� = 𝛷�𝜋0 + 𝜋2 𝛥𝐶𝐿𝐼𝑡−2 �; 1 − 𝑞𝑡 = 𝑃�𝑆𝑡 = 1�𝑆𝑡−1 = 2�,
where 𝑟𝑖𝑡 is the monthly excess return for a given decile portfolio and 𝑆𝑡𝑖 is the regime indicator. TB is the
one-month Treasury bill rate from the Federal Reserve Economic Data (FRED) at Federal Reserve Bank of St.
Louis, DEF is the spread between Moody's seasoned Baa-rated corporate bond yield and 10-year treasury
bond yield from the FRED, INFL is the 1-month log-difference in the CPI index (seasonally adjusted) from
the BLS (Bureau of Labor Statistics), DIV is the sum of dividend payments accruing to the CRSP valueweighted market portfolio over the previous 12 months divided by the contemporaneous level of the index.
𝛥𝐶𝐿𝐼𝑡−2 is the two-month lagged value of the year-on-year log difference in the Composite Leading Indicator.
Numbers in parentheses are 𝑡-statistics (parameter estimates divided by standard errors). ‘WML’ indicates
‘Winner’ portfolio minus ‘Loser’ portfolio, and ‘HML’ is the highest book-to-market decile portfolio minus
the lowest book-to-market decile portfolio. The sample period is from March 1960 to December 2007.
Parameters
Momentum portfolio (𝑖)
Loser
3
5
7
9
Winner
WML
HML
Parameters in the conditional mean equation:
Constant, state 1 (𝛽𝑖0,1 )
Constant, state 2 (𝛽𝑖0,2 )
TB, state 1 (𝛽𝑖1,1 )
TB, state 2 (𝛽𝑖1,2 )
DEF, state 1 (𝛽𝑖2,1 )
DEF, state 2 (𝛽𝑖2,2 )
INFL, state 1 (𝛽𝑖3,1 )
INFL, state 2 (𝛽𝑖3,2 )
-0.104
(-2.92)
0.023
(1.52)
-1.235
(-2.84)
0.057
(0.28)
4.154
(3.49)
-1.158
(-1.17)
-2.316
(-0.83)
-1.492
(-0.98)
-0.093
(-4.14)
0.032
(3.10)
-0.877
(-3.09)
0.102
(0.77)
3.205
(4.60)
-1.183
(-2.48)
-1.091
(-0.63)
-1.049
(-1.07)
-0.090
(-3.81)
0.001
(0.08)
-0.959
(-3.45)
-0.060
(-0.49)
1.645
(2.63)
0.547
(1.18)
-0.910
(-0.61)
-1.730
(-1.93)
47
-0.083
(-3.53)
0.002
(0.11)
-0.880
(-3.19)
-0.063
(-0.45)
1.296
(1.97)
0.555
(1.22)
-0.623
(-0.42)
-2.200
(-2.50)
-0.087
(-3.06)
0.001
(0.11)
-0.755
(-2.78)
-0.194
(-1.44)
0.914
(1.16)
0.822
(1.73)
-0.737
(-0.39)
-3.151
(-3.10)
-0.039
(-1.69)
-0.007
(-0.49)
-0.572
(-2.29)
-0.273
(-1.20)
0.749
(1.01)
1.755
(3.10)
0.099
(0.05)
-4.351
(-3.20)
0.065
(1.02)
-0.030
(1.00)
0.663
(0.81)
-0.331
(-0.91)
-3.404
(-1.65)
2.914
(2.06)
2.415
(0.46)
-2.859
(-1.08)
-0.089
(-2.33)
0.050
(1.39)
-0.246
(-0.63)
-0.460
(-4.32)
1.254
(1.17)
-0.419
(-0.61)
-0.174
(-0.07)
-1.137
(-0.84)
Table 6 (Continued)
DIV, state 1 (𝛽𝑖4,1 )
DIV, state 2 (𝛽𝑖4,1 )
3.777
(3.68)
-0.214
(-0.37)
3.028
(4.77)
-0.283
(-0.85)
3.746
(4.16)
0.306
(1.00)
3.555
(3.91)
0.367
(1.10)
3.709
(3.99)
0.583
(1.92)
2.048
(2.85)
0.712
(1.80)
0.930
(4.07)
-28.090
(-2.19)
6.488
(0.63)
1.077
(6.15)
-24.733
(-2.46)
4.450
(0.44)
1.589
(9.38)
-13.278
(-0.88)
-1.925
(-0.13)
1.636
(9.34)
-10.095
(-0.61)
-5.856
(-0.36)
1.653
(9.67)
-3.997
(-0.25)
-9.861
(-0.67)
1.516
(8.08)
-0.366
(-0.03)
-28.391
(-1.49)
0.110
0.070
0.063
0.062
0.067
0.075
(16.16)
(17.94)
(16.01)
(14.60)
(14.71)
(16.97)
0.042
(11.60)
690
0.029
(14.14)
913
0.030
(17.21)
971
0.030
(15.24)
978
0.035
(16.52)
921
0.040
(14.32)
814
-1.729
2.386
(1.00) (2.27)
0.926 -0.779
(1.01) (-1.31)
Transition probability parameters
Constant (𝜋0𝑖 )
CLI, state 1 (𝜋1𝑖 )
CLI, state 2 (𝜋2𝑖 )
0.586
0.410
27.725
-19.386
-34.879
-6.337
-0.035
0.012
-0.003
0.012
Standard deviations
σ, state 1 (σ𝑖,1 )
σ, state 2 (σ𝑖,2 )
Log likelihood value
48
Table 7 Parameter Estimates for the Univariate Markov Switching Model of Excess
Returns on Ten Decile Momentum Portfolios:
𝐌𝐏 As an Alternative Instrument in Modeling State Transition Probabilities
The following univariate Markov switching model was estimated for excess returns on each momentum
decile portfolio i:
𝑟𝑖𝑡 = 𝛽𝑖0,𝑆𝑡 + 𝛽𝑖1,𝑆𝑡 𝑇𝐵𝑡−1 + 𝛽𝑖2,𝑆𝑡 𝐷𝐸𝐹𝑡−1 + 𝛽𝑖3,𝑆𝑡 𝐼𝑁𝐹𝐿𝑡−1 + 𝛽𝑖4,𝑆𝑡 𝐷𝐼𝑉𝑡−1 + 𝜀𝑖𝑡
2
𝜀𝑖𝑡 ~𝑁(0, 𝜎𝑖,𝑆
), 𝑆𝑡 = {1, 2}
𝑡
𝑖
𝑖
𝑖
𝑖
𝑝𝑡 = 𝑃�𝑆𝑡 = 1�𝑆𝑡−1
= 1� = 𝛷�𝜋0𝑖 + 𝜋1𝑖 𝑀𝑃𝑡−1 �; 1 − 𝑝𝑡𝑖 = 𝑃�𝑆𝑡𝑖 = 2�𝑆𝑡−1
= 1�
𝑖
𝑖
𝑖
𝑖
𝑖
𝑖
𝑖
𝑖
𝑞𝑡 = 𝑃�𝑆𝑡 = 2�𝑆𝑡−1 = 2� = 𝛷�𝜋0 + 𝜋2 𝑀𝑃𝑡−1 �; 1 − 𝑞𝑡 = 𝑃�𝑆𝑡 = 1�𝑆𝑡−1 = 2�,
where 𝑟𝑖𝑡 is the monthly excess return for a given decile portfolio and 𝑆𝑡𝑖 is the regime indicator. TB is the
one-month Treasury bill rate from the Federal Reserve Economic Data (FRED) at Federal Reserve Bank of St.
Louis, DEF is the spread between Moody's seasoned Baa-rated corporate bond yield and 10-year treasury
bond yield from the FRED, INFL is the 1-month log-difference in the CPI index (seasonally adjusted) from
the BLS (Bureau of Labor Statistics), DIV is the sum of dividend payments accruing to the CRSP valueweighted market portfolio over the previous 12 months divided by the contemporaneous level of the index.
MP is the monthly growth rate of industrial production. Numbers in parentheses are t-statistics (parameter
estimates divided by standard errors). ‘WML’ indicates ‘Winner’ portfolio minus ‘Loser’ portfolio, and ‘HML’
is the highest book-to-market decile portfolio minus the lowest book-to-market decile portfolio. The sample
period is from January 1954 to December 2007.
Parameters
Momentum portfolio (𝑖)
Loser
3
5
7
9
Winner
WML
HML
Parameters in the conditional mean equation:
Constant, state 1 (𝛽𝑖0,1 )
Constant, state 2 (𝛽𝑖0,2 )
TB, state 1 (𝛽𝑖1,1 )
TB, state 2 (𝛽𝑖1,2 )
DEF, state 1 (𝛽𝑖2,1 )
DEF, state 2 (𝛽𝑖2,2 )
INFL, state 1 (𝛽𝑖3,1 )
-0.146
(-3.04)
0.007
(0.51)
-0.721
(-1.50)
-0.274
(-2.14)
5.064
(3.74)
-0.099
(-0.17)
-3.050
(-0.90)
-0.118
(-3.98)
0.007
(0.77)
-0.405
(-1.32)
-0.221
(-2.36)
3.307
(4.12)
0.377
(0.92)
-2.024
(-1.02)
-0.095
(-3.80)
-0.006
(-0.60)
-0.972
(-3.35)
-0.130
(-1.39)
1.585
(2.50)
0.939
(2.65)
-0.802
(-0.52)
49
-0.088
(-3.37)
-0.001
(-0.12)
-0.872
(-2.98)
-0.129
(-1.27)
1.347
(2.04)
0.796
(2.20)
-0.579
(-0.36)
-0.080
(-3.00)
-0.007
(-0.70)
-0.768
(-2.63)
-0.203
(-2.16)
0.720
(1.00)
1.240
(3.25)
-1.031
(-0.54)
-0.038
(-1.53)
-0.009
(-0.76)
-0.580
(-2.05)
-0.311
(-2.51)
0.817
(1.12)
1.875
(3.69)
-0.576
(-0.28)
0.108
(1.55)
-0.016
(-0.64)
0.142
(0.19)
-0.038
(-0.16)
-4.248
(-2.07)
1.974
(1.95)
2.473
(0.46)
-0.089
(-1.64)
0.041
(0.78)
-0.172
(-0.46)
-0.360
(-3.53)
1.536
(1.09)
0.315
(0.05)
0.549
(0.187)
Table 7 (Continued)
INFL, state 2 (𝛽𝑖3,2 )
-1.428
(-1.19)
3.426
(3.19)
0.424
(1.31)
-1.235
(-1.43)
2.759
(4.08)
0.318
(1.41)
-1.982
(-2.50)
3.886
(4.31)
0.432
(2.18)
-2.264
(-2.89)
3.642
(4.04)
0.393
(1.85)
-3.099
(-3.49)
3.694
(3.71)
0.536
(2.68)
-4.263
(-3.81)
2.214
(3.05)
0.588
(2.20)
1.476
(8.96)
111.546
(-4.09)
9.160
(0.48)
1.580
(8.43)
1.801
(10.32)
1.833
(9.71)
1.884
(9.06)
1.826
(5.79)
0.350
0.410
-87.749
-68.164
-66.918
-54.479
-53.979
57.567
-19.386
(-3.28)
9.440
(0.45)
(-2.29)
-1.339
(-0.06)
(-2.23)
-6.184
(-0.26)
(-1.60)
1.058
(0.05)
(-1.49)
1.983
(0.08)
-7.177
-6.337
0.121
0.076
0.065
0.063
0.068
0.078
-0.044
0.012
(15.19)
(16.26)
(14.77)
(14.55)
(13.46)
(14.93)
0.047
0.033
0.030
0.030
0.035
0.041
-0.006
0.012
(19.28)
(18.91)
(19.02)
(18.08)
(16.61)
(14.07)
816
1062
1130
1133
1070
952
DIV, state 1 (𝛽𝑖4,1 )
DIV, state 2 (𝛽𝑖4,1 )
-2.835 -0.941
(-1.32) (-1.11)
-1.212
2.117
(-0.74) (1.88)
0.164 -0.941
(0.28) (-2.90)
Transition probability parameters
Constant (𝜋0𝑖 )
MP, state 1 (𝜋1𝑖 )
MP, state 2
(𝜋2𝑖 )
Standard deviations:
σ, state 1 (σ𝑖,1 )
σ, state 2 (σ𝑖,2 )
Log likelihood value
50
Table 8 Parameter Estimates for the Univariate Markov Switching Model of Excess
Returns on Ten Decile Portfolios Formed on Industry Momentum
The following univariate Markov switching model is estimated for excess returns on each industry
momentum decile portfolio i:
𝑟𝑖𝑡 = 𝛽𝑖0,𝑆𝑡 + 𝛽𝑖1,𝑆𝑡 𝑇𝐵𝑡−1 + 𝛽𝑖2,𝑆𝑡 𝐷𝐸𝐹𝑡−1 + 𝛽𝑖3,𝑆𝑡 𝐼𝑁𝐹𝐿𝑡−1 + 𝛽𝑖4,𝑆𝑡 𝐷𝐼𝑉𝑡−1 + 𝜀𝑖𝑡
2
𝜀𝑖𝑡 ~𝑁(0, 𝜎𝑖,𝑆
), 𝑆𝑡 = {1, 2}
𝑡
𝑖
𝑖
𝑖
𝑖
𝑝𝑡 = 𝑃�𝑆𝑡 = 1�𝑆𝑡−1
= 1� = 𝛷�𝜋0𝑖 + 𝜋1𝑖 𝑇𝐵𝑡−1 �; 1 − 𝑝𝑡𝑖 = 𝑃�𝑆𝑡𝑖 = 2�𝑆𝑡−1
= 1�
𝑖
𝑖
𝑖
𝑖
𝑖
𝑖
𝑖
𝑖
𝑞𝑡 = 𝑃�𝑆𝑡 = 2�𝑆𝑡−1 = 2� = 𝛷�𝜋0 + 𝜋2 𝑇𝐵𝑡−1 �; 1 − 𝑞𝑡 = 𝑃�𝑆𝑡 = 1�𝑆𝑡−1
= 2�,
where 𝑟𝑖𝑡 is the monthly excess return for a given decile portfolio and 𝑆𝑡𝑖 is the regime indicator. TB is the
one-month Treasury bill rate from the Federal Reserve Economic Data (FRED) at Federal Reserve Bank of St.
Louis, DEF is the spread between Moody's seasoned Baa-rated corporate bond yield and 10-year treasury
bond yield from the FRED, INFL is the 1-month log-difference in the CPI index (seasonally adjusted) from
the BLS (Bureau of Labor Statistics), DIV is the sum of dividend payments accruing to the CRSP valueweighted market portfolio over the previous 12 months divided by the contemporaneous level of the index.
Numbers in parentheses are t-statistics (parameter estimates divided by standard errors). ‘WML’ indicates
‘Winner’ minus ‘Loser’ portfolio constructed from using industry portfolios, ‘WML (individuals)’ is ‘Winner’
minus ‘Loser’ portfolio constructed from using individual stocks, and ‘HML’ is the highest book-to-market
decile portfolio minus the lowest book-to-market decile portfolio. The sample period is from January 1954 to
December 2007.
Parameters
Loser
Momentum portfolio (𝑖)
3
5
7
Winner
WML
(Industries)
WML
(Stocks)
0.043
(0.78)
0.021
(1.02)
0.332
(0.52)
0.037
(0.19)
-1.339
(-0.76)
-0.170
(-0.21)
4.205
(0.91)
-3.874
(-2.12)
0.142
(1.45)
0.002
(0.08)
-0.064
(-0.06)
0.069
(0.27)
-5.116
(-1.78)
1.325
(1.33)
6.637
(0.68)
-3.766
(-1.77)
HML
Parameters in the conditional mean equation:
Constant, state 1 (𝛽𝑖0,1 )
Constant, state 2 (𝛽𝑖0,2 )
TB, state 1 (𝛽𝑖1,1 )
TB, state 2 (𝛽𝑖1,2 )
DEF, state 1 (𝛽𝑖2,1 )
DEF, state 2 (𝛽𝑖2,2 )
INFL, state 1 (𝛽𝑖3,1 )
INFL, state 2 (𝛽𝑖3,2 )
-0.088
(-2.70)
-0.021
(-2.07)
-0.820
(-2.15)
-0.259
(-2.56)
2.977
(3.04)
1.130
(2.79)
-2.874
(-1.11)
-0.434
(-0.48)
-0.094
(-3.97)
0.010
(0.91)
-0.436
(-1.62)
-0.430
(-4.54)
2.532
(3.82)
0.902
(1.70)
-1.876
(-1.11)
-1.962
(-2.16)
-0.095
(-4.13)
0.008
(0.80)
-0.572
(-2.30)
-0.369
(-3.63)
2.226
(3.43)
0.881
(1.96)
-1.614
(-0.97)
-2.310
(-2.48)
51
-0.087
(-3.79)
0.009
(0.83)
-0.550
(-2.18)
-0.307
(-3.03)
2.053
(3.22)
0.910
(1.96)
-1.737
(-1.04)
-2.607
(-2.66)
-0.045
(-1.54)
0.000
(0.03)
-0.489
(-1.30)
-0.222
(-2.20)
1.638
(1.59)
0.960
(2.23)
1.331
(0.48)
-4.308
(-4.48)
-0.079
(-1.14)
0.042
(0.78)
-0.119
(-0.25)
-0.398
(-3.09)
1.150
(0.61)
-0.032
(-0.05)
-0.036
(-0.01)
-1.090
(-0.85)
Table 8 (Continued)
DIV, state 1 (𝛽𝑖4,1 )
DIV, state 2 (𝛽𝑖4,1 )
2.529
(2.67)
0.802
(3.51)
2.612
(3.34)
0.254
(1.14)
3.068
(4.64)
0.311
(1.44)
2.964
(4.46)
0.276
(1.20)
1.703
(2.19)
0.588
(2.47)
-0.825
(-0.56)
-0.214
(-0.46)
-1.939
(-0.83)
-0.200
(0.33)
1.929
(1.39)
-0.945
(-2.35)
2.113
(7.45)
-9.565
(-2.05)
-6.890
(-1.53)
1.965
(4.64)
-6.306
(-3.28)
-7.951
(-1.34)
1.977
(5.39)
-5.521
(-2.29)
-7.203
(-1.15)
1.888
(4.65)
-3.534
(-2.23)
-4.330
(-0.59)
1.362
(4.50)
1.178
(-1.60)
6.446
(1.06)
-0.751
-0.102
0.410
10.744
15.322
-19.386
13.335
3.781
-6.337
0.096
0.068
0.067
0.066
0.084
-0.011
-0.053
0.012
(15.19)
(16.26)
(14.77)
(14.55)
(13.46)
(14.93)
0.038
0.032
0.033
0.034
0.039
0.001
-0.008
0.012
(21.92)
(12.72)
(20.36)
(20.43)
(18.86)
(17.57)
967
1050
1055
1047
1022
971
Transition probability parameters
Constant (𝜋0𝑖 )
TB, state 1 (𝜋1𝑖 )
TB, state 2 (𝜋2𝑖 )
Standard deviations:
σ, state 1 (σ𝑖,1 )
σ, state 2 (σ𝑖,2 )
Log likelihood value
52
Table 9 Averages of the Financial Ratios Proxying for Leverage and Growth Options
for Ten Decile Momentum Portfolios
This table presents the time-series averages of the financial ratios proxying for leverage (the asset-toequity and debt-to-equity ratios) and for growth options (the market-to-book equity and market-to-book
asset ratios) of each momentum portfolio over the whole period from 1963 to 2007. The asset-to-equity
ratio is defined as the ratio of the book value of assets to the market value of equity. The debt-to-equity
ratio is defined as the ratio of total assets minus book equity to market equity. The market-to-book equity
is defined as the ratio of market equity to book equity, and the market-to-book asset is defined as the
ratio of the sum of book debt and market equity to the book value of asset. Every month when
momentum portfolios are rebalanced, each ratio (A/B) of the portfolio is computed as the ratio of the
aggregate value of accounting variable A to the aggregate value of accounting variable B of all firms
included in the portfolio. ‘WML’ indicates ‘Winner’ portfolio minus ‘Loser’ portfolio.
Leverage
Loser
2
3
4
5
6
7
8
9
Winner
WML
(t-statistic)
Asset-to
-equity
2.216
1.687
1.513
1.385
1.340
1.280
1.219
1.190
1.184
1.164
Debt-to
-equity
0.778
0.526
0.443
0.394
0.379
0.361
0.344
0.340
0.348
0.360
-1.051
(-27.10)
-0.418
(-24.25)
Growth Options
Market-toMarket-tobook equity
book asset
1.429
1.183
1.653
1.304
1.731
1.347
1.854
1.409
1.907
1.431
2.001
1.473
2.146
1.534
2.242
1.572
2.385
1.610
2.681
1.716
1.252
0.532
(27.37)
(25.78)
53
Table 10 Averages of the Financial Ratios Proxying for Leverage and Growth Options
Across Business Cycles
This table presents the time-series averages of the aggregate financial ratios proxying for leverage (the
asset-to-equity and debt-to-equity ratios) and for growth options (the market-to-book equity and marketto-book asset ratios) across business cycles over the whole period from 1963 to 2007. The asset-toequity ratio is defined as the ratio of the book value of assets to the market value of equity. The debt-toequity ratio is defined as the ratio of total assets minus book equity to the market equity. The market-tobook equity is defined as the ratio of market equity to book equity, and the market-to-book asset is
defined as the ratio of the sum of book debt and market equity to the book value of asset. Every month
when momentum portfolios are rebalanced, each ratio (A/B) of the portfolio is computed as the ratio of
the aggregate value of accounting variable A to the aggregate value of accounting variable B of all firms.
Recession and expansion periods are based on historical NBER business cycle dates.
Leverage
Expansion
Recession
Asset-to
-equity
2.036
2.671
Debt-to
-equity
0.566
0.685
Expansion-Recession
(t-statistic)
-0.635
(-6.18)
-0.119
(-5.09)
54
Growth Options
Market-to
Market-to
-book equity
-book asset
2.104
1.299
1.527
1.139
0.576
(7.43)
0.159
(8.35)
Figure 1 Time-Series of the Probability of Being in High and Low Volatility States for
the Winner and Loser portfolios
For the winner portfolio, time series of the probability of being in state 1 (high volatility; Panel A) and state 2
(low volatility; Panel B) at time t conditional on information in period t − 1 in the univariate Markov
switching model are plotted. Similarly, for the loser portfolio, time series of the probability of being in state 1
(Panel C) and state 2 (Panel D) are plotted. Shaded areas indicate NBER recession periods.
Panel A: Winner, High Volatility (State 1)
Panel B: Winner, Low Volatility (State 2)
55
Figure 1 (Continued)
Panel C: Loser, High Volatility (State 1)
Panel D : Loser, Low Volatility (State 2)
56
Figure 2 Expected Excess Returns from Univariate and Bivariate Markov Switching
Models
The expected excess returns for the winner portfolio (Panel A), the loser portfolio (Panel B), and the
winner-minus-loser (Panel C) obtained from the univariate and bivariate Markov switching models
are plotted over time. The solid lines denote the expected excess returns obtained from the bivariate
Markov switching model, and the dashed lines denote the expected excess returns obtained from the
univariate model. Shaded areas indicate NBER recession periods.
Panel A: Winner portfolio
57
Figure 2 (Continued)
Panel B: Loser portfolio
Panel C: Winner-minus-Loser
58
Figure 3 Predicted Excess Returns from the Bivariate Markov Switching Model
The predicted excess returns for the loser portfolio (Panel A), the winner portfolio (Panel B), and
the winner-minus-loser portfolio (Panel C) are obtained from the bivariate Markov switching
regression model. The solid lines plot the in-sample predicted excess returns, and the dotted lines
plot the out-of-sample predicted excess returns. The out-of-sample forecasts are from January 1977
to December 2007.
Panel A: Winner portfolio
59
Figure 3 (Continued)
Panel B: Loser portfolio
Panel C: Winner-minus-Loser portfolio
60
Figure 4 Leverage and Growth Options of Loser and Winner Stocks Before and After
Portfolio Formation
For the loser and winner portfolios, we take the values of two ratios proxying for leverage (the
asset-to-equity and debt-to-equity ratios) and two ratios proxying for growth options (the market-tobook equity (ME/BE) and market-to-book asset ratios (MA/BA)) over the period from 𝑡 − 36
months to 𝑡 + 36 months, where 𝑡 is the portfolio formation month and it varies from January
1966 to December 2004. Then, we compute the averages over the period (t − 36, t + 36). ‘Month 0’
is the portfolio formation month.
Panel B: Debt-to-Equity (Leverage)
Panel A: Asset-to-Equity (Leverage)
Panel C: Market-to-Book Equity (Growth Options)
Panel D: Market-to-Book Asset (Growth Options)
61
Figure 5 Aggregate Leverage and Growth Options Across Business Cycles
These figures plot the time-series averages of the aggregate financial ratios proxying for leverage
(the asset-to-equity and debt-to-equity ratios) and for growth options (the market-to-book equity
(ME/BE) and market-to-book asset (MA/BA) ratios) across business cycles over the whole period
from 1963 to 2007. Shaded areas indicate NBER recession period.
Panel A: Asset-to-Equity (Leverage)
Panel B: Debt-to-Equity (Leverage)
Panel C: Market-to-Book Equity (Growth Options)
Panel D: Market-to-Book Asset (Growth Options)
62