Time-Varying Expected Momentum Profits
Dongcheol Kim*
Tai-Yong Roh†
Byoung-Kyu Min‡
Suk-Joon Byun§
This draft: April 2013
Abstract:
We examine 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,
whereas in the recession state the expected returns of loser stocks are more affected. Consequently,
expected momentum profits display strong procyclical variations. 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
We examine 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,
whereas in the recession state the expected returns of loser stocks are more affected. Consequently,
expected momentum profits display strong procyclical variations. 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
1
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.
2
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 according to the theory of
Gomes, Kogan, and Zhang (2003) and Zhang (2005). These authors find that the recent crosssectional 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.
3
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 growth 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 shows 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 (i.e., the
4
trough of the business cycle). The above two findings are robust to using alternative instrumental
variables in modeling state transition probabilities. 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 4 describes 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.
5
2. Sources of Time-Varying Expected Momentum Profits
In his theoretical model, Johnson (2002) argues that stock prices are a 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) show that 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 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.
6
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.
7
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
be the return of a test asset in excess of the riskless return at time
vector of conditioning variables available up to time
and let
used to predict
be a
. 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,
where
. Specifically,
denotes a normal distribution with mean zero and variance
Markov switching specification,
are either
or
. In a two-state
, meaning that the parameters to be estimated
or
.
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:
8
where
is a vector of variables publicly available at time
probabilities between times
and
and affects the state transition
. 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,
, 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
denote the vector of parameters to be estimated in the likelihood
function. The probability density function of the return, conditional on being state , is Gaussian
defined as
for
. The information set
contains
,
,
, and lagged values of these
variables. Then, the log-likelihood function is
9
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).
9
where the density function
is simply obtained by summing the probability-weighted
state probabilities across the two states. It is defined as
where
information at time
is the conditional probability of being in state
at time
given
. The conditional state probabilities can be obtained from the standard
probability theorem:
By Bayes‟ rule, the conditional state probabilities can be written as
The conditional state probabilities
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 return) on the
momentum decile portfolios as test assets. Momentum portfolios are constructed in accordance with
Jegadeesh and Titman (1993) by sorting all stocks every month into one of 10 decile portfolios
10
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 match the beginning of the sample period with Gulen, Xing,
and Zhang (2011): Our sample period is from March 1960 to December 2012.
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 0.369% per month for the past loser portfolio (Portfolio 1) to 1.127% per month for the past
winner portfolio (Portfolio 10). The mean return on the WML is quite significant: 0.758% per
month. A distinct pattern is found in skewness, which almost monotonically decreases from 1.375
for the loser portfolio to -0.661 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. Loser portfolios tend to have greater kurtosis than do winner portfolios.
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 relative three-month Treasury bill rate,
the default spread, the growth in the monetary base, and the dividend yield. These variables are
11
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 Treasury bill rate 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).
12
The growth in the money base (MB) is defined as the 12-month log-difference in the
monetary base reported by the St. Louis Federal Reserve. This variable is included in the
conditional mean equation, since this variable affects stock returns through changes in macro
liquidity (or money flow liquidity) and eventually micro liquidity (or transaction liquidity) in stock
markets.10 This variable also affects stock returns through shocks in monetary policies that can
affect aggregate economic conditions. In particular, Fama (1981) argues that it is important to
control for money supply when establishing the inflation-future real economic activity proxy story.
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
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).11
10
Instead of the growth in monetary base, we also include the inflation rate. However, the results are
qualitatively similar. The reason that the inflation rate can be included is that since 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). 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 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.
11
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.
13
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:
where
is the monthly excess return for the
decile momentum portfolio at time ,
normally distributed random error term with mean zero and variance
, and
regressors are lagged by one month. The conditional variance of excess returns,
is the
. The
, is allowed to
depend only on the state of economy:
We do not include autoregressive conditional heteroskedasticity (ARCH) effects in the conditional
volatility equation. Table 1 shows the first-order autocorrelations of the raw excess returns and the
squared raw excess returns in each of the 10 decile momentum portfolios. All ten portfolios exhibit
a significant positive first-order autocorrelation at the one percent level. Only six squared raw
excess returns out of the ten 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.
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:
14
where
is the relative three-month Treasury bill rate calculated as the difference between
the current Treasury bill rate and its 12-month backward moving average, and
is the
cumulative probability density function of a standard normal variable.12 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) for portfolios
P1 (loser), P2, P4, P6, P8, and P10 (winner).13 The constant parameter estimates in the conditional
mean equation in state 1 (
are much lower than those in state 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 estimate in state 1 (
greater than that in state 2 (
) is
) in all portfolios. All conditional volatilities are highly significantly
estimated. Schwert (1990) 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
12
Instead of the relative Treasury bill rate, we also use the one-month Treasury bill rate in the state transition
probability equation. However, the results are quite similar. The results are available upon request.
13
The estimation results for portfolios P3, P5, P7, and P9 are not reported because of space constraint. The
results are available upon request.
15
Bureau of Economic Research (NBER) business cycle dates. These results may 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
information set at time
,
and
conditional on the
, 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
2 appears obviously opposite. In sum, the results suggest that state 1 can be regarded as the
recession state and state 2 can be regarded as the expansion state.
4.2.2. Estimation of the Conditional Mean Equations
The estimation results of the conditional mean equation (11) for portfolios P1 (loser), P2, P4, P6, P8,
and P10 (winner) are presented in Table 2. The difference in the coefficient estimates between the
winner (P10) and loser (P1) portfolios is reported in the column under WML. We also estimate the
same Markov switching regression model for each of 10 decile book-to-market portfolios.14 We
14
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).
16
report only the difference in the coefficient estimates between the value (highest book-to-market)
portfolio and the growth (lowest book-to-market) portfolio in the column under HML in Table 2 to
compare the results of WML with those of HML.
The coefficient estimates on the relative three-month Treasury bill rate (RREL) are all
negative for the 10 momentum portfolios in both states 1 and 2 (
and
), which means that
when the short-term interest rate increases (relative to the average of the prior 12 months) in the
previous month, the returns of all momentum portfolios decrease in the current month. There is a
systematic pattern in these coefficients across the portfolios. In recession state, moving from the
loser to the winner portfolio, the coefficient on RREL increases monotonically from -1.300 (with statistic of -1.17) to -0.074 (with -statistic of -0.12). The difference in the coefficients between the
winner and loser portfolios (WML) is 1.226 (with -statistic of 0.71). In the expansion state,
however, the coefficient decreases monotonically from -0.153 (with -statistic of -0.50) to -1.065
(with -statistic of -3.49). The difference in the coefficients between the winner and loser portfolios
(WML) is -0.912 (with -statistic of -1.57). This evidence indicates that in the recession state,
interest rate changes have a greater negative impact on loser stocks than winner stocks, while in the
expansion state, interest rate changes have a greater negative impact on winner stocks than loser
stocks.
The coefficient estimates on the default spread (DEF) in both states (
and
)
exhibit a systematic variation across the portfolios in both states. In the recession state, moving
from the loser to winner portfolios, the coefficient estimate on DEF decreases largely monotonically
from 3.779 (with -statistic of 2.17) to -1.107 (with -statistic of -1.30). The difference in the
coefficient estimates between the winner and loser portfolios is marginally statistically significant: 4.886 (with -statistic of -1.78). In the expansion state, however, the coefficient estimate increases
17
monotonically from -0.248 (with -statistic of -0.49) to 0.580 (with -statistic of 1.23). The
difference in the coefficient estimates between the winner and loser portfolios is 0.828 (with statistic of 0.86). 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 0.782 and -0.654 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;
that is, the value premium is counter-cyclical, which is consistent with Gulen, Xing, and Zhang
(2011).
The coefficient estimates on the growth in monetary base (MB) in both states (
and
) do not show any particular pattern across the portfolios and are all statistically insignificant.
One noteworthy thing, however, is that the coefficient estimates on MB are all positive in recession
state but are mixed in sign in expansion state, which means that an increase in money supply
produces higher expected returns of the testing portfolios during recession periods but does not
during expansion periods.
18
The coefficient estimates on the dividend yield (DIV) in both states (
and
) also
exhibit a systematic variation across the portfolios in both states. In the recession state, moving
from the loser to winner portfolios, the coefficient estimate on DIV (
) decreases monotonically
from 3.047 (with -statistic of 2.93) to 0.516 (with -statistic of 0.97). The coefficient estimates are
all positive and mostly significant at the five percent level. The difference in the coefficient
estimates between the winner and loser portfolios is marginally statistically significant: -2.531 (with
-statistic of -1.61). In the expansion state, however, the coefficient estimate (
) increases
monotonically from -0.261 (with -statistic of -0.78) to 0.466 (with -statistic of 1.43). The
difference in the coefficient estimates between the winner and loser portfolios is 0.727 (with statistic of 1.24). These results indicates that dividend yields have a positively greater impact on
stock returns in recession periods than in expansion periods and that loser (winner) stocks are more
greatly affected in the recession (expansion) state by cash flow shocks from dividends. This
evidence also means that momentum profits are procyclical. Note that the differences in the
coefficient estimates on DIV between the value and growth portfolio (HML) are 1.456 (with statistic of 1.40) and -1.416 (with -statistic of -3.14) in the recession and expansion states,
respectively. These are exactly opposite in sign to those of WML and are used as evidence of the
counter-cyclicality of the value premium by Gulen, Xing, and Zhang (2011).
In sum, in the recession state, the loser portfolio has a greater sensitivity to all four
conditioning macroeconomic variables than does the winner portfolio. Specifically, the coefficient
estimates of the loser versus winner portfolios on the variables, RREL, DEF, MB, and DIV, are 1.300 ( -statistic of -1.17) vs. -0.074 ( -statistic of -0.12), 3.779 ( -statistic of 2.17) vs. -1.107 ( statistic of -1.30), 0.150 ( -statistic of 1.66) vs. 0.044 ( -statistic of 0.60), and 3.047 ( -statistic of
2.93) vs. 0.516 ( -statistic of 0.97), respectively. The coefficient estimates of the loser portfolio are
19
mostly statistically significant, while those of the winner portfolio are insignificant. In the
expansion state, however, we observe an opposite pattern in the coefficient estimates to the case of
the recession state. That is, the winner portfolio tends to have a greater sensitivity to the variables
(except for MB) than does the loser portfolio. Specifically, the coefficient estimates of the loser
versus winner portfolios on RREL, DEF, and DIV are -0.153 ( -statistic of -0.50) vs. -1.065 ( statistic of -3.49), -0.248 ( -statistic of -0.49) vs. 0.580 ( -statistic of 1.23), and -0.261 ( -statistic of
-0.78) vs. 0.466 ( -statistic of 1.43), respectively. Contrary to the case in the recession state, the
coefficient estimates of the winner portfolio are marginally statistically significant in the expansion
state, while those of the loser portfolio are insignificant. These above results indicate that loser
stocks are riskier in recession periods than winner stocks, while winner stocks are riskier in
expansion periods than loser stocks. This is consistent with the notion that momentum profits are
procyclical.
4.2.3. Tests for Equality of the Slope Coefficients Across States
The previous results show that returns on the momentum portfolios react asymmetrically to the
macroeconomic conditioning variables across 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 relative Treasury
bill rate, the default spread, the growth in monetary base, and the dividend yield) are equal across
states. We employ a likelihood ratio test for the null hypotheses:
for
for each of the 10 momentum portfolios. Table 3 reports the unrestricted and restricted
log-likelihood values and
-values from a standard
20
distribution for the 10 momentum
portfolios. Six out of the 10 portfolios have a
-value less than 5% and all portfolios have
-values
less than 10%, 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
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
be a
vector consisting of excess returns on the loser and winner portfolios,
and
,
respectively. Then, the joint conditional mean equation is specified as follows:
where
is a
is a
coefficient vector with elements
for
vector of normal residuals with mean zero and covariance matrix
21
, and
,
.
Here
is a positive semidefinite
winner portfolios‟ excess returns in state
covariance matrix of the residuals from the loser and
. For estimation convenience, we assume the form of the
conditional covariance matrix as follows:
In other words, we assume that the diagonal elements of this variance-covariance matrix,
,
depend only on the state of economy, as in the univariate case of equation (11). The off-diagonal
elements,
, assume a state-dependent correlation between the residuals, denoted
. We also
do not include ARCH effects in the conditional volatility equation. The transition probabilities from
the univariate model are maintained:
4.3.2. Estimation Results
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
22
The nulls of identical asymmetries across states for loser and winner stocks are modestly rejected at
standard significance levels for the slope coefficients on the conditioning variables, RREL, MB, and
DIV, in the conditional mean equation. These results are somewhat consistent with those reported
by Perez-Quiros and Timmermann (2000) for size portfolios and with those of Gulen, Xing, and
Zhang (2011) for book-to-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. Therefore, the state-based asymmetries in
the coefficients between the winner and loser portfolios are prominent, although the degree of the
asymmetries is weaker than in the cases of large-small and value-growth stocks.
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
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
23
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 1.2%
per month during expansions (i.e., low volatility states), while it is -0.4% 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 4.18). 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: 0.30 and 0.25 for the
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: -0.11 and
-0.17, respectively.15
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
15
Since the GDP is of quarterly frequency, the monthly excess returns are transformed into quarterly returns
by compounding monthly returns over each quarter.
24
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 March 1960 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).16 In this way, we reestimate the nonlinear model by adding one new month to the previous estimation window (starting
from March 1960) 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 2012.
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
portfolio (Panel C). For comparison, the out-of-sample predicted excess returns (with the dotted line)
are overlaid with the in-sample predicted excess returns (with the solid 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 March 1960 to December 2012. The
out-of-sample predictions are highly correlated with the in-sample predictions. Their correlation
coefficients are 0.35 for the winner portfolio, 0.73 for the loser portfolio, and 0.84 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
16
The initial sample period from 1960 to 1976 follows from Perez-Quiros and Timmermann (2000) and
Gulen, Xing, and Zhang (2011).
25
average excess returns but slightly higher standard deviation than the in-sample predictions. For the
winner portfolio, the average returns are 1.00% and 1.05% for the out-of-sample and in-sample
predictions, respectively, while the standard deviations are 2.00% and 0.96%, respectively. For the
loser portfolio, the average returns are 0.79% and 0.57% for the out-of-sample and in-sample
predictions, respectively, while the standard deviations are 3.32% and 2.26%, respectively. For the
winner-minus-loser portfolio, the average returns are 0.21% and 0.49% for the out-of-sample and
in-sample predictions, respectively, while the standard deviations are 2.45% and 1.89%,
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 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 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 outof-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 14.21% versus 10.62%, lower standard deviation of 24.38%
versus 31.41%, and thus a higher Sharpe ratio of 0.375 versus 0.177). However, the switching
26
portfolio does not outperform the buy-and-hold strategy for the winner 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
26.22%, 43.39%, and 0.345, respectively, and the corresponding statistics of the buy-and-hold
strategy are 8.35%, 50.68%, and 0.039, respectively. For the winner portfolio, the switching
portfolio similarly outperforms the buy-and-hold 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. This switching trading
strategy based on the loser portfolio performs better than that based on the winner portfolio.
4.5. Robustness Tests
Following Gray (1996), we use the relative Treasury bill rate as the instrument in modeling the state
transition probabilities. Since the estimation results of the conditional mean equation may be
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
), by following Perez-Quiros and Timmermann (2000), and the other is the one-month-
(
lagged monthly growth rate of industrial production (
where
17
), defined as
is the index level of industrial production at month .17
The industrial production indexes are obtained from the Federal Reserve Bank of St. Louis.
27
,
As in Table 2, Tables 6 and 7 present the estimation results of the univariate Markov
switching model for portfolios P1 (loser), P2, P4, P6, P8, and P10 (winner) when the instrumental
variables in modeling state transition probabilities are
and
, respectively.18 The
overall results from using these new instrumental variables are similar to those from Table 2, which
uses the relative Treasury bill rate as the instrumental variable. Therefore, the inferences from Table
2 are robust to changes in the specification of the state transition probabilities. Specifically, in the
recession state, the loser portfolio has a greater sensitivity to all four conditioning macroeconomic
variables than does the winner portfolio. Contrary to the case in the recession state, however, the
winner portfolio tends to have a greater sensitivity to the variables in the expansion state than does
the loser portfolio. These results indicate that loser stocks are riskier in recession periods than
winner stocks, while winner stocks are riskier in expansion periods than loser stocks. The null
hypotheses of identical asymmetries across states for loser and winner stocks are also modestly
rejected at standard significance levels in the slope coefficients on the four conditioning variables
(not reported in the tables).19
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.
18
The estimation results for portfolios P3, P5, P7, and P9 are not reported. The results are available upon
request.
19
The results are available upon request.
28
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
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
29
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.20 The asset-to-equity ratio of a
portfolio is computed as the median value of the asset-to-equity ratios of the firms included in the
portfolio, every month when momentum portfolios are rebalanced. Likewise, we compute the other
ratios of the portfolio.
Table 8 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 2012.21 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 2.572 for loser stocks to 1.206 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
1.181 (the loser portfolio) to 2.294 (the winner portfolio). The market-to-book asset ratio also
monotonically increases across portfolios from 1.079 (the loser portfolio) to 1.552 (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
20
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).
21
Since some of Compustat items are available from 1963, our firm characteristic analysis begins from 1963.
30
formation. To do this, we take the values of the four ratios proxying for leverage and growth options
over the period from
months to
months, where
is the portfolio formation month
and varies from January 1966 to December 2009, and compute the averages over the period
(
. Figure 4 illustrates the values of the four proxy ratios of the loser and winner
portfolios over the period (
. 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
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 8 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 the period from January 1963 to December 2012. The
aggregate leverage exhibits strong countercyclical variation. The two leverage proxy variables (in
31
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 9 shows that the averages of the aggregate leverage variables are higher during
recessions than during expansions (2.677 versus 2.054 for the asset-to-equity ratio and 0.713 versus
0.566 for the debt-to-equity ratio), while the averages of the aggregate growth option variables are
higher during expansions than during recessions (1.689 versus 1.203 for the market-to-book equity
ratio and 1.272 versus 1.075 for the market-to-book asset ratio). The differences in the averages
between expansions and recessions are all statistically significant at the 1% level.
The results in Figure 5 and Table 9 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. Conclusions
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
32
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.
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.
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 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
33
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.
34
References
Ahn, D.-H., J. Conrad, R. F. Dittmar. 2002. Risk adjustment and trading strategies. Rev. Financial
Stud. 16 459–485.
Ang, A., G. Bekaert. 2007. Stock return predictability: Is it there? Rev. Financial Stud. 20 651–707.
Avramov, D., T. Chordia. 2006. Asset pricing models and financial market anomalies. Rev.
Financial Stud. 19 1001–1040.
Bakshi, G. S., Z. Chen. 1996. Inflation, asset prices, and the term structure of interest rates in
monetary economies. Rev. Financial Stud. 9 241–275.
Bansal, R., R. F. Dittmar, C. T. Lundblad. 2005. Consumption, dividends, and the cross-section of
equity returns, J.Finance 60 1639–1672.
Barberis, N., A. Shleifer, R. Vishny. 1998. A model of investor sentiment. J. Financial Econom. 49
307–343.
Berk, J. B., R. C. Green, V. Naik. 1999. Optimal investment, growth options, and security returns. J.
Finance 54 1553–1607.
Bhandari, L. C. 1988. Debt/equity ratio and expected common stock returns: Empirical evidence. J.
Finance 43 507–528.
Boudoukh, J., M. Richardson. 1993. Stock returns and inflation: A long-horizon perspective. Amer.
Econom. Rev. 83, 1346–1355.
Boudoukh, J., M. Richardson, R. F. Whitelaw. 1994. Industry returns and the Fisher effect. J.
Finance 49 1595–1615
Campbell, J. Y. 1987. Stock returns and the term structure. J. Financial Economic 18 373–399.
Campbell, J.Y., R. J. Shiller. 1988. The dividend–price ratio and expectations of future dividends
and discount factors. Rev. Financial Stud. 1 195–228.
Chen, N. F., R. Roll, S. A. Ross. 1986. Economic forces and the stock market. J. Bus. 59(3) 383–
403.
Chordia, T., L. Shivakumar. 2002. Momentum, business cycle, and time-varying expected return, J.
Finance 57 985–1019.
Chui, A. C. W., S. Titman, K. C. J. Wei. 2010. Individualism and momentum around the world. J.
Finance 65 361–392.
35
Conrad, J., G. Kaul. 1998. An anatomy of trading strategies. Rev. Financial Stud. 11 489–519.
Cooper, M. J., R. C. Gutierrez Jr., A. Hameed. 2004. Market states and momentum. J. Finance 59
1345–1365.
Daniel, K., D. Hirshleifer, A. Subrahmanyam. 1998. Investor psychology and investor security
market under-and overreactions. J. Finance 53 1839–1886.
Danthine, J.-P., J. B. Donaldson. 1986. Inflation and asset prices in an exchange economy.
Econometrica 54 585–605.
Durland, J. M., T. H. McCurdy. 1994. Duration-dependent transitions in a Markov model of US
GNP growth. J. Bus. Econom. Statist. 12. 100–112.
Fama, E.F. 1981. Stock returns, real activity, inflation, and money. Amer. Econom. Rev. 71, 545–565.
Fama, E. F. Efficient capital markets II. 1991. J. Finance. 46 1575–1617.
Fama, E. F., K. R. French. 1988. Dividend yields and expected stock returns. J. Financial Economic
22 3–25.
Fama, E. F., K. R. French. 1989. Business conditions and expected returns on stocks and bonds, J.
Financial Econom. 25 23–49.
Fama, E. F., G. W. Schwert. 1977. Asset returns and inflation. J. Financial Economics 5 115–146.
Filardo, A. J. 1994. Business-cycle phases and their transitional dynamics. J. Bus. Econom. Statist.
12. 299–308.
Geske, R., R. Roll. 1983. The fiscal and monetary linkage between stock returns and inflation. J.
Finance 38 1–33.
Glosten, L., R. Jagannathan, D. Runkle. 1993. On the relation between expected value and the
volatility of the nominal excess return on stocks. J. Finance 48 1779–1801.
Gomes, J., L. Kogan, L. Zhang. 2003. Equilibrium cross section of returns. J. Political Econom. 111
693–732.
Goyal, V. K., K. Lehn, S. Racic. 2002. Growth opportunities and corporate debt policy: The case of
the U.S. defense industry. J. Financial Econom. 64 35–59.
Gray, Stephen F. 1996. Modeling the conditional distribution of interest rates as a regime switching
process, J. Financial Economics 42 27–62.
Griffin, J. M., X. Ji, J. S. Martin. 2003. Momentum investing and business cycle risks: Evidence
from pole to pole. J. Finance 58 2515–2547.
36
Grundy, B. D., J. S. Martin. 2001. Understanding the nature of the risks and the source of the
rewards to momentum investing. Rev. Financial Stud. 14 29–78.
Gulen, H., Y. Xing, L. Zhang. 2011. Value versus growth: Time-varying expected stock returns,
Financial Management 40, 381–407.
Hamilton, J. D. 1989. A new approach to the economic analysis of non-stationary time series and
the business cycle. Econometrica 57 357–384.
Hamilton, J. D., G. Lin. 1996. Stock market volatility and the business cycle, J. Appl. Econometrics
11 573–593.
Hong, H., J. Stein. 1999. A unified theory of underreaction, momentum trading, and overreaction in
asset markets. J. Finance 54 2143–2184.
Jagannathan, R., Z. Wang. 1996. The conditional CAPM and the cross-section of expected returns. J.
Finance 51 3–54.
Jegadeesh, N., S. Titman. 1993. Returns to buying winners and selling losers: Implications for stock
market efficiency. J. Finance 48 65–91.
Jiang, G., C. M. C. Lee, G. Y. Zhang. 2005. Information uncertainty and expected stock returns.
Rev. Accouning Stud. 10 185–221
Johnson, T. C. 2002. Rational momentum effects. J. Finance 57 585–608.
Kandel, S., R. F. Stambaugh. 1990. Expectations and volatility of consumption and asset returns.
Rev. Financial Stud. 3 207–232.
Kaul, G. 1987. Stock returns and inflation: The role of the monetary sector. J. Financial Econom.
18, 253–276.
Kaul, G. 1990. Monetary regimes and the relation between stock returns and inflationary
expectations. J. Financial Quant. Anal. 25 307–321.
Keim, D. B., R. F. Stambaugh. 1986. Predicting returns in the stock and bond markets, J. Financial
Econom. 17 357–390.
Kim, C.-J., C. R. Nelson. 1999. State-Space Models with Regime Switching, MIT Press, Cambridge,
Massachusetts.
Liu, L. X., L. Zhang. 2008. Momentum profits, factor pricing, and macroeconomic risk. Rev.
Financial Stud. 21 2417–2448.
Mahue, J. M., T. H. McCurdy. 2000 Identifying bull and bear markets in stock returns. J. Bus.
Econom. Statist. 18, 100–112.
37
Mandelker, G. N., S. G. Rhee. 1984. The impact of the degrees of operating and financial leverage
on systematic risk of common stock. J. Financial Quant. Anal. 19 45–57.
Marshall, D.A. 1992. Inflation and asset returns in a monetary economy. J. Finance 47 1315–1342.
Perez-Quiros, G., A. Timmermann. 2000. Firm size and cyclical variations in stock returns. J.
Finance 55 1229–1262.
Sagi, J. S., M. S. Seasholes. 2007. Firm-specific attributes and the cross section of momentum. J.
Financial Economics 84 389–434.
Schwert, G. W. 1990. Stock returns and real activity: A century of evidence. J. Finance 45 1237–
1257.
Stivers, C., L. Sun. 2010. Cross-sectional return dispersion and time variation in value and
momentum premiums. J. Financial Quant. Anal. 45 987–1014.
Stulz, R.M. 1986. Asset pricing and expected inflation. J. Finance 41 209–223.
Zhang, L. 2005. The value premium. J. Finance 60 67–103.
Zhang, X. F. 2006. Information uncertainty and stock returns. J. Finance 61 105–136.
38
Table 1 Moments of Monthly Excess Returns for Ten Decile
Momentum Portfolios
This table reports the mean, standard deviation, skewness, and excess 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 sixmonth 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.
and
are the first-order autocorrelations of the raw excess return and squared raw excess returns,
respectively. Numbers in brackets indicate -values of the first-order autocorrelations. The sample
period is from March 1960 to December 2012.
Momentum
portfolio
Mean
Standard
deviation
Skewness
Excess
Kurtosis
Loser
2
3
4
5
6
7
8
9
Winner
0.369
0.549
0.722
0.751
0.799
0.817
0.863
0.900
0.979
1.127
9.064
6.734
6.052
5.599
5.294
5.123
5.112
5.220
5.532
6.516
1.375
0.581
0.340
0.138
-0.108
-0.273
-0.419
-0.580
-0.709
-0.661
6.491
5.193
5.251
5.071
4.407
4.174
4.302
4.025
3.728
2.664
0.201 [0.000]
0.220 [0.000]
0.224 [0.000]
0.213 [0.000]
0.201 [0.000]
0.185 [0.000]
0.170 [0.000]
0.157 [0.000]
0.154 [0.000]
0.162 [0.000]
0.213 [0.000]
0.138 [0.001]
0.143 [0.000]
0.104 [0.009]
0.080 [0.043]
0.055 [0.163]
0.051 [0.194]
0.048 [0.224]
0.074 [0.062]
0.114 [0.004]
WML
0.758
5.999
-2.971
17.000
0.166 [0.020]
0.109 [0.006]
39
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:
,
where
is the monthly excess return for a given decile portfolio and
is the regime indicator. RREL is the relative 3-month T-bill rate calculated as
the difference between the current T-bill rate and its 12-month backward moving average, DEF is the spread between Moody's seasoned Baa-rated
corporate bond yield and 10-year treasury bond yield from the FRED, MB is defined as the 12-month log-difference in the monetary base reported by
the St. Louis Federal Reserve, 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 March 1960 to December 2012.
Parameters
Loser
Mean equation:
-0.179
Constant,
(-3.33)
-1.300
RREL,
(-1.17)
3.779
DEF,
(2.17)
0.150
MB,
(1.66)
3.047
DIV,
(2.93)
In State 1
P8
Winner
WML
HML
-0.063
-0.003
0.176
(-2.97)
-0.783
(-2.20)
-0.576
(-0.13)
-0.074
(-1.23)
0.329
(0.34)
0.046
(0.65)
2.150
(3.41)
(-0.84)
-0.347
(-0.34)
0.031
(0.32)
1.902
(3.02)
(-0.12)
-1.107
(-1.30)
0.044
(0.60)
0.516
(0.97)
P2
P4
P6
-0.116
-0.108
-0.083
(-3.17)
-0.917
(-3.40)
-0.789
(-1.18)
1.791
(1.52)
0.076
(1.15)
2.285
(3.18)
(-1.11)
1.146
(1.05)
0.050
(0.73)
2.392
(3.38)
Parameters
Winner
WML
0.007
-0.001
-0.020
0.042
(1.05)
-0.475
(0.74)
-0.582
(-0.06)
-1.065
(-0.81)
-0.912
(2.48)
0.056
(-2.30)
0.135
(0.45)
0.033
(1.40)
-0.036
(-0.18)
(-2.64)
0.142
(0.44)
0.034
(1.33)
0.090
(0.43)
(-3.49)
0.580
(1.23)
-0.004
(-0.11)
0.466
(1.43)
(-1.57)
0.828
(0.86)
0.109
(1.00)
0.727
(1.24)
(0.14)
-0.654
(-1.39)
0.044
(0.54)
-1.416
(-3.14)
P2
-0.064
0.019
0.014
0.010
0.009
(2.21)
1.226
(-1.26)
-0.124
(1.47)
-0.153
(1.38)
-0.276
(1.07)
-0.399
(0.71)
-4.886
(-1.78)
-0.106
(-0.66)
-2.531
(-1.61)
(-0.11)
0.782
(0.46)
0.006
(0.06)
1.456
(1.40)
(-0.50)
-0.248
(-0.49)
-0.113
(-1.43)
-0.261
(-0.78)
(-1.14)
-0.053
(-0.13)
0.000
(-0.00)
-0.247
(-1.00)
(-1.88)
0.125
(0.36)
0.035
(0.99)
-0.122
(-0.59)
40
P4
In State 2
P8
Loser
P6
HML
Transition probability parameters:
Constant,
RREL,
1.485
(11.1)
102.3
(3.76)
Error term volatility:
0.133
Std dev,
(13.4)
Log-likelihood value:
732.0
1.460
(10.4)
88.25
(3.32)
1.576
(11.4)
83.64
(3.03)
1.521
(10.1)
115.3
(2.92)
1.498
(10.3)
41.40
(3.03)
1.278
(8.14)
11.10
(2.41)
-0.207
-0.041
-60.90
-108.9
44.86
(2.78)
44.87
(2.40)
34.14
(1.74)
37.41
(1.79)
32.90
(1.57)
13.28
(0.84)
-31.57
62.32
0.097
0.080
0.071
0.072
0.081
-0.052
0.025
0.051
0.038
0.034
0.032
0.034
0.039
-0.013
0.014
(15.8)
(14.8)
(15.1)
(15.1)
(18.5)
(19.1)
(19.7)
(19.6)
(18.6)
(20.2)
(13.6)
906.8
1014.5
1057.2
1038.6
879.8
41
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:
,
where
is the monthly excess return for a given decile portfolio and
is the regime indicator. RREL is the relative 3-month T-bill rate calculated as
the difference between the current T-bill rate and its 12-month backward moving average, DEF is the spread between Moody's seasoned Baa-rated
corporate bond yield and 10-year treasury bond yield from the FRED, MB is defined as the 12-month log-difference in the monetary base reported by
the St. Louis Federal Reserve, 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 -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.,
for
, for
momentum portfolio i. The sample period is from March 1960 to December 2012.
Unrestricted log likelihood value
Restricted log likelihood with
Loser
Decile 2
Decile 3
Decile 4
Decile 5
732
907
973
1014
1042
723
900
965
1008
1037
[0.00]
[0.01]
[0.00]
[0.01]
[0.05]
Decile 6
Decile 7
Decile 8
Decile 9
Winner
Unrestricted log likelihood value
Restricted log likelihood with
1057
1055
1039
999
880
1052
1051
1034
994
876
[p-value]
[0.04]
[0.09]
[0.07]
[0.08]
[0.08]
[p-value]
42
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:
where
, respectively,
,
,for
as
be a
vector consisting of excess returns on the loser and winner portfolios,
and
is a
coefficient vector with elements
for
, and
. The conditional variance-covariance matrix,
, has the following form:
and
for
. The transition probabilities are defined
and
where
is the cumulative density function of a standard normal variable. RREL is the relative 3-month T-bill
rate calculated as the difference between the current T-bill rate and its 12-month backward moving average, DEF is
the spread between Moody's seasoned Baa-rated corporate bond yield and 10-year treasury bond yield from the
FRED, MB is defined as the 12-month log-difference in the monetary base reported by the St. Louis Federal Reserve,
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 -statistics (parameter
estimates divided by standard errors). The sample period is from March 1960 to December 2012.
Parameter
Mean equation:
Constant,
:
RREL,
DEF,
MB,
DIV,
Error term volatility:
Std. dev,
Loser (L)
In State 1
Winner (W)
-0.090
(-1.71)
Test for
-0.023
(-0.63)
-1.709
(-1.44)
Test for
-0.709
(-0.88)
1.428
(0.78)
Test for
-0.117
(-0.09)
0.172
(1.53)
Test for
0.015
(0.20)
2.114
(2.02)
Test for
1.095
(1.59)
0.139
(15.9)
0.092
(17.1)
Parameter
In State 2
Loser (L)
,
0.007
0.017
(0.51)
(1.41)
Log-likelihood value = 1,957 [p-value = 0.11]
,
-0.181
-0.697
(-0.58)
(-2.49)
Log-likelihood value = 1,957 [p-value = 0.07]
,
-0.173
-0.148
(-0.34)
(-0.30)
Log-likelihood value = 1,958 [p-value = 0.20]
,
0.014
-0.001
(0.27)
(-0.02)
Log-likelihood value = 1,957 [p-value = 0.06]
,
-0.248
-0.042
(-0.86)
(-0.16)
Log-likelihood value = 1,957 [p-value = 0.09]
0.052
(21.5)
43
Winner (W)
0.049
(20.6)
Parameters common to both deciles
Transition probability parameters:
Constant,
RREL,
Test for
1.251
(10.5)
66.32
(4.04)
,
31.90
(2.48)
Log-likelihood value = 1,957 [p-value = 0.05]
0.748
(22.1)
0.861
(58.5)
Correlation coefficients:
44
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 2012.
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 2012)
Average return
Std dev of return
Sharpe ratio
5.07
1.00
10.62
31.41
0.177
14.21
24.38
0.375
19.39
22.16
0.646
13.57
17.78
0.478
2.15
29.59
-0.142
12.18
25.85
0.226
22.23
20.62
0.842
13.80
16.11
0.555
Panel B: Recession states - NBER
Average return
Std dev of return
Sharpe ratio
6.35
1.41
8.35
50.68
0.039
26.22
43.39
0.345
Panel C: Expansion states - NBER
Average return
Std dev of return
Sharpe ratio
4.86
0.90
10.99
27.07
0.227
45
12.24
19.61
0.377
Table 6 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:
,
where
is the monthly excess return for a given decile portfolio and
is the regime indicator. RREL is the relative 3-month T-bill rate calculated as
the difference between the current T-bill rate and its 12-month backward moving average, DEF is the spread between Moody's seasoned Baa-rated
corporate bond yield and 10-year treasury bond yield from the FRED, MB is defined as the 12-month log-difference in the monetary base reported by
the St. Louis Federal Reserve, 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.
is the two-month lagged value of the year-on-year log difference in the
Composite Leading Indicator. Numbers in parentheses are -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 2012.
Parameters
Loser
Mean equation:
-0.084
Constant,
(-1.95)
-1.400
RREL,
(-1.53)
1.627
DEF,
(1.13)
0.130
MB,
(1.56)
1.433
DIV,
In State 1
P8
Winner
WML
HML
-0.049
0.001
0.085
(-2.39)
-0.862
(-1.77)
-0.270
(0.05)
-0.037
(-1.64)
-0.013
(-0.02)
0.069
(1.37)
1.434
(-0.37)
0.083
(0.07)
-0.036
(-0.26)
1.310
(-0.06)
-0.854
(-0.93)
0.012
(0.14)
0.307
P2
P4
P6
-0.071
-0.061
-0.055
(-2.44)
-1.144
(-2.39)
-0.796
(-1.86)
1.187
(1.22)
0.070
(1.21)
1.367
(-1.40)
0.605
(0.68)
0.069
(1.36)
1.262
Parameters
Loser
P2
-0.054
0.014
(1.12)
1.363
(-1.32)
0.033
(0.78)
-2.481
(-0.94)
-0.118
(-0.71)
-1.126
(0.03)
0.721
(0.52)
0.005
(0.06)
1.231
46
In State 2
P8
Winner
WML
HML
-0.007
-0.001
-0.015
0.042
(-0.32)
-0.324
(-0.77)
-0.500
(-0.07)
-1.047
(-0.60)
-1.338
(1.80)
0.106
(-1.40)
0.425
(1.28)
0.015
(0.43)
0.319
(-2.20)
0.486
(1.42)
0.017
(0.78)
0.545
(-3.16)
0.519
(0.97)
-0.001
(-0.02)
0.561
(-2.11)
1.146
(1.14)
-0.042
(-0.36)
0.824
(0.20)
-0.654
(-0.84)
0.035
(0.21)
-1.363
P4
P6
0.016
-0.001
-0.003
(1.02)
0.291
(1.41)
0.327
(-0.13)
-0.256
(0.77)
-0.627
(-1.10)
0.041
(0.45)
-0.262
(1.16)
-0.497
(-0.96)
0.028
(0.36)
-0.172
(-1.01)
0.435
(1.11)
-0.009
(-0.11)
0.241
(1.70)
(2.31)
(2.37)
(2.91)
(2.31)
(0.60)
(-0.74)
(1.37)
(-0.78)
(-0.63)
(0.91)
(1.25)
(2.16)
(1.77)
(1.41)
(-2.18)
1.263
(7.82)
-12.84
(-1.63)
1.493
(10.0)
-6.163
(-0.81)
1.530
(10.1)
-5.459
(-0.57)
1.464
(9.43)
6.309
(0.81)
1.176
(7.25)
7.521
(1.21)
-0.048
-0.155
20.56
-4.206
3.824
(0.44)
2.352
(0.27)
-0.280
(-0.03)
-8.278
(-0.82)
-10.97
(-1.23)
-7.029
(-0.74)
-10.85
35.94
0.091
0.075
0.067
0.068
0.080
-0.047
0.020
0.048
0.035
0.032
0.031
0.033
0.038
-0.010
0.013
(17.8)
(16.0)
(16.8)
(16.5)
(19.6)
(14.9)
(16.5)
(15.4)
(18.4)
(20.0)
(14.4)
901.8
1009.0
1052.4
1032.3
877.8
Transition probability parameters:
Constant,
CLI,
1.225
(6.39)
-13.04
(-1.43)
Error term volatility:
0.127
Std dev,
(15.9)
Log-likelihood value:
723.9
47
Table 7 Parameter Estimates for the Univariate Markov Switching Model of Excess Returns on Ten
Decile Momentum Portfolios:
As an Alternative Instrument in Modelling State Transition
Probabilities
The following univariate Markov switching model was estimated for excess returns on each momentum decile portfolio i:
,
where
is the monthly excess return for a given decile portfolio and
is the regime indicator. RREL is the relative 3-month T-bill rate calculated as
the difference between the current T-bill rate and its 12-month backward moving average, DEF is the spread between Moody's seasoned Baa-rated
corporate bond yield and 10-year treasury bond yield from the FRED, MB is defined as the 12-month log-difference in the monetary base reported by
the St. Louis Federal Reserve, 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. MP is the monthly growth rate of industrial production. Numbers in parentheses are statistics (parameter estimates divided by standard errors). „WML‟ indicates „Winner‟ portfolio minus „Loser‟ portfolio, and „HML‟ is the highest bookto-market decile portfolio minus the lowest book-to-market decile portfolio. The sample period is from March 1960 to December 2012.
Parameters
Loser
Mean equation:
-0.097
Constant,
(-1.94)
-1.339
RREL,
(-1.30)
1.822
DEF,
(1.09)
0.140
MB,
(1.42)
1.669
DIV,
(1.70)
In State 1
P8
Winner
WML
HML
-0.073
0.002
0.099
(-2.46)
-0.798
(-2.47)
-0.567
(0.10)
-0.160
(-1.47)
0.313
(0.34)
0.057
(1.00)
1.672
(2.61)
(-0.94)
0.150
(0.16)
0.052
(0.93)
1.836
(2.88)
(-0.27)
-1.027
(-1.21)
0.036
(0.49)
0.351
(0.69)
P2
P4
P6
-0.071
-0.084
-0.07
(-2.27)
-1.103
(-2.60)
-0.874
(-1.59)
1.283
(1.16)
0.074
(1.11)
1.413
(2.05)
(-1.40)
0.967
(0.93)
0.049
(0.76)
1.807
(2.44)
Parameters
In State 2
P8
Winner
WML
HML
0.007
-0.002
-0.016
0.042
(0.28)
-0.283
(0.76)
-0.525
(-0.17)
-1.068
(-0.58)
-1.225
(1.95)
0.056
(-1.20)
0.159
(0.40)
0.022
(0.69)
0.178
(0.58)
(-2.38)
0.035
(0.11)
0.008
(0.21)
0.223
(1.03)
(-3.34)
0.590
(1.19)
-0.005
(-0.14)
0.548
(1.73)
(-1.96)
1.133
(1.05)
-0.025
(-0.25)
0.810
(1.26)
(0.11)
-0.654
(-1.90)
0.044
(0.38)
-1.416
(-2.60)
Loser
P2
P4
P6
-0.064
0.014
0.011
0.010
0.003
(1.24)
1.180
(-1.26)
-0.124
(1.06)
0.157
(0.90)
0.189
(0.89)
-0.126
(0.69)
-2.849
(-1.03)
-0.104
(-0.63)
-1.318
(-0.85)
(-0.11)
0.782
(0.46)
0.006
(0.06)
1.456
(1.40)
(0.47)
-0.543
(-1.00)
0.021
(0.30)
-0.261
(-0.88)
(0.60)
-0.303
(-0.62)
0.032
(0.68)
-0.097
(-0.37)
(-0.52)
0.018
(0.04)
0.031
(0.92)
-0.078
(-0.31)
48
Transition probability parameters:
Constant,
MP,
1.400
(8.57)
-101.9
(-3.78)
Error term volatility:
0.135
Std dev,
(15.3)
Log-likelihood value:
731.0
1.364
(7.88)
-84.23
(-3.04)
1.554
(8.28)
-86.38
(-3.04)
1.571
(7.88)
-79.54
(-2.29)
1.714
(6.40)
-104.7
(-2.34)
1.160
(7.49)
11.10
(0.65)
-0.239
-0.041
113.0
-108.9
19.76
(0.77)
21.44
(0.82)
10.09
(0.36)
-0.410
(-0.01)
-25.76
(-0.64)
-0.999
(-0.04)
-20.76
62.32
0.096
0.080
0.071
0.073
0.080
-0.055
0.025
0.051
0.037
0.034
0.032
0.035
0.038
-0.013
0.014
(15.6)
(14.6)
(14.7)
(13.5)
(18.7)
(18.5)
(15.5)
(16.4)
(16.4)
(17.3)
(14.1)
907.2
1013.0
1054.3
1033.2
876.5
49
Table 8 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 2012. 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. Each ratio (A/B) of the
portfolio is computed as the median value of the ratios of accounting variable A to accounting variable B
of the firms included in the portfolio, every month when momentum portfolios are rebalanced. „WML‟
indicates „Winner‟ portfolio minus „Loser‟ portfolio.
Leverage
Loser
2
3
4
5
6
7
8
9
Winner
WML
(t-statistic)
Debt to Equity
Asset to Equity
0.742
0.527
0.462
0.413
0.397
0.372
0.353
0.338
0.323
0.291
-0.451
(-26.54)
2.572
2.038
1.862
1.713
1.642
1.563
1.493
1.426
1.349
1.206
-1.366
(-30.88)
50
Growth Option
Market-toMarket-tobook equity
book asset
1.181
1.079
1.276
1.127
1.338
1.160
1.419
1.197
1.480
1.226
1.547
1.255
1.638
1.294
1.729
1.331
1.879
1.390
2.294
1.552
1.113
0.473
(42.92)
(38.38)
Table 9 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 2012. 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. Each ratio
(A/B) of the portfolio is computed as the median value of the ratios of accounting variable A to
accounting variable B of the firms included in the portfolio, every month when momentum portfolios are
rebalanced. Recession and expansion periods are based on historical NBER business cycle dates.
Leverage
Expansion
Recession
Difference (E - R)
(t-statistic)
Debt-to-Equity
Asset-to-Equity
0.566
0.713
-0.147
(-7.11)
2.054
2.677
-0.623
(-7.17)
51
Growth Option
Market-toMarket-tobook equity
book asset
1.689
1.272
1.203
1.075
0.486
0.197
(7.24)
(8.03)
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)
52
Figure 1 (Continued)
Panel C: Loser, High Volatility (State 1)
Panel D : Loser, Low Volatility (State 2)
53
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
54
Figure 2 (Continued)
Panel B: Loser portfolio
Panel C: Winner-minus-Loser
55
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 2012.
Panel A: Winner portfolio
56
Figure 3 (Continued)
Panel B: Loser portfolio
Panel C: Winner-minus-Loser portfolio
57
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
months to
months, where is the portfolio formation month and it varies from January
1966 to December 2009. Then, we compute the averages over the period (
. „Month 0‟
is the portfolio formation month.
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)
58
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 2012. 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)
59