Download Time-varying expected momentum profits

Journal of Banking & Finance 49 (2014) 191–215
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Journal of Banking & Finance
journal homepage: www.elsevier.com/locate/jbf
Time-varying expected momentum profits
Dongcheol Kim a, Tai-Yong Roh b, Byoung-Kyu Min c,⇑, Suk-Joon Byun b
a
Business School, Korea University, Seoul, Republic of Korea
Graduate School of Finance, Korea Advanced Institute of Science and Technology (KAIST), Seoul, Republic of Korea
c
Institute of Financial Analysis, University of Neuchatel, Neuchatel, Switzerland
b
a r t i c l e
i n f o
Article history:
Received 15 November 2013
Accepted 11 September 2014
JEL classifications:
G12
G14
Keywords:
Momentum
Time-varying expected returns
Markov switching regression model
Business cycle
Procyclicality
Growth options
a b s t r a c t
This paper examines the time variations of expected momentum profits using a two-state Markov
switching model with time-varying transition probabilities to evaluate the empirical relevance of recent
rational theories of momentum profits. We find that in the expansion state the expected returns of
winner stocks are more affected by aggregate economic conditions than those of loser stocks, while in
the recession state the expected returns of loser stocks are more affected than those of winner stocks.
Consequently, expected momentum profits display strong procyclical variations. We argue that the
observed momentum profits are the realization of such expected returns and can be interpreted as the
procyclicality premium. We provide a plausible explanation for time-varying momentum profits through
the differential effect of leverage and growth options across business cycles.
Ó 2014 Elsevier B.V. All rights reserved.
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 et al., 1998; Daniel et al., 1998; Hong and Stein, 1999;
Jiang et al., 2005; Zhang, 2006; Chui et al., 2010). Another is a
rational risk-based 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 et al., 1999;
Johnson, 2002; Ahn et al., 2002; Bansal et al., 2005; Sagi and
Seasholes, 2007; Liu and Zhang, 2008).
⇑ Corresponding author. Tel.: +41 32 718 15 74; fax: +41 32 718 14 01.
E-mail addresses: [email protected] (D. Kim), [email protected]
(T.-Y. Roh), [email protected] (B.-K. Min), [email protected]
(S.-J. Byun).
http://dx.doi.org/10.1016/j.jbankfin.2014.09.004
0378-4266/Ó 2014 Elsevier B.V. All rights reserved.
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 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,
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D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
statistically insignificant profits during recessions. Their findings
uncover procyclical time variations in momentum profits. Cooper
et al. (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 et al.
(1998) and Hong and Stein (1999).1
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 discriminate whether winners are riskier than losers or vice versa from their results.2 Cooper et al. (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 et al. (2003) and Zhang (2005). These authors find that the
recent cross-sectional return dispersion is negatively related to the
subsequent momentum profits and thus suggest that momentum
profits are procyclical.3
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 et al. (2011). This flexible econometric model allows us to
combine the cross-sectional 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.
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, PerezQuiros and Timmermann (2000) examine whether a differential
response exists in expected returns to shocks to aggregate economic conditions between small and large firms. Gulen et al.
(2011) also examine a differential response in expected returns
between value and growth firms and find strong counter-cyclicality
1
Cooper et al. (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-of-sample 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
Chordia and Shivakumar (2002) admit to this weakness in their approach: ‘‘We do
not impose cross-sectional 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).
3
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.
of the value premium. Our paper is not the first to examine the procyclicality of momentum profits. Chordia and Shivakumar (2002),
Cooper et al. (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 time-variation
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 trough of the business cycle). The above two
findings are robust to estimating exogenously the state transition
probabilities and identifying the states, using alternative instrumental variables in modeling state transition probabilities, and
assuming the fat-tailed distribution of stock returns. 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 timevarying momentum profits.
The remainder of this paper proceeds as follows. Section 2 discusses the sources of time-varying 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.
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
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D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
(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
end-of-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.4
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 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.
4
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 et al. (2003) show that the model of Chen et al.
(1986) does not provide any evidence that macroeconomic risk variables can explain
momentum.
3. An econometric model of time-varying expected returns
Based on Sagi and Seasholes’ (2007) theoretical model, we
argue that momentum profits are procyclical because of the extent
of exercising growth options across business cycles. To empirically
examine the procyclical behavior of momentum profits, the
Markov switching regression framework is appropriate since it
can accommodate the time-varying behavior of momentum
profits across business cycles and business cycles can be regarded
as states. In this regard, we employ the Perez-Quiros and
Timmermann (2000) Markov switching regression framework with
time-varying transition probabilities based on Hamilton (1989)
and Gray (1996). Let rt be the return of a test asset in excess of
the riskless return at time t and let Xt1 be a vector of conditioning
variables available up to time t 1 used to predict rt. The Markov
switching specification takes all parameters (the intercept term,
slope coefficients, and volatility of excess returns) as a function
of a single, latent state variable, St. Specifically,
rt ¼ b0;St þ b0St X t1 þ et ;
et N 0; r2St ;
ð1Þ
where N 0; r2St denotes a normal distribution with mean zero and
variance r2St . In a two-state Markov switching specification, St = 1 or
St = 2, meaning that the parameters to be estimated are either
h1 ¼ b0;1 ; b01 ; r21 or h2 ¼ b0;2 ; b02 ; r22 .
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 firstorder Markov chain as follows:
pt ¼ ProbðSt ¼ 1jSt1 ¼ 1; yt1 Þ ¼ pðyt1 Þ;
ð2Þ
1 pt ¼ ProbðSt ¼ 2jSt1 ¼ 1; yt1 Þ ¼ 1 pðyt1 Þ;
ð3Þ
qt ¼ ProbðSt ¼ 2jSt1 ¼ 2; yt1 Þ ¼ qðyt1 Þ;
ð4Þ
1 qt ¼ ProbðSt ¼ 1jSt1 ¼ 2; yt1 Þ ¼ 1 qðyt1 Þ;
ð5Þ
where yt1 is a vector of variables publicly available at time t 1
and affects the state transition probabilities between times t 1
and t. Although the standard formulation of the Markov switching
model assumes the state transition probabilities to be constant, it
would be more reasonable to assume that the probability of staying
in a state depends on prior conditioning information, yt1, 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; PerezQuiros and Timmermann, 2000), interest rates (Gray, 1996; Gulen
et al., 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.5 Let h = (h1, h2) denote the
vector of parameters to be estimated in the likelihood function.
The probability density function of the return, conditional on being
state j, is Gaussian defined as
8 2 9
>
<
=
rt b0;j b0j X t1 >
1
f ðr t jXt1 ; St ¼ j; hÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffi exp >
>
2r2j
:
;
2pr2j
ð6Þ
for j = 1,2. The information set Xt1 contains Xt1,rt1,yt1, and
lagged values of these variables. Then, the log-likelihood function is
5
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).
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D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
Lðr t jXt1 ; hÞ ¼
T
X
log½/ðr t jXt1 ; hÞ;
ð7Þ
t¼1
where the density function /(rtjXt1;h) is simply obtained by summing the probability-weighted state probabilities across the two
states. It is defined as
/ðrt jXt1 ; hÞ ¼
2
X
f ðrt jXt1 ; St ¼ j; hÞProbðSt ¼ jjXt1 ; hÞ;
ð8Þ
j¼1
where Prob(St = jjXt1;h) is the conditional probability of being in
state j at time t given information at time t 1. The conditional
state probabilities can be obtained from the standard probability
theorem:
ProbðSt ¼ ijXt1 ; hÞ ¼
2
X
ProbðSt ¼ ijSt1 ¼ j; Xt1 ; hÞ
j¼1
ProbðSt1 ¼ jjXt1 ; hÞ:
ð9Þ
By Bayes’ rule, the conditional state probabilities can be written as
ProbðSt1 ¼ jjXt1 ;hÞ
f ðr t1 jSt1 ¼ j;X t1 ; yt1 ; Xt2 ;hÞProbðSt1 ¼ jjX t1 ;yt1 ; Xt2 ;hÞ
¼ P2
:
j¼1 f ðr t1 jSt1 ¼ j;X t1 ; y t1 ; Xt2 ;hÞProbðSt1 ¼ jjX t1 ;y t1 ; Xt2 ;hÞ
ð10Þ
The conditional state probabilities Prob(St = ijXt1;h) are driven by
iterating recursively Eqs. (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 time-varying.
4. Empirical results
4.1. Data and model specification
We use monthly excess returns (raw returns minus the onemonth 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 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 et al. (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 et al. (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 positively-skewed, while Portfolios 5 through 10 are negativelyskewed. These results indicate that past (short-term) 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
commonly used in the literature on the predictability of stock
returns. As in Perez-Quiros and Timmermann (2000) and Gulen
et al. (2011), we use the relative three-month Treasury bill rate
(RREL) 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
et al., 1993). Berk et al. (1999) present a theoretical model
Table 1
Moments of monthly excess returns for ten decile momentum portfolios.
Momentum portfolio
Mean
Standard deviation
Skewness
Excess Kurtosis
q1(rit)
Loser
2
3
4
5
6
7
8
9
Winner
WML
0.369
0.549
0.722
0.751
0.799
0.817
0.863
0.900
0.979
1.127
0.758
9.064
6.734
6.052
5.599
5.294
5.123
5.112
5.220
5.532
6.516
5.999
1.375
0.581
0.340
0.138
0.108
0.273
0.419
0.580
0.709
0.661
2.971
6.491
5.193
5.251
5.071
4.407
4.174
4.302
4.025
3.728
2.664
17.000
0.201
0.220
0.224
0.213
0.201
0.185
0.170
0.157
0.154
0.162
0.166
q1 r2it
[0.000]
[0.000]
[0.000]
[0.000]
[0.000]
[0.000]
[0.000]
[0.000]
[0.000]
[0.000]
[0.020]
0.213
0.138
0.143
0.104
0.080
0.055
0.051
0.048
0.074
0.114
0.109
[0.000]
[0.001]
[0.000]
[0.009]
[0.043]
[0.163]
[0.194]
[0.224]
[0.062]
[0.004]
[0.006]
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 six-month returns, and held for six months. Excess returns are calculated as the difference between monthly stock returns and the onemonth 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.q1(rit) and q1 r 2it are the first-order autocorrelations of the raw excess return and squared raw excess returns,
respectively. Numbers in brackets indicate p-values of the first-order autocorrelations. The sample period is from March 1960 to December 2012.
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
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).
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.6 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).7
To capture the movements of momentum portfolio returns, we
specify Eq. (1) by including the above-mentioned return predictable variables in the following conditional mean equation:
r it ¼ bi0;St þ bi1;St RRELt1 þ bi2;St DEF t1 þ bi3;St MBt2
þ bi4;St DIV t1 þ eit ;
ð11Þ
6
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 et al., 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.
7
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.
195
where rit is the monthly excess return for the ith decile momentum
portfolio at time t,eit is the normally distributed random error term
with mean zero and variance r2i;St , and St = {1,2}. The regressors are
lagged by one month. The conditional variance of excess returns,
r2i;St , is allowed to depend only on the state of economy:
ln
r2i;St ¼ ki;St :
ð12Þ
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 et al. (2011), we model the
time-varying state transition probabilities to be dependent on
the level of short interest rates, Treasury bill rates, as follows:
pit ¼ Prob Sit ¼ 1jSit1 ¼ 1; yt1 ¼ U pi0 þ pi1 RRELt1 ;
1 pit ¼ Prob Sit ¼ 2jSit1 ¼ 1; yt1 ;
qit ¼ Prob Sit ¼ 2jSit1 ¼ 2; yt1 ¼ U pi0 þ pi2 RRELt1 ;
1 qit ¼ Prob Sit ¼ 1jSit1 ¼ 2; yt1 ;
ð13Þ
ð14Þ
ð15Þ
ð16Þ
where RRELt1 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 U() is the cumulative
probability density function of a standard normal variable.8 For
robustness checks, we also use two alternative instrumental
variables in modeling state transition probability equations instead
of the relative three-month 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 Eqs.
(11)–(16) for portfolios P1 (loser), P2, P4, P6, P8, and P10 (winner).9
The constant parameter estimates in the conditional mean equation
^i0;1 Þ are much lower than those in state 2 (b
^i0;2 Þ in all
in state 1 (b
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
^ i;1 Þ is greater than that in state 2 ðr
^ i;2 Þ in all
estimate in state 1 ðr
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 Bureau
of Economic Research (NBER) business cycle dates. These results
may indicate that state 1 is the recession state and state 2 is the
8
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.
9
The estimation results for portfolios P3, P5, P7, and P9 are not reported because of
space constraint. The results are available upon request.
196
Table 2
Parameter estimates for the univariate Markov switching model of excess returns on ten decile momentum portfolios.
Parameters
In State 1
Parameters
P2
P4
P6
P8
Winner
WML
HML
Mean equation:
^i0;1
Constant, b
0.179
0.116
0.108
0.083
0.063
0.003
0.176
0.064
^i1;1
RREL, b
(3.33)
1.300
(3.17)
0.917
(3.40)
0.789
(2.97)
0.783
(2.20)
0.576
(0.13)
0.074
(2.21)
1.226
(1.26)
0.124
^i2;1
DEF, b
(1.17)
3.779
(1.18)
1.791
(1.11)
1.146
(1.23)
0.329
(0.84)
0.347
(0.12)
1.107
(0.71)
4.886
^i3;1
MB, b
(2.17)
0.150
(1.52)
0.076
(1.05)
0.050
(0.34)
0.046
(0.34)
0.031
(1.30)
0.044
^i4;1
DIV, b
(1.66)
3.047
(1.15)
2.285
(0.73)
2.392
(0.65)
2.150
(0.32)
1.902
(2.93)
(3.18)
(3.38)
(3.41)
1.576
(11.4)
83.64
Transition probability parameters:
1.485
1.460
^ i0
Constant, p
(11.1)
(10.4)
102.3
88.25
^ i1
RREL,p
In State 2
Loser
P2
P4
P6
P8
Winner
WML
HML
^i0;2
b
0.019
0.014
0.010
0.009
0.007
0.001
0.020
0.042
^i1;2
b
(1.47)
0.153
(1.38)
0.276
(1.07)
0.399
(1.05)
0.475
(0.74)
0.582
(0.06)
1.065
(0.81)
0.912
(2.48)
0.056
(0.11)
0.782
^i2;2
b
(0.50)
0.248
(1.14)
0.053
(1.88)
0.125
(2.30)
0.135
(2.64)
0.142
(3.49)
0.580
(1.57)
0.828
(0.14)
0.654
(1.78)
0.106
(0.46)
0.006
^i3;2
b
(0.49)
0.113
(0.13)
0.000
(0.36)
0.035
(0.45)
0.033
(0.44)
0.034
(1.23)
0.004
(0.86)
0.109
(1.39)
0.044
(0.60)
0.516
(0.66)
2.531
(0.06)
1.456
^i4;2
b
(1.43)
0.261
(0.00)
0.247
(0.99)
0.122
(1.40)
0.036
(1.33)
0.090
(0.11)
0.466
(1.00)
0.727
(0.54)
1.416
(3.02)
(0.97)
(1.61)
(1.40)
(0.78)
(1.00)
(0.59)
(0.18)
(0.43)
(1.43)
(1.24)
(3.14)
1.521
1.498
1.278
0.207
0.041
(10.1)
115.3
(10.3)
41.40
(8.14)
11.10
60.90
108.9
44.86
44.87
34.14
37.41
32.90
13.28
31.57
62.32
(2.78)
(2.40)
(1.74)
(1.79)
(1.57)
(0.84)
0.051
(19.1)
0.038
(19.7)
0.034
(19.6)
0.032
(18.6)
0.034
(20.2)
0.039
(13.6)
0.013
0.014
(3.76)
(3.32)
(3.03)
(2.92)
(3.03)
(2.41)
Error term volatility:
^ i;1
0.133
Std dev, r
(13.4)
0.097
(15.8)
0.080
(14.8)
0.071
(15.1)
0.072
(15.1)
0.081
(18.5)
Log-likelihood value:
732.0
906.8
1014.5
1057.2
1038.6
879.8
0.052
0.025
p^ i2
r^ i;2
The following univariate two-state Markov switching model is estimated for excess returns on each momentum decile portfolio i:
r it ¼ bi0;St þ bi1;St RRELt1 þ bi2;St DEF t1 þ bi3;St MBt2 þ bi4;St DIV t1 þ eit
eit N 0; r2i;St ; St ¼ f1; 2g
pit ¼ P Sit ¼ 1jSit1 ¼ 1 ¼ U pi0 þ pi1 RRELt1 ; 1 pit ¼ P Sit ¼ 2jSit1 ¼ 1
qit ¼ P Sit ¼ 2jSit1 ¼ 2 ¼ U pi0 þ pi2 RRELt1 ; 1 qit ¼ P Sit ¼ 1jSit1 ¼ 2 ;
where rit is the monthly excess return for a given decile portfolio and Sit is the regime indicator. RREL is the relative three-month Treasury-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, 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.
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
Loser
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
expansion state. A further analysis of the identification of the states
will be presented in Section 4.3.
4.2.2. Estimation of the conditional mean equations
The estimation results of the conditional mean Eq. (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 to compare with the results of WML.10 We report only the difference in the
coefficient estimates between the value (highest book-to-market)
portfolio and the growth (lowest book-to-market) portfolio 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
^i1;1 and b
^i1;2 Þ, which means that when
in both states 1 and 2 (b
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 t-statistic of 1.17) to 0.074 (with t-statistic of 0.12).
The difference in the coefficients between the winner and loser
portfolios (WML) is 1.226 (with t-statistic of 0.71). In the expansion state, however, the coefficient decreases monotonically from
0.153 (with t-statistic of 0.50) to 1.065 (with t-statistic of
3.49). The difference in the coefficients between the winner
and loser portfolios (WML) is 0.912 (with t-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
^i2;1 and b
^i2;2 Þ exhibit a systematic variation across the
states (b
portfolios in both states. In the recession state, moving from
the loser to winner portfolios, the coefficient estimate on DEF
^i2;1 Þ decreases largely monotonically from 3.779 (with t-statis(b
tic of 2.17) to 1.107 (with t-statistic of 1.30). The difference
in the coefficient estimates between the winner and loser portfolios is marginally statistically significant: 4.886 (with t-statistic of 1.78). In the expansion state, however, the coefficient
^i2;2 Þ increases monotonically from 0.248
estimate on DEF (b
(with t-statistic of 0.49) to 0.580 (with t-statistic of 1.23).
The difference in the coefficient estimates between the winner
and loser portfolios is 0.828 (with t-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
et al. (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-to-market), 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
10
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).
197
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 et al. (2011).
The coefficient estimates on the growth in monetary base (MB)
^i3;1 and b
^i3;2 Þ do not show any particular pattern
in both states (b
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.
The coefficient estimates on the dividend yield (DIV) in both
^i4;1 and b
^i4;2 Þ also exhibit a systematic variation across
states (b
the portfolios in both states. In the recession state, moving from
the loser to winner portfolios, the coefficient estimate on DIV
^i4;1 Þ decreases monotonically from 3.047 (with t-statistic of
(b
2.93) to 0.516 (with t-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 t-statistic of 1.61). In the expansion state, however, the
^i4;2 Þ increases monotonically from 0.261
coefficient estimate (b
(with t-statistic of 0.78) to 0.466 (with t-statistic of 1.43). The
difference in the coefficient estimates between the winner and
loser portfolios is 0.727 (with t-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
t-statistic of 1.40) and 1.416 (with t-statistic of 3.14) in the
recession and expansion states, respectively. These are also
exactly opposite in sign to those of WML and are used as evidence of the counter-cyclicality of the value premium by Gulen
et al. (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 (t-statistic of 1.17) vs. 0.074 (t-statistic
of 0.12), 3.779 (t-statistic of 2.17) vs. 1.107 (t-statistic of 1.30),
0.150 (t-statistic of 1.66) vs. 0.044 (t-statistic of 0.60), and 3.047 (tstatistic of 2.93) vs. 0.516 (t-statistic of 0.97), respectively. The
coefficient estimates of the loser portfolio are 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 these
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 (t-statistic of 0.50) vs. 1.065
(t-statistic of 3.49), 0.248 (t-statistic of 0.49) vs. 0.580 (t-statistic of 1.23), and 0.261 (t-statistic of 0.78) vs. 0.466 (t-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.
198
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
Table 3
Tests for equality of the slope coefficients across states in the Markov switching model.
Unrestricted log likelihood value
Restricted log likelihood with bik;St ¼1 ¼ bik;St ¼2 for k ¼ 1; 2; 3; 4
[p-value]
Unrestricted log likelihood value
Restricted log likelihood with bik;St ¼1 ¼ bik;St ¼2 for k ¼ 1; 2; 3; 4
[p-value]
Loser
Decile 2
Decile 3
Decile 4
Decile 5
732
723
[0.00]
907
900
[0.01]
973
965
[0.00]
1014
1008
[0.01]
1042
1037
[0.05]
Decile 6
Decile 7
Decile 8
Decile 9
Winner
1057
1052
[0.04]
1055
1051
[0.09]
1039
1034
[0.07]
999
994
[0.08]
880
876
[0.08]
The following univariate Markov switching model is estimated for excess returns on each momentum decile portfolio i:
rit ¼ bi0;St þ bi1;St RRELt1 þ bi2;St DEF t1 þ bi3;St MBt2 þ bi4;St DIV t1 þ eit
eit N 0; r2i;St ; St ¼ f1; 2g
pit ¼ P Sit ¼ 1jSit1 ¼ 1 ¼ U pi0 þ pi1 RRELt1 ; 1 pit ¼ P Sit ¼ 2jSit1 ¼ 1
qit ¼ P Sit ¼ 2jSit1 ¼ 2 ¼ U pi0 þ pi2 RRELt1 ; 1 qit ¼ P Sit ¼ 1jSit1 ¼ 2 ;
where rit is the monthly excess return for a given decile portfolio and Sit 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). The likelihood ratio tests are conducted on the null hypothesis that the coefficient are equal across states, i.e., bik;St ¼1 ¼ bik;St ¼2 for
k = 1,2,3,4,f or momentum portfolio i. The sample period is from March 1960 to December 2012.
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: bik;St ¼1 ¼ bik;St ¼2 for k = 1,2,3,4 for each
of the 10 momentum portfolios. Table 3 reports the unrestricted
and restricted log-likelihood values and p-values from a standard
v2 distribution for the 10 momentum portfolios. Six out of the
10 portfolios have a p-value less than 5% and all portfolios have
p-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.
Perez-Quiros and Timmermann (2000) and Gulen et al. (2011),
the bivariate
Markov
switching regression model is as follows.
0
Let r t ¼ r Lt ; rW
be a (2 1) vector consisting of excess returns
t
on the loser and winner portfolios, rLt and rW
t , respectively. Then,
the joint conditional mean equation is specified as follows:
rt ¼ b0;St þ b1;St RRELt1 þ b2;St DEF t1 þ b3;St MBt2
þ b4;St DIV t1 þ et ;
where bk;St
for k = 1,2,3,4, and et is a (2 1) vector of normal residuals with
P
P
mean zero and covariance matrix
St ; St ¼ f1; 2g. Here
St is a
positive semidefinite (2 2) covariance matrix of the residuals
from the loser and winner portfolios’ excess returns in state St.
For estimation convenience, we assume the form of the conditional
covariance matrix as follows:
!
8
X
>
>
>
¼ ki;St
ln
>
>
<
ii;St
!1=2
!1=2
>
X
X
X
>
>
>
¼ qSt
>
:
ij;St
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
ð17Þ
0
is a (2 1) coefficient vector with elements bLk;St ; bW
k;St
ii;St
for i ¼ j
ð18Þ
for i – j
jj;St
In other words, we assume that the diagonal elements of this
P
variance–covariance matrix, ii;St , depend only on the state of economy, as in the univariate case of Eq. (11). The off-diagonal elements,
P
ij;St , assume a state-dependent correlation between the residuals,
denoted qSt . We also do not include ARCH effects in the conditional
volatility equation.
As in the univariate case, we use the relative three-month
Treasury bill rate (RREL) as the instrumental variable in modeling
state transition probabilities to compare the results with those of
the univariate case. As an alternative instrumental variable, we
also use the one-month-lagged monthly growth rate of industrial
production (MPt1), defined as MPt = logIPt logIPt1, where IPt is
the index level of industrial production at month t. The purpose
of using MP as an alternative instrumental variable is
twofold. The first is to obtain the estimates of the state transition
probabilities based on the real economic variable rather than the
financial economic variable. These estimates would be more
Table 4
Estimation results of the Bivariate Markov switching model for excess returns of the loser and winner portfolios.
Parameter
In State 1
Loser (L)
Parameter
Winner (W)
WML
Panel A: Instrumental variable in modeling state transition probabilities = Relative three-month T-bill rate (RREL)
Mean equation:
^0;1 :
0.090
0.023
0.067
Constant, b
(1.71)
(0.63)
^0;2
b
(1.94)
In State 2
Loser (L)
Winner (W)
WML
0.007
0.017
0.010
(0.51)
(1.41)
(1.63)
0.181
0.697
0.516
(0.58)
(2.49)
(3.20)
0.173
0.148
0.026
(0.34)
(0.30)
(0.10)
0.014
0.001
0.015
(0.27)
(0.02)
(0.63)
0.248
0.042
0.205
(0.86)
(0.16)
(1.42)
0.052
(21.5)
0.049
(20.6)
W
Test for Hb0 : bL0;1 bL0;2 ¼ bW
0;1 b0;2 , Log-likelihood value = 1,957 [p-value = 0.11]
^1;1
RREL, b
1.709
0.709
1.000
(1.44)
(0.88)
(1.23)
^1;2
b
W
Test for Hb1 : bL1;1 bL1;2 ¼ bW
1;1 b1;2 , Log-likelihood value = 1,957 [p-value = 0.07]
^2;1
DEF, b
1.428
0.117
1.546
(0.78)
(0.09)
(1.30)
^2;2
b
W
Test for Hb2 : bL2;1 bL2;2 ¼ bW
2;1 b2;2 , Log-likelihood value = 1,958 [p-value = 0.20]
^3;1
MB, b
0.015
0.158
(1.53)
(0.20)
(2.08)
^3;2
b
W
Test for Hb3 : bL3;1 bL3;2 ¼ bW
3;1 b3;2 , Log-likelihood value = 1,957 [p-value = 0.06]
^4;1
DIV, b
2.114
1.095
1.019
(2.02)
(1.59)
(1.45)
^4;2
b
W
Test for Hb4 : bL4;1 bL4;2 ¼ bW
4;1 b4;2 , Log-likelihood value = 1,957 [p-value = 0.09]
Error term volatility:
^1
Std. dev, r
0.139
(15.9)
Transition probability parameters:
^0
Constant, p
^1
RREL, p
r^ 2
0.092
(17.1)
1.251
(10.5)
66.32
(4.04)
¼ p2 , Log-likelihood value = 1,957 [p-value = 0.05]
p^ 2
31.90
(2.48)
0.748
q^ 2
(22.1)
Panel B: Instrumental variable in modelling state transition probabilities = Monthly growth rate of industrial production (MP)
0.861
(58.5)
Test for Hp1 : p1
Correlation coefficients:
^1
q
Mean equation:
^0;1 :
Constant, b
0.068
0.015
0.053
(1.19)
(0.37)
(1.38)
^0;2
b
0.009
0.018
0.009
(0.52)
(1.15)
(1.45)
0.068
0.443
0.374
(0.19)
(1.44)
(1.84)
0.301
0.220
0.080
(0.49)
(0.39)
(0.32)
0.008
0.008
0.016
(0.15)
(0.17)
(0.66)
0.249
0.080
0.209
(0.79)
(0.23)
(1.43)
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
0.172
W
Test for Hb0 : bL0;1 bL0;2 ¼ bW
0;1 b0;2 , Log-likelihood value = 1,957 [p-value = 0.11]
^1;1
RREL, b
1.253
1.093
0.160
(1.04)
(1.29)
(0.19)
^1;2
b
W
Test for Hb1 : bL1;1 bL1;2 ¼ bW
1;1 b1;2 , Log-likelihood value = 1,958 [p-value = 0.56]
^2;1
DEF, b
0.992
0.419
1.411
(0.50)
(0.30)
(1.12)
^2;2
b
W
Test for Hb2 : bL2;1 bL2;2 ¼ bW
2;1 b2;2 , Log-likelihood value = 1,958 [p-value = 0.25]
^3;1
MB, b
0.181
0.022
0.159
(1.63)
(0.30)
(2.15)
^3;2
b
W
Test for Hb3 : bL3;1 bL3;2 ¼ bW
3;1 b3;2 , Log-likelihood value = 1,957 [p-value = 0.08]
^4;1
DIV, b
1.657
1.092
0.564
(1.51)
(1.42)
(0.70)
^4;2
b
W
Test for Hb4 : bL4;1 bL4;2 ¼ bW
4;1 b4;2 , Log-likelihood value = 1,958 [p-value = 0.33]
199
(continued on next page)
200
Table 4 (continued)
Parameter
Error term volatility:
^1
Std. dev, r
In State 1
Loser (L)
Winner (W)
0.141
(12.9)
0.091
(14.7)
Transition probability parameters
^0
Constant, p
^1
MP, p
Test for Hp1 : p1
Correlation coefficients:
^1
q
Parameter
WML
1.180
(9.32)
108.47
(3.65)
¼ p2 , Log-likelihood value = 1,951 [p-value = 0.00]
0.731
(22.9)
r^ 2
In State 2
Loser (L)
Winner (W)
0.053
(17.4)
0.051
(17.9)
p^ 2
20.29
(1.33)
^2
q
0.868
(88.9)
WML
The bivariate Markov switching regression model for the loser and winner portfolios’ excess returns is specified as follows:
0
0
P
L
W
where r t ¼ rLt ; r W
be a (2 1) vector consisting of excess returns on the loser and winner portfolios, r Lt and r W
for k = 1,2,3,4, and et Nð0; St Þ; St ¼ f1; 2g.
t
t , respectively, bk;St is a (2 1) coefficient vector with elements bk;St ; bk;St
P P 1=2 P 1=2
P
P
The conditional variance–covariance matrix, St , has the following form: ln
¼ kiSt , for i = j and ij;St ¼ qSt
for i – j. The transition probabilities are defined as
ii;St
ii;St
jj;St
pt ¼ PðSt ¼ 1jSt1 ¼ 1Þ ¼ Uðp0 þ p1 IV t1 Þ and qt ¼ PðSt ¼ 2jSt1 ¼ 2Þ ¼ Uðp0 þ p2 IV t1 Þ;
where U() is the cumulative density function of a standard normal variable, and IV is the instrumental variable in the state transition probabilities. Two instrumental variables are used. The first one is the relative three-month
Treasury-bill rate calculated as the difference between the current T-bill rate and its 12-month backward moving average (RRELt1) (Panel A), and the second one is the monthly growth rate of industrial production ( MPt1) (Panel
B). DEF is the spread between Moody’s seasoned Baa-rated corporate bond yield and 10-year treasury bond yield, 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). The sample period is from March 1960 to December 2012.
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
rt ¼ b0;St þ b1;St RRELt1 þ b2;St DEF t1 þ b3;St MBt2 þ b4;St DIV t1 þ et ;
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
informative to identify the states in relating to actual business
cycles such as the NBER cycles, since MP is more directly related
to real economic activities than is a financial variable such as the
Treasury rate. The second is to conduct a robustness check for
the sensitivity of the estimation results of the conditional mean
equation to the choice of the instrumental variable. The state
transition probabilities for the bivariate model are modeled as
follows:
pt ¼ PðSt ¼ 1jSt1 ¼ 1Þ ¼ Uðp0 þ p1 IV t1 Þ;
ð19Þ
qt ¼ PðSt ¼ 2jSt1 ¼ 2Þ ¼ Uðp0 þ p2 IV t1 Þ;
ð20Þ
201
where IVt1 is the instrumental variable which is defined as either
RRELt1 or MPt1.
4.3.2. Estimation results
Table 4 presents the estimation results of the bivariate Markov
switching regression model when the instrumental variables in
modeling state transition probabilities are the relative threemonth Treasury bill rate (Panel A) and the monthly growth rate
of industrial production (Panel B), respectively. When the instrumental variable is the relative Treasury bill rate, the coefficient
estimates on the conditioning variables for both loser and winner
Fig. 1. 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, when the instrumental variable in
the state transition probabilities is the relative three-month Treasury bill rate. 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.
202
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
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. For instance, in the recession state, the coefficient on RREL for loser stocks is greater in absolute value than that
for winner stocks. The difference in the coefficient estimate
between winner and loser stocks is 1.000 (with t-statistic of
1.23). The opposite pattern is found for the expansion state; the
coefficient on RREL for winner stocks is greater in absolute value
than that for loser stocks. The difference in the coefficient
estimates between the winner and loser portfolios is statistically
significant: 0.516 (with t-statistic of 3.20). The estimation
results for the conditional mean equation are also similar, when
the instrumental variable in the state transition probabilities is
the monthly growth rate of industrial production. That is, the coefficient estimates on the macroeconomic variables in the mean
equation show that loser (winner) stocks are more greatly affected
in the recession (expansion) state by the conditioning variables.
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
L
W for k ¼ 1; 2; 3; and 4:
bk;1 bLk;2 ¼ bW
k;1 bk;2 ð21Þ
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, when the instrumental
variable in the state transition probabilities is the relative Treasury
bill rate. These results are somewhat consistent with those reported
by Perez-Quiros and Timmermann (2000) for size portfolios and
with those of Gulen et al. (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.
Fig. 1 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), when the instrumental variable in the state transition
probabilities is the relative Treasury bill rate. 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 of Fig. 1 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 Fig. 1 show that the series obtained from the
univariate and joint bivariate models for the loser and winner portfolios are approximately similar. The expected returns of both loser
and winner portfolios tend to increase during recession periods but
decrease during expansion periods. However, the loser portfolio
tends to display this pattern more strongly than the winner
portfolio. As a result, as seen in Panel C, the expected momentum
profit (WML) tends to decrease sharply and have a negative value
during recessions but to increase and have a positive value just
Fig. 2. Time-series of the state transition probabilities of being in high and low volatility states. The figure plots the time-series of the state transition probabilities of being in
state 1 (high volatility; Panel A) and state 2 (low volatility; Panel B), which are estimated from the bivariate Markov switching model using the monthly growth rate of
industrial production as an instrumental variable in the state transition probability. Shaded areas indicate NBER recession periods.
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
203
Fig. 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.
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 et al. (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
204
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
Table 5
Trading results based on the out-of-sample prediction from the Bivariate Markov switching regression model.
Treasury bill
Loser portfolio
Winner portfolio
Switching portfolio
Buy-and-hold
Switching portfolio
Panel A: The whole period (January 1977 to December 2012)
Average return
5.07
10.62
Std dev of return
1.00
31.41
Sharpe ratio
0.177
Buy-and-hold
14.21
24.38
0.375
19.39
22.16
0.646
13.57
17.78
0.478
Panel B: Recession states – NBER
Average return
6.35
Std dev of return
1.41
Sharpe ratio
8.35
50.68
0.039
26.22
43.39
0.345
2.15
29.59
0.142
12.18
25.85
0.226
Panel C: Expansion states – NBER
Average return
4.86
Std dev of return
0.90
Sharpe ratio
10.99
27.07
0.227
12.24
19.61
0.377
22.23
20.62
0.842
13.80
16.11
0.555
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.
coefficients between the expected excess returns of HML and these
two macroeconomic variables are negative: 0.11 and 0.17,
respectively.11
4.3.3. Further identifying the states
To further identify the states, we plot the state transition probabilities over time. Fig. 2 shows the time-series state transition
probabilities estimates of being in state 1 (Panel A) and state 2
(Panel B) at time t conditional
on the information
set at time
t 1; Prob Sit ¼ 1jXt1 ; h and Prob Sit ¼ 2jXt1 ; h , respectively.
These probabilities estimates are obtained when the instrumental
variable in the state transition probabilities is the monthly growth
rate of industrial production. As mentioned earlier, these estimates
would be more informative to identify the states in relating to
actual business cycles such as the NBER cycles, since MP is more
directly related to real economic activities than is a financial variable such as the Treasury bill rate.
As shown in Fig. 2, during the NBER recession periods (shaded
areas), the state transition probabilities of being in state 1 tend
to increase and are relatively high, while the state transition probabilities of being in state 2 tend to decrease and are low. On the
other hand, during the NBER expansion periods, an opposite pattern is observed. The results suggest that state 1 can be regarded
as the recession state and state 2 can be regarded as the expansion
state.
To examine more specifically whether the estimated state transitional probabilities are associated with actual business cycles
(i.e., the NBER classification of business cycles), we conduct some
formal statistical tests. First, we compute the averages of the state
transition probabilities estimates of being in state 1 across NBER
expansionary and recessionary periods. The averages are 0.264
and 0.580, respectively. The difference between these two averages
is highly statistically significant (t-statistic of 11.57).12 Second, to
examine the extent to which the state transition probabilities are
correlated with the NBER business cycle, we estimate the following
probit regression model:
Prob½Dt ¼ 1jpt ¼ Fða þ bpt Þ;
ð22Þ
11
Since the GDP is of quarterly frequency, the monthly excess returns are
transformed into quarterly returns by compounding monthly returns over each
quarter.
12
The averages of the state transition probabilities of being in state 2 across NBER
expansionary and recessionary periods are 0.736 and 0.237, respectively. The tstatistic of the difference in the averages is 11.57.
where F is the cumulative normal distribution, Dt is a binary variable that takes a value of one during the NBER recessionary periods
and zero otherwise, pt is the state transition probability estimate of
^ is positive
being in state 1. The estimate of the slope coefficient, b,
and statistically significant; it is 2.000, with t-statistic of 9.53. This
suggests that the state transition probability of being in state 1 is
strongly positively associated with the probability of being a NBER
recession period. Finally, we compute the point-biserial correlation
coefficient, which is used to measure the degree of the association
when one variable is dichotomous (discrete) and the other is continuous.13 The computed point-biserial correlation coefficient
between the NBER recession binary variable (Dt) and the state transitional probabilities estimates of being in state 1 is also positive and
strongly statistically significant; it is 0.418, with t-statistic of 11.57.
These above results suggest that state 1 can be regarded as the
recession state and state 2 can be regarded as the expansion state.
Further evidence supporting this is the estimation results of the
slope coefficients in the state transition probabilities of Eqs. (19)
and (20). The slope coefficient estimate of the transition probabil^ 1 , is negative and statistically significant; it is
ity in state 1, p
108.47, with t-statistic of 3.65. However, the slope coefficient
^ 2 , is positive
estimate of the transition probability in state 2, p
and statistically moderately significant; it is 20.29, with t-statistic
of 1.33. These results indicates that an increase in the industrial
production growth is associated with a decrease in the probability
of being in state 1 but an increase in the probability of being in
state 2. This implies that states 1 and 2 can be regarded as the
recession and expansion states, respectively.
4.4. Trading rules based on out-of-sample predictions
To avoid potential problems from over-fitting a complex nonlinear model with a large number of parameters being estimated as in
this paper, it is necessary to examine out-of-sample predictability
of the model. Since the conditional mean Eq. (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 et al. (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
13
The point-biserial correlation coefficient is a special case of Pearson in which one
variable is quantitative and the other variable is dichotomous (see Glass and Hopkins,
1995). It has been often used in accounting and finance literature (Hagerman and
Zmijewski, 1979; Clinch et al., 2012).
Table 6
Estimation results of the univariate Markov switching model for excess returns on momentum portfolios after estimating state transition probabilities exogenously.
Parameters
In State 1
Parameters
Loser
P2
P4
P6
P8
Winner
WML
HML
Mean equation:
^i0;1
Constant, b
0.058
0.043
0.044
0.034
0.026
0.015
0.043
0.005
^i1;1
RREL, b
(2.06)
1.017
(2.08)
0.669
(2.60)
0.622
(2.17)
0.594
(1.62)
0.565
(0.77)
0.721
(0.95)
0.296
(0.17)
0.877
^i2;1
DEF, b
(1.52)
0.824
(1.35)
0.582
(1.55)
0.577
(1.61)
0.272
(1.50)
0.066
(1.51)
0.437
(0.27)
1.261
^i3;1
MB, b
(0.93)
0.142
(0.88)
0.090
(1.07)
0.086
(0.56)
0.069
(0.13)
0.050
(0.72)
0.031
^i4;1
DIV, b
(2.18)
1.114
(1.89)
0.830
(2.23)
0.911
(1.97)
0.781
(1.36)
0.718
In State 2
P2
P4
P6
P8
Winner
WML
HML
^i0;2
b
0.005
0.001
0.011
0.007
0.012
0.015
0.020
0.034
^i1;2
b
(0.24)
0.431
(0.07)
0.547
(0.87)
0.527
(0.56)
0.546
(0.92)
0.619
(0.82)
0.963
(0.57)
0.532
(1.35)
0.487
(0.91)
0.241
^i2;2
b
(0.94)
0.052
(1.64)
0.036
(1.75)
0.330
(1.84)
0.475
(1.92)
0.221
(2.02)
0.017
(0.61)
0.069
(0.75)
1.094
(0.89)
0.111
(0.26)
0.399
^i3;2
b
(0.06)
0.169
(0.06)
0.009
(0.58)
0.233
(0.88)
0.229
(0.39)
0.199
(0.02)
0.062
(0.04)
0.107
(0.95)
0.044
(0.72)
0.787
(1.06)
0.327
(0.53)
0.075
^i4;2
b
(0.52)
0.168
(0.07)
0.260
(3.94)
0.384
(3.90)
0.496
(2.55)
0.466
(0.61)
0.012
(0.30)
0.156
(0.10)
0.485
(0.41)
(0.90)
(1.48)
(1.97)
(1.67)
(0.03)
(0.21)
(1.45)
0.053
(17.5)
0.038
(16.5)
0.034
(16.3)
0.033
(18.1)
0.036
(18.2)
0.048
(18.1)
0.005
0.007
(1.68)
(1.70)
(2.28)
(2.15)
(1.92)
(1.67)
(0.30)
(0.70)
Error term volatility:
^ i;1
0.112
Std dev, r
(23.1)
0.084
(23.3)
0.067
(23.0)
0.061
(25.6)
0.062
(22.8)
0.076
(22.9)
0.036
0.016
Log-likelihood value:
683.2
865.5
978.1
1027.0
1006.9
854.9
r^ i;2
The two-state Markov switching model is estimated in a two-step approach. In the first step, we estimate the following two-state Markov switching model for the consumption growth rates.
Dct ¼ a0;St þ a1;St DEF t1 þ zt
zt N 0; r2c;St ; St ¼ f1; 2g
pt ¼ PðSt ¼ 1jSt1 ¼ 1Þ ¼ Uðp0 þ p1 RRELt1 Þ;
qt ¼ PðSt ¼ 2jSt1 ¼ 2Þ ¼ Uðp0 þ p2 RRELt1 Þ;
1 pt ¼ PðSt ¼ 2jSt1 ¼ 1Þ
1 qt ¼ PðSt ¼ 1jSt1 ¼ 2Þ;
where Dct = ln(Ct) ln(Ct3) is the quarter-on-quarter consumption growth rate, and Ct is the real personal consumption expenditures of nondurable goods and services obtained from the Bureau of Economic Analysis. In the second
^t ; qt Þ are used as given probabilities in the following Markov switching model for excess returns on decile momentum portfolios.
step, the state transition probabilities estimated in the first step ðp
rit ¼ bi0;St þ bi1;St RRELt1 þ bi2;St DEF t1 þ bi3;St MBt2 þ bi4;St DIV t1 þ eit
eit N 0; r2i;St ; St ¼ f1; 2g
pt ¼ PðSt ¼ 1jSt1 ¼ 1Þ;
1 pt ¼ PðSt ¼ 2jSt1 ¼ 1Þ
qt ¼ PðSt ¼ 2jSt1 ¼ 2Þ;
1 qt ¼ PðSt ¼ 1jSt1 ¼ 2Þ:
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-tomarket decile portfolio. The sample period is from March 1960 to December 2012.
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
Loser
205
206
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
Fig. 4. Expected momentum profits from the two-step Markov switching model. The figure displays time-series of the expected momentum profits (WML) obtained from the
two-step Markov switching model in which state transition probabilities are estimated separately in the Markov switching model for consumption growth rates and these
estimated state transition probabilities are used as given probabilities in the Markov switching model for excess returns on decile momentum portfolios. Shaded areas
indicate NBER recession periods.
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).14 In this way, we re-estimate 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.
Fig. 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 wholeperiod 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
14
The initial sample period from 1960 to 1976 follows from Perez-Quiros and
Timmermann (2000) and Gulen et al. (2011).
lower 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 PerezQuiros 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 out-of-sample predictability is particularly significant for the switching portfolio based on
the loser portfolio and during the recession state.
Table 7
Parameter estimates for the univariate Markov Switching model of excess returns on ten decile momentum portfolios using alternative instrumental variables in modeling state transition probabilities.
Parameters
In State 1
Loser
Parameters
P2
P4
P6
P8
Winner
WML
HML
Panel A: Instrumental variable (IV) in modeling state transition probabilities = Monthly growth rate of industrial production (MP)
Mean equation:
^i0;1
^i0;2
0.097
0.071
0.084
0.07
0.073
0.002
0.099
0.064
Constant, b
b
In State 2
Loser
P2
P4
P6
P8
Winner
WML
HML
0.014
0.011
0.010
0.003
0.007
0.002
0.016
0.042
(0.90)
0.189
(0.89)
0.126
(0.28)
0.283
(0.76)
0.525
(0.17)
1.068
(0.58)
1.225
(1.95)
0.056
(2.27)
1.103
(2.60)
0.874
(2.46)
0.798
(2.47)
0.567
(0.10)
0.160
(1.24)
1.180
(1.26)
0.124
^i1;2
b
(1.06)
0.157
^i2;1
DEF, b
(1.30)
1.822
(1.59)
1.283
(1.40)
0.967
(1.47)
0.313
(0.94)
0.150
(0.27)
1.027
(0.69)
2.849
(0.11)
0.782
^i2;2
b
(0.47)
0.543
(0.60)
0.303
(0.52)
0.018
(1.20)
0.159
(2.38)
0.035
(3.34)
0.590
(1.96)
1.133
(0.11)
0.654
^i3;1
MB, b
(1.09)
0.140
(1.16)
0.074
(0.93)
0.049
(0.34)
0.057
(0.16)
0.052
(1.21)
0.036
(1.03)
0.104
(0.46)
0.006
^i3;2
b
(1.00)
0.021
(0.62)
0.032
(0.04)
0.031
(0.40)
0.022
(0.11)
0.008
(1.19)
0.005
(1.05)
0.025
(1.90)
0.044
^i4;1
DIV, b
(1.42)
1.669
(1.11)
1.413
(0.76)
1.807
(1.00)
1.672
(0.93)
1.836
(0.49)
0.351
(0.63)
1.318
(0.06)
1.456
^i4;2
b
(0.30)
0.261
(0.68)
0.097
(0.92)
0.078
(0.69)
0.178
(0.21)
0.223
(0.14)
0.548
(0.25)
0.810
(0.38)
1.416
(2.44)
(2.61)
(2.88)
(0.69)
(0.85)
(1.40)
(0.88)
(0.37)
(0.31)
(0.58)
(1.03)
(1.73)
(1.26)
(2.60)
1.554
1.571
1.714
1.160
0.239
0.041
(8.28)
86.38
(7.88)
79.54
(6.40)
104.7
(7.49)
11.10
113.0
108.9
19.76
21.44
10.09
0.410
25.76
0.999
20.76
62.32
(0.77)
(0.82)
(0.36)
(0.01)
(0.64)
(0.04)
0.051
(18.5)
0.037
(15.5)
0.034
(16.4)
0.032
(16.4)
0.035
(17.3)
0.038
(14.1)
0.013
0.014
0.003
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
(1.70)
(2.05)
Transition probability parameters:
1.400
1.364
^ i0
Constant, p
(8.57)
(7.88)
101.9
84.23
^ i1
MP,p
(3.78)
Error term volatility:
^ i;1
0.135
Std dev, r
(15.3)
(3.04)
(3.04)
(2.29)
(2.34)
(0.65)
0.096
(15.6)
0.080
(14.6)
0.071
(14.7)
0.073
(13.5)
0.080
(18.7)
Log-likelihood value:
731.0
907.2
1013.0
1054.3
1033.2
876.5
0.055
0.025
p^ i2
r^ i;2
Panel B: Instrumental variable (IV) in modeling state transition probabilities = two-month lagged value of the log difference in the Composite Leading Indicator (DCLI)
Mean equation:
^i0;1
^i0;2
0.084
0.071
0.061
0.055
0.049
0.001
0.085
0.054
0.014
0.016
0.001
Constant, b
b
(1.95)
(2.44)
(2.39)
(2.39)
(1.77)
(0.05)
(1.12)
(1.32)
(1.02)
(1.41)
(0.13)
^i1;2
^i1;1
1.400
1.144
0.796
0.862
0.270
0.037
1.363
0.033
0.291
0.327
0.256
b
RREL, b
^i2;1
DEF, b
(1.53)
1.627
(1.86)
1.187
(1.40)
0.605
(1.64)
0.013
(0.37)
0.083
(0.06)
0.854
(0.78)
2.481
(0.03)
0.721
^i2;2
b
(0.77)
0.627
(1.16)
0.497
(1.01)
0.435
(1.40)
0.425
(2.20)
0.486
(3.16)
0.519
(2.11)
1.146
(0.20)
0.654
^i3;1
MB, b
(1.13)
0.130
(1.22)
0.070
(0.68)
0.069
(0.02)
0.069
(0.07)
0.036
(0.93)
0.012
(0.94)
0.118
(0.52)
0.005
^i3;2
b
(1.10)
0.041
(0.96)
0.028
(1.11)
0.009
(1.28)
0.015
(1.42)
0.017
(0.97)
0.001
(1.14)
0.042
(0.84)
0.035
^i4;1
DIV, b
(1.56)
1.433
(1.21)
1.367
(1.36)
1.262
(1.37)
1.434
(0.26)
1.310
(0.14)
0.307
(0.71)
1.126
(0.06)
1.231
^i4;2
b
(0.45)
0.262
(0.36)
0.172
(0.11)
0.241
(0.43)
0.319
(0.78)
0.545
(0.02)
0.561
(0.36)
0.824
(0.21)
1.363
(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)
Transition probability parameters:
1.225
1.263
^ i0
Constant, p
3.824
2.352
0.280
8.278
10.97
7.029
10.85
35.94
(0.44)
(0.27)
(0.03)
(0.82)
(1.23)
(0.74)
^ i1
CLI,p
1.493
1.530
1.464
1.176
0.048
0.155
(6.39)
13.04
(7.82)
12.84
(10.0)
6.163
(10.1)
5.459
(9.43)
6.309
(7.25)
7.521
20.56
4.206
(1.43)
(1.63)
(0.81)
(0.57)
(0.81)
(1.21)
p^ i2
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
^i1;1
RREL, b
(1.94)
1.339
(continued on next page)
207
Table 7 (continued)
where rit is the monthly excess return for a given decile portfolio and Sit is the regime indicator. RREL is the relative three-month Treasury-bill rate, DEF is the spread between Moody’s seasoned Baa-rated corporate bond yield and 10year treasury bond yield, MB is defined as the 12-month log-difference in the monetary base, 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. IV is the instrumental variable in the state transition probabilities. Two instrumental variables are used. The first one is the monthly growth rate of industrial production (MPt1) (Panel A), and the
second one is the two-month lagged value of the year-on-year log difference in the Composite Leading Indicator (DCLIt2) (Panel B). 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.
0.010
0.038
(14.4)
0.033
(20.0)
0.032
(15.4)
r it ¼ bi0;St þ bi1;St RRELt1 þ bi2;St DEF t1 þ bi3;St MBt2 þ bi4;St DIV t1 þ eit
eit N 0; r2i;St ; St ¼ f1; 2g
pit ¼ P Sit ¼ 1jSit1 ¼ 1 ¼ U pi0 þ pi1 IV t1 ; 1 pit ¼ P Sit ¼ 2jSit1 ¼ 1
qit ¼ P Sit ¼ 2jSit1 ¼ 2 ¼ U pi0 þ pi2 IV t1 ; 1 qit ¼ P Sit ¼ 1jSit1 ¼ 2 ;
877.8
1032.3
1052.4
1009.0
901.8
Log-likelihood value:
723.9
The following univariate Markov switching model was estimated for excess returns on each momentum decile portfolio i:
0.068
(16.5)
0.091
(17.8)
0.075
(16.0)
0.067
(16.8)
0.080
(19.6)
0.047
0.020
r^ i;2
0.048
(14.9)
0.035
(16.5)
0.031
(18.4)
WML
Winner
P8
P6
P4
Loser
P2
Error term volatility:
^ i;1
0.127
Std dev, r
(15.9)
In State 2
Parameters
HML
WML
Winner
P8
P6
P4
P2
Loser
In State 1
Parameters
0.013
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
HML
208
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 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
4.5.1. Estimating the state transition probabilities exogenously to the
model
We have interpreted the high return volatility state as recession
and the low return volatility state as expansion economy. However, this interpretation should be drawn with caution, since the
state variables may not coincide with actual business cycles.15
The states identified by stock returns could be somewhat inconsistent with the real economic conditions. Our interpretation could
mean that the returns of momentum portfolios are endogenous to
the recession/expansion states. As a robustness check for the validity
of our interpretation of the identified states, we therefore identify
exogenously the states by using a real macroeconomic variable
rather than using stock returns. As such macroeconomic variable,
we choose real consumption growth, since consumption growth
reflects the underlying fundamentals of the aggregate economy,
and thus, high and low consumption growth volatility states can
coincide with expansion and recession states.
To identify exogenously the states and the state transition probabilities, we employ a two-step estimation approach. In the first
step, we estimate the following two-state Markov switching model
for the quarter-on-quarter consumption growth rates to obtain the
state transition probabilities and the high and low consumption
growth volatility states.
Dct ¼ a0;St þ a1;St DEF t1 þ zt ;
zt N 0; r2c;St
ð23Þ
pt ¼ PðSt ¼ 1jSt1 ¼ 1Þ ¼ Uðp0 þ p1 RRELt1 Þ;
qt ¼ PðSt ¼ 2jSt1 ¼ 2Þ ¼ Uðp0 þ p2 RRELt1 Þ;
where Dct = ln(Ct) ln(Ct3) is the quarter-on-quarter consumption
growth rate, Ct is the real personal consumption expenditures of
nondurable goods and services, and St is an unobservable state
variable St = {1,2} that follows a two-state Markov chain with
time-varying transition probability matrix, as characterized in
Eqs. (13)–(16).16 In the second step, the state transition probabilities
15
For instance, stock return volatilities reached exceptionally high level in October
1987, but this period is not considered as a recession, according to the NBER
classification of the business cycle.
16
The similar specification has been used in the literature (e.g., Whitelaw, 2000;
Ozoguz, 2009).
Table 8
Parameter estimates for the univariate Markov switching model of excess returns on ten decile momentum portfolios: modelling error terms with the student t distribution.
Parameters
In State 1
Parameters
P2
P4
P6
P8
Winner
WML
HML
Mean equation:
^i0;1
Constant, b
0.218
0.141
0.112
0.048
0.031
0.005
0.223
0.080
^i1;1
RREL, b
(4.66)
0.828
(4.65)
0.665
(3.70)
0.540
(1.77)
0.039
(1.15)
0.140
(0.23)
0.205
(1.87)
1.033
(2.13)
1.069
^i2;1
DEF, b
(1.08)
3.555
(1.25)
2.486
(1.21)
1.987
(0.07)
0.351
(0.21)
1.284
(0.33)
1.380
(1.04)
4.935
^i3;1
MB, b
(2.10)
0.241
(2.62)
0.104
(2.32)
0.070
(0.34)
0.153
(1.11)
0.042
(1.47)
0.028
^i4;1
DIV, b
(2.48)
3.673
(2.12)
2.481
(1.70)
2.072
(1.78)
1.310
(0.43)
1.474
(4.68)
(4.24)
(3.54)
(2.42)
1.554
(10.3)
66.43
Loser
P2
P4
P6
P8
Winner
WML
HML
^i0;2
b
0.031
0.032
0.030
0.009
0.003
0.002
0.030
0.068
^i1;2
b
(1.41)
0.353
(2.53)
0.330
(2.49)
0.485
(0.83)
0.666
(0.58)
0.549
(0.94)
1.158
(1.57)
0.805
(3.73)
0.533
(1.38)
2.030
^i2;2
b
(1.10)
0.217
(1.26)
0.107
(1.99)
0.091
(2.37)
0.571
(2.17)
0.413
(3.71)
0.574
(1.80)
0.791
(1.02)
1.470
(2.55)
0.213
(1.59)
0.381
^i3;2
b
(0.45)
0.027
(0.26)
0.190
(0.23)
0.196
(1.61)
0.031
(1.11)
0.026
(1.18)
0.003
(1.15)
0.023
(2.43)
0.346
(0.20)
0.457
(1.25)
3.216
(3.94)
1.875
^i4;2
b
(0.34)
0.859
(3.31)
0.391
(4.34)
0.189
(1.36)
0.576
(1.13)
0.416
(0.09)
0.455
(0.28)
1.315
(6.38)
2.497
(2.39)
(0.91)
(3.45)
(1.36)
(2.28)
(1.48)
(0.74)
(1.32)
(1.28)
(1.39)
(2.64)
(4.98)
1.456
1.418
1.253
0.173
0.034
(10.0)
57.62
(10.1)
103.7
(5.02)
41.02
(3.24)
41.91
55.90
55.07
60.29
70.20
38.51
27.37
12.68
42.39
59.08
(3.08)
(1.83)
(2.57)
(2.63)
(3.15)
(3.04)
(2.64)
(1.42)
(1.41)
(0.79)
0.090
(9.22)
0.070
(11.4)
0.063
(9.18)
0.068
(11.0)
0.078
(16.0)
0.058
0.013
r^ i;2
0.051
(14.5)
0.038
(13.4)
0.033
(11.6)
0.033
(11.8)
0.033
(18.0)
0.038
(12.5)
0.013
0.019
3.917
(3.24)
4.777
(3.12)
4.576
(3.19)
4.882
(2.89)
6.488
(2.71)
3.453
8.940
ni,2
10.87
(1.05)
12.60
(0.83)
8.263
(1.60)
9.721
(1.54)
145.8
(0.38)
26.76
(0.46)
15.89
31.44
916.7
1025.6
1060.6
1044.5
886.0
Transition probability parameters:
1.426
1.473
^ i0
Constant, p
(10.5)
(9.76)
82.93
68.77
^ i1
RREL,p
(3.57)
Error term volatility:
^ i;1
0.136
Std dev, r
(4.99)
Degree of freedom:
ni,1
3.035
(3.38)
Log-likelihood value:
742.4
In State 2
p^ i2
The following univariate two-state Markov switching model is estimated for excess returns on each momentum decile portfolio i:
r it ¼ bi0;St þ bi1;St RRELt1 þ bi2;St DEF t1 þ bi3;St MBt2 þ bi4;St DIV t1 þ eit ;
eit ¼ ri;St wt ; wt Student t with degrees of freedom ni;St > 2; St ¼ f1; 2g
pit ¼ P Sit ¼ 1jSit1 ¼ 1 ¼ U pi0 þ pi1 RRELt1 ;
qit ¼ P Sit ¼ 2jSit1 ¼ 2 ¼ U pi0 þ pi2 RRELt1 ;
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
Loser
1 pit ¼ P Sit ¼ 2jSit1 ¼ 1
1 qit ¼ P Sit ¼ 1jSit1 ¼ 2 ;
where rit is the monthly excess return for a given decile momentum portfolio, and Sit 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.
209
210
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
Table 9
Averages of the financial ratios proxying for leverage and growth options for ten decile momentum portfolios.
Leverage
Loser
2
3
4
5
6
7
8
9
Winner
WML
(t-statistic)
Growth option
Debt to Equity
Asset to Equity
Market-to-book equity
Market-to-book asset
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)
1.181
1.276
1.338
1.419
1.480
1.547
1.638
1.729
1.879
2.294
1.113
(42.92)
1.079
1.127
1.160
1.197
1.226
1.255
1.294
1.331
1.390
1.552
0.473
(38.38)
This table presents the time series averages of the financial ratios proxying for leverage (the asset-to-equity 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.
^t Þ are used as given probabilities in
^t ; q
estimated in the first step ðp
estimating the coefficients on the conditional mean equation of
(17) through the likelihood function of (7) for excess returns on
ten decile momentum portfolios. This two-step approach can estimate the state transition probabilities and identify the states exogenously from the returns of momentum portfolios.
Table 6 reports the two-step estimation results of the two-state
Markov switching model. The results are similar in the pattern of
the coefficient estimates across momentum portfolios to those
obtained from the (one-step) simultaneous estimation for the
parameters of the two-state Markov switching model which are
reported in Table 2. That is, the results obtained from the two-step
estimation also lead to the same conclusions that momentum profits are procyclical and that loser stocks are riskier in recession
periods than winner stocks, while winner stocks are riskier in
expansion periods than loser stocks.
To further examine the similarity of the expected momentum
profits (WML) obtained from the two-step estimation and the
simultaneous estimation, we plot these two sets of time-series
of the expected momentum profits (WML) in Panels A and B of
Fig. 4 for the univariate and bivariate Markov switching models,
respectively. Fig. 4 shows that these two sets of the expected
momentum profits co-move very closely. Specifically, in the univariate case (Panel A), the correlation coefficients between the
two sets of the expected momentum profits are 0.561, 0.465,
and 0.580 for the whole, expansion, and recession periods,
respectively.17 The averages (in percent) of the two sets of the
expected momentum profits (the two-step vs. the simultaneous
estimation) are 0.76 vs. 0.92, 1.02 vs. 1.17, and 0.71 vs. 0.40
for the whole, expansion, and recession periods, respectively. In
the bivariate case (Panel B), the correlation coefficients are 0.813,
0.738, and 0.862 for the whole, expansion, and recession periods,
respectively. The averages (in percent) of the expected momentum
profits are 0.72 vs. 0.70, 0.95 vs. 0.89, and 0.53 vs. 0.27,
respectively.
In summary, there is no significant difference in the results
between the two-step estimation and the (one-step) simultaneous
estimation. Therefore, our interpretation of the high return
volatility state as recession and the low return volatility state as
expansion economy is valid.
17
The p-values of the correlation coefficients are all 0.0000.
4.5.2. Using alternative instruments in modeling state transition
probabilities
Following Gray (1996), we have used the relative three-month
Treasury bill rate as the instrumental variable in the state transition probabilities for the univariate Markov switching model. Since
the estimation results of the conditional mean equation may be
sensitive across the states to the choice of the instrument variable
in the state transition probabilities, it would be necessary to conduct robustness tests by using alternative instrument variables.
We choose two alternative instrument variables in the state transition probabilities: The first one is the one-month-lagged monthly
growth rate of industrial production (MPt1), and the second one is
the two-month-lagged value of the year-on-year log difference in
the Composite Leading Indicator (DCLIt2), by following PerezQuiros and Timmermann (2000).
As in Tables 2 and 7 presents 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 the state
transition probabilities are MPt1 (Panel A) and DCLIt2 (Panel B),
respectively.18 The overall results from using these new instrumental variables are similar to those from Table 2, which uses the relative three-month 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).
4.5.3. Incorporating the fat-tailed nature of returns into the model
It has been well documented that the empirical distribution of
stock returns exhibits excess kurtosis. It is worthwhile, therefore,
18
The estimation results for portfolios P3, P5, P7, and P9 are not reported. The
results are available upon request.
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
211
Fig. 5. 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-to-book equity (ME/BE) and market-to-book
asset ratios (MA/BA)) over the period from t 36 months to t + 36 months, where t is the portfolio formation month and it varies from January 1966 to December 2009. Then,
we compute the averages over the period (t 36, t + 36). ‘Month 0’ is the portfolio formation month.
to conduct a robustness check by incorporating the fat-tailed nature of returns in estimating the Markov switching model. To do
this, we assume that the error term, eit, in the conditional mean
Eq. (11) follows a Student t distribution. Specifically, the error term
is assumed to have the following form.
eit ri;St wt ;
ð24Þ
where wt is a Student-t variate with degrees of freedom ni;St . In
this specification, the return volatility and degrees of freedom are
allowed to switch regimes.
Table 8 reports the estimation results of the two-state Markov
switching model with Student t error terms. The results are quite
similar to those with normal error terms reported in Table 2. In
other words, our conclusions are robust to the fat-tailed nature
of stock returns.
5. A plausible explanation for time-varying momentum profits
We have shown that during the expansion state winner stocks
are riskier than loser stocks, while during the recession state loser
stocks are riskier than winner stocks. Consequently, the expected
momentum profits display strong procyclical variations. We now
examine the potential driving sources of time-variations in
expected momentum profits.
Other things being equal, firms with large recent positive price
moves (winners) are more likely to decrease their (financial)
leverage than firms with large recent negative price moves
(losers). Hence, a momentum sort will tend to sort firms by
recent leverage changes. Since higher leverage implies higher
systematic risk (Mandelker and Rhee, 1984), losers are riskier
than winners; hence momentum trading should have lower
expected returns. With the presence of growth options, however,
winner stocks become riskier than loser stocks, as discussed in
Section 2. Winner stocks that have had recent good performance
are more likely to increase the value of growth options than loser
stocks that have had recent bad performance. Since growth
options are riskier than assets in place, winners are riskier than
losers and hence momentum trading should have higher expected
returns. Therefore, the riskiness and expected return of momentum portfolios result from the relative importance of the leverage
and growth options effect. During expansions, when growth
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.
212
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
Fig. 6. 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.
5.1. Momentum, leverage, and growth options
This section examines how leverage and growth options differ
across momentum portfolios. We use the asset-to-equity and
debt-to-equity ratios as proxies for leverage and the market-tobook equity and market-to-book asset ratios as proxies for growth
options.19 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 9 presents the time-series averages of the asset-to-equity,
debt-to-equity, market-to-book equity, and market-to-book asset
ratios over the whole period from 1963 to 2012.20 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 debt-to-equity ratio. In contrast, we observe an opposite
pattern in the variables proxying for growth options. The market-to19
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-to-book asset is defined as the ratio of the sum of book debt and market
equity to the book value of assets, as in Goyal et al. (2002).
20
Since some of Compustat items are available from 1963, our firm characteristic
analysis begins from 1963.
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 formation. To do
this, we take the values of the four ratios proxying for leverage
and growth options over the period from t 36 months to
t + 36 months, where t is the portfolio formation month and varies
from January 1966 to December 2009, and compute the averages
over the period (t 36,t + 36). Fig. 5 illustrates the values of the
four proxy ratios of the loser and winner portfolios over the period
(t 36,t + 36). It shows that the winner portfolio has lower values
of the leverage proxy variables (asset-to-equity and debt-to-equity
ratios) and greater values of the growth option proxy variables
(market-to-book equity and market-to-book asset ratios) than
the loser portfolio does over the portfolio formation period (six
months before portfolio formation). In fact, the spread in the value
of each proxy variable between the winner and loser portfolios
sharply increases over the portfolio formation period and peaks
at the portfolio formation month (month 0). The spread begins to
decrease after the portfolio formation month but remains positive.
Overall, the results in Table 9 and Fig. 5 show that sorting firms
on past stock returns is related to sorting firms on leverage and
growth options.
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D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
Table 10
Averages of the financial ratios proxying for leverage and growth options across business cycles.
Leverage
Expansion
Recession
Difference (E - R)
(t-statistic)
Growth option
Debt-to-Equity
Asset-to-Equity
Market-to-book equity
Market-to-book asset
0.282
0.433
0.151
(6.51)
1.431
2.068
0.637
(8.14)
1.686
1.203
0.483
(12.12)
1.271
1.075
0.196
(12.38)
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 market-to-book asset ratios) across business cycles 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 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.
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 (macrolevel) leverage and growth options covary with business cycles,
since leverage and growth options are implicit driving forces in
sorting momentum portfolios.
Fig. 6 plots the aggregate values of the two proxy variables for
leverage (the asset-to-equity ratio in Panel A and the debt-toequity ratio in Panel B) and two other proxy variables for growth
options (the market-to-book ratio in Panel C and the market-tobook 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 Panels A and B) sharply increase during
recessions and tend to decrease during expansions. On the
contrary, the aggregate growth options exhibit strong procyclical
variation. The two growth option proxy variables (in Panels C
and D) sharply decrease during recessions and tend to increase
during expansions. Table 10 shows that the averages of the aggregate leverage variables are higher during recessions than during
expansions (2.068 versus 1.431 for the asset-to-equity ratio and
0.433 versus 0.282 for the debt-to-equity ratio), while the averages
of the aggregate growth option variables are higher during
expansions than during recessions (1.686 versus 1.203 for the
market-to-book equity ratio and 1.271 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 Fig. 6 and Table 10 indicate that winner stocks are
riskier during expansions, since these stocks tend to have greater
growth options and lower leverage during expansions when
growth options have a higher effect and leverage has a lower effect.
Conversely, loser stocks are riskier during recessions, since these
stocks tend to have lower growth options and greater leverage during recessions when growth options have a lower effect and leverage has a higher effect.
6. Conclusions
We examine the procyclicality of momentum profits using the
two-state Markov switching regression framework of PerezQuiros and Timmermann (2000) and find that momentum profits
display strong procyclical variation. Our results show that in the
recession state loser stocks tend to have greater loadings on the
conditioning macroeconomic variables than winner stocks, while
in the expansion state winner stocks tend to have greater loadings
on those variables than loser stocks. In other words, in recessions
loser (winner) stocks are most (least) strongly affected by aggregate economic conditions, whereas in expansions winner (loser)
stocks are most (least) strongly affected. This indicates that returns
on momentum portfolios react asymmetrically to the aggregate
economic conditions in recession and expansion states. This asymmetry across recession and expansion states for loser stocks is
identical to the asymmetry for winner stocks. This identical asymmetry for winner and loser stocks is contrasted with the results
reported by Perez-Quiros and Timmermann (2000) for size portfolios and by Gulen et al. (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 time-varying behavior of the expected momentum profit is the
opposite of the counter-cyclical behavior of the value premium
shown by Gulen et al. (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 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.
Acknowledgements
The authors would like to thank conference participants at the
2014 World Finance Conference in Venice, the 2013 11th International Paris Finance Meeting, the 2012 Financial Management
Association Annual Meeting in Atlanta, the 2012 7th International
Conference on Asia-Pacific Financial Markets in Seoul, and the
2012 Joint Conference Allied Korea Finance Associations. The paper
214
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
received the outstanding paper award at the 2012 7th International Conference on Asia-Pacific Financial Markets and the best
paper award at the 2012 Joint Conference Allied Korea Finance
Associations. Kim is supported by the Korea Research Foundation
Grant funded by the Government of Korea (NRF-2012S1A5A2A01014413). The authors thank the editor (Ike Mathur)
and an anonymous reviewer for their constructive comments and
suggestions that greatly improved the paper. Any remaining errors
are our own responsibilities.
Appendix A. Estimation methodologies of the Markov switching
regression model
Three different estimation methodologies of the two-state
Markov switching regression model are employed in this paper.
We describe each of these estimation methodologies below.
A.1. A univariate Markov switching regression model
We first specify the conditional mean Eq. (1) for the excess
return of each momentum portfolio. As exogenous variables that
affect the excess returns of the momentum portfolio, we choose
the four macroeconomic variables for the conditional mean equation as follows:
r it ¼ bi0;St þ bi1;St RRELt1 þ bi2;St DEF t1 þ bi3;St MBt2 þ bi4;St DIV t1 þ eit ;
ðA1Þ
where rit is the monthly excess return for the ith decile momentum
portfolio at time t,eit is the normally distributed random error term
with mean zero and variance r2i;St , and St = {1,2}. The conditional
variance of excess returns, r2i;St , is allowed to depend only on the
state of economy:
ln
r2i;St ¼ ki;St :
ðA2Þ
The time-varying state transition probabilities are dependent on the
level of an instrumental variable (IV) and are specified as follows:
pit ¼ Prob Sit ¼ 1jSit1 ¼ 1; yt1 ¼ U pi0 þ pi1 IV t1 ;
1 pit ¼ Prob Sit ¼ 2jSit1 ¼ 1; yt1 ;
qit ¼ Prob Sit ¼ 2jSit1 ¼ 2; yt1 ¼ U pi0 þ pi2 IV t1 ;
1 qit ¼ Prob Sit ¼ 1jSit1 ¼ 2; yt1 ;
ðA3Þ
ðA4Þ
ðA5Þ
ðA6Þ
where U() is the cumulative probability density function of a standard normal variable. We use the relative three-month Treasury bill
rate and the monthly growth rate of industrial production as the
instrumental variable.
A.2. A Bivariate Markov switching regression model
0
Let r t ¼ r Lt ; rW
be a (2 1) vector consisting of excess returns
t
on the loser and winner portfolios, r Lt and rW
t , respectively. The joint
conditional mean equation for the excess return of the loser and
winner portfolios is specified as follows:
convenience, we assume the form of the conditional covariance
matrix as follows:
!
8
X
>
>
>
ln
¼ ki;St
>
>
<
ii;St
!1=2
!1=2
>
X
X
X
>
>
>
¼
q
>
St
:
ij;St
ðA9Þ
qt ¼ PðSt ¼ 2jSt1 ¼ 2Þ ¼ Uðp0 þ p2 IV t1 Þ;
ðA10Þ
where IVt1 is the one-month lagged instrumental variable for
which the relative three-month Treasury bill rate or the monthly
growth rate of industrial production is used.
A.3. A Two-step approach to estimate a Markov switching regression
model
In addition to the simultaneous estimation of the conditional
mean equation and the state transition probabilities as in the previous cases, we also estimate the two-state Markov switching
regression model in a two-step approach. In the first step, we identify exogenously the states and the state transition probabilities as
follows: we estimate the following univariate two-state Markov
switching model for the consumption growth rates and obtain
the state transition probabilities and the high and low consumption growth volatility states.
Dct ¼ a0;St þ a1;St DEF t1 þ zt ;
ðA11Þ
ðA12Þ
qt ¼ PðSt ¼ 2jSt1 ¼ 2Þ ¼ Uðp0 þ p2 RRELt1 Þ;
ðA13Þ
where Dct = ln(Ct) ln(Ct3) is the quarter-on-quarter consumption
growth rate, Ct is the real personal consumption expenditures of
nondurable goods and services, and St is an unobservable state variable St = {1,2} that follows a two-state Markov chain with timevarying transition probability matrix.
In the second step, the state transition probabilities estimated
^t Þ are used as given probabilities in estimating
^t ; q
in the first step (p
the coefficients on the conditional mean equation for the excess
returns of each momentum portfolio. This two-step approach estimates the state transition probabilities and identifies the states
exogenously from the returns of momentum portfolios.
Appendix B . Definition of the variables
The variables used in the Markov switching regression model
are defined below.
Variable
Definition
Source
DEF
the spread between Moody’s seasoned
Baa-rated corporate bond yield (BAA) and
10-year treasury bond yield (GS10)
the monthly growth rate of industrial
production (INDPRO)
the 12-month log-difference in the
monetary base (AMBNS)
the relative three-month Treasury-bill rate
calculated as the difference between the
current T-bill rate (TB3MS) and its
12-month backward moving average
FRED
MP
MB
loser and winner portfolios’ excess returns in state St. For estimation
zt N 0; r2c;St
pt ¼ PðSt ¼ 1jSt1 ¼ 1Þ ¼ Uðp0 þ p1 RRELt1 Þ;
ðA7Þ
tive semidefinite (2 2) covariance matrix of the residuals from the
for i – j
jj;St
pt ¼ PðSt ¼ 1jSt1 ¼ 1Þ ¼ Uðp0 þ p1 IV t1 Þ;
0
is a (2 1) coefficient vector with elements bLk;St ; bW
k;St
for k = 1,2,3,4, and et is a (2 1) vector of normal residuals with
P
P
mean zero and covariance matrix St , St = {1,2}. Here St is a posi-
ðA8Þ
As in the univariate case, the time-varying transition probabilities
for the bivariate model are as follows:
rt ¼ b0;St þ b1;St RRELt1 þ b2;St DEF t1 þ b3;St MBt2 þ b4;St DIV t1 þ et ;
where bk;St
ii;St
for i ¼ j
RREL
FRED
FRED
FRED
D. Kim et al. / Journal of Banking & Finance 49 (2014) 191–215
Appendix B. (continued)
Variable
Definition
Source
DIV
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
CRSP
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