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The Expected Value Premium
Long Chen∗
Eli Broad College of Business
Michigan State University
Ralitsa Petkova†
Weatherhead School of Management
Case Western Reserve University
Lu Zhang‡
Simon School of Business
University of Rochester and NBER
November 2005
Abstract
We estimate the expected value premium using expected rates of dividend growth and
expected dividend-price ratios for value and growth portfolios. We find that during
the 1941—2002 period, the expected value premium is positive in every year, significant
in various subsamples, and countercyclical. Unlike the equity premium, there is no
noticeable trend in the expected value premium. Potential driving sources of our
results are suggested.
∗
East Lansing, MI 48821; tel: (517)353-2955, fax: (517)432-1080, email: [email protected].
10900 Euclid Avenue, Cleveland OH 44106. Tel: (216)368-8553, email: [email protected].
‡
Carol Simon Hall 3-160B, University of Rochester, Rochester, NY 14627. Tel: (585)275-3491, fax:
(585)273-1140, and email: [email protected].
†
Contents
2
1
Introduction
It is well known that value stocks outperform growth stocks in terms of realized average
returns. However, there still exists a heated controversy around the issue of whether this
value premium is due to rational compensations for risk or irrational behavior on the side
of investors. Furthermore, some recent studies (e.g., Schwert (2002)) suggest that the value
premium has declined over time, implying that academic research might have influenced the
efficiency of capital markets. In this paper we offer a fresh look at the controversy about the
value premium. We use fundamentals (dividends) to estimate the expected value premium
rather than the average realized historical return. The main advantage of this approach is
that the average realized return is at best a noisy proxy for the expected return (e.g., Elton
(1999)). The expected value premium estimates based on fundamentals will help us shed
more light on the debate about the source of the value premium and its magnitude.
In this study, we focus on three main economic questions. First, is there an ex ante
expected value premium? In the current literature, the majority of papers documenting
the existence of the value premium implicitly assume that the average realized return is an
unbiased proxy for the expected return (e.g., Fama and French (1993); Lakonishok, Shleifer,
and Vishny (1994); and Davis, Fama, and French (2000)). However, recent studies have
cast doubt on this practice. In particular, the average realized return might not converge
to the expected return in finite samples. For example, Elton (1999) observes that there are
periods longer than ten years during which the stock market return is on average lower than
the risk free rate (1973—1984), and periods longer than 50 years during which risky bonds
underperform on average the risk free rate (1927—1981).1
1
See also Brav, Lehavy, and Michaely (2003) and Campello, Chen, and Zhang (2003).
3
In light of these issues, we construct an ex ante measure of the expected value premium.
The logic behind our measure can be explained through a simple rearrangement of Gordon’s
growth formula:
R=
where R is the equity return,
D
P
D
+ g,
P
is the dividend-price ratio, and g is the dividend growth
rate. If we take expectation on both sides, the formula implies that the expected equity
return can be decomposed into an expected dividend-price ratio component and an expected
dividend growth component. Blanchard (1993) regresses the dividend-price ratio and the
future dividend growth rate on a set of conditional macroeconomic variables to analyze the
path of the expected market risk premium. The fitted values from the regressions are then
used to construct the time series of the expected equity premium. We follow the same
method in analyzing the expected value premium.
More precisely, we use the dividend growth rates of value and growth stocks to estimate
their corresponding expected rates of dividend growth, which can then be combined with the
expected dividend-price ratios to obtain expected returns. The conditional expected return
is the sum of the fitted values from time-series regressions of the realized dividend-price ratio
and a weighted average future dividend growth rate on a set of predetermined variables. This
method avoids the use of the average realized return as a proxy for the expected return. Our
work is thus in the spirit of a growing literature that uses valuation models to estimate
expected returns (e.g., Claus and Thomas (2000), Gebhardt, Lee, and Swaminathan (2000),
Fama and French (2002)).
Note that the method we use to construct the value premium lets us compute the path
of the conditional expected return spread between value and growth portfolios rather than
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the unconditional expected return spread (e.g., Fama and French (2002)). The advantage of
analyzing the conditional expected value premium is that we can answer the second research
question on our agenda: is the expected value premium countercyclical? The cyclicality of
the expected value premium might be the key to distinguishing alternative theories about
the value premium. Recent rational theories (e.g., Zhang (2003)) predict that the expected
value premium should be countercyclical. Specifically, in bad times firms usually invest at
a lower rate than the long run average. In bad times, value firms, having lower profitability
than growth firms, will start to disinvest. Since cutting capital is more costly than expanding
capital, value firms do not have enough flexibility to scale down. Therefore, value firms are
affected more by economic downturns, and are hence riskier than growth firms in bad times.
The countercyclical risk dispersion between value and growth, combined with a constant or a
countercyclical price of risk, implies a countercyclical expected value premium. In contrast,
the overreaction theory on the value premium focuses on the predictability of the abnormal
return of the value strategy, as opposed to its time-varying expected return (e.g., DeBondt
and Thaler (1985); Lakonishok, Shleifer, and Vishny (1994); and Daniel, Hirshleifer, and
Subrahmanyam (1998)).
The third question we address is: is there a long-run trend in the expected value premium?
Recent studies (e.g., Schwert (2002)) have argued that the value premium has decreased over
the past ten years, suggesting that academic research has made capital markets more efficient.
We investigate this hypothesis using an ex ante measure of the expected value premium.
Our main findings can be summarized as follows. First, there exists a significantly positive
expected value premium. Its sample average (measured by the expected HML return) is 5.1%
per year in the period from 1941 to 2002. Similar results are obtained in the subsample from
5
1963 to 2002. This evidence reinforces the view that the value premium is real and is not
due to sample selection or data-snooping biases.
Second, the expected value premium exhibits countercyclical movements. For example, in
the sample from 1941 to 2002, the correlation between the expected value premium (measured
by the expected HML return) and the default spread (a countercyclical variable) is 0.39 (pvalue 0.00). The correlation between the expected HML return and the growth rate of real
investment (a procyclical variable) is -0.28 (p-value 0.03). The results are similar over the
period from 1963 to 2002. This evidence provides support for the theoretical prediction in
Zhang (2003) that the expected value premium is countercyclical.
Third, we find that the expected value premium responds to macroeconomic shocks
significantly and in the direction predicted by rational asset pricing theory. A positive shock
to real consumption growth or real investment growth leads to lower expected value premia.
Therefore, the evidence suggests that the ex ante value premium goes up with negative
macro shocks, and it goes down with positive macro shocks. The covariance of the expected
value premium with macroeconomic factors whose risks cannot be diversified away provides
support for the view that the expected value premium is a compensation for systematic risk.
Finally, we document an increase in the expected value premium from 1950 to the early
1980s and a decrease thereafter. A combination of technological innovation and investor
learning may be responsible for this finding. In summary, we contribute to the current
literature on the value premium by providing new evidence on the magnitude and cyclical
movements in the expected value premium.
The rest of the paper is organized as follows. Section 2 describes in detail the methods we
use to construct the expected value premium. In Section 3 we document various descriptive
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statistics for the ex ante value premium and its components—the expected dividend-price ratio
and the expected dividend growth rate. Section 4 studies the trend and cyclical movements
in the expected value premium. It also examines the relation between the premium and
various macroeconomic variables. Section 5 provides a variety of robustness checks and
Section 6 offers a summary and interpretation of our results.
2
Research Design
This section describes the empirical methods we use to construct the expected value
premium. Section ?? discusses the basic idea and Section ?? presents implementation details
including the construction of our sample.
2.1
The Basic Idea
The method we use to construct expected returns follows closely that of Blanchard (1993),
who studies movements in the expected equity premium. The basic idea is to use dividend
growth rates to estimate expected rates of dividend growth, which can then be combined
with expected dividend-price ratios to obtain expected returns.
To be precise, let Rt+1 be the realized stock return from time t to t+1, i.e., 1+Rt+1 =
(Dt+1 +Pt+1 )/Pt , where Pt is the stock price known at time t, and Dt+1 is the real dividend
paid over the period from t to t+1; Dt+1 is unknown until the beginning of time t+1.
If we assume that the dividend-price ratio is stationary, we can solve for Pt as the present
value of future dividends discounted by the sequence of realized rates of return. We next
divide both sides by Dt , take conditional expectations at time t, and linearize to obtain the
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expected return at time t, denoted as Et [Rt+1 ]
Et [Rt+1 ] = Et
∙
¸
Dt+1
+ Et [Agt+1 ],
Pt
(1)
where Agt+1 is the long-run growth rate of dividends defined as the annuity value of the
growth rate of future dividends
Agt+1
¸ ∞ ∙
¸i
r̄ − ḡ X 1 + ḡ
≡
gt+i+1 ,
1 + r̄ i=0 1 + r̄
∙
(2)
with ḡ and r̄ being the average growth rate of real dividends and the average real stock
return, respectively. Finally, gt+1 denotes the growth rate of real dividends from time t to
t+1.
The interpretation of equation (??) is straightforward—the expected return conditional
on time t is the sum of the expected dividend-price ratio and the expected long-run growth
rate of dividends, both conditional on time t. Accordingly, the expected value premium
equals the sum of the dispersion in the expected dividend-price ratio and the dispersion in
the expected long-run growth rate of dividends between value and growth.
2.2
Implementation
We now discuss the details of estimating expected returns based on equations (??) and (??)
for value and growth portfolios.
Data
We obtain relevant data from three main sources. The first source is the Center for Research
in Securities Prices (CRSP) monthly stock file that contains information on stock prices,
shares outstanding, dividends, and returns for NYSE, AMEX, and Nasdaq stocks. The
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second source is the COMPUSTAT annual research file that provides accounting information
for publicly traded U.S. firms. To alleviate the potential survivorship bias due to backfilling
data, we require that firms be on COMPUSTAT for two years before using the data. The
third source is Moody’s book equity information used in Davis, Fama, and French (2000),
available from Kenneth French’s web site.2
Our sample period begins in 1941 and ends in 2002. In earlier periods, only a few firms
have data on dividends once we classify them into value and growth portfolios. Potential
problems with disclosure regulations also affect our choice of the starting date of the sample
period.
We construct value and growth portfolios by sorting on book-to-market ratios. We
implement both a one-way sort on book-to-market to obtain five quintile portfolios and
a two-way, two-by-three sort on size and book-to-market to obtain six portfolios (e.g., Fama
and French (1993)). Our timing in portfolio construction differs slightly from that commonly
used in the literature. Instead of in June, we form portfolios in December of each year t. We
use book equity from the fiscal year ending in calendar year t−1 divided by market equity
at the end of December of year t. This method avoids any look-ahead bias that might arise
as accounting information from the current fiscal year is often not available at the end of the
calendar year. Portfolio ranking is effective from January of year t+1 to December of year
t+1. This portfolio timing is lined up with the timing of dividend growth which occurs from
the beginning to the end of the calendar year. The different timing convention that we use
is not a source of concern, however. In fact, using slightly more lagged information on book
value makes it harder to find an ex ante value premium. We repeat the analysis using the
original book-to-market and size portfolios constructed by Fama and French, and we find
2
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data-library.html
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similar results.
Our definition of book equity follows that of Cohen, Polk, and Vuolteenaho (2003). In
particular, book equity is defined as stockholders’ equity plus balance sheet deferred taxes
(item 74) and investment tax credit (item 208 if available) plus post-retirement benefit
liabilities (item 330 if available) minus the book value of preferred stock. Depending on data
availability, we use redemption (item 56), liquidation (item10), or par value (item 130), in
this order, to represent the book value of preferred stock. Stockholders’ equity is equal to
Moody’s book equity (whenever available) or the book value of common equity (item 60)
plus the par value of preferred stock. If neither is available, stockholders’ equity is calculated
as the book value of assets (item 6) minus total liability (item 181).
Estimating Expected Returns
To estimate the expected returns for value and growth portfolios, we have to construct the
time series of their dividend-price ratios and dividend growth rates.
For each portfolio, we construct the real dividend-price ratio from the time series
of value-weighted realized stock returns with and without dividends and the time series
of the consumer price index from the U.S. Bureau of Labor Statistics. In particular,
¡
¢
X
(CPIt /CPIt+1 ), where Rt+1 is the nominal return with dividends
Dt+1 /Pt = Rt+1 − Rt+1
X
from time t to t + 1, Rt+1
is the nominal return without dividends over the same period,
and CPIt is the level of the consumer price index. The real dividend growth is calculated as
¢
Dt+1 /Pt ¡ X
Rt + 1 (CPIt−1 /CPIt )−1. The resulting real dividend growth rates are quite
gt+1 = D
t /Pt−1
volatile even at the portfolio level. To trim the outliers, we replace any annual observations
of dividend growth higher than 50% with 50% and those lower than −50% with −50%.
Next, we construct the long-run dividend growth rate, Agt+1 , given by equation (??), for
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each portfolio. We estimate r̄ as the sample average of the realized real equity returns, and
ḡ as the sample average of the real dividend growth rates. Agt+1 is an infinite sum of future
real dividend growth rates. In practice we use a finite sum of 100 years of future growth. We
assume that the future real dividend growth rates beyond 2002 equal the average dividend
growth rate during the 1980—2002 period. We also use the full-sample average and find the
results to be quite stable. The focus on the 1980—2002 period intends to pick up the recent
trend of dividend growth.
Finally, we regress Agt+1 and Dt+1 /Pt on a set of conditioning variables. The fitted values
from these regressions provide the two components of the expected returns. Their sum gives
us the time series of expected portfolio returns.
The set of conditioning variables includes four aggregate variables and one portfoliospecific variable. Our choice of the four aggregate conditioning variables is standard from
the time series predictability literature. These variables include: the aggregate dividend
yield, computed as the sum of dividend payments accruing to the CRSP value-weighted
portfolio over the previous 12 months, divided by the contemporaneous level of the index
(e.g., Fama and French (1988)); the default premium, defined as the yield spread between
Moody’s Baa and Aaa corporate bonds from the monthly database of the Federal Reserve
Bank of Saint Louis (e.g., Keim and Stambaugh (1986) and Fama and French (1989)); the
term premium, defined as the yield spread between a long-term and a one-year Treasury
bond from Ibbotson Associates (e.g., Campbell (1987) and Fama and French (1989)); and
the one-month Treasury bill rate from CRSP (e.g., Fama and Schwert (1977) and Fama
(1981)).
Previous studies find that the log book-to-market spread, defined as the log book-to-
11
market of portfolio ten minus the log book-to-market of portfolio one from ten deciles sorted
on book-to-market, can predict future value-minus-growth returns (e.g., Asness, Friedman,
Krail, and Liew (2000); Cohen, Polk, and Vuolteenaho (2003)). Theoretical explanation
of this predictability is provided in Gomes, Kogan, and Zhang (2003) and Zhang (2003).
Therefore, we also use the log book-to-market spread to predict the long-run dividend growth
rate and the dividend-price ratio of value strategies. We obtain data on the returns and the
year-end book-to-market ratios of all book-to-market deciles from Kenneth French’s web
site. From January to December of year t, the book-to-market of a portfolio is calculated by
dividing its book-to-market ratio at the end of December of year t−1, where book value and
market value are both measured at the end of December, by its compounded gross return
from the end of December of year t−1 to the current month of year t.
3
Preliminary Analysis
This section presents some preliminary analysis on estimating the expected value premium.
Table ?? reports descriptive statistics for returns, growth rates, dividend yields, both
realized and expected, for value and growth portfolios. We report the results for the full
sample from 1941 to 2002 and the subsample from 1963 to 2002. When we use the one-way
sort on book-to-market, we denote the resulting five portfolios as Low, 2, 3, 4, and High.
Portfolios Low and High represent the extremes in terms of book-to-market. The difference
between the returns of High and Low is denoted as p5-1 and it represent the value-minusgrowth strategy for the one-way sort of the assets.
When we use the two-way sort on size and book-to-market, we denote the resulting
six portfolios by two letters—LS, LB, MS, MB, HS, and HB. For example, portfolio LS
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contains stocks with the bottom 30% book-to-market ratios and the bottom 50% market
capitalizations. The value-minus-growth strategy for the two-way sort of the assets is defined
as 1/2*(HS+HB)-1/2*(LS+LB) and it is denoted as HML. Note that this is not the HML
factor used by Fama and French (1993) due to our different timing convention in forming
portfolios. Later in the paper we repeat the analysis using their original factor and find
similar results.
3.1
Average Returns
The first two rows of all panels in Table ?? show that value-minus-growth strategies are
profitable in our samples. For example, the average realized return of portfolio p5-1 is 5.8%
per year (t-statistic 2.88) in the full sample, and 4.6% per year (t-statistic 2.10) in the
subsample. Similarly, the average realized annual return of HML is 6% in the full sample
and 5.9% in the subsample; both are highly significant.
3.2
Growth Rates
Rows three and four in all panels of Table ?? show that the realized real dividend growth rate
of value portfolios is on average higher than that of growth portfolios, albeit the difference
is not significant. The real dividend growth rate, gt+1 , of portfolio High-minus-Low is on
average 4.5% per year in the full sample. Controlling for size increases the number further to
5.6% for HML. Moreover, the middle two rows of all panels show that the expected long-run
real dividend growth, Et [Agt+1 ], follows largely the same pattern. Except for p5-1 in the
subsample, the value portfolio has significantly higher expected long-run growth than the
growth portfolio. The average expected long-run dividend growth of HML is 3.5% per year
(t-statistic 19.90) in the full sample, and 2.8% (t-statistic 21.00) in the subsample.
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Importantly, our evidence does not contradict the conventional wisdom that growth
stocks have higher proportions of growth opportunities and grow faster than value stocks.
The reason is that our evidence is obtained from portfolios rebalanced annually, while the
conventional wisdom is based on portfolios with fixed sets of firms without rebalancing.
To illustrate this point, we study the event-time evolution of profitability (defined as the
earnings-lagged book equity), the rate of dividend payment (defined as the dividend-lagged
book equity), and the real dividend growth rate for value and growth stocks during 21 years
around the portfolio formation year. Our test design follows closely that of Fama and French
(1995). We define value and growth portfolios both from book-to-market quintiles and from
six size and book-to-market portfolios. The sets of stocks in the value and growth portfolios
are fixed throughout the event years.
The sample we use is from 1941 to 2002. We back out data on dividends from valueweighted portfolio returns with and without dividends. Since data on earnings is not available
for the pre-COMPUSTAT years we follow Cohen, Polk, and Vuolteenaho (2003) and use the
clean-surplus relation to compute earnings from data on book equity and dividends, i.e.,
earnings(t) =book value(t)−book value(t − 1)+dividends(t). To be consistent, we use this
method to compute earnings in later years even when earnings data from COMPUSTAT is
available.
Figure ?? reports the results. First, Panels A and D confirm the main result in Fama
and French (1995, Figure 1) that growth firms are persistently more profitable than value
firms. Panels B and E show that growth firms also have higher rates of dividends than value
firms. The dispersion in dividend rates appears even more persistent than the dispersion in
profitability, especially for the two-way sort. More importantly, Panels C and F show that
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the real dividend growth rates of growth stocks are higher than the real dividend growth rates
of value stocks. This dispersion is large at the portfolio formation year. It remains positive
for almost ten years for the one-way sort. However, for the two-way sort the dispersion is
much more short-lived and converges in about three years.
However, when the portfolios are rebalanced annually, the firms in the growth portfolio
next year are not the same as the firms in the growth portfolio this year. There is no
particular reason to expect the dividend growth of growth portfolios to be higher than the
dividend growth of value portfolios, given that growth rates are measured using different
sets of firms. Consistent with this observation, Figure ?? plots the time series of the annual
realized real dividend growth rates for portfolios five and one in the book-to-market quintiles
(Panel A), portfolio five-minus-one (Panel B), portfolios High and Low from the six portfolios
sorted on size and book-to-market (Panel C), and HML (Panel D). Panels B and D show that
the real dividend growth rates for the value-minus-growth strategies are frequently negative.
3.3
Expected Returns
Not surprisingly, Table ?? reports that the average expected dividend-price ratio,
Et [Dt+1 /Pt ], is higher for value firms than for growth firms. Given that the expected long-run
dividend growth and the expected dividend-price ratio are both higher for value firms, their
expected returns are even higher than the expected returns of growth firms. The last two
rows of Panels A and B show that the expected return of p5-1 is 3.7% per year (t-statistic
14.91) in the full sample, and 2.8% (t-statistic 12.05) in the subsample. Similarly, the last
two rows of Panels C and D show that the expected return on HML is 5.1% (t-statistic 40.89)
in the full sample, and 5.1% (t-statistic 31.73) in the subsample. To sum up, the expected
value premium is significantly positive on average.
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An interesting result that emerges from Table 1 is that the expected return for each
portfolio in the one-way and two-way sorts is substantially lower than the average realized
return of the portfolio. Such a result has been documented before for the excess return on the
market portfolio. For example, Fama and French (2002) report that for the sample period
1951-2002 the expected equity premium is 4.32% per year, while the average realized equity
premium is 7.43% per year. Our evidence suggests that this result holds in the cross-section
of portfolios sorted by size and book-to-market. However, the expected value premium within
each sorting procedure is not far away from the average realized return. This indicates that
the difference between the expected return and the average realized return cancels out at
the cross-sectional level.
We next study the time series of the expected p5-1 return, the expected HML return,
their respective expected long-run dividend growth rates, and their expected dividend-price
ratios. Figure ?? plots the sample paths. Panels A and C show that the expected p51 and HML returns are positive throughout the sample. Both series of expected returns
display positive spikes during most of the recessions in the sample. They also seem to covary
positively with the default premium, a well-known countercyclical variable commonly used to
predict the business cycle and the excess return of the market (e.g., Jagannathan and Wang
(1996)). Therefore, this informal analysis suggests that the expected value premium is also
countercyclical. Panels A and C also suggest that the expected value premium increased in
the period 1950-1980 and decreased after that.
Panels B and D of Figure ?? show that the expected dividend growth rates and the
expected dividend-price ratios of p5-1 and HML covary negatively with each other. There
has been a noticeable decline in the expected long-run dividend growth from the early 1940s
16
to the early 1980s, but an increase thereafter. The expected dividend-price ratios display the
opposite long-term movements. As a result, these graphs do not show an obvious trend in
the expected value premium, although the expected p5-1 return appears to decline slightly
over time.
Finally, Table ?? reports the results of autocorrelations and MacKinnon (1994) unit
root tests for the expected p5-1 return, the expected HML return, and their respective
components—the expected dividend-growth rate and the expected dividend-price ratio. The
table shows that the two separate components are more persistent than the expected value
premium itself. The unit root tests fail to reject the unit-root null for the expected dividendprice ratios of both p5-1 and HML, but the null is rejected for the expected dividend growth
rates and the expected value premium.
3.4
The Equity Premium
The method we use to derive the expected value premium is the dynamic extension of the
method used in Fama and French (2002) to compute the equity premium. It is therefore
interesting to compare the properties of the equity premium constructed our way to those
documented in Fama and French. First, our estimates are close to those obtained by Fama
and French (2002). During the 1951—2000 period studied by Fama and French, our estimates
of the expected long-run real dividend growth rate, the expected real dividend yield, the
expected real equity return, and the average realized real market return are 0.84%, 4.07%,
4.91%, and 10.21%, respectively. These numbers are reasonably close to their counterparts
in Fama and French—1.05%, 3.70%, 4.75%, and 9.62%, respectively. The differences are likely
due to the way we use to construct our sample.
Second, consistent with Fama and French (2002), the equity premium that we estimate
17
is much lower than the average realized real equity return. The expected equity premium
from 1941 to 2002 is 4.25% per year, compared to the realized real equity premium of 7.76%
per year for the same period.
More importantly, our estimates also show that the equity premium has declined over
time. The time series plot in Figure ?? shows that the equity premium reaches its peak of
about 10% in the early 1950s, declines over the next two decades to about 2.5% in the mid
1970s, climbs up to about 5% in the mid 1980s, then declines over the next one and half
decades to about 1% in the early 2000s. Using the equity premium in a trend regression
yields a negative slope of -0.077% (t-statistic -8.46) in the 1941—2002 sample. Although the
slope is only an insignificant -0.021% (t-statistic -1.65) in the post-1963 sample, it increases
in magnitude to -0.142% (t-statistic -7.73) in the sample after 1980.
4
The Expected Value Premium
We now study the dynamics, including both trend and cyclical movements, of the expected
value premium in more details.
4.1
Trend Dynamics
We use two methods to isolate the cyclical component of the expected value premium from
the low-frequency, trend component. The first method is to regress the expected value
premium on a time trend
ValuePremiumt = a + b t + εt ,
(3)
where the fitted component including the intercept is defined as the trend component and
the residual is defined as the cyclical component. The second method is to pass the expected
18
value premium through the Hodrick-Prescott (1997) (HP) filter that separates the trend and
the cyclical components of the premium.
The results from regression (??) indicate that the expected p5-1 return exhibits a
downward trend in the 1941—2002 and the 1963—2002 samples, but the expected HML return
does not. The expected HML return exhibits a downward trend in the 1980—2002 sample,
but the expected p5-1 return does not. Specifically, the slope coefficient, b, is -0.078% for the
expected p5-1 return (t-statistic -8.01) in the 1941—2002 sample, -0.052% (t-statistic -2.83)
in the post-1963 sample, and -0.091% (t-statistic -1.86) in the 1980—2002 sample. The slope
is -0.010% (t-statistic -1.47) for the expected HML return in 1941—2002, -0.011% (t-statistic
-0.76) in 1963—2002, and -0.097% (t-statistic -3.12) in the 1980—2002 sample.
Using the HP filter yields somewhat different results. Figure ?? plots the trend and
cyclical components of the expected p5-1 return and the expected HML return. Panel B
shows a downward movement in the HP-filtered, low frequency component of the expected
p5-1 and HML returns. Panel A shows the patterns for the low frequency movements of
p5-1 and HML in the case of the time-trend regression (estimated using the full sample). As
pointed out before, there is a downward trend for p5-1 and no clear trend for HML. Overall,
the evidence based on the time-trend regression and the HP filter indicates a downward
trend in the low-frequency movements of the expected value premium.
4.2
Cyclical Dynamics
Panels C and D of Figure ?? trace the cyclical components of the expected p5-1 and HML
return. Panel C is based on the residuals from the time-trend regression, while Panel D is
based on the HP filter. Both panels plot the NBER recession dummy which takes the value
of one in recessions and zero otherwise. The graphs show that the expected value premium
19
is countercyclical—the expected value premium rises in recessions and drops in expansions.
We use two additional methods to study the cyclical properties of the expected value
premium. As an informal test, we first report the lead-lag cross-correlation structure of
the expected value premium with a list of cyclical indicators. We then supplement the
cross-correlations with a more formal VAR analysis.
Cross Correlations
Table ?? reports the cross-correlations. The list of cyclical indicators includes the default
premium, the NBER recession dummy, the real investment growth rate, and the real
consumption growth rate. Among the four variables, the default premium and the recession
dummy are countercyclical, while the real investment and consumption growth rates are
procyclical.
The middle column in Panel A of Table ?? shows that the contemporaneous correlation
between the expected p5-1 return and the default premium is 0.49 and that between the
expected p5-1 return and the recession dummy is 0.50. Both correlations are significant.
Further, the contemporaneous correlation between the expected p5-1 return and real
investment growth is -0.36 and that between the expected p5-1 return and real consumption
growth is -0.32 (both are significant). The middle column in Panel B shows that using the
expected HML return yields similar results. To sum up, the evidence suggests that the
expected value premium is countercyclical.
The middle columns of the two panels also show that the countercyclical property of the
expected value premium derives mostly from a similar property of the expected dividendprice ratio. The dividend price ratios of p5-1 and HML have significantly positive correlations
with the default premium. Their correlations with the other cyclical variables are not
20
significant. In contrast, the expected dividend growth rates of p5-1 and HML exhibit only
weak comovements with the business cycle, relative to the expected dividend-price ratio.
VAR Analysis
We next supplement the lead-lag correlations with a more formal VAR analysis. The VAR
contains one measure of the expected value premium (either the expected p5-1 or the HML
return) and one cyclical indicator. We use two cyclical variables separately in the VAR—the
investment growth and the real consumption growth. Using other cyclical variables yields
similar results (not reported). The lag in the VAR is one, which is chosen according to the
Akaike information criterion. In some specifications, we also include the one-month T-bill
rate in the VAR in an effort to isolate the business cycle shock from the monetary policy
shocks.
Table ?? reports the results. Panel A shows that the coefficients of real investment growth
in the VAR are negative and significant (in the full sample period for portfolio p5-1 they are
marginally significant). The result holds with and without controlling for the T-bill. Panel
B shows that the coefficients of real consumption growth rate are all negative; they are all
significant for both the long and the short sample periods. These results suggest that the
expected value premium responds negatively to aggregate shocks. A positive shock to real
consumption growth or real investment leads to lower expected value premia.
To interpret the magnitudes of the VAR coefficients and to quantify the business cycle
sensitivity of the expected value premium, we study the impulse response functions implied
by the estimated VARs. Figure 6 plots these impulse response functions of the expected
value premium in the presence of a one standard deviation shock to the explanatory variable.
Panels A—C report the response of the expected value premium to a one-standard deviation
21
positive shock to real investment growth, while Panels E—H report the results relative to
real consumption growth. In all cases, the premium responds negatively to good news about
investment and consumption.
4.3
Predictive Regressions
Previous studies (e.g., Asness et al. (2000) and Cohen, Polk, and Vuolteenaho (2003))
document that the realized value premium is predictable using the log book-to-market spread,
suggesting that the expected value premium is time-varying. We investigate this issue further
within our empirical framework. Specifically, we regress the value premium defined as the
sum of the long-run dividend growth, Agt+1 , and the dividend-price ratio, Dt+1 /Pt , on a set
of conditioning variables including the aggregate dividend yield, the default premium, the
term premium, the log book-to-market spread, and the one-month T-bill rate. To study the
predictability of the two separate components of the value premium, we also regress Agt+1
and Dt+1 /Pt on the same set of conditioning variables. These regressions have been used to
construct the expected dividend growth and the expected dividend-price ratio. To adjust
for the small-sample bias in the slopes and their standard errors (e.g., Stambaugh (1999)),
we use the simulation method of Nelson and Kim (1993).
Table ?? reports the results. The first three rows of all panels show that the value
premium is indeed predictable. The adjusted R2 clusters around 30% except for the HML
return in the 1941—2002 sample that has an adjusted R2 of 15.60%. Although not reported
in the table, the null hypothesis that all the slopes are jointly zero is strongly rejected
in all cases. The results suggest that the aggregate dividend yield has reliable predictive
power with positive slopes, except for the HML return in the period from 1941 to 2002.
Further, the term premium has significant predictive ability for the expected value premium
22
with negative slopes in all cases and across both sample periods. Consistent with previous
studies, we find that the log book-to-market spread is largely a positive predictor of the value
premium, except for predicting the p5-1 return in the post-1963 sample (Panel B). However,
our results also show that its predictive power is weak, given that all corresponding p-values
are close to 0.20. The default premium and the short-term rate do not have significant
predictive ability for the expected value premium in the presence of the other conditioning
variables.
The rest of Table ?? reports the results of predicting the two separate components of
the value premium including the dividend growth and dividend-price ratio. The set of
conditioning variables does an overall better job in predicting the separate components than
the value premium itself. This is reflected in much higher adjusted R2 s; in some cases the
goodness-of-fit coefficients are more than doubled. Moreover, in some cases, the conditioning
variables have different signs when predicting the two components, consistent with the results
in Figure ?? that the expected dividend growth and the expected dividend-price ratio are
negatively correlated. Specifically, the short rate is a significantly positive predictor of the
long-run dividend growth, and a significantly negative predictor of the dividend-price ratio.
A similar result holds for the term premium variable.
5
Robustness
This section reports a set of robustness checks such as incorporating stock repurchases into
the computation of the payout, using an alternative set of instruments to model expectations,
and using a different set of portfolios to measure the value premium.
23
5.1
Stock Repurchases
Fama and French (2001) document that the proportion of firms paying dividends has fallen
dramatically over the years regardless of their levels of earnings. Grullon and Michaely
(2002) show that dividends have been the dominant form of payout in earlier periods, but
repurchases have become more important in recent years.
In light of this evidence, we add net stock repurchases as part of the dividends. We
replicate the results on ḡt+1 , Et [Agt+1 ], Et [Dt+1 /Pt ], and Et [Rt+1 ] in Table ??, the time series
plots in Figure ??, and the VAR analysis in Table ?? and Figure 6.
Table 6 reports the descriptive statistics for returns, dividend growth rates, and dividend
yields, both realized and expected, for value and growth portfolios.
Including stock
repurchases in the payout makes all numbers in the table larger than their counterparts
in Table ??. The value-minus-growth strategies are even more profitable in the case of
repurchases—for example, the average realized return of p5-1 is 6.4% per year in the full
sample and 5.7% per year in the subsample. Both values are significant. A similar result
holds for the average realized return of HML. The expected value premium remains positive
and significant—for example, the average expected return of p5-1 is 4.5% per year in the full
sample and 4.2% per year in the subsample. The average expected return of HML is higher—
it is 5.5% per year for 1941—2002 and 5.7% for 1963—2002. The table also shows that value
portfolios have significantly higher expected long-run dividend growth rates and expected
dividend-price ratios than growth portfolios.
Figure 7 plots the trend and cyclical components of the expected p5-1 and HML returns,
taking into account stock repurchases. Panel B shows that there is no obvious downward
or upward movement in the HP-filtered, low-frequency component of both HML and p524
1. Panel A, however, shows a downward trend for the p5-1 portfolio using the time trend
regression method. There is a slight upward trend for HML in Panel A.
Table 7 reports the results from the VAR analysis of the expected value premium in the
case of stock repurchases. In Panel A, the coefficients of real investment growth in the VAR
are negative and significant. The result holds with and without controlling for monetary
policy. Panel B shows that the coefficients of real consumption growth rate are all negative;
they are all significant for both the long and the short sample periods. These results suggest
that the expected value premium responds negatively to aggregate shocks.
Finally, Figure 8 presents the impulse response functions implied by the estimated VAR in
Table 7. Panels A—C report the response of the expected value premium to a one-standard
deviation positive shock to real investment growth, while Panels E—H report the results
relative to real consumption growth. In all cases, the expected value premium responds
negatively to good news about investment and consumption.
In summary, when stock repurchases are included in the computation of the expected
value premium, the results are qualitatively very similar to the ones reported for our base
case. The expected value premium is still significantly positive and countercyclical. The
new result that emerges is that there does not exist a clear trend pattern in the movement
of the expected value premium.
5.2
Alternative Instruments
We also study the robustness of our benchmark results with respect to alternative sets of
instruments. We first exclude the log book-to-market spread from the list of conditioning
variables. The results look qualitatively very similar to the ones reported for our base case
(they are not reported for the sake of brevity, but are readily available upon request).
25
Alternatively, we include Lettau and Ludvigson’s (2001, 2004) cay and cdy variables in
the set of conditioning variables used to predict the dividend growth and the dividend-price
ratio of value and growth. The new results indicate that there is a significant downward
trend in the expected HML return once we use the HP filter. This is consistent with our
previous findings based on the benchmark case. The impulse response functions from the
VAR analysis indicate that, once we control for monetary policy, the value premium responds
negative to good news about investment and consumption in the first year, but the response
is positive in the second year.
5.3
Alternative Portfolios
Finally, we use the portfolios already constructed by Fama and French to measure the
expected value premium and its components.
These portfolios include the group of 6
constructed by a 2x3 sort on size and book-to-market, as well as the 25 portfolios constructed
by a 5x5 sort on size and book-to-market. The results reported before are robust to the choice
of these portfolio. They are not reported for the sake of brevity, but are readily available
upon request.
6
Summary and Interpretation
In this paper we use fundamentals (dividends) to estimate the expected value premium. We
show that over the period from 1941 to 2002, the expected value premium is positive and
significant. Its magnitude is 5.1% per year based on the expected HML return. In addition,
we document that the expected value premium is countercyclical—it rises in recessions and
decreases in expansions. Furthermore, the expected value premium tends to be positively
correlated with countercyclical variables like the default spread and negatively correlated
26
with procyclical variables like the growth in real consumption. We also report that the
expected value premium increases with negative macroeconomic shocks and decreases as a
result of positive macroeconomic shocks. There is some evidence that the expected value
premium has decreased over time.
What might explain the time variation in the expected value premium? Why is the
expected value premium be countercyclical? Why do the dividend growth rates of value
firms respond more to shocks to aggregate economic conditions? The rational asset pricing
model of Zhang (2003) provides a unified explanation for the cyclical properties of the value
premium and the spread in dividend growth between value and growth.
Using a neoclassical framework, Zhang (2003) argues that the expected value premium is
countercyclical. There are two key features in Zhang’s model. First, costly reversibility
implies that it is more costly for firms to cut their productive assets than to expand
them (e.g., Abel and Eberly (1994)). Second, a countercyclical price of risk implies that
shareholders’ discount rates are higher in bad times than in good times (e.g., Fama and
French (1989) and Campbell and Cochrane (2000)).
Specifically, firms usually invest at a lower rate than the long run average rate in
recessions. Value firms, having lower profitabilities than growth firms, will start to disinvest.
Without costly reversibility, value firms will have enough flexibility to scale down. With
costly reversibility, disinvestment implies higher adjustment costs which prevents value firms
from scaling down as much as they would have otherwise.
A countercyclical price of risk further complements and propagates the effects of costly
reversibility. When the price of risk is countercyclical, firms’ discount rates are higher in
recessions. Higher discount rates lead to lower conditional expectation of the firm value as
27
the net present value of its future dividends. As future prospects become gloomier, value
firms would scrap even more capital than in the case with a constant price of risk. Because
costly reversibility prevents value firms from disinvesting, these firms are burdened with
more unproductive assets in bad times.
Growth firms, on the other hand, are relatively immune to the effects of costly reversibility
and a countercyclical price of risk in recessions. Growth firms have more productive assets,
and hence have fewer incentives to scale down in recessions. The net effect is a countercyclical
value premium. In summary, we test the following refutable hypothesis formulated by Zhang
(2003): The expected value premium is countercyclical; in particular, the expected returns
of value firms are more sensitive to negative shocks to aggregate economic conditions than
those of growth firms.
28
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30
30
Et [Rt+1 ]
Et [Dt+1 /Pt ]
Et [Agt+1 ]
gt+1
Rt+1
Et [Rt+1 ]
Et [Dt+1 /Pt ]
Et [Agt+1 ]
gt+1
Rt+1
0.085
2.89
0.009
0.33
0.003
3.54
0.032
10.66
0.035
10.36
LS
0.071
3.12
0.013
0.59
0.017
14.85
0.031
17.55
0.048
23.63
Low
0.090
4.59
0.026
1.22
0.019
16.15
0.048
28.63
0.067
27.54
3
0.105
4.58
0.036
1.53
0.028
32.58
0.054
28.01
0.082
35.67
4
0.129
4.60
0.058
2.12
0.031
12.14
0.054
30.08
0.085
21.69
High
0.073
3.29
0.014
0.69
0.018
15.19
0.033
18.97
0.051
25.22
LB
0.123
4.47
0.050
2.07
0.041
44.34
0.042
16.87
0.083
28.45
MS
0.080
3.96
0.015
0.72
0.008
5.47
0.048
28.23
0.056
19.86
MB
0.157
5.12
0.087
3.45
0.062
33.75
0.040
25.51
0.102
32.67
HS
0.120
4.56
0.048
1.99
0.030
19.64
0.057
29.86
0.087
27.13
HB
Panel C: 1941–2002, two-way sort
0.075
3.60
0.016
0.60
0.017
17.70
0.041
22.22
0.058
24.36
2
Panel A: 1941–2002, one-way sort
0.060
4.24
0.056
2.07
0.035
19.90
0.016
9.90
0.051
40.89
HML
0.058
2.88
0.045
1.42
0.014
4.65
0.023
15.10
0.037
14.91
p5-1
0.066
1.83
0.005
0.14
0.001
1.156
0.017
13.77
0.018
8.78
LS
0.058
2.01
0.016
0.53
0.019
11.06
0.023
29.97
0.041
20.64
Low
0.077
3.58
0.023
0.79
0.014
11.59
0.042
27.47
0.056
31.94
3
0.103
4.41
0.039
1.30
0.025
24.64
0.050
21.17
0.075
29.53
4
0.104
4.03
0.036
1.04
0.019
13.18
0.050
23.14
0.069
23.21
High
0.059
2.12
0.015
0.51
0.020
11.51
0.025
31.06
0.045
20.81
LB
0.113
3.57
0.053
1.52
0.039
30.08
0.031
21.42
0.070
29.99
MS
0.066
3.00
0.011
0.41
0.001
1.12
0.042
25.33
0.044
20.00
MB
0.143
4.43
0.077
2.49
0.054
40.33
0.034
21.82
0.088
39.60
HS
0.101
4.11
0.033
1.13
0.023
25.60
0.054
22.51
0.077
24.49
HB
Panel D: 1963–2002, two-way sort
0.058
2.33
0.013
0.34
0.016
11.69
0.033
30.36
0.049
23.19
2
Panel B: 1963–2002, one-way sort
0.059
3.81
0.045
1.22
0.028
21.00
0.023
14.74
0.051
31.73
HML
0.046
2.10
0.020
0.48
0.000
0.12
0.028
14.60
0.028
12.05
p5-1
This table reports the sample averages of the realized return, Rt+1 , the realized dividend growth, gt+1 , the expected long-run dividend growth,
Et [Agt+1 ], the expected dividend yield, Et [Dt+1 /Pt ], and the expected return, Et [Rt+1 ] for various value and growth portfolios. The corresponding
t-statistics are reported in the rows below the sample averages. The results for both the full sample from 1941 to 2002 and for the subsample from
1963 to 2002 are reported. Panels A and B contain the results for five quintiles sorted on book-to-market, while Panels C and D contain the results
for six portfolios based on a two-by-three sort on size and book-to-market. In Panels A and B, p5-1 denotes the difference between portfolio High
and portfolio Low in the five book-to-market quintiles. In Panels C and D, portfolios are denoted by two letters, for example, portfolio LS contains
stocks with the bottom 30% book-to-market ratios and the bottom 50% market capitalization. Significant average realized and expected returns
for p5-1 and HML and their t-statistics are highlighted.
Table 1 : Descriptive Statistics for The Realized Returns, Realized Dividend Growth, Expected Long-run Dividend
Growth, Expected Dividend Yield, and Expected Returns of Value and Growth Portfolios
Table 2 : Autocorrelations and Unit Root Tests for the Expected Value Premium
and Its Two Components (1941—2002)
This table reports autocorrelations of orders 1, 2, 3, 4, 5, and 10, MacKinnon unit root test statistics, and
their corresponding p-values for the expected return of portfolio 5-1 (Panel A), the expected return of HML
(Panel B), and their respective two components—the expected long-run dividend growth rate, Et [Agt+1 ] and
the expected dividend-price ratio, Et [Dt+1 /Pt ].
Panel A: p5-1
Autocorrelations (order)
Et [Agt+1 ]
Et [Dt+1 /Pt ]
Premium
Unit-Root Test
1
2
3
4
5
10
Statistic
p-value
0.811
0.873
0.638
0.731
0.720
0.473
0.675
0.608
0.368
0.615
0.583
0.295
0.599
0.574
0.323
0.496
0.235
0.275
-2.889
-1.992
-5.180
0.047
0.290
0.000
Panel B: HML
Autocorrelations (order)
Et [Agt+1 ]
Et [Dt+1 /Pt ]
Premium
Unit-Root Test
1
2
3
4
5
10
Statistic
p-value
0.740
0.916
0.534
0.576
0.808
0.188
0.513
0.728
0.028
0.468
0.705
-0.066
0.459
0.688
0.010
0.440
0.423
0.070
-3.740
-1.705
-6.058
0.004
0.428
0.002
31
32
Real gCON
Real gINV
Cycle
DEF
Real gCON
Real gINV
Cycle
DEF
Real gCON
Real gINV
Cycle
DEF
0.43
0.00
-0.08
0.56
-0.01
0.93
-0.14
0.29
-0.32
0.02
0.06
0.64
0.01
0.92
0.03
0.85
0.09
0.50
-0.13
0.32
-0.01
0.92
-0.07
0.61
-5
-3
-2
-1
0
1
2
3
4
0.13
0.32
0.03
0.83
0.18
0.18
0.17
0.19
0.42
0.00
0.28
0.03
-0.14
0.27
0.07
0.62
-0.39
0.00
0.20
0.13
0.25
0.06
0.23
0.08
-0.34
0.01
0.15
0.27
0.24
0.07
0.15
0.26
-0.27
0.03
0.14
0.28
-0.02
0.91
0.09
0.48
-0.27
0.04
0.13
0.30
-0.16
0.21
-0.10
0.42
-0.38
0.00
- 0.03
0.80
-0.11
0.40
-0.12
0.38
-0.58
0.00
-0.15
0.26
-0.04
0.74
0.03
0.81
-0.54
0.00
-0.08
0.54
0.04
0.76
0.14
0.28
-0.46
0.00
-0.02
0.86
0.05
0.70
0.14
0.31
0.49 0.28 -0.22 -0.29 -0.13
0.00 0.03 0.10 0.02 0.33
0.50 0.12 -0.34 -0.24 -0.03
0.00 0.37 0.01 0.06 0.83
-0.36 -0.35 -0.15 0.15 0.18
0.00 0.01 0.26 0.25 0.19
-0.32 -0.54 -0.26 0.13 0.21
0.01 0.00 0.04 0.32 0.11
Expected Dividend Growth, Et [Agt+1 ]
-0.03
0.83
-0.03
0.82
0.13
0.34
0.20
0.13
0.47
0.00
-0.10
0.45
-0.02
0.86
-0.12
0.38
0.45
0.00
-0.14
0.29
-0.05
0.71
-0.11
0.42
0.46
0.00
-0.11
0.40
0.03
0.82
-0.02
0.88
0.59
0.00
-0.03
0.84
-0.00
0.99
0.01
0.92
0.77 0.92 0.69 0.49 0.41
0.00 0.00 0.00 0.00 0.00
0.19 0.27 0.07 -0.05 -0.08
0.14 0.04 0.62 0.69 0.55
-0.04 -0.20 -0.23 -0.09 0.03
0.75 0.12 0.08 0.48 0.82
-0.04 -0.26 -0.36 -0.22 -0.01
0.74 0.04 0.01 0.09 0.96
Expected Dividend-Price Ratio, Et [Dt+1 /Pt ]
-0.36
0.01
0.19
0.15
0.26
0.05
0.23
0.08
0.05
0.71
-0.01
0.94
0.22
0.10
0.22
0.10
Expected Value Premium, Et [Agt+1 ] + Et [Dt+1 /Pt ]
-4
Panel A: p5-1
-5
-3
-2
-1
0
1
2
3
4
-0.22
0.10
-0.28
0.03
-0.24
0.06
-0.31
0.02
-0.31
0.02
-0.25
0.06
0.07
0.63
0.04
0.74
-0.48
0.00
-0.08
0.54
0.08
0.57
0.18
0.18
-0.33
0.01
0.05
0.72
0.01
0.92
0.07
0.61
-0.21
0.11
-0.15
0.26
0.21
0.12
0.24
0.08
0.43 0.49 0.52 0.49 0.50 0.60 0.70
0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.01 -0.08 -0.16 -0.19 -0.12 0.01 0.16
0.93 0.56 0.23 0.14 0.36 0.93 0.21
0.07 -0.02 -0.14 -0.16 -0.07 -0.00 0.00
0.58 0.86 0.29 0.24 0.58 0.97 0.98
0.04 -0.10 -0.18 -0.16 -0.04 -0.02 -0.03
0.78 0.45 0.19 0.23 0.74 0.91 0.84
0.77
0.00
0.17
0.18
-0.12
0.35
-0.18
0.16
0.60
0.00
0.02
0.89
-0.14
0.28
-0.26
0.05
0.44
0.00
-0.06
0.65
-0.03
0.83
-0.14
0.30
0.38
0.00
-0.06
0.67
0.05
0.71
0.03
0.83
Expected Dividend-Price Ratio, Et [Dt+1 /Pt ]
-0.42 -0.34 -0.38 -0.38 -0.27 -0.17 -0.19 -0.42 -0.57
0.00 0.01 0.00 0.00 0.04 0.20 0.14 0.00 0.00
0.03 -0.01 0.17 0.18 0.14 0.15 0.13 -0.11 -0.22
0.82 0.96 0.20 0.17 0.29 0.24 0.30 0.41 0.09
0.04 0.00 0.34 0.31 0.26 -0.07 -0.26 -0.14 0.00
0.77 0.99 0.01 0.02 0.04 0.62 0.04 0.28 0.99
0.12 -0.01 0.30 0.29 0.15 0.05 -0.21 -0.20 0.05
0.40 0.96 0.02 0.03 0.25 0.70 0.10 0.13 0.71
Expected Dividend Growth, Et [Agt+1 ]
0.36
0.00
0.18
0.16
-0.34
0.01
-0.49
0.00
Expected Value Premium, Et [Agt+1 ] + Et [Dt+1 /Pt ]
-4
-0.05 0.13 0.10 0.02 0.08 0.28 0.39
0.69 0.33 0.47 0.88 0.53 0.03 0.00
0.06 -0.07 0.08 0.09 0.05 0.16 0.41
0.68 0.62 0.56 0.51 0.68 0.21 0.00
0.14 0.04 0.19 0.11 0.14 -0.16 -0.28
0.32 0.77 0.15 0.43 0.28 0.23 0.03
0.12 -0.05 0.16 0.11 0.08 0.04 -0.24
0.39 0.72 0.24 0.43 0.56 0.77 0.06
5
Panel B: HML
0.42
0.00
0.02
0.90
0.05
0.72
0.03
0.85
-0.34
0.01
0.02
0.89
-0.00
0.98
0.02
0.86
-0.08
0.55
0.07
0.61
0.21
0.11
0.22
0.10
5
This table reports the lead-lag correlations between the expected value premium, expected dividend growth, and expected dividend-price ratio of
portfolios p5-1 and HML and a list of cyclical indicators including the default premium, DEF, the NBER recession dummy, Cycle, real investment
growth, gINV , and real consumption growth, gCON . Panel A reports the results for portfolio 5-1, and Panel B does the same for HML. p-values are
reported in the rows below the correlations.
Table 3 : Lead-Lag Correlations between the Expected Value Premium, Expected Dividend Growth, and Expected
Dividend-Price Ratio and Cyclical Indicators (1941–2002)
33
Expected HML Return
Expected p5-1 Return
-0.008
0.071
-0.016
0.021
no T-bill
-0.007
0.066
-0.017
0.020
with T-bill
1941–2002
-0.031
0.015
-0.042
0.029
no T-bill
-0.024
0.006
-0.041
0.026
with T-bill
1963–2002
Panel A: Real gINV
-0.158
0.000
-0.239
0.000
no T-bill
-0.122
0.000
-0.231
0.000
with T-bill
1941–2002
-0.255
0.000
-0.342
0.001
no T-bill
-0.143
0.005
-0.295
0.004
with T-bill
1963–2002
Panel B: Real gCON
This table reports the results from a first-order VAR that includes the expected value premium and one of two cyclical indicators–real investment
growth, gINV , and real consumption growth, gCON . The table reports the equation for the expected value premium for the 1941–2002 and 1963–2002
samples. We also report the results with and without controlling for monetary policy as captured by the one-month T-bill rate. The lag in the VAR
is one, which is chosen based on the Akaike information criterion. p-values associated with Newey-West t-statistics adjusted for heteroscedasticity
and autocorrelation of up to six lags are reported in the rows below the coefficients.
Table 4 : VAR Analysis
34
0.055
0.038
0.016
Agt+1
bias
p
Dt+1 /Pt
bias
p
-0.002
0.000
0.221
0.005
-0.001
0.036
0.003
-0.001
0.128
div
intercept
Premium
bias
p
0.001
-0.001
0.346
0.028
Dt+1 /Pt
bias
p
0.007
-0.001
0.071
0.007
-0.002
0.034
0.017
0.045
div
Agt+1
bias
p
Premium
bias
p
intercept
0.001
-0.001
0.252
-0.017
0.001
0.002
-0.016
0.001
0.002
term
0.006
0.002
0.290
0.006
-0.001
0.310
0.012
0.001
0.213
ls
0.008
-0.001
0.004
-0.004
0.000
0.183
0.004
-0.001
0.099
def
0.002
-0.001
0.150
-0.008
0.000
0.008
-0.006
0.000
0.049
term
-0.003
0.002
0.263
0.014
0.000
0.076
0.011
0.003
0.180
ls
Panel C: 1941–2002, HML
0.013
-0.001
0.000
0.003
0.000
0.385
0.015
0.000
0.015
def
Panel A: 1941–2002, p5-1
0.009
0.000
0.007
-0.006
-0.001
0.112
0.003
-0.001
0.212
rf
0.007
0.000
0.044
-0.020
0.000
0.006
-0.013
0.001
0.025
rf
0.395
0.794
0.156
adj. R
2
0.581
0.540
0.301
adj. R
2
0.008
0.058
0.066
intercept
0.011
0.028
0.039
intercept
-0.006
-0.001
0.118
0.023
-0.001
0.000
0.017
-0.003
0.002
div
-0.006
-0.002
0.226
0.024
-0.003
0.001
0.018
-0.005
0.010
div
0.004
0.000
0.099
-0.020
0.001
0.001
-0.016
0.001
0.012
term
-0.013
0.001
0.101
0.002
0.002
0.490
-0.011
0.003
0.249
ls
0.007
-0.001
0.009
-0.005
-0.001
0.207
0.002
-0.002
0.255
def
0.004
-0.001
0.063
-0.013
0.000
0.001
-0.009
0.000
0.035
term
-0.011
0.002
0.057
0.023
0.001
0.042
0.012
0.003
0.260
ls
Panel D: 1963–2002, HML
0.008
-0.001
0.021
-0.002
-0.001
0.425
0.005
-0.002
0.206
def
Panel B: 1963–2002, p5-1
0.013
0.000
0.005
-0.019
-0.002
0.008
-0.006
-0.001
0.294
rf
0.015
0.001
0.010
-0.030
-0.002
0.007
-0.015
0.000
0.103
rf
0.452
0.721
0.309
adj. R2
0.575
0.485
0.315
adj. R2
This table reports predictive regressions for the value premium, Agt+1 + Dt+1 /Pt , and its two components–the long-run dividend growth rate,
Agt+1 , and the dividend-price ratio, Dt+1 /Pt . We report results for both portfolio 5-1 from the one-way sort on book-to-market and HML from
the two-way sort on size and book-to-market. There are five regressors, (i) dividend yield, div, computed as the sum of dividends accruing to the
CRSP value-weighted portfolio over the previous 12 months divided by the current index level; (ii) default premium, def, which is the yield spread
between Baa and Aaa corporate bonds; (iii) term premium, term, computed as the yield spread between ten-year and one-year government bonds;
(iv) log book-to-market spread, ls, defined as the log book-to-market of decile ten minus that of decile one from ten book-to-market portfolios; and
(v) one-month Treasury bill, rf. All regressors are standardized to have zero mean and unit variance. We report intercepts, slopes, bias in slopes,
adjusted R2 s, and p-values adjusted for small-sample problems using Nelson and Kim (1993) methods.
Table 5 : Predictive Regressions for the Value Premium and Its Two Components–the Long-Run Dividend Growth
Rate and the Dividend-Price Ratio
35
Et [Rt+1 ]
Et [Dt+1 /Pt ]
Et [Agt+1 ]
gt+1
Rt+1
Et [Rt+1 ]
Et [Dt+1 /Pt ]
Et [Agt+1 ]
gt+1
Rt+1
0.090
3.10
0.035
1.19
0.029
19.67
0.038
15.58
0.067
43.64
LS
0.073
3.22
0.032
1.31
0.036
21.75
0.036
24.68
0.072
44.90
Low
0.097
4.92
0.048
2.09
0.041
81.97
0.055
40.58
0.096
91.64
3
0.111
4.84
0.050
1.92
0.039
61.09
0.061
38.82
0.100
71.03
4
0.137
4.90
0.080
2.77
0.054
46.70
0.064
43.64
0.117
50.45
High
0.076
3.41
0.029
1.27
0.031
19.71
0.038
26.48
0.069
39.04
LB
0.131
4.76
0.074
2.79
0.064
62.56
0.051
26.94
0.115
98.70
MS
0.086
4.26
0.033
1.37
0.024
60.71
0.055
40.19
0.079
58.78
MB
0.166
5.40
0.109
4.06
0.088
139.33
0.050
50.42
0.138
118.80
HS
HB
0.127
4.82
0.064
2.59
0.044
47.01
0.066
42.03
0.110
47.63
Panel C: 1941–2002, two-way sort
0.080
3.83
0.029
1.03
0.024
24.72
0.047
34.71
0.071
40.28
2
Panel A: 1941–2002, one-way sort
0.063
4.43
0.054
2.22
0.036
24.58
0.020
11.69
0.055
52.55
HML
0.064
3.19
0.048
1.47
0.017
8.08
0.028
18.18
0.045
27.09
p5-1
0.074
2.05
0.039
1.02
0.036
26.64
0.026
22.51
0.062
48.34
LS
0.060
2.10
0.044
1.30
0.043
23.90
0.030
29.62
0.073
32.04
Low
0.086
3.99
0.054
1.76
0.043
90.52
0.053
31.95
0.095
68.91
3
0.110
4.72
0.062
1.71
0.042
72.73
0.060
28.80
0.102
51.73
4
0.117
4.56
0.068
1.89
0.051
59.91
0.064
34.11
0.115
48.08
High
0.063
2.25
0.038
1.15
0.037
19.79
0.032
29.51
0.069
27.80
LB
0.123
3.90
0.086
2.25
0.069
98.13
0.043
33.24
0.111
88.47
MS
0.074
3.36
0.039
1.17
0.023
42.71
0.053
30.58
0.076
46.92
MB
0.155
4.80
0.112
3.30
0.088
148.01
0.049
39.48
0.137
116.08
HS
HB
0.111
4.56
0.058
1.90
0.044
51.17
0.065
32.46
0.109
41.20
Panel D: 1963–2002, two-way sort
0.065
2.59
0.034
0.83
0.026
18.74
0.042
34.82
0.068
30.94
2
Panel B: 1963–2002, one-way sort
0.064
4.07
0.047
1.47
0.030
23.57
0.027
19.27
0.057
57.38
HML
0.057
2.59
0.024
0.61
0.008
5.23
0.034
22.36
0.042
36.72
p5-1
This table reports the sample averages of the realized return, Rt+1 , the realized dividend growth, gt+1 , the expected long-run dividend growth,
Et [Agt+1 ], the expected dividend yield, Et [Dt+1 /Pt ], and the expected return, Et [Rt+1 ] for various value and growth portfolios. The computation
of dividends includes net stock repurchases. The corresponding t-statistics are reported in the rows below the sample averages. The results for
both the full sample from 1941 to 2002 and for the subsample from 1963 to 2002 are reported. Panels A and B contain the results for five quintiles
sorted on book-to-market, while Panels C and D contain the results for six portfolios based on a two-by-three sort on size and book-to-market. In
Panels A and B, p5-1 denotes the difference between portfolio High and portfolio Low in the five book-to-market quintiles. In Panels C and D,
portfolios are denoted by two letters, for example, portfolio LS contains stocks with the bottom 30% book-to-market ratios and the bottom 50%
market capitalization. Significant average realized and expected returns for p5-1 and HML and their t-statistics are highlighted.
Table 6 : Including Repurchases: Descriptive Statistics for The Realized Returns, Realized Dividend Growth,
Expected Long-run Dividend Growth, Expected Dividend Yield, and Expected Returns of Value and Growth
Portfolios
36
Expected HML Return
Expected p5-1 Return
-0.009
0.012
-0.014
0.006
no T-bill
-0.009
0.013
-0.014
0.009
with T-bill
1941–2002
-0.034
0.000
-0.035
0.000
no T-bill
-0.030
0.000
-0.034
0.000
with T-bill
1963–2002
Panel A: Real gCON
-0.149
0.000
-0.177
0.000
no T-bill
-0.142
0.000
-0.178
0.000
with T-bill
1941–2002
-0.203
0.000
-0.210
0.000
no T-bill
-0.159
0.000
-0.195
0.000
with T-bill
1963–2002
Panel B: Real gCON
This table reports the results from a first-order VAR that includes the expected value premium and one of two cyclical indicators–real investment
growth, gINV , and real consumption growth, gCON . The computation of dividends includes net stock repurchases. The table reports the equation
for the expected value premium for the 1941–2002 and 1963–2002 samples. We also report the results with and without controlling for monetary
policy as captured by the one-month T-bill rate. The lag in the VAR is one, which is chosen based on the Akaike information criterion. p-values
associated with Newey-West t-statistics adjusted for heteroscedasticity and autocorrelation of up to six lags are reported in the rows below the
coefficients.
Table 7 : Including Repurchases: VAR Analysis
37
−10
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
−10
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
−6
−4
−2
0
2
4
6
−8
−6
−4
−2
0
2
4
6
Panel D: Et+1 /Bt , two-way sort
−8
8
8
S/H
B/H
S/L
B/L
Value Stocks
Growth Stocks
Panel A: Et+1 /Bt , one-way sort
10
10
0.01
−10
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.02
−10
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
−6
−4
−2
0
2
4
6
−8
−6
−4
−2
0
2
4
6
Panel E: Dt+1 /Bt , two-way sort
−8
8
8
S/H
B/H
S/L
B/L
Value Stocks
Growth Stocks
Panel B: Dt+1 /Bt , one-way sort
10
10
0.9
−10
0.95
1
1.05
1.1
1.15
0.9
−10
0.95
1
1.05
1.1
1.15
−8
−8
−4
−2
0
2
4
6
−6
−4
−2
0
2
4
6
Panel F: gt+1 , two-way sort
−6
Panel C: gt+1 , one-way sort
8
8
S/H
B/H
S/L
B/L
Value Stocks
Growth Stocks
10
10
This figure plots the event time evolution of dividend growth rates and the ratios of dividend and lagged book equity for value and growth portfolios.
We construct value and growth portfolios using a one-way sort on book-to-market into five quintiles and using a two-by-three sort on size and
book-to-market into six portfolios as Fama and French (1993). Panel A plots profitability defined as earnings divided by lagged book value, Et+1 /Bt ,
for portfolios five (value) and one (growth) from the quintiles, and Panel D does the same for four portfolios including small-high (S/H), big-high
(B/H), small-low (S/L), and big-low (B/L) from the six portfolios sorted on size and book-to-market. Panel plots the dividend-lagged book equity,
Dt+1 /Bt , for portfolios five and one from the quintiles, and Panel E does the same for S/H, B/H, S/L, and B/L. Panel C plots the real dividend
growth rates, gt+1 , for portfolios five and one from the quintiles, and Panel F does the same for S/H, B/H, S/L, and B/L.
Figure 1 : Event-Time Evolution of Profitability, Real Dividend/Lagged Book Equity, and Real Dividend Growth
for Value and Growth Portfolios (1941–2003)
Figure 2 : Times Series of the Annual Realized Real Dividend Growth Rates of
Value and Growth Portfolios (1941—2002)
This figure plots the sample paths of the annual realized real dividend growth rates of value and growth
portfolios. We construct value and growth portfolios using a one-way sort on book-to-market (into five
quintiles) and using a two-by-three sort on size and book-to-market (into six portfolios). Panel A plots the
annual real dividend growth rates for portfolios five and one from the quintiles, and Panel B plots that for
portfolio five-minus-one, p5-1. Panel C plots the annual real dividend growth rates for portfolios High and
Low defined from the six portfolios after controlling for size as in Fama and French (1993). Panel D does the
same for HML. In Panels A and C, the solid lines represent the value portfolios, and the broken lines represent
Dt+1 /Pt
the growth portfolios. The real dividend growth is measured as gt+1 = D
(RtX + 1)(CPIt−1 /CPIt ) − 1,
t /Pt−1
where RtX is the nominal value-weighted portfolio return without dividend from year t−1 to t, and CPIt is the
X
consumer price index at year t. The dividend yield is constructed as Dt+1 /Pt = (Rt+1 −Rt+1
)(CPIt /CPIt+1 ),
where Rt+1 is the nominal value-weighted portfolio return with dividend from year t to t+1.
Panel B: p5-1
Panel A: Portfolios 5 and 1
0.5
Dividend Growth Diff. between Value and Growth Stocks
Value Stocks
Growth Stocks
0.6
0.4
0.3
0.4
0.2
0.2
0.1
0
0
−0.1
−0.2
−0.2
−0.4
−0.3
−0.4
−0.6
−0.5
1950
1960
1970
1980
1990
2000
1950
Panel A: Portfolios High and Low
1960
1970
1980
1990
2000
Panel B: HML
0.5
Value Stocks
Growth Stocks
Dividend Growth Diff. between Value and Growth Stocks
0.4
0.4
0.3
0.2
0.2
0.1
0
0
−0.2
−0.1
−0.2
−0.4
−0.3
−0.4
−0.5
−0.6
1950
1960
1970
1980
1990
2000
38
1950
1960
1970
1980
1990
2000
Figure 3 : Times Series of the Expected Value Premium and Its Two
Components—the Expected Long-Run Dividend Growth and the Expected
Dividend-Price Ratio (1941—2002)
This figure plots the times series of the expected value premium, Et [Rt+1 ], and its two components—the
expected long-run dividend growth, Et [Agt+1 ], and the expected dividend-price ratio, Et [Dt+1 /Pt ]. Panel
A plots the expected return of portfolio five-minus-one, p5-1, and Panel B plots its two components—the
expected long-run dividend growth and the expected dividend-price ratio. Panel C plots the expected HML
return, and Panel D plots its two components—the expected long-run dividend growth and the expected
dividend-price ratio. In Panels A and C, we also plot the scaled default spread defined as the Baa yield over
the Aaa yield scaled by two. In all panels, the shadowed rectangles represent the NBER recession dummy,
which takes the value of one in recessions and zero otherwise.
Panel A: p5-1: Expected Return
Panel B: p5-1: Et [Agt+1 ] and Et [Dt+1 /Pt ]
0.12
The Expected Value Premium
Scaled Default Spread
Expected Dividend Growth
Expected Dividend Price Ratio
0.06
0.1
0.04
0.08
0.02
0.06
0
0.04
−0.02
0.02
−0.04
0
1950
1960
1970
1980
1990
2000
1950
Panel C: HML: Expected Return
1960
1970
1980
1990
2000
Panel D: HML: Et [Agt+1 ] and Et [Dt+1 /Pt ]
0.09
0.08
Expected Dividend Growth
Expected Dividend Price Ratio
0.08
The Expected Value Premium
Scaled Default Spread
0.07
0.07
0.06
0.06
0.05
0.04
0.05
0.03
0.04
0.02
0.03
0.01
0.02
0.01
0
1950
1960
1970
1980
1990
2000
39
−0.01
1950
1960
1970
1980
1990
2000
Figure 4 : Time Series of the Expected Equity Premium (1941—2002)
This figure plots the time series of the constructed expected equity premium.
The Expected Market Equity Premium
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1950
1960
1970
40
1980
1990
2000
Figure 5 : Trend and Cyclical Components of the Expected Value Premium
(1941—2002)
This figure reports trend and cyclical components of the expected value premium including both that from
a one-way sort on book-to-market (broken line) and that from a two-way sort on size and book-to-market
(solid line). Panel A reports the time trend and Panel B reports the trend components from a Hodrock
and Prescott (HP) filter. Panel C reports the cyclical component after the time trend is removed from the
expected value premium, and Panel D reports the cyclical component from the HP-filter. In all panels, the
shadowed rectangles represent the NBER recession dummy, which takes the value of one in recessions and
zero otherwise.
Panel A: Time Trend
Panel B: HP-filtered Trend
0.1
Two−way sort
One−way sort
Two−way sort
One−way sort
0.06
0.09
0.055
0.05
0.08
0.045
0.04
0.07
0.035
0.03
0.06
0.025
0.02
0.05
0.015
0.01
1950
1960
1970
1980
1990
2000
0.04
1950
Panel C: Cyclical Component with Time Trend
1960
1970
1980
1990
2000
Panel D: HP-filtered Cyclical Component
0.05
0.04
Two−way sort
One−way sort
Two−way sort
One−way sort
0.04
0.03
0.03
0.02
0.02
0.01
0.01
0
0
−0.01
−0.01
−0.02
−0.02
−0.03
1950
1960
1970
1980
1990
2000
41
−0.03
1950
1960
1970
1980
1990
2000
42
−5
−4
−3
−2
−1
0
1
0
x 10
−3
1
2
3
4
5
6
7
8
9
Panel A: p5-1: gINV , no T-bill
0
x 10
−3
1
2
3
4
5
6
7
8
9
Panel E: p5-1: gCON , no T-bill
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
10
10
0
1
2
3
4
5
6
7
8
9
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
0
−3
x 10
1
2
3
4
5
6
7
8
9
Panel F: p5-1: gCON , with T-bill
−2.5
10
10
−3
0
1
2
3
4
5
6
7
8
9
Panel C: HML: gINV , no T-bill
x 10
−7
−6
−5
−4
−3
−2
−1
0
1
0
−3
x 10
1
2
3
4
5
6
7
8
9
Panel G: HML: gCON , no T-bill
−5
−4
−2
−2
−1
0
1
−3
−3
x 10
−1.5
−1
−0.5
0
0.5
Panel B: p5-1: gINV , with T-bill
10
10
0
−3
x 10
1
2
3
4
5
6
7
8
9
10
−6
−5
−4
−3
−2
−1
0
1
0
−3
x 10
1
2
3
4
5
6
7
8
9
10
Panel H: HML: gCON , with T-bill
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Panel D: HML: gINV , with T-bill
This figure plots the impulse response functions for the expected return of p5-1 and the expected HML return in the presence of a one-standarddeviation positive shock to real investment growth, gINV (Panels A–D), and to real consumption growth, gCON (Panels E–H). We report the results
with and without controlling for the one-month T-bill rate in the VAR. In all panels, a two-standard-error band is also plotted.
Figure 6 : Impulse Response Functions for the Expected Value Premium After A One-Standard-Deviation Positive
Shock to Real Investment Growth or the Real Consumption Growth
Figure 7 : Including Repurchases: Trend and Cyclical Components of the Expected
Value Premium (1941—2002)
This figure reports trend and cyclical components of the expected value premium including both that from
a one-way sort on book-to-market (broken line) and that from a two-way sort on size and book-to-market
(solid line). The computation of dividends includes net stock repurchases. Panel A reports the time trend
and Panel B reports the trend components from a Hodrock and Prescott (HP) filter. Panel C reports the
cyclical component after the time trend is removed from the expected value premium, and Panel D reports
the cyclical component from the HP-filter. In all panels, the shadowed rectangles represent the NBER
recession dummy, which takes the value of one in recessions and zero otherwise.
Panel A: Time Trend
Panel B: HP-filtered Trend
0.065
0.08
Two−way sort
One−way sort
Two−way sort
One−way sort
0.075
0.06
0.07
0.055
0.065
0.06
0.05
0.055
0.045
0.05
0.045
0.04
0.04
0.035
0.035
0.03
1950
1960
1970
1980
1990
2000
0.03
Panel C: Cyclical Component with Time Trend
1950
1960
1970
1980
1990
2000
Panel D: HP-filtered Cyclical Component
0.04
Two−way sort
One−way sort
0.05
Two−way sort
One−way sort
0.03
0.04
0.02
0.03
0.01
0.02
0
0.01
−0.01
0
−0.02
−0.01
−0.02
1950
1960
1970
1980
1990
2000
43
−0.03
1950
1960
1970
1980
1990
2000
44
0
x 10
−3
1
2
3
4
5
6
7
8
9
Panel A: p5-1: gINV , no T-bill
−3
1
2
3
4
5
6
7
8
9
10
−4
−2.5
−2.5
−4
−2
−2
−3
−1.5
−1.5
−3.5
−1
−1
−3
−0.5
−0.5
−3.5
0
1
0.5
0
0
1
2
3
4
5
6
7
8
9
0
−3
x 10
1
2
3
4
5
6
7
8
9
Panel F: p5-: gCON , with T-bill
−2.5
0
x 10
10
−2
−1.5
−1
−0.5
0
−3
x 10
Panel B: p5-1: gINV , with T-bill
0.5
0.5
1
Panel E: p5-1: gCON , no T-bill
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
10
10
0
−3
x 10
1
2
3
4
5
6
7
8
9
−5
−4
−3
−2
−1
0
1
0
x 10
−3
1
2
3
4
5
6
7
8
9
Panel G: HML: gCON , no T-bill
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Panel C: HML: gINV , no T-bill
10
10
0
−3
x 10
1
2
3
4
5
6
7
8
9
10
−5
−4
−3
−2
−1
0
1
0
−3
x 10
1
2
3
4
5
6
7
8
9
10
Panel H: HML: gCON , with T-bill
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
Panel D: HML: gINV , with T-bill
This figure plots the impulse response functions for the expected return of p5-1 and the expected HML return in the presence of a one-standarddeviation positive shock to real investment growth, gINV (Panels A–D), and to real consumption growth, gCON (Panels E–H). The computation
of dividends includes net stock repurchases. We report the results with and without controlling for the one-month T-bill rate in the VAR. In all
panels, a two-standard-error band is also plotted.
Figure 8 : Including Repurchases: Impulse Response Functions for the Expected Value Premium After A
One-Standard-Deviation Positive Shock to Real Investment Growth and Real Consumption Growth