Expected Value Premium and Implied Cost of Capital
Yan Li, David T. Ng, and Bhaskaran Swaminathan1
May 28, 2012
In this paper, we use an implied cost of capital approach to construct the expected value premium. We show that the expected value premium strongly predicts future realized value premium,
both in sample and out of sample. The expected value premium remains a statistically and economically significant predictor after controlling for the value spread and business cycle variables.
We investigate both risk-based theories and behavioral theories on value premium, and find that
the expected value premium contains both a risk component and a mispricing component.
JEL Classification: G12
Keywords: Expected Value Premium, Implied Cost of Capital, Predictability, Value Spread, Term
Spread, Default Spread, Risk and Mispricing
1
Yan Li, [email protected], Department of Finance, Fox School of Business, Temple University, Philadelphia,
PA 19122; David T. Ng, [email protected], Dyson School of Applied Economics and Management, Cornell University,
252 Warren Hall, Ithaca, NY 14853; and Bhaskaran Swaminathan, [email protected], LSV Asset Management, 155
North Wacker Dr., Chicago, IL 60606. We thank Francis X. Diebold, Amy Dittmar, Paul Gao, Marcin Kacperczyk,
Nagpurnanand R. Prabhala, Michael R. Roberts, Oleg Rytchkov, Yuzhao Zhang, and seminar participants at the
Johnson School of Management for their helpful comments. Any errors are our own.
1
1
Introduction
One of the most robust findings in empirical asset pricing studies is the value premium—value
stocks (stocks with high book-to-market ratios) earn higher average returns than growth stocks
(stocks with low book-to-market ratios) (e.g., Rosenberg, Reid, and Lanstein (1985), Fama and
French (1992), Lakonishok, Shleifer, and Vishny (1994)). Since the seminal contributions of Fama
and French (1992, 1993, 1996), the value premium has become increasingly important in portfolio
management, asset allocation, capital budgeting, and many other applications. Most existing studies studying the value premium use average realized returns to proxy expected returns. However,
realized returns are notoriously noisy proxy of expected returns (e.g., Elton (1999), Chen, Petkova,
and Zhang (2008)). In this paper, we propose an ex-ante value premium based on an implied cost
of capital approach.
Our method of constructing the expected value premium closely follows Pastor, Sinha, and
Swaminathan (2008) and Li, Ng, and Swaminathan (2012), which use an implied cost of capital
approach to construct ex-ante equity premium. Basically, each month, we construct the value
and growth portfolios following the 3-by-2 two-way sorting of Fama and French (1993). We then
obtain the expected returns for the value and growth portfolios by value-weighting the firm-level
implied cost of capital within each portfolio, respectively, and then obtain the difference between
the expected returns for the value and growth portfolios as the expected value premium.
We find that our expected value premium measure is an excellent ex-ante measure of realized
value premium. Over the time period of 1976.01-2011.12, the annual expected value premium has
a mean of 3.78% and a standard deviation of 2.39%. The univariate regression indicates that the
expected value premium significantly predicts future realized value premium at all horizons ranging
from one month to three years, regardless of whether predicting the realized value premium from
our constructed value/growth portfolios, or the HML factor of the Fama-French three-factor model.
The expected value premium explains 15% of our constructed value premium at the 1-year horizon
and 44% at the 3-year horizon. The economic significance associated with the expected value
premium is also large.
We further investigate the predictive power of our expected value premium in the presence of
other predictors. One important predictor of the value premium that has been discovered in the
literature is the value spread, which is the difference between book-to-market ratios of growth and
value portfolios (Asness, Friedman, Krail, and Liew (2000), Cohen, Polk, and Vuolteenaho (2003)).
2
In particular, Cohen, Polk, and Vuolteenaho (2003) find that the value spread is a significant
predictor of the HML factor in Fama and French (1993). In the multivariate regressions which
control for the value spread, our expected value premium still strongly predicts future realized
value premium and the HML factor of the Fama-French three-factor model. The predictive power
of the expected value premium remains strong in the presence of well-known business cycle variables,
including the term spread and the default spread.
In light of the well-known issues with predictive regressions (e.g., Stambaugh (1999)), we use a
rigorous Monte Carlo simulation procedure to assess the statistical significance of our regressions.
Our results hold under the stringent simulated p-values. We further investigate the robustness
of our results to alternative measures of expected value premium, including a measure based on
different forecast horizons of the implied cost of capital model, and a measure based on one-way
sorting to form value/growth portfolio. Our results are robust to these alternative specifications.
Because we estimate the expected value premium using analyst forecasted earnings which might
contain errors, we further show that our results are not driven by analyst forecast optimism bias.
We also show that an important component of the expected value premium is the forecast of future
dividend growth rates.
Our in-sample analysis therefore establishes the expected value premium as an excellent predictor of future realized value premium. Recently, the out-of-sample test has captured much attention
in the return predictability literature (e.g., Campbell (2000), Campbell and Thompson (2008), and
Welch and Goyal (2008)). Given that the “style investing” is the widely used trading strategies
among practitioners, it is particularly interesting to know whether trading on the expected value
premium can help generate real profits. We examine the out-of-sample performance of the expected
value premium, and our results show that in the two forecast periods we examine (1988.01-2011.12
and 1994.01-2011.12), the expected value premium is also a reliable predictor in predicting future realized value premium. The expected value premium outperforms the value spread and the business
cycle variables, and it also contains distinct and important information beyond these variables.
Finally, in light the debate on the interpretation of value premium, we examine whether the
expected value premium stands for risk or mispricing. Our results show that the expected value
premium is largely countercyclical, which is consistent with the general prediction of risk-based
theories. We also design a novel test using the expected value premium to forecast the difference of
cumulative abnormal returns between value and growth stocks around the earnings announcement
window. Since risk and the price of risk do not change within small windows, our finding that the
3
expected value premium strongly forecasts this cumulative abnormal returns shows that mispricing
is also an important component of the ex-ante value premium.
Our work contributes to the growing literature that uses valuation models to estimate expected
stock returns (e.g., Blanchard, Shiller, and Siegel (1993), Lee, Myers, and Swaminathan (1999),
Claus and Thomas (2001), Gebhardt, Lee, and Swaminathan (2001), Jagannathan, McGrattan,
and Scherbina (2000), Constantinides (2002), Fama and French (2002), Chen, Petkova, and Zhang
(2008), van Binsbergen and Koijen (2010), Rytchkov (2010), Li, Ng, and Swaminathan (2012), and
Wu and Zhang (2011)). Our paper is most related to Chen, Petkova, and Zhang (2008) which
also estimates the expected value premium. While Chen, Petkova, and Zhang (2008) separately
estimates the expected dividend price ratio and expected dividend growth rate and obtain the
expected value premium as the sum of the two, we use a theoretically justifiable approach—the
implied cost of capital approach—to incorporate the two simultaneously. We also systematically
investigate the forecasting performance of the expected value premium for future realized value
premium in univariate and multivariate regressions, both in sample and out of sample, which are
not studied in Chen, Petkova, and Zhang (2008). In another related paper, Wu and Zhang (2011)
use the same method as ours to obtain expected returns for dollar neutral long-short trading
strategies formed on a wide array of anomaly variables, including the value premium. We differ
from them in that while they emphasize the level of the expected value premium, our focus is on the
dynamics of the expected value premium. Finally, our study contributes to the debate on whether
ex-ante value premium stands for risk or mispricing. We bring new evidence to the literature that
both risk and mispricing are important components of the ex-ante value premium.
Our paper proceeds as follows. We describe the methodology to construct the expected value
premium in Section 2. Section 3 discusses data and summary statistics. Section 4 presents the
in-sample and out-of-sample analysis of the expected value premium for predicting future realized
value premium, and Section 5 investigates the interpretation of the expected value premium. Section
6 concludes the paper.
2
Construction of Expected Value Premium
In this section, we first describe the method to construct the firm-level implied cost of capital. We
then discuss how to construct the value and growth portfolios and obtain their respective expected
returns using the firm-level implied cost of capital. The expected value premium is defined as the
4
difference between the expected returns for the value and growth portfolios.
2.1
Firm-level Implied Cost of Capital
Our construction of firm-level implied cost of capital follows the approach of Pastor, Sinha, and
Swaminathan (2008), Lee, Ng, and Swaminathan (2009), and Li, Ng, and Swaminathan (2012).
According to the free cash flow model, the firm-level implied cost of capital (ICC) is constructed
as the internal rate of return that equates the present value of future dividends to current stock
price. That is,
Pt =
∞
∑
Et (Dt+k )
k=1
(1 + re )k
,
(1)
To implement equation (1), we need to explicitly forecast free cash flows for a finite horizon.
More specifically, we forecast the free cash flows in two parts: i) the present value of free cash flows
up to a terminal period t + T , and ii) a continuing value that captures free cash flows beyond the
terminal period. We estimate free cash flows up to year t + T , as the product of annual earnings
forecasts and one minus the plowback rate:
Et (F CF Et+k ) = F Et+k × (1 − bt+k ) ,
(2)
where F Et+k and bt+k are the earnings forecasts and the plowback rate forecasts for year t + k,
respectively.
We forecast earnings up to year t + T in three stages: i) we explicitly forecast earnings (in
dollars) for years t + 1 and t + 2. IBES analysts supply a one-year ahead F E1 and a two-year-ahead
F E2 earnings per share (EPS) forecast for each firm in the IBES database. ii) we then use the
growth rate implicit in the forecasts in years t + 1 and t + 2 to forecast earnings in year t + 3; that is,
g3 = F E2 /F E1 − 1, and the three-year-ahead earnings forecast is given by F E3 = F E2 (1 + g3 ).2
Firms with growth rates above 100% (below 2%) are given values of 100% (2%). iii) we forecast
earnings from year t + 4 to year t + T + 1 implicitly by assuming that the year t + 3 earnings growth
rate g3 reverts to steady-state values by year t + T + 2. We assume the steady-state growth rate
starting in year t + T + 2 is equal to the long-run nominal GDP growth rate, g, computed as the
sum of the long-run real GDP growth rate (a rolling average of annual real GDP growth) and the
long-run average rate of inflation based on the implicit GDP deflator. Specifically, earnings growth
2
In addition to F E1 and F E2 , I/B/E/S analysts forecast the long-term earnings growth rate (Ltg). An alternative
way of obtaining g3 is to use Ltg. In untabulated results, we show that g3 = F E2 /F E1 − 1 is a better measure than
g3 = Ltg, because the former better predicts the actual earnings in year t + 3.
5
rates and earnings forecasts using the exponential decline are computed as follows for years t + 4
to t + T + 1 (k = 4, ..., T + 1):
gt+k = gt+k−1 × exp [log (g/g3 ) / (T − 1)] and
(3)
F Et+k = F Et+k−1 × (1 + gt+k ) .
We forecast plowback rates using a two-stage approach: i) we explicitly forecast plowback rates
for years t + 1 and t + 2. For each firm, the plowback rate is computed as one minus that firm’s
dividend payout ratio. We estimate the dividend payout ratio by dividing actual dividends from the
most recent fiscal year by earnings over the same time period.3 We exclude share repurchases due
to the practical problems associated with determining the likelihood of their recurrence in future
periods.4 Payout ratios of less than zero (greater than one) are assigned a value of zero (one). ii)
we assume that the plowback rate in year t + 2, b2 reverts linearly to a steady-state value by year
t + T + 1 computed from the sustainable growth rate formula. This formula assumes that, in the
steady state, the product of the return on new investments and the plowback rate ROE ∗ b is equal
to the growth rate in earnings g. We further impose the condition that, in the steady state, ROE
equals re for new investments, because competition will drive returns on these investments down
to the cost of equity.
Substituting ROE with cost of equity re in the sustainable growth rate formula and solving for
plowback rate b provides the steady-state value for the plowback rate, which equals the steady-state
growth rate divided by cost of equity g/re . The intermediate plowback rates from t + 3 to t + T
(k = 3, ..., T ) are computed as follows:
bt+k = bt+k−1 −
b2 − b
.
T −1
(4)
The terminal value T V is computed as the present value of a perpetuity equal to the ratio of
the year t + T + 1 earnings forecast divided by the cost of equity:
T Vt+T =
F Et+T +1
,
re
(5)
where F Et+T +1 is the earnings forecast for year t + T + 1.5 It is easy to show that the Gordon
growth model for T V will simplify to equation (5) when ROE equals re .
3
If earnings are negative, the plowback rate is computed as the median ratio across all firms in the corresponding
industry-size portfolio. The industry-size portfolios are formed each year by first sorting firms into 49 industries
based on the Fama–French classification and then forming three portfolios with an equal number of firms based on
their market cap within each industry.
4
Our results are robust to including share issuances and repurchases.
5
Note that the use of the no-growth perpetuity formula does not imply that earnings or cash flows do not grow
after period t + T . Rather, it simply means that any new investments after year t + T earn zero economic profits. In
other words, any growth in earnings or cash flows after year T is value-irrelevant.
6
Substituting equations (2) to (5) into the infinite-horizon free cash flow valuation model in
equation (1) provides the following empirically tractable finite horizon model:
Pt =
T
∑
F Et+k × (1 − bt+k )
k
k=1
(1 + re )
+
F Et+T +1
re (1 + re )T
.
(6)
In this paper, following Pastor, Sinha, and Swaminathan (2008) and Li, Ng, and Swaminathan
(2012), we use a 15-year horizon (T = 15) to implement the model in (6) and compute re as the
rate of return that equates the present value of free cash flows to the current stock price. The
resulting re is the firm level ICC measure used in our empirical analyses.
2.2
Value and Growth Portfolios
We construct value and growth portfolios by implementing a two-way, two-by-three sort on size
and book-to-market to obtain six portfolios as in Fama and French (1993). The book equity is
stockholder equity plus balance sheet-deferred taxes and investment tax credits plus post-retirement
benefit liabilities minus the book value of preferred stock. Depending on data availability, we use
redemption, liquidation, or par value, in this order, to represent the book value of preferred stock.
Stockholder equity is the book value of common equity. If the book value of common equity is not
available, stockholder equity is calculated as the book value of assets minus total liabilities.
Our two-way sorting procedure follows Fama and French (1993). In June of each year from 1980
to 2009, all NYSE stocks on CRSP are ranked on size (price times shares). The median NYSE size
is then used to split NYSE, Amex. and NASDAQ stocks into two groups, small and big (S and B).
We also break NYSE, Amex, and NASDAQ stocks into three book-to-market equity groups based
on the breakpoints for the bottom 30% (L), middle 40% (M). and top 30% (H) of the ranked values
of BE/ME for NYSE stocks. Following Fama and French (1992), we do not use negative-BE firms,
when calculating the breakpoints for BE/ME, or when forming the size portfolios. The two-way
sorting generates six portfolios (denoted S/L, B/L, S/M, B/M, S/H, and B/H). In particular, the
S/L portfolio contains the stocks in the small size group that are also in the low book-to-market
group, and the B/H portfolio contains the large stocks that are also in the high book-to-market
group. The value portfolio is (S/H + B/H)/2, and the growth portfolio is (S/L + B/L)/2. The
value strategy of longing in the value portfolio and shorting the growth portfolio, denoted as HML,
is defined as (S/H + B/H)/2 - (S/L + B/L)/2.
7
2.3
Expected and Realized Value Premium
To construct the expected value premium, we first construct the expected returns for the value
and growth portfolios, respectively. More specifically, the value portfolio is (S/H + B/H)/2, and
the growth portfolio is (S/L + B/L)/2. Each month, the expected return for the value portfolio,
denoted as ICCH
t , is obtained by first value-weighting the firm-level implied cost of capital in the
portfolio of S/H and B/H, respectively, to obtain their respective expected returns, and then take
the average expected returns of S/H and B/H. We do the same for the growth portfolio to obtain
its expected return, denoted as ICCL
t .
The value premium at each month t is defined as
ICCtHM L = ICCtH − ICCtL .
The realized returns for the value portfolio and growth portfolio and the realized value premium
are obtained in a similar way. The realized return of the value portfolio at month t, denoted as RH
t ,
is the average return of the portfolio returns of S/H and B/H, where the portfolio returns of S/H
and B/H are obtained by value-weighting the firm-level returns within each portfolio. The realized
return of the growth portfolio at month t, denoted as RL
t , is the average return of the portfolio
returns for S/L and B/L. The realized value premium is defined as
RtHM L = RtH − RtL .
3
Data and Summary Statistics
3.1
Data
To construct ICC, we obtain return data from CRSP, accounting data from COMPUSTAT, and
analyst forecasts from I/B/E/S. We only include firms with positive market value and book value
of common equity. To ensure we only use publicly available information, we obtain these items
from the most recent fiscal year ending at least 3 months prior to the month in which the implied
cost of capital is computed. Data on nominal GDP growth rates are obtained from the Bureau of
Economic Analysis. Our GDP data begin in 1930. Each year, we compute the steady-state GDP
growth rate as the historical average of the GDP growth rates using annual data up to that year.
To establish the expected value premium as a good ex-ante measure, we run a host of regressions
to control for other predictors of realized value premium. One important control variable we
8
examine is the value spread (VS ), or the difference between book-to-market ratios of growth and
value portfolios. The value spread has been documented as an important predictor of realized value
premium (e.g., Cohen, Polk, and Vuolteenaho (2003), Asness, Friedman, Krail, and Liew (2000)).
We obtain the monthly time series of the value spread based on our constructed value and growth
portfolios, similar to the way we construct the expected value premium in Subsection 2.3. More
specifically, we obtain the book-to-market ratio for the value (growth) portfolio as the average of
book-to-market ratios of S/H and B/H (S/L and B/L), where the book-to-market ratio of S/H and
B/H (S/L and B/L) are obtained by value-weighting the firm-level book-to-market ratios within
each portfolio. The value spread is defined as the log difference of the book-to-market ratio for the
value portfolio and that for the growth portfolio.
Another set of controls are the business cycle variables including the term spread (Term) and
the default spread (Default), which have been identified as predictors of realized value premium
(e.g., Kao and Shumaker (1999)). More importantly, since the term spread and default spread are
typically considered as risk proxies (e.g., Fama and French (1989)), controlling for these variables
will shed light on whether expected value premium stands purely for risk. The term spread is the
yield difference between Moody’s Aaa bonds and the 1-month T-bill rate representing the slope of
the treasury yield curve. The 1-month T-bill rate is obtained from WRDS. The default spread is
the difference between yields on BAA and AAA-rated corporate bonds obtained from the economic
research database at the Federal Reserve Bank at St. Louis (FRED).
If our expected value premium is a good ex-ante measure, ideally, it should be able to predict
not only our constructed value premium described in Subsection 2.3, but also the HML factor in
the Fama-French three-factor model (Fama and French (1993, 1996)). Our data on the HML factor
comes from Kenneth French’s website. Based on the 6 size and book-to-market ratio portfolios
provided on Kenneth French’s website (S/L, B/L, S/M, B/M, S/H, and B/H), we can also obtain
the Fama-French value portfolio as (S/H + B/H)/2, and the growth portfolio is (S/L + B/L)/2.
3.2
Summary Statistics
Table 1 provides summary statistics for the relevant variables. All values except the value spread are
expressed in annual percentage terms. Panel A of Table 1 presents the summary statistics for the
expected returns of the value portfolio (ICCH ), the growth portfolio (ICCL ), and the expected value
premium (ICCHM L ). The average annual expected returns are 16.03%, 12.25%, and 3.78% for H,
L, and HML, respectively, and their standard deviations are 4.33%, 2.55%, and 2.39%, respectively.
9
Panel B provides the realized returns of H, L, and HML. The average annual realized returns for
our constructed H, L, and HML are 17.41%, 13.23%, and 4.18%, respectively, with the standard
deviations being 16.92%, 18.83%, and 9.82%, respectively. Therefore, compared with the realized
returns, the expected returns are much more stable, indicated by their substantially lower standard
deviations. Interestingly, although the value portfolio has a lower standard deviation for realized
returns than the growth portfolio (16.92% vs. 18.83%), its ex-ante expected returns are more
volatile (4.33% vs. 2.55%). Moreover, expected returns are much more persistent than realized
returns. The first-order autocorrelations for ICCH and ICCL are both 0.97, and for ICCHM L it
is 0.92. Among these three expected return measure, the expected value premium ICCHM L has
the fastest mean reversion, with its autocorrelation dropping to 0.08 after three years. Lee, Myers,
and Swaminathan (1999) show that lower standard deviation and fast mean reversion are superior
properties of expected return measures. Therefore, we expect our ICCHM L to be a better ex-ante
measure than ICCH and ICCL .
Among other predictors of the realized value premium, the value spread has mean of 1.38 and
standard deviation of 0.16, with a first-order autocorrelation of 0.97; the term spread is 3.07%, and
the default spread is 1.11% (Panel C). Panel D shows that the expected value premium is weakly
correlated with the value spread and the term spread, with the corresponding correlations being
−0.08 (p-value 0.10), and 0.08 (p-value 0.10), and it has a higher correlation with the default spread
(correlation 0.40 with p-value 0.00).
The average number of firms used to construct the relevant portfolios is provided in Panel E.
It is worth mentioning that due to the coverage of I/B/E/S, the firms we use to construct our
portfolios include relatively larger firms in the universe. To the extent that value premium is
more pronounced among smaller firms, this sample selection issue tends to bias against finding
predictability of our expected value premium for realized value premium. More importantly, our
constructed portfolio returns are highly correlated with the Fama-French portfolios which use the
entire universe of the firms: the correlations of the returns for our constructed H, L and HML with
the corresponding Fama-French portfolio return are 0.98, 0.99, and 0.95, respectively (Panel B).
Figure 1 plots the time series of expected value premium from January 1976 to December 2011.
To examine its low-frequency movements, we also use all historical data starting from January of
1981 to obtain its moving average mean and standard deviation. The mean and two-standarddeviation bands are also plotted in Figure 1. This figure is line with Schwert (2003) which notes
that the magnitude of the value premium has declined over the 1990s. Figure 1 also provides the
10
NBER recession periods in shaded areas, the discussion of which we defer to Section 5 when we
investigate whether the expected value premium stands for risk or mispricing.
As can be seen from Equation (6), our expected value premium is obtained from a valuation
model by assuming a specific horizon T . Therefore, we do not emphasize the absolute level of
expected value premium; rather, we emphasize its dynamics. That is, our criteria of an excellent
ex-ante measure of value premium is that it should reliably predict future realized value premium.
We now turn our discussion to examining the predictive power of our expected value premium
measure for the realized value premium.
4
Predictability of Expected Value Premium
We conduct two predictability analysis for the expected value premium. First, in Subsection 4.1,
we examine whether the expected value premium is a reliable predictor of future realized value
premium using the entire sample. In this in-sample analysis, we will conduct both univariate
regression analysis on expected value premium, as well as multivariate regression analysis controlling
for other predictors including the value spread, the term spread, and the default spread. To check
the robustness of our results, we will construct alternative measures of expected value premium
by entertaining a different horizon T in Equation (6), and by using a different sorting procedure
to form value and growth portfolios. We also provide more economic intuition of our expected
value premium by investigating the role of dividend growth forecasts. Finally, we show that the
predictive power of expected value premium is not driven by analyst forecast optimism bias.
After establishing the predictive power of expected value premium in sample, we further examine
its out-of-sample performance. More specifically, we break the entire sample into an estimation
period and a forecast period. Based on the information of expected value premium in the estimation
period, we can generate forecasted value premium in the forecast period, and examine how close
these forecasts are to the actual realized value premium. We conduct a variety of tests including
the out-of-sample R2 test, the utility gains from forming mean-variance portfolios, and the forecast
encompassing tests—the details of which to be discussed in Subsection 4.2—and we show that the
information embedded in expected value premium is also valuable in real-time scenarios. Given
that many mutual fund investors adopt the “style investing” strategy by buying the value portfolio
and shorting the growth portfolio, the out-of-sample analysis is particularly interesting.
11
4.1
In Sample Predictability Analysis
In this subsection, we systematically analyze the predictive power of the expected value premium for
both our constructed value premium and the HML factor in the Fama-French three-factor model.
4.1.1
Univariate Regressions
To set the stage, we first examine the predictive power of the expected value premium in univariate
regressions. Because the expected value premium is the difference between the expected returns
for the value and growth portfolios, it is interesting to know whether the expected return for the
value/growth portfolio has predictive power for the realized value/growth portfolio return. To
investigate these issues, we conduct the following univariate regression:
K Y
∑
t+k
= a + b × Xt + ut+K .
K
k=1
(7)
H , X = ICC H , (2) Y
L
Here, we consider three specifications of (7): (1) Yt+k = Rt+k
t
t+k = Rt+k ,
t
HM L , X = ICC HM L . Here RH , RL , and RHM L are the realized
Xt = ICCtL , and (3) Yt+k = Rt+k
t
t
t+k
t+k
t+k
returns for portfolio H, L, and HML, respectively, at month t + k, and ICCH , ICCL , and ICCHM L
are the expected returns for H, L, and HML, respectively at month t, b is the slope coefficient, K
is the forecasting horizon in months, and ut+k is the regression residual.
If the expected return measure is an ex-ante measure of future realized returns, then we expect
to obtain a positive value for b in all three specifications of (7). Therefore, a one-sided test is
appropriate. In addition to predicting the constructed portfolios returns, we also examine the
forecastability of our expected return measures for the corresponding Fama-French portfolios.
For each regression specification, we consider different horizons: K = 1, 6, 12, 24, and 36
months. One problem with the regressions in (7) is the use of overlapping observations, which
induces serial correlation in the regression residuals. Specifically, under both the null hypothesis of
no predictability and alternative hypotheses that fully account for time-varying expected returns,
the regression residuals are autocorrelated up to certain lags. As a result, the regression standard
errors from ordinary least squares (OLS) would be too low and the t-statistics too high. Moreover,
the regression residuals are likely to be conditionally heteroskedastic. We correct for both the
induced autocorrelation and the conditional heteroskedasticity using the Generalized Method of
Moments (GMM) standard errors with the Newey-West correction (see Hansen (1982), Hansen and
Hodrick (1980), and Newey and West (1987)). We use K − 1 lags to calculate the Newey-West
standard errors, and we call the resulting statistic the Z-statistic.
12
While the GMM standard errors consistently estimate the asymptotic variance-covariance matrix, Richardson and Smith (1991) show these standard errors are biased in small samples due to
the sampling variation in estimating the autocovariances. To avoid these problems, we generate
small sample distributions of the test statistics using a Monte Carlo simulation (see Hodrick (1992),
Nelson and Kim (1993), Swaminathan (1996) and Lee, Myers, and Swaminathan (1999)). The Appendix describes our Monte Carlo simulation methodology. Finally, since the forecasting regressions
use the same data at various horizons, the regression slopes will be correlated. It is, therefore, not
correct to draw inferences about predictability based on any one regression. To address this issue,
Richardson and Stock (1989) propose a joint test based on the average slope coefficient. Following
their paper, we compute the average slope statistic, which is the arithmetic average of regression
slopes across different horizons, to test the null hypothesis that the slopes at different horizons are
jointly zero. We also conduct Monte Carlo simulations to compute the statistical significance of
the average slope estimate.
Tables 2 presents the regression results of (7). Panel A presents the results for predicting the
constructed portfolio returns, and Panel B presents the results for predicting the Fama-French
portfolio returns. These results show that the expected returns of the value and growth portfolios
predict future realized returns. When predicting the return of our constructed value portfolio, the
slope coefficients associated with ICCH are all positive, and highly significant at the 5% significant
levels; the adjusted R2 is 2% at the 1-month horizon, 15% at the 1-year horizon and it increases
to 44% at the 3-year horizon; the average slope coefficient across all horizons is 1.47%, and it is
significant at the 5% level. We observe similar results when using ICCH to predict the Fama-French
value portfolio: ICCH reliably predicts future returns for the value portfolio in all horizons from one
month to three years, and it is statistically significant at the 5% level at all horizons; the adjusted
R2 increases from 14% at the 1-year horizon to 39% at the 3-year horizon; the average slope is
significant at the 1% level. ICCL , the expected return measure for the growth portfolio, also
positively predicts future returns for the growth portfolio, but its predictive power is weaker than
that of ICCH . When predicting our constructed growth portfolio, ICCL is statistically significant
at the 1-month, 2-year and 3-year horizons, and when predicting the Fama-French growth portfolio,
it is statistically significant in all horizons except the 6-month horizon. The p-values of the average
slope coefficient of ICCL are marginally significant, 0.12 for the constructed portfolio and 0.08 for
the Fama-French portfolio.
HM L .
Now let us turn to our most interested regression when using ICCtHM L to predict Rt+k
13
The regression results provide strong evidence that the expected value premium is a remarkable
predictor of future realized value premium. This is true regardless of whether we predict the
constructed value premium (Panel A), or the HML factor of the Fama-French three-factor model
(Panel B). The slope coefficients of ICCHM L are uniformly positive, they are significant at the 1%
level, for all horizons from one month to three years, as indicated by both the Z-statistic and the
more stringent bootstrapped p-values. The adjusted R2 associated with these regressions are also
large. For example, in Panel A, the R2 of ICCHM L for predicting the constructed value premium
is 2% at the 1-month horizon, 22% at the 1-year horizon, and 33% at the 3-year horizon; in Panel
B, the adjusted R2 of ICCHM L for predicting the Fama-French HML factor is similar, with 1%
at the 1-month horizon, 18% at the 1-year horizon, and 24% at the 3-year horizon. Moreover,
the economic magnitude associated with ICCHM L is also substantial. At the 1-month horizon,
a one-standard-deviation increase in ICCHM L (2.39%) translates into an annualized increase of
4.52% (2.39% × 1.89) for the constructed value premium, and an annualized increase of 4.35%
(2.39% × 1.82) for the HML factor.
Overall, the expected value premium strongly predict future realized value premium in univariate regressions. The fact that the expected value premium also strongly predicts the HML factor
of the Fama-French model serves as additional out-of-sample supporting evidence.
4.1.2
Multivariate Regressions
In this section, we examine whether the predictability of the expected value premium still exists in
the presence of other variables that can potentially predict future realized value premium. Before
looking at the multivariate regressions, Table A1 provides the univariate regression results of the
value spread, the term spread, and the default spread for forecasting future realized value premium.
Consistent with Cohen, Polk, and Vuolteenaho (2003) the value spread is significant predictor of
the Fama-French HML factor (Panel B). It is significant at the 10% at the 2-year and 3-year
horizons, and the p-value across horizons is 0.05. Among the business cycle variables, the term
spread positively predicts constructed value premium, although it is not statistically significant;
the default spread also has some statistical power at individual forecasting horizon. Overall, these
results are substantially weaker than those associated with the expected value premium in Table 2.
We then conduct the following multivariate regression:
K Y
∑
t+k
= a + b × ICCtHM L + A × Xt + ut+K .
k=1 K
14
(8)
HM L , X =VS , and (2) Y
HM L
We consider two specifications of (8): (1) Yt+k = Rt+k
t
t
t+k = Rt+k ,
HM L is the realized value premium at month t + k, ICCHM L
Xt = [Term t , Default t ]. Again, Rt+k
t
is the expected value premium at month t, VS t , Term t , and Default t are the value spread, term
spread, and default spread at month t. If the expected value premium contains information beyond
the value spread and business cycle variables, we should obtain a positive and significant coefficient
for b.
Panels A and B of Table 3 present the regression results of (8) for the above two specifications,
respectively. The dependent variable is our constructed value premium in Panels A1 and B1, and
it is the Fama-French HML factor in Panels A2 and B2.
Table 3 shows that the expected value premium strongly predicts future realized value premium,
even after controlling for the value spread and the business cycle variables. In the presence of the
value spread, the slope coefficients associated with ICCHM L are all positive for all forecasting
horizons, and they are all statistically significant at the 1% level. This is true for predicting both
the constructed value premium (Panel A1) and the Fama-French HML factor (Panel A2). In the
presence of ICCHM L , VS is also significant at horizons longer than two years, suggesting that these
two variables contain distinct information. For the constructed value premium, ICCHM L and VS
together explain 29% at the 1-year horizon, and 43% at the 3-year horizon.
Panel B shows that the expected value premium remains statistically significant at the 5% level
in the presence of the business cycle variables. On the other hand, the business cycle variables
are largely driven out by the expected value premium. Notably, the signs of the term spread and
default spread even become negative in most horizons.
To summarize, the multivariate regression results show that the expected value premium remains
a strong predictor of future realized value premium in the presence of the value spread and the
commonly used business cycle variables.
4.1.3
Further Analyses
This subsection provides robustness checks on alternative measures of expected value premium,
investigates the role of dividend growth forecasts in the expected value premium measure, and
analyzes the impact of analyst forecast biases on our results. To save space, we report the results
for our constructed value premium only, as the results for predicting the HML factor of Fama-French
model are very similar.
15
Alternative Measures of Expected Value Premium
Our main measure of expected value
premium forecasts future cash flows up to 15 years, i.e., T = 15 in Equation (6). To examine the
robustness of our results, we estimate the expected value premium with T = 10 and T = 20. When
T = 10, the expected return for the value portfolio is 14.61%, and the expected return for the
growth portfolio is 10.83%, so both expected returns are lower than those when T = 15 (Panel A
of Table 1). However, the expected value premium as a difference of the two expected returns are
largely unaffected, with a mean of 3.77% and a standard deviation of 2.22%. When T = 20, the
expected return for the value portfolio is 17.11%, and the expected return for the growth portfolio is
13.39%, so both expected returns are higher than those when T = 15. The expected value premium
has a mean of 3.72% and a standard deviation of 2.54%, which are still comparable to those when
T = 15.
Panel A1 of Table 4 provides the univariate regression results of (7) for ICCHM L constructed
based on T = 10 (the results for T = 20 are very similar and thus not reported). We observe that
the expected value premium is significant at the 1% level at all forecasting horizons and, and the
average slope coefficient across all horizons is also significant (p-value 0.00). The adjusted R2 is
2% at the 1-month horizon, and it increases to 34% at the 3-year horizon. These results are very
similar to those in Table 2, when the expected value premium is constructed based on T = 15.
Our main measure of constructing the expected value premium is based on a 3-by-2 two-way
sorting procedure. As another robustness check, we construct an alternative measure based on a
5-by-1 one-way sorting.6 That is, in June of each year from 1975 to 2011, using the breakpoints for
the 20%, 40%, 60%, and 80% of the ranked values of BE/ME for NYSE stocks, we sort all CRSP
common stocks into five quintile L, 2, 3, 4, and H. Portfolio H is the value portfolio containing
stocks with the highest book-to-market ratio, and portfolio L is the growth portfolio containing
stocks with the lowest book-to-market ratio. The expected value premium is the difference between
the ICC for the value portfolio and the ICC for the growth portfolio.
The mean and the standard deviation of the expected value premium are 5.25% and 3.34% for
the 5-by-1 sorting. Because the one-way sorting does not control for the size effect, the expected
value premium from the one-way sorting is slightly higher than from the two-way sorting. Panel
A2 of Table 4 provides the univariate regression results using this alternative measure of expected
value premium. We find that the expected value premium still reliably forecasts future realized
6
In unreported results, we perform a 5-by-5 two-way sorting to construct the expected value premium, and our
results remain robust as well.
16
value premium: it is still significant at all horizons at the 1% level, the p-value of the overall slope
coefficient is 0.00; it explains 2% at the 1-month horizon and increases to 19% at the 1-year horizon.
Overall, the statistical significance of ICCHM L based on a one-way sorting provides similar results
to those based on a two-way sorting.
The Role of Dividend Growth Asness, Friedman, Krail, and Liew (2000) finds that both the
value spread and the earnings growth spread between value and growth firms help predict future
value premium. Chen, Petkova, and Zhang (2008) estimates dividend-price ratio and dividend
growth separately to obtain the expected value premium measure. Our measure of expected value
premium, on the other hand, is based on the implied cost of capital approach, which naturally
incorporates the information on both dividend-price ratio and dividend growth. We now examine
the role of dividend growth forecasts in predicting future realized value premium.
More specifically, we construct an alternative measure of expected value premium which does
not incorporate the information of dividend growth. That is, rather than estimating ICC as in
Subsection 2.1, which imposes reasonable assumptions about earnings (dividend) growth, we simply
assume that there is no earnings (dividend) growth. Therefore, ICC is estimated as a return to a
perpetuity and equals to F E1/P , with F E1 being the analyst forecasted earnings in the next year.
We then value-weight F E1/P for the firms in the value portfolio and growth portfolio to obtain
the portfolio-level ICC, and calculate the difference between the ICC for the value portfolio and the
growth portfolio as the new expected value premium FEHM L . Therefore, FEHM L is the expected
value premium measure without incorporating the information of dividend growth. If dividend
growth is an important ingredient of expected value premium, we expect ICCHM L to outperform
FEHM L in forecasting future realized value premium.
Panel B of Table 4 provides the univariate regression results for FEHM L (Panel B1), as well
as the bivariate regression results involving both ICCHM L and FEHM L (Panel B2). We find that
FEHM L also positively predicts future realized value premium, and it is statistically significant at
the 5% level for both individual horizons tests as well as the joint horizon test. But the explanatory
power of FEHM L is weaker than ICCHM L , judged by both statistical significance and the adjusted
R2 . In the bivariate regression, FEHM L loses its predictive power in the presence of ICCHM L : it
is not longer significant at any horizon, and it even becomes negative at the 1-year horizon. On
the other hand, ICCHM L continues to be significant in both the individual horizon tests and the
joint horizon test. Therefore, the dividend growth rate is an indispensable component of ICCHM L ,
17
which makes ICCHM L a better measure of expected value premium than FEHM L .
Analyst Forecast Biases
Our calculation of expected value premium uses analysts’ forecast of
future earnings, which might be biased. Notably, several studies find that analyst forecasts tend to
be optimistic. We now show that the predictive power of ICCHM L is not driven by analyst forecast
optimism. Our main finding is that higher ICCHM L predicts higher realized value premium. All
else equal, higher analyst forecasts lead to higher ICC; if our result is driven by analyst optimism,
then we expect analysts to be more optimistic for value stocks than for growth stocks, and thus,
the difference in analyst optimism bias between value and growth stocks should positively predict
future value premium.
To investigate the predictive power of analyst optimism bias, for each firm in each month, we
compute the ratio of the difference between the consensus 1-year-ahead analyst forecast of earnings
per share (EPS) and the corresponding actual EPS to the 1-year-ahead forecast.7 We valueweight the optimism biases across firms in each month to compute the overall analyst optimism
for the value portfolio and the growth portfolio, respectively, and obtain the difference as our
main measure of optimism bias (AE HM L ) for the expected value premium. For our results to be
explained by analyst optimism, a high value of AE HM L should predict high realized value premium
and weaken/eliminate the predictive power of ICCHM L .
Panel C of Table 4 provides the predictive power of AE HM L for realized value premium. In
the univariate regression (Panel C1), analyst forecast optimism negatively predicts future realized
value premium at most horizons, and the adjusted R2 is close to 0. In Panel C2, we run a bivariate
regression on both ICCHM L and AE HM L . We find that ICCHM L continues to positively forecast
realized value premium at all horizons, and the statistical significance is also comparable to that
provided in the univariate regression of Table 2. On the other hand, AE HM L is not significant at
either individual horizons or across horizons. These results provide strong evidence that our results
are not driven by analyst forecast optimism.
4.2
Out-of-Sample Analysis
So far, we have established the expected value premium as a good ex-ante measure of value premium.
It is interesting to know whether the predictive power of expected value premium still holds in the
out-of-sample setting. In this subsection, we explore the out-of-sample performance of the expected
7
Note that this is just the negative of the forecast error.
18
value premium.
4.2.1
Econometric Specification
We start with the following predictive regression model:
HM L
Rt+1
= αi + βi xi,t + εi,t+1 ,
(9)
HM L is the monthly realized value premium from our constructed portfolios, x
where Rt+1
i,t is the ith
monthly predictive variable. Our main interest is the expected value premium measure ICCHM L ,
but we also examine the predictive power of other variables including the value spread, the term
spread, and the default spread. εi,t+1 is the error term.
Following Welch and Goyal (2008), we use a recursive method to estimate the model and
generate out-of-sample forecasts of the value premium. More specifically, we divide the entire
sample T into two periods: an estimation period composed of the first m observations and an
out-of-sample forecast period composed of the remaining q = T − m observations. The initial
out-of-sample forecast based on the predictive variable xi,t is generated by
HM L
R̂i,m+1
= α̂i,m + β̂i,m xi,m ,
where α̂i,m and β̂i,m are obtained using ordinary least squares (OLS) by estimating (9) using
observations from 1 to m. The second out-of-sample forecast is generated according to
HM L
R̂i,m+2
= α̂i,m+1 + β̂i,m+1 xi,m+1 ,
where α̂i,m+1 and β̂i,m+1 are obtained by estimating (9) using observations from 1 to m + 1. So
when generating the next-period forecast, the forecaster uses all information up to the current
period, which mimics the real-time forecasting situation. Proceeding in this manner through the
end of the forecast period, for each predictive variable xi , we can obtain a time series of predicted
}T −1
{
HM L
.8
value premium R̂i,t+1
t=m
Following Campbell and Thompson (2008), Welch and Goyal (2008) and Rapach, Strauss,
∑
HM L
HM L
as a
and Zhou (2010), we use the historical average realized value premium Rt+1 = tj=1 Rj
benchmark forecasting model. If the predictive variable xi contains useful information in forecasting
8
Alternatively, we can use a rolling method to estimate (9) and obtain the out-of-sample forecast. Specifically, we
use observations from 1 to m to estimate the model and generate forecast at time m + 1, and use observations from
2 to m + 1 to estimate the model, and generate forecast at time m + 2, and proceed in this manner. This rolling
method is less sensitive to structural breaks in the data. Using the rolling method yields similar results to using the
recursive method.
19
HM L
HM L should be closer to the true market returns than R
the future value premium, then R̂i,t+1
i,t+1 .
We now introduce the forecast evaluation method.
4.2.2
Forecast evaluation
Our first measure to compare the performance of alternative predictive variables is the out-of-sample
2 . This is akin to the familiar in-sample R2 , and is defined as
R2 statistics, Ros
)2
∑q ( HM L
HM L
R
−
R̂
m+k
i,m+k
k=1
2
Ros
=1− ∑
)2 .
(
q
HM L − RHM L
R
t+1
k=1
m+k
2 statistic measures the reduction in mean squared prediction error (MSPE) for the predictive
The Ros
regression (9) using a particular forecasting variable relative to the historical average forecast. For
HM L and thus
different predictive variables xi , we can obtain different out-of-sample forecast R̂i,m+k
2 . If a forecast variable beats the historical average forecast, then R2 > 0. A predictive
different Ros
os
2 performs better in the out-of-sample forecasting context.
variable that has a higher Ros
We formally test whether a predictive regression model using xi has a statistically lower MSPE
2 ≤ 0 against the
than the historical average model. This is equivalent to testing the null of Ros
2 > 0. To evaluate the statistical significance of R2 , we first calculate the following
alternative of Ros
os
statistic as defined by Clark and West (2007):
[((
)
)2 ) ((
)2 )]
(
HM L 2
HM L
HM L
HM L
HM L
HM L
−
Rt+1 − R̂i,t+1
−
Rt+1 − R̂i,t+1
.
ft+1 = Rt+1 − Rt+1
We then regress ft+1 on a constant and obtain the t-statistic corresponding to the constant
2 is obtained from the one-sided t-statistic (upper-tail) based on the
estimated. The p-value of Ros
standard normal distribution.9
To explicitly account for the risk borne by an investor over the out-of-sample period, we also
calculate the realized utility gains for a mean-variance investor, following Campbell and Thompson
(2008), Welch and Goyal (2008), and Rapach, Strauss, and Zhou (2010). More specifically, based
on the forecasts of expected return and expected variance of the value strategy, a mean-variance
investor with relative risk aversion parameter γ makes her optimal portfolio decision by allocating
her portfolio monthly between investing in the stocks using the value strategy and risk-free asset.
Her allocation to stock in period t + 1 is:
w1,t
( ) ( HM L )
Rt+1
1
,
=
2
γ
σ̂t+1
9
(10)
See Clark and West (2007) for superior performance of this test over the popular test in Diebold and Mariano
(1995).
20
if she forecasts the expected return using historical average, and is:
( ) ( HM L )
R̂i,t+1
1
w2,t =
.
2
γ
σ̂t+1
(11)
if she forecasts the expected return using a particular predictive variable. In both portfolio decisions,
2
σ̂t+1
is the forecast for the variance of stock returns. Similar to Campbell and Thompson (2008)
2
and Rapach, Strauss, and Zhou (2010), we assume that the investor obtains σ̂t+1
by using a ten-year
rolling window of monthly returns.10
The investor’s average utility level in the out-of-sample period is (the utility level can also be
viewed as the certainty equivalent return for the mean-variance investor) is:
1
U1 = µ1 − γ σ̂12 ,
2
(12)
if she uses historical average to make her portfolio decision, and is
1
U2 = µ2 − γ σ̂22 ,
2
(13)
if she uses a predictive variable to make her portfolio decision. Here, µ1 (µ2 ) and σ̂12 (σ̂22 ) correspond
to the sample mean and variance for the return on the portfolio formed based on historical average
(a predictive variable) over the out-of-sample period.
We measure the utility gain of using a particular predictive variable as the difference between
(13) and (12). We multiply this difference by 1200 to express it in average annualized percentage
return. This utility gain can be viewed as the portfolio management fee that an investor with
mean-variance preferences would be willing to pay to access a particular forecasting variable. We
report the results based on γ = 3 .
Finally, in order to explore the information content of ICCHM L compared with other forecasting
variables, we also conduct a forecasting encompassing test due to Harvey, Leybourne, and Newbold
(1998). The null hypothesis is that the model i forecast encompasses the model j forecast against
the one-sided alternative that the model i forecast does not encompass the model j forecast. Define
gt+1 = (ε̂i,t+1 − ε̂j,t+1 ) ε̂i,t+1 , where ε̂i,t+1 (ε̂j,t+1 ) is the forecasting error based on predictive variable
HM L − R̂HM L , and ε̂
HM L − R̂HM L . The Harvey, Leybourne, and
i (j), i.e, ε̂i,t+1 = Rt+1
j,t+1 = Rt+1
i,t+1
j,t+1
Newbold (1998) test can be conducted as follows:
]
[
HLN = q/ (q − 1) V̂ (g)−1/2 g,
10
Following Campbell and Thompson (2008), we constrain the portfolio weight on stocks to lie between 0% and
150% (inclusive) each month.
21
where g = 1/q
q
∑
q
(
)∑
gt+k , and V̂ (g) = 1/q 2
(gt+k − g)2 . The statistical significance of the test
k=1
k=1
statistic is assessed according to the tq−1 distribution.
4.2.3
Out-of-sample forecasting results
In the out-of-sample forecasting scenario, how to choose the estimation and forecast periods is
ultimately an ad-hoc choice, but the criteria are clear: it is important to have enough observations
in the evaluation period to obtain reliable estimates of the predictive model, and it is also important
to have a long-enough period for the model to be evaluated. In our experiment, we examine two
specifications: in the first case, the forecast period is from 1988.01 to 2011.12, and in the second
case, the forecast period is from 1994.01 to 2011.12. So the two forecast periods correspond to 2/3
and 1/2 of the entire sample.
2 test results and the utility gains based on different forePanel A of Table 5 provides the Ros
casting variables. We observe that the expected value premium ICCHM L is the best out-of-sample
2 is 2.14% in the first forecast period, and 1.54% in the
predictor in both forecasting periods. Its Ros
2 is statistically significant at the 1% level. Campbell and Thompsecond forecast period. Both Ros
2 values such as 0.5% can signal an economically
son (2008) argue that for monthly data, positive Ros
meaningful degree of return predictability for a mean-variance investor, which provides a simple
assessment of forecastability in practice. Therefore, the out-of-sample forecasting performance of
the expected value premium is very impressive. Contrary to ICCHM L , all other variables produce
2 , indicating that they cannot beat the naive predictor of historical average.
negative Ros
To get a visual impression of how each model performs over the forecasting period, Figure 2
plots the differences between cumulative squared prediction error for the historical average forecast
and the cumulative squared prediction error for the forecasting models using different predictive
variables, for the forecast period of 1994.01-2011.12. If a curve lies above the horizon line of zero,
then the forecasting model using a particular predictive variable outperforms the historical average
model. As pointed out by Welch and Goyal (2008), the units on these plots are not intuitive, what
matters is the slope of the curves: a positive slope indicates that a particular forecasting model
consistently outperforms the historical average model, while a negative slope indicates the opposite.
If a forecasting model consistently beats the historical average model, then the corresponding curve
will have a slope that is always positive; the closer a forecasting model is to this ideal, the better
the performance of this model. In Figure 2, the difference between the prediction error of the
historical average forecast and the cumulative squared prediction error for the ICCHM L forecast
22
stays above zero for the vast majority of the time, and the slope of the difference stays positive
for longer periods than for other forecasting variables. Overall, the other variables tend to beat
the historical average towards the end of 1990s, but seriously underperform at the beginning of the
2000s.
Panel A of Table 5 also reports the utility gains using a specific forecasting model versus
using the historical average. The expected value premium produces positive utility gains in both
forecasting periods, indicating that investors are willing to pay for access to the information in
ICCHM L to form their optimal portfolios. The value spread, term spread, and default spread also
produce positive utility gains, albeit smaller than those of ICCHM L .
We further examine whether ICCHM L contains distinct information than that contained in
existing variables such as the value spread. The Harvey, Leybourne, and Newbold (1998) test
results are presented in Panel B of Table 5. In both forecast periods, the expected value premium
ICCHM L contains distinct information from existing forecasting variables. We strongly reject the
null hypothesis that ICCHM L is encompassed by another variable for all variables at the 1% level
in the first forecast period, and at the 2% level in the second forecast period. On the other hand,
we cannot reject the null hypothesis that ICCHM L encompasses other forecasting variables at
conventional levels. Among other variables, the term spread and default spread contain different
information from the value spread, especially in the first forecast period.
So far, our out-of-sample analysis has shown that the expected value premium has excellent
predictive power for the realized value premium in a real-time fashion. More importantly, it contains
important and distinct information than in other commonly used forecasting variables of the value
premium.
5
Risk or Mispricing?
After decades of research, the literature has reached some agreement that the value strategy has
produced superior returns. The interpretation of why it has done so, however, remains controversial
and is still an on-going debate. On the one hand, rational risk-based theories argue that value stocks
are fundamentally riskier than growth stocks, especially during economic downturns, and therefore,
the value premium is a compensation for investors who bear higher fundamental risk (e.g., Fama
and French (1992),Lettau and Ludvigson (2001), Gomes, Kogan, and Zhang (2003), Campbell and
Vuolteenaho (2004), Zhang (2005), and Kiku (2006)). Several recent empirical work study the
23
countercyclical variation of the value premium, and provides supportive evidence for the risk-based
explanation (e.g., Petkova and Zhang (2005), Chen, Petkova, and Zhang (2008), Gulen, Xing,
and Zhang (2008), Kiku (2006), Santos and Veronesi (2010)). On the other hand, Lakonishok,
Shleifer, and Vishny (1994) argue that the value strategy produces higher returns because investors
follow contrarian strategies. These strategies might be due to investors’ extrapolating past earnings
growth too far into the future, or due to investors’ overreaction to good or bad news, or some other
behavioral reasons. Regardless of the underlying reason, investors tend to be over-excited with
growth stocks and over-pessimistic with value stocks, generating overpricing in growth stocks and
underpricing in value stocks. They also find supportive evidence for this mispricing explanation.
In this section, we address the intriguing question of whether the value premium stands for risk
or mispricing. The advantage of our study is that rather than using realized value premium as
in most existing studies, we use the expected value premium, which we have shown as a reliable
ex-ante measure of realized value premium both in sample and out of sample. Since theories impose
restrictions only on the ex-ante value premium, our study represents a clean test and provides new
evidence towards this debate.
5.1
Risk Interpretation
If values stocks are riskier than growth stocks during economic recessions when the marginal utility
of investors is especially high, then risk-based theories generally predict a countercyclical behavior
of the expected value premium. We first examine whether this is the case. Figure 1 plots the NBER
recession periods in shaded areas. During our sample period, there are five recessions: 1980.011980.07, 1981.07-1982.11, 1990.7-1991.3, 2001.03-2001.11, and 2007.12-2009.06. The second and
last recession periods span longer (16 and 18 months, respectively), while the other recessions are
generally less than a year. A casual impression is that the expected value premium captures the
last recession very well: it is very low at the peak of the business cycle and becomes very high at
the trough of the business cycle. The behavior of the expected value premium in other recessions
is not as good, but the general pattern is there. In fact, the contemporaneous correlation between
the expected value premium and a dummy variable for the NBER recession is 0.20 (p-value 0.00).
This indicates that the expected value premium is largely countercyclical.
Since the NBER recessions last for only a few months, we further investigate how the expected
value premium relates to economic condition. The contemporaneous correlation between ICCHM L
and the industrial production growth is −0.014 (p-value 0.77). Therefore, although high ICCHM L
24
coincides with bad economic condition, the evidence is weak. Furthermore, if value stocks are riskier
than growth stocks in economic downturns, then the expected value premium should positively
predict future economic condition. We thus run the univariate regression of (7) with the dependent
variable Yt+k being the industrial production growth rate at month t+k, and Xt being our expected
value premium measure ICCHM L at month t. The industrial production growth rate is calculated
based on the industrial production index from the economic research database at the Federal
Reserve Bank at St. Louis (FRED). If the expected value premium contains a risk component,
then we expect a positive sign for b. As a comparison, we also run the univariate regression using
the value spread, the term spread, or the default spread (Xt = V St , or T erm, or Def ault).
Panel A of Table 6 provides the regression results. Overall, the expected value premium positively predicts future industrial production growth rates for all horizons except the 1-month horizon.
Although it is not statistically significant at individual horizons, the overall slope is significant at
the 10% level. This suggests that the expected value premium indeed contains a risky component.
Among the other variables, consistent with Chen (1991), the term spread is a significant positive
predictor of future economic condition. The intuition is as follows. When future output is expected
to be high, individuals desire to smooth consumption by borrowing more today, bidding up the
interests. The strong negative signs for the default spread in shorter horizons (within a year) indicates that the default spread better serves as an indicator of current economic condition rather
than a predictor of future economic condition (Chen (1991)). The value spread has no predictive
power for future industrial production growth.
Because the expected value premium is positive correlated with the term and default spread,
we further run multivariate regression to examine whether the predictive power of ICCHM L still
exists after controlling for other variables. This is done by setting Yt+k as the industrial production
growth rate at month t + k, and Xt = [V St , T erm, Def ault] in (8). Panel B of Table 6 provides the
results. In the presence of other variables, the predictive power of ICCHM L become even stronger.
The slope is positive across horizons and is statistically significant at horizons within a year; the
p-value of the overall slope coefficient is 0.00.
Our analysis so far suggests that the expected value premium indeed contains a risk component,
although the evidence is not particularly strong. This is largely consistent with extant evidence on
the tests of risk-based theories (e.g., Chen, Petkova, and Zhang (2008)).
25
5.2
Mispricing Interpretation
The term spread and default spread are typically considered as proxies for risk. In the multivariate
regression of Subsection 4.1.2, we find that that expected value premium continues to predict future
realized value premium after controlling for these risk proxies (Panel B of Table 3), which suggests
that the expected value premium also contains a mispricing component. This evidence is only
suggestive rather than conclusive because we might miss other risk proxies in the multivariate
regression. To show that the expected value premium contains a mispricing component, we need
to design a clean test which holds the risk component of expected value premium constant. We
take up this task in this subsection.
Earnings announcements are important events which bring new information to the market
regarding the fundamental values of firms. Therefore, if there is mispricing in value and growth
stocks, it is most likely to be resolved during the earnings announcements. Moreover, the abnormal
returns around earnings announcements have been documented as an important source of the
realized value premium. For example, Porta, Lakonishok, Shleifer, and Vishny (1997) find that
size-adjusted abnormal returns around earnings announcements are substantially higher for value
stocks than for growth stocks and the return differential accounts for up to about 30% of the
annual value premium reported in prior studies. Therefore, our test on the mispricing component
of expected value premium focuses on the abnormal returns around the earnings announcement.
More specifically, for each stock at each quarter, we obtain its quarterly abnormal returns,
defined as the cumulative market-adjusted abnormal returns (CAR) within the window of [-2,
2], with 0 standing for the announcement date. We then average the CARs of value stocks and
the CARs of growth stocks and take the difference as CAR HM L . Because CAR HM L is the return
within only a few days, neither risk (e.g., factor loadings) nor the price of risk (e.g., risk factors) will
change. Therefore, if the expected value premium predicts future CAR HM L , then the prediction
must be due to its mispricing component.
The quarterly earnings announcement dates are taken from the quarterly COMPUSTAT file.
The daily stock return for stocks and the market are taken from the daily CRSP files. The quarterly
values of expected value premium and other variables are obtained by taking the average of their
corresponding monthly values within a quarter. The data span from the first quarter of 1976 to the
second quarter of 2010. To examine whether the expected value premium and other variables predict
future CAR HM L , we first conduct the univariate regression analysis of (7) with the dependent
26
L
variable Yt+k being the CAR HM
t+k , i.e., the difference between cumulative abnormal return between
value and growth at quarter t + k. The explanatory variable Xt is either one of the following:
ICCHM L V St , T erm, Def ault. Since we conduct quarterly regressions, K is the forecasting horizon
in quarters, and we consider K = (1, 2, 3, 4).
The results are provided in Panel A of Table 7. If value stocks are cheaper than usual relative
to growth stocks, i.e., ICCHM L is higher than average, then they should outperform growth stocks
more than usual in the future. Similarly, if growth stocks are more expensive than usual relative to
value stocks, i.e., ICCHM L is lower than average, then they should underperform value stocks more
than usual in the future. Therefore, we expect a higher ICCHM L to predict a higher CAR HM L .
This is indeed what we find. The slope coefficients of ICCHM L are positive across all horizons, and
they are also statistically significant with p-values all equal to 0.00. The average slope coefficient
also has a p-value of 0.00. Moreover, the expected value premium explains 8% of CAR HM L at the
1-quarter horizon, and 25% of CAR HM L at the 1-year horizon. Again, since risk and the price of
risk do not change within a few days, the strong prediction of ICCHM L for CAR HM L suggests that
there is an important mispricing component in ICCHM L . Among other variables, notably the term
spread and default spread, there is no predictive power for CAR HM L . None of the slope coefficients
are statistically significant at any horizons, and the adjusted R2 associated with these regressions
are close to 0%.
Panel B of Table 7 presents the multivariate regression of results of (8) with the dependent variL
HML , V S , T erm, Def ault].
able Yt+k being the CAR HM
t
t+k , and the independent variable Xt = [ICC
In the presence of all other variables, the predictive power of ICCHM L for CAR HM L is still strong,
and the results are similar to those in the univariate regression case (Panel A). This further confirms
that the expected value premium contains a mispricing component.
6
Conclusion
In this paper, we use a theoretically justifiable approach—the implied cost of capital approach—
to construct the expected value premium. We conduct comprehensive analysis to show that our
expected value premium is an excellent ex-ante measure: it reliably predicts future realized value
premium, both in univariate regressions and in multivariate regressions, both in sample and out of
sample. Using this ex-ante measure, we revisit the question of whether the expected value premium
stands risk or mispricing. We find that both risk and mispricing are important components of the
27
expecte value premium.
We take the first step to test the broad implications of risk-versus-mispricing theories in this
paper. Since all theories on value premium impose restrictions on expected returns, our ex-ante
measure can also be useful for testing specific theories that have been proposed in the literature
(e.g, Barberis, Shleifer, and Vishny (1998), Daniel, Hirshleifer, and Subrahmanyam (1998), Hong
and Stein (1999), Lettau and Ludvigson (2001), Gomes, Kogan, and Zhang (2003), Campbell and
Vuolteenaho (2004), Zhang (2005), and Kiku (2006)).
Our study can also be useful for other research contexts. Numerous studies are relying on the
Fama-French three-factor model as a benchmark model to conduct asset-pricing tests, portfolio
allocation, performance evaluation, capital budeting, etc. Since our expected value premium is a
reliable predictor of the HML factor and theoretically, we should use ex-ante measures in applying
the Fama-French model, it is potentially interesting to replace the HML factor with our expected
value premium measure in these applications.
28
7
Appendix
For each regression, we conduct a Monte Carlo simulation using a VAR procedure to assess the
statistical significance of relevant statistics. We illustrate our procedure for the univariate regression
involving ICCHM L . The simulation method is conducted in the same way for other regressions.
Define Zt = (RtHM L , ICCtHM L )′ , where Zt is a 2 × 1 column vector. We first fit a first-order VAR
to Zt using the following specification:
Zt+1 = A0 + A1 Zt + ut+1 ,
(14)
where A0 is a 2 × 1 vector of intercepts and A1 is a 2 × 2 matrix of VAR coefficients, and ut+1 is a
2 × 1 vector of VAR residuals. The estimated VAR is used as the data generating process (DGP)
for the simulation. The point estimates in (14) are used to generate artificial data for the Monte
Carlo simulations. We impose the null hypothesis of no predictability on RtHM L in the VAR. This
is done by setting the slope coefficients on the explanatory variables to zero, and by setting the
intercept in the equation of RtHM L to be its unconditional mean. We use the fitted VAR under
the null hypothesis of no predictability to generate T observations of the state variable vector,
(RtHM L , ICCtHM L ). The initial observation for this vector is drawn from a multivariate normal
distribution with mean equal to the historical mean and variance-covariance matrix equal to the
historical estimated variance-covariance matrix of the vector of state variables. Once the VAR
is initiated, shocks for subsequent observations are generated by randomizing (sampling without
replacement) among the actual VAR residuals. The VAR residuals for RtHM L are scaled to match
its historical standard errors. These artificial data are then used to run bivariate regressions and
generate regression statistics. This process is repeated 1, 000 times to obtain empirical distributions
of regression statistics. The Matlab numerical recipe mvnrnd is used to generate standard normal
random variables.
29
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Table 1. Summary Statistics (1976.01-2011.12.)
Panel A reports the summary statistics for the expected return of the value portfolio (ICCH ), the
expected return of the growth portfolio (ICCL ), and the expected value premium (ICCHM L ). Panel
B reports the summary statistics for the realized return of the value portfolio (RH ), the realized
return of the growth portfolio (RL ), and the realized value premium (RHM L ). The last column
of Panel B also reports the correlation of the constructed portfolio returns with the respective
Fama-French portfolios obtained from Kenneth French’s website. Panel C reports the summary
statistics of other predictors of the value premium, including the value spread (VS ), the term spread
(Term), and the default spread (Default). Panel D reports the correlations among the expected
value premium and other predictors of the value premium; the p-values of the correlations are
provided in parentheses. Panel E reports the average number of firms in the portfolios used to
construct the expected and realized return measures in Panels A and B. The value and growth
portfolios in this table are obtained using a two-way sorting procedure in Fama and French (1993).
That is, we perform a two-by-three sorting of all stocks based on the size and book-to-market ratio
to obtain six portfolios, namely, S/H, S/M, S/L, B/H, B/M, and B/L. The value portfolio is defined
as (S/H+B/H)/2, and the growth portfolio is defined as (S/L+B/L)/2. The realized returns of the
value and growth portfolios, RH and RL , respectively, are the corresponding portfolio-level returns.
The expected returns of the value and growth portfolios, ICCH and ICCL , respectively, are the
portfolio-level implied cost of capital. The realized value premium RHM L and the expected value
premium ICCHM L are calculated as RH -RL and ICCH -ICCL , respectively. The data are monthly
data from January 1976 to December 2011. All variables except the value spread are expressed in
annual percentage terms.
34
Panel A: Expected Returns
Autocorrelation at Lag
Mean
Std
Max
Min
1
12
24
36
48
60
H
ICC
16.03
4.33
27.66
9.38
0.97
0.83
0.69
0.57
0.49
0.41
ICCL
12.25
2.55
19.25
8.45
0.97
0.87
0.82
0.76
0.68
0.59
3.78
2.39
10.89
0.09
0.92
0.57
0.31
0.08
0.04
-0.02
HM L
ICC
Panel B: Realized Returns
Autocorrelation at Lag
Mean
Std
Max
Min
1
12
24
36
48
60
with FF
RH
17.41
16.92
280.19
-297.19
0.16
0.08
0.01
0.00
0.04
-0.04
0.98
RL
13.23
18.83
181.14
-337.70
0.10
-0.02
0.05
-0.04
0.01
-0.05
0.99
4.18
9.82
132.94
-127.53
0.14
-0.03
-0.05
0.01
0.04
-0.03
0.95
HM L
R
Panel C: Other Predictors
Autocorrelation at Lag
Mean
Std
Max
Min
1
12
24
36
48
60
VS
1.38
0.16
1.96
1.08
0.97
0.67
0.47
0.39
0.30
0.18
Term
3.07
1.57
6.73
-3.01
0.91
0.43
0.07
-0.29
-0.37
-0.16
Default
1.11
0.48
3.38
0.55
0.96
0.47
0.28
0.19
0.09
0.10
Panel D: Correlation Matrix
ICCHM L
VS
Term
Default
1.00
-0.08
0.08
0.40
(0.10)
(0.10)
(0.00)
1.00
0.10
-0.24
(0.04)
(0.00)
1.00
0.16
HM L
ICC
VS
Term
(0.00)
Default
1.00
Panel E: Number of Firms
Mean
S/L
S/M
S/H
B/L
B/M
B/H
364
501
368
323
271
115
35
Table 2. Univariate Regressions on Expected Return Measures.
This table presents the univariate regression of the realized return of the value portfolio RH on its expected
return measure ICCH , the univariate regression of the realized return of the growth portfolio RL on its
expected return measure ICCL , and the univariate regression of the realized value premium RHM L on the
expected value premium ICCHM L , using monthly data from 1976.01 to 2011.12. Panel A presents the
results for predicting our constructed portfolio returns, and Panel B presents the results for predicting the
corresponding Fama-French portfolios obtained from Kenneth French’s website. The expected and realized
return measures used in this table are constructed based on a 3-by-2 two-way sorting procedure. K is the
forecasting horizon in months. b is the slope coefficient from the OLS regressions. avg. is the average
slope coefficient across all horizons. Z(b) is the asymptotic Z-statistics computed using the GMM standard
errors with Newey-West correction. These standard errors correct for the autocorrelation in regressions due
to overlapping observations and for generalized heteroskedasticity. The adj.R2 is obtained from the OLS
regression. The p-values of the Z-statistics (pval ) and the average slope coefficient are simulated using data
generated under the null of no predictability from 1,000 trials of a Monte Carlo simulation. The artificial
data for the simulation are generated under the null using the VAR approach described in the Appendix.
36
Panel A: Predicting Constructed Portfolio Returns
ICCH
ICCL
ICCHM L
K
b
Z(b)
pval
adj.R2
b
Z(b)
pval
adj.R2
b
Z(b)
pval
adj.R2
1
1.62
2.78
0.01
0.02
1.90
1.59
0.10
0.01
1.89
3.20
0.00
0.02
6
1.55
2.94
0.02
0.07
1.66
1.60
0.17
0.03
1.89
3.67
0.00
0.08
12
1.51
2.89
0.03
0.15
1.75
1.97
0.14
0.07
2.27
5.14
0.00
0.22
24
1.40
3.76
0.02
0.31
1.61
2.97
0.06
0.16
1.71
4.06
0.00
0.26
36
1.27
4.58
0.01
0.44
1.53
3.45
0.05
0.26
1.37
4.52
0.00
0.33
avg.
1.47
0.02
1.69
0.12
1.83
0.00
Panel B: Predicting Fama-French Portfolio Returns
ICCH
ICCL
ICCHM L
K
b
Z(b)
pval
adj.R2
b
Z(b)
pval
adj.R2
b
Z(b)
pval
adj.R2
1
1.59
2.89
0.01
0.01
2.06
1.69
0.07
0.01
1.82
2.91
0.00
0.01
6
1.53
3.07
0.02
0.07
1.93
1.75
0.12
0.03
1.88
3.14
0.01
0.07
12
1.45
2.86
0.04
0.14
2.05
2.12
0.09
0.08
2.28
4.61
0.00
0.18
24
1.31
3.44
0.04
0.27
1.91
2.87
0.05
0.18
1.67
3.78
0.01
0.21
36
1.17
4.13
0.04
0.39
1.83
3.44
0.05
0.29
1.26
4.11
0.01
0.24
avg.
1.41
0.01
1.96
0.08
37
1.78
0.00
Table 3. Multivariate Regressions for the Realized Value Premium.
This table reports the multivariate regression of the realized value premium on the expected value premium (ICCHM L )
and the value spread (VS ) in Panel A, and the multivariate regression of the realized value premium on the expected
value premium (ICCHM L ) and the business cycle variables, including the term spread (Term) and the default spread
(Default), in Panel B, using monthly data from 1976.01 to 2011.12. The dependent variable in Panels A1 and B1 is
the constructed value premium, obtained as the difference between the return of the value portfolio and the return of
the growth portfolio based on a 3-by-2 sorting as in Fama and French (1993), and the dependent variable in Panels
A2 and B2 is the HML factor of the Fama-French three-factor model, obtained from Kenneth French’s website.
K is the forecasting horizon in months. b is the slope coefficient from the OLS regressions. avg. is the average
slope coefficient across all horizons. Z(b) is the asymptotic Z-statistics computed using the GMM standard errors
with Newey-West correction. These standard errors correct for the autocorrelation in regressions due to overlapping
observations and for generalized heteroskedasticity. The adj.R2 is obtained from the OLS regression. The p-value
of the Z-statistics (pval ) is simulated using data generated under the null of no predictability from 1,000 trials of a
Monte Carlo simulation. The artificial data for the simulation are generated under the null using the VAR approach
described in the Appendix.
38
Panel A: Multivariate Regressions with ICCHM L and VS
Panel A1: Predicting the Constructed Value Premium
HM L
ICC
Panel A2: Predicting the Fama-French HML Factor
ICCHM L
VS
VS
K
b
Z(b)
pval
c
Z(c)
pval
adj.R2
b
Z(b)
pval
c
Z(c)
pval
adj.R2
1
1.95
3.21
0.00
1.00
0.73
0.37
0.02
1.92
2.95
0.00
1.55
0.99
0.27
0.02
6
1.98
4.02
0.00
1.50
1.25
0.27
0.11
2.01
3.60
0.00
2.23
1.67
0.17
0.12
12
2.36
5.65
0.00
1.62
1.60
0.21
0.29
2.42
5.37
0.00
2.42
2.21
0.10
0.31
24
1.78
4.15
0.00
1.39
2.68
0.09
0.37
1.79
4.04
0.01
2.17
4.08
0.02
0.44
36
1.43
4.48
0.00
0.99
3.30
0.08
0.43
1.37
4.26
0.01
1.73
5.59
0.01
0.52
avg.
1.90
0.00
1.30
0.00
2.02
0.14
1.90
Panel B: Multivariate Regression with ICCHM L and Business Cycle Variables
Panel B1: Predicting the Constructed Value Premium
ICCHM L
Term
Default
K
b
Z(b)
pval
c
Z(c)
pval
d
Z(d)
pval
adj.R2
1
2.46
3.44
0.00
0.09
0.10
0.47
-7.17
-1.67
0.95
0.02
6
2.29
4.24
0.00
0.62
0.84
0.26
-5.36
-1.62
0.89
0.10
12
2.46
4.96
0.00
0.98
1.39
0.15
-3.15
-1.39
0.85
0.24
24
1.61
3.42
0.01
0.75
1.42
0.15
0.66
0.26
0.41
0.28
36
1.19
3.41
0.02
-0.04
-0.11
0.51
2.44
1.12
0.22
0.35
avg.
2.00
0.00
0.48
0.25
-2.52
0.79
Panel B2: Predicting the Fama-French HML Factor
ICCHM L
Term
Default
K
b
Z(b)
pval
c
Z(c)
pval
d
Z(d)
pval
adj.R2
1
2.50
3.21
0.00
-0.03
-0.03
0.48
-8.49
-1.67
0.95
0.02
6
2.33
3.94
0.01
0.43
0.47
0.32
-5.92
-1.54
0.88
0.08
12
2.54
4.53
0.00
1.04
1.05
0.19
-4.03
-1.43
0.84
0.21
24
1.62
3.06
0.03
0.95
1.39
0.16
-0.09
-0.03
0.53
0.23
36
1.19
2.88
0.04
0.03
0.07
0.46
0.97
0.40
0.39
0.24
avg.
2.03
0.00
0.48
0.24
-3.51
39
0.87
0.02
Table 4. Further Analyses on the Expected Value Premium.
This table conducts several further analyses on our expected value premium measure. Panel A provides the univariate
regressions of realized value premium on two alternative measures of expected value premium. In Panel A1, the
expected value premium is constructed based on T = 10, and in Panel A2, the expected value premium measure is
obtained based on a 5-by-1 one-way sorting procedure. In the 5-by-1 sorting, in June of each year from 1975 to 2011,
using the breakpoints for the 20%, 40%, 60%, and 80% of the ranked values of BE/ME for NYSE stocks, we sort all
CRSP common stocks into five quintiles L, 2, 3, 4, and H. Portfolio H is the value portfolio containing stocks with the
highest book-to-market ratio, and portfolio L is the growth portfolio containing stocks with the lowest book-to-market
ratio. The expected value premium is the difference between the ICC for the value portfolio and the ICC for the
growth portfolio. Panel B analyzes the role of dividend growth component of expected value premium in forecasting
future realized value premium. FEHM L is the expected value premium which does not incorporate forecasts of future
dividend growth. Panel B1 provides the univariate regression of realized value premium on FEHM L , and Panel B2
provides the bivariate regression of realized value premium on ICCHM L and FEHM L . To construct FEHM L , we first
estimate the firm-level ICC as a return to a perpetuity and equals to F E1/P . We then obtain the portfolio-level ICC
by value-weighting F E1/P for the firms in the value portfolio and growth portfolio, respectively, and calculate the
difference between the ICC for the value portfolio and the growth portfolio as FEHM L . Panel C presents investigates
whether the predictive power of expected value premium is due to analyst being more optimistic for value stocks
than for growth stocks. AE HM L is the difference of analyst forecast optimism for the value portfolio and growth
portfolio. Panel C1 reports the univariate regression of realized value premium on analyst forecast optimism AE HM L ,
and Panel C2 reports the bivariate regression of realized value premium on AE HM L and ICCHM L . To construct
AE HM L , for each firm at each month, we calculate the firm-level analyst forecast optimism as the ratio of the
difference between the consensus 1-year-ahead analyst forecast of earnings per share (EPS) and the corresponding
actual EPS to the 1-year-ahead forecast. We value-weight the forecast optimism biases across firms within the value
and growth portfolios in each month to compute the corresponding portfolio-level analyst forecast optimism bias.
The analyst forecast optimism (AE HM L ) used in these regressions is the difference between the analyst forecast
optimism bias for the value portfolio and the analyst forecast optimism bias for the growth portfolio. Panels A-C use
monthly data from 1976.01 to 2011.12, and Panel D uses monthly data from 1976.01 to 2010.12. The realized value
premium used in this table is our constructed value premium. b is the slope coefficient from the OLS regressions. avg.
is the average slope coefficient across all horizons. K is the forecasting horizon in months. Z(b) is the asymptotic
Z-statistics computed using the GMM standard errors with Newey-West correction. These standard errors correct for
the autocorrelation in regressions due to overlapping observations and for generalized heteroskedasticity. The adj.R2
is obtained from the OLS regression. The p-value of the Z-statistics (pval ) is simulated using data generated under
40
the null of no predictability from 1,000 trials of a Monte Carlo simulation. The artificial data for the simulation are
generated under the null using the VAR approach described in the Appendix.
Panel A: Alternative Measures of Expected Value Premium
Panel A1: T = 10
Panel A2: 5-by-1 Sorting
ICCHM L
ICCHM L
K
b
Z(b)
pval
adj.R2
b
Z(b)
pval
adj.R2
1
2.05
3.23
0.00
0.02
1.74
3.07
0.00
0.02
6
2.01
3.40
0.00
0.08
1.73
3.11
0.01
0.08
12
2.37
4.65
0.00
0.20
1.91
4.14
0.00
0.19
24
1.84
3.84
0.01
0.26
1.47
3.81
0.01
0.21
36
1.51
4.57
0.00
0.34
1.19
4.30
0.01
0.24
avg.
1.95
0.00
1.61
0.00
Panel B: The Role of Dividend Growth
Panel B1: Univariate Regression
Panel B2: Bivariate Regression
FEHM L
ICCHM L
FEHM L
K
b
Z(b)
pval
adj.R2
b
Z(b)
pval
c
Z(c)
pval
adj.R2
1
1.98
2.20
0.02
0.01
1.65
2.14
0.02
0.58
0.50
0.38
0.01
6
1.83
2.11
0.07
0.04
1.75
2.99
0.02
0.35
0.36
0.48
0.08
12
1.87
2.42
0.06
0.07
2.31
4.78
0.00
-0.10
-0.13
0.63
0.21
24
2.03
3.00
0.05
0.17
1.40
3.32
0.01
0.74
1.23
0.25
0.27
36
1.90
3.53
0.04
0.27
0.98
3.64
0.01
0.92
1.74
0.18
0.36
avg.
1.92
0.00
0.50
0.04
1.62
0.40
Panel C: Analyst Forecast Optimism
Panel C1: Univariate Regression
AE
Panel C2: Bivariate Regression
HM L
HM L
AE HM L
ICC
K
b
Z(b)
pval
adj.R2
b
Z(b)
pval
c
Z(c)
pval
adj.R2
1
-0.01
-0.76
0.75
0.00
1.85
3.07
0.00
0.00
-0.43
0.65
0.01
6
-0.01
-1.76
0.91
0.01
1.83
3.51
0.00
-0.01
-1.12
0.83
0.08
12
-0.01
-1.91
0.94
0.01
2.26
5.12
0.00
0.00
-1.26
0.84
0.22
24
0.00
-0.30
0.58
0.00
1.73
4.03
0.01
0.00
0.25
0.44
0.27
36
0.00
0.54
0.35
0.00
1.38
4.39
0.01
0.01
1.31
0.18
0.33
avg.
0.00
0.00
0.00
0.85
1.81
41
0.64
Table 5. Out-of-sample Analysis.
This table summarizes the out-of-sample analysis of forecasting models using different forecasting variables. Panel A
2
reports the Ros
statistic of Campbell and Thompson (2008), as well as the utility gain. Panel B reports the p-values of
the forecasting encompassing test statistic of Harvey, Leybourne, and Newbold (1998) (HLN statistic). We consider
two forecast periods, namely, 1988.01-2011.12 in Panels A1 and B2 and 1994.01-2011.12 in Panels A2 and B2. The
2
dependent variable is our the constructed value premium. Statistical significance of Ros
is obtained based on the
p-value for the Clark and West (2007) out-of-sample adjusted-MSPE statistic; the statistic corresponds to a onesided test of the null hypothesis that the competing forecasting model using a specific forecasting variable has equal
expected squared prediction error relative to the historical average forecasting model against the alternative that the
competing model has a lower expected squared prediction error than the historical average benchmark model. Utility
gain (Ugain) is the portfolio management fee (in annualized percentage return) that an investor with mean-variance
preferences and risk aversion coefficient of three would be willing to pay to have access to the forecasting model using
a particular forecasting variable relative to the historical average benchmark forecasting model; the weight on stocks
in the investor’s portfolio is constrained to lie between zero and 1.5 (inclusive). The HLN statistic corresponds to
a one-sided (upper-tail) test of the null hypothesis that the forecast given in the column heading encompasses the
forecast given in the row heading against the alternative hypothesis that the forecast given in the column heading
does not encompass the forecast given in the row heading.
42
2
Panel A: Ros
Tests and Utility Gains Analysis
Panel A1: Forecast Period
Panel A2: Forecast Period
1988.01-2011.12
1994.01-2011.12
2
Ros
HM L
ICC
pval
2.14
0.01
Ugain
HM L
1.73
ICC
2
Ros
pval
Ugain
1.54
0.01
2.10
VS
-3.66
1.14
VS
-2.32
1.62
Term
-0.08
0.08
Term
-0.41
0.11
Default
-0.20
0.78
Default
-0.20
1.09
Panel B: Forecasting Encompassing Test
Panel B1: Forecast Period (1988.01-2011.12)
ICCHM L
HM L
ICC
VS
0.83
VS
0.00
Term
0.00
0.85
Default
0.00
0.83
Term
Panel B2: Forecast Period (1994.01-2011.12)
ICCHM L
Default
HM L
ICC
VS
Term
Default
0.64
0.74
0.67
0.14
0.10
0.84
0.83
0.02
0.01
VS
0.02
0.48
Term
0.00
0.68
Default
0.01
0.72
0.29
43
0.25
0.59
Table 6. Predicting Future Industrial Production Growth Rates.
This table provides the regression analysis of future industrial production growth rates on the expected value premium
(ICCHM L ), the value spread (VS ), the term spread (Term), and the default spread (Default), using data from 1976.01
to 2011.12. Panel A provides the univariate regression results, and Panel B provides the multivariate regression results.
b is the slope coefficient from the OLS regressions. avg. is the average slope coefficient across all horizons. K is
the forecasting horizon in months. Z(b) is the asymptotic Z-statistics computed using the GMM standard errors
with Newey-West correction. These standard errors correct for the autocorrelation in regressions due to overlapping
observations and for generalized heteroskedasticity. The adj.R2 is obtained from the OLS regression. The p-value of
the Z-statistics (pval ) is bootstrapped using data generated under the null of no predictability from 1,000 trials of a
Monte Carlo simulation. The artificial data for the simulation are generated under the null using the VAR approach
described in the Appendix.
44
Panel A: Univariate Regressions
ICCHM L
VS
b
Z(b)
pval
adj.R2
0.00
-0.04
-0.22
0.58
0.00
0.31
0.01
-0.14
-0.57
0.66
0.00
0.67
0.32
0.01
-0.23
-0.86
0.73
0.01
0.25
0.89
0.29
0.04
-0.17
-0.74
0.70
0.01
36
0.25
1.15
0.22
0.06
-0.01
-0.02
0.54
0.00
avg.
0.17
K
b
Z(b)
pval
adj.R
1
-0.02
-0.13
0.55
6
0.17
0.70
12
0.20
24
2
0.09
-0.12
0.73
Term
Default
b
Z(b)
pval
adj.R2
0.01
-5.77
-5.43
1.00
0.11
0.07
0.04
-3.15
-2.34
0.97
0.08
3.10
0.02
0.08
-1.40
-1.13
0.83
0.02
0.93
4.25
0.01
0.20
-0.09
-0.09
0.58
0.00
36
0.85
3.66
0.02
0.25
0.10
0.10
0.53
0.00
avg.
0.74
K
b
Z(b)
pval
adj.R
1
0.49
1.57
0.07
6
0.63
2.09
12
0.80
24
2
0.00
-2.06
1.00
Panel B: Multivariate Regression
ICCHM L
VS
Term
K
b
Z(b)
pval
c
Z(c)
pval
d
Z(d)
pval
1
0.51
2.85
0.00
-0.51
-2.98
1.00
0.85
3.05
0.00
6
0.50
1.87
0.07
-0.46
-2.13
0.93
0.86
3.32
0.01
12
0.36
1.19
0.17
-0.44
-2.01
0.91
0.95
4.01
0.00
24
0.29
1.28
0.18
-0.30
-1.95
0.89
1.00
5.29
0.00
36
0.28
1.58
0.15
-0.10
-0.54
0.63
0.86
4.26
0.01
avg.
0.39
0.00
-0.36
0.99
0.91
Default
K
b
Z(b)
pval
adj.R2
1
-7.73
-6.81
1.00
0.15
6
-5.04
-3.52
1.00
0.20
12
-3.07
-2.61
0.98
0.18
24
-1.53
-2.07
0.93
0.27
36
-0.96
-1.20
0.81
0.31
avg.
-3.67
1.00
45
0.00
Table 7. Predicting Future Cumulative Abnormal Returns Around Earnings
Announcements.
This table provides the regression analysis of the expected value premium (ICCHM L ), the value spread (VS ), the
term spread (Term), and the default spread (Default), for predicting future abnormal returns around earnings announcements, using quarterly data from 1976.Q4 to 2010.Q2. Panel A provides the univariate regression results,
and Panel B provides the multivariate regression results. For each stock, its abnormal return is calculated as the
market-adjusted returns around [-2, 2], with 0 being the quarterly earnings’ announcement date. Each quarter, we
calculate the value-weighted abnormal returns for the value portfolio and growth portfolio, respectively, and use their
differences as the dependent variable in the regressions. K is the forecasting horizon in quarters. b is the slope
coefficient from the OLS regressions. avg. is the average slope coefficient across all horizons. Z(b) is the asymptotic
Z-statistics computed using the GMM standard errors with Newey-West correction. These standard errors correct for
the autocorrelation in regressions due to overlapping observations and for generalized heteroskedasticity. The adj.R2
is obtained from the OLS regression. The p-value of the Z-statistics (pval ) is simulated using data generated under
the null of no predictability from 1,000 trials of a Monte Carlo simulation. The artificial data for the simulation are
generated under the null using the VAR approach described in the Appendix. For presentation purposes, we multiply
the coefficients of b by 100.
46
Panel A: Univariate Regressions
ICCHM L
VS
b
Z(b)
pval
adj.R2
0.08
6.86
0.67
0.37
0.00
0.00
0.14
3.92
0.36
0.50
0.00
4.38
0.00
0.22
1.64
0.15
0.57
0.00
4.35
0.00
0.25
-0.76
-0.07
0.63
0.00
K
b
Z(b)
pval
adj.R
1
1.96
3.92
0.00
2
2.24
4.05
3
2.40
4
2.32
avg.
2.23
2
0.00
2.92
0.48
Term
Default
K
b
Z(b)
pval
adj.R2
b
Z(b)
pval
adj.R2
1
-0.60
-0.57
0.73
0.00
3.79
0.85
0.19
0.01
2
-0.23
-0.23
0.60
0.00
1.79
0.42
0.32
0.00
3
0.10
0.09
0.51
0.00
0.37
0.08
0.42
0.00
4
0.30
0.28
0.45
0.00
-0.88
-0.19
0.49
0.00
avg.
-0.11
0.59
1.27
0.27
Panel B: Multivariate Regression
ICCHM L
VS
Term
K
b
Z(b)
pval
b
Z(b)
pval
b
Z(b)
pval
1
2.83
4.08
0.00
5.41
0.54
0.45
-0.47
-0.55
0.76
2
2.95
3.96
0.00
3.46
0.36
0.53
-0.27
-0.35
0.67
3
2.99
3.93
0.00
1.79
0.21
0.58
-0.05
-0.07
0.59
4
2.76
3.73
0.01
0.09
0.01
0.61
0.03
0.04
0.56
avg.
2.88
0.00
2.69
0.52
-0.19
Default
K
b
Z(b)
pval
adj.R2
1
8.55
1.81
0.05
0.11
2
7.32
1.92
0.07
0.18
3
6.50
1.72
0.11
0.26
4
5.06
1.34
0.18
0.27
avg.
6.86
0.01
47
0.67
Table A1. Univariate Regressions of Other Forecasting Variables for the Value Premium.
This table presents the univariate regressions of the realized value premium (RHM L ) on three forecasting
variables, namely, the value spread (VS ), the term spread (Term), and the default spread (Default), using
monthly data from 1976.01 to 2011.12. Panel A presents the results for predicting our constructed value
premium, and Panel B presents the results for predicting the HML factor of Fama-French three-factor model.
The constructed value premium is the difference between the return of the value portfolio and the return of
the growth portfolio based on a 3-by-2 sorting as in Fama and French (1993), and the HML factor of FamaFrench three-factor model is obtained from Kenneth French’s website. K is the forecasting horizon in months.
b is the slope coefficient from the OLS regressions. avg. is the average slope coefficient across all horizons.
Z(b) is the asymptotic Z-statistics computed using the GMM standard errors with Newey-West correction.
These standard errors correct for the autocorrelation in regressions due to overlapping observations and
for generalized heteroskedasticity. The adj.R2 is obtained from the OLS regression. The p-values of the
Z-statistics (pval ) and the average slope coefficient are simulated using data generated under the null of
no predictability from 1,000 trials of a Monte Carlo simulation. The artificial data for the simulation are
generated under the null using the VAR approach described in the Appendix.
48
Panel A: Predicting Constructed Portfolio Returns
VS
Term
K
b
Z(b)
pval
adj.R
1
0.81
0.59
0.43
6
1.31
1.05
12
1.41
24
2
Default
b
Z(b)
pval
adj.R2
0.00
-2.21
-0.57
0.73
0.00
0.27
0.00
-0.45
-0.13
0.55
0.00
1.25
0.16
0.02
2.30
0.85
0.27
0.01
1.02
1.42
0.13
0.04
4.29
1.43
0.16
0.07
0.21
0.37
0.36
0.00
4.98
1.99
0.09
0.15
b
Z(b)
pval
adj.R
0.00
0.05
0.06
0.48
0.32
0.03
0.65
0.73
1.22
0.29
0.05
1.17
1.24
1.79
0.20
0.09
36
0.85
1.84
0.23
0.08
avg.
1.12
0.16
0.62
2
0.19
1.78
0.29
Panel B: Predicting Fama-French Portfolio Returns
VS
Term
K
b
Z(b)
pval
adj.R
1
1.36
0.87
0.32
6
2.04
1.47
12
2.20
24
2
Default
b
Z(b)
pval
adj.R2
0.00
-3.51
-0.77
0.79
0.00
0.35
0.00
-1.03
-0.27
0.60
0.00
1.08
0.18
0.02
1.61
0.58
0.34
0.00
1.19
1.45
0.12
0.04
3.68
1.30
0.18
0.04
0.23
0.39
0.35
0.00
3.53
1.36
0.20
0.06
b
Z(b)
pval
adj.R
0.01
-0.12
-0.12
0.53
0.22
0.05
0.45
0.43
1.78
0.18
0.11
1.20
2.02
2.86
0.07
0.20
36
1.59
3.40
0.06
0.24
avg.
1.84
0.05
0.59
0.22
49
2
0.86
0.42
50
Figure 1: Expected Value Premium (1976.01-2011.12). This figure plots the time series of expected value premium constructed from a
two-way sort. The three horizontal dashed curves correspond to the rolling mean and the two-standard-deviation bands calculated using
all historic data starting from January 1981.
−3
4
x 10
Expected Value Premium
−3
4
3
2
2
0
1
−2
0
−4
−1
94.01 97.01 00.01 03.01 06.01 09.01
−3
1
x 10
Value Spread
−6
94.01 97.01 00.01 03.01 06.01 09.01
Term Spread
0.5
x 10
−4
20
x 10
Default Spread
15
10
0
5
−0.5
−1
94.01 97.01 00.01 03.01 06.01 09.01
0
−5
94.01 97.01 00.01 03.01 06.01 09.01
Figure 2: Cumulative Prediction Errors for forecasting Constructed Portfolios (1994.01-2011.12).
This figure plots the cumulative square prediction error of the realized value premium for the
historical average benchmark forecasting model minus the cumulative square prediction error for
the forecasting model using expected value premium, value spread, term spread, and default spread,
respectively. Expected value premium and the realized value premium are constructed from a twoway sort.
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