Expected Return and Asset Pricing♣
Alon Brav
Duke University
Reuven Lehavy
University of Michigan
Roni Michaely
Cornell University and IDC
First Draft: September 2002
This Draft: June 2003
♣
We thank Yakov Amihud, Malcolm Baker, Brad Barber, Larry Blume, Kobi Boudoukh, Markus Brunnermeier,
Peter Demerjian, John Graham, Campbell Harvey, J.B. Heaton, Matt Richardson, Jay Shanken, Hersh Shefrin,
Andrei Shleifer, Meir Statman, Avanidhar Subrahmanyam, seminar participants at Cornell University, Duke
University, Emory University, Harvard Business School, the Interdisciplinary Center Herzlyia, Israel, New York
University, Rice University, Tel-Aviv University, Tulane University, The Wharton School, University of British
Columbia, University of Chicago, University of Florida, University of Haifa, University of Washington, Seattle,
Vanderbilt University, Yale University, the 2003 Utah Winter Finance Conference, and John Nugent from Value
Line Institutional Services for their comments. We owe special thanks to Ravi Bansal and Michael Roberts for their
suggestions. Please address correspondence to either Brav at Fuqua School of Business, Duke University, Box
90120, Durham, North Carolina 27708, phone: (919) 660-2908, email: [email protected], Lehavy at University
of Michigan, 701 Tappan Street, Ann Arbor, Michigan 48109, phone: (734) 763-1508, email: [email protected],
or Michaely at Cornell University, Johnson Graduate School of Management, Ithaca, NY 14853-6201, phone: (607)
255-7209, email [email protected].
Expected Return and Asset Pricing
ABSTRACT
Asset pricing models predict relations between assets’ expected rates of return and
their risks or behavioral attributes. However, almost all tests of these models use
realized return as a proxy for expected return. This paper extracts proxies for the
market’s estimate of expected return using Value Line analysts’ forecasts (for
approximately 3,800 stocks over 27 years), and tests the relation between these
ex-ante expected return estimates and factors proposed in the literature to explain
them. Consistent with the predictions of the CAPM, we find that market beta is
positively related to expected returns. As with tests that employ ex-post realized
returns, we find that expected return is negatively related to the firm’s market
value of equity. Contrary to the implication of Fama and French (1993) and
others, we find that high book-to-market firms are not expected to earn higher
returns than low book-to-market firms. We also find that prior one-year returns
are negatively related to expected returns. Finally, we use an additional source of
expected return (based on First Call sell-side analysts’ estimates for 7,000 stocks
over five years) and find that the basic relation between expected returns and
stocks’ characteristics are similar for both sets of expectations.
Expected Return and Asset Pricing
Asset pricing models seek to establish the determinants of financial assets’ expected rates
of return. Classic asset pricing models, such as Sharpe (1964), Lintner (1965), and Black (1972),
predict that an asset’s expected return should be positively related to its systematic market risk.
Based on Merton’s (1973) ICAPM, Fama and French ((1992), (1993)) explain their findings of
higher returns for high book-to-market stocks and low capitalization stocks as a compensation
for risk. That is, those stocks are expected to earn higher rates of return because they are riskier.
While these models aim at explaining cross-sectional variation in expected returns,
researchers have been forced to use realized return as a proxy for expected return. This necessary
working assumption (due to the absence of data revealing market’s expectations of future
outcomes) implies that tests of asset pricing models are conditioned on both the hypothesis of
rational expectations in the sense that the average realization is a good proxy for expectation and
that the null asset pricing model describes the relation between expectations and asset pricing
attributes.
Using realized returns, however, it is difficult (if not impossible) to disentangle whether
some of the results in the empirical literature are driven by the inadequacy of the rational
expectations assumption or because expectations are not in line with those generated by the null
asset pricing model. In other words, an investigation into the relation between expected returns
and assets’ characteristics is a way to test asset pricing models without the restrictive assumption
that realized return is an unbiased proxy for ex-ante expected asset returns.
In this paper we complement existing research by providing direct tests of the relation
between ex-ante measures of expected return and several factors that have been shown to explain
cross-sectional variation in ex-post realized returns such as beta, book-to-market, market
capitalization, and price momentum. For example, the prediction emanating from a rational
asset-pricing model such as the CAPM is that beta is priced by the market. Therefore, the
expected return on high beta stocks should be higher than the expected return on low beta stocks.
Likewise, following Fama and French (1993), if stocks with larger loadings on a book-to-market
factor are riskier than firms with lower loadings, then this implies that the former stocks should
earn higher expected returns then the latter. We provide evidence on these predictions and the
importance of these factors that is not subject to the criticism mentioned above.
Why might realized return not be a good proxy for expected return? For one, the extent of
the noise in this proxy is likely to be large (Froot (1989)). In his presidential address Elton
(1999) argues that average realized return may be a biased proxy for expected return since the
use of this proxy relies on the belief that information surprises tend to cancel out over the period
of study. Realized returns may also be noisy and biased estimates of expected returns due to
more complex learning effects that have been studied by researchers who were interested in the
possibility of convergence to rational expectations equilibrium (e.g., Blume and Easley (1982,
1984), and Bray and Kreps (1987)). More recently, Lewellen and Shanken (2002) and Brav and
Heaton (2002) argue that rational learning can generate price paths that, ex-post, yield “value-”
and “momentum-” type effects in stock returns.
There are several additional advantages to the use of expectations rather than realizations
of returns. First, the requirement for a long time series of returns is not critical, since we do not
need to average realizations through time to get a good proxy for expectations. We use
expectations directly. Thus, each cross-sectional regression is a reliable representation of the
cross-sectional relation between expected return and assets’ characteristics. Second, data on
expectations is not subject to the over-lapping observations problem that hinders inferences when
long run realized return data is used (see Froot (1989)). Third, since we conduct asset pricing
tests on a new and untested data source, our results can shed light on the possibility that size,
book-to-market, or momentum are merely an outcome of data snooping (Black (1986)).1
Our expected return data is drawn primarily from Value Line, an independent research
provider which covers approximately 3,800 stocks over the period 1975-2001 (92 percent of the
NYSE, AMEX, Nasdaq in terms of market value). While the coverage is wide and not likely to
suffer from incentives-related biases, it is likely to proxy for market expectations with some
1
Evidence from earlier sample period (Davis, Fama, French (2000)) as well as international evidence (Rouwenhorst
(1998)) suggests that these effects are not likely to be an outcome of data snooping. Indeed, the current debate is
mainly on why these factors are priced and not whether they are priced.
2
noise for which we take account following Froot (1989). In basing our tests on Value Line
expectations, however, we are implicitly assuming they reflect market expectations. Though
market expectations are unobservable, there are several reasons to believe Value Line
expectations represent at least a significant portion of market’s expectations. First, Value Line
estimates impact market prices. Indeed, Huberman and Kandel (1990) argue that Value Line
analysts are able to predict time-variation in state variables that are important in the
determination of assets’ expected returns. Second, researchers and practitioners have been using
analysts’ earnings and growth forecasts as a proxy of the market’s estimates of these variables.
Third, subscribers to Value Line data (which include individual investors, brokerage and asset
management firms, and corporations) have been paying for this service for a while, and it is
likely that they would adopt these expectations.
To ensure that our results are not simply unique to Value Line expectations we augment
our analysis by using an additional source of market expectations, namely a large sample of
expected returns obtained from sell-side analysts for as many as 7,000 firms during the period
1997 through 2001. We find that the basic relation between expected returns and stocks’
characteristics is similar for both sets of expectations.2
Our first finding concerns one of the most important questions in the asset pricing
literature: Is beta priced? Using realized returns as a proxy for expected returns, the evidence is
mixed. Early tests, such as Black, Jensen, and Scholes (1972), Fama and MacBeth (1973) find
that firms’ betas are positively related to their realized returns. Using later data and monthly
return intervals Fama and French ((1992), (1993)) and others do not find a significant relation.
But when annual return intervals are used (Kothari, Shanken, and Sloan (1995)) beta is found to
be significantly related to realized returns.
Unlike many of the tests that use ex-post return data we find that beta risk is priced,
consistent with traditional asset pricing models. That is, a stock’s expected return is positively
2
It is possible that analysts’ long-term expected returns may also reflect perceived mispricing. Thus, the relation that
we document between expected returns and firms attributes, at least to some extent, might be driven by
characteristics that reflect misvaluation. We examine this possibility (Section IV.D) using a sub-set of securities that
analysts define as correctly priced and find results that are almost identical to those when the entire sample is used.
3
related to its beta. The price of beta risk is statistically significant and within a reasonable
economic range. The results indicate a market risk premium of around seven percent per year.
This is important. It may point out that the lack of beta-risk pricing using realized returns is
simply due to the noisiness of this proxy for expectations.
With respect to market capitalization, we find that investors expect smaller capitalization
stocks to earn significantly higher returns than large capitalization stocks. This result is
consistent with what other studies find using realized returns. The findings that investors expect
higher rates of return and on average receive higher returns on small capitalization stocks,
suggests that size is a risk factor (over and above market beta).
Using Value Line expectations, we do not find evidence that high book-to-market stocks
have higher expected returns than low book-to-market stocks. When using sell-side analysts
expected returns (from First Call) we find that the coefficient on book-to-market is negative and
significant. Regardless of the source of expectations, these results are inconsistent with the
notion that high book-to-market stocks are perceived by the market to be riskier and therefore
command higher expected returns. Combined, one result emerges clearly: High book-tomarket/low growth firms are not perceived as riskier investments relative to low book-to-market
firms.
We also examine whether the momentum factor (or prior return characteristic) is priced.
We find that it is generally priced, but that the sign is negative. This suggests that investors
expect stocks with high past returns to have lower returns in the future than stocks with low or
negative returns. In other words, investors do not consider recent well-performing (“winner”)
stocks as riskier and thus require higher expected returns than recent “loser” stocks.
Though most asset pricing tests utilize realized return as a proxy for expected return,
there are a few studies that employ other proxies. Froot (1989) uses interest-rate expectations
from a survey of financial market participants to test the expectations hypothesis. Examining the
impact of information surprises on bond pricing, Elton (1999) uses information from a survey by
the Money Market Survey, and Welch (2000) uses a survey of 226 academics to estimate the
expected market risk premium. Shefrin and Statman (2002) use ordinal ranking of
4
recommendations to proxy for expected returns and relate them to firm characteristics such as
book-to-market and market capitalization. They find that stocks with buy recommendations are
more likely to be low book-to-market stocks which they interpret as an indication of higher
expected return for those types of stocks, consistent with our findings. They also report that large
stocks are more likely to receive buy recommendations than small stocks. In contrast, we find
that, holding risk and other factors constant, small stocks exhibit higher expected return. Ang and
Peterson (1985) investigate the relation between expected return and dividend yields using yearend expected return data constructed from Value Line during the 1973-1983 period. They also
report a negative relation between return and size and positive relation between return and beta.
Finally, using Value Line forecasts of dividends and target prices, Botosan and Plumlee (2001)
obtain estimates of firm cost of capital and ask whether these estimates are correlated with firm
characteristics. They also find a positive relation between market beta and cost of equity.
Contrary to the findings in Ang and Peterson (1985) and this paper, Botosan and Plumlee find no
association between market capitalization and Value Line estimates of the cost of equity.
A related literature uses sell-side analysts’ earnings forecasts within a DCF model, like
the Gordon growth model to extract the firm’s implied cost of capital. Gebhardt, Lee, and
Swaminathan (2001) and Claus and Thomas (2001) are able to infer expectations of returns using
analysts’ estimates of earnings as well as long term growth rates, and a specific discounted
cashflow formulation. Gebhardt, Lee, and Swaminathan then examine the relation between these
implied estimates and firm characteristics, whereas Claus and Thomas examine the relation
between the implied expected rate of return on the market portfolio and historical realizations.
Both of these papers are important as they reveal information regarding patterns in implied
expectations. In contrast to the current paper, however, both papers rely on a specific discounted
cashflow formulation and are required to make strong assumptions about the forecast horizon,
terminal values, growth rates, and the methods by which analysts estimate firms' terminal values.
By using analysts' expected returns directly, our work differs from these papers as we are able to
5
avoid having to make these critical auxiliary assumptions.3
The rest of the paper is organized as follows. We present the data and variable definitions
in Section I. The relations between expected return and factor loadings or firm characteristics are
presented in Section II. In Section III we introduce an additional proxy for expected returns and
examine how these expectations are related to stocks’ attributes. In Section IV we conduct
various robustness checks. We conclude the paper in Section V.
I. Data and Variable Descriptions
A. Data Description
We construct estimates of ex-ante expected returns using analysts’ target prices. Data on
target prices is obtained primarily from Value Line (hereafter “VL”). VL publishes weekly
research reports for individual companies. Each company is analyzed on a quarterly cycle such
that a typical firm receives four reports per year. Among the various statistics published in the
report, VL includes a four-year range of low and high target prices. VL arrives at the high and
low ranges by first calculating an expected four-year target price and then, based on historical
price volatility, calculates symmetric high and low deviations about the expected target.
The target price information provided by the VL service has several distinguishing
features. First, VL is an independent research service with no affiliation to investment banking
activity. Hence, analysts’ optimism bias (Rajan and Servaes (1997)) or conflict of interest bias
(Michaely and Womack (1999)) is less likely to affect its expected return estimates.4 In fact,
given the structure of VL recommendations, there is no reason to believe that they have either
positive or negative bias. VL has as many reports with very positive recommendations
(timeliness rank equals 1) as reports with very negative recommendations (timeliness rank equals
3
Similar methodology is used by Harris, Marston, Mishra, and O'Brien (2002) to estimate the cost of capital of S&P
500 firms.
4
In the lawsuit New York State brought against five executives in connection with IPO spinning by Salomon Smith
Barney, one of the Salomon brokers is quoted to say: “Perhaps, as brokers, we should subscribe to Morningstar or
Value Line where research opinions are not based on influence by underwriting fees or the interest of the firm but on
the best interest.” (Filed to the supreme court of the State of New York, by State of New York and Elliot Spitzer,
September 30th 2002.)
6
5). Second, VL’s historical file contains a long time series of historical expected returns which
we collect starting from 1975 through 2001. Third, VL estimates cover approximately 90 percent
of U.S. traded firms in terms of their market value. These features (independence, long time
series of returns, and wide cross-sectional coverage) distinguish the VL estimates from other
available estimates of expected returns. While it is likely that VL estimates of expected return are
correlated with the market’s expectation, it is also likely that they do not precisely represent
those expectations. Following Froot (1989) we assume that VL expectations are equal to the
market expectations plus a random measurement error.
While there is a clear parallel between our setting and the standard asset pricing tests that
use realized returns as a proxy for expected returns, the direct measure of expected return has
several advantages over realized return. First, the standard error of the measure used in this paper
is smaller than the one produced by using realized returns as a proxy for expected returns.
Second, the overlapping observations problem is not as severe. For example, when one uses
realized returns and a one-year investment horizon revised every month, the number of
independent observations is rather limited since month t observation has 11 months overlap with
the observation in month t+1. This is not the case with expected returns since expectations are
reformed at time t and t+1 and are independent of future realizations. We do, however, account
for possible serial correlation induced by revisions in these expectations. Third, using
expectations directly does not necessitate the assumptions that expectations are rational, constant
over time or that there is no learning. When using realized returns to test asset pricing models,
we implicitly test jointly the validity of these auxiliary assumptions along with the underlying
asset pricing model. In addition, even when abstracting from these advantages, the use of
expectations provide an independent test of those models. This test is not subject to the data
mining problem described in Black (1986).
The use of expected rates of return in lieu of realized returns raises an additional subtle
issue related to the distinction between the predictions generated by the null asset pricing model
and the auxiliary assumptions that are needed in order to “take the model to the data.”
Specifically, the models that we are testing here, be it the CAPM or Merton’s ICAPM, lean
7
heavily on Milton Friedman’s (1953) “as if” methodological approach to modeling in economics
with the (only) goal generating predictions which link expected rates of returns on financial
assets with their risks. Indeed, the models intentionally abstract from the manner with which
investors actually derive the necessary knowledge regarding the underlying economy to form
these expectations. As a result, there is no explicit concern with investors’ learning (which is in
fact impossible to consider in a one-period CAPM). Similarly, the manner with which investors
form their beliefs regarding future cashflows and risks is not held important for testing the
resulting models. Thus, whether investors form their expectations using simple rules of thumb or
sophisticated models is not a feature that the model is designed to explain.
It is therefore important to recognize that the added assumption of rational expectations is
just that, an assumption, which is internally consistent with the other model assumptions, and
enables the use of realized returns in asset pricing tests. Similarly, our use of Value Line and
First Call expectations embodies the assumption that these expectations reflect market-wide
expectations. It allows us to falsify the model predictions. Ultimately, and as stated in the
introduction, we view the tests in this paper as complementary to those that have been conducted
in the literature using realized returns. Those who believe that quick convergence to a rational
expectations equilibrium is an unreasonable assumption may be persuaded with the view that
complex learning dynamics can lead realized returns to be a biased and noisy proxy for
investors’ expectations (Lewellen and Shanken (2002)). On the other hand, some may criticize
our assumption that the set of expectations we use represent the market’s expectations. On the
whole, by replacing one auxiliary assumption with another we believe that our tests can only
increase our knowledge regarding the viability of the null models. Given the potential advantages
of using expectations directly and especially given the extensive use of realized returns in testing
those models over the last four decades, it is worthwhile to carefully examine the results we
present here.
A1. The Value Line Database
Panel A of Table I provides descriptive statistics on the Value Line database for the
8
period 1975 through 2001.5 Coverage over time is fairly stable starting with approximately 5,600
price target reports in 1975, reaching about 5,800 reports in 2001 with an annual average of
5,361 price targets. The target price database includes reports for 3,779 distinct firms and we
include in our analysis only firms with common shares (CRSP share codes 10 and 11). Finally,
firms covered by VL account for approximately 92 percent of the total market value of all
securities with common shares on CRSP.6
Panels B through D of Table I describe the size, book-to-market, and prior price
momentum characteristics of our sample firms relative to the universe of firms available on
CRSP. Size in month t is computed as the market capitalization as of the end of the prior month
(expressed in millions of dollars). Book-to-market is computed as the ratio of annual common
shareholders’ equity (Compustat item #60) to market capitalization as of the end of the fiscal
year. This ratio is applied to the 12-month period beginning six months subsequent to the end of
the fiscal year. Prior price momentum for month t is the buy and hold return for the 11-month
period ending one month prior to month t.
We present statistics for each characteristic within decile portfolios. Size deciles are
based on NYSE capitalization cutoffs, while book-to-market and momentum cutoffs are based
on the universe of available firms excluding those with non-common shares. For each decile, we
report the mean of the monthly averages of the respective characteristic.
The statistics in Panel B indicate that there is a tendency for VL analysts to follow large
capitalization stocks. For example, 54.4 percent of companies fall into the smallest size decile
compared to only about 14.9 percent for the sample firms. Similarly, while the top three size
deciles comprise 6.9 percent of the universe of firms, 18.8 percent of the sample firms are
represented in these deciles.
At the same time, Value Line analysts also tend not to follow high book-to-market firms.
5
The data for the period 1987-2001 was purchased directly from Value Line Institutional Services. We handcollected individual reports for the period 1975-1986 and for several weeks in which we found missing data in the
period 1987-2001.
6
In unreported analysis we assess the frequency of quarterly target price revisions. We find that the probability that,
on average, a VL analyst revises his price forecast in the ensuing 1, 2, 3, and 4 quarters are 61, 87, 95, and 97
percent, respectively.
9
The greater representation of growth firms is evident from Panel C. For example, only about 22
percent of the sample firms are concentrated in the highest three book-to-market deciles (relative
to 30 percent in the universe). Finally, VL analysts show only a slight tendency to follow high
momentum stocks. About 32 percent of the sample firms comprise the top three momentum
deciles.
B. Proxy for Ex-Ante Expected Return
Our measure of expected return is based on VL’s four-year target price. As we mentioned
earlier, VL provides high and low ranges of prices which we average to obtain the expected fouryear target price. We then divide this target price by the firm’s market price which was
outstanding nine days prior to the Value Line report date, since analysts actually send their report
for publication for a given Friday on the Wednesday of the preceding week (all prices are
converted to the same split-adjusted basis).
To account for the dividend yield component in the expected return calculation we also
add a prospective dividend yield term as follows. For the period of 1975 through 1986, we first
calculate for each firm in the sample estimates of the annual dividend yield and growth rates of
dividends immediately prior to the calculation of the expected return. Dividends are calculated as
the sum of the dividends paid in the fiscal year before the price target is issued (Compustat data
item #21). Dividend growth rate is the ratio of current to prior year dividend per share given by
Compustat (Compustat data item #26), adjusted for stock splits. The dividend yield is then
calculated as the estimated dividend next year relative to the end of year stock price. Interim
dividends are reinvested at the firm’s cost of capital, the variable we actually want to compute.
We therefore begin by calculating the following expression for the expected return, which
includes the future value of dividends, assuming that dividends will continue to grow at the same
historical rate, gH, in the following four years:
(1 + ER )
VL 4
(
)
 1 + ERVL 4 − (1 + g )4 
TPt
D
H
=
+   ⋅ (1 + g H ) ⋅ 
,
VL
ER
g
Pt − 9  P  H
−
H


10
(1)
where
TPt
Pt − 9
is the expected return without the dividends. We then simply solve for the annual
expected annual return ERVL that satisfies this equality.
Since, for the period of 1987 through 2001, we were able to obtain VL analysts’ forecasts
of dividend growth rates as well as forecasts of the next year dividends we use those estimates in
calculating the value of dividends:
(1 + ER )
VL 4
Div next year
=
TPt
+
Pt −9
(
)
 1 + ER VL 4 − (1 + g )4 

⋅


ER VL − g


,
Pt −9
(2)
where g is the VL’s forecasted dividend growth rate, Divnext year is the VL forecast of next year
dividends. We then solve for the annual expected return ERVL as in equation (1) above.7
Table II presents statistics for the Value Line-based measure of expected return, ERVL.
For the stocks represented in this sample, the grand mean annual expected return (across stocks
and across years) is 20.9 percent. Over the same time period, the median annual expected return
is 18.8 percent, the same order of magnitude as the average.8
The annual expected returns provided in Table II are all based on equal-weighting of
individual firm forecasts. Since value-weighting provides for a natural alternative to describe our
data, we present in Figure I the time-series of value-weighted market expected returns. (The
expected return on this value-weighted portfolio can be viewed as an estimate of the expected
return on the market portfolio.) Specifically, each period we value-weight all firms’ expected
return by their prior period market value of equity. Since firms receive coverage only once a
quarter we present estimates of market expectations on a quarterly basis for each end of quarter
from 1975 through 2001. For example, the value-weighted expected return for March 1999 is the
weighted average expected return of all stocks with a valid expected return on VL from January
7
We winsorize individual target price to stock price ratio at the 1st and 99th percentiles to mitigate the possible effect
of extreme observations.
8
In unreported tests we have examined whether these expected returns have “reasonable” properties that are “model
free”, that is, that they correspond to a variation in a factor that is unambiguously related to expected returns. To this
end we examined the cross sectional correlation of our expected return measure with leverage. Clearly, high
leverage should lead to higher expected return, and that is what we find. The correlation between debt to equity ratio
and expected return is positive and highly significant.
11
1, 1999, to March 31, 1999. We observe that the value-weighted expected return is lower than its
equal-weighted equivalent. For the VL series the average value-weighted expected return is 17.8
percent compared to the equal weighted return of 20.9 percent.
II. Cross-sectional Variation in Ex-Ante Expected Returns
In this section we ask whether variation in expected equity returns is related to the firms’
factor loadings on different sets of factors that have been proposed in the literature. We examine
three such asset-pricing model specifications: the Sharpe-Lintner CAPM, the Fama and French
(1993) three-factor model, and Carhart’s (1997) four-factor model. 9,10
The basic asset-pricing question is whether expected returns are determined by firm
factor loadings. There are two common approaches to estimate the factor loadings. The first is to
directly estimate the covariance between the factor return and stock return using historical data.
The second is to proxy for the covariance by stocks’ characteristics such as size and book-tomarket. This approach assumes that a better proxy for actual covariances is obtained by the
characteristics rather than by historical estimation.11 Since it is not our objective here to take a
stand on which approach is preferred, we simply present the results using both factors and
characteristics.12
Our tests are based on Fama and MacBeth’s (1973) methodology. First, firm specific
9
To establish a useful benchmark, in unreported analysis, we follow Fama and French (1992) and Jegadeesh and
Titman (1993), and conduct Fama MacBeth-type cross-sectional regressions in which the dependent variable is realized
monthly return for the Value Line sample for the period 1975 through 2001. As in Fama and French (1992) we find that
market beta is associated with an insignificant premium while there is a statistically significant negative (positive)
relation between firm size (book-to-market ratio) and realized returns. Realized returns have a positive and significant
association with prior price momentum.
10
Fama and French’s empirical model includes RMRF, the excess return on the value-weighted market portfolio,
SMB, the return on a zero investment portfolio formed by subtracting the return on a large firm portfolio from the return
on a small firm portfolio, and HML, the return on a portfolio of high book-to-market stocks less the return on a portfolio
of low book-to-market stocks. Carhart’s model includes a fourth factor, PR12, which is formed by taking the return on
high return stocks minus the return on low return stocks over the preceding year.
11
See Reisman ((1992), (2001)) for a theoretical justification for this approach.
12
A potential shortcoming of the second approach is that those stock’s characteristics have nothing to do with the
covariance structure of returns and they are correlated with expected returns for other reasons. See Daniel and
Titman (1997) for a thorough discussion of this issue. As Daniel, Titman and Wei (2001) point out, whether factors
or characteristics are better able to explain stocks’ return has no direct implication to whether investors’ behavior is
rational or irrational.
12
factor loadings on the size and book-to-market factor in a given month are estimated using the
preceding 60 months. We require a minimum of 24 months with valid data for the estimation.
Since VL provides their estimated market betas for each stock, we use this beta rather than the
one estimated with the procedure described above.13 Second, for every month from January 1975
through December 2001, we regress firms’ expected annual excess return (ERVL-Rf) on their
estimated factor loadings and on their characteristics. We report the average of the estimated
intercept and slope coefficients and their associated t-statistics for various model specifications.
The t-statistics are computed using standard errors that allow for the possibility that the estimated
factor premia are serially correlated (see Pontiff (1996)). Specifically, we first estimate an
autoregressive model for the time-series of the factor premia and then derive standard errors that
are consistent with the estimated structure. Appendix A provides the details.14
We begin with tests of the CAPM by regressing the expected excess return on the firm’s
market beta. The results, presented in Table III, column 1, indicate that the estimate of the annual
market premium is positive and highly significant (t-statistic=5.1). Its magnitude is economically
reasonable with a point estimate of 6.7 percent. The positive and significant coefficient on beta is
particularly important in light of recent evidence that beta is not able to explain variation in expost returns (Fama and French (1992)). The evidence here is that stocks’ expected returns are in
fact related to their systematic exposure to the market, and that the relation is both economically
and statistically significant.
Next, consider column 2 in which we present estimates of factor premiums for the Fama
and French three-factor model. We find that the estimate of the market premium drops slightly to
5.8 percent on an annual basis but is still highly significant. The factor loading on size is priced
as well with a slope coefficient equal to 1.6 percent annually (t-statistic=2.1). Thus, even after
controlling for market beta, the size premium is positive and significant, consistent with the
findings of many studies that use ex-post returns. In other words, holding market risk constant,
13
In the next section we examine the sensitivity of our results to alternative measures of market beta.
In unreported results we conduct inferences on the regression estimates using Newey-West standard errors.
Inferences using these standard errors yield lower coefficient standard errors in all regression specifications and thus
more powerful rejections. We chose to present inferences that are more conservative.
14
13
investors expect small firms to generate higher returns than large firms.
The estimated premium related to covariation with the book-to-market factor, γHML, is
statistically insignificant. This estimate equals 0.2 percent (t-statistic=0.4), suggesting that
investors do not expect value stocks (high book-to-market) to yield higher returns than growth
(low book-to-market) stocks because they are riskier. This result is inconsistent with the
implications drawn from empirical studies that employ securities’ ex-post returns (Fama and
French (1992)) which interpret the positive relation between book-to-market ratio and ex-post
returns as evidence in favor of an additional systematic risk factor.
Column 3 of table III presents results for the four-factor model (including momentum as
the fourth factor). We find that market and size factors are still associated with positive and
significant premiums. Consistent with the results presented earlier, the premium demanded on
the book-to-market factor is insignificant. The estimate of the premium associated with the
momentum factor, γPMOM, is negative (-2.1 percent) and significant. This implies that investors
condition their expected return on past price trajectory, but exactly in the opposite direction of
what has been found when ex-post returns are used (e.g., Jegadeesh and Titman (1993)). Our
results indicate that investors, or at least Value Line analysts, do not consider recent wellperforming (“winner”) stocks as riskier and thus require higher expected returns than recent
“loser” stocks.
The regression results also reveal a systematic pattern in expectations that cannot be
entirely explained by any of the factor models that we employ. The positive and significant
intercept (about 6.8 percent in all regressions) is consistent with the notion that there are other
factors missing from our specifications. It is also interesting to note that the R2 of the factor
regressions are larger by order of magnitude than what has been reported when the dependent
variable is realized return. On the other hand, the fact that our R2 does not equal 100 percent
indicates that we have likely measured the market’s expectations with noise and/or that we have
an incomplete model of how expectations are formed. But overall, current factor models are
better able to explain the cross-sectional variation in expected returns than in realized returns.
As we discussed before, a priori it is unclear whether covariations of firm returns with
14
systematic factors are better captured by estimates of factor loadings using historical data or by
proxies for the factor loadings using firms’ characteristics. Panel B in Table III presents the
regression results with firms’ characteristics as the independent variables.
We estimate the three-characteristics model (Model II in panel B) using market beta
along with (log of) size and (log of) book-to-market. The results reveal that investors expect
small firms’ returns to be significantly higher than that of large firms. This result is consistent
with what we find when the factors loadings are the explanatory variables. They are also
consistent with what Banz (1981), Reinganum (1981), and Fama and French (1992) find when
the dependent variable is the ex-post realized return – that is, on average, small stocks’ returns
turn out to be higher than large stocks’ returns.
One possible interpretation of these results (market beta positively associated with
expected returns and market size negatively related with these expectations) is that they arise
simply because analysts have actually been taught the CAPM in business schools as well as size
related differences in realized returns. Examining the regression when only the first half of the
sample is used (1975-1986) reveals that this is not likely to be the case. Our results from this
subsample are similar to what is presented in Table III. In particular, the beta coefficient is
positive and highly significant. It is unlikely that the results in the early part of the sample are
simply an outcome of Business school education. In addition, Value Line manuals (Value Line
Methods (1979)), and conversations with Value Line analysts, reveal that they compute target
prices using a product of a forecasted earning multiple and earnings per share. Thus, beta is
priced even though it may not be accounted for explicitly.
Contrary to the finding of studies that use ex-post returns, but just like the results in Panel
A (when factors loadings are the independent variable), the premium associated with book-tomarket is insignificant. The coefficient on prior return is negative and significant (Model III in
panel B). The negative coefficient suggests that annual expected returns are higher for past losers
than for past winners. This result is surprising. First, it contrasts with what we observe when
realized returns are used for this short time horizon (e.g., Jegadeesh and Titman (1993)). Second,
it is not intuitive to reconcile this result with Jegadeesh, Kim, Krische, and Lee (2002) who find
15
that analysts tend to follow momentum stocks. If analysts tend to follow momentum stocks, it is
probably because they believe those stocks will generate higher returns. But then why do we
observe that their expected return is negatively affected by momentum? One possibility is that
our expected return measure also contains expectations from the beginning of the month, which
may artificially create this negative relation. We check for this possibility in the robustness
section and find no evidence that this is the cause of the relation.
To summarize, the use of expected return derived from independent analysts’ estimates
leads to three important results. First, beta is priced. Cross-sectionally, expected return is
positively related to beta. Second, holding market beta constant, the expected return on small
stocks is significantly higher than the expected return on large capitalization stocks.15 Third,
book-to-market is not a risk factor in the traditional sense. We do not find evidence that high
book-to-market stocks are associated with a higher expected return. If indeed they were riskier, it
must be the case that investors would have expected them to earn higher returns.
These findings are also illustrated in Figure II. For each firm over the period 1975 to
2001 we calculate an average of all outstanding expected annual excess returns as well as
average of market betas and slopes on the Fama and French size factor, SMB. We then form
independent decile breakpoints by beta and slopes on SMB and allocate all firms into one of the
resulting 100 portfolios. A cross-sectional regression is then estimated in which portfolio
expected excess returns are regressed against their average betas and size-related factor loadings.
The plot provides the portfolio expected excess returns as well as the fitted plane from the least
squares regression. Our finding that market-related risk is positively priced can be easily seen.
Furthermore, the higher premium for size-related risk is evident as well, controlling for market
risk.
15
In unreported results we have examined whether alternative measures of firm size such as sales and total assets are
also negatively related to firm expected returns. We obtain similar evidence using these alternative measures in the
characteristics-based regressions.
16
III. An Alternative Set of Expected Returns
While Value Line services are being subscribed to by a wide array of both individuals
and institutions, we find it prudent to employ another set of expectations and to examine whether
the relation between expected return and the factors are unique to the Value Line estimates of
expected returns, or whether they hold when other estimates of expectations are used. To this
end, we construct an expected return measure based on First Call’s database (hereafter FC),
which gathers target prices issued by sell-side analysts. Specifically, we employ FC’s one year
ahead target price forecasts for over 7,000 firms during the period 1997 through 2001. Using
these target price forecasts we calculate analysts’ expected returns for these stocks.
The information provided by FC is widely disseminated to all major institutional
investors as well as a wide array of other investors, including individuals. Indeed, empirical
evidence shows that analysts’ opinions affect prices in a significant way (Womack (1996),
Barber, Lehavy, McNichols, and Trueman (2001), Brav and Lehavy (2002), Boni and Womack
(2003)). Another advantage of this set of expectations is that a typical stock receives a target
price from more than one analyst (on average there is a target price from eight analysts per
stock). As a result, the average (or the median) First Call target price is likely to be less noisy
and thus better reflect the opinion of market participants. One the other hand, a potential concern
with sell-side analysts’ expectations and recommendations is that they are biased, since it is
known they are overly optimistic in their forecasts (e.g., Rajan and Servaes (1997), Michaely and
Womack (1999)) and that it may not accurately represent market expectations. (As we show
below, this bias has a significant impact on the regression intercept but not on the relation
between expected return and stock’s attributes).
Panel A of Table IV provides descriptive statistics on the target price database. The year
1997 is the first one with complete target price data (coverage begins in November 1996 with
3,862 target price reports for that year). Coverage increases over time substantially from about
49,000 price target reports in 1997 to about 92,000 reports in 2001. The average number of price
targets per covered firm (column 3) also increases from 11 in 1997 to 23 in 2001. The target
price database is quite comprehensive and includes reports for 7,073 firms. The number of
17
participating brokerage houses remains fairly constant over the years, with an increase from 125
in 1997 to 140 in 2001 (column 5), with 229 distinct brokerage houses issuing price target
reports across all years. Over the time period we analyze, each firm in the sample is covered, on
average, by eight brokerage houses. We include in our analysis only firms with ordinary shares
(CRSP share codes 10 and 11). Finally, we find that these firms account for approximately 97
percent of the total market value of all securities with ordinary shares on CRSP.
Panels B through D describe the size, book-to-market, and prior price momentum
characteristics of our sample firms relative to the universe of firms available on CRSP. (the
variables’ definitions and the construction of the decile portfolios are described in section I.A.).
The statistics in panel B indicate that FC analysts cover fewer small firms (decile 1) than is
representative from the broad universe. For example, 49 percent of companies fall into the
smallest size decile compared to only 32.5 percent for the sample firms. In all other deciles the
percentage of firms from FC is slightly larger than the broad universe. This evidence indicates
that this sample is tilted toward larger firms.16 In panel C we report similar statistics but for
book-to-market sorts. There is some tendency for analysts to follow high-growth, low book-tomarket firms and avoid value, high book-to-market firms. Finally, in panel D it can be seen that
there is a slight tendency for analysts to avoid “losers,” low momentum stocks with 8.6 percent
of the sample firms falling in the bottom decile.17
To construct the expected return, ERFC, we begin by excluding individual target prices
outstanding for more than 30 days. Next, in any given month over the period 1997 through 2001
we calculate the ratio of each individual analyst target price to the stock price outstanding two
days prior to the announcement of the individual target price (all prices are converted to the same
split-adjusted basis). Finally, in any given month we average the individual analysts’
expectations to obtain the consensus expected return.
To account for the dividend yield component in the expected return calculation we also
add a prospective dividend yield term as follows. For every firm in the sample we first calculate
16
Womack (1996) reports similar evidence for sell-side stock recommendations.
Stickel (2001) and Jegadeesh, Kim, Krische, and Lee (2002) find similar characteristics for analyst stock
recommendations.
17
18
estimates of annual dividends and growth rates of dividends immediately prior to the calculation
of the expected return. Dividends are calculated as the sum of the dividends paid in the fiscal
year before the price target is issued (Compustat data item #21). Dividend growth rate is the ratio
of current to prior year dividend per share (Compustat data item #26), adjusted for stock splits.
The dividend yield is then calculated as the estimated dividend next year relative to the price two
days prior to the issuance date of the price target. The adjustment to the expected return is then
the product of the dividend yield and (one plus) the growth rate, g, of dividends:18
1 + ER FC =
where
TPt
Pt − 2
TPt
Div current (1 + g )
+
Pt − 2
Pt − 2
(3)
is the stock’s consensus expected return without the dividends.19
A. Regression results based on First Call Expected Returns
Before reporting the regression results, we provide some summary statistics regarding the
First Call expected return (unreported in tables). For the period of 1997-2001 the average annual
expected return across all stocks is 45 percent. Average ERFC increases from 33 percent in 1997
to 56 percent in 2000, but decreases to 46 percent in 2001. This trend coincides with the general
market increase and subsequent reversal. Comparing the annual expected return of FC analysts to
that of VL suggests that VL analysts have significantly lower expectations of returns than sellside analysts. As we show later, this difference is not driven by the slightly different array of
covered companies nor by the different time frames of the samples.20
Table V presents regression results for a four-factor model specification using First Call
expected returns.21 The results indicate that the estimate of the annual market premium is
positive and significant (t-statistic=4.3). Its point estimate of 7.0 percent is very similar to what
we find using the Value Line expected return. The estimate of the intercept implies an abnormal
18
The average dividend growth rate for the firms in our sample period is 6 percent.
We winsorize individual target price to stock price ratio at the 1st and 99th percentiles to mitigate the possible
effect of extreme observations.
20
Brav, Lehavy and Michaely (2003) analyze the incentive bias of sell-side analysts relative to independent
analysts.
21
To save space we do not report regression results for the CAPM and the Fama and French three factor model since
estimated factor premiums within these models are similar to those with a four factor model specification.
19
19
annual return of 20.3 percent, much higher than what has been found using Value Line expected
return (an intercept of 6.7 percent). It is likely that the difference captures sell-side analysts’
systematic tendency to issue overly optimistic expected return estimates.
The factor loading on size is priced as well with a slope coefficient equal to 11.1 percent
annually (t-statistic=7.1). The estimated premium related to covariation with the book-to-market
factor, γHML, is negative and insignificant. This estimate equals –6.3 percent (t-statistic=-1.5).
The estimate of the premium associated with the momentum factor, γPMOM, is negative (-1.3
percent) but insignificant with a t-statistic of -1.1. The characteristic regressions yield similar
results: (i) positive market beta coefficient, (ii) negative size coefficient, indicating that small cap
stocks have higher expected return than large cap stocks, (iii) negative and significant book-tomarket coefficient, implying that growth stocks (low BM) have higher expected returns than
value stocks (high BM), and (iv) negative coefficient on prior return.
The similarities between the Value Line expected return results and the First Call
expected return results are striking. Perhaps the most important feature is that beta is priced for
both sets of expectations. Both data bases also indicate that investors expect small stocks to earn
higher return than large stocks. We can also infer that at a minimum, book-to-market is not
priced (VL results), and if anything, it is priced in a way opposite of what has been predicted by
Fama and French: value stocks have lower expected return than growth stocks (FC results).
To ensure that our comparison of the results in Tables III (Value Line) and those in Table
V (First Call) is a valid one and not driven by differences in the time period examined or in the
firms covered, we repeat the experiment using only firms that are covered by both databases, and
for the time period they both covered those stocks. Naturally, this constraint restricts the sample
to the period 1997-2001. The results however, did not change materially: the market beta and
SMB beta coefficients are positive and significant, the HML beta is negative, but significant only
for the First Call sample and the momentum beta is negative.22
22
To ensure the integrity of the First Call results we conducted several additional robustness tests: (i) we use only
the most recent issued price target in any given month to ensure that staleness of price target does not affect our
results; (ii) we use the median, instead of consensus price target to calculate expected return, (iii) we use only new
price targets and not reiteration of old price targets. None of our results changed.
20
IV. Robustness
In this section we perform additional robustness tests. First, we examine the impact of
dividends on our results. Recall that we calculate expected return by taking the forecasted price,
adding the dividends, and dividing by the current market price. Underlying this procedure is the
assumption that the target prices published by analysts do not include the value of the interim
dividends and that our computations capture the market’s expected dividends. We examine the
effect of this assumption on our results. Second, we examine the sensitivity of our results to
different empirical methodology. Specifically, we re-examine our results when the regressions
are conducted on portfolios and not on individual securities. Third, rather than using VL
expected betas, we use realized returns to estimate market betas, and see how sensitive the
results are to this change. Fourth, it is possible that the firm characteristics that we employ proxy
for perceived mispricing by VL analysts. While such a hypothesis is inconsistent with the models
that we estimate, we nevertheless attempt to document whether this interpretation is a viable
alternative. Fifth, it is possible that a more accurate proxy for the market’s expected return is to
use prices subsequent to the announcement of the target price (and not beforehand, as we have
done so far). Using the post announcement price to calculate expected return proxies for
market’s expectations after the new information was announced, and hence better reflects
equilibrium expected return. Finally, to examine whether the results can be attributed to only one
particular type of firm, we sort our sample firms into terciles based on size, book-to-market, and
momentum and examine the robustness of our earlier regression results within these sorts.23
A. The Use of Dividends in the Formation of the Proxy for Expected Returns
One possible concern for our derivation of estimates of expected returns is the addition of
23
We have also examined the robustness of our inferences to potential staleness in the formation of target prices
and the inclusion of low priced stocks. To address staleness, that is, the possibility that VL analysts use the
previously issued three-month-old target price in lieu of a fresh target price, we delete from the VL sample all
observations in which the analyst’s target price was identical to that issued in the previous report. We have then
repeated the regressions in Table III and found that the use of this limited sample does not alter any of our
conclusions. We have also checked whether the inclusion of low priced stocks (price lower than 5 dollars) alters our
inferences by deleting altogether all such cases. Again, our conclusions remain unchanged.
21
the dividend yield component to the VL estimates (see equations (1) and (2)). Recall that our
goal is to test the predictions of various asset-pricing models and as such it is valid to examine
how one portion of expected returns – expected price appreciation – varies with measures such
as beta and market capitalization. It may be argued that the addition of an expectation for the
prospective dividend yield may introduce noise, and perhaps even bias our estimation.
In addition, it is not completely clear whether the target price provided by analysts
accounts for the dividends to be paid during the investment horizon. For example, if analysts
value the stocks using the Free Cash Flow method, the interim dividend payments will not affect
the total value, and the target price can be viewed as the future value of all cash flows. In that
case adding the dividends, as we did, results in some double counting, certainly adds noise, and
even biases the estimate as long as dividends are correlated with firms’ characteristics, which
they are. In this section we therefore repeat some of the regressions that we undertook earlier in
Section II and ask whether our conclusions are sensitive to the inclusion of dividends.
Table VI presents regression results for the factor and characteristic models. Comparing
the factor model results with those presented in Table III indicates that the omission of dividends
from the expected return estimates reduces the intercept from 6.7 percent with dividends (Table
III) to 0.4 percent without dividends, which is insignificantly different from zero. At the same
time the R2 increases from 13.8 percent to 20.9 percent, a 50 percent increase. Likewise, the
loadings on all factors other than momentum increase (in absolute value). The market premium
increases from 5.7 percent on an annual basis to 8.9 percent. A similar increase is observed for
the size premium. Finally, the second column indicates that there are no significant changes in
the characteristics regression coefficients with (Table III) and without (Table VI) dividends,
though the explanatory power of the regression increases as well.
In summary, the decision whether to include estimated future dividends on the left-hand
side (when expected return is calculated) does not change the nature of most of our conclusions.
Perhaps the most important change is the insignificant intercept and the higher R2 when
dividends are excluded. The sign of the market beta coefficient as well as the size coefficient do
not change and remain positive and significant.
22
B. Regression Results Based on Portfolio Expected Returns
The results we have obtained so far are based on ex-ante expectations of individual firm
returns. We now ask whether our results are robust to an alternative set of tests, namely
characteristics-sorted portfolios of the individual firms, which are less likely to suffer from noise
in the formation of firm expected returns.
The empirical procedure is similar to what has been used in many of the studies that
utilize realized returns. Specifically, beginning in June 1970, we form quartile breakpoints based
on firms’ market beta, and tercile breakpoints based on market capitalization, and book-tomarket ratio. The breakpoints are based on the universe of firms with available data on the CRSP
and Compustat tapes.24 We then allocate the sample firms into one of the resulting 4x3x3=36
portfolios and calculate the portfolios’ equal-weighted realized return for the next 12 months as
well as the average annual expected return, market capitalization, book-to-market, and one-year
momentum characteristics. We then re-form these portfolios annually. Next, beginning in
January 1975, we estimate for each of the 36 portfolios factor loadings given Carhart’s fourfactor model specification using the five-year data preceding this month. We repeat this
estimation for all months until December 2001. Finally, for each month in the sample period, we
employ the portfolios’ annual expected returns and their estimated factor loadings/characteristics
as in Section II and estimate factor premia using the Fama-MacBeth regression setup.
Table VII provides the regression results. The estimated factor premia are all consistent
with those reported in Table III. Market beta is associated with a positive annual premium in
both regression specifications. Consistent with our previous results, we find that market
capitalization is inversely related to expected returns, and that expected return is not affected by
book-to-market. Finally, prior-year return is negatively related to our portfolio expected return.
(Interestingly, the momentum coefficient is not significant in the factor model regression.) Using
the portfolio approach the average R2 is 35 percent, more than double the R2 when we employ
individual security returns as the independent variable.
24
We use Fama and French’s breakpoints for market capitalization and book-to-market obtained from Ken French’s
web site. The market beta breakpoints are obtained by estimating individual firms’ betas using the five years of
monthly returns leading to the portfolio formation month.
23
C. Using Historical and Full Sample Market Betas
Since Value Line reports its estimated market beta, we have used it as our market beta
rather than estimating it using historical data (based on past realized returns)—as we have done
for all the other factor loadings. In this subsection we use two alternative procedures to estimate
betas. In the first procedure we estimate a firm’s market beta in a given month t using the prior
60 months of realized return data. All other factor loadings (and characteristics) are estimated as
before. In the second procedure we estimate, for each firm, full-sample factor loadings using all
available returns data. We then use these sets of estimates in the Fama MacBeth regressions.
Thus, this procedure affects not only market beta, but size, book-to market and momentum
loadings as well.
The results based on historical data are reported in panel A of Table VIII. Several points
are worth mentioning. First, the use of historical beta rather than VL beta reduces the estimated
market risk premium by almost 50 percent, (from 6.7 percent to 2.9 percent). Second, the use of
historical beta increases the intercept significantly. Third, all other slope coefficients are not
impacted significantly by the alternate estimation procedure for the market beta. Next, panel B
provides results based on full-sample factor loadings. Here too we find that estimates of the
market premium are lower than those reported in Table III (3.7 percent in the four-factor
specification) while the premium associated with firm size is slightly larger than that reported in
Table III. We conclude that our results are robust to the measurement of market beta although it
is evident that use of historical realized return data to estimate market betas seems to introduce
measurement error in our second-stage regressions thus possibly leading to an error in the
variables problem and a lower estimate of the market risk premium.25
25
We have also explored an alternative approach to the estimation of factor loadings that does not rely on the use of
realized return data. Specifically, following Fama and French (1993) we have constructed market, size and book-tomarket “factors” using firm expected returns. We then used the 36 portfolios discussed in section IV.B and
estimated the portfolios’ factor loadings by regressing each portfolio’s time-series of expected returns on the factor
expected return series. We find the following results: First, the three factors explain, on average, 60 percent of the
time-series variation of the portfolios’ expected return over the full sample period. Second, the estimated factor
loadings (which are now based exclusively on expected return data) yield similar results to those reported in panel A
of Table III. Third, the portfolio factor loadings estimated using ex-ante data are highly positively correlated with
portfolio factor loadings estimated using realized return data.
24
D. The Impact of a Potential Value Line Mispricing Bias
Based on the number of subscriptions, VL is the largest advisory service in the world.
Presumably, investors are willing to pay for its services because it is able to identify mispricing
in the market. In this subsection we examine an alternative hypothesis, namely, that the relation
between expected return and the models that we have documented earlier are actually due to
these variable’s implicit correlation with measures of under and over valuation. For example,
perhaps small firms are perceived to be more undervalued than large firms thus leading to our
estimate of a negative relation between expected return and market capitalization. While such a
robustness test is inconsistent with the null equilibrium models that we have entertained, we
believe that it can shed light on the viability of mispricing as a competing hypothesis.
To address this issue we conduct two separate tests. In the first we use another piece of
information provided by VL, namely, the timeliness rank. VL ranks the stocks it follows into five
categories, where one is the best and five is the worst, based on Value Line prediction of future
stock price performance. Unlike sell-side analysts recommendations VL recommendations are
symmetrically distributed around “timeliness equals three”, the equivalent of a hold
recommendation.26 To examine the extent to which our results are affected by potential
mispricing we rerun the factor and characteristic regressions only on stocks with a timeliness
rank of three. Those stocks (nearly 50 percent of the sample) are supposed to be fairly priced. In
the second test we allow for the possibility that upon the announcement of the analyst’s report
investors immediately react to the hypothesized mispricing component in the forecast.
Subsequent to such a reaction any remaining appreciation between the stock price and the VL
target price should reflect the expected return component. We therefore recompute all our
estimates of expected return as in equations (1) and (2) using in the denominator market prices as
of five days after the release of the report. We then repeat the analysis of Table III.
These results based on the timeliness rank (post-event price) are reported in panel A
(panel B) of Table IX. In general the results are rather similar to what has been reported in Table
26
VL conducts a cross-sectional comparison of every stock in its large 1,700-based company universe. 100 firms are
allocated to group one, 300 into group two, 800 into group three, 300 into group four, and 100 into group five.
25
III when the entire sample is used: (i) The beta slope coefficient is positive and significant for
both the factor and characteristic models; (ii) The size coefficient indicates that small
capitalization stocks are expected to earn higher returns than large cap stocks; (iii) both the factor
and characteristic models suggest that growth stocks (low book-to-market) have an expected
return that is not significantly different than value stocks, and (iv) both models indicate that past
losers are expected to have higher returns than past winners. Overall, these results clearly suggest
that the relation we find between expected return and stocks’ attributes are not because of
perceived mispricing.
E. Size, Book-to-market, and Price Momentum Sorts
In this subsection we ask whether the results reported in the previous section can be
attributed to only one particular type of firm. In Table X we sort our sample firms into terciles
based on size, book-to-market, and momentum and examine the robustness of our earlier
regression results within these sorts. Specifically, using tercile breakpoints for each of these
three variables we repeat, each month, the Fama-MacBeth estimation. The regression results,
based on the firm factor loadings, are presented in panel A of Table X.27
Beginning with the size sorts, the table reveals that, within each sort expected return is
positively related to beta, high book-to-market stocks are not expected to have lower expected
returns than low book-to-market stocks, and recent “losers” are expected to earn higher returns
than recent “winners”. The sorts by book-to-market and momentum reveal, again, that the
regression results reported in Table III are not driven by a particular characteristic of the sample
firms since the estimates of the premiums are stable across both kinds of sorts. The market
premium is consistently positive and significant, the size premiums are generally positive and
significant, the book-to-market coefficients are insignificant, and the price momentum
coefficient is negative and significant for any of the sorts. The size (momentum) sort also
27
We obtained the breakpoints from Ken French’s website: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/.
To save space we do not present regression results based on firm characteristics that yield similar conclusions.
26
indicates that the intercept for small (“loser”) stocks is about double that of large (“winner”)
stocks.
V. Discussion and Conclusions
Asset pricing theories generate predictions linking assets’ expected returns and their
attributes. This paper complements and extends existing research by using direct measures of
expected return and provides evidence on the relation between expected returns and the pricing
of assets in financial markets. We are able to provide new evidence on how expected return is
related to market beta, firm capitalization, book-to-market, and price momentum.
The evidence in this paper is relevant to these important issues along several dimensions.
Perhaps the most significant contribution of the paper is the evidence that when using expected
returns rather than realized returns, beta and expected returns are positively related. While we
find evidence inconsistent with the CAPM (for example, that firm size is related to expected
returns), the fact that beta is associated with an economically significant premium is important as
the model is taught in virtually every basic finance class around the world and survey results
show that over 80 percent of managers use the CAPM beta to calculate the cost of equity capital
(Graham and Harvey (2001)). This result is robust to the time period examined. It is also robust
to whether we use expected return produced by independent analysts or sell-side analysts.
We also find that in addition to market beta, the firm’s market value of equity is
negatively related to its expected return. This finding is consistent with many other studies that
utilize average realized return. Combining these two results suggest that the size factor is risk:
Investors expect small capitalization stocks to generate higher returns and indeed realized returns
show that small capitalization stocks on average outperform large capitalization stocks,
consistent with a risk-return tradeoff.
Several researchers argue that value stocks (high book-to-market stocks) are riskier than
growth stocks and hence their ex-post return is higher, on average. A clear implication of this
argument is that investors should expect such stocks to earn higher returns. Thus, our third
27
important finding is that there is little evidence in favor of the hypothesis that book-to-market is
a risk factor. There is no evidence that investors expect high book-to-market stocks to generate
higher returns than low book-to-market stocks, as this argument implies. This contrasts with the
evidence discovered using realized returns, where the book-to-market factor is found to be
positively related to realized return. Taken together, the combined evidence is potentially
consistent with behavioral and rational learning explanations. Due to incorrect beliefs regarding
persistence of past performance, investors might irrationally expect glamour stocks to
outperform value stocks (De Bondt and Thaler (1985) and Lakonishok, Shleifer, and Vishny
(1994)). It also may be possible that investors attempt to rationally learn the rate with which firm
fundamentals change and have, ex-post, expected more changes than actually occurred.
Finally, as we pointed out in the introduction, market expectations are unobservable; yet,
it is reassuring that all of our major findings can be corroborated by using an additional set of
expectations. There are several reasons to believe the expectations we employ here represent at
least a significant portion of market’s expectations. Both sources of expectations are followed by
a wide array of investors, and it is common practice to use analysts’ earnings forecasts for
companies’ valuation, which directly affect prices. As long as the set of expectations we use are
correlated with the market’s expectations, our results are informative about the relation between
the market’s expected returns and asset pricing. Given the literature’s ubiquitous reliance on
realized returns in tests of assets pricing models over the past three decades, our results should
provide a valuable alternative source of evidence on the viability of our assets pricing models.
28
Appendix A: Computation of Adjusted T-Statistics
In this Appendix we describe the manner with which we derive the standard errors for the
estimates reported in Tables IV and V. Consider, for simplicity, the Fama-MacBeth regression in
which the null asset-pricing model is the CAPM. The regression for each period takes the form:
(A1)
ER − R f = γ 0 + γ 1 β + v
We focus, without loss of generality, on the vector of time series of estimated slopes γˆ1,t ,
and drop the subscript for convenience. Note that t=1,…,60 in the case of First Call based
estimates of expected return and t=1,…,180 in the case of Value Line. Our goal is to derive the
standard error for the time-series average
1
T
T
∑γ
t
.
t =1
We begin with the estimation of a pth-order autoregression for the demeaned vector γˆ .
γˆ t = ρ 1γˆ t −1 + ρ 2 γˆ t − 2 + K ρ p γˆ t − p + ε t .
(A2)
For the First-Call based regressions we adopt an AR(3) specification, while for the Value-Line
based regressions we specify an AR(6). We have verified, for both databases, that higher orders
of the autoregression process do not lead to any changes in the reported inferences.
To ease the derivation, we convert the p-order autoregression in (A2) into a first-order
vector autoregression. Denote the (1xp) vector of estimated autoregressive parameters by ρ̂ and
let
y t = Ay t −1 + η t ,
(A3)
 ρˆ

where y t = [γˆ t γˆ t −1 K γˆ t − p +1 ]' is a (px1) vector of estimated slopes, A = 
 is a (p x p) matrix
 I p −1 0
and Ip-1 is a (p-1 x p-1) identity matrix. The disturbance term η t = [ε t 0] is a (p x 1) vector with the
scalar ε t in the (1,1) position, and 0 is a (p-1 x 1) vector of zeros. We assume that
η t ~ IID N (0, Σ ) , where
Σ is a (p x p) matrix with the scalar σ ε2 , namely the residual variance
from (A2), in the (1,1) position and zeros otherwise.
The covariance matrix of the sample average
1
VAR
T

T
∑
t =1
1
T
T
∑y
t
may be expressed as:
t =1
T −1
 1


T −i   i
' i
y t  = ⋅ I p2 − A ⊗ A −1 vec(Σ ) ⋅  I p +

 ⋅  A + A  
 T

 
T  
i =1 


[
]
∑
( )
(A4)
(See Hamilton (1994), p. 265 for a formal derivation of the second moment for a vector
autoregression process). The square root of the (1,1) element of the covariance matrix is the
standard error we are interested in.
Figure I: Value-Weighted Annual Expected Return
This figure provides the time-series of the value weighted market expected return using individual firm expected return data obtained from Value Line. We calculate for
each end of quarter beginning in 1975 through 2001 (March, June, September, and December) the annual value weighted return using all expected returns issued
within a given quarter. The resulting time-series of quarterly market expectations are given below.
40%
35%
25%
20%
15%
10%
5%
0%
1975Q1
1975Q3
1976Q1
1976Q3
1977Q1
1977Q3
1978Q1
1978Q3
1979Q1
1979Q3
1980Q1
1980Q3
1981Q1
1981Q3
1982Q1
1982Q3
1983Q1
1983Q3
1984Q1
1984Q3
1985Q1
1985Q3
1986Q1
1986Q3
1987Q1
1987Q3
1988Q1
1988Q3
1989Q1
1989Q3
1990Q1
1990Q3
1991Q1
1991Q3
1992Q1
1992Q3
1993Q1
1993Q3
1994Q1
1994Q3
1995Q1
1995Q3
1996Q1
1996Q3
1997Q1
1997Q3
1998Q1
1998Q3
1999Q1
1999Q3
2000Q1
2000Q3
2001Q1
2001Q3
Annual Expected Return
30%
Month
Figure II: Relation between expected return and firm exposures to marketand size-related factors
The figure provides cross-sectional regression results based on 100 portfolios sorted by market
beta and factor loadings on a size-related factor mimicking portfolio. Using Value Line
expectations we average individual firm annual expected excess return over the sample period
1975-2001 as well as their market betas and factor loadings on the Fama and French SMB
portfolio (denoted "SMB"). We then form decile cutoffs based on the latter factor loadings and
allocated all firms into one of the resulting 100 portfolios. A cross-sectional regression is then
estimated where the dependent variable is the portfolios' annual expected excess return and the
independent variables are the portfolios' average loadings. The regression yields the following
estimates: E(r)-Rf=0.06+0.04β+0.02*SMB. We plot the portfolio expected excess returns as
well as the regression plane associated with the least squares estimation.
Table I
Description of Value Line Target Price Database, 1975-2001
This table reports statistics on the Value Line target price database. Panel A presents, by year, the
number of target price observations, number of firms, and the percent market capitalization of the
sample firms relative to the universe of firms listed on the NYSE, AMEX, and NASDAQ. The Value
Line sample and the benchmark universe of firms are restricted to companies with common shares
(CRSP share codes 10 or 11). The last row in panel A presents statistics for the 27 -year sample
period. Panels B through D describe the size, book-to-market, and prior price momentum
characteristics of our sample firms relative to the universe of firms available on CRSP. Size in month
t is computed as the market capitalization as of the end of the prior month (expressed in millions of
dollars). Book-to-market is computed as the ratio of annual common shareholders equity (Compustat
item #60) to market capitalization as of the end of the fiscal year. This ratio is applied to the 12-month
period beginning six months subsequent to the end of the fiscal year. Prior price momentum for
month t is the buy and hold return for the 11-month period ending one month prior to month t. We
present statistics for each characteristic within decile portfolios. Size deciles are based on NYSE
capitalization cutoffs while book-to-market and momentum cutoffs are based on the universe of
available firms. For each decile, we report the mean of the monthly averages of the respective
characteristic.
Panel A: Value Line Target Price Database
Year
Number of
Target Prices
Number of
Firms
% of Market
Capitalization
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
5,579
5,059
5,862
5,871
5,968
5,805
5,985
6,017
5,062
5,616
4,993
5,195
5,291
5,395
5,245
5,196
5,202
5,184
5,292
5,184
5,174
5,067
5,100
5,100
5,328
5,645
5,771
1,526
1,554
1,604
1,600
1,612
1,634
1,615
1,609
1,543
1,555
1,510
1,452
1,487
1,478
1,421
1,395
1,371
1,370
1,368
1,392
1,387
1,365
1,424
1,436
1,451
1,569
1,575
95%
95%
95%
95%
95%
95%
94%
94%
93%
93%
93%
93%
93%
93%
93%
94%
94%
93%
92%
91%
90%
90%
90%
91%
92%
91%
92%
1975-2001
144,761
3,779
92%
Table I (Continued )
Panel B: Size characteristics
Small
2
3
4
5
Size
6
7
8
9
Large
Universe:
Average Market Cap
Percentage of firms
23
54.4%
104
14.6%
212
8.0%
360
5.6%
564
4.2%
867
3.4%
1,359
3.0%
2,264
2.6%
4,199
2.2%
18,317
2.1%
688
100.0%
Value Line Population:
Average Market Cap
Percentage of firms
36
14.9%
110
16.1%
216
13.2%
362
11.2%
567
9.5%
870
8.5%
1,362
7.8%
2,267
7.1%
4,203
6.1%
18,320
5.6%
2,025
100.0%
Overall
Panel C: Book-to-market characteristics
Low
2
3
4
5
Book-to-market
6
7
8
9
High
Overall
Universe:
Average book-to-market
0.18
0.34
0.48
0.61
0.74
0.87
1.03
1.22
1.53
2.44
0.94
Value Line Population:
Average book-to-market
Percentage of firms
0.19
8.9%
0.34
11.4%
0.48
11.9%
0.61
11.9%
0.74
12.0%
0.87
11.5%
1.02
10.5%
1.22
9.1%
1.53
7.4%
2.34
5.3%
0.83
100.0%
7
8
9
Winner
Panel D: Momentum characteristics
Loser
2
3
4
5
Momentum
6
Universe:
Average prior 11 month
-49.9%
-27.7%
-14.7%
-4.6%
4.2%
13.0%
22.5%
34.7%
53.8%
109.4%
14.1%
Value Line Population:
Average prior 11 month
Percentage of firms
-48.0%
4.9%
-27.3%
7.6%
-14.5%
9.4%
-4.6%
10.6%
4.3%
11.5%
13.0%
12.0%
22.5%
12.1%
34.7%
11.8%
53.5%
11.1%
104.8%
9.1%
18.1%
100.0%
Overall
Table II
Descriptive Statistics of Value Line's Annual Expected Return
This table provides summary statistics on the distribution of Value Line’s expected annual return
by year. Expected return is calculated as the ratio of the Value Line target price and the firm’s
market price which was outstanding nine days prior to the Value Line report date. We use market
price nine days prior the Value Line report date since analysts actually send their report for
publication on a given Friday on the Wednesday of the preceding week. For the period 1975-1986
we adjust for expected dividend payments by calculating the growth rate of dividends as well as
the dividend yield immediately preceding the report date. Dividends are assumed to be reinvested
at the firm cost of capital. For the period 1987-2001 we employ Value Line analysts’ forecasts of
dividend growth rates as well as forecasts of the next year dividends, again reinvesting interim
dividends at the firm cost of capital. Target price to stock price ratio at the 1st and the 99th
percentiles are winsorized to mitigate the potential effects of extreme observations. Equations (1)
and (2) in the text provide the calculation of the expected return.
Year
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
Mean
0.340
0.300
0.286
0.283
0.311
0.313
0.289
0.313
0.198
0.233
0.198
0.153
0.147
0.188
0.168
0.211
0.191
0.175
0.148
0.156
0.149
0.136
0.121
0.128
0.155
0.187
0.174
Q1
0.329
0.296
0.281
0.263
0.303
0.298
0.265
0.282
0.189
0.229
0.191
0.144
0.131
0.183
0.164
0.188
0.182
0.170
0.143
0.149
0.139
0.132
0.109
0.110
0.146
0.186
0.171
Median
0.347
0.299
0.285
0.281
0.308
0.312
0.291
0.325
0.204
0.235
0.201
0.157
0.132
0.188
0.167
0.201
0.186
0.175
0.148
0.161
0.149
0.135
0.120
0.127
0.152
0.186
0.177
Q3
0.352
0.303
0.290
0.303
0.318
0.327
0.314
0.345
0.207
0.237
0.205
0.162
0.164
0.192
0.172
0.233
0.200
0.179
0.154
0.164
0.160
0.140
0.132
0.145
0.165
0.188
0.177
Std Deviation
0.019
0.006
0.006
0.025
0.010
0.017
0.033
0.048
0.015
0.006
0.010
0.013
0.032
0.007
0.006
0.030
0.015
0.006
0.007
0.013
0.013
0.006
0.016
0.021
0.013
0.002
0.005
N
5,579
5,058
5,862
5,871
5,968
5,804
5,984
6,015
5,060
5,615
4,993
5,194
5,292
5,395
5,245
5,196
5,202
5,184
5,292
5,184
5,174
5,067
5,101
5,100
5,328
5,645
5,771
1975-2001
0.209
0.153
0.188
0.286
0.068
146,179
Table III
Fama-MacBeth Regressions Based on Value Line Estimates of One-Year Expected
Excess Return, 1975-2001
This table presents time-series averages of 324 slopes from month -by-month regressions of one-year
expected excess returns on a set of estimated factor loadings (panel A) or firm characteristics (panel B).
One-year expected excess return is equal to the di fference between the Value Line expected return
VL
estimate ER , which was described in Section I.B, and the one-year risk free rate obtained from the
Fama-Bliss files on CRSP. Factor loadings for month t are estimated using data from month t -61
through t-1 (requiring a minimum of 24 valid monthly returns). Since Value Line provides in each
report a firm-specific market beta, we replace the estimated market beta with the one reported by Value
Line. Size in month t is computed as the log of market capitalizati on measured as of the end of the
previous June (expressed in millions of dollars). Book -to-market is computed as the ratio of annual
common shareholders equity (Compustat item #60) to market capitalization as of the end of the fiscal
year. The log of this ratio is applied to the 12-month period beginning six months subsequent to the end
of the fiscal year. Prior price momentum for month t is the buy and hold return for the 11-month period
ending one month prior to month t. We winsorize monthly observations at the 1st and 99th percentiles to
mitigate the possible effect of extreme observations. T-statistics adjusted for the overlapping nature of
the data are the ratio of the time-series average divided by the estimated time-series standard error (see
Appendix A for the details).
Panel A: Regressions of one-year expected excess return on factor loadings
Model I
Model II
Intercept
0.068
0.068
β
Model III
0.067
3.2
3.7
4.2
0.067
0.058
0.057
5.1
4.7
4.4
0.016
0.016
2.1
2.1
0.002
0.001
0.4
0.2
γSMB
γHML
γPMOM
-0.021
-3.0
Average Adjusted R2
Average Monthly N
5.9%
443
12.2%
435
Panel B: Regressions of one-year expected excess return on firm characteristics
Model I
Model II
Intercept
0.068
0.119
β
13.8%
428
Model III
0.125
3.2
5.3
5.3
0.067
0.078
0.076
5.1
7.3
6.1
-0.010
-0.009
-2.6
-2.5
0.012
0.010
1.8
1.8
log(Size)
log(Book-to-Market)
Prior Return
-0.096
-11.5
Average Adjusted R2
Average Monthly N
5.9%
443
13.4%
406
26.0%
401
Table IV
Description of First Call Target Price Database, 1997-2001
This table reports statistics on the First Call target price database. To ensure accurate dating of analysts’ reports, we include only observations
coded as “Real-time” (i.e., reports received from live feeds such as the broker note and are dated as the date the report was published). Panel
A presents, by year, the number of observations, the average number of reports per firm, the number of firms, the number of brokerage houses
issuing reports, the mean number of brokerage houses per firm, and the percent market capitalization of the sample firms relative to the
universe of firms listed on the NYSE, AMEX, and NASDAQ. The last row in panel A presents statistics for the five -year sample period.
Panels B-D describe the size, book-to-market, and prior price momentum characteristics of our sample firms relative to the universe of firms
available on CRSP. Size in month t is computed as the market capitalization as of the end of the prior month (expressed in millions of
dollars). Book-to-market is computed as the ratio of annual common shareholders equity (Compustat item #60) to market capitalization as of
the end of the fiscal year. This ratio is applied to the 12-month period beginning six months subsequent to the end of the fiscal year. Prior
price momentum for month t is the buy and hold return for the 11-month period ending one month prior to month t. We present statistics for
each characteristic within decile portfolios. Size deciles are based on NYSE capitalization cutoffs while book -to-market and momentum
cutoffs are based on the universe of available firms excluding those with non-common shares (CRSP share codes different from 10 or 11). For
each decile, we report the mean of the monthly averages of the respective characteristic.
Panel A: First Call Individual Target Price Database
Year
Price Targets
Avg. No.
N
Per Firm
Brokers
Avg. No.
N
Per Firm
Number
of Firms
% of Market Cap
(1)
(2)
(3)
(4)
(5)
(6)
(7)
1997
1998
1999
2000
2001
1997-2001
49,369
78,894
93,832
93,144
91,781
407,020
11
16
19
20
23
18
4,621
4,824
4,953
4,691
4,038
7,073
125
137
151
152
140
229
4
5
5
6
6
8
95%
96%
98%
99%
98%
97%
Panel B: Size characteristics
Universe:
Average Market Cap
Percentage of firms
First Call Population:
Average Market Cap
Percentage of firms
Small
2
3
4
5
Size
6
7
8
9
Large
Overall
40
49.0%
178
14.9%
360
9.1%
609
6.6%
953
4.8%
1,473
4.0%
2,377
3.5%
4,262
3.2%
8,963
2.6%
50,570
2.2%
1,841
100.0%
54
32.5%
179
18.4%
360
11.8%
609
8.9%
953
6.7%
1,473
5.5%
2,378
4.9%
4,262
4.5%
8,958
3.7%
50,520
3.2%
2,548
100.0%
8
9
High
Overall
0.70
0.86
1.10
1.81
0.65
0.70
9.6%
0.85
8.8%
1.09
7.6%
1.77
6.2%
0.57
100.0%
8
9
Winner
Overall
6.4%
17.8% 32.4% 55.9% 130.9%
9.0%
-64.4% -42.1% -26.9% -14.7% -3.9% 6.4%
8.6%
9.3%
9.6%
9.8% 9.9% 10.1%
17.8% 32.4% 56.0% 131.1%
10.4% 10.6% 10.8% 10.8%
12.3%
100.0%
Panel C: Book-to-market characteristics
Universe:
Average book-to-market
First Call Population:
Average book-to-market
Percentage of firms
Book-to-market
6
7
Low
2
3
4
5
0.10
0.21
0.30
0.39
0.48
0.11
11.2%
0.21
11.8%
0.30
11.7%
0.58
0.39
0.48
0.58
11.3% 11.2% 10.6%
Panel D: Momentum characteristics
Loser
Universe:
Average prior 11 month
First Call Population:
Average prior 11 month
Percentage of firms
2
3
4
5
-65.2% -42.1% -27.0% -14.7% -3.9%
Momentum
6
7
Table V
Fama-MacBeth Regressions Based on First Call Estimates of One-Year
Expected Return, 1997-2001
This table presents time-series averages of 60 slopes from month-by-month regressions
of one-year expected excess returns on a set of estimated factor loadings and/or firm
characteristics. One-year expected excess return is equal to the difference between the
FC
First Call expected return estimate ER , which was described in Section III, and the
one-year risk free rate obtained from the Fama-Bliss files on CRSP. Factor loadings for
month t are estimated using data from month t-61 through t-1 (requiring a minimum of
24 valid monthly returns). Size in month t is computed as the log of market
capitalization measured as of the end of the previous June (expressed in million s of
dollars). Book-to-market is computed as the ratio of annual common shareholders
equity (Compustat item #60) to market capitalization as of the end of the fiscal year.
The log of this ratio is applied to the 12-month period beginning six months subsequent
to the end of the fiscal year. Prior price momentum for month t is the buy and hold
return for the 11-month period ending one month prior to month t. We winsorize
monthly observations at the 1st and 99th percentiles to mitigate the possible effect of
extreme observations. T-statistics adjusted for the overlapping nature of the data are the
ratio of the time-series average divided by the estimated time-series standard error (see
Appendix A for the details).
Factor Loadings
0.203
Intercept
Characteristics
Intercept
0.592
5.8
β
γSMB
11.2
β
0.109
4.3
3.7
0.111
log(Size)
-0.058
0.070
7.1
γHML
-0.063
-12.8
log(Book-to-Market)
-1.5
γPMOM
-0.013
-4.5
-0.091
Prior Return
-1.1
2
Average Adjusted R
Average Monthly N
13.8%
1,779
-0.035
-5.1
2
Average Adjusted R
Average Monthly N
16.5%
1,626
Table VI
Fama-MacBeth Regressions Based on Value Line Estimates of One-Year Expected
Excess Return Excluding Expected Dividends , 1975-2001
This table presents time-series averages of 324 slopes from month-by-month regressions of
one-year expected price appreciation on a set of estimated factor loadings and firm
characteristics. One-year expected price appreciation is equal to the difference between the
Value Line price forecast to stock price ratio and the one-year risk free rate obtained from
the Fama-Bliss files on CRSP. Factor loadings for month t are estimated using data from
month t-61 through t-1 (requiring a minimum of 24 valid monthly returns). Since Value
Line provides in each report a firm-specific market beta, we replace the estimated market
beta with the one reported by Value Line. Size in month t is computed as the log of market
capitalization measured as of the end of the previous June (expressed in millions of dollars).
Book-to-market is computed as the ratio of annual common shareholders equity (Compustat
item #60) to market capitalization as of the end of the fiscal year. The log of this ratio is
applied to the 12-month period beginning six months subsequent to the end of the fiscal
year. Prior price momentum for month t is the buy and hold return for the 11-month period
ending one month prior to month t. We winsorize monthly observations at the 1st and 99th
percentiles to mitigate the possible effect of extreme observations. T-statistics adjusted for
the overlapping nature of the data are the ratio of the time -series average divided by the
estimated time-series standard error (see Appendix A for the details).
Factor Loadings
Intercept
0.004
Characteristics
Intercept
0.3
β
0.089
3.1
β
4.7
γSMB
0.024
-0.003
log(Size)
-0.019
log(Book-to-Market)
20.9%
428
0.001
0.1
Prior Return
-3.0
Average Adjusted R2
Average Monthly N
-0.013
-4.4
-0.5
γPMOM
0.115
7.0
3.4
γHML
0.084
-0.094
-11.9
Average Adjusted R2
Average Monthly N
31.3%
400
Table VII
Fama-MacBeth Portfolio Regressions Based on Value Line Expected Return
Estimates, 1975-2001
This table presents time-series averages of 324 slopes from monthly regressions of portfolio one-year
expected excess return on a set of estimated portfolio factor loadings and portfolio characteristics.
Portfolios are formed as follows. Beginning in June 1974 we form quartile breakpoints based on
firms’ market beta, and tercile breakpoints based on market capitalization and book-to-market ratio.
The breakpoints are based on the universe of firms with available data on the CRSP and Compustat
tapes. Sample firms are allocated into one of the resulting 4x3x3=36 portfolios. We calculate the
portfolios’ equal weighted realized return for the next 12 months, the average monthly expected
return, market capitalization, book-to-market, and one-year momentum characteristics. The portfolios
are then re-formed annually. Beginning in January 1975 we estimate for each of the 36 portfolios
factor loadings given Carhart’s four-factor model using the five-year data preceding this month. We
repeat this estimation monthly until December 2001. We employ the portfolios’ monthly expected
returns and their estimated factor loadings/characteristics as in Table III and estimate the factor
premia. Since Value Line provides in each report a firm -specific market beta, we calculate a
portfolio’s market beta for month t as the equal weighted average of the firms’ Value Line betas for
that month. All other factor loadings are derived from a regression over the preceding 60 months.
Prior to the portfolio formation we winsorize individual firm monthly observations at the 1st and 99th
percentiles to mitigate the possible effect of extreme observations . T-statistics are adjusted for the
overlapping nature of the data (see Appendix A for the details).
Factor Loadings
Intercept
0.056
Characteristics
Intercept
2.6
β
0.060
7.2
β
4.8
γSMB
0.033
0.004
log(Size)
-0.015
log(Book-to-Market)
34.7%
35
0.005
1.2
Prior Return
-1.8
Average Adjusted R2
Average Monthly N
-0.007
-2.2
0.4
γPMOM
0.044
5.6
3.2
γHML
0.137
-0.059
-4.1
Average Adjusted R2
Average Monthly N
35.0%
31
Table VIII
Fama-MacBeth Regressions Based on Value Line Estimates of One-Year Expected Excess
Return, Using Historical Betas (Panel A) and Full Sample Betas (Panel B), 1975-2001
This table presents time-series averages of 324 slopes from month-by-month regressions of one-year
expected excess returns on a set of estimated factor loadings using historical betas (panel A) and factor
loadings using full sample betas (panel B). One-year expected excess return is equal to the difference
between the Value Line expected return estimate ERVL, which was described in Section I.B, and the oneyear risk free rate obtained from the Fama-Bliss files on CRSP. Factor loadings for month t are estimated
using data from month t-61 through t-1 (requiring a minimum of 24 valid monthly returns). Size in month t
is computed as the log of market capitalization measured as of the end of the previous June (expressed in
millions of dollars). Book-to-market is computed as the ratio of annual common shareholders equity
(Compustat item #60) to market capitalization as of the end of the fiscal year. The log of this ratio is applied
to the 12-month period beginning six months subsequent to the end of the fiscal year . Prior price
momentum for month t is the buy and hold return for the 11-month period ending one month prior to month
t. We winsorize monthly observations at the 1st and 99th percentiles to mitigate the possible effect of
extreme observations. T-statistics adjusted for the overlapping nature of the data are the ratio of the time series average divided by the estimated time-series standard error (see Appendix A for the details).
Panel A: Regressions of one-year expected excess return on factor loadings (using historical betas)
Model I
Model II
Model III
Intercept
0.093
0.093
0.094
βHISTORICAL
3.4
3.4
3.7
0.039
0.031
0.029
4.5
4.4
4.8
0.019
0.019
2.7
2.8
-0.007
-0.007
-1.1
-1.2
γSMB
γHML
γPMOM
-0.019
-2.6
Average Adjusted R2
Average Monthly N
5.9%
429
11.7%
429
13.0%
430
Panel B: Regressions of one-year expected excess return on factor loadings (using full sample betas)
Model I
Model II
Model III
Intercept
0.079
0.078
0.082
βFULL SAMPLE
3.1
4.1
3.9
0.053
0.043
0.037
4.6
4.6
4.2
0.026
0.025
3.4
3.1
-0.007
-0.007
-1.2
-1.3
γSMB
γHML
γPMOM
-0.025
-1.6
Average Adjusted R2
Average Monthly N
7.5%
421
13.2%
422
14.8%
422
Table IX
Fama-MacBeth Regressions Based on Value Line Estimates of Expected Return
for Observations with a Timeliness Rank 3 (Panel A) and Expected Returns using
Post-Announcement Stock Price (Panel B), 1975-2001
This table presents time-series averages of 324 slopes from month-by-month regressions of one-year expected
excess returns on a set of estimated factor loadings and firm characteristics for observations with a timeliness
ranking equal to 3 (Panel A) and regression results using expected returns in which target prices are scaled by the
stock price outstanding 5 trading days after the Value Line price forecast announcement to compute the expected
return (Panel B). Value Line timeliness rank is the rank for a stock's probable relative market performance in the
year ahead. The rank ranges from 1 (highest) to 5 (lowest). Stocks ranked 3 (average) will probably advance or
decline with the market in the year ahead. One-year expected excess return is equal to the difference between the
Value Line expected return estimate ERVL, described in Section I.B, and the one-year risk free rate obtained from
the Fama-Bliss files on CRSP. Factor loadings for month t are estimated using data from month t-61 through t-1
(requiring a minimum of 24 valid monthly returns). Since Value Line provides in each report a firm-specific
market beta, we replace the estimated market beta with the one reported by Value Line. Size in month t is
computed as the log of market capitalization measured as of the end of the previous June (expressed in millions
of dollars). Book-to-market is computed as the ratio of annual common shareholders equity (Compustat item
#60) to market capitalization as of the end of the fiscal year. The log of this ratio is applied to the 12-month
period beginning six months subsequent to the end of the fiscal year. Prior price momentum for month t is the
buy and hold return for the 11 -month period ending one month prior to month t. We winsorize monthly
observations at the 1 st and 99th percentiles to mitigate the possible effect of extreme observations. T -statistics
adjusted for the overlapping nature of the data are the ratio of the time -series average divided by the estimated
time-series standard error (see Appendix A for the details).
Panel A: Regressions based on observations with a Timeliness Rank 3
Factor Loadings
Characteristics
Intercept
0.058
Intercept
4.3
β
0.059
7.8
β
5.9
γSMB
0.015
-0.002
log(Size)
-0.016
log(Book-to-Market)
14.4%
195
0.005
1.1
Prior Return
-2.4
Average Adjusted R2
Average Monthly N
-0.007
-2.9
-0.5
γPMOM
0.068
6.5
1.7
γHML
0.115
-0.096
-7.2
Average Adjusted R2
Average Monthly N
22.2%
185
Panel B: Regressions based on post-announcement stock price expected returns
Factor Loadings
Characteristics
Intercept
0.066
Intercept
0.125
4.2
β
0.057
γSMB
0.016
5.6
β
0.075
log(Size)
-0.009
4.5
6.3
2.1
γHML
0.001
-2.5
log(Book-to-Market)
0.2
γPMOM
-0.020
1.8
Prior Return
-2.8
Average Adjusted R2
Average Monthly N
13.4%
428
0.010
-0.095
-11.2
Average Adjusted R2
Average Monthly N
25.1%
401
Table X
Fama-MacBeth Regressions Within Sorts of Size, Book-to-Market,
and Price Momentum, 1975-2001
This table presents in panel A (panel B) time -series averages of 324 slopes from month-by-month
regressions of one-year expected excess returns on a set of estimated factor loadings (panel A) or firm
characteristics (panel B). The calculation of expected excess returns is described in Section I and Table III.
Firms are sorted, each month, into tercile portfolios based on three characteristics: size, book -to-market,
and momentum. Breakpoints are obtained from: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/.
For each portfolio we estimate month-by-month Fama-MacBeth cross-sectional regressions and report the
resulting time series averages of the parameter estimates. The factor loadings used in the regressions for
each month t are estimated using data from month t-61 through t-1 (requiring a minimum of 24 valid
monthly returns). Since Value Line provides in each report a firm -specific market beta, we replace the
estimated market beta with the one reported by Value Line. We winsorize monthly observations at the 1st
and 99th percentiles to mitigate the possible effect of extreme observations . T-statistics adjusted for the
overlapping nature of the data are the ratio of the time-series average divided by the estimated time-series
standard error (see Appendix A for the details).
Panel A: Regressions of Value Line one-year expected excess return on factor loadings
Size
Book-to-market
Price Momentum
Small Medium Large
Low Medium High
Loser Medium Winner
Intercept
0.090
0.055 0.038
0.073 0.053 0.098
0.106 0.060 0.059
β
γSMB
γHML
γPMOM
Average Adjusted R2
Average Monthly N
3.9
3.3
2.8
4.3
4.4
4.2
8.0
5.2
3.2
0.073
0.073
0.073
0.046
0.072
0.053
0.060
0.063
0.036
7.0
6.7
3.5
3.2
5.4
4.5
4.2
4.4
5.5
0.005
0.004
-0.002
0.012
0.015
0.009
0.015
0.017
0.010
1.2
0.7
-0.2
1.5
2.2
1.4
4.9
2.7
2.2
-0.002
0.001
0.005
-0.002
-0.001
-0.006
-0.002
0.002
0.004
-0.8
0.1
0.8
-0.4
-0.2
-1.7
-0.5
0.7
1.5
-0.012
-0.016
-0.024
-0.018
-0.020
-0.014
-0.014
-0.017
-0.017
-4.7
-2.5
-2.7
-2.6
-3.3
-3.3
-4.8
-3.8
-4.6
8.5%
102
13.9%
183
15.0%
142
13.0%
155
15.9%
179
9.4%
80
10.7%
123
15.3%
173
9.0%
131
Panel B: Regressions of Value Line one-year expected excess Return on firm characteristics
Size
Book-to-market
Price Momentum
Small Medium Large
Low Medium High
Loser Medium Winner
Intercept
0.109
0.075 0.060
0.139 0.112 0.127
0.119 0.114 0.117
β
log(Size)
log(Book-to-Market)
Prior Return
Average Adjusted R2
Average Monthly N
5.0
6.4
3.0
5.5
7.4
5.8
5.2
6.9
5.3
0.073
0.078
0.079
0.060
0.087
0.071
0.063
0.080
0.059
6.6
7.1
4.0
5.7
5.2
6.4
5.4
5.5
8.5
-0.002
-0.001
-0.001
-0.009
-0.009
-0.006
-0.005
-0.008
-0.008
-0.4
-0.9
-0.3
-2.0
-3.5
-2.2
-2.5
-2.8
-3.0
0.006
0.010
0.013
0.004
0.014
0.008
0.010
0.011
0.013
1.6
1.8
2.1
1.1
1.2
1.5
1.9
2.1
2.7
-0.093
-0.093
-0.094
-0.098
-0.097
-0.095
-0.147
-0.120
-0.047
-12.7
-11.1
-11.3
-12.0
-12.3
-13.6
-17.9
-10.8
-7.3
17.7%
93
23.9%
174
24.4%
133
25.4%
149
26.2%
178
19.5%
74
15.4%
114
17.8%
165
12.7%
122
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