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The Market Premium is Borne Again:
Using Economic Forecasts of Betas in CAPM Tests
First draft: March 2001
This version: December 2003
Abstract
This paper re-examines the significance of beta along with size and book-to-market ratio in the
cross-section of stock returns. We make use of a novel instrumental variables approach to correct
for the Errors in Variables bias due to the measurement errors in beta estimates. Our results show
that average stock returns indeed reflect a significant compensation for beta risk of about 6 to 9
percent annually. The value premium is also robust, although the size premium tends to disappear.
Our conclusions are insensitive to changes in data selection criteria, as well as different portfolio
formation procedures and different estimation techniques.
JEL Classification: G11, G12, G13.
Key words: Assets in place, beta, CAPM, growth opportunities, market risk premium, size, value.
∗
The Market Premium is Borne Again:
Using Economic Forecasts of Betas in CAPM Tests
Abstract
This paper re-examines the significance of beta along with size and book-to-market ratio in the
cross-section of stock returns. We make use of a novel instrumental variables approach to correct
for the Errors in Variables bias due to the measurement errors in beta estimates. Our results show
that average stock returns indeed reflect a significant compensation for beta risk of about 6 to 9
percent annually. The value premium is also robust, although the size premium tends to disappear.
Our conclusions are insensitive to changes in data selection criteria, as well as different portfolio
formation procedures and different estimation techniques.
Empirical asset pricing research has long been trying to gauge the ability of beta to explain
the cross-sectional variation in security returns. There is now ample evidence showing that several
asset specific factors besides beta are statistically related to the cross-section of average returns.
Although these results contradict even the most general form of the Capital Asset Pricing Model
(CAPM; Sharpe (1964), Lintner (1965), and Black (1972)), they nevertheless are consistent with
multifactor asset pricing models such as variants of the Intertemporal Capital Asset Pricing Model
(ICAPM, Merton (1973)) or the Arbitrage Pricing Theory (APT, Ross(1976)).
Multi-factor beta pricing models have received wide attention since the seventies. For example, Banz (1981) and Reinganum (1981) use firm size in addition to beta to help explain the
cross-section of security returns. More recently, Fama and French (1992, 1995) show that, among a
battery of variables that have been shown to be related to security returns, size and book-to-market
equity ratio are the most capable ones. Market risk, measured by the CAPM beta, seems to have
no effect on average returns once these variables are taken into account.
The real controversy now is not if size and book-to-market ratio are statistically related to
the cross-section of average returns, but why these characteristics seem to matter. There are several
competing explanations for size and value (book-to-market) premiums. The first apparent conclusion is that the documented results may be due to sample specific chance outcomes1 . However,
evidence contradicting this view comes from out-of-sample tests utilizing U.S. and international
data. For instance, Davis, Fama and French (2000) show that there is a strong value premium and
a relatively smaller size premium on NYSE industrial firms using an extended data sample going
back to 1926.
An alternative risk-based explanation is that size and book-to-market ratio are proxies for
undiversifiable systematic risk factors, and therefore the observed premiums reflect compensation
for extra systematic risk. Specifically, Fama and French (1993, 1996) show that a three-factor
model, which incorporates size and book-to-market ratio factor portfolios as well as a market
portfolio, explains almost all reported “anomalies”, with the notable exception of Jegadeesh and
Titman (1993) momentum portfolio returns. Challenging this hypothesis, Daniel and Titman
(1997) argue that high returns on small capitalization and high book-to-market stocks are due to
the characteristics rather than co-movements of these stocks with risk factors. However, Davis,
1
See Black (1993), and MacKinlay (1995).
1
Fama and French (2000) argue that the evidence of Daniel and Titman (1997) is specific to their
relatively short sample period and that more powerful tests over a much longer sample period
provide evidence in the direction of a risk story.
The third category views book-to-market ratio as a mispricing proxy, suggesting that high
returns associated with value (i.e., high book-to-market) stocks are generated by investors who
overreact and extrapolate past performance too far into the future. Advocates of this view include
DeBondt and Thaler (1985, 1987) and Lakonishok, Shleifer and Vishny (1994), among others. In
addition, Lakonishok et. al. argue that value strategies appear to be no riskier than others on
average, and reject risk-based explanations of the value premium.
In the wake of these controversial arguments, we re-examine the significance of market beta,
book-to-market equity ratio, and size in the cross-section of average security returns over the 1963
to 1999 period. We present results using the traditional Fama-MacBeth (1973) monthly crosssectional regression approach as well as a more efficient weighted average estimator. Our results
document a statistically and economically significant risk premium for beta on the order of 6 to
9 percent annually in the presence of size and book-to-market equity ratio. This is in contrast to
the results of Fama and French (1992) and others who find that estimated betas are not priced.
However, beta alone still does not account for all of the cross-sectional variation in average returns;
book-to-market equity ratio seems to be priced, and continues to be a significant factor.
The errors in variables (EIV) bias due to measurement errors in market beta estimates is a
major econometric problem for linear beta pricing model tests. It is well known that, in a multivariable relationship where one independent variable has measurement error among others that
are measured without error, the estimate of the coefficient for that variable will be biased towards
zero. This suggests that, unless the EIV problem is corrected, the price of beta risk will be biased
towards zero. On the other hand, the direction of bias on the coefficients for the variables measured
without error depends on the covariance matrix of observations.
Several methods have been proposed to account for the EIV bias. Grouping, i.e. working
with portfolios rather than individual assets, and re-estimation of portfolio betas reduces measurement errors in market beta estimates used in cross-sectional tests. For example, the traditional
Fama-MacBeth approach allocates securities to portfolios based on estimates of betas from non-
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overlapping time-series of returns that precede the time-series of returns used to re-estimate portfolio betas. Litzenberger and Ramaswamy (1979) use sample estimates of variances of observed betas
to arrive at maximum likelihood estimators of risk premiums. Shanken (1992) modifies the traditional two-pass approach and provides an adjustment for the standard errors of CSR estimators.
Instead of two-pass estimators, Gibbons (1982) uses a maximum likelihood approach to eliminate
the EIV problem by simultaneously estimating betas and risk premiums. However, Shanken notes
that the advantage of simultaneous estimation is lost in linearizing a non-linear constraint, and
that Gibbons’s estimator is still subject to the EIV bias. Kim (1995) suggests an N-consistent
(consistent when the time series-sample is fixed, and the number of securities is allowed to increase
without bound) estimation method within the two-pass estimation methodology. Kim’s correction
is most useful when individual securities, rather than portfolios, are used in CSR tests.
We propose a simple approach within the spirit of the traditional two-pass methodology
to reduce the impact of the EIV bias on risk premium estimates. To reduce time-series data
requirements and to be able to use the power of grouping in reducing inconsistency, we utilize
an instrumental variable methodology. In this approach, an instrumental variable that is highly
correlated with market beta estimates but uncorrelated with measurement errors is used in portfolio
formation and cross-sectional tests. Following Jacquier, Titman and Yalçın (2003), we use a simple
economic model to help us come up with the instrumental variable. This model assumes that firms
are portfolios of growth opportunities and assets in place, and defines the stock beta as a weighted
average of the growth opportunity and the assets in place betas. In the first stage of this procedure,
individual security betas, estimated over the portfolio formation period, are regressed on accounting
variables which are hypothesized to reflect growth opportunities. The fitted part of this regression
is then used in the second stage cross-sectional tests to determine risk premium estimates.
The rest of the paper is organized as follows. Section I explains the empirical methodology
in considerable detail. Section II describes the data. The empirical results are presented in Section
III. Section IV discusses the robustness of our results to different portfolio formation procedures
and data selection criteria. The final section gives the conclusions.
3
I.
The Empirical Methodology
A.
Linear Beta Pricing Models and Cross-Sectional Tests
In its most general form, the Capital Asset Pricing Model (CAPM) predicts a positive linear
relation between an asset’s expected rate of return and its covariance risk (beta):
E(Ri ) = γ0 + γ1 βi
where E(Ri ) is the expected rate of return on asset i, and βi is the asset’s beta defined as
cov(Ri , Rm )/var(Rm ), and Rm is the rate of return on the market portfolio.
In order to test for model misspecification in the CAPM, we can employ an extended form
of the above equation which takes asset specific factors into account:
E(Ri ) = γ0 + γ1 βi + γ2 Fi
(1)
where E(Ri ) and βi are as defined before, and Fi is a vector of asset specific factors. The estimate
of γ2 should be a vector of zeros if the CAPM is valid. For example, Fama and French (1992) use
market capitalization and book-to-market equity ratio in such an extended form to conclude that
the market beta bore no relation to the cross-section of average stock returns in the presence of
these firm characteristics.
The two-stage cross-sectional regression (CSR) method of Fama and MacBeth (1973) is one
of the most popular testing methods in empirical analysis of linear beta pricing models. However, a
major econometric problem with this methodology is the presence of heteroscedastic and correlated
errors stemming from using stock returns as the dependent variable. Under this error structure, the
ordinary least squares (OLS) is inefficient relative to the generalized least squares (GLS) estimator.
On the other hand, it is not practical to have a full GLS estimator when the number of assets is
large relative to the time series of observations, as this would reduce the precision with which the
covariance matrix is estimated. To overcome this problem, Litzenberger and Ramaswamy (1979),
and more recently Ferson and Harvey (1999,) suggest using a restricted covariance matrix in a
general GLS framework.
4
Consider a pooled time-series cross-sectional regression method to test the model in (1):
E(U U 0 ) = Ω
R = Xγ + U,
(2)
where R is a TNx1 vector of excess returns, T is the number of periods, and N is the number of
assets. The first N rows of R are the excess returns of N assets for the first time period of the
sample, followed by the second period, etc. X is a TNxK matrix of independent variables, with
a column of ones and columns of betas and asset specific factors such as size or book-to-market
equity ratio, stacked up like the excess returns. The Kx1 vector γ is the set of risk premiums we
would like to estimate. Ω is the TNxTN covariance matrix.
One way to estimate the risk premium vector γ in (2) is to use OLS regression. However,
under the heteroscedastic and correlated error structure, OLS estimator of risk premium vector is
inefficient relative to GLS which provides the best linear unbiased estimator. If Ω is known, then
the GLS estimator of γ and its covariance matrix are given as:
γ̂GLS = (X 0 Ω−1 X)−1 X 0 Ω−1 R
var(γ̂GLS ) = (X 0 Ω−1 X)−1
If we assume that the excess rates of returns are serially uncorrelated, then the covariance
matrix Ω has a block-diagonal structure with Ωt ’s on the diagonal, and zeros elsewhere. Under this
assumption, we can run a separate cross-sectional GLS regression in each period t=1,...,T:
−1 0 −1
γ̂t(GLS) = (Xt0 Ω−1
t X t ) X t Ωt Rt
−1
var(γ̂t(GLS) ) = (Xt0 Ω−1
t Xt )
where Ωt is the NxN covariance matrix of the error term for period t. Huang and Litzenberger (1988)
and Ferson and Harvey (1999) show that the pooled time-series cross-sectional GLS estimator γ
can be written as a weighted average of the individual period estimates. The weights are inversely
5
proportional to their variances, discounting less precise estimates:
−1
0 −1
γ̂GLS = (Σt Xt0 Ω−1
t Xt ) (Σt Xt Ωt Rt )
where Σt indicates summation over different time periods.
Ferson and Harvey (1999) show that the above “efficient-weighted Fama-MacBeth estimator”, is equal to:
γ̂GLS = Σt wt γ̂t(GLS)
−1
0 −1
where wt = (Σt Xt0 Ω−1
t Xt ) (Xt Ωt Xt ), and variances are given as:
2
var(γ̂GLS ) = (1/T )(T −1 Σt wt2 γ̂t(GLS)
− (Σt wt γ̂t(GLS) )2 )
We use several different estimation techniques in our asset pricing tests. Every month in the
test period, stock returns are regressed on variables which are hypothesized to explain the crosssection of average stock returns; i.e. estimates of beta, size and book-to-market equity ratio. The
time series averages of monthly regression slopes then allow us to test whether different explanatory
variables actually do explain average returns. First, in order to have comparability with earlier
empirical studies, we estimate the risk premiums using the traditional Fama-MacBeth procedure,
and take the arithmetic average of monthly regression slopes to come up with the final estimate of γ.
Second, we use Ferson and Harvey’s efficient weighted Fama-MacBeth procedure using WLS, where
the monthly regression slopes are weighted in inverse proportion to their variances. We assume a
diagonal structure for Ωt where the diagonal elements are set equal to estimates of the variance
of monthly portfolio excess returns. As a robustness check, we repeat our cross-sectional tests
using an efficient weighted full covariance matrix GLS estimation, where the covariance matrix is
estimated using daily return observations. The results we report are robust to different estimation
techniques.
B.
Errors in Variables Bias and Instrumental Variables Approach
Independent variables which contain measurement errors cause yet another source of econo-
metric problem for testing linear beta pricing models. As we have mentioned earlier, using estimated
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values instead of unobservable true betas in equation (1) introduces an Errors in Variables (EIV)
bias. Consequently, both OLS and GLS estimators of risk premiums become biased and inconsistent
in the second stage CSR tests2 .
Several solutions have been proposed to overcome the EIV problem within the two-pass
methodology3 . One way, when security residual variances are exactly known, is to use an MLE or
adjusted GLS approach that takes variance of measurement errors directly into account. Another
widely used approach is to form portfolios of securities and use grouped data to reduce measurement
errors in explanatory variables. Under the assumption that measurement errors are uncorrelated
across assets, the bias would approach zero as the number of assets in each portfolio increases.
However, if the rankings of securities based on estimated betas are correlated with measurement
errors, then variances of measurement errors will not go down to zero, even in the limit. The earlier
empirical studies that take the EIV bias into account include Litzenberger and Ramaswamy (1979),
Gibbons (1982), Shanken (1992)and Kim (1995, 1997), among others.
Ex ante determinants of beta have received wide attention since the seventies, although
the interest have waned somewhat recently. For instance, Beaver, Kettler and Scholes (1970) use
accounting variables as instrumental variables in the prediction of future market betas, and show
that this approach provides better predictions. Rosenberg and McKibben (1973) and Rosenberg
(1984, 1985) combine fundamental firm characteristics with past market data to provide efficient
predictions of both diversifiable and firm specific components of risk in common stocks. In a study
concerned with a multi-factor model of security returns, Rosenberg (1974) shows that the loadings
of individual security returns on risk factors are determined by observable characteristics of the
firm, such as accounting data, industry membership and historical behavior of returns. Moreover,
he also shows that the conventional market beta is a function of these same characteristics. More
recently, Berk, Green and Naik (1999) model the firm value in terms of fundamental state variables
that summarize the systematic risk of existing assets and the importance of growth options.
We propose an instrumental variables approach that eliminates the need for non-overlapping
portfolio formation and re-estimation periods in the classical two-pass methodology. By grouping
securities into portfolios using a criteria that is highly correlated with true betas but uncorrelated
2
Levi (1973) shows that the direction of bias can be calculated once the covariance matrix of observations is
known.
3
For a thorough analysis on this subject see Chapter 10 in Huang and Litzenberger (1988).
7
with measurement errors, we are able to substantially reduce the impact of the EIV bias on risk
premium estimates.
To get an instrumental variable for market beta estimates, we make use of the notion that
a firm can be considered as a portfolio of assets in place and growth opportunities (see Myers
(1977)). Following this logic, Jacquier, Titman and Yalçın (2003) hypothesize that equity betas are
related to the amount of growth opportunities imbedded in the assets of the firm, and capture this
relation by projecting estimated betas on proxies of growth opportunities. Their evidence reveals
that fundamental firm characteristics that are naturally related to growth opportunities explain up
to 66% of the cross-sectional variation in betas of portfolios based on past performance.
II.
The Data
Our data sample consists of non-financial firms in the intersection of the NYSE, AMEX, and
NASDAQ monthly returns file from the Center for Research in Securities Prices (CRSP) and the
merged COMPUSTAT annual industrial file. The data selection and portfolio formation criterion
are similar to those in Fama and French (1992). Every June from 1963 to 1998, one-hundred
size-beta portfolios are formed by first allocating individual stocks to ten groups along their equity
market capitalization, and second, subdividing each group into ten groups along estimated market
betas. The equal weighted returns of these portfolios are then tracked for twelve months from July
to next June. The market model regression, with the CRSP value weighted index as the market
proxy, is used to estimate individual security and portfolio market betas4 .
To ensure that the information is available at the time of portfolio formation, we measure
accounting data as of the fiscal year end prior to the test period. In order to be included in the
sample, a firm must have a market beta estimate, equity market capitalization values on June and
the prior December, as well as the required accounting data. We use the market value of a firm on
December of the prior year to calculate its book-to-market equity ratio, as well as other accounting
data based ratios5 .
4
We use sixty monthly returns ending on June in the market model regression. In section IV, we discuss the case
where we relax this data selection criteria by requiring firms to have at least twenty-four non-missing monthly returns
over the sixty month portfolio formation period.
5
Book equity is defined as the book value of shareholders’ equity, plus balance sheet deferred taxes and investment
tax credit, minus the book value of preferred stock. Depending on data availability, the value of preferred stock is
8
The grouping procedure discussed above reduces the magnitude of inconsistency in risk
premium estimates by reducing the variance of the measurement error in beta estimates. However,
as Huang and Litzenberger (1988) note, any type of grouping results in a loss of efficiency in
parameter estimates. Therefore, it is important to group assets in a way to maximize the between
group variation of expected returns in order to minimize the efficiency loss. Size-beta sorting is a
proven way to accomplish this without sacrificing too much information.
Table I summarizes the characteristics of size-beta sorted portfolios. We report, for each
portfolio, its estimated beta, average monthly excess return and size over the 36 distinct twelvemonth testing periods. Panel A shows that there is a positive relationship between estimated betas
and size among low-beta groups; betas get larger as we move on to larger capitalization groups. In
contrast, there is a negative relationship between estimated betas and size among high-beta groups.
The size premium in average security returns is evident in Panel B; among all beta groups, small
capitalization portfolios have higher returns than large capitalization ones. Panel C shows that
size-beta sorting achieves its goal of producing variation in betas that is unrelated to size; in any
given size group, the average size across different beta deciles are quite similar.
III.
A.
Empirical Results
Beta Sorting and Measurement Errors
The negative correlation between average excess returns and beta estimates among small
beta portfolio groups observed in Table I clearly contradicts the CAPM. However, the empirical
pattern we observe may be far from reflecting the relationship between the true beta and expected
returns.
We can write the estimated beta, β̂p , as a sum of the true beta, βp , and a measurement
error, ṽp : β̂p = βp + ṽp . The beta of a portfolio in Table I is estimated using the market model OLS
regression over the sixty-month portfolio formation period. Assuming that the measurement error
is uncorrelated with the true beta, we can find an estimate of the variance of the measurement error,
var(ṽp ), using the variance of the estimate of the true beta: var(β̂p ) = var(βp + ṽp ) = var(ṽp ).
estimated as redemption, liquidation or par value.
9
Table II shows estimates of variances of measurement errors for size-beta portfolios. Panel A
reports average standard errors from thirty-six market model regressions. As the results show, we
observe a u-shaped pattern for almost all size deciles; conditional on size, estimates for extreme-beta
portfolios are much less precise. We have an even stronger pattern for beta groups; conditional
on beta, measurement errors monotonically decline with size. The most precise estimation is
for large capitalization portfolios with betas close to that of the market. To illustrate, standard
errors of large capitalization portfolios in beta groups 3 through 8 are, on average, equal to 0.06;
compared to standard errors of 0.246 and 0.273 for small capitalization low and high beta portfolios.
These conclusions are also confirmed in Panel B, which reports time-series standard deviations of
portfolio betas as proxies for measurement errors. The smallest standard deviation of 0.063 is for
high capitalization portfolios in beta groups 5 and 6, suggesting that these portfolios’ betas are
much more stable over time than that of, for instance, the low capitalization-low beta portfolio,
which has a time-series standard deviation of 0.624.
Not surprisingly perhaps, low and high beta portfolio betas are estimated with larger errors
compared to those of average-beta portfolios, and small market capitalization portfolio betas are
estimated less precisely compared to betas of large capitalization ones. To the extent that the
ranking of securities are correlated with measurement errors in betas; firms with large negative
errors in betas will be clustered in low beta groups, and those with large positive errors will be
concentrated among high beta groups. This is evident in the u-shaped pattern in estimates of the
variance of the measurement error in both panels in Table II.
Although the proposed size-beta grouping reduces the magnitude of inconsistency in portfolio
beta estimates, variances of measurement errors are not close to zero because rankings of assets
based on estimated betas are correlated with measurement errors. As a result of these large
measurement errors documented in Table II, OLS and GLS risk premium estimates in the secondstage cross-sectional tests no longer will have the desired properties; risk premium estimates will
be biased and inconsistent due to the EIV problem. As a result, grouping assets into portfolios
using estimated betas will not completely solve the EIV problem.
A better way to form portfolios is to use a grouping procedure that is highly correlated with
true betas, but uncorrelated with measurement errors. Such a procedure will create substantial
variation in the true beta across portfolios. Moreover, variances of measurement errors of portfolio
10
betas should approach zero as the number of assets in those portfolios increase. Accordingly, we
describe, in the next section, a special case of the Instrumental Variables (IV) approach to further
reduce the bias caused by the EIV problem.
B.
Instrumental Variables
Every June from 1963 to 1998, we estimate the market beta of each firm in the sample
using the market model regression. We also form firm level descriptors that proxy for growth
opportunities using accounting data as of the prior fiscal year end, and equity market value as
of the prior December. Following Jacquier et. al. (2003), the descriptors are defined as capital
expenditures-to-net fixed assets ratio, dividend yield, earnings-to-price ratio, debt-to-equity ratio,
and book-to-market equity value6 . In the first stage of the Instrumental Variables (IV) approach,
market beta estimates which are measured with errors are projected onto growth proxies in a
cross-sectional OLS regression:
β̂it = α0,t +
K
X
αk,t dk,it + ũit ,
i = 1, 2, ..., Nt
k=1
where β̂it is the market beta estimate for firm i, dk,it are the descriptors which proxy for growth
opportunities, and Nt is the number of firms. In order to reduce the impact of outliers on the
regression, we windsorize the descriptors using the 1st and 99th percentile values7 . Even though
the above regression is a firm-level cross-sectional regression, the r-square averages almost 18%, a
testament to the ability of growth opportunities to explain betas.
In the second stage, fitted betas from the above model, β̄it , are used to group assets into
one-hundred equally weighted size-fitted beta portfolios. First, we place firms into ten groups based
on their equity market capitalization at the end of June, then, subdivide each decile into ten groups
along fitted betas, β̄it . We calculate the portfolio fitted beta as the average of individual security
fitted betas in that portfolio, and estimate the portfolio market beta using sixty-month formation
period returns.
6
Please see Jacquier et. al. (2003) for a more detailed explanation of these growth proxies.
For capital expenditures, book-to-market, earnings-to-price and estimated betas we use 1st and 99th percentile
values as cutoffs. For dividend yield and debt-to-equity ratio we only use the 99th percentile cutoff.
7
11
Table III mirrors Table I for size-fitted beta portfolios; we report, for each portfolio, its
average monthly excess return, fitted beta and estimated beta. Panel A shows that fitted betas
monotonically decline with size for almost all beta groups. The size premium in returns is still
reflected in Panel B; small capitalization portfolio returns are significantly higher within all beta
subgroups. However, the premium now is in line with the systematic risk pattern in Panel A. The
estimated beta, reported in Panel C, exhibits almost the same relationship with size as the fitted
beta, except for the largest-beta subgroup.
Our interest in this paper is centered on measurement errors in betas and on the impact of
the EIV bias on risk premium estimates. Before we illustrate the reduction in the measurement
error in betas directly; however, it might be instructive to analyze the effect of grouping using betas
measured with error. To the extent that the ranking of securities are correlated with measurement
errors in betas, the advantage of grouping will be nullified. This is best reflected in the difference
between portfolio beta estimates reported in panels A and C of tables I and III, respectively. Remember that in Table I, we rank securities into groups using estimated betas; which have larger
measurement errors compared to fitted betas that are used for ranking in Table III. The range of
estimated portfolio betas in Table I, within any size subgroup, is considerably wider than its counterpart in Table III; which supports our earlier claim that securities with large negative (positive)
errors are clustered among low (high) beta subgroups. For instance, for Size-5 group in Table I,
estimated portfolio beta varies from 0.35 to 2.19; a range of 1.84. In comparison, even though there
is a healthy spread in estimated portfolio beta from 0.81 to 1.51 in Table III for Size-5 group, the
range is down to 0.7.
To have a more direct sense for the reduction in measurement errors, Table IV reports timeseries standard deviations of fitted and estimated portfolio betas in panels A and B, respectively.
Estimated betas contain higher measurement errors compared to fitted betas in almost all size
groups, except the large capitalization ones. This is to be expected however, since measurement
errors in these portfolios are very low to begin with.
Panel A suggests that, compared to Table II-Panel B, there is a significant reduction in the
variance of measurement errors overall when we use fitted betas to measure portfolio systematic risk;
size-fitted beta portfolios seem to have much more stable betas overtime relative to size-estimated
beta portfolios. In addition, the pronounced u-shape we have observed in Table II diminishes,
12
suggesting that we are able to reduce the correlation between measurement errors and ranking of
individual securities. One can claim that the comparison between fitted and estimated betas is a
bit unfair. A more relevant question, may be, is how precisely portfolio betas are estimated for
size-fitted beta groups compared to size-estimated beta groups. A comparison of panels B in tables
IV and II suggests that, in general, the precision of estimated betas is better for size-fitted beta
portfolios than size-estimated beta portfolios.
The results we report in this section document that ranking stocks using fitted betas is not as
highly correlated with measurement errors as ranking utilizing estimated betas. Therefore, the EIV
bias will be least problematic if we were to use fitted betas both for the relative ranking of stocks in
the formation period and as proxies for market risk in the 2nd stage cross-sectional regression. In
the following section, we examine the potential impact of the EIV bias on risk premium estimates
in more detail.
C.
Price of Risk and Measurement Errors
The empirical finance research suggests that several asset specific factors in addition to
beta are statistically related to security returns. For example, Banz (1981), Reinganum (1981)
and Keim (1983) show that small firms have greater returns than those predicted by the CAPM.
In addition, Fama and French (1992, 1993, 1996), Davis (1994), and Kothari and Shanken (1995),
among others, report that firms with greater book-to-market equity ratios earn greater risk adjusted
returns. Furthermore, other fundamental value-price ratios, such as dividend yield and earningsto-price ratio, are also shown to have significant power in explaining the cross-section of stock
returns.
To examine whether beta is priced, we use the extended form in equation (1) which takes
size and book-to-market ratio into account. For every month in the one year test period from July
to next June, portfolio returns are regressed on variables hypothesized to explain the cross-section
of average stock returns:
Rt = γ0t + γ1t βt−1 + γ2t St−1 + γ3t BMt−1 + t
(3)
where Rt is the vector of monthly size-beta portfolio returns in excess of the risk-free rate, βt−1
13
is the vector of portfolio betas, and St−1 and BMt−1 are natural logarithms of average size and
book-to-market ratio of stocks in portfolios. Using time series averages of regression slopes from
432 monthly (36 one-year test periods) cross-sectional regressions, we are able to test whether these
variables indeed explain average returns.
Table V summarizes two different versions of the estimation methodology that we employ
for two different grouping techniques. In the traditional Fama-MacBeth approach, we use OLS in
the cross-sectional regression. Estimates γ̂t over time are assumed to be independent and identical
sampled values of the price of risk, γ. The time-series sample average is regarded as the final
estimate of γ. The standard error of the estimate is also calculated from the time-series sample. In
the second procedure, we use the restricted covariance matrix GLS estimation (i.e., WLS), where
the covariance matrix is assumed to be diagonal. The final estimate of γ is calculated as the efficient
weighted average of the time-series sample, γ̂t(W LS) , where coefficients of each month are weighted
in inverse proportion to their variances. The standard error of the final estimate is also calculated
as a weighted average.
The results for size-estimated beta portfolio returns are presented in Panel A. Consistent
with Fama and French (1992), the results show that both size and book-to-market ratio seem
to have significant explanatory power in the cross-section of average asset returns. Market risk
represented by beta estimates, on the other hand, does not appear to have any role at all. In
addition, the statistically significant positive intercept term indicates that the model is inadequate
in explaining the cross-section of average returns.
The results above, like most of their previous empirical counterparts, suffer from the EIV
bias. The CSR model in equation (3) has one independent variable measured with error and two
measured without error. Beta estimates have been used instead of the true unobservable beta,
which causes OLS and WLS estimators of risk premiums to become biased and inconsistent in the
second stage CSR.
In order to reduce the impact of the EIV bias on risk premium estimates, we repeat the
analysis using the IV approach outlined in the previous section. Cross-sectional tests for size-fitted
beta portfolio returns are reported in Panel B of Table V. The results suggest an economically
and statistically significant compensation of about 0.67 to 0.74 percent per month for beta risk.
14
Furthermore, the intercept term is not significantly different from zero, which provides additional
support for the ability of fitted betas to capture the cross-sectional variation in asset returns. In
addition to the beta risk, it seems that book-to-market plays an important role in the cross-section
of asset returns; the average regression coefficient on this factor varies from 0.43 to 0.60 percent
per month. In contrast, the risk premium for size tends to disappear; the coefficient on size, using
either estimation technique, is both statistically and economically insignificant.
Ideally, we would like to estimate the risk premium vector using a full scale generalized
least squares (GLS) approach. However, a precise estimate of the covariance matrix of returns is
not available when the number of assets is large relative to the time series observations, which is
sixty months in our case. To overcome this problem, we estimate the covariance matrix of the
100 size-estimated beta portfolios using daily returns over the sixty month formation period. Risk
premiums are estimated using the same structure as in (3). The outcomes of both the classical
and the efficient weighted Fama-MacBeth approach using GLS are reported in Table VI. Since the
daily data has a shorter time-series on the CRSP files, we now have 32 one-year test periods, which
corresponds to 384 monthly cross-sectional regressions. Table VI confirms our previous conclusions;
there is a significant compensation of about 0.49 to 0.57 percent per month for beta risk, the value
premium is a robust 0.41 to 0.64 percent per month, whereas the size premium disappears.
To summarize, the evidence we report so far suggests that beta risk demands a significant
positive risk premium after the EIV correction and that book-to-market ratio, unlike size, continues
to be an important factor in the cross-section of stock returns. Kim (1997) reports similar results
and suggests that the robust value premium is partly due to the weaker correlation between bookto-market ratio and beta relative to the correlation between size and beta.
IV.
Robustness of The Results
In this section, we conduct additional experiments to check the robustness of our results to
different data selection criteria and portfolio formation procedures. We first analyze the sensitivity
of risk premium estimates to data selection by expanding the sample and allowing stocks with
twenty-four or more valid monthly returns over the sixty-month portfolio formation period to be
included in the final analysis.
15
Table VII reports the results on size-beta portfolios for this expanded sample. The portfolio
formation procedure and estimation techniques used are the same as those described in the previous section. In panels A and B, risk premiums are estimated from 432 monthly cross-sectional
regressions using monthly size-estimated beta and size-fitted beta portfolio returns, respectively. In
Panel C, risk premiums are estimated from 384 monthly cross-sectional regressions using monthly
size-fitted beta portfolio returns, and daily returns to compute the covariance matrix used in the
GLS. The conclusions drawn are similar to what we have already outlined earlier. In Panel A,
the EIV bias forces estimates of risk premium on beta towards zero in both estimation methods.
Correcting for the EIV bias, we show in Panels B and C that fitted beta demands a risk premium of
0.73 to 0.88 using the traditional Fama-MacBeth approach, and a monthly premium of 0.73 percent
using the efficient weighted GLS estimation methodology.
So far in our results, even after correcting for the EIV bias, the value premium has a robust
impact on the cross-section of average stock returns. For example, in panels B and C of Table
VII, the estimate of value premium varies from 0.47 to 0.78 percent per month. The robust value
premium suggests that another way to maximize the between group variation in expected returns
would be to form portfolios based on book-to-market (BM) rankings. Following this logic, on June
of every year, we place individual common stocks into ten groups based on their book-to-market
equity ratios instead of size. Common stocks in each of these deciles are further divided into ten
groups based on either their estimated or fitted betas.
We repeat our asset pricing tests using these BM-beta portfolio monthly returns. The results,
which are presented in Table VIII, are qualitatively similar to the results on size-beta portfolios.
Namely, the risk premium estimate on estimated beta is insignificant in the presence of size and
BM ratio due to the EIV bias; but once the bias is corrected for, the fitted beta commands a risk
premium anywhere from 0.42 to 0.82 percent per month depending on the estimation procedure.
As before, book-to-market ratio continues to be an important factor in the cross-section of stock
returns, whereas the size effect disappears.
16
V.
Conclusions
Previous studies suggest that several asset specific factors - especially price-scaled ratios -
seem to matter in the cross-section of average stock returns. Moreover, these studies document
that the compensation for market risk represented by beta is indistinguishable from zero. In fact,
in a multivariate relationship where one independent variable has measurement error among the
rest measured without error, the estimate of the coefficient for that variable will unambiguously
be downward biased. This observation is critical for linear beta pricing models used to test model
misspecification in CAPM, as the true beta in unobservable and some sort of an estimate has
to be used instead. Therefore, without correcting the Errors in Variables (EIV) bias due to the
measurement errors in betas, it is premature to announce the death of beta.
We add to the literature on the effects of the EIV bias on risk premium estimates in linear
beta pricing models by using economic forecasts of betas in cross-sectional asset pricing tests. The
procedure we propose is in the spirit of the traditional two-pass methodology and makes use of an
Instrumental Variables approach which eliminates the requirement of non-overlapping time-series
data requirement for portfolio formation and beta re-estimation periods.
Following Jacquier, Titman and Yalçın (2003), we use a simple economic model to come
up with an instrumental variable that is highly correlated with the true beta, but uncorrelated
with the measurement error. Using size-beta sorted portfolio returns and efficient weighted FamaMacBeth procedure to estimate risk premiums, we document that the cross-section of expected
returns reveals a compensation of roughly about 6 to 9 percent per year for beta risk. However,
beta does not appear to be the only factor that affects the cross-section of average returns. In line
with previous empirical studies, the book-to-market ratio is priced and accounts for a significant
portion of the total variation in cross-sectional returns. The results we document seem to be robust
to several different data selection and portfolio formation criterion.
17
References
Ahn, Seugn C., and C. Gadarowski, 1999, Two-Pass Cross-Sectional Regression of Factor Pricing
Models: Minimum Distance Approach, Working Paper, Arizona State University.
Amihud, Yakov, Christensen J. Bent and Haim Mandelson, 1993, Further Evidence on the RiskReturn Relationship, Working Paper, New York University.
Banz, Rolf W., 1981, The Relationship Between Return and Market Value of Common Stocks,
Journal of Financial Economics, 9, 3-18.
Beaver, W., Kettler, P. and Scholes, M., 1970, The Association Between Market Determined and
Accounting Determined Risk Measures , Accounting Review 45, 654-682.
Berk, Jonathan B., 1995, A Critique of Size Related Anomalies, Review of Financial Studies, 8,
275-286.
Berk, J.B., C.G. Green and V. Naik, 1999, Optimal Investment, Growth Options, and Security
Returns, Journal of Finance 54, 1553-1607.
Chan, K.C. and Nai-Fu Chen, 1988, An Unconditional Asset Pricing Test and the Role of Firm
Size as an Instrumental Variable for Risk, Journal of Finance, 43, 309-325.
Chung, K.H., Charoenwong, C., Autumn 1991, Investment Options, Assets in Place, and the Risk
of Stocks, Financial Management, 21-33.
Daniel, Kent and Sheridan Titman, 1997, Evidence on the Characteristics of Cross Sectional Variation in Stock Returns, Journal of Finance, 52, 1-33.
Davis, James L., Eugene F. Fama and Kenneth R. French, 1992, The Cross-Section of Expected
Stock Returns, Journal of Finance, 47, 427-465.
De Bondt, W.F.M. and Robert Thaler, 1985, Does the Stock Market Overreact?, Journal of Finance
40, 793-805.
De Bondt, W.F.M. and Robert Thaler, 1987, Further evidence on investor overreaction and stock
market seasonality, Journal of Finance 42, 557-581.
Elton, Edwin J., 1999, Presidential Address: Expected Return, Realized Return, and Asset Pricing
Tests, Journal of Finance, 54, 1199-1220.
Fama, Eugene F. and Kenneth R. French, 1992, The Cross-Section of Expected Stock Returns,
Journal of Finance, 47, 427-465.
Fama, Eugene F. and Kenneth R. French, 1993, Common Risk Factors in the Returns on Stocks
and Bonds, Journal of Financial Economics, 33, 3-56.
Fama, Eugene F. and Kenneth R. French, 1995, Size and Book-to-Market Factors in Earnings and
Returns, Journal of Finance, 50, 131-155.
Fama, Eugene F. and Kenneth R. French, 1996, Multifactor Explanations of Asset Pricing Anomolies,
Journal of Finance, 51, 55-87.
Fama, Eugene F. and James D. MacBeth, 1973, Risk, Return and Equilibrium: Empirical Tests,
Journal of Political Economy, 81, 607-636.
Ferson, Wayne E. and Campbell R. Harvey, 1991, The Variation of Economic Risk Premiums,
Journal of Political Economy , 99, 385-415.
Ferson, Wayne E. and Campbell R. Harvey, 1999, Conditioning Variables and the Cross-Section of
18
Stock Returns, Journal of Finance, 54, 1325-1360.
Ferson, Wayne E. and Robert Korajczyk, 1995, Do Arbitrage Pricing Models Explain the Predictability of Stock Returns?, Journal of Business, 68, 309-349.
Griliches, Zvi, 1974, Errors in Variables and Other Unobservables, Econometrica, 42, 971-998.
Handa, Puneet, S.P. Kothari and Charles Wasley, 1989, The Relation Between the Return Intervals
and Betas: Implications for the Size Effect, Journal of Financial Economics, 23, 79-100.
Handa, Puneet, S.P. Kothari and Charles Wasley, 1993, Sensitivity of Multivariate Tests of the
Capital Asset Pricing Model to the Return Measurement Interval, Journal of Finance, 48, 15431551.
Huang, Chi-fu and Robert H. Litzenberger, 1988, Foundations for Financial Economics, Prentice
Hall.
Jacquier, Eric, Sheridan Titman and Atakan Yalçın, 2003, Growth Opportunities and Assets in
Place: Implications for Equity Betas, Working Paper, Boston College.
Jagannathan, Ravi and Zhenwu Wang, 1996, The Conditional CAPM and the Cross-Section of
Expected Returns, Journal of Finance, 51, 3-54.
Jagannathan, Ravi and Zhenwu Wang, 1998, Asymptotic Theory for Estimating Beta Pricing
Models using Cross Sectional Regressions, Journal of Finance, 53, 1285-1309.
Kim, Dongcheol, 1995, The Errors in the Variables Problem in the Cross-Section of Expected Stock
Returns, Journal of Finance, 50, 1605-1634.
Kim, Dongcheol, 1997, A Reexamination of Firm Size, Book-to-Market, and Earnings Price in
the Cross-Section of Expected Stock Returns, Journal of Financial and Quantitative Analysis, 32,
463-489.
Kester, W.C., 1984, Today’s Options for Tomorrow’s Growth, Harvard Business Review, MarchApril, 153-160.
Kester, W.C., 1986, An Options Approach to Corporate Finance, in Edward I. Altman, ed.: Handbook of Corporate Finance(New York: Wiley), 5.1-5.35.
Kothari, S.P., Jay Shanken and Richard G. Sloan, 1995, Another Look at the Cross-Section of
Expected Stock Returns, Journal of Finance, 50, 185-224.
Leamer, Edward E., 1987, Errors in the Variables in Linear systems, Econometrica, 55, 893-909.
Levhari, David and Haim Levy, 1973, The Capital Asset Pricing Model and the Investment Horizon,
The Review of Economics and Statistics, 59, 92-104.
Levi, Maurice D., 1973, Errors in the Variables Bias in the Presence of Correctly Measured Variables, Econometrica, 41, 985-986.
Litzenberger, Robert H. and Krishna Ramaswamy, 1979, The Effects of Personal Taxes and Dividends on Capital Asset Prices, Journal of Financial Economics, 7, 163-195.
Louis, K.C. and Josef Lakonishok, 1992, Robust Measurement of Beta Risk , Journal of Financial
and Quantitative Analysis, 27, 265-282.
McCallum, B.T., 1972, Relative Asymptotic Bias from Errors of Omission and Measurement,
Econometrica, 40, 757-758.
Myers, S., 1977, Determinants of Corporate Borrowing, Journal of Financial Economics 5, 147-175.
19
Pindyck, R., 1988, Irreversible Investment, Capacity Choice, and the Value of the Firm, American
Economic Review 78, 969-985.
Raymond, Kan and Chu Zhang, 1999, Two-Pass Tests of Asset Pricing Models with Useless Factors,
Journal of Finance, 54, 203-235.
Reinganum, Marc R., 1981, Misspecification of Capital Asset Pricing: Empirical Anomalies Based
on Earning Yield and Market Value , Journal of Financial Economics, 9, 19-46.
Rosenberg, B. and McKibben, W., 1973, The Prediction of Systematic and Specific Risk in Common
Stocks, Journal of Financial and Quantitative Analysis, 317-333.
Rosenberg, B., 1974, Extra-Market Components of Covariance in Security Returns, Journal of
Financial and Quantitative Analysis, 263-274.
Rosenberg, B., 1984, Prediction of Common Stock Investment Risk, Journal of Portfolio Management Fall, 44-53.
Rosenberg, B., 1985, Prediction of Common Stock Betas, Journal of Portfolio Management Winter,
5-14.
Shanken, Jay, 1992, On the Estimation of Beta Pricing Models, Review of Financial Studies, 5,
1-34.
Skinner, D.J., 1993, The Investment Opportunity Set and Accounting Procedure Choice, Journal
of Accounting and Economics 16, 407-445.
20
Table I: Characteristics of size / beta portfolios
Every June from 1963 to 1998, one-hundred equally weighted size-beta portfolios are formed by
first allocating individual stocks to ten groups along their equity market capitalization, and second,
subdividing each group into ten groups along estimated market betas. The equal weighted monthly
returns of these portfolios are then tracked for the following year from July to next June. We report,
for each portfolio, its estimated beta, average monthly excess return and size over the 36 years (432
months)
A: Estimated betas
βL
β2
β3
β4
β5
β6
β7
β8
β9
βH
0.78
0.89
0.93
0.96
0.97
1.02
0.99
0.96
0.93
0.87
0.93
1.04
1.07
1.09
1.10
1.14
1.10
1.07
1.04
0.95
1.07
1.18
1.21
1.24
1.23
1.26
1.21
1.18
1.13
1.04
1.24
1.34
1.35
1.38
1.38
1.39
1.34
1.30
1.23
1.13
1.44
1.53
1.52
1.55
1.54
1.55
1.48
1.45
1.37
1.23
1.71
1.76
1.75
1.79
1.76
1.76
1.70
1.66
1.54
1.36
2.27
2.29
2.22
2.24
2.19
2.17
2.12
2.04
1.88
1.65
β5
β6
β7
β8
β9
βH
1.56
0.97
0.94
0.85
0.83
1.05
0.72
0.70
0.69
0.56
1.81
1.00
0.95
0.84
0.95
0.81
0.63
0.59
0.72
0.81
1.80
1.11
0.74
0.96
0.91
0.80
0.67
0.53
0.62
0.45
1.97
1.05
0.76
0.66
0.78
0.67
0.79
0.55
0.60
0.43
1.73
0.83
0.85
0.54
0.70
0.73
0.59
0.59
0.54
0.63
2.08
1.03
0.70
0.68
0.43
0.64
0.68
0.40
0.31
0.27
C: Market capitalization (ln(ME))
βL
β2
β3
β4
β5
β6
β7
β8
β9
βH
1.58
2.51
3.13
3.66
4.16
4.71
5.29
5.96
6.73
8.36
1.57
2.52
3.14
3.65
4.15
4.71
5.27
5.94
6.73
8.30
1.59
2.51
3.13
3.65
4.16
4.69
5.26
5.94
6.72
8.26
1.60
2.53
3.13
3.65
4.16
4.70
5.28
5.94
6.72
8.14
1.58
2.52
3.14
3.65
4.15
4.70
5.29
5.90
6.70
7.96
Small
Size-2
Size-3
Size-4
Size-5
Size-6
Size-7
Size-8
Size-9
Large
0.05
0.16
0.23
0.27
0.35
0.42
0.41
0.44
0.43
0.45
0.42
0.55
0.59
0.63
0.69
0.72
0.72
0.68
0.65
0.64
0.61
0.74
0.78
0.80
0.84
0.88
0.87
0.84
0.81
0.77
B: % Monthly Excess Returns
βL
β2
β3
β4
Small
Size-2
Size-3
Size-4
Size-5
Size-6
Size-7
Size-8
Size-9
Large
Small
Size-2
Size-3
Size-4
Size-5
Size-6
Size-7
Size-8
Size-9
Large
1.34
0.92
1.07
0.61
0.52
0.74
0.76
0.92
0.56
0.42
1.47
2.50
3.11
3.65
4.13
4.71
5.26
5.94
6.74
8.60
1.49
0.97
1.07
0.67
0.59
0.66
0.82
0.76
0.59
0.56
1.52
2.50
3.12
3.66
4.16
4.70
5.28
5.94
6.73
8.65
1.37
1.00
1.07
0.98
0.91
0.69
0.85
0.73
0.66
0.73
1.54
2.50
3.14
3.65
4.17
4.69
5.28
5.95
6.72
8.73
1.59
0.96
1.03
0.73
0.64
0.73
0.64
0.75
0.78
0.51
1.56
2.49
3.13
3.66
4.17
4.70
5.27
5.95
6.73
8.63
1.55
2.50
3.13
3.65
4.17
4.70
5.28
5.96
6.73
8.53
21
Table II: Measurement errors in betas for size / beta portfolios
One-hundred equally weighted size-beta portfolios are formed annually as indicated in table I. The
monthly returns of these stock portfolios are then computed for the sixty-month formation period,
and portfolio betas are re-estimated from the market model regression:
rpt − rf t = αp + βp ∗ (rmt − rf t ) + pt ,
t = 1, ..., 60,
where rpt is the monthly return on portfolio p, rf t is the monthly risk-free rate, and rmt is the
monthly return on the value weighted CRSP index. Panel A reports the average OLS standard error
of βp from the above regression, and panel B reports its time-series standard deviation.
A: Average OLS standard errors of estimated betas
βL
β2
β3
β4
β5
β6
β7
β8
β9
βH
0.203
0.184
0.166
0.154
0.134
0.122
0.111
0.098
0.081
0.066
0.211
0.195
0.174
0.167
0.147
0.137
0.118
0.101
0.087
0.066
0.230
0.210
0.195
0.178
0.152
0.149
0.133
0.120
0.094
0.075
0.273
0.248
0.228
0.215
0.201
0.184
0.172
0.155
0.129
0.111
B: Time-series standard deviations of estimated betas
βL
β2
β3
β4
β5
β6
β7
β8
β9
βH
0.302
0.229
0.199
0.176
0.153
0.151
0.138
0.087
0.070
0.072
0.269
0.228
0.203
0.183
0.174
0.174
0.132
0.117
0.093
0.084
0.289
0.273
0.272
0.244
0.248
0.263
0.220
0.218
0.161
0.158
Small
Size-2
Size-3
Size-4
Size-5
Size-6
Size-7
Size-8
Size-9
Large
Small
Size-2
Size-3
Size-4
Size-5
Size-6
Size-7
Size-8
Size-9
Large
0.246
0.159
0.145
0.137
0.105
0.098
0.098
0.082
0.081
0.077
0.624
0.471
0.385
0.320
0.272
0.218
0.218
0.168
0.130
0.142
0.172
0.161
0.129
0.118
0.109
0.095
0.090
0.076
0.070
0.071
0.415
0.327
0.276
0.258
0.209
0.175
0.154
0.137
0.116
0.129
0.171
0.151
0.136
0.131
0.110
0.101
0.091
0.078
0.071
0.058
0.382
0.296
0.260
0.232
0.192
0.168
0.144
0.122
0.100
0.097
0.185
0.158
0.144
0.127
0.117
0.107
0.095
0.082
0.070
0.060
0.358
0.271
0.244
0.219
0.176
0.160
0.133
0.108
0.083
0.080
0.183
0.165
0.149
0.137
0.124
0.115
0.094
0.086
0.072
0.060
0.336
0.257
0.234
0.213
0.172
0.148
0.129
0.091
0.064
0.063
22
0.188
0.170
0.154
0.139
0.132
0.115
0.100
0.093
0.074
0.058
0.326
0.249
0.220
0.203
0.175
0.138
0.128
0.086
0.053
0.063
0.314
0.238
0.208
0.183
0.156
0.140
0.137
0.087
0.053
0.065
Table III: Characteristics of size / fitted beta portfolios
Every June from 1963 to 1998, one-hundred equally weighted size-beta portfolios are formed by
first allocating individual stocks to ten groups along their equity market capitalization, and second,
subdividing each group into ten groups along fitted betas. The equal weighted monthly returns of
these portfolios are then tracked for the following year from July to next June. Portfolio estimated
betas are calculated as indicated in table II. Individual security fitted betas are formed by projecting
estimated betas on growth opportunity proxies. The portfolio fitted beta is calculated as the simple
average of individual security fitted betas in that portfolio. We report, for each portfolio, its fitted
beta, average monthly excess return and estimated beta over the 36 years (432 months).
A: Fitted betas
βL
β2
Small
Size-2
Size-3
Size-4
Size-5
Size-6
Size-7
Size-8
Size-9
Large
0.81
0.80
0.77
0.77
0.76
0.76
0.76
0.75
0.72
0.73
1.00
0.98
0.96
0.95
0.93
0.94
0.94
0.91
0.87
0.89
β3
β4
β5
β6
β7
β8
β9
βH
1.10
1.07
1.05
1.04
1.02
1.02
1.01
1.00
0.97
0.99
1.17
1.14
1.11
1.09
1.08
1.08
1.07
1.05
1.03
1.05
1.22
1.19
1.17
1.15
1.13
1.13
1.11
1.11
1.08
1.09
1.26
1.24
1.22
1.20
1.18
1.18
1.16
1.15
1.13
1.13
1.30
1.28
1.27
1.25
1.23
1.22
1.21
1.19
1.17
1.16
1.34
1.32
1.32
1.30
1.29
1.27
1.26
1.24
1.22
1.20
1.40
1.38
1.38
1.35
1.35
1.34
1.32
1.30
1.28
1.26
1.52
1.50
1.49
1.47
1.47
1.45
1.42
1.41
1.38
1.36
β5
β6
β7
β8
β9
βH
B: % Monthly Excess Returns
βL
β2
β3
β4
Small
Size-2
Size-3
Size-4
Size-5
Size-6
Size-7
Size-8
Size-9
Large
1.55
1.19
1.27
0.93
0.95
0.89
0.74
0.61
0.70
0.63
1.48
1.37
1.09
0.95
0.95
0.94
0.81
0.80
0.73
0.56
1.92
0.97
1.06
0.82
0.99
0.79
0.79
0.68
0.50
0.62
1.87
0.87
0.90
0.98
0.76
0.87
0.78
0.94
0.64
0.66
1.74
0.98
0.95
0.57
0.74
0.77
0.80
0.78
0.45
0.60
1.64
0.54
0.69
0.64
0.57
0.50
0.67
0.49
0.72
0.44
1.38
0.95
0.54
0.56
0.30
0.46
0.52
0.50
0.65
0.38
2.01
1.00
0.66
0.38
0.39
0.68
0.45
0.34
0.41
0.48
C: Estimated betas
βL
β2
β3
β4
β5
β6
β7
β8
β9
βH
0.95
1.09
1.10
1.13
1.07
1.14
1.07
1.06
1.04
0.99
1.06
1.14
1.14
1.16
1.19
1.22
1.15
1.10
1.07
1.01
1.07
1.22
1.18
1.23
1.19
1.23
1.21
1.17
1.14
1.01
1.11
1.23
1.28
1.29
1.34
1.31
1.26
1.21
1.14
1.04
1.20
1.29
1.37
1.38
1.42
1.39
1.37
1.32
1.22
1.09
1.23
1.30
1.39
1.41
1.52
1.53
1.45
1.45
1.33
1.16
1.16
1.36
1.38
1.53
1.51
1.60
1.63
1.62
1.53
1.35
Small
Size-2
Size-3
Size-4
Size-5
Size-6
Size-7
Size-8
Size-9
Large
1.49
0.90
0.76
0.84
0.76
0.87
0.91
0.60
0.64
0.47
0.83
0.82
0.79
0.80
0.81
0.77
0.75
0.69
0.70
0.66
1.59
1.05
1.25
0.84
0.90
0.77
0.68
0.81
0.67
0.58
0.93
0.96
0.98
0.94
0.93
1.00
0.96
0.93
0.83
0.81
0.95
1.01
0.99
1.03
1.02
1.08
1.07
1.06
0.97
0.93
23
Table IV: Measurement errors in betas for size / fitted beta portfolios
One-hundred equally weighted size-fitted beta portfolios are formed annually and their estimated
and fitted betas are calculated as explained in table III. Panel A reports the time-series standard
deviations of the fitted portfolio betas, whereas panel C reports the time-series standard deviations
of the estimated (market) betas.
A: Time-series standard deviations of fitted betas
βL
β2
β3
β4
β5
β6
β7
β8
β9
βH
0.221
0.204
0.184
0.159
0.142
0.128
0.126
0.122
0.123
0.137
0.222
0.204
0.193
0.173
0.163
0.148
0.138
0.135
0.131
0.145
0.230
0.210
0.199
0.182
0.181
0.170
0.155
0.154
0.146
0.156
0.245
0.228
0.223
0.195
0.202
0.191
0.177
0.180
0.160
0.164
B: Time-series standard deviations of estimated betas
βL
β2
β3
β4
β5
β6
β7
β8
β9
βH
0.367
0.319
0.307
0.256
0.253
0.184
0.158
0.150
0.106
0.090
0.409
0.298
0.300
0.272
0.248
0.204
0.162
0.161
0.159
0.088
0.357
0.340
0.315
0.317
0.237
0.251
0.247
0.152
0.165
0.159
Small
Size-2
Size-3
Size-4
Size-5
Size-6
Size-7
Size-8
Size-9
Large
Small
Size-2
Size-3
Size-4
Size-5
Size-6
Size-7
Size-8
Size-9
Large
0.184
0.166
0.142
0.139
0.156
0.157
0.155
0.191
0.197
0.202
0.270
0.204
0.155
0.141
0.146
0.143
0.172
0.204
0.223
0.219
0.200
0.144
0.112
0.102
0.109
0.108
0.110
0.123
0.138
0.159
0.354
0.253
0.235
0.162
0.154
0.145
0.146
0.134
0.110
0.132
0.221
0.158
0.120
0.103
0.099
0.103
0.106
0.108
0.107
0.129
0.351
0.247
0.179
0.234
0.204
0.150
0.135
0.109
0.109
0.084
0.224
0.180
0.128
0.110
0.101
0.102
0.110
0.107
0.104
0.127
0.354
0.228
0.219
0.216
0.211
0.140
0.148
0.125
0.128
0.092
0.224
0.193
0.150
0.121
0.106
0.108
0.116
0.112
0.109
0.133
0.329
0.287
0.235
0.267
0.250
0.154
0.165
0.121
0.098
0.080
24
0.222
0.200
0.170
0.142
0.120
0.116
0.120
0.116
0.115
0.134
0.369
0.327
0.317
0.305
0.167
0.153
0.158
0.118
0.107
0.102
0.443
0.316
0.264
0.269
0.242
0.190
0.163
0.119
0.103
0.091
Table V: Price of (Beta) risk in the cross-section of stock returns
Risk premia are estimated from 432 monthly cross-sectional regressions (CSR) using monthly sizeestimated beta and size-fitted beta portfolio returns. For portfolio formation details please see tables I
and III. The average coefficients are reported as percentage per month. The dependent variable in Panel
A is the equal weighted monthly return series on the size-estimated beta portfolios. The independent
variables are a constant, estimated portfolio betas, and natural logarithms of average portfolio size and
book-to-market ratios. The dependent variable in Panel B is the equal weighted monthly return series
on the size-fitted beta portfolios. The independent variables are a constant, fitted portfolio betas, and
natural logarithms of average portfolio size and book-to-market ratios. We report results using two
different estimation techniques. First rows of the panels report the Fama-MacBeth OLS cross sectional
regression (OLS-CSR) results. The estimates γ̂t in each period are assumed to be the independent and
identical sampled values of the price of risk, γ. The time-series sample mean of γ̂t is the final estimate of
γ. The standard error of the estimate is calculated as the time-series standard deviation. Second rows
report the results from GLS cross-section regression, with covariance matrix restricted to be diagonal,
i.e., WLS-CRS. The diagonal elements are set equal to estimates of the variance of the monthly portfolio
returns in excess of the 1-month T-Bill rate. The final estimate of γ reported results from the efficient
weighted average of the γ̂t,W LS series, where weights for each date t are inversely proportional to their
variances (Efficient Weighted WLS-CSR). The standard error of the estimate is also calculated as a
weighted average of the time-series sample. t-ratios are reported below the average coefficients.
A: Size / beta portfolios
γ0 (int)
γ1 (mkt)
γ2 (size)
γ3 (BM)
Adj. R2
1.23
(4.17)
0.89
(2.51)
0.02
(0.15)
0.07
(0.56)
-0.07
(-1.81)
-0.05
(-1.29)
0.44
(3.07)
0.24
(2.46)
0.27
B: Size / fitted-beta portfolios
γ0 (int)
γ1 (mkt)
γ2 (size)
γ3 (BM)
Adj. R2
0.74
(2.11)
0.67
(1.89)
-0.05
(-1.30)
-0.02
(-0.65)
0.60
(4.91)
0.43
(4.61)
0.26
OLS-CSR
Efficient weighted WLS-CSR
OLS-CSR
Efficient weighted WLS-CSR
0.47
(1.33)
0.08
(0.19)
25
Table VI: Price of fitted-beta risk in the cross-section of stock returns
Risk premia are estimated from 384 monthly cross-sectional regressions (CSR) using monthly sizefitted beta portfolio returns. For portfolio formation details please see Table III. The average coefficients are reported as percentage per month. The dependent variable is the equal weighted monthly
return series on the size-fitted beta portfolios. The independent variables are a constant, fitted portfolio betas, and natural logarithms of average portfolio size and book-to-market ratios. We report
results using two different estimation techniques. First row reports the Fama-MacBeth OLS cross
sectional regression (FM OLS CSR) results. The estimates γ̂t in each period are assumed to be the
independent and identical sampled values of the price of risk, γ. The time-series sample mean is
regarded as the final estimate of γ. The standard error of the estimate is calculated as the timeseries standard deviation. Second row reports the results on the efficient weighted GLS estimation
procedure. The covariance matrix of the 100 portfolios used in the GLS estimation is calculated
from the daily returns over the 60 month portfolio formation period. The final estimate of γ is found
from the efficient weighted average of the γ̂t(GLS) series, where weights for each date t are inversely
proportional to their variances (Efficient Weighted GLS CSR). The standard error of the estimate
is also calculated as a weighted average of the time-series sample. t-ratios are reported below the
average coefficients.
Size / fitted-beta portfolios
OLS-CSR
Efficient weighted GLS-CSR
γ0 (int)
γ1 (mkt)
γ2 (size)
γ3 (BM)
Adj. R2
0.60
(1.62)
0.54
(1.76)
0.49
(1.30)
0.57
(2.08)
-0.04
(-0.78)
-0.04
(-1.12)
0.64
(4.77)
0.41
(3.76)
0.27
26
Table VII: Robustness to stock selection criteria
We analyze the sensitivity of risk premia estimates to the stock selection criteria. In this table, stock
with 24 or more valid monthly returns over the 60-month portfolio formation period are included
in the sample. In Panels A and B, risk premia are estimated from 432 monthly cross-sectional
regressions (CSR) using monthly size / beta and size / fitted-beta portfolio returns, respectively.
Estimation techniques used in Panels A and B are explained in Table V. In Panel C, risk premia
are estimated from 384 monthly cross-sectional regressions using monthly size / fitted-beta portfolio
returns, and daily returns to compute the covariance matrix used in the GLS. The estimation
technique used in Panel C is explained in Table VI.
A: Size / beta portfolios (432 cross-sectional regressions)
γ0 (int) γ1 (mkt) γ2 (size) γ3 (BM)
OLS-CSR
Efficient weighted WLS-CSR
1.20
(3.88)
0.88
(2.32)
0.04
(0.39)
0.16
(1.47)
-0.06
(-1.51)
-0.03
(-0.81)
0.76
(4.58)
0.29
(2.56)
B: Size / fitted-beta portfolios (432 cross-sectional regressions)
γ0 (int) γ1 (mkt) γ2 (size) γ3 (BM)
OLS-CSR
Efficient weighted WLS-CSR
0.34
(1.00)
-0.06
(-0.16)
0.88
(2.71)
1.17
(3.15)
-0.07
(-1.57)
-0.04
(-1.03)
0.69
(5.50)
0.47
(5.28)
C: Size / fitted-beta portfolios (384 cross-sectional regressions)
γ0 (int) γ1 (mkt) γ2 (size) γ3 (BM)
OLS-CSR
Efficient weighted GLS-CSR
0.35
(0.94)
0.47
(1.65)
0.73
(2.09)
0.73
(2.86)
27
-0.05
(-0.96)
-0.06
(-1.78)
0.78
(5.59)
0.50
(4.81)
Adj. R2
0.29
Adj. R2
0.28
Adj. R2
0.30
Table VIII: Robustness to sorting criterion, book-to-market ratio instead of size
We analyze the sensitivity of risk premia estimates to the portfolio formation procedure. We report
risk premia estimates using book-to-market equity ratio (BM) and beta sorted portfolios. In forming
this table, any stock with 24 or more valid monthly returns over the 60-month portfolio formation
period are included in the data sample. In Panels A and B, risk premia are estimated from 432
monthly cross-sectional regressions (CSR) using monthly BM-estimated beta and BM-fitted beta
portfolio returns, respectively. For estimation techniques used in Panels A and B please refer to
Table V. In Panel C, risk premia are estimated from 384 monthly cross-sectional regressions (CSR)
using monthly BM-fitted beta portfolio returns. For estimation techniques used in Panel C please
refer to Table VI.
A: BM / beta portfolios (432 cross-sectional regressions)
γ0 (int) γ1 (mkt) γ2 (size) γ3 (BM)
OLS-CSR
Efficient weighted WLS-CSR
1.25
(4.05)
1.44
(4.13)
-0.04
(-0.33)
0.00
(0.00)
-0.04
(-1.31)
-0.07
(-1.91)
0.35
(5.09)
0.17
(3.95)
B: BM / fitted-beta portfolios (432 cross-sectional regressions)
γ0 (int) γ1 (mkt) γ2 (size) γ3 (BM)
OLS-CSR
Efficient weighted WLS-CSR
0.32
(0.91)
-0.02
(-0.04)
0.63
(2.01)
0.82
(2.51)
-0.01
(-0.22)
-0.03
(-0.68)
0.43
(6.51)
0.13
(3.90)
C: BM / fitted-beta portfolios (384 cross-sectional regressions)
γ0 (int) γ1 (mkt) γ2 (size) γ3 (BM)
OLS-CSR
Efficient weighted GLS-CSR
0.39
(1.01)
0.25
(0.81)
0.42
(1.26)
0.62
(2.81)
28
0.01
(0.26)
-0.02
(-0.60)
0.47
(6.47)
0.28
(4.56)
Adj. R2
0.25
Adj. R2
0.25
Adj. R2
0.26