Description
Estimates of systematic risk or beta are an important determinant of the cost of capital.
The standard technique used to compile beta estimates is an ordinary least squares regression of stock
returns on market returns using four to five years of monthly data. This convention assumes that a
longer time series of data will not adequately capture risks associated with existing assets. This paper
seeks to address this issue
Accounting Research Journal
Bias, stability, and predictive ability in the measurement of systematic risk
Stephen Gray J ason Hall Drew Klease Alan McCrystal
Article information:
To cite this document:
Stephen Gray J ason Hall Drew Klease Alan McCrystal, (2009),"Bias, stability, and predictive ability in the
measurement of systematic risk", Accounting Research J ournal, Vol. 22 Iss 3 pp. 220 - 236
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Bias, stability, and predictive
ability in the measurement of
systematic risk
Stephen Gray and Jason Hall
UQ Business School, The University of Queensland, St Lucia, Australia
Drew Klease
Queensland Investment Corporation, Brisbane, Australia, and
Alan McCrystal
UQ Business School, The University of Queensland, St Lucia, Australia
Abstract
Purpose – Estimates of systematic risk or beta are an important determinant of the cost of capital.
The standard technique used to compile beta estimates is an ordinary least squares regression of stock
returns on market returns using four to ?ve years of monthly data. This convention assumes that a
longer time series of data will not adequately capture risks associated with existing assets. This paper
seeks to address this issue.
Design/methodology/approach – Each year from 1980 to 2004, equity betas are estimated for
1,717 Australian ?rms over periods of four to 45 years, and form equal value portfolios of high,
medium and low beta stocks. The paper compares expected returns – derived from the capital asset
pricing model (CAPM) and subsequent realised market returns – and actual returns over subsequent
annual and four-year periods.
Findings – The paper shows that the ability of beta estimates to predict future stock returns
systematically increases with the length of the estimation window and when the Vasicek bias
correction is applied. However, estimation error is insigni?cantly different from that associated with a
na? ¨ve assumption that beta equals one for all stocks.
Research limitations/implications – The implication is that using all available returns data in
beta estimation, along with the Vasicek bias correction, reduces the imprecision of expected returns
estimates derived from the CAPM. A limitation of the method is the use of conditional realised returns
as a proxy for expected returns, given that it is not possible directly to observe expected returns
incorporated into share prices.
Originality/value – The paper contributes to the understanding of corporate ?nance practitioners
and academics, who routinely use beta estimates derived from ordinary least squares regression.
Keywords Beta factor, Capital asset pricing model, Australia, United States of America
Paper type Research paper
Introduction
The Capital Asset Pricing Model (CAPM) is the pre-dominant framework for
estimating the cost of equity capital[1]. Yet considerable debate remains as to the
appropriate parameter inputs, especially regarding beta estimation techniques.
The standard technique to estimate a ?rm’s beta is an ordinary least squares (OLS)
regression using four or ?ve years of monthly data. This method is so well accepted
that other estimates have been evaluated on the basis of how close they are to the OLS
estimate (Klemkosky and Martin, 1975; Elton et al., 1978; Eubank and Zumwalt, 1979).
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/1030-9616.htm
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Accounting Research Journal
Vol. 22 No. 3, 2009
pp. 220-236
qEmerald Group Publishing Limited
1030-9616
DOI 10.1108/10309610911005563
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Recent research by Hooper et al. (2005) ?nds that an autoregressive model with three
lags, estimated on 20 years of quarterly data, minimises the error associated with
forecasting the next period’s beta.
Other research examines the association between contemporaneous beta estimates,
computed from different techniques. For example, Bowman and Graves (2004) evaluate
the usefulness of comparable ?rm analysis by testing the association between the beta
estimates of Australian ?rms and the mean beta estimates of comparable ?rms in the
USA. Their conclusion, that comparable ?rm beta estimates under-estimate actual
betas (emphasis added), could easily be interpreted in the reverse manner. We could
just as easily conclude that estimating the systematic risk of the ?rm using its own
time-series of returns results in an over-estimate of systematic risk, compared to the
actual beta estimated as the mean of a sample of comparable ?rms. In other words, we
have two estimates of the systematic risk of sample ?rms, but no means of
distinguishing which is most relevant for valuation.
Systematic risk is not directly observable using any technique; it can only be
estimated. Thus, it is not possible to evaluate the merits of one technique purely on the
basis of its ability to predict the beta derived from an alternative technique. We can
however, evaluate the ability of alternative estimation techniques to predict future
stock returns, when incorporated into the CAPM, an approach advocated by Rosenberg
and McKibbin (1973). In this paper, we examine whether beta estimates generated from
short estimation windows have a relatively greater ability to predict future stock
returns, compared to beta estimates derived from longer estimation periods and with
the application of the Vasicek (1973) bias correction.
It is puzzling that the short window, OLS technique has become standard practice.
In almost all ?nance research, the convention is to rely upon all available data, unless
there is persuasive evidence that the data is unreliable for addressing the research
question. In contrast, when estimating systematic risk, researchers ignore returns data
outside of a short estimation period unless presented with persuasive evidence that it
should be included. Researchers implicitly assume that there is an optimal estimation
period, which involves a trade-off between the timeliness of a short returns window
versus the reduction in standard error associated with a long returns window. This
trade-off view is perhaps motivated by Blume’s (1975) rationalisation that
mean-reversion in beta estimates is due to the changing nature of management’s
risk preferences in their investment decisions (p. 795).
Our results show that the ability of beta estimates to predict future stock returns
systematically increases with the length of the estimation window, the implication
being that all available returns data should be used in beta estimation. In addition, for
all estimation periods, there is an increase in returns predictability when the Vasicek
adjustment is applied. However, there is no signi?cant difference in estimation error
associated with long-window, Vasicek-adjusted beta estimates and the na? ¨ve
assumption that all ?rms have an equity beta equal to one. Our conclusion is that
all beta estimates from historical returns data should be treated with caution,
especially those, which are substantially different from one. Nevertheless, if regression
analysis on historical returns is used to measure systematic risk, we ?nd no reason to
exclude returns outside an arbitrary window, and additional bene?ts result from the
Vasicek bias correction.
Bias, stability,
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221
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Beta adjustment techniques
Vasicek adjustment
Data providers such as the Centre for Research in Finance (CRIF), Bloomberg and
Datastream provide beta estimates generated from an OLS regression of stock returns
on market returns from four to ?ve years of monthly data. CRIF will supply a beta
estimate if there is a minimum of 24 months of returns data within the most recent
four-year period.
The default beta estimates of Bloomberg and Datastream (b
adjusted
) both make
adjustments to the OLS estimate (b
OLS
), placing some weight (1 – w) on a prior
estimate of one, according to the following equation:
b
adjusted
¼ w £ b
OLS
þ 1 2w ð Þ £ 1 ð1Þ
In the case of Bloomberg, the default weight placed on the prior estimate of one is a
constant one-third. In the case of Datastream, the weight varies according to the
precision of the beta estimate. These adjustments are speci?c applications of the
Vasicek adjustment to beta computations. Vasicek (1973) recognised that OLS beta
estimates are biased in the sense that the more the sample estimate deviates from an
unconditional expectation, the greater the chance that the estimate results from
sampling error. Using Bayesian techniques, Vasicek showed that an unbiased beta
estimate is given by the following equation:
b
Vasicek
¼
s
2
prior
s
2
prior
þs
2
estimate
£ b
estimate
þ 1 2
s
2
prior
s
2
prior
þs
2
estimate
!
£ b
prior
ð2Þ
where:
.
b
estimate
and s
estimate
¼ the beta estimate and its standard error; and
.
b
prior
and s
prior
¼ the prior expectations of beta and its standard error.
This result is conditional upon knowing the population mean and standard error. In
reality we cannot directly observe these parameters, but they can be estimated. In our
study, we assume a prior beta estimate of one, as opposed to an industry mean for
example, under the reasoning that beta estimation is an incremental process. Prior to
any company- or industry-speci?c information being available, one is the unbiased
beta estimate for all ?rms. Subsequent to an OLS regression of stock returns against
market returns, and without any additional company- or industry-speci?c information,
the unconditional expectation is re?ned.
Filtering adjustments
There are also instances in which practitioners use ?ltering techniques to exclude
sample beta estimates considered to be so unreliable that they should carry zero weight
in comparable ?rm analysis. One such ?ltering adjustment is to exclude beta estimates
where the regression has little explanatory power (low R-squared statistic) as adopted
by Bowman and Bush (2005). Another adjustment is to eliminate beta estimates from
the sample with t-statistics below two relative to a null hypothesis of zero.
In our descriptive statistics we illustrate the substantial upward bias associated
with these ?ltering techniques, and consequently do not evaluate their predictive
ability. Consider the decision rule whereby we eliminate observations where the
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R-squared statistic is less than 10 per cent. We have the following criteria for selection,
whereby we are more likely to retain high beta estimates and discard low beta
estimates:
R
2
. 0:10
b
i
¼
COV r
i
;r
m ð Þ
s
2
m
¼
r
i;m
s
i
s
m
s
2
m
¼
r
i;m
s
i
s
m
r
i;m
¼ b
i
£
s
m
s
i
r
2
i;m
¼ R
2
¼ b
2
i
£
s
2
m
s
2
i
. 0:10
ð3Þ
where:
b
i
= the OLS beta estimate for stock i;
s
i
= the standard deviation of returns for stock i;
s
m
= the standard deviation of returns on the market portfolio; and
r
i,m
= the correlation between returns on stock i and the market portfolio.
The series of equations presented previously shows that, for stocks with the same
volatility, the R-squared ?lter is more likely to retain stocks with high beta estimates.
The t-statistic ?lter for excluding beta estimates also has an upward bias. Suppose the
decision-rule was to retain only stocks whose beta estimate was at least two times the
standard error of the estimate, according to the following criteria:
t
i
. 2
t
i
¼
b
i
s
error;i
. 2
: ð4Þ
For stocks with the same standard error, we are more likely to retain the stocks with
high beta estimates. The problem with the adoption of this technique in practice is that
the t-statistic is computed relative to a mean estimate of zero. An unbiased ?lter would
be to compute a t-statistic relative to a mean estimate of one, and discard beta estimates
where the absolute value of the t-statistic is greater than the cut-off point. However,
this alternative is less useful than the Vasicek bias correction. In comparable ?rm
analysis, the Vasicek adjustment will place greater aggregate weight on beta estimates
with low standard errors. Adopting a t-statistic ?lter places zero weight on
observations outside an arbitrary cut-off and equal weight on observations within the
arbitrary cut-off.
Research method
We estimated the equity beta for stocks in the CRIF database of Australian listed
companies at 31 December each year from 1979-2003. Estimates end in 2003 because
we evaluate returns predictability over a four-year forecast window and our returns
data ends in 2007. We computed the beta estimates by performing OLS regressions of
monthly stock returns versus market returns, where the estimation window ranged
from four to 45 years. For beta estimates relating to 1979, the longest estimation
Bias, stability,
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window is 21 years, which increases to 45 years for the 2003 estimates. Provided a
stock has 24 months of returns data in the estimation window and 48 months of returns
data in the four subsequent years, its beta estimate is retained for estimation periods of
four years or more. For example, if stock A is listed for just four years, we retain the
same beta estimate for longer estimation windows of ?ve, six years and so on. In
contrast, if stock B is listed for ten years, we compute different beta estimates over four
to ten years, and then retain the ten-year beta estimate for all future estimation windows.
We then computed Vasicek-adjusted beta estimates. These estimates are a weighted
average of the OLS estimate and a prior estimate of one, where the weights are
determined by the standard error of the OLS estimate and the standard deviation of
OLS estimates across the entire sample in each period. The equation for our
Vasicek-adjusted beta estimates is as follows:
b
Vasicek
¼
s
2
market
s
2
market
þs
2
OLS
£ b
OLS
þ 1 2
s
2
market
s
2
market
þs
2
OLS
!
£ 1 ð5Þ
where:
b
OLS
¼ the beta estimate from the OLS regression of stock returns on market
returns.
s
OLS
¼ the standard error of the beta estimate from the OLS regression.
s
market
¼ the standard deviation of beta estimates across the sample ?rms in each
period.
We examined whether these beta estimates had any association with future stock
returns over forecast horizons of one and four years. First, we formed low, medium and
high beta portfolios of equal value by ranking stocks from the minimum to maximum
four-year OLS beta estimate in each time -eriod. If beta estimates have an association
with future returns in the anticipated direction, we should observe low beta portfolios
outperform the market during periods when market returns are below the risk-free
rate, and high beta portfolios outperform the market when market returns are above
the risk-free rate. Second, consistent with Rosenberg and McKibben (1973)[2], we
computed the expected return on each portfolio implied by the CAPM, conditional upon
the realised performance of the market, according to the following equation:
E r
p;t;i;j
À Á
¼ r
f ;t
þb
p;t;i;j
r
m;t
2r
f ;t
À Á
ð6Þ
where:
E(r
p,t,i, j
) ¼ the expected return on the portfolio in period t according to beta
estimation period i and estimation technique j (OLS, Vasicek-adjusted
or na? ¨ve assumption of one);
r
f,t
¼ the risk-free rate at the start of the forecast period, as measured by the
yield-to-maturity on three month Treasury notes;
r
m,t
¼ the realised return on the equity market during the forecast period, as
measured by the market capitalisation weighted return on stocks in
the CRIF database; and
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b
p,t,i, j
¼ the estimated portfolio beta according to estimation period i and
estimation technique j (OLS, Vasicek-adjusted or na? ¨ve assumption of
one), computed as a market capitalisation weighted average of the
individual stocks’ beta estimates.
We conducted our analysis on conditional expected returns for the following reason.
Ideally, we would perform this evaluation with reference to an alternative direct
estimate of expected returns, something, which cannot be directly observed. What we
can observe is realised returns, for which we need an appropriate benchmark. That
benchmark is the conditional expected returns model presented above.
In practice, we would observe beta estimates being incorporated into the CAPM
with an expected market risk premium, which has been empirically estimated at
around 4-6 per cent, and which ?uctuates according to economic conditions[3]. One
alternative would be to compare the cost of capital implied by this assumption with
subsequent realised returns. This presents two problems.
First, it expands the scope of the research question to the joint estimation of both an
appropriate beta estimate and market risk premium. Our research question is whether
estimates of equity beta from historical data provide a reliable indicator of future
exposure to market movements, and therefore the systematic risk of a ?rm or project at
the end of the estimation period. Another way to phrase our research question is to ask,
“If we had perfect foresight of market returns, would OLS beta estimates provide a
reliable measure of the ?rm’s opportunity cost of funds?”. If this cost of capital estimate
is unreliable assuming perfect knowledge of subsequent market returns, it must be
unreliable in the real-world case where the market risk premium is estimated.
Second, even if beta estimates were a perfect predictor of market risk in a
subsequent time-period, the noise associated with this alternative approach is so large
that our tests would lack power. We attempted this alternative approach and found
that expected returns derived from this approach have no ability to explain the
variation in realised returns for both individual stocks and portfolios, and over holding
periods of one and four years.
We use portfolios in our analysis in order to minimise the impact of
company-speci?c information on stock returns, thereby isolating the impact of
market returns[4]. In our dataset, the cross-sectional standard deviation of annual
stock returns is 108 per cent, which is ?ve times the 21 per cent volatility of annual
returns on the broader market. In contrast, the cross-sectional standard deviation of
annual returns on our portfolios is 22 per cent, through the reduction of
company-speci?c risk. Our portfolios are value-weighted because small stocks have
relatively greater volatility than large stocks so require more stocks in the portfolio to
achieve the same degree of diversi?cation. For robustness, we repeat our regression
analysis at the ?rm level, on both an equal and market capitalisation weighted basis.
For each estimation technique and period, there are 75 portfolios of one-year returns
and overlapping four-year returns. This is the product of 25 periods of returns data
(comprising annual returns from 1980-2004, and four-year returns from 1980-2007) and
three portfolios of high, medium and low beta stocks. We also report results for 21
portfolios of non-overlapping four-year returns (ending each four years from 1983 to
2007). Our four-year returns are computed according to the following equation:
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Four-year returns ¼
Y
4
t¼1
1 þ r
t
ð Þ 21 ð7Þ
where r
t
¼ annual return on the portfolio or the market during year t.
We use two alternative techniques to measure the association between actual
portfolio returns and expected returns. First, we measure the difference between actual
and expected returns, and report the mean absolute error and square root of the mean
squared error. We perform paired t-tests to determine whether there is a statistically
signi?cant reduction in estimation error associated with the Vasicek adjustment and
with increasing estimation windows. Second, we perform a regression of actual returns
versus expected returns. This allows us to evaluate bias in expected returns – which is
present if the regression coef?cient on expected returns is signi?cantly different from
one – and predictive ability as measured by the adjusted R-squared statistic.
Data
Beta estimates
Our sample consists of 384,240 beta estimates, which is the product of 1,717 unique
?rms and an average 224 beta estimates for each ?rm. There are 12,031 beta estimates
derived from four years of historical returns which represents an average of seven
four-year estimates for each ?rm[5]. As shown in Panel A of Table I, the variation in
beta estimates decreases signi?cantly under longer estimation periods and with the
application of the Vasicek adjustment. When a four-year estimation window is used,
the standard deviation of beta estimates is 1.14, which decreases to 0.81 when all
returns data is used and to 0.51 when we also incorporate the Vasicek adjustment.
We illustrate the bias in beta estimates resulting from applying ?lters, which
required either a t-statistic of at least two, computed relative to a null hypothesis of
zero, or an R-squared statistic of at least 10 per cent. For these two ?lters, the mean beta
estimates computed over four years of returns are 1.57 and 1.58, respectively, due to the
elimination of 51 and 61 per cent of observations. Even when all available data is used
to compute beta estimates, the bias is present. In the case of the t-statistic ?lter, the
mean estimate is 1.23, despite the elimination of just 26 per cent of observations. In the
case of the R-squared ?lter, the mean estimate is 1.33.
We also observe that stocks with high or low beta estimates are relatively small and
exhibit high returns volatility. The average market capitalisation of medium beta
stocks is $1.1 billion, compared to $400 million for low and high beta stocks. High beta
stocks had an equally weighted standard deviation of annual returns of 133 per cent,
compared to 89 per cent for low beta stocks and 84 per cent for medium beta stocks. On
a market capitalisation weighted basis, the standard deviation of annual returns for
high beta stocks was 48 per cent, compared to 38 per cent for low beta stocks and 36
per cent for medium beta stocks. In aggregate, the descriptive statistics suggest that
volatility of returns is associated with market capitalisation, rather than beta
estimates, and that extreme beta estimates are more prevalent amongst small ?rms.
Cost of capital estimates
Beta estimates are primarily used to estimate the required return to equity holders in a
?rmor project. In Table II we compare the distributionof annual cost of capital estimates
whichwouldresult fromour alternative betaestimates, assumingamarket riskpremium
of 6 per cent. The variation in cost of capital estimates across the sample declines as more
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Table I.
Descriptive statistics
Bias, stability,
and predictive
ability
227
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data are usedinestimationandwiththe application of the Vasicekadjustment. The issue
is whether these adjustments merelyreduce noise inthe estimationof the cost of capital or
whether they mask true differences in systematic risk across ?rms.
The distribution of estimates, which are conditional upon realised market returns is
much greater. Despite this increase in variance, in the results which follow, we observe
that conditional expected returns have the ability to explain realised returns on
portfolios and individual stocks.
Results
Portfolio analysis
Our evaluation of one-year returns is presented in Table III. In the left-hand section of
Panel A we present the mean absolute error of actual versus expected portfolio returns
derived from OLS beta estimates, Vasicek-adjusted estimates and the na? ¨ve
assumption that equity beta equals one. The mean absolute error implied by OLS
estimates is 10.04 per cent, which decreases to 9.59 per cent if the Vasicek adjustment is
applied and 7.56 per cent assuming beta equals one. These differences are signi?cant at
levels of 0.02 per cent and 1 per cent, respectively. We also observe a reduction in the
mean absolute error as we increase the estimation period to ten, 20 and up to 45 years.
For OLS estimates, the mean absolute error decreases to 8.13 per cent, while the mean
absolute error associated with the Vasicek adjustment decreases to 7.99 per cent, and is
signi?cantly lower in every estimation period.
Mean absolute errors for increasing estimation periods are illustrated in Figure 1.
The ?gure illustrates the substantial reduction in estimation error associated with
increasing the estimation period and the consistently lower error attributed to the
Vasicek adjustment. It also shows that the mean absolute error is minimised by simply
assuming a beta estimate of one for all stocks. For long estimation windows this
difference is not statistically signi?cant.
We observe the same result when we examine squared errors rather than absolute
errors. The square root of the mean squared error is 13.81 per cent for the four-year
OLS estimates, which decreases to 13.12 per cent when the Vasicek adjustment is
applied, and to 10.68 per cent if we also incorporate all available returns information.
Again, the na? ¨ve assumption that beta equals one has the lowest estimation error, with
the square root of the mean squared error equal to 9.68 per cent.
Mean Std dev. Min. Tenth perc. Median Ninetieth perc. Max.
Risk-free rate 8.1 3.9 4.1 4.6 6.1 14.9 19.4
Cost of capital estimates under CAPM and assuming market risk premium ¼ 6 per cent
OLS four-year 15.0 7.7 251.7 7.1 13.9 24.6 136.4
OLS all-year 14.5 6.1 223.5 7.7 13.7 22.6 60.1
Vasicek four-year 14.1 5.3 23.8 8.0 13.4 21.7 34.1
Vasicek all-year 14.0 4.9 20.7 8.3 13.3 20.9 35.3
Beta ¼ 1 14.1 3.9 10.1 10.6 12.1 20.9 25.4
Expected returns conditional on realised market returns
OLS four-year 16.1 33.0 2326.3 211.8 11.3 48.4 485.0
OLS all-year 15.5 27.2 2174.7 210.2 11.5 45.4 326.9
Vasicek four-year 15.1 24.4 280.6 29.5 11.6 42.4 185.0
Vasicek all-year 14.8 22.6 262.2 29.0 11.8 41.6 165.4
Beta ¼ 1 14.9 20.0 216.5 29.5 12.3 42.2 64.9
Table II.
Unconditional versus
conditional cost of capital
estimates (%)
ARJ
22,3
228
D
o
w
n
l
o
a
d
e
d
b
y
P
O
N
D
I
C
H
E
R
R
Y
U
N
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R
S
I
T
Y
A
t
2
1
:
0
9
2
4
J
a
n
u
a
r
y
2
0
1
6
(
P
T
)
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e
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(
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Table III.
Estimation error in
annual returns across
high, medium and low
beta portfolios (%)
Bias, stability,
and predictive
ability
229
D
o
w
n
l
o
a
d
e
d
b
y
P
O
N
D
I
C
H
E
R
R
Y
U
N
I
V
E
R
S
I
T
Y
A
t
2
1
:
0
9
2
4
J
a
n
u
a
r
y
2
0
1
6
(
P
T
)
The results are clearly in favour of estimating equity beta using all available returns
data with the Vasicek adjustment, versus the standard approach of short window OLS
regression. The limitation of these results is that there is insuf?cient evidence that
Vasicek-adjusted beta estimates outperform the na? ¨ve assumption that beta equals one.
Hence, while the use of regression analysis to estimate beta is questionable, if it is to be
used, then the appropriate presumption is that all returns information is value-relevant
and a bias correction is useful. This presumption could be rebutted if, for example,
there is clear evidence of a change in the ?rm’s assets or leverage.
In Panel B we present the results of regression analysis of actual versus expected
returns across the 75 portfolios. Explanatory power increases with the length of the
estimation window and with application of the Vasicek adjustment, with the adjusted
R-squared statistic rising from 72 to 80 per cent. The coef?cient on expected returns
also increases towards one and the intercept systematically declines towards zero.
We repeated this analysis with reference to total returns computed over four years
subsequent to portfolio formation, resulting in 75 overlapping periods beginning in
1979 and ending in 2007. This analysis is presented in Table IV. The implications of
this analysis are identical to those from our analysis of annual returns. Incorporating
longer estimation windows signi?cantly reduces mean absolute error and mean
squared error, as does the application of the Vasicek adjustment. Regression
coef?cients on expected returns increase towards one and explanatory power improves
as the estimation window increases and the Vasicek adjustment is applied but the
market-return itself remains the better predictor of four-year returns, compared to
either beta estimation technique.
Finally, we performed the same analysis on seven non-overlapping four-year
periods ending in 2007, resulting in 21 observations, and present the results in Table V.
Figure 1.
Mean absolute annual
error according to beta
estimation period
ARJ
22,3
230
D
o
w
n
l
o
a
d
e
d
b
y
P
O
N
D
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C
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a
n
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a
r
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6
(
P
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)
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(
9
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r
c
e
p
t
(
%
)
(
p
-
v
a
l
u
e
v
s
¼
0
)
C
o
e
f
?
c
i
e
n
t
(
p
-
v
a
l
u
e
v
s
¼
1
)
A
d
j
R
2
(
%
)
O
L
S
4
4
2
.
3
(
0
.
2
)
0
.
4
3
3
(
1
)
2
4
4
-
1
0
2
7
.
3
(
9
)
0
.
6
4
1
(
1
8
)
4
1
4
-
2
0
2
7
.
3
(
1
1
)
0
.
6
5
4
(
2
3
)
3
9
A
l
l
(
4
-
4
5
)
2
6
.
8
(
1
2
)
0
.
6
5
9
(
2
4
)
4
0
V
a
s
i
c
e
k
4
3
9
.
7
(
1
)
0
.
4
7
1
(
2
)
2
6
4
-
1
0
2
5
.
4
(
1
2
)
0
.
6
6
9
(
2
2
)
4
3
4
-
2
0
2
5
.
5
(
1
4
)
0
.
6
8
1
(
2
7
)
4
1
A
l
l
(
4
-
4
5
)
2
5
.
0
(
1
5
)
0
.
6
8
7
(
2
8
)
4
1
B
e
t
a
¼
1
8
.
6
(
5
0
)
0
.
9
0
7
(
6
5
)
6
2
N
o
t
e
s
:
W
e
f
o
r
m
e
d
e
q
u
a
l
-
v
a
l
u
e
p
o
r
t
f
o
l
i
o
s
o
f
l
o
w
,
m
e
d
i
u
m
,
a
n
d
h
i
g
h
b
e
t
a
s
t
o
c
k
s
a
c
c
o
r
d
i
n
g
t
o
t
h
e
i
r
f
o
u
r
-
y
e
a
r
O
L
S
b
e
t
a
e
s
t
i
m
a
t
e
s
a
t
3
1
D
e
c
e
m
b
e
r
o
f
e
a
c
h
y
e
a
r
f
r
o
m
1
9
7
9
-
2
0
0
3
.
O
v
e
r
t
h
e
s
u
b
s
e
q
u
e
n
t
f
o
u
r
y
e
a
r
s
,
w
e
m
e
a
s
u
r
e
d
t
h
e
t
o
t
a
l
r
e
t
u
r
n
s
o
f
t
h
e
s
e
p
o
r
t
f
o
l
i
o
s
,
a
n
d
c
o
m
p
a
r
e
d
t
h
e
i
r
p
e
r
f
o
r
m
a
n
c
e
w
i
t
h
t
o
t
a
l
e
x
p
e
c
t
e
d
r
e
t
u
r
n
s
d
e
r
i
v
e
d
f
r
o
m
t
h
e
i
r
b
e
t
a
e
s
t
i
m
a
t
e
s
a
n
d
t
h
e
p
e
r
f
o
r
m
a
n
c
e
o
f
t
h
e
m
a
r
k
e
t
p
o
r
t
f
o
l
i
o
.
E
x
p
e
c
t
e
d
r
e
t
u
r
n
s
a
r
e
c
o
m
p
u
t
e
d
a
c
c
o
r
d
i
n
g
t
o
e
q
u
a
t
i
o
n
(
7
)
,
w
h
e
r
e
b
y
t
h
e
b
e
t
a
e
s
t
i
m
a
t
e
i
s
i
n
c
o
r
p
o
r
a
t
e
d
i
n
t
o
t
h
e
C
A
P
M
,
a
l
o
n
g
w
i
t
h
r
e
a
l
i
s
e
d
m
a
r
k
e
t
r
e
t
u
r
n
s
a
n
d
t
h
e
y
i
e
l
d
t
o
m
a
t
u
r
i
t
y
o
n
t
h
r
e
e
-
m
o
n
t
h
t
r
e
a
s
u
r
y
n
o
t
e
s
a
t
t
h
e
s
t
a
r
t
o
f
t
h
e
p
o
r
t
f
o
l
i
o
f
o
r
m
a
t
i
o
n
p
e
r
i
o
d
.
O
v
e
r
a
l
l
,
t
h
e
r
e
a
r
e
7
5
o
v
e
r
l
a
p
p
i
n
g
f
o
u
r
-
y
e
a
r
r
e
t
u
r
n
s
o
b
s
e
r
v
a
t
i
o
n
s
,
t
h
e
p
r
o
d
u
c
t
o
f
2
5
f
o
u
r
-
y
e
a
r
r
e
t
u
r
n
s
a
n
d
t
h
r
e
e
s
t
o
c
k
p
o
r
t
f
o
l
i
o
s
(
l
o
w
,
m
e
d
i
u
m
,
a
n
d
h
i
g
h
b
e
t
a
s
)
.
I
n
P
a
n
e
l
A
,
w
e
r
e
p
o
r
t
m
e
a
n
a
b
s
o
l
u
t
e
e
r
r
o
r
s
a
n
d
t
h
e
s
q
u
a
r
e
r
o
o
t
o
f
m
e
a
n
s
q
u
a
r
e
d
e
r
r
o
r
s
a
c
r
o
s
s
t
h
e
l
o
w
,
m
e
d
i
u
m
a
n
d
h
i
g
h
b
e
t
a
p
o
r
t
f
o
l
i
o
s
w
h
e
r
e
t
h
e
e
r
r
o
r
i
s
t
h
e
d
i
f
f
e
r
e
n
c
e
i
n
a
c
t
u
a
l
v
e
r
s
u
s
e
x
p
e
c
t
e
d
r
e
t
u
r
n
s
.
I
n
P
a
n
e
l
B
w
e
r
e
p
o
r
t
t
h
e
r
e
s
u
l
t
s
o
f
a
r
e
g
r
e
s
s
i
o
n
o
f
a
c
t
u
a
l
v
e
r
s
u
s
e
x
p
e
c
t
e
d
r
e
t
u
r
n
s
a
c
r
o
s
s
t
h
e
7
5
p
o
r
t
f
o
l
i
o
s
.
S
t
a
n
d
a
r
d
e
r
r
o
r
s
a
r
e
a
d
j
u
s
t
e
d
f
o
r
h
e
t
e
r
o
s
k
e
d
a
s
t
i
c
i
t
y
Table IV.
Estimation error in 75
overlapping four-year
holding periods across
high, medium, and low
beta portfolios (%)
Bias, stability,
and predictive
ability
231
D
o
w
n
l
o
a
d
e
d
b
y
P
O
N
D
I
C
H
E
R
R
Y
U
N
I
V
E
R
S
I
T
Y
A
t
2
1
:
0
9
2
4
J
a
n
u
a
r
y
2
0
1
6
(
P
T
)
M
e
a
n
a
b
s
o
l
u
t
e
e
r
r
o
r
(
p
-
v
a
l
u
e
s
v
s
O
L
S
)
S
q
u
a
r
e
r
o
o
t
o
f
m
e
a
n
s
q
u
a
r
e
e
r
r
o
r
O
L
S
V
a
s
i
c
e
k
B
e
t
a
¼
1
O
L
S
V
a
s
i
c
e
k
B
e
t
a
¼
1
P
a
n
e
l
A
:
C
o
m
p
a
r
i
s
o
n
o
f
m
e
a
n
a
b
s
o
l
u
t
e
a
n
d
m
e
a
n
s
q
u
a
r
e
d
e
r
r
o
r
s
(
n
¼
2
1
p
o
r
t
f
o
l
i
o
s
f
r
o
m
1
9
8
0
-
2
0
0
7
)
E
s
t
i
m
a
t
i
o
n
y
e
a
r
s
4
2
4
.
9
2
4
.
4
(
3
3
)
2
0
.
9
(
1
3
)
3
1
.
9
3
0
.
5
(
2
5
)
2
5
.
9
(
1
4
)
4
-
1
0
2
0
.
1
1
9
.
9
(
3
8
)
2
0
.
9
(
2
2
)
2
5
.
0
2
5
.
1
(
3
9
)
2
5
.
9
(
2
3
)
4
-
2
0
1
9
.
9
1
9
.
8
(
4
1
)
2
0
.
9
(
1
8
)
2
5
.
0
2
5
.
4
(
2
0
)
2
5
.
9
(
2
5
)
A
l
l
(
4
-
4
5
)
1
9
.
8
1
9
.
7
(
4
0
)
2
0
.
9
(
1
5
)
2
4
.
9
2
5
.
3
(
2
0
)
2
5
.
9
(
2
3
)
t
-
t
e
s
t
s
f
o
r
m
e
a
n
s
i
m
p
l
i
e
d
b
y
a
l
l
y
e
a
r
s
O
L
S
e
s
t
i
m
a
t
e
s
v
e
r
s
u
s
m
e
a
n
s
i
m
p
l
i
e
d
b
y
f
o
u
r
-
y
e
a
r
O
L
S
e
s
t
i
m
a
t
e
s
:
O
n
e
-
t
a
i
l
e
d
p
-
v
a
l
u
e
(
%
)
8
1
0
1
3
1
0
1
2
1
4
P
a
n
e
l
B
:
R
e
g
r
e
s
s
i
o
n
o
f
r
e
a
l
i
s
e
d
r
e
t
u
r
n
s
v
e
r
s
u
s
e
x
p
e
c
t
e
d
r
e
t
u
r
n
s
T
e
c
h
n
i
q
u
e
E
s
t
i
m
a
t
i
o
n
y
e
a
r
s
I
n
t
e
r
c
e
p
t
(
%
)
(
p
-
v
a
l
u
e
v
s
¼
0
)
C
o
e
f
?
c
i
e
n
t
(
p
-
v
a
l
u
e
v
s
¼
1
)
A
d
j
R
2
(
%
)
O
L
S
4
2
8
.
4
(
4
)
0
.
6
2
3
(
3
)
5
2
4
-
1
0
1
1
.
1
(
4
1
)
0
.
8
7
1
(
4
6
)
5
6
4
-
2
0
7
.
3
(
6
1
)
0
.
9
4
2
(
7
6
)
5
5
A
l
l
(
4
-
4
5
)
7
.
2
(
6
1
)
0
.
9
3
9
(
7
5
)
5
5
V
a
s
i
c
e
k
4
2
4
.
7
(
9
)
0
.
6
7
5
(
7
)
4
2
4
-
1
0
9
.
7
(
4
9
)
0
.
8
9
1
(
5
5
)
5
5
4
-
2
0
6
.
7
(
6
5
)
0
.
9
4
9
(
8
0
)
5
3
A
l
l
(
4
-
4
5
)
6
.
7
(
6
5
)
0
.
9
4
6
(
7
8
)
5
3
B
e
t
a
¼
1
6
.
2
(
6
9
)
0
.
9
3
7
(
7
2
)
5
1
N
o
t
e
s
:
W
e
f
o
r
m
e
d
e
q
u
a
l
-
v
a
l
u
e
p
o
r
t
f
o
l
i
o
s
o
f
l
o
w
,
m
e
d
i
u
m
,
a
n
d
h
i
g
h
b
e
t
a
s
t
o
c
k
s
a
c
c
o
r
d
i
n
g
t
o
t
h
e
i
r
f
o
u
r
-
y
e
a
r
O
L
S
b
e
t
a
e
s
t
i
m
a
t
e
s
a
t
3
1
D
e
c
e
m
b
e
r
o
f
1
9
7
9
,
1
9
8
3
,
1
9
8
7
,
1
9
9
1
,
1
9
9
5
,
1
9
9
9
a
n
d
2
0
0
3
.
O
v
e
r
t
h
e
s
u
b
s
e
q
u
e
n
t
f
o
u
r
y
e
a
r
s
,
w
e
m
e
a
s
u
r
e
d
t
h
e
t
o
t
a
l
r
e
t
u
r
n
s
o
f
t
h
e
s
e
p
o
r
t
f
o
l
i
o
s
,
a
n
d
c
o
m
p
a
r
e
d
t
h
e
i
r
p
e
r
f
o
r
m
a
n
c
e
w
i
t
h
t
o
t
a
l
e
x
p
e
c
t
e
d
r
e
t
u
r
n
s
d
e
r
i
v
e
d
f
r
o
m
t
h
e
i
r
b
e
t
a
e
s
t
i
m
a
t
e
s
a
n
d
t
h
e
p
e
r
f
o
r
m
a
n
c
e
o
f
t
h
e
m
a
r
k
e
t
p
o
r
t
f
o
l
i
o
.
E
x
p
e
c
t
e
d
r
e
t
u
r
n
s
a
r
e
c
o
m
p
u
t
e
d
a
c
c
o
r
d
i
n
g
t
o
e
q
u
a
t
i
o
n
(
7
)
,
w
h
e
r
e
b
y
t
h
e
b
e
t
a
e
s
t
i
m
a
t
e
i
s
i
n
c
o
r
p
o
r
a
t
e
d
i
n
t
o
t
h
e
C
A
P
M
,
a
l
o
n
g
w
i
t
h
r
e
a
l
i
s
e
d
m
a
r
k
e
t
r
e
t
u
r
n
s
a
n
d
t
h
e
y
i
e
l
d
t
o
m
a
t
u
r
i
t
y
o
n
t
h
r
e
e
-
m
o
n
t
h
t
r
e
a
s
u
r
y
n
o
t
e
s
a
t
t
h
e
s
t
a
r
t
o
f
t
h
e
p
o
r
t
f
o
l
i
o
f
o
r
m
a
t
i
o
n
p
e
r
i
o
d
.
O
v
e
r
a
l
l
,
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Table V.
Estimation error in 21
non-overlapping
four-year holding periods
across high, medium, and
low beta portfolios (%)
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Compared with the results for overlapping periods, coef?cients in the regression
analysis increase towards one and intercept terms decrease towards zero, but our
overall conclusions are unchanged.
In sum, our results show that predictability of returns from OLS beta estimates
systematically increases with an increase in the estimation window and with
application of the Vasicek adjustment. The inability to outperform a na? ¨ve estimate
that beta equals one may be due to a lack of power in our statistical tests, which rely on
realised returns as a measure of expectations, or limitations of the CAPM itself.
However, we can reject the conventional practice that OLS beta estimates improve by
truncating returns data to only the most recent years.
Firm analysis
We repeated our regression analysis with reference to the sample of 12,031 ?rm-years,
rather than portfolios of high, medium, and low beta stocks. Results for unweighted
regressions are presented in Panel A of Table VI. Explanatory power has fallen to
around one-tenth of that reported in the portfolio results. However, we still observe that
explanatory power increases for longer versus shorter estimation periods, and with the
use of Vasicek-adjusted beta estimates relative to OLS estimates.
We also observe that the coef?cients on expected returns derived from OLS
estimates approximate one, while coef?cients on expected returns derived from
Technique Estimation years
Intercept
(%) ( p-value vs ¼ 0) Coef?cient ( p-value vs ¼ 1)
Adj R
2
(%)
Panel A: Regression of realised returns versus expected returns (equal weight)
OLS 4 14.3 (,0.01) 0.833 (1) 6.5
4-10 11.4 (,0.01) 1.038 (50) 7.0
4-20 11.1 (,0.01) 1.070 (22) 7.2
All (4-45) 11.0 (,0.01) 1.075 (19) 7.3
Vasicek 4 9.6 (,0.01) 1.205 (0.05) 7.4
4-10 7.5 (,0.01) 1.348 (,0.01) 7.9
4-20 8.1 (,0.01) 1.359 (,0.01) 8.1
All (4-45) 7.5 (,0.01) 1.362 (,0.01) 8.1
Beta ¼ 1 5.3 (,0.01) 1.507 (,0.01) 7.8
Panel B: Regression of realised returns versus expected returns (weighted by market capitalization)
OLS 4 3.30 (0.1) 0.743 (,0.01) 14.7
4-10 2.11 (4) 0.855 (0.2) 16.1
4-20 1.97 (5) 0.888 (2) 16.6
All (4-45) 1.79 (8) 0.898 (3) 16.8
Vasicek 4 2.77 (1) 0.794 (0.02) 15.3
4-10 1.77 (9) 0.888 (2) 16.5
4-20 1.63 (11) 0.919 (10) 17.0
All (4-45) 1.43 (17) 0.931 (17) 17.2
Beta ¼ 1 0.43 (69) 1.002 (96) 18.1
Notes: We regressed annual actual returns against expected returns for 12,031 ?rm-years, with
holding periods beginning at 31 December each year from 1979-2003. Expected returns are computed
according to equation (7), whereby the beta estimate is incorporated into the CAPM, along with
realised market returns and the yield to maturity on three-month Treasury notes at the start of the
portfolio formation period
Table VI.
Estimation error in
annual returns across
individual ?rm-years (%)
Bias, stability,
and predictive
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233
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Vasicek-adjusted estimates exceed one, and the coef?cient on market returns is highest
of all. This occurred because small, high beta stocks earned the highest returns in the
time-period under study. As shown in Panel B, in a market capitalisation weighted
least squares regression, as we increase the estimation window or apply the Vasicek
adjustment to the same estimation window, the coef?cients systematically increase
towards one and the intercept terms decline towards zero.
We reach the same conclusions when we examine four-year holding periods, as
shown in Table VII. However, explanatory power has fallen substantially with an
increase in the dispersion of individual stock returns[6]. Unfortunately, as with the
portfolio analysis, the explanatory power of these regressions is entirely due to the
in?uence of market returns. There is no evidence that incorporating beta estimates
from historical data improved the predictability of stock returns, relative to a na? ¨ve
assumption that all stocks have a beta estimate equal to one. However, if historical
returns are to be used in this manner, estimation error is minimised by using all
available data and applying the Vasicek adjustment.
Conclusion
There is no evidence that eliminating returns observations in beta computations
increases their ability to predict returns. Mean absolute forecast errors systematically
decrease with an increase in the number of monthly returns used to compute beta
Technique Estimation years
Intercept
(%) ( p-value vs ¼ 0) Coef?cient ( p-value vs ¼ 1)
Adj R
2
(%)
Panel A: Regression of realised returns versus expected returns (equal weight)
OLS 4 89.8 (,0.01) 0.382 (,0.01) 0.3
4-10 76.1 (,0.01) 0.570 (,0.01) 0.4
4-20 67.4 (,0.01) 0.691 (1) 0.6
All (4-45) 66.4 (,0.01) 0.701 (2) 0.6
Vasicek 4 65.5 (,0.01) 0.735 (6) 0.5
4-10 58.7 (,0.01) 0.975 (87) 0.7
4-20 42.4 (,0.01) 1.067 (70) 0.9
All (4-45) 41.4 (,0.01) 1.079 (65) 0.9
Beta ¼ 1 12.9 (7) 1.475 (0.1) 1.0
Panel B: Regression of realised returns versus expected returns (weighted by market capitalisation)
OLS 4 36.0 (,0.01) 0.500 (,0.01) 4.4
4-10 24.2 (,0.01) 0.687 (,0.01) 6.1
4-20 25.1 (,0.01) 0.691 (1) 5.6
All (4-45) 21.1 (0.3) 0.749 (,0.01) 6.6
Vasicek 4 31.5 (,0.01) 0.572 (,0.01) 5.0
4-10 20.2 (0.03) 0.751 (1) 6.5
4-20 21.4 (0.04) 0.750 (1) 5.9
All (4-45) 16.8 (0.3) 0.819 (5) 7.0
Beta ¼ 1 9.3 (10) 0.915 (36) 7.5
Notes: We regressed actual returns over four-year holding periods against expected returns for 12,031
stocks, with holding periods beginning at 31 December each year from 1979-2003. Expected returns
are computed according to equation (7), whereby the beta estimate is incorporated into the CAPM,
along with realised market returns and the yield to maturity on three-month Treasury notes at the
start of the portfolio formation period
Table VII.
Estimation error in
overlapping four-year
returns across individual
?rm-years (%)
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estimates. Application of the Vasicek adjustment, which minimises the impact of
imprecise beta estimates, results in further reductions in forecast error. Over forecast
horizons of one and four years, these techniques reduce mean absolute error by
19-20 per cent, compared with the use of four-year, OLS beta estimates. Despite this
improvement in predictive ability, the na? ¨ve assumption that all stocks have a beta
estimate equal to one generates the smallest estimation error.
Notes
1. In a survey paper of US-listed ?rms, Graham and Harvey (2001) report that 73 per cent of
respondents always or almost-always use the CAPM to estimate the cost of equity capital. In
a similar paper on Australian-listed ?rms, Truong et al. (2008) report that 72 per cent of
respondents use the CAPM to estimate the cost of capital.
2. Rosenberg and McKibbin (1973) employ the same overarching criteria for determining the
reliability of beta estimates, speci?cally whether these estimates generate useful predictions
of future returns, conditional upon market returns.
3. The average annual market return relative to the yield on short-term government securities
is estimated at 6.6 per cent in Australia (Brailsford et al., 2008) and 7.2 per cent in the USA
(Dimson et al., 2003). Estimates of the equity risk premium derived from equity prices
include 4.3 per cent assuming a constant growth dividend discount model (Fama and French,
2002) and 5.3 per cent where cost of equity and long-term growth are jointly estimated
(Easton et al., 2002).
4. The use of portfolios is consistent with Fama and French (1993) who used 25 portfolios
sorted on market capitalisation and book-to-market ratio to measure the association between
stock returns and factor-mimicking portfolios which measure market returns relative to the
risk-free rate, excess returns to small relative to large market capitalisation stocks and
excess returns to high versus low book-to-market stocks. As also highlighted by Fama and
French, the use of value-weighted portfolios is consistent with variance minimisation.
5. The 1,717 unique ?rms in our dataset represent 48 per cent of all ?rms in the CRIF database
from 1980-2004. Our annual returns analysis relies upon 144,372 months of returns data,
which represents 57 per cent of monthly returns in the database during the same time-period.
This primarily results from the requirement that monthly returns data is available for all
?rms for a continuous 48-month period subsequent to portfolio formation. Hence, our results
are only generalisable to ?rms with relatively long trading histories, which by construction
are relatively successful. However, we note that our sample is spread across large and small
stocks, and there is no reason to expect that beta estimates would generate more accurate
returns expectations for non-surviving ?rms. Firms which de-list as a result of takeover, or
bankruptcy are likely to have relatively more months of returns data which is affected by
company-speci?c factors, thereby introducing even more noise into the dataset.
6. As mentioned in section 3, we also performed our ?rm analysis using a constant market risk
premium with assumed values of 4, 5 and 6 per cent. In this analysis, there is zero association
between actual and expected returns on individual stocks under all beta assumptions.
References
Blume, M.E. (1975), “Betas and their regression tendencies”, Journal of Finance, Vol. 30,
pp. 785-95.
Bowman, R.G. and Bush, S.R. (2005), “A test of the usefulness of comparable company analysis”,
working paper, University of Auckland, Auckland.
Bias, stability,
and predictive
ability
235
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in Australia”, Accounting Research Journal, Vol. 17, pp. 121-35.
Brailsford, T., Handley, J.C. and Maheswaran, K. (2008), “Re-examination of the historical equity
risk premium in Australia”, Accounting and Finance, Vol. 48, pp. 73-97.
Dimson, E., Marsh, P. and Staunton, M. (2003), “Global evidence on the equity risk premium”,
Journal of Applied Corporate Finance, Vol. 15, pp. 8-19.
Easton, P., Taylor, G., Shroff, P. and Sougiannis, T. (2002), “Using forecasts of earnings to
simultaneously estimate growth and the rate of return on equity investment”, Journal of
Accounting Research, Vol. 40, pp. 657-76.
Elton, E.J., Gruber, M.J. and Urich, T.J. (1978), “Are betas best?”, Journal of Finance, Vol. 33,
pp. 1375-84.
Eubank, A.A. and Zumwalt, J.J. (1979), “An analysis of the forecast error impact of alternative
beta adjustment techniques and risk classes”, Journal of Finance, Vol. 34, pp. 761-76.
Fama, E.F. and French, K.R. (1993), “Common risk factors in the returns on stocks and bonds”,
Journal of Financial Economics, Vol. 33, pp. 3-56.
Fama, E.F. and French, K.R. (2002), “The equity premium”, Journal of Finance, Vol. 57, pp. 637-59.
Graham, J.R. and Harvey, C.R. (2001), “The theory and practice of corporate ?nance: evidence
from the ?eld”, Journal of Financial Economics, Vol. 60, pp. 187-243.
Hooper, V.J., Ng, K. and Reeves, J.J. (2005), “Beta forecasting: a two-decade evaluation”,
University of New South Wales, working paper, Sydney and Stanford University,
Stanford, CA.
Klemkosky, R.C. and Martin, J.D. (1975), “The adjustment of beta forecasts”, Journal of Finance,
Vol. 30, pp. 1123-8.
Rosenberg, B. and McKibbin, W. (1973), “The prediction of systematic and speci?c risk in
common stocks”, Journal of Financial and Quantitative Analysis, Vol. 8, pp. 317-33.
Truong, G., Partington, G. and Peat, M. (2008), “Cost-of-capital estimation and capital-budgeting
practice in Australia”, Australian Journal of Management, Vol. 33, pp. 95-121.
Vasicek, O.A. (1973), “A note on using cross-sectional information in Bayesian estimation of
security betas”, Journal of Finance, Vol. 28, pp. 1233-9.
About the authors
Stephen Gray is a Professor in Finance at UQ Business School, The University of Queensland,
Queensland, Australia.
Jason Hall is a Lecturer in Finance at UQ Business School, the University of Queensland,
Queensland, Australia. Jason Hall is the corresponding author and can be contacted at:
[email protected]
Drew Klease is a University of Queensland Graduate and now works at Queensland
Investment Corporation.
Alan McCrystal is a Research Analyst at UQ Business School, The University of Queensland,
Queensland, Australia.
ARJ
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This article has been cited by:
1. Tristan Fitzgerald, Stephen Gray, Jason Hall, Ravi Jeyaraj. 2013. Unconstrained estimates of the equity
risk premium. Review of Accounting Studies 18, 560-639. [CrossRef]
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doc_566273434.pdf
Estimates of systematic risk or beta are an important determinant of the cost of capital.
The standard technique used to compile beta estimates is an ordinary least squares regression of stock
returns on market returns using four to five years of monthly data. This convention assumes that a
longer time series of data will not adequately capture risks associated with existing assets. This paper
seeks to address this issue
Accounting Research Journal
Bias, stability, and predictive ability in the measurement of systematic risk
Stephen Gray J ason Hall Drew Klease Alan McCrystal
Article information:
To cite this document:
Stephen Gray J ason Hall Drew Klease Alan McCrystal, (2009),"Bias, stability, and predictive ability in the
measurement of systematic risk", Accounting Research J ournal, Vol. 22 Iss 3 pp. 220 - 236
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Bias, stability, and predictive
ability in the measurement of
systematic risk
Stephen Gray and Jason Hall
UQ Business School, The University of Queensland, St Lucia, Australia
Drew Klease
Queensland Investment Corporation, Brisbane, Australia, and
Alan McCrystal
UQ Business School, The University of Queensland, St Lucia, Australia
Abstract
Purpose – Estimates of systematic risk or beta are an important determinant of the cost of capital.
The standard technique used to compile beta estimates is an ordinary least squares regression of stock
returns on market returns using four to ?ve years of monthly data. This convention assumes that a
longer time series of data will not adequately capture risks associated with existing assets. This paper
seeks to address this issue.
Design/methodology/approach – Each year from 1980 to 2004, equity betas are estimated for
1,717 Australian ?rms over periods of four to 45 years, and form equal value portfolios of high,
medium and low beta stocks. The paper compares expected returns – derived from the capital asset
pricing model (CAPM) and subsequent realised market returns – and actual returns over subsequent
annual and four-year periods.
Findings – The paper shows that the ability of beta estimates to predict future stock returns
systematically increases with the length of the estimation window and when the Vasicek bias
correction is applied. However, estimation error is insigni?cantly different from that associated with a
na? ¨ve assumption that beta equals one for all stocks.
Research limitations/implications – The implication is that using all available returns data in
beta estimation, along with the Vasicek bias correction, reduces the imprecision of expected returns
estimates derived from the CAPM. A limitation of the method is the use of conditional realised returns
as a proxy for expected returns, given that it is not possible directly to observe expected returns
incorporated into share prices.
Originality/value – The paper contributes to the understanding of corporate ?nance practitioners
and academics, who routinely use beta estimates derived from ordinary least squares regression.
Keywords Beta factor, Capital asset pricing model, Australia, United States of America
Paper type Research paper
Introduction
The Capital Asset Pricing Model (CAPM) is the pre-dominant framework for
estimating the cost of equity capital[1]. Yet considerable debate remains as to the
appropriate parameter inputs, especially regarding beta estimation techniques.
The standard technique to estimate a ?rm’s beta is an ordinary least squares (OLS)
regression using four or ?ve years of monthly data. This method is so well accepted
that other estimates have been evaluated on the basis of how close they are to the OLS
estimate (Klemkosky and Martin, 1975; Elton et al., 1978; Eubank and Zumwalt, 1979).
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/1030-9616.htm
ARJ
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Accounting Research Journal
Vol. 22 No. 3, 2009
pp. 220-236
qEmerald Group Publishing Limited
1030-9616
DOI 10.1108/10309610911005563
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Recent research by Hooper et al. (2005) ?nds that an autoregressive model with three
lags, estimated on 20 years of quarterly data, minimises the error associated with
forecasting the next period’s beta.
Other research examines the association between contemporaneous beta estimates,
computed from different techniques. For example, Bowman and Graves (2004) evaluate
the usefulness of comparable ?rm analysis by testing the association between the beta
estimates of Australian ?rms and the mean beta estimates of comparable ?rms in the
USA. Their conclusion, that comparable ?rm beta estimates under-estimate actual
betas (emphasis added), could easily be interpreted in the reverse manner. We could
just as easily conclude that estimating the systematic risk of the ?rm using its own
time-series of returns results in an over-estimate of systematic risk, compared to the
actual beta estimated as the mean of a sample of comparable ?rms. In other words, we
have two estimates of the systematic risk of sample ?rms, but no means of
distinguishing which is most relevant for valuation.
Systematic risk is not directly observable using any technique; it can only be
estimated. Thus, it is not possible to evaluate the merits of one technique purely on the
basis of its ability to predict the beta derived from an alternative technique. We can
however, evaluate the ability of alternative estimation techniques to predict future
stock returns, when incorporated into the CAPM, an approach advocated by Rosenberg
and McKibbin (1973). In this paper, we examine whether beta estimates generated from
short estimation windows have a relatively greater ability to predict future stock
returns, compared to beta estimates derived from longer estimation periods and with
the application of the Vasicek (1973) bias correction.
It is puzzling that the short window, OLS technique has become standard practice.
In almost all ?nance research, the convention is to rely upon all available data, unless
there is persuasive evidence that the data is unreliable for addressing the research
question. In contrast, when estimating systematic risk, researchers ignore returns data
outside of a short estimation period unless presented with persuasive evidence that it
should be included. Researchers implicitly assume that there is an optimal estimation
period, which involves a trade-off between the timeliness of a short returns window
versus the reduction in standard error associated with a long returns window. This
trade-off view is perhaps motivated by Blume’s (1975) rationalisation that
mean-reversion in beta estimates is due to the changing nature of management’s
risk preferences in their investment decisions (p. 795).
Our results show that the ability of beta estimates to predict future stock returns
systematically increases with the length of the estimation window, the implication
being that all available returns data should be used in beta estimation. In addition, for
all estimation periods, there is an increase in returns predictability when the Vasicek
adjustment is applied. However, there is no signi?cant difference in estimation error
associated with long-window, Vasicek-adjusted beta estimates and the na? ¨ve
assumption that all ?rms have an equity beta equal to one. Our conclusion is that
all beta estimates from historical returns data should be treated with caution,
especially those, which are substantially different from one. Nevertheless, if regression
analysis on historical returns is used to measure systematic risk, we ?nd no reason to
exclude returns outside an arbitrary window, and additional bene?ts result from the
Vasicek bias correction.
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Beta adjustment techniques
Vasicek adjustment
Data providers such as the Centre for Research in Finance (CRIF), Bloomberg and
Datastream provide beta estimates generated from an OLS regression of stock returns
on market returns from four to ?ve years of monthly data. CRIF will supply a beta
estimate if there is a minimum of 24 months of returns data within the most recent
four-year period.
The default beta estimates of Bloomberg and Datastream (b
adjusted
) both make
adjustments to the OLS estimate (b
OLS
), placing some weight (1 – w) on a prior
estimate of one, according to the following equation:
b
adjusted
¼ w £ b
OLS
þ 1 2w ð Þ £ 1 ð1Þ
In the case of Bloomberg, the default weight placed on the prior estimate of one is a
constant one-third. In the case of Datastream, the weight varies according to the
precision of the beta estimate. These adjustments are speci?c applications of the
Vasicek adjustment to beta computations. Vasicek (1973) recognised that OLS beta
estimates are biased in the sense that the more the sample estimate deviates from an
unconditional expectation, the greater the chance that the estimate results from
sampling error. Using Bayesian techniques, Vasicek showed that an unbiased beta
estimate is given by the following equation:
b
Vasicek
¼
s
2
prior
s
2
prior
þs
2
estimate
£ b
estimate
þ 1 2
s
2
prior
s
2
prior
þs
2
estimate
!
£ b
prior
ð2Þ
where:
.
b
estimate
and s
estimate
¼ the beta estimate and its standard error; and
.
b
prior
and s
prior
¼ the prior expectations of beta and its standard error.
This result is conditional upon knowing the population mean and standard error. In
reality we cannot directly observe these parameters, but they can be estimated. In our
study, we assume a prior beta estimate of one, as opposed to an industry mean for
example, under the reasoning that beta estimation is an incremental process. Prior to
any company- or industry-speci?c information being available, one is the unbiased
beta estimate for all ?rms. Subsequent to an OLS regression of stock returns against
market returns, and without any additional company- or industry-speci?c information,
the unconditional expectation is re?ned.
Filtering adjustments
There are also instances in which practitioners use ?ltering techniques to exclude
sample beta estimates considered to be so unreliable that they should carry zero weight
in comparable ?rm analysis. One such ?ltering adjustment is to exclude beta estimates
where the regression has little explanatory power (low R-squared statistic) as adopted
by Bowman and Bush (2005). Another adjustment is to eliminate beta estimates from
the sample with t-statistics below two relative to a null hypothesis of zero.
In our descriptive statistics we illustrate the substantial upward bias associated
with these ?ltering techniques, and consequently do not evaluate their predictive
ability. Consider the decision rule whereby we eliminate observations where the
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R-squared statistic is less than 10 per cent. We have the following criteria for selection,
whereby we are more likely to retain high beta estimates and discard low beta
estimates:
R
2
. 0:10
b
i
¼
COV r
i
;r
m ð Þ
s
2
m
¼
r
i;m
s
i
s
m
s
2
m
¼
r
i;m
s
i
s
m
r
i;m
¼ b
i
£
s
m
s
i
r
2
i;m
¼ R
2
¼ b
2
i
£
s
2
m
s
2
i
. 0:10
ð3Þ
where:
b
i
= the OLS beta estimate for stock i;
s
i
= the standard deviation of returns for stock i;
s
m
= the standard deviation of returns on the market portfolio; and
r
i,m
= the correlation between returns on stock i and the market portfolio.
The series of equations presented previously shows that, for stocks with the same
volatility, the R-squared ?lter is more likely to retain stocks with high beta estimates.
The t-statistic ?lter for excluding beta estimates also has an upward bias. Suppose the
decision-rule was to retain only stocks whose beta estimate was at least two times the
standard error of the estimate, according to the following criteria:
t
i
. 2
t
i
¼
b
i
s
error;i
. 2
: ð4Þ
For stocks with the same standard error, we are more likely to retain the stocks with
high beta estimates. The problem with the adoption of this technique in practice is that
the t-statistic is computed relative to a mean estimate of zero. An unbiased ?lter would
be to compute a t-statistic relative to a mean estimate of one, and discard beta estimates
where the absolute value of the t-statistic is greater than the cut-off point. However,
this alternative is less useful than the Vasicek bias correction. In comparable ?rm
analysis, the Vasicek adjustment will place greater aggregate weight on beta estimates
with low standard errors. Adopting a t-statistic ?lter places zero weight on
observations outside an arbitrary cut-off and equal weight on observations within the
arbitrary cut-off.
Research method
We estimated the equity beta for stocks in the CRIF database of Australian listed
companies at 31 December each year from 1979-2003. Estimates end in 2003 because
we evaluate returns predictability over a four-year forecast window and our returns
data ends in 2007. We computed the beta estimates by performing OLS regressions of
monthly stock returns versus market returns, where the estimation window ranged
from four to 45 years. For beta estimates relating to 1979, the longest estimation
Bias, stability,
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window is 21 years, which increases to 45 years for the 2003 estimates. Provided a
stock has 24 months of returns data in the estimation window and 48 months of returns
data in the four subsequent years, its beta estimate is retained for estimation periods of
four years or more. For example, if stock A is listed for just four years, we retain the
same beta estimate for longer estimation windows of ?ve, six years and so on. In
contrast, if stock B is listed for ten years, we compute different beta estimates over four
to ten years, and then retain the ten-year beta estimate for all future estimation windows.
We then computed Vasicek-adjusted beta estimates. These estimates are a weighted
average of the OLS estimate and a prior estimate of one, where the weights are
determined by the standard error of the OLS estimate and the standard deviation of
OLS estimates across the entire sample in each period. The equation for our
Vasicek-adjusted beta estimates is as follows:
b
Vasicek
¼
s
2
market
s
2
market
þs
2
OLS
£ b
OLS
þ 1 2
s
2
market
s
2
market
þs
2
OLS
!
£ 1 ð5Þ
where:
b
OLS
¼ the beta estimate from the OLS regression of stock returns on market
returns.
s
OLS
¼ the standard error of the beta estimate from the OLS regression.
s
market
¼ the standard deviation of beta estimates across the sample ?rms in each
period.
We examined whether these beta estimates had any association with future stock
returns over forecast horizons of one and four years. First, we formed low, medium and
high beta portfolios of equal value by ranking stocks from the minimum to maximum
four-year OLS beta estimate in each time -eriod. If beta estimates have an association
with future returns in the anticipated direction, we should observe low beta portfolios
outperform the market during periods when market returns are below the risk-free
rate, and high beta portfolios outperform the market when market returns are above
the risk-free rate. Second, consistent with Rosenberg and McKibben (1973)[2], we
computed the expected return on each portfolio implied by the CAPM, conditional upon
the realised performance of the market, according to the following equation:
E r
p;t;i;j
À Á
¼ r
f ;t
þb
p;t;i;j
r
m;t
2r
f ;t
À Á
ð6Þ
where:
E(r
p,t,i, j
) ¼ the expected return on the portfolio in period t according to beta
estimation period i and estimation technique j (OLS, Vasicek-adjusted
or na? ¨ve assumption of one);
r
f,t
¼ the risk-free rate at the start of the forecast period, as measured by the
yield-to-maturity on three month Treasury notes;
r
m,t
¼ the realised return on the equity market during the forecast period, as
measured by the market capitalisation weighted return on stocks in
the CRIF database; and
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b
p,t,i, j
¼ the estimated portfolio beta according to estimation period i and
estimation technique j (OLS, Vasicek-adjusted or na? ¨ve assumption of
one), computed as a market capitalisation weighted average of the
individual stocks’ beta estimates.
We conducted our analysis on conditional expected returns for the following reason.
Ideally, we would perform this evaluation with reference to an alternative direct
estimate of expected returns, something, which cannot be directly observed. What we
can observe is realised returns, for which we need an appropriate benchmark. That
benchmark is the conditional expected returns model presented above.
In practice, we would observe beta estimates being incorporated into the CAPM
with an expected market risk premium, which has been empirically estimated at
around 4-6 per cent, and which ?uctuates according to economic conditions[3]. One
alternative would be to compare the cost of capital implied by this assumption with
subsequent realised returns. This presents two problems.
First, it expands the scope of the research question to the joint estimation of both an
appropriate beta estimate and market risk premium. Our research question is whether
estimates of equity beta from historical data provide a reliable indicator of future
exposure to market movements, and therefore the systematic risk of a ?rm or project at
the end of the estimation period. Another way to phrase our research question is to ask,
“If we had perfect foresight of market returns, would OLS beta estimates provide a
reliable measure of the ?rm’s opportunity cost of funds?”. If this cost of capital estimate
is unreliable assuming perfect knowledge of subsequent market returns, it must be
unreliable in the real-world case where the market risk premium is estimated.
Second, even if beta estimates were a perfect predictor of market risk in a
subsequent time-period, the noise associated with this alternative approach is so large
that our tests would lack power. We attempted this alternative approach and found
that expected returns derived from this approach have no ability to explain the
variation in realised returns for both individual stocks and portfolios, and over holding
periods of one and four years.
We use portfolios in our analysis in order to minimise the impact of
company-speci?c information on stock returns, thereby isolating the impact of
market returns[4]. In our dataset, the cross-sectional standard deviation of annual
stock returns is 108 per cent, which is ?ve times the 21 per cent volatility of annual
returns on the broader market. In contrast, the cross-sectional standard deviation of
annual returns on our portfolios is 22 per cent, through the reduction of
company-speci?c risk. Our portfolios are value-weighted because small stocks have
relatively greater volatility than large stocks so require more stocks in the portfolio to
achieve the same degree of diversi?cation. For robustness, we repeat our regression
analysis at the ?rm level, on both an equal and market capitalisation weighted basis.
For each estimation technique and period, there are 75 portfolios of one-year returns
and overlapping four-year returns. This is the product of 25 periods of returns data
(comprising annual returns from 1980-2004, and four-year returns from 1980-2007) and
three portfolios of high, medium and low beta stocks. We also report results for 21
portfolios of non-overlapping four-year returns (ending each four years from 1983 to
2007). Our four-year returns are computed according to the following equation:
Bias, stability,
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Four-year returns ¼
Y
4
t¼1
1 þ r
t
ð Þ 21 ð7Þ
where r
t
¼ annual return on the portfolio or the market during year t.
We use two alternative techniques to measure the association between actual
portfolio returns and expected returns. First, we measure the difference between actual
and expected returns, and report the mean absolute error and square root of the mean
squared error. We perform paired t-tests to determine whether there is a statistically
signi?cant reduction in estimation error associated with the Vasicek adjustment and
with increasing estimation windows. Second, we perform a regression of actual returns
versus expected returns. This allows us to evaluate bias in expected returns – which is
present if the regression coef?cient on expected returns is signi?cantly different from
one – and predictive ability as measured by the adjusted R-squared statistic.
Data
Beta estimates
Our sample consists of 384,240 beta estimates, which is the product of 1,717 unique
?rms and an average 224 beta estimates for each ?rm. There are 12,031 beta estimates
derived from four years of historical returns which represents an average of seven
four-year estimates for each ?rm[5]. As shown in Panel A of Table I, the variation in
beta estimates decreases signi?cantly under longer estimation periods and with the
application of the Vasicek adjustment. When a four-year estimation window is used,
the standard deviation of beta estimates is 1.14, which decreases to 0.81 when all
returns data is used and to 0.51 when we also incorporate the Vasicek adjustment.
We illustrate the bias in beta estimates resulting from applying ?lters, which
required either a t-statistic of at least two, computed relative to a null hypothesis of
zero, or an R-squared statistic of at least 10 per cent. For these two ?lters, the mean beta
estimates computed over four years of returns are 1.57 and 1.58, respectively, due to the
elimination of 51 and 61 per cent of observations. Even when all available data is used
to compute beta estimates, the bias is present. In the case of the t-statistic ?lter, the
mean estimate is 1.23, despite the elimination of just 26 per cent of observations. In the
case of the R-squared ?lter, the mean estimate is 1.33.
We also observe that stocks with high or low beta estimates are relatively small and
exhibit high returns volatility. The average market capitalisation of medium beta
stocks is $1.1 billion, compared to $400 million for low and high beta stocks. High beta
stocks had an equally weighted standard deviation of annual returns of 133 per cent,
compared to 89 per cent for low beta stocks and 84 per cent for medium beta stocks. On
a market capitalisation weighted basis, the standard deviation of annual returns for
high beta stocks was 48 per cent, compared to 38 per cent for low beta stocks and 36
per cent for medium beta stocks. In aggregate, the descriptive statistics suggest that
volatility of returns is associated with market capitalisation, rather than beta
estimates, and that extreme beta estimates are more prevalent amongst small ?rms.
Cost of capital estimates
Beta estimates are primarily used to estimate the required return to equity holders in a
?rmor project. In Table II we compare the distributionof annual cost of capital estimates
whichwouldresult fromour alternative betaestimates, assumingamarket riskpremium
of 6 per cent. The variation in cost of capital estimates across the sample declines as more
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Table I.
Descriptive statistics
Bias, stability,
and predictive
ability
227
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data are usedinestimationandwiththe application of the Vasicekadjustment. The issue
is whether these adjustments merelyreduce noise inthe estimationof the cost of capital or
whether they mask true differences in systematic risk across ?rms.
The distribution of estimates, which are conditional upon realised market returns is
much greater. Despite this increase in variance, in the results which follow, we observe
that conditional expected returns have the ability to explain realised returns on
portfolios and individual stocks.
Results
Portfolio analysis
Our evaluation of one-year returns is presented in Table III. In the left-hand section of
Panel A we present the mean absolute error of actual versus expected portfolio returns
derived from OLS beta estimates, Vasicek-adjusted estimates and the na? ¨ve
assumption that equity beta equals one. The mean absolute error implied by OLS
estimates is 10.04 per cent, which decreases to 9.59 per cent if the Vasicek adjustment is
applied and 7.56 per cent assuming beta equals one. These differences are signi?cant at
levels of 0.02 per cent and 1 per cent, respectively. We also observe a reduction in the
mean absolute error as we increase the estimation period to ten, 20 and up to 45 years.
For OLS estimates, the mean absolute error decreases to 8.13 per cent, while the mean
absolute error associated with the Vasicek adjustment decreases to 7.99 per cent, and is
signi?cantly lower in every estimation period.
Mean absolute errors for increasing estimation periods are illustrated in Figure 1.
The ?gure illustrates the substantial reduction in estimation error associated with
increasing the estimation period and the consistently lower error attributed to the
Vasicek adjustment. It also shows that the mean absolute error is minimised by simply
assuming a beta estimate of one for all stocks. For long estimation windows this
difference is not statistically signi?cant.
We observe the same result when we examine squared errors rather than absolute
errors. The square root of the mean squared error is 13.81 per cent for the four-year
OLS estimates, which decreases to 13.12 per cent when the Vasicek adjustment is
applied, and to 10.68 per cent if we also incorporate all available returns information.
Again, the na? ¨ve assumption that beta equals one has the lowest estimation error, with
the square root of the mean squared error equal to 9.68 per cent.
Mean Std dev. Min. Tenth perc. Median Ninetieth perc. Max.
Risk-free rate 8.1 3.9 4.1 4.6 6.1 14.9 19.4
Cost of capital estimates under CAPM and assuming market risk premium ¼ 6 per cent
OLS four-year 15.0 7.7 251.7 7.1 13.9 24.6 136.4
OLS all-year 14.5 6.1 223.5 7.7 13.7 22.6 60.1
Vasicek four-year 14.1 5.3 23.8 8.0 13.4 21.7 34.1
Vasicek all-year 14.0 4.9 20.7 8.3 13.3 20.9 35.3
Beta ¼ 1 14.1 3.9 10.1 10.6 12.1 20.9 25.4
Expected returns conditional on realised market returns
OLS four-year 16.1 33.0 2326.3 211.8 11.3 48.4 485.0
OLS all-year 15.5 27.2 2174.7 210.2 11.5 45.4 326.9
Vasicek four-year 15.1 24.4 280.6 29.5 11.6 42.4 185.0
Vasicek all-year 14.8 22.6 262.2 29.0 11.8 41.6 165.4
Beta ¼ 1 14.9 20.0 216.5 29.5 12.3 42.2 64.9
Table II.
Unconditional versus
conditional cost of capital
estimates (%)
ARJ
22,3
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(
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5
p
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s
Table III.
Estimation error in
annual returns across
high, medium and low
beta portfolios (%)
Bias, stability,
and predictive
ability
229
D
o
w
n
l
o
a
d
e
d
b
y
P
O
N
D
I
C
H
E
R
R
Y
U
N
I
V
E
R
S
I
T
Y
A
t
2
1
:
0
9
2
4
J
a
n
u
a
r
y
2
0
1
6
(
P
T
)
The results are clearly in favour of estimating equity beta using all available returns
data with the Vasicek adjustment, versus the standard approach of short window OLS
regression. The limitation of these results is that there is insuf?cient evidence that
Vasicek-adjusted beta estimates outperform the na? ¨ve assumption that beta equals one.
Hence, while the use of regression analysis to estimate beta is questionable, if it is to be
used, then the appropriate presumption is that all returns information is value-relevant
and a bias correction is useful. This presumption could be rebutted if, for example,
there is clear evidence of a change in the ?rm’s assets or leverage.
In Panel B we present the results of regression analysis of actual versus expected
returns across the 75 portfolios. Explanatory power increases with the length of the
estimation window and with application of the Vasicek adjustment, with the adjusted
R-squared statistic rising from 72 to 80 per cent. The coef?cient on expected returns
also increases towards one and the intercept systematically declines towards zero.
We repeated this analysis with reference to total returns computed over four years
subsequent to portfolio formation, resulting in 75 overlapping periods beginning in
1979 and ending in 2007. This analysis is presented in Table IV. The implications of
this analysis are identical to those from our analysis of annual returns. Incorporating
longer estimation windows signi?cantly reduces mean absolute error and mean
squared error, as does the application of the Vasicek adjustment. Regression
coef?cients on expected returns increase towards one and explanatory power improves
as the estimation window increases and the Vasicek adjustment is applied but the
market-return itself remains the better predictor of four-year returns, compared to
either beta estimation technique.
Finally, we performed the same analysis on seven non-overlapping four-year
periods ending in 2007, resulting in 21 observations, and present the results in Table V.
Figure 1.
Mean absolute annual
error according to beta
estimation period
ARJ
22,3
230
D
o
w
n
l
o
a
d
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d
b
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P
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t
2
1
:
0
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2
4
J
a
n
u
a
r
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2
0
1
6
(
P
T
)
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e
a
n
a
b
s
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l
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t
e
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r
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r
(
p
-
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¼
1
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a
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:
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d
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a
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u
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r
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e
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r
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(
n
¼
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5
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1
9
8
0
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2
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)
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5
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(
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(
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2
6
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3
(
4
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3
(
5
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4
7
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6
4
6
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3
(
1
2
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3
4
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8
(
9
)
4
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2
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.
7
2
6
.
4
(
6
)
2
3
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3
(
5
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4
7
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6
4
6
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5
(
1
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)
3
4
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8
(
9
)
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l
l
(
4
-
4
5
)
2
6
.
5
2
6
.
2
(
7
)
2
3
.
3
(
6
)
4
7
.
3
4
6
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2
(
1
5
)
3
4
.
8
(
9
)
t
-
t
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t
s
f
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e
(
%
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1
5
5
6
P
a
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B
:
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m
a
t
i
o
n
y
e
a
r
s
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n
t
e
r
c
e
p
t
(
%
)
(
p
-
v
a
l
u
e
v
s
¼
0
)
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o
e
f
?
c
i
e
n
t
(
p
-
v
a
l
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e
v
s
¼
1
)
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d
j
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2
(
%
)
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L
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4
4
2
.
3
(
0
.
2
)
0
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4
3
3
(
1
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2
4
4
-
1
0
2
7
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3
(
9
)
0
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6
4
1
(
1
8
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4
1
4
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2
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2
7
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3
(
1
1
)
0
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6
5
4
(
2
3
)
3
9
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l
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(
4
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4
5
)
2
6
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(
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)
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9
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4
)
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0
V
a
s
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c
e
k
4
3
9
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7
(
1
)
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4
7
1
(
2
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2
6
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5
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4
(
1
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6
9
(
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3
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2
5
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5
(
1
4
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0
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1
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2
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1
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l
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(
4
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5
)
2
5
.
0
(
1
5
)
0
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6
8
7
(
2
8
)
4
1
B
e
t
a
¼
1
8
.
6
(
5
0
)
0
.
9
0
7
(
6
5
)
6
2
N
o
t
e
s
:
W
e
f
o
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m
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d
e
q
u
a
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-
v
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l
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m
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a
n
d
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a
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n
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r
f
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r
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y
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a
r
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L
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b
e
t
a
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s
t
i
m
a
t
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s
a
t
3
1
D
e
c
e
m
b
e
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o
f
e
a
c
h
y
e
a
r
f
r
o
m
1
9
7
9
-
2
0
0
3
.
O
v
e
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t
h
e
s
u
b
s
e
q
u
e
n
t
f
o
u
r
y
e
a
r
s
,
w
e
m
e
a
s
u
r
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d
t
h
e
t
o
t
a
l
r
e
t
u
r
n
s
o
f
t
h
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s
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p
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f
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s
,
a
n
d
c
o
m
p
a
r
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d
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h
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i
r
p
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r
f
o
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d
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f
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h
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m
a
r
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p
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.
E
x
p
e
c
t
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d
r
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t
u
r
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s
a
r
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c
o
m
p
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d
a
c
c
o
r
d
i
n
g
t
o
e
q
u
a
t
i
o
n
(
7
)
,
w
h
e
r
e
b
y
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h
e
b
e
t
a
e
s
t
i
m
a
t
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o
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n
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h
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A
P
M
,
a
l
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n
g
w
i
t
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r
e
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Table IV.
Estimation error in 75
overlapping four-year
holding periods across
high, medium, and low
beta portfolios (%)
Bias, stability,
and predictive
ability
231
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Table V.
Estimation error in 21
non-overlapping
four-year holding periods
across high, medium, and
low beta portfolios (%)
ARJ
22,3
232
D
o
w
n
l
o
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d
b
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P
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N
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a
r
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6
(
P
T
)
Compared with the results for overlapping periods, coef?cients in the regression
analysis increase towards one and intercept terms decrease towards zero, but our
overall conclusions are unchanged.
In sum, our results show that predictability of returns from OLS beta estimates
systematically increases with an increase in the estimation window and with
application of the Vasicek adjustment. The inability to outperform a na? ¨ve estimate
that beta equals one may be due to a lack of power in our statistical tests, which rely on
realised returns as a measure of expectations, or limitations of the CAPM itself.
However, we can reject the conventional practice that OLS beta estimates improve by
truncating returns data to only the most recent years.
Firm analysis
We repeated our regression analysis with reference to the sample of 12,031 ?rm-years,
rather than portfolios of high, medium, and low beta stocks. Results for unweighted
regressions are presented in Panel A of Table VI. Explanatory power has fallen to
around one-tenth of that reported in the portfolio results. However, we still observe that
explanatory power increases for longer versus shorter estimation periods, and with the
use of Vasicek-adjusted beta estimates relative to OLS estimates.
We also observe that the coef?cients on expected returns derived from OLS
estimates approximate one, while coef?cients on expected returns derived from
Technique Estimation years
Intercept
(%) ( p-value vs ¼ 0) Coef?cient ( p-value vs ¼ 1)
Adj R
2
(%)
Panel A: Regression of realised returns versus expected returns (equal weight)
OLS 4 14.3 (,0.01) 0.833 (1) 6.5
4-10 11.4 (,0.01) 1.038 (50) 7.0
4-20 11.1 (,0.01) 1.070 (22) 7.2
All (4-45) 11.0 (,0.01) 1.075 (19) 7.3
Vasicek 4 9.6 (,0.01) 1.205 (0.05) 7.4
4-10 7.5 (,0.01) 1.348 (,0.01) 7.9
4-20 8.1 (,0.01) 1.359 (,0.01) 8.1
All (4-45) 7.5 (,0.01) 1.362 (,0.01) 8.1
Beta ¼ 1 5.3 (,0.01) 1.507 (,0.01) 7.8
Panel B: Regression of realised returns versus expected returns (weighted by market capitalization)
OLS 4 3.30 (0.1) 0.743 (,0.01) 14.7
4-10 2.11 (4) 0.855 (0.2) 16.1
4-20 1.97 (5) 0.888 (2) 16.6
All (4-45) 1.79 (8) 0.898 (3) 16.8
Vasicek 4 2.77 (1) 0.794 (0.02) 15.3
4-10 1.77 (9) 0.888 (2) 16.5
4-20 1.63 (11) 0.919 (10) 17.0
All (4-45) 1.43 (17) 0.931 (17) 17.2
Beta ¼ 1 0.43 (69) 1.002 (96) 18.1
Notes: We regressed annual actual returns against expected returns for 12,031 ?rm-years, with
holding periods beginning at 31 December each year from 1979-2003. Expected returns are computed
according to equation (7), whereby the beta estimate is incorporated into the CAPM, along with
realised market returns and the yield to maturity on three-month Treasury notes at the start of the
portfolio formation period
Table VI.
Estimation error in
annual returns across
individual ?rm-years (%)
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Vasicek-adjusted estimates exceed one, and the coef?cient on market returns is highest
of all. This occurred because small, high beta stocks earned the highest returns in the
time-period under study. As shown in Panel B, in a market capitalisation weighted
least squares regression, as we increase the estimation window or apply the Vasicek
adjustment to the same estimation window, the coef?cients systematically increase
towards one and the intercept terms decline towards zero.
We reach the same conclusions when we examine four-year holding periods, as
shown in Table VII. However, explanatory power has fallen substantially with an
increase in the dispersion of individual stock returns[6]. Unfortunately, as with the
portfolio analysis, the explanatory power of these regressions is entirely due to the
in?uence of market returns. There is no evidence that incorporating beta estimates
from historical data improved the predictability of stock returns, relative to a na? ¨ve
assumption that all stocks have a beta estimate equal to one. However, if historical
returns are to be used in this manner, estimation error is minimised by using all
available data and applying the Vasicek adjustment.
Conclusion
There is no evidence that eliminating returns observations in beta computations
increases their ability to predict returns. Mean absolute forecast errors systematically
decrease with an increase in the number of monthly returns used to compute beta
Technique Estimation years
Intercept
(%) ( p-value vs ¼ 0) Coef?cient ( p-value vs ¼ 1)
Adj R
2
(%)
Panel A: Regression of realised returns versus expected returns (equal weight)
OLS 4 89.8 (,0.01) 0.382 (,0.01) 0.3
4-10 76.1 (,0.01) 0.570 (,0.01) 0.4
4-20 67.4 (,0.01) 0.691 (1) 0.6
All (4-45) 66.4 (,0.01) 0.701 (2) 0.6
Vasicek 4 65.5 (,0.01) 0.735 (6) 0.5
4-10 58.7 (,0.01) 0.975 (87) 0.7
4-20 42.4 (,0.01) 1.067 (70) 0.9
All (4-45) 41.4 (,0.01) 1.079 (65) 0.9
Beta ¼ 1 12.9 (7) 1.475 (0.1) 1.0
Panel B: Regression of realised returns versus expected returns (weighted by market capitalisation)
OLS 4 36.0 (,0.01) 0.500 (,0.01) 4.4
4-10 24.2 (,0.01) 0.687 (,0.01) 6.1
4-20 25.1 (,0.01) 0.691 (1) 5.6
All (4-45) 21.1 (0.3) 0.749 (,0.01) 6.6
Vasicek 4 31.5 (,0.01) 0.572 (,0.01) 5.0
4-10 20.2 (0.03) 0.751 (1) 6.5
4-20 21.4 (0.04) 0.750 (1) 5.9
All (4-45) 16.8 (0.3) 0.819 (5) 7.0
Beta ¼ 1 9.3 (10) 0.915 (36) 7.5
Notes: We regressed actual returns over four-year holding periods against expected returns for 12,031
stocks, with holding periods beginning at 31 December each year from 1979-2003. Expected returns
are computed according to equation (7), whereby the beta estimate is incorporated into the CAPM,
along with realised market returns and the yield to maturity on three-month Treasury notes at the
start of the portfolio formation period
Table VII.
Estimation error in
overlapping four-year
returns across individual
?rm-years (%)
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estimates. Application of the Vasicek adjustment, which minimises the impact of
imprecise beta estimates, results in further reductions in forecast error. Over forecast
horizons of one and four years, these techniques reduce mean absolute error by
19-20 per cent, compared with the use of four-year, OLS beta estimates. Despite this
improvement in predictive ability, the na? ¨ve assumption that all stocks have a beta
estimate equal to one generates the smallest estimation error.
Notes
1. In a survey paper of US-listed ?rms, Graham and Harvey (2001) report that 73 per cent of
respondents always or almost-always use the CAPM to estimate the cost of equity capital. In
a similar paper on Australian-listed ?rms, Truong et al. (2008) report that 72 per cent of
respondents use the CAPM to estimate the cost of capital.
2. Rosenberg and McKibbin (1973) employ the same overarching criteria for determining the
reliability of beta estimates, speci?cally whether these estimates generate useful predictions
of future returns, conditional upon market returns.
3. The average annual market return relative to the yield on short-term government securities
is estimated at 6.6 per cent in Australia (Brailsford et al., 2008) and 7.2 per cent in the USA
(Dimson et al., 2003). Estimates of the equity risk premium derived from equity prices
include 4.3 per cent assuming a constant growth dividend discount model (Fama and French,
2002) and 5.3 per cent where cost of equity and long-term growth are jointly estimated
(Easton et al., 2002).
4. The use of portfolios is consistent with Fama and French (1993) who used 25 portfolios
sorted on market capitalisation and book-to-market ratio to measure the association between
stock returns and factor-mimicking portfolios which measure market returns relative to the
risk-free rate, excess returns to small relative to large market capitalisation stocks and
excess returns to high versus low book-to-market stocks. As also highlighted by Fama and
French, the use of value-weighted portfolios is consistent with variance minimisation.
5. The 1,717 unique ?rms in our dataset represent 48 per cent of all ?rms in the CRIF database
from 1980-2004. Our annual returns analysis relies upon 144,372 months of returns data,
which represents 57 per cent of monthly returns in the database during the same time-period.
This primarily results from the requirement that monthly returns data is available for all
?rms for a continuous 48-month period subsequent to portfolio formation. Hence, our results
are only generalisable to ?rms with relatively long trading histories, which by construction
are relatively successful. However, we note that our sample is spread across large and small
stocks, and there is no reason to expect that beta estimates would generate more accurate
returns expectations for non-surviving ?rms. Firms which de-list as a result of takeover, or
bankruptcy are likely to have relatively more months of returns data which is affected by
company-speci?c factors, thereby introducing even more noise into the dataset.
6. As mentioned in section 3, we also performed our ?rm analysis using a constant market risk
premium with assumed values of 4, 5 and 6 per cent. In this analysis, there is zero association
between actual and expected returns on individual stocks under all beta assumptions.
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About the authors
Stephen Gray is a Professor in Finance at UQ Business School, The University of Queensland,
Queensland, Australia.
Jason Hall is a Lecturer in Finance at UQ Business School, the University of Queensland,
Queensland, Australia. Jason Hall is the corresponding author and can be contacted at:
[email protected]
Drew Klease is a University of Queensland Graduate and now works at Queensland
Investment Corporation.
Alan McCrystal is a Research Analyst at UQ Business School, The University of Queensland,
Queensland, Australia.
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This article has been cited by:
1. Tristan Fitzgerald, Stephen Gray, Jason Hall, Ravi Jeyaraj. 2013. Unconstrained estimates of the equity
risk premium. Review of Accounting Studies 18, 560-639. [CrossRef]
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