ESTIMATING THE MARKET RISK PREMIUM USING DATA FROM
MULTIPLE MARKETS
Martin Lally*
School of Economics and Finance
Victoria University of Wellington
John Randal**
School of Economics and Finance
Victoria University of Wellington
Abstract
This paper has developed an estimator for a country’s market risk premium that involves
optimally combining an estimator based upon only local data and the cross-country
average of these estimators. The analysis suggests that the usual practice of invoking
only local data to estimate a country’s market risk premium is significantly inferior to the
use of a cross-country average or a combined estimator with high weighting on the crosscountry average.
1. Introduction
The market risk premium for an individual market is a critical parameter in portfolio
analysis of the Markowitz type (Markowitz, 1952, 1959) and the widely used standard
version of the Capital Asset Pricing Model (CAPM: Sharpe, 1964; Lintner, 1965; Mossin,
1966). In turn, the CAPM is used to estimate the cost of equity capital, which in turn is
used for equity valuations of the Discounted Cash Flow type and in setting allowed
output prices for firms subject to price control. In estimating the market risk premium for
a particular market, a widely used and long-standing method involves a simple average
over annual ex-post outcomes (market returns net of the risk free rate) for a long period.
Following Ibbotson and Sinquefield’s (1976) estimate for the US, such estimates have
been generated for a wide range of markets and a range of alternative periods (Siegel,
1992; Dimson and Marsh, 1982; Officer, 1989; Chay et al, 1993).
More recently,
Dimson et al (2002) have presented estimates for 17 markets over the common period
from 1900-2000, and have subsequently updated these results and expanded the coverage
to 19 markets (Dimson et al, 2011). In addition, a wide range of alternative estimation
techniques have been proposed and estimates presented for various markets (Merton,
1980; Siegel, 1992; Scruggs, 1998; Cornell, 1999; Pastor and Stambaugh, 2001; Claus
and Thomas, 2001; Jagannathan et al, 2001; Fama and French, 2002; Mayfield, 2004;
Maheu and McCurdy, 2009; Donaldson et al, 2010).
For any given methodology, some of the variation in the estimates across markets will be
estimation error and these estimation errors across markets are less than perfectly
2
correlated, which points to using the cross-country average. Furthermore, in respect of
the historical averaging method, this approach has been proposed by Dimson et al (2003,
pp. 13-14). However, true market risk premiums are likely to differ across markets and
this argues for using local data for each market. Taking account of both points suggests
that one should form a weighted average of the estimator based upon only local data and
the cross-country average of these estimators. At the firm level, such an approach has
been adopted for estimating betas (Vasicek, 1973), variances (Karolyi, 1993), and
expected returns (Jorion, 1986; Grauer and Hakansson, 1995). More generally, the use of
combined estimators of this type is well established in the statistical literature (James and
Stein, 1961; Efron and Morris, 1975; Efron, 2010).
In light of all this, the present paper has three goals. The first is to develop an estimator
of the market risk premium for an individual market that optimally combines the estimate
based upon data for only that market and the cross-country average, i.e., to determine the
optimal weight on local data; we call this the “combined estimator”. The second goal is
to empirically assess the statistical advantage from using the combined estimator over
each of its two components.
The third goal is to estimate the confidence interval
surrounding the point estimate for the optimal weight on local data.
2. Theory
The relative merits of using a cross-country average rather than exclusively local data
depend upon the extent of cross-country variation in true market risk premiums. Such
3
variation can exist if markets are completely segmented and if markets are completely
integrated, although the causes will differ. If markets were completely segmented and
(consistent with this) the standard version of the CAPM (Sharpe, 1964; Lintner, 1965;
Mossin, 1966) applied, then the market risk premium for market j would be the product
of the variance for market j and the coefficient of relative risk aversion for the
representative investor in market j (Guo and Whitelaw, 2006). Both of these underlying
parameters can vary across markets and estimates of market variances suggest that there
is significant variation across markets (Cavaglia et al, 2000, Table 1). Thus, if markets
were completely segmented, market risk premiums could differ significantly across
markets. On the other hand, if markets were completely integrated and (consistent with
this) the Solnik (1974) version of the CAPM applied, then the market risk premium for
market j would be the product of the world market risk premium and the beta of market j
against the world market. The latter coefficient can vary across markets and estimates of
such betas suggest that there may be significant variation across countries (Harvey, 2000,
Exhibits 1A and 1B). Thus, if markets were completely integrated, market risk premiums
could differ significantly across markets. So, regardless of the extent to which markets
are segmented, true market risk premiums could differ significantly across markets. This
would tend to undercut the value from a cross-country average of estimates of the market
risk premium. However, due to estimation error, some combination of a cross-country
average and the estimate using only data from the country of interest might still be
superior to the latter. We now seek to determine the optimal combination.
4
Define  as the mean of the population from which the true market risk premiums for a
set of countries are drawn,  j as the true market risk premium for country j, ˆ j as the
estimate for  j using only local data, and ˆ as the cross-country average of the ˆ j . We
express  j as  j     j , where  j is a random and independent drawing from the
cross-sectional distribution of countries’ true market risk premiums (with mean zero and
variance denoted  2 ). Letting ej denote the estimation error for country j, being the
estimate using only local data less the true value, it follows that the estimated market risk
premium for country j using only local data is as follows.1
ˆ j   j  e j     j  e j
We now consider one of these (randomly selected) countries (designated as country 1).
Defining k as the weighting applied to ˆ1 , the combined estimator for the market risk
premium of country 1 is then as follows
(1)
1
ˆc  kˆ1  (1  k )ˆ
This approach, in which the true market risk premiums are treated as random variables, follows Efron and
Morris (1975). The alternative approach (James and Stein, 1961) is to treat the true market risk premiums
as a set of fixed numbers, but the resulting estimators are the same. In addition the derivations of results
within the statistics literature assume that the parameters in question are estimated using time-series data.
By contrast, we do not make this assumption in this section, and therefore our results are more general.
5
and the estimation error associated with this estimator for 1 is as follows.
ˆc  1  kˆ1  (1  k )ˆ  1
In choosing the weighting k, and consistent with the literature (Efron and Morris, 1975),
we choose k to minimise the MSE of a randomly selected country (country 1) with the
expectation over both estimation errors and countries. If the estimation errors ej have the
same variance  e2 for all markets and all distinct pairs of the random variables e1,….,eN
have the same correlation coefficient denoted ρ, then the optimal country weights within
the cross-country average ˆ will be equal.2 Letting N denote the number of countries
used in forming the cross-country average, the MSE from the combined estimator (with
the expectation over both estimation errors and countries) is then as follows.3
(2)
2
1
1
1 
1



MSE   2 (1  k ) 2 1     e2   k 2 1     e2 1  k 2 1  
 N
 N 
 N
N


The weights are optimal in the sense of minimising the variance of the estimator. This remains true even if
there is cross-country variation in the market values of the countries’ market portfolios. However, ceteris
paribus, larger markets will tend to have smaller
estimates for
3
 e2 and this will justify a higher weighting if differential
 e2 are acknowledged. Section 4 will address this issue.
The proof for equation (2) appears in the Appendix.
6
To find the value for k that minimises this function, denoted k0, we set the first derivative
to zero and this implies the following.4
(3)
k0 
 2
 2   e2 (1   )
The intuition for this result is apparent by consideration of extreme cases. If the variance
in the cross-sectional distribution of true market risk premiums (  2 ) is zero, then
equation (3) implies that the optimal weight on local data (k0) should be zero and hence
the cross-country average should be used, i.e., if all countries have the same market risk
premium then the best estimator for each country’s market risk premium would be the
cross-country average (because there is no disadvantage in using data from other
countries whilst gaining the ‘diversification’ benefit from doing so). On the other hand,
if  2 is positive but the variance in the estimation error for any country (  e2 ) is zero,
then equation (3) implies that k0 = 1, i.e., if countries have different market risk premiums
and there is no estimation error when using local data, then the best estimator for a
country’s market risk premium would use only local data (so as to avoid the ‘bias’ from
using data from other markets, which is drawn from distributions with different means to
that of the local market). Finally, if  2 is positive and  e2 is positive but the correlation
coefficient in the estimation errors of any pair of countries (  ) is 1, then equation (3) still
implies that k0 = 1, i.e., if countries have different market risk premiums and there is
estimation error when using only local data but perfect correlation in such errors across
4
The second derivative is positive, and therefore the second order condition for a minimum is satisfied.
7
countries, then the best estimator for a country’s market risk premium would still involve
using only local data because there is no diversification benefit from using data from
other markets (due to perfect correlation) whilst the bias from using data from other
markets still holds (the data from other markets are drawn from distributions with
different means).
Turning now to the implementation of equation (3), this requires estimates for  2 ,  e2
and  . The first of these cannot be directly estimated, but can be deduced from the
cross-sectional distribution of the locally-based estimates of the market risk premiums for
the N countries. Defining V as the expectation of the cross-sectional sample variance in
these locally-based estimates of the market risk premiums, it follows that5
(4)
 2  V   e2 (1   )
Substitution of this into equation (3) yields the following value for the optimal weight on
local data:
(5)
k0 
V   e2 (1   )
V
Reference to equations (3) and (4) also reveals that k0 is the ratio of  2 to V, i.e., the
proportion of cross-sectional variance in the local-data based estimates of the market risk
premiums that is attributable to cross-sectional variance in true market risk premiums.
5
The proof for equation (4) appears in the Appendix.
8
To estimate k0, equation (5) is invoked along with estimates for V,  e2 and ρ, and
estimates of the latter three parameters arise from the set of estimates for the market risk
premiums of the N countries or from the underlying data.
3. An Example
Before proceeding to look at empirical data, we first consider an example to illustrate the
advantages of the combined estimator over that based on only local data or the crosscountry average.
Suppose that there are 20 countries (N = 20) with  e2  .03 2 and ρ = .30. If only local
data are used (k = 1) then the MSE (equivalent to the variance) of the estimator is .03 2 .
By contrast, if equal weight is placed on all markets (k = 0), then the MSE of the
estimator (given by equation (2) with k = 0) differs from this. If  2  0 , in which case
all markets have the same true market risk premium, the MSE of the estimator is .017 2 ,
and this is naturally less than from using only local data because there is no disadvantage
(bias) from using data from other markets but there is a diversification benefit from using
data from a set of imperfectly correlated markets. However, as  2 increases, the use of
foreign data introduces growing bias and therefore the MSE of the cross-country average
will eventually exceed that from using only local data because the bias effect eventually
outweighs the diversification effect. Furthermore, as  2 increases, the optimal value for
9
k given by equation (3) rises. Consequently, the MSE of the combined estimator (given
by equation (2) with the optimal value for k) is always below that from using only local
data. In addition, apart from when  2 is zero in which case the MSEs are the same, the
combined estimator always has a lower MSE than the cross-country average. All of this
is shown in Figure 1.
This example demonstrates the advantages of the combined estimator over that using
only local data and the cross-country average. As the disadvantage from foreign data
decreases, in the form of lower cross-sectional variation in true market risk premiums, the
optimal weight on local data goes to zero and the MSE from the combined estimator will
be significantly less than that from using only local data. Also, as the disadvantage from
foreign data increases, in the form of greater cross-sectional variation in true market risk
premiums, the optimal weight on local data goes to 1 and the MSE from the combined
estimator will be significantly less than that from using the cross-country average. The
key issue is then the extent of cross-sectional variation in true market risk premiums and
the next section uses empirical data to estimate this.
4. Comparison of Estimators
Section 2 has developed a formula for the MSE of the combined estimator, the optimal
weight on local data, and procedures for estimating other relevant parameters. This
section now seeks to estimate this optimal weight, the MSE of the resulting combined
estimator, and to compare it with the MSE arising from use of the cross-country average
10
(k = 0) and from the use of only local data (k = 1). To do this, we consider methodologies
for estimating the market risk premium for which estimates have been generated for a
wide range of markets and the methodology readily allows for estimating both the
standard deviation of the point estimate and the correlation coefficient between point
estimates for different countries. At the present time, the only methodology that appears
to meet both requirements is the historical averaging methodology, and the natural choice
of estimates are those provided for 19 markets over the common period 1900-2010 by
Dimson et al (2011).6 Table 1 reports the estimates for these markets along with their
estimated standard errors.7 In respect of the expectation of the cross-sectional sample
variance in the locally-based estimates of the market risk premiums (V), an unbiased
estimate will arise from the cross-sectional sample variance in the estimated market risk
premiums shown in the first column of Table 1, and this is .0182 2 . In respect of the
variance in the estimation error using only local data (  e2 ), an unbiased estimate arises by
averaging over the estimates in the last column of Table 1, and this yields .0224 2 .
Finally, in respect of the correlation coefficient between the estimation errors for any pair
of markets (), an unbiased estimate arises by averaging over the estimated correlations
between all pairs of markets, using 111 years of data in each case, and the resulting
6
The more limited information available from other methodologies will be considered in a later section.
7
The estimates are arithmetic means on stock returns net of the yield on bills, supplied by Mike Staunton
from the Dimson, Marsh and Staunton data. The figures differ slightly from those given in Dimson et al
(2011, Table 9) in that the latter employ geometric rather than arithmetic differencing.
11
estimate is .355.8 Substituting these parameter estimates into equation (5) then yields an
estimate for the optimal weight on local data as follows.
(6)
k0 
.0182 2  .0224 2 (1  .355)
 .021
.0182 2
The numerator here is .0026 2 and a comparison of equations (4) and (5) reveals that it is
an estimate of the variance of the distribution from which the true market risk premiums
are drawn (  2 ).9 Following equation (2), with these estimates for  2 and k0, the RMSE
of the combined estimator is .014. The result is identical (to three decimal points) from
using a cross-country average (k = 0). By comparison, and again following equation (2),
the RMSE when only local data are utilised (k = 1) is .022. Thus, using the optimal value
for k or a cross-country average lowers the RMSE of the estimator by almost 40%. These
results and the underlying parameter estimates are summarised in Table 2.
8
For each pair of markets, the correlation in the point estimates for the market risk premium is equal to the
correlation in the annual equity returns net of the risk free rate, and is estimated using the time series of
these annual outcomes. The returns must be in local currency units for each of the two markets, consistent
with the definition of the correlation coefficient. The correlations estimated in this way were provided by
Mike Staunton using the Dimson, Marsh and Staunton data, and are shown in Table 4.
9
If the estimate of k0 were negative, and therefore the estimate of
 2
would also be negative, then
resetting the estimate for k0 to zero is recommended to reduce the MSE of the combined estimator (see
Efron and Morris, 1975, p. 312).
12
The impact from using a combined estimator rather than an estimator based upon only
local data varies across countries. As shown in Table 1, at one extreme (Italy), use of the
combined estimator rather than the estimator based upon only local data lowers the point
estimate by almost 300 basis points from .102 to .073. At the other extreme (Denmark),
use of the combined estimator rather than the estimator based upon only local data raises
the point estimate by 240 basis points from .048 to .072. On the other hand, the point
estimate would not change much for many markets such as the US (.074 versus .072).
However, even when the point estimate would not significantly change, the statistical
reliability of the estimator is dramatically improved by use of the combined estimator or
the cross-country average, i.e., a 40% reduction in the RMSE of the estimator.
5. Alternative Assumptions
All of the analysis so far is premised on the assumption that the standard deviation in the
estimation error ej is the same for all countries (  e ) and all distinct pairs of the random
variables e1,….,eN have the same correlation coefficient (ρ). Neither of these assumptions
is likely to be true, and there is also considerable variation across countries or country
pairs in the estimates for both of them (as shown in Tables 1 and 3). In view of this, we
now consider the implications of allowing the standard deviations in the estimation errors
to vary across countries.
This requires equations (2) to (5) to be modified. In addition, the country weights within
the cross-country average may require adjustment.
With country weights wi, the
13
Appendix shows that the MSE for a randomly selected country (country 1) from use of
the combined estimator (with the expectation over both estimation errors and countries) is
now as follows.
(7)
N


2
MSE   2 (1  k ) 2 (1  w1 ) 2  (1  k ) 2  w j    e21 k  (1  k ) w1 
2


 (1  k )
N
2
w 
2
j
N
2
ej
 2  e1 k  (1  k ) w1 (1  k ) w j  ej
2
2
N
N
  (1  k ) 2   wi w j  ei ej
i  2 j  2,
j i
Differentiating this with respect to k, and setting this derivative to zero, yields the optimal
value for k in respect of country 1 as follows:10
(8)
k0 
X1
Y1
where
N
N


X 1   2 (1  w1 ) 2   w 2j    e21 w1 (1  w1 )   w 2j  ej2
2
2


N N

N

   wi w j  ei ej   e1 (1  2w1 ) w j  ej 
 i 2 j 2

2
 j i

10
Since any country can be designated as country 1, the formula can be applied to any country.
14
N
N


Y1   2 (1  w1 ) 2   w 2j    e21 (1  w1 ) 2   w 2j  ej2
2
2


N N

N

   wi w j  ei ej  2 e1 (1  w1 ) w j  ej 
 i 2 j 2

2
 j i

With V defined in the same way as in the previous section, the Appendix also shows that
N
N

(9)
 2  V 
1
N
N

1
2
ej

i 1 j 1
j i
ei
 ej
N ( N  1)
In respect of the country weights within the cross-country average, the variance of the
estimation error arising from the cross-country average is minimised using weights that
are proportional to the inverse of the individual market variances.11 These weights are
shown in Table 4. The unbiased estimates for V and  are still .0182 2 and .355
respectively (as in the previous section) whilst the unbiased estimates for  ei are shown
in Table 1. Substitution of the relevant parameter estimates into equation (9) yields an
estimate for  2 of .0004 2 . Substitution of this estimate for  2 and other relevant
parameter values into equation (8) then yields the optimal value for k for each country,
designated k0, and the results are shown in the penultimate column of Table 4. These k0
values range from -0.117 for Germany to 0.192 for Canada, and are monotonically
11
See Granger (1989, section 8.3) for a discussion of combining estimators and the use of this particular
weighting scheme.
15
decreasing in the estimated standard deviation for the market.12 Interestingly the weight
of the US in the cross-country average is only fifth largest (at .065), because its estimated
standard deviation is only fifth smallest, and the optimal weight on the estimator using
only US data (as part of the combined estimator) is only .062.
Substitution of these values for k0 along with other relevant parameter values into
equation (7) then yields the MSE of the estimation error for each country. Despite the
cross-country variation in the k0 values, the resulting MSEs are .013 in all cases (to three
decimal points). By comparison, in the previous section, k0 was estimated at .021 for all
countries and the MSE of the estimator was .014. Thus, the recognition of differential
standard deviations across markets gives rise to variation in k0 values across countries but
it does not materially change the MSE of the estimator. This holds even for the market
with the smallest estimated standard deviation (Canada) and therefore the largest value
for k0. Furthermore, the cross-country variation in the estimated standard deviations is
likely to exceed the true cross-country variation and therefore the benefits from both
12
Negative k0 values arise from the coefficient on p in the numerator of equation (8) because the first term
in the numerator (involving
 2 ) is positive and the next two terms net to zero under the scheme for setting
wi described above. Thus, a necessary condition for k0 being negative is that p is non-zero. The intuition
for negative k0 values is apparent by considering an extreme case in which
 2
is zero (implying that all
countries have the same market risk premium), p = 1, N = 2 and country 1 has the larger variance.
Following equation (8), k0 is negative because this reduces the MSE (equivalent to the variance) of the
combined estimator to zero. Similarly, for a two-asset portfolio with a correlation coefficient between the
two assets of 1, a sufficiently negative weight on the higher variance asset will also reduce the portfolio
variance to zero.
16
differentially weighting markets in forming the cross-country average and from different
values for k0 across countries are likely to be exaggerated.13 In short, assuming all
markets have the same standard deviation on the estimation error, and therefore both
equally weighting markets in forming the cross-country average and applying the same k0
value to each market (.021), is only slightly inferior to recognising differential crosscountry standard deviations for the estimation errors, and even the slight apparent
advantage of the latter approach will be overestimated. This suggests that one should
equally weight all markets in forming the cross-country average and apply the same k0
value to all markets. Furthermore, as shown in the previous section, this common k0
value (.021) is so close to zero that there is no material benefit to departing from a zero
value. This suggests that one should equally weight all markets in forming the crosscountry average then use this cross-country average to estimate the market risk premium
for each market. This conclusion springs from the fact that the estimated cross-country
variation in true market risk premiums is very low.
6. Statistical Inference
The analysis so far utilises point estimates for the parameters in equations (5) and (8).
However the estimates of these parameters are subject to sampling variation and
consequently the estimate for the optimal weight on local data is also subject to sampling
13
The cross-sectional distribution of the estimates is the cross-sectional distribution of true values subject
to adding an estimation error to each true value and the latter are uncorrelated with the former. So, the
cross-sectional variation in the estimates is likely to exceed the cross-sectional variation in the true values.
17
variation. Accordingly it is desirable to estimate the confidence interval around the point
estimate of the optimal weight on local data. However there are difficulties in doing so
directly via equation (5) or (8). Consequently, we turn to an alternative approach that
readily lends itself to determination of a confidence interval around the point estimate.
Let ˆ1 j and ˆ2 j denote the estimates of the market risk premium for country j based upon
only local data for two non-overlapping time periods. In the presence of sampling errors,
these estimates will exhibit mean reversion, i.e., high (low) estimates in the first period
relative to the cross-country average will tend to be followed by lower (higher) estimates
in the second period. This is expressed in the following regression model
(10)
ˆ2 j  a  bˆ1 j   j
with b less than 1 in general. If the true market risk premiums in the two subperiods were
equal, the slope coefficient in a regression of ˆ2 j on ˆ1 j would be an unbiased estimate
of the true value b, i.e.,
(11)
 1 N ˆ
ˆ ˆ
ˆ 
 N  1  (1 j  1 )( 2 j   2 ) 
j 1

b  E
1 N ˆ


(1 j  ˆ1 ) 2



N  1 j 1


In general, it cannot be shown that the RHS of the last equation matches that of equation
(3), and therefore it cannot be shown in general that the coefficient b is equal to the
18
optimal weight k0.14 However, consideration of certain extreme cases suggests that there
is a close correspondence between these two parameters. For example, if the crosssectional variation in true market risk premiums were zero, meaning that the population
mean  characterised each country, then the estimates ˆ1 j and ˆ2 j would cease to be
positively correlated, i.e.,
ˆ2 j    e2 j
ˆ1 j    e1 j
So the parameter b in equation (10) would be zero. Under the same scenario, k0 would
also be zero as discussed following equation (3).
Alternatively, if cross-sectional
variation in the true market risk premiums were positive and sampling error went to zero,
then
ˆ2 j  ˆ1 j
Following equation (10), b then becomes 1. Under the same scenario, k0 = 1 as discussed
following equation (3).
Finally, if cross-sectional variation in the true market risk
premiums is positive and sampling error exists but the correlation in sampling errors
across countries goes to 1, then
ˆ2 j  ˆ1 j  e2  e1
14
By contrast, it can be shown that the expectation of the numerator in equation (11) matches that in
equation (3) and that the expectation of the denominator in (11) matches that in (3).
19
where e1 (e2) is the estimation error common to all firms in the first (second) subperiod.
Again, following equation (10), this implies that b becomes 1. Under the same scenario,
k0 = 1 as discussed following equation (3).
Thus, in all three extreme cases, the
regression model parameter b matches the optimal weight k0.
In view of all this, we investigate the question of whether an estimate of b derived from a
regression of the type shown in equation (10) would be an unbiased estimator of k0. If it
is unbiased, or the degree of bias is slight, then this provides an alternative approach to
estimating k0 with the advantage that such an approach allows the confidence interval
around the point estimate to be determined.
simulation.
We therefore conduct the following
We draw 19 true market risk premiums independently from a normal
distribution with mean .07 and standard deviation .005; these two parameter values
approximate the estimates obtained in section 4. The estimated market risk premiums are
then determined by adding a normally distributed estimation error to each true market
risk premium, with mean zero, standard deviation .031 and a correlation coefficient
between all pairs of the estimation errors of .35; the latter figure matches the averages of
the individual country pairs in section 4 whilst the standard deviation of .031 is the
average of the individual countries in section 4 (.022) subject to scaling up by
2 to
reflect the halving of the time period in the regression. These estimated market risk
premiums are designated as those for the first subperiod. The process is then repeated to
yield the estimated market risk premiums for the second subperiod, with the estimation
errors for the second subperiod being independent of those in the first subperiod. The
20
cross-sectional regression of estimates for the second subperiod on those for the first
subperiod is then run, and both the estimate of b and its standard error are recorded. This
process is then repeated 5000 times. Consequently, the average of these estimates for b
can be treated as closely approximating the true value for b.
Since the correlations are positive, the Cholesky decomposition of the correlation matrix
is computed and the resulting triangular matrix is used to form linear combinations of
independent normal random variables with the required correlation structure. In addition,
the fact that the estimation errors for countries are correlated implies that the regression
residuals will have the same property.
So, an HAC (heteroscedasticity and
autocorrelation consistent) estimate for the standard error of b̂ is presented (Andrews,
1991) as well as the conventional estimate. Finally, and consequent upon the use in
section 5 of differential estimates across countries for the standard deviation of the
estimation errors, we repeat the simulation using the differential point estimates of these
standard deviations.
The results of the simulation are shown in Table 5. The second column shows the k0
value determined in accordance with equation (3) or the cross-country average of the k0
values determined in accordance with equation (8). The third column shows the average
of the b̂ values across the 5000 simulation runs, which should closely approximate the
expected value. The fourth column shows the average across the 5000 simulation runs of
the conventionally estimated standard error of b̂ . Finally, the last column shows the
average across the 5000 simulation runs of the HAC estimates for the standard error of
21
b̂ . The difference between the values in the second and third columns is very small in
both cases. Accordingly, the value of b̂ from the regression shown in equation (10) can
be treated as an approximately unbiased estimator of k0 or the cross-country average of
the k0. In addition the values in the last column are materially different to those in the
third column, implying that the HAC approach should be adopted in estimating the
standard error of b̂ .
We therefore conduct the cross-sectional regression in equation (10), with ˆ1 j and ˆ2 j
equal to the estimates of the market risk premium for country j based upon only local data
for 1900-1950 and 1951-2010 respectively.15 This data is shown in Table 6. The OLS
regression yields b̂ = 0.048 with an HAC estimated standard error of 0.095. The 95%
confidence interval on b is then from -0.152 to 0.248. Accordingly, one cannot reject the
hypothesis that b = 0 whilst one can clearly reject the hypothesis that b = 1.16 It follows
that one cannot reject the hypothesis that k0 = 0 whilst one can clearly reject the
hypothesis that k0 = 1. Thus, the use of the cross-country average would seem to be
clearly superior to the use of only local data in estimating the market risk premium for a
particular market, and any weighting given to local data in a combined estimator should
not exceed 0.25. This point estimate here for b, and hence k0, of .048 is similar to the
point estimate of .021 obtained in section 4, but with the advantage of now providing a
15
As with the estimates in section 4, the estimates are arithmetic means on stock returns net of the yield on
bills, supplied by Mike Staunton from the Dimson, Marsh and Staunton data.
16
The conventional estimate for the standard error of the slope coefficient is .121, giving rise to a 95%
confidence interval from -0.21 to 0.30. Even in this case, the hypothesis that b = 1 is clearly rejected.
22
confidence interval around the point estimate. In addition, following equation (3), an
estimate for k0 equal to 0.048 in conjunction with the earlier estimates for  e and ρ of
0.0224 and 0.355 respectively implies an estimate for   of 0.4%, which is larger than
the estimates of 0.26% in section 4 and 0.04% in section 5.
In summary, using data from each market from both the first and second half of the data
series to generate estimates of the market risk premiums for each market, and crosssectionally regressing the later estimates on the earlier ones, generates an estimate for k0
of .048, which is very similar to the estimate obtained in section 4 but with the advantage
of now providing a 95% confidence interval on the estimate, from -0.152 to 0.248.
Furthermore, this new estimate for k0 implies an estimate for the standard deviation on
the cross-country distribution of true market risk premiums of 0.4%.
7. Forecasting
Although the focus of this paper is estimation rather than forecasting, some techniques
useful for forecasting can also be applied to estimation. One such approach would be to
use estimates of the market risk premiums in the first half of the time series examined
earlier to generate forecasts of the market risk premiums in the second half of the time
series under various forecasting models, use estimates of the market risk premiums in the
second half of the series to proxy for the true values (because the latter are unobservable),
and then examine the comparative accuracy of various forecasting models. The natural
forecasting models to examine would be the use of only local data, the cross-country
23
average, and the optimal combination (from the first half of the time series). However,
the evidence obtained so far indicates that the optimal combination is virtually
indistinguishable from the cross-country average. So, we compare the forecasting ability
of local data and the cross-country average. In respect of proxies for the market risk
premiums in the second half of the time series, standard practice is to use the ex-post
outcome (such as Welch and Goyal, 2008) and this corresponds to the use of only local
data. However, having judged that the cross-country average is a superior estimator, we
also consider the cross-country average (from the second half of the time series) as a
proxy for the true market risk premium in each market in that period.
The RMSE of these forecasts is shown in Table 7. The first row of results arises when
using only local data to proxy for the market risk premiums in the second half of the
series. Forecasting these proxy values using the cross-country average in the first half of
the series yields a smaller RMSE than using only local data in the first half of the series
(2.64% versus 4.01%). Similarly, the last row of the table shows the results arising using
the cross-country average from the second half of the series to proxy for the true market
risk premiums in that period. Again, forecasting using the cross-country average in the
first half of the series yields a smaller RMSE than using only local data in the first half of
the series (2.11% versus 3.81%). It is also noticeable that, for any given method of
forecasting (i.e., either column in Table 7), the better proxy for the market risk premiums
in the second half of the series is the cross-country average rather than the use of only
local data. So, consistent with the previous analysis, the cross-country average is clearly
24
superior to the use of only local data for both forecasting and proxying for the true market
risk premiums in the second half of the time series.
By contrast, in a pure forecasting exercise, the model presented earlier would require
extension to include the time series properties of the estimation errors ej.
In the
simulation exercise reported in section 6, we assumed that estimation errors in the
successive periods were independent, an assumption that seems reasonable when
estimates are each based upon 55 years of data. However, in a pure forecasting exercise,
estimates might be annual and the assumption of independence would be less reasonable.
For example, a stationary ARMA process might be assumed.
In summary, application of a forecasting technique to the data in section 6 also reveals
the clear superiority of the cross-country average over the use of only local data.
Furthermore, irrespective of the forecasting model used to generate the forecasts, the
assessment of the forecasts requires a procedure for estimating the unobservable market
risk premiums and again the cross-country average is superior to the use of only local
data.
8. Alternative Methodologies
The use of multi-country estimates of the market risk premium has so far been limited to
the historical averaging methodology because estimates for a wide range of markets are
available and the historical averaging methodology also readily allows for estimating
25
both the standard error of the point estimates and the correlation coefficient between
point estimates for different markets.
Nevertheless, it is possible to draw some
conclusions about the results from using multi-country estimates from estimation
methodologies other than historical averaging. Amongst these alternative methodologies,
the lack of estimated market risk premiums for a wide range of countries prevents us
from estimating the variance in the cross-sectional distribution of true market risk
premiums (  2 ). However the true value for  2 is invariant to the methodology used to
estimate the market risk premium and our estimates for it from the historical averaging
methodology range from .0004 2 to .004 2 .17 The same extrapolation is not necessarily
justified for ρ but for the present purposes we use the estimate obtained in section 4 from
the historical averaging methodology (0.355). Finally, regarding the variance of the
estimator (  e2 ), some empirical work using these alternative methodologies provides
estimates of  e2 . Furthermore, equation (3) reveals that the optimal weight on local data
is inversely related to  e2 and positively related to  2 . So, providing we can extrapolate
the estimate for ρ from the historical averaging methodology to these alternative
methodologies, the highest weight on local data will arise from the highest estimate for
 2 ( .004 2 ) along with the use of that methodology for estimating the market risk
premium that provides the lowest estimate of  e2 .
17
Section 4 estimates this variance at .0026
2
whilst section 5 estimates it as .0004
2
and section 7
2
estimates it as .004 .
26
Amongst these alternative methodologies, the lowest estimate for  e2 appears to be that
in Fama and French (2002), where the market risk premium is estimated from the average
dividend yield plus the average expected capital gain (estimated from the average growth
rate in dividends) less the risk free rate; when using this methodology, Fama and French
(ibid, page 644) estimate  e2 at .0074 2 . With this estimate and the highest estimate for
 2 ( .004 2 ), application of equation (3) shows that the optimal weight on local data
would be as follows
k0 
 2
.004 2

 .31
 2   e2 (1   ) .004 2  .0074 2 (1  .355)
Using other estimation methodologies or lower estimates of  2 , the estimate of k0 would
be lower. Thus, even for estimation methodologies other than historical averaging, the
optimal weight on local data is still low. Furthermore such results are not particularly
sensitive to ρ; for example, if ρ rises from .355 to .50, then the estimate of k0 in the last
equation only rises to .37.
In summary, preliminary application of the combined estimator to a range of
methodologies other than historical averaging indicates that the optimal weight on local
data would still be low.
9. Conclusions
27
This paper has developed an estimator for a country’s market risk premium, which
involves optimally combining an estimator based upon only local data with the crosscountry average of these estimators.
This paper has also compared this combined
estimator to that of its two components, using historical data, and the conclusions are as
follows.
Firstly, using the historical averaging methodology and assuming that the standard
deviations for the estimation errors arising from the use of only local data are the same
for all countries, very little of the cross-country variation in estimated market risk
premiums appears to be due to cross-country variation in true market risk premiums, and
therefore the combined estimator places very little weight upon the estimator based upon
only local data. Secondly, both the combined estimator and the cross-country average
have RMSEs that are almost 40% less than that arising from using only local data.
Thirdly, recognition of differential standard deviations in estimation errors across
countries, which leads to differential country weights in the cross-country average as well
as differences across countries in the optimal weight on local data, produces only small
reductions in the RMSE of the combined estimator. Fourthly, using historical data from
each market from both the first and second half of the data series to generate estimates of
the market risk premiums for each market, and regressing the later estimates on the
earlier ones, also generates an estimate for the optimal weight on local data that is close
to zero, but with the advantage of also generating a 95% confidence interval around the
point estimate with an upper bound of 0.25. Fifthly, using data from the first half of the
data series to generate forecasts for the market risk premiums for each market, and
28
estimating the latter from data in the second half of the data series, the cross-country
average is again clearly superior to the use of only local data both for generating the
forecasts and estimating the subsequent unobservable market risk premiums in the course
of assessing the accuracy of the forecasts.
Lastly, preliminary application of the
combined estimator to a range of methodologies other than historical averaging indicates
that the optimal weight on local data would still be low.
All of this suggests that, regardless of the methodology for estimating the market risk
premium, the usual practice of invoking only local data is significantly inferior to the use
of a cross-country average or a combined estimator with very high weighting on the
cross-country average.
29
APPENDIX
This Appendix provides the proofs for equations (2), (4), (7) and (9) in the paper. We
start with the scenario examined in section 2, in which the estimation errors arising from
using only local data for each market have the same standard deviation, all country pairs
have the same correlation coefficient for the estimation errors, and therefore the optimal
country weights within the cross-country average are equal. In respect of country j, the
estimator for the market risk premium using only data for that market ( ˆ j ) is the sum of
the true value  j and an estimation error e j , and  j is the sum of the mean of the crosssectional distribution of countries’ true market risk premiums (  ) and a random drawing
 j from that distribution:
ˆ j   j  e j     j  e j
The combined estimator places weight k on ˆ j and the balance on the cross-country
average of these estimates ( ˆ ). For a randomly selected country (designated 1), the
estimation error associated with the combined estimator is therefore as follows.
ˆc  1  kˆ1  (1  k )ˆ  1
In choosing the weighting k, and consistent with the literature (Efron and Morris, 1975),
we minimise the sum across countries of the mean squared errors. This is equivalent to
minimising the MSE of a randomly selected country (country 1) with the expectation
30
over both estimation errors and countries.
Furthermore, across both countries and
estimation errors, the estimation error shown in the last equation is mean zero and
therefore minimising the MSE (with the expectation over both countries and estimation
errors) is equivalent to minimising the variance of the estimation error for the randomly
selected country 1. Letting N denote the number of countries used in forming the crosscountry average, recognising that the random variables 1.... N are mutually independent,
that  j and ej are independent for each j, and that the random variables e1….eN are
correlated with correlation coefficient denoted ρ, the MSE using the combined estimator
is as follows.

MSE  Var ˆc  1



1 N
 Var k (   1  e1 )  (1  k )  (   j  e j )  (   1 )
N j 1


  1 k 
1 N
1 N 
 1 k 
 Var 1  k 
 1  (1  k )  j  e1  k 
  (1  k )  e j 
N
N j 2
N 
N j 2 


 
2
2




1 k
(1  k ) 2
1 k 
(1  k ) 2

2 
    k 
 1 
(
N

1
)


k


( N  1)



e 
2
2
N
N
N 
N





2

2


1  k  1  k 
2 (1  k )
 2  e2  k 
(
N

1
)


( N  1) 2  ( N  1)


e

2
N  N 
N



1
1
1 
1



  2 (1  k ) 2 1     e2   k 2 1     e2 1  k 2 1  
 N
 N 
 N
N


This is equation (2). Defining V as the expectation of the cross-sectional sample variance
in these locally-based estimates of the market risk premiums, it follows that
31
 N ˆ
ˆ 2
  ( j   ) 
j 1

V  E

N 1 





1 N
 E (ˆ j  ˆ ) 2
N  1 j 1

N
E (ˆi  ˆ ) 2
N 1
ˆ 1 N ˆ 
N

E i   j 
N 1 
N j 1 
2

N
1

E    i  ei 
N 1 
N

(   j  e j )

j 1

N
2
  N 1 

N
1
 N 1  1

E  i 
  ei 
   j   e j 
N 1   N 
N j i 
 N  N j i

2
2
2


 2
 e2
N  2  N 1
N  2  N 1


(
N

1
)



( N  1)


 

 e
2
2
N  1   N 
N
N
 N  1   N 


  N 1  1
N
( N  1) 2  ( N  1) 
 e2  2
 ( N  1) 

N 1
N2
  N N

  2   e2   e2
Solving this equation for  2 yields equation (4) as follows:
 2  V   e2 (1   )
32
We now turn to the scenario examined in section 5, in which the standard deviations of
the estimation errors from using only local data differ across countries and therefore the
country weights within the cross-country average (wj) may differ.
As before, we
minimise the MSE with the expectation over both countries and estimation errors and this
is equivalent to minimising the variance of the estimation error for randomly selected
country 1 arising from the combined estimator:

MSE  Var ˆc  1

N


 Var k (   1  e1 )  (1  k ) w j (   j  e j )  (   1 )
j 1


N
N


 Var  1 (1  k )( w1  1)  (1  k ) w j  j  e1 k  w1 (1  k )   (1  k ) w j e j 
j 2
j 2


N
N


2
  2 (1  k ) 2 ( w1  1) 2  (1  k ) 2  w 2j    e21 k  (1  k ) w1   (1  k ) 2  w 2j  ej2
j 2
j 2


N
N
N
 2  e1 k  (1  k ) w1 (1  k ) w j  ej   (1  k ) 2  wi w j  ei ej
j 2
i 2 j 2
j i
This is equation (7), which reduces to equation (2) in the special case in which the wi are
equal and the  ei are also equal. Turning to V, still defined with the cross-country
average equally weighted, this is now as follows:
33
 N ˆ
ˆ 2
  ( j   ) 

V  E  i 1
N 1 




1 N
 E (ˆi  ˆ ) 2
N  1 i 1
1 N ˆ 1 N ˆ 

j 
 E i  N 
N  1 i 1 
j 1

2

1 N 
1 N

E




e

(   j  e j )



i
s
N  1 i 1 
N j 1

2


1
1 N 
 N 1
 N 1 1 N


  ei 
   j   e j 
 E i 
N  1 i 1   N 
N j 1
 N  N j 1
j i
j i


2
N
Since the random variables  j and ej are independent, and  1 ,....,  N are mutually
independent, and the correlation coefficient between the estimation errors for country
pairs is the same for all pairs (ρ), it follows that

2
1 N  2  ( N  1) 2  N  1 
1
2  N 1
   ei 
V
  
 2


2
N  1 i 1  
N
N
 N 



 
1



N  1 i 1  N 2

N
N
N

j 1 k 1
j i k i
k j
ej
 ek
( N  1)
 N 1 N
  2   2  ei2 
( N  1) N 2
 N  i 1

 


j 1
j i

N
2
ej


 N 1
 2 2   ei   ej 
 N 
j 1

j i

N
N

j 1
2
ej
34

( N  2) 
( N  1) N 2
  2 
1
N
N

i 1
ei
N
N
  ei ej 
i 1 j 1
j i


N
2
N2
N
 
i 1 j 1
j i
N
 
( N  1) N
i 1 j 1
j i
N
ei
ei
 ej
 ej
Solving this equation for  2 yields the following:
N
N

 2  V 
1
N
N

1
2
ej
p
i 1 j 1
j i
ei
 ej
N ( N  1)
This is equation (9), which reduces to equation (4) in the special case in which the  ei are
equal.
35
Table 1: Estimated Market Risk Premiums and Standard Deviations
_____________________________________________________________________
ˆ
ˆ (ˆ)
_____________________________________________________________________
Australia
.0851
.0167
Belgium
.0572
.0234
Canada
.0573
.0163
Denmark
.0485
.0195
Finland
.0995
.0287
France
.0896
.0233
Germany
.0998
.0305
Ireland
.0558
.0204
Italy
.1020
.0303
Japan
.0952
.0263
Netherlands
.0663
.0216
New Zealand
.0596
.0174
Norway
.0611
.0251
S Africa
.0861
.0209
Spain
.0557
.0207
Sweden
.0694
.0209
Switzerland
.0520
.0179
UK
.0631
.0189
US
.0738
.0188
Mean
.0725
.0220
_____________________________________________________________________
This table shows estimates of the market risk premiums for 19 markets using the
historical averaging methodology and only local data for each market ( ˆ ), along with the
estimated standard deviations of these estimators ˆ (ˆ) .
36
Table 2: The RMSE of Competing Estimators
________________________________________________________________________
N
Vˆ
ˆ e2
̂
ˆ 2
k0
RMSE RMSE
RMSE
k=1
k = .021
k=0
________________________________________________________________________
19 .0182 2 .0224 2
.355 .0026 2
.021
.022
.014
.014
________________________________________________________________________
This table shows the estimated RMSE of competing estimators of the market risk
premium, arising from a range of weights on local data (k). The underlying parameter
values are also shown.
37
Table 3: Estimated Correlation Coefficients
_____________________________________________________________________
UK US Ge Ja Ne Fr It Swi Au Ca Sw De Sp Be Ire SA Nor NZ
_____________________________________________________________________
US
.55
Ger .28 .28
Jap
.21 .19 .31
Ne
.45 .40 .39 .15
Fr
.43 .29 .37 .21 .45
It
.23 .30 .14 .14 .24 .47
Swi .60 .53 .41 .22 .53 .55 .33
Aus .54 .44 .36 .15 .41 .47 .33 .51
Can .47 .83 .31 .22 .38 .35 .29 .55 .55
Swe .45 .47 .31 .26 .53 .48 .30 .53 .40 .51
Den .38 .35 .24 .34 .48 .37 .27 .47 .36 .44 .61
Sp
.23 .21 .09 .12 .26 .47 .34 .36 .30 .28 .40 .31
Bel .34 .36 .44 .13 .57 .62 .21 .49 .34 .37 .41 .33 .37
Ire
.65 .39 .31 .17 .47 .48 .27 .64 .61 .45 .47 .57 .37 .42
SAf .32 .45 .16 .24 .19 .22 .23 .26 .42 .55 .28 .26 .11 .19 .32
Nor .16 .24 .11 .17 .36 .38 .27 .38 .31 .43 .42 .46 .22 .28 .23 .36
NZ
.37 .28 .15 .19 .35 .39 .27 .31 .62 .34 .49 .42 .40 .22 .49 .38 .33
Fin .23 .29 .16 .24 .32 .32 .07 .35 .23 .47 .56 .45 .38 .33 .31 .30 .37 .32
_____________________________________________________________________
This table shows estimated correlation coefficients for the estimation errors for market
pairs, based upon annual data over the period 1900-2010.
38
Table 4: Optimal Market Weights
_____________________________________________________________________
w
k0
_____________________________________________________________________
Australia
.082
.166
Belgium
.042
-.061
Canada
.086
.192
Denmark
.061
.036
Finland
.028
-.110
France
.042
-.058
Germany
.025
-.117
Ireland
.055
.005
Italy
.025
-.116
Japan
.033
-.094
Netherlands
.049
-.027
New Zealand
.076
.126
Norway
.036
-.083
S Africa
.052
-.010
Spain
.053
-.005
Sweden
.052
-.010
Switzerland
.071
.099
UK
.064
.056
US
.065
.062
_____________________________________________________________________
This table shows the optimal country weights within the cross-country average when
differential standard deviations across countries for the estimation errors using local data
are recognised (w). In addition the optimal weight on local data in the combined
estimator for each country (k0) is also shown.
39
Table 5: Simulation Results
_____________________________________________________________________
e
AV (bˆ)
k0
AV (sbOLS
)
AV (sbHAC
)
ˆ
ˆ
_____________________________________________________________________
.031
.038
.038
.242
.219
Varying
.042
.036
.242
.225
_____________________________________________________________________
This table shows the average results from 5000 simulation trials, for each of two
scenarios relating to the standard deviation of the estimation errors for individual markets
when only local data is used. The first column shows the true value for the standard
deviation of the estimation error for each market. The second column shows the true
value for k0 or the cross-country average of the true values if they differ across countries.
The third column shows the average regression slope, the fourth column shows the
average estimate of the OLS standard error on the regression slope, and the last column
shows the average estimate of the HAC standard error.
40
Table 6: Sub-period Estimates of the Market Risk Premiums
_____________________________________________________________________
ˆ1
ˆ2
_____________________________________________________________________
Australia
.0993
.0730
Belgium
.0634
.0519
Canada
.0606
.0544
Denmark
.0249
.0686
Finland
.0971
.1014
France
.0821
.0960
Germany
.0873
.1090
Ireland
.0210
.0854
Italy
.1355
.0736
Japan
.0857
.1032
Netherlands
.0388
.0898
New Zealand
.0604
.0589
Norway
.0314
.0864
S Africa
.0838
.0880
Spain
.0290
.0784
Sweden
.0356
.0982
Switzerland
.0199
.0792
UK
.0320
.0896
US
.0718
.0756
_____________________________________________________________________
This table shows estimates of the market risk premiums for 19 markets using the
historical-averaging methodology and local data for each market for both 1900-1950 ( ˆ1 )
and for 1951-2010 ( ˆ ).
2
41
Table 7: Forecasting Results
_____________________________________________________________________
Forecasting Using
Forecasting Using
MRP Proxy
Local Data
Cross-Country Average
_____________________________________________________________________
Local Data
4.01%
2.64%
Cross-Country Average
3.81%
2.11%
_____________________________________________________________________
This table shows the RMSE from forecasting the market risk premiums in the period
1951-2010 using either local data or the cross-country average from 1900-1950, and
proxying the market risk premiums in the period 1951-2010 using either the local data or
the cross-country average from that period.
42
This figure shows the relationship between the RMSE of various estimators and
the standard deviation of the cross-sectional distribution of true MRPs. The
estimators considered involve placing all weight upon local data (k = 1), placing
all weight on the cross-country average (k = 0), and the optimal combination of
these two estimators. The figure shows the extent of superiority of the combined
estimator over each of its components.
43
REFERENCES
Andrews, D. “Heteroscedasticity and Autocorrelation Consistent Covariance Matrix
Estimation.” Econometrica, 59 (1991), 817-858.
Cavaglia, S.; C. Brightman; and M. Aked. “The Increasing Importance of Industry
Factors.” Financial Analysts Journal, Sept-Oct (2000), 41-54.
Chay, J.; A. Marsden; and R. Stubbs. “Historical Rates of Return to Equities, Bonds, and
the Equity Risk Premium: New Zealand Evidence.” Pacific Accounting Review, 5 (1993),
27-46.
Claus, J., and J. Thomas. “Equity Premia as Low as Three Percent?: Evidence from
Analysts’ Earnings Forecasts for Domestic and International Stocks.” The Journal of
Finance, 56 (2001), 1629-1666.
Cornell, B. The Equity Risk Premium, New York: John Wiley & Sons (1999).
Dimson, E., and P. Marsh. “Calculating the Cost of Capital.” Long Range Planning, 15
(1982), 112-120.
______________________ M. Staunton. Triumph of the Optimists, New Jersey:
Princeton University Press (2002).
__________________________________ “Global Evidence on the Equity Risk
Premium.” Journal of Applied Corporate Finance, 15 (4), Fall (2003), 27-38.
____________________________________ Credit Suisse Global Investment Returns
Sourcebook 2011. Credit Suisse (2011).
Donaldson, G.; M. Kamstra; and L. Kramer. “Estimating the Equity Premium.” Journal
of Financial and Quantitative Analysis, 45 (2010), 813-846.
Efron, B. Large Scale Inference: Empirical Bayes Methods for Estimation, Testing and
Prediction, Cambridge: Cambridge University Press (2010).
_______ and C. Morris. “Data Analysis Using Stein’s Estimator and its Generalisations.”
Journal of the American Statistical Association, 70 (1975), 311-319.
Fama, E., and K. French. “The Equity Premium.” The Journal of Finance, 57 (2002),
637-659.
Granger, C. Forecasting in Business and Economics, 2nd edition, Boston: Academic Press
Inc (1989).
44
Grauer, R., and N. Hakannson. “Stein and CAPM Estimators of the Means in Asset
Allocation.” International Review of Financial Analysis, 4 (1995), 35-66.
Guo, H., and R. Whitelaw. “Uncovering the Risk-Return Relation in the Stock Market.”
The Journal of Finance, 61 (2006), 1433-1463.
Harvey, C. “Drivers of Expected Returns in International Markets.” Emerging Markets
Quarterly, Fall (2000), 1-17.
Ibbotson, R., and R. Sinquefield. “Stocks, Bonds, Bills and Inflation; Year by Year
Historical Returns (1926-1974).” Journal of Business, 49 (1976), 11–47.
Jagannathan, R.; E. McGrattan; and A. Scherbina. “The Declining US Equity Premium.”
Federal Bank of Minneapolis Quarterly Review, 24 (2001), 3-19.
James, W., and C. Stein. “Estimation with Quadratic Loss.” Proceedings of the Fourth
Berkeley Symposium on Mathematical Statistics and Probability, 1 (1961), 361-379.
Jorion, P. “Bayes-Stein Estimation for Portfolio Analysis.” Journal of Financial and
Quantitative Analysis, 21 (1986), 279-292.
Karolyi, A. “A Bayesian Approach to Modelling Stock Return Volatility for Option
Valuation.” Journal of Financial and Quantitative Analysis, 28 (1993), 579-594.
Lehmann, E., and G. Casella. Theory of Point Estimation, 2nd edition, New York:
Springer (1998).
Lintner, J. “The Valuation of Risky Assets and the Selection of Investments in Stock
Portfolios and Capital Budgets.” Review of Economics and Statistics, 47 (1965), 13-37.
Maheu, J., and T. McCurdy. “How Useful are Historical Data for Forecasting the LongRun Equity Return Distribution.” Journal of Business & Economic Statistics, 27 (2009),
95-112.
Markowitz, H. “Portfolio Selection.” The Journal of Finance, 7 (1952), 77-91.
___________ Portfolio Selection, New Haven: Yale University Press (1959).
Mayfield, S. “Estimating the Market Risk Premium.” Journal of Financial Economics, 73
(2004), 465-496.
Merton, R. “On Estimating the Expected Return on the Market.” Journal of Financial
Economics, 8 (1980), 323-61.
Mossin, J. “Equilibrium in a Capital Asset Market.” Econometrica, 24 (1966), 768-783.
45
Officer, R. “Rates of Return to Equities, Bond Yields and Inflation Rates: An Historical
Perspective.” In Share Markets and Portfolio Theory, 2nd edition, Ball, R., F. Finn, P.
Brown, and R. Officer, eds. University of Queensland Press (1989).
Pastor, L., and R. Stambaugh. “The Equity Premium and Structural Breaks.” The Journal
of Finance, 56 (2001), 1207-1245.
Scruggs, J. “Resolving the Puzzling Intertemporal Relation between the Market Risk
Premium and Conditional Market Variance: A Two-Factor Approach”. The Journal of
Finance, 53 (1998), 575-603.
Sharpe, W. “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of
Risk.” The Journal of Finance, 19 (1964), 425-442.
Siegel, J. “The Equity Premium: Stock and Bond Returns since 1802.” Financial Analysts
Journal, Jan-Feb (1992), 28-38.
Solnik, B. “An Equilibrium Model of the International Capital Market.” Journal of
Economic Theory, 8 (1974), 500-524.
Vasicek, O. “A Note on Using Cross-Sectional Information in Bayesian Estimation of
Security Betas.” The Journal of Finance, 28 (1973), 1233-1239.
Welch, I., and A. Goyal. “A Comprehensive Look at the Empirical Performance of
Equity Premium Prediction.” The Review of Financial Studies, 21 (2008), 1455-1508.
46