Associate Professor Martin Lally presented Forward Looking Estimates of the Market Risk Premium at the half-day Regulatory Cost of Capital II: What is the Market Risk Premium? Copies of Martins underlying papers on the topics can be obtained by contacting him at martin.lally@vuw.ac.n

Lally, Martin

ResearchArchive at Victoria University of Wellington

FORWARD LOOKINGESTIMATES OF THEMARKET RISKPREMIUMASSOCIATE PROFESSOR MARTIN LALLYSCHOOL OF ECONOMICS & FINANCEVICTORIA UNIVERSITY OF WELLINGTON21 JUNE 2005TheoryThe market value of equities is equal to thepresent value of future dividends to existingshareholders in these companies, discounted atE(Rm):.....))(1()1)(1()(1)1(2211 +++++++=mmmmmREggDIVREgDIVP  .....))(1()1)(1()(1)1(12211 +++++++=mmmmREggDREgD i.e.,Estimation of E(Rm) from this equationrequires• Short-run expected dividend growth ratesg1, g2, …• Market cash dividend yield Dm • Long-run expected dividend growth  rate (g)The estimates of the MRP in twoversions of the CAPM are then asfollows••! MRPS = E(Rm )" Rf)33.1()( !!= fmT RREMRP  EstimatesThe upper bound on g is the long-runexpected growth rate in aggregatedividends for existing coys, and …the upper bound on this is the long-runexpected growth rate in aggregatedividends for all companies (present andfuture), and….this matches the long-run expectedgrowth rate in GDP⇒g = long-run expected GDP growth          – effect of share issues          – effect of new coysParameter Estimates• Cash dividend yield = 5%• Short-run expected dividend growth rates:    5%, 5%, 5%      5.1%, 5.8%      9.9%, 8.6%, 7.3%• Long-run expected growth rate in GDP:CPI = 2.1%, Real GDP = 2.9% ⇒ 5%• Deduction from GDP growth rate = 1%  ⇒g = 4%• Risk free rate = ten year government bonds = 5.9% (May 2005)ResultsSuppose the highest set of short-run growth ratesare invoked (9.9%, 8.6%, 7.3%)  and… suppose the long-run expected growth rate ofdividends per share = 5%   and… suppose convergence takes 10 years⇒expected growth rates in dividends per share overthe next 10 years are.099, .086, .073, .070, .0664, .0631, .060, .0566, .0533, .050Results cont⇒   E(Rm) = .111⇒   MRPS = .111 - .059 = .052           MRPT = .111 - .059(1-.33) = .071Table 1: Estimated Market RiskPremiums: High EPS Forecasts_____________________________________________                        N = 10          N = 20          N = 30_____________________________________________g = .030 .056 (.037) .063 (.044) .067 (.048)g = .040 .063 (.044) .069 (.050) .072 (.053)g = .050 .071 (.052) .075 (.056) .077 (.058)_____________________________________________Table 2: Estimated Market RiskPremiums: Low EPS Forecasts_______________________________________________                       N = 10            N = 20          N = 30_______________________________________________   g = .030 .047 (.028) .050 (.031) .053 (.034)   g = .040 .055 (.036) .056 (.037) .057 (.038)   g = .050 .063 (.044) .063 (.044) .063 (.044)_______________________________________________Conclusions• the central estimates for MRPT are .056and .069, averaging .062• the range of estimates is .047 to .077• the central estimate for MRPS is .043 with a range from .028 to .058• these central estimates are below thosefrom historical averaging• the central estimates are similar to those in Lally (2001)Limitations• the long-run expected growth rate in dividends per share for existing companies is unclear• the short-run expected growth rates in EPS for existing companies are unclear• the period over which short-run growth ratesconverge on the long-run growth rate is unclear• the current market value of equities is assumed to be “rational”• the market’s pricing model is consistent with the model into which the MRP estimates are insertedTHE OPTIMALESTIMATOR OF THEMARKET RISKPREMIUMASSOCIATE PROFESSOR MARTIN LALLYSCHOOL OF ECONOMICS & FINANCEVICTORIA UNIVERSITY OF WELLINGTON21 JUNE 2005Theory:Methods for estimating the market riskpremium for a particular market• Historical averaging for that market (Ibbotson)• Historical averaging of the reward-to-riskratio coupled with a current estimate ofrisk (Merton)• Forward-looking methods• Historical averaging for other markets(Ibbotson)Some estimators dominate others, but not when theydraw upon different sources of information.  Examples:• Historical averaging for that market (Ibbotson)• Forward-looking methods involving dividends• Historical averaging for other markets (Ibbotson)• Forward-looking methods involving accountingnumbersIn this case, the best estimator will be somecombination of the individual estimatorsIf estimators are unbiased, the best combination(weights w1, w2….wn) minimises standarddeviation, and these weights satisfy the followingsystem of equations.!""" =++nnwww1122111........! wn"n1+ w2"n2+ ........wn"nn= #1.......21=++nwww  ………………………………...If some estimators are biased (with biases B1,B2….Bn), the best set of weights minimises rootmean squared error (RMSE), and these weightssatisfy the following system of equations.!"" =+++ )(..........)( 1111111 nnn BBwBBw………………………………………………………………!"" =+++ )(..........)( 11 nnnnnnnn BBwBBw! w1+ w2+ .................wn=1EXAMPLE: TWO ESTIMATORSSuppose the estimators are as follows:• Ibbotson estimator for local market:     standard deviation = .02• Forward-looking estimator involving dividends:standard deviation = .02• These estimators are uncorrelatedIf the two estimators are unbiased, the weights onthe two estimators are chosen to minimise standarddeviation• Weights are 50% on each estimator• Standard deviation of the combined estimatoris .014• Reduction in standard deviation compared tothe best individual estimator is 30% of eitherindividual estimatorIf the Ibbotson estimator is biased by .02, thenweights are chosen to minimise RMSE• Weights are 33% on the Ibbotson estimator and67% on the forward-looking estimator• RMSE of the combined estimator is .0164• Reduction in RMSE compared to the best individual estimator is 18%Table 1: Optimal Estimator WeightsWith Two Estimators________________________________________________________        σ                                       Ibbotson Bias                                       0                         .02                          .04________________________________________________________.01 .20, .80 (10%) .11, .89 (6%) .05, .95 (2%).02 .50, .50 (30%) .33, .67 (18%) .17, .83 (9%).03 .69, .31 (17%) .53, .47 (27%) .31, .69 (17%)________________________________________________________Table 1: Optimal Estimator WeightsWith Two Estimators contThis table shows the optimal weights on theIbbotson estimator and an unbiased forward-lookingestimator respectively, with the bracketed figurebeing the reduction in RMSE from this optimalestimator relative to the best individual estimator.The optimal weights depend upon the bias in theIbbotson estimator and the standard deviation of theforward-looking estimator (σ).  The standarddeviation on the Ibbotson estimator is assumed tobe .02EXAMPLE: THREE ESTIMATORSSuppose the estimators are as follows:• Ibbotson estimator for the local market: standard deviation = .02• Forward-looking estimator involving dividends:standard deviation = .02• Ibbotson estimator for the rest of the world:standard deviation = .014• The forward-looking estimator is uncorrelatedwith the other two• The correlation between the Ibbotson estimators= .60If all three estimators are unbiased, the weightson the three estimators are chosen to minimisestandard deviation:• Weights are 7%, 32% and 61%• Standard deviation of the combined estimator is .0114• Reduction in standard deviation compared to the best individual estimatoris 19% of the best individual estimatorIf the Ibbotson estimator for the rest of the world isbiased by .02, the weights are then chosen tominimise RMSE:• Weights are 36%, 45% and 19%• RMSE of the combined estimator is .0134• Reduction in RMSE compared to the best individual estimator is 33%Table 2: Optimal Estimator WeightsWith Three Estimators___________________________________________________________ σ                                             World Ibbotson Bias                       0                                    .02                                  .04___________________________________________________________.01 .03, .66, .31 (19%) .16, .76, .08 (13%)     .18, .79, .03 (11%).02 .07, .32, .61 (19%) .36, .45, .19 (33%)     .41, .53, .06 (30%).03 .08, .18, .74 (10%) .48, .26, .26 (23%)     .62, .29, .09 (19%)___________________________________________________________This table shows the optimal weights on theIbbotson estimator for the local market, the forward-looking estimator, and the Ibbotson estimator for theworld market respectively, with the bracketed figurebeing the reduction in RMSE from this optimalestimator relative to the best individual estimator.The optimal weights depend upon the bias in theIbbotson estimator for the world market and thestandard deviation of the forward-looking estimator(σ).  The standard deviation on the Ibbotsonestimator for the local market is assumed to be .02,that on the Ibbotson estimator for the world marketis assumed to be .014, and the last two estimatorsare assumed to have a correlation coefficient of .60.CONCLUSIONS• With two estimators, the reduction in RMSE isup to 30% of the best individual estimator• With three estimators, the reduction in RMSE isup to 35% of the best individual estimator• With four estimators, the reduction in RMSE isup to 38% of the best individual estimator• These maximum reductions in RMSE occur when the RMSE for the individual estimatorsare equal• These diversification benefits are similar in principle to those arising in portfolio analysis

Forward Looking Estimates of the Market Risk Premium

Victoria University of Wellington

FORWARD LOOKING
ESTIMATES OF THE
MARKET RISK
PREMIUM
ASSOCIATE PROFESSOR MARTIN LALLY
SCHOOL OF ECONOMICS & FINANCE
VICTORIA UNIVERSITY OF WELLINGTON
21 JUNE 2005
Theory
The market value of equities is equal to the
present value of future dividends to existing
shareholders in these companies, discounted at
E(Rm):
.....
))(1(
)1)(1(
)(1
)1(
2
211 +
+
++
+
+
+
=
m
m
m
m
m
RE
ggDIV
RE
gDIV
P  
.....
))(1(
)1)(1(
)(1
)1(
1
2
211 +
+
++
+
+
+
=
m
m
m
m
RE
ggD
RE
gD
 
i.e.,
Estimation of E(Rm) from this equation
requires
• Short-run expected dividend growth rates
g1, g2, …
• Market cash dividend yield Dm
 
• Long-run expected dividend growth  
rate (g)
The estimates of the MRP in two
versions of the CAPM are then as
follows
•
•
! 
MRPS = E(Rm )" Rf
)33.1()( !!= fmT RREMRP  
Estimates
The upper bound on g is the long-run
expected growth rate in aggregate
dividends for existing coys, and …
the upper bound on this is the long-run
expected growth rate in aggregate
dividends for all companies (present and
future), and….
this matches the long-run expected
growth rate in GDP
⇒g = long-run expected GDP growth
          – effect of share issues
          – effect of new coys
Parameter Estimates
• Cash dividend yield = 5%
• Short-run expected dividend growth rates:
    5%, 5%, 5%      5.1%, 5.8%      9.9%, 8.6%, 7.3%
• Long-run expected growth rate in GDP:
CPI = 2.1%, Real GDP = 2.9% ⇒ 5%
• Deduction from GDP growth rate = 1%  ⇒
g = 4%
• Risk free rate = ten year government bonds 
= 5.9% (May 2005)
Results
Suppose the highest set of short-run growth rates
are invoked (9.9%, 8.6%, 7.3%)  and
… suppose the long-run expected growth rate of
dividends per share = 5%   and
… suppose convergence takes 10 years
⇒expected growth rates in dividends per share over
the next 10 years are
.099, .086, .073, .070, .0664, .0631, .060, .0566, .0533, .050
Results cont
⇒   E(Rm) = .111
⇒   MRPS = .111 - .059 = .052    
       MRPT = .111 - .059(1-.33) = .071
Table 1: Estimated Market Risk
Premiums: High EPS Forecasts
_____________________________________________
                        N = 10          N = 20          N = 30
_____________________________________________
g = .030 .056 (.037) .063 (.044) .067 (.048)
g = .040 .063 (.044) .069 (.050) .072 (.053)
g = .050 .071 (.052) .075 (.056) .077 (.058)
_____________________________________________
Table 2: Estimated Market Risk
Premiums: Low EPS Forecasts
_______________________________________________
                       N = 10            N = 20          N = 30
_______________________________________________
   g = .030 .047 (.028) .050 (.031) .053 (.034)
   g = .040 .055 (.036) .056 (.037) .057 (.038)
   g = .050 .063 (.044) .063 (.044) .063 (.044)
_______________________________________________
Conclusions
• the central estimates for MRPT are .056
and .069, averaging .062
• the range of estimates is .047 to .077
• the central estimate for MRPS is .043 
with a range from .028 to .058
• these central estimates are below those
from historical averaging
• the central estimates are similar to 
those in Lally (2001)
Limitations
• the long-run expected growth rate in 
dividends per share for existing companies 
is unclear
• the short-run expected growth rates in EPS 
for existing companies are unclear
• the period over which short-run growth rates
converge on the long-run growth rate is 
unclear
• the current market value of equities is 
assumed to be “rational”
• the market’s pricing model is consistent 
with the model into which the MRP 
estimates are inserted
THE OPTIMAL
ESTIMATOR OF THE
MARKET RISK
PREMIUM
ASSOCIATE PROFESSOR MARTIN LALLY
SCHOOL OF ECONOMICS & FINANCE
VICTORIA UNIVERSITY OF WELLINGTON
21 JUNE 2005
Theory:
Methods for estimating the market risk
premium for a particular market
• Historical averaging for that market 
(Ibbotson)
• Historical averaging of the reward-to-risk
ratio coupled with a current estimate of
risk (Merton)
• Forward-looking methods
• Historical averaging for other markets
(Ibbotson)
Some estimators dominate others, but not when they
draw upon different sources of information.  Examples:
• Historical averaging for that market (Ibbotson)
• Forward-looking methods involving dividends
• Historical averaging for other markets (Ibbotson)
• Forward-looking methods involving accounting
numbers
In this case, the best estimator will be some
combination of the individual estimators
If estimators are unbiased, the best combination
(weights w1, w2….wn) minimises standard
deviation, and these weights satisfy the following
system of equations.
!""" =++
nn
www
1122111
........
! 
w
n
"
n1
+ w
2
"
n2
+ ........w
n
"
nn
= #
1.......
21
=++
n
www  
………………………………...
If some estimators are biased (with biases B1,
B2….Bn), the best set of weights minimises root
mean squared error (RMSE), and these weights
satisfy the following system of equations.
!"" =+++ )(..........)( 1111111 nnn BBwBBw
………………………………………………………………
!"" =+++ )(..........)( 11 nnnnnnnn BBwBBw
! 
w
1
+ w
2
+ .................w
n
=1
EXAMPLE: TWO ESTIMATORS
Suppose the estimators are as follows:
• Ibbotson estimator for local market:     
standard deviation = .02
• Forward-looking estimator involving dividends:
standard deviation = .02
• These estimators are uncorrelated
If the two estimators are unbiased, the weights on
the two estimators are chosen to minimise standard
deviation
• Weights are 50% on each estimator
• Standard deviation of the combined estimator
is .014
• Reduction in standard deviation compared to
the best individual estimator is 30% of either
individual estimator
If the Ibbotson estimator is biased by .02, then
weights are chosen to minimise RMSE
• Weights are 33% on the Ibbotson estimator and
67% on the forward-looking estimator
• RMSE of the combined estimator is .0164
• Reduction in RMSE compared to the best 
individual estimator is 18%
Table 1: Optimal Estimator Weights
With Two Estimators
________________________________________________________
        σ                                       Ibbotson Bias
                                       0                         .02                          .04
________________________________________________________
.01 .20, .80 (10%) .11, .89 (6%) .05, .95 (2%)
.02 .50, .50 (30%) .33, .67 (18%) .17, .83 (9%)
.03 .69, .31 (17%) .53, .47 (27%) .31, .69 (17%)
________________________________________________________
Table 1: Optimal Estimator Weights
With Two Estimators cont
This table shows the optimal weights on the
Ibbotson estimator and an unbiased forward-looking
estimator respectively, with the bracketed figure
being the reduction in RMSE from this optimal
estimator relative to the best individual estimator.
The optimal weights depend upon the bias in the
Ibbotson estimator and the standard deviation of the
forward-looking estimator (σ).  The standard
deviation on the Ibbotson estimator is assumed to
be .02
EXAMPLE: THREE ESTIMATORS
Suppose the estimators are as follows:
• Ibbotson estimator for the local market: 
standard deviation = .02
• Forward-looking estimator involving dividends:
standard deviation = .02
• Ibbotson estimator for the rest of the world:
standard deviation = .014
• The forward-looking estimator is uncorrelated
with the other two
• The correlation between the Ibbotson estimators
= .60
If all three estimators are unbiased, the weights
on the three estimators are chosen to minimise
standard deviation:
• Weights are 7%, 32% and 61%
• Standard deviation of the combined 
estimator is .0114
• Reduction in standard deviation 
compared to the best individual estimator
is 19% of the best individual estimator
If the Ibbotson estimator for the rest of the world is
biased by .02, the weights are then chosen to
minimise RMSE:
• Weights are 36%, 45% and 19%
• RMSE of the combined estimator is .0134
• Reduction in RMSE compared to the best 
individual estimator is 33%
Table 2: Optimal Estimator Weights
With Three Estimators
___________________________________________________________
 σ                                             World Ibbotson Bias
                       0                                    .02                                  .04
___________________________________________________________
.01 .03, .66, .31 (19%) .16, .76, .08 (13%)     .18, .79, .03 (11%)
.02 .07, .32, .61 (19%) .36, .45, .19 (33%)     .41, .53, .06 (30%)
.03 .08, .18, .74 (10%) .48, .26, .26 (23%)     .62, .29, .09 (19%)
___________________________________________________________
This table shows the optimal weights on the
Ibbotson estimator for the local market, the forward-
looking estimator, and the Ibbotson estimator for the
world market respectively, with the bracketed figure
being the reduction in RMSE from this optimal
estimator relative to the best individual estimator.
The optimal weights depend upon the bias in the
Ibbotson estimator for the world market and the
standard deviation of the forward-looking estimator
(σ).  The standard deviation on the Ibbotson
estimator for the local market is assumed to be .02,
that on the Ibbotson estimator for the world market
is assumed to be .014, and the last two estimators
are assumed to have a correlation coefficient of .60.
CONCLUSIONS
• With two estimators, the reduction in RMSE is
up to 30% of the best individual estimator
• With three estimators, the reduction in RMSE is
up to 35% of the best individual estimator
• With four estimators, the reduction in RMSE is
up to 38% of the best individual estimator
• These maximum reductions in RMSE occur 
when the RMSE for the individual estimators
are equal
• These diversification benefits are similar in 
principle to those arising in portfolio analysis


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Forward Looking Estimates of the Market Risk Premium

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