第五章 CAPM的应用
利用 Markowitz 模型进行积极证券组合管理
市场模型在消极证券组合管理中的应用
利用 Beta去得到好的协方差估计
利用 Beta去得到好的期望回报率估计
CAPM在消极证券组合管理中的应用
Black-Litterman 方法
例子,Global Portfolio Optimization
1,利用 Markowitz 模型进行积极证券组合管理
经典 Markowitz 模型的缺点
待估计的期望值、协方差参数数量大
利用历史数据得到的最优证券组合权重不合理
When investors impose no constraints,the models
almost always ordain large short positions in many
assets,
When constraints rule out short positions,the models
often prescribe corner solutions with zero weights in
many assets,as well as unreasonably large weights in
the assets of markets with small capitalizations.
These unreasonable results stem from two well
recognized problems:
由历史数据得到的期望值估计对将来回报率预测能力很差
最优证券组合权重对于期望回报率假设非常敏感
例,100种证券形成的证券组合例,let us look at the sort of portfolio allocation we get
if we use historical returns and volatilities as inputs:
Historical correlations:
If you use our procedures and calculate and optimal
portfolio,with,you will get portfolio
weights of,%7.10?P?
We can make a number of points about these
optimal portfolios.
They illustrate what we mean when we claim that
standard mean-variance optimization models often
generate unreasonable portfolios.
The use of past excess returns to represent a neutral set
of views is equivalent to assuming that the constant
portfolio weights that would have performed best
historically are in some sense neutral,In reality,of
course,they are not neutral at all,but rather are a very
special set of weights that go short assets that have
done poorly and go long assets that have done well in
the particular historical period.
A remedy for both of these problems is to use
(1) market model to calculate asset covariance,
(2) and use the CAPM to determine what market
expectation must be,and then combine your,view”
with the CAPM derived estimates to get portfolio
weights.
The key input we will need for both of these
is the set of asset betas,so,first,we must
consider the problem of estimating betas
2,Beta值的估计利用市场模型估计
An approach to estimating and is to
assume that a market model ( and the CAPM)
describes returns
The market model
To get expected returns use:
To get covariances /correlations,use:
For standard deviations:
fMifi rrErrE
irEij?
2Mjiij
ji
Mji
ij
2?
2222 iMiiii
itftMtiiftit rrrr
待估计的参数数量大大减少（例如,100种证券形成的证券组合）
i?
3,CAPM与积极的证券组合管理
为了得到最优证券组合，我们需要估计证券组合前沿，有效集
方法一：完全忽略市场的观点，而估计所有证券的期望回报率、协方差
例如：利用历史的数据
该方法存在问题
方法二：接受市场的观点，简单的持有市场证券组合
如果你拥有市场价格还没有反映的信息，该方法并不告诉该如何处理
The approach we will instead take is to
1,Calculate beta’s for the securities we plan to hold
2,Using these beta’s,calculate and,
assuming the CAPM holds exactly
3,Then,incorporate our information by,perturbing”
the values away from the CAPM-calculated
values
4,Finally,using these estimates and the Markowitz
portfolio optimization tools determine our
optimal portfolio weights,
ijirE
Note that if
We start with all of the assets in market portfolio
We use the unmodified and we get from
step 2 above
Then the weights we calculate in step 4 will be
exactly those of the market portfolio
ijirE
An example
Using monthly data for GE,IBM,Exxon (XON),and GM for 94:01-98:12
---VW is the Value- Weighted index of all NYSE,AMEX,and NASDAQ common stocks.
Rf is the (nominal) 1-month T-Bill yield,which was 4.394%/year (0.359%/month) in
January’ 99.
Excess Returns
Mean(%) Std(%) alpha(%) beta Std(%) (%)
IBM 3.22 8.44 1.72 1.14 7.13 28.5
XON 1.41 4.03 0.63 0.59 3.28 33.7
GM 0.64 7.34 -0.69 1.02 6.14 30.0
GE 2.26 5.86 0.90 1.04 4.15 49.9
VW 1.31 4.02
Rf 0.39 0.05
2adjR
To get the correlation structure,we have,
ji
Mji
ij
2?
IBM XON GM GE
IBM 1 0.32 0.30 0.39
XON 0.32 1 0.33 0.42
GM 0.30 0.33 1 0.40
GE 0.39 0.42 0.40 1
Plugging the (1) expected return,(2) return standard deviation,
and (3) correlation matrix into the Excel spreadsheet we get the
following weights for the tangency portfolio:
Is this a reasonable portfolio? Why?
Weight(%)
IBM 29.6
XON 49.4
GM -21.4
GE 42.4
It seems unreasonable that we should
hold such extreme portfolio positions.
The equilibrium arguments we used in
developing the CAPM suggest that the
market knows something we don’t about
future expected returns!
Use the CAPM as a way of getting around this
problem:
Use the SML:
and the past (average) return on the market to
get equilibrium estimates of the expected returns,
fMifi rrErrE
(%)
IBM 1.91
XON 0.99
GM 1.70
GE 1.73
irE
With this equilibrium set of expected returns,we now get the
portfolio weights:
Is this a more reasonable portfolio?
Now,what is driving the portfolio weights?
Why are these not the market weights?
When would these be the actual market weights?
Is this the portfolio you want to hold,given that you were
constrained to hold these four assets?
Weight(%)
IBM 13.6
XON 33.3
GM 16.4
GE 36.6
However,there may be times when we think that the
market is a little wrong along one or more
dimensions ( a very dangerous assumption!)
1,First,suppose that I think that the market has
underestimated the earnings that IBM will announce in the
next month,and that IBM’s expected return is 2% higher
than the market expects.
Also,I have no information on the other three securities that
would lead me to think that they are mispriced,
and I believe that the past betas,and residual std dev’s are
good indications of the relative future values.
2,In this case,we would use the same variance and covariance
inputs,but would change the expected returns to:
and gives portfolio weights of:
(%)
IBM 3.91
XON 0.99
GM 1.70
GE 1.73
irE
Weight(%)
IBM 54.1
XON 17.7
GM 8.7
GE 19.5
3,Alternatively,suppose that I think the risk ( )
of Exxon is increasing.
4,I guess that Exxon’s will rise from 0.59 to 0.8
First,I should recalculate almost everything using the
equations:
fMifi rrErrE
2Mjiij
ji
Mji
ij
2?
2222 iMiiii
The new correlations we come up with are
as opposed to the old correlation matrix of
IBM XON GM GE
IBM 1 0.44 0.30 0.39
XON 0.44 1 0.45 0.57
GM 0.30 0.45 1 0.40
GE 0.39 0.57 0.40 1
IBM XON GM GE
IBM 1 0.32 0.30 0.39
XON 0.32 1 0.33 0.42
GM 0.30 0.33 1 0.40
GE 0.39 0.42 0.40 1
If,I believe that these are the new correlations,but that
the market still believes that the past correlations
represent the future (and will not discover this
information over the next several months) then I would
use the old,giving new portfolio
weights of
%99.0?X O NrE
Weight(%)
IBM 17.0
XON 16.7
GM 20.5
GE 45.7
If,however,I believe that market knows that the of
Exxon is higher,and the expected return on Exxon is
higher now to compensate for the increased risk,then
the expected returns and weights become:
Why is the weight on Exxon higher than initially,given
that the of Exxon is now higher?
(%) Weight(%)
IBM 1.91 9.2
XON 1.34 54.8
GM 1.70 11.1
GE 1.73 24.8
irE
One other alternative is that I do not believe
that the market yet knows that the risk of
Exxon is higher,and will discover this in the
next few months.
What should I do in this case?
Finally,if I am only somewhat confident of my
belief that the market will not adjust the price
of Exxon properly,I might want to adjust the
portfolio weights only part way
This is what the Black-Litterman method does,as we
shall see next
Black and Litterman,"Global
Asset Allocation"
A,Introduction
1,Black & Litterman develop a model analogous
to what we just did with the CAPM.
2,Black and Litterman use an international
version of the CAPM (developed by Fischer Black)
to get the "baseline"expected return.
3,They assume that the estimated variances and
covariances are correct,and calculate the
"equilibrium" expected returns based on these.
This is appropriate,since covariances can be
estimated accurately,especially using daily data.
4,Then,they allow the portfolio manager to
specify any number of market \views" in the form
of expected returns or differences in expected
returns,and a variance (measure of uncertainty)
for each of the views
If the manager holds no views,she will hold the
equilibrium/market portfolio.
Or,if her views are high variance (low certainty),she
will hold close to the equilibrium portfolio.
When her views are low variance,she will move
considerably away from the market portfolio.
5,This method is useful in that it tells you how to
optimally incorporate your information/views to
"tilt" your portfolio,taking advantage of the
correlation structure to hedge large positions.
NOTE!
By itself,the equilibrium concept is
interesting but not particularly useful,
Its real value is to provide a neutral
framework the investor can adjust
according to his own views,
optimization objectives and constraints.
Expressing views
Investor trying to use quantitative asset allocation models must
translate their views into a complete set of expected excess
returns on assets that can be used as a basis for portfolio
optimization,The problem is that optimal portfolio weights from
a mean-variance model are incredibly sensitive to minor
changes in expected excess returns,The advantage of
incorporating a global equilibrium will become apparent when
we show how to combine it with an investor’s views to generate
well-behaved portfolios,without requiring the investor to
express a complete set of expected excess returns.
We should emphasize that the distinction we are making-----
between investor views on the one hand and a complete set of
expected excess returns for all asset on the other ----is not
usually recognized.
Most views are relative.
How B-L approach translates a few views into
expected excess returns for all assets is one
of its more complex features,but also one of
its most innovative
There are two distinct sources of information
about future excess returns---investor views and
market equilibrium.
Assume both sources of information are uncertain
and are best expressed as probability distributions.
Choose expected excess returns that are as
consistent as possible with both sources of
information.
利用 Markowitz 模型进行积极证券组合管理
市场模型在消极证券组合管理中的应用
利用 Beta去得到好的协方差估计
利用 Beta去得到好的期望回报率估计
CAPM在消极证券组合管理中的应用
Black-Litterman 方法
例子,Global Portfolio Optimization
1,利用 Markowitz 模型进行积极证券组合管理
经典 Markowitz 模型的缺点
待估计的期望值、协方差参数数量大
利用历史数据得到的最优证券组合权重不合理
When investors impose no constraints,the models
almost always ordain large short positions in many
assets,
When constraints rule out short positions,the models
often prescribe corner solutions with zero weights in
many assets,as well as unreasonably large weights in
the assets of markets with small capitalizations.
These unreasonable results stem from two well
recognized problems:
由历史数据得到的期望值估计对将来回报率预测能力很差
最优证券组合权重对于期望回报率假设非常敏感
例,100种证券形成的证券组合例,let us look at the sort of portfolio allocation we get
if we use historical returns and volatilities as inputs:
Historical correlations:
If you use our procedures and calculate and optimal
portfolio,with,you will get portfolio
weights of,%7.10?P?
We can make a number of points about these
optimal portfolios.
They illustrate what we mean when we claim that
standard mean-variance optimization models often
generate unreasonable portfolios.
The use of past excess returns to represent a neutral set
of views is equivalent to assuming that the constant
portfolio weights that would have performed best
historically are in some sense neutral,In reality,of
course,they are not neutral at all,but rather are a very
special set of weights that go short assets that have
done poorly and go long assets that have done well in
the particular historical period.
A remedy for both of these problems is to use
(1) market model to calculate asset covariance,
(2) and use the CAPM to determine what market
expectation must be,and then combine your,view”
with the CAPM derived estimates to get portfolio
weights.
The key input we will need for both of these
is the set of asset betas,so,first,we must
consider the problem of estimating betas
2,Beta值的估计利用市场模型估计
An approach to estimating and is to
assume that a market model ( and the CAPM)
describes returns
The market model
To get expected returns use:
To get covariances /correlations,use:
For standard deviations:
fMifi rrErrE
irEij?
2Mjiij
ji
Mji
ij
2?
2222 iMiiii
itftMtiiftit rrrr
待估计的参数数量大大减少（例如,100种证券形成的证券组合）
i?
3,CAPM与积极的证券组合管理
为了得到最优证券组合，我们需要估计证券组合前沿，有效集
方法一：完全忽略市场的观点，而估计所有证券的期望回报率、协方差
例如：利用历史的数据
该方法存在问题
方法二：接受市场的观点，简单的持有市场证券组合
如果你拥有市场价格还没有反映的信息，该方法并不告诉该如何处理
The approach we will instead take is to
1,Calculate beta’s for the securities we plan to hold
2,Using these beta’s,calculate and,
assuming the CAPM holds exactly
3,Then,incorporate our information by,perturbing”
the values away from the CAPM-calculated
values
4,Finally,using these estimates and the Markowitz
portfolio optimization tools determine our
optimal portfolio weights,
ijirE
Note that if
We start with all of the assets in market portfolio
We use the unmodified and we get from
step 2 above
Then the weights we calculate in step 4 will be
exactly those of the market portfolio
ijirE
An example
Using monthly data for GE,IBM,Exxon (XON),and GM for 94:01-98:12
---VW is the Value- Weighted index of all NYSE,AMEX,and NASDAQ common stocks.
Rf is the (nominal) 1-month T-Bill yield,which was 4.394%/year (0.359%/month) in
January’ 99.
Excess Returns
Mean(%) Std(%) alpha(%) beta Std(%) (%)
IBM 3.22 8.44 1.72 1.14 7.13 28.5
XON 1.41 4.03 0.63 0.59 3.28 33.7
GM 0.64 7.34 -0.69 1.02 6.14 30.0
GE 2.26 5.86 0.90 1.04 4.15 49.9
VW 1.31 4.02
Rf 0.39 0.05
2adjR
To get the correlation structure,we have,
ji
Mji
ij
2?
IBM XON GM GE
IBM 1 0.32 0.30 0.39
XON 0.32 1 0.33 0.42
GM 0.30 0.33 1 0.40
GE 0.39 0.42 0.40 1
Plugging the (1) expected return,(2) return standard deviation,
and (3) correlation matrix into the Excel spreadsheet we get the
following weights for the tangency portfolio:
Is this a reasonable portfolio? Why?
Weight(%)
IBM 29.6
XON 49.4
GM -21.4
GE 42.4
It seems unreasonable that we should
hold such extreme portfolio positions.
The equilibrium arguments we used in
developing the CAPM suggest that the
market knows something we don’t about
future expected returns!
Use the CAPM as a way of getting around this
problem:
Use the SML:
and the past (average) return on the market to
get equilibrium estimates of the expected returns,
fMifi rrErrE
(%)
IBM 1.91
XON 0.99
GM 1.70
GE 1.73
irE
With this equilibrium set of expected returns,we now get the
portfolio weights:
Is this a more reasonable portfolio?
Now,what is driving the portfolio weights?
Why are these not the market weights?
When would these be the actual market weights?
Is this the portfolio you want to hold,given that you were
constrained to hold these four assets?
Weight(%)
IBM 13.6
XON 33.3
GM 16.4
GE 36.6
However,there may be times when we think that the
market is a little wrong along one or more
dimensions ( a very dangerous assumption!)
1,First,suppose that I think that the market has
underestimated the earnings that IBM will announce in the
next month,and that IBM’s expected return is 2% higher
than the market expects.
Also,I have no information on the other three securities that
would lead me to think that they are mispriced,
and I believe that the past betas,and residual std dev’s are
good indications of the relative future values.
2,In this case,we would use the same variance and covariance
inputs,but would change the expected returns to:
and gives portfolio weights of:
(%)
IBM 3.91
XON 0.99
GM 1.70
GE 1.73
irE
Weight(%)
IBM 54.1
XON 17.7
GM 8.7
GE 19.5
3,Alternatively,suppose that I think the risk ( )
of Exxon is increasing.
4,I guess that Exxon’s will rise from 0.59 to 0.8
First,I should recalculate almost everything using the
equations:
fMifi rrErrE
2Mjiij
ji
Mji
ij
2?
2222 iMiiii
The new correlations we come up with are
as opposed to the old correlation matrix of
IBM XON GM GE
IBM 1 0.44 0.30 0.39
XON 0.44 1 0.45 0.57
GM 0.30 0.45 1 0.40
GE 0.39 0.57 0.40 1
IBM XON GM GE
IBM 1 0.32 0.30 0.39
XON 0.32 1 0.33 0.42
GM 0.30 0.33 1 0.40
GE 0.39 0.42 0.40 1
If,I believe that these are the new correlations,but that
the market still believes that the past correlations
represent the future (and will not discover this
information over the next several months) then I would
use the old,giving new portfolio
weights of
%99.0?X O NrE
Weight(%)
IBM 17.0
XON 16.7
GM 20.5
GE 45.7
If,however,I believe that market knows that the of
Exxon is higher,and the expected return on Exxon is
higher now to compensate for the increased risk,then
the expected returns and weights become:
Why is the weight on Exxon higher than initially,given
that the of Exxon is now higher?
(%) Weight(%)
IBM 1.91 9.2
XON 1.34 54.8
GM 1.70 11.1
GE 1.73 24.8
irE
One other alternative is that I do not believe
that the market yet knows that the risk of
Exxon is higher,and will discover this in the
next few months.
What should I do in this case?
Finally,if I am only somewhat confident of my
belief that the market will not adjust the price
of Exxon properly,I might want to adjust the
portfolio weights only part way
This is what the Black-Litterman method does,as we
shall see next
Black and Litterman,"Global
Asset Allocation"
A,Introduction
1,Black & Litterman develop a model analogous
to what we just did with the CAPM.
2,Black and Litterman use an international
version of the CAPM (developed by Fischer Black)
to get the "baseline"expected return.
3,They assume that the estimated variances and
covariances are correct,and calculate the
"equilibrium" expected returns based on these.
This is appropriate,since covariances can be
estimated accurately,especially using daily data.
4,Then,they allow the portfolio manager to
specify any number of market \views" in the form
of expected returns or differences in expected
returns,and a variance (measure of uncertainty)
for each of the views
If the manager holds no views,she will hold the
equilibrium/market portfolio.
Or,if her views are high variance (low certainty),she
will hold close to the equilibrium portfolio.
When her views are low variance,she will move
considerably away from the market portfolio.
5,This method is useful in that it tells you how to
optimally incorporate your information/views to
"tilt" your portfolio,taking advantage of the
correlation structure to hedge large positions.
NOTE!
By itself,the equilibrium concept is
interesting but not particularly useful,
Its real value is to provide a neutral
framework the investor can adjust
according to his own views,
optimization objectives and constraints.
Expressing views
Investor trying to use quantitative asset allocation models must
translate their views into a complete set of expected excess
returns on assets that can be used as a basis for portfolio
optimization,The problem is that optimal portfolio weights from
a mean-variance model are incredibly sensitive to minor
changes in expected excess returns,The advantage of
incorporating a global equilibrium will become apparent when
we show how to combine it with an investor’s views to generate
well-behaved portfolios,without requiring the investor to
express a complete set of expected excess returns.
We should emphasize that the distinction we are making-----
between investor views on the one hand and a complete set of
expected excess returns for all asset on the other ----is not
usually recognized.
Most views are relative.
How B-L approach translates a few views into
expected excess returns for all assets is one
of its more complex features,but also one of
its most innovative
There are two distinct sources of information
about future excess returns---investor views and
market equilibrium.
Assume both sources of information are uncertain
and are best expressed as probability distributions.
Choose expected excess returns that are as
consistent as possible with both sources of
information.
