第四章 最优投资组合（实际）
Estimation of the inputs
A computer exercise using real data
Implementation problem and some
solutions
Estimation results in uncertainty
Shrinkage and Bayesian inference to deal
with uncertainty
Imposing restrictions.
1.投资者的最优投资组合过程
投资者的最优投资组合过程可以分为三步：
数据的估计
寻找最优投资集
选择最优证券组合第一步：数据的估计
数据的估计：如何得到证券回报率的期望值、方差、协方差
标准差相对稳定，利用历史数据估计
但期望值不一定
仅仅关注过去可能导致错误的结果
将来和过去可能完全不同期望值的估计方法
样本均值
在如下条件下合理
股票指标保持不变
将来的回报率和过去的回报率由同一个模型产生
否则，需要根据现在的环境调整
寻找特殊变量来预测回报率
寻找现在可以观测，可以预测将来回报率的变量
一般不容易找到
已有的几个例子：红利收益、红利价格比、市盈率、一些利率变量
回归
例如
t
t
t p
dr
1
期望值的估计方法
利用 CAPM,APT等模型
A model weighting game in estimating
expected returns (By Lubos Pastor)
It is hard to overstate the importance of expected
returns in investment,(money manager,corporate
manager,ordinary consumer).
Unfortunately,expected returns are as elusive as
they are important,There is no absolute agreement
among finance professionals on how expected
returns should be estimated.
The best estimates are produced by combining
finance theory with historical returns data and our
own judgment.
Does history repeat itself?
Unless we suspect that expected return
changes nontrivially over time,the sample
average return is an unbiased estimator of
expected return,The unbiasedness of the
sample average return is its main advantage.
The main disadvantage of the sample
average is its imprecision.
GM,1991-2000,the sample average return is
14% per year,The standard error is 10% per year,
With 95% confidence,the true expected return is
within two standard errors of the sample average,
[-6%,34%].
Does history repeat itself?
Would the precision increase if we used
weekly instead of monthly data? Although
higher-frequency data helps in estimating
variances and covariance of returns,it does
not help in estimating expected returns.
Intuitively,what matters for expected return is the
beginning and ending levels of prices over a given
period,but not what happens in between.
Does history repeat itself?
The only way to get a more precise sample
average is to collect more data further back
in time,But as we add older data,we gain
precision at the expense of introducing
potential bias.
GM,1925-2000,historical average is 15.5%,
standard error is 3.4%,[8.7%,22.3%],The
interval is still too wide for comfort.
Moreover,GM is very different from seventy years
ago,so the current estimate could be
contaminated by old data.
Does history repeat itself?
In general,as we add older data,we gain precision
at the expense of introducing potential bias,Striking
the right balance is difficult and requires sound
judgment.
Despite its drawbacks,the long-run average return is
a popular estimator for expected returns on
aggregate market indices,Unfortunately,we have no
theory for what the expected market return should
be.
Luckily,for individual stocks and most portfolios,we
can rely on estimates produced by theoretical asset
pricing models,Those estimates tend to be
substantially more precise than sample averages.
Theory is good
CAPM
In order to use CAPM
Riskless rate,
beta,
equity premium
Theory is good
What value we choose for the risk-free rate depend
on our objectives.
If we want to forecast expected stock returns over the next
month,the appropriate risk-free rate is the yield to maturity
on a Treasury bill that matures in one month,
If we want to estimate the firm's cost of capital in order to
value the firm's future cash flows,the risk-free rate should
be derived from a longer-term Treasury bond,such as a ten-
year bond,The bond's duration should come close to the
duration of the firm's cash flows,
Very long-term bonds should be avoided,because their
yields might also reflect premiums for risks such as inflation.
Theory is good
Beta
Beta is typically estimated by regressing
the most recent five to ten years worth of
monthly stock returns on market returns.
the estimate of GM's beta using its monthly
data in January 1996 through December 2000
is 1.11,the 95% confidence interval for GM's
beta based on the monthly data is 0.65 to 1.57.
Theory is good
How much data should we use to estimate beta?
The further we go back in time,the higher the statistical
precision of the estimate,but the bigger the possibility of
introducing some old data bias.
Unlike with sample averages,however,here it often pays
to use more frequent data,For example,whereas the
95% confidence interval for GM's beta based on the
monthly data is 0.65 to 1.57,this interval based on
weekly data is tighter,0.69 to 1.08,GM's beta estimated
using weekly data is 0.88.
Theory is good
However,going from monthly to weekly data is
recommended only for the most liquid and volatile stocks.
For other stocks,some week-to-week price changes are
simply movements between the bid and ask prices around
the same true price,which introduces additional error into
the estimation,Also,it may take a while for market-wide
news to get into the prices of illiquid stocks,which biases
the usual beta estimates downward,Conveniently,betas of
illiquid stocks can be estimated using an alternative
approach developed by the Chicago economists Myron
Scholes and Joseph Williams (1977,J,F,E).
Theory is good
The shrinkage estimate of the beta
The usual estimates of beta contain a fair amount of
noise,
A common and useful way of reducing that noise is to
"shrink" the usual estimates to a reasonable value,such
as one,The value of one is reasonable because the
average beta across all stocks is one,by construction.
The shrinkage estimate of beta is the weighted of the
sample estimate and of the shrinkage target,
Example,the "adjusted" betas reported by Merrill Lynch
put a 2/3 weight on the sample estimate and a 1/3 weight
on the value of one,The adjusted weekly beta for GM is
therefore (2/3)*0.88 + (1/3)*1 =0.92.
Theory is good
Shrinkage betas can be justified as so-
called "Bayesian" estimators
They reflect not only data but also prior
knowledge or judgment,
Bayesian estimators have rock-solid axiomatic
foundations in statistics and decision theory,
unlike many other estimators commonly used
by statisticians.
Theory is good
Beside,one”,when we have more information for the stock,
we choose the more reasonable prior guess for betas
Example,we know which industry it operates in,Since the
average beta among auto-manufacturing companies is around
1.2,a reasonable prior guess for GM's beta is 1.2.
How much weight we put on the prior guess and how much
on the estimate from the data depends on the precision of
the sample estimate and on the strength of our prior beliefs,
Example,Those beliefs can be based on the dispersion of the
betas of other auto manufacturers,the stronger the
concentration of auto company betas around 1.2,the more
weight we put on the prior guess,With equal weights on the
prior guess and the weekly sample estimate,GM's industry-
adjusted beta is (1/2)*0.88 + (1/2)*1.2 = 1.04.
Risk premium
Estimating the equity premium is more difficult
than estimating betas：
More frequent data does not help and there is no
obvious prior guess.
The most common approach is to average a long
series of excess market returns,which leads to
equity premium estimates anywhere between 5%
and 9% per year,depending on the sample period.
Example,Equity premium is 4.8%,GM,
6%+1.04*4.8%=11%
But our models are not perfect
The CAPM is just a model,not a perfect description
of reality,Indeed,many academic studies reject the
validity of the CAPM,since some stock return
patterns seem inconsistent with the model,Does this
mean that we should throw the model away and rely
only on model-free estimators,such as the sample
average return?
No! Every model is "wrong",almost by definition,
because it makes simplifying assumptions about our
complex world,But even a model that is not exactly
right can be useful.
But our models are not perfect
The study finds that even if we have
only modest confidence in a pricing
model such as the CAPM,our cost of
capital estimates should be heavily
weighted toward the model,Average
stock returns are noisy,so they should
receive small weights,In other words,
theory is more powerful than data when
estimating expected stock returns.
But our models are not perfect
The CAPM is not the only theoretical model of
expected returns,A sensible solution to construct a
weighted average of expected return estimates from
all models that we are willing to consider,including
the no-theory model that produces the sample
average estimate,Each estimate should be weighted
by the probability that its parent model is correct.
The CAPM is the model with the strongest theoretical
foundation,and should therefore receive the largest
weight,Other models should receive weights
commensurate to their theoretical support as well as
their empirical success.
But our models are not perfect
Not knowing which model is right turns
out to be less important on average
than not knowing the parameters within
each model,We should therefore spend
less time searching for the right model,
and more time trying to improve the
estimates within each model.
The premium is the most uncertainty
Shooting at a moving target
There is an emerging consensus in the academia that expected
returns vary over time.
For example,expected stock returns seem to be related to the
business cycle -- they tend to be higher in recessions and lower in
expansions.
Among the variables that have been found useful in explaining
the time-variation in expected stock market returns are the
aggregate dividend-price ratio (D/P) and earnings-price ratio
(E/P),Low values of these ratios have historically predicted low
returns,In other words,when prices are high relative to the
fundamentals,future returns are on average low,especially at
longer horizons such as ten years ahead.
data mining
Only predictors with a solid theoretical justification have a chance
of working not only in the past but also in the future.
Shooting at a moving target
if it is hard to estimate expected returns
when they are constant,it is even
harder to estimate them when they
change through time.
There is no simple recipe on how to estimate
expected returns,Since data is noisy and no
theory is strictly flawless,judgment enters
the process at numerous points.
Given the importance of expected returns and
the huge uncertainty associated with them,
the finance profession clearly needs to invest
more into their estimation,Such an
investment will undoubtedly provide a high
expected return,But please don't ask me for
an exact number.
方差、协方差估计方法
样本方差、样本协方差
BUT if the number of assets is large
there is a potential problem here,Why?
第二步：寻找投资机会集
是否有买、卖空限制
if yes,then no additional restrictions on the
will are required
if no,then we need the additional
restrictions
第三步：选择最优证券组合
在实际中，不利用效用函数
确定投资者能够忍受的最大风险水平
在有效集上确定具有该风险的证券组合
可以这样看待风险
如果证券组合的回报率服从正态分布，且期望回报率和标准差各为 和
以 68%的概率获得的回报率属于
以 95%的概率获得的回报率属于
E S
SESE,
SESE 2,2
The Optimal Portfolio in Practice
Implication of the Separation Principle,A portfolio manager will offer the same
risky portfolio to all clients,In practice,different managers focus on different
subsets of the whole universe of financial assets,derive different efficient
frontiers,and offer different,optimal” portfolios to their clients,Why?
The theory of portfolio selection builds on many simplifying assumptions:
No Market Frictions (tax,transactions costs,limited divisibility of financial assets,
market segmentation).
No Heterogeneity in Investors (e.g,rich vs,poor,informed vs,uninformed,young vs,
old).
Static expected returns and variance - no forecast ability in returns or volatility (e.g,
financial analysts,accounting information,macroeconomic variables do not play any
role in making an investment decision).
Our next step toward reality,link the following two bodies of ideas,
Security Analysis,subjective,judgmental
Portfolio Selection,objective,statistical.
Some important questions to think about:
Can security analysis improve portfolio performance?
How do analysts’ opinions enter in security selection?
The Optimal Portfolio in Practice
Passive strategies
describes a portfolio decision that avoids
any direct or indirect security analysis.
The reason
the alternative active strategy is not free
free-rider benefit
Active strategies
GIGO
the optimization technique is the easiest
part of the portfolio construction problem,
The real arena of competition among
portfolio managers is security analysis.
2,具体的例子
3,均值 -方差分析中的问题
由均值 -方差得出的最优证券组合有些并不合理：一些资产大量卖空，一些资产的权为 0：
最优证券组合对参数的值非常敏感
一些参数的估计非常困难，特别是期望值的估计
完全刻画投资者关于每个数据估计的置信度非常困难
4,几种特殊不完善市场
借贷利率不相等的情形
不存在无风险债券
非流动性资产
5.计算量
6。结论
Asset allocation requires good estimates
of expected returns
However,expected returns are hard to get,
so asset allocation is difficult
We can provide some discipline and
organization through some models of
asset pricing
Estimation of the inputs
A computer exercise using real data
Implementation problem and some
solutions
Estimation results in uncertainty
Shrinkage and Bayesian inference to deal
with uncertainty
Imposing restrictions.
1.投资者的最优投资组合过程
投资者的最优投资组合过程可以分为三步：
数据的估计
寻找最优投资集
选择最优证券组合第一步：数据的估计
数据的估计：如何得到证券回报率的期望值、方差、协方差
标准差相对稳定，利用历史数据估计
但期望值不一定
仅仅关注过去可能导致错误的结果
将来和过去可能完全不同期望值的估计方法
样本均值
在如下条件下合理
股票指标保持不变
将来的回报率和过去的回报率由同一个模型产生
否则，需要根据现在的环境调整
寻找特殊变量来预测回报率
寻找现在可以观测，可以预测将来回报率的变量
一般不容易找到
已有的几个例子：红利收益、红利价格比、市盈率、一些利率变量
回归
例如
t
t
t p
dr
1
期望值的估计方法
利用 CAPM,APT等模型
A model weighting game in estimating
expected returns (By Lubos Pastor)
It is hard to overstate the importance of expected
returns in investment,(money manager,corporate
manager,ordinary consumer).
Unfortunately,expected returns are as elusive as
they are important,There is no absolute agreement
among finance professionals on how expected
returns should be estimated.
The best estimates are produced by combining
finance theory with historical returns data and our
own judgment.
Does history repeat itself?
Unless we suspect that expected return
changes nontrivially over time,the sample
average return is an unbiased estimator of
expected return,The unbiasedness of the
sample average return is its main advantage.
The main disadvantage of the sample
average is its imprecision.
GM,1991-2000,the sample average return is
14% per year,The standard error is 10% per year,
With 95% confidence,the true expected return is
within two standard errors of the sample average,
[-6%,34%].
Does history repeat itself?
Would the precision increase if we used
weekly instead of monthly data? Although
higher-frequency data helps in estimating
variances and covariance of returns,it does
not help in estimating expected returns.
Intuitively,what matters for expected return is the
beginning and ending levels of prices over a given
period,but not what happens in between.
Does history repeat itself?
The only way to get a more precise sample
average is to collect more data further back
in time,But as we add older data,we gain
precision at the expense of introducing
potential bias.
GM,1925-2000,historical average is 15.5%,
standard error is 3.4%,[8.7%,22.3%],The
interval is still too wide for comfort.
Moreover,GM is very different from seventy years
ago,so the current estimate could be
contaminated by old data.
Does history repeat itself?
In general,as we add older data,we gain precision
at the expense of introducing potential bias,Striking
the right balance is difficult and requires sound
judgment.
Despite its drawbacks,the long-run average return is
a popular estimator for expected returns on
aggregate market indices,Unfortunately,we have no
theory for what the expected market return should
be.
Luckily,for individual stocks and most portfolios,we
can rely on estimates produced by theoretical asset
pricing models,Those estimates tend to be
substantially more precise than sample averages.
Theory is good
CAPM
In order to use CAPM
Riskless rate,
beta,
equity premium
Theory is good
What value we choose for the risk-free rate depend
on our objectives.
If we want to forecast expected stock returns over the next
month,the appropriate risk-free rate is the yield to maturity
on a Treasury bill that matures in one month,
If we want to estimate the firm's cost of capital in order to
value the firm's future cash flows,the risk-free rate should
be derived from a longer-term Treasury bond,such as a ten-
year bond,The bond's duration should come close to the
duration of the firm's cash flows,
Very long-term bonds should be avoided,because their
yields might also reflect premiums for risks such as inflation.
Theory is good
Beta
Beta is typically estimated by regressing
the most recent five to ten years worth of
monthly stock returns on market returns.
the estimate of GM's beta using its monthly
data in January 1996 through December 2000
is 1.11,the 95% confidence interval for GM's
beta based on the monthly data is 0.65 to 1.57.
Theory is good
How much data should we use to estimate beta?
The further we go back in time,the higher the statistical
precision of the estimate,but the bigger the possibility of
introducing some old data bias.
Unlike with sample averages,however,here it often pays
to use more frequent data,For example,whereas the
95% confidence interval for GM's beta based on the
monthly data is 0.65 to 1.57,this interval based on
weekly data is tighter,0.69 to 1.08,GM's beta estimated
using weekly data is 0.88.
Theory is good
However,going from monthly to weekly data is
recommended only for the most liquid and volatile stocks.
For other stocks,some week-to-week price changes are
simply movements between the bid and ask prices around
the same true price,which introduces additional error into
the estimation,Also,it may take a while for market-wide
news to get into the prices of illiquid stocks,which biases
the usual beta estimates downward,Conveniently,betas of
illiquid stocks can be estimated using an alternative
approach developed by the Chicago economists Myron
Scholes and Joseph Williams (1977,J,F,E).
Theory is good
The shrinkage estimate of the beta
The usual estimates of beta contain a fair amount of
noise,
A common and useful way of reducing that noise is to
"shrink" the usual estimates to a reasonable value,such
as one,The value of one is reasonable because the
average beta across all stocks is one,by construction.
The shrinkage estimate of beta is the weighted of the
sample estimate and of the shrinkage target,
Example,the "adjusted" betas reported by Merrill Lynch
put a 2/3 weight on the sample estimate and a 1/3 weight
on the value of one,The adjusted weekly beta for GM is
therefore (2/3)*0.88 + (1/3)*1 =0.92.
Theory is good
Shrinkage betas can be justified as so-
called "Bayesian" estimators
They reflect not only data but also prior
knowledge or judgment,
Bayesian estimators have rock-solid axiomatic
foundations in statistics and decision theory,
unlike many other estimators commonly used
by statisticians.
Theory is good
Beside,one”,when we have more information for the stock,
we choose the more reasonable prior guess for betas
Example,we know which industry it operates in,Since the
average beta among auto-manufacturing companies is around
1.2,a reasonable prior guess for GM's beta is 1.2.
How much weight we put on the prior guess and how much
on the estimate from the data depends on the precision of
the sample estimate and on the strength of our prior beliefs,
Example,Those beliefs can be based on the dispersion of the
betas of other auto manufacturers,the stronger the
concentration of auto company betas around 1.2,the more
weight we put on the prior guess,With equal weights on the
prior guess and the weekly sample estimate,GM's industry-
adjusted beta is (1/2)*0.88 + (1/2)*1.2 = 1.04.
Risk premium
Estimating the equity premium is more difficult
than estimating betas：
More frequent data does not help and there is no
obvious prior guess.
The most common approach is to average a long
series of excess market returns,which leads to
equity premium estimates anywhere between 5%
and 9% per year,depending on the sample period.
Example,Equity premium is 4.8%,GM,
6%+1.04*4.8%=11%
But our models are not perfect
The CAPM is just a model,not a perfect description
of reality,Indeed,many academic studies reject the
validity of the CAPM,since some stock return
patterns seem inconsistent with the model,Does this
mean that we should throw the model away and rely
only on model-free estimators,such as the sample
average return?
No! Every model is "wrong",almost by definition,
because it makes simplifying assumptions about our
complex world,But even a model that is not exactly
right can be useful.
But our models are not perfect
The study finds that even if we have
only modest confidence in a pricing
model such as the CAPM,our cost of
capital estimates should be heavily
weighted toward the model,Average
stock returns are noisy,so they should
receive small weights,In other words,
theory is more powerful than data when
estimating expected stock returns.
But our models are not perfect
The CAPM is not the only theoretical model of
expected returns,A sensible solution to construct a
weighted average of expected return estimates from
all models that we are willing to consider,including
the no-theory model that produces the sample
average estimate,Each estimate should be weighted
by the probability that its parent model is correct.
The CAPM is the model with the strongest theoretical
foundation,and should therefore receive the largest
weight,Other models should receive weights
commensurate to their theoretical support as well as
their empirical success.
But our models are not perfect
Not knowing which model is right turns
out to be less important on average
than not knowing the parameters within
each model,We should therefore spend
less time searching for the right model,
and more time trying to improve the
estimates within each model.
The premium is the most uncertainty
Shooting at a moving target
There is an emerging consensus in the academia that expected
returns vary over time.
For example,expected stock returns seem to be related to the
business cycle -- they tend to be higher in recessions and lower in
expansions.
Among the variables that have been found useful in explaining
the time-variation in expected stock market returns are the
aggregate dividend-price ratio (D/P) and earnings-price ratio
(E/P),Low values of these ratios have historically predicted low
returns,In other words,when prices are high relative to the
fundamentals,future returns are on average low,especially at
longer horizons such as ten years ahead.
data mining
Only predictors with a solid theoretical justification have a chance
of working not only in the past but also in the future.
Shooting at a moving target
if it is hard to estimate expected returns
when they are constant,it is even
harder to estimate them when they
change through time.
There is no simple recipe on how to estimate
expected returns,Since data is noisy and no
theory is strictly flawless,judgment enters
the process at numerous points.
Given the importance of expected returns and
the huge uncertainty associated with them,
the finance profession clearly needs to invest
more into their estimation,Such an
investment will undoubtedly provide a high
expected return,But please don't ask me for
an exact number.
方差、协方差估计方法
样本方差、样本协方差
BUT if the number of assets is large
there is a potential problem here,Why?
第二步：寻找投资机会集
是否有买、卖空限制
if yes,then no additional restrictions on the
will are required
if no,then we need the additional
restrictions
第三步：选择最优证券组合
在实际中，不利用效用函数
确定投资者能够忍受的最大风险水平
在有效集上确定具有该风险的证券组合
可以这样看待风险
如果证券组合的回报率服从正态分布，且期望回报率和标准差各为 和
以 68%的概率获得的回报率属于
以 95%的概率获得的回报率属于
E S
SESE,
SESE 2,2
The Optimal Portfolio in Practice
Implication of the Separation Principle,A portfolio manager will offer the same
risky portfolio to all clients,In practice,different managers focus on different
subsets of the whole universe of financial assets,derive different efficient
frontiers,and offer different,optimal” portfolios to their clients,Why?
The theory of portfolio selection builds on many simplifying assumptions:
No Market Frictions (tax,transactions costs,limited divisibility of financial assets,
market segmentation).
No Heterogeneity in Investors (e.g,rich vs,poor,informed vs,uninformed,young vs,
old).
Static expected returns and variance - no forecast ability in returns or volatility (e.g,
financial analysts,accounting information,macroeconomic variables do not play any
role in making an investment decision).
Our next step toward reality,link the following two bodies of ideas,
Security Analysis,subjective,judgmental
Portfolio Selection,objective,statistical.
Some important questions to think about:
Can security analysis improve portfolio performance?
How do analysts’ opinions enter in security selection?
The Optimal Portfolio in Practice
Passive strategies
describes a portfolio decision that avoids
any direct or indirect security analysis.
The reason
the alternative active strategy is not free
free-rider benefit
Active strategies
GIGO
the optimization technique is the easiest
part of the portfolio construction problem,
The real arena of competition among
portfolio managers is security analysis.
2,具体的例子
3,均值 -方差分析中的问题
由均值 -方差得出的最优证券组合有些并不合理：一些资产大量卖空，一些资产的权为 0：
最优证券组合对参数的值非常敏感
一些参数的估计非常困难，特别是期望值的估计
完全刻画投资者关于每个数据估计的置信度非常困难
4,几种特殊不完善市场
借贷利率不相等的情形
不存在无风险债券
非流动性资产
5.计算量
6。结论
Asset allocation requires good estimates
of expected returns
However,expected returns are hard to get,
so asset allocation is difficult
We can provide some discipline and
organization through some models of
asset pricing
