8-1
第 8章最优风险资产组合
Optimal Risky Portfolios
8-2
最优风险资产组合
Optimal Risky Portfolios
8.1 分散 化与资产组合风险
8.2 两种风险资产的资产组合
8.3 在股票、债券与国库券之间的资产配置
8.4 马科维茨的 资产组合选择模型
8.5 电子表格模型
8.6 具有无风险资产限制的最优资产组合
8-3
最优风险资产组合
Optimal Risky Portfolios
从分散化如何降低资产组合投资回报的风险开始。在建立这一基点之后，我们将从资产配置和证券选择的两方面考察有效分散化策略。我们将首先考察一个不包含无风险资产的资产配置
，我们将运用两个有风险的共同基金：一个是长期债券基金，
一个是股票基金。然后我 们将加上一个无风险资产来决定一个最优资产组合。
The efficient diversification strategies at the asset allocation and security
selection levels,We start with a simple example of asset allocation that
excludes the risk-free asset,To that effect we use two risky mutual funds,a
long-term bond fund and a stock fund,With this example we investigate
the relationship between investment proportions and the resulting
portfolio expected return and standard deviation,We then add a risk-
free asset to the menu and determine the optimal asset allocation,
8-4
分散化与风险
Risk Reduction with Diversification
股票数量
Number of
Securities
标准方差 St,Deviation
市场风险 (系统风险 )
Market Risk
独特风险 (非系统风险 )
Unique Risk
8-5
多样化与组合风险
Diversification and Portfolio risk
一种股票,One-security,
风险来自宏观经济
Risks come from macro econ.
风险来自企业自己
Risks come from company self
两种股票,Two - security
股票组合降低风险
Portfolio will reduce risk
8-6
R=[Dt+(Pt-P0)]/P0
E( R ) = ∑ RiPi (I=1 to n)
Ri,预期收益率 expected return
Pi:预期收益的概率 probability of expected return
For example:
E( R ) = 15%* 0.25 +10%* 0.5 +8% *0.25 =10.75%
R =(1/M)*∑ Ri (I=1 to n) 如果假设未来各年的收益都相等 when all expected returns are same
E( R ) =R =（ 15%+10%+8%） /3=11%
单个股票收益
Single Security Return
8-7
σR2 = ∑ Pi（ Ri - E( R )） 2 (I=1 to n)
= (1/4)(15-11)2 +(1/2)(10-11)2 +(1/4)(8-11)2 =6.75
σR= （ 6.75） （ 1/2） =2.6
σ（ R） 2 均方差
σ（ R） 标准方差单个股票风险
Single Security Risk
8-8
我们将考察一个包括两个共同基金的资产组合，一个是专门投资于长期债券的债券资产组合 D，一个是专门投资于股权证券的股票基金 E，表 8-1列出了影响这些基金收益率的参数，这些参数可以从真实的基金中估计得出。
We will consider a portfolio comprised of two mutual funds,a bond
portfolio specializing in long-term debt securities,denoted D,and a stock
fund that specializes in equity securities,E,Table 8.1 lists the parameters
describing the rate-of-return distribution of these funds,These parameters
are representative of those that can be estimated from actual funds.
两种股票组合：收益
Two-Security Portfolio,Return
8-9
组合收益率 rp = WDrD + WErE
WD =投资与债券中的部分基金
rD =投资债券的收益
WＥ = Proportion of funds in Security（股票）
rＥ = Expected return on Security （股票）
两种股票组合：收益
Two-Security Portfolio,Return
WiS
i=1
n
= 1
8-10
资产组合的期望收益是资产组合中各种证券的期望收益的加权平均值
The expected return on the portfolio is a
weighted average of expected returns on
the component securities with portfolio
proportions as weights:
E(rp)=WDE(rD)+WEE(rE)
两种股票组合：收益
Two-Security Portfolio,Return
8-11
sp2 = wＤ 2sＤ 2 + wＥ 2sE2 + 2wDwE
Cov(rD,rE)s
D2 = 债券的方差 Variance of Security Ｄ
sE2 =股票的方差 Variance of SecurityＥ
Cov(rＤ,rＥ ) =债券和股票收益的 协方差
Cov(rＤ,rＥ ) =Covariance of returns for
Security Ｄ and Security Ｅ
两种股票组合：风险
Two-Security Portfolio,Risk
8-12
协方差 Covariance
ρDE = 收益的相关系数 Correlation coefficient of
returns
Cov(rＤ,rE) = ρ DE sDsE
Cov(rＤ,rD) = σD2
sD = 证券 D收益的标准方差
Standard deviation of returns for Security D
sE = Standard deviation of returns for Security E
sp2 = wＤ 2sＤ 2 + wＥ 2sE2 + 2wDwE ρDE σD σ E
8-13
相关系数,取值范围
Correlation Coefficients,Possible Values
如果 r = 1.0,证券组合将是正相关
If r = 1.0,the securities would be perfectly positively
correlated
如果 r = -1.0,证券组合将是负相关
If r = - 1.0,the securities would be perfectly negatively
correlated
r D,E 取值范围
Range of values for r D,E
-1.0 < r < 1.0
8-14
相关系数,取值范围
Correlation Coefficients,Possible Values
如果 r = 1.0 If r = 1.0
sp2 = wＤ 2sＤ 2 + wＥ 2sE2 + 2wDwE σD σ E
sp = wＤ sＤ + wＥ sE
sp2 = (wＤ sＤ + wＥ sE)2
8-15
相关系数,取值范围
Correlation Coefficients,Possible Values
如果 r = -1.0 If r = -1.0
sp2 = wＤ 2sＤ 2 + wＥ 2sE2 - 2wDwE σD σ E
sp = ︳ wＤ sＤ - wＥ sE ︳
sp2 = (wＤ sＤ - wＥ sE)2
wＤ sＤ - wＥ sE = 0
wＤ =σE/(σD + σE) w E=σ D / (σD+σE) =1-WD
8-16
s2p = W12s12+ W22s22
+ 2W1W2
rp = W1r1 + W2r2 + W3r3
Cov(r1r2)
+ W32s32
Cov(r1r3)+ 2W1W3
Cov(r2r3)+ 2W2W3
三种证券组合
Three-Security Portfolio
8-17
rp =多种证券的加权平均
rp = Weighted average of the n securities
sp2 = (考虑全部成双量的 协方差）
sp2 = (Consider all pair-wise
covariance measures)
多种证券组合的一般性
In General,For an n-Security Portfolio:
8-18
E(rp) = W1r1 + W2r2
两种股票组合
Two-Security Portfolio
sp2 = w12s12 + w22s22 + 2W1W2 Cov(r1r2)
sp = [w12s12 + w22s22 + 2W1W2
Cov(r1r2)]1/2
8-19
r = 0
期望收益
E(r)
r = 1r = -1
r = -1
r =,3
13%
8%
12% 20% 标准差St,Dev
不同相关系数的两种股票组合
TWO-SECURITY PORTFOLIOS WITH
DIFFERENT CORRELATIONS
8-20
组合风险 /收益 两种股票组合：相关有效性
Portfolio Risk/Return Two Securities,Correlation Effects
相关性取决于相关系数
Relationship depends on correlation coefficient
-1.0 < r < +1.0
相关性愈小，降低风险的可能性就大
The smaller the correlation,the greater the risk
reduction potential
如果 r = +1.0,降低风险是不可能的
If r = +1.0,no risk reduction is possible
8-21
1
1 2
- Cov(r1r2)
W1 =
+ - 2Cov(r1r2)
2
W2 = (1 - W1)
最小方差组合
Minimum Variance Combination
s 2
s 2
2E(r2) =,14 =,20Sec 2
12 =,2
E(r1) =,10 =,15Sec 1 s
s r
s 2
8-22
W1 =
(.2)2 - (.2)(.15)(.2)
(.15)2 + (.2)2 - 2(.2)(.15)(.2)
W1 =,6733
W2 = (1 -,6733) =,3267
最小方差组合 r =,2
Minimum Variance Combination,r =,2
8-23
rp =,6733(.10) +,3267(.14) =,1131
p = [(.6733)2(.15)2 + (.3267)2(.2)2 +
2(.6733)(.3267)(.2)(.15)(.2)]1/2
p= [.0171]
1/2 =,1308
最小方差,在 r =,2 时的收益和风险
Minimum Variance,Return and Risk with r =,2
s
s
8-24
W1 =
(.2)2 - (.2)(.15)(.2)
(.15)2 + (.2)2 - 2(.2)(.15)(-.3)
W1 =,6087
W2 = (1 -,6087) =,3913
最小方差组合 r = -.3
Minimum Variance Combination,r = -.3
8-25
rp =,6087(.10) +,3913(.14) =,1157
p = [(.6087)2(.15)2 + (.3913)2(.2)2 +
2(.6087)(.3913)(.2)(.15)(-.3)]1/2
p= [.0102]
1/2 =,1009
最小方差,在 r =,3 时的收益和风险
Minimum Variance,Return and Risk with r = -.3
s
s
8-26
多种证券组合
Extending Concepts to All Securities
在给定收益率时优化组合将使风险降到最低水平
The optimal combinations result in lowest level
of risk for a given return
优化替换被描绘为有效边界
The optimal trade-off is described as the efficient
frontier
这些组合是重要的 These portfolios are dominant
8-27
马科维茨的资产组合选择模型,证券选择
The Markowitz Portfolio Selection Model Security Selection
我们可在多种风险证券和无风险资产中间进行资产组合的构造。
在两种风险资产 的例子中，问题分为三个部分，第一，我们要从可能的风险资产组合中识别出风险 - 收益组合。第二，
我们通过资产组合权重的计算，找出最优风险资产组合，此时有最大 斜率的资本配置线。最后，我们通过加入无风险资产，
找到完整的资产组合。
We can generalize the portfolio construction problem to the case
of many risky securities and a risk-free asset,As in the two
risky assets example,the problem has three parts,First,we
identify the risk return combinations available from the set
of risky assets,Next,we identify the optimal portfolio of
risky assets by finding the portfolio weights that result in the
steepest CAL,Finally,we choose an appropriate complete
portfolio by mixing the risk- free asset with the optimal risky
portfolio,
8-28
E(r) 风险资产的最小方差边界The minimum-variance frontier of risky assets
有效率边界
Efficient
frontier
Global
minimum
variance
portfolio Minimum
variance
frontier
单个资产
Individual assets
St,Dev.
8-29
无风险资产组合
Extending to Include Riskless Asset
优化组合转变为线性
The optimal combination becomes linear
风险和无风险资产单一组合将是主要部分
A single combination of risky and riskless
assets will dominate
8-30
组合预期收益 E(r)
CAL (全球最小方差
Global minimum variance)
CAL (A)CAL (P)
M
P
A
F
P P&F A&FM
A
G
P
M
s
供选择的资本分配线
ALTERNATIVE CALS
8-31
最优资本分配线与无风险投资
Dominant CAL with a Risk-Free Investment (F)
资本分配线优与其他线 – 它有最好的风险 /收益或最大的斜率
CAL(P) dominates other lines -- it has the
best risk/return or the largest slope
斜率 =（预期收益率 -无风险利率） /标准差
Slope = (E(R) - Rf) / s
[ E(RP) - Rf) / s P ] > [E(RA) - Rf) / sA]
8-32
最优资产组合
Optimal Risky Portfolio
8-33
最优资产组合
Optimal Risky Portfolio
8-34
最优资产组合
Optimal Risky Portfolio
8-35
最优资产组合
Optimal Risky Portfolio
8-36
最优资产组合
Optimal Risky Portfolio
8-37
最优资产组合
Optimal Risky Portfolio
8-38
最优资产组合：举例
Optimal Risky Portfolio,Exam
8-39
最优资产组合：举例
Optimal Risky Portfolio,Exam
8-40
多种证券组合
Extending Concepts to All Securities
一个完整的资产组合的步骤：
1) 确定所有各类证券的回报特征（例如期望收益、方差、斜方差等）。
2) 建造风险资产组合：
a,计算最优风险资产组合 P（ 8 - 7式）；
b,运用步骤（ a）中确定的权重和 8 - 1式与 8 - 2式来计算资产组合 P的资产。
3) 把基金配置在风险资产组合和无风险资产上：
a,计算资产组合 P（风险资产组合）和国库券（无风险资产）
的权重（ 8 - 8式）；
b,计算出完整的资产组合中投资于每一种资产和国库券上的投资份额。
8-41
多种证券组合
Extending Concepts to All Securities
The steps we followed to arrive at the complete portfolio.
1,Specify the return characteristics of all securities
(expected returns,variances,covariances).
2,Establish the risky portfolio:
a,Calculate the optimal risky portfolio,P (equation 8.7).
b,Calculate the properties of Portfolio P using the
weights determined in step (a) and equations 8.1 and 8.2.
3,Allocate funds between the risky portfolio and the risk-free
asset:
a,Calculate the fraction of the complete portfolio
allocated to Portfolio P (the risky portfolio) and to T-bills
(the risk-free asset) (equation 8.8).
b,Calculate the share of the complete portfolio invested
in each asset and in T-bills.
8-42
小结
Summary
资产组合的期望收益是资产组合中各项资产期望收益的以各项资产为权重的加权平均值。
The expected return of a portfolio is the
weighted average of the component security
expected returns with the investment
proportions as weights.
8-43
小结
Summary
资产组合的方差是协方差矩阵各元素与投资比例为权重相乘的加权总值。因此,每一资产的方差以其投资比例的平方进行加权，任一对资产的协方差在协方差矩阵中 出现两次。所以，资产组合方差中包含着二次斜方差权重，这是由两项资产的每一项资产投资比例的乘积构成的。
The variance of a portfolio is the weighted sum of the
elements of the covariance matrix with the product of the
investment proportions as weights,Thus the variance of
each as- set is weighted by the square of its investment
proportion,Each covariance of any pair of assets appears
twice in the covariance matrix; thus the portfolio variance
includes twice each covariance weighted by the product of
the investment proportions in each of the two assets.
8-44
小结
Summary
即便协方差为正，只要资产不是完全正相关的，资产组合的标准差就仍小于组合中各项资产的标准差的加权平均值。因此，只要资产不是完全正相关的，
分散化的 资产组合就是有价值的。
Even if the covariances are positive,the portfolio
standard deviation is less than the weighted
average of the component standard deviations,as
long as the assets are not perfectly positively
correlated,Thus portfolio diversification is of value
as long as assets are less than perfectly correlated.
8-45
小结
Summary
资产组合中一项资产的斜方差相对于其他资产的协方差越大，
它对资产组合方差的作用就越大。一项资产如与资产组合完全负相关，它将具有完全对冲的功能。一 项完全对冲资产可以降低资产组合的方差至零。
The greater an asset’s covariance with the other
assets in the portfolio,the more it contributes to
portfolio variance,An asset that is perfectly
negatively correlated with a portfolio can serve as
a perfect hedge,The perfect hedge asset can reduce
the portfolio variance to zero.
8-46
小结
Summary
有效率边界是利用图表来表示在某 — 特定风险水平上，有最大收益的一组资产组合。理性投资者将在有效边界上选择资产组合。
The efficient frontier is the graphical
representation of a set of portfolios that
maximize expected return for each level of
portfolio risk,Rational investors will choose
a portfolio on the efficient frontier,
8-47
小结
Summary
一个资产组合经理在确定有效边界时，首先要估计资产的期望收益与协方差矩阵。这个输入清单被输入优化程序中，得到在有效率边界上资产组合中各项资产的比例、期望收益与标准差等。
A portfolio manager identifies the efficient frontier
by first establishing estimates for the asset
expected returns and the covariance matrix,This
input list is then fed into an optimization program
that reports as outputs the investment proportions,
expected returns,and standard deviations of the
portfolios on the efficient frontier.
8-48
小结
Summary
一般来说，资产组合经理会得到不同的有效资产组合，因为他们的证券分析方法与质量是不同的。经理们主要在证券分析质量，而不是在管理费上展开竞争。
In general,portfolio managers will arrive at
different efficient portfolios because of
differences in methods and quality of
security analysis,Managers compete on the
quality of their security analysis relative to
their management fees.
8-49
小结
Summary
如果无风险资产存在，输入清单亦可以确定，所有投资者都将选择在有效率边 界上同样的资产组合：与资本配置线相切的资产组合。具有相同输入清单的所有投资者将持有相同的风险资产组合，不同的是在风险资产组合和无风险资产之间的资金分配。这一结果就是资产组合构造的分离原则。
If a risk-free asset is available and input lists are
identical,all investors will choose the same portfolio on the
efficient frontier of risky assets,the portfolio tangent to
the CAL,All investors with identical input lists will hold an
identical risky portfolio,differing only in how much each
allocates to this optimal portfolio and to the risk-free asset,
This result is characterized as the separation principle of
portfolio construction.
8-50
小结
Summary
当无风险资产不存在时，每个投资者在有效率边界上选择风险资产组合。如果 有无风险资产，但借入是受限制的，那么，只有冒险型投资者会受到影响，他们将根据其愿意冒险的程度在有效率边界上选择资产组合。
When a risk-free asset is not available,each
investor chooses a risky portfolio on the efficient
frontier,If a risk-free asset is available but borrowing
is restricted,only aggressive investors will be
affected,They will choose portfolios on the
efficient frontier according to their degree of risk
tolerance.
第 8章最优风险资产组合
Optimal Risky Portfolios
8-2
最优风险资产组合
Optimal Risky Portfolios
8.1 分散 化与资产组合风险
8.2 两种风险资产的资产组合
8.3 在股票、债券与国库券之间的资产配置
8.4 马科维茨的 资产组合选择模型
8.5 电子表格模型
8.6 具有无风险资产限制的最优资产组合
8-3
最优风险资产组合
Optimal Risky Portfolios
从分散化如何降低资产组合投资回报的风险开始。在建立这一基点之后，我们将从资产配置和证券选择的两方面考察有效分散化策略。我们将首先考察一个不包含无风险资产的资产配置
，我们将运用两个有风险的共同基金：一个是长期债券基金，
一个是股票基金。然后我 们将加上一个无风险资产来决定一个最优资产组合。
The efficient diversification strategies at the asset allocation and security
selection levels,We start with a simple example of asset allocation that
excludes the risk-free asset,To that effect we use two risky mutual funds,a
long-term bond fund and a stock fund,With this example we investigate
the relationship between investment proportions and the resulting
portfolio expected return and standard deviation,We then add a risk-
free asset to the menu and determine the optimal asset allocation,
8-4
分散化与风险
Risk Reduction with Diversification
股票数量
Number of
Securities
标准方差 St,Deviation
市场风险 (系统风险 )
Market Risk
独特风险 (非系统风险 )
Unique Risk
8-5
多样化与组合风险
Diversification and Portfolio risk
一种股票,One-security,
风险来自宏观经济
Risks come from macro econ.
风险来自企业自己
Risks come from company self
两种股票,Two - security
股票组合降低风险
Portfolio will reduce risk
8-6
R=[Dt+(Pt-P0)]/P0
E( R ) = ∑ RiPi (I=1 to n)
Ri,预期收益率 expected return
Pi:预期收益的概率 probability of expected return
For example:
E( R ) = 15%* 0.25 +10%* 0.5 +8% *0.25 =10.75%
R =(1/M)*∑ Ri (I=1 to n) 如果假设未来各年的收益都相等 when all expected returns are same
E( R ) =R =（ 15%+10%+8%） /3=11%
单个股票收益
Single Security Return
8-7
σR2 = ∑ Pi（ Ri - E( R )） 2 (I=1 to n)
= (1/4)(15-11)2 +(1/2)(10-11)2 +(1/4)(8-11)2 =6.75
σR= （ 6.75） （ 1/2） =2.6
σ（ R） 2 均方差
σ（ R） 标准方差单个股票风险
Single Security Risk
8-8
我们将考察一个包括两个共同基金的资产组合，一个是专门投资于长期债券的债券资产组合 D，一个是专门投资于股权证券的股票基金 E，表 8-1列出了影响这些基金收益率的参数，这些参数可以从真实的基金中估计得出。
We will consider a portfolio comprised of two mutual funds,a bond
portfolio specializing in long-term debt securities,denoted D,and a stock
fund that specializes in equity securities,E,Table 8.1 lists the parameters
describing the rate-of-return distribution of these funds,These parameters
are representative of those that can be estimated from actual funds.
两种股票组合：收益
Two-Security Portfolio,Return
8-9
组合收益率 rp = WDrD + WErE
WD =投资与债券中的部分基金
rD =投资债券的收益
WＥ = Proportion of funds in Security（股票）
rＥ = Expected return on Security （股票）
两种股票组合：收益
Two-Security Portfolio,Return
WiS
i=1
n
= 1
8-10
资产组合的期望收益是资产组合中各种证券的期望收益的加权平均值
The expected return on the portfolio is a
weighted average of expected returns on
the component securities with portfolio
proportions as weights:
E(rp)=WDE(rD)+WEE(rE)
两种股票组合：收益
Two-Security Portfolio,Return
8-11
sp2 = wＤ 2sＤ 2 + wＥ 2sE2 + 2wDwE
Cov(rD,rE)s
D2 = 债券的方差 Variance of Security Ｄ
sE2 =股票的方差 Variance of SecurityＥ
Cov(rＤ,rＥ ) =债券和股票收益的 协方差
Cov(rＤ,rＥ ) =Covariance of returns for
Security Ｄ and Security Ｅ
两种股票组合：风险
Two-Security Portfolio,Risk
8-12
协方差 Covariance
ρDE = 收益的相关系数 Correlation coefficient of
returns
Cov(rＤ,rE) = ρ DE sDsE
Cov(rＤ,rD) = σD2
sD = 证券 D收益的标准方差
Standard deviation of returns for Security D
sE = Standard deviation of returns for Security E
sp2 = wＤ 2sＤ 2 + wＥ 2sE2 + 2wDwE ρDE σD σ E
8-13
相关系数,取值范围
Correlation Coefficients,Possible Values
如果 r = 1.0,证券组合将是正相关
If r = 1.0,the securities would be perfectly positively
correlated
如果 r = -1.0,证券组合将是负相关
If r = - 1.0,the securities would be perfectly negatively
correlated
r D,E 取值范围
Range of values for r D,E
-1.0 < r < 1.0
8-14
相关系数,取值范围
Correlation Coefficients,Possible Values
如果 r = 1.0 If r = 1.0
sp2 = wＤ 2sＤ 2 + wＥ 2sE2 + 2wDwE σD σ E
sp = wＤ sＤ + wＥ sE
sp2 = (wＤ sＤ + wＥ sE)2
8-15
相关系数,取值范围
Correlation Coefficients,Possible Values
如果 r = -1.0 If r = -1.0
sp2 = wＤ 2sＤ 2 + wＥ 2sE2 - 2wDwE σD σ E
sp = ︳ wＤ sＤ - wＥ sE ︳
sp2 = (wＤ sＤ - wＥ sE)2
wＤ sＤ - wＥ sE = 0
wＤ =σE/(σD + σE) w E=σ D / (σD+σE) =1-WD
8-16
s2p = W12s12+ W22s22
+ 2W1W2
rp = W1r1 + W2r2 + W3r3
Cov(r1r2)
+ W32s32
Cov(r1r3)+ 2W1W3
Cov(r2r3)+ 2W2W3
三种证券组合
Three-Security Portfolio
8-17
rp =多种证券的加权平均
rp = Weighted average of the n securities
sp2 = (考虑全部成双量的 协方差）
sp2 = (Consider all pair-wise
covariance measures)
多种证券组合的一般性
In General,For an n-Security Portfolio:
8-18
E(rp) = W1r1 + W2r2
两种股票组合
Two-Security Portfolio
sp2 = w12s12 + w22s22 + 2W1W2 Cov(r1r2)
sp = [w12s12 + w22s22 + 2W1W2
Cov(r1r2)]1/2
8-19
r = 0
期望收益
E(r)
r = 1r = -1
r = -1
r =,3
13%
8%
12% 20% 标准差St,Dev
不同相关系数的两种股票组合
TWO-SECURITY PORTFOLIOS WITH
DIFFERENT CORRELATIONS
8-20
组合风险 /收益 两种股票组合：相关有效性
Portfolio Risk/Return Two Securities,Correlation Effects
相关性取决于相关系数
Relationship depends on correlation coefficient
-1.0 < r < +1.0
相关性愈小，降低风险的可能性就大
The smaller the correlation,the greater the risk
reduction potential
如果 r = +1.0,降低风险是不可能的
If r = +1.0,no risk reduction is possible
8-21
1
1 2
- Cov(r1r2)
W1 =
+ - 2Cov(r1r2)
2
W2 = (1 - W1)
最小方差组合
Minimum Variance Combination
s 2
s 2
2E(r2) =,14 =,20Sec 2
12 =,2
E(r1) =,10 =,15Sec 1 s
s r
s 2
8-22
W1 =
(.2)2 - (.2)(.15)(.2)
(.15)2 + (.2)2 - 2(.2)(.15)(.2)
W1 =,6733
W2 = (1 -,6733) =,3267
最小方差组合 r =,2
Minimum Variance Combination,r =,2
8-23
rp =,6733(.10) +,3267(.14) =,1131
p = [(.6733)2(.15)2 + (.3267)2(.2)2 +
2(.6733)(.3267)(.2)(.15)(.2)]1/2
p= [.0171]
1/2 =,1308
最小方差,在 r =,2 时的收益和风险
Minimum Variance,Return and Risk with r =,2
s
s
8-24
W1 =
(.2)2 - (.2)(.15)(.2)
(.15)2 + (.2)2 - 2(.2)(.15)(-.3)
W1 =,6087
W2 = (1 -,6087) =,3913
最小方差组合 r = -.3
Minimum Variance Combination,r = -.3
8-25
rp =,6087(.10) +,3913(.14) =,1157
p = [(.6087)2(.15)2 + (.3913)2(.2)2 +
2(.6087)(.3913)(.2)(.15)(-.3)]1/2
p= [.0102]
1/2 =,1009
最小方差,在 r =,3 时的收益和风险
Minimum Variance,Return and Risk with r = -.3
s
s
8-26
多种证券组合
Extending Concepts to All Securities
在给定收益率时优化组合将使风险降到最低水平
The optimal combinations result in lowest level
of risk for a given return
优化替换被描绘为有效边界
The optimal trade-off is described as the efficient
frontier
这些组合是重要的 These portfolios are dominant
8-27
马科维茨的资产组合选择模型,证券选择
The Markowitz Portfolio Selection Model Security Selection
我们可在多种风险证券和无风险资产中间进行资产组合的构造。
在两种风险资产 的例子中，问题分为三个部分，第一，我们要从可能的风险资产组合中识别出风险 - 收益组合。第二，
我们通过资产组合权重的计算，找出最优风险资产组合，此时有最大 斜率的资本配置线。最后，我们通过加入无风险资产，
找到完整的资产组合。
We can generalize the portfolio construction problem to the case
of many risky securities and a risk-free asset,As in the two
risky assets example,the problem has three parts,First,we
identify the risk return combinations available from the set
of risky assets,Next,we identify the optimal portfolio of
risky assets by finding the portfolio weights that result in the
steepest CAL,Finally,we choose an appropriate complete
portfolio by mixing the risk- free asset with the optimal risky
portfolio,
8-28
E(r) 风险资产的最小方差边界The minimum-variance frontier of risky assets
有效率边界
Efficient
frontier
Global
minimum
variance
portfolio Minimum
variance
frontier
单个资产
Individual assets
St,Dev.
8-29
无风险资产组合
Extending to Include Riskless Asset
优化组合转变为线性
The optimal combination becomes linear
风险和无风险资产单一组合将是主要部分
A single combination of risky and riskless
assets will dominate
8-30
组合预期收益 E(r)
CAL (全球最小方差
Global minimum variance)
CAL (A)CAL (P)
M
P
A
F
P P&F A&FM
A
G
P
M
s
供选择的资本分配线
ALTERNATIVE CALS
8-31
最优资本分配线与无风险投资
Dominant CAL with a Risk-Free Investment (F)
资本分配线优与其他线 – 它有最好的风险 /收益或最大的斜率
CAL(P) dominates other lines -- it has the
best risk/return or the largest slope
斜率 =（预期收益率 -无风险利率） /标准差
Slope = (E(R) - Rf) / s
[ E(RP) - Rf) / s P ] > [E(RA) - Rf) / sA]
8-32
最优资产组合
Optimal Risky Portfolio
8-33
最优资产组合
Optimal Risky Portfolio
8-34
最优资产组合
Optimal Risky Portfolio
8-35
最优资产组合
Optimal Risky Portfolio
8-36
最优资产组合
Optimal Risky Portfolio
8-37
最优资产组合
Optimal Risky Portfolio
8-38
最优资产组合：举例
Optimal Risky Portfolio,Exam
8-39
最优资产组合：举例
Optimal Risky Portfolio,Exam
8-40
多种证券组合
Extending Concepts to All Securities
一个完整的资产组合的步骤：
1) 确定所有各类证券的回报特征（例如期望收益、方差、斜方差等）。
2) 建造风险资产组合：
a,计算最优风险资产组合 P（ 8 - 7式）；
b,运用步骤（ a）中确定的权重和 8 - 1式与 8 - 2式来计算资产组合 P的资产。
3) 把基金配置在风险资产组合和无风险资产上：
a,计算资产组合 P（风险资产组合）和国库券（无风险资产）
的权重（ 8 - 8式）；
b,计算出完整的资产组合中投资于每一种资产和国库券上的投资份额。
8-41
多种证券组合
Extending Concepts to All Securities
The steps we followed to arrive at the complete portfolio.
1,Specify the return characteristics of all securities
(expected returns,variances,covariances).
2,Establish the risky portfolio:
a,Calculate the optimal risky portfolio,P (equation 8.7).
b,Calculate the properties of Portfolio P using the
weights determined in step (a) and equations 8.1 and 8.2.
3,Allocate funds between the risky portfolio and the risk-free
asset:
a,Calculate the fraction of the complete portfolio
allocated to Portfolio P (the risky portfolio) and to T-bills
(the risk-free asset) (equation 8.8).
b,Calculate the share of the complete portfolio invested
in each asset and in T-bills.
8-42
小结
Summary
资产组合的期望收益是资产组合中各项资产期望收益的以各项资产为权重的加权平均值。
The expected return of a portfolio is the
weighted average of the component security
expected returns with the investment
proportions as weights.
8-43
小结
Summary
资产组合的方差是协方差矩阵各元素与投资比例为权重相乘的加权总值。因此,每一资产的方差以其投资比例的平方进行加权，任一对资产的协方差在协方差矩阵中 出现两次。所以，资产组合方差中包含着二次斜方差权重，这是由两项资产的每一项资产投资比例的乘积构成的。
The variance of a portfolio is the weighted sum of the
elements of the covariance matrix with the product of the
investment proportions as weights,Thus the variance of
each as- set is weighted by the square of its investment
proportion,Each covariance of any pair of assets appears
twice in the covariance matrix; thus the portfolio variance
includes twice each covariance weighted by the product of
the investment proportions in each of the two assets.
8-44
小结
Summary
即便协方差为正，只要资产不是完全正相关的，资产组合的标准差就仍小于组合中各项资产的标准差的加权平均值。因此，只要资产不是完全正相关的，
分散化的 资产组合就是有价值的。
Even if the covariances are positive,the portfolio
standard deviation is less than the weighted
average of the component standard deviations,as
long as the assets are not perfectly positively
correlated,Thus portfolio diversification is of value
as long as assets are less than perfectly correlated.
8-45
小结
Summary
资产组合中一项资产的斜方差相对于其他资产的协方差越大，
它对资产组合方差的作用就越大。一项资产如与资产组合完全负相关，它将具有完全对冲的功能。一 项完全对冲资产可以降低资产组合的方差至零。
The greater an asset’s covariance with the other
assets in the portfolio,the more it contributes to
portfolio variance,An asset that is perfectly
negatively correlated with a portfolio can serve as
a perfect hedge,The perfect hedge asset can reduce
the portfolio variance to zero.
8-46
小结
Summary
有效率边界是利用图表来表示在某 — 特定风险水平上，有最大收益的一组资产组合。理性投资者将在有效边界上选择资产组合。
The efficient frontier is the graphical
representation of a set of portfolios that
maximize expected return for each level of
portfolio risk,Rational investors will choose
a portfolio on the efficient frontier,
8-47
小结
Summary
一个资产组合经理在确定有效边界时，首先要估计资产的期望收益与协方差矩阵。这个输入清单被输入优化程序中，得到在有效率边界上资产组合中各项资产的比例、期望收益与标准差等。
A portfolio manager identifies the efficient frontier
by first establishing estimates for the asset
expected returns and the covariance matrix,This
input list is then fed into an optimization program
that reports as outputs the investment proportions,
expected returns,and standard deviations of the
portfolios on the efficient frontier.
8-48
小结
Summary
一般来说，资产组合经理会得到不同的有效资产组合，因为他们的证券分析方法与质量是不同的。经理们主要在证券分析质量，而不是在管理费上展开竞争。
In general,portfolio managers will arrive at
different efficient portfolios because of
differences in methods and quality of
security analysis,Managers compete on the
quality of their security analysis relative to
their management fees.
8-49
小结
Summary
如果无风险资产存在，输入清单亦可以确定，所有投资者都将选择在有效率边 界上同样的资产组合：与资本配置线相切的资产组合。具有相同输入清单的所有投资者将持有相同的风险资产组合，不同的是在风险资产组合和无风险资产之间的资金分配。这一结果就是资产组合构造的分离原则。
If a risk-free asset is available and input lists are
identical,all investors will choose the same portfolio on the
efficient frontier of risky assets,the portfolio tangent to
the CAL,All investors with identical input lists will hold an
identical risky portfolio,differing only in how much each
allocates to this optimal portfolio and to the risk-free asset,
This result is characterized as the separation principle of
portfolio construction.
8-50
小结
Summary
当无风险资产不存在时，每个投资者在有效率边界上选择风险资产组合。如果 有无风险资产，但借入是受限制的，那么，只有冒险型投资者会受到影响，他们将根据其愿意冒险的程度在有效率边界上选择资产组合。
When a risk-free asset is not available,each
investor chooses a risky portfolio on the efficient
frontier,If a risk-free asset is available but borrowing
is restricted,only aggressive investors will be
affected,They will choose portfolios on the
efficient frontier according to their degree of risk
tolerance.
