10-1
第 10章单一指数和多因素模型
Single Index and Multifactor Models
10-2
单一指数和多因素模型
Single Index and Multifactor Models
10.1 单一指数证券市场
10.2 CAPM模型与指数模型
10.3 指数模型的行业性
10.4 多因素模型
10-3
假定证券分析人员能详细地分析 50种股票,这意味着需要输入如下这些数据：
Suppose your security analysts can thoroughly analyze
50 stocks,This means that your input list will include
the following:
n＝ 50个期望收益的估计 (estimates of expected returns)
n＝ 50个方差估计 (estimates of variances)
(n2-n)/2＝ 1225 个协方差估计 (estimates of covariances)
1325个估计值 (estimates)
单一指数模型的优势
Advantages of the Single Index Model
10-4
减少输入数量
Reduces the number of inputs for
diversification.
简化证券分析
Easier for security analysts to specialize.
单一指数模型的优势
Advantages of the Single Index Model
10-5
证券 i的持有期收益,The holding-period return on security i is:
ri = E(ri)+mi +ei
E(ri )是证券持有期期初的期望收益,mi 是在证券持有期间非预期的宏观事件对证券收益的影响,ei 是非预期的公司特有事件的影响。
E(ri ) is the expected return on the security as of the beginning
of the holding period,mi is the impact of unanticipated macro
events on the security’s return during the period,and ei is the
impact of unanticipated firm-specific events,
单一指数模型
Single Factor Model
10-6
i =证券 i 对宏观因素的敏感度
the responsiveness of security i to macro events
F=宏观因素的非预测成分,与证券收益有关
some macro factor; in this case F is unanticipated movement;
F is commonly related to security returns
假设：主要证券市场指数，譬如标准普尔 500指数的收益率，是一般宏观因素的有效代表。
Assumption,a broad market index like the S&P500 is the
common factor,
ri = E(ri)+βiF +ei
单一指数模型
Single Factor Model
10-7
根据指数模型，依照与等式 10-2相似的原理，我们可以把实际的或已实现的证券收益率区分成宏观（系统）
的与微观（公司特有）的两部分。我们把每个证券的收益率写成三个部分的总和：
According to the index model,we can separate the
actual or realized rate of return on a security into
macro (systematic) and micro (firm-specific)
components in a manner similar to that in equation
10.2,We write the rate of return on each security as
a sum of three components:
单一指数模型
Single Factor Model
10-8
单一指数模型
Single Factor Model
10-9
(ri - rf) = i +?i(rm - rf) + eia
风险溢价 Risk Prem 市场风险溢价 Market Risk Prem
或指数 风险溢价 or Index Risk Prem
单一指数模型
Single Factor Model
10-10
Let,Ri = (ri - rf)
Rm = (rm - rf)
风险溢价格式
Risk premium
format
Ri = ai +?i(Rm) + ei
风险溢价格式
Risk Premium Format
R代表超过无风险收益的超额收益
excess returns over the risk-free rate
10-11
单一指数模型
Single Factor Model
每种证券有两种风险来源：市场的或系统的风险，它们的区别源于它们对宏观经济因素的敏感度，这个差异反映在 RM上，以及对公司特有风险的敏感度，
这个差异反映在 e上。如果我们记市场超额收益 RM的方差为 σ2M，则我们可以把每个股票收益率的方差拆分成两部分：
each security has two sources of risk,market or systematic risk,attributable
to its sensitivity to macroeconomic factors as reflected in RM,and firm-
specific risk,as reflected in e,If we denote the variance of the excess return
on the market,RM,as σ2M,then we can break the variance of the rate of
return on each stock into two components:
10-12
单一指数模型
Single Factor Model
RM 和 ei的协方差为零，因为 ei定义为公司特有的，即独立于市场的运动。因此证券 i的收益率的方差为：
The covariance between RM and e i is zero because e i is
defined as firm specific,that is,independent of movements
in the market,Hence the variance of the rate of return on
security i equals the sum of the variances due to the
common and the firm-specific components
10-13
单一指数模型
Single Factor Model
两个股票超额收益率的协方差，譬如 Ri与 R j 的协方差，仅仅来自于一般因素 RM,因为 e i和 e j 都是每个公司特有的，它们显然不相关。所以，两个股票的协方差为,
The covariance between RM and ei is zero because e i is
defined as firm specific,that is,independent of movements in
the market,Hence the variance of the rate of return on
security i equals the sum of the variances due to the common
and the firm-specific components,
10-14
单一指数模型
Single Factor Model
n个期望超额收益 E(Ri)的估计，
n个敏感度协方差 i的估计，
n个公司特有方差 σ2(ei)的估计，
1个（一般）宏观经济因素的方差 σ2 M的估计，
那么，这一计算式就表明这些（ 3n＋ 1）个估计值将为我们的单指数证券模型准备好输入的数据。这样，对于有 50种证券的资产组合，我们将需要 151个估计值，而不是 1325 个；对整个纽约证券交易所的大约 3000 个证券，我们将需要 9001个估计值,而不是大约 450万个！
These calculations show that if we have
n estimates of the expected excess returns,E(Ri)
n estimates of the sensitivity coefficients,i
n estimates of the firm-specific variances,σ2(ei)
1 estimate for the variance of the (common) macroeconomic factor,σ2 M
then these (3n+1) estimates will enable us to prepare the input list for this single-
index security universe,Thus for a 50-security portfolio we will need 151 estimates
rather than 1,325; for the entire New York Stock Exchange,about 3,000 securities,
we will need 9,001 estimates rather than approximately 4.5 million!
10-15
证券特征线
Security Characteristic Line
证券 i超额收益 Excess Returns (i)
SCL
.
.
...,.,
.,.
.,
.
.,
.,.
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.,.,
.,.
.
.
.,
.,
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市场指数超额收益
Excess returns
on market index
Ri = a i +?iRM + ei
...
10-16
证券特征线
Security Characteristic Line
在上图中，横轴测度了市场指数（超过无风险利率的）的超额收益，竖轴测度了 GM的超额收益。一对超额收益（一个是市场超额收益，一个是 GM的超额收益）组成了散点图中的一点。这些点从第 1到第 12，代表着从 1月份到 12月份每月的标准普尔 500
指数和 GM的超额收益。单指数模型表明,GM的超 额收益与标准普尔 500指数的超额收益之间的关系由下式给定
The horizontal axis in Figure 10.1 measures the excess return (over the risk-free rate)
on the market index,whereas the vertical axis measures the excess return of GM
stock,A pair of excess returns (one for the market index,one for GM stock)
constitutes one point on this scatter diagram,The points are numbered 1 through 12,
representing excess returns for the S&P 500 and GM for each month from January
through December,The single-index model states that the relationship between the
excess returns on GM and the S&P 500 is given by
RGM t = αGM + βGM RMt + eGM t
10-17
证券特征线
Security Characteristic Line
这一关系类似于回归方程。 在一个单变量的线性回归等式中，依赖变量标在一条截距为 α、斜率为 β 的直线周围。假定这条线的偏差 e 与独立变量不相关；同样
，它们相互之间也不相关。
The resemblance of this relationship to a regression
equation,In a single-variable regression equation,
the dependent variable plots around a straight line
with an intercept αand a slope β,The deviations from
the line,e,are assumed to be mutually uncorrelated
as well as uncorrelated with the independent variable,
10-18
证券特征线
Security Characteristic Line
我们通过 β GM来 测度的 GM对市场的敏感度，它是回归直线的斜率。回归直线的截距是 αGM，它代表了 平均的公司特有收益。在任一时期里，回归直线的特定观测偏差记为 e GMt,称为残值。每一个残值都是实际股票收益与由描述股票同市场之间一般关系的回归等式所预测出的股票收益之间的差异。因此，它们测度了特定期间公司特有事件的影响。利息参数 α,β和 Var（ e），可以用标准回归技术来估计。
The sensitivity of GM to the market,measured by β GM,is the slope of the
regression line,The intercept of the regression line is αGM,representing the
average firm-specific return when the market’s excess return is zero,Deviations of
particular observations from the regression line in any period are denoted eGMt,and
called residuals,Each of these residuals is the difference between the actual stock
return and the return that would be predicted from the regression equation
describing the usual relationship between the stock and the market; therefore,
residuals measure the impact of firm-specific events during the particular month,
The parameters of interest,α,β,and Var(e),can be estimated using standard
regression techniques.
10-19
Using the Text Example from Table 10-1
10-20
Using the Text Example from Table 10-1
10-21
市场或系统风险,Market or systematic risk,
风险来自宏观经济因素或市场指数
risk related to the macro economic factor or market
index.
非系统风险或公司特有风险,
Unsystematic or firm specific risk,
风险与宏观经济因素或市场指数无关
risk not related to the macro factor or market index.
总风险 = 系统 +非系统
Total risk = Systematic + Unsystematic
风险的构成
Components of Risk
10-22
风险测量
Measuring Components of Risk
每个证券的超额收益率为：
Ri = a i +?iRm + ei
资产组合的 超额收益率为：
Rp = a p +?pRM + eP (10-5)
10-23
风险测量
Measuring Components of Risk
我们注意到等权重（每种资产权重 w i＝ 1/n）资产组合的超额收益率为,
The excess rate of return on this equally weighted
portfolio,for which each portfolio weight w i =1/n,is
10-24
风险测量
Measuring Components of Risk
比较等式 10-5和 10-6，我们看到资产组合对市场的敏感度由下式给出,
Comparing equations 10.5 and 10.6,we see that the
portfolio has a sensitivity to the market given by
10-25
风险测量
Measuring Components of Risk
资产组合有一个常数（截距）的非市场收益成分,
It has a nonmarket return component of a constant
(intercept)
10-26
风险测量
Measuring Components of Risk
零均值变量
The zero mean variable
10-27
P2 =?P2?M2 +?2(eP)
where;
P2 = 资产组合总方差 total variance
P2?M2 = 系统方差 systematic variance
2(eP) = 非系统方差 unsystematic variance
风险测量
Measuring Components of Risk
10-28
资产组合方差的非系统成分是?2(eP)，它来源于公司特有成分 ei。因为这 些 ei是独立的，都具有零期望值，所以平均法则可以被用来得出这样的结论：随着越来越多的股票加入到资产组合中，公司特有风险倾向于被消除掉，结果只剩下越来越小的非市场风险，这些风险被认为是可分散的。为更准确地理解这一点，考虑有公司特有成分的等权重“资产组合”的方差公式。因为 ei是不相关的，则风险测量
Measuring Components of Risk
10-29
这里 σ2(e)是公司特有方差的均值。由于这一均值独立于 n，所以当 n 变大时,σ2(eP)就变得小得可以忽略了。
简而言之，随着分散化程度的加强，资产组合的方差接近于系统方差。系统方差定义为市场因素的方差乘以资产组合敏感系数的平方 β２ P 。图 10-2对此作了说明。
where σ2(e) is the average of the firm-specific
variances,Because this average is independent of n,
when n gets large,σ2(eP) becomes negligible.
To summarize,as diversification increases,the total
variance of a portfolio approaches the systematic
variance,defined as the variance of the market factor
multiplied by the square of the portfolio sensitivity
coefficient,β２ P,This is shown in Figure 10.2.
风险测量
Measuring Components of Risk
10-30
风险系数与资产组合方差
The variance of a portfolio with risk coefficient
证券数量
Number of
Securities
标准方差 St,Deviation
市场风险
Market Risk
独有风险 Unique Risk
2(eP)=?2(e) / n
P2?M2
10-31
资本资产定价模型与指数模型
The CAPM and the index model
资本资产定价模型是关于预期收益的论断，然而实际上，任何人都可以直接观察到已实现的收益。为了使期望收益变成已实现收益，我们可以运用指数模型。我们把超额收益写成下列形式,
The CAPM is a statement about expected returns,
whereas in practice all anyone can observe directly
are ex post or realized returns,To make the leap
from expected to realized returns,we can employ
the index model,which we will use in excess return
form as:
10-32
资本资产定价模型与指数模型
The CAPM and the index model
从股票 i的收益与市场指数收益之间的协方差开始我们的分析。通过定义，公司特有的或非系统的成分独立于整个市场的或系统的成分，即 Cov(R M,ei)＝ 0，从这一关系导出证券 i的超额收益率与市场指数的协方差为
Deriving the covariance between the returns on stock i and the
market index,By definition,the firm-specific or nonsystematic
component is independent of the market wide or systematic
component,that is,Cov(R M,ei)＝ 0,From this relationship,it
follows that the covariance of the excess rate of return on
security i with that of the market index is
10-33
资本资产定价模型与指数模型
The CAPM and the index model
因为 Cov(Ri,RM)＝ βi σM2，等式 10-9中的敏感度系数 i代表指数模型的回归线的斜率，它等于
Because Cov(Ri,RM)＝ βi σM2,the sensitivity
coefficient,i,in equation 10.9,which is the
slope of the regression line representing the
index model,equals
10-34
资本资产定价模型与指数模型
The CAPM and the index model
指数模型贝塔系数的结果与资本资产定价模型期望收益
- 贝塔关系的贝塔相同,除非我们重新安排带有特定的可观测市场指数（理论的）的 CAPM市场资产组合。
The index model beta coefficient turns out to be the
same beta as that of the CAPM expected return–
beta relationship,except that we replace the
(theoretical) market portfolio of the CAPM with the
well-specified and observable market index.
10-35
指数模型与期望收益 - 贝塔关系
The index model and the expected return - beta relationship
对任意资产 i和（理论的）市场资产组合，有
for any asset i and the (theoretical) market portfolio,
这里 βi ＝ Cov(R i,RM)/σM2 。这显示了相对于（理论的）市场资产组合平均超额收益的资产平均期望超额收益的情况。
where βi ＝ Cov(R i,RM)/σM2,This is a statement about the
mean of expected excess returns of assets relative to the mean
excess return of the (theoretical) market portfolio.
10-36
指数模型与期望收益 - 贝塔关系
The index model and the expected return - beta relationship
如果等式 10-9中的指数 M代表了真实的市场资产组合，我们可以对等式每边取期望，以此来说明指数模型的详细内容,即
If the index M in equation 10.9 represents the true
market portfolio,we can take the expectation of each
side of the equation to show that the index model
specification is
10-37
指数模型与期望收益 - 贝塔关系
The index model and the expected return - beta relationship
指数模型关系与资本资产定价模型的期望收益 -贝塔关系
（等式 10-8）的比较表明,资本资产定价模型预言 αi对所有资产都将为零。一个股票的阿尔法值是它超过（或者低于）通过资本资产定价模型预测的可能期望收益的部分。如果股票公平定价，则其 阿尔法必定为零。
A comparison of the index model relationship to the
CAPM expected return–beta relationship (equation
10.8) shows that the CAPM predicts that αi should be
zero for all assets,The alpha of a stock is its expected
return in excess of (or below) the fair expected return
as predicted by the CAPM,If the stock is fairly priced,
its alpha must be zero.
10-38
指数模型与期望收益 - 贝塔关系
The index model and the expected return - beta relationship
在指数模型的直观形式 —市场模型 （ market model） 中,
还有另一合适的方差 。 正规的说,市场模型表明,任意证券的,意外,收益是市场的,意外,收益的一个比例,加上一个公司特有的,意外,收益,有：
There is yet another applicable variation on the intuition
of the index model,the market model,Formally,the
market model states that the return ―surprise‖ of any
security is proportional to the return surprise of the
market,plus a firm-specific surprise:
10-39
指数模型与期望收益 - 贝塔关系
The index model and the expected return - beta relationship
这个等式与指数模型不同,它把收益分成公司特有的和系统的两部分 。 然而,如果资本资产定价模型是有效的,那可以看到,把
E(ri)从等式 10-8中消掉,则市场模型等式变成了指数模型等式 。
由于这个原因,,指数模型,和,市场模型,可以相互变换着用 。
This equation divides returns into firm-specific and
systematic components somewhat differently from the index
model,If the CAPM is valid,however,you can see that,
substituting for E(ri) from equation 10.8,the market model
equation becomes identical to the index model,For this reason
the terms ―index model‖ and ―market model‖ are used
interchangeably.
10-40
多因素模型
Multifactor Models
根据市场收益使用因素
Use factors in addition to market return
– 举例含有行业生产,预期通货膨胀等
Examples include industrial production,expected inflation
etc.
– 用多元回归为每一个因素估计 β值
Estimate a beta for each factor using multiple regression.
法马与弗伦奇建立了一个多因素模型。 Fama and French
返回一个帐面价值 - 市值比涵数和市场收益高低涵数
– Returns a function of size and book-to-market value as well
as market returns.
10-41
summary
经济的单因素模型把不确定性来源分成系统
（宏观经济）的因素或公司特有（微观经济）
的因素。指数模型假设宏观因素可以由股票收益的一个公开指数所代表。
A single-factor model of the economy classifies
sources of uncertainty as systematic
(macroeconomic) factors or firm-specific
(microeconomic) factors,The index model
assumes that the macro factor can be
represented by a broad index of stock returns.
10-42
summary
单指数模型大大降低了马克维茨资产组合选择程序的数据数量，它把精力放在 了对证券的专门分析中。
The single-index model drastically reduces the
necessary inputs in the Markowitz port- folio
selection procedure,It also aids in
specialization of labor in security analysis.
10-43
summary
根据指数模型的详细内容，资产组合或资产的系统风险等于 β2M，而两项资产的协方差为
βi βj σ M2。
According to the index model specification,
the systematic risk of a portfolio or asset
equals β2M and the covariance between two
assets equals βi βj σ M2。
10-44
summary
指数模型通过运用对超额收益率的回归分析来估计。回归曲线的斜率是资产的贝塔值，而截距是样本期间的资产的阿尔法。回归线也称为证券特征线
（ SCL）。回归贝塔等于资本资产定价模型的贝塔，除非回归运用的是实际收益，而资本资产定价模 型根据的是期望收益。该模型预言，由指数模型回归测度的阿尔法的平均值将为零。
The index model is estimated by applying regression analysis to excess rates
of return,The slope of the regression curve is the beta of an asset,whereas
the intercept is the asset’s alpha during the sample period,The regression
line is also called the security characteristic line,The regression beta is
equivalent to the CAPM beta,except that the regression uses actual returns
and the CAPM is specified in terms of expected returns,The CAPM predicts
that the average value of alphas measured by the index model regression
will be zero.
10-45
summary
操盘手习惯于用总的而不是超额的收益率来估计指数模型。这使他们的阿尔法估计值等于 α＋ rf (1-β)。
Practitioners routinely estimate the index
model using total rather than excess rates
of return,This makes their estimate of
alpha equal to α＋ rf (1-β)。
10-46
summary
贝塔显示了一个沿时间趋向于 1的趋势。贝塔的预测法试图预言这一趋势。另外，其他的财务变量也可以被用来帮助预测贝塔。
Betas show a tendency to evolve toward 1
over time,Beta forecasting rules attempt
to predict this drift,Moreover,other
financial variables can be used to help
forecast betas.
10-47
Summary
多因素模型试图通过在更多的细节上把系统部分模型化来提高单指数模型的解释能力。这些模型利用指示器，试图把握一个范围广泛的宏观经济的风险因素和某些时候的公司特征变量，譬如公司的规模或帐面价值 -市值比率。
Multifactor models seek to improve the explanatory
power of the single-index model by modeling the
systematic component of returns in greater detail,These
models use indicators intended to capture a wide range
of macroeconomic risk factors and,sometimes,firm-
characteristic variables such as size or book-to-market
ratio.
10-48
Summary
ICAPM 是单因素 CAPM的一个扩展，它也是一种证券收益的多因素模型，但它不必指定一定要考虑哪些风险因素。
An extension of the single-factor CAPM,
the ICAPM,is a multifactor model of
security returns,but it does not specify
which risk factors need to be considered.
第 10章单一指数和多因素模型
Single Index and Multifactor Models
10-2
单一指数和多因素模型
Single Index and Multifactor Models
10.1 单一指数证券市场
10.2 CAPM模型与指数模型
10.3 指数模型的行业性
10.4 多因素模型
10-3
假定证券分析人员能详细地分析 50种股票,这意味着需要输入如下这些数据：
Suppose your security analysts can thoroughly analyze
50 stocks,This means that your input list will include
the following:
n＝ 50个期望收益的估计 (estimates of expected returns)
n＝ 50个方差估计 (estimates of variances)
(n2-n)/2＝ 1225 个协方差估计 (estimates of covariances)
1325个估计值 (estimates)
单一指数模型的优势
Advantages of the Single Index Model
10-4
减少输入数量
Reduces the number of inputs for
diversification.
简化证券分析
Easier for security analysts to specialize.
单一指数模型的优势
Advantages of the Single Index Model
10-5
证券 i的持有期收益,The holding-period return on security i is:
ri = E(ri)+mi +ei
E(ri )是证券持有期期初的期望收益,mi 是在证券持有期间非预期的宏观事件对证券收益的影响,ei 是非预期的公司特有事件的影响。
E(ri ) is the expected return on the security as of the beginning
of the holding period,mi is the impact of unanticipated macro
events on the security’s return during the period,and ei is the
impact of unanticipated firm-specific events,
单一指数模型
Single Factor Model
10-6
i =证券 i 对宏观因素的敏感度
the responsiveness of security i to macro events
F=宏观因素的非预测成分,与证券收益有关
some macro factor; in this case F is unanticipated movement;
F is commonly related to security returns
假设：主要证券市场指数，譬如标准普尔 500指数的收益率，是一般宏观因素的有效代表。
Assumption,a broad market index like the S&P500 is the
common factor,
ri = E(ri)+βiF +ei
单一指数模型
Single Factor Model
10-7
根据指数模型，依照与等式 10-2相似的原理，我们可以把实际的或已实现的证券收益率区分成宏观（系统）
的与微观（公司特有）的两部分。我们把每个证券的收益率写成三个部分的总和：
According to the index model,we can separate the
actual or realized rate of return on a security into
macro (systematic) and micro (firm-specific)
components in a manner similar to that in equation
10.2,We write the rate of return on each security as
a sum of three components:
单一指数模型
Single Factor Model
10-8
单一指数模型
Single Factor Model
10-9
(ri - rf) = i +?i(rm - rf) + eia
风险溢价 Risk Prem 市场风险溢价 Market Risk Prem
或指数 风险溢价 or Index Risk Prem
单一指数模型
Single Factor Model
10-10
Let,Ri = (ri - rf)
Rm = (rm - rf)
风险溢价格式
Risk premium
format
Ri = ai +?i(Rm) + ei
风险溢价格式
Risk Premium Format
R代表超过无风险收益的超额收益
excess returns over the risk-free rate
10-11
单一指数模型
Single Factor Model
每种证券有两种风险来源：市场的或系统的风险，它们的区别源于它们对宏观经济因素的敏感度，这个差异反映在 RM上，以及对公司特有风险的敏感度，
这个差异反映在 e上。如果我们记市场超额收益 RM的方差为 σ2M，则我们可以把每个股票收益率的方差拆分成两部分：
each security has two sources of risk,market or systematic risk,attributable
to its sensitivity to macroeconomic factors as reflected in RM,and firm-
specific risk,as reflected in e,If we denote the variance of the excess return
on the market,RM,as σ2M,then we can break the variance of the rate of
return on each stock into two components:
10-12
单一指数模型
Single Factor Model
RM 和 ei的协方差为零，因为 ei定义为公司特有的，即独立于市场的运动。因此证券 i的收益率的方差为：
The covariance between RM and e i is zero because e i is
defined as firm specific,that is,independent of movements
in the market,Hence the variance of the rate of return on
security i equals the sum of the variances due to the
common and the firm-specific components
10-13
单一指数模型
Single Factor Model
两个股票超额收益率的协方差，譬如 Ri与 R j 的协方差，仅仅来自于一般因素 RM,因为 e i和 e j 都是每个公司特有的，它们显然不相关。所以，两个股票的协方差为,
The covariance between RM and ei is zero because e i is
defined as firm specific,that is,independent of movements in
the market,Hence the variance of the rate of return on
security i equals the sum of the variances due to the common
and the firm-specific components,
10-14
单一指数模型
Single Factor Model
n个期望超额收益 E(Ri)的估计，
n个敏感度协方差 i的估计，
n个公司特有方差 σ2(ei)的估计，
1个（一般）宏观经济因素的方差 σ2 M的估计，
那么，这一计算式就表明这些（ 3n＋ 1）个估计值将为我们的单指数证券模型准备好输入的数据。这样，对于有 50种证券的资产组合，我们将需要 151个估计值，而不是 1325 个；对整个纽约证券交易所的大约 3000 个证券，我们将需要 9001个估计值,而不是大约 450万个！
These calculations show that if we have
n estimates of the expected excess returns,E(Ri)
n estimates of the sensitivity coefficients,i
n estimates of the firm-specific variances,σ2(ei)
1 estimate for the variance of the (common) macroeconomic factor,σ2 M
then these (3n+1) estimates will enable us to prepare the input list for this single-
index security universe,Thus for a 50-security portfolio we will need 151 estimates
rather than 1,325; for the entire New York Stock Exchange,about 3,000 securities,
we will need 9,001 estimates rather than approximately 4.5 million!
10-15
证券特征线
Security Characteristic Line
证券 i超额收益 Excess Returns (i)
SCL
.
.
...,.,
.,.
.,
.
.,
.,.
.
.,.,
.,.
.
.
.,
.,
.
.,,.
.,.,..
.
.,.
.
市场指数超额收益
Excess returns
on market index
Ri = a i +?iRM + ei
...
10-16
证券特征线
Security Characteristic Line
在上图中，横轴测度了市场指数（超过无风险利率的）的超额收益，竖轴测度了 GM的超额收益。一对超额收益（一个是市场超额收益，一个是 GM的超额收益）组成了散点图中的一点。这些点从第 1到第 12，代表着从 1月份到 12月份每月的标准普尔 500
指数和 GM的超额收益。单指数模型表明,GM的超 额收益与标准普尔 500指数的超额收益之间的关系由下式给定
The horizontal axis in Figure 10.1 measures the excess return (over the risk-free rate)
on the market index,whereas the vertical axis measures the excess return of GM
stock,A pair of excess returns (one for the market index,one for GM stock)
constitutes one point on this scatter diagram,The points are numbered 1 through 12,
representing excess returns for the S&P 500 and GM for each month from January
through December,The single-index model states that the relationship between the
excess returns on GM and the S&P 500 is given by
RGM t = αGM + βGM RMt + eGM t
10-17
证券特征线
Security Characteristic Line
这一关系类似于回归方程。 在一个单变量的线性回归等式中，依赖变量标在一条截距为 α、斜率为 β 的直线周围。假定这条线的偏差 e 与独立变量不相关；同样
，它们相互之间也不相关。
The resemblance of this relationship to a regression
equation,In a single-variable regression equation,
the dependent variable plots around a straight line
with an intercept αand a slope β,The deviations from
the line,e,are assumed to be mutually uncorrelated
as well as uncorrelated with the independent variable,
10-18
证券特征线
Security Characteristic Line
我们通过 β GM来 测度的 GM对市场的敏感度，它是回归直线的斜率。回归直线的截距是 αGM，它代表了 平均的公司特有收益。在任一时期里，回归直线的特定观测偏差记为 e GMt,称为残值。每一个残值都是实际股票收益与由描述股票同市场之间一般关系的回归等式所预测出的股票收益之间的差异。因此，它们测度了特定期间公司特有事件的影响。利息参数 α,β和 Var（ e），可以用标准回归技术来估计。
The sensitivity of GM to the market,measured by β GM,is the slope of the
regression line,The intercept of the regression line is αGM,representing the
average firm-specific return when the market’s excess return is zero,Deviations of
particular observations from the regression line in any period are denoted eGMt,and
called residuals,Each of these residuals is the difference between the actual stock
return and the return that would be predicted from the regression equation
describing the usual relationship between the stock and the market; therefore,
residuals measure the impact of firm-specific events during the particular month,
The parameters of interest,α,β,and Var(e),can be estimated using standard
regression techniques.
10-19
Using the Text Example from Table 10-1
10-20
Using the Text Example from Table 10-1
10-21
市场或系统风险,Market or systematic risk,
风险来自宏观经济因素或市场指数
risk related to the macro economic factor or market
index.
非系统风险或公司特有风险,
Unsystematic or firm specific risk,
风险与宏观经济因素或市场指数无关
risk not related to the macro factor or market index.
总风险 = 系统 +非系统
Total risk = Systematic + Unsystematic
风险的构成
Components of Risk
10-22
风险测量
Measuring Components of Risk
每个证券的超额收益率为：
Ri = a i +?iRm + ei
资产组合的 超额收益率为：
Rp = a p +?pRM + eP (10-5)
10-23
风险测量
Measuring Components of Risk
我们注意到等权重（每种资产权重 w i＝ 1/n）资产组合的超额收益率为,
The excess rate of return on this equally weighted
portfolio,for which each portfolio weight w i =1/n,is
10-24
风险测量
Measuring Components of Risk
比较等式 10-5和 10-6，我们看到资产组合对市场的敏感度由下式给出,
Comparing equations 10.5 and 10.6,we see that the
portfolio has a sensitivity to the market given by
10-25
风险测量
Measuring Components of Risk
资产组合有一个常数（截距）的非市场收益成分,
It has a nonmarket return component of a constant
(intercept)
10-26
风险测量
Measuring Components of Risk
零均值变量
The zero mean variable
10-27
P2 =?P2?M2 +?2(eP)
where;
P2 = 资产组合总方差 total variance
P2?M2 = 系统方差 systematic variance
2(eP) = 非系统方差 unsystematic variance
风险测量
Measuring Components of Risk
10-28
资产组合方差的非系统成分是?2(eP)，它来源于公司特有成分 ei。因为这 些 ei是独立的，都具有零期望值，所以平均法则可以被用来得出这样的结论：随着越来越多的股票加入到资产组合中，公司特有风险倾向于被消除掉，结果只剩下越来越小的非市场风险，这些风险被认为是可分散的。为更准确地理解这一点，考虑有公司特有成分的等权重“资产组合”的方差公式。因为 ei是不相关的，则风险测量
Measuring Components of Risk
10-29
这里 σ2(e)是公司特有方差的均值。由于这一均值独立于 n，所以当 n 变大时,σ2(eP)就变得小得可以忽略了。
简而言之，随着分散化程度的加强，资产组合的方差接近于系统方差。系统方差定义为市场因素的方差乘以资产组合敏感系数的平方 β２ P 。图 10-2对此作了说明。
where σ2(e) is the average of the firm-specific
variances,Because this average is independent of n,
when n gets large,σ2(eP) becomes negligible.
To summarize,as diversification increases,the total
variance of a portfolio approaches the systematic
variance,defined as the variance of the market factor
multiplied by the square of the portfolio sensitivity
coefficient,β２ P,This is shown in Figure 10.2.
风险测量
Measuring Components of Risk
10-30
风险系数与资产组合方差
The variance of a portfolio with risk coefficient
证券数量
Number of
Securities
标准方差 St,Deviation
市场风险
Market Risk
独有风险 Unique Risk
2(eP)=?2(e) / n
P2?M2
10-31
资本资产定价模型与指数模型
The CAPM and the index model
资本资产定价模型是关于预期收益的论断，然而实际上，任何人都可以直接观察到已实现的收益。为了使期望收益变成已实现收益，我们可以运用指数模型。我们把超额收益写成下列形式,
The CAPM is a statement about expected returns,
whereas in practice all anyone can observe directly
are ex post or realized returns,To make the leap
from expected to realized returns,we can employ
the index model,which we will use in excess return
form as:
10-32
资本资产定价模型与指数模型
The CAPM and the index model
从股票 i的收益与市场指数收益之间的协方差开始我们的分析。通过定义，公司特有的或非系统的成分独立于整个市场的或系统的成分，即 Cov(R M,ei)＝ 0，从这一关系导出证券 i的超额收益率与市场指数的协方差为
Deriving the covariance between the returns on stock i and the
market index,By definition,the firm-specific or nonsystematic
component is independent of the market wide or systematic
component,that is,Cov(R M,ei)＝ 0,From this relationship,it
follows that the covariance of the excess rate of return on
security i with that of the market index is
10-33
资本资产定价模型与指数模型
The CAPM and the index model
因为 Cov(Ri,RM)＝ βi σM2，等式 10-9中的敏感度系数 i代表指数模型的回归线的斜率，它等于
Because Cov(Ri,RM)＝ βi σM2,the sensitivity
coefficient,i,in equation 10.9,which is the
slope of the regression line representing the
index model,equals
10-34
资本资产定价模型与指数模型
The CAPM and the index model
指数模型贝塔系数的结果与资本资产定价模型期望收益
- 贝塔关系的贝塔相同,除非我们重新安排带有特定的可观测市场指数（理论的）的 CAPM市场资产组合。
The index model beta coefficient turns out to be the
same beta as that of the CAPM expected return–
beta relationship,except that we replace the
(theoretical) market portfolio of the CAPM with the
well-specified and observable market index.
10-35
指数模型与期望收益 - 贝塔关系
The index model and the expected return - beta relationship
对任意资产 i和（理论的）市场资产组合，有
for any asset i and the (theoretical) market portfolio,
这里 βi ＝ Cov(R i,RM)/σM2 。这显示了相对于（理论的）市场资产组合平均超额收益的资产平均期望超额收益的情况。
where βi ＝ Cov(R i,RM)/σM2,This is a statement about the
mean of expected excess returns of assets relative to the mean
excess return of the (theoretical) market portfolio.
10-36
指数模型与期望收益 - 贝塔关系
The index model and the expected return - beta relationship
如果等式 10-9中的指数 M代表了真实的市场资产组合，我们可以对等式每边取期望，以此来说明指数模型的详细内容,即
If the index M in equation 10.9 represents the true
market portfolio,we can take the expectation of each
side of the equation to show that the index model
specification is
10-37
指数模型与期望收益 - 贝塔关系
The index model and the expected return - beta relationship
指数模型关系与资本资产定价模型的期望收益 -贝塔关系
（等式 10-8）的比较表明,资本资产定价模型预言 αi对所有资产都将为零。一个股票的阿尔法值是它超过（或者低于）通过资本资产定价模型预测的可能期望收益的部分。如果股票公平定价，则其 阿尔法必定为零。
A comparison of the index model relationship to the
CAPM expected return–beta relationship (equation
10.8) shows that the CAPM predicts that αi should be
zero for all assets,The alpha of a stock is its expected
return in excess of (or below) the fair expected return
as predicted by the CAPM,If the stock is fairly priced,
its alpha must be zero.
10-38
指数模型与期望收益 - 贝塔关系
The index model and the expected return - beta relationship
在指数模型的直观形式 —市场模型 （ market model） 中,
还有另一合适的方差 。 正规的说,市场模型表明,任意证券的,意外,收益是市场的,意外,收益的一个比例,加上一个公司特有的,意外,收益,有：
There is yet another applicable variation on the intuition
of the index model,the market model,Formally,the
market model states that the return ―surprise‖ of any
security is proportional to the return surprise of the
market,plus a firm-specific surprise:
10-39
指数模型与期望收益 - 贝塔关系
The index model and the expected return - beta relationship
这个等式与指数模型不同,它把收益分成公司特有的和系统的两部分 。 然而,如果资本资产定价模型是有效的,那可以看到,把
E(ri)从等式 10-8中消掉,则市场模型等式变成了指数模型等式 。
由于这个原因,,指数模型,和,市场模型,可以相互变换着用 。
This equation divides returns into firm-specific and
systematic components somewhat differently from the index
model,If the CAPM is valid,however,you can see that,
substituting for E(ri) from equation 10.8,the market model
equation becomes identical to the index model,For this reason
the terms ―index model‖ and ―market model‖ are used
interchangeably.
10-40
多因素模型
Multifactor Models
根据市场收益使用因素
Use factors in addition to market return
– 举例含有行业生产,预期通货膨胀等
Examples include industrial production,expected inflation
etc.
– 用多元回归为每一个因素估计 β值
Estimate a beta for each factor using multiple regression.
法马与弗伦奇建立了一个多因素模型。 Fama and French
返回一个帐面价值 - 市值比涵数和市场收益高低涵数
– Returns a function of size and book-to-market value as well
as market returns.
10-41
summary
经济的单因素模型把不确定性来源分成系统
（宏观经济）的因素或公司特有（微观经济）
的因素。指数模型假设宏观因素可以由股票收益的一个公开指数所代表。
A single-factor model of the economy classifies
sources of uncertainty as systematic
(macroeconomic) factors or firm-specific
(microeconomic) factors,The index model
assumes that the macro factor can be
represented by a broad index of stock returns.
10-42
summary
单指数模型大大降低了马克维茨资产组合选择程序的数据数量，它把精力放在 了对证券的专门分析中。
The single-index model drastically reduces the
necessary inputs in the Markowitz port- folio
selection procedure,It also aids in
specialization of labor in security analysis.
10-43
summary
根据指数模型的详细内容，资产组合或资产的系统风险等于 β2M，而两项资产的协方差为
βi βj σ M2。
According to the index model specification,
the systematic risk of a portfolio or asset
equals β2M and the covariance between two
assets equals βi βj σ M2。
10-44
summary
指数模型通过运用对超额收益率的回归分析来估计。回归曲线的斜率是资产的贝塔值，而截距是样本期间的资产的阿尔法。回归线也称为证券特征线
（ SCL）。回归贝塔等于资本资产定价模型的贝塔，除非回归运用的是实际收益，而资本资产定价模 型根据的是期望收益。该模型预言，由指数模型回归测度的阿尔法的平均值将为零。
The index model is estimated by applying regression analysis to excess rates
of return,The slope of the regression curve is the beta of an asset,whereas
the intercept is the asset’s alpha during the sample period,The regression
line is also called the security characteristic line,The regression beta is
equivalent to the CAPM beta,except that the regression uses actual returns
and the CAPM is specified in terms of expected returns,The CAPM predicts
that the average value of alphas measured by the index model regression
will be zero.
10-45
summary
操盘手习惯于用总的而不是超额的收益率来估计指数模型。这使他们的阿尔法估计值等于 α＋ rf (1-β)。
Practitioners routinely estimate the index
model using total rather than excess rates
of return,This makes their estimate of
alpha equal to α＋ rf (1-β)。
10-46
summary
贝塔显示了一个沿时间趋向于 1的趋势。贝塔的预测法试图预言这一趋势。另外，其他的财务变量也可以被用来帮助预测贝塔。
Betas show a tendency to evolve toward 1
over time,Beta forecasting rules attempt
to predict this drift,Moreover,other
financial variables can be used to help
forecast betas.
10-47
Summary
多因素模型试图通过在更多的细节上把系统部分模型化来提高单指数模型的解释能力。这些模型利用指示器，试图把握一个范围广泛的宏观经济的风险因素和某些时候的公司特征变量，譬如公司的规模或帐面价值 -市值比率。
Multifactor models seek to improve the explanatory
power of the single-index model by modeling the
systematic component of returns in greater detail,These
models use indicators intended to capture a wide range
of macroeconomic risk factors and,sometimes,firm-
characteristic variables such as size or book-to-market
ratio.
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Summary
ICAPM 是单因素 CAPM的一个扩展，它也是一种证券收益的多因素模型，但它不必指定一定要考虑哪些风险因素。
An extension of the single-factor CAPM,
the ICAPM,is a multifactor model of
security returns,but it does not specify
which risk factors need to be considered.
