Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,10-,*,McGraw-Hill/Irwin,2007,The McGraw-Hill Companies,Inc.,All Rights Reserved.,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,指数模型,Index Models,01,02,03,概况三,点击此处输入,相关文本内容,整体概况,概况一,点击此处输入,相关文本内容,概况二,点击此处输入,相关文本内容,指数模型,Index Models,1,证券市场的单因素模型,2,单指数模型,3,估计单指数模型,4,投资组合的构建与单指数模型,5,指数模型在投资组合管理中的实际运用,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),(,n,2,-,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,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,证券,i,的持有期收益：,The holding-period return on security,i i,s:,r,i,=E(r,i,)+m,i,+e,i,E,(,r,i,),是证券持有期期初的期望收益,m,i,是在证券持有期间非预期的宏观事件对证券收益的影响，,e,i,是非预期的公司特有事件的影响。,E,(,r,i,),is the expected return on the security as of the beginning of the holding period,m,i,is the impact of unanticipated macro events on the securitys return during the period,and,e,i,is the impact of unanticipated firm-specific events.,单一指数模型,Single Factor Model,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.,r,i,=E(r,i,)+,i,F,+e,i,单一指数模型,Single Factor Model,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,单一指数模型,Single Factor Model,(r,i,-r,f,),=,i,+,i,(r,m,-r,f,),+e,i,a,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,Let:R,i,=(r,i,-r,f,),R,m,=(r,m,-r,f,),风险溢价格式,Risk premium,format,R,i,=,i,+,i,(R,m,),+e,i,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,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,R,M,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,R,M,的方差为,2,M,，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,R,M,and,firm-specific risk,as reflected in,e,.If we denote the variance of the excess return on the market,R,M,as,2,M,then we can break the variance of the rate of return on each stock into two components:,单一指数模型,Single Factor Model,R,M,和,e,i,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,R,M,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,单一指数模型,Single Factor Model,两个股票超额收益率的协方差，譬如,R,i,与,R,j,的协方差，仅仅来自于一般因素,R,M,，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,R,M,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:,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,2,(,ei),的估计，,1,个（一般）宏观经济因素的方差,2,M,的估计，,那么，这一计算式就表明这些（,3,n,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,2,(,ei),1 estimate for the variance of the(common)macroeconomic factor,2,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!,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,R,i,=,i,+,i,R,M,+e,i,.,.,.,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