1、指数模型 Index Models,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:n50个期望收益的估计 (estimates of expected ret
2、urns)n50个方差估计 (estimates of variances)(n2-n)/21225 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 M
3、odel,证券i的持有期收益： The holding-period return on security i is:ri = E(ri)+mi +eiE(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 securitys ret
4、urn during the period, and ei 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
5、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)+iF +ei,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
6、 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,风险溢价Risk Prem,市场风险溢价Ma
7、rket Risk Prem,或指数风险溢价or Index Risk Prem,单一指数模型 Single Factor Model,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2M，则我们可以把每个股票收益率的方差拆分成两部分： e
8、ach 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 2M , then we can break the variance of the rat
9、e of return on each stock into two components:,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
10、 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, ind
11、ependent 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（ 3n1）个
12、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(e
13、i) 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
14、, 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,Ri
15、= i + iRM + ei,.,.,.,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
16、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&
17、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 byRGM t = GM + GM RMt + eGM t,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、斜率为 的直线周围。假定这条线的偏差 e 与独立变量
18、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 and a slope . The deviations from the line, e, are assumed to be mutually uncorrelated as well as uncorrel
19、ated with the independent variable.,证券特征线 Security Characteristic Line,我们通过 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GM，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、和Var（e），可以用标准回归技术来估计。 The sensitivity of GM to the market, measu
20、red by GM, is the slope of the regression line. The intercept of the regression line is GM, representing the average firm-specific return when the markets excess return is zero. Deviations of particular observations from the regression line in any period are denoted eGMt, and called residuals. Each
21、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. Th
22、e parameters of interest, ，, and Var(e), can be estimated using standard regression techniques.,Using the Text Example from Table 10-1,Using the Text Example from Table 10-1,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:Uns
23、ystematic 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,风险测量 Measuring Components of Risk,每个证券的超额收益率为： Ri = i + iRm + ei 资产组合的超额收益率为：,Rp = p + pRM + eP (10-5),风险测量 Measuring Comp
24、onents of Risk,我们注意到等权重（每种资产权重 w i1/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,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 sensi
25、tivity to the market given by,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),风险测量 Measuring Components of Risk,零均值变量 The zero mean variable,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
26、) = 非系统方差unsystematic variance,风险测量 Measuring Components of Risk,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,这里2(e)是公司特有方差的均值。由于这
27、一均值独立于n，所以当 n 变大时，2(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P 。图10-2对此作了说明。where 2(e) is the average of the firm-specific variances. Because this average is independent of n, when n gets large, 2(eP) becomes negligible. To summarize, as diversification increases, the
28、 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, P . This is shown in Figure 10.2.,风险测量 Measuring Components of Risk,风险系数与资产组合方差 The variance of a portfolio with risk coe
29、fficient,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 s2(eP)=s2(e) / n,bP2sM2,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 practic
30、e 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:,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
31、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
32、the covariance of the excess rate of return on security i with that of the market index is,资本资产定价模型与指数模型 The CAPM and the index model,因为 Cov(Ri，RM) i M2，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) i M2, the sensitivity coefficient, i, in equation 10.9, which is the slope of the regression li
33、ne representing the index model, equals,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 returnbeta relationship, except that we replace t
34、he (theoretical) market portfolio of the CAPM with the well-specified and observable market index.,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i Cov(R i ，RM)/M2 。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
35、均期望超额收益的情况。where i Cov(R i ，RM)/M2 . This is a statement about the mean of expected excess returns of assets relative to the mean excess return of the (theoretical) market portfolio.,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
36、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,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
37、比较表明， 资本资产定价模型预言i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 returnbeta relationship (equation 10.8) shows that the CAPM predicts that i should be zero for all assets. The alpha of a stock is its expected r
38、eturn 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.,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
39、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:,指数模型与期望收益 - 贝塔关系 The index mod
40、el 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 v
41、alid, 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.,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
42、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.,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
43、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.,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2M，而两项资产的协方差为i j M2。According to the index model specification, the systematic ri
44、sk of a portfolio or asset equals 2M and the covariance between two assets equals i j M2。,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
45、analysis to excess rates of return. The slope of the regression curve is the beta of an asset, whereas the intercept is the assets 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
46、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.,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rf (1-)。Practitioners routinely estimate the index model using
47、 total rather than excess rates of return. This makes their estimate of alpha equal to rf (1-)。,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.,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.,
