1、1,Chapter10,Single index model 单一指数模型,The main goal of this chapter is to introduce Sharpes Single Index Model of Capital Asset Pricing.,According to the Single Index Model, by using observable realized returns on a security to regress a relationship between realized returns on that security and the 。

2、 market index, we can examine how the returns on a particular asset or portfolio changes with respect to the returns of the market.,3,10.1 Guidelines,证券的收益与方差 “均值-方差模型“的局限性 单一指数模型的主要假设 估计单一指数模型 单一指数模型与多样化 多样化组合对比单一资产,Securitys returns and its variance The limitation of the “mean, variance approach”。

3、Main assumptions for single-index model Estimating the single index model Single index model and diversification Well-diversified portfolio versus single assets,.单一指数证券市场 . A single-index security market,4,1. Securitys Returns and its Variance,1) The limitation of the “mean, variance approach” “The。

4、mean, variance approach” to portfolio analysis involves estimating and then selecting the portfolio that offers the best mean-variance combination. With “The mean, variance approach”, Consider what we need to do: (1) Mean and variance of each asset. (2) Correlation between each asset. (3) Mean and v 。

5、ariance differing combinations of assets.,5,This is fine when the portfolio consists of only two assets. What if there are 20 assets to construct? you need the information: n =20 estimates of E(Ri) n =20 estimates of variances n(n - 1)/2 =190 estimates of covariances 230 estimates So for each combin 。

6、ation we need 230 estimates.,6,In light of the fact that a 20-security portfolio is relatively small, if n = 200 we need 20,300 estimates per combination. Single-index model enables us to dramatically reduce the number of parameters required to perform portfolio analysis.,7,2) Main Assumptions of Si 。

7、ngle-index model,(1) The returns on stocks tend to change in a similar fashion as an average return on the market, because the same economic factors affect almost all firms. This implies the excess return on stock i over risk-free rate can be decomposed into three components: rirf = i + i(rmrf) + ei 。

8、 (10.1),8,Three components of the excess return on stock i over risk-free rate,(1)i :A constant常数, which is different for each stock. (2)i(rm-rf) ：A component proportional to the excess return on market index, rm-rf. (3) ei :A random and unpredictable component due to unexpected events that are rele 。

9、vant only to this stock (firm specific).,9,We further denote excess returns over the risk-free rate by R（因为股票市场的收益超出或低于无风险资产收益的那部分的大小可以代表宏观经济状况）. rewrite this equation as Ri = i + iRM + ei(10.2) It implies that each security has two sources of risk: 1. market risk, attributable to its sensitivity to 。

10、 macroeconomic factors as reflected in RM, 2. firm-specific risk, as reflected in ei.,10,2) Assumptions of Single-index model _continued,(2) All the macroeconomic factors affect the security market as a whole. Assumption (2) indicates that the firm-specific component of returns ei and ej of stocks i 。

11、 and j are uncorrelated with each other, therefore Cov(ei, ej) = 0. Because i and j are constants, Cov(Ri,Rj) = Cov(i + iRm + ei, j + jRm + ej) = Cov(iRM, jRM) = ijM2 (10.5),11,Specific uncertainty ei is independent of the market movements. Covariance (ei, ej) = 0;
Covariance (ei, Rm) = 0 E(Ri) = i。

12、+ iRM(10.3) i2 =i2M2 +2 (ei) (10.4),12,This means that the variance of security is return also has two components: (1) The variance of returns attributable to the economy (stock market) volatility as a whole i2M2. (2) The variance of returns attributable to firm-specific uncertainty 2(ei).,13,2. Est 。

【单一|单一指数模型】13、imating the Single Index Model,In Equation 10.2, Ri = i + iRM + ei Applying Ordinary Least Squares one regression technique that estimates parameters from available data and applying to a linear function: Y = a + bX + , we can estimate alpha and beta.,14,The deduction steps are as following: By gath 。

14、er data on RM and Ri for a same time period, and running the regression analysis The squared correlation coefficient, r2 should be the least to meet the condition for Ri is greatly explained by RM. (10.6),15,The estimated parameters i, and i, can be: i =iM/ M2 (10.7) and (10.8) Therefore, Betas here 。

15、 are historical estimates based on historical data.,Table 10.1 Monthly excess returns on AT&T stock and S&P500 during 2001/1 - 2001/12,Figure 10.1Security characteristic line for AT&T stock during 2001,18,use S AT&T = 0.08620.89 0.0911 = 0.0051 Therefore, the relationship equation between excess ret 。

16、urns on AT&T stock and on the market index is: RAT&T = 0.0051 + 0.89RM + eAT&T,19,The estimated regression equation is the security characteristic line (SCL). That is,. The SCL represents the typical return on a security is a function of the return on the market. In our example, the SCL for Stock A。

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标题：单一|单一指数模型