『易坊知识库摘要_单一|单一指数模型( 二 )』17、is:,20,For each month, t, the residual, et, is the deviation of ATT,t = RATT + ATT with A = 0.89. We can find that the undiversified stock is subject to nonsystematic risk, as a result it is seen...
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17、is:,20,For each month, t, the residual, et, is the deviation of AT&T stocks excess return from the predict value of the SCL for the stock, eAT&T,t = RAT&T,t(AT&T + AT&TRM,t) (10.9) The variance of firm-specific component returns is (10.10) where n-2 is the degrees of freedom of the regression.,21,3。
18、. Single Index Model and Diversification,The single index model, first suggested by Sharpe, can also be applied to portfolio diversification. Rp = p + pRM + ep (10-11) And 2p = p22M +2 (ep)(10-12) Suppose all stocks are held in the same proportion in the portfolio, 1/N:,22,For the equal proportion p 。
19、ortfolio, the portfolio beta is given by Portfolio return also has a nonmarket return component of a constant which is the average of the individual alphas: and the average of the firm-specific components:,23,Equation (10.12) can be rewritten as (10.14) Then as n increases, the second term on the ri 。
20、ght hand side of Equation 10.14 becomes negligible.,24,Conclusion:,The systematic risk component of the portfolio variance, p2M2, depends on marketwide risk and the average of the sensitivity coefficients of the individual securities, which cannot be diversified away, no matter how many stocks are h 。
21、eld in the portfolio.,25,The portfolio variance of the residual component of return, 2 (ep), tend to be negligible in an ever-larger portfolio.,26,4 . Well-diversified Portfolio versus Single assets,A well-diversified portfolio refers to a portfolio that includes enough numbers of securities so that 。
22、 the risk of the portfolio closely approximates the systematic risk of the overall market. It has following characters: Rp = p + pRM and, 2p = p2M2 The characteristic line for that portfolio can be shown as in Figure 10.2.,27,Figure 10.2 Security characteristic line for a well-diversified portfolio, 。
23、28,Figure 10.1 shows the characteristic line for a particular stock, AT&T with A = 0.89. We can find that the undiversified stock is subject to nonsystematic risk, as a result it is seen in a scatter of points around the line, with positive or negative firm-specific disturbances, eis. In contrast, t 。
24、he well-diversified portfolios excess return, is determined completely by the systematic (market) excess return.,29,10.2 Guidelines,The basic version of the single index model is shown as equation 10.2, which is: rirf = i + i(rmrf) + ei Because eis are independent, and all have zero expected value a 。
25、nd the value of constant variances equals to zero on average. It also can be expressed as: E(Ri)Rf =i + iE(RM)Rf (10.13) It looks very similar to the CAPM equation introduced in the last chapter: E(Ri)- Rf = iE(RM)Rf (10.14),. CAMP and Single Index Model CAPM与单一指数模型,30,If the stock market index can。
26、represent the market, the CAPM relationship is known as the single index model (Equation 10.13), with a prediction that alpha should be zero in equilibrium.In an equilibrium condition excess return is made only for the systematic risk (market risk).Therefore if the stock is fairly priced, its alpha。
27、must be zero.,31,The Index Model give rise to the use of extent to which asset i is overpriced (i0) in comparison with the value predicted by the CAPM.,32,In long run, the average alpha of a stock should be zero. i equals to realized average return minus required return predicted in CAPM However, in 。
28、 practice, some stock prices are not fairly priced at times.,33,Moreover, CAPM is difficult to implement due to the following reasons: First, the CAPM describes the equilibrium relationship between expected returns and the market, however, the market return is not observable. Second, the CAPM descri 。
29、bes the relationship between expected returns and betas, but asset betas are not known with certainty, and empirical results show that beta values vary with time.,34,Single index model has solved the complications of the CAPM in three ways. First, it uses an observable index as a proxy for the marke 。
30、t portfolio. Second, the index model uses observable realized returns on a security. Third, the index model is a regression model specifying a relationship between realized returns and a market index, which enable us to estimate asset betas using historical prices data.,35,10.3 Adjusting Betas,As we 。
31、 stated earlier, betas vary with time and historical betas are not reliable predictors of future betas. Adjusting historical betas is necessary to make forecasting for future betas. One way to accomplish it is to collect historical data to calculate betas over different sample periods and then estim 。
32、ate a regression model: Current beta = a + b (Past beta)(10.14),36,Given estimates of a and b, we could then forecast future betas using the model: Future beta = a + b (Current beta) However, this simple technique assumes that beta move in a continued trend. There is no reason to believe beta will c 。
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标题：单一|单一指数模型( 二 )