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M p p q β β β 52 239 464 e 2 2 m 2 q 2 q q 2 σ β

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M P P Q = + × + × = β + β = β 52 . 239 ) 400 75 . 0 ( 52 . 464 ) e ( 2 2 M 2 Q 2 Q Q 2 = × = σ β σ = σ 300 400 75 . 0 ) r , r ( Cov 2 M Q M Q = × = σ β = 11. a. Merrill Lynch adjusts beta by taking the sample estimate of beta and averaging it with 1.0, using the weights of 2/3 and 1/3, as follows: adjusted beta = [(2/3) × 1.24] + [(1/3) × 1.0] = 1.16 b. If you use your current estimate of beta to be β t–1 = 1.24, then β t = 0.3 + (0.7 × 1 . 24) = 1.168 8-5
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8-6 12. The regression results provide quantitative measures of return and risk based on monthly returns over the five-year period. β for ABC was 0.60, considerably less than the average stock’s β of 1.0. This indicates that, when the S&P 500 rose or fell by 1 percentage point, ABC’s return on average rose or fell by only 0.60 percentage point. Therefore, ABC’s systematic risk (or market risk) was low relative to the typical value for stocks. ABC’s alpha (the intercept of the regression) was 3.2%, indicating that when the market return was 0%, the average return on ABC was –3.2%. ABC’s unsystematic risk (or residual risk), as measured by σ (e), was 13.02%. For ABC, R 2 was 0.35, indicating closeness of fit to the linear regression greater than the value for a typical stock. β for XYZ was somewhat higher, at 0.97, indicating XYZ’s return pattern was very similar to the β for the market index. Therefore, XYZ stock had average systematic risk for the period examined. Alpha for XYZ was positive and quite large, indicating a return of almost 7.3%, on average, for XYZ independent of market return. Residual risk was 21.45%, half again as much as ABC’s, indicating a wider scatter of observations around the regression line for XYZ. Correspondingly, the fit of the regression model was considerably less than that of ABC, consistent with an R 2 of only 0.17. The effects of including one or the other of these stocks in a diversified portfolio may be quite different. If it can be assumed that both stocks’ betas will remain stable over time, then there is a large difference in systematic risk level. The betas obtained from the two brokerage houses may help the analyst draw inferences for the future. The three estimates of ABC’s β are similar, regardless of the sample period of the underlying data. The range of these estimates is 0.60 to 0.71, well below the market average β of 1.0. The three estimates of XYZ’s β vary significantly among the three sources, ranging as high as 1.45 for the weekly data over the most recent two years. One could infer that XYZ’s β for the future might be well above 1.0, meaning it might have somewhat greater systematic risk than was implied by the monthly regression for the five-year period. These stocks appear to have significantly different systematic risk characteristics. If these stocks are added to a diversified portfolio, XYZ will add more to total volatility. 13. For Stock A: α A = r A [ r f + β A (r M r f )] = 11 [6 + 0.8(12 6)] = 0.2% For stock B: α B = 14 [6 + 1.5(12 6)] = 1% Stock A would be a good addition to a well-diversified portfolio. A short position in Stock B may be desirable.
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8-7 14. The R 2 of the regression is: 0.70 2 = 0.49 Therefore, 51% of total variance is unexplained by the market; this is nonsystematic risk.
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