Chapter 8 - Index Models
?P?wM?wA??A?(1?0.0931)?(0.0931?0.9)?0.99E(RP)??P??P?E(RM)?(0.0931?2.8%)?(0.99?8%)?8.18"2?P??P??M??2(eP)?(0.99?23)2?(0.09312?1969.03)?535.54?P?23.14%
With A = 2.8, the optimal position in this portfolio is:
y?8.18?0.5455
0.01?2.8?535.54The final positions in each asset are: Bills M A C
b.
The mean and variance of the optimized complete portfolios in the
unconstrained and short-sales constrained cases, and for the passive strategy are:
Unconstrained Constrained Passive
2 σC0.5685 × 8.42% = 4.79 0.56852 × 528.94 = 170.95 0.5455 × 8.18% = 4.46 0.54552 × 535.54 = 159.36 0.5401 × 8.00% = 4.32 0.54012 × 529.00 = 154.31
1 – 0.5455 =
0.5455 ? (1 ? 0.0931) = 0.5455 ? 0.0931 ? 0.3352 = 0.5455 ? 0.0931 ? 0.6648 =
45.45% 49.47 1.70 3.38 100.00
E(RC )
2The utility levels below are computed using the formula: E(rC)?0.005AσC
Unconstrained Constrained Passive
8% + 4.79% – (0.005 × 2.8 × 170.95) = 10.40% 8% + 4.46% – (0.005 × 2.8 × 159.36) = 10.23% 8% + 4.32% – (0.005 × 2.8 × 154.31) = 10.16%
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Chapter 8 - Index Models
19. All alphas are reduced to 0.3 times their values in the original case. Therefore, the
relative weights of each security in the active portfolio are unchanged, but the alpha of the active portfolio is only 0.3 times its previous value: 0.3 × ?16.90% = ?5.07% The investor will take a smaller position in the active portfolio. The optimal risky portfolio has a proportion w* in the active portfolio as follows:
?/?2(e)?0.0507/21,809.6w0????0.01537 22E(rM?rf)/?M0.08/23The negative position is justified for the reason given earlier. The adjustment for beta is:
w*?w0?0.01537???0.0151
1?(1?β)w01?[(1?2.08)?(?0.01537)]Since w* is negative, the result is a positive position in stocks with positive alphas
and a negative position in stocks with negative alphas. The position in the index portfolio is: 1 – (–0.0151) = 1.0151
To calculate the Sharpe ratio for the optimal risky portfolio we compute the information ratio for the active portfolio and the Sharpe ratio for the market portfolio. The information ratio of the active portfolio is 0.3 times its previous value:
A =
??5.07= –0.0343 and A2 =0.00118 ??(e)147.68Hence, the square of the Sharpe ratio of the optimized risky portfolio is:
S2 = S2M + A2 = (8%/23%)2 + 0.00118 = 0.1222 S = 0.3495
Compare this to the market’s Sharpe ratio: SM = The difference is: 0.0017
Note that the reduction of the forecast alphas by a factor of 0.3 reduced the squared information ratio and the improvement in the squared Sharpe ratio by a factor of:
20. If each of the alpha forecasts is doubled, then the alpha of the active portfolio will
also double. Other things equal, the information ratio (IR) of the active portfolio also doubles. The square of the Sharpe ratio for the optimized portfolio (S-square) equals the square of the Sharpe ratio for the market index (SM-square) plus the square of the information ratio. Since the information ratio has doubled, its square quadruples. Therefore: S-square = SM-square + (4 × IR) Compared to the previous S-square, the difference is: 3IR Now you can embark on the calculations to verify this result.
0.32 = 0.09
8%= 0.3478 23%8-12
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Chapter 8 - Index Models
CFA PROBLEMS 1.
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, R2 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 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 R2 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.
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Chapter 8 - Index Models
2.
The R2 of the regression is: 0.702 = 0.49
Therefore, 51% of total variance is unexplained by the market; this is nonsystematic risk.
9 = 3 + ? (11 ? 3) ? ? = 0.75 d. b.
3. 4. 5.
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McGraw-Hill Education.