Chapter 8 – Index Models
CHAPTER 8: INDEX MODELS
PROBLEM SETS
1. The advantage of the index model, compared to the Markowitz procedure, is the
vastly reduced number of estimates required. In addition, the large number of
2. The trade-off entailed in departing from pure indexing in favor of an actively
3. The answer to this question can be seen from the formulas for w 0 (equation 8.20)
and w* (equation 8.21). Other things held equal, w 0 is smaller the greater the
4. The total risk premium equals: + ( × Market risk premium). We call alpha a
nonmarket return premium because it is the portion of the return premium that is
independent of market performance.
Chapter 8 – Index Models
8-2
5. a. To optimize this portfolio one would need:
b. In a single index model: ri − rf = α i + β i (r M – rf ) + e i
Equivalently, using excess returns: R i = α i + β i R M + e i
The variance of the rate of return can be decomposed into the components:
(l) The variance due to the common market factor:
22 Mi
(2) The variance due to firm specific unanticipated events:
)(σ2i
e
6. a. The standard deviation of each individual stock is given by:
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8-3
b. The expected rate of return on a portfolio is the weighted average of the
expected returns of the individual securities:
The beta of a portfolio is similarly a weighted average of the betas of the
individual securities:
The variance of this portfolio is:
)(σβσ 2222 PMPP e+=
where
22 σβ MP
is the systematic component and
)(
2P
e
is the nonsystematic
component. Since the residuals (ei ) are uncorrelated, the nonsystematic
variance is:
7. a. The two figures depict the stocks’ security characteristic lines (SCL). Stock A
has higher firm-specific risk because the deviations of the observations from
the SCL are larger for Stock A than for Stock B. Deviations are measured by
the vertical distance of each observation from the SCL.
Chapter 8 – Index Models
8-4
c. The R2 (or squared correlation coefficient) of the SCL is the ratio of the
explained variance of the stock’s return to total variance, and the total
variance is the sum of the explained variance plus the unexplained variance
(the stock’s residual variance):
d. Alpha is the intercept of the SCL with the expected return axis. Stock A has a
small positive alpha whereas Stock B has a negative alpha; hence, Stock A’s
alpha is larger.
e. The correlation coefficient is simply the square root of R2, so Stock B’s
8. a. Firm-specific risk is measured by the residual standard deviation. Thus, stock
A has more firm-specific risk: 10.3% > 9.1%
b. Market risk is measured by beta, the slope coefficient of the regression. A has
a larger beta coefficient: 1.2 > 0.8
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9. The standard deviation of each stock can be derived from the following
equation for R2:
10. The systematic risk for A is:
11. The covariance between the returns of A and B is (since the residuals are assumed
to be uncorrelated):
Chapter 8 – Index Models
12. Note that the correlation is the square root of R2:
2
ρR=
13. For portfolio P we can compute:
14. Note that the variance of T-bills is zero, and the covariance of T-bills with any asset
is zero. Therefore, for portfolio Q:
15. a. Beta Books 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:
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16. For Stock A:
17. a.
Alpha (α)
Expected excess return
αi = ri – [rf + βi × (rM – rf ) ]
E(ri ) – rf
αA = 20% – [8% + 1.3 × (16% – 8%)] = 1.6%
20% – 8% = 12%
αB = 18% – [8% + 1.8 × (16% – 8%)] = – 4.4%
18% – 8% = 10%
αC = 17% – [8% + 0.7 × (16% – 8%)] = 3.4%
αD = 12% – [8% + 1.0 × (16% – 8%)] = – 4.0%
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8-8
b. To construct the optimal risky portfolio, we first determine the optimal active
portfolio. Using the Treynor-Black technique, we construct the active portfolio:
a
2(e)
a /
2(e)
Sa /
2(e)
A
0.000476
–0.6142
With these weights, the forecast for the active portfolio is:
The levered position in B [with high 2(e)] overcomes the diversification
effect and results in a high residual standard deviation. The optimal risky
portfolio has a proportion w* in the active portfolio, computed as follows:
The negative position is justified for the reason stated earlier.
The adjustment for beta is:
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c. To calculate the Sharpe ratio for the optimal risky portfolio, we compute the
information ratio for the active portfolio and Sharpe’s measure for the market
portfolio. The information ratio for the active portfolio is computed as follows:
d. To calculate the makeup of the complete portfolio, first compute the beta, the
mean excess return, and the variance of the optimal risky portfolio:
βP = wM + (wA × βA ) = 1.0486 + [(–0.0486) 2.08] = 0.95
The final positions are (M may include some of stocks A through D):
Bills
1 – 0.5685 =
43.15%
M
0.5685 l.0486 =
59.61
A
0.5685 (–0.0486) (–0.6142) =
1.70
B
0.5685 (–0.0486) 1.1265 =
C
0.5685 (–0.0486) (–1.2181) =
3.37
D
0.5685 (–0.0486) 1.7058 =
100.00%
18. a. If a manager is not allowed to sell short, he will not include stocks with negative
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alphas in his portfolio, so he will consider only A and C:
The forecast for the active portfolio is:
α = (0.3352 × 1.6) + (0.6648 × 3.4) = 2.80%
The weight in the active portfolio is:
Adjusting for beta:
The information ratio of the active portfolio is:
Hence, the square of the Sharpe ratio is:
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With A = 2.8, the optimal position in this portfolio is:
b. The mean and variance of the optimized complete portfolios in the
unconstrained and short-sales constrained cases, and for the passive strategy
are:
E(RC )
2
σC
Unconstrained
0.5685 × 8.42% = 4.79
0.56852 × 528.94 = 170.95
Constrained
0.5455 × 8.18% = 4.46
Passive
0.5401 × 8.00% = 4.32
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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:
The negative position is justified for the reason given earlier.
The adjustment for beta is:
Hence, 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
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)
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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
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.
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2. The R2 of the regression is: 0.702 = 0.49