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
The Sharpe ratio indicates that a higher alpha makes a security more desirable.
Alpha, the numerator of the Sharpe ratio, is a fixed number that is not affected by
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CHAPTER 8: INDEX MODELS
5. a. To optimize this portfolio one would need:
CHAPTER 8: INDEX MODELS
b. The expected rate of return on a portfolio is the weighted average of the
expected returns of the individual securities:
E(rP ) = wA × E(rA ) + wB × E(rB ) + wf × rf
The beta of a portfolio is similarly a weighted average of the betas of the
individual securities:
βP = wA × βA + wB × βB + wf × β f
The variance of this portfolio is:
)(σβσ
2222
PMPP
e
where
22
σβ
MP
is the systematic component and
)(
2
P
e
is the nonsystematic
component. Since the residuals (ei ) are uncorrelated, the nonsystematic
variance is:
2 2 2 2 2 2 2
( ) ( ) ( ) ( )
P A A B B f f
e w e w e w es s s s= ´ + ´ + ´
where σ2(eA ) and σ2(eB ) are the firm-specific (nonsystematic) variances of
Stocks A and B, and σ2(e f ), the nonsystematic variance of T-bills, is zero. The
residual standard deviation of the portfolio is thus:
The total variance of the portfolio is then:
47.699405)2278.0(σ 222
P
7. a. The two figures depict the stocks’ security characteristic lines (SCL). Stock A
b. Beta is the slope of the SCL, which is the measure of systematic risk. The
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CHAPTER 8: INDEX MODELS
c. The R2 (or squared correlation coefficient) of the SCL is the ratio of the
)(σσβ
σβ
222
22
2
iMi
Mi
e
R
Since the explained variance for Stock B is greater than for Stock A (the
2
( )
B
es
d. Alpha is the intercept of the SCL with the expected return axis. Stock A has a
8. a. Firm-specific risk is measured by the residual standard deviation. Thus, stock
c. R2 measures the fraction of total variance of return explained by the market
d. Rewriting the SCL equation in terms of total return (r) rather than excess
return (R):
( )
(1 )
A f M f
A f M
r r r r
r r r
a b
a b b
– = + ´ – Þ
= + ´ – + ´
The intercept is now equal to:
(1 ) 1% (1 1.2)
f f
r ra b+ ´ – = + ´ –
Since rf = 6%, the intercept would be:
1% 6%(1 1.2) 1% 1.2% 0.2%+ – = – =-
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CHAPTER 8: INDEX MODELS
9. The standard deviation of each stock can be derived from the following
equation for R2:
2
22
2
σ
σβ
i
Mi
i
R
Therefore:
%30.31σ
980
20.0
207.0
σβ
σ
22
2
22
2
A
A
MA
A
R
CHAPTER 8: INDEX MODELS
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CHAPTER 8: INDEX MODELS
12. Note that the correlation is the square root of R2:
2
ρR
1/2
,
1/2
,
( ) 0.20 31.30 20 280
( ) 0.12 69.28 20 480
A M A M
B M B M
Cov r r
Cov r r
r s s
r s s
= = ´ ´ =
= = ´ ´ =
13. For portfolio P we can compute:
CHAPTER 8: INDEX MODELS
b. If you use your current estimate of beta to be βt–1 = 1.24, then
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CHAPTER 8: INDEX MODELS
16. For Stock A:
[ ( )] .11 [.06 0.8 (.12 .06)] 0.2%
A A f A M f
r r r ra b= – + ´ – = – + ´ – =
For stock B:
[ ( )] .14 [.06 1.5 (.12 .06)] 1%
B B f B M f
r r r ra b= – + ´ – = – + ´ – =-
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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