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Chapter 08  Index Models
81
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 estimates
required for the Markowitz procedure can result in large aggregate estimation errors
when implementing the procedure.
The disadvantage of the index model arises from the
model’s assumption that return residuals are uncorrelated.
This assumption will be
incorrect if the index used omits a significant risk factor.
2.
The tradeoff entailed in departing from pure indexing in favor of an actively managed
portfolio is between the probability (or possibility) of superior performance against the
certainty of additional management fees.
3.
The answer to this question can be seen from the formulas for w
0
and w*.
Other things
held equal, w
0
is smaller the greater the residual variance of a candidate asset for
inclusion in the portfolio.
Further, we see that regardless of beta, when w
0
decreases, so
does w*.
Therefore, other things equal, the greater the residual variance of an asset, the
smaller its position in the optimal risky portfolio.
That is, increased firmspecific risk
reduces the extent to which an active investor will be willing to depart from an indexed
portfolio.
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.
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 the standard
deviation of returns, the denominator of the Sharpe ratio.
Hence, an increase in alpha
increases the Sharpe ratio.
Since the portfolio alpha is the portfolioweighted average of
the securities’ alphas, then, holding all other parameters fixed, an increase in a
security’s alpha results in an increase in the portfolio Sharpe ratio.
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82
5.
a.
To optimize this portfolio one would need:
n = 60 estimates of means
n = 60 estimates of variances
770
,
1
2
n
n
2
estimates of covariances
Therefore, in total:
890
,
1
2
n
3
n
2
estimates
b.
In a single index model: r
i
r
f
=
i
+
i
(r
M
– r
f
) + e
i
Equivalently, using excess returns: R
i
=
i
+
i
R
M
+ e
i
The variance of the rate of return on each stock can be decomposed into the
components:
(l)
The variance due to the common market factor:
2
M
2
i
(2)
The variance due to firm specific unanticipated events:
)
e
(
i
2
In this model:
j
i
j
i
)
r
,
r
(
Cov
The number of parameter estimates is:
n = 60 estimates of the mean E(r
i
)
n = 60 estimates of the sensitivity coefficient
i
n = 60 estimates of the firmspecific variance
2
(e
i
)
1 estimate of the market mean E(r
M
)
1 estimate of the market variance
2
M
Therefore, in total, 182 estimates.
Thus, the single index model reduces the total number of required parameter
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This note was uploaded on 10/11/2011 for the course FBE 441 taught by Professor Callahan during the Fall '07 term at USC.
 Fall '07
 Callahan

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