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 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 trade-off entailed in departing from pure indexing in favor of an actively
managed portfolio is between the probability (or the 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 (equation 8.20)
and w* (equation 8.21). 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 firm-specific 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 portfolio-weighted 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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Chapter 8 - Index Models
5. a. To optimize this portfolio one would need:
n = 60 estimates of means n = 60 estimates of variances
n2?n?1,770estimates of covariances 2n2?3n?1,890estimates Therefore, in total: 2
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)
2The variance due to the common market factor: ?i2?M
(2) The variance due to firm specific unanticipated events: σ2(ei) In this model: Cov(ri,rj)?βiβjσ The number of parameter estimates is:
n = 60 estimates of the mean E(ri )
n = 60 estimates of the sensitivity coefficient β i n = 60 estimates of the firm-specific variance σ2(ei ) 1 estimate of the market mean E(rM )
21 estimate of the market variance?M
Therefore, in total, 182 estimates.
The single index model reduces the total number of required estimates from 1,890 to 182. In general, the number of parameter estimates is reduced from:
?n2?3n???2?? to (3n?2) ??
6.
a.
The standard deviation of each individual stock is given by:
2σi?[βi2?M??2(ei)]1/2
Since βA = 0.8, βB = 1.2, σ(eA ) = 30%, σ(eB ) = 40%, and σM = 22%, we get:
σA = (0.82 × 222 + 302 )1/2 = 34.78% σB = (1.22 × 222 + 402 )1/2 = 47.93%
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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
E(rP ) = (0.30 × 13%) + (0.45 × 18%) + (0.25 × 8%) = 14% 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
βP = (0.30 × 0.8) + (0.45 × 1.2) + (0.25 × 0.0) = 0.78 The variance of this portfolio is:
222σ2?β??σ(eP) PPM22whereβ2is the systematic component andσ?(eP)is the nonsystematic PMcomponent. Since the residuals (ei ) are uncorrelated, the nonsystematic variance is:
222?2(eP)?wA??2(eA)?wB??2(eB)?w2f??(ef)
= (0.302 × 302 ) + (0.452 × 402 ) + (0.252 × 0) = 405
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:
σ(eP ) = (405)1/2 = 20.12% The total variance of the portfolio is then:
22σ2.47 P?(0.78?22)?405?699The total standard deviation is 26.45%.
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.
Beta is the slope of the SCL, which is the measure of systematic risk. The SCL for Stock B is steeper; hence Stock B’s systematic risk is greater.
b.
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Chapter 8 - Index Models
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):
βi2σ2 R?22M2βiσM?σ(ei)2
d.
e. 8.
b.
c.
d.
Since the explained variance for Stock B is greater than for Stock A (the
2explained variance isβ2BσM, which is greater since its beta is higher), and its residual variance ?2(eB) is smaller, its R2 is higher than Stock A’s. 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.
The correlation coefficient is simply the square root of R2, so Stock B’s correlation with the market is higher.
Firm-specific risk is measured by the residual standard deviation. Thus, stock A has more firm-specific risk: 10.3% > 9.1%
Market risk is measured by beta, the slope coefficient of the regression. A has a larger beta coefficient: 1.2 > 0.8
R2 measures the fraction of total variance of return explained by the market return. A’s R2 is larger than B’s: 0.576 > 0.436
Rewriting the SCL equation in terms of total return (r) rather than excess return (R):
a.
rA?rf?????(rM?rf)?rA???rf?(1??)???rMThe intercept is now equal to:
??rf?(1??)?1%?rf?(1?1.2)
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:
Explained varianceβi2σ2M R??2Total varianceσi2iTherefore:
2β20.72?202AσMσ???9802 0.20RA2AσA?31.30%For stock B:
1.22?202σ??4,800 0.12σB?69.28+
10. The systematic risk for A is:
22?A??M?0.702?202?196
The firm-specific risk of A (the residual variance) is the difference between A’s total risk and its systematic risk:
980 – 196 = 784 The systematic risk for B is:
22?B??M?1.202?202?576
B’s firm-specific risk (residual variance) is:
4,800 – 576 = 4,224
11. The covariance between the returns of A and B is (since the residuals are assumed
to be uncorrelated):
Cov(rA,rB)?βAβBσ2M?0.70?1.20?400?336 The correlation coefficient between the returns of A and B is:
ρAB?
Cov(rA,rB)336??0.155
σAσB31.30?69.288-5
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McGraw-Hill Education.