CHAPTER 8
AN INTRODUCTION TO ASSET PRICING MODELS
Q. Explain the difference between systematic and unsystematic risk?
Systematic risk refers to that portion of total variability of returns caused by factors
affecting the prices of all securities, e.g., economic, political and sociological changes factors that are uncontrollable, external, and broad in their effect on all securities.
Unsystematic risk refers to factors that are internal and “unique” to the industry or
company, e.g., management capability, consumer preferences, labor strikes, etc. Notably,
it is not possible to get rid of the overall systematic risk, but it is possible to eliminate the
“unique” risk for an individual asset in a diversified portfolio.
Q: The CAPM contends that there is a systematic and unsystematic risk for an
individual security? Which is the relevant risk variable and why is it relevant? Why
is the other risk variable not relevant?
In a capital asset pricing model (CAPM) world the relevant risk variable is the security’s
systematic risk - its covariance of return with all other risky assets in the market. This
risk cannot be eliminated. The unsystematic risk is not relevant because it can be
eliminated through diversification - for instance, when you hold a large number of
securities, the poor management capability, etc., of some companies will be offset by the
above average capability of others.
Q: What are the similarities and dissimilarities between SML and CML as models
of the risk-return trade-off?
For plotting, the SML the vertical axis measures the rate of return while the horizontal
axis measures normalized systematic risk (the security’s covariance of return with the
market portfolio divided by the variance of the market portfolio). By definition, the beta
(normalized systematic risk) for the market portfolio is 1.0 and is zero for the risk-free
asset. It differs from the CML where the measure of risk is the standard deviation of
return (referred to as total risk).
Q. Identify and briefly discuss three criticisms of beta as used in the CAPM.
Any three of the following are criticisms of beta as used in CAPM.
1. Theory does not measure up to practice. In theory, a security with a zero beta should
give a return exactly equal to the risk-free rate. But actual results do not come out that
way, implying that the market values something besides a beta measure of risk.
2. Beta is a fickle short-term performer. Some short-term studies have shown risk and
return to be negatively related. For example, Black, Jensen and Scholes found that from
April 1957 through December 1965, securities with higher risk produced lower returns
than less risky securities. This result suggests that (1) in some short periods, investors
1
may be penalized for taking on more risk, (2) in the long run, investors are not rewarded
enough for high risk and are overcompensated for buying securities with low risk, and (3)
in all periods, some unsystematic risk is being valued by the market.
3. Estimated betas are unstable. Major changes in a company affecting the character of
the stock or some unforeseen event not reflected in past returns may decisively affect the
security’s future returns.
4. Beta is easily rolled over. Richard Roll has demonstrated that by changing the market
index against which betas are measured, one can obtain quite different measures of the
risk level of individual stocks and portfolios. As a result, one would make different
predictions about the expected returns, and by changing indexes, one could change the
risk-adjusted performance ranking of a manager.
Q. Briefly explain whether investors should expect a higher return from holding
portfolio A versus portfolio B under CAPM. Assume that both portfolios are fully
diversified.
Under CAPM, the only risk that investors should be compensated for bearing is the risk
that cannot be diversified away (systematic risk). Because systematic risk (measured by
beta) is equal to one for both portfolios, an investor would expect the same return for
Portfolio A and Portfolio B.
Since both portfolios are fully diversified, it doesn’t matter if the specified risk for each
individual security is high or low. The specific risk has been diversified away for both
portfolios.
Q. Explain the concepts of specific risk, systematic risk, variance, covariance,
standard deviation, and beta as they relate to investment.
Systematic risk is market-related risk that cannot be diversified away. Because systematic
risk cannot be diversified away, investors are rewarded for assuming this risk.
The variance of an individual security is the sum of the probability-weighted average of
the squared differences between the security’s expected return and its possible returns.
The standard deviation is the square root of the variance. Both variance and standard
deviation measure total risk, including both systematic and specific risk. Assuming the
rates of return are normally distributed, the likelihood for a range of rates may be
expressed using standard deviations. For example, 68 percent of returns may be
expressed using standard deviations. Thus, 68 percent of returns can be expected to fall
within + or -1 standard deviation of the mean, and 95 percent within 2 standard
deviations of the mean.
Covariance measures the extent to which two securities tend to move, or not move,
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together. The level of covariance is heavily influenced by the degree of correlation
between the securities (the correlation coefficient) as well as by each security’s standard
deviation. As long as the correlation coefficient is less than 1, the portfolio standard
deviation is less than the weighted average of the individual securities’ standard
deviations. The lower the correlation, the lower the covariance and the greater the
diversification benefits (negative correlations provide more diversification benefits than
positive correlations).
The capital asset pricing model (CAPM) asserts that investors will hold only fully
diversified portfolios. Hence, total risk as measured by the standard deviation is not
relevant because it includes specific risk (which can be diversified away).
Under the CAPM, beta measures the systematic risk of an individual security or portfolio.
Beta is the slope of the characteristic line that relates a security’s returns to the returns of
the market portfolio. By definition, the market itself has a beta of 1.0. The beta of a
portfolio is the weighted average of the betas of each security contained in the portfolio.
Portfolios with betas greater than 1.0 have systematic risk higher than that of the market;
portfolios with betas less than 1.0 have lower systematic risk. By adding securities with
betas that are higher (lower), the systematic risk (beta) of the portfolio can be increased
(decreased) as desired.
CHAPTER 8
Answers to Problems
8.
E(Ri) = RFR + i(RM - RFR)
= .10 + i(.14 - .10)
= .10 + .04i
Stock
U
N
D
Beta
85
1.25
-.20
(Required Return) E(Ri) = .10 + .04i
.10 + .04(.85) = .10 + .034 = .134
.10 + .04(1.25)= .10 + .05 = .150
.10 + .04(-.20) = .10 - .008 = .092
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Stock
Current
Price
Expected
Price
Expected
Dividend
U
22
24
0.75
24  22  0.75
 .1250
22
N
48
51
2.00
51  48  2.00
 .1042
48
D
37
40
1.25
40  37  1.25
 .1149
37
Stock
U
N
D
Beta
.85
1.25
-.20
Required
.134
.150
.092
Estimated
.1250
.1042
.1149
Estimated Return
Evaluation
Overvalued
Overvalued
Undervalued
If you believe the appropriateness of these estimated returns, you would buy stocks D
and sell stocks U and N.
E(R)
N
14%
U
*U’
* N’
*D’
D
-0.5
-0.2
0.5
.085
1.0
1.25
4. (a).
Bi 
COVi,m

2
m
and ri,m 
COVi,m
 i  m 
then COVi,m = (ri,m)(i)( m)
4
For Intel:
COV i,m = (.72)(.1210)(.0550) = .00479
Beta 
.00479 .00479

 1.597
.0030
(.055) 2
For Ford:
COV i,m = (.33)(.1460)(.0550) = .00265
Beta 
.00265
 .883
.0030
For Anheuser Busch:
COV i,m = (.55)(.0760)(.0550) = .00230
Beta 
.00230
 .767
.0030
For Merck:
COV i,m = (.60)(.1020)(.0550) = .00337
Beta 
.00337
 1.123
.0030
4.(b). E(Ri) = RFR + Bi(RM - RFR)
= .08 + Bi(.15 - .08)
= .08 + .07Bi
Stock
Intel
Ford
Anheuser Busch
Merck
Beta
1.597
.883
.767
1.123
E(Ri) = .08 + .07Bi
.08 + .1118 = .1918
.08 + .0618 = .1418
.08 + .0537 = .1337
.08 + .0786 = .1586
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11(c).
.20
*Intel
*AB
RM = .15
.10
RFR=.08
5(a).
*Ford
*Merck
Security Market Line
i.
Fair-value plot. The following template shows, using the CAPM, the expected
return, ER, of Stock A and Stock B on the SML. The points are consistent with
the following equations:
ER on stock = Risk-free rate + Beta x (Market return – Risk-free rate)
ER for A = 4.5% + 1.2(14.5% - 4.5%)
= 16.5%
ER for B = 4.5% + 0.8(14.5% - 4.5%)
= 12.5%
ii.
Analyst estimate plot. Using the analyst’s estimates, Stock A plots below the
SML and Stock B, above the SML.
*Stock A
14.5%
*Stock B
4.5%
5(b).
0.8
1.2
Over vs. Undervalue
Stock A is overvalued because it should provide a 16.5% return according to the CAPM
whereas the analyst has estimated only a 16.0% return.
Stock B is undervalued because it should provide a 12.5% return according to the
CAPM whereas the analyst has estimated a 14% return.
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7(a).
The security market line (SML) shows the required return for a given level of systematic
risk. The SML is described by a line drawn from the risk-free rate: expected return is 5
percent, where beta equals 0 through the market return; expected return is 10 percent,
where beta equal 1.0.
15%
Security Market Line (SML)
12%
10%
9%
*Stock Y
*Market
*Stock Y
5%
.5
.7
1.0
1.3
1.5
2.0
Beta()
7 (b). The expected risk-return relationship of individual securities may deviate from that
suggested by the SML, and that difference is the asset’s alpha. Alpha is the difference
between the expected (estimated) rate of return for a stock and its required rate of return
based on its systematic risk Alpha is computed as
ALPHA () = E(ri) - [rf + (E(rM) - rf)]
where
E(ri)
rf
i
E(rM)
= expected return on Security i
= risk-free rate
= beta for Security i
= expected return on the market
Calculation of alphas:
Stock X:
Stock Y:
= 12% - [5% + 1.3% (10% - 5%)] = 0.5%
= 9% - [5% + 0.7%(10% - 5%)] = 0.5%
In this instance, the alphas are equal and both are positive, so one does not dominate the
other.
Another approach is to calculate a required return for each stock and then subtract that
required return from a given expected return. The formula for required return (k) is
k = rf + i (rM - rf ).
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Calculations of required returns:
Stock X: k = 5% + 1.3(10% - 5%) = 11.5%
= 12% - 11.5% = 0.5%
Stock Y: k = 5% + 0.7(10% - 5%) = 8.5%
= 9% - 8.5% = 0.5%
7 (c). By increasing the risk-free rate from 5 percent to 7 percent and leaving all other factors
unchanged, the slope of the SML flattens and the expected return per unit of incremental
risk becomes less. Using the formula for alpha, the alpha of Stock X increases to 1.1
percent and the alpha of Stock Y falls to -0.1 percent. In this situation, the expected
return (12.0 percent) of Stock X exceeds its required return (10.9 percent) based on the
CAPM. Therefore, Stock X’s alpha (1.1 percent) is positive. For Stock Y, its expected
return (9.0 percent) is below its required return (9.1 percent) based on the CAPM.
Therefore, Stock Y’s alpha (-0.1 percent) is negative. Stock X is preferable to Stock Y
under these circumstances.
Calculations of revised alphas:
Stock X = 12% - [7% + 1.3 (10% - 7%]
= 12% - 10.95% = 1.1%
Stock Y = 9% - [7% + 0.7(10% - 7%)]
= 9% - 9.1% = -00.1%
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