Chapter 10  Arbitrage Pricing Theory and Multifactor Models of Risk and Return
CHAPTER 10: ARBITRAGE PRICING THEORY
AND MULTIFACTOR MODELS OF RISK AND RETURN
PROBLEM SETS
1.
The revised estimate of the expected rate of return on the stock would be the old
estimate plus the sum of the products of the unexpected change in each factor times the
respective sensitivity coefficient:
revised estimate = 12% + [(1
×
2%) + (0.5
×
3%)] = 15.5%
2.
The APT factors must correlate with major sources of uncertainty, i.e., sources of
uncertainty that are of concern to many investors. Researchers should investigate factors
that correlate with uncertainty in consumption and investment opportunities. GDP, the
inflation rate, and interest rates are among the factors that can be expected to determine
risk premiums. In particular, industrial production (IP) is a good indicator of changes in
the business cycle. Thus, IP is a candidate for a factor that is highly correlated with
uncertainties that have to do with investment and consumption opportunities in the
economy.
3.
Any pattern of returns can be “explained” if we are free to choose an indefinitely large
number of explanatory factors. If a theory of asset pricing is to have value, it must
explain returns using a reasonably limited number of explanatory variables (i.e.,
systematic factors).
4.
Equation 10.9 applies here:
E(r
p
) = r
f
+
β
P1
[E(r
1
)

r
f
] +
β
P2
[E(r
2
) – r
f
]
We need to find the risk premium (RP) for each of the two factors:
RP
1
= [E(r
1
)

r
f
] and RP
2
= [E(r
2
)

r
f
]
In order to do so, we solve the following system of two equations with two unknowns:
31 = 6 + (1.5
×
RP
1
) + (2.0
×
RP
2
)
27 = 6 + (2.2
×
RP
1
) + [(–0.2)
×
RP
2
]
The solution to this set of equations is:
RP
1
= 10% and RP
2
= 5%
Thus, the expected returnbeta relationship is:
E(r
P
) = 6% + (
β
P1
×
10%) + (
β
P2
×
5%)
101
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View Full DocumentChapter 10  Arbitrage Pricing Theory and Multifactor Models of Risk and Return
5.
The expected return for Portfolio F equals the riskfree rate since its beta equals 0.
For Portfolio A, the ratio of risk premium to beta is: (12

6)/1.2 = 5
For Portfolio E, the ratio is lower at: (8 – 6)/0.6 = 3.33
This implies that an arbitrage opportunity exists. For instance, you can create a Portfolio
G with beta equal to 0.6 (the same as E’s) by combining Portfolio A and Portfolio F in
equal weights. The expected return and beta for Portfolio G are then:
E(r
G
) = (0.5
×
12%) + (0.5
×
6%) = 9%
β
G
= (0.5
×
1.2) + (0.5
×
0) = 0.6
Comparing Portfolio G to Portfolio E, G has the same beta and higher return. Therefore,
an arbitrage opportunity exists by buying Portfolio G and selling an equal amount of
Portfolio E. The profit for this arbitrage will be:
r
G
– r
E
=[9% + (0.6
×
F)]

[8% + (0.6
×
F)] = 1%
That is, 1% of the funds (long or short) in each portfolio.
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 Spring '10
 AttilaOdabaşı
 Arbitrage, Capital Asset Pricing Model, Financial Markets, Modern portfolio theory, Arbitrage Pricing Theory, Portfolio G

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