CHAPTER 12 B-
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CHAPTER 12
AN ALTERNATIVE VIEW OF RISK AND
RETURN: THE ARBITRAGE PRICING
THEORY
Answers to Concept Questions
1. Systematic risk is risk that cannot be diversified away through formation of a portfolio. Generally,
systematic risk factors are those factors that affect a large number of firms in the market, however,
2. Any return can be explained with a large enough number of systematic risk factors. However, for a
3. The market risk premium and inflation rates are probably good choices. The price of wheat, while a
risk factor for Ultra Bread, is not a market risk factor and will not likely be priced as a risk factor
4. a. Real GNP was higher than anticipated. Since returns are positively related to the level of GNP,
returns should rise based on this factor.
b. Inflation was exactly the amount anticipated. Since there was no surprise in this announcement,
it will not affect Lewis-Striden returns.
c. Interest rates are lower than anticipated. Since returns are negatively related to interest rates, the
5. The main difference is that the market model assumes that only one factor, usually a stock market
6. The fact that APT does not give any guidance about the factors that influence stock returns is a
commonly-cited criticism. However, in choosing factors, we should choose factors that have an
8. It is the weighted average of expected returns plus the weighted average of each security’s beta times
a factor F plus the weighted average of the unsystematic risks of the individual securities.
9. Choosing variables because they have been shown to be related to returns is data mining. The relation
found between some attribute and returns can be accidental, thus overstated. For example, the
10. Using a benchmark composed of British stocks is wrong because the stocks included are not of the
same style as those in a U.S. growth stock fund.
Solutions to Questions and Problems
NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this solutions
manual, rounding may appear to have occurred. However, the final answer for each problem is found
without rounding during any step in the problem.
Basic
1. Since we have the expected return of the stock, the revised expected return can be determined using
2. a. If m is the systematic risk portion of return, then:
m = GDPΔGDP + InflationΔInflation + rΔInterest rates
m = .0000734($19,843 – 19,571) – .90(.0270 – .0260) – .32(.0320 – .0340)
CHAPTER 12 B-
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b. The unsystematic return is the return that occurs because of a firm specific factor such as the bad
news about the company. So, the unsystematic return of the stock is –.85 percent. The total return
is the expected return, plus the two components of unexpected return: the systematic risk portion
of return and the unsystematic portion. So, the total return of the stock is:
3. a. If m is the systematic risk portion of return, then:
b. The unsystematic return is the return that occurs because of a firm specific factor such as the
increase in market share. If is the unsystematic risk portion of the return, then:
c. The total return is the expected return, plus the two components of unexpected return: the
systematic risk portion of return and the unsystematic portion. So, the total return of the stock is:
4. The beta for a particular risk factor in a portfolio is the weighted average of the betas of the assets.
This is true whether the betas are from a single factor model or a multi-factor model. So, the betas of
the portfolio are:
F1 = .20(1.55) + .20(.81) + .60(.73)
F1 = .91
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Intermediate
5. We can express the multifactor model for each portfolio as:
E(RP ) = RF + 1F1 + 2F2
where F1 and F2 are the respective risk premiums for each factor. Expressing the return equation for
each portfolio, we get:
6. a. The market model is specified by:
R =
R
+ (RM –
M
R
M
R
) +
so applying that to each stock:
Stock A:
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b. Since we don’t have the actual market return or unsystematic risk, we will get a formula with
those values as unknowns:
RP = .30RA + .45RB + .25RC
c. Using the market model, if the return on the market is 15 percent and the systematic risk is
zero, the return for each individual stock is:
RA = 10.5% + 1.20(15% – 14.2%)
RA = 11.46%
RP = 13.84%
Alternatively, to find the portfolio return, we can use the return of each asset and its portfolio
RP = 13.84%
7. a. Since the five stocks have the same expected returns and the same betas, the portfolio also has
the same expected return and beta. However, the unsystematic risks might be different, so the
expected return of the portfolio is:
P
R
= 11% + .84F1 + 1.69F2 + (1/5)(1 + 2 + 3 + 4 + 5)
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8. To determine which investment an investor would prefer, you must compute the variance of portfolios
created by many stocks from either market. Because you know that diversification is good, it is
reasonable to assume that once an investor has chosen the market in which she will invest, she will
buy many stocks in that market.
Known:
In our case:
Var(RP) = E[RP – E(RP)]2
Note however, to use this, first we must find RP and E(RP). So, using the assumption about equal
weights and then substituting in the known equation for Ri:
Z
X
Y
)YE( )X)E(E( )ZE( ~~
a
~+=
and
E(a) = a
Now use the above to find E(RP):
E(RP) = E
++ i
F
N
1
β .10
Next, substitute both of these results into the original equation for variance:
Var(RP) = E[RP – E(RP)]2
Var(RP) = E
2
.10 –
N
1
β .10
++ i
εF
Var(RP) =
2
222 ),Cov(
N
1
– 1 σ
N
1
σβ
++ ji
So now, to summarize what we have so far:
R1i = .10 + 1.5F + 1i
R2i = .10 + .5F + 2i
Finally we can begin answering parts a, b, and c for various values of the correlations:
a. Substitute (1i,1j) = (2i,2j) = 0 into the respective variance formulas:
Var(R1P) = .0225
b. If we assume (1i,1j) = .9, and (2i,2j) = 0, the variance of each portfolio is:
Var(R1P) = .0225 + .04(1i,1j)
Var(R1P) = .0225 + .04(.9)
Var(R1P) = .0585
c. If we assume (1i,1j) = 0, and (2i,2j) = .5, the variance of each portfolio is:
Var(R1P) = .0225 + .04(1i,1j)
Var(R1P) = .0225 + .04(0)
Var(R1P) = .0225
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d. Since the expected returns are equal, indifference implies that the variances of the portfolios in
the two markets are also equal. So, set the variance equations equal, and solve for the correlation
of one market in terms of the other:
Var(R1P) = Var(R2P)
9. a. In order to find standard deviation, , you must first find the variance, since =
Var
. Recall
from statistics a property of variance:
If:
Y X Z ~~
a
~+=
where a is a constant, and
Z
~
,
X
~
, and
Y
~
are random variables, then:
)YVar( )XVar( )ZVar( 2~~
a
~+=
A
σ
= .702(.0121) + .01 = .015929
A
σ
=
.015929
.031824
.049725
= .1262, or 12.62%
b. From the above formula for variance, note that as N → ,
N
)Var(εi
→ 0, so you get:
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c. We can use the model:
i
R
= RF + i(
M
R
– RF)
which is the CAPM (or APT Model when there is one factor and that factor is the Market). So,
the expected return of each asset is:
d. If short selling is allowed, rational investors will sell short Asset C, causing the price of Asset C
10. a. Let:
X1 = the proportion of Security 1 in the portfolio, and
X2 = the proportion of Security 2 in the portfolio
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Now, apply the condition given in the hint that the return of the portfolio does not depend on
F1. This means that the portfolio beta for that factor will be 0, so:
P1 = 0 = X111 + (1 – X1)21
P1 = 0 = X1(1.0) + (1 – X1)(.5)
and solving for X1 and X2:
b. Following the same logic as in part a, we have
P2 = 0 = X331 + (1 – X3)41
P2 = 0 = X3(1) + (1 – X3)(1.5)
and
c. The portfolio in part b provides a risk-free return of 10%, which is higher than the 5% return
provided by the risk-free security. To take advantage of this opportunity, borrow at the risk-free
d. First assume that the risk-free security will not change. The price of Security 4 (that everyone is
trying to sell short) will decrease, and the price of Security 3 (that everyone is trying to buy) will
increase. Hence the return of Security 4 will increase and the return of Security 3 will decrease.
The alternative is that the prices of Securities 3 and 4 will remain the same, and the price of the
risk-free security drops until its return is 10%.