Elton, Gruber, Brown, and Goetzmann
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions to Text Problems: Chapter 16
Chapter 16: Problem 1
From the text we know that three points determine a plane. The APT equation for
a plane is:
R i = λ 0 + λ1bi1 + λ 2 bi 2
Assuming that the three portfolios given in the problem are in equilibrium (on the
plane), then their expected returns are determined by:
12 = λ0 + λ1 + 0.5λ2
(a)
13.4 = λ0 + 3λ1 + 0.2λ2
(b)
12 = λ0 + 3λ1 − 0.5λ2
(c)
The above set of linear equations can be solved simultaneously for the three
unknown values of λ0, λ1 and λ2. There are many ways to solve a set of
simultaneous linear equations. One method is shown below.
Subtract equation (a) from equation (b):
1.4 = 2λ1 − 0.3λ 2
(d)
Subtract equation (a) from equation (c):
0 = 2λ1 − λ2
(e)
Subtract equation (e) from equation (d):
1.4 = 0.7λ 2 or λ 2 = 2
Substitute λ2 = 2 into equation (d):
1.4 = 2λ1 − 0.6 or λ1 = 1
Substitute λ1 = 1 and λ 2 = 2 into equation (a):
12 = λ 0 + 1+ 1 or λ 0 = 10
Thus, the equation of the equilibrium APT plane is:
R i = 10 + bi1 + 2bi 2
Elton, Gruber, Brown, and Goetzmann
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions To Text Problems: Chapter 16
16-1
Chapter 16: Problem 2
According to the equilibrium APT plane derived in Problem 1, any security with
b1 = 2 and b2 = 0 should have an equilibrium expected return of 12%:
R i = 10 + bi1 + 2bi 2 = 10 + 2 + 2 × 0 = 12%
Assuming the derived equilibrium APT plane holds, since portfolio D has bD1 = 2 and
bD2 = 0 with an expected return of 10%, the portfolio is not in equilibrium and an
arbitrage opportunity exists.
The first step is to use portfolios in equilibrium to create a replicating equilibrium
investment portfolio, call it portfolio E, that has the same factor loadings (risk) as
portfolio D. Using the equilibrium portfolios A, B and C in Problem 1 and recalling
that an investment portfolio’s weights sum to 1 and that a portfolio’s factor
loadings are weighted averages of the individual factor loadings we have:
bE1 = X A b A1 + X B bB1 + (1− X A − X B ) bC1 = 1X A + 3 X B + 3(1− X A − X B ) = bD1 = 2
bE 2 = X A b A2 + X B bB 2 + (1− X A − X B ) bC2 = 0.5 X A + 0.2 X B − 0.5(1− X A − X B ) = bD2 = 0
Simplifying the above two equations, we have:
− 2 X A = −1 or X A =
X A + 0.7 X B =
Since X A =
1
2
1
2
1
1
, X B = 0 and X C = 1− X A − X B = .
2
2
Since portfolio E was constructed from equilibrium portfolios, portfolio E is also on
the equilibrium plane. We have seen above that any security with portfolio E’s
factor loadings has an equilibrium expected return of 12%, and that is the
expected return of portfolio E:
RE = X A R A + X B RB + X C RC =
1
1
× 12 + 0 × 13.4 + × 12 = 12%
2
2
So now we have two portfolios with exactly the same risk: the target portfolio D
and the equilibrium replicating portfolio E. Since they have the same risk (factor
loadings), we can create an arbitrage portfolio, combining the two portfolios by
going long in one and shorting the other. This will create a self-financing (zero net
investment) portfolio with zero risk: an arbitrage portfolio.
Elton, Gruber, Brown, and Goetzmann
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions To Text Problems: Chapter 16
16-2
In equilibrium, an arbitrage portfolio has an expected return of zero, but since
portfolio D is not in equilibrium, neither is the arbitrage portfolio containing D and E,
and an arbitrage profit may be made. We need to short sell either portfolio D or E
and go long in the other. The question is: which portfolio do we short and which do
we go long in? Since both portfolios have the same risk and since portfolio E has a
higher expected return than portfolio D, we want to go long in E and short D; in
other words, we want X EARB = 1 and X DARB = −1. This gives us:
∑X
ARB
i
= X EARB + X DARB = 1− 1 = 0 (zero net investment)
i
But since portfolio E consists of a weighted average of portfolios A, B and C,
1
1
X EARB = 1 is the same thing as X AARB = , X BARB = 0 and X CARB = , so we have:
2
2
∑X
ARB
i
= X AARB + X BARB + X CARB + X DARB =
i
∑X
b ARB1 =
1
1
+ 0 + − 1 = 0 (zero net investment)
2
2
ARB
bi1
i
i
= X AARB b A1 + X BARB bB1 + X CARB bC1 + X DARB bD1
1
1
= × 1+ 0 × 3 + × 3 − 1× 2
2
2
=0
b ARB 2 =
∑X
(zero factor 1 risk)
ARB
bi 2
i
i
= X AARB b A2 + X BARB bB 2 + X CARB bC2 + X DARB bD2
1
1
= × 0.5 + 0 × 0.2 − × 0.5 − 1× 0
2
2
=0
R ARB =
∑X
(zero factor 2 risk)
ARB
Ri
i
i
= X AARB R A + X BARB R B + X CARB R C + X DARB R D
1
1
= × 12 + 0 × 13.4 + × 12 − 1× 10
2
2
= 2%
(positive arbitrage return)
As arbitrageurs exploit the opportunity by short selling portfolio D, the price of
portfolio D will drop, thereby pushing portfolio D’s expected return up until it
reaches its equilibrium level of 12%, at which point the expected return on the
arbitrage portfolio will equal 0. There is no reason to expect any price effects on
portfolios A, B and C, since the arbitrage with portfolio D can be accomplished
using other assets on the equilibrium APT plane.
Elton, Gruber, Brown, and Goetzmann
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions To Text Problems: Chapter 16
16-3
Chapter 16: Problem 3
From the text we know that three points determine a plane. The APT equation for
a plane is:
R i = λ 0 + λ1bi1 + λ 2 bi 2
Assuming that the three portfolios given in the problem are in equilibrium (on the
plane), then their expected returns are determined by:
12 = λ 0 + λ1 + λ 2
(a)
13 = λ0 + 1.5λ1 + 2λ2
(b)
17 = λ 0 + 0.5λ1 − 3λ 2
(c)
Solving for the three unknowns in the same way as in Problem 1, we obtain the
following solution to the above set of simultaneous linear equations:
λ0 = 8 ; λ1 = 6 ; λ2 = −2 ;
Thus, the equation of the equilibrium APT plane is:
R i = 8 + 6bi1 − 2bi 2
Chapter 16: Problem 4
According to the equilibrium APT plane derived in Problem 3, any security with
b1 = 1 and b2 = 0 should have an equilibrium expected return of 12%:
R i = 8 + 6bi1 − 2bi 2 = 8 + 6 − 2 × 0 = 14%
Assuming the derived equilibrium APT plane holds, since portfolio D has bD1 = 1 and
bD2 = 0 with an expected return of 15%, the portfolio is not in equilibrium and an
arbitrage opportunity exists.
The first step is to use portfolios in equilibrium to create a replicating equilibrium
investment portfolio, call it portfolio E, that has the same factor loadings (risk) as
portfolio D. Using the equilibrium portfolios A, B and C in Problem 3 and recalling
that an investment portfolio’s weights sum to 1 and that a portfolio’s factor
loadings are weighted averages of the individual factor loadings we have:
bE1 = X A b A1 + X B bB1 + (1− X A − X B ) bC1 = 1X A + 1.5 X B + 0.5(1− X A − X B ) = bD1 = 1
bE 2 = X A b A2 + X B bB 2 + (1− X A − X B ) bC2 = X A + 2 X B − 3(1− X A − X B ) = bD2 = 0
Elton, Gruber, Brown, and Goetzmann
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions To Text Problems: Chapter 16
16-4
Simplifying the above two equations, we have:
1
1
XA + XB =
2
2
4X A + 5X B = 3
Solving the above two simultaneous equations we have:
XA =
1
1
1
, X B = and X C = 1− X A − X B = .
3
3
3
Since portfolio E was constructed from equilibrium portfolios, portfolio E is also on
the equilibrium plane. We have seen above that any security with portfolio E’s
factor loadings has an equilibrium expected return of 14%, and that is the
expected return of portfolio E:
RE = X A R A + X B RB + XC RC =
1
1
1
× 12 + × 13 + × 17 = 14%
3
3
3
So now we have two portfolios with exactly the same risk: the target portfolio D
and the equilibrium replicating portfolio E. Since they have the same risk (factor
loadings), we can create an arbitrage portfolio, combining the two portfolios by
going long in one and shorting the other. This will create a self-financing (zero net
investment) portfolio with zero risk: an arbitrage portfolio. In equilibrium, an
arbitrage portfolio has an expected return of zero, but since portfolio D is not in
equilibrium, neither is the arbitrage portfolio containing D and E, and an arbitrage
profit may be made.
We need to short sell either portfolio D or E and go long in the other. The question
is: which portfolio do we short and which do we go long in? Since both portfolios
have the same risk and since portfolio D has a higher expected return than
portfolio E, we want to go long in D and short E; in other words, we want X DARB = 1
and X EARB = −1. This gives us:
∑X
ARB
i
= X DARB + X EARB = 1− 1 = 0 (zero net investment)
i
Elton, Gruber, Brown, and Goetzmann
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions To Text Problems: Chapter 16
16-5
But since portfolio E consists of a weighted average of portfolios A, B and C,
1
1
1
X EARB = −1 is the same thing as X AARB = − , X BARB = − and X CARB = − , so we have:
3
3
3
∑X
ARB
i
= X AARB + X BARB + X CARB + X DARB = −
i
∑X
b ARB1 =
1 1 1
− − + 1 = 0 (zero net investment)
3 3 3
ARB
bi1
i
i
= X AARB b A1 + X BARB bB1 + X CARB bC1 + X DARB bD1
1
1
1
= − × 1− × 1.5 − × 0.5 + 1× 1
3
3
3
=0
b ARB 2 =
∑X
(zero factor 1 risk)
ARB
bi 2
i
i
= X AARB b A2 + X BARB bB 2 + X CARB bC2 + X DARB bD2
1
1
1
= − × 1− × 2 + × 3 + 1× 0
3
3
3
=0
R ARB =
∑X
(zero factor 2 risk)
ARB
Ri
i
i
= X AARB R A + X BARB R B + X CARB R C + X DARB R D
1
1
1
= − × 12 − × 13 − × 17 + 1× 15
3
3
3
= 1%
(positive arbitrage return)
As arbitrageurs exploit the opportunity by buying portfolio D, the price of portfolio
D will rise, thereby pushing portfolio D’s expected return down until it reaches its
equilibrium level of 14%, at which point the expected return on the arbitrage
portfolio will equal 0. There is no reason to expect any price effects on portfolios A,
B and C, since the arbitrage with portfolio D can be accomplished using other
assets on the equilibrium APT plane.
Chapter 16: Problem 5
The general K-factor APT equation for expected return is:
R i = λ0 +
K
∑λ b
k
ik
k =1
where λ0 is the return on the riskless asset, if it exists.
Elton, Gruber, Brown, and Goetzmann
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions To Text Problems: Chapter 16
16-6
Given the data in the problem and in Table 16.1 in the text, along with a riskless
rate of 8%, the Sharpe multi-factor model for the expected return on a stock in the
construction industry is:
R i = 8 + 5.36 × 1.2 + 0.24 × 6 − 5.56 × 0.4 − 0.12 × 0.2 − 2 × 1− 1.59
= 10.034%
The last number, −1.59, enters because the stock is a construction stock.
Chapter 16: Problem 6
A.
From the text we know that, for a 2-factor APT model to be consistent with the
standard CAPM, λ j = R m − RF β λj . Given that R m − RF = 4 and using results from
(
(
)
)
Problem 1, we have:
1 = 4 β λ1 or β λ1 = 0.25 ; 2 = 4 β λ 2 or β λ 2 = 0.5 .
B.
From the text we know that β i = bi1β λ1 + bi 2 β λ 2 . So we have:
β A = 1× 0.25 + 0.5 × 0.5 = 0.5
β B = 3 × 0.25 + 0.2 × 0.5 = 0.85
β C = 3 × 0.25 − 0.5 × 0.5 = 0.5
C.
Assuming all three portfolios in Problem 1 are in equilibrium, then we can use any
one of them to find the risk-free rate. For example, using portfolio A gives:
(
)
(
)
R A = Rf + R m − RF β A or RF = R A − R m − RF β A
(
)
Given that R A = 12% , β A = 0.5 and R m − RF = 4% , we have:
RF = 12 − 4 × 0.5 = 10%
Elton, Gruber, Brown, and Goetzmann
Modern Portfolio Theory and Investment Analysis, 7th Edition
Solutions To Text Problems: Chapter 16
16-7