CHAPTER 12 B-
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:
EF = 0 and = .10
E = 0 and Si = .20 for all i
If we assume the stocks in the portfolio are equally-weighted, the weight of each stock is
N
1
, that is:
Xi =
N
1
for all i
If a portfolio is composed of N stocks each forming 1/N proportion of the portfolio, the return on the
portfolio is 1/N times the sum of the returns on the N stocks. To find the variance of the respective
portfolios in the 2 markets, we need to use the definition of variance from statistics:
Var(x) = E[x – E(x)]2
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:
RP =
R
N
1
i
RP =
N
1
(.10 + F + i)
RP = .10 + F +
N
1
i
Also, recall from statistics a property of expected value, that is:
If:
Y X Z ~~
a
~
where a is a constant, and
Z
~
,
X
~
, and
Y
~
are random variables, then:
1

CHAPTER 12 B-
and
E(a) = a
Now use the above to find E(RP):
E(RP) = E
 i
F
N
1
β .10
E(RP) = .10 + E(F) +
)E(
N
1
i
E(RP) = .10 + (0) +
0
N
1
E(RP) = .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) = E
2
N
1
β
εF
Var(RP) = E
2
2
2
22
N
1
N
1
2β β

εFF
Var(RP) =
2
222 ),Cov(
N
1
– 1 σ
N
1
σβ
 ji

Finally, since we can have as many stocks in each market as we want, in the limit, as N ,
N
1
0, so we get:
Var(RP) = 22 + Cov(i,j)
and, since:
Cov(i,j) = ij(i,j)
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CHAPTER 12 B-
and the problem states that 1 = 2 = .10, so:
Var(RP) = 22 + 12(i,j)
Var(RP) = 2(.01) + .04(i,j)
So now, to summarize what we have so far:
R1i = .10 + 1.5F + 1i
R2i = .10 + .5F + 2i
E(R1P) = E(R2P) = .10
Var(R1P) = .0225 + .04(1i,1j)
Var(R2P) = .0025 + .04(2i,2j)
Finally we can begin answering the questions a, b, & c for various values of the correlations:
a. Substitute (1i,1j) = (2i,2j) = 0 into the respective variance formulas:
Var(R1P) = .0225
Var(R2P) = .0025
Since Var(R1P) > Var(R2P), and expected returns are equal, a risk averse investor will prefer to
invest in the second market.
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
Var(R2P) = .0025 + .04(2i,2j)
Var(R2P) = .0025 + .04(0)
Var(R2P) = .0025
Since Var(R1P) > Var(R2P), and expected returns are equal, a risk averse investor will prefer to
invest in the second market.
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
Var(R2P) = .0025 + .04(2i,2j)
Var(R2P) = .0025 + .04(.5)
Var(R2P) = .0225
Since Var(R1P) = Var(R2P), and expected returns are equal, a risk averse investor will be
indifferent between the two markets.
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CHAPTER 12 B-
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)
.0225 + .04(1i,1j) = .0025 + .04(2i,2j)
(2i,2j) = (1i,1j) + .5
Therefore, for any set of correlations that have this relationship (as found in part c), a risk
adverse investor will be indifferent between the two markets.
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
~
and:
Var(a) = 0
The problem states that return-generation can be described by:
Ri,t = i + i(RM) + i,t
Realize that Ri,t, RM, and i,t are random variables, and i and i are constants. Then, applying
the above properties to this model, we get:
Var(Ri) =
2
βi
Var(RM) + Var(i)
and now we can find the standard deviation for each asset:
2
A
σ
= .702(.0121) + .01 = .015929
A
σ
=
.015929
= .1262, or 12.62%
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CHAPTER 12 B-
2
B
σ
= 1.22(.0121) + .0144 = .031824
B
σ
=
.031824
= .1784, or 17.84%
2
C
σ
= 1.52(.0121) + .0225 = .049725
C
σ
=
.049725
= .2230, or 22.30%
b. From the above formula for variance, note that as N ,
N
)Var(εi
0, so you get:
Var(Ri) =
2
βi
Var(RM)
So, the variances for the assets are:
2
A
σ
= .72(.0121) = .005929
2
B
σ
= 1.22(.0121) = .017424
2
C
σ
= 1.52(.0121) = .027225
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:
A
R
= 3.3% + .70(10.6% – 3.3%) = 8.41%
B
R
= 3.3% + 1.2(10.6% – 3.3%) = 12.06%
C
R
= 3.3% + 1.5(10.6% – 3.3%) = 14.25%
We can compare these results for expected asset returns as per CAPM or APT with the expected
returns given in the table. This shows that Assets A & B are accurately priced, but Asset C is
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CHAPTER 12 B-
overpriced (the model shows the return should be higher). Thus, rational investors will not hold
Asset C.
d. If short selling is allowed, rational investors will sell short Asset C, causing the price of Asset C
to decrease until no arbitrage opportunity exists. In other words, the price of Asset C should
decrease until the return becomes 14.25 percent.
10. a. Let:
X1 = the proportion of Security 1 in the portfolio, and
X2 = the proportion of Security 2 in the portfolio
and note that since the weights must sum to 1.0:
X1 = 1 – X2
Recall from Chapter 11 that the beta for a portfolio (or in this case the beta for a factor) is the
weighted average of the security betas, so
P1 = X111 + X221
P1 = X111 + (1 – X1)21
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:
X1 = – 1
X2 = 2
Thus, sell short Security 1 and buy Security 2.
To find the expected return on that portfolio, use
RP = X1R1 + X2R2
so applying the above:
E(RP) = –1(20%) + 2(20%)
E(RP) = 20%
P1 = –1(1) + 2(.5)
P1 = 0
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)
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CHAPTER 12 B-
and
X3 = 3
X4 = –2
Thus, sell short Security 4 and buy Security 3. Then,
E(RP2) = 3(10%) + (–2)(10%)
E(RP2) = 10%
P2 = 3(.5) – 2(.75)
P2 = 0
Note that since both P1 and P2 are 0, this is a risk free portfolio!
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
rate of 5% and invest the funds in a portfolio built by selling short Security 4 and buying
Security 3 with weights (3,–2) as in part b.
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%.
Finally, a combined movement of all security prices is also possible. The prices of Security 4
and the risk-free security will decrease and the price of Security 3 will increase until the
opportunity disappears.
7