Download Ch 4

CHAPTER 4
ANSWERS
4-1
a.
The probability distribution for complete certainty is a vertical line.
b.
The probability distribution for total uncertainty is the X axis from -∞ to +∞.
4-2
Security A is less risky if held in a diversified portfolio because of its negative correlation with other
stocks. In a single-asset portfolio, Security A would be riskier because σA > σB and CVA > CVB.
4-3
a.
No, it is not riskless. The portfolio would be free of default risk and liquidity risk, but
inflation could erode the portfolio’s purchasing power. If the actual inflation rate is greater
than that expected, interest rates in general will rise to incorporate a larger inflation premium
(IP) and—as we shall see in Chapter 5—the value of the portfolio would decline.
b.
No, you would be subject to interest rate reinvestment rate risk. You might expect to “roll
over” the Treasury bills at a constant (or even increasing) rate of interest, but if interest rates
fall, your investment income will decrease.
c.
A U.S. government-backed bond that provided interest with constant purchasing power (that
is, an indexed bond) would be close to riskless (risk free). No such bond exists in the United
States, however.
a.
The expected return on a life insurance policy is calculated just as for a common stock. Each
outcome is multiplied by its probability of occurrence, and then these products are summed.
For example, suppose a one-year term policy pays $10,000 at death, and the probability of the
policyholder’s death in that year is 2 percent. Then, there is a 98 percent probability of zero
return and a 2 percent probability of $10,000:
4-4
Expected return = 0.98($0) + 0.02($10,000) = $200.
This expected return could be compared to the premium paid. Generally, the premium will be
larger because of sales and administrative costs, and insurance company profits, indicating a
negative expected rate of return on the investment in the policy.
4-5
b.
There is a perfect negative correlation between the returns on the life insurance policy and the
returns on the policyholder’s human capital. In fact, these events (death and future lifetime
earnings capacity) are mutually exclusive, because a person has no future earnings when he or
she dies.
c.
People are generally risk averse. Therefore, they are willing to pay a premium to decrease the
uncertainty of their future cash flows. A life insurance policy guarantees an income (the face
value of the policy) to the policyholder’s beneficiaries when the policyholder’s future
earnings capacity drops to zero.
The risk premium on a high beta stock would increase more.
RPj = Risk Premium for Stock j = (kM - kRF)ßj.
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If risk aversion increases, the slope of the SML will increase, and so will the market risk premium
(kM – kRF). The product (kM – kRF)ßj is the risk premium of the jth stock. If ßj is low (say, 0.5), then
the product will be small because RPj will increase by only half the increase in RPM. However, if ßj
is large (say, 2.0), then its risk premium will rise by twice the increase in RPM.
____________________________________________________________
SOLUTIONS
4-1
a.
($1 million)(0.5) + ($0)(0.5) = $0.5 million.
b.
You would probably take the sure $0.5 million.
c.
Risk averter. The expected payoff of both alternatives is the same, but Alternative (1) offers a
sure $0.5 million.
d.
(1) ($1,150,000)(0.5) + ($0)(0.5) = $575,000, or an expected profit of $75,000.
(2) $75,000/$500,000 = 15%.
(3) This depends on the your degree of risk aversion.
(4) Again, this depends on the individual.
(5) The situation would be unchanged if the stocks’ returns were perfectly positively
correlated. Otherwise, the stock portfolio would have the same expected return as the
single stock (15 percent) but a lower standard deviation. If the correlation coefficient (r)
between each pair of stocks was negative one, the portfolio would be virtually riskless.
Because r for stocks is generally in the range of +0.4 to +0.6, investing in a portfolio of
stocks would definitely be an improvement over investing in the single stock.
4-2
a.
k̂ M = 0.1(10%) + 0.2(12%) + 0.4(13%) + 0.2(16%) + 0.1(17%) = 13.5%.
b.
To determine the fund's beta, ßF, the weight for the amount invested in each stock needs to be
computed.
A = $160/$500 =
B = $120/$500 =
C = $80/$500 =
D = $80/$500 =
E = $60/$500 =
0.32
0.24
0.16
0.16
0.12
ßF = 0.32(0.5) + 0.24(2.0) + 0.16(4.0) + 0.16(1.0) + 0.12(3.0)
= 0.16 + 0.48 + 0.64 + 0.16 + 0.36 = 1.8.
c.
kRF = 8% (given)
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Therefore, the SML equation is
kF = kRF + (kM - kRF)ßF = 8% + (13.5% - 8%)ßF = 8% + (5.5%)ßF.
d.
Use ßF = 1.8 in the SML determined in Part b:
k̂ F = 8% + (13.5% - 8%)1.8 = 8% + 9.90% = 17.90%.
e.
kN = Required rate of return on new stock = 8% + (5.5%)2.0 = 19%.
An expected return of 18 percent on the new stock is below the 19 percent required rate of
return on an investment with a risk of ß = 2.0. Because kN = 19% > k̂ N  18%, the new stock
should not be purchased. The expected rate of return that would make McAlhany indifferent
to purchasing the stock is 19 percent.
4-3
a.
k̂ A  0.1(33%) + 0.2(20%) + 0.4(15%) + 0.3(0%) = 13.3%
k̂ B  0.1(60%) + 0.2(30%) + 0.4(5%) + 0.3(-20%) = 8.0%
b.
k̂ P = 0.5(13.3%) + 0.5(8%) = 10.65%
Probability
0.1
0.2
0.4
0.3
Stock A
33%
20
15
0
Stock B
60%
30
5
-20
50/50 Portfolio
46.5%
25.0
10.0
-10.0
= 0.5(33%) + 0.5(60%)
= 0.5(20%) + 0.5(30%)
= 0.5(15%) + 0.5(5%)
= 0.5(0%) + 0.5(-20%)
k̂ P = 0.1(46.5%) + 0.2(25.0%) + 0.4(10.0%) 0.3(-10.0%) = 10.65%
c.
 A  0.1(33  13.3) 2  0.2(20  13.3) 2  0.4(15  13.3) 2  0.3(0  13.3) 2
 102 .01  10.1%
 B  0.1(60  8) 2  0.2(30  8) 2  0.4(5  8) 2  0.3(20  8) 2
 606  24.6%
 P  0.1(46.5  10.65) 2  0.2(25  10.65) 2  0.4(10  10.65) 2  0.3(10  10.65) 2
 297.8025  17.3%
d.
CVA = 10.1/13.3 = 0.76
CVB = 24.6/8.0 = 3.1
Stock B is riskier; it has a higher coefficient of variation.
e.
CVP = 17.3/10.65 = 1.6
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f.
4-4
If the only options are to invest in Stock A only, Stock B only, or a portfolio containing 50
percent of Stock A and 50 percent of Stock B, a risk averse investor would prefer Stock A,
because it has the best risk/return relationship. If a portfolio can be formed by combining
different proportions of Stock A and Stock B, then a portfolio with a better risk/return
relationship probably can be formed.
The answers to a, b, c, and d are given below:
kA
________
2000
2001
2002
2003
2004
Mean
Std. Dev.
Coeff. of Var.
kB
________
50/50
Portfolio
_________
-18.00%
33.00
15.00
- 0.50
27.00
-14.50%
21.80
30.50
- 7.60
26.30
-16.25%
27.40
22.75
- 4.05
26.65
11.30
20.79
1.84
11.30
20.78
1.84
11.30
20.13
1.78
The computations for the average return, standard deviation, and coefficient of variation for Stock A
are:
kA
-18.0%
33.0
15.0
-0.5
27.0
56.5%
kA =
kA
kA - k A
(kA - k A )2
11.3%
11.3
11.3
11.3
11.3
-29.3%
21.7
3.7
-11.8
15.7
858.49
470.89
13.69
139.24
246.49
1,728.80
56.5%
= 11.30%
5
1,728.80
= 20.79%
5 -1
20.79%
CV =
= 1.84
11.30%
sA =
e.
A risk-averse investor would choose the portfolio over either Stock A or Stock B alone,
because the portfolio offers the same expected return but with less risk. This result occurs
because returns on A and B are not perfectly positively correlated (rAB = 0.88).
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4-5
Investment
Stock D
Stock E
Stock F
Expected
Standard
Coefficient of
Return, k̂
10.0%
36.0
12.0
Deviation
8.0%
24.0
10.0
Variation, σ
0.80
0.67
0.83
= 8/10
= 24/36
= 10/12
Stock F has the greatest relative risk—that is, the greatest risk per unit (%) of return.
4-6
a.
k̂ M = (0.3)(15%) + (0.4)(9%) + (0.3)(18%) = 13.5%.
k̂ S = (0.3)(20%) + (0.4)(5%) + (0.3)(12%) = 11.6%.
b.
 M = (0.3)(15 - 13.5 )2 + (0.4)(9 - 13.5 )2 + (0.3)(18 - 13.5 )2  14.85  3.85%
 S = (0.3)(20 - 11.6 )2 + (0.4)(5 - 11.6 )2 + (0.3)(12 - 11.6 )2  38.64  6.22%
4-7
3.85%
= 0.29
13.5%
6.22%
= 0.54
CVS =
11.6%
c.
CVM =
a.
k̂ Y = 0.1(-35%) + 0.2(0%) + 0.4(20%) + 0.2(25%) + 0.1(45%) = 14%
k̂ X = 12%
b.
 2X = (–10% – 12%)2(0.1) + (2% – 12%)2(0.2) + (12% – 12%)2(0.4)
+ (20% – 12%)2(0.2) + (38% – 12%)2(0.1) = 148.8
 X  148 .8  12.20%
 Y  20.35%
CVX = 12.20%/12% = 1.02
CVY = 20.35%/14% = 1.45.
If Stock Y is less highly correlated with the market than Stock X, then it might have a lower
beta than Stock X, and hence be less risky in a portfolio sense.
4-8
a.
kB = kRF + (kM – kRF)ßB
14% = 8% + (11% – 8%)ßB
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14% = 8% + 3%(ßB)
6% = 3%(ßB)
2 = ßB.
b.
kB = 8% + 3%(ßB)
kB = 8% + 3%(1.5)
kB = 12.5%.
4-9
a.
kX = kRF + (kM – kRF)ßX = 9% + (14% – 9%)1.3 = 15.5%.
b.
RPM = kM – kRF = 14% - 9% = 5%
(1) kRF increases to 10%; RPM does not change:
kX = kRF + (RPM)ßX = 10% + (5%)1.3 = 16.5%.
(2) kRF decreases to 8%; RPM does not change:
kX = kRF + (RPM)ßX = 8% + (5%)1.3 = 14.5%.
c.
(1) kM increases to 16%; RPM = 16% – 9% = 7%
kX = kRF + (RPM)ßX = 9% + (7%)1.3 = 18.1%.
(2) kM decreases to 13%; RPM = 13% – 9% = 4%
kX = kRF + (RPM)ßX = 9% + (4%)1.3 = 14.2%.
4-10
Alternative solutions:
1.
Old portfolio beta = 1.12 = (0.05)ß1 + (0.05)ß2 +...+ (0.05)ß20
1.12 = (Σßj)(0.05)
Σßj = 1.12/0.05 = 22.4 = 20(1.12)
New portfolio beta = (22.4 - 1.0 + 1.75)/20 = 1.1575 = 1.16
2.
Σßj excluding the stock with the beta equal to 1.0 is 22.4 – 1.0 = 21.4, so the beta of the
portfolio excluding this stock is ß = 21.4/19 = 1.1263. The beta of the new portfolio is:
1.1263(0.95) + 1.75(0.05) = 1.1575 = 1.16
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4-11
Portfolio beta:
Stock
(1)
Investment
(2)
Weight
(3)
A
B
C
D
$ 400,000
600,000
1,000,000
2,000,000
$4,000,000
0.10
0.15
0.25
0.50
1.00
Beta
(4)
1.50
-0.50
1.25
0.75
Portfolio beta
(5) = (3) x (4)
0.1500
-0.0750
0.3125
0.3750
0.7625
kp = kRF + (kM – kRF)(ßp) = 6% + (14% – 6%)(0.7625) = 12.1%.
Alternative solution: First compute the return for each stock using the CAPM equation [kRF + (kM –
kRF)ßj], and then compute the weighted average of these returns.
kRF = 6% and kM – kRF = 8%.
Stock
A
B
C
D
Total
Investment
Beta
kj=kRF+(kM–kRF)ßj
Weight
$ 400,000
600,000
1,000,000
2,000,000
$4,000,000
1.50
–0.50
1.25
0.75
18%
2
16
12
0.10
0.15
0.25
0.50
1.00
kp = 18%(0.10) + 2%(0.15) + 16%(0.25) + 12%(0.50) = 12.1%.
4-12
We know that ßT = 1.50, ßV = 0.75, kM = 15%, kRF = 9%.
kj = kRF + (kM – kRF)ßj = 9% + (15% - 9%)ßj.
kT = 9% + 6%(1.50) = 18.0%
kV = 9% + 6%(0.75) = 13.5
4.5%
4-13
Before adding the new stock, ßP = 1.5 and the amount invested in the portfolio is $200,000. An
additional $50,000 will be invested in a stock with a beta equal to 2.5. The beta of the portfolio
after the new stock is added will be:
ßP = (200,000/250,000)1.5 + (50,000/250,000)2.5 = 0.8(1.5) + 0.2(2.5) = 1.7
4-14
The solution is given in the Instructor's Manual, Solutions to Integrative Problems.
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4-15
a.
Computer-Related Problem
INPUT DATA:
Probability:
0.1
0.2
0.4
0.3
1.0
^
k =
Std Dev =
CV =
KEY OUTPUT:
Stock A
33.00%
20.00%
15.00%
0.00%
13.30%
10.10%
0.76
Percent in: Stock A
50.00%
b.
Stock B
60.00%
30.00%
5.00%
-20.00%
8.00%
24.62%
3.08
Stock B
50.00%
INPUT DATA:
Probability:
0.1
0.2
0.4
0.3
1.0
^
k =
Std Dev =
CV =
15.20%
17.31%
1.14
Stock C
0.00%
Portfolio
46.50%
25.00%
10.00%
-10.00%
10.65%
17.26%
1.62
Portfolio
100.00%
KEY OUTPUT:
Stock A
33.00%
20.00%
15.00%
0.00%
Stock B
60.00%
30.00%
5.00%
-20.00%
Stock C
50.00%
10.00%
25.00%
-6.00%
13.30%
10.10%
0.76
8.00%
24.62%
3.08
15.20%
17.31%
1.14
Percent in: Stock A
33.33%
c. – d.
Stock C
50.00%
10.00%
25.00%
-6.00%
Stock B
33.33%
Stock C
33.34%
Portfolio
47.67%
20.00%
15.00%
-8.67%
12.17%
16.48%
1.35
Portfolio
100.00
The correlation among the stocks is positive, but not perfect. As a result, it would be a good
idea to include all three stocks in a portfolio so as to achieve diversification.
87