Solution manual financial management 10e by keown chapter 06

CHAPTER 6

Risk and
Rates of Return
CHAPTER ORIENTATION
This chapter introduces the concepts that underlie the valuation of securities and their rates
of return. We are specifically concerned with common stock, preferred stock, and bonds.
We also look at the concept of the investor's expected rate of return on an investment.

CHAPTER OUTLINE
I.

II.

III.

The relationship between risk and rates of return
A.

Data have been compiled by Ibbotson and Sinquefield on the actual returns
for various portfolios of securities from 1926-2002.

B.

The following portfolios were studied.
1.

Common stocks of small firms

2.

Common stocks of large companies

3.

Long-term corporate bonds

4.

Long-term U.S. government bonds

5.

U.S. Treasury bills

C.

Investors historically have received greater returns for greater risk-taking
with the exception of the U.S. government bonds.

D.

The only portfolio with returns consistently exceeding the inflation rate has
been common stocks.

Effects of Inflation on Rates of Return
A.

When a rate of interest is quoted, it is generally the nominal or, observed
rate. The real rate of interest represents the rate of increase in actual
purchasing power, after adjusting for inflation.

B.

Consequently, the nominal rate of interest is equal to the sum of the real rate
of interest, the inflation rate, and the product of the real rate and the
inflation rate.

Term Structure of Interest Rates
144


The relationship between a debt security’s rate of return and the length of time until
the debt matures is known as the term structure of interest rates or the yield to
maturity.
IV.

Expected Return
A.

The expected benefits or returns to be received from an investment come in
the form of the cash flows the investment generates.

B.

Conventionally, we measure the expected cash flow, X , as follows:
X = XiP(Xi)
where

N

=

the number of possible states of the economy.

Xi

=

the cash flow in the ith state of the economy.

P(Xi) =
V.

the probability of the ith cash flow.

Riskiness of the cash flows
A.

Risk can be defined as the possible variation in cash flow about an expected
cash flow.

B.

Statistically, risk may be measured by the standard deviation about the
expected cash flow.

C.

Risk and diversification
1.

Total variability can be divided into:
a.

The variability of returns unique to the security (diversifiable
or unsystematic risk)

b.

The risk related to market movements (nondiversifiable or
systematic risk)

2.

By diversifying, the investor can eliminate the "unique" security risk.
The systematic risk, however, cannot be diversified away.

3.

The market rewards diversification. We can lower risk without
sacrificing expected return, and/or we can increase expected return
without having to assume more risk.

4.

Diversifying among different kinds of assets is called asset
allocation. Compared to diversification within the different asset
classes, the benefits received are far greater through effective asset
allocation.

5.

Risk and being patient
a.

An investor in common stocks must often wait longer to earn
the higher returns than those provided by bonds.

b.

The capital markets reward us not just for diversifying, but
also for being patient. The returns tend to converge toward
the average as we lengthen our holding period.

145


VI.

6.

The characteristic line tells us the average movement in a firm's
stock price in response to a movement in the general market, such as
the stock market. The slope of the characteristic line, which has
come to be called beta, is a measure of a stock's systematic or
market risk. The slope of the line is merely the ratio of the "rise" of
the line relative to the "run" of the line.

7.

If a security's beta equals one, a 10 percent increase (decrease) in
market returns will produce on average a 10 percent increase
(decrease) in security returns.

8.

A security having a higher beta is more volatile and thus more risky
than a security having a lower beta value.

9.

A portfolio's beta is equal to the average of the betas of the stocks in
the portfolio.

Required rate of return
A.

The required rate of return is the minimum rate necessary to compensate an
investor for accepting the risk he or she associates with the purchase and
ownership of an asset.

B.

Two factors determine the required rate of return for the investor:

C.

1.

The risk-free rate of interest which recognizes the time value of

money.

2.

The risk premium which considers the riskiness (variability of
returns) of the asset and the investor's attitude toward risk.

Capital asset pricing model-CAPM
1.

The required rate of return for a given security can be expressed as
= + beta x
or
kj = krf + βj (km - krf)

2.

Security market line
a.

Graphically illustrates the CAPM.

b.

Designates the risk-return trade-off existing in the market,
where risk is defined in terms of beta according to the CAPM
equation.

146

ANSWERS TO
END-OF-CHAPTER QUESTIONS
6-1.

Data have been compiled by Ibbotson and Sinquefield on the actual returns for the
following portfolios of securities from 1926-2002.
1.

U.S. Treasury bills

2.

U.S. government bonds

3.

Corporate bonds

4.

Common stocks for large firms

5.

Common stocks for small firms

Investors historically have received greater returns for greater risk-taking with the
exception of the U.S. government bonds. Also, the only portfolio with returns
consistently exceeding the inflation rate has been common stocks.

6.2

When a rate of interest is quoted, it is generally the nominal or, observed rate. The
real rate of interest represents the rate of increase in actual purchasing power, after
adjusting for inflation. Consequently, the nominal rate of interest is equal to the
sum of the real rate of interest, the inflation rate, and the product of the real rate and
the inflation rate.

6-3

The relationship between a debt security’s rate of return and the length of time until
the debt matures is known as the term structure of interest rates or the yield to
maturity. In most cases, longer terms to maturity command higher returns or yields.

6-4.

(a)

The investor's required rate of return is the minimum rate of return
necessary to attract an investor to purchase or hold a security.

(b)

Risk is the potential variability in returns on an investment. Thus, the
greater the uncertainty as to the exact outcome, the greater is the risk. Risk
may be measured in terms of the standard deviation or by the variance term,
which is simply the standard deviation squared.

(c)

A large standard deviation of the returns indicates greater riskiness
associated with an investment. However, whether the standard deviation is
large relative to the returns has to be examined with respect to other
investment opportunities. Alternatively, probability analysis is a meaningful
approach to capture greater understanding of the significance of a standard
deviation figure. However, we have chosen not to incorporate such an
analysis into our explanation of the valuation process.

(a)

Unique risk is the variability in a firm's stock price that is associated with
the specific firm and not the result of some broader influence. An employee
strike is an example of a company-unique influence.

(b)

Systematic risk is the variability in a firm's stock price that is the result of
general influences within the industry or resulting from overall market or
economic influences. A general change in interest rates charged by banks is
an example of systematic risk.

6-5.

147


6-6.

Beta indicates the responsiveness of a security's returns to changes in the market
returns. Beta is multiplied by the market risk premium and added to the risk-free

rate of return to calculate a required rate of return.

6-7.

The security market line is a graphical representation of the risk-return trade-off
that exists in the market. The line indicates the minimum acceptable rate of return
for investors given the level of risk. Since the security market line results from
actual market transactions, the relationship not only represents the risk-return
preferences of investors in the market but also represents the investors' available
opportunity set.

6-8.

The beta for a portfolio is equal to the weighted average of the individual stock
betas, weighted by the percentage invested in each stock.

6-9.

If a stock has a great amount of variability about its characteristic line (the graph of
the stock's returns against the market's returns), then it has a high amount of
unsystematic or company-unique risk. If, however, the stock's returns closely
follow the market movements, then there is little unsystematic risk.

SOLUTIONS TO
END-OF-CHAPTER PROBLEMS
Solutions to Problems Set A
6-1A.
krf = .045 + .073 + (.045 x .073)
krf = .1213
or

12.13% = nominal rate of interest
6-2A.
krf = .064 + .038 + (.064 x .038)
krf = .1044
or
10.44% = nominal rate of interest

148


6-3A.
(A)
Probability
P(ki)
.15
.30
.40
.15

(B)
Return
(ki)
-1%
2
3
8

(A) x (B)
Expected Return
k

-.15%
0.60%
1.20%
1.20%
2.85%
2
k=



Weighted
Deviation
(ki - k )2P(ki)
2.223%
0.217%
0.009%
3.978%
= 6.427%
=

2.535%

No, Pritchard should not invest in the security. The level of risk is excessive for a
return which is less than the rate offered on treasury bills.
6-4A.
Common Stock A:
(A)
Probability
P(ki)
0.3

0.4
0.3

(B)
Return
(ki)
11%
15
19

(A) x (B)
Expected Return
k
3.3%
6.0
5.7
2
k = 15.0%



Weighted
Deviation
(ki - k )2P(ki)
4.8%
0.0
4.8
= 9.6%
=

3.10%

Common Stock B
(A)
Probability
P(ki)
0.2
0.3
0.3
0.2

(B)
Return
(ki)
-5%
6
14
22

(A) x (B)
Weighted
Expected Return
Deviation
(ki - k )2P(ki)
k
-1.0%
41.472%
1.8
3.468
4.2

6.348
4.4
31.752
2 = 83.04%
k = 9.4%
 = 9.11%

Common Stock A is better. It has a higher expected return with less risk.

149


6-5A.
Common Stock A:
(A)
Probability
P(ki)
0.2
0.5
0.3

(B)
Return
(ki)
- 2%
18
27

(A) x (B)
Weighted

Expected Return
Deviation
(ki - k )2P(ki)
k
-0.4%
69.9%
9.0
0.8
8.1
31.8
2
=
16.7%
 = 102.5%

k

 =

10.12%

Common Stock B:
(A)
Probability
P(ki)
0.1
0.3
0.4
0.2

(B)
Return
(ki)
4%
6
10
15
k

(A) x (B)
Weighted
Expected Return
Deviation
(ki - k )2P(ki)
k
0.4%
2.704%
1.8
3.072
4.0
0.256
3.0
6.728
2
=
9.2%
 =
12.76%

 =

Common Stock A
k = 16.7%
 = 10.12%

3.57%

Common Stock B
k = 9.2%
 = 3.57%

We cannot say which investment is "better." It would depend on the investor's
attitude toward the risk-return tradeoff.
6-6A.
(a)

= + Beta
= 6 % + 1.2 (16% - 6%)
= 18%

(b)

The 18 percent "fair rate" compensates the investor for the time value of
money and for assuming risk. However, only nondiversifiable risk is being
considered, which is appropriate.

6-7A. Eye balling the characteristic line for the problem, the rise relative to the run is
about 0.5. That is, when the S & P 500 return is eight percent Aram's expected
return would be about four percent. Thus, the beta is also approximately 0.5 (4 ÷
8).

150


6-8A.
A
B
C
D
6-9A.`=

+
+
+
+
+

6.75%
6.75%
6.75%
6.75%
+

(12%
(12%
(12%
(12%

-

x

x
x
x
x

6.75%)
6.75%)
6.75%)
6.75%)

Beta =
1.50 =
0.82 =
0.60 =
1.15 =

14.63%
11.06%
9.90%
12.79%

(Market Return - Risk-Free Rate) X Beta
=

7.5% + (11.5% - 7.5%) x 0.765

=

10.56%

6-10A. If the expected market return is 12.8 percent and the risk premium is 4.3 percent, the
riskless rate of return is 8.5 percent (12.8% - 4.3%). Therefore;
Tasaco

=

8.5% + (12.8% - 8.5%) x 0.864 = 12.22%

LBM

=

8.5% + (12.8% - 8.5%) x 0.693 = 11.48%

Exxos

=

8.5% + (12.8% - 8.5%) x 0.575 = 10.97%

6-11A.
Asman
Time
1
2
3
4

Price
$10

12
11
13

Salinas
Return

20.00%
-8.33
18.18

Price
$30
28
32
35

Return
-6.67%
14.29
9.38

A holding-period return indicates the rate of return you would earn if you bought a
security at the beginning of a time period and sold it at the end of the period, such
as the end of the month or year.

151


6-12A.a.

Month

kb

1
2
3
4
5
6
Sum

Zemin
(kb - k )2

6.00%
3.00
1.00
-3.00
5.00
0.00
12.00

16.00%
1.00
1.00
25.00
9.00
4.00
56.00

kb

Market
(kb - k )2

4.00%
2.00
-1.00
-2.00
2.00
2.00
7.00

2.00%

1.17%

24.00%

14.04%

8.03%
0.69
4.69
10.03
0.69
0.69
24.82

(Sum ÷ 6)

Variance
(Sum  5)

11.20%
3.35%

b.

=

4.97%
2.23%

+ (Market Return - Risk-Free Rate) X Beta
=

8%

+

[(14% - 8%) X 1.54]

= 17.24%

c.

Zemin's historical return of 24 percent exceeds what we would consider a
fair return of 17.24 percent, given the stock's systematic risk.

a.

The portfolio expected return, k p, equals a weighted average of the
individual stock's expected returns.

6-13A.

kp

=

(0.20)(16%) + (0.30)(14%) + (0.15)(20%) + (0.25)(12%) +
(0.10)(24%)

=

15.8%

152


b.

The portfolio beta, ßp, equals a weighted average of the individual stock
betas
ßp

c.

=

(0.20)(1.00) + (0.30)(0.85) + (0.15)(1.20) + (0.25)(0.60) +
(0.10)(1.60)

=

0.95

Plot the security market line and the individual stocks

25.00

5
3

Expected Return

20.00
P 1
M
2

15.00
4

10.00
5.00
0.00
0.00

0.50

1.00

1.50

2.00

Beta
d.

A "winner" may be defined as a stock that falls above the security market
line, which means these stocks are expected to earn a return exceeding what
should be expected given their beta or systematic risk. In the above graph,
these stocks include 1, 3, and 5. "Losers" would be those stocks falling
below the security market line, which are represented by stocks 2 and 4 ever
so slightly.

e.

Our results are less than certain because we have problems estimating the
security market line with certainty. For instance, we have difficulty in
specifying the market portfolio.

153


6-14A a.
Market

Month

Price

kt

Jul-02
Aug-02
Sep-02
Oct-02
Nov-02
Dec-02
Jan-03
Feb-03
Mar-03
Apr-03
May-03
Jun-03
Jul-03

1328.72
1320.41
1282.71
1362.93
1388.91
1469.25
1394.46
1366.42
1498.58
1452.43

1420.60
1454.60
1430.83

Sum

Mathews
(kt - k )2

-0.63%
-2.86%
6.25%
1.91%
5.78%
-5.09%
-2.01%
9.67%
-3.08%
-2.19%
2.39%
-1.63%

0.0002
0.0013
0.0031
0.0001
0.0026
0.0034
0.0007
0.0080

0.0014
0.0008
0.0003
0.0005

8.52%

0.0225

Price

kt

34.50
41.09
37.16
38.72
38.34
41.16
49.47
56.50
65.97
63.41
62.34
66.84
66.75

(kt - k )2

19.10%

-9.56%
4.20%
-0.98%
7.36%
20.19%
14.21%
16.76%
-3.88%
-1.69%
7.22%
-0.13%
72.79%

0.0170
0.0244
0.0003
0.0050
0.0002
0.0199
0.0066
0.0114
0.0099
0.0060
0.0001
0.0038
0.1048

b)
Average monthly return
Standard deviation

0.71%

6.07%
4.52%

9.76%

c)
s
25.00% Mathew

20.00%
15.00%
10.00%
5.00%

-10.00%

0.00%
-5.00%
0.00%
-5.00%
-10.00%
-15.00%

154

Market Index
5.00%

10.00%

15.00%


d.

Mathews returns seem to correlate to the market returns during the majority
of the year, but show great volatility.

6-15A
Stock 1
(A)
Probability
P(ki)
0.15
0.40
0.30
0.15

(B)
Return
(ki)
2%
7
10
15

(A) x (B)

Weighted
Expected Return
Deviation
(ki - k )2P(ki)
k
0.30%
6.048%
2.80
0.729
3.00
0.817
2.25
6.633
2
8.35%  = 14.227%
k =
 =
3.77%

(B)
Return
(ki)
-3%
20
25

(A) x (B)
Weighted
Expected Return
Deviation

2
(k
k
i - k ) P(ki)
-0.75%
85.56%
10.00
10.13
6.25
22.56
2
 = 118.25%
k = 15.50%
 =
10.87%

(B)
Return
(ki)
-5%
10
15
30

(A) x (B)
Weighted
Expected Return
Deviation
(k
k

i - k )2P(ki)
-0.50%
36.1%
4.00
6.4
4.50
0.3
6.00
51.2
2
 = 94.0%
k = 14.00%
 =
9.7%

Stock 2
(A)
Probability
P(ki)
0.25
0.50
0.25

Stock 3
(A)
Probability
P(ki)
0.10
0.40
0.30

0.20

We cannot say which investment is "better." It would depend on the investor's
attitude toward the risk-return tradeoff.

155


6-16A
H
T
P
W

+
+
+
+
+

5.5%
5.5%
5.5%
5.5%

(11%
(11%
(11%
(11%

-

x
x
x
x
x

5.5%)
5.5%)
5.5%)
5.5%)

Beta
0.75
1.40
0.95
1.25

6-17A
Williams
Time
1
2
3
4

Price
$33
27

35
39

Return
-18.18%
29.63
11.43

Davis
Price
$19
15
14
23

Return
-21.05%
-6.67
64.29

6-18A
(a)

= + Beta
= 5 % + 1.2 (9% - 5%)
= 9.8%

(b)

= + Beta

= 5 % + 0.85 (9% - 5%)
= 8.4%

(c)

If beta is 1.2:
Required rate
of return

= 5 % + 1.2 (12% - 5%)
= 13.4%

If beta is 0.85:
Required rate
of return

= 5 % + 0.85 (12% - 5%)
= 10.95%

156

=
=
=
=
=

9.63%
13.20%
10.73%

12.38%


SOLUTION TO INTEGRATIVE PROBLEM
1.

Holding-period returns for Market, Reynolds Computer, and Andrews
Price

01May
June
July
Aug
Sept
Oct
Nov
Dec
02Jan
Feb
Mar
Apr
May
June
July
Aug
Sept
Oct
Nov
Dec
03Jan

Febr
Mar
Apr
May
Sum

2.

Market
kt
(kt - k )2

1090.82
1133.84
3.94%
1120.67 -1.16%
957.28 -14.58%
1017.01
6.24%
1098.67
8.03%
1163.63
5.91%
1229.23
5.64%
1279.64
4.10%
1238.33 -3.23%
1286.37
3.88%

1335.18
3.79%
1301.84 -2.50%
1372.71
5.44%
1328.72 -3.20%
1320.41 -0.63%
1282.71 -2.86%
1362.93
6.25%
1388.91 1.91%
1469.25 5.78%
1394.46 -5.09%
1366.42 -2.01%
1498.58
9.67%
1452.43 -3.08%
1420.60 -2.19%
30.07%

Average
Monthly
Return
Standard
Deviation

0.0007
0.0006
0.0251
0.0025

0.0046
0.0022
0.0019
0.0008
0.0020
0.0007
0.0006
0.0014
0.0018
0.0020
0.0004
0.0017
0.0025
0.0000
0.0021
0.0040
0.0011
0.0071
0.0019
0.0012
.0689

Reynolds Computer
Price
kt
(kt - k )2 Price
20.60
23.20
27.15
25.00

32.88
32.75
30.41
36.59
50.00
40.06
40.88
41.19
34.44
37.00
40.88
48.81
41.81
40.13
43.00
51.00
38.44
40.81
53.94
50.13
43.13

12.62%
17.03%
-7.92%
31.52%
-0.40%
-7.15%
20.32%
36.65%

-19.88%
2.05%
0.76%
-16.39%
7.43%
10.49%
19.40%
-14.34%
-4.02%
7.15%
18.60%
-24.63%
6.17%
32.17%
-7.06%
-13.96%
106.62%

0.0067
0.0158
0.0153
0.0733
0.0023
0.0134
0.0252
0.1037
0.0592
0.0006
0.0014
0.0434

0.0009
0.0037
0.0224
0.0353
0.0072
0.0007
0.0201
0.0845
0.0003
0.0769
0.0132
0.0339

24.00
26.72
20.94
15.78
18.09
21.69
23.06
28.06
26.03
26.44
28.06
36.94
36.88
37.56
23.25
22.88
24.78

27.19
26.56
24.25
32.00
35.13
44.81
30.23
34.00

Andrews
kt
(kt - k )2
11.33%
-21.63%
-24.64%
14.64%
19.90%
6.32%
21.68%
-7.23%
1.58%
6.13%
31.65%
-0.16%
1.84%
-38.10%
-1.59%
8.30%
9.73%
-2.32%

-8.70%
31.96%
9.78%
27.55%
-32.54%
12.47%
77.95%

1.25%

4.44%

3.25%

5.47%

16.93%

18.60%

157

0.0065
0.0619
0.0778
0.0130
0.0277
0.0009
0.0340
0.0110

0.0003
0.0008
0.0806
0.0012
0.0002
0.1710
0.0023
0.0026
0.0042
0.0031
0.0143
0.0824
0.0043
0.0591
0.1281
0.0085
.7958


3.

158


Reynolds vs Market

0.4
0.3
0.2

Market

0.1
0
-0.2

-0.1

0

0.1

0.2

-0.1
-0.2
-0.3
Reynolds
Andrews vs. Market
0.4
0.3
0.2

Andrews

0.1
0
-0.2

-0.1

-0.1

0

-0.2
-0.3
-0.4
-0.5
Marke t

159

0.1

0.2


4

Reynolds’s returns have a great amount of volatility with some correlation to the
market returns.
The same can be said of Andrews. The returns show a great amount of volatility
that followed the market returns only part of the time.

5.

Monthly returns of a portfolio of equal amounts of Reynolds and Andrews.

2001 June

July
August
September
October
November
December
2002 January
February
March
April
May
June
July
August
September
October
November
December
2003 January
February
March
April
May
Average
return
Standard
deviation

160

Monthly
Returns
11.98%
-2.32%
-16.27%
23.08%
9.74%
-0.41%
21.02%
14.70%
-9.16%
4.09%
16.20%
-8.28%
4.65%
-13.81%
8.90%
-3.00%
2.84%
2.43%
4.95%
3.66%
7.97%
29.87%
-19.80%
-0.75%
3.84%
12.29%

6.
Reynolds and Andrews
40.00%

50% Reynolds 50% Andrews

30.00%

20.00%

10.00%

0.00%
-20.00%

-10.00%

0.00%

10.00%

20.00%

-10.00%

-20.00%

-30.00%
Market

We see in this new graph where both stocks are included as a single portfolio that the
relationship of the stocks with the market approximates an average of the relationships
taken alone. Note the reduction in volatility that occurs when risk is diversified even
between just two stocks.

161


7.

Monthly holding-period returns for long-term government bonds
(ki - k )2
2001 June
5.70%
0.48%
0.000000%
July
5.68%
0.47%
0.000001%
August
5.54%
0.46%
0.000004%
September
5.20%
0.43%
0.000023%
October
5.01%

0.42%
0.000041%
November
5.25%
0.44%
0.000020%
December
5.06%
0.42%
0.000036%
2002 January
5.16%
0.43%
0.000027%
February
5.37%
0.45%
0.000012%
March
5.58%
0.47%
0.000003%
April
5.55%
0.46%
0.000004%
May
5.81%
0.48%
0.000000%

June
6.04%
0.50%
0.000005%
July
5.98%
0.50%
0.000003%
August
6.07%
0.51%
0.000006%
September
6.07%
0.51%
0.000006%
October
6.26%
0.52%
0.000016%
November
6.15%
0.51%
0.000009%
December
6.35%
0.53%
0.000022%
2003 January
6.63%

0.55%
0.000050%
February
6.23%
0.52%
0.000014%
March
6.05%
0.50%
0.000005%
April
5.85%
0.49%
0.000000%
May
6.15%
0.51%
0.000009%
Average
Monthly
Return

0.48%

Standard
Deviation

0.04%

162

8.

Monthly portfolio returns when portfolio consists of equal amounts invested in
Reynolds, Andrews, and long-term government bonds.

2001 June
July
August
September
October
November
December
2002 January
February
March
April
May
June
July
August
September
October
November
December
2003 January
February
March
April

May
Sum

8.14%
-1.39%
-10.69%
15.53%
6.63%
-0.13%
14.15%
9.94%
-5.95%
2.88%
10.95%
-5.36%
3.27%
-9.04%
6.10%
-1.83%
2.07%
1.79%
3.48%
2.63%
5.49%
20.08%
-13.04%
-0.33%
65.36%

Average Monthly

Return

(ki - k )2
0.0029
0.0017
0.0180
0.0164
0.0015
0.0008
0.0131
0.0052
0.0075
0.0000
0.0068
0.0065
0.0000
0.0138
0.0011
0.0021
0.0000
0.0001
0.0001
0.0000
0.0008
0.0301
0.0248
0.0009
0.1542

2.72%

Std. Dev..

8.19%

163


9.

Comparison of average returns and standard deviations
Average
Returns
4.44%
3.25%
0.48%
3.84%
2.72%

Reynolds
Andrews
Government security
Reynolds & Andrews
Reynolds, Andrews,
& government security
Market

1.25%

Standard

Deviations
16.93%
18.60%
0.04%
12.29%
8.19%
5.47%

From the findings above, we see that higher average returns are associated with
higher risk (standard deviations), and that by diversification we can reduce risk,
possibly without reducing the average return.
10.

Based on the standard deviations, Andrews has more risk than Reynolds, 18.60
percent standard deviation versus 16.93 percent standard deviation. However, when
we only consider systematic risk, Andrews is slightly less risky--Reynolds's beta is
1.96 compared to Andrews’ beta of 1.49. (The betas given here for Reynolds and
Andrews come from financial services who calculate firms' betas. These are not
consistent with the graphs above where we see Andrews' returns as being more
responsive to the general market. We are seeing the problem of using only 24
months of returns as we have done.)

11.

= + (Market Return - Risk-Free Rate) X Beta
Market Return = 1.25 % Average Monthly Return X 12 Months = 15%.
(The average returns for the market over a two-year period may be high or low
relative to the longer-term past, and as a result should not be considered as
“typical” investor expectations. For instance, if we used information from Ibbotson
& Sinquefield for the years 1926-2002, the market risk premium—market return

less risk-free rate—was 8.4 percent, and not the 19 percent that we use below. The
point: Do not think two years fairly captures what we can expect in the future?)
Reynolds:
23.64% = 6% + (15% - 6%) X 1.96
Andrews:
19.41% = 6% + (15% - 6%) X 1.49
And if we used the market premium of 8.4 percent:
Reynolds:
22.46% = 6% + 8.4% X 1.96
Andrews:
18.52% = 6% + 8.4% X 1.49

164


Solutions to Problem Set B
6-1B.
krf = .05 + .07 + (.05 x .07)
krf = .1235
or
12.35% = nominal rate of interest
6-2B.
krf = .03 + .05 + (.03 x .05)
krf = .0815
or
8.15% = nominal rate of interest
6-3B.
(A)
Probability
P(ki)

.15
.30
.40
.15

(B)
Return
(ki)
-3%
2
4
6

(A) x (B)
Weighted
Expected Return
Deviation
2
(k
k
i - k ) P(ki)
-0.45%
4.788
0.60
0.127
1.60
0.729
0.90
1.683
2

 =
7.327%
k = 2.65%
 =
2.707%

No, Gautney should not invest in the security. The security’s expected rate of
return is less than the rate offered on treasury bills.
6-4B.
Security A:
(A)
Probability
P(ki)
0.2
0.5
0.3

(B)
Return
(ki)
- 2%
19
25

(A) x (B)
Weighted
Expected Return
Deviation
2
(k

k
i - k ) P(ki)
-0.4%
69.19%
9.5
2.88
7.5
21.17
16.6%
2 = 93.24%
k =
 =
9.66%

165


Security B:
(A)
Probability
P(ki)
0.1
0.3
0.4
0.2

(B)
Return
(ki)
5%

7
12
14

(A) x (B)
Weighted
Expected Return
Deviation
(ki - k )2P(ki)
k
0.5%
2.704%
2.1
3.072
4.8
1.296
2.8
2.888
2
 =
9.96%
k = 10.2%
 =
3.16%

Security A
k = 16.6%
 = 9.66%

Security B

k = 10.2%
 = 3.16%

We cannot say which investment is "better." It would depend on the investor's
attitude toward the risk-return tradeoff.
6-5B.
Common Stock A:
(A)
Probability
P(ki)
0.2
0.6
0.2

(B)
Return
(ki)
10%
13
20

(A) x (B)
Expected Return
k
2.0%
7.8
4.0
k = 13.8%

Weighted

Deviation
(ki - k )2P(ki)
2.89%
0.38
7.69
2
 = 10.96%

 = 3.31%
Common Stock B
(A)
Probability
P(ki)
0.15
0.30
0.40
0.15

(B)
Return
(ki)
6%
8
15
19

(A) x (B)
Weighted
Expected Return
Deviation

(ki - k )2P(ki)
k
0.9%
5.67%
2.4
5.17
6.0
3.25
2.85
7.04
2
 =
21.13%
k = 12.15%
 =
4.60%
Common Stock A is better. It has a higher expected return with less risk.

166


6-6B.
(a)

= + Beta
= 8 % + 1.5 (16% - 8%)
= 20%

(b)

The 20 percent "fair rate" compensates the investor for the time value of
money and for assuming risk. However, only nondiversifiable risk is being
considered, which is appropriate.

6-7B. Eye balling the characteristic line for the problem, the rise relative to the run is
about 1.75. That is, when the S & P 500 return is four percent Bram's expected
return would be about seven percent. Thus, the beta is also approximately 1.75 (7 ÷
4).
6-8B.
A
B
C
D
6-9B.

6.75%
6.75%
6.75%
6.75%

+
+
+
+
+

(12%
(12%
(12%
(12%

-

6.75%)
6.75%)
6.75%)
6.75%)

x
x
x
x
x

Beta
1.40
0.75
0.80
1.20

=

+ (Market Return - Risk-Free Rate) X Beta

=

7.5% + (10.5% - 7.5%) x 0.85

=
=

=
=
=

14.10%
10.69%
10.95%
13.05%

=
10.05%
6-10B. If the expected market return is 12.8 percent and the risk premium is 4.3 percent,
the riskless rate of return is 8.5 percent (12.8% - 4.3%). Therefore;
Dupree
= 8.5% + (12.8% - 8.5%) x 0.82 = 12.03%
Yofota
= 8.5% + (12.8% - 8.5%) x 0.57 = 10.95%
MacGrill = 8.5% + (12.8% - 8.5%) x 0.68 = 11.42%
6-11B.
O'Toole
Time Price
1
$22
2
24
3
20
4
25

Return
9.09%
-16.67%
25.00%

Baltimore
Price
Return
$45
50
11.11%
48
-4.00%
52
8.33%

A holding-period return indicates the rate of return you would earn if you bought a
security at the beginning of a time period and sold it at the end of the period, such
as the end of the month or year,

167


6-12B.
(a)

Sugita
kt
(kt - k )2
1.80%

0.01%
-0.50
5.68
2.00
0.01
-2.00
15.08
5.00
9.71
5.00
9.71
11.30
40.20

Month
1
2
3
4
5
6
Sum

Market
kt
(kt - k )2
1.50%
0.06%
1.00
0.06

0.00
1.56
-2.00
10.56
4.00
7.56
3.00
3.06
7.50
22.86

1.88%

1.25%

22.60%

15.00%

(Sum ÷ 6)

Variance
(Sum ÷ 5)

8.04%

4.58%

2.84%

2.14%

b.
= + (Market Return - Risk-Free Rate) X Beta
=

8%

+

[(15% - 8%) X 1.18] = 16.26%

c.

Sugita's historical return of 22.6 percent exceeds what we would consider a
fair return of 16.26 percent, given the stock's systematic risk.

a.

The portfolio expected return, k p, equals a weighted average of the
individual stock's expected returns.
(0.10)(12%) + (0.25)(11%) + (0.15)(15%) + (0.30)(9%) +
kp =
(0.20)(14%)

6-13B

=

11.7%

168