Financial managment Solution Manual: Risk and Rates of Return

After reading this chapter, students should be able to:
• Define dollar return and rate of return.
• Define risk and calculate the expected rate of return, standard
deviation, and coefficient of variation for a probability distribution.
• Specify how risk aversion influences required rates of return.
• Graph diversifiable risk and market risk; explain which of these is
relevant to a well-diversified investor.
• State the basic proposition of the Capital Asset Pricing Model (CAPM)
and explain how and why a portfolio’s risk may be reduced.
• Explain the significance of a stock’s beta coefficient, and use the
Security Market Line to calculate a stock’s required rate of return.
• List changes in the market or within a firm that would cause the
required rate of return on a firm’s stock to change.
• Identify concerns about beta and the CAPM.
• Explain how stock price volatility is more likely to imply risk than
earnings volatility.
Learning Objectives: 5 - 1
Chapter 5
Risk and Rates of Return
LEARNING OBJECTIVES
Risk analysis is an important topic, but it is difficult to teach at the
introductory level. We just try to give students an intuitive overview of how
risk can be defined and measured, and leave a technical treatment to advanced
courses. Our primary goals are to be sure students understand (1) that
investment risk is the uncertainty about returns on an asset, (2) the concept
of portfolio risk, and (3) the effects of risk on required rates of return.
What we cover, and the way we cover it, can be seen by scanning
Blueprints, Chapter 5. For other suggestions about the lecture, please see
the “Lecture Suggestions” in Chapter 2, where we describe how we conduct our
classes.
DAYS ON CHAPTER: 3 OF 58 DAYS (50-minute periods)

Lecture Suggestions: 5 - 2
LECTURE SUGGESTIONS
5-1 a. The probability distribution for complete certainty is a vertical
line.
b. The probability distribution for total uncertainty is the X-axis from
-∞ to +∞.
5-2 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 more risky because σ
A
> σ
B
and CV
A
> CV
B
.
5-3 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 7 the value
of the portfolio would decline.
b. No, you would be subject to 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. The U.S. Treasury currently issues indexed bonds.

5-4 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 1-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:
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.
b. There is a perfect negative correlation between the returns on the
life insurance policy and the returns on the policyholder’s human
Answers and Solutions: 5 - 3
ANSWERS TO END-OF-CHAPTER QUESTIONS
capital. In fact, these events (death and future lifetime earnings
capacity) are mutually exclusive.
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.
5-5 The risk premium on a high-beta stock would increase more.
RP
j
= Risk Premium for Stock j = (k
M
- k
RF

)b
j
.
If risk aversion increases, the slope of the SML will increase, and so
will the market risk premium (k
M
- k
RF
). The product (k
M
- k
RF
)b
j
is the
risk premium of the jth stock. If b
j
is low (say, 0.5), then the
product will be small; RP
j
will increase by only half the increase in
RP
M
.
However, if b
j
is large (say, 2.0), then its risk premium will rise by
twice the increase in RP
M
.

5-6 According to the Security Market Line (SML) equation, an increase in
beta will increase a company’s expected return by an amount equal to the
market risk premium times the change in beta. For example, assume that
the risk-free rate is 6 percent, and the market risk premium is 5
percent. If the company’s beta doubles from 0.8 to 1.6 its expected
return increases from 10 percent to 14 percent. Therefore, in general,
a company’s expected return will not double when its beta doubles.
5-7 Yes, if the portfolio’s beta is equal to zero. In practice, however, it
may be impossible to find individual stocks that have a nonpositive
beta. In this case it would also be impossible to have a stock portfolio
with a zero beta. Even if such a portfolio could be constructed,
investors would probably be better off just purchasing Treasury bills,
or other zero beta investments.
5-8 No. For a stock to have a negative beta, its returns would have to
logically be expected to go up in the future when other stocks’ returns
were falling. Just because in one year the stock’s return increases
when the market declined doesn’t mean the stock has a negative beta. A
stock in a given year may move counter to the overall market, even
though the stock’s beta is positive.
Answers and Solutions: 5 - 4
5-1
k
ˆ
= (0.1)(-50%) + (0.2)(-5%) + (0.4)(16%) + (0.2)(25%) + (0.1)(60%)
= 11.40%.
σ
2
= (-50% - 11.40%)
2
(0.1) + (-5% - 11.40%)

2
(0.2) + (16% - 11.40%)
2
(0.4)
+ (25% - 11.40%)
2
(0.2) + (60% - 11.40%)
2
(0.1)
σ
2
= 712.44; σ = 26.69%.
CV =
11.40%
26.69%
= 2.34.
5-2 Investment Beta
$35,000 0.8
40,000 1.4
Total $75,000
b
p
= ($35,000/$75,000)(0.8) + ($40,000/$75,000)(1.4) = 1.12.
5-3 k
RF
= 5%; RP
M
= 6%; k
M
= ?

k
M
= 5% + (6%)1 = 11%.
k when b = 1.2 = ?
k = 5% + 6%(1.2) = 12.2%.
5-4 k
RF
= 6%; k
M
= 13%; b = 0.7; k = ?
k = k
RF
+ (k
M
- k
RF
)b
= 6% + (13% - 6%)0.7
= 10.9%.
5-5 a. k = 11%; k
RF
= 7%; RP
M
= 4%.
k = k
RF
+ (k
M
– k
RF

)b
11% = 7% + 4%b
4% = 4%b
b = 1.
Answers and Solutions: 5 - 5
SOLUTIONS TO END-OF-CHAPTER PROBLEMS
b. k
RF
= 7%; RP
M
= 6%; b = 1.
k = k
RF
+ (k
M
– k
RF
)b
k = 7% + (6%)1
k = 13%.
5-6 a.
∑
=
=
n
1i
ii
kPk
ˆ
.

Y
k
ˆ
= 0.1(-35%) + 0.2(0%) + 0.4(20%) + 0.2(25%) + 0.1(45%)
= 14% versus 12% for X.
b. σ =
∑
=
−
n
1i
i
2
i
P)k
ˆ
k(
.
2
X
σ
= (-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%.
σ
X
= 12.20% versus 20.35% for Y.
CV
X
= σ
X
/
k
ˆ
X
= 12.20%/12% = 1.02, while
CV
Y
= 20.35%/14% = 1.45.
If Stock Y is less highly correlated with the market than X, then it
might have a lower beta than Stock X, and hence be less risky in a
portfolio sense.
5-7 a. k
i
= k
RF
+ (k
M
- k
RF
)b

i
= 9% + (14% - 9%)1.3 = 15.5%.
b. 1. k
RF
increases to 10%:
k
M
increases by 1 percentage point, from 14% to 15%.
k
i
= k
RF
+ (k
M
- k
RF
)b
i
= 10% + (15% - 10%)1.3 = 16.5%.
2. k
RF
decreases to 8%:
k
M
decreases by 1%, from 14% to 13%.
k
i
= k
RF
+ (k

M
- k
RF
)b
i
= 8% + (13% - 8%)1.3 = 14.5%.
c. 1. k
M
increases to 16%:
k
i
= k
RF
+ (k
M
- k
RF
)b
i
= 9% + (16% - 9%)1.3 = 18.1%.
2. k
M
decreases to 13%:
k
i
= k
RF
+ (k
M
- k

RF
)b
i
= 9% + (13% - 9%)1.3 = 14.2%.
5-8 Old portfolio beta =
$150,000
$142,500
(b) +
$150,000
$7,500
(1.00)
1.12 = 0.95b + 0.05
1.07 = 0.95b
1.1263 = b.
New portfolio beta = 0.95(1.1263) + 0.05(1.75) = 1.1575 ≈ 1.16.
Alternative Solutions:
1. Old portfolio beta = 1.12 = (0.05)b
1
+ (0.05)b
2
+ + (0.05)b
20
1.12 =
∑
)b(
i
(0.05)
∑
i
b

= 1.12/0.05 = 22.4.
New portfolio beta = (22.4 - 1.0 + 1.75)(0.05) = 1.1575 ≈ 1.16.
2.
∑
i
b
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 b =
21.4/19 = 1.1263. The beta of the new portfolio is:
1.1263(0.95) + 1.75(0.05) = 1.1575 ≈ 1.16.
5-9 Portfolio beta =
$4,000,000
$400,000
(1.50) +
$4,000,000
$600,000
(-0.50)
+
$4,000,000
$1,000,000
(1.25) +
$4,000,000
$2,000,000
(0.75)
b
p
= (0.1)(1.5) + (0.15)(-0.50) + (0.25)(1.25) + (0.5)(0.75)
= 0.15 - 0.075 + 0.3125 + 0.375 = 0.7625.
k
p

= k
RF
+ (k
M
- k
RF
)(b
p
) = 6% + (14% - 6%)(0.7625) = 12.1%.
Alternative solution: First, calculate the return for each stock using
the CAPM equation [k
RF
+ (k
M
- k
RF
)b], and then calculate the weighted
average of these returns.
k
RF
= 6% and (k
M
- k
RF
) = 8%.
Answers and Solutions: 5 - 7
Stock Investment Beta k = k
RF
+ (k
M

- k
RF
)b Weight
A $ 400,000 1.50 18% 0.10
B 600,000 (0.50) 2 0.15
C 1,000,000 1.25 16 0.25
D 2,000,000 0.75 12 0.50
Total $4,000,000 1.00
k
p
= 18%(0.10) + 2%(0.15) + 16%(0.25) + 12%(0.50) = 12.1%.
5-10 We know that b
R
= 1.50, b
S
= 0.75, k
M
= 13%, k
RF
= 7%.
k
i
= k
RF
+ (k
M
- k
RF
)b
i

= 7% + (13% - 7%)b
i
.
k
R
= 7% + 6%(1.50) = 16.0%
k
S
= 7% + 6%(0.75) = 11.5


4.5 %
5-11
X
k
ˆ
= 10%; b
X
= 0.9; σ
X
= 35%.
Y
k
ˆ
= 12.5%; b
Y
= 1.2; σ
Y
= 25%.
k

RF
= 6%; RP
M
= 5%.
a. CV
X
= 35%/10% = 3.5. CV
Y
= 25%/12.5% = 2.0.
b. For diversified investors the relevant risk is measured by beta.
Therefore, the stock with the higher beta is more risky. Stock Y has
the higher beta so it is more risky than Stock X.
c. k
X
= 6% + 5%(0.9)
k
X
= 10.5%.
k
Y
= 6% + 5%(1.2)
k
Y
= 12%.
d. k
X
= 10.5%;
X
k
ˆ

= 10%.
k
Y
= 12%;
Y
k
ˆ
= 12.5%.
Stock Y would be most attractive to a diversified investor since its
expected return of 12.5% is greater than its required return of 12%.
e. b
p
= ($7,500/$10,000)0.9 + ($2,500/$10,000)1.2


= 0.6750 + 0.30


= 0.9750.
k
p
= 6% + 5%(0.975)
k
p
= 10.875%.
f. If RP
M
increases from 5% to 6%, the stock with the highest beta will
have the largest increase in its required return. Therefore, Stock Y
will have the greatest increase.

Check:
k
X
= 6% + 6%(0.9)


= 11.4%. Increase 10.5% to 11.4%.
k
Y
= 6% + 6%(1.2)


= 13.2%. Increase 12% to 13.2%.
5-12 k
RF
= k* + IP = 2.5% + 3.5% = 6%.
k
s
= 6% + (6.5%)1.7 = 17.05%.
5-13 Using Stock X (or any stock):
9% = k
RF
+ (k
M
– k
RF
)b
X
9% = 5.5% + (k
M

– k
RF
)0.8
(k
M
– k
RF
) = 4.375%.
5-14 In equilibrium:
k
J
=
J
k
ˆ
= 12.5%.
k
J
= k
RF
+ (k
M
- k
RF
)b
12.5% = 4.5% + (10.5% - 4.5%)b
b = 1.33.
5-15 b
HRI
= 1.8; b

LRI
= 0.6. No changes occur.
k
RF
= 6%. Decreases by 1.5% to 4.5%.
k
M
= 13%. Falls to 10.5%.
Now SML: k
i
= k
RF
+ (k
M
- k
RF
)b
i
.
k
HRI
= 4.5% + (10.5% - 4.5%)1.8 = 4.5% + 6%(1.8) = 15.3%
k
LRI
= 4.5% + (10.5% - 4.5%)0.6 = 4.5% + 6%(0.6) = 8.1%
Difference 7.2%
5-16 An index fund will have a beta of 1.0. If k
M
is 12.5 percent (given in
the problem) and the risk-free rate is 5 percent, you can calculate the

market risk premium (RP
M
) calculated as k
M
- k
RF
as follows:
k = k
RF
+ (RP
M
)b
12.5% = 5% + (RP
M
)1.0
7.5% = RP
M
.
Now, you can use the RP
M
, the k
RF
, and the two stocks’ betas to calculate
their required returns.
Answers and Solutions: 5 - 9
Bradford:
k
B
= k
RF

+ (RP
M
)b
= 5% + (7.5%)1.45
= 5% + 10.875%
= 15.875%.
Farley:
k
F
= k
RF
+ (RP
M
)b
= 5% + (7.5%)0.85
= 5% + 6.375%
= 11.375%.
The difference in their required returns is:
15.875% - 11.375% = 4.5%.
5-17 Step 1: Determine the market risk premium from the CAPM:


0.12 = 0.0525 + (k
M
- k
RF
)1.25
(k
M
- k

RF
) = 0.054.
Step 2: Calculate the beta of the new portfolio:
The beta of the new portfolio is ($500,000/$5,500,000)(0.75) +
($5,000,000/$5,500,000)(1.25) = 1.2045.
Step 3: Calculate the required return on the new portfolio:
The required return on the new portfolio is:
5.25% + (5.4%)(1.2045) = 11.75%.
5-18 After additional investments are made, for the entire fund to have an
expected return of 13%, the portfolio must have a beta of 1.5455 as shown
below:
13% = 4.5% + (5.5%)b
b = 1.5455.
Since the fund’s beta is a weighted average of the betas of all the
individual investments, we can calculate the required beta on the
additional investment as follows:
1.5455 =
0$25,000,00
00)(1.5)($20,000,0
+
0$25,000,00
X$5,000,000
1.5455 = 1.2 + 0.2X
0.3455 = 0.2X
X = 1.7275.
5-19 a. ($1 million)(0.5) + ($0)(0.5) = $0.5 million.
b. You would probably take the sure $0.5 million.
c. Risk averter.
d. 1. ($1.15 million)(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 individual’s 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 between each pair of stocks was a negative one, the
portfolio would be virtually riskless. Since r for stocks is
generally in the range of +0.6 to +0.7, investing in a portfolio
of stocks would definitely be an improvement over investing in the
single stock.
5-20 a.
k
ˆ
M
= 0.1(7%) + 0.2(9%) + 0.4(11%) + 0.2(13%) + 0.1(15%) = 11%.
k
RF
= 6%. (given)
Therefore, the SML equation is
k
i
= k
RF
+ (k
M
- k
RF
)b

i
= 6% + (11% - 6%)b
i
= 6% + (5%)b
i
.
b. First, determine the fund’s beta, b
F
. The weights are the percentage
of funds invested in each stock.
A = $160/$500 = 0.32
B = $120/$500 = 0.24
C = $80/$500 = 0.16
D = $80/$500 = 0.16
E = $60/$500 = 0.12
b
F
= 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.
Next, use b
F
= 1.8 in the SML determined in Part a:
F
k
ˆ
= 6% + (11% - 6%)1.8 = 6% + 9% = 15%.
c. k
N
= Required rate of return on new stock = 6% + (5%)2.0 = 16%.
An expected return of 15 percent on the new stock is below the 16

percent required rate of return on an investment with a risk of b =
2.0. Since k
N
= 16% >
k
ˆ
N
= 15%, the new stock should not be
purchased. The expected rate of return that would make the fund
indifferent to purchasing the stock is 16 percent.
Answers and Solutions: 5 - 11
5-21 The answers to a, b, c, and d are given below:
k
A
k
B
Portfolio
1998 (18.00%) (14.50%) (16.25%)
1999 33.00 21.80 27.40
2000 15.00 30.50 22.75
2001 (0.50) (7.60) (4.05)
2002 27.00 26.30 26.65
Mean 11.30 11.30 11.30
Std. Dev. 20.79 20.78 20.13
Coef. Var. 1.84 1.84 1.78
e. A risk-averse investor would choose the portfolio over either Stock A
or Stock B alone, since 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 (r
AB

= 0.88).
5-22 The detailed solution for the spreadsheet problem is available both on
the instructor’s resource CD-ROM and on the instructor’s side of South-
Western’s web site, .
Spreadsheet Problem: 5 - 13
SPREADSHEET PROBLEM
Merrill Finch Inc.
Risk and Return
5-23 ASSUME THAT YOU RECENTLY GRADUATED WITH A MAJOR IN FINANCE, AND YOU
JUST LANDED A JOB AS A FINANCIAL PLANNER WITH MERRILL FINCH INC., A
LARGE FINANCIAL SERVICES CORPORATION. YOUR FIRST ASSIGNMENT IS TO
INVEST $100,000 FOR A CLIENT. BECAUSE THE FUNDS ARE TO BE INVESTED
IN A BUSINESS AT THE END OF ONE YEAR, YOU HAVE BEEN INSTRUCTED TO
PLAN FOR A ONE-YEAR HOLDING PERIOD. FURTHER, YOUR BOSS HAS
RESTRICTED YOU TO THE FOLLOWING INVESTMENT ALTERNATIVES IN THE TABLE
BELOW, SHOWN WITH THEIR PROBABILITIES AND ASSOCIATED OUTCOMES.
(DISREGARD FOR NOW THE ITEMS AT THE BOTTOM OF THE DATA; YOU WILL FILL
IN THE BLANKS LATER.)
RETURNS ON ALTERNATIVE INVESTMENTS

ESTIMATED RATE OF RETURN

STATE OF THE T- HIGH COLLEC- U.S. MARKET 2-STOCK
ECONOMY PROB. BILLS TECH TIONS RUBBER PORTFOLIO
PORTFOLIO
RECESSION 0.1 8.0% -22.0% 28.0% 10.0%* -13.0%
3.0%
BELOW AVG 0.2 8.0 -2.0 14.7 -10.0 1.0
AVERAGE 0.4 8.0 20.0 0.0 7.0 15.0 10.0
ABOVE AVG 0.2 8.0 35.0 -10.0 45.0 29.0

BOOM 0.1 8.0 50.0 -20.0 30.0 43.0 15.0
k-HAT (
k
ˆ
) 1.7% 13.8% 15.0%
STD DEV (σ) 0.0 13.4 18.8 15.3
3.3
COEF OF VAR (CV) 7.9 1.4 1.0 0.3
BETA (b) -0.87 0.89
*NOTE THAT THE ESTIMATED RETURNS OF U.S. RUBBER DO NOT ALWAYS MOVE IN
THE SAME DIRECTION AS THE OVERALL ECONOMY. FOR EXAMPLE, WHEN THE
ECONOMY IS BELOW AVERAGE, CONSUMERS PURCHASE FEWER TIRES THAN THEY
WOULD IF THE ECONOMY WAS STRONGER. HOWEVER, IF THE ECONOMY IS IN A
FLAT-OUT RECESSION, A LARGE NUMBER OF CONSUMERS WHO WERE PLANNING TO
Integrated Case: 5 - 14
INTEGRATED CASE
PURCHASE A NEW CAR MAY CHOOSE TO WAIT AND INSTEAD PURCHASE NEW TIRES
FOR THE CAR THEY CURRENTLY OWN. UNDER THESE CIRCUMSTANCES, WE WOULD
EXPECT U.S. RUBBER’S STOCK PRICE TO BE HIGHER IF THERE IS A RECESSION
THAN IF THE ECONOMY WAS JUST BELOW AVERAGE.
MERRILL FINCH’S ECONOMIC FORECASTING STAFF HAS DEVELOPED
PROBABILITY ESTIMATES FOR THE STATE OF THE ECONOMY, AND ITS SECURITY
ANALYSTS HAVE DEVELOPED A SOPHISTICATED COMPUTER PROGRAM, WHICH WAS
USED TO ESTIMATE THE RATE OF RETURN ON EACH ALTERNATIVE UNDER EACH
STATE OF THE ECONOMY. HIGH TECH INC. IS AN ELECTRONICS FIRM;
COLLECTIONS INC. COLLECTS PAST-DUE DEBTS; AND U.S. RUBBER
MANUFACTURES TIRES AND VARIOUS OTHER RUBBER AND PLASTICS PRODUCTS.
MERRILL FINCH ALSO MAINTAINS A “MARKET PORTFOLIO” THAT OWNS A MARKET-
WEIGHTED FRACTION OF ALL PUBLICLY TRADED STOCKS; YOU CAN INVEST IN
THAT PORTFOLIO, AND THUS OBTAIN AVERAGE STOCK MARKET RESULTS. GIVEN

THE SITUATION AS DESCRIBED, ANSWER THE FOLLOWING QUESTIONS.
A. 1. WHY IS THE T-BILL’S RETURN INDEPENDENT OF THE STATE OF THE ECONOMY?
DO T-BILLS PROMISE A COMPLETELY RISK-FREE RETURN?
ANSWER: [SHOW S5-1 THROUGH S5-7 HERE.] THE 8 PERCENT T-BILL RETURN DOES NOT
DEPEND ON THE STATE OF THE ECONOMY BECAUSE THE TREASURY MUST (AND
WILL) REDEEM THE BILLS AT PAR REGARDLESS OF THE STATE OF THE ECONOMY.
THE T-BILLS ARE RISK-FREE IN THE DEFAULT RISK SENSE BECAUSE THE
8 PERCENT RETURN WILL BE REALIZED IN ALL POSSIBLE ECONOMIC STATES.
HOWEVER, REMEMBER THAT THIS RETURN IS COMPOSED OF THE REAL RISK-FREE
RATE, SAY 3 PERCENT, PLUS AN INFLATION PREMIUM, SAY 5 PERCENT. SINCE
THERE IS UNCERTAINTY ABOUT INFLATION, IT IS UNLIKELY THAT THE
REALIZED REAL RATE OF RETURN WOULD EQUAL THE EXPECTED 3 PERCENT. FOR
EXAMPLE, IF INFLATION AVERAGED 6 PERCENT OVER THE YEAR, THEN THE
REALIZED REAL RETURN WOULD ONLY BE 8% - 6% = 2%, NOT THE EXPECTED 3
PERCENT. THUS, IN TERMS OF PURCHASING POWER, T-BILLS ARE NOT
RISKLESS.
ALSO, IF YOU INVESTED IN A PORTFOLIO OF T-BILLS, AND RATES THEN
DECLINED, YOUR NOMINAL INCOME WOULD FALL; THAT IS, T-BILLS ARE
EXPOSED TO REINVESTMENT RATE RISK. SO, WE CONCLUDE THAT THERE ARE NO
TRULY RISK-FREE SECURITIES IN THE UNITED STATES. IF THE TREASURY
Integrated Case: 5 - 15
SOLD INFLATION-INDEXED, TAX-EXEMPT BONDS, THEY WOULD BE TRULY
RISKLESS, BUT ALL ACTUAL SECURITIES ARE EXPOSED TO SOME TYPE OF RISK.
A. 2. WHY ARE HIGH TECH’S RETURNS EXPECTED TO MOVE WITH THE ECONOMY WHEREAS
COLLECTIONS’ ARE EXPECTED TO MOVE COUNTER TO THE ECONOMY?
Integrated Case: 5 - 16
ANSWER: [SHOW S5-8 HERE.] HIGH TECH’S RETURNS MOVE WITH, HENCE ARE
POSITIVELY CORRELATED WITH, THE ECONOMY, BECAUSE THE FIRM’S SALES,
AND HENCE PROFITS, WILL GENERALLY EXPERIENCE THE SAME TYPE OF UPS AND
DOWNS AS THE ECONOMY. IF THE ECONOMY IS BOOMING, SO WILL HIGH TECH.

ON THE OTHER HAND, COLLECTIONS IS CONSIDERED BY MANY INVESTORS TO BE
A HEDGE AGAINST BOTH BAD TIMES AND HIGH INFLATION, SO IF THE STOCK
MARKET CRASHES, INVESTORS IN THIS STOCK SHOULD DO RELATIVELY WELL.
STOCKS SUCH AS COLLECTIONS ARE THUS NEGATIVELY CORRELATED WITH (MOVE
COUNTER TO) THE ECONOMY. (NOTE: IN ACTUALITY, IT IS ALMOST
IMPOSSIBLE TO FIND STOCKS THAT ARE EXPECTED TO MOVE COUNTER TO THE
ECONOMY. EVEN COLLECTIONS SHARES HAVE POSITIVE (BUT LOW) CORRELATION
WITH THE MARKET.)
B. CALCULATE THE EXPECTED RATE OF RETURN ON EACH ALTERNATIVE AND FILL IN
THE BLANKS ON THE ROW FOR
k
ˆ
IN THE TABLE ABOVE.
ANSWER: [SHOW S5-9 AND S5-10 HERE.] THE EXPECTED RATE OF RETURN,
k
ˆ
, IS
EXPRESSED AS FOLLOWS:
∑
=
=
n
1i
ii
kPk
ˆ
.
HERE P
i
IS THE PROBABILITY OF OCCURRENCE OF THE iTH STATE, k

i
IS THE
ESTIMATED RATE OF RETURN FOR THAT STATE, AND n IS THE NUMBER OF
STATES. HERE IS THE CALCULATION FOR HIGH TECH:
k
ˆ
HIGH TECH
= 0.1(-22.0%)

+

0.2(-2.0%)

+

0.4(20.0%)

+

0.2(35.0%)

+
0.1(50.0%)
= 17.4%.
WE USE THE SAME FORMULA TO CALCULATE k’S FOR THE OTHER ALTERNATIVES:


k
ˆ
T-BILLS

= 8.0%.
k
ˆ
COLLECTIONS
= 1.7%.
k
ˆ
U.S.RUBBER
= 13.8%.

k
ˆ
M
= 15.0%.
Integrated Case: 5 - 17
C. YOU SHOULD RECOGNIZE THAT BASING A DECISION SOLELY ON EXPECTED
RETURNS IS ONLY APPROPRIATE FOR RISK-NEUTRAL INDIVIDUALS. SINCE YOUR
CLIENT, LIKE VIRTUALLY EVERYONE, IS RISK AVERSE, THE RISKINESS OF
EACH ALTERNATIVE IS AN IMPORTANT ASPECT OF THE DECISION. ONE
POSSIBLE MEASURE OF RISK IS THE STANDARD DEVIATION OF RETURNS.
1. CALCULATE THIS VALUE FOR EACH ALTERNATIVE, AND FILL IN THE BLANK ON
THE ROW FOR σ IN THE TABLE ABOVE.
ANSWER: [SHOW S5-11 THROUGH S5-13 HERE.] THE STANDARD DEVIATION IS
CALCULATED AS FOLLOWS:
σ =
∑
=
−
n
1i

i
2
i
P)k
ˆ
k(
.
σ
HIGH TECH
= [(-22.0

-

17.4)
2
(0.1)

+

(-2.0

-

17.4)
2
(0.2)

+

(20.0

-
17.4)
2
(0.4)
+ (35.0 - 17.4)
2
(0.2) + (50.0 - 17.4)
2
(0.1)]½
=
401.4
= 20.0%.
HERE ARE THE STANDARD DEVIATIONS FOR THE OTHER ALTERNATIVES:
σ
T-BILLS
= 0.0%.
σ
COLLECTIONS
= 13.4%.
σ
U.S. RUBBER
= 18.8%.
σ
M
= 15.3%.
C. 2. WHAT TYPE OF RISK IS MEASURED BY THE STANDARD DEVIATION?
ANSWER: [SHOW S5-14 AND S5-15 HERE.] THE STANDARD DEVIATION IS A MEASURE OF
A SECURITY’S (OR A PORTFOLIO’S) STAND-ALONE RISK. THE LARGER THE
STANDARD DEVIATION, THE HIGHER THE PROBABILITY THAT ACTUAL REALIZED

RETURNS WILL FALL FAR BELOW THE EXPECTED RETURN, AND THAT LOSSES
RATHER THAN PROFITS WILL BE INCURRED.
Integrated Case: 5 - 18
C. 3. DRAW A GRAPH THAT SHOWS ROUGHLY THE SHAPE OF THE PROBABILITY
DISTRIBUTIONS FOR HIGH TECH, U.S. RUBBER, AND T-BILLS.
Integrated Case: 5 - 19
ANSWER:
ON THE BASIS OF THESE DATA, HIGH TECH IS THE MOST RISKY INVESTMENT,
T-BILLS THE LEAST RISKY.
D. SUPPOSE YOU SUDDENLY REMEMBERED THAT THE COEFFICIENT OF VARIATION
(CV) IS GENERALLY REGARDED AS BEING A BETTER MEASURE OF STAND-ALONE
RISK THAN THE STANDARD DEVIATION WHEN THE ALTERNATIVES BEING
CONSIDERED HAVE WIDELY DIFFERING EXPECTED RETURNS. CALCULATE THE
MISSING CVs, AND FILL IN THE BLANKS ON THE ROW FOR CV IN THE TABLE
ABOVE. DOES THE CV PRODUCE THE SAME RISK RANKINGS AS THE STANDARD
DEVIATION?
ANSWER: [SHOW S5-16 THROUGH S5-19 HERE.] THE COEFFICIENT OF VARIATION (CV)
IS A STANDARDIZED MEASURE OF DISPERSION ABOUT THE EXPECTED VALUE; IT
SHOWS THE AMOUNT OF RISK PER UNIT OF RETURN.
CV =
k
ˆ
σ
.
CV
T-BILLS
= 0.0%/8.0% = 0.0.
CV
HIGH TECH
= 20.0%/17.4% = 1.1.

CV
COLLECTIONS
= 13.4%/1.7% = 7.9.
CV
U.S. RUBBER
= 18.8%/13.8% = 1.4.
Integrated Case: 5 - 20
Probability of
Occurrence
Rate of Return (%)
T-Bills
U.S. Rubber
High Tech
-60 -45 -30 -15 0 15 30 45 60
CV
M
= 15.3%/15.0% = 1.0.
WHEN WE MEASURE RISK PER UNIT OF RETURN, COLLECTIONS, WITH ITS LOW
EXPECTED RETURN, BECOMES THE MOST RISKY STOCK. THE CV IS A BETTER
MEASURE OF AN ASSET’S STAND-ALONE RISK THAN σ BECAUSE CV CONSIDERS
BOTH THE EXPECTED VALUE AND THE DISPERSION OF A DISTRIBUTION A
SECURITY WITH A LOW EXPECTED RETURN AND A LOW STANDARD DEVIATION
COULD HAVE A HIGHER CHANCE OF A LOSS THAN ONE WITH A HIGH σ BUT A
HIGH
k
ˆ
.
E. SUPPOSE YOU CREATED A 2-STOCK PORTFOLIO BY INVESTING $50,000 IN HIGH
TECH AND $50,000 IN COLLECTIONS.
1. CALCULATE THE EXPECTED RETURN (

p
k
ˆ
), THE STANDARD DEVIATION (σ
p
), AND
THE COEFFICIENT OF VARIATION (CV
p
) FOR THIS PORTFOLIO AND FILL IN THE
APPROPRIATE BLANKS IN THE TABLE ABOVE.
ANSWER: [SHOW S5-20 THROUGH S5-23 HERE.] TO FIND THE EXPECTED RATE OF RETURN
ON THE TWO-STOCK PORTFOLIO, WE FIRST CALCULATE THE RATE OF RETURN ON
THE PORTFOLIO IN EACH STATE OF THE ECONOMY. SINCE WE HAVE HALF OF
OUR MONEY IN EACH STOCK, THE PORTFOLIO’S RETURN WILL BE A WEIGHTED
AVERAGE IN EACH TYPE OF ECONOMY. FOR A RECESSION, WE HAVE: k
p
=
0.5(-22%) + 0.5(28%) = 3%. WE WOULD DO SIMILAR CALCULATIONS FOR THE
OTHER STATES OF THE ECONOMY, AND GET THESE RESULTS:
STATE PORTFOLIO
RECESSION 3.0%
BELOW AVERAGE 6.4
AVERAGE 10.0
ABOVE AVERAGE 12.5
BOOM 15.0
NOW WE CAN MULTIPLY PROBABILITIES TIMES OUTCOMES IN EACH STATE TO
GET THE EXPECTED RETURN ON THIS TWO-STOCK PORTFOLIO, 9.6 PERCENT.
ALTERNATIVELY, WE COULD APPLY THIS FORMULA,
k = w
i

× k
i
= 0.5(17.4%) + 0.5(1.7%) = 9.6%,
Integrated Case: 5 - 21
WHICH FINDS k AS THE WEIGHTED AVERAGE OF THE EXPECTED RETURNS OF THE
INDIVIDUAL SECURITIES IN THE PORTFOLIO.
IT IS TEMPTING TO FIND THE STANDARD DEVIATION OF THE PORTFOLIO AS
THE WEIGHTED AVERAGE OF THE STANDARD DEVIATIONS OF THE INDIVIDUAL
SECURITIES, AS FOLLOWS:
σ
p
≠ w
i
(σ
i
) + w
j
(σ
j
) = 0.5(20%) + 0.5(13.4%) = 16.7%.
HOWEVER, THIS IS NOT CORRECT IT IS NECESSARY TO USE A DIFFERENT
FORMULA, THE ONE FOR σ THAT WE USED EARLIER, APPLIED TO THE TWO-STOCK
PORTFOLIO’S RETURNS.
THE PORTFOLIO’S σ DEPENDS JOINTLY ON (1) EACH SECURITY’S σ AND
(2) THE CORRELATION BETWEEN THE SECURITIES’ RETURNS. THE BEST WAY TO
APPROACH THE PROBLEM IS TO ESTIMATE THE PORTFOLIO’S RISK AND RETURN
IN EACH STATE OF THE ECONOMY, AND THEN TO ESTIMATE σ
p
WITH THE σ
FORMULA. GIVEN THE DISTRIBUTION OF RETURNS FOR THE PORTFOLIO, WE CAN

CALCULATE THE PORTFOLIO’S σ AND CV AS SHOWN BELOW:
σ
p
= [(3.0 - 9.6)
2
(0.1) + (6.4 - 9.6)
2
(0.2) + (10.0 - 9.6)
2
(0.4)
+ (12.5 - 9.6)
2
(0.2) + (15.0 - 9.6)
2
(0.1)]
½
= 3.3%.
CV
p
= 3.3%/9.6% = 0.34.
E. 2. HOW DOES THE RISKINESS OF THIS 2-STOCK PORTFOLIO COMPARE WITH THE
RISKINESS OF THE INDIVIDUAL STOCKS IF THEY WERE HELD IN ISOLATION?
ANSWER: [SHOW S5-24 THROUGH S5-27 HERE.] USING EITHER σ OR CV AS OUR STAND-
ALONE RISK MEASURE, THE STAND-ALONE RISK OF THE PORTFOLIO IS
SIGNIFICANTLY LESS THAN THE STAND-ALONE RISK OF THE INDIVIDUAL
STOCKS. THIS IS BECAUSE THE TWO STOCKS ARE NEGATIVELY CORRELATED
WHEN HIGH TECH IS DOING POORLY, COLLECTIONS IS DOING WELL, AND VICE
VERSA. COMBINING THE TWO STOCKS DIVERSIFIES AWAY SOME OF THE RISK
INHERENT IN EACH STOCK IF IT WERE HELD IN ISOLATION, i.e., IN A 1-
STOCK PORTFOLIO.

OPTIONAL QUESTION
Integrated Case: 5 - 22
DOES THE EXPECTED RATE OF RETURN ON THE PORTFOLIO DEPEND ON THE PERCENTAGE OF
THE PORTFOLIO INVESTED IN EACH STOCK? WHAT ABOUT THE RISKINESS OF THE
PORTFOLIO?
ANSWER: USING A SPREADSHEET MODEL, IT’S EASY TO VARY THE COMPOSITION OF THE
PORTFOLIO TO SHOW THE EFFECT ON THE PORTFOLIO’S EXPECTED RATE OF
RETURN AND STANDARD DEVIATION:
HIGH TECH PLUS COLLECTIONS
% IN HIGH TECH
k
ˆ
p
σ
p
0% 1.7% 13.4%
10 3.3 10.0
20 4.9 6.7
30 6.4 3.3
40 8.0 0.0
50 9.6 3.3
60 11.1 6.7
70 12.7 10.0
80 14.3 13.4
90 15.8 16.7
100 17.4 20.0
THE EXPECTED RATE OF RETURN ON THE PORTFOLIO IS MERELY A LINEAR
COMBINATION OF THE TWO STOCK’S EXPECTED RATES OF RETURN. HOWEVER,
PORTFOLIO RISK IS ANOTHER MATTER. σ
p

BEGINS TO FALL AS HIGH TECH AND
COLLECTIONS ARE COMBINED; IT REACHES ZERO AT 40 PERCENT HIGH TECH;
AND THEN IT BEGINS TO RISE. HIGH TECH AND COLLECTIONS CAN BE
COMBINED TO FORM A NEAR ZERO RISK PORTFOLIO BECAUSE THEY ARE VERY
CLOSE TO BEING PERFECTLY NEGATIVELY CORRELATED; THEIR CORRELATION
COEFFICIENT IS
-0.9998. (NOTE: UNFORTUNATELY, WE CANNOT FIND ANY ACTUAL STOCKS
WITH r = -1.0.)
F. SUPPOSE AN INVESTOR STARTS WITH A PORTFOLIO CONSISTING OF ONE
RANDOMLY SELECTED STOCK. WHAT WOULD HAPPEN (1) TO THE RISKINESS AND
(2) TO THE EXPECTED RETURN OF THE PORTFOLIO AS MORE AND MORE RANDOMLY
SELECTED STOCKS WERE ADDED TO THE PORTFOLIO? WHAT IS THE IMPLICATION
FOR INVESTORS? DRAW A GRAPH OF THE TWO PORTFOLIOS TO ILLUSTRATE YOUR
ANSWER.
Integrated Case: 5 - 23
ANSWER: [SHOW S5-28 AND S5-29 HERE.]
Density
0
Portfolio of Stocks
with k
p
= 16%
One Stock
16
%
THE STANDARD DEVIATION GETS SMALLER AS MORE STOCKS ARE COMBINED IN
THE PORTFOLIO, WHILE k
p
(THE PORTFOLIO’S RETURN) REMAINS CONSTANT.
THUS, BY ADDING STOCKS TO YOUR PORTFOLIO, WHICH INITIALLY STARTED AS

A
1-STOCK PORTFOLIO, RISK HAS BEEN REDUCED.
IN THE REAL WORLD, STOCKS ARE POSITIVELY CORRELATED WITH ONE
ANOTHER IF THE ECONOMY DOES WELL, SO DO STOCKS IN GENERAL, AND VICE
VERSA. CORRELATION COEFFICIENTS BETWEEN STOCKS GENERALLY RANGE FROM
+0.5 TO +0.7. A SINGLE STOCK SELECTED AT RANDOM WOULD ON AVERAGE
HAVE A STANDARD DEVIATION OF ABOUT 35 PERCENT. AS ADDITIONAL STOCKS
ARE ADDED TO THE PORTFOLIO, THE PORTFOLIO’S STANDARD DEVIATION
DECREASES BECAUSE THE ADDED STOCKS ARE NOT PERFECTLY POSITIVELY
CORRELATED. HOWEVER, AS MORE AND MORE STOCKS ARE ADDED, EACH NEW
STOCK HAS LESS OF A RISK-REDUCING IMPACT, AND EVENTUALLY ADDING
ADDITIONAL STOCKS HAS VIRTUALLY NO EFFECT ON THE PORTFOLIO’S RISK AS
MEASURED BY σ. IN FACT, σ STABILIZES AT ABOUT 20.4 PERCENT WHEN 40
OR MORE RANDOMLY SELECTED STOCKS ARE ADDED. THUS, BY COMBINING
STOCKS INTO WELL-DIVERSIFIED PORTFOLIOS, INVESTORS CAN ELIMINATE
ALMOST ONE-HALF THE RISKINESS OF HOLDING INDIVIDUAL STOCKS. (NOTE:
IT IS NOT COMPLETELY COSTLESS TO DIVERSIFY, SO EVEN THE LARGEST
INSTITUTIONAL INVESTORS HOLD LESS THAN ALL STOCKS. EVEN INDEX FUNDS
GENERALLY HOLD A SMALLER PORTFOLIO THAT IS HIGHLY CORRELATED WITH AN
Integrated Case: 5 - 24
INDEX SUCH AS THE S&P 500 RATHER THAN HOLD ALL THE STOCKS IN THE
INDEX.)
THE IMPLICATION IS CLEAR: INVESTORS SHOULD HOLD WELL-DIVERSIFIED
PORTFOLIOS OF STOCKS RATHER THAN INDIVIDUAL STOCKS. (IN FACT,
INDIVIDUALS CAN HOLD DIVERSIFIED PORTFOLIOS THROUGH MUTUAL FUND
INVESTMENTS.) BY DOING SO, THEY CAN ELIMINATE ABOUT HALF OF THE
RISKINESS INHERENT IN INDIVIDUAL STOCKS.
G. 1. SHOULD PORTFOLIO EFFECTS IMPACT THE WAY INVESTORS THINK ABOUT THE
RISKINESS OF INDIVIDUAL STOCKS?
ANSWER: [SHOW S5-30 HERE.] PORTFOLIO DIVERSIFICATION DOES AFFECT INVESTORS’

VIEWS OF RISK. A STOCK’S STAND-ALONE RISK AS MEASURED BY ITS σ OR
CV, MAY BE IMPORTANT TO AN UNDIVERSIFIED INVESTOR, BUT IT IS NOT
RELEVANT TO A WELL-DIVERSIFIED INVESTOR. A RATIONAL, RISK-AVERSE
INVESTOR IS MORE INTERESTED IN THE IMPACT THAT THE STOCK HAS ON THE
RISKINESS OF HIS OR HER PORTFOLIO THAN ON THE STOCK’S STAND-ALONE
RISK. STAND-ALONE RISK IS COMPOSED OF DIVERSIFIABLE RISK, WHICH CAN
BE ELIMINATED BY HOLDING THE STOCK IN A WELL-DIVERSIFIED PORTFOLIO,
AND THE RISK THAT REMAINS IS CALLED MARKET RISK BECAUSE IT IS PRESENT
EVEN WHEN THE ENTIRE MARKET PORTFOLIO IS HELD.
G. 2. IF YOU DECIDED TO HOLD A 1-STOCK PORTFOLIO, AND CONSEQUENTLY WERE
EXPOSED TO MORE RISK THAN DIVERSIFIED INVESTORS, COULD YOU EXPECT TO
BE COMPENSATED FOR ALL OF YOUR RISK; THAT IS, COULD YOU EARN A RISK
PREMIUM ON THAT PART OF YOUR RISK THAT YOU COULD HAVE ELIMINATED BY
DIVERSIFYING?
ANSWER: [SHOW S5-31 HERE.] IF YOU HOLD A ONE-STOCK PORTFOLIO, YOU WILL BE
EXPOSED TO A HIGH DEGREE OF RISK, BUT YOU WON’T BE COMPENSATED FOR
IT. IF THE RETURN WERE HIGH ENOUGH TO COMPENSATE YOU FOR YOUR HIGH
RISK, IT WOULD BE A BARGAIN FOR MORE RATIONAL, DIVERSIFIED INVESTORS.
THEY WOULD START BUYING IT, AND THESE BUY ORDERS WOULD DRIVE THE
PRICE UP AND THE RETURN DOWN. THUS, YOU SIMPLY COULD NOT FIND STOCKS
IN THE MARKET WITH RETURNS HIGH ENOUGH TO COMPENSATE YOU FOR THE
STOCK’S DIVERSIFIABLE RISK.
Integrated Case: 5 - 25