Answers and Solutions: 6 – 1
Chapter 6
Risk and Return
ANSWERS TO END-OF-CHAPTER QUESTIONS
6-1 a. Stand-alone risk is only a part of total risk and pertains to the risk an investor takes by
holding only one asset. Risk is the chance that some unfavorable event will occur.
For instance, the risk of an asset is essentially the chance that the asset’s cash flows
will be unfavorable or less than expected. A probability distribution is a listing, chart
or graph of all possible outcomes, such as expected rates of return, with a probability
assigned to each outcome. When in graph form, the tighter the probability
distribution, the less uncertain the outcome.
d. The standard deviation (σ) is a statistical measure of the variability of a set of
observations. The variance (σ2) of the probability distribution is the sum of the
squared deviations about the expected value adjusted for deviation.
e. A risk averse investor dislikes risk and requires a higher rate of return as an
inducement to buy riskier securities. A realized return is the actual return an investor
receives on their investment. It can be quite different than their expected return.
Answers and Solutions: 6 – 2
coefficient (ρ) of +1.0 means that the two variables move up and down in perfect
synchronization, while a coefficient of -1.0 means the variables always move in
opposite directions. A correlation coefficient of zero suggests that the two variables
are not related to one another; that is, they are independent.
k. The beta coefficient is a measure of a stock’s market risk, A stock with a beta greater
than 1 has stock returns that tend to be higher than the market when the market is up
but tend to be below the market when the market is down. The opposite is true for a
stock with a beta less than 1..
n. Equilibrium is the condition under which the expected return on a security is just
equal to its required return,
r
= r, and the market price is equal to the intrinsic value.
The Efficient Markets Hypothesis (EMH) states (1) that stocks are always in
equilibrium and (2) that it is impossible for an investor to consistently “beat the
market.” In essence, the theory holds that the price of a stock will adjust almost
o. The Fama-French 3-factor model has one factor for the excess market return (the
market return minus the risk free rate), a second factor for size (defined as the return
on a portfolio of small firms minus the return on a portfolio of big firms), and a third
factor for the book–to-market effect (defined as the return on a portfolio of firms with
a high book-to-market ratio minus the return on a portfolio of firms with a low book–
to-market ratio).
p. Most people don’t behave rationally in all aspects of their personal lives, and
behavioral finance assumes that investors have the same types of psychological
6-2 a. The probability distribution for complete certainty is a vertical line.
b. The probability distribution for total uncertainty is the X axis from – to +.
6-3 Security A is less risky if held in a diversified portfolio because of its lower beta and
6-4 The risk premium on a high beta stock would increase more.
RPj = Risk Premium for Stock j = (rM – rRF)bj.
6-5 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
SOLUTIONS TO END-OF-CHAPTER PROBLEMS
6-1 Investment Beta
$20,000 0.7
6-2 rRF = 4%; rM = 12%; b = 0.8; rs = ?
6-3 rRF = 5%; RPM = 7%; rM = ?
rM = 5% + (7%)1 = 12% = rs when b = 1.0.
rs when b = 1.7 = ?
6-5
r
= (0.1)(-50%) + (0.2)(-5%) + (0.4)(16%) + (0.2)(25%) + (0.1)(60%)
= 11.40%.
σ2 = (-50% – 11.40%)2(0.1) + (-5% – 11.40%)2(0.2) + (16% – 11.40%)2(0.4)
Answers and Solutions: 6 – 6
6-6 a.
r
m= (0.3)(15%) + (0.4)(9%) + (0.3)(18%) = 13.5%.
6-7 a. rA = rRF + (rM – rRF)bA
12% = 5% + (10% – 5%)bA
12% = 5% + 5%(bA)
7% = 5%(bA)
6-8 a. ri = rRF + (rM – rRF)bi = 5% + (12% – 5%)1.4 = 14.8%.
b. 1. rRF increases to 6%:
c. 1. rM increases to 14%:
ri = rRF + (rM – rRF)bi = 5% + (14% – 5%)1.4 = 17.6%.
Answers and Solutions: 6 – 8
6-9 Old portfolio beta =
(b) +
(0.8)
Alternative Solutions:
1. Old portfolio beta = 1.2 = (0.0667)b1 + (0.0667)b2 +…+ (0.0667)b20
1.2 = (bi)(0.0667)
bi = 1.2/0.0667 = 18.0.
Answers and Solutions: 6 – 9
6-10 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)
Stock Investment Beta r = rRF + (rM – rRF)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
rp = 18%(0.10) + 2%(0.15) + 16%(0.25) + 12%(0.50) = 12.1%.
6-12 We know that bR = 1.50, bS = 0.75, rM = 13%, rRF = 7%.
ri = rRF + (rM – rRF)bi = 7% + (13% – 7%)bi.
rR = 7% + 6%(1.50) = 16.0%
rS = 7% + 6%(0.75) = 11.5
4.5%
6-13 The answers to a, b, and c are given below:
Answers and Solutions: 6 – 10
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 (ρAB =
-0.13).
6-14 a. bX = 1.3471; bY = 0.6508. These can be calculated with a spreadsheet.
b. rX = 6% + (5%)1.3471 = 12.7355%.
Answers and Solutions: 6 – 12
SOLUTION TO SPREADSHEET PROBLEM
6-14 The detailed solution for the spreadsheet problem is available in the file Ch06-P15 Build
a Model Solution.xls at the textbook’s Web site.
Mini Case: 6 – 13
Assume that you recently graduated with a major in finance, and you just landed a job as a
financial planner with Barney Smith 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 new
business the client plans to start 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 investment
alternatives shown in the table below. (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- Alta Repo Am. Market 2-stock
economy Prob. Bills Inds Men Foam 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
MINI CASE
Mini Case: 6 – 14
Barney Smith’s economic forecasting staff has developed probability estimates for the
state of the economy, and its security analysts have developed a sophisticated computer
program that was used to estimate the rate of return on each alternative under each state
of the economy. Alta Industries is an electronics firm; Repo Men collects past-due debts;
and American Foam manufactures mattresses and various other foam products. Barney
Smith also maintains an “index fund” which owns a market-weighted fraction of all
publicly traded stocks; you can invest in that fund, and thus obtain average stock market
results. Given the situation as described, answer the following questions.
a. What are investment returns? What is the return on an investment that costs
$1,000 and is sold after one year for $1,100?
Answer: Investment return measures the financial results of an investment. They may be
b. 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: 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.
Mini Case: 6 – 15
b. 2. Why are Alta Ind.’s returns expected to move with the economy whereas Repo
Men’s are expected to move counter to the economy?
Answer: Alta Industries’ 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
c. Calculate the expected rate of return on each alternative and fill in the blanks on
the row for
r
in the table.
Answer: The expected rate of return,
r
, is expressed as follows:
Mini Case: 6 – 16
d. You should recognize that basing a decision solely on expected returns is
appropriate only for risk-neutral individuals. Because 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.
Answer: The standard deviation is calculated as follows:
d. 2. What type of risk is measured by the standard deviation?
Answer: The standard deviation is a measure of a security’s (or a portfolio’s) stand-alone risk.
Mini Case: 6 – 17
d. 3. Draw a graph that shows roughly the shape of the probability distributions for
Alta Industries, American Foam, and T-bills.
Answer:
Mini Case: 6 – 18
e. 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 in the row for
CV in the table. Does the CV produce the same risk rankings as the standard
deviation?
Answer: 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.
f. Suppose you created a 2-stock portfolio by investing $50,000 in Alta Industries
and $50,000 in Repo Men.
1. Calculate the expected return (
p
r
), the standard deviation (σp), and the
coefficient of variation (CVp) for this portfolio and fill in the appropriate blanks
in the table.
Mini Case: 6 – 19
Answer: 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
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 σp
with the σ formula. Given the distribution of returns for the portfolio, we can
Mini Case: 6 – 20
f. 2. How does the risk of this 2-stock portfolio compare with the risk of the
individual stocks if they were held in isolation?
Answer: Using either σ or CV as our stand-alone risk measure, the stand-alone risk of the
g. Suppose an investor starts with a portfolio consisting of one randomly selected
stock. What would happen (1) to the risk 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.
Answer:
Density
Portfolio of stocks
with rp= 16%
Density
Portfolio of stocks
with rp= 16%
Mini Case: 6 – 21
h. 1. Should portfolio effects impact the way investors think about the risk of
individual stocks?
Answer: 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
Mini Case: 6 – 22
h. 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: 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
i. How is market risk measured for individual securities? How are beta
coefficients calculated?
Answer: Market risk, which is relevant for stocks held in well-diversified portfolios, is defined
j. Suppose you have the following historical returns for the stock market and for
another company, P.Q. Unlimited. Explain how to calculate beta, and use the
historical stock returns to calculate the beta for PQU. Interpret your results.
YEAR
MARKET
PQU
1
25.7%
40.0%
2
8.0%
-15.0%
3
-11.0%
-15.0%
4
15.0%
35.0%
5
32.5%
10.0%
6
13.7%
30.0%
7
40.0%
42.0%
8
10.0%
-10.0%
9
-10.8%
-25.0%
10
-13.1%
25.0%
Answer: Betas are calculated as the slope of the “characteristic” line, which is the regression
line showing the relationship between a given stock and the general stock market.
Show the graph with the regression results. Point out that the beta is the slope
coeeficient, which is 0.83. State that an average stock, by definition, moves with the
market. Beta coefficients measure the relative volatility of a given stock relative to
the stock market. The average stock’s beta is 1.0. Most stocks have betas in the
Mini Case: 6 – 24
k. The expected rates of return and the beta coefficients of the alternatives as
supplied by Barney Smith’s computer program are as follows:
Security Return (
r
) Risk (Beta)
Alta Inds 17.4% 1.30
Market 15.0 1.00
Am. Foam 13.8 0.89
T-Bills 8.0 0.00
Repo Men 1.7 (0.87)
(1) Do the expected returns appear to be related to each alternative’s market risk?
(2) Is it possible to choose among the alternatives on the basis of the information
developed thus far?
Answer: The expected returns are related to each alternative’s market risk—that is, the higher
l. 1. Write out the Security Market Line (SML) equation, use it to calculate the
required rate of return on each alternative, and then graph the relationship
between the expected and required rates of return.
Answer: Here is the SML equation:
ri = rrf + (rm – rrf)bi.
10%
15%
20%
25%
Return
Market
Alta Inds.
l. 2. How do the expected rates of return compare with the required rates of return?
Answer: We have the following relationships:
Expected Required
Return Return
(Note: the plot looks somewhat unusual in that the x axis extends to the left of zero.
–10%
–5%
0%
5%
-3
-2
-1
0
1
2
3
Beta
Repo Men
T-Bills
Mini Case: 6 – 26
l. 3. Does the fact that Repo Men has an expected return that is less than the T-bill
rate make any sense?
Answer: Repo Men is an interesting stock. Its negative beta indicates negative market risk—
including it in a portfolio of “normal” stocks will lower the portfolio’s risk.
l. 4. What would be the market risk and the required return of a 50-50 portfolio of
Alta Industries and Repo Men? Of Alta Industries and American Foam?
Answer: Note that the beta of a portfolio is simply the weighted average of the betas of the
stocks in the portfolio. Thus, the beta of a portfolio with 50 percent Alta Inds and 50
percent Repo Men is:
Mini Case: 6 – 27
m. 1. Suppose investors raised their inflation expectations by 3 percentage points over
current estimates as reflected in the 8 percent T-bill rate. What effect would
higher inflation have on the SML and on the returns required on high– and low-
risk securities?
Answer:
m. 2. Suppose instead that investors’ risk aversion increased enough to cause the
market risk premium to increase by 3 percentage points. (Inflation remains
constant.) What effect would this have on the SML and on returns of high– and
low-risk securities?
Answer: When investors’ risk aversion increases, the SML is rotated upward about the y–
(%)rM
Web Appendix 6B
Calculating Beta Coefficients With a Financial Calculator
Solutions to Problems
6B-1 a.
b. Because b = 0.62, Stock Y is about 62% as volatile as the market; thus, its relative risk is
about 62% that of an average stock.
c. 1. Stand-alone risk as measured by would be greater, but beta and hence systematic
(relevant) risk would remain unchanged. However, in a 1-stock portfolio, Stock Y
would be riskier under the new conditions.
2. CAPM assumes that company-specific risk will be eliminated in a portfolio, so the
d. 1. The stock‘s variance and would not change, but the risk of the stock to an investor
holding a diversified portfolio would be greatly reduced, because it would now have
a negative correlation with rM.
(%)rY
40
30
e. The following figure shows a possible set of probability distributions. We can be
reasonably sure that the 100-stock portfolio comprised of b = 0.62 stocks as described in
Condition 2 will be less risky than the “market.” Hence, the distribution for Condition 2
will be more peaked than that of Condition 3.
Stock Y has b = 0.62, while the average stock (M) has b = 1.0; therefore,
ry = rRF + (rM – rRF)0.62 < rM = rRF + (rM – rRF)1.0.
A disequilibrium exists—Stock Y should be bid up to drive its yield down. More likely,
however, the data simply reflect the fact that past returns are not an exact basis for
expectations of future returns.
Web Solutions: 6B – 30
periods of transition, when the risk of the firm is changing, the beta can yield conclusions
that are exactly opposite to the actual facts. Once the company‘s risk stabilizes, the
calculated beta should rise and should again approximate the true beta.
6B-2 a.
The slope of the characteristic line is the stock’s beta coefficient.
Slope =
M
i
r
r
Run
Rise
=
.
The graph of the SML is as follows:
30
35
ri(%)
Stock A
30
35
ri(%)
Stock A
Web Solutions: 6B – 31
The equation of the SML is thus:
ri = rRF + (rM – rRF)bi = 9% + (14% – 9%)bi = 9% + (5%)bi.
c. Required rate of return on Stock A:
rA = rRF + (rM – rRF)bA = 9% + (14% – 9%)1.0 = 14%.
Required rate of return on Stock B:
rB = 9% + (14% – 9%)0.5 = 11.50%.
SML
ri
E(r M) = 14%
SML
ri
E(r M) = 14%
SML
ri
E(r M) = 14%E(r M) = 14%