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Dollar return (Profit) $60
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Poor Case 0.20 −4%
Most Likely 0.40 6%
Good Case 0.20 16%
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A B C D E F G H I J
1/6/2015
Amount invested $1,000
Worst Case 0.10 −14%
CHAPTER 2 MINI CASE
a. What are investment returns? What is the return on an investment that costs $1,000 and is sold after 1
year for $1,060?
b. Graph the probability distribution for the 5 scenarios during the next year for the 10-year zero coupon
bonds. What might the graph of the probability distribution look like if there were an infinite number of
scenarios (i.e., if it were a continuous distribution and not a discrete distribution)?
Assume that you recently graduated and landed a job as a financial planner with Cicero Services, an
investment advisory company. Your first client recently inherited some assets and has asked you to evaluate
them. The client presently owns a bond portfolio with $1 million invested in zero coupon Treasury bonds
that mature in 10 years. The client also has $2 million invested in the stock of Blandy, Inc., a company that
produces meat-and-potatoes frozen dinners. Blandy’s slogan is “Solid food for shaky times.”
Unfortunately, Congress and the President are engaged in an acrimonious dispute over the budget and the
Scenario
Return on a
10-Year
ZeroCoupon
Treasury
Bond During
Next Year
Probability of
Scenario
Discrete Probability Distribution for 5 Scenarios
0.4
0.4
0.5
Probability of
Scenario
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You have also gathered historical returns for the past 10 years for Blandy, Gourmange Corporation (a
producer of gourmet specialty foods), and the stock market.
The risk-free rate is 4% and the market risk premium is 5%.
portfolio. After consulting with the economists at your firm, you have specified 5 possible scenarios for the
resolution of the dispute at the end of the year. For each scenario, you have estimated the probability of the
scenario occurring and the impact on interest rates and bond prices if the scenario occurs. Given this
information, you have calculated the rate of return on 10-year zero coupon for each scenario. The
probabilities and returns are shown further below.
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Poor Case 0.20 −4% −0.8%
Most Likely 0.40 6% 2.4%
Good Case 0.20 16% 3.2%
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Worst Case 0.10 −14% −1.4% −20% 4.0% 0.4%
Poor Case 0.20 −4% −0.8% −10% 1.0% 0.2%
Most Likely 0.40 6% 2.4% 0% 0.0% 0.0%
Good Case 0.20 16% 3.2% 10% 1.0% 0.2%
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A B C D E F G H I J
Calculating Expected Returns
Inputs: Expected Return
Scenario
Rate of
Return
(2)
Product of
Probability and
Return
(1) x (2) = (3)
Worst Case 0.10 −14% −1.4%
Excel function for finding expected return of discrete events:
Calculating Expected Returns and Standard Deviations: Discrete Probabilities
Inputs: Expected Return
Scenario
Rate of
Return
(2)
Product of
Probability and
Return
(1) x (2) = (3)
Deviation from
Expected Return
(2) − Exp. r = (4)
Squared
Deviation
(4)2 = (5)
Sq. Dev. ×
Prob.
(1) x (5) = (6)
10.95%
Excel functions for finding expected return and standard deviation of discrete events
e. Your client has decided that the risk of the bond portfolio is acceptable and wishes to leave it as it is. Now
your client has asked you to use historical returns to estimate the standard deviation of Blandy’s stock
returns. (Note: Many analysts use 4 to 5 years of monthly returns to estimate risk and many use 52 weeks of
weekly returns; some even use a year or less of daily returns. For the sake of simplicity, use Blandy’s 10
annual returns.)
Continuous Probability Distribution for Infinite Number of Scenarios
d. What is stand-alone risk? Use the scenario data to calculate the standard deviation of the bond’s return for
the next year.
Std. Dev. = Square root of var. =
Standard Deviation
c. Use the scenario data to calculate the expected rate of return for the 10-year zero coupon Treasury bonds
during the next year.
0.12
0.14
0.16
0.18
Probability
Continuous Probability Distribution
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A B C D E F G H I J
Year Market Blandy Gourmange
130% 26% 47%
27% 15% -54%
Blandy Gourmange
Year Blandy Gourmange Portfolio
126% 47% 31.3%
215% -54% -2.3%
Notice that the historical returns for Blandy and Gourmange do not move in perfect lockstep.
g. Explain correlation to your client. Calculate the estimated correlation between Blandy and Gourmange.
Does this explain why the portfolio standard deviation was less than Blandy’s standard deviation?
Historical Stock Returns for Blandy and Gourmange
Stock Returns
e. Your client has decided that the risk of the bond portfolio is acceptable and wishes to leave it as it is. Now
your client has asked you to use historical returns to estimate the standard deviation of Blandy’s stock
returns. (Note: Many analysts use 4 to 5 years of monthly returns to estimate risk and many use 52 weeks of
weekly returns; some even use a year or less of daily returns. For the sake of simplicity, use Blandy’s 10
annual returns.)
f. Your client is shocked at how much risk Blandy stock has and would like to reduce the level of risk. You
suggest that the client sell 25% of the Blandy stock and create a portfolio with 75% Blandy stock and 25% in
the high-risk Gourmange stock. How do you suppose the client will react to replacing some of the Blandy
stock with high-risk stock? Show the client what the proposed portfolio return would have been in each of
year of the sample. Then calculate the s average return and standard deviation using the portfolio’s annual
returns. How does the risk of this two-stock portfolio compare with the risk of the individual stocks if they
were held in isolation?
Stock Returns
Gourmange
60%
75%
Rate of Return
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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: See Ch 02 Mini Case Show
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stock contributes to the standard deviation of a well-diversified portfolio.
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rRF The risk-free rate. It varies over time, but is constant for all firms at a given time.
j. According to the Capital Asset Pricing model, what measures the amount of risk that an individual stock
contributes to a well-diversified portfolio? Define this measurement.
The SML predicts stock i’s required return to be:
Loosely speaking, correlation measures the tendency of two variables to move together.
h. Suppose an investor starts with a portfolio consisting of one randomly selected stock. As more and more
randomly selected stocks are added to the portfolio, what happens to the portfolio’s risk?
The market risk premium. It is the amount above and beyond the risk-free rate that
k. What is the Security Market Line? How is beta related to a stock’s required rate of return?
The SML shows the relationship between the stock’s beta and its required return, as predicted by the CAPM.
i. (1.) Should portfolio effects impact the way investors think about the risk of individual stocks? Answer:
See Ch 02 Mini Case Show
(2.) If you decided to hold a 1-stock portfolio and consequently were exposed to more risk than diversified
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318% -14% 15%
610% -18% 40%
726% 42% 17%
Standard deviation of returns: 20.1% 25.2% 38.6%
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A B C D E F G H I J
Year Market Blandy Gourmange
130% 26% 47%
27% 15% -54%
Risk-free rate = 4%
Market risk premium = 5%
Blandy Gourmange
Stock Returns of Blandy and the Market: Estimating Beta
Beta can also be calculated as the slope of a regression of the stock (on the y-axis) and the market (on the x-
axis). This can be done using the SLOPE function or by plotting the returns and specifying that the chart show
the TRENDLINE.
Calculating Beta as the Slope of a Regression Using Excel Functions (See Excel explanations to right)
Stock Returns
l. Calculate the correction coefficient between Blandy and the market. Use this and the previously calculated
(or given) standard deviations of Blandy and the market to estimate Blandy’s beta. Does Blandy contribute
more or less risk to a well-diversified portfolio than does the average stock? Use the SML to estimate
Blandy’s required return. Assume that the risk-free rate is 4% and the market risk premium is 5%. Use the
SML to estimate Blandy’s required return.
m. Show how to estimate beta using regression analysis.
y = 0.6027x + 0.0158
R² = 0.2316
45.0%
y-axis: Historical
Blandy Returns
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Base Case
Higher Risk-
Free Rate
Higher Market
Risk Premium
Beta
SML: Base
Case
Base Case Risk-
Free Rate
SML: Higher Risk-
Free Rate
SML: Higher
Market Risk
Premium
Stock
Portfolio
Weight
Beta Weighted Beta
o. Your client decides to invest $1.4 million in Blandy stock and $0.6 million in Gourmange stock. What are
the weights for this portfolio? What is the portfolio’s beta? What is the required return for this portfolio?
Changes to Inputs for the Security Market Line
The Security Market Line shows the projected changes in expected return, due to changes in the beta
coefficient. However, we can also look at the potential changes in the required return due to variations in
other factors, for example the market return and risk-free rate. In other words, we can see how required
returns can be influenced by changing inflation and risk aversion. The level of investor risk aversion is
measured by the market risk premium (rM – rRF), which is also the slope of the SML. Hence, an increase in the
market return results in an increase in the maturity risk premium, other things held constant.
n. (1) Suppose interest rates go up by 3 percentage points over the current 4% risk-free rate. What effect
would higher interest rates have on the SML and on the returns required on high- and low-risk securities?
(2) Suppose instead that investors’ risk aversion increased enough to cause the market risk premium to
increase by 3 percentage points. (Assume the risk-free rate remains constant.) What effect would this have
on the SML and on returns of high- and low-risk securities?
SML: Base
Case
SML: Higher
Risk-Free Rate
12%
14%
16%
18%
Impact of Increase in Risk-Free Rate
SML: Base
SML: Higher
Market Risk
Premium
14%
16%
18%
20%
22%
Impact of Increase in Market Risk Premium
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A B C D E F G H I J
Alternative Approach to Find Required Return on Porfolio
Stock
Portfolio
Required Return Weighted Return
JJ CC
The required return on a portfolio is a weighted average of the required returns of the individual assets in
the portfolio.
Portfolio Manager
Additonal data for graph
p. Jordan Jones (JJ) and Casey Carter (CC) are portfolio managers at your firm. Each manages a well-
diversified portfolio. Your boss has asked for your opinion regarding their performance in the past year. JJ’s
portfolio has a beta of 0.6 and had a return of 8.5%; CC’s portfolio has a beta of 1.4 and had a return of 9.5%.
Which manager had better performance? Why?
q. What does market equilibrium mean? If equilibrium does not exist, how will it be established? Answer:
r. What is the Efficient Markets Hypothesis (EMH) and what are its three forms? What evidence supports the
SML
12%
14%
16%
Performance Evaluation
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