BUS 365: Investments
Solution to Practice Problems
Risk and Return
1) For this question, please assume that markets are efficient. BRIEFLY answer the following.
Suppose that a risk-averse investor wants to form a well-diversified portfolio of ten stocks. How might the investor go about choosing those stocks? What factors are important in that choice?
Answer: The investor should choose stocks that are not highly correlated with each other.
It may even be desirable to find stocks that are negatively correlated with each other.
Industry, type of product (retail, raw materials, luxury, necessity etc.), and geographic location are a few of the many factors the investor should consider.
2) Draw and label the Capital Market Line under the assumption that investors cannot short the risk-free asset.
Answer:
Expected
Return
Market
Portfolio
Risk-
Free
Rate
Standard
Deviatio n
3) Consider three major drug companies. Eli Lilly has a beta of 0.31; Johnson & Johnson has a beta of 0.24; Merck has a beta of 0.35. Currently, you have invested $5,000 in Merck and you do not want to change that position. You have an additional $20,000 to invest and wish to form a market-neutral portfolio of the three stocks. a) How should you allocate the $20,000 between Eli Lilly stock and Johnson & Johnson stock?

Answer: We want w
LLY

0.31 + w
JNJ

0.24 + w
MRK

0.35 = 0 for a market-neutral portfolio.
Since we have $5,000 invested in Merck, we know that w
MRK
= $5,000/$25,000 = 0.2. We also know that w
LLY
+ w
JNJ
+ w
MRK
= 1, or w
LLY
+ w
JNJ
+ 0.2 = 1. This gives w
LLY
= 0.8w
JNJ
. Substituting this into our original equation gives
(0.8-w
JNJ
)

0.31 + w
JNJ

0.24 + 0.2

0.35 = 0
Solving this gives w
JNJ
= 4.543 and w
LLY
= -3.743. Thus we would need to purchase
4.543

$25,000 = $113,571 worth of Johnson and Johnson stock, and short 3.743

$25,000 =
$93,571 worth of Eli Lilly stock. b) Suppose that the risk-free rate is 4% and that the expected market return is 9%. What is the expected return on your portfolio?
Answer: A beta of zero gives an expected return equal to the risk-free rate, so our expected return would be 4%.
4) The following graph depicts the expected returns and standard deviations of 5 different assets.
You are a risk-averse investor who plans to invest in one (and only one) of those assets. You have no other information on the assets and no way to get additional information on them.
Which assets would you definitely NOT choose? BRIEFLY explain your intuition. What investor-specific factor would help you determine which of the remaining assets you would choose? BRIEFLY explain.
Expected
Return
A
E
C
B
D
Risk
(Standard Deviation)
Answer: Asset E has a higher expected return and lower risk than assets C and D. Asset A has a higher expected return and lower risk than assets B and C. We would therefore not choose B, C, or D. The decision to choose asset A or asset E would be based on our level of risk aversion, with more risk averse people choosing asset E and less risk averse people choosing asset A.

5) You have $100,000 to invest and wish to invest $50,000 in each of two stocks. Assuming that you want to diversify as much as possible and that you have no other investments, how would you go about choosing the two companies? What company characteristics are important in the choice? Give an example of two companies that you might choose.
Answer: The greatest diversification will come from holding two negatively-correlated stocks and/or two stocks that are well-diversified themselves. Important factors include industry, cyclical nature, sensitivity to market movements, etc. You might choose, for example, a bank and a collection agency. Alternatively, you might choose Tyco and GE, which are both highly diversified companies. As such, they have less firm-specific risk than the typical stock.
6) BRIEFLY discuss the implication(s) of using historical return data to estimate beta for a stock.
Answer: We know that historical data is not always a good predictor of the future. For stocks, this is true in several respects. First, approximating the market portfolio is difficult and may lead to error. Second, betas are not stationary (i.e., they aren’t, on average, constant over time). In fact, we know that betas tend toward one over time, so we should adjust for that tendency. Finally, it is difficult to estimate beta for privately-held stocks because we do not have market data on the stock’s price.
7) Assume that the CAPM is correct and applicable to markets. Assume also that markets are equilibrium. Explain the collapse of Enron’s stock in light of the two primary sources of stock value--expected cash flows and risk.
Answer: Criminal misrepresentation by firm managers is diversifiable risk, so the discount rate estimated using the CAPM probably did not increase greatly. Rather, the collapse was due to a dramatic decrease in expected cash flows to shareholders.
8) Consider three major drug companies. Eli Lilly has a beta of 0.31; Johnson & Johnson has a beta of 0.24; Merck has a beta of 0.35. Currently, you have invested $5,000 in Johnson &
Johnson and you do not want to change that position. You have an additional $25,000 to invest and wish to form a market-neutral portfolio of the three stocks. a) How should you allocate the $25,000?
We desire a portfolio beta of zero. We know immediately that we will have to short a stock.
Otherwise, the weighted average of the positive betas would also be positive.
(5,000/30,000)

0.24 + (X
M
/30,000)

0.35 + (X
L
/30,000)

0.31 = 0
Substituting X
M
+ X
L
= $25,000 and solving gives X
L
= $248,750 and X
M
= -$223,750.

b) Suppose that the risk-free rate is 3.5% and that the expected market return is 7.5%. What is the expected return on your portfolio?
According to the CAPM, the expected return would be 3.5% since beta would be zero. Of course, we would only form the portfolio in the first place if we had higher expectations.
9) Consider the graph below. What (approximately) is the beta of the stock? BRIEFLY explain your logic.
40.00%
30.00%
20.00%
10.00%
0.00%
-10.00%
-20.00%
-30.00%
-40.00%
-50.00%
-60.00%
Market Return
Stock Return
Time
Answer: The stock beta is about 1.5. Notice first that a strong positive correlation exists between the market and the stock. Notice also that when the market moves X%, the stock tends to move about 1.5 times that amount. This is, roughly speaking, what beta is.
10) BRIEFLY evaluate the validity of the following argument. “The CAPM is largely useless because it relies on beta, which in turn relies on volatility. For example, a stock may go way up during a period in which the market goes up by a smaller amount. Our estimate of beta would be higher as a result. Therefore, this highly successful stock would be deemed to be more risky even though it has done really well!”

Answer: If we simply calculating the historical beta and blindly use it in the CAPM formula, the statement is entirely valid (this is Buffett's criticism of the CAPM). However, the intelligent investor computes the beta and carefully considers the possibility that the historical calculation is biased, either by firm-specific events or by significant changes in the nature of the company. After making adjustments for these things (albeit subjective ones in some cases), the CAPM can be successfully applied.
11) a) In simple terms, explain what a discount rate is. In doing so, be sure to mention the specific factors that should (in theory) affect the discount rate.
Answer: A discount rate is the minimum return needed to satisfy investors. Said differently, it is the return that would exactly compensate investors for the level of systematic risk associated with the investment. Said differently, it is the investor's opportunity cost (i.e., the expected return on assets of similar risk). In theory, the discount rate is determined entirely by the risk-free rate, the expected market return, and the level of systematic risk (

).
b) How might we go about estimating one for use in valuing a stock?
Answer: The appropriate discount rate for valuing a stock is the cost of equity, which can be estimated using the CAPM.
c) How might we go about estimating one for use in valuing a company?
Answer: The appropriate discount rate for valuing a company (i.e., stock+preferred stock+debt) is the Weighted Average Cost of Capital (WACC), which can be estimated using the company's capital structure in conjunction with the costs of debt, preferred stock, and common stock.
12) BRIEFLY explain how the Weighted Average Cost of Capital is used in financial analysis and BRIEFLY outline how to estimate the various components of it.
Answer: The WACC is used as a hurdle rate when assessing a potential investment in a company. Most often, the concept is applied by discounting the expected free cash flows of the firm at the firm’s WACC, which gives us an estimate of the value of the firm. The various components and how we might estimate them are as follows.
W d
Weight on debt
R
T d
Required return on debt
Tax rate
Ratio of the market value of the company’s debt to the total of the market values of the company’s debt, preferred stock, and equity.
We often use the yield-to-maturity of the company’s outstanding debt.
We can estimate the tax rate from the

company’s financial statements, but 35% is the IRS mandated rate for companies with taxable income over $1,000,000
W ps
Weight on preferred stock Ratio of the market value of the company’s preferred stock to the total of the market values of the company’s debt, preferred stock, and equity.
R ps
Required return on preferred stock We typically use the ratio of the
W e
Weight on equity promised dividend to the current market price.
Ratio of the market value of the company’s equity to the total of the market values of the company’s debt, preferred stock, and equity.
We typically use the CAPM. R e
Required return on equity
13) In recent years, Stock A has suffered from a series of firm-specific events that have gradually driven the stock price down while the market was moving up. You wish to estimate the beta of
Stock A for use in discounting future cash flows. Below is a table showing recent historical betas for Stock A and its closest peers.
Stock Historical Beta
A 0.35
B
C
0.95
1.04
D
E
1.13
0.98
What beta would you use for Stock A? BRIEFLY explain your choice.
Answer: The historical beta of Stock A is unreliable because the data is tainted by the series of firm-specific events. Our best bet is probably to use the average of the peers, which is 1.025.
14) You wish to estimate the Weighted Average Cost of Capital for a company. Using the information found below, what is your best estimate of the company’s WACC?
Book value of debt $2,000,000
Book value of equity
Book value of preferred stock
Current market value of preferred stock (per share)
Expected market risk premium
Historical correlation between equity returns and market returns
Market value of debt
Market value of equity
Market value of preferred stock
$1,000,000
$500,000
$11.50
4.2%
0.64
$1,900,000
$6,300,000
$485,000

Promised annual dividend on preferred stock
Standard deviation of historical equity returns
Standard deviation of historical market returns
Yield-to-maturity on 5-year Treasury bonds
$1.15
53%
21%
4.8%
Yield-to-maturity on company’s existing debt
Tax Rate
Answer: w d
= $1,900,000/($1,900,000+$6,300,000+$485,000) = 0.219
R d
= 8.2% APR, or 1.041
2 -1 = 8.37% EAR
T = 35% w ps
= $485,000/($1,900,000+$6,300,000+$485,000) = 0.056
8.2%
35%
R ps
= $1.15/$11.50 = 10% w e
= $6,300,000/($1,900,000+$6,300,000+$485,000) = 0.725

= 0.64×0.53/0.21 = 1.62
R e
= 4.8%+1.62×4.2% = 11.6%
WACC = 0.219×8.37%×(1-0.35) + 0.056×10% + 0.725×11.6% = 10.16%
15) You wish to estimate an appropriate discount rate for use in valuing Stock B. Consider the chart found in the Problem Set, which depicts one year of weekly, historical stock prices
(normalized to $100 initial values) of the S&P 500 index, Stock A, and Stock B. Stock A and B are both very similar companies in the same industry. Stock A has had no major firm-specific developments over the period shown in the chart. In contrast, Stock B’s company has had a series of firm-specific problems during the year. What is your best estimate of the beta of Stock
B? BRIEFLY explain your intuition.
Answer: Firm-specific events can affect the historical beta of a company, thereby making it unusable as an estimate of the future beta of the stock. In this case, we cannot reasonably use the historical data for Stock B as a basis for estimating its future beta. Since Stock B is similar to Stock A, we might use the beta of Stock A as our estimate of the future beta for
Stock B. Looking at the chart, we see that the prices of Stock A are positively correlated with the market but move roughly twice the magnitude. It follows that Stock A will have a beta of roughly 2.
16) You wish to form a portfolio of three companies such that the portfolio has a beta of 1.0.
Stock A has a beta of 1.5. Stock B has a beta of 0.8. Stock C has a beta of 0.6. You currently have $50,000 invested in Stock A and you do not want to change that position. You have another
$100,000 to invest. a) What positions should you take in Stock B and Stock C?
Answer:
Since the investments in B and C must total $100,000, we have
($50/$150)×1.5 + ($X/$150)×0.8 + (($100-$X)/$150)×0.6,

where X is the amount invested in B. Solving gives X = $75,000. Therefore,
Investment in B = $75,000
Investment in C = $25,000 b) Suppose that the risk-free rate is 4.5% and that the expected market return is 8.5%. What is the expected return on each of the three positions? What is the expected return on your portfolio?
Answer:
Exp. return on A = 4.5%+1.5×(8.5%-4.5%) = 10.5%
Exp. return on B = 4.5%+0.8×(8.5%-4.5%) = 7.7%
Exp. return on C = 4.5%+0.6×(8.5%-4.5%) = 6.9%
Exp. portfolio return=($50/$150)×10.5%+($75/$150)×7.7%+($25/$150)×6.9%= 8.5%