# prac riskret sol by 8869Er

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```									                                    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.

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 wLLY0.31 + wJNJ0.24 + wMRK0.35 = 0 for a market-neutral portfolio.
Since we have \$5,000 invested in Merck, we know that wMRK = \$5,000/\$25,000 = 0.2. We
also know that wLLY + wJNJ + wMRK = 1, or wLLY + wJNJ + 0.2 = 1. This gives wLLY = 0.8-
wJNJ. Substituting this into our original equation gives
(0.8-wJNJ)0.31 + wJNJ0.24 + 0.20.35 = 0
Solving this gives wJNJ = 4.543 and wLLY = -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

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
A
A

E                     B
A
A

C
D            A

A

Risk
(Standard Deviation)
A

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 + (XM/30,000)0.35 + (XL/30,000)0.31 = 0

Substituting XM + XL = \$25,000 and solving gives XL = \$248,750 and XM = -\$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

40.00%
30.00%
20.00%
10.00%
0.00%
Return

Market Return
-10.00%
Stock Return
-20.00%
-30.00%
-40.00%
-50.00%
-60.00%
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.

Wd     Weight on debt                            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.
Rd     Required return on debt                   We often use the yield-to-maturity of the
company’s outstanding debt.
T      Tax rate                                  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
Wps    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.
Rps    Required return on preferred stock        We typically use the ratio of the
promised dividend to the current market
price.
We     Weight on equity                          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.
Re     Required return on equity                 We typically use the CAPM.

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            0.95
C            1.04
D            1.13
E            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                                                  \$1,000,000
Book value of preferred stock                                           \$500,000
Current market value of preferred stock (per share)                       \$11.50
Historical correlation between equity returns and market returns            0.64
Market value of debt                                                  \$1,900,000
Market value of equity                                                \$6,300,000
Market value of preferred stock                                         \$485,000
Promised annual dividend on preferred stock                                 \$1.15
Standard deviation of historical equity returns                              53%
Standard deviation of historical market returns                              21%
Yield-to-maturity on 5-year Treasury bonds                                  4.8%
Yield-to-maturity on company’s existing debt                                8.2%
Tax Rate                                                                     35%

wd = \$1,900,000/(\$1,900,000+\$6,300,000+\$485,000) = 0.219
Rd = 8.2% APR, or 1.0412-1 = 8.37% EAR
T = 35%
wps = \$485,000/(\$1,900,000+\$6,300,000+\$485,000) = 0.056
Rps = \$1.15/\$11.50 = 10%
we = \$6,300,000/(\$1,900,000+\$6,300,000+\$485,000) = 0.725
 = 0.64×0.53/0.21 = 1.62
Re = 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

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?
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?