MØA 155 - Fall 2011
PROBLEM SET: Hand in 2
Exercise 1.
A levered corporation, firm XYZ, is going to take on a new project. It may choose either a high-risk project
or a low-risk project. The table below shows the value of the firm depending on the project it chooses and
the state of the economy. The firm has debt equal to $300.
State
Probability
Value of firm (dollars)
Low-risk project
Recession
Boom
0.4
0.6
600
900
High-risk project
Recession
Boom
0.4
0.6
300
1000
Which of the following statements is correct?
1. Bondholders prefer the low-risk project.
2. Shareholders are indifferent to either project.
3. Shareholders have an incentive to choose the high-risk project.
4. None of the above statements are correct.
5. I choose not to answer.
Exercise 2.
A company is evaluating a project, developing a new computer design. The project requires an investment
of 2 million. The revenues in later years depend on the price the computer achieves in the market. The
company calculates expected cash flows for the next 5 years as 550,000 per year. However, this assumes
that the company sells computers in all the next five years. The company has the option of closing down
production early if the computer price develops unfavourably.
The project is to be evaluated at a required rate of return of 10%.
Which of the following statements is correct?
1. The NPV of the project is above 84,933.
2. The NPV of the project equals 84,933.
3. The NPV of the project is below 84,933.
4. There is not enough information to state whether the NPV of the project is above, equal to, or below
84,933.
5. I choose not to answer.
Exercise 3. CAPM (RWJ 10.29) [3]
The risk free rate is 6.3% and the market portfolio has an expected rate of return of 14.8%. The market
portfolio has a variance of 0.0121. Portfolio Z has a correlation coefficient with the market of 0.45 and a
variance of 0.0169.
1
1. According to the CAPM, what is the expected rate of return on portfolio Z ?
Exercise 4. SML deviations
Consider the following illustration of the expected return – beta relation for two stocks, A and B,
E[r̃p ]
6
rB
SML
m
r
E[r̃m ]
r rf
A
1
β
Note that the two stocks do not plot on the SML. Stock A plots below and stock B plots above the SML.
1. How would you combine the two in a portfolio?
Exercise 5. [3]
The expected rate of return on the market portfolio is 14% and the risk-free rate is 6%.
1. Find the β for a portfolio that has an expected rate of return equal to 10%.
Exercise 6.
Assume that the CAPM holds. The riskless rate of return is 15%, the expected return on the market is 23%
with a standard deviation of 18%.
1. What is the expected return on a stock with a β of 1.5?
2. What is the expected return if the β is -2.0?
3. What is the standard deviation on an efficient portfolio with a β of 1.5?
4. What is the standard deviation on an efficient portfolio with an expected return of 19%?
Exercise 7. XZY [4]
The current price of XZY stock is 50. XZY is not expected to pay dividends the next few years. It has a
cost of equity capital of 10% and a cost of debt capital of 5%. The beta of the equity is one, and the market
risk premium is 6%. Interest rates are stated with annual compounding.
1. What is the one year forward price for XZY stock?
Exercise 8. Rank and File [6]
The Rank and File Company is considering a rights issue to raise $50 million. An underwriter offers to
"stand by" (i.e., to guarantee the success of the issue by buying any unwanted stock at the issue price). The
underwriter’s fee is $2 million.
2
1. What kind of option does Rank and File acquire if it accepts the underwriter’s offer?
2. What determines the value of the option?
3. How would you calculate whether the underwriter’s offer is a fair deal?
Exercise 9.
Stock A is expected next period to have either a price of 10 or 30. The current stock price for A is 25 and
the per period risk free interest rate is 2%.
What is the price of a one period call option on A with exercise price of 15?
(a) 0.00
(b) 7.65
(c) 8.72
(d) 11.40
(e) I choose not to answer.
Exercise 10. Musical CAPM [5]
After leaving school, you have started in your new job in the music industry, where you are analyzing new
bands for possible record contracts. Since you majored in finance, you want to use your knowledge of finance
to figure out the expected return. You assume that a CAPM-like relation holds, the risk of a band is
measured by its covariance with the Billboard index of music.
You are considering three bands
1. K.Dozer, a band that does pop deconstruction (After hearing this band’s destruction of “American
Pie,” you will never want to hear the original again.) Since pop deconstruction will depend on the
general hit lists, you estimate the beta (relative to the music market) of K.Dozer to be 2.
2. B.Livers, a punk bluegrass band. (What other bluegrass band will cover Pink Floyd, Motorhead and
Hank Williams sr in the same set?) This brand of music is only known to the conosseiurs, which can
be viewed as a captive market. Hence the beta of B.Livers is estimated at 0.2.
3. The Rev.H.H. does speed rockabilly, and you view this band’s beta as 1.
You also have information about how much debt these bands have had to use to buy their equipment.
1. K.Dozer has been borrowing from a loan shark at 20%, buying half of their equipment with borrowed
funds.
2. B.Livers won enough in the Texas lottery to buy all their instruments, including the extra banjo’s.
3. The Rev has been borrowing from his mother, who’s financed 80% of his equipment at 8%.
The current risk free interest rate is 5%, and the expected music market return is 15%.
1. Find the cost of equity capital for these three bands, assuming all their earnings are not taxed. (These
bands ususally get paid with a brown bag of used twenties’ after each gig.)
Exercise 11.
Suppose a firm owns assets worth $10 mill with a beta of 1.20 and variance of 0.25. The firm is contemplating
purchasing an asset at fair market value for $10 million. The beta of the new asset is 0.80 and its variance
is 0.16. The correlation between the returns on the new asset and the firm’s existing assets is zero.
3
1. What is the required return on the firm’s existing assets and the new asset if rf = 8% and E[rm ] = 15%?
2. What is the variance of the firm after it acquires the new asset?
3. What is the new company-wide cost of capital after the firm acquires the new asset?
4. Does the acquisition of the new asset increase the welfare of the firm’s existing shareholders? Under
what conditions would the acquisition be a good one?
Exercise 12. Leveraging [4]
Consider the following market-value balance sheet.
Assets 200
50 Debt (D)
150 Equity (E)
200 200 Market value (V )
βE = 0.8 and the debt is risk free. Now suppose this firm announces an issue of $120 of additional debt. All
the proceeds of the debt issue are paid out as a special dividend. After these changes, the market value of
old debt falls to $40, and βD = 0.3. What is the new βE ? Ignore taxes.
Exercise 13. Leverage (WC 15.9) [6]
You are provided with the following information: The firm’s expected net operating income (X) is $600. Its
value as an unlevered firm (VU ) is $2,000. The tax rate is 40%. The ratio of debt to equity for the levered
firm, when it is levered, is 1. The cost of debt capital is 10%. Use the MM propositions to:
1. Calculated the after–tax cost of equity capital for both the levered and the unlevered firm.
2. Calculate the after–tax weighed average cost of capital for each.
3. Why is the cost of equity capital higher for the levered firm, but the weighted average cost of capital
lower?
Exercise 14. WWI
The overall firm beta for Wild Widget, Inc. (WWI), is 0.9. WWI has a target debt equity ratio of 12 . The
expected return on the market is 16%, and Treasury Bills are currently selling to yield 8%. WWI one year
bonds that carry a 7% coupon are selling for $972.72. The corporate tax rate is 34%.
1. What is WWI’s cost of equity?
2. What is WWI’s cost of debt?
3. What is WWI’s weighted average cost of capital?
Exercise 15. Hedging [3]
Consider the following statement:
Firms should not diversify firm–specific risk because investors can do so themselves.
1. Comment on this statement. Is it correct or incorrect?
2. If you agree with the statement, what factors can then explain the levels of corporate hedging we
observe in the economy?
Exercise 16. JB [4]
JB Manufacturing is currently an all-equity firm. The equity of firm is worth $2 million. The cost of that
equity is 18%. JB pays no taxes. JB plans to issue $400,000 in debt and use the proceeds to repurchase
equity. The cost of debt is 10%.
4
1. After the repurchase the stock, what will the overall cost of capital be?
2. After the repurchase, what will the cost of equity be?
5
Empirical
Solutions
MØA 155 - Fall 2011
PROBLEM SET: Hand in 2
Exercise 1.
Note that the bondholders are indifferent, the value of the firm never goes below 300, so there is always
enough to cover the bonds.
To look at the shareholders preferences, look at the expected value of the cash flow to the shareholders:
Low risk
0.4 × 600 + 0.6 × 900 − 300 = 480
High risk:
0.4 × 300 + 0.6 × 1000 − 300 = 420
The shareholders certainly won’t go after the high risk, low return project, so they prefer the low risk project.
Correct is “None of the above statements are correct.”
Exercise 2.
Calculate the NPV assuming the company maintains production for five years
NPV
= −2
0.55
(1 + 0.1)1
0.55
+
(1 + 0.1)2
0.55
+
(1 + 0.1)3
0.55
+
(1 + 0.1)4
0.55
+
(1 + 0.1)5
= 0.0849327,
+
or 84.933. The option to abandon early can only add to this NPV. Hence
(a) is correct
Exercise 3. CAPM (RWJ 10.29) [3]
6
= ρ(rz , rm ) · σz · σm
√
√
= 0.45 0.0121 0.0169
cov(rz , rm )
=
var(rm )
0.006435
=
0.0121
cov(rz , rm )
βz =
var(rm )
0.006435
=
= 0.53
0.0169
E[rz ] = rf + (E[rm ] − rf )βz
=
0.063 + (0.148 − 0.063) · 0.53
=
10.8%.
Exercise 4. SML deviations
Buy the underpriced and sell the overpriced stock: A long position in stock A and a short position in stock
B.
Exercise 5. [3]
1. SML:
E[ri ] = rf + βi (E[rm ] − rf )
Using
rf = 6
E[rm ] = 14
want
10 = 6 + βi · 8
therefore
βi =
1
10 − 6
=
8
2
Exercise 6.
rf = 15%
E[r̃m ] = 23%
σm = 18%
1.
= rf + (E[rm ] − rf )β
r
=
0.15 + (0.23 − 0.15)1.5 = 27%
2.
r
=
rf + (E[rm ] − rf )β
=
0.15 + (0.23 − 0.15)(−2.0) = −1%
7
3. Here we need to find the portfolio variance as a function of beta. The standard deviation of an efficient
portfolio (on the CML) is σp = ωσm , where ω was the fraction invested in the risky asset. What is the
relationship between ω and β?
We know
E[r̃p ] = (1 − ω)rf + ωE[r̃m ]
and
E[r̃p ] = rf + (E[rm ] − rf )β
Let us expand the second expression.
E[r̃p ] = rf + E[r̃m ]β − βrf
⇒ E[r̃p ] = (1 − β)rf + βE[r̃m ]
Hence
σp = ωσm = βσm = 1.5 · 18% = 27%
Note: This relationship is only true when β > 0.
4.
19% = rf + (E[rm ] − rf )β
⇒ 0.19 = 0.15 + (0.23 − 0.15)β
⇒ β=
0.19 − 0.15
=
0.23 − 0.15
1
2
and
σp = 12 σm = 9%
Exercise 7. XZY [4]
A forward price on non-dividend paying stock will be found as
F = 50(1 + rf )
where rf is the risk free interest rate (with annual compounding, as stated).
Need to find the risk free interest rate. We don’t know the debt is risk free, hence recover the risk free rate
from the CAPM and the equity cost of capital
rE = rf + βE (E[rm ] − rf )
0.1 = rf + 1 × 0.06
rf = 0.04
F = 50 × 1.04 = 52
Note though that assuming debt was risk free was a not unreasonable assumption, so people who stated that
they made that assumption got close to full score. But this assumption must have been made explicitly.
Would then have gotten an estimate of
F = 50 × 1.05 = 52.5
Exercise 8. Rank and File [6]
1. Option to put stock on underwriter.
8
2. i. P/Ex = value of stock (ex rights) / issue price
ii. t = time from rights agreement to final exercise date for right.
iii. σ 2 = variance of stock returns.
iv. rf = interest rate.
3. NPV of deal to underwriter should be zero. Npv = $2 million - PV(Put). [Note: Our answer to
(b) ignores dilution. In Brealey& Myers chapter 22, we discuss how dilution affects the valuation of
warrants and convertibles. It has a similar effect on the valuation of standby underwriting, e.g., if the
option is exercised, the underwriter pays over the issue price, but he also obtains an equity stake in
this new money. After reading Chapter 22, students might like to return and think about the effect of
dilution on the value of standby agreements.]
Exercise 9.
The stock has price movements:
* Su = 30
H S0 = 25
HH
H
HH
H
j Sd = 10
H
u = 1.2, d = 0.4.
The Call has payoffs:
H C0 =?
HH
H
HH
q=
C0 =
* Cu = (30 − 15) = 15
H
j Cd = 0
H
1.02 − 0.4
= 0.775
1.2 − 0.4
1
(0.775 · 15 + 0) = 11.40
1.02
(d) is correct
Exercise 10. Musical CAPM [5]
1. We are given the asset beta for the three bands, cost of debt, and the debt/equity ratio:
Band
KD
BL
RHH
β∗
2
0.2
1
rD
20%
–
8%
9
Equity
50%
100%
20%
Debt
50%
–
80%
If we knew βD , we could use this info to calculate βE :
βE = β ∗ + (β ∗ − βD )
D
E
However, we are not given this, so we have to use
rE = r∗ + (r∗ − rD )
D
E
Use β ∗ to find r∗ :
r∗ = rf + (E[rm ] − rf )β ∗
KD
BL
RHH
0.05 + (0.15 − 0.05) · 2
0.05 + (0.15 − 0.05) · 0.2
0.05 + (0.15 − 0.05) · 2
=
=
=
25%
7%
15%
Then plug into
rE = r∗ + (r∗ − rD )
KD
BL
RHH
D
E
rE = 0.25 + (0.25 − 0.20) 0.5·V
0.5·V
= 0.25 + 0.05 = 30%
rE = r∗ = 7%
rE = 0.15 + (0.15 − 0.08) 0.8·V
0.2·V
= 0.15 + 0.07 · 4 = 0.15 + 0.28 = 43%
For the curious: The actual band names are Killdozer, Bad Livers, and the Reverend Horton Heat.
Exercise 11.
1.
rold
rnew
=
0.08 + (0.15 − 0.08) · 1.20
=
16.4%
=
0.08 + (0.15 − 0.08) · 0.80
=
13.6%
2.
σp2 = 0.52 · 0.25 + 0.52 · 0.16 = 0.1025
σp = 0.32 = 32%
3.
βp = 0.5 · 1.20 + 0.5 · 0.80 = 1.0
rp = 0.08 + (0.15 − 0.08) · 1.0 = 15%
4. No. The price paid was the fair market value of the assets. Therefore, the shareholders are neither
better or worse off. The fact that the variance of the firm’s assets declined substantially after the
acquisition does not benefit the shareholders. This is diversification that they could have gotten
themselves in the capital markets. Only if the project has a positive NPV (based on the project beta)
is it a good one.
10
Exercise 12. Leveraging [4]
The old firm has an asset beta β ∗ of
β∗ =
=
D
E
βE +
βD
D+E
D+E
150
50
0.8 +
0.0
150 + 50
150 + 50
= 0.75 · 0.8 = 0.6
The net value of the firm does not change, all the proceeds of the new debt is used to pay dividends. Hence
the asset beta does not change, β ∗ = 0.6. After the debt is issued the following relation hold:
E = 200 − 120 − 40 = 40
E
D
βE +
βD
D+E
D+E
E
120 + 40
0.6 =
βE +
βD
200
200
160
40
βE +
· 0.3
0.6 =
200
200
0.6 = 0.2 · βE + 0.24
0.6 − 0.24
βE =
= 1.8
0.2
β∗
=
Exercise 13. Leverage (WC 15.9) [6]
1. Let us use the data for the unlevered firm to find r∗ . We are given the (before-tax) operating income
X. The value of the unlevered firm is
VU =
after-tax income
r∗
after-tax income
VU
X(1
−
τ
)
600(1
− 0.4)
r∗ =
= 18%
=
VU
2, 000
For the unlevered firm, this is also the cost of equity capital, rE = r∗ = 18%.
→ r∗ =
Use this to find the value of the levered firm.
rE
D
E
0.18 + (0.18 − 0.10)(1 − 0.4) · 1
22.8%.
= r∗ + (r∗ − rD )(1 − τ )
=
=
2.
W ACCU = 18%
W ACCL
1
1
rE + (1 − τ )rD
2
2
1
1
=
0.228 + (1 − 0.4)0.10
2
2
= 14.4%
=
11
3. The equity is riskier, hence the return on equity for the levered firm is higher. The lower W ACC
reflects the tax savings from the leverage.
Exercise 14. WWI
rf = 0.08
∗
r = 0.08 + 0.9(0.16 − 0.08) = 0.152
972.72 =
1000 × 1.07
1 + rD
→ rD = 10%
D
1
=
E
2
D
D
→1+
=1+
E
E
E+D
D
→
=1+
E
E
1
1
→ E =
1+ D
E+D
E
2
E
=
D+E
3
E
D
r∗ =
rD +
rE
D+E
E+D
1
2
→ 0.152 = 0.1 + rE
3
3
1
2
→ 0.152 − 0.1 = rE
3
3
1
2
→ 0.152 − 0.1 = rE
3
3
rE = 17.8%
2
1
W ACC = (1 − 0.34)0.1 + 0.178 = 14.07%
3
3
Exercise 15. Hedging [3]
1. In itself, the statement is correct. Since the owners of stocks in a firm only care about systematic risk,
firm–specific risk is by definition irrelevant.
2. Possible factors to mention: (Others may be relevant)
• Firm’s managers are not diversified as regards their labour income. They, not the owners, may
want to hedge firm–specific risks.
• In order to monitor the managers, the owners may want them to hedge uncontrollable risk,
increasing the probability that a bad result was due to bad management.
Exercise 16. JB [4]
12
1. When the firm is an all-equity firm, the cost of equity capital (18%) is the same as the cost of capital
for the firm. When the firm does not pay taxes, the cost of capital for the firm is not affected by the
debt-equity mix.
The cost of capital thus remains the same, 18%.
2. To find equity capital, use
rE
D
E
=
r∗ + (r∗ − rD )
=
0.18 + (0.18 − 0.10)
D
E
=
0.18 + (0.18 − 0.10)
400
2, 000 − 400
=
0.18 + 0.08
=
400
1, 600
0.18 + 0.02 = 20%.
13