FIN 819
Homework
Chapter 3
Problems 1 – 4, 7, 9, 11, 14, 16, 19
Chapter 6
Problems 3, 4, 6, 7, 11 – 14, 17 – 19, 23, 24, 29, 31, 32
Chapter 9
Problems 2, 3, 7, 8, 10 – 13, 16, 19, 20, 24, 25
Chapter 12
Problems 1 – 4, 8, 9, 15, 17 – 21, 26
Chapter 10
Problems 4 – 6, 11, 22, 23, 26, 29 – 32, 35 – 37
Chapter 11
Problems 1, 4 – 6, 9, 17, 19, 22 – 25, 27, 30, 33, 36, 37, 42, 48 – 50
Chapter 20
Problems 6 – 10, 12, 15 – 18, 21, 23, 29
Chapter 21
Problems 2, 5, 7, 8, 11, 12, 14, 15
Chapter 22
Problems 1, 5, 6(a,b,c), 7
Chapter 3
Arbitrage and Financial Decision Making
3-1.
Honda Motor Company is considering offering a $2000 rebate on its minivan, lowering the
vehicle’s price from $30,000 to $28,000. The marketing group estimates that this rebate will
increase sales over the next year from 40,000 to 55,000 vehicles. Suppose Honda’s profit margin
with the rebate is $6000 per vehicle. If the change in sales is the only consequence of this decision,
what are its costs and benefits? Is it a good idea?
The benefit of the rebate is tat Honda will sell more vehicles and earn a profit on each additional
vehicle sold:
Benefit = Profit of $6,000 per vehicle × 15,000 additional vehicles sold = $90 million.
The cost of the rebate is that Honda will make less on the vehicles it would have sold:
Cost = Loss of $2,000 per vehicle × 40,000 vehicles that would have sold without rebate = $80 million.
Thus, Benefit – Cost = $90 million – $80 million = $10 million, and offering the rebate looks
attractive.
(Alternatively, we could view it in terms of total, rather than incremental, profits. The benefit as
$6000/vehicle × 55,000 sold = $330 million, and the cost is $8,000/vehicle × 40,000 sold = $320
million.)
3-2.
You are an international shrimp trader. A food producer in the Czech Republic offers to pay you
2 million Czech koruna today in exchange for a year’s supply of frozen shrimp. Your Thai
supplier will provide you with the same supply for 3 million Thai baht today. If the current
competitive market exchange rates are 25.50 koruna per dollar and 41.25 baht per dollar, what
is the value of this deal?
Czech buyer’s offer = 2,000,000 CZK / (25.50 CZK/USD) = 78,431.37 USD
Thai supplier’s offer = 3,000,000 THB / (41.25 THB/USD) = 72,727.27 USD
The value of the deal is $78,431 – 72,727 = $5704 today.
3-3.
Suppose the current market price of corn is $3.75 per bushel. Your firm has a technology that
can convert 1 bushel of corn to 3 gallons of ethanol. If the cost of conversion is $1.60 per bushel,
at what market price of ethanol does conversion become attractive?
The price in which ethanol becomes attractive is ($3.75 + $1.60 / bushel of corn) / (3 gallons of ethanol
/ bushel of corn) = $1.78 per gallon of ethanol.
3-4.
Suppose your employer offers you a choice between a $5000 bonus and 100 shares of the
company stock. Whichever one you choose will be awarded today. The stock is currently trading
for $63 per share.
a.
Suppose that if you receive the stock bonus, you are free to trade it. Which form of the bonus
should you choose? What is its value?
b. Suppose that if you receive the stock bonus, you are required to hold it for at least one year.
What can you say about the value of the stock bonus now? What will your decision depend
on?
a.
Stock bonus = 100 × $63 = $6,300
Cash bonus = $5,000
Since you can sell (or buy) the stock for $6,300 in cash today, its value is $6,300 which is better
than the cash bonus.
1
b.
3-7.
Because you could buy the stock today for $6,300 if you wanted to, the value of the stock bonus
cannot be more than $6,300. But if you are not allowed to sell the company’s stock for the next
year, its value to you could be less than $6,300. Its value will depend on what you expect the stock
to be worth in one year, as well as how you feel about the risk involved. You might decide that it
is better to take the $5,000 in cash then wait for the uncertain value of the stock in one year.
You have an investment opportunity in Japan. It requires an investment of $1 million today and
will produce a cash flow of ¥ 114 million in one year with no risk. Suppose the risk-free interest
rate in the United States is 4%, the risk-free interest rate in Japan is 2%, and the current
competitive exchange rate is ¥ 110 per $1. What is the NPV of this investment? Is it a good
opportunity?
Cost = $1 million today
Benefit = ¥114 million in one year
 ¥1.02 in one year 
=
 ¥111.76 million today
¥ today


= ¥114 million in one year ÷ 
 110¥ 
= ¥111.76 million today ÷  =
 $1.016 million today
 $ today 
NPV
= $1.016 million − $1 million
= $16,000
The NPV is positive, so it is a good investment opportunity.
3-9.
You run a construction firm. You have just won a contract to construct a government office
building. Constructing it will take one year and require an investment of $10 million today and
$5 million in one year. The government will pay you $20 million upon the building’s completion.
Suppose the cash flows and their times of payment are certain, and the risk-free interest rate is
10%.
a.
What is the NPV of this opportunity?
b. How can your firm turn this NPV into cash today?
a. =
NPV PVBenefits − PVCosts
PVBenefits = $20 million in one year ÷
 $1.10 in one year 


$ today


= $18.18 million
PVThis year's cost = $10 million today
PVNext year's cost = $5 million in one year ÷
 $1.10 in one year 


$ today


= $4.55 million today
NPV= 18.18 − 10 − 4.55
= $3.63 million today
b.
The firm can borrow $18.18 million today, and pay it back with 10% interest using the $20 million
it will receive from the government (18.18 × 1.10 = 20). The firm can use $10 million of the 18.18
million to cover its costs today and save $4.55 million in the bank to earn 10% interest to cover its
cost of 4.55 × 1.10 = $5 million next year.
This leaves 18.18 – 10 – 4.55 = $3.63 million in cash for the firm today.
2
3-11.
Your computer manufacturing firm must purchase 10,000 keyboards from a supplier. One
supplier demands a payment of $100,000 today plus $10 per keyboard payable in one year.
Another supplier will charge $21 per keyboard, also payable in one year. The risk-free interest
rate is 6%.
a.
What is the difference in their offers in terms of dollars today? Which offer should your firm
take?
b. Suppose your firm does not want to spend cash today. How can it take the first offer and not
spend $100,000 of its own cash today?
a.
Supplier 1: PVCosts = 100,000 + $10 ×
Supplier 2: PVCosts = 21 ×
10,000
1.06
10,000
1.06
= $194,339.62
= $198,113.21
Costs are lower under the first supplier’s offer, so it is better choice.
b.
3-14.
The firm can borrow $100,000 at 6% from a bank for one year to make the initial payment to the
first supplier. One year later, the firm will pay back the bank $106,000 (100,000 × 1.06) and the
first supplier $100,000 (10 × 10,000), for a total of $206,000. This amount is less than the
$210,000 (21 × 10,000) the second supplier asked for.
An American Depositary Receipt (ADR) is security issued by a U.S. bank and traded on a U.S.
stock exchange that represents a specific number of shares of a foreign stock. For example,
Nokia Corporation trades as an ADR with symbol NOK on the NYSE. Each ADR represents one
share of Nokia Corporation stock, which trades with symbol NOK1V on the Helsinki stock
exchange. If the U.S. ADR for Nokia is trading for $6.74 per share, and Nokia stock is trading on
the Helsinki exchange for 6.20 € per share, use the Law of One Price to determine the current $/€
exchange rate.
We can trade one share of Nokia stock for $6.74 per share in the U.S. and €6.20 per share in Helsinki.
By the Law of One Price, these two competitive prices must be the same at the current exchange rate.
Therefore, the exchange rate must be:
today.
3-16.
An Exchange-Traded Fund (ETF) is a security that represents a portfolio of individual stocks.
Consider an ETF for which each share represents a portfolio of two shares of Hewlett-Packard
(HPQ), one share of Sears (SHLD), and three shares of General Electric (GE). Suppose the
current stock prices of each individual stock are as shown here:
a.
What is the price per share of the ETF in a normal market?
b. If the ETF currently trades for $120, what arbitrage opportunity is available? What trades
would you make?
c.
If the ETF currently trades for $150, what arbitrage opportunity is available? What trades
would you make?
a.
We can value the portfolio by summing the value of the securities in it:
Price per share of ETF = 2 × $28 + 1 × $40 + 3 × $14 = $138
3
3-19.
b.
If the ETF currently trades for $120, an arbitrage opportunity is available. To take advantage of it,
one should buy ETF for $120, sell two shares of HPQ, sell one share of SHLD, and sell three
shares of GE. Total profit for such transaction is $18.
c.
If the ETF trades for $150, an arbitrage opportunity is also available. It can be realized by buying
two shares of HPQ, one share of SHLD, and three shares of GE, and selling one share of the ETF
for $150. Total profit would be $12.
Xia Corporation is a company whose sole assets are $100,000 in cash and three projects that it
will undertake. The projects are risk-free and have the following cash flows:
Xia plans to invest any unused cash today at the risk-free interest rate of 10%. In one year, all
cash will be paid to investors and the company will be shut down.
a.
What is the NPV of each project? Which projects should Xia undertake and how much cash
should it retain?
b. What is the total value of Xia’s assets (projects and cash) today?
c.
What cash flows will the investors in Xia receive? Based on these cash flows, what is the
value of Xia today?
d. Suppose Xia pays any unused cash to investors today, rather than investing it. What are the
cash flows to the investors in this case? What is the value of Xia now?
e.
Explain the relationship in your answers to parts (b), (c), and (d).
a.
NPVA =
−20, 000 +
NPVB =
−10, 000 +
NPVC =
−60, 000 +
30, 000
1.1
25, 000
1.1
80, 000
1.1
=
$7, 272.73
=
$12, 727.27
=
$12, 727.27
All projects have positive NPV, and Xia has enough cash, so Xia should take all of them.
b.
Total value today = Cash + NPV(projects) = 100,000 + 7,272.73 + 12,727.27 + 12,727,27 =
$132,727.27
c.
After taking the projects, Xia will have 100,000 – 20,000 – 30,000 – 60,000 = $10,000 in cash left
to invest at 10%. Thus, Xia’s cash flows in one year = 30,000 + 25,000 + 80,000 + 10,000 × 1.1 =
$146,000.
Value of Xia today =
146,000
1.1
= $132,727.27
The same as calculated in b.
d.
Unused cash = 100,000 – 20,000 – 30,000 – 60,000 = $10,000
Cash flows today = $10,000
Cash flows in one year = 30,000 + 25,000 + 80,000 = $135,000
4
Value of Xia today = 10,000 +
e.
135,000
1.1
= $132,727.27
Results from b, c, and d are the same because all methods value Xia’s assets today. Whether Xia
pays out cash now or invests it at the risk-free rate, investors get the same value today. The point
is that a firm cannot increase its value by doing what investors can do by themselves (and is the
essence of the separation principle).
5
Chapter 6
Valuing Bonds
The following table summarizes prices of various default-free, zero-coupon bonds (expressed as a
percentage of face value):
a.
Compute the yield to maturity for each bond.
b. Plot the zero-coupon yield curve (for the first five years).
c.
Is the yield curve upward sloping, downward sloping, or flat?
a.
Use the following equation.
1/ n
 FVn 
1 + YTM n =
 P 


1/1
 100 
1 + YTM=
1


 95.51 
4.70%
⇒ YTM=
1
1/ 3
 100 
1 + YTM=
3


 86.38 
⇒ YTM=
5.00%
3
1/ 4
 100 
1 + YTM=
4


 81.65 
5.20%
⇒ YTM=
4
1/ 5
 100 
1 + YTM=
5


 76.51 
b.
⇒ YTM=
5.50%
5
The yield curve is as shown below.
Zero Coupon Yield Curve
Yield to Maturity
6-3.
5.6
5.4
5.2
5
4.8
4.6
0
2
4
6
Maturity (Years)
c.
The yield curve is upward sloping.
1
6-4.
Suppose the current zero-coupon yield curve for risk-free bonds is as follows:
a.
What is the price per $100 face value of a two-year, zero-coupon, risk-free bond?
b. What is the price per $100 face value of a four-year, zero-coupon, risk-free bond?
c.
What is the risk-free interest rate for a five-year maturity?
a.
P = 100/(1.055)2 = $89.85
4
b. P 100/(1.0595)
=
=
$79.36
c.
6-6.
6.05%
Suppose a 10-year, $1000 bond with an 8% coupon rate and semiannual coupons is trading for a
price of $1034.74.
a.
What is the bond’s yield to maturity (expressed as an APR with semiannual compounding)?
b. If the bond’s yield to maturity changes to 9% APR, what will the bond’s price be?
a.
=
$1, 034.74
40
40
40 + 1000
+
+ +
YTM
YTM 2
YTM 20
(1 +
) (1 +
)
(1 +
)
2
2
2
Using the annuity spreadsheet:
NPER Rate
PV
PMT
Given:
20
-1,034.74 40
Solve For Rate:
3.75%
⇒=
YTM 7.5%
FV
1,000
Excel Formula
=RATE(20,40,-1034.74,1000)
Therefore, YTM = 3.75% × 2 = 7.50%
PV
b. =
40
40
40 + 1000
+
+L+
=
.09
.09 2
.09 20
(1 +
) (1 +
)
(1 +
)
2
2
2
$934.96.
Using the spreadsheet
With a 9% YTM = 4.5% per 6 months, the new price is $934.96
NPER
Rate
PV
PMT
FV
Excel Formula
Given:
20
4.50%
40
1,000
Solve For PV:
(934.96)
=PV(0.045,20,40,1000)
6-7.
Suppose a five-year, $1000 bond with annual coupons has a price of $900 and a yield to maturity
of 6%. What is the bond’s coupon rate?
=
900
C
C
C + 1000
+
++
⇒
=
C $36.26, so the coupon rate is 3.626%.
(1 + .06) (1 + .06) 2
(1 + .06)5
We can use the annuity spreadsheet to solve for the payment.
NPER
Rate
PV
PMT
FV
Given:
5
6.00% -900.00
1,000
Solve For PMT:
36.26
Excel Formula
=PMT(0.06,5,-900,1000)
2
Therefore, the coupon rate is 3.626%.
6-11.
Suppose that Ally Financial Inc. issued a bond with 10 years until maturity, a face value of
$1000, and a coupon rate of 7% (annual payments). The yield to maturity on this bond when it
was issued was 6%.
a.
What was the price of this bond when it was issued?
b. Assuming the yield to maturity remains constant, what is the price of the bond immediately
before it makes its first coupon payment?
c.
Assuming the yield to maturity remains constant, what is the price of the bond immediately
after it makes its first coupon payment?
a.
When it was issued, the price of the bond was
=
P
b.
70
70 + 1000
+ ... +
= $1073.60.
(1 + .06)
(1 + .06)10
Before the first coupon payment, the price of the bond is
70
70 + 1000
P=
70 +
... +
=
$1138.02.
(1 + .06)
(1 + .06)9
c.
After the first coupon payment, the price of the bond will be
P=
6-12.
70
70 + 1000
= $1068.02.
... +
(1 + .06)
(1 + .06)9
Suppose you purchase a 10-year bond with 6% annual coupons. You hold the bond for four
years, and sell it immediately after receiving the fourth coupon. If the bond’s yield to maturity
was 5% when you purchased and sold the bond,
a.
What cash flows will you pay and receive from your investment in the bond per $100 face
value?
b. What is the internal rate of return of your investment?
a.
First, we compute the initial price of the bond by discounting its 10 annual coupons of $6 and final
face value of $100 at the 5% yield to maturity.
Given:
Solve For PV:
NPER
10
Rate
5.00%
PV
PMT
6
FV
100
(107.72)
Excel Formula
= PV(0.05,10,6,100)
Thus, the initial price of the bond = $107.72. (Note that the bond trades above par, as its coupon
rate exceeds its yield.)
Next we compute the price at which the bond is sold, which is the present value of the bonds cash
flows when only 6 years remain until maturity.
Given:
Solve For PV:
NPER
6
Rate
5.00%
PV
(105.08)
PMT
6
FV
100
Excel Formula
= PV(0.05,6,6,100)
3
Therefore, the bond was sold for a price of $105.08. The cash flows from the investment are
therefore as shown in the following timeline.
b.
6-13.
Year
0
Purchase Bond
Receive Coupons
Sell Bond
Cash Flows
–$107.72
–$107.72
1
2
3
4
$6
$6
$6
$6.00
$6.00
$6.00
$6
$105.08
$111.08
We can compute the IRR of the investment using the annuity spreadsheet. The PV is the purchase
price, the PMT is the coupon amount, and the FV is the sale price. The length of the investment N
= 4 years. We then calculate the IRR of investment = 5%. Because the YTM was the same at the
time of purchase and sale, the IRR of the investment matches the YTM.
NPER
Rate
PV
PMT FV
Excel Formula
Given:
4
–107.72
6
105.08
Solve For Rate:
5.00%
= RATE(4,6,-107.72,105.08)
Consider the following bonds:
a.
What is the percentage change in the price of each bond if its yield to maturity falls from 6%
to 5%?
b. Which of the bonds A–D is most sensitive to a 1% drop in interest rates from 6% to 5% and
why? Which bond is least sensitive? Provide an intuitive explanation for your answer.
a.
We can compute the price of each bond at each YTM using Eq. 6.5. For example, with a 6%
YTM, the price of bond A per $100 face value is
P(bond A, 6% YTM)
=
100
= $41.73.
1.0615
The price of bond D is
1 
1
P(bond D, 6% YTM) =
8×
1 −
.06  1.0610
 100
=
$114.72.
+
10
 1.06
One can also use the Excel formula to compute the price: –PV(YTM, NPER, PMT, FV).
Once we compute the price of each bond for each YTM, we can compute the % price change as
Percent change =
( Price at 5% YTM ) − ( Price at 6% YTM )
.
( Price at 6% YTM )
The results are shown in the table below.
4
Bond
A
B
C
D
b.
6-14.
Coupon Rate
(annual payments)
0%
0%
4%
8%
Maturity
(years)
15
10
15
10
Price at
6% YTM
$41.73
$55.84
$80.58
$114.72
Price at
5% YTM
$48.10
$61.39
$89.62
$123.17
Percentage Change
15.3%
9.9%
11.2%
7.4%
Bond A is most sensitive, because it has the longest maturity and no coupons. Bond D is the least
sensitive. Intuitively, higher coupon rates and a shorter maturity typically lower a bond’s interest
rate sensitivity.
Suppose you purchase a 30-year, zero-coupon bond with a yield to maturity of 6%. You hold the
bond for five years before selling it.
a.
If the bond’s yield to maturity is 6% when you sell it, what is the internal rate of return of
your investment?
b. If the bond’s yield to maturity is 7% when you sell it, what is the internal rate of return of
your investment?
c.
If the bond’s yield to maturity is 5% when you sell it, what is the internal rate of return of
your investment?
d. Even if a bond has no chance of default, is your investment risk free if you plan to sell it
before it matures? Explain.
a.
Purchase price = 100 / 1.0630 = 17.41. Sale price = 100 / 1.0625 = 23.30. Return = (23.30 / 17.41)1/5
– 1 = 6.00%. I.e., since YTM is the same at purchase and sale, IRR = YTM.
b.
Purchase price = 100 / 1.0630 = 17.41. Sale price = 100 / 1.0725 = 18.42. Return = (18.42 / 17.41)1/5
– 1 = 1.13%. I.e., since YTM rises, IRR < initial YTM.
c.
Purchase price = 100 / 1.0630 = 17.41. Sale price = 100 / 1.0525 = 29.53. Return = (29.53 / 17.41)1/5
– 1 = 11.15%. I.e., since YTM falls, IRR > initial YTM.
d.
Even without default, if you sell prior to maturity, you are exposed to the risk that the YTM may
change.
For Problems 17–19, assume zero-coupon yields on default-free securities are as summarized in the following
table:
6-17.
What is the price today of a two-year, default-free security with a face value of $1000 and an
annual coupon rate of 6%? Does this bond trade at a discount, at par, or at a premium?
=
P
CPN
CPN
CPN + FV
60
60 + 1000
+
+ ... +
=
+
= $1032.09
1 + YTM 1 (1 + YTM 2 ) 2
(1 + YTM N ) N (1 + .04) (1 + .043) 2
This bond trades at a premium. The coupon of the bond is greater than each of the zero coupon yields,
so the coupon will also be greater than the yield to maturity on this bond. Therefore it trades at a
premium
6-18.
What is the price of a five-year, zero-coupon, default-free security with a face value of $1000?
The price of the zero-coupon bond is
5
=
P
6-19.
FV
1000
=
=
$791.03
N
(1 + YTM N )
(1 + 0.048)5
What is the price of a three-year, default-free security with a face value of $1000 and an annual
coupon rate of 4%? What is the yield to maturity for this bond?
The price of the bond is
=
P
CPN
CPN
CPN + FV
40
40
40 + 1000
+
+ ... +
=
+
+
= $986.58.
N
2
2
1 + YTM 1 (1 + YTM 2 )
(1 + .04) (1 + .043) (1 + .045)3
(1 + YTM N )
The yield to maturity is
=
P
CPN
CPN
CPN + FV
+
+ ... +
1 + YTM (1 + YTM ) 2
(1 + YTM ) N
$986.58
=
6-23.
40
40
40 + 1000
+
+
⇒ YTM
= 4.488%
(1 + YTM ) (1 + YTM ) 2 (1 + YTM )3
Prices of zero-coupon, default-free securities with face values of $1000 are summarized in the
following table:
Suppose you observe that a three-year, default-free security with an annual coupon rate of 10%
and a face value of $1000 has a price today of $1183.50. Is there an arbitrage opportunity? If so,
show specifically how you would take advantage of this opportunity. If not, why not?
First, figure out if the price of the coupon bond is consistent with the zero coupon yields implied by the
other securities.
970.87
=
1000
(1 + YTM 1 )
938.95
=
1000
(1 + YTM 2 ) 2
→ YTM
=
3.2%
2
904.56
=
1000
(1 + YTM 3 )3
→ YTM
=
3.4%
3
→ YTM
=
3.0%
1
According to these zero coupon yields, the price of the coupon bond should be:
100
100
100 + 1000
+
+
=
$1186.00.
(1 + .03) (1 + .032) 2 (1 + .034)3
The price of the coupon bond is too low, so there is an arbitrage opportunity. To take advantage of it:
Buy 10 Coupon Bonds
Short Sell 1 One-Year Zero
Short Sell 1 Two-Year Zero
Short Sell 11 Three-Year Zeros
Net Cash Flow
Today
-11835.00
+970.87
+938.95
+9950.16
24.98
1 Year
+1000
-1000
2 Years
+1000
0
0
-1000
3 Years
+11,000
-11,000
0
6
6-24.
Assume there are four default-free bonds with the following prices and future cash flows:
Do these bonds present an arbitrage opportunity? If so, how would you take advantage of this
opportunity? If not, why not?
To determine whether these bonds present an arbitrage opportunity, check whether the pricing is
internally consistent. Calculate the spot rates implied by Bonds A, B, and D (the zero coupon bonds),
and use this to check Bond C. (You may alternatively compute the spot rates from Bonds A, B, and C,
and check Bond D, or some other combination.)
=
934.58
1000
⇒ YTM
=
7.0%
1
(1 + YTM 1 )
881.66
=
1000
⇒ YTM
=
6.5%
2
(1 + YTM 2 ) 2
839.62
=
1000
⇒ YTM
=
6.0%
3
(1 + YTM 3 )3
Given the spot rates implied by Bonds A, B, and D, the price of Bond C should be $1,105.21. Its price
really is $1,118.21, so it is overpriced by $13 per bond. Yes, there is an arbitrage opportunity.
To take advantage of this opportunity, you want to (short) Sell Bond C (since it is overpriced). To
match future cash flows, one strategy is to sell 10 Bond Cs (it is not the only effective strategy; any
multiple of this strategy is also arbitrage). This complete strategy is summarized in the table below.
Sell 10 Bond C
Buy Bond A
Buy Bond B
Buy 11 Bond D
Net Cash Flow
Today
11,182.10
–934.58
–881.66
–9,235.82
130.04
1 Year
–1,000
1,000
0
0
0
2Years
–1,000
0
1,000
0
0
3Years
–11,000
0
0
11,000
0
Notice that your arbitrage profit equals 10 times the mispricing on each bond (subject to rounding
error).
6-29.
The following table summarizes the yields to maturity on several one-year, zero-coupon
securities:
7
a.
What is the price (expressed as a percentage of the face value) of a one-year, zero-coupon
corporate bond with a AAA rating?
b. What is the credit spread on AAA-rated corporate bonds?
c.
What is the credit spread on B-rated corporate bonds?
d. How does the credit spread change with the bond rating? Why?
a.
The price of this bond will be
=
P
6-31.
100
= 96.899.
1 + .032
b.
The credit spread on AAA-rated corporate bonds is 0.032 – 0.031 = 0.1%.
c.
The credit spread on B-rated corporate bonds is 0.049 – 0.031 = 1.8%.
d.
The credit spread increases as the bond rating falls, because lower rated bonds are riskier.
HMK Enterprises would like to raise $10 million to invest in capital expenditures. The company
plans to issue five-year bonds with a face value of $1000 and a coupon rate of 6.5% (annual
payments). The following table summarizes the yield to maturity for five-year (annual-pay)
coupon corporate bonds of various ratings:
a.
Assuming the bonds will be rated AA, what will the price of the bonds be?
b. How much total principal amount of these bonds must HMK issue to raise $10 million today,
assuming the bonds are AA rated? (Because HMK cannot issue a fraction of a bond, assume
that all fractions are rounded to the nearest whole number.)
c.
What must the rating of the bonds be for them to sell at par?
d. Suppose that when the bonds are issued, the price of each bond is $959.54. What is the likely
rating of the bonds? Are they junk bonds?
a.
The price will be
=
P
b.
65
65 + 1000
+ ... +
= $1008.36.
(1 + .063)
(1 + .063)5
Each bond will raise $1008.36, so the firm must issue:
$10, 000, 000
= 9917.13 ⇒ 9918 bonds.
$1008.36
This will correspond to a principle amount of 9918 × $1000 =
$9,918, 000.
c.
For the bonds to sell at par, the coupon must equal the yield. Since the coupon is 6.5%, the yield
must also be 6.5%, or A-rated.
d.
First, compute the yield on these bonds:
959.54
=
65
65 + 1000
YTM 7.5%.
+ ... +
⇒=
(1 + YTM )
(1 + YTM )5
Given a yield of 7.5%, it is likely these bonds are BB rated. Yes, BB-rated bonds are junk bonds.
6-32.
A BBB-rated corporate bond has a yield to maturity of 8.2%. A U.S. Treasury security has a
yield to maturity of 6.5%. These yields are quoted as APRs with semiannual compounding. Both
bonds pay semiannual coupons at a rate of 7% and have five years to maturity.
8
a.
What is the price (expressed as a percentage of the face value) of the Treasury bond?
b. What is the price (expressed as a percentage of the face value) of the BBB-rated corporate
bond?
c.
What is the credit spread on the BBB bonds?
a. P
=
b.=
P
c.
35
(1 + .0325)
35
(1 + .041)
+ ... +
+ ... +
35 + 1000
= $1, 021.06
=
(1 + .0325)10
102.1%
35 + 1000
= $951.58
= 95.2%
(1 + .041)10
credit spread = 8.2% - 6.5% = 1.7%
9
Chapter 9
Valuing Stocks
9-2.
Anle Corporation has a current price of $20, is expected to pay a dividend of $1 in one year, and
its expected price right after paying that dividend is $22.
a.
What is Anle’s expected dividend yield?
b. What is Anle’s expected capital gain rate?
9-3.
c.
What is Anle’s equity cost of capital?
a.
Div yld = 1/20 = 5%
b.
Cap gain rate = (22-20)/20 = 10%
c.
Equity cost of capital = 5% + 10% = 15%
Suppose Acap Corporation will pay a dividend of $2.80 per share at the end of this year and $3
per share next year. You expect Acap’s stock price to be $52 in two years. If Acap’s equity cost
of capital is 10%:
a.
What price would you be willing to pay for a share of Acap stock today, if you planned to
hold the stock for two years?
b. Suppose instead you plan to hold the stock for one year. What price would you expect to be
able to sell a share of Acap stock for in one year?
9-7.
c.
Given your answer in part (b), what price would you be willing to pay for a share of Acap
stock today, if you planned to hold the stock for one year? How does this compare to you
answer in part (a)?
a.
P(0) = 2.80 / 1.10 + (3.00 + 52.00) / 1.102 = $48.00
b.
P(1) = (3.00 + 52.00) / 1.10 = $50.00
c.
P(0) = (2.80 + 50.00) / 1.10 = $48.00
Dorpac Corporation has a dividend yield of 1.5%. Dorpac’s equity cost of capital is 8%, and its
dividends are expected to grow at a constant rate.
a.
What is the expected growth rate of Dorpac’s dividends?
b. What is the expected growth rate of Dorpac’s share price?
9-8.
a.
Eq 9.7 implies rE = Div Yld + g , so 8% – 1.5% = g = 6.5%.
b.
With constant dividend growth, share price is also expected to grow at rate g = 6.5% (or we can
solve this from Eq 9.2).
In mid-2018, some analysts recommended that General Electric (GE) suspend its dividend
payments to preserve cash needed for investment. Suppose you expected GE to stop paying
dividends for two years before resuming an annual dividend of $1 per share, paid 3 years from
now, growing by 3% per year. If GE's equity cost of capital is 9%, estimate the value of GE's
shares today.
The price in two years is P(t+2) = Div(t+3)/(r – g) = $1/(.09 – .03) = $16.67
The price today is P(t) = P(t+2)/(1+r) = $16.67/1.092 = $14.03
9-10.
DFB, Inc., expects earnings this year of $5 per share, and it plans to pay a $3 dividend to
shareholders. DFB will retain $2 per share of its earnings to reinvest in new projects with an
expected return of 15% per year. Suppose DFB will maintain the same dividend payout rate,
1
retention rate, and return on new investments in the future and will not change its number of
outstanding shares.
a.
What growth rate of earnings would you forecast for DFB?
b. If DFB’s equity cost of capital is 12%, what price would you estimate for DFB stock?
9-11.
c.
Suppose DFB instead paid a dividend of $4 per share this year and retained only $1 per
share in earnings. That is, it chose to pay a higher dividend instead of reinvesting in as many
new projects. If DFB maintains this higher payout rate in the future, what stock price would
you estimate now? Should DFB follow this new policy?
a.
Eq 9.12: g = retention rate × return on new invest = (2/5) × 15% = 6%
b.
P = 3 / (12% – 6%) = $50
c.
g = (1/5) × 15% = 3%, P = 4 / (12% – 3%) = $44.44. No, projects are positive NPV (return
exceeds cost of capital), so don’t raise dividend.
Cooperton Mining just announced it will cut its dividend from $4 to $2.50 per share and use the
extra funds to expand. Prior to the announcement, Cooperton’s dividends were expected to grow
at a 3% rate, and its share price was $50. With the new expansion, Cooperton’s dividends are
expected to grow at a 5% rate. What share price would you expect after the announcement?
(Assume Cooperton’s risk is unchanged by the new expansion.) Is the expansion a positive NPV
investment?
Estimate rE: rE = Div Yield + g = 4 / 50 + 3% = 11%
New Price: P = 2.50/(11% – 5%) = $41.67
In this case, cutting the dividend to expand is not a positive NPV investment.
9-12.
Procter and Gamble (PG) paid an annual dividend of $2.87 in 2018. You expect PG to increase
its dividends by 8% per year for the next five years (through 2023), and thereafter by 3% per
year. If the appropriate equity cost of capital for Procter and Gamble is 8% per year, use the
dividend-discount model to estimate its value per share at the end of 2018.
Since dividends are growing at the cost of capital over the next 5 years, the present value of the next 5
years’ dividends are:
PV1−5 =
2.87 × 1.08 2.87 × 1.082
2.87 × 1.085
...
+
+
+
=5 × 2.87 =$14.35.
1.08
1.082
1.085
Value on date 5 of the rest of the dividend payments:
=
PV5
2.87(1.08)51.03
= 86.87
0.08 − 0.03
Discounting this value to the present gives
PV0
=
86.87
$59.12
=
5
(1.08)
So the value of P&G is: P = PV1−5 + PV0 = 14.35 + 59.12 = $73.47.
9-13.
Colgate-Palmolive Company has just paid an annual dividend of $1.50. Analysts are predicting
dividends to grow by $0.12 per year over the next five years. After then, Colgate’s earnings are
expected to grow 6% per year, and its dividend payout rate will remain constant. If Colgate’s
equity cost of capital is 8.5% per year, what price does the dividend-discount model predict
Colgate stock should sell for today?
2
PV of the first 5 dividends:
PV of the remaining dividends in year 5:
Discounting back to the present
Thus the price of Colgate is
9-16.
Alphabet (GOOGL) has yet to pay a dividend, but in spring 2018 it announced it would
repurchase $8.5 billion worth of shares over the year. (Actually, the exact amount was
$8,589,869,056—known as a “perfect” number because it is equal to the sum of its divisors). If
the amount spent on share repurchases were expected to grow by 7% per year, and Alphabet’s
equity cost of capital is 8%, estimate Alphabet’s market capitalization. If Alphabet has 700
million shares outstanding, what stock price does this correspond to?
Equity Value = 8.5 / (8% – 7%) = $850 billion
Share price = 850 / 0.700 = $1214.29
9-19.
Heavy Metal Corporation is expected to generate the following free cash flows over the next five
years:
After then, the free cash flows are expected to grow at the industry average of 4% per year.
Using the discounted free cash flow model and a weighted average cost of capital of 14%:
a.
Estimate the enterprise value of Heavy Metal.
b. If Heavy Metal has no excess cash, debt of $300 million, and 40 million shares outstanding,
estimate its share price.
a.
V(4) = 82 / (14% – 4%) = $820
V(0) = 53 / 1.14 + 68/1.142 + 78 / 1.143 + (75 + 820) / 1.144 =$681
b.
9-20.
P = (681 + 0 – 300)/40 = $9.53
IDX Technologies is a privately held developer of advanced security systems based in Chicago.
As part of your business development strategy, in late 2008 you initiate discussions with IDX’s
founder about the possibility of acquiring the business at the end of 2008. Estimate the value of
IDX per share using a discounted FCF approach and the following data:
■
Debt: $30 million
■
Excess cash: $110 million
■
Shares outstanding: 50 million
3
■
Expected FCF in 2009: $45 million
■
Expected FCF in 2010: $50 million
■
Future FCF growth rate beyond 2010: 5%
■
Weighted-average cost of capital: 9.4%
From 2010 on, we expect FCF to grow at a 5% rate. Thus, using the growing perpetuity formula, we
can estimate IDX’s Terminal Enterprise Value in 2009 = $50/(9.4% – 5%) = $1136.
Adding the 2009 cash flow and discounting, we have
Enterprise Value in 2008 = ($45 + $1136)/(1.094) = $1080.
Adjusting for Cash and Debt (net debt), we estimate an equity value of
Equity Value = $1080 + 110 – 30 = $1160.
Dividing by number of shares:
Value per share = $1160/50 = $23.20.
9-24.
You notice that PepsiCo (PEP) has a stock price of $108.55 and EPS of $6.44. Its competitor, the
Coca-Cola Company (KO), has EPS of $2.48. Estimate the value of a share of Coca-Cola stock
using only this data.
PepsiCo P/E = $108.55/$6.44 = 16.9x. Apply to Coca-Cola: $2.48 ×16.9 = $41.91.
9-25.
Suppose that in January 2006, Kenneth Cole Productions had EPS of $1.65 and a book value of
equity of $12.05 per share.
a.
Using the average P/E multiple in Table 9.1, estimate KCP’s share price.
b. What range of share prices do you estimate based on the highest and lowest P/E multiples in
Table 9.1?
c.
Using the average price to book value multiple in Table 9.1, estimate KCP’s share price.
d. What range of share prices do you estimate based on the highest and lowest price to book
value multiples in Table 9.1?
a.
Share price = Average P/E × KCP EPS = 15.01 × $1.65 = $24.77
b.
Minimum = 8.66 × $1.65 = $14.29, Maximum = 22.62 × $1.65 = $37.32
c.
2.84 × $12.05 = $34.22
d.
1.12 × $12.05 = $13.50, 8.11 × $12.05 = $97.73
4
Chapter 12
Estimating the Cost of Capital
12-1.
Suppose Pepsico’s stock has a beta of 0.57. If the risk-free rate is 3% and the expected return of
the market portfolio is 8%, what is Pepsico’s equity cost of capital?
3% + 0.57 × (8%-3%) = 5.85%
12-2.
Suppose the market portfolio has an expected return of 10% and a volatility of 20%, while
Microsoft’s stock has a volatility of 30%.
a.
Given its higher volatility, should we expect Microsoft to have an equity cost of capital that is
higher than 10%?
b. What would have to be true for Microsoft’s equity cost of capital to be equal to 10%?
12-3.
a.
No, volatility includes diversifiable risk, and so it cannot be used to assess the equity cost of
capital.
b.
Microsoft stock would need to have a beta of 1.
Aluminum maker Alcoa has a beta of about 2.0, whereas Hormel Foods has a beta of 0.45. If the
expected excess return of the market portfolio is 5%, which of these firms has a higher equity
cost of capital, and how much higher is it?
Alcoa is 5% × (2-0.45) = 7.75% higher.
12-4.
Suppose all possible investment opportunities in the world are limited to the five stocks listed in
the table below. What does the market portfolio consist of (what are the portfolio weights)?
Total value of the market = 10 × 10 + 20 × 12 + 8 × 3 + 50 × 1 + 45 × 20 = $1.314 billion
Stock
A
B
C
D
E
12-8.
Portfolio Weight
10 × 10
= 7.61%
1314
20 × 12
= 18.26%
1314
8× 3
= 1.83%
1314
50
= 3.81%
1314
45 × 20
= 68.49%
1314
Suppose that in place of the S&P 500, you wanted to use a broader market portfolio of all U.S.
stocks and bonds as the market proxy. Could you use the same estimate for the market risk
1
premium when applying the CAPM? If not, how would you estimate the correct risk premium to
use?
No, expected return of this portfolio would be lower due to bonds. Compute the historical excess return
of this new index.
12-9.
From the start of 1999 to the start of 2009, the S&P 500 had a negative return. Does this mean
the market risk premium we should use in the CAPM is negative?
No! Investors were not expecting a negative return. To estimate an expected return, we need much
more data.
12-15.
In mid-2009, Rite Aid had CCC-rated, 6-year bonds outstanding with a yield to maturity of
17.3%. At the time, similar maturity Treasuries had a yield of 3%. Suppose the market risk
premium is 5% and you believe Rite Aid’s bonds have a beta of 0.31. The expected loss rate of
these bonds in the event of default is 60%.
a.
What annual probability of default would be consistent with the yield to maturity of these
bonds in mid-2009?
b. In mid-2015, Rite-Aid’s bonds had a yield of 7.1%, while similar maturity Treasuries had a
yield of 1.5%. What probability of default would you estimate now?
a.
Rd = 3% + 0.31(5%) = 4.55% = y – pL = 17.3% – p(0.60)
p = (17.3% – 4.55%)/0.60 = 21.25%
b.
12-17.
p = (7.1% – 1.5% – 0.31(5%))/0.60 = 6.75%
The Dunley Corp. plans to issue 5-year bonds. It believes the bonds will have a BBB rating.
Suppose AAA bonds with the same maturity have a 4% yield. Assume the market risk premium
is 5% and use the data in Table 12.2 and Table 12.3.
a.
Estimate the yield Dunley will have to pay, assuming an expected 60% loss rate in the event
of default during average economic times. What spread over AAA bonds will it have to pay?
b. Estimate the yield Dunley would have to pay if it were a recession, assuming the expected
loss rate is 80% at that time, but the beta of debt and market risk premium are the same as
in average economic times. What is Dunley’s spread over AAA now?
c.
In fact, one might expect risk premia and betas to increase in recessions. Redo part (b)
assuming that the market risk premium and the beta of debt both increase by 20%; that is,
they equal 1.2 times their value in recessions.
a.
Use CAPM to estimate expected return, using AAA rate as rf rate: r+.1 × rp=4% + .10(5%) =
4.5%
So, y – p × l = 4.5%
y = 4.5% + p(60%) = 4.5% + .5%(60%) = 4.8%
Spread = 4.8% – 4% = 0.8%
b.
Use CAPM to estimate expected return, using AAA rate as rf rate: r+.1 × rp=4% + .10(5%) =
4.5%
y= 4.5% + 3%(80%) = 6.90%
Spread = 6.90% – 4% = 2.9%
2
c.
Use CAPM to estimate expected return, using AAA rate as rf rate: r+.1 × 1.2 × rp × 1.2=4% +
.10(5%)1.22 = 4.72%
So, y – p × l = 4.72%
y = 4.72% + p(80%) = 4.72% + 3%(80%) = 7.12%
Spread = 7.12% – 4% = 3.12%
12-18.
Your firm is planning to invest in an automated packaging plant. Harburtin Industries is an
allequity firm that specializes in this business. Suppose Harburtin’s equity beta is 0.85, the
riskfree rate is 4%, and the market risk premium is 5%. If your firm’s project is all equity
financed, estimate its cost of capital.
Project beta = 0.85 (using all equity comp)
Thus, rp = 4% + 0.85(5%) = 8.25%
12-19.
Consider the setting of Problem 18. You decided to look for other comparables to reduce
estimation error in your cost of capital estimate. You find a second firm, Thurbinar Design,
which is also engaged in a similar line of business. Thurbinar has a stock price of $20 per share,
with 15 million shares outstanding. It also has $100 million in outstanding debt, with a yield on
the debt of 4.5%. Thurbinar’s equity beta is 1.00.
a.
Assume Thurbinar’s debt has a beta of zero. Estimate Thurbinar’s unlevered beta. Use the
unlevered beta and the CAPM to estimate Thurbinar’s unlevered cost of capital.
b. Estimate Thurbinar’s equity cost of capital using the CAPM. Then assume its debt cost of
capital equals its yield, and using these results, estimate Thurbinar’s unlevered cost of
capital.
c.
Explain the difference between your estimate in part (a) and part (b).
d. You decide to average your results in part (a) and part (b), and then average this result with
your estimate from Problem 18. What is your estimate for the cost of capital of your firm’s
project?
a.
E = 20 × 15 = 300
E+D = 400
Bu = 300/400 × 1.00 + 100/400 × 0 = 0.75
Ru = 4% + .75(5%) = 7.75%
b.
Re = 4% + 1.0 × 5% = 9%
Ru = 300/400 × 9% + 100/400 × 4.5% = 7.875%
c.
In the first case, we assumed the debt had a beta of zero, so rd = rf = 4%
In the second case, we assumed rd = ytm = 4.5%
d.
Thurbinar Ru = (7.75 + 7.875)/2 = 7.8125%
Harburtin Ru = 8.25%
Estimate = (8.25% + 7.8125%)/2 = 8.03%
12-20.
IDX Tech is looking to expand its investment in advanced security systems. The project will be
financed with equity. You are trying to assess the value of the investment, and must estimate its
cost of capital. You find the following data for a publicly traded firm in the same line of business:
3
What is your estimate of the project’s beta? What assumptions do you need to make?
Assume debt is risk-free and market value = book value. Assume comparable assets have same risk as
project.
Be = 1.20, Bd = 0
E = 15 × 80 = 1,200 million
D = 400 million
Bu = (1200/1600) × 1.20 + (400/1600) × 0 = 0.90
12-21.
In mid-2015, Cisco Systems had a market capitalization of $130 billion. It had A-rated debt of
$25 billion as well as cash and short-term investments of $60 billion, and its estimated equity
beta at the time was 1.11.
a.
What is Cisco’s enterprise value?
b. Assuming Cisco’s debt has a beta of zero, estimate the beta of Cisco’s underlying business
enterprise.
a.
EV = E + D – C = 130 + 25 – 60 = $95 billion
b.
Net Debt = 25 – 60 = –35
Ru = (130/95) × 1.11 + (–35/95) × 0 = 1.52
12-26.
Unida Systems has 40 million shares outstanding trading for $10 per share. In addition, Unida
has $100 million in outstanding debt. Suppose Unida’s equity cost of capital is 15%, its debt cost
of capital is 8%, and the corporate tax rate is 40%.
a.
What is Unida’s unlevered cost of capital?
b. What is Unida’s after-tax debt cost of capital?
c.
What is Unida’s weighted average cost of capital?
a.
E = 40 × $10 = $400 million
D = $100 million
Ru = 400/500 × 15% + 100/500 × 8% = 13.6%
b.
Rd=8% × (1-40%) = 4.8%
c.
Rwacc = 400/500 × 15% + 100/500 × 4.8% = 12.96%
4
Chapter 10
Capital Markets and the Pricing of Risk
10-4.
You bought a stock one year ago for $50 per share and sold it today for $55 per share. It paid a
$1 per share dividend today.
a.
What was your realized return?
b. How much of the return came from dividend yield and how much came from capital gain?
Compute the realized return and dividend yield on this equity investment.
R
a. =
b.
Rdiv
=
1 + (55 − 50)
= 0.12
= 12%
50
1
= 2%
50
R=
capital gain
55 − 50
= 10%
50
The realized return on the equity investment is 12%. The dividend yield is 2% and the capital gain
is 10%.
10-5.
Repeat Problem 4 assuming that the stock fell $5 to $45 instead.
a.
Is your capital gain different? Why or why not?
b. Is your dividend yield different? Why or why not?
Compute the capital gain and dividend yield under the assumption the stock price has fallen to $45.
a.
Rcapital gain =
45 − 50 / 50 =
−10%. Yes, the capital gain is different, because the difference between
the current price and the purchase price is different than in Problem 4.
b.
The dividend yield does not change, because the dividend is the same as in Problem 4.
The capital gain changes with the new lower price; the dividend yield does not change.
10-6.
Using the data in the following table, calculate the return for investing in Boeing stock (BA) from
January 2, 2008, to January 2, 2009, and also from January 3, 2011, to January 3, 2012,
assuming all dividends are reinvested in the stock immediately.
Historical Stock and Dividend Data for Boeing
Date
Price
1/2/2008
86.62
2/6/2008
79.91
5/7/2008
Dividend
Date
Price
Dividend
1/3/2011
66.40
0.40
2/9/2011
72.63
0.42
84.55
0.40
5/11/2011
79.08
0.42
8/6/2008
65.40
0.40
8/10/2011
57.41
0.42
11/5/2008
49.55
0.40
11/8/2011
66.65
0.42
1/2/2009
45.25
1/3/2012
74.22
1
Price
86.62
79.91
84.55
65.4
49.55
45.25
Price
66.4
72.63
79.08
57.41
66.65
74.22
Dividend
0.4
0.4
0.4
0.4
Dividend
0.42
0.42
0.42
0.42
R
-7.28%
6.31%
-22.18%
-23.62%
-8.68%
R
10.02%
9.46%
-26.87%
16.83%
11.36%
1+R
0.92715308
1.06307095
0.77823773
0.76376147
0.91321897
0.535006
-46.50%
1+R
1.1001506
1.09458901
0.73128477
1.16826337
1.11357839
1.14564829
14.56%
10-11.
Consider an investment with the following returns over four years:
a.
What is the compound annual growth rate (CAGR) for this investment over the four years?
b. What is the average annual return of the investment over the four years?
c.
Which is a better measure of the investment’s past performance?
d. If the investment’s returns are independent and identically distributed, which is a better
measure of the investment’s expected return next year?
a.
10-22.
1
10%
2
20%
3
-5%
4
15%
1.10
1.20
0.95
1.15
b.
see table above
c.
CAGR
d.
Arithmetic average
Ave
10.00%
CAGR
9.58%
Consider the following two, completely separate, economies. The expected return and volatility
of all stocks in both economies is the same. In the first economy, all stocks move together—in
good times all prices rise together and in bad times they all fall together. In the second economy,
2
stock returns are independent—one stock increasing in price has no effect on the prices of other
stocks. Assuming you are risk-averse and you could choose one of the two economies in which to
invest, which one would you choose? Explain.
A risk-averse investor would choose the economy in which stock returns are independent because this
risk can be diversified away in a large portfolio.
10-23.
Consider an economy with two types of firms, S and I. S firms all move together. I firms move
independently. For both types of firms, there is a 60% probability that the firms will have a 15%
return and a 40% probability that the firms will have a −10% return. What is the volatility
(standard deviation) of a portfolio that consists of an equal investment in 20 firms of (a) type S,
and (b) type I?
a.
E [ R ] = 0.15 ( 0.6 ) − 0.1( 0.4 ) = 0.05
Standard Deviation
=
( 0.15 − 0.05)
2
0.6 + ( −0.1 − 0.05 ) =
0.4 0.12247
2
Because all S firms in the portfolio move together there is no diversification benefit. So the
standard deviation of the portfolio is the same as the standard deviation of the stocks—12.25%.
b.
E [R ] = 0.15 ( 0.6) − 0.1( 0.4) = 0.05
Standard Deviation
=
( 0.15 − 0.05)
2
0.6 + ( −0.1 − 0.05 ) =
0.4 0.12247
2
Type I stocks move independently. Hence the standard deviation of the portfolio is
SD ( Portfolio of 20 Type I=
stocks )
10-26.
0.12247
= 2.74%.
20
Identify each of the following risks as most likely to be systematic risk or diversifiable risk:
a.
The risk that your main production plant is shut down due to a tornado.
b. The risk that the economy slows, decreasing demand for your firm’s products.
c.
The risk that your best employees will be hired away.
d. The risk that the new product you expect your R&D division to produce will not materialize.
10-29.
a.
diversifiable risk
b.
systematic risk
c.
diversifiable risk
d.
diversifiable risk
What is an efficient portfolio?
An efficient portfolio is any portfolio that only contains systemic risk; it contains no diversifiable risk.
10-30.
What does the beta of a stock measure?
Beta measures the amount of systemic risk in a stock
10-31.
You turn on the news and find out the stock market has gone up 10%. Based on the data in
Table 10.6, by how much do you expect each of the following stocks to have gone up or down: (1)
Starbucks, (2) Tiffany & Co., (3) Hershey, and (4) McDonald’s.
Beta*10%
Starbucks 8%
3
Tiffany & Co. 17.7%
Hershey 3.3%
McDonald’s 6.3%
10-32.
Based on the data in Table 10.6, estimate which of the following investments you expect to lose
the most in the event of a severe market down turn: (1) A $2000 investment in Hershey, (2) a
$1500 investment in Macy’s, or (3) a $1000 investment in Amazon
For each 10% market decline,
Hershey down 10%*0.33 = 3.3%,
3.3% × 2,000 = $66 loss;
Macy’s down 10%*.75 = 7.5%,
7.5% × 1,500 = $112.50 loss;
Amazon down 10%*1.62 = 16.2%,
16.2% × 1,000 = $162 loss;
Amazon investment will lose most.
10-35.
Suppose the market risk premium is 5% and the risk-free interest rate is 4%. Using the data in
Table 10.6, calculate the expected return of investing in
a.
Starbucks’ stock.
b. Hershey’s stock.
10-36.
c.
Autodesk’s stock.
a.
4% + 0.80 × 5% = 8%
b.
4% + 0.33 × 5% = 5.65%
c.
4% + 1.76 × 5% = 12.8%
Given the results to Problem 35, why don’t all investors hold Autodesk’s stock rather than
Hershey’s stock?
Hershey’s stock has less market risk, so investors don’t need as high an expected return to hold it.
Hershey’s stock will perform much better in a market downturn.
10-37.
Suppose the market risk premium is 6.5% and the risk-free interest rate is 5%. Calculate the
cost of capital of investing in a project with a beta of 1.2.
Cost of Capital =
rf + β ( E [ R m ] − rf ) =
5 + 1.2 ( 6.5 ) =
12.8%
4
Chapter 11
Optimal Portfolio Choice and the Capital Asset Pricing Model
11-1.
You are considering how to invest part of your retirement savings. You have decided to put
$200,000 into three stocks: 50% of the money in GoldFinger (currently $25/share), 25% of the
money in Moosehead (currently $80/share), and the remainder in Venture Associates (currently
$2/share). If GoldFinger stock goes up to $30/share, Moosehead stock drops to $60/share, and
Venture Associates stock rises to $3 per share,
a.
What is the new value of the portfolio?
b. What return did the portfolio earn?
c.
If you don’t buy or sell shares after the price change, what are your new portfolio weights?
a.
Let
ni
be the number of share in stock I, then
nG
=
200, 000 × 0.5
= 4, 000
25
=
nM
200, 000 × 0.25
= 625
80
=
nV
200, 000 × 0.25
= 25, 000.
2
The new value of the portfolio is
p = 30nG + 60nM + 3nv
= $232,500.
232,500
=
− 1 16.25%
200, 000
b.
Return
=
c.
The portfolio weights are the fraction of value invested in each stock.
nG × 30
= 51.61%
232,500
n × 60
Moosehead: M
= 16.13%
232,500
n ×3
Venture: V
= 32.26%
232,500
GoldFinger:
11-4.
There are two ways to calculate the expected return of a portfolio: either calculate the expected
return using the value and dividend stream of the portfolio as a whole, or calculate the weighted
average of the expected returns of the individual stocks that make up the portfolio. Which return
is higher?
Both calculations of expected return of a portfolio give the same answer.
11-5.
Using the data in the following table, estimate (a) the average return and volatility for each stock,
(b) the covariance between the stocks, and (c) the correlation between these two stocks.
1
−10 + 20 + 5 − 5 + 2 + 9
= 3.5%
6
21 + 30 + 7 − 3 − 8 + 25
RB =
6
= 12%
=
a. R A
( −0.1 − 0.035 )2 +



2
2
1 ( 0.2 − 0.035 ) + ( 0.05 − 0.035 ) + 

Variance of A = 
5 ( −0.05 − 0.035 )2 + ( 0.02 − 0.035 )2 


 + ( 0.09 − 0.035 )2

= 0.01123
SD( RA )
Volatility of A = =
=
Variance
of A
=
.01123 10.60%
( 0.21 − 0.12 )2 + ( 0.3 − 0.12 )2 + 

1
2
2
Variance of B = ( 0.07 − 0.12 ) + ( −0.03 − 0.12 ) + 

5
( −0.08 − 0.12 )2 + ( 0.25 − 0.12 )2 


= 0.02448
Volatility of B = =
SD( RB )
Variance
=
of B
=
.02448 15.65%
b.
c.
Correlation
=
Covariance
SD(R A )SD(R B )
= 6.27%
11-6.
Use the data in Problem 5, consider a portfolio that maintains a 50% weight on stock A and a
50% weight on stock B.
a.
What is the return each year of this portfolio?
b. Based on your results from part a, compute the average return and volatility of the portfolio.
c.
Show that (i) the average return of the portfolio is equal to the average of the average
returns of the two stocks, and (ii) the volatility of the portfolio equals the same result as from
the calculation in Eq. 11.9.
d. Explain why the portfolio has a lower volatility than the average volatility of the two stocks.
a, b, and c. See table below.
2
d.
11-9.
The portfolio has a lower volatility than the average volatility of the two stocks because some of
the idiosyncratic risk of the stocks in the portfolio is diversified away.
Suppose two stocks have a correlation of 1. If the first stock has an above average return this
year, what is the probability that the second stock will have an above average return?
Because the correlation is perfect, they move together (always) and so the probability is 1.
11-17.
What is the volatility (standard deviation) of an equally weighted portfolio of stocks within an
industry in which the stocks have a volatility of 50% and a correlation of 40% as the portfolio
becomes arbitrarily large?
ave cov = (0.5 × 0.5 × 0.4)0.5 = 31.62%
11-19.
Stock A has a volatility of 65% and a correlation of 10% with your current portfolio. Stock B
has a volatility of 30% and a correlation of 25% with your current portfolio. You currently hold
both stocks. Which will increase the volatility of your portfolio: (i) selling a small amount of
stock B and investing the proceeds in stock A, or (ii) selling a small amount of stock A and
investing the proceeds in stock B?
From Eq. 11.13, marginal contribution to risk is SD(Ri) × Corr(Ri,Rp)
For A: 65% × 10% = 6.5%;
For B: 30% × 25% = 7.5%.
So, volatility increases if we sell A and add B.
11-22.
Suppose Intel’s stock has an expected return of 26% and a volatility of 50%, while Coca-Cola’s
has an expected return of 6% and volatility of 25%. If these two stocks were perfectly negatively
correlated (i.e., their correlation coefficient is −1),
a.
Calculate the portfolio weights that remove all risk.
b. If there are no arbitrage opportunities, what is the risk-free rate of interest in this economy?
a.
If the two stocks are perfectly correlated negatively, they fluctuate due to the same risks, but in
opposite directions. Because Intel is twice as volatile as Coke, we will need to hold twice as much
Coke stock as Intel in order to offset Intel’s risk. That is, our portfolio should be 2/3 Coke and 1/3
Intel.
We can check this using Eq. 11.9.
Var ( RP ) = (2 / 3) 2 SD( RCoke ) 2 + (1/ 3) 2 SD( RIntel ) 2 + 2(2 / 3)(1/ 3)Corr( RCoke , RIntel ) SD( RCoke ) SD( RIntel )
= (2 / 3) 2 (0.252 ) + (1/ 3) 2 (0.502 ) + 2(2 / 3)(1/ 3)(−1)(.25)(.50)
=0
b.
From Eq. 11.3, the expect return of the portfolio is
=
E[ RP ] (2 / 3) E[ RCoke ] + (1/ 3) E[ RIntel ]
= (2 / 3)6% + (1/ 3)26%
= 12.67%.
Because this portfolio has no risk, the risk-free interest rate must also be 12.67%.
For Problems 23–25, suppose Johnson & Johnson and the Walgreen Company have expected returns and
volatilities shown below, with a correlation of 22%.
3
11-23.
Calculate (a) the expected return and (b) the volatility (standard deviation) of a portfolio that is
equally invested in Johnson & Johnson’s and Walgreen’s stock.
In this case, the portfolio weights are xj = xw = 0.50. From Eq. 11.3,
=
E[ RP ] x j E[ R j ] + xw E[ Rw ]
= 0.50(7%) + 0.50(10%)
= 8.5%.
We can use Eq. 11.9.
x j 2 SD( R j ) 2 + xw 2 SD( Rw ) 2 + 2 x j xw Corr ( R j , Rw ) SD( R j ) SD( Rw )
SD( RP ) =
.502 (.162 ) + .502 (.20) 2 + 2(.50)(.50)(.22)(.16)(.20)
=
= 14.1%
11-24.
For the portfolio in Problem 23, if the correlation between Johnson & Johnson’s and Walgreen’s
stock were to increase,
a.
Would the expected return of the portfolio rise or fall?
b. Would the volatility of the portfolio rise or fall?
11-25.
a.
The expected return would remain constant, assuming only the correlation changes, 0.5 × 0.07 +
0.5 × 0.10 = 0.085.
b.
The volatility of the portfolio would increase (due to the correlation term in the equation for the
volatility of a portfolio).
Calculate (a) the expected return and (b) the volatility (standard deviation) of a portfolio that
consists of a long position of $10,000 in Johnson & Johnson and a short position of $2000 in
Walgreen’s.
In this case, the total investment is $10,000 – 2,000 = $8,000, so the portfolio weights are xj =
10,000/8,000 = 1.25, xw = –2,000/8,000 = –0.25. From Eq. 11.3,
=
E[ RP ] x j E[ R j ] + xw E[ Rw ]
= 1.25(7%) − 0.25(10%)
= 6.25%.
We can use Eq. 11.9,
x j 2 SD( R j ) 2 + xw 2 SD( Rw ) 2 + 2 x j xw Corr ( R j , Rw ) SD( R j ) SD( Rw )
SD( RP ) =
1.252 (.162 ) + (−0.25) 2 (.20) 2 + 2(1.25)(−0.25)(.22)(.16)(.20)
=
= 19.5%.
11-27.
A hedge fund has created a portfolio using just two stocks. It has shorted $35,000,000 worth of
Oracle stock and has purchased $85,000,000 of Intel stock. The correlation between Oracle’s and
Intel’s returns is 0.65. The expected returns and standard deviations of the two stocks are given
in the table below:
4
a.
What is the expected return of the hedge fund’s portfolio?
b. What is the standard deviation of the hedge fund’s portfolio?
a.
The total value of the portfolio is $50m (=-$35+$85). This means that the weight on Oracle is
–70% and the weight on Intel is 170%. The expected return is
Expected return =
−0.7 × 12% + 1.7 × 14.5%
= 16.25%.
b.
Variance =
( −0.7 ) × ( 0.45) + (1.7 ) × ( 0.40 )
2
2
2
2
+ 2 × ( −0.7 ) × (1.7 ) × 0.65 × 0.45 × 0.40
= 0.283165
Std dev = (.283165)^.5 = 53.2%
11-30.
Suppose Target’s stock has an expected return of 20% and a volatility of 40%, Hershey’s stock
has an expected return of 12% and a volatility of 30%, and these two stocks are uncorrelated.
a.
What is the expected return and volatility of an equally weighted portfolio of the two stocks?
Consider a new stock with an expected return of 16% and a volatility of 30%. Suppose this new
stock is uncorrelated with Target’s and Hershey’s stock.
b. Is holding this stock alone attractive compared to holding the portfolio in (a)?
c.
Can you improve upon your portfolio in (a) by adding this new stock to your portfolio?
Explain.
a.
ER
Vol
XA
11-33.
A
50%
XB
20%
40%
50%
B
Vol
12%
30%
25.0%
Corr
ER
0%
16.0%
b.
No, it has the same expected return with higher volatility.
c.
Yes, adding this stock and reducing weight on the others will reduce risk while leaving expected
return unchanged.
Suppose you have $100,000 in cash, and you decide to borrow another $15,000 at a 4% interest
rate to invest in the stock market. You invest the entire $115,000 in a portfolio J with a 15%
expected return and a 25% volatility.
a.
What is the expected return and volatility (standard deviation) of your investment?
b. What is your realized return if J goes up 25% over the year?
c.
What return do you realize if J falls by 20% over the year?
5
a. x
=
115, 000
= 1.15
100, 000
(
)
E [ R ] = rf + x E  R j  − r = 4% + 1.15 (11% ) = 16.65%
= x=
=
SD  R j  1.15
Volatility
25% 28.75%
b. R
=
c.
11-36.
115, 000(1.25) − 15, 000(1.04)
=
− 1 28.15%
100, 000
115, 000(0.80) − 15, 000(1.04)
R=
− 1 =−23.6%
100, 000
Assume the risk-free rate is 4%. You are a financial advisor, and must choose one of the funds
below to recommend to each of your clients. Whichever fund you recommend, your clients will
then combine it with risk-free borrowing and lending depending on their desired level of risk.
Which fund would you recommend without knowing your client’s risk preference?
Sharpe ratios of A, B, and C are 0.6, 0.5 and 1, so you would choose C; it is the best choice no matter
what your clients’ risk preferences are.
11-37.
Assume all investors want to hold a portfolio that, for a given level of volatility, has the
maximum possible expected return. Explain why, when a risk-free asset exists, all investors will
choose to hold the same portfolio of risky stocks.
Investors who want to maximize their expected return for a given level of volatility will pick portfolios
that maximize their Sharpe ratio. The set of portfolios that do this is a combination of a risk-free
asset—a single portfolio of risk assets—the tangential portfolio.
11-42.
Calculate the Sharpe ratio of each of the three portfolios in Problem 41. What portfolio weight in
Hannah stock maximizes the Sharpe ratio?
Initial Portfolio 60-40 Portfolio 85-15 Portfolio
Natasha Fund
Expected Return
Volatility
0.14
0.2
0.14
0.2
0.14
0.2
Hannah Stock
Expected Return
Volatilty
0.2
0.6
0.2
0.6
0.2
0.6
Risk Free Rate
0.038
0.038
0.038
Portfolio weight in Hannah
Expected Return of Portfolio
Volatility of Portfolio
Sharpe Ratio
0
0.14
0.2
0.51
0.4
0.164
0.268328157
0.469574275
0.15
0.149
0.192353841
0.577061522
6
The Sharpe Ratio is maximized at 15% in Hannah Stock.
11-48.
Suppose the risk-free return is 4% and the market portfolio has an expected return of 10% and
a volatility of 16%. Merck & Co. (Ticker: MRK) stock has a 20% volatility and a correlation
with the market of 0.06.
a.
What is Merck’s beta with respect to the market?
b. Under the CAPM assumptions, what is its expected return?
a.
b.
11-49.
Consider a portfolio consisting of the following three stocks:
The volatility of the market portfolio is 10% and it has an expected return of 8%. The risk-free
rate is 3%.
a.
Compute the beta and expected return of each stock.
b. Using your answer from part a, calculate the expected return of the portfolio.
c.
What is the beta of the portfolio?
d. Using your answer from part c, calculate the expected return of the portfolio and verify that
it matches your answer to part b.
HEC Corp
Green Midget
AliveAndWell
Portfolio
Weight
Volatility
Correlation with Beta (Part
the
Market answer)
Portfolio
0.25
0.35
0.4
12%
25%
13%
0.4
0.6
0.5
a Expected
Return (Part a
answer)
0.48
1.5
0.65
5.4
10.5
6.25
Part c answer Part b answer
0.905
7.525
Part d answer
Expected Return calculated from porfolio
7.525
Porfolio
11-50.
Suppose Autodesk stock has a beta of 2.16, whereas Costco stock has a beta of 0.69. If the riskfree interest rate is 4% and the expected return of the market portfolio is 10%, what is the
expected return of a portfolio that consists of 60% Autodesk stock and 40% Costco stock,
according to the CAPM?
7
β = ( 0.6)( 2.16) + ( 0.4)( 0.69) = 1.572
E [ R] =
4 + (1.572)(10 − 4) =
13.432%
8
Chapter 20
Financial Options
20-6.
You own a call option on Intuit stock with a strike price of $40. The option will expire in exactly
three months’ time.
a.
If the stock is trading at $55 in three months, what will be the payoff of the call?
b. If the stock is trading at $35 in three months, what will be the payoff of the call?
c.
Draw a payoff diagram showing the value of the call at expiration as a function of the stock
price at expiration.
Long call option: value at expiration:
20-7.
a.
$15
b.
0$
c.
Draw graph:
Assume that you have shorted the call option in Problem 6.
a.
If the stock is trading at $55 in three months, what will you owe?
b. If the stock is trading at $35 in three months, what will you owe?
c.
Draw a payoff diagram showing the amount you owe at expiration as a function of the stock
price at expiration.
Short call: value at expiration date:
a.
You owe $15.
b.
You owe nothing.
c.
Draw the payoff diagram:
1
20-8.
You own a put option on Ford stock with a strike price of $10. The option will expire in exactly
six months’ time.
a.
If the stock is trading at $8 in six months, what will be the payoff of the put?
b. If the stock is trading at $23 in six months, what will be the payoff of the put?
c.
Draw a payoff diagram showing the value of the put at expiration as a function of the stock
price at expiration.
Long put value at expiration:
20-9.
a.
$2
b.
$0
c.
Draw payoff diagram:
Assume that you have shorted the put option in Problem 8.
a.
If the stock is trading at $8 in three months, what will you owe?
b. If the stock is trading at $23 in three months, what will you owe?
c.
Draw a payoff diagram showing the amount you owe at expiration as a function of the stock
price at expiration.
2
Short put: value at expiration:
20-10.
a.
You owe $2.
b.
You owe nothing.
c.
Draw payoff diagram:
What position has more downside exposure: a short position in a call or a short position in a put?
That is, in the worst case, in which of these two positions would your losses be greater?
Downside exposure is larger with a short call (the downside is unlimited) than with a short put (the
downside cannot be larger than the strike price).
20-12.
You are long both a call and a put on the same share of stock with the same exercise date. The
exercise price of the call is $40 and the exercise price of the put is $45. Plot the value of this
combination as a function of the stock price on the exercise date.
⇓
3
20-15.
You own a share of Costco stock. You are worried that its price will fall and would like to insure
yourself against this possibility. How can you purchase insurance against this possibility?
To protect against a fall in the price of Costco, you can buy a put with Costco as the underlying asset.
By doing this, over the life of the option you are guaranteed to get at least the strike price from selling
the stock you already have.
20-16.
It is November 9, 2018, and you own IBM stock. You would like to insure that the value of your
holdings will not fall significantly. Using the data in Problem 3, and expressing your answer in
terms of a percentage of the current value of your portfolio:
a.
What will it cost to insure that the value of your holdings will not fall below $115 per share
between now and the third Friday in November?
b. What will it cost to insure that the value of your holdings will not fall below $115 per share
between now and the third Friday in December?
20-17.
c.
What will it cost to insure that the value of your holdings will not fall below $120 per share
between now and the third Friday in December?
a.
To ensure that the value of your IBM does not fall significantly, you would purchase a protective
put. The current ask price for a protective put with a strike price of $115 that expires the third
Friday of November is $0.17 per share. As a percentage of your portfolio this cost to insure is
$0.17/$122.26 = 0.0014 or 0.14% of the value of your portfolio.
b.
The ask price of a protective put with a strike price of $115 that expires on the third Friday of
December is $1.36. The cost to insure the value of your holdings will not fall is $1.36/$122.26 =
0.0111 or 1.11%.
c.
The ask price of a protective put with a strike price of $120 that expires on the third Friday of
December is $2.71. The cost to insure the value of your holdings will not fall is $2.71/$122.26 =
0.0222 or 2.22%.
Dynamic Energy Systems stock is currently trading for $33 per share. The stock pays no
dividends. A one-year European put option on Dynamic with a strike price of $35 is currently
trading for $2.10. If the risk-free interest rate is 10% per year, what is the price of a one-year
European call option on Dynamic with a strike price of $35?
Put-call parity:
C = P+S−
K
35
= 2.10 + 33 −
= 3.282
1+ r
1.1
4
20-18.
You happen to be checking the newspaper and notice an arbitrage opportunity. The current
stock price of Intrawest is $20 per share and the one-year risk-free interest rate is 8%. A one
year put on Intrawest with a strike price of $18 sells for $3.33, while the identical call sells for $7.
Explain what you must do to exploit this arbitrage opportunity.
The arbitrage opportunity exists because:
$7 > $3.33 + $20 −
$18
= $6.66 .
1 + 0.08
(
)
So the call is overpriced compared to the portfolio of a put, the stock, and risk-free borrowing.
As a result, the strategy would be to sell the call option, buy the put, buy the stock, and borrow $16.67
(the present value of $18).
The net amount left immediately after doing this is $.34 (=$7 – $6.66), with no cash flows when the
options expire.
20-21.
Suppose Amazon stock is trading for $2000 per share, and Amazon pays no dividends.
a.
What is the maximum possible price of a call option on Amazon?
b. What is the maximum possible price of a put option on Amazon with a strike price of $2250?
c.
What is the minimum possible value of a call option on Amazon stock with a strike price of
$1750?
d. What is the minimum possible value of an American put option on Amazon stock with a
strike price of $2250?
20-23.
a.
$2000 (stock price)
b.
$2250 (strike price)
c.
Intrinsic value = $250
d.
Intrinsic value = $250
You are watching the option quotes for your favorite stock, when suddenly there is a news
announcement. Explain what type of news would lead to the following effects:
a.
Call prices increase, and put prices fall.
b. Call prices fall, and put prices increase.
20-29.
c.
Both call and put prices increase.
a.
Good news about the stock, which raises its stock price
b.
Bad news, which lowers the stock’s price
c.
News that increases the volatility of the stock
Wesley Corp. stock is trading for $25/share. Wesley has 20 million shares outstanding and a
market debt-equity ratio of 0.5. Wesley’s debt is zero-coupon debt with a 5-year maturity and a
yield to maturity of 10%.
a.
Describe Wesley’s equity as a call option. What is the maturity of the call option? What is
the market value of the asset underlying this call option? What is the strike price of this call
option?
b. Describe Wesley’s debt using a call option.
c.
Describe Wesley’s debt using a put option.
a.
Maturity = 5 years
5
Assets = E + D = $25 per share × 20 million shares + .5($25 × 20) = $500 + 250 = $750 million
The market value of the asset underlying this call option is $750.00 million.
The strike price is equal to the debt. Therefore,
Face Value of Debt = 0.50 × ($25 per share × 20 million shares) × (1+0.1)5 = $402.63 million
The strike price of this call option is $402.63 million.
b.
Long the firm’s assets and short the equity call option above
c.
Long risk-free debt and short a put option on Wesley’s assets with a 5-yr maturity and $402.63
million face value
6
Chapter 21
Option Valuation
21-2.
Using the information in Problem 1, use the Binomial Model to calculate the price of a one year
put option on Estelle stock with a strike price of $25.
The parameters are the same as in 21-1, but the payoff of the put is 0 if the stock goes up and 5 if the
stock goes down. Therefore, the replicating portfolio is
∆ = (0 − 5)/(30 − 20) = –0.5 and B = (5 − 20×(–0.5))/1.06 = 14.15.
Therefore, P = –0.5 × 25 + 14.15 = $1.65.
21-5.
Suppose the option in Example 21.1 actually sold in the market for $8. Describe a trading strategy
that yields arbitrage profits.
In Example 21.1, the theoretical put price is $3.30. If it actually sells for a higher price, it is overvalued,
and we can sell it and buy the replicating portfolio to earn an arbitrage profit. This means that at t = 0,
you will sell the put, short 0.3333 shares of stock, and invest $23.30 in Treasury Bills.
Following this strategy, you will earn the put price less (60 × (–0.3333) + 23.30) = $3.30, that is $8 –
$3.30 = $4.7 upfront.
Then, if the stock goes up, our portfolio of the put, shares, and borrowing is worth $0 – 0.3333 × 72 +
23.30 × 1.03 = $0. If it goes down, our portfolio is worth –$6 – 0.3333 × $54 + 23.30 × 1.03 = $0, so
that at maturity, the payoff of the option and the value of the replicating portfolio cancel out.
(Note that this is not the only arbitrage strategy one can follow—e.g., we could invest more in T-Bills
initially—but it is the strategy that generates the most cash upfront without any risk of loss in the future.)
21-7.
Eagletron’s current stock price is $10. Suppose that over the current year, the stock price will
either increase by 100% or decrease by 50%. Also, the risk-free rate is 25% (EAR).
a.
What is the value today of a one-year at-the-money European put option on Eagletron stock?
b. What is the value today of a one-year European put option on Eagletron stock with a strike
price of $20?
c.
a.
Suppose the put options in parts a and b could either be exercised immediately, or in one year.
What would their values be in this case?
0
1
S
Put (K=10)
20
0
5
5
10
Pu = 0, Pd = 5, delta = (0 – 5)/(20 – 5) = –1/3, B = (5 – 5(–1/3))/1.25 = 5.33
P = –1/3(10) + 5.33 = $2.00
b.
0
1
S
Put (K=20)
20
0
5
15
10
1
The payoffs are Pu = 0, Pd = 15. Because these payoffs are three times the payoffs in (a), the put
must be worth 3 × $2 = $6.
c.
We can exercise the put immediately and get its intrinsic value. In (a), intrinsic value = 0 because
the strike price is $10, so not relevant, put value is still $2
In (b) intrinsic value = 20 – 10 = $10, so it is better to exercise now → value is $10 (not $6).
21-8.
What is the highest possible value for the delta of a call option? What is the lowest possible value?
(Hint: See Figure 21.1.)
Delta <= 1, Delta >= 0
21-11.
Roslin Robotics stock has a volatility of 30% and a current stock price of $60 per share. Roslin
pays no dividends. The risk-free interest is 5%. Determine the Black-Scholes value of a one-year,
at-the-money call option on Roslin stock.
𝑑𝑑1 =
ln[𝑆𝑆/𝑃𝑃𝑃𝑃(𝐾𝐾)]
𝜎𝜎√𝑇𝑇
60
𝜎𝜎√𝑇𝑇 ln[60/(1.05)] 0.3√1
+
=
+
= 0.313
2
2
0.3√1
𝑑𝑑2 = 𝑑𝑑1 − 𝜎𝜎√𝑇𝑇 = 0.313 − 0.3√1 = 0.013
𝑁𝑁(𝑑𝑑1 ) = 𝑁𝑁(0.313) = 0.623; 𝑁𝑁(𝑑𝑑2 ) = 𝑁𝑁(0.013) = 0.505
21-12.
𝐶𝐶 = 𝑆𝑆 × 𝑁𝑁(𝑑𝑑1 ) − 𝑃𝑃𝑃𝑃(𝐾𝐾) × 𝑁𝑁(𝑑𝑑2 ) = 60 × 0.623 −
60
× 0.505 = $8.50
1.05
Rebecca is interested in purchasing a European call on a hot new stock, Up, Inc. The call has a
strike price of $100 and expires in 90 days. The current price of Up stock is $120, and the stock
has a standard deviation of 40% per year. The risk-free interest rate is 6.18% per year.
a.
Using the Black-Scholes formula, compute the price of the call.
b. Use put-call parity to compute the price of the put with the same strike and expiration date.
a.
Using the Black-Sholes formula:
𝑃𝑃𝑃𝑃(𝐾𝐾) =
𝑑𝑑1 =
100
90
(1.0618)365
ln[𝑆𝑆/𝑃𝑃𝑃𝑃(𝐾𝐾)]
𝜎𝜎√𝑇𝑇
+
= 98.532
𝜎𝜎√𝑇𝑇 ln[120/98.532] 0.4�90/365
=
+
= 1.092
2
2
0.4�90/365
𝑑𝑑2 = 𝑑𝑑1 − 𝜎𝜎√𝑇𝑇 = 1.092 − 0.4�90/365 = 0.893
𝑁𝑁(𝑑𝑑1 ) = 𝑁𝑁(1.092) = 0.863; 𝑁𝑁(𝑑𝑑2 ) = 𝑁𝑁(0.893) = 0.814
b.
21-14.
𝐶𝐶 = 𝑆𝑆 × 𝑁𝑁(𝑑𝑑1 ) − 𝑃𝑃𝑃𝑃(𝐾𝐾) × 𝑁𝑁(𝑑𝑑2 ) = 120 × 0.863 − 98.532 × 0.814 = $23.29
Using put-call parity:
𝑃𝑃 = 𝐶𝐶 − 𝑆𝑆 + 𝑃𝑃𝑃𝑃(𝐾𝐾) = 23.29 − 120 + 98.532 = $1.82
Using the market data in Figure 20.10 and a risk-free rate of 0.25% per annum, calculate the
implied volatility of Google stock in September 2012, using the bid price of the 700 January 2014
call option.
The bid of the call is $87.60, the time to expiration is approximately 495 / 365 = 1.356 years (September
10, 2012 to Jan 18, 2014 = 20 + 92 +365 + 18 = 495 days), and the price of the stock is $700.77.
With PV(K) = 700/(1+0.0025)1.356 = 697.63,
2
𝑑𝑑1 =
ln[𝑆𝑆/𝑃𝑃𝑃𝑃(𝐾𝐾)]
𝜎𝜎√𝑇𝑇
+
𝜎𝜎√𝑇𝑇 ln[770.77/697.63] 𝜎𝜎√1.356
=
+
2
2
𝜎𝜎√1.356
𝑑𝑑2 = 𝑑𝑑1 − 𝜎𝜎√𝑇𝑇 = 𝑑𝑑1 − 𝜎𝜎√1.356
The Black-Scholes formula, Eq. 21.7, implies:
You can find the σ by guessing or using a calculator or spreadsheet program. Thus, the implied volatility
for Google derived from this call option (using the bid price) is about 26.6%.
21-15.
Using the implied volatility you calculated in Problem 14, and the information in that problem,
use the Black-Scholes option pricing formula to calculate the value of the 750 January 2014 call
option.
Implied volatility is 26.6% from Problem 14. The strike price = 750, current stock price is 700.77, and
the risk-free rate is 0.25%, with a time to maturity of 1.356 years.
The Black-Scholes formula gives:
With PV(K) = 750/(1+0.0025)1.356 =747.46,
𝑑𝑑1 =
ln[𝑆𝑆/𝑃𝑃𝑃𝑃(𝐾𝐾)]
𝜎𝜎√𝑇𝑇
+
𝜎𝜎√𝑇𝑇 ln[700.77/747.46] 0.266√1.356
=
+
= −0.054
2
2
0.266√1.356
𝑑𝑑2 = 𝑑𝑑1 − 𝜎𝜎√𝑇𝑇 = −0.054 − 0.266√1.356 = −0.363
Therefore,
𝐶𝐶 = 𝑆𝑆 × 𝑁𝑁(𝑑𝑑1 ) − 𝑃𝑃𝑃𝑃(𝐾𝐾) × 𝑁𝑁(𝑑𝑑2 ) = 700.77 × 0.479 − 747.46 × 0.358 = $67.65
Note that the call price is in the range of the quotes provided in Table 20.10.
3
Chapter 22
Real Options
22-1.
Your company is planning on opening an office in Japan. Profits depend on how fast the economy
in Japan recovers from its current recession. There is a 50% chance of recovery this year. You are
trying to decide whether to open the office now or in a year. Construct the decision tree that shows
the choices you have to open the office either today or one year from now.
Decision Tree
22-5.
Describe the benefits and costs of delaying an investment opportunity.
By delaying, you delay the benefits of taking on the project and your competitors might take advantage
of this delay. However, by delaying, uncertainty can be resolved, so you can become better informed
and make better decisions.
22-6.
You are a financial analyst at Global Conglomerate and are considering entering the shoe business.
You believe that you have a very narrow window for entering this market. Because of Christmas
demand, the time is right today and you believe that exactly a year from now would also be a good
opportunity. Other than these two windows, you do not think another opportunity will exist to
break into this business. It will cost you $35 million to enter the market. Because other shoe
manufacturers exist and are public companies, you can construct a perfectly comparable
company. Hence, you have decided to use the Black-Scholes formula to decide when and if you
should enter the shoe business. Your analysis implies that the current value of an operating shoe
company is $40 million and it has a beta of 1. However, the flow of customers is uncertain, so the
1
value of the company is volatile—your analysis indicates that the volatility is 25% per year. Fifteen
percent of the value of the company is attributable to the value of the free cash flows (cash available
to you to spend how you wish) expected in the first year. If the one-year risk-free rate of interest
is 4%:
a.
Should Global enter this business and, if so, when?
b. How will the decision change if the current value of a shoe company is $36 million instead of
$40 million?
c.
Plot the value of your investment opportunity as a function of the current value of a shoe
company.
a.
PV(Div) = $40 × 0.15 = $6 million
Note: There is no need to discount $6 million because, according to the problem, it is already the
present value of cash flows in the first year.
Sx= S – PV(Div) = $40 – $40 × 0.15 = $34 million
K = 35 million, T = 1, σ = 0.25, rf = 0.04
𝑑𝑑1 =
ln[𝑆𝑆 𝑥𝑥 /𝑃𝑃𝑃𝑃(𝐾𝐾)]
𝜎𝜎√𝑇𝑇
35
𝜎𝜎√𝑇𝑇 ln[34/(1.04)] 0.25√1
+
=
+
= 0.166
2
2
0.25√1
𝑑𝑑2 = 𝑑𝑑1 − 𝜎𝜎√𝑇𝑇 = 0.166 − 0.25√1 = −0.084
𝑁𝑁(𝑑𝑑1 ) = 𝑁𝑁(0.166) = 0.566; 𝑁𝑁(𝑑𝑑2 ) = 𝑁𝑁(−0.084) = 0.467
𝐶𝐶 = 𝑆𝑆 𝑥𝑥 × 𝑁𝑁(𝑑𝑑1 ) − 𝑃𝑃𝑃𝑃(𝐾𝐾) × 𝑁𝑁(𝑑𝑑2 ) = 34 × 0.566 −
35
× 0.467 = $3.54
1.04
So the value of waiting is $3.54 million. The value of investing today is $40 – $35 = $5 million.
So they should enter the business now.
b.
Sx = S – PV(Div) = $36 – $36 × 0.15 = $30.6 million
K = 35 million, T = 1, σ = 0.25, rf = 0.04
𝑑𝑑1 =
ln[𝑆𝑆 𝑥𝑥 /𝑃𝑃𝑃𝑃(𝐾𝐾)]
𝜎𝜎√𝑇𝑇
35
𝜎𝜎√𝑇𝑇 ln[30.6/(1.04)] 0.25√1
+
=
+
= −0.256
2
2
0.25√1
𝑑𝑑2 = 𝑑𝑑1 − 𝜎𝜎√𝑇𝑇 = −0.256 − 0.25√1 = −0.506
𝑁𝑁(𝑑𝑑1 ) = 𝑁𝑁(−0.256) = 0.399; 𝑁𝑁(𝑑𝑑2 ) = 𝑁𝑁(−0.506) = 0.307
𝐶𝐶 = 𝑆𝑆 𝑥𝑥 × 𝑁𝑁(𝑑𝑑1 ) − 𝑃𝑃𝑃𝑃(𝐾𝐾) × 𝑁𝑁(𝑑𝑑2 ) = 30.6 × 0.399 −
35
× 0.307 = $1.90
1.04
So the value of waiting is $1.90 million. The value of investing today is $36 – $35 = $1 million.
So they should not enter the business now.
c.
2
22-7.
It is the beginning of September and you have been offered the following deal to go heli-skiing. If
you pick the first week in January and pay for your vacation now, you can get a week of heli-skiing
for $2500. However, if you cannot ski because the helicopters cannot fly due to bad weather, there
is no snow, or you get sick, you do not get a refund. There is a 40% probability that you will not
be able to ski. If you wait until the last minute and go only if you know that the conditions are
perfect and you are healthy, the vacation will cost $4000. You estimate that the pleasure you get
from heli-skiing is worth $6000 per week to you (if you had to pay any more than that, you would
choose not to go). If your cost of capital is 8% per year, should you book ahead or wait?
Decision Tree
If you book now, your expected benefit from skiing in 4 months is:
6, 000(0.60) + 0(0.40) =
$3, 600
The NPV of booking today is therefore:
3
NPV =
3, 600
4
− 2,500 = $1, 008.82
1.0812
If you wait to book the expected benefit in 4 months is:
2,000 (0.60) + 0 (0.40) = $1,200
The PV of this today is
$1, 200
4
1.0812
= $1,169.61
So you should wait.
4