75
Chapter 6
Valuing Bonds
6-1. A five-year bond with a face value of $1000 has a coupon rate of 6.5%, with semiannual
payments.
a. What is the coupon payment for this bond?
b. Draw the cash flows for the bond on a timeline.
a. The coupon payment is:
Coupon Rate Face Value 0.065 $1000 $32.50.
6-2. Assume that a bond will make payments every six months as shown on the following timeline:
a. What is the maturity of the bond (in years)?
b. What is the coupon rate (in percent)?
c. What is the face value?
6-3. The following table summarizes prices of various default-free, zero-coupon bonds (expressed as a
percentage of face value):
1
$32.50
0
2
$32.50
3
$32.50
10
$32.50 +
$1000
76 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition
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?
b. The yield curve is as shown below.
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 three-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 four-year maturity?
6-5. In the Global Financial Crisis box in Section 6.1, www.Bloomberg.com reported that the three-
month Treasury bill sold for a price of $100.002556 per $100 face value. What is the yield to
maturity of this bond, expressed as an EAR?
4
100 1 0.01022%
100.002556
− = −
78 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition
Bonds trading at a discount generate a return both from receiving the coupons and from receiving a
face value that exceeds the price paid for the bond. As a result, the yield to maturity of discount bonds
exceeds the coupon rate.
6-10. Suppose a seven-year, $1000 bond with a 10.46% coupon rate and semiannual coupons is trading
with a yield to maturity of 8.78%.
a. Is this bond currently trading at a discount, at par, or at a premium? Explain.
b. If the yield to maturity of the bond rises to 9.54% (APR with semiannual compounding), at
what price will the bond trade?
80 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition
©2017 Pearson Education, Ltd.
16 16
1 1 100
P(bond C, 7% YTM) 2 1 $52.77.
0.07 1.07 1.07
= − + =
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 6% YTM Price at 7% YTM .
Price at 7% YTM
−
The results are shown in the table below.
Bond
Coupon Rate
Maturity
Price at 7%
Price at 6%
Change
A
0%
16
$33.87
$39.36
16.2%
B
0%
12
$44.40
$49.70
11.9%
C
2%
16
$52.77
$59.58
12.9%
D
7%
12
$100.00
$108.38
8.4%
b. Bond A is most sensitive, because it has the longest maturity and no coupons, so its future cash
flows have the highest discount factors. Bond D is the least sensitive. Intuitively, higher coupon
rates and a shorter maturity mean that relatively more of the bond’s cash flows happen early and
thus cannot be as greatly affected by changes in interest rates as bonds with low coupon rates and
longer maturities.
6-14. Suppose you purchase a 30-year, zero-coupon bond with a yield to maturity of 4%. You hold the
bond for five years before selling it.
a. If the bond’s yield to maturity is 4% when you sell it, what is the internal rate of return of
your investment?
b. If the bond’s yield to maturity is 5% when you sell it, what is the internal rate of return of
your investment?
c. If the bond’s yield to maturity is 3% 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.
6-15. Suppose you purchase a 30-year Treasury bond with a 6% annual coupon, initially trading at
par. In 10 years’ time, the bond’s yield to maturity has risen to 7% (EAR).
a. If you sell the bond now, what internal rate of return will you have earned on your
investment in the bond?
b. If instead you hold the bond to maturity, what internal rate of return will you earn on your
investment in the bond?
c. Is comparing the IRRs in (a) versus (b) a useful way to evaluate the decision to sell the bond?
Explain.
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6-16. Suppose the current yield on a one-year, zero coupon bond is 4%, while the yield on a five-year,
zero coupon bond is 5%. Neither bond has any risk of default. Suppose you plan to invest for one
year. You will earn more over the year by investing in the five-year bond as long as its yield does
not rise above what level?
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?
2
60 60 1000 $1,018.81
(1 0.046) (1 0.05)
++
This bond trades at a premium. The coupon of the bond is greater than each of the zero-coupon yields,
Chapter 6/Valuing Bonds 83
The yield to maturity is:
2…
1(1 ) (1 )
50 50 1000
1010.05 … 4.77%.
(1 ) (1 )
N
N
CPN CPN CPN FV
PYTM YTM YTM
YTM
YTM YTM
+
= + + +
+++
+
= + + =
++
c. If the yield increased to 5.2%, the new price would be:
2…
1(1 ) (1 )
50 50 1000
… $991.39.
(1 .052) (1 .052)
N
N
CPN CPN CPN FV
PYTM YTM YTM
+
= + + +
+++
+
= + + =
++
6-23. 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 $1183.95. Is there an arbitrage opportunity? If so,
show specifically how you would take advantage of this opportunity. If not, why not?
Chapter 6/Valuing Bonds 85
b. What is the zero-coupon yield curve for years 1 through 4?
a. We can construct a two-year zero coupon bond using the one– and two-year coupon bonds as
23
(1.05832) (1.05832) (1.05832)
By the Law of One Price:
Price(Three-year zero) = Price(Three-year coupon bond) – Price(One-year zero) – Price(Two-year
86 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition
Finally, we can do the same for the four-year zero:
Cash Flow in Year:
1
2
3
4
Four-year coupon bond ($1000 face value)
140
140
140
1,140
Less: one-year zero ($140 face value)
(140)
Less: two-year zero ($140 face value)
—
(140)
Less: three-year zero ($140 face value)
—
—
(140)
Four-year zero ($1140 face value)
—
—
—
1,140
Now, Price(Four-year coupon bond) =
2 3 4
140 140 140 1,140 $1,289.76.
(1.05696) (1.05696) (1.05696) (1.05696)
+ + + =
By the Law of One Price:
Solving for the YTM:
1/4
4
1,140 1 5.86%.
907.70
YTM
= − =
Thus, we have computed the zero-coupon yield curve as shown.
6-26. Explain why the expected return of a corporate bond does not equal its yield to maturity.
The yield to maturity of a corporate bond is based on the promised payments of the bond. But there is
6-27. In the Data Case in Chapter 5, we suggested using the yield on Florida State bonds to estimate
the State of Florida’s cost of capital. Why might this estimate overstate the actual cost of capital?
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6-28. Grummon Corporation has issued zero-coupon corporate bonds with a five-year maturity.
Investors believe there is a 20% chance that Grummon will default on these bonds. If Grummon
does default, investors expect to receive only 50 cents per dollar they are owed. If investors
require a 6% expected return on their investment in these bonds, what will be the price and yield
to maturity on these bonds?
100((1 ) ( )) 67.25
d d r−+ =
67.25
6-29. The following table summarizes the yields to maturity on several one-year, zero-coupon
securities:
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?
6-30. Andrew Industries is contemplating issuing a 30-year bond with a coupon rate of 6.87% (annual
coupon payments) and a face value of $1000. Andrew believes it can get a rating of A from
Standard and Poor’s. However, due to recent financial difficulties at the company, Standard and
Poor’s is warning that it may downgrade Andrew Industries bonds to BBB. Yields on A-rated,
long-term bonds are currently 6.37%, and yields on BBB-rated bonds are 6.77%.
a. What is the price of the bond if Andrew maintains the A rating for the bond issue?
b. What will the price of the bond be if it is downgraded?
a. When originally issued, the price of the bond was
30
68.7 68.7 1,000 $1,066.18.
(1 0.0637) (1 0.0637)
P+
= + + =
++
88 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition
b. If the bond is downgraded, its price will fall to
30
68.7 68.7 1,000 $1,012.70.
(1 0.0677) (1 0.0677)
P+
= + + =
++
6-31. HMK Enterprises would like to raise $12 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 5.1% (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 $12 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 $966.21. What is the likely
rating of the bonds? Are they junk bonds?
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6-33. The Isabelle Corporation rents prom dresses in its stores across the southern United States. It
has just issued a five-year, zero-coupon, $100 face value corporate bond at a price of $65. You
have purchased this bond and intend to hold it until maturity.
a. What is the yield to maturity of the bond?
b. What is the expected return on your investment (expressed as an EAR) if there is no chance
of default?
c. What is the expected return (expressed as an EAR) if there is a 100% probability of default
and you will recover 90% of the face value?
d. What is the expected return (expressed as an EAR) if the probability of default is 50%, the
likelihood of default is higher in bad times than good times, and, in the case of default, you
will recover 90% of the face value?
e. For parts (b–d), what can you say about the five-year, risk-free interest rate in each case?
1/5
100 1 9.00%
6-34. What does it mean for a country to “inflate away” its debt? Why might this be costly for
investors even if the country does not default?
6-35. Suppose the yield on German government bonds is 1.3%, while the yield on Spanish government
bonds is 6.1%. Both bonds are denominated in euros. Which country do investors believe is more
likely to default? Why?
90 Berk/DeMarzo, Corporate Finance, Fourth Edition, Global Edition
Appendix
Problems A.1–A.4 refer to the following table:
6-A.1. What is the forward rate for year 2 (the forward rate quoted today for an investment that begins
in one year and matures in two years)?
= − = − =
1
(1 ) 1.04
+
6-A.2. What is the forward rate for year 3 (the forward rate quoted today for an investment that begins
in two years and matures in three years)? What can you conclude about forward rates when the
yield curve is flat?
From Eq 6A.2,
33
3
322
2
(1 ) 1.055
1 1 5.50%
(1 ) 1.055
YTM
fYTM
+
= − = − =
+
When the yield curve is flat (spot rates are equal), the forward rate is equal to the spot rate.
6-A.3. What is the forward rate for year 5 (the forward rate quoted today for an investment that begins
in four years and matures in five years)?
From Eq 6A.2,
55
5
544
4
(1 ) 1.045
1 1 2.52%
(1 ) 1.050
YTM
fYTM
+
= − = − =
+
When the yield curve is flat (spot rates are equal), the forward rate is equal to the spot rate.
6-A.4. Suppose you wanted to lock in an interest rate for an investment that begins in one year and
matures in five years. What rate would you obtain if there are no arbitrage opportunities?
Call this rate f1,5. If we invest for one-year at YTM1, and then for the four years from year 1 to 5 at rate
1,5 1.19825 1 4.625%.f= − =
6-A.5. Suppose the yield on a one-year, zero-coupon bond is 5%. The forward rate for year 2 is 4%,
and the forward rate for year 3 is 3%. What is the yield to maturity of a zero-coupon bond that
matures in three years?
Chapter 6/Valuing Bonds 91
©2017 Pearson Education, Ltd.
We can invest for three years with risk by investing for one year at 5%, then locking in a rate of 4% for
the second year, and 3% for the third year. The return from this strategy must equal the return from
investing in a three-year, zero-coupon bond (see Eq 6A.3):
(1 + YTM3)3 = (1.05)(1.04)(1.03) = 1.12476
Therefore: YTM3 = 1.124761/3 – 1 = 3.997%.