Chapter 6
Valuing Bonds
6-1. A 30-year bond with a face value of $1000 has a coupon rate of 5.5%, 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:
b. The timeline for the cash flows for this bond is (the unit of time on this timeline is six-month
periods):
6-2. Assume that a bond will make payments every six months as shown on the following timeline
(using six-month periods):
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):
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.
b. The yield curve is as shown below.
Zero Coupon Yield Curve
4.6
5
5.2
0 2 4 6
Maturity (Years)
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?
6-5. In the Global Financial Crisis box in Section 6.1, 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?
6-6. 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?
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?
We can use the annuity spreadsheet to solve for the payment.
6-8. The prices of several bonds with face values of $1000 are summarized in the following table:
6-9. Explain why the yield of a bond that trades at a discount exceeds the bond’s coupon rate.
6-10. Suppose a seven-year, $1000 bond with an 8% coupon rate and semiannual coupons is trading
with a yield to maturity of 6.75%.
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 7% (APR with semiannual compounding), what
price will the bond trade for?
a. Because the yield to maturity is less than the coupon rate, the bond is trading at a premium.
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?
Chapter 6/Valuing Bonds 81
a. When it was issued, the price of the bond was
6-12. 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.
82 Berk/DeMarzo, Corporate Finance, Fourth Edition
b. We can compute the IRR of the investment using the annuity spreadsheet. The PV is the purchase
6-13. 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. 8.5. For example, with a 6%
6-14. 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.
6-15. Suppose you purchase a 30-year Treasury bond with a 5% 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.
6-16. Suppose the current yield on a one-year, zero coupon bond is 3%, 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?
The return from investing in the one-year is the yield. The return for investing in the five-year for
84 Berk/DeMarzo, Corporate Finance, Fourth Edition
So you break even when
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?
6-18. What is the price of a five-year, zero-coupon, default-free security with a face value of $1000?
N
6-19. 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
The yield to maturity is
6-20. What is the maturity of a default-free security with annual coupon payments and a yield to
maturity of 4%? Why?
6-21. Consider a four-year, default-free security with annual coupon payments and a face value of
$1000 that is issued at par. What is the coupon rate of this bond?
Solve the following equation:
6-22. Consider a five-year, default-free bond with annual coupons of 5% and a face value of $1000.
a. Without doing any calculations, determine whether this bond is trading at a premium or at a
discount. Explain.
b. What is the yield to maturity on this bond?
c. If the yield to maturity on this bond increased to 5.2%, what would the new price be?
b. To compute the yield, first compute the price.
c. If the yield increased to 5.2%, the new price would be:
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 $1183.50. Is there an arbitrage opportunity? If so,
show specifically how you would take advantage of this opportunity. If not, why not?
86 Berk/DeMarzo, Corporate Finance, Fourth Edition
According to these zero-coupon yields, the price of the coupon bond should be:
The price of the coupon bond is too low, so there is an arbitrage opportunity. To take advantage of it:
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?
Chapter 6/Valuing Bonds 87
To take advantage of this opportunity, you want to (short) Sell Bond C (since it is overpriced). To
6-25. Suppose you are given the following information about the default-free, coupon-paying yield
curve:
a. Use arbitrage to determine the yield to maturity of a two-year, zero-coupon bond.
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
follows.
Cash Flow in Year:
Two-year coupon bond ($1,000 Face Value)
Less: one-year bond ($100 Face Value)
Two-year zero ($1,100 Face Value)
By the Law of One Price:
88 Berk/DeMarzo, Corporate Finance, Fourth Edition
b. We already know YTM(1) = 2%, YTM(2) = 4%. We can construct a three-year zero as follows:
Cash Flow in Year:
By the Law of One Price:
Price(Three-year zero) = Price(Three-year coupon bond) – Price(One-year zero) – Price(Two-year
zero)
Finally, we can do the same for the four-year zero:
Cash Flow in Year:
By the Law of One Price:
Price(Four-year zero) = Price(Four-year coupon bond) – PV(coupons in years 1–3)
Chapter 6/Valuing Bonds 89
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
some chance the corporation will default and pay less. Thus, the bond’s expected return is typically
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?
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?
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?
a. The price of this bond will be
6-30. Andrew Industries is contemplating issuing a 30-year bond with a coupon rate of 7% (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 6.5%, and yields on BBB-rated bonds are 6.9%.
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
6-31. 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?
Chapter 6/Valuing Bonds 91
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
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.
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?
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 corporate bond at a price of $74. 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?
92 Berk/DeMarzo, Corporate Finance, Fourth Edition
e. For parts (b–d), what can you say about the five-year, risk-free interest rate in each case?
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%, while the yield on Spanish government
bonds is 6%. Both bonds are denominated in euros. Which country do investors believe is more
likely to default? How can you tell?
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)?
From Eq 6A.2,
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,
Chapter 6/Valuing Bonds 93
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,
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
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?