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CHAPTER 8
INTEREST RATES AND BOND
VALUATION
Answers to Concept Questions
2. All else the same, the Treasury security will have lower coupons because of its lower default risk, so
it will have greater interest rate risk.
do?
4. Prices and yields move in opposite directions. Since the bid price must be lower, the bid yield must
be higher.
5. Bond issuers look at outstanding bonds of similar maturity and risk. The yields on such bonds are
used to establish the coupon rate necessary for a particular issue to initially sell for par value. Bond
6. Yes. Some investors have obligations that are denominated in dollars; i.e., they are nominal. Their
8. Treasury bonds have no credit risk since they are backed by the U.S. government, so a rating is
9. The term structure is based on pure discount bonds. The yield curve is based on coupon-bearing
issues.
11. As a general constitutional principle, the federal government cannot tax the states without their
consent if doing so would interfere with state government functions. At one time, this principle was
12. Lack of transparency means that a buyer or seller can’t see recent transactions, so it is much harder
to determine what the best bid and ask prices are at any point in time.
13. When the bonds are initially issued, the coupon rate is set at auction so that the bonds sell at par
value. The wide range of coupon rates shows the interest rate when each bond was issued. Notice
that interest rates have evidently declined. Why?
14. Companies charge that bond rating agencies are pressuring them to pay for bond ratings. When a
15. A 100-year bond looks like a share of preferred stock. In particular, it is a loan with a life that almost
certainly exceeds the life of the lender, assuming that the lender is an individual. With a junk bond,
16. a. The bond price is the present value of the cash flows from a bond. The YTM is the interest rate
used in valuing the cash flows from a bond.
b. If the coupon rate is higher than the required return on a bond, the bond will sell at a premium,
since it provides periodic income in the form of coupon payments in excess of that required by
17. A long-term bond has more interest rate risk compared to a short-term bond, all else the same. A low
coupon bond has more interest rate risk than a high coupon bond, all else the same. When comparing
a high coupon, long-term bond to a low coupon, short-term bond, we are unsure which has more
Solutions to Questions and Problems
NOTE: All end-of-chapter problems were solved using a spreadsheet. Many problems require multiple
steps. Due to space and readability constraints, when these intermediate steps are included in this solutions
manual, rounding may appear to have occurred. However, the final answer for each problem is found
without rounding during any step in the problem.
Basic
1. The price of a pure discount (zero coupon) bond is the present value of the par value. Remember, even
though there are no coupon payments, the periods are semiannual to stay consistent with coupon bond
2. The price of any bond is the PV of the interest payments, plus the PV of the par value. Notice this
problem assumes a semiannual coupon. The price of the bond at each YTM will be:
a. P = $35({1 – [1/(1 + .035)]30}/.035) + $1,000[1/(1 + .035)30]
3. Here we are finding the YTM of a semiannual coupon bond. The bond price equation is:
R = 3.215%
4. Here we need to find the coupon rate of the bond. All we need to do is to set up the bond pricing
equation and solve for the coupon payment as follows:
P = $1,055 = C(PVIFA3.4%,23) + $1,000(PVIF3.4%,23)
5. The price of any bond is the PV of the interest payments, plus the PV of the par value. The fact that
the bond is denominated in euros is irrelevant. Notice this problem assumes an annual coupon. The
6. Here we are finding the YTM of an annual coupon bond. The fact that the bond is denominated in yen
is irrelevant. The bond price equation is:
7. To find the price of a zero coupon bond, we need to find the value of the future cash flows. With a
zero coupon bond, the only cash flow is the par value at maturity. We find the present value assuming
8. To find the price of this bond, we need to find the present value of the bond’s cash flows. So, the price
9. To find the price of this bond, we need to find the present value of the bond’s cash flows. So, the price
of the bond is:
10. The approximate relationship between nominal interest rates (R), real interest rates (r), and inflation
(h) is:
R r + h
11. The Fisher equation, which shows the exact relationship between nominal interest rates, real interest
rates, and inflation, is:
12. The Fisher equation, which shows the exact relationship between nominal interest rates, real interest
rates, and inflation, is:
13. The Fisher equation, which shows the exact relationship between nominal interest rates, real interest
rates, and inflation, is:
14. The coupon rate, located in the second column of the quote, is 5.250 percent. The bid price is:
Bid price = 129.1328 = 129.1328%
Bid price = (129.1328/100)($10,000)
15. This is a premium bond because it sells for more than 100 percent of face value. The dollar asked price
is:
Price = (128.4688/100)($10,000)
Price = $12,846.88
The current yield is the annual coupon payment divided by the price, so:
16. Zero coupon bonds are priced with semiannual compounding to correspond with coupon bonds. The
price of the bond when purchased was:
P0 = $1,000/(1 + .0285)40
17. Here we are finding the YTM of annual coupon bonds for various maturity lengths. The bond price
equation is:
P = C(PVIFAR%,t) + $1,000(PVIFR%,t)
Bond Miller:
P0 = $32.50(PVIFA2.65%,26) + $1,000(PVIF2.65%,26) = $1,111.71
P1 = $32.50(PVIFA2.65%,24) + $1,000(PVIF2.65%,24) = $1,105.55
Bond Modigliani:
P0 = $26.50(PVIFA3.25%,26) + $1,000(PVIF3.25%,26) = $895.76
P1 = $26.50(PVIFA3.25%,24) + $1,000(PVIF3.25%,24) = $901.07
$1,200
$1,300
Maturity and Bond Price
All else held equal, the premium over par value for a premium bond declines as maturity approaches,
and the discount from par value for a discount bond declines as maturity approaches. This is called
“pull to par.” In both cases, the largest percentage price changes occur at the shortest maturity lengths.
18. Any bond that sells at par has a YTM equal to the coupon rate. Both bonds sell at par, so the initial
YTM on both bonds is the coupon rate, 5.8 percent. If the YTM suddenly rises to 7.8 percent:
PLaurel = $29(PVIFA3.9%,6) + $1,000(PVIF3.9%,6) = $947.41
PHardy = $29(PVIFA3.9%,40) + $1,000(PVIF3.9%,40) = $799.09
The percentage change in price is calculated as:
19. Initially, at a YTM of 9 percent, the prices of the two bonds are:
PFaulk = $35(PVIFA4.5%,24) + $1,000(PVIF4.5%,24) = $855.05
PYoo = $55(PVIFA4.5%,24) + $1,000(PVIF4.5%,24) = $1,144.95
If the YTM rises from 9 percent to 11 percent:
20. The bond price equation for this bond is:
R = 2.733%
This is the semiannual interest rate, so the YTM is:
YTM = 2 2.733% = 5.47%
21. The company should set the coupon rate on its new bonds equal to the required return. The required
return can be observed in the market by finding the YTM on outstanding bonds of the company. So,
the YTM on the bonds currently sold in the market is:
22. Accrued interest is the coupon payment for the period times the fraction of the period that has passed
since the last coupon payment. Since we have a semiannual coupon bond, the coupon payment per six
months is one-half of the annual coupon payment. There are two months until the next coupon
payment, so four months have passed since the last coupon payment. The accrued interest for the bond
is:
Accrued interest = $64/2 × 4/6
23. Accrued interest is the coupon payment for the period times the fraction of the period that has passed
since the last coupon payment. Since we have a semiannual coupon bond, the coupon payment per six
months is one-half of the annual coupon payment. There are four months until the next coupon
24. To find the number of years to maturity for the bond, we need to find the price of the bond. Since we
already have the coupon rate, we can use the bond price equation, and solve for the number of years
to maturity. We are given the current yield of the bond, so we can calculate the price as:
Current yield = .0842 = $90/P0
P0 = $90/.0842 = $1,068.88
Now that we have the price of the bond, the bond price equation is:
25. The bond has 12 years to maturity, so the bond price equation is:
R = 3.852%
This is the semiannual interest rate, so the YTM is:
26. We found the maturity of a bond in Problem 24. However, in this case, the maturity is indeterminate.
27. The price of a zero coupon bond is the PV of the par value, so:
a. P0 = $1,000/1.029550
P0 = $233.71
b. In one year, the bond will have 24 years to maturity, so the price will be:
P1 = $1,000/1.029548
28. a. The coupon bonds have a 7 percent coupon which matches the 7 percent required return, so they
will sell at par. The number of bonds that must be sold is the amount needed divided by the bond
price, so:
Number of coupon bonds to sell = $50,000,000/$1,000
Number of coupon bonds to sell = 50,000
so:
Zeroes repayment = 393,905($1,000)
Zeroes repayment = $393,904,545
c. The total coupon payment for the coupon bonds will be the number of bonds times the coupon
payment. For the cash flow of the coupon bonds, we need to account for the tax deductibility of
the interest payments. To do this, we will multiply the total coupon payment times one minus the
Year 1 interest deduction per bond = $135.98 – 126.93
Year 1 interest deduction per bond = $9.04
The total cash flow for the zeroes will be the interest deduction for the year times the number of
zeroes sold, times the tax rate. The cash flow for the zeroes in Year 1 will be:
Cash flows for zeroes in Year 1 = (393,905)($9.04)(.21)
29. To find the capital gains yield and the current yield, we need to find the price of the bond. The current
price of Bond P and the price of Bond P in one year is:
P: P0 = $80(PVIFA6.5%,10) + $1,000(PVIF6.5%,10) = $1,107.83
P1 = $80(PVIFA6.5%,9) + $1,000(PVIF6.5%,9) = $1,099.84
Current yield = $80/$1,107.83
30. a. The rate of return you expect to earn if you purchase a bond and hold it until maturity is the YTM.
The bond price equation for this bond is:
b. To find our HPY, we need to find the price of the bond in two years. The price of the bond in two
years, at the new interest rate, will be:
