CHAPTER 7
INTEREST RATES AND BOND
VALUATION
Answers to Concepts Review and Critical Thinking Questions
3. No. If the bid price were higher than the ask price, the implication would be that a dealer was willing
5. There are two benefits. First, the company can take advantage of interest rate declines by calling in
an issue and replacing it with a lower coupon issue. Second, a company might wish to eliminate a
6. 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
7. Yes. Some investors have obligations that are denominated in dollars; that is, they are nominal. Their
8. Companies pay to have their bonds rated simply because unrated bonds can be difficult to sell; many
9. Junk bonds often are not rated because there would be no point in an issuer paying a rating agency to
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10. The term structure is based on pure discount bonds. The yield curve is based on coupon-bearing
11. Bond ratings have a subjective factor to them. Split ratings reflect a difference of opinion among
12. 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
13. Lack of transparency means that a buyer or seller can’t see recent transactions, so it is much harder
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,
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 yield to maturity is the required rate of return on a bond expressed as a nominal annual interest
rate. For noncallable bonds, the yield to maturity and required rate of return are interchangeable
2. Price and yield move in opposite directions; if interest rates rise, the price of the bond will fall. This
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3. The price of any bond is the PV of the interest payment, plus the PV of the par value. Notice this
problem assumes an annual coupon. The price of the bond will be:
We would like to introduce shorthand notation here. Rather than write (or type, as the case may be)
which stands for Present Value Interest Factor
which stands for Present Value Interest Factor of an Annuity
These abbreviations are shorthand notation for the equations in which the interest rate and the
4. Here we need to find the YTM of a bond. The equation for the bond price is:
Notice the equation cannot be solved directly for R. Using a spreadsheet, a financial calculator, or
trial and error, we find:
If you are using trial and error to find the YTM of the bond, you might be wondering how to pick an
interest rate to start the process. First, we know the YTM has to be higher than the coupon rate since
Approximate YTM = [Annual interest payment + (Price difference from par / Years to maturity)] /
[(Price + Par value) / 2]
Solving for this problem, we get:
This is not the exact YTM, but it is close, and it will give you a place to start.
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5. 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:
Solving for the coupon payment, we get:
The coupon payment is the coupon rate times par value. Using this relationship, we get:
6. To find the price of this bond, we need to realize that the maturity of the bond is 14 years. The bond
7. Here we are finding the YTM of a semiannual coupon bond. The bond price equation is:
Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator, or trial
and error, we find:
Since the coupon payments are semiannual, this is the semiannual interest rate. The YTM is the APR
of the bond, so:
8. 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:
Solving for the coupon payment, we get:
Since this is the semiannual payment, the annual coupon payment is:
CHAPTER 27 – 5
And the coupon rate is the annual coupon payment divided by par value, so:
9. 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 semiannual compounding to keep the YTM of a zero coupon bond equivalent to the YTM
of a coupon bond, so:
10. 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:
11. 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:
12. The approximate relationship between nominal interest rates (R), real interest rates (r), and inflation
(h) is:
R r + h
The Fisher equation, which shows the exact relationship between nominal interest rates, real interest
rates, and inflation is:
(1 + R) = (1 + r)(1 + h)
13. The Fisher equation, which shows the exact relationship between nominal interest rates, real interest
rates, and inflation is:
(1 + R) = (1 + r)(1 + h)
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14. The Fisher equation, which shows the exact relationship between nominal interest rates, real interest
rates, and inflation is:
(1 + R) = (1 + r)(1 + h)
15. The Fisher equation, which shows the exact relationship between nominal interest rates, real interest
rates, and inflation is:
(1 + R) = (1 + r)(1 + h)
16. The coupon rate, located in the first column of the quote, is 4.375 percent. The bid price is:
Bid price = 112.6016 = 112.6016%
The previous day’s ask price is found by:
The previous day’s asked price in dollars was:
Previous day’s asked price = 113.3828 = 113.3828%
17. This is a premium bond because it sells for more than 100 percent of face value. The dollar asked
price is:
The current yield is the annual coupon payment divided by the price, so:
Current yield = Annual coupon payment / Price
The YTM is located under the “Asked Yield” column, so the YTM is 3.604 percent.
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The bid-ask spread is the difference between the bid price and the ask price, so:
In dollars, the bid-ask spread is:
Intermediate
18. 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)
X: P0 = $42.50(PVIFA3.5%,26) + $1,000(PVIF3.5%,26) = $1,126.68
P1 = $42.50(PVIFA3.5%,24) + $1,000(PVIF3.5%,24) = $1,120.44
P13 = $1,000
Y: P0 = $35(PVIFA4.25%,26) + $1,000(PVIF4.25%,26) = $883.33
P1 = $35(PVIFA4.25%,24) + $1,000(PVIF4.25%,24) = $888.52
Also, notice that the price of each bond when no time is left to maturity is the par value, even though
19. 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, 6.5 percent. If the YTM suddenly rises to 8.5 percent:
The percentage change in price is calculated as:
Percentage change in price = (New price – Original price) / Original price
If the YTM suddenly falls to 4.5 percent:
PSam% = ($1,055.54 – 1,000) / $1,000 = .0555, or 5.55%
All else the same, the longer the maturity of a bond, the greater is its price sensitivity to changes in
interest rates.
0%
1%
2%
3%
4%
5%
6%
7%
8%
9%
10%
$500
$1,000
$1,500
$2,000
$2,500
YTM and Bond Price
Bond Sam
Bond Dave
Yield to Maturity
Bond Price
CHAPTER 27 – 9
20. Initially, at a YTM of 6 percent, the prices of the two bonds are:
PJ = $15(PVIFA3%,38) + $1,000(PVIF3%,38) = $662.61
If the YTM rises from 6 percent to 8 percent:
PJ = $15(PVIFA4%,38) + $1,000(PVIF4%,38) = $515.80
The percentage change in price is calculated as:
Percentage change in price = (New price – Original price) / Original price
PJ% = ($515.80 – 662.61) / $662.61 = –.2216, or –22.16%
If the YTM declines from 6 percent to 4 percent:
PJ = $15(PVIFA2%,38) + $1,000(PVIF2%,38) = $867.80
All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to changes
in interest rates.