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CHAPTER 5
INTEREST RATES AND BOND
VALUATION
Answers to Concept Questions
1. No. As interest rates fluctuate, the value of a Treasury security will fluctuate. Long-term Treasury
2. All else the same, the Treasury security will have lower coupons because of its lower default risk, so
5. There are two benefits. First, the company can take advantage of interest rate declines by calling in an
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 issuers
are equal only if the bond sells for exactly at par.
example, often must plan for pension payments many years in the future. If those payments are fixed
in dollar terms, then it is the nominal return on an investment that is important.
8. Companies pay to have their bonds rated because unrated bonds can be difficult to sell; many large
9. Junk bonds often are not rated because there would be no point in an issuer paying a rating agency to
assign its bonds a low rating (it’s like paying someone to kick you!).
10. The term structure is based on pure discount bonds. The yield curve is based on coupon-bearing issues.
11. Bond ratings have a subjective factor to them. Split ratings reflect a difference of opinion among credit
agencies.
if doing so would interfere with state government functions. At one time, this principle was thought
to provide for the tax-exempt status of municipal interest payments. However, modern court rulings
make it clear that Congress can revoke the municipal exemption, so the only basis now appears to be
historical precedent. The fact that the states and the federal government do not tax each other’s
13. 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.
The wide range coupon of coupon rates shows the interest rate when the bond was issued. Notice that
interest rates have evidently declined. Why?
company pays for a rating, it has the opportunity to make its case for a particular rating. With an
unsolicited rating, the company has no input.
16. 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,
the credit risk can be so high that the borrower is almost certain to default, meaning that the creditors
are very likely to end up as part owners of the business. In both cases, the “equity in disguise” has a
used in valuing the cash flows from a bond.
since it provides periodic income in the form of coupon payments in excess of that required by
investors on other similar bonds. If the coupon rate is lower than the required return on a bond,
the bond will sell at a discount since it provides insufficient coupon payments compared to that
required by investors on other similar bonds. For premium bonds, the coupon rate exceeds the
premium bonds, the current yield exceeds the YTM, for discount bonds the current yield is less
than the YTM, and for bonds selling at par value, the current yield is equal to the YTM. In all
cases, the current yield plus the expected one-period capital gains yield of the bond must be equal
18. 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
interest rate risk. Generally, the maturity of a bond is a more important determinant of the interest rate
Solutions to Questions and Problems
1. The price of a pure discount (zero coupon) bond is the present value of the par. Remember, even
though there are no coupon payments, the periods are semiannual to stay consistent with coupon bond
payments. So, the price of the bond for each YTM is:
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)]46 }/.035) + $1,000[1/(1 + .035)46]
P = $1,000.00
b. P = $35({1 – [1/(1 + .045)]46 }/.045) + $1,000[1/(1 + .045)46]
P = $807.12
c. P = $35({1 – [1/(1 + .025)]46 }/.025) + $1,000[1/(1 + .025)46]
P = $1,271.54
We would like to introduce shorthand notation here. Rather than write (or type, as the case may be)
the entire equation for the PV of a lump sum, or the PVA equation, it is common to abbreviate the
equations as:
3. Here we are finding the YTM of a semiannual coupon bond. The bond price equation is:
P = $1,050 = $25.50(PVIFAR%,26) + $1,000(PVIFR%,26)
Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator, or trial
and error, we find:
4. Here we need to find the coupon rate of the bond. We need to set up the bond pricing equation and
solve for the coupon payment as follows:
P = $1,040 = C(PVIFA3.2%,25) + $1,000(PVIF3.2%,25)
Solving for the coupon payment, we get:
And the coupon rate is the annual coupon payment divided by par value, so:
5. The price of any bond is the PV of the interest payment, 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 price
of the bond will be:
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:
P = ¥103,250 = ¥4,900(PVIFAR%,18) + ¥100,000(PVIFR%,18)
Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator, or trial
and error, we find:
R = 4.63%
(h) is:
R = r + h
Approximate r = .048 – .027
rates, and inflation, is:
(1 + R) = (1 + r)(1 + h)
R = (1 + .018)(1 + .034) – 1
rates, and inflation, is:
(1 + R) = (1 + r)(1 + h)
h = [(1 + .121)/(1 + .076)] – 1
10. The Fisher equation, which shows the exact relationship between nominal interest rates, real interest
rates, and inflation, is:
(1 + R) = (1 + r)(1 + h)
r = [(1 + .114)/(1 + .039)] – 1
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:
P = $4,391.30
the price of the bond is:
P = $49(PVIFA1.90%,26) + $2,000(PVIF1.90%,26)
the price of the bond is:
P = $92.50(PVIFA1.95%,32) + $5,000(PVIF1.95%,32)
Bid price = (139.5156/100)
$1,000
Bid price = $1,395.156
The previous day’s ask price is found by:
Previous day’s asked price = Today’s asked price – Change
Previous day’s asked price = 139.5781 – .1406
Previous day’s dollar price = (139.4375/100)
$1,000
Previous day’s dollar price = $1,394.375
based on the asked price, so the current yield is:
Current yield = Annual coupon payment/Price
Current yield = $45/$1,381.250
Current yield = .0326, or 3.26%
price equation is:
P = C(PVIFAR%,t) + $1,000(PVIFR%,t)
P3 = $41(PVIFA3.1%,20) + $1,000(PVIF3.1%,20) = $1,147.41
P8 = $41(PVIFA3.1%,10) + $1,000(PVIF3.1%,10) = $1,084.87
P12 = $41(PVIFA3.1%,2) + $1,000(PVIF3.1%,2) = $1,019.11
P1 = $31(PVIFA4.1%,24) + $1,000(PVIF4.1%,24) = $849.08
P3 = $31(PVIFA4.1%,20) + $1,000(PVIF4.1%,20) = $865.29
P8 = $31(PVIFA4.1%,10) + $1,000(PVIF4.1%,10) = $919.29
P12 = $31(PVIFA4.1%,2) + $1,000(PVIF4.1%,2) = $981.17
is called “pull to par.” In both cases, the largest percentage price changes occur at the shortest maturity
lengths.
17. 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:
PLaurel = $32.50(PVIFA4.25%,8) + $1,000(PVIF4.25%,8) = $933.36
PHardy = $32.50(PVIFA4.25%,46) + $1,000(PVIF4.25%,46) = $799.39
The percentage change in price is calculated as:
$700
$800
$900
$1,000
$1,100
$1,200
$1,300
13 12 11 10 9876543210
Bond Price
Maturity (Years)
Maturity and Bond Price
Miller Bond
Modigliani Bond
18. Initially, at a YTM of 9 percent, the prices of the two bonds are:
PFaulk = $28.50(PVIFA4.5%,28) + $1,000(PVIF4.5%,28) = $740.24
PGonas = $61.50(PVIFA4.5%,28) + $1,000(PVIF4.5%,28) = $1,259.76
PFaulk = $28.50(PVIFA5.5%,28) + $1,000(PVIF5.5%,28) = $625.78
PGonas = $61.50(PVIFA5.5%,28) + $1,000(PVIF5.5%,28) = $1,091.79
$500
$700
$900
$1,100
$1,300
$1,500
$1,700
$1,900
$2,100
$2,300
$2,500
0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%
Bond Price
Yield to Maturity
YTM and Bond Price
Bond Laurel
Bond Hardy
The percentage change in price is calculated as:
Percentage change in price = (New price – Original price)/Original price
19. The current yield is:
Current yield = Annual coupon payment/Price
Current yield = $64/$1,080
Current yield = .0593, or 5.93%
The bond price equation for this bond is:
This is the semiannual interest rate, so the YTM is:
