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Business

BUSI 2504

Robert Riordan

Fall

Description

CHAPTER 7
INTEREST RATES AND BOND VALUATION
Learning Objectives
LO1 Important bond features and types of bonds.
LO2 Bond values and yields and why they fluctuate.
LO3 Bond ratings and what they mean.
LO4 How are bond prices quoted.
LO5 The impact of inflation on interest rates.
LO6 The term structure of interest rates and the determinants of bond yields.
Answers to Concepts Review and Critical Thinking Questions
2. (LO2) All else the same, the government security will have lower coupons because of its lower
default risk, so it will have greater interest rate risk.
4. (LO4) Prices and yields move in opposite directions. Since the bid price must be lower, the bid yield
must be higher.
6. (LO1) 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 also simply ask potential purchasers what coupon rate would be necessary to attract
them. The coupon rate is fixed and simply determines what the bond’s coupon payments will be. The
required return is what investors actually demand on the issue, and it will fluctuate through time. The
coupon rate and required return are equal only if the bond sells for exactly par.
8. (LO3) Companies pay to have their bonds rated simply because unrated bonds can be difficult to sell;
many large investors are prohibited from investing in unrated issues.
10. (LO6) The term structure is based on pure discount bonds. The yield curve is based on coupon-
bearing issues.
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. (LO2) 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 terms. Unlike YTM and required return, the coupon rate is not a return used as the
interest rate in bond cash flow valuation, but is a fixed percentage of par over the life of the bond used
to set the coupon payment amount. For the example given, the coupon rate on the bond is still 10
percent, and the YTM is 7 percent.
2. (LO2) Price and yield move in opposite directions; if interest rates fall, the price of the bond will rise.
This is because the fixed coupon payments determined by the fixed coupon rate are more valuable
when interest rates fall —hence, the price of the bond increases when interest rates drop to 3 percent.
S7-1 NOTE: Most problems do not explicitly list a par value for bonds. Even though a bond can have any par
value, in general, corporate bonds in Canada will have a par value of $1,000. We will use this par value in
all problems unless a different par value is explicitly stated.
4. (LO2) Here we need to find the YTM of a bond. The equation for the bond price is:
P = $1,080 = $70(PVIFA R%,9 + $1,000(PVIF R%,9)
Notice the equation cannot be solved directly for R. Using a spreadsheet, a financial calculator, or trial
and error, we find:
R = YTM = 5.83%
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
the bond is a discount bond. That still leaves a lot of interest rates to check. One way to get a starting
point is to use the following equation, which will give you an approximation of the YTM:
Approximate YTM = [Annual interest payment + (Price difference from par / Years to maturity)] /
[(Price + Par value) / 2]
Solving for this problem, we get:
Approximate YTM = [$70 + (–$80 / 9] / [($1,080 + 1,000) / 2] = 5.88%
This is not the exact YTM, but it is close, and it will give you a place to start
5. (LO2) 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 = $870 = C(PVIFA 7.5%,16+ $1,000(PVIF 7.5%,16
Solving for the coupon payment, we get:
C = $60.78
The coupon payment is the coupon rate times par value. Using this relationship, we get:
Coupon rate = $60.78 / $1,000 = .0608 or 6.08%
6. (LO2) To find the price of this bond, we need to realize that the maturity of the bond is 10 years. The
bond was issued one year ago, with 11 years to maturity, so there are 10 years left on the bond. Also,
the coupons are semiannual, so we need to use the semiannual interest rate and the number of
semiannual periods. The price of the bond is:
P = $39(PVIFA 4.3%,20 + $1,000(PVIF 4.3%,20= $947.05
7. (LO2) Here we are finding the YTM of a semiannual coupon bond. The bond price equation is:
P = $1,040 = $46(PVIFA ) + $1,000(PVIF )
R%,20 R%,20
Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator, or trial
and error, we find:
R = 4.298%
S7-2 Since the coupon payments are semiannual, this is the semiannual interest rate. The YTM is the APR
of the bond, so:
YTM = 2 × 4.298% = 8.60%
8. (LO2) 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 = $1,136.50 = C(PVIFA 3.4%,29+ $1,000(PVIF 3.4%,29
Solving for the coupon payment, we get:
C = $41.48
Since this is the semiannual payment, the annual coupon payment is:
2 × $41.48 = $82.95
And the coupon rate is the annual coupon payment divided by par value, so:
Coupon rate = $82.95 / $1,000
Coupon rate = .08295 or 8.30%
9. (LO5) The approximate relationship between nominal interest rates (R), real interest rates (r), and
inflation (h) is:
R = r + h
Approximate r = .08 – .045 =.035 or 3.50%
The Fisher equation, which shows the exact relationship between nominal interest rates, real interest
rates, and inflation is:
(1 + R) = (1 + r)(1 + h)
(1 + .08) = (1 + r)(1 + .045)
Exact r = [(1 + .08) / (1 + .045)] – 1 = .0335 or 3.35%
10. (LO5) 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 + .058)(1 + .04) – 1 = .1003 or 10.03%
12. (LO5) 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 + .142) / (1.053)] – 1 = .0845 or 8.45%
14. (LO2) There is a negative relationship between bond yields and bond prices. If an investment
manager thinks that yields on Quebec provincial bonds will decrease then (s)he should buy them
S7-3 because they will increase in price and any investor who buys the bonds at today’s price will receive a
capital gain.
Intermediate
16. (LO2) 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, 8 percent. If the YTM suddenly rises to 10 percent:
PSam = $40(PVIFA 5%,4 + $1,000(PVIF 5%,4 = $964.54
PDave = $40(PVIFA 5%,30+ $1,000(PVIF 5%,30 = $846.28
The percentage change in price is calculated as:
Percentage change in price = (New price – Original price) / Original price
ΔP Sam = ($964.54 – 1,000) / $1,000 = – 3.55%
ΔP Dave = ($846.28 – 1,000) / $1,000 = – 15.37%
If the YTM suddenly falls to 6 percent:
PSam = $40(PVIFA 3%,4 + $1,000(PVIF 3%,4 = $1,037.17
PDave = $40(PVIFA 3%,30+ $1,000(PVIF 3%,30 = $1,196.00
ΔP Sam = ($1,037.17 – 1,000) / $1,000 = + 3.72%
ΔP Dave = ($1,196.00 – 1,000) / $1,000 = + 19.60%
All else the same, the longer the maturity of a bond, the greater is its price sensitivity to changes in
interest rates.
17. (LO2) Initially, at a YTM of 7 percent, the prices of the two bonds are:
PJ = $20(PVIFA 3.5%,16+ $1,000(PVIF 3.5%,16 = $818.59
PK = $60(PVIFA 3.5%,16+ $1,000(PVIF 3.5%,16 = $1,302.35
If the YTM rises from 7 percent to 9 percent:
PJ = $20(PVIFA 4.5%,16+ $1,000(PVIF 4.5%,16 = $719.15
PK = $60(PVIFA 4.5%,16+ $1,000(PVIF 4.5%,16 $1,168.51
The percentage change in price is calculated as:
Percentage change in price = (New price – Original price) / Original price
ΔP J = ($719.15 – 818.59) / $818.59 = – 12.15%
ΔP % = ($1,168.51 – 1,302.35) / $1,302.35 = – 10.28%
K
If the YTM declines from 7 percent to 5 percent:
PJ = $20(PVIFA 2.5%,16+ $1,000(PVIF 2.5%,16 $934.72
S7-4 PK = $60(PVIFA 2.5%,16+ $1,000(PVIF 2.5%,16 = $1,456.93
ΔP J = ($934.72 – 818.59) / $818.59 = + 14.19%
ΔP K = ($1,456.93 – 1,302.35) / $1,302.35 = + 11.87%
All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to changes in
interest rates.
18. (LO2) The bond price equation for this bond is:
P0= $955 = $42(PVIFA R%,18 + $1,000(PVIF R%,18
Using a spreadsheet, financial calculator, or trial and error we find:
R = 4.572%
This is the semiannual interest rate, so the YTM is:
YTM = 2 × 4.572% = 9.14%
The current yield is:
Current yield = Annual coupon payment / Price = $84 / $955 = .0880 or 8.80%
The effective annual yield is the same as the EAR, so using the EAR equation from the previous
chapter:
Effective annual yield = (1 + 0.04572) – 1 = .0935 or 9.35%
20. (LO2) 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 five months until the next coupon
payment, so one month has passed since the last coupon payment. The accrued interest for the bond
is:
Accrued interest = $86/2 × 1/6 = $7.17
And we calculate the clean price as:
Clean price = Dirty price – Accrued interest = $1,090 – 7.17 = $1,082.83
22. (LO2) 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 = .0710 = $90/P 0
P0= $90/.0710 = $1,267.61
Now that we have the price of the bond, the bond price equation is:
P = $1,267.61 = $90[(1 – (1/1.063) ) / .063 ] + $1,000/1.063 t
S7-5 We can solve this equation for t as follows:
$1,267.61(1.063) = $1,428.57 (1.063) – 1,428.57 + 1,000
428.57 = 160.96(1.063) t
t
2.6626 = 1.063
t = log 2.6626 / log 1.063 = 16.03 ≈ 16 years
The bond has 16 years to maturity.
24. (LO2)
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
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
YTM; for discount bonds, the YTM exceeds the coupon rate, and for bonds selling at par, the
YTM is equal to the coupon rate.
c. Current yield = annual coupon payment / price. Yield to maturity (YTM) is the interest rate
required in the market on a bond, and this yield value is the discount rate used in the valuation
formula for a bond. A premium bond sells above par value, and the current yield is always
greater than YTM for a premium bond. A discount bond sells below par value, and the current
yield is always lower than the YTM for a discount bond. For bonds selling at par, the current
yield and YTM are equal.
26. (LO2)
a. The coupon bonds have a 7% coupon which matches the 7% 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 = $20,000,000 / $1,000 = 20,000
The number of zero coupon bonds to sell would be:

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