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
1.
(LO1) No. As interest rates fluctuate, the value of a government security will fluctuate. Long-term government
securities have substantial interest rate risk.
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.
3.
(LO4) No. If the bid were higher than the ask, the implication would be that a dealer was willing to sell a bond
and immediately buy it back at a higher price. How many such transactions would you like to do?
4.
(LO4) Prices and yields move in opposite directions. Since the bid price must be lower, the bid yield must be
higher.
5.
(LO1) 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 covenant for
some reason. Calling the issue does this. The cost to the company is a higher coupon. A put provision is
desirable from an investor’s standpoint, so it helps the company by reducing the coupon rate on the bond. The
cost to the company is that it may have to buy back the bond at an unattractive price.
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.
7.
(LO5) Yes. Some investors have obligations that are denominated in dollars; i.e., they are nominal. Their
primary concern is that an investment provides the needed nominal dollar amounts. Pension funds, for
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.
(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.
9.
(LO3) Government bonds have no credit risk, so a rating is not necessary. Junk bonds often are not rated
because there would 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.
(LO6) The term structure is based on pure discount bonds. The yield curve is based on coupon-bearing issues.
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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 8 percent.
2.
(LO2) Price and yield move in opposite directions; if interest rates fall, the price of the bond will rise and if
interest rates rise, the price of the bond will decrease. 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 decrease
when interest rates rose to15 percent.
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.
3.
(LO2) 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:
P = $75({1 – [1/(1 + .0875)10 ] } / .0875) + $1,000[1 / (1 + .0875)10] = $918.89
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:
PVIFR,t = 1 / (1 + r)t
which stands for Present Value Interest Factor
PVIFAR,t = ({1 – [1/(1 + r)t ]} / r )
which stands for Present Value Interest Factor of an Annuity
These abbreviations are short hand notation for the equations in which the interest rate and the number of
periods are substituted into the equation and solved. We will use this shorthand notation in remainder of the
solutions key.
4.
(LO2) Here we need to find the YTM of a bond. The equation for the bond price is:
P = $934 = $90(PVIFAR%,9) + $1,000(PVIFR%,9)
Notice the equation cannot be solved directly for R. Using a spreadsheet, a financial calculator, or trial and
error, we find:
R = YTM = 10.153%
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
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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 = [$90 + ($66 / 9] / [($934 + 1,000) / 2] = 10.07%
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 = $1,045 = C(PVIFA7.5%,13) + $1,000(PVIF7.5%,13)
Solving for the coupon payment, we get:
C = $80.54
The coupon payment is the coupon rate times par value. Using this relationship, we get:
Coupon rate = $80.54 / $1,000 = .08054 or 8.054%
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 = $34.5(PVIFA3.7%,20) + $1,000(PVIF3.7%,20) = $965.10
7.
(LO2) Here we are finding the YTM of a semiannual coupon bond. The bond price equation is:
P = $1,050 = $42(PVIFAR%,20) + $1,000(PVIFR%,20)
Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator, or trial and error,
we find:
R = 3.837%
Since the coupon payments are semiannual, this is the semiannual interest rate. The YTM is the APR of the
bond, so:
YTM = 2  3.837% = 7.674%
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 = $924 = C(PVIFA3.4%,29) + $1,000(PVIF3.4%,29)
Solving for the coupon payment, we get:
C = $29.84
Since this is the semiannual payment, the annual coupon payment is:
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2 × $29.84 = $59.68
And the coupon rate is the annual coupon payment divided by par value, so:
Coupon rate = $59.68 / $1,000
Coupon rate = .05968 or 5.968%
9.
(LO5) The approximate relationship between nominal interest rates (R), real interest rates (r), and inflation (h)
is:
R=r+h
Approximate r = .07 – .038 =.032 or 3.20%
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 + .07) = (1 + r)(1 + .038)
Exact r = [(1 + .07) / (1 + .038)] – 1 = .0308 or 3.08%
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 + .03)(1 + .047) – 1 = .07841 or 7.841%
11. (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)
h = [(1 + .14) / (1 + .09)] – 1 = .0459 or 4.59%
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 + .114) / (1.048)] – 1 = .063 or 6.3%
13.
(LO2) To calculate the value of the bond, we must recognize that the maturity of the bond is 3.4 years; 26
days remaining in January 2012, 29 days in February (leap year), 31 days in March and 31 days in May, for a
subtotal of 0.4 years, and then 3 years until June 1, 2015. For a $1,000 par value with a 4.5% coupon, a
111.23% bid price, and a current yield of 1.12%, the bond value is calculated as:
Semi-annual interest rate = 0.0112 / 2 = .0056
Semi-annual time periods = 3.4 x 2 = 6.8
$22.50 (PVIFA0.0056%, 6.8) + $1,000 (PVIF0.0056%, 6.8) = $1,123.63. Small differences are possible due
to rounding.
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This solution uses the shorthand notation:
PVIFR,t = 1 / (1 + r)t
which stands for Present Value Interest Factor
PVIFAR,t = ({1 – [1/(1 + r)]t } / r )
which stands for Present Value Interest Factor of an Annuity
Using the bond price equation, the solution is:
P = $22.50({1 – [1 / (1 + .0056)]6.8 } / .0056) + $1,000[1 / (1 + .0056)6.8] = $1,109.85
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 because they will increase in
price and any investor who buys the bonds at today’s price will receive a capital gain.
Intermediate
15. (LO2) Here we are finding the prices of the annual coupon bonds for various maturity lengths. The bond price
equation is:
P = C(PVIFAR%,t) + $1,000(PVIFR%,t)
X:
Y:
P0
P1
P3
P8
P12
P13
P0
P1
P3
P8
P12
P13
= $80(PVIFA6%,13) + $1,000(PVIF6%,13)
= $80(PVIFA6%,12) + $1,000(PVIF6%,12)
= $80(PVIFA6%,10) + $1,000(PVIF6%,10)
= $80(PVIFA6%,5) + $1,000(PVIF6%,5)
= $80(PVIFA6%,1) + $1,000(PVIF6%,1)
= $60(PVIFA8%,13) + $1,000(PVIF8%,13)
= $60(PVIFA8%,12) + $1,000(PVIF8%,12)
= $60(PVIFA8%,10) + $1,000(PVIF8%,10)
= $60(PVIFA8%,5) + $1,000(PVIF8%,5)
= $60(PVIFA8%,1) + $1,000(PVIF8%,1)
= $1,177.05
= $1,167.68
= $1,147.20
= $1,084.25
= $1,018.87
= $1,000
= $841.92
= $849.28
= $865.80
= $920.15
= $981.48
= $1,000
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.
Also, notice that the price of each bond when no time is left to maturity is the par value, even though the
purchaser would receive the par value plus the coupon payment immediately. This is because we calculate the
clean price of the bond.
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, 9 percent. If the YTM suddenly rises to 11 percent:
PSam
= $45(PVIFA5.5%,6) + $1,000(PVIF5.5%,6)
= $950.04
PDave
= $45(PVIFA5.5%,40) + $1,000(PVIF5.5%,40)
= $839.54
The percentage change in price is calculated as:
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Percentage change in price = (New price – Original price) / Original price
PSam%
= ($950.04-$1,000)/$1,000 = -5.00%
PDave%
= ($839.54-$1,000)/$1,000 = -16.05%
If the YTM suddenly falls to 7 percent:
PSam
= $$45 (PVIFA
3.5%,6)
+ $1,000 (PVIF
3.5%,6)
PDave
= $45(PVIFA3.5%,40) + $1,000(PVIF3.5%,40)
PSam%
= ($1,053.29-$1,000)/$1,000 = +5.33%
PDave%
= $1,053.29
= $1,213.55
= ($1,213.55-$1,000)/$1,000 = +21.36%
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 8 percent, the prices of the two bonds are:
PJ
= $20(PVIFA4%,18) + $1,000(PVIF4%,18)
= $746.81
PK
= $60(PVIFA4%,18) + $1,000(PVIF4%,18)
= $1,253.19
If the YTM rises from 8 percent to 10 percent:
PJ
= $20(PVIFA5%,18) + $1,000(PVIF5%,18) = $649.31
PK
= $60(PVIFA5%,18) + $1,000(PVIF5%,18)
= $1,116.89
The percentage change in price is calculated as:
Percentage change in price = (New price – Original price) / Original price
PJ%
PK%
= ($649.31 – 746.81) / $746.81
= ($1,116.89 – 1,253.19) / $1,253.19
= – 13.05%
= – 10.88%
If the YTM declines from 8 percent to 6 percent:
PJ
= $20(PVIFA3%,18) + $1,000(PVIF3%,18)
= $862.46
PK
= $60(PVIFA3%,18) + $1,000(PVIF3%,18)
= $1,412.61
PJ%
= ($862.46 – 746.81) / $746.81
= + 15.48%
PK%
= ($1,412.61 – 1,253.19) / $1,253.19
= + 12.72%
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 = $1,068 = $46(PVIFAR%,18) + $1,000(PVIFR%,18)
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Using a spreadsheet, financial calculator, or trial and error we find:
R = 4.0602132
This is the semiannual interest rate, so the YTM is:
YTM = 2  4.06% = 8.12%
The current yield is:
Current yield = Annual coupon payment / Price = $92 / $1,068 = .08614 or 8.614%
The effective annual yield is the same as the EAR, so using the EAR equation from the previous chapter:
Effective annual yield = (1 + 0.0406)2 – 1 = .0828 or 8.28%
19. (LO2) 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:
P = $930 = $40(PVIFAR%,40) + $1,000(PVIFR%,40)
Using a spreadsheet, financial calculator, or trial and error we find:
R = 4.37%
This is the semiannual interest rate, so the YTM is:
YTM = 2  4.37% = 8.74%
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 = $74/2 × 1/6 = $6.17
And we calculate the clean price as:
Clean price = Dirty price – Accrued interest = $968 – 6.17 = $961.83
21. (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 three months until the next coupon payment, so three
months have passed since the last coupon payment. The accrued interest for the bond is:
Accrued interest = $68/2 × 3/6 = $17
And we calculate the dirty price as:
Dirty price = Clean price + Accrued interest = $1,073 + 17 = $1,090
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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 = .075 = $80/P0
P0 = $80/.075 = $1,066.67
Now that we have the price of the bond, the bond price equation is:
P = $1,066.67 = $80[(1 – (1/1.072)t ) / .072 ] + $1,000/1.072t
We can solve this equation for t as follows:
$1,066.67(1.072)t = $1,111.11 (1.072)t – 1,111.11 + 1,000
111.11 = 44.44(1.072)t
2.5 = 1.072t
t = log 2.5 / log 1.072 = 13.18 or 13 years 2 months approximatley.
The bond has 13 years two months to maturity.
23. (LO4) The bond has 10 years to maturity, and like a typical corporate or government bond pays semi-annually,
so the bond price equation is:
P = $1,089.60 = $36(PVIFAR%,20) + $1,000(PVIFR%,20)
Using a spreadsheet, financial calculator, or trial and error we find:
R = 2.9978% or 3%
This is the semiannual interest rate, so the YTM is:
YTM = 2  3% = 6%
The “EST Spread” column shows the difference between the YTM of the bond quoted and the YTM of the
Government of Canada bond with a similar maturity. The column lists the spread in basis points. One basis
point is one-hundredth of one percent, so 100 basis points equals one percent. The spread for this bond is 468
basis points, or 4.68%. This makes the equivalent Treasury yield:
Equivalent Government bond yield = 6% – 4.68% = 1.32%
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.
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c.
25.
The current yield only refers to the yield of the bond at the current moment. It does not reflect the total
return over the life of the bond. 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.
(LO2) The price of a zero coupon bond is the PV of the par, so:
a.
P0 = $1,000/1.0925 = $115.97
b.
In one year, the bond will have 24 years to maturity, so the price will be:
P1 = $1,000/1.0924 = $126.40
The interest deduction is the price of the bond at the end of the year, minus the price at the beginning of
the year, so:
Year 1 interest deduction = $126.40 – 115.97 = $10.43
The price of the bond when it has one year left to maturity will be:
P29 = $1,000/1.09 = $917.43
Year 29 interest deduction = $1,000 – 917.43 = $82.57
c..
The company would prefer straight-line methods if allowed because the valuable interest deductions
occur earlier in the life of the bond. Under the zero coupon method they can only deduct the accrued
interest of $10.43 in the first year compared to $82.57 in the last year.
26. (LO2)
a.
The coupon bonds have a 8% coupon which matches the 8% required return, so they will sell at par
$1,000 Face Value or maturity value. The number of bonds that must be sold is the amount needed
divided by the bond price, so:
Number of coupon bonds to sell = $30,000,000 / $1,000 = 30,000
The number of zero coupon bonds to sell would be:
Price of zero coupon bonds = $1,000/1.0830 = $99.38
Number of zero coupon bonds to sell = $30,000,000 / $99.38 = 301,871.604
Note: In this case, the price of the bond was rounded to the number of cents when calculating the number
of bonds to sell.
b.
The repayment of the coupon bond will be the par value plus the last coupon payment times the number
of bonds issued. So:
Coupon bonds repayment = 30,000($1,080) = $32,400,000
The repayment of the zero coupon bond will be the par value times the number of bonds issued, so:
Zeroes: repayment = 301,871.604($1,000) = $301,871,604
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c.
The total coupon payment for the coupon bonds will be the number 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 tax rate. So:
Coupon bonds: (30,000)($80)(1–.35) = $1,560,000 cash outflow
Note that this is cash outflow since the company is making the interest payment.
For the zero coupon bonds, the first year interest payment is the difference in the price of the zero at the
end of the year and the beginning of the year. The price of the zeroes in one year will be:
P1 = $1,000/1.0829 = $107.32
The year 1 interest deduction per bond will be this price minus the price at the beginning of the year,
which we found in part b, so:
Year 1 interest deduction per bond = $107.32 – 99.38 = $7.94
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 = (301,871.604)($7.94)(.35) = $839,695.62
Notice the cash flow for the zeroes is a cash inflow. This is because of the tax deductibility of the
imputed interest expense. That is, the company gets to write off the interest expense for the year even
though the company did not have a cash flow for the interest expense. This reduces the company’s tax
liability, which is a cash inflow.
During the life of the bond, the zero generates cash inflows to the firm in the form of the interest tax
shield of debt. We should note an important point here: If you find the PV of the cash flows from the
coupon bond and the zero coupon bond, they will be the same. This is because of the much larger
repayment amount for the zeroes.
27. (LO2) We found the maturity of a bond in Problem 22. However, in this case, the maturity is indeterminate. A
bond selling at par can have any length of maturity. In other words, when we solve the bond pricing equation
as we did in Problem 22, the number of periods can be any positive number.
28. (LO5) We first need to find the real interest rate on the savings. Using the Fisher equation, the real interest rate
is:
(1 + R) = (1 + r)(1 + h)
1 + .11 = (1 + r)(1 + .038)
r = .0694 or 6.94%
Now we can use the future value of an annuity equation to find the annual deposit. Doing so, we find:
FVA = C{[(1 + r)t – 1] / r}
$1,500,000 = $C[(1.069440 – 1) / .0694]
C = $7,637.76
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Challenge
29. (LO2) 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 = $120(PVIFA9%,5) + $1,000(PVIF9%,5) = $1,116.69
P1 = $120(PVIFA9%,4) + $1,000(PVIF9%,4) = $1,097.19
Current yield = $120 / $1,116.69 = .1075 or 10.75%
The capital gains yield is:
Capital gains yield = (New price – Original price) / Original price
Capital gains yield = ($1,097.19 – 1,116.69) / $1,116.69 = –.0175 or –1.75%
The current price of Bond D and the price of Bond D in one year is:
D:
P0 = $60(PVIFA9%,5) + $1,000(PVIF9%,5) = $883.31
P1 = $60(PVIFA9%,4) + $1,000(PVIF9%,4) = $902.81
Current yield = $60 / $883.31 = .0679 or 6.79%
Capital gains yield = ($902.81 – 883.31) / $883.31 = +0.0221 or +2.21%
All else held constant, premium bonds pay high current income while having price depreciation as maturity
nears; discount bonds do not pay high current income but have price appreciation as maturity nears. For either
bond, the total return is still your Yield to Maturity 9%, but this return is distributed differently between
current income and capital gains.
30. (LO2)
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:
P0 = $1,060 = $70(PVIFAR%,10) + $1,000(PVIF R%,10)
Using a spreadsheet, financial calculator, or trial and error we find:
R = YTM = 6.178%
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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:
P2 = $70(PVIFA5.178%,8) + $1,000(PVIF5.178%,8) = $1,116.92
To calculate the HPY, we need to find the interest rate that equates the price we paid for the bond with
the cash flows we received. The cash flows we received were $80 each year for two years, and the price
of the bond when we sold it. The equation to find our HPY is:
P0 = $1,060 = $70(PVIFAR%,2) + $1,116.92 (PVIFR%,2)
Solving for R, we get:
R = HPY = 9.17%
The realized HPY is greater than the expected YTM when the bond was bought because interest rates dropped
by 1 percent; bond prices rise when yields fall.
31. (LO2) The price of any bond (or financial instrument) is the PV of the future cash flows. Even though Bond M
makes different coupons payments, to find the price of the bond, we just find the PV of the cash flows. The PV
of the cash flows for Bond M is:
PM
PM
= $1,100(PVIFA3.5%,16)(PVIF3.5%,12) + $1,400(PVIFA3.5%,12)(PVIF3.5%,28) + $20,000(PVIF3.5%,40)
= $19,018.78
Notice that for the coupon payments of $1,400, we found the PVA for the coupon payments, and then
discounted the lump sum back to today.
Bond N is a zero coupon bond with a $20,000 par value, therefore, the price of the bond is the PV of the par,
or:
PN
= $20,000(PVIF3.5%,40) = $5,051.45
S7-12
Calculator Solutions
Financial calculators typically require the PV function to have the opposite sign of the PMT and FV functions. For
the sake of consistency, values entered below were selected to generate positive solutions. Although this is an
arbitrary decision, it is strongly recommended that students get used to standardizing their calculator inputs to avoid
obtaining unexpected or incorrect outcomes. Note that while negative signs are used extensively in these solutions,
negative numbers are commonly entered by pressing the +/- key on a financial calculator after the number.
3. (LO2)
Enter
10
N
8.75%
I/Y
Solve for
4. (LO2)
Enter
9
N
Solve for
5. (LO2)
Enter
I/Y
10.153%
13
N
7.5%
I/Y
PV
$918.89
±$934
PV
±$1045
PV
Solve for
Coupon rate = $80.54 / $1,000 = 8.054%
6. (LO2)
Enter
20
N
3.70%
I/Y
Solve for
7. (LO2)
Enter
20
N
Solve for
3.84%  2 = 7.68%
8. (LO2)
Enter
29
N
I/Y
3.84%
3.40%
I/Y
PV
$965.10
±$1,050
PV
±$924
PV
Solve for
$29.84  2 = $59.68
$59.68 / $1,000 = 5.968%
15. (LO2)
P0
Enter
Solve for
$1,000
FV
$90
PMT
$1,000
FV
PMT
$80.54
$1,000
FV
$34.50
PMT
$1,000
FV
$42
PMT
$1,000
FV
PMT
$29.84
$1,000
FV
Bond X
13
N
6%
I/Y
12
N
6%
I/Y
Solve for
P1
Enter
$75
PMT
PV
$1,177.05
PV
$1167.68
S7-13
$80
PMT
$1,000
FV
$80
PMT
$1,000
FV
P3
Enter
10
N
6%
I/Y
5
N
6%
I/Y
1
N
6%
I/Y
Solve for
P8
Enter
Solve for
P12
Enter
Solve for
PV
$1,147.20
PV
$1,084.25
PV
$1,018.87
$80
PMT
$1,000
FV
$80
PMT
$1,000
FV
$80
PMT
$1,000
FV
$80
PMT
$1,000
FV
$60
PMT
$1,000
FV
$60
PMT
$1,000
FV
$60
PMT
$1,000
FV
$60
PMT
$1,000
FV
Bond Y
P0
Enter
13
N
8%
I/Y
12
N
8%
I/Y
10
N
8%
I/Y
5
N
8%
I/Y
1
N
8%
I/Y
Solve for
P1
Enter
Solve for
P3
Enter
Solve for
P8
Enter
Solve for
P12
Enter
Solve for
PV
$841.92
PV
$849.28
PV
$865.80
PV
$920.15
PV
$981.48
16. (LO2) If both bonds sell at par, the initial YTM on both bonds is the coupon rate, 9 percent. If the YTM
suddenly rises to 11 percent:
PSam
Enter
6
N
5.5%
I/Y
40
N
5.5%
I/Y
PV
Solve for
$950.04
PSam% = (926.24 – 1,000) / $1,000 = – 7.38%
PDave
Enter
PV
$839.54
Solve for
PDave% = ($839.54 - 1000) / $1000 = -16.05%
S7-14
$45
PMT
$1,000
FV
$45
PMT
$1,000
FV
If the YTM suddenly falls to 7 percent:
PSam
Enter
6
N
3.5%
I/Y
$45
PMT
$1,000
FV
40
N
3.5%
I/Y
$45
PMT
$1,000
FV
PV
Solve for
$1,053.29
PSam% = = ($1053.29-1000)/$1000 = +5.33%
PDave
Enter
PV
Solve for
$1,213.55
PDave% = =($1213.55-1000)/$1000 = +21.36%
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 8 percent, the prices of the two bonds are:
PJ
Enter
18
N
4%
I/Y
18
N
4%
I/Y
Solve for
PK
Enter
Solve for
PV
$746.81
PV
$1,253.19
$20
PMT
$1,000
FV
$60
PMT
$1,000
FV
If the YTM rises from 8 percent to 10 percent:
PJ
Enter
18
N
5%
I/Y
$20
PMT
$1,000
FV
18
N
5%
I/Y
$60
PMT
$1,000
FV
$20
PMT
$1,000
FV
$60
PMT
$1,000
FV
PV
Solve for
$649.31
PJ% = ($649.31 – 746.81) / $746.81 = – 13.05%
PK
Enter
PV
Solve for
$1,116.89
PK% = ($1,116.89 – 1,253.19) / $1,253.19 = – 10.88%
If the YTM declines from 8 percent to 6 percent:
PJ
Enter
18
N
3%
I/Y
PV
Solve for
$862.46
PJ% = ($862.46 – 746.81) / $746.81 = + 15.48
PK
Enter
18
3%
N
I/Y
PV
Solve for
$1,412.61
PK% = ($1,412.613 – 1,253.19) / $1,253.19 = + 12.72%
All else the same, the lower the coupon rate on a bond, the greater is its price sensitivity to
S7-15
changes in interest rates.
18. (LO2)
Enter
18
N
Solve for
I/Y
4.06%
±$1,068
PV
46
PMT
$1,000
FV
4.06  2 = 8.12%
Enter
8.12 %
NOM
Solve for
EFF
8.28%
2
C/Y
19. (LO2) 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.
Enter
40
N
Solve for
4.37%  2 = 8.74%
I/Y
4.37%
±$930
PV
40
PMT
$1,000
FV
22. (LO2) Current yield = .075 = $90/P0 ; P0 = $80/0.75 = $1,066.67
Enter
7.5%
N
I/Y
Solve for
13.83
13.83 or  13 years 10 months
23. (LO4)
Enter
20
N
Solve for
3% × 2 = 6%
I/Y
3%
±$1,066.67
PV
80
PMT
$1,000
FV
±1089.60
PV
36
PMT
$1,000
FV
PMT
$1,000
FV
PMT
$1,000
FV
PMT
$1,000
FV
Equivalent Government Bond Yield = 6.00% - 4.68% = 1.32%
25. (LO2)
a. Po
Enter
25
N
9%
I/Y
24
N
9%
I/Y
1
N
9%
I/Y
Solve for
b. P1
Enter
PV
$115.97
PV
Solve for
$126.40
year 1 interest deduction = $126.40 – 115.97 = $10.43
P24
Enter
Solve for
PV
$917.43
S7-16
year 25 interest deduction = $1,000 – 917.43 = $82.57
S7-17
26. (LO2)
a.
The coupon bonds have a 8% coupon rate, which matches the 8% required return, so
they will sell at par;
Number of bonds = $30M/$1000 = 30,000.
For the zeroes:
Enter
30
N
8%
I/Y
Solve for
$30M/$99.38 = 301,871.604 will be issued.
b.
PV
$99.38
PMT
$1,000
FV
Coupon bonds: repayment = 30,000($1,080) = $32.4M
Zeroes: repayment = 301,871.604($1,000) = $301,871,604
Coupon bonds: (30,000)($80)(1 –.35) = $1,560,000 cash outflow
Zeroes:
c.
Enter
$1,000
PV
PMT
FV
Solve for
$107.32
year 1 interest deduction = $107.32 – 99.38 = $7.94
(301,871,604)($7.94)(.35) = $839,695.62 cash inflow
During the life of the bond, the zero generates cash inflows to the firm in the form of the
interest tax shield of debt.
29. (LO2)
Bond P
P0
Enter
29
N
8%
I/Y
5
N
9%
I/Y
4
N
9%
I/Y
5
N
9%
I/Y
4
N
9%
I/Y
$120
PMT
$1,000
FV
$120
PV
PMT
Solve for
$1,097.19
Current yield = $120 / $1,116.69 = 10.75%
Capital gains yield = ($1,097.19 – 1,116.69) / $1,116.69 = –1.75%
$1,000
FV
Solve for
P1
Enter
Bond D
P0
Enter
Solve for
P1
Enter
PV
$1,116.69
PV
$883.31
PV
$902.81
$60
PMT
$1,000
FV
$60
PMT
$1,000
FV
Solve for
Current yield = $60 / $883.31 = 6.79%
Capital gains yield = ($902.81 – 883.31) / $883.31 = 2.21%
All else held constant, premium bonds pay high current income while having price depreciation as maturity
nears; discount bonds do not pay high current income but have price appreciation as maturity nears. For
S7-18
either bond, the total return is still 7%, but this return is distributed differently between current income and
capital gains.
30. (LO2)
a.
Enter
±$1,060
$70
$1,000
I/Y
PV
PMT
FV
Solve for
6.178%
This is the rate of return you expect to earn on your investment when you purchase the bond.
b.
Enter
10
N
8
N
5.178%
I/Y
Solve for
PV
$1,116.92
$70
PMT
$1,000
FV
The HPY is:
Enter
2
N
±$1,060
$80
$1,116.92
I/Y
PV
PMT
FV
Solve for
9.171%
The realized HPY is greater than the expected YTM when the bond was bought because interest rates
dropped by 1 percent; bond prices rise when yields fall.
31. (LO2)
PM
CFo
C01
F01
C02
F02
C03
F03
C04
F04
I = 3.5%
NPV CPT
$19,018.78
PN
Enter
Solve for
40
N
$0
$0
12
$1,100
16
$1,400
11
$21,400
1
3.5%
I/Y
PV
$5,501.45
S7-19
PMT
$20,000
FV
APPENDIX 7A
A.1 Portfolio managers use duration since bonds with differing coupons will respond differently to
interest rate changes. Duration measures will reflect these differences even if the bonds have equal
time to maturity.
A.2
Year
1
2
3
4
5
6
7
Payment
Present Value
Relative Value
Weighted Value
8
8
8
8
8
8
108
7.692308
7.39645
7.111971
6.838434
6.575417
6.322516
82.07112
124.0082
0.076923
0.073964
0.07112
0.068384
0.065754
0.063225
0.820711
0.076923
0.147929
0.213359
0.273537
0.328771
0.379351
5.744979
7.164849
The duration of the bond is 7.16 years using the YTM of 4% as the discount factor.
A.3 This is most simply understood by considering the case of the 1-year zero coupon bonds versus 5-year
zero coupon bonds. If interest rates rise, the 5-year bond will fall much more in price than the 1-year
bond, making the latter a more attractive purchase if interest rates are expected to rise.
APPENDIX 7B
B.1 NPV = [$1,000(.10 – .06)/.06] – $250 = $416.67
B.2 NPV = 0 = [$1,000(.10– r)/r] – $250; r = .10/1.250 = 8.00%
B.3 P = $1,000 = [$C + {0.5($C/.095) + 0.5($1,092)}]/1.07; C = $83.66, or 8.366%
B.4 P = $1,000 = [$60 + {0.5($60/.095) + 0.5($1,000 + x)}]/1.07; x = Call premium = $388.42
B.5 P = [$60 + {0.5($60/.095) + 0.5($1,080)}]/1.07= $855.88
B.6 P = [$60 + {0.5($60/.095) + 0.5($60/.055)}]/1.07 = $860.98 if no call provision present.
Cost of the call provision = $860.98 – $855.88 = $5.10
B.7
a. Under the assumption that the bond is callable at the stated premium of $68 in two years we need
to solve for the value of C such that:
P = $1,000 =
[$C + {0.30($C × PVIFA(7.75%, 18) + 1,000 × PVIF (7.75%, 18)) + 0.70($1,068)}]/(1.0625)2;
C = $78.49, or 7.85%
b. Solve for the value of C such that:
P = $1,000 =
[$C + {0.2($C × PVIFA(7.75%, 18) + 1,000 × PVIF (7.75%, 18)) +
0.5($C × PVIFA(7.00%, 18) + 1,000 × PVIF (7.00%, 18)) + 0.3($1,068)}]/(1.0625)2;
C = $76.65, or 7.67%
S2-21
B.8 # bonds outstanding = $53M/$1,000 = 53,000
NPV = 0 = [$53M(.0925 – r)/r] – $9M – 53,000($140); r = 7.06%
B.9 NPV = [$53M(.0925 – .0765)](PVIFA (7.65%, 16)) – $9M – 53,000($140) =
$7,676,886 – $9M – 53,000($140) = -$8,743,113
Challenge
B.10
NPV =
0 = [$53M(.0925 – r)](PVIFAr%, 16) – $9M – 53,000($140)
0 = -$16.42 + $4.903M(PVIFAr%, 16) – ($53M)r[{1 – (1/[1+r]16}/r]
0 = -$16.42M + $4.093M(PVIFAr%, 16) – ($53M) + $53M(1/[1+r]16)
0 = -$69.42M + $4.093M(PVIFAr%, 16) + $53M(PVIFr%, 16)
Using Solver in Excel: r = IRR = 6.1522%
[$1,000(.10 – .06)(1 – .39)/{.06(1– .39)}] – $250(1– .39)
= $666.67 – $152.50 = $514.17
The net effect of the taxes is to make refunding more attractive because of the tax deductibility of the
call premium, which is a cost. The difference between the NPV here and in Problem 1 is mainly due
to the $97.50 tax savings generated by the call premium.
B.11 NPV =
S2-21