Bonds and Their Valuation
? List the four main classifications of bonds and differentiate among them. After reading this chapter, students should be able to: ? Identify the key characteristics common to all bonds.
? Calculate the value of a bond with annual or semiannual interest
? Explain why the market value of an outstanding fixed-rate bond will fall
when interest rates rise on new bonds of equal risk, or vice versa.
? Calculate the current yield, the yield to maturity, and/or the yield to
call on a bond.
? Differentiate between interest rate risk, reinvestment rate risk, and
? List major types of corporate bonds and distinguish among them.
? Explain the importance of bond ratings and list some of the criteria
used to rate bonds.
? Differentiate among the following terms: Insolvent, liquidation, and
? Read and understand the information provided on the bond market page of
Learning Objectives: 7 - 1
This chapter serves two purposes. First, it provides important and useful
information on bonds per se. Second, it provides a good example of the use of
time value concepts, so it reinforces the topics covered in Chapter 6.
We begin our lecture with a discussion of the different types of bonds
and their characteristics. Then we move on to how bond values are established,
how yields are determined, the effects of changing interest rates on bond
prices, and the riskiness inherent in different types of bonds.
The details of what we cover, and the way we cover it, can be seen by
scanning Blueprints, Chapter 7. For other suggestions about the lecture,
please see the “Lecture Suggestions” in Chapter 2, where we describe how we
conduct our classes.
DAYS ON CHAPTER: 4 OF 58 DAYS (50-minute periods)
Lecture Suggestions: 7 - 2
ANSWERS TO END-OF-CHAPTER QUESTIONS
7-1 Yes, the statement is true.
7-2 False. Short-term bond prices are less sensitive than long-term bond
prices to interest rate changes because funds invested in short-term
bonds can be reinvested at the new interest rate sooner than funds tied
up in long-term bonds.
7-3 The price of the bond will fall and its YTM will rise if interest rates
rise. If the bond still has a long term to maturity, its YTM will
reflect long-term rates. Of course, the bond’s price will be less
affected by a change in interest rates if it has been outstanding a long
time and matures shortly. While this is true, it should be noted that
the YTM will increase only for buyers who purchase the bond after the
change in interest rates and not for buyers who purchased previous to
If the bond is purchased and held to maturity, the bondholder’s YTM will
not change, regardless of what happens to interest rates.
7-4 If interest rates decline significantly, the values of callable bonds
will not rise by as much as those of bonds without the call provision.
It is likely that the bonds would be called by the issuer before
maturity, so that the issuer can take advantage of the new, lower rates.
7-5 From the corporation’s viewpoint, one important factor in establishing a
sinking fund is that its own bonds generally have a higher yield than do
government bonds; hence, the company saves more interest by retiring its
own bonds than it could earn by buying government bonds. This factor
causes firms to favor the second procedure. Investors also would prefer
the annual retirement procedure if they thought that interest rates were
more likely to rise than to fall, but they would prefer the government
bond purchase program if they thought rates were likely to fall. In
addition, bondholders recognize that, under the government bond purchase
scheme, each bondholder would be entitled to a given amount of cash from
the liquidation of the sinking fund if the firm should go into default,
whereas under the annual retirement plan, some of the holders would
receive a cash benefit while others would benefit only indirectly from
the fact that there would be fewer bonds outstanding.
On balance, investors seem to have little reason for choosing one
method over the other, while the annual retirement method is clearly
more beneficial to the firm. The consequence has been a pronounced
trend toward annual retirement and away from the accumulation scheme.
7-6 a. If a bond’s price increases, its YTM decreases.
Answers and Solutions: 7 - 3
b. If a company’s bonds are downgraded by the rating agencies, its YTM
c. If a change in the bankruptcy code made it more difficult for
bondholders to receive payments in the event a firm declared
bankruptcy, then the bond’s YTM would increase.
d. If the economy entered a recession, then the possibility of a firm
defaulting on its bond would increase; consequently, its YTM would
e. If a bond were to become subordinated to another debt issue, then the
bond’s YTM would increase.
7-7 As an investor with a short investment horizon, I would view the 20-year
Treasury security as being more risky than the 1-year Treasury security.
If I bought the 20-year security, I would bear a considerable amount of
interest rate risk. Since my investment horizon is only one year, I
would have to sell the 20-year security one year from now, and the price
I would receive for it would depend on what happened to interest rates
during that year. However, if I purchased the 1-year security I would
be assured of receiving my principal at the end of that one year, which
is the 1-year Treasury’s maturity date.
Answers and Solutions: 7 - 4
SOLUTIONS TO END-OF-CHAPTER PROBLEMS
7-1 With your financial calculator, enter the following:
N = 10; I = YTM = 9%; PMT = 0.08 ? 1,000 = 80; FV = 1000; PV = V = ? BPV = $935.82.
7-2 With your financial calculator, enter the following to find YTM:
N = 10 ? 2 = 20; PV = -1100; PMT = 0.08/2 ? 1,000 = 40; FV = 1000; I = YTM = ?
YTM = 3.31% ? 2 = 6.62%.
With your financial calculator, enter the following to find YTC:
N = 5 ? 2 = 10; PV = -1100; PMT = 0.08/2 ? 1,000 = 40; FV = 1050; I = YTC = ?
YTC = 3.24% ? 2 = 6.49%.
7-3 The problem asks you to find the price of a bond, given the following
facts: N = 16; I = 8.5/2 = 4.25; PMT = 45; FV = 1000.
With a financial calculator, solve for PV = $1,028.60.
= $985; M = $1,000; Int = 0.07 ? $1,000 = $70. B
a. Current yield = Annual interest/Current price of bond
b. N = 10; PV = -985; PMT = 70; FV = 1000; YTM = ?
Solve for I = YTM = 7.2157% ? 7.22%.
c. N = 7; I = 7.2157; PMT = 70; FV = 1000; PV = ?
Solve for V
= PV = $988.46. B
7-5 a. 1. 5%: Bond L: Input N = 15, I = 5, PMT = 100, FV = 1000, PV = ?,
PV = $1,518.98.
Bond S: Change N = 1, PV = ? PV = $1,047.62.
2. 8%: Bond L: From Bond S inputs, change N = 15 and I = 8, PV = ?,
PV = $1,171.19.
Bond S: Change N = 1, PV = ? PV = $1,018.52. Answers and Solutions: 7 - 5
3. 12%: Bond L: From Bond S inputs, change N = 15 and I = 12, PV = ?,
PV = $863.78.
Bond S: Change N = 1, PV = ? PV = $982.14.
b. Think about a bond that matures in one month. Its present value is
influenced primarily by the maturity value, which will be received in
only one month. Even if interest rates double, the price of the bond
will still be close to $1,000. A 1-year bond’s value would fluctuate
more than the one-month bond’s value because of the difference in the
timing of receipts. However, its value would still be fairly close
to $1,000 even if interest rates doubled. A long-term bond paying
semiannual coupons, on the other hand, will be dominated by distant
receipts, receipts that are multiplied by 1/(1 + kt/2), and if k ddincreases, these multipliers will decrease significantly. Another
way to view this problem is from an opportunity point of view. A 1-
month bond can be reinvested at the new rate very quickly, and hence
the opportunity to invest at this new rate is not lost; however, the
long-term bond locks in subnormal returns for a long period of time.
N7-6 a. VINTM = ?B?tN(1?k)(1?k)t1?dd
M = $1,000. I = 0.09($1,000) = $90.
1. V = $829: Input N = 4, PV = -829, PMT = 90, FV = 1000, I = ? I = B14.99%.
2. V = $1,104: Change PV = -1104, I = ? I = 6.00%. B
b. Yes. At a price of $829, the yield to maturity, 15 percent, is
greater than your required rate of return of 12 percent. If your
required rate of return were 12 percent, you should be willing to buy
the bond at any price below $908.88.
7-7 The rate of return is approximately 15.03 percent, found with a
calculator using the following inputs:
N = 6; PV = -1000; PMT = 140; FV = 1090; I = ? Solve for I = 15.03%.
7-8 a. Using a financial calculator, input the following:
N = 20, PV = -1100, PMT = 60, FV = 1000, and solve for I = 5.1849%.
However, this is a periodic rate. The nominal annual rate =
5.1849%(2) = 10.3699% ? 10.37%.
b. The current yield = $120/$1,100 = 10.91%. Answers and Solutions: 7 - 6 c. YTM = Current Yield + Capital Gains (Loss) Yield
10.37% = 10.91% + Capital Loss Yield
d. Using a financial calculator, input the following:
N = 8, PV = -1100, PMT = 60, FV = 1060, and solve for I = 5.0748%. -0.54% = Capital Loss Yield.
However, this is a periodic rate. The nominal annual rate =
5.0748%(2) = 10.1495% ? 10.15%.
7-9 The problem asks you to solve for the YTM, given the following facts:
N = 5, PMT = 80, and FV = 1000. In order to solve for I we need PV.
However, you are also given that the current yield is equal to 8.21%.
Given this information, we can find PV.
Current yield = Annual interest/Current price
0.0821 = $80/PV
PV = $80/0.0821 = $974.42.
Now, solve for the YTM with a financial calculator:
N = 5, PV = -974.42, PMT = 80, and FV = 1000. Solve for I = YTM = 8.65%.
7-10 The problem asks you to solve for the current yield, given the following
facts: N = 14, I = 10.5883/2 = 5.29415, PV = -1020, and FV = 1000. In
order to solve for the current yield we need to find PMT. With a
financial calculator, we find PMT = $55.00. However, because the bond
is a semiannual coupon bond this amount needs to be multiplied by 2 to
obtain the annual interest payment: $55.00(2) = $110.00. Finally, find
the current yield as follows:
Current yield = Annual interest/Current price = $110/$1,020 = 10.78%.
7-11 The bond is selling at a large premium, which means that its coupon rate
is much higher than the going rate of interest. Therefore, the bond is
likely to be called--it is more likely to be called than to remain
outstanding until it matures. Thus, it will probably provide a return
equal to the YTC rather than the YTM. So, there is no point in
calculating the YTM--just calculate the YTC. Enter these values:
N = 10, PV = -1353.54, PMT = 70, FV = 1050, and then solve for I.
The periodic rate is 3.2366 percent, so the nominal YTC is 2 ? 3.2366% =
6.4733% ? 6.47%. This would be close to the going rate, and it is about
what the firm would have to pay on new bonds.
7-12 a. To find the YTM:
N = 10, PV = -1175, PMT = 110, FV = 1000
I = YTM = 8.35%.
b. To find the YTC, if called in Year 5:
Answers and Solutions: 7 - 7
N = 5, PV = -1175, PMT = 110, FV = 1090
I = YTC = 8.13%.
c. The bonds are selling at a premium which indicates that interest rates
have fallen since the bonds were originally issued. Assuming that
interest rates do not change from the present level, investors would
expect to earn the yield to call. (Note that the YTC is less than the
If called in Year 6:
d. Similarly from above, YTC can be found, if called in each subsequent N = 6, PV = -1175, PMT = 110, FV = 1080
year. I = YTM = 8.27%.
If called in Year 7:
N = 7, PV = -1175, PMT = 110, FV = 1070
I = YTM = 8.37%.
If called in Year 8:
N = 8, PV = -1175, PMT = 110, FV = 1060
I = YTM = 8.46%.
If called in Year 9:
N = 9, PV = -1175, PMT = 110, FV = 1050
I = YTM = 8.53%.
According to these calculations, the latest investors might expect a
call of the bonds is in Year 6. This is the last year that the
expected YTC will be less than the expected YTM. At this time, the
firm still finds an advantage to calling the bonds, rather than
seeing them to maturity.
7-13 First, we must find the amount of money we can expect to sell this bond
for in 5 years. This is found using the fact that in five years, there
will be 15 years remaining until the bond matures and that the expected
YTM for this bond at that time will be 8.5%.
N = 15, I = 8.5, PMT = 90, FV = 1000
PV = -$1,041.52. V = $1,041.52. B
This is the value of the bond in 5 years. Therefore, we can solve for
the maximum price we would be willing to pay for this bond today,
subject to our required rate of return of 10%.
N = 5, I = 10, PMT = 90, FV = 1041.52
PV = -$987.87. V = $987.87. B
We are willing to pay up to $987.87 for this bond today.
7-14 Before you can solve for the price, we must find the appropriate
semiannual rate at which to evaluate this bond.
Answers and Solutions: 7 - 8
2 EAR = (1 + NOM/2) - 1 20.0816 = (1 + NOM/2) - 1
NOM = 0.08.
Semiannual interest rate = 0.08/2 = 0.04 = 4%.
Solving for price:
N = 20, I = 4, PMT = 45, FV = 1000
PV = -$1,067.95. V = $1,067.95. B
7-15 a. The current yield is defined as the annual coupon payment divided by
the current price.
CY = $80/$901.40 = 8.875%.
b. Solving for YTM:
N = 9, PV = -901.40, PMT = 80, FV = 1000
I = YTM = 9.6911%.
c. Expected capital gains yield can be found as the difference between
YTM and the current yield.
CGY = YTM - CY = 9.691% - 8.875% = 0.816%.
Alternatively, you can solve for the capital gains yield by first
finding the expected price next year.
N = 8, I = 9.6911, PMT = 80, FV = 1000
PV = -$908.76. V = $908.76. B
Hence, the capital gains yield is the percent price appreciation over
the next year.
CGY = (P - P)/P = ($908.76 - $901.40)/$901.40 = 0.816%. 100
7-16 Using the TIE ratio, we can solve for the firm's current operating
TIE = EBIT/Int Exp
3.2 = EBIT/$10,500,000
EBIT = $33,600,000.
Using the same methodology, you can solve for the maximum interest expense the firm can bear without violating its covenant.
2.5 = $33,600,000/Int Exp
Max Int Exp = $13,440,000.
Therefore, the firm can raise debt to the point that its interest expense increases by $2.94 million ($13.44 ? $10.50). The firm can
raise $25 million at 8%, which would increase the cost of debt by $25 ?
0.08 = $2 million. Additional debt will be issued at 10%, and the amount of debt to be raised can be found, since we know that only an additional $0.94 million in interest expense can be incurred.
Answers and Solutions: 7 - 9
Hence, the firm may raise up to $34.4 million in additional debt without Additional Int Exp = Additional Debt ? Cost of debt violating its bond covenants. $0.94 million = Additional Debt ? 0.10 7-17 First, we must find the price Baili paid for this bond. Additional Debt = $9.40 million.
N = 10, I = 9.79, PMT = 110, FV = 1000
PV = -$1,075.02. V = $1,075.02. B
Then to find the one-period return, we must find the sum of the change
in price and the coupon received divided by the starting price.
Ending price - Beginning price ? Coupon receivedOne-period return = Beginning price
One-period return = ($1,060.49 - $1,075.02 + $110)/$1,075.02
One-period return = 8.88%.
7-18 The answer depends on when one works the problem. We used The Wall
Street Journal, February 3, 2003:
a. AT&T’s 8.625%, 2031 bonds had an 8.6 percent current yield. The
bonds sold at a premium, 100.75% of par, so the coupon interest rate
would have to be set lower than 8.625% for the bonds to sell at par.
If we assume the bonds aren’t callable, we can do a rough calculation
of their YTM. Using a financial calculator, we input the following
N = 29 ? 2 = 58, PV = 1.0075 ? -1,000 = -1007.50, PMT =
0.08625 ? 21,000 = 86.25/2 = 43.125, FV = 1000, and then solve for YTM = k = d
4.2773% ? 2 = 8.5546%.
Thus, AT&T would have to set a rate of 8.55 percent on new long-term
b. The return on AT&T’s bonds is the current yield of 8.6 percent, less
a small capital loss in 2031. The total return is about 8.55 percent.
7-19 a. Yield to maturity (YTM):
With a financial calculator, input N = 28, PV = -1165.75, PMT = 95, FV
= 1000, I = ? I = k = YTM = 8.00%. d
Yield to call (YTC):
With a calculator, input N = 3, PV = -1165.75, PMT = 95, FV = 1090,
I = ? I = k = YTC = 6.11%. d
Answers and Solutions: 7 - 10