Bonds and Their Valuation
After reading this chapter, students should be able to:
; List the four main classifications of bonds and differentiate among them. ; Identify the key characteristics common to all bonds.
; Calculate the value of a bond with annual or semiannual interest payments. ; Calculate the yield to maturity, the yield to call, and the current yield on a bond. ; 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.
; Differentiate between interest rate risk, reinvestment rate risk, and default risk. ; 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 reorganization. ; Read and understand the information provided on the bond market page of your newspaper.
Chapter 7: Bonds and Their Valuation Learning Objectives 141
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 2.
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.
What we cover, and the way we cover it, can be seen by scanning the slides and Integrated Case solution for Chapter 7, which appears at the end of this chapter solution. 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: 3 OF 58 DAYS (50-minute periods)
142 Lecture Suggestions Chapter 7: Bonds and Their Valuation
Answers to End-of-Chapter Questions
7-1 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.
7-2 Yes, the statement is true.
7-3 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.
For example, consider two bonds, both with a 10% annual coupon and a $1,000 par value. The
only difference between them is their maturity. One bond is a 1-year bond, while the other is a 20-
year bond. Consider the values of each at 5%, 10%, 15%, and 20% interest rates.
5% $1,047.62 $1,623.11
10% 1,000.00 1,000.00
15% 956.52 687.03
20% 916.67 513.04
As you can see, the price of the 20-year bond is much more volatile than the price of the 1-year bond.
7-4 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 the change. If the bond is purchased and held to maturity,
the bondholder’s YTM will not change, regardless of what happens to interest rates. For example,
consider two bonds with an 8% annual coupon and a $1,000 par value. One bond has a 5-year maturity,
while the other has a 20-year maturity. If interest rates rise to 15% immediately after issue the value of
the 5-year bond would be $765.35, while the value of the 20-year bond would be $561.85.
7-5 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.
7-6 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-
Chapter 7: Bonds and Their Valuation Integrated Case 143
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.
7-7 a. If a bond’s price increases, its YTM decreases.
b. If a company’s bonds are downgraded by the rating agencies, its YTM increases.
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 increase.
e. If a bond were to become subordinated to another debt issue, then the bond’s YTM would increase.
7-8 If a company sold bonds when interest rates were relatively high and the issue is callable, then the
company could sell a new issue of low-yielding securities if and when interest rates drop. The
proceeds of the new issue would be used to retire the high-rate issue, and thus reduce its interest
expense. The call privilege is valuable to the firm but detrimental to long-term investors, who will be
forced to reinvest the amount they receive at the new and lower rates.
7-9 A sinking fund provision facilitates the orderly retirement of the bond issue. Although sinking funds
are designed to protect investors by ensuring that the bonds are retired in an orderly fashion, sinking
funds can work to the detriment of bond holders. On balance, however, bonds that have a sinking
fund are regarded as being safer than those without such a provision, so at the time they are issued
sinking fund bonds have lower coupon rates than otherwise similar bonds without sinking funds.
7-10 A call for sinking fund purposes is quite different from a refunding call- a sinking fund call requires no
call premium, but only a small percentage of the issue is normally callable in a given year. A refunding
call gives the issuer the right to call all the bond issue for redemption. The call provision generally states
that the issuer must pay the bondholders an amount greater than the par value if they are called.
7-11 Convertibles and bonds with warrants are offered with lower coupons than similarly-rated straight
bonds because both offer investors the chance for capital gains as compensation for the lower coupon
rate. Convertible bonds are exchangeable into shares of common stock, at a fixed price, at the option
of the bondholder. On the other hand, bonds issued with warrants are options that permit the holder
to buy stock for a stated price, thereby providing a capital gain if the stock’s price rises.
7-12 This statement is false. Extremely strong companies can use debentures because they simply do not
need to put up property as security for their debt. Debentures are also issued by weak companies
that have already pledged most of their assets as collateral for mortgage loans. In this latter case, the
debentures are quite risky, and that risk will be reflected in their interest rates.
7-13 The yield spread between a corporate bond over a Treasury bond with the same maturity reflects
both investors’ risk aversion and their optimism or pessimism regarding the economy and corporate
profits. If the economy appeared to be heading into a recession, the spread should widen. The
change in spread would be even wider if a firm’s credit strength weakened.
7-14 Assuming a bond issue is callable, the YTC is a better estimate of a bond’s expected return when
interest rates are below an outstanding bond’s coupon rate. The YTM is a better estimate of a bond’s
expected return when interest rates are equal or above an outstanding bond’s coupon rate.
144 Integrated Case Chapter 7: Bonds and Their Valuation
Solutions to End-of-Chapter Problems
9-1 With your financial calculator, enter the following:
N = 10; I/YR = YTM = 9%; INT = 0.08 ？ 1,000 = 80; FV = 1000; PV = V = ? B
PV = $935.82.
9-2 V = $985; M = $1,000; Int = 0.07 ？ $1,000 = $70. B
Current yield = Annual interest/Current price of bond
a. N = 10; PV = -985; INT = 70; FV = 1000; YTM = ?
Solve for Rd = YTM = 7.2157% ， 7.22%.
b. N = 7; Rd = 7.2157; PMT = 70; FV = 1000; PV = ?
Solve for V = PV = $988.46. B
9-3 The problem asks you to find the price of a bond, given the following facts: N = 2 ？ 8 = 16; Rd =
8.5/2 = 4.25; INT = 45; FV = 1000.
With a financial calculator, solve for PV = $1,028.60.
9-4 With your financial calculator, enter the following to find YTM:
N = 10 ？ 2 = 20; PV = -1100; INT = 0.08/2 ？ 1,000 = 40; FV = 1000; Rd = 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/YR = YTC = ?
YTC = 3.24% ？ 2 = 6.49%.
Since the YTC is less than the YTM, investors would expect the bonds to be called and to earn the
YTC.( most likely yield=6.49%)
9-5 a. 1. 5%: Bond L: Input N = 15, Rd = 5, INT = 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 Rd = 8, PV = ?, PV = $1,171.19.
Bond S: Change N = 1, PV = ? PV = $1,018.52.
3. 12%: Bond L: From Bond S inputs, change N = 15 and Rd = 12, PV = ?, PV = $863.78.
Bond S: Change N = 1, PV = ? PV = $982.14.
Chapter 7: Bonds and Their Valuation Integrated Case 145
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 t/2), and if r increases, these multipliers will decrease significantly. multiplied by 1/(1 + rdd
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.
9-6 a. Years to Maturity Price of Bond C Price of Bond Z
4 $1,012.79 $ 693.04
3 1,010.02 759.57
2 1,006.98 832.49
1 1,003.65 912.41
0 1,000.00 1,000.00
b. Bond Price PathsBond Price Paths
Bond CBond C $1,000$1,000
$800$800 Bond PriceBond PriceBond ZBond Z
Price at 8% Price at 7% Change
10-year, 10% annual coupon $1,134.20 $1,210.71 6.75%
10-year zero 463.19 508.35 9.75
5-year zero 680.58 712.99 4.76
30-year zero 99.38 131.37 32.19
$100 perpetuity 1,250.00 1,428.57 14.29
7-8 The rate of return is approximately 15.03%, N = 6; PV = 1000; INT =100X0.14= 140; FV = 1090;
I/YR = ? Solve for I/YR = 15.03%.
146 Integrated Case Chapter 7: Bonds and Their Valuation
Despite a 15% return on the bonds, investors are not likely to be happy that they were called. Because if the bonds have been called, this indicates that interest rates have fallen sufficiently that the YTC is less than the YTM. (Since they were originally sold at par, the YTM at issuance= 14%.) Rates are sufficiently low to justify the call. Now investors must reinvest their funds in a much lower interest rate environment.
NINTM9-9 a. V = ; B！tN(1;r)(1;r)？t1dd
M = $1,000. INT = 0.09($1,000) = $90.
1. V = $829: Input N = 4, PV = 829, INT = 90, FV = 1000, YTM = Rd = ? Rd = 14.99%. B
2. V = $1,104: Change PV = 1104, YTM = Rd = ? I/YR = 6.00%. B
b. Yes. At a price of $829, the yield to maturity, 15%, is greater than your required rate of return of
12%. If your required rate of return were 12%, you should be willing to buy the bond at any
price below $908.88.
9-10 a. Solving for YTM:
N = 9, PV = -901.40, PMT = 80, FV = 1000
I/YR = YTM = 9.6911%.
b. The current yield is defined as the annual coupon payment divided by the current price.
CY = $80/$901.40 = 8.875%.
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/YR = 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
c. As long as promised coupon payments are made, the current yield will not change as a result of
changing interest rates. However, as rates change they will cause the end-of-year price to
change and thus the realized capital gains yield to change. As a result, the realized return to
investors will differ from the YTM.
7-11 a. Using a financial calculator, input the following to solve for YTM:
N = 20, PV = -1100, PMT = 60, FV = 1000, and solve for YTM = I/YR = 5.1849%.
However, this is a periodic rate. The nominal YTM = 5.1849%(2) = 10.3699% ， 10.37%.
Chapter 7: Bonds and Their Valuation Integrated Case 147
For the YTC, input the following:
N = 8, PV = -1100, PMT = 60, FV = 1060, and solve for YTC = I/YR = 5.0748%.
However, this is a periodic rate. The nominal YTC = 5.0748%(2) = 10.1495% ， 10.15%.
So the bond is likely to be called, and investors are most likely to earn a 10.15% yield.
b. The current yield = $120/$1,100 = 10.91%. The current yield will remain the same; however, if
the bond is called the YTC reflects the total return (rather than the YTM) so the capital gains yield
will be different.
c. YTM = Current yield + Capital gains (loss) yield
10.37% = 10.91% + Capital loss yield
-0.54% = Capital loss yield.
This is the capital loss yield if the YTM is expected.
However, based on our calculations in part a the total return expected would actually be the YTC
= 10.15%. So, the expected capital loss yield = 10.15% – 10.91% = -0.76%.
7-12 a. Yield to maturity (YTM):
With a financial calculator, input N = 28, PV = -1165.75, PMT = 95, FV = 1000, I/YR = ? I/YR =
YTM = 8.00%.
Yield to call (YTC):
With a calculator, input N = 3, PV = -1165.75, PMT = 95, FV = 1090, I/YR = ? I/YR = YTC =
b. Knowledgeable investors would expect the return to be closer to 6.1% than to 8%. If interest
rates remain substantially lower than 9.5%, the company can be expected to call the issue at the
call date and to refund it with an issue having a coupon rate lower than 9.5%.
c. If the bond had sold at a discount, this would imply that current interest rates are above the
coupon rate (i.e., interest rates have risen). Therefore, the company would not call the bonds, so
the YTM would be more relevant than the YTC.
7-13 The problem asks you to solve for the YTM and Price, given the following facts: N = 5 ？ 2 = 10, PMT = 80/2 = 40, and FV = 1000. In order to solve for I/YR we need PV.
However, you are also given that the current yield is equal to 8.21%. Given this information, we can find PV (Price).
Current yield = Annual interest/Current price
0.0821 = $80/PV
PV = $80/0.0821 = $974.42.
148 Integrated Case Chapter 7: Bonds and Their Valuation
Now, solve for the YTM with a financial calculator:
N = 10, PV = -974.42, PMT = 40, and FV = 1000. Solve for I/YR = YTM = 4.32%. However, this is a periodic rate so the nominal YTM = 4.32%(2) = 8.64%.
7-14 The problem asks you to solve for the current yield, given the following facts: N = 14, I/YR = 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%.
7-15 a. 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. Therefore, the likely life remaining on these bonds is 5
years (the time to call).
b. Since the bonds are likely to be called, they 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
N = 2 ？ 5 = 10, PV = -1353.54, PMT = 0.14/2 ？ 1,000 = 70, FV = 1050, and then solve for YTC
The periodic rate is 3.2366%, 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.
7-16 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/YR = 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/YR = 10, PMT = 90, FV = 1041.52
PV = -$987.87. V = $987.87. B
You would be willing to pay up to $987.87 for this bond today.
7-17 Before you can solve for the price, we must find the appropriate semiannual rate at which to evaluate this bond.
2 EAR = (1 + I/2) – 1 NOM20.0816 = (1 + I/2) – 1 NOM
I = 0.08. NOM
Chapter 7: Bonds and Their Valuation Integrated Case 149
Semiannual interest rate = 0.08/2 = 0.04 = 4%.
Solving for price:
N = 2 ？ 10 = 20, I/YR = 4, PMT = 0.09/2 ？ 1,000 = 45, FV = 1000
PV = -$1,067.95. V = $1,067.95. B
7-18 First, we must find the price Joan paid for this bond.
N = 10, I/YR = 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%.
7-19 a. According to Table 7-4, the yield to maturities for Albertson’s and Ford Motor Co. bonds are
6.303% and 8.017%, respectively. So, Albertson’s would need to set a coupon of 6.3% to sell its
bonds at par, while Ford would need to set a coupon of 8%.
b. Current investments in Albertson’s and Ford Motor Co. would be expected to earn returns equal
to their expected present yields. The return is safer for Albertson’s. Looking at the table, we see
that the Ford Motor Co. bonds were originally issued with a lower coupon but their yields have
increased greatly (resulting in a spread of 320 basis points, compared to Albertson’s spread of
149 basis points).
7-20 a. Find the YTM as follows:
N = 10, PV = -1175, PMT = 110, FV = 1000
I/YR = YTM = 8.35%.
b. Find the YTC, if called in Year 5 as follows:
N = 5, PV = -1175, PMT = 110, FV = 1090
I/YR = 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 YTM.)
d. Similarly from above, YTC can be found, if called in each subsequent year.
If called in Year 6:
N = 6, PV = -1175, PMT = 110, FV = 1080
I/YR = YTC = 8.27%.
150 Integrated Case Chapter 7: Bonds and Their Valuation