CHAPTER 14
THE VALUATION OF FIXED-INCOME SECURITIES
Teaching Guides for Questions and Problems in the Text
QUESTIONS
14-1. Bond prices fluctuate in response to changes in interest rates. Interest rates allocate
the supply of credit. When the supply relative to the demand declines, interest rates rise.
14-2. The current yield is the annual interest divided by the price of the bond. It is a
measure of the current flow of income produced by the bond.
14-3. Bonds selling at a discount offer the investor current income plus some capital gain.
If the investor wants to accumulate an amount as of a particular time (e.g., at retirement),
that individual can purchase discounted bonds that mature at the desired time.
If a bond is selling for a premium, that means interest rates have fallen. The firm may
14-4. The prices of bonds with lower coupons or longer terms to maturity tend to be more
volatile. Riskier securities tend to experience more price volatility than quality bonds if the
14-5. The yield to call is the return the bondholder earns from the present until the bond is
called at a specific date and at a specific call price. It differs from the yield to maturity in
two ways. First, the date of a call is not known while the maturity date is known. Second,
14-6. The term of a bond is the maturity date of a bond. The duration is the average time it
takes the bondholder to receive the interest and principal payments. A bond’s term and
duration will be equal only for a zero coupon bond. For all coupon bonds, the duration is
less than the term, because the early interest payments reduce the average time for the
14-7. If the investor anticipates a decline in interest rates, the individual wants to move out
of short-term debt instruments and purchase long-term bonds. Such a strategy will produce
capital gains when interest rates decline. (It may produce substantial capital losses if
14-8. This question asks the student to compare price volatility of various pairs of bonds.
The question is an easy means to review the factors that affect price volatility (e.g., term to
maturity or size of coupon). This question also makes a good (and easy to grade) test
question.
a. For a given term to maturity (e.g., five years), the bond with the smaller coupon should
b. For a given coupon (10 percent), the bond with the longer term to maturity should
c. Bond B has both the smaller coupon (6 versus 10 percent) and the longer term (eight
d. Zero coupon bonds tend to have the greatest fluctuations in price. In this case Bond B is
a zero coupon bond with ten years to maturity. Its price should be considerably more
volatile than a 10 percent bond with one year to maturity.
PROBLEMS
In the first problem, interest is assumed to be paid semi-
annually. In subsequent problems, interest is assumed to be paid annually. This assumption
eases the calculations using interest tables, since the tables do not have fractions such as
3.5 percent and all possible years. If you do not use interest tables, all the problems can be
worked assuming semi-annual interest payments. Answers are provided for both annual and
semi-annual payments.
14-1. a.
For annual compounding:
Using the present value tables, the price of the bond is
b.
PB = $20 + 20 + … + 20 + 1,000
(1+.06/2) (1+.06/2)2 (1+.06/2)10×2 (1+.06/2)10×2
Using the present value tables, the price of the bond is
For annual compounding:
Using the present value tables, the price of the bond is
PB = $40(7.360) + 1,000(0.558) = $852
c. The current yield: in a) $40/$1,000 = 4%
d. In (b) the price of the bond is lower because the current rate of interest is higher (6
percent versus 6 percent). The prices of existing bonds with lower coupons are marked
down until their yields to maturity are comparable to the yields on bonds being issued.
14-2. While problem #1 is designed to familiarize the student with bond pricing, problem
#2 shows the potential impact of a call feature.
a. PB = $70 + 70 + … + 100 + 1,000
b. PB = $70 + 70 + 1,000
c. PB = $70 + 70 + 1,070
d. The answers to (b) and (c) differ from (a) because antici-pation of the call reduces the
value of the bond. Instead of collecting $70 for eight years, the investor expects to collect
14-3. This problem illustrates the impact of the term to maturity on the bond’s price
volatility.
The price of the bond with the longer term to maturity experiences the larger price
fluctuation.
14-4. In the previous problems, the term of the bond, the coupon rate, and the yields were
given. The unknown was the price. In this problem the price of the bond is given, and the
problem asks for the yield to maturity.
To solve the equation using interest tables, select a rate such as 8 percent. The student
should automatically discard the suggestion to use 4 percent, since the bond is selling for a
discount and not a premium. At 8 percent, the present value of the payments is less than the
price, which indicates that 12 percent is too high. The student should then select a lower
interest rate and repeat the calculation until 7 percent is found to equate both sides of the
equation:
The current yield is $50/$859 = 5.82 percent.
The mechanics of this problem are tedious using interest tables and are an argument to use
a financial calculator or a computer program, such as Excel. The issue I have with using
these aids is that the student often does not understand the concept of the yield to maturity
14-5. Since the bond does not pay any interest, there is no current yield. The yield to
maturity is determined as follows:
Using interest tables:
To use a financial calculator, enter N = 5; PMT = 0; FV = 1000; PV = -519; I = ? = 14.02.
(The answer for semi-annual payments would be N = 10, PMT = 0, FV = 1000, PV = -519
and I = 6.78 per period or 13.56 annually.)
The answer may also be derived as follows:
14-7. This problem contrasts two bonds with perceptibly different coupons.
a.
b. The current yields: Bond XY = $50/$574.70 = 8.7%
Bond AB = $140/$1340.96 = 10.4%
c. Bond AB may be called because yields have fallen since the bond was issued (from 14
d. If a bond is riskier, its value should be less. If Bond CD has the same coupon and term
14-8. a and b. The cash flows from the two investments are the same. The problem
illustrates that the calculation of returns on bonds is the same for stocks.
14-9. a. Price of the preferred stock: $6/.1 = $60
b.
c. The price at 12 percent: $6/.12 = $50
14-10. a. $4/.06 = $66.67
b. Ppf = $4 + $4 + … + $4 + $100
(1+.06) (1+.06)2 (1+.06)20 (1+.06)20
The reason for the differences in price is the fact that one preferred stock is perpetual and
the other must be retired. The retirement of the stock means the investor can anticipate
receiving the par value of the stock after 20 years and that affects the stock’s price.
14-11. This problem contrasts two bonds, one of which has a variable coupon that changes
with current interest rates.
a. (Two years have lapsed; the term is now eight years.)
PA = $80 + 80 + … + 80 + 1,000
b. (The term is now six years, and yields are 7 percent.)
PA = $80 + 80 + … + 80 + 1,000
I = 3.5, PMT = 40, FV = 1000, and P = -1048.32.)
c. The price of Bond A moved inversely with changes in interest rates. The price of Bond B
did not fluctuate, because its coupon adjusted for changes in interest rates. (For
14-12 . In this problem the call feature is relevant in parts d and e.
a. PB = $115 + 115 + … + 115 + 1,000
b. Interest rates have risen (from 11.5 to 12 percent), so there is no reason to anticipate that
the bond will be called.
c. PB = $115 + 115 + … + 115 + 1,000
d. While the company may want to call the bond, it cannot currently call it. The indenture
specifies that the bond cannot be called for two years.
expected life of the bond is three years, the value of the bond if the comparable interest rate
is 8 percent is
Notice the inclusion of the call penalty. Be certain that the student realizes the inclusion of
the call penalty reduces the probability that the firm will call the bond, since the penalty is
a cost associated with premature refunding.
14-13. The bond pays no interest for three years and then pays $100 annually for nine
years after which the $1,000 principal is retired. The total number of years is twelve. The
price of the bond is the present value of these annual cash flows:
Present value of the interest: