d. How is the value of a bond determined? What is the value of a 10-year, $1,000
par value bond with a 10 percent annual coupon if its required rate of return is
10 percent?
Answer: A bond has a specific cash flow pattern consisting of a stream of constant interest
0 10% 1 2 3 9 10
.
38.55
385.54
1,000.00
Mini Case: 5 – 22
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website, in whole or in part.
e. 1. What would be the value of the bond described in part d if, just after it had been
issued, the expected inflation rate rose by 3 percentage points, causing investors
to require a 13 percent return? Would we now have a discount or a premium
bond?
Answer: With a financial calculator, just change the value of rd = I/YR from 10% to 13%, and
press the PV button to determine the value of the bond:
rate, the bonds’ values fall below par, so they sell at a discount.
e. 2. What would happen to the bonds’ value if inflation fell, and rd declined to 7
percent? Would we now have a premium or a discount bond?
rises above par, or to a premium. Further, the longer the maturity, the greater the
price effect of any given interest rate change.
Mini Case: 5 – 23
e. 3. What would happen to the value of the 10-year bond over time if the required
rate of return remained at 13 percent, or if it remained at
7 percent? (Hint: with a financial calculator, enter PMT, I/YR, FV, and N, and
then change (override) n to see what happens to the PV as the bond approaches
maturity.)
Answer: Assuming that interest rates remain at the new levels (either 7% or 13%), we could
find the bond‘s value as time passes, and as the maturity date approaches. If we then
approaches, so the value of a premium bond decreases to $1,000, and the value of a
discount bond increases to $1,000 (barring default).
Mini Case: 5 – 24
f. 1. What is the yield to maturity on a 10-year, 9 percent annual coupon, $1,000 par
value bond that sells for $887.00? That sells for $1,134.20? What does the fact
that a bond sells at a discount or at a premium tell you about the relationship
between rd and the bond’s coupon rate?
Answer: The yield to maturity (YTM) is that discount rate which equates the present value of a
PV1
PVM
SUM = PV = 887
We want to find r in this equation:
.
)r + (1
M
+
)r + (1
INT
+ … +
)r + (1
INT
= PV =
VNN1
B
We know n = 10, PV = -887, PMT = 90, and FV = 1000, so we have an equation with
one unknown, rd. We can solve for rd by entering the known data into a financial
calculator and then pressing the I/YR = rd button. The YTM is found to be 10.91%.
Alternatively, we could use present value interest factors:
We can tell from the bond’s price, even before we begin the calculations, that the
YTM must be above the 9% coupon rate. We know this because the bond is selling at
a discount, and discount bonds always have r > coupon rate.
If the bond were priced at $1,134.20, then it would be selling at a premium. In
that case, it must have a YTM that is below the 9 percent coupon rate, because all
Mini Case: 5 – 25
website, in whole or in part.
f. 2. What are the total return, the current yield, and the capital gains yield for the
discount bond? (Assume the bond is held to maturity and the company does not
default on the bond.)
Answer: The current yield is defined as follows:
.
bond theof priceCurrent
paymentinterest coupon Annual
= YieldCurrent
priceyear –of–Beginning
The total expected return is the sum of the current yield and the expected capital gains
yield:
.
yield gains
capital Expected
+
yieldcurrent
Expected
=
Return Total
Expected
For our 9% coupon, 10-year bond selling at a price of $887 with a YTM of 10.91%,
the current yield is:
887$
Knowing the current yield and the total return, we can find the capital gains yield:
YTM = current yield + capital gains yield
And
The capital gains yield calculation can be checked by asking this question: “What is
the expected value of the bond 1 year from now, assuming that interest rates remain at
current levels?” This is the same as asking, “What is the value of a 9–year, 9 percent
Mini Case: 5 – 27
website, in whole or in part.
g. How does the equation for valuing a bond change if semiannual payments are
made? Find the value of a 10-year, semiannual payment, 10 percent coupon
bond if nominal rd = 13%.
Answer: In reality, virtually all bonds issued in the U.S. have semiannual coupons and are
valued using the setup shown below:
1 2 N YEARS
M
PV1
.
.
.
PVN
website, in whole or in part.
We would use this equation to find the bond’s value:
.
)
2/
r
+ (1
M
+
)
2/
r
+ (1
2INT/
=
VN2
d
t
d
N2
1 =t
B
The payment stream consists of an annuity of 2N payments plus a lump sum equal to
the maturity value.
website, in whole or in part.
h. Suppose a 10-year, 10 percent, semiannual coupon bond with a par value of
$1,000 is currently selling for $1,135.90, producing a nominal yield to maturity
of 8 percent. However, the bond can be called after 5 years for a price of $1,050.
h. 1. What is the bond’s nominal yield to call (YTC)?
50 50 50 50 50 50 50 50 50 50
1,050
PV1
.
.
PV4
PV5C
PV5CP
1,135.90 = sum of PVs
Mini Case: 5 – 30
website, in whole or in part.
h. 2. If you bought this bond, do you think you would be more likely to earn the YTM
or the YTC? Why?
i. Write a general expression for the yield on any debt security (rd) and define
these terms: real risk-free rate of interest (r*), inflation premium (IP), default
risk premium (DRP), liquidity premium (LP), and maturity risk premium
(MRP).
Answer: rd = r* + IP + DRP + LP + MRP.
r* is the real risk-free interest rate. It is the rate you see on a riskless security if
marketability.