Mini Case: 5 – 21
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
payments plus the return of par at maturity. The annual coupon payment is the cash
The bond consists of a 10-year, 10% annuity of $100 per year plus a $1,000 lump sum
payment at t = 10:
PV Annuity = $ 614.46
PV Maturity Value = 385.54
Value Of Bond = $1,000.00
Mini Case: 5 – 22
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:
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?
Answer: In the second situation, where rd falls to 7 percent, the price of the bond rises above par.
Just change rd from 13% to 7%. We see that the 10-year bond’s value rises to $1,210.71.
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 plotted
the data, we would find the situation shown below:
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
bond’s cash flows to its price. In other words, it is the promised rate of return on the
bond. (Note that the expected rate of return is less than the YTM if some probability
of default exists.) On a time line, we have the following situation when the bond sells
for $887:
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.
Mini Case: 5 – 25
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
Mini Case: 5 – 26
Capital Gains Yield =
V
)/
V
–
V
(BBB 001
= ($893.87 – $887)/$887 = 0.0077 = 0.77%,
This agrees with our earlier calculation (except for rounding). When the bond is selling
for $1,134.20 and providing a total return of rd = YTM = 7.08%, we have this situation:
Current Yield = $90/$1,134.20 = 7.94%
Mini Case: 5 – 27
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:
Mini Case: 5 – 28
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
V10-YEAR = $50 ((1- 1/(1+0.05)20)/0.065) + $1,000 (1/(1+0.05)20)
= $50(12.4622) + $1,000(0.37689) = $623.11 + $376.89 = $1,000.00.
At a 13 percent required return:
Mini Case: 5 – 29
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)?
Answer: If the bond were called, bondholders would receive $1,050 at the end of year 5. Thus,
the time line would look like this:
The easiest way to find the YTC on this bond is to input values into your calculator: n
= 10; PV = -1135.90; PMT = 50; and FV = 1050, which is the par value plus a call
premium of $50; and then press the rd = I/YR button to find I/YR = 3.765%. However,
this is the 6-month rate, so we would find the nominal rate on the bond as follows:
Mini Case: 5 – 30
h. 2. If you bought this bond, do you think you would be more likely to earn the YTM
or the YTC? Why?
Answer: Since the coupon rate is 10% versus YTC = rd = 7.53%, it would pay the company to
call the bond, get rid of the obligation to pay $100 per year in interest, and sell
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
there were no inflation.
Mini Case: 5 – 31
j. Define the real risk-free rate (r*). What security can be used as an estimate of r*?
What is the nominal risk-free rate (rRF)? What securities can be used as estimates
of rRF?
Answer: The real risk-free rate, r*, is the rate that a hypothetical riskless security pays each
moment if zero inflation were expected. The real risk-free rate is not constant—r*
k. Describe a way to estimate the inflation premium (IP) for a T-Year bond.
Answer: Treasury Inflation-Protected Securities (TIPS) are indexed to inflation. The IP for a