-kickoff
Chapter 8
Valuing Bonds
8-1.
A 30-year bond with a
face value of $$1000 has a coupon rate of
5.5%
, with semiannual payments.
a.
What is the coupon
payment for this bond?
b.
Draw the cash flows for the bond on a
timeline.
a.
The coupon
payment is:
C
PN<
/p>
?
C
oupon
R
ate
?
Face V
alue
N
um
ber
of C
oupons per Y
ear
?
0.055
?
$$1000
2
?
$$27.50.
b.
The
timeline
for
the
cash
flows
for
this
bond
is
(the
unit
of
time
on
this
timeline
is
six-month
periods):
0
1
2
3
60
2
$$27.50
$$27.50
$$27.50
$$27.50 +
$$1000
P
?
< br>100/(1.055)
?
$$89.85
8-2.
Assume
that
a
bond
will
make
payments
every
six
months
as
shown
on
the
following
timeline
(using six-month periods):
a.
What is the maturity of
the bond (in years)?
b.
What
is the coupon rate (in percent)?
c.
What is the face
value?
a.
The
maturity is 10 years.
b.
(20/1000) x 2 = 4%, so the coupon rate
is 4%.
c.
The face value is
$$1000.
?2011 Pearson Education
Berk/DeMarzo
?
Corporate
Finance, Second Edition
107
8-3.
The following table summarizes prices
of various default-free, zero-coupon bonds
(expressed as a
percentage of face
value):
a.
Compute the yield to maturity for each
bond.
b.
Plot the zero-
coupon yield curve (for the first five years).
c.
Is the yield curve upward
sloping, downward sloping, or flat?
a.
Use the following
equation.
1
?
YTM
n
?
FV
n
?
?
?
?
?
P
?
1
/
n
1
/
1
?
100
?
1
?
Y
T
M
1
?
?<
/p>
?
?
95.51
?
?
Y
T
M<
/p>
1
?
4.70%
1
/
2
?
100
?
1
?
Y
T
M
1
?
?
?
?
91.05
?
1
?
Y
T
M
3
?
Y
T
M
< br>1
?
4.80%
1
/
3
?
< br>100
?
?
?
< br>?
?
86.38
?
?
Y
T
M
< br>3
?
5.00%
1
/
4
1
< br>?
Y
T
M
4
?
100
?
?
?
?
?
8
1.65
?
?
100
< br>?
?
?
?
?
76.51
?
?
< br>Y
T
M
4
?
5.20%
1
< br>/
5
1
?
Y
T
M
5
?
Y
T
M
5
?
5.50%
b.
The yield curve is as
shown below.
Zero Coupon Yield Curve
5.6
5.4
5.2
5
4.8
4.6
0
2
4
6
Maturity (Ye
ars)
Y
i
e
l
d
t
o
M
a
t
u
r
i
t
y
c.
The yield curve is upward
sloping.
?2011 Pearson Education
108
Berk/DeMarzo
?
Corporate
Finance, Second Edition
8-4.
Suppose the current zero-coupon yield
curve for risk-free bonds is as follows:
a.
What is the
price per $$100 face value of a two-year, zero-
coupon, risk-free bond?
b.
What is the price per $$100 face value
of a four
-year, zero-coupon, risk-free
bond?
c.
What is
the risk-free interest rate for a five-year
maturity?
a.
b.
P
?
100(1.055)
?
$$89.85
2
P
?
100/(1.0595)
?
$$79.36
4
c.
6.05%
8-5.
In the
box in Section 8.1, reported that the
three
-month
Treasury
bill sold for a
price of
$$100.002556 per $$100 face value. What is the yield
to maturity of this bond, expressed as
an EAR?
100
?
?
?
?
< br>?
1
?
?
0.01022%
?
100.00
2556
?
4
8-6.
Suppose a 10-year, $$1000 bond with an
8%
coupon rate and semiannual coupons
is trading for a
price of $$1034.74.
a.
What is the bond’s yield
to maturity (expressed as an APR with semiannual
compounding)?
b.
If the bond’s yield to maturity changes
to 9%
APR, what will the bond’s price
be?
a.
$$1,
034.74
?
(1
?
40
YTM
2
)
?
(1
?
40
YTM
2
)
2
?
?
?
40
?
1000
(1
?<
/p>
YTM
2
)
20
?
YTM
?
7
.5%
Using the annuity
spreadsheet:
NPER
Rate
PV
PMT
Given:
20
-1,034.74
40
Solve For Rate:
3.75%
Therefore, YTM = 3.75% ×
2 =
7.50%
b.
PV
?
40
(1
?
.09
2
)
?
(1
?
40
.09
2
)
2
FV
1,000
Excel
Formula
=RATE(20,40,-1034.74,1000)
?
L
?
40
?<
/p>
1000
(1
?
.09
2
)
20
?
$$934.96.
Using
the spreadsheet
With a 9% YTM = 4.5%
per 6 months, the new price is $$934.96
NPER
Rate
PV
PMT
FV
Excel
Formula
Given:
20
4.50%
40
1,000
Solve For
PV:
(934.96)
=PV(0.045,20,40,1000)
?2011 Pearson Education
Berk/DeMarzo
?
Corporate
Finance, Second Edition
109
8-7.
Suppose a five-year, $$1000 bond with
annual coupons has a price of $$900 and a yield to
maturity
of 6%. What is the bond’s
coupon rate?
900
?
C
(1
?
.0
6)
?
C
(1
?
.06)
2
?
?
?
C
?
1
000
(1
?
.06)
5
?
C
?
$$36.26, so the coupon rate is 3.626%.
We can use the annuity spreadsheet to
solve for the payment.
NPER
Rate
PV
PMT
FV
Given:
5
6.00%
-900.00
1,000
Solve For
PMT:
36.26
Therefore, the coupon rate is 3.626%.
8-8.
Excel Formula
=PMT(0.06,5,-900,1000)
The prices of several bonds with face
values of $$1000 are summarized in the following
table:
For each bond, state whether it trades
at a discount, at par, or at a premi
um.
Bond A trades at a discount. Bond D
trades at par. Bonds B and C trade at a
premium.
8-9.
Explain why the yield of a bond that
trades at a discount exceeds the bond’s coupon
rate.
Bonds trading at a
discount generate a return both
from
receiving the coupons and from receiving a
face value that exceeds the price paid
for the bond. As a result, the yield to maturity
of discount bonds
exceeds the coupon
rate.
8-10.
Suppose
a seven-year, $$1000 bond
with
an 8%
coupon
rate and semiannual coupons is trading
with a yield to maturity of 6.75%.
a.
Is this bond currently
trading at a discount, at par, or at a premium?
Explain.
b.
If
the yield to
maturity of the
bond rises to 7%
(APR
with semiannual compounding),
what
price will the bond
trade for?
a.
Because the
yield to maturity is less than the coupon rate,
the bond is trading at a premium.
b.
NPER
Given:
14
Solve For PV:
8-11.
Suppose
that
General
Motors
Acceptance
Corporation
issued
a
bond
with
10
years
until
maturity,
a
face
value
of
$$1000,
and
a
coupon
rate
of
7%
(annual
payments).
The
yield
to
maturity
on this bond when it was issued was 6%.
a.
What was the price of
this bond when it was issued?
b.
Assuming the
yield to
maturity remains
constant,
what is the
price
of
the
bond immediately
before it makes its first coupon
payment?
c.
Assuming the
yield to
maturity remains constant,
what is the
price of
the
bond immediately
after it makes its first coupon
payment?
Rate
3.50%
PV
PMT
40
(1,054.60)
FV
1,000
Excel
Formula
=PV(0.035,14,40,1000)
40
(1
?
.035)
?<
/p>
40
(1
?
.0
35)
2
?
?
?
40
?
1000
(1
?
.035)
14
?
$$1,
054.60
?2011 Pearson Education
110
Berk/DeMarzo
?
Corporate
Finance, Second Edition
a.
When it was issued, the price of the
bond was
P
?
70
(1
?
.06)
?
...
?
70
?<
/p>
1000
(1
?
.06)
10
?
$$1073.60.<
/p>
b.
Before the
first coupon payment, the price of the bond
is
P
?
7
0
?
70
(1
?
.06)
...
?
< br>70
?
1000
(1
?
.06)
9
?
$$1138.02.
c.
After the first coupon payment, the
price of the bond will be
P<
/p>
?
70
(1
?<
/p>
.06)
...
?
70
?
1000
(1
?
.06)
9
?
$$1068.02.
8-12.
Suppose
you
purchase
a
10-year
bond
with
6%
annual
coupons.
You
hold
the
bond
for
four
years,
and
sell it immediately after receiving the fourth
coupon.
If
the
bond’s yield
to
maturity
was 5%
when you purchased and sold the bond,
a.
What cash
flows
will
you
pay and receive
from
your investment in
the
bond
per $$100
face
value?
b.
What is the internal rate of return of
your investment?
a.
First, we compute the initial price of
the bond by discounting its 10 annual coupons of
$$6 and final
face value of $$100 at the
5% yield to maturity.
Given:
Solve For PV:
Thus, the initial price of
the bond = $$107.72.
(Note that the bond
trades above par, as its coupon
rate
exceeds its yield.)
Next we compute the
price at which the bond is sold, which is the
present value of the bonds cash
flows
when only 6 years remain until maturity.
NPER
Given:
6
Solve
For PV:
Therefore,
the
bond
was
sold
for
a
price
of
$$105.08.
The
cash
flows
from
the
investment
are
therefore as shown in
the following timeline.
Year
Purchase Bond
Receive
Coupons
Sell Bond
Cash
Flows
0
–
$$107.72
–
$$107.72
1
$$6
$$6.00
2
$$6
$$6.00
3
$$6
$$6.00
4
$$6
$$105.08
$$111.08
Rate
5.00%
PV
(105.08)
PMT
6
FV
100
Excel
Formula
= PV(0.05,6,6,100)
NPER
10
Rate
5.00%
PV
(107.72)
PMT
6
FV
100
Excel Formula
=
PV(0.05,10,6,100)
?2011 Pearson
Education
Berk/DeMarzo
?
Corporate
Finance, Second Edition
111
b.
We can compute the IRR of the
investment using the annuity spreadsheet. The PV
is the purchase
price, the PMT is the
coupon amount, and the FV is the sale
price.
The length of the investment N
= 4 years. We then calculate the IRR of
investment = 5%.
Because the
YTM was the same at the
time of
purchase and sale, the IRR of the investment
matches the YTM.
NPER
Rate
PV
PMT
FV
Excel
Formula
Given:
4
–
107.72
6
105.08
Solve
For Rate:
5.00%
= RATE(4,6,-107.72,105.08)
8-13.
Consider the following
bonds:
a.
What
is the percentage change in the price of each bond
if its yield to maturity falls from 6%
to 5%
?
b.
Which of the bonds
A
–
D is most sensitive to a
1%
drop in interest rates from
6%
to 5%
and
why?
Which bond is least sensitive? Provide an
intuitive explanation for your
ans
wer.
a.
We can compute the price of each bond
at each YTM using Eq. 8.5. For example, with a 6%
YTM,
the price of bond A per $$100 face
value is
P(bond A, 6%
YTM
)
?
100
1.06
15
?
$$41.73.
The price of bond D is
P
(bond D
,
6%
Y
T
M
)
?
8
?
1<
/p>
?
1
?
100<
/p>
1
?
?
?
$$114.72.
?
10
?
10
.06
< br>?
1.06
?
1.06
One can also use the Excel formula to
compute the price:
–
PV(YTM,
NPER, PMT, FV).
Once we compute the
price of each bond for each YTM, we can compute
the % price
change as
Percent change =
?
Price at 5%
YTM<
/p>
?
?
?
Pric
e at 6%
YTM
?
.
?
Price at 6%
YTM
?
The results are shown in
the table below.
Bond
A
B
C
D
Coupon Rate
(annual payments)
0%
0%
4%
8%
Maturity
(years)
15
10
15
10
< br>Price at
6% YTM
$$41.73
$$55.84
$$80.58
$$114.72
Price at
5% YTM
$$48.1
0
$$61.39
$$89.62
$$123
.17
Percentage Change
15.3%
9.9%
11.2%
7.4%
b.
Bond A is most sensitive,
because it has the longest maturity and no
coupons. Bond D is the least
sensitive.
Intuitively, higher
coupon rates and a
shorter maturity typically lower a bond’s interest
rate sensitivity.
?2011
Pearson Education
112
Berk/DeMarzo
?
Corporate
Finance, Second Edition
8-14.
Suppose you purchase a 30-year, zero-
coupon bond with a yield to maturity of 6%. You
hold the
bond for five years before
selling it.
a.
If the
bond’s yield
to
maturity is 6%
when you sell
it,
what is the internal rate
of return
of
your
investment?
b.
If
the
bond’s yield
to
maturity is 7%
when you sell
it,
what is the internal rate
of return
of
your
investment?
c.
If
the
bond’s yield
to
maturity is 5%
when you
se
ll it,
what is the
internal rate
of return
of
your investment?
d.
Even
if
a
bond
has
no
chance
of
default,
is
your
investment
risk
free
if
you
plan
to
sell
it
before it matures?
E
xplain.
a.
Purchase price = 100 /
1.06
30
= 17.41. Sale price =
100 / 1.06
25
= 23.30. Return
= (23.30 / 17.41)
1/5
–
1 = 6.00%. I.e., since YTM
is the same at purchase and sale, IRR =
YTM.
b.
Purchase
price = 100 / 1.06
30
=
17.41. Sale price = 100 /
1.07
25
= 18.42. Return =
(18.42 / 17.41)
1/5
–
1 = 1.13%. I.e., since YTM
rises, IRR < initial YTM.
c.
Purchase price = 100 /
1.06
30
= 17.41. Sale price =
100 / 1.05
25
= 29.53. Return
= (29.53 / 17.41)
1/5
–
1 = 11.15%. I.e., since
YTM falls, IRR > initial YTM.
d.
Even without default, if you sell prior
to
maturity, you are exposed to the
risk that the YTM
may
change.
8-15.
Suppose
you
purchase
a
30-year
Treasury
bond
with
a
5%
annual
coupon,
initially
trading
at
par.
In 10 years’ time, the bond’s yield to maturity
has risen to 7%
(EAR).
a.
If
you
sell
the
bond
now,
what
internal
rate
of
return
will
you
have
earned
on
your
investment in the
bond?
b.
If
instead
you
hold
the
bond to
maturity,
what internal rate
of return
will
you earn on your
investment
in the bond?
c.
Is comparing the IRRs in (a) versus (b)
a useful way to evaluate the decision to sell the
bond?
Explain.
a.
3.17%
b.
5%
c.
We can’t simply compare
IRRs.
By not selling the bond for
its current price of $$78.81,
we will
earn the current
market return of 7% on that amount going
forward.
8-16.
Suppose the current yield
on
a one-year, zero coupon
bond
is 3%
, while the yield
on
a five
-year,
zero
coupon bond is 5%. Neither bond has any risk of
default. Suppose you plan to invest for one
year. You will earn more over the year
by investing in the five-year bond as long as its
yield does
not rise above what
level?
The return from
investing in the
1 year
is
the yield. The return
for
investing in the
5 year for
initial
price
p
0
and selling after one
year at price p1 is
1
(1.05)
1
(1
?
y
)
5
5
p
1
p
0
?
1
. We have
p
0
?
p
1
?
,
.
?2011
Pearson Education
Berk/DeMarzo
?
Corporate
Finance, Second Edition
113
So you break
even when
1
p
1
p
0
?
1
?
(1
?
y
)
1
(1.05)
(1.05
)
(1
?
y
)
y
?
5
4
5
4
?
1
?
y
1
?
0.03
?
1.03
5
/
4
1
/
4
(1.05)
(1
.03)
?
1
?
5.51%.
For Problems
17
–
22, assume zero-coupon
yields on default-free securities are as
summarized in the following
table:
8-17.
What
is
the
price
today
of
a
two-year,
default-free
security
with
a
face
value
of
$$1000
and
an
annual
coupon rate of 6%
? Does this bond trade
at a discount, at par, or at a premium?
P
?
C
PN<
/p>
1
?
YTM
1<
/p>
?
C
PN
(1<
/p>
?
YTM
2
)<
/p>
2
?
...
?<
/p>
C
PN
?
FV<
/p>
(1
?
YTM
N
)
N
?
60<
/p>
(1
?
.04)
?
60
?
1000
(1
?
.043)
2
?
$$1032.09
This bond trades at a premium. The
coupon of the bond is greater than each of the
zero coupon yields,
so
the
coupon
will
also
be
greater
than
the
yield
to
maturity
on
this
bond.
Therefore
it
trades
at
a
premium
8-18.
What is the price of a five-year, zero-
coupon, default-free security with a face value of
$$1000?
The price of the
zero
-coupon bond is
P
?
FV
(1
?
YTM
N
N
)
?
1000
(1
?
0.048)
5
?
$$79
1.03
8
-19.
What is the
price of a
three-year,
default-free
security
with
a face
value
of $$1000 and
an
annual
coupon
rate of 4%? What is the yield to maturity for this
bond?
The price of the bond
is
P
?
C
PN
1
?
YTM
1
?
C
PN
(1
?
YTM
2
)
2
?
...
?
C
PN
?
FV
(1
?
YTM
N
)
N
?
40
(1
?
.04)
< br>?
40
(1
?
< br>.043)
2
?
40
?
1000
(1
?
.045)
3
?
$$9
86.58.
The yield to maturity
is
P
?
C
PN
1
?
Y
TM
?
C
PN
(1
?
YTM
)
40
(1
?
YTM
< br>)
?
2
?
...
?
C
PN
?
FV
(1
?
YTM
)
?
N
8-20.
$$986.58
?
40
(1
?
YTM
)
2
4
0
?
1000
(1
?
YTM
)
3
?
YTM
?
4.488%
What
is
the
maturity
of
a
default-free
security
with
annual
coupon
payments
and
a
yield
to
maturity
of 4%
? Why?
The maturity
must be one year. If the maturity were
longer than one year, there would be an arbitrage
opportunity.
?2011 Pearson
Education
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