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Concept Questions
◆
Define pure discount bonds,
level-coupon bonds, and consols.
A pure
discount bond is one that makes no intervening
interest payments. One receives a single lump sum
payment at maturity. A level-coupon
bond is a combination of an annuity and a lump sum
at maturity. A
consol is a bond that
makes interest payments forever.
◆
Contrast the state interest
rate and the effective annual interest rate for
bonds paying semi-annual interest.
Effective annual interest rate on a
bond takes into account two periods of compounding
per year received
on the coupon
payments. The state rate does not take this into
account.
◆
What is the
relationship between interest rates and bond
prices?
There is an inverse
relationship. When one goes up, the other goes
down.
◆
How does one
calculate the yield to maturity on a bond?
One finds the discount rate that
equates the promised future cash flows with the
price of the bond.
◆
What are
the three factors determining a firm's P/E ratio?
Today's expectations of future growth
opportunities.
The discount rate.
The accounting method.
◆
What is the closing price
of General Data?
The closing price of
General Data is 6 3/16.
◆
What is the PE of General
House?
The PE of General House is 29.
◆
What is the annual dividend
of General Host?
The annual dividend of
General Host is zero.
Concept Questions - Appendix To Chapter
5
◆
What is the difference
between a spot interest rate and the yield to
maturity?
The yield to maturity is the
geometric average of the spot rates during the
life of the bond.
◆
Define
the forward rate.
Given a one-year bond
and a two-year bond, one knows the spot rates for
both. The forward rate is the rate
of
return implicit on a one-year bond purchased in
the second year that would equate the terminal
wealth of
purchasing the one-year bond
today and another in one year with that of the
two-year bond.
◆
What is the
relationship between the one-year spot rate, the
two-year spot rate and the forward rate over
the second year?
The forward
rate f2 = [(1+r2)2 /(1+r1 )] - 1
◆
What is the expectation
hypothesis?
Investors set interest
rates such that the forward rate over a given
period equals the spot rate for that period.
◆
What is the liquidity-
preference hypothesis?
This hypothesis
maintains that investors require a risk premium
for holding longer-term bonds (i.e. they
prefer to be liquid or short-term
investors). This implies that the market sets the
forward rate for a given
period above
the expected spot rate for that period.
Questions And Problems
How
to Value Bonds
5.1 What is the present
value of a 10-year, pure discount bond that pays
$$1,000 at maturity and is priced to
yield the following rates?
a. 5 percent
b. 10 percent
c. 15 percent
Solutions
a.
$$1,000 / 1.0510 = $$613.91
b.
$$1,000 / 1.1010 =
$$385.54
c.
$$1,000 / 1.1510 = $$247.18
5.2 Microhard has issued a bond with
the following characteristics:
Principal: $$1,000
Term to
maturity: 20 years
Coupon rate: 8 percent
Semiannual payments
Calculate the price of the Microhard
bond if the stated annual interest rate is:
a. 8 percent
b. 10 percent
c. 6 percent
Solutions
The amount of the semi-annual interest
payment is $$40 (=$$1,000
?
0.08 / 2). There are a
total of 40 periods; i.e., two half
years in each of the twenty years in the term to
maturity.
The annuity
factor tables can be used to price these bonds.
The appropriate discount rate to
use is the semi-annual rate. That rate
is simply the annual rate divided by two. Thus,
for
part b
the rate to be
used is 5% and for part c is it 3%.
a.
$$40 (19.7928) + $$1,000 /
1.0440 = $$1,000
Notice that whenever the coupon rate
and the market rate are the same, the bond is
priced at par.
b.
$$40
(17.1591) + $$1,000 / 1.0540 = $$828.41
Notice that whenever the
coupon rate is below the market rate, the bond is
priced
below
par.
c.
$$40
(23.1148) + $$1,000 / 1.0340 = $$1,231.15
Notice that
whenever the coupon rate is above the market rate,
the bond is priced
above par.
5.3
Consider a bond with a face value of $$1,000. The
coupon is paid semiannually and the market
interest
rate (effective annual
interest rate) is 12 percent. How much would you
pay for the bond if a. the coupon
rate
is 8 percent and the remaining time to maturity is
20 years?
b. the coupon rate is 10
percent and the remaining time to maturity is 15
years?
Solutions
Semi-
annual discount factor = (1.12)1/2 - 1 =
0.05830 = 5.83%
a.
Price
= $$40
40
?
0<
/p>
.
0583
+ $$1,000 / 1.058340
=
$$614.98 + $$103.67
= $$718.65
b.
Price
= $$50
+ $$1,000 / 1.058330
=
$$700.94 + $$182.70
= $$883.64
5.4 Pettit Trucking has issued an
8-percent, 20-year bond that pays interest
semiannually. If the market
prices the
bond to yield an effective annual rate of 10
percent, what is the price of the bond?
Solutions
Effective annual
rate of 10%:
Semi-annual
discount factor = (1.1)0.5 - 1 = 0.04881 = 4.881%
Price
= $$40
+ $$1,000 / 1.0488140
=
$$846.33
5.5 A bond is sold
at $$923.14 (below its par value of $$1,000). The
bond has 15 years to maturity and
investors require a 10-percent yield on
the bond. What is the coupon rate for the bond if
the coupon is paid
semiannually?
Solutions
$$923.14 =
C
+ $$1,000 / 1.0530
= (15.37245) C + $$231.38
C = $$45
The annual coupon rate =
$$45
?
2 / $$1,000 = 0.09 = 9%
5.6 You have just purchased
a newly issued $$1,000 five-year Vanguard Company
bond at par. This five-
year bond pays
$$60 in interest semiannually. You are also
considering the purchase of another Vanguard
Company bond that returns $$30 in
semiannual interest payments and has six years
remaining before it
matures. This bond
has a face value of $$1,000.
40
?
0
.
04881
?
30
0
.
< br>0583
?
30
0
.
05
a. What is effective annual
return on the five-year bond?
b. Assume
that the rate you calculated in part (a) is the
correct rate for the bond with six years remaining
before it matures. What should you be
willing to pay for that bond?
c. How
will your answer to part (b) change if the five-
year bond pays $$40 in semiannual interest?
Solutions
a.
The semi-annual interest rate is $$60 /
$$1,000 = 0.06. Thus, the effective annual rate is
1.062 - 1 =
0.1236 = 12.36%.
b.
Price = $$30
+ $$1,000 / 1.0612
= $$748.48
?
12
0
.
06
?
12
0
.
04
c.
Price = $$30
+ $$1,000 /
1.0412
= $$906.15
Note:
In parts b and c we are implicitly
assuming that the yield curve is flat. That is,
the yield in year 5
applies for year 6
as well.
Bond Concepts
5.7
Consider two bonds, bond A and bond B, with equal
rates of 10 percent and the same face values of
$$1,000. The coupons are paid annually
for both bonds. Bond A has 20 years to maturity
while bond B has
10 years to maturity.
a. What are the prices of the two bonds
if the relevant market interest rate is 10
percent?
b. If the market interest rate
increases to 12 percent, what will be the prices
of the two bonds?
c. If the market
interest rate decreases to 8 percent, what will be
the prices of the two bonds?
Solutions
a.
b.
c.
PA = $$100
20
?
0<
/p>
.
12
20
?<
/p>
0
.
10
+
$$1,000 / 1.1020 = $$1,000
PB = $$100
?
10
0
.
10
+ $$1,000 / 1.1010 = $$1,000
PA = $$100
+ $$1,000 / 1.1220 = $$850.61
PB = $$100
?
10
0
.
12
+
$$1,000 / 1.1210 = $$887.00
PA = $$100
20
?
0
.
08
+ $$1,000 / 1.0820 = $$1,196.36
PB
= $$100
+ $$1,000 / 1.0810 = $$1,134.20
5.8 a. If the market
interest rate (the required rate of return that
investors demand) unexpectedly increases,
what effect would you expect this
increase to have on the prices of long-term bonds?
Why?
b. What would be the effect of the
rise in the interest rate on the general level of
stock prices? Why?
Solutions
a.
The price of long-term
bonds should fall. The price is the PV of the
cash flows
associated with the bond. As the
interest rate rises, the PV of those flows falls.
This can be easily seen by
looking at a one-year, pure discount bond.
Price = $$1,000 / (1 + i)
As i. increases, the
denominator rises. This increase causes the price
to fall.
b.
The effect upon stocks is not as
certain as that upon the bonds. The nominal
interest
rate is a function of both the real
interest rate and the inflation rate; i.e.,
(1
+ i) = (1 + r) (1 + inflation)
From this relationship it is easy to
conclude that as inflation rises, the nominal
interest
rate rises. Stock prices are a
function of dividends and future prices as
well as the interest rate.
Those dividends and future prices are determined
by the
earning
power of the firm. When inflation occurs, it may
increase or decrease
firm earnings. Thus, the effect of a
rise in the level of general prices upon the
level of stock
prices is uncertain.
?
10
0
.
08
5.9 Consider a
bond that pays an $$80 coupon annually and has a
face value of $$1,000. Calculate the yield to
maturity if the bond has
a.
20 years remaining to maturity and it is sold at
$$1,200.
b. 10 years remaining to
maturity and it is sold at $$950.
Solutions
a.
20
?
$$1,200 = $$80
r
+ $$1,000 / (1 + r)20
r = 0.0622 = 6.22%
b.
r = 0.0877 = 8.77%
5.10 The Sue Fleming
Corporation has two different bonds currently
outstanding. Bond A has a face value
of
$$40,000 and matures in 20 years. The bond makes no
payments for the first six years and then pays
$$2,000 semiannually for the subsequent
eight years, and finally pays $$2,500 semiannually
for the last six
years. Bond B also has
a face value of $$40,000 and a maturity of 20
years; it makes no coupon payments
over
the life of the bond. If the required rate of
return is 12 percent compounded semiannually, what
is the
current price of Bond A? of Bond
B?
Solutions
PA =
($$2,000
) / (1.06)12 +
($$2,500
) / (1.06)28 + $$40,000 /
(1.06)40
= $$18,033.86
PB = $$ 40,000 / (1.06)40 = $$3,888.89
The Present
Value of Common Stocks
5.11 Use the
following February 11, 2000, WSJ quotation for
AT&T Corp. Which of the following
statements is false?
a. The
closing price of the bond with the shortest time
to maturity was $$1,000.
b. The annual
coupon for the bond maturing in year 2016 is
$$90.00.
c. The price on the day before
this quotation (i.e., February 9) for the ATT bond
maturing in year 2022 was
$$1.075 per
bond contract.
d. The current yield on
the ATT bond maturing in year 2002 was 7.125%
e. The ATT bond maturing in year 2002
has a yield to maturity less than 7.125%.
Bonds Cur Yld Vol Close Net Chg
ATT 9s 16 ? 10 117 _ 1/4
ATT
5 1/8 01 ? 5 100 _ 3/4
ATT 7 1/8 02 ?
193 104 1/8 _ 1/4
ATT 8 1/8 22 ? 39 107
3/8 _ 1/8
Solutions
a.
True
True
False
False
True
5.12 Following are selected quotations
for New York Exchange Bonds from the Wall Street
Journal. Which
of the following
statements about Wilson
’
s
bond is false?
a. The bond maturing in
year 2000 has a yield to maturity greater
th
an 63?8%.
b.
The closing price of the bond with the shortest
time to maturity on the day before this quotation
was
$$1,003.25.
c. This
annual coupon for the bond maturing in year 2013
is $$75.00.
d. The current yield on the
Wilson
’
s bond with the
longest time to maturity was 7.29%.
e.
None of the above.
Quotations as of 4
P.M. Eastern Time
Friday, April 23,
1999
Bonds Current Yield Vol Close Net
WILSON 6 3/8 99 ? 76 100 3/8 _ 1/8
WILSON 6 3/8 00 ? 9 98 1/2
10
?
$$950 = $$80
r
+ $$1,000 / (1 + r)10
?
16
0
.
06
?
12
0
.
06
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