CHAPTER 7
1.
Assume that the one-period spot interest rate is 3 percent and
the two-period spot interest rate is 6 percent. Answer the
following questions:
(a) What is the present value of $100 received one year from
now?
(b) What is the present value of $100 received two years from
now?
(c) You are going to receive $100 two years from now. What is
its time 1 value? What is the forward interest rate?
(d) Suppose you invest $1 today at the two-period spot interest
rate; what is its value at time 2? Alternatively you invest $1
at the one-period spot rate and reinvest at the forward
interest rate; what is the value at time 2? How do these two
investments compare?
R0,1 = 3%
R0,2 = 6%
a. PV =
100
= $97.09
1.03
b. PV =
100
= $89.00
(1.06 )2
c. (1 + R0,2)2 = (1 + R0,1)(1 + f0,2)
(1 + 0.06)2 = (1.03)(1 + f0,2)
f0,2 = 9.087%
Time 1 value =
100
= $91.67
1.0909
d. 1(1 + R0,2)2 = 1(1+ 0.06)2 = $1.12
1(1 + R0,1)(1 + f0,2) = 1(1.03)(1.0909) = $1.12
2.
Treasury strips with $100 par values have the following prices:
one-period, $90; two-period, $80. Answer the following
questions:
(a) What are the one-period and two-period spot interest rates?
(b) What is the forward interest rate?
(c) If you invest $1 in one-period strips, what is the value after
one period?
(d) If you invest $1 in two-period strips, what is the value after
two periods?
(e) If you invest $1 at time 1 at the forward interest rate implied
by the strips, what is the value of this dollar at time 2?
a. 90 =
80 =
100
1 + R 0,1
R 0,1 = 11.11%
100
(1 + R 0,2 )2
R 0,2 = 11.80%
b. (1 + R0,2)2 = (1 + R0,1)(1 + f0,2)
(1.1180)2 = (1.1111)(1 + f0,2)
f0,2 = 12.50%
c. 1(1.1111) = $1.11
d. 1(1.1180)2 = $1.25
e. 1(1.1250) = $1.1250
3.
Treasury strips with $100 par values have the following prices:
one-period, $94; two-period, $87. You are going to receive an
annuity of $100 for the next two periods. What is the present
value of this annuity? What is the time 1 value of this annuity?
What is the time 2 value of this annuity?
PVA = (100)(0.94) + (100)(.87) = $181.00
4.
Time 1:
181
= $192.55 = (181)(1 + R 0,1)
0.94
Time 2:
181
= $208.05 = (181)(1 + R 0,2 )2
0.87
An annuity of $100 per period for two periods has a present
value of $178.33. If the term structure of interest rates is flat,
compute the interest rate.
178.33 =
100
100
+
1 + R (1 + R )2
R = 8%
In financial calculator:
N = 2; PV = -178.33; PMT = 100; FV = 0; I/YR = ?
I/YR = 8%
5.
Suppose the term structure in problem 1 applies. A two-period
coupon-bearing bond has an annual coupon of $5.00 and par
value of $100. Answer the following questions:
(a) What is the bond’s price?
(b) What is the bond’s yield to maturity?
(c) Suppose you arrange (at time 0) to buy this bond in the
forward market for delivery at time 1 immediately after the
coupon is paid. What should the forward price be?
a. P =
5
105
+
= $98.30
1.03 (1.06 )2
b. In financial calculator:
N = 2; PV = -98.30; PMT = 5; FV = 100; I/YR = YTM
I/YR = YTM= 5.93%
c. (1.06)2 = 1.03(1 + f0,2) f0,2 = 9.087%
F=
6.
105
105
=
= $96.25
1 + f 0,2 1.0909
As a reward for reading this book, you are given a choice
between $100 received one year from now and $115 received
two years from now. How should you go about deciding which
of these choices is better?
1 + f 0,2 =
7.
115
100
f 0,2 = 15%
f0,2 > 15%
prefer $100 in one year
f0,2 = 15%
indifferent
f0,2 < 15%
prefer $115 in two years
Suppose that the prices of one- through four-period strips per
$100 of par are $95, $90, $85, $80. Compute the spot and
forward interest rates and show these on a graph.
95 =
100
1 + R 0,1
90 =
100
(1 + R 0,2 )2
R 0,2 = 5.410%
85 =
100
(1 + R 0,3 )3
R 0,3 = 5.567%
80 =
100
(1 + R 0,4 )4
R 0,4 = 5.737%
R 0,1 = 5.263%
(1.0541)2 = (1.0526)(1 + f0,2) f0,2 = 5.556%
(1.0567)3 = (1.0541)2(1 + f0,3) f0,3 = 5.882%
(1.0574)4 = (1.0557)3(1 + f0,4) f0,4 = 6.250%
8.
Compute the spot and forward interest rates if the prices of oneperiod and two-period strips are each $92 per $100 of par value.
100
100
=
= 92
1 + R 0,1 (1 + R 0,2 )2
R0,1 = 8.70% R0,2 = 4.26%
(1.0426)2 = (1.087)(1 + f0,2)
f0,2 = 0%
9.
Using the information in problem 7, compute the yield to maturity
on a two-period par bond.
y par2 =
10.
11.
1  0.90
 5.41%
0.95  0.90
A one-period par bond has a price of $100, par value of $100,
and coupon of $6. A two-period par bond has a price of $100,
par value of $100, and coupon of $8. What is the two-period
spot interest rate?
100 =
106
1 + R 0,1
100 =
8
108
+
1.06 (1 + R 0,2)2
R 0,1 = 6%
R 0,2 = 8.08%
Suppose that the one-period spot interest rate is 10 percent.
What is the minimum value for the two-period spot interest
rate? (Hint: Express the two-period spot rate in terms of the
one-period spot rate and the forward rate.)
(1 + R0,2)2 = (1 + R0,1)(1 + f0,2)
let f0,2 = 0
1 + R0,2 = 1.10
12.
R0,2 = 4.88%
You observe that a one-year bond with annual coupon of $6 and
par value of $100 has a current price of $102.91. A two-year
bond with annual coupon of $6.50 and par value of $100 has a
current market price of $101.10. Compute the one-year spot
interest rate, the two-year spot interest rate, and the prices of
one-year and two-year strips with $100 par values.
102.91 
106
 1.030026
102.91
 3%
1  R0,1 
R0,1
106
1  R0,1
101.10 
6.50
106.50

1  R 0,1 (1  R 0, 2 ) 2
101.10 
6.50
106.50

1.03 (1  R 0, 2 ) 2
94.7893 
106.50
(1  R 0, 2 ) 2
R 0, 2  6%
13.
P1 
100
 97.08
1  R0,1
P2 
100
 89
1  R0,2 2
There are two two-year bonds. One bond has an annual coupon
of $4.50, par value $100, current price of $97.37. The other
bond has an annual coupon of $5.00, par value of $100, and
price of $98.30. (a) Find the present value of two-year annuity
of 1 dollar per year. (b) Compute the price of a two-year strip
and the two-year spot interest rate. (c) Compute the price of a
one-year strip and the one-year of spot interest rate. Hint: the
present value of a two-year annuity of 1 dollar per year equals
the price of a one-year strip divided by $100 plus the price of a
two-year strip divided by $100.
97.37 
4.50
104.50

1  R 0,1 (1  R 0, 2 ) 2
98.30 
5
105

1  R 0,1 (1  R 0, 2 ) 2
PVA s  slope 
98.30  97.37
5  4.50
 1.86
S 2  97.37 - (4.50)(1.86)
S 2  $89,
R 0,2  6%
S1
S
 2
100 100
S
89
1.86  1 
100 100
S1  $97,
R 0,1  3.09%
PVA 2 
P
98.30
97.37
S2
4.50
14.
5.00
C
Assume that the one-period spot interest rate is 6% and the twoperiod spot interest rate is 10%. A two-year bond with annual
coupon was of $8.50 and par value of $100 is traded. You
make a forward purchase for delivery in one year of this bond.
What is the forward price of this bond?
0
|
1
|
$95.05 F =
1 + f0,2
=
(1  R 0, 2 ) 2
1  R 0,1
= 1.1415
F = 108.50/1.1415 = $95.05
Also,
2
|
108.50
1  f 0, 2
F=
15.
108.50
(1  R 0,1 ) = 95.05
(1  R 0, 2 ) 2
Suppose a 4-year Treasury strip with $100 par value has a price
of $75.00 and a 5-year Treasury strip with $100 par value has a
price of $70.00. What is the forward interest rate for period 5?
1  f 0,5 
S4 75

 1.0714
S5 70
f 0,5  7.14%
16.
Suppose there is a forward market for Treasury strips. The
forward price for a strip with a delivery date in 2 years and
maturity in 3 years is $92.00. The forward price for a strip with
a delivery date in 1 year and maturity in 2 years is $94. The
spot price of a 1-period strip is 95.00. Determine the spot price
of a three-period strip.
0
|
1
|
95
0.95
94
0.94
100
1|
S3 = 100(0.95)(0.94)(0.92) = 82.156
=
100
(1  R 0,1 )(1  f 0, 2 )(1  f 0,3 )
2
|
92
0.92
100
1|
3
|
100
1|
17.
Suppose that a one-period Treasury strip with $100 par value
has a price of $97.05 and a two-period strip has a price of
$93.75. A two-period coupon bearing bond has an annual
coupon of $7.50 and par value of $100. Determine the yield to
maturity on this coupon-bearing bond.
0
|
97.05
0.9705
93.75
0.9375
1
|
2
|
100
1
100
1|
P2 = 7.50(0.9705) + 107.50(0.9375)
= 108.06
y = 3.27%.
18.
Consider a bond with an annual coupon of $6, par value of
$100, and maturity of 3 years. A forward contract for delivery
of this bond at time 2 has a price of $96.36 and a forward
contract for delivery of this bond at time 1 has a price of
$96.57. Determine the forward interest for time period 2 (i.e.,
f0,2).
0
|
1
|
2
|
96.36
6
106

1  f 0, 2 (1  f 0, 2 )(1  f 0,3 )
6
96.36
96.57= 

1  f 0, 2 1  f 0, 2
102.36
96.57
=
1  f 0, 2
96.57=
f0,2 = 6%
3
|
=
106
1  f 0,3
f0,3 = 10%
19.
Suppose that the price of 1-, 2-, and 3-year strips are $97, $93,
$88 per $100 of par. A 3-year bond has an annual coupon of $5
and par value of $100. Determine its yield to maturity.
P3 = (5)(0.97) + (5)(0.93) + 105(0.88) = 101.90
y = 4.31%