CHAPTER 8
1.
A one-period strip has a price of $86 and par value of $100. A two-period strip has a
price of $88 and par value of $100. Show the arbitrage opportunity.
Long
Short
Net
Cum Net
2.
1
|
+100
+2
+2
+100
+102
2
|
-100
-100
+2
A two-period bond has an annual coupon of $7, par value of $100, and price of $98.
Another two-period bond has an annual coupon of $8, par value of $100, and price of
$98. Are there any arbitrage opportunities?
Short
Long
Net
Cum Net
3.
0
|
-86
+88
0
|
+98
-98
1
|
-7
+8
2
|
-107
+108
0
0
+1
+1
+1
+2
In problem 2, is there an arbitrage opportunity if the $8-coupon bond has a price of
$98.25?
0
|
1
|
2
|
Short x
+98x
-7x
-107x
Long
-98.25
+8
+108
0
0
+0.98
+0.98
+0.73
+1.71
Net
Cum Net
x
98.25
 1.0026
98
4.
In problem 2, suppose the $8-coupon bond has a price of $101, are there arbitrage
opportunities?
Long
Short
Net
Cum Net
0
|
-98
+101
1
|
+7
-8
2
|
+107
-108
+3
+3
-1
+2
-1
+1
5.
The coupon on a two-period par bond is $9. A one-period strip has a price of $93.46 and
a two-period strip has a price of $87.34. Are there arbitrage opportunities? Explain.
0
1
2
|____________|____________|
Long
-100
+9
+109
Short .09
+8.41
-9
Short 1.09
+95.20
-109
___________________________
Net
+3.61
0
0
Cum Net
+3.61
+3.61
+3.61
6.
Suppose that R0,1 is 3 percent and R0,2 is 6 percent. For one-period and two-period strips
with $100 par values, show the operations to create:
(a) a long forward position; (b) a short forward position.
R0,1 = 3%
R0,2 = 6%
S1 =
100
= $97.09
1.03
x=
89
= 0.9167
97.09
S2 =
100
= $89.00
(1.06 )2
a.
0
1
2
|________________________|________________________|
Long
-89.00
+100
Short .9167 +97.09(.9167)
-100(.9167)
_________________________________________________
Long Forward 0
-91.67
+100
b.
0
1
2
|________________________|________________________|
Short
+89.00
-100
Long .9167 -97.09(.9167)
+100(.9167)
_________________________________________________
Short Forward 0
+91.67
-100
7.
Assuming the term structure in the preceding problem, show the arbitrage operations if
the actual forward interest rate is: (a) 15 percent, (b) 5 percent.
(1.06)2 = (1.03)(1 + f0,2)
f0,2 = 9.09%
a. Implied f0,2 = 9.09%
Actual f0,2 = 15%
Borrow at 9.09% and lend at 15%
a)
|____________|____________|
Short
+89.00
-100
Long .9167 -89.00
+91.67
Lend at 15% 0
-86.96
+100
___________________________
Net
0
+4.71
0
Cum Net
0
+4.71
+4.71
b. Implied f0,2 = 9.09%
Actual f0,2 = 5%
Borrow at 5% and lend at 9.09%
0
1
2
|____________|____________|
Long
-89.00
+100
Short .9167 +89.00
-91.67
Borrow at 5% 0
+95.25
-100
___________________________
Net
0
+3.57
0
Cum Net
0
+3.57
+3.57
8.
Suppose that you buy a three-year bond with annual coupons of $7 and par value of $100
for a price of $102. One-year, two-year, and three-year strips with $100 par values sell
for $95, $90, and $85 respectively. Suppose that you shortsell 7% of the one-period
strip, shortsell 100% of the 3-period strip, and shortsell 11.50% of the two-period strip.
Is this an arbitrage position? What would describe this position?
Long
Short 7%
Short 11.5%
Short 100%
Net
0
1
2
|
|
|
-102
+7
+7
+(95)(0.07) -(100)(0.07)
+6.65
-7.00
+(90)(0.115)
-(100)(0.115)
+10.35
-11.5
+85
0
This is the long forward position.
0
-4.50
3
|
+107
-100
+7
9.
A five-year bond has a price of $100, annual coupon of $10, and par value of $100. An
eight-year bond has a price of $100, annual coupon of $6, and par value of $100. Are
there any arbitrage opportunities?
0
1
|
|
Long
-100 +10
Short
+100 -6
Net
0
+4
Cum Net 0
+4
2
|
+10
-6
+4
+8
3
|
+10
-6
+4
+12
4
5
6
7
8 Total Inflows
|
|
|
|
|
|
+10 +110
150
-6
-6
-6
-6 -106 148
+4 +104 -6
-6 -106
+16 +120 +114 +108 +2
The $10 coupon bond has more total inflows ($150) and receives them sooner than the $6
coupon bond ($148).
10.
Suppose there are three 2-year bonds with par values of $100 and with coupons of $3, $4,
$5 and that the prices of one-year and two-year strips are $94 and $89. In a perfect
market, what should be the prices of bonds with coupons of $3, $4, and $5? If the bond
with the $4 coupon sells for $97, describe the arbitrage opportunities. If the bond with
the $4 coupon sells for $96, describe the arbitrage opportunities.
$3 coupon:
P = (3)(0.94) + (103)(0.89) = 94.49
PVA = 1.83
P = (3)(1.83) + (100)(0.89) = 94.49
$4 coupon:
P = (4)(1.83) + (100)(0.89) = 96.32
$5 coupon:
P = (5)(1.83) + (100)(0.89) = 98.15
If Pc=4 = $97, short this bond and buy 0.50 of $3 coupon and 0.50 of $5 coupon. Profit =
97 – 96.32.
If Pc=4 = $96, buy this bond, short 0.50 of $3 coupon and 0.50 of $5 coupon for a profit of
0.32.
11.
Suppose that there is a spot market for Treasury strips and a forward market. A oneperiod strip with $100 par value sells for $95.50 and a two-period strip sells for $92.25
per $100 of par in the spot market. In the forward market for strips for delivery in one
year, strips have quotes of $97.00 per $100 of par. Please describe any arbitrage
opportunities available?
Long
Short x
12.
0
|
92.25
1
|
x(95.50)
(100)x
x
Long Forward
Short Forward
0
0
-96.60
+97.00
+100
-100
Net
0
+0.40
0
92.25
 0.9660
95.50
A one-period strip has a spot interest rate of 8% and par value of $100. A two-period strip
has a spot interest rate of 3.80% and par value of $100. Are there any arbitrage
opportunities?
0
|
13.
2
|
+100
1
|
Short
+92.81
Long
-92.59
+100
Net
Cum Net
+0.22
+0.22
+100
+100.22
2
|
-100
100
(1.038) 2
100
92.59 
1.08
92.81 
-100
+0.22
Arbitrage
Bond G is a two-period bond with annual coupon of $5.25, par value of $100, and price
of $101. Bond H is a two-period bond with annual coupon of $5.50, par value of $100,
and price of $101. Are there any arbitrage opportunities?
Short G
Long H
Net
Cum Net
0
|
+101
-101
1
|
-5.25
+5.50
2
|
-105.25
+105.50
0
0
+0.25
+0.25
+0.25
+0.50
14.
Bond G is a two-period bond with annual coupon of $5.25, par value of $100, and price
of $97.50. Bond H is a two-period bond with annual coupon of $6.50, par value of $100,
and price of $101.00. Are there any arbitrage opportunities?
Long G
Short H
Net
Cum Net
15.
1
|
+5.25
-6.50
2
|
+105.25
+106.50
+3.50
+3.50
-1.25
+2.25
-1.25
+1.00
Suppose that one-period strips sell at $95 and two-period strips sell at $91 per $100 of
par. A two-period bond has an annual coupon of $8, $100 par value, and price of $106.
Are there any arbitrage opportunities?
Short
Long (1.08)
Long (0.08)
Net
16.
0
|
-97.50
+101.00
0
|
+106
-91(1.08)
-(98.28)
95(0.08)
(7.6)
0.12
1
|
-8
2
|
-108
+100(1.08)
100(0.08)
0
0
Suppose that one-period strips sell at $97.50 and two-period strips sell at $94.00 per $100
of par. A two-period bond has an annual coupon of $6, $100 par value, and price of
$106. Are there are any arbitrage opportunities?
Short
Long (1.06)
Long (0.06)
Net
0
|
+106
-94
-(99.64)
-97.50
-5.85
0.51
1
|
-6
2
|
-106
+100(1.06)
100(0.06)
0
0
17.
Suppose that a 1-period Treasury strip with $100 par value has a price of $91, a 2-period
strip with $100 par value has a price of $88, and a 3-year bond with annual coupon of $8
has a price of $100. Suppose an investor decides to buy the 3 period bond and short 8%
of a 1-period strip. What is the correct way to describe this position?
(a)
(b)
(c)
(d)
(e)
Long forward position.
Short forward position.
An arbitrage position.
An extremely risky position.
None of the above.
The correct answer is e. This is a long spot position.
Long
Short (0.08)
Net
0
|
-100
+91(0.08)
+7.28
1
|
+8
-100(0.08)
2
|
+8
3
|
+108
-92.72
0
+8
+108