Chapter 8
Swaps
Question 8.1.
We first solve for the present value of the cost per two barrels:
$22
1
.
06
+
$23
(
1
.
065
)
2
=
41
.
033
.
We then obtain the swap price per barrel by solving:
x
1
.
06
+
x
(
1
.
065
)
2
=
41
.
033
⇔
x
=
22
.
483
,
which was to be shown.
Question 8.2.
a)
We first solve for the present value of the cost per three barrels, based on the forward prices:
$20
1
.
06
+
$21
(
1
.
065
)
2
+
$22
(
1
.
07
)
3
=
55
.
3413
.
We then obtain the swap price per barrel by solving:
x
1
.
06
+
x
(
1
.
065
)
2
+
x
(
1
.
07
)
3
=
55
.
341
⇔
x
=
20
.
9519
b)
We first solve for the present value of the cost per two barrels (year 2 and year 3):
$21
(
1
.
065
)
2
+
$22
(
1
.
07
)
3
=
36
.
473
.
We then obtain the swap price per barrel by solving:
x
(
1
.
065
)
2
+
x
(
1
.
07
)
3
=
36
.
473
⇔
x
=
21
.
481
104
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Chapter 8 Swaps
Question 8.3.
Since the dealer is paying fixed and receiving floating, she generates the cashflows depicted in
column 2. Suppose that the dealer enters into three short forward positions, one contract for each
year of the active swap. Her payoffs are depicted in columns 3, and the aggregate net cash flow
position is in column 4.
Year
Net Swap Payment
Short Forwards
Net Position
1
S
1
−
$20
.
9519
$20
−
S
1
−
0.9519
2
S
1
−
$20
.
9519
$21
−
S
1
+
0.0481
3
S
1
−
$20
.
9519
$22
−
S
1
+
1.0481
We need to discount the net positions to year zero. We have:
P V (netCF)
=
−
0
.
9519
1
.
06
+
0
.
0481
(
1
.
065
)
2
+
1
.
0481
(
1
.
07
)
3
=
0
.
Indeed, the present value of the net cash flow is zero.
Question 8.4.
The fair swap rate was determined to be $20.952. Therefore, compared to the forward curve price
of $20 in one year, we are overpaying $0.952. In year two, this overpayment has increased to
$0
.
952
×
1
.
070024
=
1
.
01866, where we used the appropriate forward rate to calculate the interest
payment. In year two, we underpay by $0.048, so that our total accumulative underpayment is
$0.97066. In year three, this overpayment has increased again to $0
.
97066
×
1
.
08007
=
1
.
048.
However, in year three, we receive a fixed payment of 20.952, which underpays relative to the
forward curve price of $22 by $22
−
$20
.
952
=
1
.
048. Therefore, our cumulative balance is indeed
zero, which was to be shown.
Question 8.5.
Since the dealer is paying fixed and receiving floating, she generates the cashflows depicted in
column 2. Suppose that the dealer enters into three short forward positions, one contract for each
year of the active swap. Her payoffs are depicted in columns 3, and the aggregate net position is
summarized in column 4.
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 Spring '06
 Adam
 Interest Rates, Forward price, swap price

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