Chapter 8
Swaps
n
Question 8.1
We first solve for the present value of the cost per two barrels:
2
$22
$23
41.033.
1.06
(1.065)
+
=
We then obtain the swap price per barrel by solving:
2
41.033
(1.065)
1.06
22.483,
x
x
x
+
=
⇒
=
which was to be shown.
n
Question 8.2
1.
We first solve for the present value of the cost per three barrels, based on the forward prices:
2
3
$20
$21
$22
55.3413.
1.06
(1.065)
(1.07)
+
+
=
Hence we could spend $55.3413 today to receive 1 barrel in each of the next three years. We then
obtain the swap price per barrel by solving:
2
3
55.3413
1.06
(1.065)
(1.07)
20.9519
x
x
x
x
+
+
=
⇒
=
2.
We first solve for the present value of the cost per two barrels (Year 2 and Year 3):
2
3
$21
$22
36.473.
(1.065)
(1.07)
+
=
Hence we could spend $36.473 today and receive 1 barrel of oil in Year 2 and Year 3. We obtain the
swap price per barrel by finding two equal payments we would make in Years 2 and 3 that have the
same present value:
2
3
36.473
(1.065)
(1.07)
21.481
x
x
x
+
=
⇒
=
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McDonald •
Fundamentals of Derivatives Markets
n
Question 8.3
Since the dealer is paying fixed and receiving floating, each year she has a cash flow
$20.9519.
T
S

She
can hedge this risk by selling 1 barrel forward (i.e., short one forward) in each of the three years. Her
payoffs from the swap, the short forward contracts, and the net are summarized in the following table:
Year
Net Swap Payment
Short Forwards
Net Position
1
1
$20.9519
S

1
$20
S


0.9519
2
2
$20.9519
S

2
$21
S

+
0.0481
3
3
$20.9519
S

3
$22
S

+
1.0481
We need to discount the net cash flows to year zero. We have:
2
3
0.9519
0.0481
1.0481
PV(net CF)
0.
1.06
(1.065)
(1.07)

=
+
+
=
Indeed, the present value of the net cash flow is zero.
n
Question 8.4
The fair swap rate was determined to be $20.9519. Therefore, compared to the forward curve price of
$20 in one year, we are overpaying $0.9519. In year two, with interest, this overpayment increases
to $0.9519 1.070024
$1.01853,
×
=
where we used the appropriate forward rate to calculate the interest
payment.
In year two, we underpay by $0.0481, so that our total accumulative underpayment is $1.01856

$0.0481
=
$0.97042. In year three, using the appropriate 1year forward rate of 8.007%, this net overpayment
increases to$0.97046 1.08007
$1.0481.
×
=
However, in year three, we receive a fixed payment of 20.9519,
which underpays relative to the forward curve price of $22 by$22
$20.9519
$1.0481.

=
Therefore, our
cumulative balance is indeed zero, which was to be shown.
n
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 Spring '10
 Haan
 Interest Rates, swap price

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