FM for Actuaries

92 CHAPTER 3
Figure 3.5:
Illustration of cash flows of an interest rate swap
(a) Cash flows for the two companies without interest rate swap
LIBOR
Company
A
Company
B
5%
(b) Cash flows for the two companies with interest rate swap
LIBOR
Company
A
5%
LIBOR
Company
B
5%
(c) Net cash flows for the two companies with interest rate swap
5%
Company
A
Company
B
LIBOR
Suppose the 12-month US dollar LIBOR after one year is 4.75%. According to
the swap agreement, Company B has to pay $10,000,000 × 4.75% = $475,000
to Company A at the end of the second year, while Company A needs to pay
$10,000,000 × 5% = $500,000 to Company B at the same time. In order to avoid
unnecessary transactions, most swap contracts require both parties to settle the netted
amount only. In this example, the interest rate swap net payment in that
settlement date is $25,000, payable from Company A to Company B.
The fixed interest rate in the interest rate swap contract is called the swap rate,
which will be denoted by R S . We now discuss a theoretical framework for deriving
R S such that the swap contract is equally attractive to both parties. 8
Consider a n-year interest rate swap contract with notional amount m t for year
t =1, ··· ,n. The settlement period is one year. At time 0, when the contract
is initiated, the spot rates of interest i S t are known. 9 The floating interest rate is
defined as the realized one-year spot rate i ∗ t observed at the beginning of the tth
future settlement period. It is obvious that i ∗ 1 = iS 1 .
8 It is assumed that the capital market is perfectly competitive (see Footnote 1 in this Chapter) and
frictionless, where all costs and restraints associated with transactions are non-existent.
9 These can be US Treasury spot rates, LIBOR spot rates or other types of spot rates specified in
the swap agreement.