FM for Actuaries

Spot Rates, Forward Rates and the Term Structure 93
Figure 3.6:
Cash flows of an interest rate swap with a market maker
Fixed-Rate Payer
Floating-Rate Payer
(Market Maker)
If the swap rate is R S , at the end of year t, the fixed rate payer will receive
m t (i ∗ t − R S ) net interest rate swap payment from the counterparty. At the same
time, the company has to repay a floating interest amount of m t i ∗ t to its outside
lender. Under this arrangement, the fixed rate payer has converted its floating rate
debt to a fixed rate one using the swap with annual fixed rate payment m t R S .
Suppose the counterparty (i.e., the floating rate payer) is a market maker who determines
the swap rate R S . Figure 3.6 illustrates the cash flows of these parties.
To understand the pricing of an interest rate swap (i.e., determination of R S ),
we examine the cash flows of the market maker. At the end of year t, themarket
maker (floating-rate payer) pays the floating rate amount m t i ∗ t and receives the
fixed rate amount m t R S . It should be noted that i ∗ t is unknown at time 0 when
R S is determined. The market maker can hedge this interest rate risk by entering
into forward agreements with another financial institution and makes payments to
it based on the forward rates of interest i F t . Table 3.1 summarizes the cash flows
faced by the market maker.
Table 3.1: Cash flows of the market maker in Figure 3.6
Net receipt on Net receipt on
Year forward contract swap contract Overall
1 m 1 (i ∗ 1 − iF 1 ) m 1(R S − i ∗ 1 ) m 1(R S − i F 1 )
2 m 2 (i ∗ 2 − iF 2 ) m 2(R S − i ∗ 2 ) m 2(R S − i F 2 )
.
.
.
.
t m t (i ∗ t − i F t ) m t (R S − i ∗ t ) m t (R S − i F t )
.
.
.
.
n m n (i ∗ n − iF n ) m n(R S − i ∗ n ) m n(R S − i F n )
Note: At t =1, i ∗ 1 = iF 1 and hence m 1(i ∗ 1 − iF 1 )=0.