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2.2.3 Interest rate swaps and forward rate agreements
[ SLIDE
swap 
equilibrium rate 
FRA 
VIDEO
modem 
LAN 
DSL]
A plain vanilla interest rate swap is a
contract whereby two parties agree to exchange, at known dates in the
future, a fixed for a floating set of interest rate payments without ever
exchanging the notional principal A.
The fixed leg of the swap replicates the
coupons (1.3#eq.5) payed at the end
of every accrual period spanning
from the reset time to the
payment time [t_{i};t_{i}+
t_{i}]

(2.2.3#eq.1) 
using a fixed interest rate K that is initially agreed upon when the
swap is purchased.
The floating leg consists of payments
that also occur at a time t_{i+1}=t_{i}+
t_{i}

(2.2.3#eq.2) 
using however the unknown spot rate r_{i}=r(t_{i},t_{i+1})
that prevails at some future times t_{i}.
Figure 2.2.3#fig.1:
Sketch of an example showing the cash flows when a bank takes the
credit risk from a loan taker and agrees to pay 2% in excess of the
floating spot rate in exchange of a fixed interet payments of 6%.

The present value of both legs can be discounted back in time using
discount bonds to get
where a simple compounding has been assumed to substitute the
spot rate for the discount bond using (2.2.2#eq.1).
Now compare the latter with a portfolio long one bond P(0,t_{i})
and short another with longer maturity P(0,t_{i+1}). At time _{i},
the portfolio value is

(2.2.3#eq.4) 
or indeed the same, to a normalizing constant A, as the floating leg
in (2.2.3#eq.3)

(2.2.3#eq.5) 
After identification with the definition of simply compounded forward
rates (2.2.2#eq.2), this shows that the a priori unknown values of
future spot rates have the same value today as the projected forward
rates r_{i}=F_{i}=
F(0,t_{i},t_{i+1}).
An equilibrium swap rate
can therefore be calculated in the form of a weighted average of forward
rates, making the values of the floating and the fixed legs equal when
the contract is initially written at t=0

(2.2.3#eq.6) 

(2.2.3#eq.7) 
Notice that no assumption about the random evolution of spot rates has
been made, the combination of long and short bonds being amenable to a
purely deterministic evaluation in a manner similar to what was has
been found for the putcall parity relation (2.1.3#eq.2).
By definition, a one period swap is sometimes called
forward rate agreement: an X's/Y's FRA refers
to an interest rate swap starting in X and finishing in Y months and has
a present value given by the difference between the floating and the
fixed legs

(2.2.3#eq.8) 
The total amount of cash payed after each accrual period [t_{i};t_{i+1}]
depends on the difference between the settlement rate R_{i}
and the forward rate K; after a simply compounded discounting, the cash flow
at a time t_{i}
from the seller to the buyer amounts to

(2.2.3#eq.9) 
Here is an example showing the entire sequence of events:
Wed 02Feb00 
2's/5's FRA contract written at 6% for EUR 1 Mio 
Fri 31Mar00 
settlement rate determined at 5% (3 months forward 

LIBOR for the period Tue 04Apr00 to Wed 05Jul00) 

Settlement amount given by (2.2.3#eq.9) 

1000000
(0.01
92/360)/(1+0.05
92/360)
=  2523.31 
Wed 05Jul00 
buyer pays seller EUR 2523.31 
Beware of the dealers jargon, which is opposite for bonds and swaps:
bid means to buy fixed in bonds and sell
fixed in swaps, whereas offer means to
sell fixed in bonds and buy fixed in swaps.
To conclude this section with a little review, it should now be clear that
for the holder of a swap, the earnings increase (alt. drops) when the spot
rate evolves above (alt. drops below) the projected forward rates.
At the same time, the market data in (1.3#tab.1) illustrates how an
increasing spot rate produces a rise in the par coupon (particular
coupon that prices the bond today exactly at par
 i.e. for a present value equal to the nominal principal) when the bond
trades at a discount
(alt. premium).
SYLLABUS Previous: 2.2.2 Underlying discount bonds
Up: 2.2 The credit market
Next: 2.2.4 Bond options: caps,