Question
3. An interest-rate swap is an exchange of a fixed rate against a floating rate. One party agrees to pay the other a fixed rate
3. An interest-rate swap is an exchange of a fixed rate against a floating rate. One party
agrees to pay the other a fixed rate in regular intervals until a maturity T. The other
party agrees to pay a variable rate in the same intervals. Payments are multiplied by
a fixed amount, called the notional.
Consider a swap with annual payments and notional $1. The fixed-rate payments are
pT in annual intervals until T, where pT denotes the swap rate, also known as the par
rate. The first variable-rate payment is r1, the one-year spot rate at the time the swap
is agreed. The second variable-rate payment is 1r1, the one-year spot rate one year
after the swap is agreed. The variable-rate payment in year T is T-1r1, the one-year
spot rate T-1 years after the swap is agreed.
A swap involves no cash flows at the time of agreement. Therefore, the present value
of the fixed-rate payments must equal that of the variable-rate payments.
Question:
(a) Compute the present value of a cash-ow stream that consists of the variable-rate
payments plus a payment of $1 in year T.
Hint: Find an investment policy that generates the cash-ow stream.
(b) Determine the par rate pT .
(c) Suppose that the 1-year spot rate is 5%, the 2-year spot rate is 4.5%, and the
3-year spot rate is 4%. Determine p1, p2 and p3.
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