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.