Question

Under the terms of an interest rate swap, a financial institution has agreed to pay 9% annual rate and to receive three-month LIBOR in return on a notional principal of $100 million with payment being exchanged every three months. The swap has a remaining life of 17 months. The average of the bid and offer fixed rates currently being swapped for three-month LIBOR is 10% for all maturities. The three-month LIBOR rate one month ago was 10.6%. All rates are compounded quarterly

How can you decompose the swap into different futures contacts? What are the values of those futures contracts?

Answer 1

Solution:

Firstly, we find the value of swap using the bond approach. For the discount rate, we convert the 10% rate with quarterly compounding to 9.876% with continuous compounding using the formula

r = (1 - exp^(-r*1/4)) x 4

r = (1 – exp^(-0.10*1/4)) x 4

r = 0.09876 or 9.876%

The floating bond has its next payment in two months’ time. The payment is in the amount of 100(0.09876) 1/4 = $2.469 million since its interest rate is determined by last month’s 3-month LIBOR rate.

Immediately after this payment, the floating bond’s value is $100 million. The value of the floating bond is therefore

(100 + 2.469)e^−0.09876*(2/12) = $100.80 million

For the fixed bond, each coupon payment is in the amount of 100(0.09) 1/4 = $2.25 million. The value of the fixed bond is given by

2.25e^(-0.09876 x 2/12) + 2.25e^(-0.09876 x 5/12) + 2.25e^(-0.09876 x 8/12) + 102.25e^(-0.09876 x 11/12) = $99.88

The value of the swap is then 100.80 – 99.88 = $0.92 million