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A semi-annual pay interest rate swap where the fixed rate is 5% (with semi-annual compounding) has a remaining life of ten months. The six-month LIBOR rate observed two months ago was 4.85% with semi-annual compounding. Today’s four and ten month zero rates are 5.3% and 5.8%, respectively, per annum continuously compounded. The swap has a principal value of $1,000,000. a) For this interest rate swap, the party receiving a fixed rate of interest is the (swap buyer or swap seller), and effectively holds (a short position or a long position) in a fixed-rate bond and (a short position or a long position) in a floating-rate bond. b) The value of the interest rate swap to the buyer is ? . Note: Please provide your answer as an integer without commas in the format of xxxxxx (for example, if the answer is $123,456.00, type in 123456; if the answer is -$123,456.00, type in -123456).

Question

A semi-annual pay interest rate swap where the fixed rate is 5% (with semi-annual compounding) has a remaining life of ten months. The six-month LIBOR rate observed two months ago was 4.85% with semi-annual compounding. Today’s four and ten month zero rates are 5.3% and 5.8%, respectively, per annum continuously compounded. The swap has a principal value of $1,000,000.

a) For this interest rate swap, the party receiving a fixed rate of interest is the
(swap buyer or swap seller), and effectively holds (a short position or a long position) in a fixed-rate bond and (a short position or a long position) in a floating-rate bond.

b) The value of the interest rate swap to the buyer is ? .

Note: Please provide your answer as an integer without commas in the format of xxxxxx (for example, if the answer is 123,456.00,typein123456;iftheansweris123,456.00, type in 123456; if the answer is -123,456.00, type in -123456).

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Solution

a) For this interest rate swap, the party receiving a fixed rate of interest is the swap buyer, and effectively holds a long position in a fixed-rate bond and a short position in a floating-rate bond.

b) To calculate the value of the interest rate swap to the buyer, we need to calculate the present value of the fixed and floating payments.

The fixed payments are based on the fixed rate of 5% per annum, semi-annually compounded. The next fixed payment will be 1,000,00051,000,000 * 5% / 2 = 25,000 in four months, and the one after that will be $25,000 in ten months.

The present value of the fixed payments is calculated using the given zero rates. The present value of the first fixed payment is 25,000e(5.325,000 * e^(-5.3% * 4/12) = 24,456. The present value of the second fixed payment is 25,000e(5.825,000 * e^(-5.8% * 10/12) = 23,456. The total present value of the fixed payments is 24,456+24,456 + 23,456 = $47,912.

The floating payment is based on the six-month LIBOR rate observed two months ago, which is 4.85% with semi-annual compounding. The next floating payment will be 1,000,0004.851,000,000 * 4.85% / 2 = 24,250 in four months.

The present value of the floating payment is 24,250e(5.324,250 * e^(-5.3% * 4/12) = 23,712.

The value of the interest rate swap to the buyer is the present value of the fixed payments minus the present value of the floating payment, which is 47,91247,912 - 23,712 = $24,200.

So, the correct answer is: 24200.

This problem has been solved

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