第九版期权期货及其他衍生品课后答案

CHAPTER 7

Swaps

Practice Questions

Problem 7.1.

Companies A and B have been offered the following rates per annum on a $20 million five-year loan:

Company A requires a floating-rate loan; company B requires a fixed-rate loan. Design a swap that will net a bank, acting as intermediary, 0.1% per annum and that will appear equally attractive to both companies.

A has an apparent comparative advantage in fixed-rate markets but wants to borrow floating.

B has an apparent comparative advantage in floating-rate markets but wants to borrow fixed. This provides the basis for the swap. There is a 1.4% per annum differential between the fixed rates offered to the two companies and a 0.5% per annum differential between the floating rates offered to the two companies. The total gain to all parties from the swap is therefore 1 4 0 5 0 9% per annum. Because the bank gets 0.1% per annum of this gain, the swap should make each of A and B 0.4% per annum better off. This means that it should lead to A borrowing at LIBOR 0 3% and to B borrowing at 6.0%. The appropriate arrangement is therefore as shown in Figure S7.1.

Figure S7.1: Swap for Problem 7.1

Problem 7.2.

Company X wishes to borrow U.S. dollars at a fixed rate of interest. Company Y wishes to borrow Japanese yen at a fixed rate of interest. The amounts required by the two companies are roughly the same at the current exchange rate. The companies have been quoted the following interest rates, which have been adjusted for the impact of taxes:

第九版期权期货及其他衍生品课后答案

Design a swap that will net a bank, acting as intermediary, 50 basis points per annum. Make the swap equally attractive to the two companies and ensure that all foreign exchange risk is assumed by the bank.

X has a comparative advantage in yen markets but wants to borrow dollars. Y has a

comparative advantage in dollar markets but wants to borrow yen. This provides the basis for the swap. There is a 1.5% per annum differential between the yen rates and a 0.4% per annum differential between the dollar rates. The total gain to all parties from the swap is therefore 1 5 0 4 1 1% per annum. The bank requires 0.5% per annum, leaving 0.3% per annum for each of X and Y. The swap should lead to X borrowing dollars at 9 6 0 3 9 3% per

annum and to Y borrowing yen at 6 5 0 3 6 2% per annum. The appropriate arrangement is therefore as shown in Figure S7.2. All foreign exchange risk is borne by the bank.

Figure S7.2: Swap for Problem 7.2

Problem 7.3.

A $100 million interest rate swap has a remaining life of 10 months. Under the terms of the swap, six-month LIBOR is exchanged for 7% per annum (compounded semiannually). The average of the bid–offer rate being exchanged for six-month LIBOR in swaps of all maturities is currently 5% per annum with continuous compounding. The six-month LIBOR rate was

4.6% per annum two months ago. What is the current value of the swap to the party paying floating? What is its value to the party paying fixed?

In four months $3.5 million ( 0 5 0 07 $100 million) will be received and $2.3 million ( 0 5 0 046 $100 million) will be paid. (We ignore day count issues.) In 10 months $3.5 million will be received, and the LIBOR rate prevailing in four months’ time will be paid. The value of the fixed-rate bond underlying the swap is

3 5e 0 05 4 12 103 5e 0 05 10 12 $102 718million

The value of the floating-rate bond underlying the swap is

(100 2 3)e 0 05 4 12 $100 609million

The value of the swap to the party paying floating is $102 718 $100 609 $2 109 million. The value of the swap to the party paying fixed is $2 109 million.

These results can also be derived by decomposing the swap into forward contracts. Consider the party paying floating. The first forward contract involves paying $2.3 million and receiving $3.5 million in four months. It has a value of 1 2e 0 05 4 12 $1 180 million. To value the second forward contract, we note that the forward interest rate is 5% per annum with continuous compounding, or 5.063% per annum with semiannual compounding. The

第九版期权期货及其他衍生品课后答案

value of the forward contract is

100 (0 07 0 5 0 05063 0 5)e 0 05 10 12 $0 929million

The total value of the forward contracts is therefore $1 180 $0 929 $2 109 million.

Problem 7.4.

Explain what a swap rate is. What is the relationship between swap rates and par yields?

A swap rate for a particular maturity is the average of the bid and offer fixed rates that a market maker is prepared to exchange for LIBOR in a standard plain vanilla swap with that maturity. The swap rate for a particular maturity is the LIBOR/swap par yield for that maturity.

Problem 7.5.

A currency swap has a remaining life of 15 months. It involves exchanging interest at 10% on £20 million for interest at 6% on $30 million once a year. The term structure of interest rates in both the United Kingdom and the United States is currently flat, and if the swap were negotiated today the interest rates exchanged would be 4% in dollars and 7% in sterling. All interest rates are quoted with annual compounding. The current exchange rate (dollars per pound sterling) is 1.5500. What is the value of the swap to the party paying sterling? What is the value of the swap to the party paying dollars?

The swap involves exchanging the sterling interest of 20 0 10 or £2 million for the dollar interest of 30 0 06 $1 8 million. The principal amounts are also exchanged at the end of the life of the swap. The value of the sterling bond underlying the swap is

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