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Options, Futures, and Other Derivatives

John C. Hull

Chapter 7

Swaps - all with Video Answers

Educators

Chapter Questions

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Problem 1

Companies A and B have been offered the following rates per annum on a $$\$ 20$$ million five-year loan:
$$
\begin{array}{lcc}
\hline & \text { Fixed rate } & \text { Floating rate } \\
\hline \text { Company A } & 5.0 \% & \text { LIBOR }+0.1 \% \\
\text { Company B } & 6.4 \% & \text { LIBOR }+0.6 \% \\
\hline
\end{array}
$$
Company A requires a floating-rate loan; Company $\mathrm{B}$ 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.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
02:34

Problem 2

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 $4 \%$ per annum (compounded semiannually). Sixmonth LIBOR forward rates for all maturities are $3 \%$ (with semiannual compounding). The six-month LIBOR rate was $2.4 \%$ two months ago. OIS rates for all maturities are $2.7 \%$ with continuous compounding. What is the current value of the swap to the party paying floating? What is the value to the party paying fixed?

Breanna Ollech
Breanna Ollech
Numerade Educator
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Problem 3

Company $\mathrm{X}$ wishes to borrow U.S. dollars at a fixed rate of interest. Company $\mathrm{Y}$ 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:
$$
\begin{array}{lcr}
\hline & \text { Yen } & \text { Dollars } \\
\hline \text { Company X } & 5.0 \% & 9.6 \% \\
\text { Company Y } & 6.5 \% & 10.0 \% \\
\hline
\end{array}
$$
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.

Lainey Roebuck
Lainey Roebuck
Numerade Educator

Problem 4

Explain what a seven-year swap rate is.

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Problem 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 riskfree interest rates in the United Kingdom is flat at $7 \%$ and the term structure of risk-free interest rates in the United States is flat at $4 \%$ (both 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?

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Problem 6

Explain the difference between the credit risk and the market risk in a financial contract.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
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Problem 7

A corporate treasurer tells you that he has just negotiated a five-year loan at a competitive fixed rate of interest of $5.2 \%$. The treasurer explains that he achieved the $5.2 \%$ rate by borrowing at six-month LIBOR plus 150 basis points and swapping LIBOR for $3.7 \%$. He goes on to say that this was possible because his company has a comparative advantage in the floating-rate market. What has the treasurer overlooked?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
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Problem 8

A bank enters into an interest rate swap with a nonfinancial counterparty using bilaterally clearing where it is paying a fixed rate of $3 \%$ and receiving LIBOR. No collateral is posted and no other transactions are outstanding between the bank and the counterparty. What credit risk is the bank subject to? Discuss whether the credit risk is greater when the yield curve is upward sloping or when it is downward sloping.

Victor Salazar
Victor Salazar
Numerade Educator
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Problem 9

Companies $X$ and $Y$ have been offered the following rates per annum on a $$\$ 5$$ million 10-year investment:
$$
\begin{array}{lcc}
\hline & \text { Fixed rate } & \text { Floating rate } \\
\hline \text { Company X } & 8.0 \% & \text { LIBOR } \\
\text { Company Y } & 8.8 \% & \text { LIBOR } \\
\hline
\end{array}
$$
Company $\mathrm{X}$ requires a fixed-rate investment; company $\mathrm{Y}$ requires a floating-rate investment. Design a swap that will net a bank, acting as intermediary, $0.2 \%$ per annum and will appear equally attractive to $\mathrm{X}$ and $\mathrm{Y}$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
02:34

Problem 10

A financial institution has entered into an interest rate swap with company $X$. Under the terms of the swap, it receives $4 \%$ per annum and pays six-month LIBOR on a principal of $$\$ 10$$ million for five years. Payments are made every six months. Suppose that company $\mathrm{X}$ defaults on the sixth payment date (end of year 3) when six-month forward LIBOR rates for all maturities are $2 \%$ per annum. What is the loss to the financial institution? Assume that six-month LIBOR was $3 \%$ per annum halfway through year 3 and that at the time of the default all OIS rates are $1.8 \%$ per annum. OIS rates are expressed with continuous compounding; other rates are expressed with semiannual compounding.

Breanna Ollech
Breanna Ollech
Numerade Educator

Problem 11

A financial institution has entered into a 10-year currency swap with company Y. Under the terms of the swap, the financial institution receives interest at $3 \%$ per annum in Swiss francs and pays interest at $8 \%$ per annum in U.S. dollars. Interest payments are exchanged once a year. The principal amounts are 7 million dollars and 10 million francs. Suppose that company Y declares bankruptcy at the end of year 6 , when the exchange rate is $$\$ 0.80$$ per franc. What is the cost to the financial institution? Assume that, at the end of year 6 , riskfree interest rates are 3\% per annum in Swiss francs and $8 \%$ per annum in U.S. dollars for all maturities. All interest rates are quoted with annual compounding.

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Problem 12

Companies $\mathrm{A}$ and $\mathrm{B}$ face the following interest rates (adjusted for the differential impact of taxes):
$$
\begin{array}{lcc}
\hline & \text { Company A } & \text { Company B } \\
\hline \text { U.S. dollars (floating rate) } & \text { LIBOR }+0.5 \% & \text { LIBOR }+1.0 \% \\
\text { Canadian dollars (fixed rate) } & 5.0 \% & 6.5 \% \\
\hline
\end{array}
$$
Assume that A wants to borrow U.S. dollars at a floating rate of interest and B wants to borrow Canadian dollars at a fixed rate of interest. A financial institution is planning to arrange a swap and requires a 50-basis-point spread. If the swap is equally attractive to $\mathrm{A}$ and $B$, what rates of interest will $A$ and $B$ end up paying?

Lainey Roebuck
Lainey Roebuck
Numerade Educator

Problem 13

After it hedges its foreign exchange risk using forward contracts, is the financial institution's average spread in Figure 7.11 likely to be greater than or less than 20 basis points? Explain your answer.

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Problem 14

"Nonfinancial companies with high credit risks are the ones that cannot access fixed-rate markets directly. They are the companies that are most likely to be paying fixed and receiving floating in an interest rate swap." Assume that this statement is true. Do you think it increases or decreases the risk of a financial institution's swap portfolio? Assume that companies are most likely to default when interest rates are high.

Victor Salazar
Victor Salazar
Numerade Educator

Problem 15

Why is the expected loss to a bank from a default on a swap with a counterparty less than the expected loss from the default on a loan to the counterparty when the loan and swap have the same principal? Assume that there are no other derivatives transactions between the bank and the counterparty, that the swap is cleared bilaterally, and that no collateral is provided by the counterparty in the case of either the swap or the loan.

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00:00

Problem 16

A bank finds that its assets are not matched with its liabilities. It is taking floating-rate deposits and making fixed-rate loans. How can swaps be used to offset the risk?

Jennifer Stoner
Jennifer Stoner
Numerade Educator
00:00

Problem 16

A bank finds that its assets are not matched with its liabilities. It is taking floating-rate deposits and making fixed-rate loans. How can swaps be used to offset the risk?

Jennifer Stoner
Jennifer Stoner
Numerade Educator
03:06

Problem 17

Explain how you would value a swap that is the exchange of a floating rate in one currency for a fixed rate in another currency.

Jennifer Stoner
Jennifer Stoner
Numerade Educator
04:44

Problem 18

OIS rates have been estimated as $3.4 \%$ for all maturities. The three-month LIBOR rate is $3.5 \%$. For a six-month swap where payments are exchanged every three months the swap rate is $3.6 \%$. All rates are expressed with quarterly compounding. What is the LIBOR forward rate for the 3- to 6-month period.

Narayan Hari
Narayan Hari
Numerade Educator
04:44

Problem 19

Six-month LIBOR is $5 \%$. LIBOR forward rates for the 6- to 12 -month period and for the 12- to 18 -month period are $5.5 \%$. Swap rates for 2- and 3-year semiannual pay swaps are $5.4 \%$ and $5.6 \%$, respectively. Estimate the LIBOR forward rates for for 18 months to 2 years, 2 to 2.5 years, and 2.5 to 3 years. Assume that the 2.5 -year swap rate is the average of the 2- and 3-year swap rates and that OIS zero rates for all maturities are $4.5 \%$. OIS rates are expressed with continuous compounding; all other rates are expressed with semiannual compounding.

Narayan Hari
Narayan Hari
Numerade Educator

Problem 20

(a) Company A has been offered the swap quotes in Table 7.3. It can borrow for three years at $3.45 \%$. What floating rate can it swap this fixed rate into? (b) Company B has been offered the swap quotes in Table 7.3. It can borrow for five years at LIBOR plus 75 basis points. What fixed rate can it swap this rate into? (c) Explain the rollover risks that Company B is taking.

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02:34

Problem 21

(a) Company $\mathrm{X}$ has been offered the swap quotes in Table 7.3. It can invest for four years at $2.8 \%$. What floating rate can it swap this fixed rate into? (b) Company Y has been offered the swap quotes in Table 7.3. It is confident that it will be able to invest at LIBOR minus 50 basis points for the next ten years. What fixed rate can it swap this floating rate into?

Breanna Ollech
Breanna Ollech
Numerade Educator
02:34

Problem 22

The one-year LIBOR rates is $3 \%$, and the LIBOR forward rate for the 1- to 2 -year period is $3.2 \%$, respectively. The three-year swap rate for a swap with annual payments is $3.2 \%$. What is the LIBOR forward rate for the 2- to 3-year period if OIS zero rates for maturities of one, two, and three years are $2.5 \%, 2.7 \%$, and $2.9 \%$, respectively. What is the value of a three-year swap where $4 \%$ is received and LIBOR is paid on a principal of $$\$ 100$$ million. All rates are annually compounded

Breanna Ollech
Breanna Ollech
Numerade Educator
02:34

Problem 23

In an interest rate swap, a financial institution has agreed to pay $3.6 \%$ per annum and to receive three-month LIBOR in return on a notional principal of $$\$ 100$$ million with payments being exchanged every three months. The swap has a remaining life of 14 months. Three-month forward LIBOR for all maturities is currently $4 \%$ per annum. The three-month LIBOR rate one month ago was $3.2 \%$ per annum. OIS rates for all maturities are currently $3.8 \%$ with continuous compounding. All other rates are compounded quarterly. What is the value of the swap?

Breanna Ollech
Breanna Ollech
Numerade Educator
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Problem 24

Company A, a British manufacturer, wishes to borrow U.S. dollars at a fixed rate of interest. Company B, a U.S. multinational, wishes to borrow sterling at a fixed rate of interest. They have been quoted the following rates per annum:
$$
\begin{array}{lcc}
\hline & \text { Sterling } & \text { U.S. Dollars } \\
\hline \text { Company A } & 11.0 \% & 7.0 \% \\
\text { Company B } & 10.6 \% & 6.2 \% \\
\hline
\end{array}
$$
(Rates have been adjusted for differential tax effects.) Design a swap that will net a bank, acting as intermediary, 10 basis points per annum and that will produce a gain of 15 basis points per annum for each of the two companies.

Lainey Roebuck
Lainey Roebuck
Numerade Educator

Problem 25

Suppose that the term structure of risk-free interest rates is flat in the United States and Australia. The USD interest rate is $7 \%$ per annum and the AUD rate is $9 \%$ per annum. The current value of the AUD is 0.62 USD. Under the terms of a swap agreement, a financial institution pays $8 \%$ per annum in AUD and receives $4 \%$ per annum in USD. The principals in the two currencies are $$\$ 12$$ million USD and 20 million AUD. Payments are exchanged every year, with one exchange having just taken place. The swap will last two more years. What is the value of the swap to the financial institution? Assume all interest rates are continuously compounded.

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Problem 26

The five-year swap rate when cash flows are exchanged semiannually is $4 \%$. A company wants a swap where it receives payments at $4.2 \%$ per annum on a principal of $$\$ 10$$ million. The OIS zero curve is flat at $3.6 \%$. How much should a derivatives dealer charge the company. All rates are expressed with semiannual compounding. (Ignore bid-offer spreads.)

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