An Introduction to Derivatives and Risk Management: With Stock-Trak Coupon

Don M. Chance, Robert Brooks

Chapter 13 Interest Rate Forwards and Options - all with Video Answers

Educators

Chapter Questions

00:23

Problem 1

How are the payment terms of an FRA different from those of most other interest rate derivatives?

Amy Jiang

Numerade Educator

01:05

Problem 2

Explain how FRAs are like swaps and how they are different.

Nick Johnson

Numerade Educator

02:48

Problem 3

Compare the use of interest rate options with forward rate agreements. Explain why a financial manager might prefer one type of contract over another.

Jennifer Stoner

Numerade Educator

Problem 4

Show how a combination of interest rate caps and floors can be equivalent to an interest rate swap.

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01:07

Problem 5

What are the advantages and disadvantages of an interest rate collar over an interest rate cap?

Nick Johnson

Numerade Educator

Problem 6

Explain how the Black model, which is designed for pricing options on futures contracts, can be used for pricing interest rate options.

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Problem 7

Explain how a swaption can be terminated at expiration by either exercising it or settling it in cash. Why are these procedures financially equivalent?

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00:40

Problem 8

Explain how the two types of swaptions are like interest rate options and how they are different.

Rae Xin

Numerade Educator

Problem 9

Explain how a bank could use a swaption to hedge the possibility that it will enter into a pay-floating, receive-fixed swap at a later date.

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Problem 10

Explain how a forward swap is like a swaption and how it is different.

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03:26

Suppose a firm plans to borrow $$\$ 5$$ million in 180 days. The loan will be taken out at whatever LIBOR is on the day the loan begins and will be repaid in one lump sum, 90 days later. The firm would like to lock in the rate it pays so it enters into a forward rate agreement with its bank. The bank agrees to lock in a rate of 12 percent. Determine the annualized cost of the loan for each of the following outcomes. Interest is based on 90 days and a 360 -day year.
a. LIBOR in 180 days is 14 percent.
b. LIBOR in 180 days is 8 percent.

James Kiss

Numerade Educator

03:26

Problem 12

The following term structure of LIBOR is given
$$
\begin{array}{ll}
\hline \text { Term } & \text { Rate } \\
\hline 90 \text { diys } & 6.00 \% \\
180 \text { days } & 6.90 \% \\
270 \text { days } & 6.30 \% \\
360 \text { days } & 6.35 \% \\
\hline
\end{array}
$$
$$
\text { a. Find the rate on a new } 6 \times 9 \text { FRA. }
$$
b. Consider an FRA that was established previously at a rate of 5.2 percent with a notional principal of $$\$ 30$$ million. The FRA expires in 180 days, and the underlying is 180-day LIBOR. Find the value of the FRA from the perspective of the party paying fixed and receiving floating as of the point in time at which the above term structure applies.

James Kiss

Numerade Educator

Problem 13

You are the treasurer of a firm that will need to borrow $$\$ 10$$ million at LIBOR plus 2.5 points in 45 days. The loan will have a maturity of 180 days, at which time all the interest and principal will be repaid. The interest will be determined by LIBOR on the day the loan is taken out. To hedge the uncertainty of this future rate, you purchase a call on LIBOR with a strike of 9 percent for a premium of $$\$ 32,000$$. Determine the amount you will pay back and the annualized cost of borrowing for LIBORs of 6 percent and 12 percent. Assume the payoff is based on 180 days and a 360 -day year. The current LIBOR is 9 percent.

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04:50

Problem 14

A large, multinational bank has committed to lend a firm $$\$ 25$$ million in 30 days at LIBOR plus $100 \mathrm{bps}$. The loan will have a maturity of 90 days, at which time the principal and all interest will be repaid. The bank is concerned about falling interest rates and decides to buy a put on LIBOR with a strike of 9.5 percent and a premium of $$\$ 60,000$$. Determine the annualized loan rate for LIBORs of 6.5 percent and 12.5 percent. Assume the payoff is based on 90 days and a 360-day year. The current LIBOR is 9.5 percent.

James Kiss

Numerade Educator

Problem 15

As the assistant treasurer of a large corporation, your job is to look for ways your company can lock in its cost of borrowing in the financial markets. The date is June 28. Your firm is taking out a loan of $$\$ 20$$ million, with interest to be paid on September 28, December 31 , March 31, and June 29. You will pay the LIBOR in effect at the beginning of the interest payment period. The current LIBOR is 10 percent. You recommend that the firm buy an interest rate cap with a strike of 10 percent and a premium of $$\$ 70,000$$. Determine the cash flows over the life of this loan if LIBOR turns out to be 11 percent on September $28,11.65$ percent on December 81 , and 12.04 percent on March 31. The payoff is based on the exact number of days and a 360-day year. If you have a financial calculator or a spreadsheet with an IRR function, solve for the internal rate of return and annualize it to determine the effective cost of borrowing.

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04:51

Problem 16

You are a funds manager for a large bank. On April 15, your bank lends a corporation $$\$ 35$$ million, with interest payments to be made on July 16 , October 15 , January 16 , and next April 16 . The amount of interest will be determined by LIBOR at the beginning of the interest payment period. On April 15, LIBOR is 8.0 percent. Your forecast is for declining interest rates, so you anticipate lower loan interest revenues. You decide to buy an interest rate floor with a strike set at 8 percent and a premium of $$\$ 60,000$$. Determine the cash flows associated with the loan if LIBOR turns out to be 7.9 percent on July $16,7.7$ percent on October 15, and 8.1 percent next January 16. The payoff is based on the exact number of days and a 360-day year. If you have a financial calculator or spreadsheet with an IRR function, determine the internal rate of return and annualize it to determine your annualized return on the loan.

Narayan Hari

Numerade Educator

Problem 17

On January 15, a firm takes out a loan of $$\$ 30$$ million, with interest payments to be made on April 16, July 15, October 14, and the following January 15, when the principal will be repaid. Interest will be paid at LIBOR based on the rate at the beginning of the interest payment period, using the exact number of days and a 360 -day year. The firm wants to buy a cap with an exercise rate of 10 percent and a premium of $$\$ 125,000$$ but is concerned about the cost. Its bank suggests that the frrm sell a floor with an exercise rate of 9 percent for the same premium. The current LIBOR is I0 percent. Determine the firm's cash flows on the loan if LIBOR turns out to be 11.35 percent on April 16, 10.2 percent on July 15 , and 8.86 percent on October 14. If you have a financial calculator or spreadsheet, determine the internal rate of return and annualize it to determine the cost of borrowing.

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04:50

Problem 18

A bank is offering an interest rate call with an expiration of 45 days. The call pays off based on 180-day LIBOR. The volatility of forward rates is 17 percent. The 45-day forward rate for 180 -day LIBOR is 0.1322 and the exercise rate is 12 percent. The risk-free rate for 45 days is 11.28 percent. All rates are continuously compounded. Use the Black model to determine how much the bank should receive for selling this call for every $$\$1$$ million of notional principal.

James Kiss

Numerade Educator

03:26

Problem 19

A firm is interested in purchasing an interest rate cap from a bank. It has received an offer price from the bank but would like to determine if the price is fair. The cap will consist of two caplets, one expiring in 91 days and the other in 182 days. They will both have strikes of 7 percent. The forward rate applicable to the first caplet is 8 percent and the forward rate applicable to the second caplet is 8.2 percent. The 91-day risk-free rate is 7.1 percent and the 182-day risk-free rate is 7.3 percent. All rates are continuously compounded. The firm's best estimate of the volatility of forward rates is 16.6 percent. The notional principal is $$\$ 10$$ million, and the payoff is based on 90-day LIBOR. Use the Black model to determine a fair price for the cap.

James Kiss

Numerade Educator

02:34

Problem 20

Consider a three-year receiver swaption with an exercise rate of 11.75 percent, in which the underlying swap is a $$\$ 20$$ million notional principal fouryear swap. The underlying rate is LIBOR. At the expiration of the swaption, the LIBOR rates are 10 percent ( 360 days), 10.5 percent ( 720 days), 10.9 percent ( 1,080 days), and 11.2 percent ( 1,440 days). Assume 360 days in a year. Determine the payoff value of the swaption.

Breanna Ollech

Numerade Educator

02:34

Problem 21

A company wants to enter into a commitment to initiate a swap in 90 days. The swap would consist of four payments 90 days apart with the underlying being LIBOR. Use the term structure of LIBOR as given below to solve for the rate on this forward swap.
$$
\begin{array}{lc}
\hline \text { Term } & \text { Rate } \\
\hline 90 \text { days } & 10.2 \% \\
180 \text { days } & 11.0 \% \\
270 \text { days } & 11.6 \% \\
350 \text { days } & 11.9 \% \\
450 \text { days } & 12.2 \% \\
\hline
\end{array}
$$

Breanna Ollech

Numerade Educator

Problem 22

Suppose your firm had issued a 12 percent annual coupon, 15 -year bond, callable at par at the 8 th year. It is now two years later, so the bonds are not callable for another 6 years. At this time, new bonds could be issued at 8 percent, which is historically quite low, especially relative to the 12 percent coupon on the bond you issued two years ago. To provide a better matching of the interest-sensitivities of your assets and liabilities, you want to lengthen the duration of the bonds. How could you use swaptions to restructure the debt? Explain what happens assuming two subsequent future possibilities: rates going up and rates going down.

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Problem 23

A firm has previously issued fixed rate non-callable debt. Because interest rates are perceived to be temporarily high, the firm would like to have the flexibility of calling the debt later when rates are expected to fall and replacing it with floating-rate debt. Explain how a frm can use swaptions to achieve this desired result. Also, identify and compare an alternative method that can be used to convert fixed-rate debt to floating-rate debt.

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03:26

Problem 24

(Concept Problem) Use the Black model to determine a fair price for an interest rate put that expires in 74 days. The forward rate is 9.79 percent, and the exercise rate is 10 percent. The appropriate risk-free rate is 8.38 percent. All rates are continuously compounded. The volatility of forward rates is 14.65 percent. The put is based on $$\$ 22$$ million notional principal and pays off based on 90-day LIBOR.

James Kiss

Numerade Educator

Problem 25

(Concept Problem) Consider a call option with an exercise rate of $\mathrm{x}$ on an interest rate, which we shall denote as simply $L$. The underlying rate is an $m$-day rate and pays off based on 360 days in a year. Now consider a put option on a $$\$ 1$$ face value zero coupon bond that pays interest in the add-on manner (as in Eurodollars) based on the rate $L$. The exercise rate is X. Show that the interest rate call option with a notional principal of $$\$ 1$$ provides the same payoffs as the interest rate put option if the notional principal on the put is $$\$ 1(1+x(m / 360))$$ and its exercise price, $X$, is $$\$ 1 /(1+x(m / 360))$$. If these two options have the same payoffs, what does that tell us about how to price the options?

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