The 6-month and 1-year zero rates are both 5% per annum. For a bond that has a life of 18 months and pays a coupon of 4% per annum (with semiannual payments and one having just been made), the yield is 5.2% per annum. What is the bond’s price? What is the 18-month zero rate? All rates are quoted with semiannual compounding.
Assuming that SOFR rates are as follows, what is the value of an FRA where the holder will pay SOFR and receive 4.5% (quarterly compounded) for a three-month period starting in one year on a principal of $1,000,000?
| Maturity (months) | Rate (% per annum) |
|---|---|
| 3 | 3.0 |
| 6 | 3.2 |
| 9 | 3.4 |
| 12 | 3.5 |
| 15 | 3.6 |
| 18 | 3.7 |
The term structure of interest rates is upward sloping. Put the following in order of magnitude:
What is the answer when the term structure of interest rates is downward sloping?
What rate of interest with continuous compounding is equivalent to 8% per annum with monthly compounding?
A deposit account pays 4% per annum with continuous compounding, but interest is actually paid quarterly. How much interest will be paid each quarter on a $10,000 deposit?
Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are, respectively, 4%, 4.2%, 4.4%, 4.6%, and 4.8% per annum, with continuous compounding. Estimate the cash price of a bond with a face value of 100 that will mature in 30 months and pay a coupon of 4% per annum semiannually.
A 3-year bond provides a coupon of 8% semiannually and has a cash price of 104. What is the bond’s yield?
A 10-year 8% coupon bond currently sells for $90. A 10-year 4% coupon bond currently sells for $80. What is the 10-year zero rate? (Hint: Consider taking a long position in two of the 4% coupon bonds and a short position in one of the 8% coupon bonds.)
A 5-year bond with a yield of 7% (continuously compounded) pays an 8% coupon at the end of each year.
(a) What is the bond’s price?
(b) What is the bond’s duration?
(c) Use the duration to calculate the effect on the bond’s price of a 0.2% decrease in its yield.
(d) Recalculate the bond’s price on the basis of a 6.8% per annum yield and verify that the result is in agreement with your answer to (c).
The cash prices of 6-month and 1-year Treasury bills are 94.0 and 89.0. A 1.5-year Treasury bond that will pay coupons of $4 every 6 months currently sells for $94.84. A 2-year Treasury bond that will pay coupons of $5 every 6 months currently sells for $97.12. Calculate the 6-month, 1-year, 1.5-year, and 2-year Treasury zero rates.
The table below gives Treasury zero rates and cash flows on a Treasury bond. Zero rates are continuously compounded.
(a) What is the bond’s theoretical price?
(b) What is the bond’s yield assuming it sells for its theoretical price?
| Maturity (years) | Zero rate | Coupon payment | Principal |
|---|---|---|---|
| 0.5 | 2.0% | $20 | |
| 1.0 | 2.3% | $20 | |
| 1.5 | 2.7% | $20 | |
| 2.0 | 3.2% | $20 | $1,000 |
Companies A and B have been offered the following rates per annum on a $20 million five-year loan:
| Fixed rate | Floating rate | |
|---|---|---|
| Company A | 5.0% | SOFR + 0.1% |
| Company B | 6.4% | SOFR + 0.6% |
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 $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). Six-month 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?
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 a six-month floating reference rate plus 150 basis points and swapping the floating reference rate 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?
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 floating. 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.
Companies X and Y have been offered the following rates per annum on a $5 million 10-year investment:
| Fixed rate | Floating rate | |
|---|---|---|
| Company X | 8.0% | LIBOR |
| Company Y | 8.8% | LIBOR |
Company X requires a fixed-rate investment; company Y requires a floating-rate invest-ment. Design a swap that will net a bank, acting as intermediary, 0.2% per annum and will appear equally attractive to X and Y.
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 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 nstitution? 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.
“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.
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.
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?
OIS rates are 3.4% for all maturities. What is the value of an OIS swap with two years to maturity where 3% is received and the floating reference rate is paid. Assume annual compounding, annual payments, and $100 million principal.
A financial institution has entered into a swap where it agreed to make quarterly payments at a rate of 3% per annum and receive the SOFR three-month reference rate on a notional principal of $100 million. The swap now has a remaining life of 7.5 months. Assume the risk-free rates with continuous compounding (calculated from SOFR) for 1.5, 4.5, and 7.5
months are 2.8%, 3.0%, and 3.1%, respectively. Assume also that the continuously compounded risk-free rate observed for the last 1.5 months is 2.7%. Estimate the value of the swap.
(a) Company A has been offered the swap quotes in follows. 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 follows. It can borrow for five years at floating plus 75 basis points. What fixed rate can it swap this rate into? (c) Explain the rollover risks that Company B is taking.
| Maturity (years) | Bid | Ask | Swap rate |
|---|---|---|---|
| 2 | 2.97 | 3.00 | 2.985 |
| 3 | 3.05 | 3.08 | 3.065 |
| 4 | 3.15 | 3.19 | 3.170 |
| 5 | 3.26 | 3.30 | 3.280 |
| 7 | 3.40 | 3.44 | 3.420 |
| 10 | 3.48 | 3.52 | 3.500 |
The one-year LIBOR rates is 3%, and the LIBOR forward rate for the 1- to 2-year period is 3.2%. 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.
A financial institution has entered into a swap where it agreed to receive quarterly payments at a rate of 2% per annum and pay the SOFR three-month reference rate on a notional principal of $100 million. The swap now has a remaining life of 10 months. Assume the risk-free rates with continuous compounding (calculated from SOFR) for 1 month, 4 months, 7 months, and 10 months are 1.4%, 1.6%, 1.7%, and 1.8%, respectively. Assume also that the continuously compounded risk-free rate observed for the last two months is 1.1%. Estimate the value of the swap.
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 notional principal of $10 million. The OIS zero curve is flat at 3.6%. How much should a derivatives dealer charge the company. Assume that all rates are expressed with semiannual compounding and ignore bid–ask spreads.