For the 1 year bond we must have where is the 1 year zero rate. It follows that or 1 1. 5%. For the 2-year bond we must have where is the 2-year zero rate. It follows that or 11. 3%. Problem 4. 9. What rate Of interest With continuous compounding is equivalent to per annum with monthly compounding? The rate of interest is where: The rate of interest is therefore 14. 91% per annum. Problem 4,10_ A deposit account pays 12% per annum with continuous compounding, but interest is actually paid quarterly. How much interest will be paid each quarter on a $10,000 deposit?
The equivalent rate of interest with quarterly compounding is where The amount of interest paid each quarter is therefore: or $304. 55. Problem 4. 11. Suppose that 6-month, 12-month, 18-month, 24-month, and 30-month zero rates are 4%, 4. 2%, 44%, 4. 6%, and 4. 8% per annum with continuous compounding respectively, Estimate the cash price off bond with a face value of 100 that will mature in 30 months and pays a coupon of 4% per annum semiannually. The bond pays $2 in 6, 12, 18, and 24 months, and $102 in 30 months. The cash price Problem 4. 12. A three-year bond provides a coupon Of 8% semiannually and has a cash price Of 04.
What is the bond’s yield? The bond pays $4 in 6, 12, 18, 24, and 30 months, and $104 in 36 months. The bond yield is the value Of that solves using the Goal seek tool in Excel or 6. 407%. Problem 4. 13. Suppose that the 6-month, 12-month, 18-month, and 24-month zero rates are 5%, 6%, 6. 5%, and 7% respectively. What is the two-year par yield? Using the notation in the text, , .
Also The formula in the text gives the par yield as To that this is correct we calculate the value of a bond that pays a coupon of 7. 072% per year (that is 3. 5365 every six months), The value is refrying that 7. 72% is the par yield. Problem 414 _ Suppose that zero interest rates with continuous compounding are as follows: Maturity(years) Rate (% per annum) Calculate forward interest rates tort the second, third, fourth, and fifth years.
The forward rates with continuous compounding are as follows: to Year 2: 4. 0% Year 3: 5. 1% Year 4: 5. 7% Years: 5. 7% Problem 4. 15. Use the rates in Problem 4. 14 to value an FAR ever you will pay 5% for the third year on $1 million. The forward rate is 5. 1% With continuous compounding or With annual compounding. The 3-year interest rate is 3. With continuous compounding. From equation (4. 10), the value Of the FAR is therefore or $1,964. 67.
Problem 4. 16. A 10-year, 8% coupon bond currently sells for SIS. A 10-year, 4% coupon bond currently sells for $80. What is the Ill-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. ) Taking a long position in two of the 4% coupon bonds and a short position in one of the coupon bonds leads to the following cash flows because the coupons cancel out. $100 in 10 years time is equivalent to $70 today,
The 10-year rate, , (continuously compounded) is therefore given by The rate is or per annum_ Problem 4. 17 _ Explain carefully why liquidity preference theory is consistent with the observation that the term structure Of interest rates tends to be upward sloping more often than it is downward sloping. If long-term rates were simply a reflection of expected future short-term rates, we would expect the term Structure to be downward sloping as often as it is upward sloping. (This is based on the assumption that half of the time investors expect rates to increase and half of the time investors expect rates to decrease).
Liquidity preference theory argues that long term rates are high relative to expected future shorter rates. This means that the term structure should be upward sloping more often than it is downward sloping. Problem 4, 18. M/hen the zero curve is upward sloping, the zero rate tort a particular maturity is greater than the par yield for that maturity. When the zero curve is downward sloping the reverse is true. ” Explain why this is so. The par yield is the Yield on a coupon-bearing bond. The zero rate is the yield on a zero-coupon bond.
When the yield curve is upward sloping, the yield on n -year coupon-bearing bond is less than the yield on an -year zero-coupon bond. This is because the coupons are discounted at a lower rate than the -year rate and drag the yield down below this rate. Similarly, when the yield curve is downward sloping, the yield on an -year coupon bearing bond is higher than the yield on an ;year zero-coupon bond. Problem 4. 19. Why are US. Treasury rates significantly lower than Other rates that are close to risk free? There are three reasons (see Business Snapshot 4. 1). 1.
Treasury bills and Treasury bonds must be purchased by financial institutions o fulfill a variety of regulatory requirements. This increases demand for these Treasury instruments driving the price up and the yield down. 2. The amount of capital a bank is required to hold to support an investment in Treasury bills and bonds is substantially smaller than the capital required to support a similar investment in other very-low-risk instruments. 3. In the United States, Treasury instruments are given a debatable tax treatment compared with most other fixed- income investments because they are not taxed at the state level.
Problem 4,20_ Why does a loan in the reap market involve very little credit risk? A reap is a contract where an investment dealer who owns securities agrees to sell them to another company now and buy them back later at a slightly higher price. The other company is providing a loan to the investment dealer. This loan involves very little credit risk. If the borrower does not honor the agreement, the lending company simply keeps the securities. If the lending Company does not keep to its side of the agreement, the original owner of the securities keeps the cash.
Problem 4. 21. Explain why an FAR is equivalent to the exchange of a floating rate of interest for fixed rate of interest? A FAR is an agreement that a certain specified interest rate, , will apply to a certain principal, , for a certain specified future time period. Suppose that the rate observed in the market for the future time period at the beginning of the time period proves to be. Fifth FAR is an agreement that will apply when the principal is invested, the holder tooth FAR can borrow the principal at and then invest it at.
The net cash flow at the end of the period is then an inflow of and an outflow of If the FAR is an agreement that will apply when the principal is arrowed, the holder of the FAR can invest the borrowed principal at The net cash flow at the end of the period is then an inflow and an outflow of. In either case we see that the VERA involves the exchange of a fixed rate of interest on the principal of for a floating rate of interest on the principal. Problem 4. 22. “An interest rate swap where six-month LABOR is exchanged for a fixed rate on a principal of $100 million is a portfolio of FRATS. ” Explain.