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3. The annual yields of zero-coupon bonds are as given below. What is the swap rate for a four-yearinterest rate swap with level notional amount and annual settlement?
Length of term in years 1 2 3 4 5
Annual yield 1.5% 2.5% 3.3% 4.2% 4.9%
t (years) 1 2 3
rt 1.28% 2.03% 2.84%
Spot interest rates at the beginning of the second year were:
t (years) 1 2 3
rt 1.54% 2.61% 3.25%
5. Compute the Macaulay duration of a ten-year 6% $1,000 bond having annual coupons and a redemption of $ 1,200 if the yield to maturity is 8%.
6. Calculate the Macaulay duration D(.05,∞) and the modified duration D(.05,2) of a preferred stock that pays dividends forever of $ 50 each six months, with the next dividend in exactly six months.
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7. A bond has Macaulay duration D(i,∞) = 5.8 and Macaulay convexity C(i,∞) = 1.2. Determine C(i,4) as a function of i.
8. Providence Health Care is obligated to make a payment of $300,000 in exactly three years. In order to provide for this obligation, their financial officer decides to purchase a combination of oneyear zero-coupon bonds and four-year zero-coupon bonds. Each of these is sold to yield an annual effective yield of 4%. How much of each type of bond should be purchased so that the present value and duration conditions of Redington immunization are satisfied? Is the convexity condition also satisfied at i = 4%?
9. The price of Ada’s bond is $1416.89 when calculated using an annual yield of 4.8%. Using a first-order modified approximation, Ada calculated that the change in price of her bond would be $32.01 if the annual yield increased from 4.80% to 4.95%. Calculate the first-order Macaulay approximation of the price of this bond for an interest rate of 4.95%.
10. A bond without a call provision has Macaulay duration of 4.5381, Macaulay convexity of X , and price of $35,328.70 when calculated using an annual interest rate of 5.7%. Using a second-order Macaulay approximation, Leta estimated that the change in price of this bond would be $305.03 if the interest rate were to decrease to 5.5%. Find the value of X .
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