The following are today\'s yields on default free zero-coupon bonds: 1 year--2%;

Question

The following are today's yields on default free zero-coupon bonds: 1 year--2%; 2 year--2.5%; 3 year--3.5%; 4 year--5%.

A. What are the one-year forward rates for 1 year, 2 years and 3 years out?

B. What is the expected yield curve for one, two and three year zero coupon bonds one year from now?

A bond currently sells at a price of 104 and has a yield to maturity of 7%. Suppose the yield to maturity falls by 25 basis points and the price increases to 105. What is the modified duration of the bond?

A ten-year bond has a yield of 5% and a duration of 7.1 years. If the bond's yield changes by 50 basis points, what is the percentage change in the bond's price?

Explanation / Answer

1a) Using expectation theory,

(1 + S2)^2 = (1 + S1) x (1 + 1F1)

1-year forward rate 1-year from now, 1F1 = (1 + 2.5%)^2 / (1 + 2%) - 1 = 3.00%

Similarly, 2F1 = (1 + 3.5%)^3 / (1 + 2.5%)^2 - 1 = 5.53%

3F1 = (1 + 5%)^4 / (1 + 3.5%)^3 - 1 = 9.63%

1b) Expected yield for one year zero coupon bond = 1F1 = 3.00%

Expected yield for two year zero coupon bond = ((1 + 1F1) x (1 + 2F1))^(1/2) - 1 = 4.26%

Expected yield for three year zero coupon bond = ((1 + 1F1) x (1 + 2F1) x (1 + 3F1))^(1/3) - 1 = 6.02%

2) % Change in bond price = % Change in yield x Duration

=> Duration = (105/104 - 1) / 0.25% = 3.85

3) % Change in bond price = Duration x % change in yield = 7.1 x 0.5% = 3.55%

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