The 0.5-year zero coupon rate is 2% and the 1-year zero coupon rate is 3%.
(a) What is the modified duration of (i) a 0.5-year zero, and (ii) a 1-year zero?
(b) Consider a forward contract to buy at time t=0.5 a zero-coupon maturing at time 1, and with face value $1.
(i) What is the forward price to pay at time t=0.5 to make this contract worth exactly zero?
(ii) This forward contract can be described as a portfolio of zero coupon bonds, with face value $1, maturing at time t=0.5 and 1. How many of the zero coupon bonds maturing at time t=0.5 do we have in the portfolio? How many of the zero coupon bonds maturing at time t=1 do we have in the portfolio?
(iii) What is the dollar duration of the forward contract? (Hint: think of the forward contract as equivalent to the bond portfolio determined above. Compute the dollar duration of the portfolio as) DDP = $position in 0.5-year zero × MD0.5 + $position in 1-year zero × MD1
(c) Suppose your assets have value $100,000 and duration 5 while your liabilities have value $100,000 and duration 2.
(i) What is the dollar duration of your net position?
(ii) How many forward contracts do you need to buy or sell to immunize your net position against parallel shifts in interest rates, based on the dollar durations computed in (b.iii) and (c.i)? (Ignore convexity in this question.)