Assume the term structure of interest rates is flat and consider a 1-factor model with a factor equal to that interest rate. Assume also the current interest rate is 8%. Your portfolio consists of $100,000 investment in a 10-year zero-coupon bonds and $200,000 investment in perpetuities that pay semi-annual coupons.

a) (2 points) Find the Duration and convexity of 10-year zero-coupon bonds.
b) (2 point) Find the Duration convexity of a perpetuity
c) (1 point) Find Duration of your portfolio
d) (1 point) Find Convexity of your portfolio

e) (2 points) Assume you want to use the securities described in this question to hedge $1,000,000 worth of securities with Duration=25 and Convexity=100. To do so, you have decided spend $Vbond to buy bonds and $Vperp to buy perpetuities (note that negative values means you are selling them). Write down the system of equatuions for $Vbond and $Vperp and find $Vbond and $Vperp.

NOTE: All interest rates are annual interest rates with semi-annual compounding. All coupon rates are annual rates paid semi-annually. All bonds have $100 face values. All Durations are “modified Duration” (not “Macauley Duration).