An investment company buys $$100$ million $($par value$)$ of a $12-$year coupon bond that pays a $6\%$ annual coupon at the date of issue. Assume that the term structure of the interest rates is flat at the rate of $7\%.$ The investment company is worried about the losses that its portfolio may suffer from an upward shift in the term structure of interest rates. It considers a duration hedging strategy and decides to enter into a position of k $($million$)12-$year zero coupon bond Pz$(0,$T$)$ to hedge away the interest rate risk. The position k in the zero$-$coupon bond is such that the portfolio value is insensitive to a parallel shift in the yield curve is $($hint: use excel financial functions to calculate prices and durations$).$

a$,$ How would you determine your position k in the zero?

b$,$ How would you hedge your interest rate exposure by using IRS transactions for hedging? The IRS available in the market now has a Dollar Duration of $920$ years for the fixed leg and $25$ for the variable leg per $100$ of par value.

c$,$ What is the convexity of your original bond portfolio?

d$,$ If you had another bond available at the market at the same yield and with an identical duration but with a convexity of $70$ which would you choose? Why?