$2.(40$ points$)$ The yield curve is flat at $7\%.$ You are forecasting that you will need to

make $25$ annual payments of $$2$M $.$ The next payment is due one year from today.

$($a$)$ Suppose you wish to immunize this liability against interest rate risk using a

single zero$-$coupon bond. What should be the bond$$s face value and maturity

to achieve this?

$($b$)$ Now, suppose you want to use a combination of $5-$year and $20-$year zero$-$coupon

bonds to immunize the liability against interest rate risk. Describe the immu$-$

nization strategy in this case.

$($c$)$ After six months have passed, explain what happened to your portfolio and

what adjustments, if any, would need to be made to the immunization strategy.

$($d$)$ If the yield curve shifts downward by $1$ basis point, how would the value of your

portfolio from part $($c$)$ change? What actions would be necessary to hedge again

against interest rate risk? Provide intuition