Suppose that you are managing an endowment fund with obligations to make annual payments of $$2$ million per year for $20$ years $($t$=1$ to t$=20).$ The yield to maturity on ALL bonds is $8\%.$ Assume an annual compounding frequency. Par value for all coupon bonds is $$1,000.$

Note: this endowment fund is not a perpetuity. Its payments last only $20$ years, so don't use formulas for PV and Duration based on perpetuities. If you use the perpetuity formula for the duration of the pension fund, you will not be able to work this problem That is don't use D $=$ y$/1+$y or PV $=$ Cash Flow$/$y

a$.$ You plan to use both $5-$year maturity bonds with coupon rates of $12\%($paid annually$)$ and $20-$year maturity bonds with coupon rates of $6\%($paid annually$)$ to immunize this obligation. How much of each of these coupon bonds $($in market value$)$ will you want to hold to both fully fund and immunize the endowment's obligation?

b$.$ What will be the par value of your holdings in the $20-$year and $5-$year coupon bonds? $($Par value can also be interpreted as face value$)$

c$.$ At t$=1$ show in detail how you can make the required $$2$ milllion payment. You must label everything clearly in order to receive full credit for this part. Use the bonds from part $($a$).$ Assume that the yield to maturity for ALL bonds is now $7\mathrm{.}5\%.$ Some of the trades from part $($d$)$ might generate cash for this part.

d$.$ Explain in detail what you should do at t$=1$ to maintain an immunized position. You must label everything clearly in order to receive full credit for this part.