Suppose, you are the CEO of a company and you just made a big investment in a new production facility. Your Chief Risk Officer provides you with the following information:

Your liabilities consist of four $$10$m payments in years $9,10,11,$ and $12$ from now.

The current interest rate is r $=1\mathrm{.}5\%.$

Your assets are currently worth $$40$m$.$

Your Chief Risk Officer is concerned that interest rates may go up given that the central bank announced an end to its quantitative easing program. You are worried that changes in interest rates will affect your net worth. Answer the following questions:

You want to immunize your liabilities. Using your $$40$ million in assets, you can invest in $4-$year and $12-$year zero coupon bonds that pay $$100$ at maturity. How much do you invest into the $4-$ and $12-$year bond, respectively?

a$)$ x$^12$y $=300,000,$ x$^4$y $=150,284$

b$)$ x$^12$y $=296,855,$ x$^4$y $=161,023$

c$)$ x$^12$y $=264,270,$ x$^4$y $=150,597$

d$)$ x$^12$y $=216,180,$ x$^4$y $=114,340$

e$)$ x$^12$y $=-216,180,$ x$^4$y $=-114,340$

f$)$ x$^12$y $=-264,270,$ x$^4$y $=-150,597$

g$)$ x$^12$y $=-296,855,$ x$^4$y $=-161,023$

h$)$ x$^12$y $=-300,000,$ x$^4$y $=-150,284$