$(12$ points$)$ Apple, Inc have two liabilities of its balance sheet. $1.$ $$100$mm face value coupon bonds that pay $5\%$ coupon and have a maturity of $5$ years $2.$ Zero coupon bonds of $$100$mm face value that have a maturity of $10$ years. The interest rate is $8\%$ at all maturities.

a$.(2$p$)$ What are the total liabilities of Apple, Inc today?

b$.(4$p$)$ What is the duration of Apple, Inc$$s total liabilities? $[$hint: remember you can think of any set of cash flows as a portfolio of zero$-$coupon bonds$]$

c$.(2$p$)$ If Apple, Inc wants to minimize its exposure to interest rate risk by adjusting the composition of its assets, what fractions should it invest in one$-$year zero$-$coupon bonds and $10-$year zero$-$coupon bonds? What would be the dollar amount?

d$.(2$p$)$ If the interest rates go up to $9\%,$ what would be the new values of assets and liabilities? Are the liabilities well immunized?

e$.(2$p$)$ Assuming Apple, Inc$$s liabilities have higher convexity than its assets, will its actual net worth after the interest rate rise be higher or lower than the approximation in $($d$)?$