College tuition is expensive, and Michael is concerned about the discouraging expenses of sending his daughter Erin to college, although Erin is currently just $3$ years old. Michael is convinced that she will go to MIT and study Quantum Physics. He estimates that, in $15$ years, MIT tuition plus cost of living will be $$75,000$ per year for the period of her degree $($four years$).$ Assume that the current term structure is at flat $2\mathrm{.}0\%($annual APR$).$ Michael has $$250,000$ in his savings account and, considering the future college tuition payments as liabilities, he plans to use that amount $($or part of it$)$ to invest in $10-$year and $20-$year maturity zero$-$coupon bonds with face value $$1,000$ in order to protect his net worth from interest rate fluctuations.

$($a$)[4$ marks$]$ What is the present value of Michael's liabilities?

$($b$)[6$ marks$]$ How many units of $10-$year and $20-$year zero$-$coupon bonds with face value $$1000$ does Michael need to invest in to immunize his net worth? Use approach $1$ to find the answer by solving a system of equations.

$($c$)[6$ marks$]$ Redo $($b$)$ with approach $2$ by matching the duration of assets and liabilities. Do you arrive at the same results?

$($d$)[6$ marks$]$ Suppose that Michael has immunized his net worth by appropriately investing in the two bonds above. What would his net worth be if the interest rate went up or down by $75$ basis points $(0\mathrm{.}75\%)$ respectively? Do not use duration or duration$-$convexity approximation.