You will be paying $$20,000$ a year in tuition expenses at the end of the next four years. Bonds currently yield $5\%.$

$($a$)$ What is the modified duration of your obligation?

$($b$)$ To immunize this obligation, you can choose from two bonds: a zero$-$coupon bond with maturity $10$ years and a perpetuity, each currently yielding $5\%.$ How much of the zero$-$coupon bond and the perpetuity will you hold in your portfolio?

$($c$)$ Suppose that $1$ year passed, and you pay the first year$$s tuition. What is the duration of your obligation after one year? How the immunized bond portfolio should be rebalanced if the yield of two bond changes to $8\%?$