Consider an insurance company with liabilities of $20$m in years $14,15,17,$ and$18$

years. Your assets consist of $35$m in cash. You are uncertain about your net worth

and would like to immunize your net worth by investing in two zero$-$coupon bonds

with maturities $10$ and $20$ years. The price of the $10-$year bond is $55\mathrm{.}84$ and the price

of $20-$year bond is $31\mathrm{.}18.$ Assume that the current term structure is flat at $6\%.$

a$)$ What is the present value of your liabilities?

b$)$ What are the duration and the modified duration of your liabilities?