$1.$ You deposit $$12,000$ annually into a life insurance fund for the next $30$ year, after which time you plan to retire. $(6$ points in total$)$

a$.$ If the deposits are made at the beginning of the year and earn an interest rate of $6\mathrm{.}5$ percent. What will be the value of the retirement fund at the end of year $30?$

b$.$ Instead of a lump sum, you wish to receive annuities for the next $20$ years $($years $31$ to $50).$ What is the constant annual payment you expect to receive at the beginning of each year if you assume an interest rate of $6\mathrm{.}5$ percent during the distribution period?

c$.$ Repeats $($a$)$ and $($b$)$ assuming an earning rate of $6\mathrm{.}5$ percent during the deposit period and an earning rate of $8\mathrm{.}5$ percent during the distribution period.