An annuity pays $$20,000$ per month every month for $20$ years. The payments are made at the end of each month. The first payment is made at the end of the first month. If the interest rate is $12$ percent compounded monthly for the first eight years, and $9$ percent compounded monthly thereafter, what is the present value of the annuity?

Note that you can solve this problem in $2$ steps:

i$)$ Use the present value of annuity formula to find the value of $$20,000$ per month for the period after year $8$ to year $20$ using $0\mathrm{.}0075($or $0\mathrm{.}75\%=\frac{9}{12})$ monthly interest rate after year $8$ and $144$ months $(20-8)12.$ Let's call the answer in this step A$.$ This answer, A$,$ is the value at year $8.$

ii$)$ Since the answer in the first step, A$,$ is in year $8,$ we need to find its value in year $0($now$)$ using the present value of a single cash flow formula. The interest rate for this step is the interest rate per month for the first eight years or $0\mathrm{.}01$