Homework $2.$ Present Value of Ordinary Annuity

Assume you and your employer each contribute HKD $500$ to your Mandatory Provident Fund

$($MPF$)$ at the end of every month $($HKD $1,$ooo in total$).$ Suppose the beginning account balance

is zero, and the return is compounded monthly. Write a Python program to calculate the

present value of the MPF$.$

Let the user set a target monthly rate of return and the number of months.

Round the answer to the nearest dollar.

The output should resemble the following.

Enter a target MONTHLY rate of return $($without the $\%$ sign$)$: $0\mathrm{.}05$

Enter the number of months: $5$

The total MPF of $$5000$ in $5$ months is worth $$4329$ today.

Follow the same submission requirement $(.$py file and screenshot$)$ as before.

Hint: You may apply the formula below or use loops.

An ordinary annuity is a series of future cash flows paid out evenly at the end of every period

$($e$.$g$.,$ monthly, annually$).$ Let the month$-$end contribution amount be $c,$ the monthly rate be $r.$

Assume the return is compounded monthly. Using the summation of geometric progressions ${?}^1,$

the total value in $n$ months is

$PV=\frac{c}{(1+r)}+\frac{c}{(1+r{)}^2}+$cdots$+\frac{c}{(1+r{)}^n}=c*\frac{(1+r{)}^{-1}-(1+r{)}^{-n-1}}{1-(1+r{)}^{-1}}=c*\frac{1-(1+r{)}^{-n}}{r}$

Illustration: Present value of an ordinary annuity$^2$

${?}^1$ Recall that $q+q^2+$cdots$+q^n=\frac{q-q^{n+1}}{1-q}.$ Then let $q=(1+r{)}^{-1}.$

${?}^{2}https\frac{\text{:}}{\frac{?}{?}}www$

investopedia.com$/$retirement$/$calculating$-$present$-$and$-$future$-$value$-$of$-$annuities$/$