I am trying to discover the formula behind the calculations in this website(among others similar ones)
https://moneysmart.gov.au/budgeting/compound-interest-calculator
The formula for compound interest without any contributions is pretty simple, but I am having trouble getting the right results for the scenario in which there are monthly contributions.
For example, what is the formula to use in order to calculate this scenario:
Initial contribution = $1,000
Annual interest rate compound once a year = 15%
Amount of years to compound for = 10
Extra monthly contribution = $200 (but not compounded monthly, just the annual 15% compound)
This should come back with a total of about 52k+ at the end of year 10.
Which formula do I use to calculate the balance for every of the 10 years?
Thanks !
Best Answer
The initial one-off payment is compounded for all $10$ periods, so at the end this becomes
$$\$1000 \times(1+15\%)^{10} = \$4045.56$$
The $10$ annual contributions of $\$200\times 12$ each are compounded for $9$ to $0$ years respectively, so at the end
$$\begin{align*} \$2400\times1.15^9 + \$2400\times 1.15^8 + \cdots + \$2400\times 1.15^0 &= \$2400\times\frac{1.15^{10}-1}{1.15-1}\\ &= \$2400\times \frac{1.15^{10}-1}{15\%}\\ &= \$48728.92 \end{align*}$$
Total that would be $\$52774.48$ at the end.
Using common financial calculator variable names,
$$FV = PV(1+i)^n + \frac{PMT}{i}[(1+i)^n-1]$$
where $PV = \$1000$, $PMT = \$2400$, $i = 15\%$, $n=10$.