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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 !

RobPratt
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Spyros
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    The calculator seems to assume that the monthly contributions earn no interest for their first end of year, and only earn interest after later full years. The result is the same as annual contributions of $$2400$. – peterwhy Jun 18 '22 at 19:07
  • that makes sense yes. I am trying to picture a scenario where everything gets compounded once a year, for simplicity purposes – Spyros Jun 18 '22 at 19:32
  • Well monthly compounding does not complicate the calculation much, if you know to calculate per-period interest rate and the total number of periods. But contributing monthly while compounding annually means those contributions are made earlier (vs. EOY contributions) without benefit. – peterwhy Jun 18 '22 at 19:42

1 Answers1

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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 $PV = \$1000$, $PMT = \$2400$, $i = 15\%$, $n=10$.

peterwhy
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  • Thank you, but is there a formula that I can use for this ? I am trying to use it programmatically and it would be great if there is a direct one. – Spyros Jun 18 '22 at 19:25
  • I guess it seems as if it's the n=10 one plus 200*12 to the (1+15%)^year for the rest. I think I can adapt that, but if there was a simpler formula, it would be great. – Spyros Jun 18 '22 at 19:27
  • Thanks a lot !! – Spyros Jun 18 '22 at 19:32