Eli5: How do derive compound interest formula
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So, compound interest is a way to make money grow over time.
If A is the final amount, and P is the principal (the initial amount you put in), then we know that
A = P * [something]
The *something* includes the interest rate, or the fraction of the amount that you gain over time.
A = P * [1+interest] * [something]
We also have to consider how many times the interest rate is applied. If you have an interest rate expressed in an annual form, but it’s *compounded monthly*, that means that every month, you gain one-twelfth of the rate.
A = P * [1 + (APR/12)]^12
We include the “to the twelfth power” at the end because we’re multiplying the same expression twelve times to arrive at the final value at the end of one year.
If we were calculating this for two years,
A = P * [1 + (APR/12)]^(12*2)
because we’d be multiplying 24 times.
There you have it:
A = P (1+(r/n))^n*t
where:
* A = final amount
* P = initial principal
* r = annual interest rate
* n = number of times per year interest is applied
* t = number of years under discussion.
PS: There is a concept in finance called “continuous compounding,” or “What would happen if we compounded and recalculated the interest over shorter and shorter timeframes until the timeframe reached 0?”
For this, we use Euler’s number, *e*, and the formula becomes:
A = Pe^rt
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