Eli5: How do derive compound interest formula

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Eli5: How do derive compound interest formula

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Consider for a moment that your grandmother is a firm believer in saving money and refuses to give gift cards or cash at holidays. Instead, she gives all of the grandchildren \$500 invested in five-year CDs (Certificates of Deposit) with compound interest figured twice a year and a 2% interest rate. Set aside the delayed gratification for a moment — The question is, how much money will the CD be worth in 5 years? That is where compound interest kicks in.

Filling in the compound interest formula of A=P(1+r/n)^nt, your holiday gift looks like this: A = 500(1+.022)^25. Following the order of operations, we calculate the amount in the parenthesis first to get A = 500(1.01)^25. From there, we input the exponents to get A = 500(1.10). Now in our final calculation, the final value is A = \$550. At the end of 5 years, your CD will be worth \$550.

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