If you've heard that compound interest is "the eighth wonder of the world," or wondered how an amount invested every month can turn into a much larger sum than what you actually deposited, this article shows exactly how that calculation works: the basic formula, the version with monthly contributions, and an example comparing the result with simple interest.
What is compound interest
With compound interest, the return from each period becomes part of the capital that earns a return in the next period. You don't just earn interest on the amount you invested at the start, you also earn interest on the interest you've already received. That's why growth accelerates over time: the more periods pass, the larger the base on which new interest is calculated.
This is how most fixed-income investments in Brazil work, such as CDB, Tesouro Direto (Brazilian treasury bonds) and savings accounts, and it's also how unpaid debt grows, such as Brazilian credit card revolving debt. The period used in the calculation can be monthly, daily or annual depending on the product: a savings account compounds monthly, for example, while some bonds compound daily. The math is the same either way, only the value of n and the rate i used per period changes.
In practice, this also means time matters more than the initial amount invested. Starting early, even with a small amount, tends to produce a larger final result than starting later with a much larger amount, because compound interest has more periods to act on the accumulated capital.
The compound interest formula
For an amount invested once, with no additional contributions, the formula is:
M = P × (1 + i)ⁿ
Where M is the final amount, P is the initial capital, i is the interest rate per period (as a decimal) and n is the number of periods.
For example, R$ 1,000 invested at a rate of 1% per month, for 12 months, results in an amount of approximately R$ 1,126.83, of which R$ 126.83 is interest accumulated on the capital and on the interest already earned.
Compound interest with a monthly contribution
Most searches for this formula come from a more practical question: what if, besides the initial amount, I also deposit money every month? The full formula, with a constant monthly contribution, looks like this:
M = P × (1 + i)ⁿ + PMT × (((1 + i)ⁿ − 1) / i)
PMT is the monthly contribution. Assuming, just as an example, an initial capital of R$ 1,000, monthly contributions of R$ 300 and a rate of 0.8% per month, over 24 months, the final amount comes to around R$ 9,112. Of that total, R$ 8,200 came out of your pocket (initial capital plus the 24 contributions), and about R$ 912 is interest.
Compound interest vs. simple interest: the effect over time
The difference between compound and simple interest is small over short periods, but it grows substantially over time. Simple interest uses the formula M = P × (1 + i × n), without compounding interest on interest already earned.
Comparing R$ 1,000 at 1% per month over 5 years (60 months): with simple interest, the amount reaches R$ 1,600. With compound interest, it reaches approximately R$ 1,816.70, a difference of over R$ 200 generated purely by interest earning interest on interest. The longer the term and the higher the rate, the larger that gap gets, because simple interest grows linearly (the same interest amount each period), while compound interest grows exponentially (the interest amount increases each period, because the base it's calculated on also increases).
To simulate your own case, with the initial capital, monthly contribution and term you choose, use the compound interest calculator. It shows the final amount, the total invested and the interest generated, with a month-by-month growth chart.
If you want to compare the same amount without the compounding effect, the simple interest calculator runs that simulation.
This content is educational and does not replace personalized financial advice.