What compound interest is
Compound interest is interest that earns interest. Each period the return is added to the amount invested and starts earning alongside it in the next period. That is why the balance curve does not rise in a straight line: it gets steeper as time goes by.
At first the difference looks small. In the early months almost all the growth comes from your own contributions. The turning point arrives halfway through, when a single month of interest becomes larger than that month’s contribution — from there on your money works harder than you do.
The same mechanism works against you when it is debt that is compounding. Credit card revolving balances and overdrafts charge compound interest, which is why a small balance snowballs in a few months.
The compound interest formula
For a single lump sum, with no contributions along the way, the formula is this:
M = C × (1 + i)^t
- M — the final amount at the end of the period
- C — the capital you invested at the start
- i — the interest rate for the period, as a decimal (0.8% per month = 0.008)
- t — time, in the same unit as the rate (a monthly rate needs a term in months)
The detail that causes the most mistakes is the unit: rate and term must use the same period. Converting an annual rate into a monthly one is not dividing by 12 — it is taking the twelfth root, because interest also compounds within the year. 12% per year equals 0.9489% per month, not 1% per month. This calculator uses the correct conversion: monthly rate = (1 + annual rate)^(1/12) − 1.
M = C × (1 + i)^t + A × [ ((1 + i)^t − 1) ÷ i ]
When there are monthly contributions, the second term adds the future value of every deposit, where A is the contribution. Each deposit earns for less time than the one before it: the first compounds for almost the whole term, the last for almost nothing. That is why starting early matters more than contributing a lot.
Simple and compound interest: the difference
With simple interest the return is always calculated on the amount invested at the start. Investing $10,000 at 1% per month earns $100 every month, forever. Over ten years that is $12,000 in interest and a balance of $22,000.
With compound interest the return joins the base for the next period. The same $10,000 at 1% per month grows to roughly $33,000 in ten years — almost twice the interest of the simple regime. In the first month both regimes give exactly the same result; the gap opens from the second month on and widens fast.
In practice almost every investment and every loan uses compound interest. Simple interest shows up only in specific cases, such as some penalties and very short-term instalments.
Practical examples
Three scenarios to give a sense of scale. Use the calculator above to swap in your own numbers.
$300 a month for 10 years
With $1,000 up front, $300 a month and 0.8% per month, you deposit $37,000 and end with about $62,667. Roughly 41% of the result came from interest, not from your pocket.
The cost of starting five years later
$500 a month at 0.8% per month for 30 years reaches about $1,038,000. Starting five years later, with 25 years of contributions, the result drops to about $620,000. A five-year delay costs more than $418,000 — far more than the $30,000 that were never deposited.
When compound interest is against you
A $2,000 credit card revolving balance at 13% per month, with no payment at all, becomes about $8,669 in twelve months. It is the same mechanism as investing, only pointed the other way.
Common mistakes when simulating
Dividing the annual rate by 12
Dividing 12% per year by 12 gives 1% per month, which overstates the return. The correct conversion is compound and gives 0.9489% per month. Over long terms that gap changes the result by tens of percent.
Forgetting inflation
A million in thirty years does not buy what a million buys today. Turn on the inflation adjustment to see the result in current purchasing power — that is the number that tells you whether the investment actually beat inflation.
Ignoring taxes and fees
On taxable fixed income, income tax applies to the interest, from 22.5% down to 15% depending on how long you stay invested. Add custody and management fees where they exist: they come out of the gross return before it becomes your balance.
Frequently asked questions
- What is the difference between simple and compound interest?
- With simple interest the return is always calculated on the amount invested at the start. With compound interest, each period’s return is added to the amount invested and starts earning too. In the first period both give the same result; from the second period on compound interest pulls ahead, and the gap widens faster and faster.
- How do you calculate compound interest with monthly contributions?
- The formula gains a second term: M = C × (1 + i)^t + A × [ ((1 + i)^t − 1) ÷ i ], where A is the monthly contribution. In practice the month-by-month calculation is easier to follow: multiply the balance by the rate, add the return, then add that month’s contribution. That is exactly what the calculator above does and shows in the table.
- Do savings accounts pay compound interest?
- Yes. Interest credited to a savings account joins the balance and earns in the following months. What changes is the rate, which is usually lower than other fixed income products. In Brazil, savings accounts pay 0.5% per month plus TR whenever the Selic rate is above 8.5% per year, and they are exempt from income tax for individuals.
- How do you convert an annual rate into a monthly rate?
- Take the twelfth root instead of dividing by 12: monthly rate = (1 + annual rate)^(1/12) − 1. A rate of 12% per year equals 0.9489% per month. Dividing by 12 would give 1% per month and overstate the return, because it ignores that interest also compounds within the year.
- How much does $1,000 a month grow to in 10 years?
- At 0.8% per month, $1,000 a month for 120 months adds up to $120,000 in contributions and reaches about $200,000 — roughly $80,000 of it coming from interest. The result changes a lot with the rate, so it is worth simulating with the real rate of the product you use.
- How long does it take for money to double with compound interest?
- The rule of 72 gives a good mental estimate: divide 72 by the rate in percent and you get the number of periods until it doubles. At 1% per month money doubles in roughly 72 months; at 10% per year, in about 7 years. For the exact figure, use the calculator and adjust the term until the balance is twice the initial amount.
- Does the calculator subtract income tax?
- Only if you turn the option on. When it is on, it applies the Brazilian regressive fixed income table to the interest: 22.5% up to 180 days, 20% up to 360 days, 17.5% up to 720 days and 15% beyond that. It applies to CDB, Tesouro Direto and fixed income funds. Savings accounts, LCI and LCA are exempt for individuals, and stocks follow their own rule.
- Does compound interest apply to debt as well?
- It does, and that is where it hurts. Credit card revolving balances, overdrafts and most loans compound. A rate that looks small per month becomes a very large number per year: 13% per month is more than 330% per year. Before investing, paying off expensive debt is usually the best return available.