Compound interest is simple to define and easy to underestimate. Interest earns interest, and over a long enough stretch the interest ends up bigger than the money you put in. The formula fits on one line, yet small choices inside it (how often interest compounds, whether you deposit at the start or end of a month, whether the rate is before or after inflation) move the answer by thousands.
This guide works through the formula by hand with numbers you can check on a phone calculator, then shows how regular contributions, compounding frequency and inflation change the result. The free compound interest calculator runs every example below, so you can follow along and then swap in your own figures.
TL;DR: Compound interest grows a balance by A = P(1 + r/n)^(nt), where P is the starting amount, r the annual rate as a decimal, n the compounding periods per year and t the years. $1,000 at 5% compounded monthly for 10 years becomes $1,647.01. Regular deposits add C × ((1 + i)^N - 1) / i on top. The Toolz compound interest calculator handles both, shows the effective annual rate and a year-by-year schedule, and runs client-side, free, no signup. Results are before tax and inflation, and the rate is held constant, so treat them as scenarios, not forecasts.
What is compound interest?
Compound interest is interest calculated on the original deposit plus all the interest already added to it. In the first period you earn interest only on what you put in. In every later period you earn interest on a bigger balance, because last period's interest is now part of it.
Simple interest, by contrast, is always calculated on the original deposit alone. The difference is small at first and large later. Take $10,000 at 5% a year for three years:
| Year | Simple interest balance | Compound interest balance (annual) |
|---|---|---|
| Start | $10,000.00 | $10,000.00 |
| 1 | $10,500.00 | $10,500.00 |
| 2 | $11,000.00 | $11,025.00 |
| 3 | $11,500.00 | $11,576.25 |
After one year the two are identical. In year two the compound balance earns 5% on $10,500, which is $525 instead of $500. By year three the gap is $76.25. Stretch the same comparison to 20 years at 6% and the gap on $10,000 grows to about $10,000: simple interest reaches $22,000 while annual compounding reaches about $32,071.
The U.S. Securities and Exchange Commission's investor education site explains the same idea and offers its own compound interest calculator on Investor.gov, which is a useful second opinion for checking results.
What is the compound interest formula?
The standard formula for a single deposit is:
A = P × (1 + r/n)^(n × t)
A = final amount
P = principal (starting amount)
r = annual interest rate as a decimal (5% = 0.05)
n = number of times interest compounds per year
t = number of years
The piece in brackets, r/n, is the rate per compounding period. The exponent, n × t, is the total number of periods.
Worked example: $1,000 at 5%, monthly, 10 years
- Rate per period: 0.05 / 12 = 0.0041667
- Number of periods: 12 × 10 = 120
- Growth factor: 1.0041667^120 = 1.647009
- Final amount: 1,000 × 1.647009 = $1,647.01
Interest earned is $647.01. Type 1000, 5, 10 and Monthly into the calculator with no recurring contribution and the future value reads 1,647.01.
To check step 3 without a scientific calculator, raise 1.0041667 to the 12th power first (1.051162), then raise that to the 10th power.
How does compounding frequency change the result?
More frequent compounding earns slightly more, but the effect is smaller than most people expect. What changes is the effective annual rate, the return you get over a full year once compounding is counted. Banks in the U.S. publish it as the annual percentage yield, or APY, under the Truth in Savings Act; the calculation rules are in Regulation DD.
The effective annual rate for n periods is (1 + r/n)^n - 1. For continuous compounding it is e^r - 1. At a 5% nominal rate:
| Compounding | Periods per year | Effective annual rate | $10,000 after 1 year |
|---|---|---|---|
| Annually | 1 | 5.00% | $10,500.00 |
| Semi-annually | 2 | 5.06% | $10,506.25 |
| Quarterly | 4 | 5.09% | $10,509.45 |
| Monthly | 12 | 5.12% | $10,511.62 |
| Daily | 365 | 5.13% | $10,512.67 |
| Continuously | infinite | 5.13% | $10,512.71 |
Going from annual to monthly adds about $11.62 a year on $10,000. Going from monthly to daily adds about a dollar. The rate and the number of years matter far more than the compounding schedule.
Frequency matters more at high rates, which is why it shows up on debt. A credit card charging 24% APR compounded monthly has an effective annual rate of 1.02^12 - 1 = 26.82%. The same math that makes savings grow makes an unpaid balance grow.
The calculator shows the effective annual rate next to every result, so you can compare two accounts that quote the same nominal rate with different compounding.
How do regular contributions change the formula?
Regular deposits are handled with the future value of an annuity, added to the growth of the starting amount:
FV = P × (1 + i)^N + C × ((1 + i)^N - 1) / i
i = rate per period (r / n)
N = total number of periods
C = contribution per period
That version assumes deposits land at the end of each period, called an ordinary annuity. When deposits land at the start of each period, called an annuity due, every deposit earns one extra period of interest, so the contribution part is multiplied by (1 + i).
Worked example: a 30-year retirement projection
Starting amount 25,000, annual rate 6%, compounded monthly, 30 years, plus 500 a month deposited at the end of each month.
- Rate per period: 0.06 / 12 = 0.005
- Periods: 12 × 30 = 360
- Growth factor: 1.005^360 = 6.022575
- Starting amount grows to 25,000 × 6.022575 = 150,564.38
- Contributions grow to 500 × (6.022575 - 1) / 0.005 = 500 × 1,004.515 = 502,257.52
- Future value: 150,564.38 + 502,257.52 = 652,821.90
Total deposited is 25,000 + (500 × 360) = 205,000, so total interest is 447,821.90. The effective annual rate is 1.005^12 - 1 = 6.17%.
Switch the contribution timing to Start of period and step 5 is multiplied by 1.005, giving 504,768.81 and a future value of 655,333.19. Depositing on the first of the month instead of the last is worth about 2,500 over 30 years in this scenario.
The year-by-year schedule makes the acceleration visible. Interest credited in year 1 is 1,709.73. Interest credited in year 30 is 37,734.93, more than the 6,000 deposited that year six times over.
What does the calculator's default example show?
The calculator opens with a starting amount of 10,000, a 7% annual rate, 20 years, monthly compounding and 200 a month at the end of each month. The result is a future value of 144,572.72 from 58,000 deposited, so 86,572.72 is interest.
Change only the Compound frequency to Annually and the result drops to 140,204.12. That is a difference of about 4,369, bigger than in the single-deposit table above because the balance is larger and the term longer, but still modest next to the effect of the rate or the years.
Change only Years from 20 to 30 and the future value more than doubles. Time is the strongest lever in the formula because it sits in the exponent.
How does the Toolz calculator compute contributions on a different schedule?
The calculator steps through the term one month at a time. Each month the balance is multiplied by the monthly equivalent of the chosen compounding: (1 + r/n)^(n/12) for discrete compounding and e^(r/12) for continuous. Contributions are added before that month's growth for start-of-period timing, or after it for end-of-period timing, at the frequency you pick: monthly, quarterly or yearly.
Two consequences are worth knowing:
- At every year boundary, a lump sum matches the closed formula exactly, whatever the compounding frequency.
- When contributions are more frequent than compounding, such as monthly deposits with annual compounding, each deposit earns interest for the fraction of the year it is in the account, at the equivalent rate. Some real accounts only credit interest on the balance at the compounding date, which pays slightly less. If you are modeling a specific account, match its compounding and deposit dates as closely as the options allow.
Inputs are checked before anything is calculated. The starting amount and contribution must be zero or more and no larger than one trillion, the rate must be between 0% and 1,000%, and the term must be more than zero and at most 200 years. Fractional years are allowed and rounded to the nearest whole month.
What is the rule of 72, and how accurate is it?
The rule of 72 estimates how many years money takes to double: divide 72 by the annual percentage rate. It works because doubling time is ln(2) / ln(1 + r), and for rates in the single digits that is close to 72 / rate.
| Rate | Rule of 72 estimate | Exact doubling time (annual compounding) |
|---|---|---|
| 3% | 24 years | 23.4 years |
| 6% | 12 years | 11.9 years |
| 9% | 8 years | 8.0 years |
| 12% | 6 years | 6.1 years |
It is a mental shortcut, good for comparing options in conversation. For an actual plan, use the exact figure: enter the rate and a term in the calculator and read off the year the schedule's closing balance first passes double the start.
How do taxes and inflation affect a projection?
The calculator's results are nominal and pre-tax. They describe the number in the account, not what it will buy or what you keep.
Inflation is the bigger distortion over long terms. To see a projection in today's money, enter a real rate instead of the nominal one. The exact conversion is:
real rate = (1 + nominal rate) / (1 + inflation rate) - 1
At 7% nominal and 3% inflation that is 1.07 / 1.03 - 1 = 3.88%, a little less than the 4% you get by subtracting. Run the default example at 3.88% and the 20-year result shrinks sharply, which is closer to the purchasing power you would have.
Taxes depend on the account. Interest in a taxable savings account is typically taxed each year, which reduces what compounds. Tax-advantaged retirement accounts defer or exempt that tax. The calculator does not model either, so for a taxable account, a rough approach is to enter the after-tax rate.
What are the most common compound interest mistakes?
These are the errors that produce wrong answers most often, by hand or in a tool.
- Entering a monthly rate in an annual field. An account quoting 1% a month is 12% a year nominal. The rate field takes the annual figure, so enter 12, not 1.
- Using a percentage instead of a decimal in the formula. By hand, 5% is 0.05. Plugging in 5 gives an absurd result.
- Confusing APR and APY. APR is the nominal rate; APY already includes compounding. Entering an APY and then choosing monthly compounding double-counts it slightly. If you only know the APY, enter it with Annually selected.
- Forgetting contribution timing. Start and end of period differ by one period of interest on every deposit.
- Treating the projection as a promise. The rate is fixed for the whole term. Real investment returns vary year to year, so run a low, middle and high rate to see a range.
- Ignoring inflation over decades. A million in 30 years is not a million today.
How does compound interest relate to other calculators?
Compound interest sits in the middle of several money questions, and each one has a better-suited tool.
| Question | Tool | Why |
|---|---|---|
| How much will my savings grow? | Compound interest calculator | Future value with deposits |
| What will a loan cost me each month? | Loan calculator | Amortized payments and total interest |
| What percentage did my balance grow? | Percentage calculator | Percentage change between two values |
| How much margin does a price leave? | Profit margin calculator | Cost, price and margin |
The loan side is the mirror image of this guide: the same periodic-rate math, applied to a balance you are paying down. The loan and EMI guide works through it, and the percentage calculation guide covers the percentage-change arithmetic used to describe growth. The full set is on the calculators hub.
Is the compound interest calculator private?
Yes. Every calculation runs client-side in your browser in plain JavaScript, and the amounts, rate and contribution plan are never sent anywhere. Results update as you type. The only thing recorded is a usage count when you press Copy summary, which carries the compounding frequency and the number of years, not any amounts. The calculator keeps working without a connection once the page has loaded.
When you have your own numbers, run a compound interest projection with a cautious rate first, then an optimistic one, and plan around the gap between them.
