Compound Interest Calculator

How compound interest works

Simple interest pays you on the money you deposited and nothing else. Compound interest pays you on the deposit plus every dollar of interest that has already been credited. The interest joins the principal, and from that point it earns interest too. That single change is why a 30 year horizon produces numbers that look like a typo.

The formula is A = P(1 + r/n)^(n t). P is the starting amount, r is the annual rate written as a decimal, n is how many times a year interest is credited, and t is the number of years. The exponent, n times t, is the total number of compounding periods. Every period multiplies the balance by (1 + r/n), and multiplication compounds.

Watch what the exponent does. At 6% compounded monthly for 20 years, n times t is 240 periods, and 1.005 to the power of 240 is 3.3102. The $10,000 becomes $33,102.04. Double the time to 40 years and the multiplier is not double, it is 10.96, because the exponent doubled rather than the base.

That is the whole mechanism. Nothing about it is complicated, and nothing about it is optional: any account that credits interest and leaves it in place compounds, whether the bank calls it that or not.

Compounding frequency: annual, monthly, daily

Frequency sets how often the interest is credited. Annual compounding applies the full 6% once a year. Monthly compounding applies 0.5% twelve times. The nominal rate is 6% either way, but the monthly version earns more because each month's interest starts earning in the next month.

Notice the shape. Moving from annual to monthly is worth $1,030.69 over 20 years. Moving from monthly to daily is worth another $95.86. The gains shrink with each step, because the difference between 12 and 365 periods is a smaller jump in effect than the difference between 1 and 12. There is a ceiling: continuous compounding, where n goes to infinity, would produce $33,201.17, which is only $3.27 more than daily.

This is why a savings account advertising daily compounding is not automatically better than one advertising monthly. A 4.5% account compounded daily pays less than a 4.6% account compounded monthly. Compare the effective annual rate, not the frequency.

APR and APY are not the same number

The nominal annual rate, often called APR on a loan or just the stated rate on a savings account, is the rate before compounding. The effective annual rate, or APY, is what you get after compounding is applied for a year. APY is always the higher of the two when interest is credited more than once a year.

The conversion is APY = (1 + r/n)^n - 1. At a 6% nominal rate, annual compounding gives 6.0000%, monthly gives 6.1678% and daily gives 6.1831%. On a $10,000 balance that is a difference of $18.31 in the first year between monthly and daily, which is small. Over 20 years the same gap grows to $95.86, because the extra fraction of a percent compounds along with everything else.

The gap widens with the rate. At 12% nominal, annual compounding gives 12.0000% and monthly gives 12.6825%. That is 0.68 percentage points, which on a credit card balance is real money. The Consumer Financial Protection Bureau explanation of APR covers how lenders are required to disclose the annualized cost, and it is the number to compare when you are borrowing rather than saving.

For a savings account, the APY is the number to compare across offers, because it already includes the compounding. For a loan or a credit card, the APR usually does not include compounding on the balance, so the cost of carrying a balance can run higher than the quoted APR suggests.

Worked example: $10,000 at 6% for 20 years

Start with $10,000, a 6% nominal rate, monthly compounding and 20 years. The monthly rate is 0.06 / 12 = 0.005. The number of periods is 12 x 20 = 240. The multiplier is 1.005 to the power of 240, which is 3.3102, and 10,000 x 3.3102 = $33,102.04.

Total interest is $23,102.04, which is more than twice the original deposit. The account earned $616.78 in year one and $1,923.05 in year 20. The final year alone pays more than three times the first year, on the same balance and the same rate, because by then the balance is more than three times larger.

Now change one variable at a time and see which one matters.

• Frequency: annual instead of monthly, and the total drops to $32,071.35, a loss of $1,030.69.
• Rate: 5% instead of 6%, and the total drops to $27,126.40, a loss of $5,975.64.
• Time: 10 years instead of 20, and the total drops to $18,193.97, a loss of $14,908.07.

Time moves the answer more than anything else on the list. That is not a slogan, it is what the exponent does. Cutting the horizon in half does not halve the result, it removes more than half of it, because the years you drop are the ones with the largest balances.

The rule of 72 and where it drifts

The rule of 72 is a mental shortcut for doubling time: divide 72 by the annual rate as a whole number and you get the years it takes to double. At 6% that is 12 years. At 8% it is 9 years. At 12% it is 6 years. No calculator needed, which is why it survives.

It comes from the fact that the natural log of 2 is 0.693, and for small rates the natural log of (1 + r) is close to r. So the exact doubling time is about 0.693 / r, and 72 / 100r is the same expression scaled for whole-number percentages. The approximation is tuned to sit near the middle of the useful range rather than being exact anywhere.

The error is under a month at 6% and under three months at 24%, which is fine for a napkin. It is not fine for a loan comparison or a retirement projection, where a 0.2 year difference at a high rate compounds into real money. Use the rule to sanity-check a number, then use the formula for the number you act on.

One more limit: the rule assumes a fixed rate and no additional deposits. It also breaks down above about 30%, where the linear approximation behind it stops working and the shortcut starts understating the doubling time by years.

What time does to the curve

Compounding is slow at the start and fast at the end, and the numbers below show why starting early beats contributing more later.

At 5 years, three quarters of the balance is still your own money. At 30 years, more than four fifths of it is interest you never deposited. The crossover where interest exceeds contributions happens around year 12 at this rate, and it arrives sooner at higher rates: at 8% it takes about 9 years, and at 12% about 6 years.

This is also the argument against interrupting the curve. A withdrawal at year 15 does not just remove the dollars, it removes every future period those dollars would have compounded through. Taking $5,000 out of the 6% account at year 15 leaves roughly $22,300 less at year 40, because that $5,000 would have grown to about $22,300 over the remaining 25 years.

What the formula leaves out

The formula is exact for a fixed rate with no fees, no taxes and no inflation. Real accounts have all three, and each one cuts the result.

• Fees. A 1% annual expense ratio on a 6% return leaves 5% net, which over 20 years turns $33,102 into $27,126, a loss of about $5,976 on a $10,000 deposit.
• Taxes. Interest in a regular savings account is taxed as ordinary income in the year it is credited, so the compounding happens on an after-tax balance. The IRS guidance on interest received covers which forms report it.
• Inflation. At 3% inflation, a dollar in 20 years buys about 55 cents of what it buys today. A 6% nominal return is closer to 2.9% real, and the doubling time stretches from 12 years to about 24.

Run the same $10,000 at 6% for 20 years with 1% fees and 3% inflation and the purchasing power of the final balance is roughly $18,300 in today's dollars, not $33,102. The nominal number is still correct, it just measures something other than what you can buy.

For a rough check, subtract the fee and the inflation rate from the nominal rate and compound the remainder. It is not exact, but it lands close enough to keep a projection honest. The SEC investor education compound interest tool lets you add monthly contributions, which this page does not.

How to use the number without over-trusting it

This calculator assumes the rate never changes, which no savings account guarantees. A high yield account at 4.5% today can be 3.2% in two years, and the projection built on 4.5% will be wrong by a wide margin over a decade. Use a rate you can defend, and run the calculation a second time at a rate two points lower to see how much of the plan depends on the rate holding.

It also assumes no deposits and no withdrawals. Most real savings plans add money monthly, and contributions usually matter more than the compounding frequency in the early years. At 6% for 20 years, $10,000 alone reaches $33,102. The same $10,000 with $100 added each month reaches $79,306, because the contributions keep buying new periods to compound through.

Finally, compare like with like. A 20 year projection at 6% is not comparable to a 5 year certificate of deposit at 4.8%, and a taxable account is not comparable to a tax-advantaged one. The math on this page is the same math every bank uses, and the difference between two offers is almost always in the rate, the fee and the tax treatment rather than in how the interest is calculated.

This is an estimate for planning, not investment, tax or legal advice. Rates change and returns are not guaranteed.