The quick answer: the formula and one example
Compound interest means each time interest is added, it joins the balance and starts earning interest too. The balance after t years is A = P × (1 + r ÷ n)^(n × t), where P is the starting amount, r is the yearly rate as a decimal (5% is 0.05), n is how many times a year interest is added and t is the number of years.
Take 10,000 at 5%, an example rate and not one you will get, with interest added once a year for 10 years: 10,000 × 1.05^10 = 16,288.95. The compound interest calculator does the sum for yearly, twice a year, quarterly, monthly, weekly, daily or continuous compounding, adds regular deposits and shows a chart and a year by year table.
Compound Interest CalculatorEnter a starting amount, a rate, a term and how often interest is added, and see the balance with the working.Open itSimple versus compound, side by side
Simple interest is paid only on the starting amount, so it adds the same sum every year (I = P × r × t, which the simple interest calculator works out). Compound interest is paid on the starting amount plus the interest already added. For 10,000 at 5%, with compounding once a year:
| Year | Balance with simple interest | Balance with compound interest | Difference |
|---|---|---|---|
| 1 | 10,500.00 | 10,500.00 | 0.00 |
| 5 | 12,500.00 | 12,762.82 | 262.82 |
| 10 | 15,000.00 | 16,288.95 | 1,288.95 |
| 20 | 20,000.00 | 26,532.98 | 6,532.98 |
| 30 | 25,000.00 | 43,219.42 | 18,219.42 |
In year one the two are identical. The gap opens because each year's interest earns interest in the next. Ten years at 5% multiplies the balance by 1.05^10 = 1.6289, a rise of 62.89%, not the 50% that adding 5% ten times would suggest. It is the same rule as in percentage calculations explained: percentages in a row multiply.
Compounding frequency matters less than you expect
The n in the formula is how many times a year interest is added. Here is the same 10,000 at 5% for 10 years, changing only n:
| Interest added | Balance after 10 years | Gain over yearly |
|---|---|---|
| Yearly (n = 1) | 16,288.95 | 0.00 |
| Twice a year (n = 2) | 16,386.16 | 97.21 |
| Quarterly (n = 4) | 16,436.19 | 147.24 |
| Monthly (n = 12) | 16,470.09 | 181.14 |
| Weekly (n = 52) | 16,483.25 | 194.30 |
| Daily (n = 365) | 16,486.65 | 197.70 |
| Continuously (e^(r × t)) | 16,487.21 | 198.26 |
Going from yearly to daily adds 197.70. Letting the same money run for 10 more years at yearly compounding adds 26,532.98 − 16,288.95 = 10,244.03, about 52 times as much. Continuous compounding, A = P × e^(r × t), is the ceiling that more and more frequent compounding approaches. The rate and the time matter far more than the frequency.
If your term is defined by two dates, count the days first (the guide to days between two dates shows how) and divide by 365 to get years, unless your agreement says to use 360. The compound interest calculator takes whole years and extra months, so round the term to the nearest month there.
Stated rate versus effective annual rate
A stated yearly rate with frequent compounding earns a little more than the number suggests. The effective annual rate is (1 + r ÷ n)^n − 1, and it puts rates with different frequencies on equal terms. For a stated 5%:
| Compounded | Effective annual rate |
|---|---|
| Yearly | 5.0000% |
| Twice a year | 5.0625% |
| Quarterly | 5.0945% |
| Monthly | 5.1162% |
| Daily | 5.1267% |
| Continuously | 5.1271% |
That is why a rate quoted as 5% compounded monthly (5.1162% effective) grows faster than one quoted as 5.05% compounded yearly (5.05% effective). The calculator shows the effective rate for whatever you enter.
Adding regular deposits
Each deposit earns interest from the day it goes in, so the earliest ones do the most work. With deposits of D made c times a year, the calculator adds up every deposit's growth using the standard annuity formula, D × (u^m − 1) ÷ (u − 1), where u is the growth over one deposit period and m is the number of deposits.
An example with an assumed rate: 200 a month at 6% compounded monthly for 30 years reaches 200,903.01 when each deposit goes in at the end of the month. You paid in 200 × 360 = 72,000, so 128,903.01 is interest, more than the deposits. If each deposit goes in at the start of the month instead, every one earns an extra month and the total is 201,907.52.
The rule of 72, and how wrong it is
Divide 72 by the yearly rate in percent to estimate how many years it takes money to double. At 6%, 72 ÷ 6 = 12 years. It is an approximation, and the error depends on the rate. Here is the rule against the exact time with yearly compounding, which is ln 2 ÷ ln(1 + r):
| Rate | Rule of 72 | Exact | The rule is |
|---|---|---|---|
| 2% | 36.0 years | 35.0 years | 2.8% high |
| 4% | 18.0 years | 17.7 years | 1.9% high |
| 6% | 12.0 years | 11.9 years | 0.9% high |
| 8% | 9.0 years | 9.0 years | 0.1% low |
| 10% | 7.2 years | 7.3 years | 1.0% low |
| 12% | 6.0 years | 6.1 years | 1.9% low |
| 20% | 3.6 years | 3.8 years | 5.3% low |
Check the 6% row: 10,000 × 1.06^12 = 20,121.96, so the money has already passed 20,000 a little before year 12. The exact doubling time is 11.9 years with yearly compounding and 11.6 years with monthly compounding. The rule is closest near 8% and drifts at very low and very high rates. It uses 72 because it is accurate near 8% and divides evenly by 2, 3, 4, 6, 8, 9 and 12.
What these numbers are not
Everything above holds the rate steady for the whole term. Real savings and borrowing rates move, and investment returns are not guaranteed. Fees, tax and inflation are not taken off: when prices rise, 100 in 20 years buys less than 100 today. Accounts also differ on the dates when interest is credited, so a statement can differ by a little.
These are worked examples of a formula. They are not predictions, and they say nothing about where to put money. Compounding also works against borrowers whenever unpaid interest is added to the balance they owe; how a loan payment and its total interest work shows that side. The loan calculator and the percentage calculator help check the sums.