How compound interest works
Compound interest is interest earned on interest. After each compounding period the interest is added to the balance, and the next period's interest is calculated on that larger balance. Over short spans it looks like simple interest; over decades the curve bends sharply upward, which is why starting early matters more than the exact rate. The balance after t years is:
A = P (1 + r/n)^(n t) P = principal, r = annual rate as a decimal, n = compounding periods per year, t = years
With a regular contribution PMT each month, the contributions form an annuity that grows alongside the principal. Using the effective monthly rate i = (1 + r/n)^(n/12) - 1 over m months:
A = P (1 + i)^m + PMT x ((1 + i)^m - 1) / i (contributions at the end of each month) A = P (1 + i)^m + PMT x ((1 + i)^m - 1) / i x (1 + i) (contributions at the start)
Worked example
The form defaults are $10,000 at 5% compounded monthly for 10 years with no contributions. Monthly compounding means n = 12, so the monthly rate is 0.05 / 12 = 0.0041667 and there are 120 periods:
A = 10,000 x (1 + 0.05/12)^120 = 10,000 x 1.0041667^120 = 10,000 x 1.647009 = $16,470.09
Interest earned is $6,470.09. Simple interest over the same span would be 10 x $500 = $5,000, so compounding added $1,470.09. The table the calculator produces shows the balance at each year end; year 1 ends at $10,511.62 and year 10's interest alone is $801.63, more than one and a half times year 1's.
Growth of $10,000 at 5%, 7%, and 10%
The rate compounds too. Ten percent is twice five percent, but over 30 years it produces more than four times the growth. Compounded monthly:
| Rate | 10 years | 20 years | 30 years | 30 years, plus $200/month |
|---|---|---|---|---|
| 5% | $16,470 | $27,126 | $44,677 | $211,129 |
| 7% | $20,097 | $40,387 | $81,165 | $325,159 |
| 10% | $27,070 | $73,281 | $198,374 | $650,472 |
The last column shows what regular contributions do. At 7%, adding $200 a month (which is $72,000 over 30 years) turns an $81,000 result into roughly $325,000; the contributions themselves account for less than a quarter of the final balance.
The rule of 72
To estimate how long money takes to double, divide 72 by the annual rate in percent. At 6% the answer is 12 years; at 9% it is 8 years. The rule comes from the exact formula t = ln 2 / ln(1 + r), which is close to 0.693 / r for small rates, and 72 is chosen over 69.3 because it has many divisors and slightly corrects for the annual compounding. It is very accurate between 4% and 12%.
| Rate | Rule of 72 | Exact doubling time |
|---|---|---|
| 2% | 36 years | 35 years |
| 4% | 18 years | 17.67 years |
| 6% | 12 years | 11.9 years |
| 8% | 9 years | 9.01 years |
| 10% | 7.2 years | 7.27 years |
| 12% | 6 years | 6.12 years |
The rule also works in reverse: to double money in 10 years you need about 7.2% a year, and inflation at 3% halves purchasing power in about 24 years.
How much does compounding frequency matter?
More frequent compounding earns a little more, but the effect is small compared with rate and time. $10,000 at 5% for 10 years:
| Compounded | Periods per year | Balance after 10 years |
|---|---|---|
| annually | 1 | $16,288.95 |
| quarterly | 4 | $16,436.19 |
| monthly | 12 | $16,470.09 |
| daily | 365 | $16,486.65 |
Daily versus annual compounding is a difference of about $198 on $16,000, or 1.2%. Raising the rate from 5% to 6% instead adds more than $1,700. Banks advertise APY, the annual yield after compounding, precisely so accounts with different frequencies can be compared on one number.
Why starting early beats saving more
Because growth compounds on growth, the years at the front of a savings plan do the most work. Compare two savers earning 7% compounded monthly. The first puts in $200 a month from age 25 to 35, then never adds another dollar: $24,000 in total. The second waits until 35 and then contributes $200 a month for 30 years, until 65: $72,000 in total.
early saver at 35: $34,617 (then left alone for 30 years) early saver at 65: $280,968 from $24,000 contributed late saver at 65: $243,994 from $72,000 contributed
The early saver contributed a third as much and finishes ahead, because the money had 40 years to double roughly four times (the rule of 72 at 7% gives a doubling every 10.3 years). The lesson is not that saving later is pointless, but that each year of delay costs more than the contributions skipped that year.
Nominal rate, periodic rate, and APY
Three rates describe the same account. The nominal rate is the quoted annual figure, 5% in the example. The periodic rate is the nominal rate divided by the number of compounding periods, 5% / 12 = 0.4167% per month. The APY, or effective annual rate, is what the account actually earns in a year once compounding is included: (1 + 0.05/12)^12 - 1 = 5.116%. The calculator reports the APY for whatever combination you enter so you can compare it directly with a bank's advertised yield.
Where compound interest shows up
- Savings and money market accounts: usually compounded daily and credited monthly. Enter the APY with annual compounding, or the nominal rate with daily.
- Certificates of deposit: fixed rate, usually daily or monthly compounding, no contributions.
- Index funds and retirement accounts: returns are not fixed, but a long-run average (7% nominal is a common planning figure for a stock-heavy portfolio) with monthly contributions models the outcome well.
- Debt: credit cards compound daily against you. A $5,000 balance at 24% with no payments grows to $6,356 in a year.