What Is Compound Interest?
Simple interest is paid only on the original sum. Compound interest is paid on the original sum and on the interest already earned, so the balance grows on an ever-larger base. The effect is small at first and then accelerates, which is why compounding is described as exponential rather than linear growth.
Put $1,000 at 10% for 30 years. With simple interest you earn $100 a year, ending with $4,000. With annual compounding you end with $17,449. The rate and the term are identical; the only difference is that interest was allowed to earn interest.
The Compound Interest Formula
A = P × (1 + r ÷ n)n×t
where A is the final amount, P the principal, r the annual rate as a decimal, n the number of compounding periods per year and t the number of years.
With regular contributions the future value of the deposits is added. For a payment PMT made at the end of each period:
A = P(1 + i)N + PMT × ((1 + i)N − 1) ÷ i
where i is the rate per period and N the total number of periods. If deposits are made at the beginning of each period instead, the second term is multiplied by (1 + i), which is why the "contribute at" setting changes the result slightly.
For continuous compounding the formula becomes A = P × ert, where e is approximately 2.71828.
How Much Does Compounding Frequency Matter?
$10,000 at 6% for 10 years, under each frequency:
| Compounding | Periods per year | Effective annual rate (APY) | Final balance |
|---|---|---|---|
| Annually | 1 | 6.0000% | $17,908.48 |
| Semi-annually | 2 | 6.0900% | $18,061.11 |
| Quarterly | 4 | 6.1364% | $18,140.18 |
| Monthly | 12 | 6.1678% | $18,193.97 |
| Daily | 365 | 6.1831% | $18,220.29 |
| Continuously | ∞ | 6.1837% | $18,221.19 |
Notice the diminishing returns. Moving from annual to monthly compounding adds about $370; moving from daily to continuous adds under a dollar. Frequency matters, but far less than the rate or the term.
Nominal Rate, APR and APY
| Term | What it means | Includes fees? |
|---|---|---|
| Nominal rate | The stated annual rate, ignoring compounding | No |
| APR | Annual rate including lender fees, used for borrowing | Yes |
| APY / EAR | Annual rate including the effect of compounding, used for saving | No |
A savings account advertising "5% APY, compounded monthly" has a nominal rate of about 4.889%. When comparing accounts, compare APY; when comparing loans, compare APR.
The Effect of Rate and Time
$10,000 invested, no further contributions:
| Years | at 4% | at 6% | at 8% | at 10% |
|---|---|---|---|---|
| 5 | $12,210 | $13,489 | $14,898 | $16,453 |
| 10 | $14,908 | $18,194 | $22,196 | $27,070 |
| 15 | $18,203 | $24,541 | $33,069 | $44,539 |
| 20 | $22,226 | $33,102 | $49,268 | $73,281 |
| 25 | $27,138 | $44,650 | $73,402 | $120,569 |
| 30 | $33,135 | $60,226 | $109,357 | $198,374 |
| 35 | $40,458 | $81,236 | $162,925 | $326,387 |
| 40 | $49,399 | $109,575 | $242,734 | $537,007 |
Two things stand out. Time does more work than rate: $10,000 at 6% for 40 years beats the same amount at 10% for 20 years. And the growth is heavily back-loaded — at 8%, more is earned in the final ten years than in the first twenty-five.
The Rule of 72
Dividing 72 by the annual interest rate gives a close approximation of the number of years needed to double the money. It is accurate to within a few months for rates between about 4% and 12%.
| Rate | Rule of 72 estimate | Exact |
|---|---|---|
| 1% | 72.0 years | 69.7 years |
| 2% | 36.0 years | 35.0 years |
| 3% | 24.0 years | 23.4 years |
| 4% | 18.0 years | 17.7 years |
| 5% | 14.4 years | 14.2 years |
| 6% | 12.0 years | 11.9 years |
| 7% | 10.3 years | 10.2 years |
| 8% | 9.0 years | 9.0 years |
| 9% | 8.0 years | 8.0 years |
| 10% | 7.2 years | 7.3 years |
| 12% | 6.0 years | 6.1 years |
| 15% | 4.8 years | 5.0 years |
| 20% | 3.6 years | 3.8 years |
The rule works because ln(2) ≈ 0.693, and 69.3 divided by the rate is the exact continuous answer; 72 is used instead because it divides evenly by more numbers and compensates for discrete compounding.
Compounding Works Against You Too
Credit card balances compound daily. A $5,000 balance at 22% APR, with no payments, grows to roughly $6,230 in one year and $9,700 in three — the same mathematics that builds savings destroys the borrower. This is why paying down high-interest debt reliably beats investing: eliminating a guaranteed 22% cost is better than chasing an uncertain 8% return.
Inflation and Tax
Two forces erode compound growth. Inflation reduces what the final balance can buy: at 3% inflation, $100,000 in 20 years has the purchasing power of about $55,400 today. Tax on interest reduces the amount that compounds each period, which compounds the loss. Both are available in the More Options section above, and a nominal return should always be checked against the real, after-tax return before it means anything.
Frequently Asked Questions
Is it better to contribute at the start or the end of each period?
The start. Each deposit then earns one extra period of interest, which over decades typically adds 3–6% to the final balance at ordinary rates.
Why does my bank's figure differ slightly?
Usually the day-count convention. Banks may use 360 or 365 days, credit interest on specific dates, and round each period. These produce small differences that grow noticeable over long terms.
What return should I assume for investments?
The S&P 500 has averaged roughly 10% nominal and about 7% after inflation over the long run, but with severe year-to-year variation and long flat stretches. A compound interest calculator assumes a constant rate, which no real investment delivers — treat the output as an illustration of the mechanism, not a forecast.
How do I reach a specific target?
Adjust the contribution until the ending balance matches your goal. Working backwards from the target is usually more useful than accepting whatever the current contribution produces.