Compound interest means earning interest on interest. The idea is simple; the consequences are not intuitive, because the growth is slow for years and then suddenly large. This article works through concrete numbers: the same deposit compounded yearly, monthly and daily, the rule of 72 against the exact doubling times, and a monthly saving plan over 30 years. It is arithmetic, not financial advice, and every figure was produced by our compound interest calculator and checked independently with Python.
The formula
A starting amount P at a yearly rate r, compounded n times a year for t years, grows to
P × (1 + r/n)^(n × t)
Each period, the interest for that period is added to the balance, and the next period's interest is calculated on the larger balance. The calculator carries the balance with far more digits than a cent and rounds once, at the end, so its results match the formula exactly.
Does compounding frequency matter?
Take 10,000 at 6% a year for 10 years:
| Compounding | Periods a year | Balance after 10 years | Interest |
|---|---|---|---|
| Yearly | 1 | 17,908.48 | 7,908.48 |
| Half-yearly | 2 | 18,061.11 | 8,061.11 |
| Quarterly | 4 | 18,140.18 | 8,140.18 |
| Monthly | 12 | 18,193.97 | 8,193.97 |
| Weekly | 52 | 18,214.89 | 8,214.89 |
| Daily | 365 | 18,220.29 | 8,220.29 |
Going from yearly to monthly adds 285.49, about 3.6% more interest. Going from monthly to daily adds only 26.32. The limit, compounding continuously, is 10,000 × e^0.6 ≈ 18,221.19, just 0.90 more than daily. The jump that matters is from yearly to something more frequent; after monthly, the difference is small change.
This is why banks quote two rates. The nominal rate is the 6%. The effective annual rate (EAR, or APY in the US) is what you actually earn in a year: (1 + 0.06/12)^12 − 1 ≈ 6.168% for monthly compounding, and about 6.183% for daily. When you compare two products, compare effective rates; a 6% account compounded monthly beats a 6.1% account compounded yearly.
The rule of 72
To estimate how long money takes to double, divide 72 by the yearly rate in percent. At 6%, 72 ÷ 6 = 12 years. How good is that? Here are the exact doubling times with yearly compounding, ln 2 ÷ ln(1 + r):
| Rate | Rule of 72 | Exact (yearly compounding) |
|---|---|---|
| 2% | 36.0 years | 35.0 years |
| 4% | 18.0 | 17.7 |
| 6% | 12.0 | 11.9 |
| 8% | 9.0 | 9.0 |
| 10% | 7.2 | 7.3 |
| 12% | 6.0 | 6.1 |
| 18% | 4.0 | 4.2 |
| 24% | 3.0 | 3.2 |
Between about 4% and 12%, the rule is within a few months, and it is almost exact at 8%. That accuracy, and the fact that 72 divides evenly by 2, 3, 4, 6, 8, 9 and 12, is why 72 is the number used. At very low rates it overestimates a little (69 or 70 works better), and at high rates it underestimates. As a check: 10,000 at 6% for 12 years gives 20,121.96 with yearly compounding and 20,507.51 with monthly, so "doubles in about 12 years" holds either way.
The rule works in reverse too, and that is where it is most useful: a debt at 24% a year doubles in about three years if nothing is paid. For how repayments interact with interest, see loan amortization schedules explained.
Regular saving: 200 a month at 6%
Most people do not deposit one lump sum; they save every month. Starting from zero, 200 at the end of every month, 6% compounded monthly:
| Year | Deposited | Interest earned | Balance |
|---|---|---|---|
| 1 | 2,400.00 | 67.11 | 2,467.11 |
| 5 | 12,000.00 | 1,954.01 | 13,954.01 |
| 10 | 24,000.00 | 8,775.87 | 32,775.87 |
| 15 | 36,000.00 | 22,163.74 | 58,163.74 |
| 20 | 48,000.00 | 44,408.18 | 92,408.18 |
| 25 | 60,000.00 | 78,598.79 | 138,598.79 |
| 30 | 72,000.00 | 128,903.01 | 200,903.01 |
In the first five years, interest is small next to the deposits: 1,954.01 against 12,000. A little after year 20 the interest overtakes everything deposited, and over the last ten years the balance more than doubles while only 24,000 more goes in. That shape is the whole argument for starting early: the last years do most of the work, and you only get them if the first years happened.
Paying at the start of each month instead of the end gives 92,870.22 after 20 years, 462.04 more, because every deposit earns one extra month.
Inflation: what it is worth
A balance in 20 years is not worth the same as the same number today. With 3% inflation a year, the 92,408.18 above is worth about 51,164.17 in today's money (92,408.18 ÷ 1.03²⁰). The real growth rate is roughly the interest rate minus inflation. At 6% interest and 3% inflation you still gain, but far less than the headline number suggests. The calculator shows both columns.
Common mistakes
- Comparing nominal rates with different compounding. Compare effective annual rates.
- Treating the rule of 72 as exact at 2% or 24%. It is a mental shortcut for the middle of the range.
- Ignoring fees and taxes. A 1% yearly fee on a 6% return leaves 5%, and over 30 years that changes the result by tens of percent, not 1%.
- Assuming interest is added to the balance. Some products pay interest out, or pay it only at maturity; without reinvesting, there is no compounding.
- Forgetting inflation when judging whether a future sum is "enough".
- Rounding every period and expecting to match a formula, or the other way round. Banks round interest to the cent each period; over decades that can differ from the formula by cents or more.
Try your own numbers in the compound interest calculator: change the compounding, the timing of deposits and the inflation rate, and export the year-by-year table as CSV. For the borrowing side of the same arithmetic, the loan calculator builds a full repayment schedule.