Compound interest is interest earned on interest. It is the reason savings can grow faster over long periods than a simple percentage suggests, and the same maths is why debt that is left unpaid can grow quickly. Once you understand the formula and its moving parts, you can check any savings projection yourself.
In this article you will learn:
- the difference between simple and compound interest
- the compound interest formula and what each variable means
- how compounding frequency (annual, monthly, daily) affects the result
- how to handle regular contributions, and the difference between payments at the end and the start of each period
- the rule of 72 and how accurate it is
- how to adjust for inflation to see what your money is worth in today's terms
All numbers in this article are illustrative examples, computed and checked to the cent. They assume a constant rate of return, which real savings and investments rarely deliver.
Simple vs compound interest
With simple interest, you earn interest only on the original amount. $10,000 at 5% simple interest earns $500 every year. After 10 years you have $15,000.
With compound interest, each period's interest is added to the balance, and the next period's interest is calculated on the larger balance. $10,000 at 5% compounded annually earns $500 in year one, $525 in year two ($10,500 × 5%), $551.25 in year three, and so on. After 10 years you have $16,288.95.
The gap of $1,288.95 in this example is the interest earned on interest. It is modest over 10 years and grows much larger over 30 or 40.
The compound interest formula
For a single lump sum:
A = P × (1 + r/n)^(n × t)
Where:
- A is the final amount
- P is the starting principal
- r is the annual interest rate as a decimal (5% = 0.05)
- n is the number of compounding periods per year (1 for annual, 12 for monthly, 365 for daily)
- t is the number of years
The interest earned is A − P.
Worked example. $10,000 at 5% compounded annually for 10 years:
A = 10,000 × (1.05)^10 = $16,288.95
How compounding frequency changes the result
The more often interest is compounded, the sooner each bit of interest starts earning its own interest. Here is the same $10,000 at a 5% annual rate for 10 years with different compounding frequencies:
| Compounding | n | Final amount | Effective annual rate |
|---|---|---|---|
| Annually | 1 | $16,288.95 | 5.000% |
| Quarterly | 4 | $16,436.19 | 5.095% |
| Monthly | 12 | $16,470.09 | 5.116% |
| Daily | 365 | $16,486.65 | 5.127% |
Two observations:
- More frequent compounding helps, but the effect shrinks. Moving from annual to monthly adds $181.14 over 10 years in this example; moving from monthly to daily adds only $16.56.
- The effective annual rate (sometimes shown as APY or AER, depending on the country and product) converts any compounding frequency into an equivalent annual rate: (1 + r/n)^n − 1. Comparing effective rates is the fairest way to compare accounts that compound differently.
Adding regular contributions
Most people save by adding money regularly rather than depositing one lump sum. The future value of a series of equal regular contributions is:
FV = C × ((1 + i)^N − 1) / i
Where:
- C is the contribution each period
- i is the interest rate per period (annual rate ÷ periods per year)
- N is the total number of contributions
This formula assumes each contribution is made at the end of the period, which is called an ordinary annuity.
If contributions are made at the start of each period (an annuity due), each one earns one extra period of interest. Multiply the result by (1 + i):
FV (due) = C × ((1 + i)^N − 1) / i × (1 + i)
Worked example: $200 a month for 20 years at 5%
- C = $200
- i = 0.05 ÷ 12
- N = 20 × 12 = 240
| Timing | Future value | Total contributed | Interest earned |
|---|---|---|---|
| End of each month (ordinary annuity) | $82,206.73 | $48,000.00 | $34,206.73 |
| Start of each month (annuity due) | $82,549.26 | $48,000.00 | $34,549.26 |
Paying in at the start of the month adds $342.53 over 20 years here. The timing difference is small compared with the effect of the rate, the amount and, above all, the number of years.
Combining a lump sum and contributions
To project a starting balance plus regular contributions, calculate each part separately and add them. Starting with $10,000 and adding $200 at the end of each month for 20 years at 5% compounded monthly:
- Lump sum: 10,000 × (1 + 0.05/12)^240 = $27,126.41
- Contributions (ordinary annuity): $82,206.73
- Total: $109,333.14
Working backwards: how much to save for a goal
Often the question is not "what will I end up with?" but "how much do I need to put away?". Rearranging the annuity formula gives the regular contribution needed to reach a target:
C = FV × i / ((1 + i)^N − 1)
Worked example. To reach $50,000 in 5 years at 4.5% compounded monthly, with deposits at the end of each month:
- FV = $50,000, i = 0.045 ÷ 12, N = 60
- C = $744.65 per month
Our savings goal calculator does this rearrangement for you and lets you include an existing balance.
The rule of 72
The rule of 72 is a mental shortcut for how long money takes to double at a given annual compound rate:
Years to double ≈ 72 ÷ annual rate (as a percentage)
It is an approximation. The exact doubling time is ln(2) ÷ ln(1 + r). Here is how close the rule gets:
| Annual rate | Rule of 72 estimate | Exact doubling time (annual compounding) |
|---|---|---|
| 2% | 36 years | 35.00 years |
| 6% | 12 years | 11.90 years |
| 9% | 8 years | 8.04 years |
| 12% | 6 years | 6.12 years |
The rule is most accurate for rates around 6% to 10% and drifts at very low or very high rates. It is useful for quick comparisons, not for planning to the dollar. It also works in reverse for debt: a balance growing at 18% a year with no repayments would roughly double in about four years.
Inflation: nominal vs real returns
A projection in future dollars can look impressive, but prices also rise over time. To understand what a future amount can buy, convert it to today's dollars.
Real rate of return: (1 + nominal rate) ÷ (1 + inflation rate) − 1
With a 7% nominal return and 3% inflation: 1.07 ÷ 1.03 − 1 = 3.88% real. The quick approximation of subtracting inflation (7% − 3% = 4%) is close but slightly overstates it.
Worked example. $10,000 growing at 7% a year for 20 years becomes $38,696.84. Discounted by 3% inflation a year, that is worth about $21,425.50 in today's dollars. The investment still roughly doubles in purchasing power, but not the nearly fourfold increase the nominal figure suggests.
Put another way, at 3% annual inflation, $100,000 in 20 years' time buys about what $55,367.58 buys today.
Taxes and fees reduce returns further. A 1% annual fee on an investment returning 7% leaves roughly 6% compounding for you, which makes a noticeable difference over decades. Include fees and tax in your own projections where they apply.
Common mistakes
Using the annual rate per period. If you compound monthly, the per-period rate is the annual rate ÷ 12, and the number of periods is years × 12.
Mixing up nominal and effective rates. A 5% rate compounded monthly is an effective 5.116% a year. Compare accounts on the same basis.
Assuming constant returns. Savings account rates change, and investment returns vary from year to year. A constant-rate projection is a planning tool, not a forecast.
Ignoring contribution timing. The ordinary annuity vs annuity due difference is small, but if your numbers do not match a calculator, check which assumption each is using.
Forgetting inflation. A large future number can still buy less than you expect.
Checklist: checking a savings projection
- Is the rate annual, and what is the compounding frequency?
- Are contributions assumed at the start or end of each period?
- Is the result in future dollars or today's dollars?
- Are fees and taxes included?
- Is the rate realistic for the type of account or investment?
- Have you tested a lower-rate scenario to see the range of outcomes?
Summary
- Compound interest adds each period's interest to the balance, so interest earns interest.
- For a lump sum, A = P × (1 + r/n)^(nt). For regular contributions, FV = C × ((1 + i)^N − 1) / i, multiplied by (1 + i) if payments are at the start of each period.
- More frequent compounding increases growth slightly; time and rate matter far more.
- The rule of 72 gives a quick doubling estimate that is close for mid-range rates.
- Convert projections into today's dollars to see real purchasing power, and allow for fees and tax.
Experiment with your own figures in the compound interest calculator. This article is general information, not financial advice. See our financial disclaimer.