Compound interest is interest earned on interest. Each time interest is added to your balance, the next round of interest is worked out on the bigger total, so money grows faster and faster over time. The formula is A = P(1 + r/n)nt: 10,000 at 5% a year compounded yearly becomes 16,288.95 after 10 years, compared with 15,000 if only simple interest were paid. This guide explains the formula symbol by symbol, works through examples, shows why compounding frequency matters less than people think, and covers regular saving, the rule of 72, inflation, fees and debt.
- Simple interest is paid only on the original amount; compound interest is also paid on interest already earned.
- The formula A = P(1 + r/n)nt uses the starting amount P, the annual rate r, the number of compounding periods a year n and the years t.
- Time and the rate matter far more than compounding frequency. APY or AER shows the true yearly rate once compounding is included.
- The rule of 72 (72 ÷ rate = years to double) is a quick approximation, not an exact answer.
- Inflation and fees compound too, and so does interest on debt such as credit cards.
Simple interest vs compound interest
With simple interest you earn the same amount every year, because interest is only ever calculated on the original sum: interest = P × r × t. With compound interest the interest is added to the balance, and from then on it earns interest of its own.
Take 10,000 at 5% a year. Simple interest pays 500 every year. Compound interest also pays 500 in the first year, but in the second year it pays 5% of 10,500, which is 525, then 5% of 11,025, which is 551.25, and so on.
| 10,000 at 5% a year | Simple interest | Compounded yearly | Difference |
|---|---|---|---|
| After 1 year | 10,500.00 | 10,500.00 | 0.00 |
| After 10 years | 15,000.00 | 16,288.95 | 1,288.95 |
| After 30 years | 25,000.00 | 43,219.42 | 18,219.42 |
The gap is small at first and then widens quickly. That is the whole point of compounding: the growth curve bends upwards, so the later years do most of the work.
The compound interest formula
The standard formula for a single sum is:
A = P(1 + r/n)nt
| Symbol | Meaning | Example |
|---|---|---|
| A | The amount at the end (balance including interest) | What you want to find |
| P | The principal, the amount you start with | 5,000 |
| r | The annual interest rate as a decimal | 4% = 0.04 |
| n | How many times a year interest is added | 12 for monthly |
| t | The number of years | 8 |
The interest earned is A − P. If interest is compounded continuously, the limit as n grows without end, the formula becomes A = Pert, where e ≈ 2.71828.
Worked example
You put 5,000 into an account paying 4% a year, compounded monthly, for 8 years.
- Rate per period: r/n = 0.04 ÷ 12 ≈ 0.003333.
- Number of periods: nt = 12 × 8 = 96.
- Growth factor: (1.003333)96 ≈ 1.3764.
- A = 5,000 × 1.3764 ≈ 6,881.98, so the interest earned is about 1,881.98.
Rounding the growth factor too early makes the answer drift by a few units, which is why it helps to use a calculator. The compound interest calculator shows this formula worked through with your own numbers.
Compounding frequency, APY, AER and EAR
Interest can be added yearly, half-yearly, quarterly, monthly or daily. More frequent compounding means interest starts earning interest sooner, so the same nominal rate grows slightly faster. Here is 10,000 at 6% for 10 years:
| Compounded | Balance after 10 years | Effective annual rate |
|---|---|---|
| Yearly (n = 1) | 17,908.48 | 6.000% |
| Quarterly (n = 4) | 18,140.18 | 6.136% |
| Monthly (n = 12) | 18,193.97 | 6.168% |
| Daily (n = 365) | 18,220.29 | 6.183% |
| Continuously | 18,221.19 | 6.184% |
The last column is the effective annual rate: (1 + r/n)n − 1, or er − 1 for continuous compounding. It folds the compounding into a single yearly figure. In the US it is called the APY (annual percentage yield), in the UK the AER (annual equivalent rate), and in finance generally the EAR. Comparing savings accounts on this figure puts them on equal terms, whatever their compounding schedule.
Notice how small the differences are. Going from yearly to daily compounding adds about 312 over ten years here, while raising the rate from 6% to 7% would add well over 1,700. Frequency is a tie-breaker; the rate and the time are what count.
Regular contributions: the future value of a series
Most people save a little at a time rather than one lump sum. Equal deposits made at regular intervals form an annuity, and their value at the end is:
FV = PMT × ((1 + i)N − 1) / i
Here PMT is each deposit, i is the interest rate per deposit period and N is the number of deposits. This assumes each deposit is made at the end of its period. If you pay at the start, every deposit earns one extra period of interest, so multiply the result by (1 + i).
Example: 200 a month for 30 years at 6% compounded monthly. Then i = 0.06 ÷ 12 = 0.005 and N = 360, so FV = 200 × (1.005360 − 1) ÷ 0.005 ≈ 200,903.01. You paid in 72,000; the other 128,903.01 is interest. Paying at the start of each month instead gives about 201,907.52.
If you save monthly but the account compounds at a different frequency, first convert the rate to an equivalent rate per deposit period: i = (1 + r/n)n/m − 1, where m is the number of deposits a year. For 5% compounded yearly with monthly deposits, i = 1.051/12 − 1 ≈ 0.4074% a month.
Why starting early matters
Two savers earn 6% compounded monthly. Saver A puts in 200 a month for 15 years, then stops and leaves the money for another 15 years. Saver B waits 15 years, then puts in 400 a month for 15 years. Both finish at the same time.
| Paid in | Balance after 30 years | |
|---|---|---|
| Saver A: 200 a month, years 1 to 15 | 36,000 | 142,739.27 |
| Saver B: 400 a month, years 16 to 30 | 72,000 | 116,327.48 |
Saver A pays in half as much and still ends up with more, because their money compounds for longer. These are illustrative figures with a constant rate; real investment returns go up and down.
The rule of 72
To estimate how long money takes to double, divide 72 by the annual interest rate in percent. At 6%, 72 ÷ 6 = 12 years. It is a mental shortcut and only an approximation. The exact doubling time with yearly compounding is ln(2) ÷ ln(1 + r):
| Rate | Rule of 72 | Exact (yearly compounding) | Error |
|---|---|---|---|
| 2% | 36.0 years | 35.00 years | 1.00 year too long |
| 4% | 18.0 years | 17.67 years | 0.33 year too long |
| 6% | 12.0 years | 11.90 years | 0.10 year too long |
| 8% | 9.0 years | 9.01 years | 0.01 year too short |
| 10% | 7.2 years | 7.27 years | 0.07 year too short |
| 12% | 6.0 years | 6.12 years | 0.12 year too short |
The rule is most accurate around 8% and drifts further off at very low or very high rates. It works the other way too: at 3% inflation, prices double in roughly 72 ÷ 3 = 24 years (exactly about 23.45).
Inflation and real returns
A bigger number is not the same as more buying power. If prices rise, each unit of money buys less, so you need to look at the real value of your balance: real value = nominal value ÷ (1 + inflation)t.
Example: 10,000 growing at 5% a year for 20 years becomes 26,532.98. If inflation averages 3% a year over that time, its real value in today's prices is 26,532.98 ÷ 1.0320 ≈ 14,690.67. The real rate of return is (1.05 ÷ 1.03) − 1 ≈ 1.94% a year, a little less than the simple difference of 2 percentage points.
When interest is lower than inflation, savings lose buying power even as the balance rises. The calculator lets you enter an inflation rate to show the real value year by year.
How fees reduce compounding
Annual charges on funds, pensions and investment platforms come off your balance every year, so they compound against you. A fee that sounds tiny can take a large share of long-term growth. The figures below are illustrative examples with a constant return, not a forecast for any real product.
| 10,000 for 30 years at 7% before fees | Net growth | Balance |
|---|---|---|
| No fees | 7.0% a year | 76,122.55 |
| 0.2% annual fee | 6.8% a year | 71,967.69 |
| 1% annual fee | 6.0% a year | 57,434.91 |
In this example a 1% yearly fee costs 18,687.64, about a quarter of the final balance, even though it is only one-seventh of the yearly return. Taxes on interest or gains work in a similar way; the rules differ from country to country, so check those that apply to you.
Compounding on debt
Compounding works just as hard for lenders. Credit cards usually quote an APR and charge interest monthly on the unpaid balance. A balance of 2,000 at 24% APR (2% a month), with no payments and no new spending, grows to 2,000 × 1.0212 ≈ 2,536.48 in a year, an effective rate of about 26.82%. Card terms vary, and some charge interest daily, but the lesson is the same: paying more than the minimum stops interest piling on interest. To plan repayments, see the debt payoff calculator or the loan calculator, which shows how much interest a fixed-rate loan costs in total.
Try it with your own numbers
The compound interest calculator handles an initial amount, monthly or yearly deposits at the start or end of each period, any compounding frequency, yearly increases in contributions and inflation, and shows a chart, a yearly table and the formula with your figures. For retirement saving in the US, the 401(k) calculator adds employer matching and contribution limits. For quick percentage changes, use the percentage calculator.
This guide is general information, not financial advice. Rates, fees and tax rules change, and investments can fall as well as rise.
Frequently asked questions
What is the compound interest formula?
A = P(1 + r/n)nt, where P is the starting amount, r the annual rate as a decimal, n the number of compounding periods a year and t the number of years. For regular deposits add the future value of a series, PMT × ((1 + i)N − 1) / i.
How much will 10,000 grow in 10 years?
It depends on the rate. As an example, at 5% a year compounded monthly 10,000 becomes about 16,470.09; at 7% compounded yearly it becomes about 19,671.51. Use the calculator to try your own rate and term.
What is the difference between APR and APY?
APR is a nominal yearly rate (for loans it can also include some fees), while APY, called AER in the UK, is the effective yearly rate after compounding. 6% APR compounded monthly is an APY of about 6.168%.
Is daily compounding much better than monthly?
Only slightly. 10,000 at 6% for 10 years becomes 18,193.97 with monthly compounding and 18,220.29 with daily compounding, a difference of about 26.
How accurate is the rule of 72?
It is close for typical rates: at 6% it says 12 years against an exact 11.90. It is less accurate at very low or very high rates, for example 36 years against 35.00 at 2%.
Does compound interest work against me on loans?
Yes, when unpaid interest is added to what you owe, as on credit cards. Most fixed-rate personal loans and mortgages are repaid in instalments that cover each month's interest, so interest does not build up as long as you keep up the payments.