Key numbers at a glance
What compound interest actually is
The word compound comes from the Latin componere, meaning to put together. In financial terms it means that interest earned in one period is added to the principal, and the next period's interest is calculated on that enlarged amount. Interest earns interest.
Consider 1.000 at 10 percent annual interest. With simple interest you earn 100 every year, always 10 percent of the original 1.000. After 10 years you have 2.000. With compound interest, the first year you earn 100 and your balance becomes 1.100. The second year you earn 10 percent of 1.100, which is 110, not 100. Your balance becomes 1.210. The third year you earn 10 percent of 1.210, which is 121. Each year the interest payment is larger than the last because the base keeps growing.
After 10 years of compound interest at 10 percent annually you have 2.594, not 2.000. That extra 594 came entirely from interest earning interest. After 30 years the compound result is 17.449 versus the simple interest result of 4.000. The gap widens every single year because compound growth is exponential while simple growth is linear.
The compound interest formula
Compounding frequency and how much it actually matters
When compounding occurs more than once per year the formula becomes FV = PV x (1 + r/m)^(m x t), where m is the number of compounding periods per year and t is the number of years.
Take 10.000 at 5 percent annual rate for 10 years. Annual compounding produces 16.289. Monthly compounding produces 16.470. Daily compounding produces 16.487. The difference between annual and daily compounding is only 198 on 10.000 over 10 years, less than 1,2 percent. The interest rate and the time period matter enormously more than the compounding frequency.
A savings account offering 4,5 percent compounding monthly beats one offering 4,4 percent compounding daily by a far larger margin than any change in compounding frequency could produce. The Annual Percentage Yield (APY) standardises comparisons by expressing any compounding frequency as a single equivalent annual rate. A 5 percent nominal rate compounded monthly has an APY of 5,12 percent. This is the number to use when comparing accounts with different compounding frequencies.
Worked examples with full calculations
FV = 10.000 x (1,07)^10 = 10.000 x 1,9672 = 19.672. The total interest earned is 9.672. Only 7.000 of that came from the 7 percent rate on the original 10.000. The remaining 2.672 came from interest compounding on previously earned interest. That extra 2.672 required no additional contribution. It is the pure mechanical benefit of compounding over time.
Investor A contributed for only 10 years but started earlier. By age 35 the 10.000 has grown to 19.672. That amount then compounds for another 30 years: 19.672 x (1,07)^30 = 149.745. Investor B compounds 10.000 for 30 years: 10.000 x (1,07)^30 = 76.123. Investor A ends up with nearly double despite contributing for one third of the time. The 10-year head start produced a 73.622 advantage.
The real rate of return is not simply 7 minus 3 = 4 percent. The precise calculation uses the Fisher equation: Real rate = (1 + nominal) / (1 + inflation) - 1 = (1,07 / 1,03) - 1 = 0,0388 = 3,88 percent. Over 20 years, 10.000 growing at 7 percent nominal becomes 38.697. In today's purchasing power that 38.697 is worth only 21.435 in real terms. Always distinguish between nominal and real returns when evaluating long-term investments.
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The Rule of 72
The Rule of 72 states: divide 72 by the annual interest rate to get the approximate number of years for money to double.
At 6 percent: 72 / 6 = 12 years to double. At 8 percent: 72 / 8 = 9 years. At 12 percent: 72 / 12 = 6 years.
The mathematical basis: for money to double, (1+r)^n = 2. Taking the natural logarithm of both sides gives n = ln(2) / ln(1+r). For small values of r this approximates to 69,3 / r. The number 72 is used instead of 69,3 because it has more integer divisors, making mental division easier for common rates like 2, 3, 4, 6, 8, 9, and 12.
The rule also works in reverse. To find the rate needed to double money in a target number of years, divide 72 by the years. To double in 9 years you need 8 percent annual return. The Rule of 72 applies equally to inflation — at 3 percent inflation, purchasing power halves in 24 years — and to debt: credit card debt at 20 percent APR doubles in 3,6 years if unpaid.
Growth of 10.000 over 20 years at different annual rates
Annual compounding, no additional contributions. At 10 percent the result is 4,5x larger than at 2 percent, not 5x as linear thinking would predict. The gap widens each year as exponential growth accelerates.
Why time beats contribution size
Most people assume that doubling their savings rate will double their eventual wealth. Under compound interest, starting earlier is more powerful than contributing more.
Two investors both target retirement at age 65. Investor A starts at 25, contributes 3.000 per year for 40 years at 7 percent. Total contributed: 120.000. Final balance: approximately 640.000. Investor B starts at 35, contributes 6.000 per year for 30 years at 7 percent, double the annual amount. Total contributed: 180.000. Final balance: approximately 566.000.
Investor A contributes 60.000 less and ends up with 74.000 more. The 10-year head start is worth more than doubling the contribution.
The reason is the exponent n in the formula. Increasing the principal PV produces linear benefits. Increasing n produces exponential benefits because every additional year multiplies the entire accumulated balance by the growth factor. The 40th year of compounding multiplies a very large number. This is why starting to invest early, even with a small amount, matters more than optimising the contribution size later.
When compound interest works against you
Compound interest is mathematically neutral. It works identically whether you are the lender receiving interest or the borrower paying it. When you carry a credit card balance at 20 percent APR compounded daily, the bank receives compound growth and you pay it.
3.000 on a credit card at 20 percent APR: after 1 year unpaid the balance is approximately 3.664. After 3 years it is approximately 5.465. After 5 years it is approximately 8.144. The balance more than doubles in under 4 years.
A common error is to think that paying the minimum controls the debt. If the minimum payment equals only the monthly interest charge, the principal never decreases and compound interest continues indefinitely. The highest guaranteed return available to most people carrying high-interest debt is paying it off. Paying off a 20 percent APR credit card is mathematically equivalent to a guaranteed 20 percent investment return.
Compound vs simple interest — 10.000 at 5% annually
| Year | Simple Interest Total | Compound Interest Total | Compound Advantage |
|---|---|---|---|
| 1 | 10.500 | 10.500 | 0 |
| 5 | 12.500 | 12.763 | 263 |
| 10 | 15.000 | 16.289 | 1.289 |
| 20 | 20.000 | 26.533 | 6.533 |
| 30 | 25.000 | 43.219 | 18.219 |
| 40 | 30.000 | 70.400 | 40.400 |
Common mistakes that cost real money
Advanced cases and adjustments
10.000 x e^(0,07 x 10) = 20.138 versus 19.672 for annual compounding200 per month added to 10.000 initial at 7 percent for 20 years produces approximately 141.000 total100.000 at -0,5 percent for 5 years: 100.000 x (0,995)^5 = 97.524Methodology and formula sources
All calculations use the standard compound interest formula FV = PV x (1 + r)^n with annual compounding unless otherwise stated. The multi-period formula FV = PV x (1 + r/m)^(m x t) is used where compounding frequency is specified. Real return calculations use the Fisher equation: (1 + real) = (1 + nominal) / (1 + inflation). All arithmetic uses exact values with rounding applied only to final displayed results.
Formula sources: standard financial mathematics as used in CFA curriculum and academic finance textbooks. No results constitute investment or financial advice.
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Formula based on standard mathematical and financial methods. Results are for informational purposes. Last reviewed May 2026. Version 3.