What is the compound interest formula?

The compound interest formula is A = P(1 + r/n)^(nt), where A is the final amount, P is the principal (initial investment), r is the annual interest rate as a decimal, n is the number of times interest compounds per year, and t is the number of years. For example, £10,000 at 7% compounded monthly for 20 years: A = 10000(1 + 0.07/12)^(12×20) = £40,188.

What is the Rule of 72?

The Rule of 72 is a quick mental calculation to estimate how long it takes to double an investment. Divide 72 by the annual interest rate: 72 ÷ 7% = approximately 10.3 years to double. It works because of logarithmic properties of exponential growth and is accurate to within a year for rates between 4% and 12%. At 8% annual growth, money doubles roughly every 9 years.

How does compounding frequency affect returns?

More frequent compounding produces higher returns. £10,000 at 10% for 10 years: annually = £25,937; monthly = £27,070; daily = £27,179. The difference between monthly and daily compounding is small, but the difference between annual and monthly compounding is significant for large amounts or long time horizons. Most savings accounts compound daily; most mortgages compound monthly.

What is the difference between APR and APY?

APR (Annual Percentage Rate) is the stated interest rate for a year without accounting for compounding within the year. APY (Annual Percentage Yield) reflects actual returns after compounding — it is always higher than APR unless compounding happens annually. A savings account advertising 5% APR compounded monthly actually yields 5.12% APY. Always compare APY when evaluating savings accounts.

Compound Interest Explained — With a Free Calculator and Real Examples

The Difference Between Simple and Compound Interest

Simple interest is calculated only on the original principal. If you deposit $1,000 at 5% simple interest for 10 years, you earn $50 per year — $500 total, ending with $1,500. Compound interest is calculated on the principal plus the accumulated interest from prior periods. At 5% compounded annually for 10 years, you earn $628.89 in interest — ending with $1,628.89. The difference of $128.89 is the effect of compounding: earning interest on your interest. Over longer periods and higher rates, this gap widens dramatically. At 10% compounded annually over 30 years, $1,000 grows to $17,449 — nearly 10 times more than the $4,000 simple interest would produce.

The Compound Interest Formula

The standard formula is: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal (starting amount), r is the annual interest rate expressed as a decimal (e.g., 0.05 for 5%), n is the number of compounding periods per year (1 = annual, 12 = monthly, 365 = daily), and t is the time in years. For example: $5,000 at 7% compounded monthly for 20 years: A = 5000 × (1 + 0.07/12)^(12×20) = 5000 × (1.005833...)^240 ≈ $19,898. The allio.tools/tools/calculators/compound-interest-calculator/ handles this calculation instantly and lets you model different rates, periods, and compounding frequencies.

Why Compounding Frequency Matters

The more frequently interest compounds, the more you earn. The difference between annual and monthly compounding is meaningful, while the difference between monthly and daily is small. Example: $10,000 at 6% for 10 years. Annual compounding: $17,908. Monthly compounding: $18,194. Daily compounding: $18,221. The gap from annual to monthly ($286) is significant. The gap from monthly to daily ($27) is negligible for most purposes. Most savings accounts compound monthly or daily. Most investment calculations use annual compounding as a simplification.

The Rule of 72 — A Mental Math Shortcut

The Rule of 72 lets you estimate how long it takes to double your money at a given interest rate: divide 72 by the annual interest rate. At 6%: 72 / 6 = 12 years to double. At 8%: 72 / 8 = 9 years. At 10%: 72 / 10 = 7.2 years. The rule also works in reverse: if you want to double your money in 6 years, you need 72 / 6 = 12% annual return. The Rule of 72 is an approximation — most accurate for rates between 6–10%. For precise calculations, use the compound interest formula or the allio.tools calculator.

Compounding Works Against You With Debt

The same mathematics that grows your savings also grows your debt. A $5,000 credit card balance at 22% APR, compounded monthly, grows to approximately $46,000 after 10 years if you make no payments. Mortgage debt compounds more slowly (typical rate 6–7%), but the larger principal makes the total interest cost significant. A $300,000 mortgage at 6.5% over 30 years results in total interest payments of approximately $382,000 — more than the original loan amount. Use the allio.tools/tools/calculators/mortgage-calculator/ to see the exact amortisation schedule for your loan.

How to Use the Free Compound Interest Calculator

The Compound Interest Calculator at allio.tools/tools/calculators/compound-interest-calculator/ lets you: set a starting principal and optional recurring contribution (monthly or annual), choose interest rate and compounding frequency, select a time period, and view the breakdown of principal, contributions, and interest earned. You can also model different scenarios by changing one variable at a time — comparing, for example, what happens if you save $200/month vs $500/month over 30 years. No sign-up is required and all calculations happen instantly in your browser.

The Most Important Takeaway

Time is the most powerful variable in the compound interest formula. Starting early matters more than earning a higher rate. A 25-year-old who invests $3,000/year at 7% for 10 years (total contribution: $30,000) then stops, will end up with more at age 65 than a 35-year-old who invests $3,000/year for 30 years (total contribution: $90,000) at the same rate. The first investor contributed a third as much but started 10 years earlier — compounding did the rest. The lesson: the best time to start is always as early as possible, even with small amounts.