Compound Interest Formula, With Examples and the Rule of 72

What is the compound interest formula?

The compound interest formula is A = P(1 + r/n)^(nt). It tells you what a single deposit grows to when the interest it earns is added back to the balance and starts earning interest too. Compound interest is interest calculated on the principal plus all interest already credited, as opposed to simple interest, which is always figured on the original principal only.

Each piece of the formula has a specific job. The term r/n is the rate charged per compounding period, so 6% compounded monthly is 0.06 ÷ 12 = 0.005, or 0.5% a month. The exponent nt counts how many periods pass: 12 periods a year for 10 years is 120. Raising (1 + r/n) to that power stacks up all the periods of growth.

If you want the interest alone rather than the ending balance, use I = P(1 + r/n)^(nt) − P. You can run any of these numbers in our interest calculator, which also handles regular deposits, taxes and inflation.

• A: ending balance (principal plus interest)
• P: principal, the amount you start with
• r: nominal annual interest rate, written as a decimal (6% = 0.06)
• n: compounding periods per year (1 yearly, 4 quarterly, 12 monthly, 365 daily)
• t: time in years

How do you calculate compound interest step by step?

Convert the rate to a decimal, divide it by the number of compounding periods, add 1, raise the result to the total number of periods, and multiply by the principal. Here is the full calculation for $10,000 invested at 6% compounded monthly for 10 years.

Step 1: r/n = 0.06 ÷ 12 = 0.005. Step 2: 1 + 0.005 = 1.005. Step 3: nt = 12 × 10 = 120 periods. Step 4: 1.005^120 ≈ 1.8194. Step 5: $10,000 × 1.8194 = $18,193.97. The interest earned is $18,193.97 − $10,000 = $8,193.97.

Compare that with simple interest on the same money. Simple interest uses I = P × r × t, so $10,000 × 0.06 × 10 = $6,000 of interest and an ending balance of $16,000. Compounding adds $2,193.97 more over the decade, and the gap widens every year after that because each year's interest is calculated on a bigger base.

How does compounding frequency change the result?

More frequent compounding produces slightly more interest because each credit starts earning sooner, but the extra amount shrinks quickly as frequency rises. The table below shows $10,000 at a 6% nominal rate for 10 years with no added deposits, as calculated by our interest calculator. The APY column is the effective annual rate: the actual one-year growth after compounding, found with APY = (1 + r/n)^n − 1.

Going from yearly to monthly compounding adds about $285 over 10 years. Going from monthly to daily adds only about $26 more, and continuous compounding, the theoretical limit where A = Pe^(rt), adds less than a dollar beyond daily. When you compare savings accounts or CDs, compare their APYs; US banks are required to disclose APY under the Truth in Savings rules, and it already accounts for compounding. Our compound interest calculator converts any rate between frequencies so you can line up offers quoted different ways.

How much does time change compound growth?

Time has a much larger effect than compounding frequency, because the exponent nt grows with every year you leave the money alone. The same $10,000 at 6% compounded monthly grows as follows, with no further deposits.

Notice that the balance gains about $8,200 in the first 10 years but about $27,100 in the third decade. Nothing about the rate changed; the base that the rate applies to simply got larger. This is the main reason starting early matters so much for retirement saving, which our guide on how much to save for retirement covers in detail.

How do you calculate compound interest with monthly contributions?

When you add money regularly, each deposit grows by the same formula from the date it goes in, so the total is the lump-sum result plus the future value of the series of deposits. For deposits made at the end of each period, that second part is PMT × [((1 + r/n)^(nt) − 1) ÷ (r/n)], where PMT is the deposit per period.

Example: start with $10,000, add $200 at the end of every month, and earn 6% compounded monthly for 20 years. The starting $10,000 grows to $33,102.04. The 240 deposits of $200 total $48,000 and grow to $92,408.18. Together the account holds $125,510.22, of which $58,000 is money you put in and $67,510.22 is interest.

The formula assumes the deposit schedule matches the compounding schedule. When they differ, such as yearly deposits into an account that compounds daily, it is easier to let a calculator step through each period. The savings calculator handles deposits that rise each year, and the future value calculator solves for the ending balance of any payment stream.

What is the rule of 72?

The rule of 72 says that money compounding at a fixed annual rate doubles in roughly 72 ÷ rate years. At 6% a year, that is 72 ÷ 6 = 12 years; at 9%, 8 years. It also works backward: to double your money in 10 years, you need about 72 ÷ 10 = 7.2% a year.

The exact doubling time for annual compounding is ln(2) ÷ ln(1 + r). The rule of 72 is a mental-math stand-in for that expression, and the table shows how close it gets. It runs a little long at low rates and a little short at high ones, and it is almost exact near 8%. Some people use 70 or 69 instead of 72 for low rates or continuous compounding, but 72 is popular because it divides evenly by 2, 3, 4, 6, 8, 9 and 12.

The same shortcut shows how fast costs grow against you. Credit card debt at 24% APR doubles in about three years if nothing is paid, and prices rising at 3% inflation double in about 24 years, halving what a fixed sum can buy.

What mistakes do people make with the compound interest formula?

The most common error is using the rate as a whole number. Plugging in 6 instead of 0.06 turns a reasonable answer into an absurd one. The second is mixing time units: if t is in years, n must be periods per year, so 18 months is t = 1.5, not 18.

Another mistake is treating an investment's average return as if it were a guaranteed interest rate. The formula assumes the same rate every period. Stock and bond returns vary from year to year, so a projection at 7% is an estimate, not a promise. Finally, the formula ignores taxes and inflation. Interest in a taxable account is usually taxed each year, which lowers the effective growth rate, and inflation reduces what the ending balance buys. Our interest calculator lets you enter both to see an after-tax, inflation-adjusted figure.

How is compound interest different for loans?

The same formula applies to debt, but it works against you. A balance you don't pay grows at the loan's rate, and interest charged on unpaid interest is what makes credit card balances climb. Lenders in the US quote an APR, which for most credit cards and loans is a nominal rate that does not show the effect of compounding, so the effective cost is a bit higher than the stated APR. Our guide to APR vs APY explains how to convert one to the other.

Installment loans such as mortgages and car loans are amortized: each payment covers that month's interest first and the rest reduces the principal, so interest is charged only on the remaining balance and does not snowball as long as you pay on schedule.