- A — the final amount after t years.
- P — the principal: the amount you start with.
- r — the annual interest rate as a decimal (5% = 0.05).
- n — how many times interest compounds per year (1 = yearly, 12 = monthly, 365 = daily).
- t — time in years.
The engine inside the formula is the term (1 + r/n): the growth factor for one compounding period. Raising it to the power nt applies that growth once per period, for every period — which is precisely what "interest on interest" means in symbols.
Worked example 1: a lump sum
$1,000 at 5%, compounded monthly, for 10 years. Here P = 1000, r = 0.05, n = 12, t = 10:
Of that final amount, $647.01 is interest — earned without a single additional deposit, purely because each month's interest joined the balance and earned interest of its own.
Adding monthly contributions
Regular deposits need a second formula — the future value of an annuity. If C is deposited at the end of every month and j is the monthly rate, the deposits accumulate to:
When compounding is monthly, j is simply r/12. For other frequencies this site first converts to the equivalent monthly rate, j = (1 + r/n)n/12 − 1, so the two parts of the calculation always agree with each other. The total balance is the grown principal plus the grown deposits.
Worked example 2: deposits only
$200 a month at 6%, compounded monthly, for 20 years, starting from $0. Here C = 200, j = 0.005, m = 240:
Total deposits were $48,000.00; the remaining $44,408.18 is interest. Note the convention: deposits land at the end of each month (an "ordinary annuity"). Deposits at the start of each month would each earn one extra month of interest and finish slightly higher.
The APY conversion
Rates with different compounding frequencies are not directly comparable, so they are converted to one effective annual rate — the APY:
A nominal 5% compounded monthly has an APY of about 5.12%: over one year, $1 grows to $1.0512 rather than $1.05. Every calculator result on this site shows the APY alongside the balance so frequencies can be compared honestly.
Continuous compounding
Push n towards infinity and (1 + r/n)nt converges to ert, giving A = P·ert. The difference from daily compounding is tiny at ordinary rates and time spans, which is why daily is the most frequent option the calculators offer.
Frequently asked questions
Why is the rate divided by n inside the bracket?
Because the annual rate is shared across the year's compounding periods. Monthly compounding at a 12% nominal rate means 1% is applied each month — twelve small multiplications that together beat one 12% multiplication, which is the whole effect.
What if the rate is 0%?
The compound formula returns exactly P, and the annuity formula is replaced by plain addition (C × m) — the calculators on this site handle both cases.
Which formula do banks use?
Deposit accounts typically credit interest on a stated schedule (often daily accrual, monthly crediting) using the same mathematics; the details are in each account's terms. This site presents the standard textbook model.
Where can I apply the formula without doing the algebra?
The compound interest calculator runs it with a year-by-year table, and the savings goal calculator runs it backwards to find a required monthly deposit.