The compound interest formula, explained term by term
The compound interest formula is FV = P(1 + r/n)^(nt): future value equals the principal multiplied by one plus the periodic rate, raised to the power of the total number of compounding periods. Every symbol in it does a specific job, and the formula only works if you get each one right. This guide takes it apart piece by piece, then rebuilds it with a real example.
The formula, term by term
P is the principal — the amount you start with. If you open an account with €1,000, P is 1,000. It never changes within the formula; it’s the base everything else multiplies against.
r is the annual rate, written as a decimal. A rate of 5% enters the formula as 0.05, not 5. This single conversion is where most calculation errors begin, and it’s worth burning into memory before you touch anything else.
n is the number of compounding periods per year. Interest paid annually has n = 1. Paid monthly, n = 12. Paid daily, n = 365. This number tells the formula how finely the year gets sliced.
t is the number of years the money is left to grow. It measures time in whole years, not months or periods — that distinction matters later.
FV is the future value — what your principal has become once compounding has done its work. It’s the answer the formula is built to produce.
The exponent nt is where the formula earns its name. It’s the total count of compounding periods across the whole term — not years, not compounding frequency alone, but the two multiplied together. Multiply r/n (the interest rate applied at each single period) by itself, compounded, nt times, and you get exponential growth rather than a flat, one-off calculation. That exponent is the entire reason compound interest outpaces simple interest: each period’s interest gets added to the balance, and the next period’s interest is calculated on that larger balance too. The formula doesn’t just add growth — it multiplies it, period after period.
A worked example
Take €1,000 (P = 1,000) at an annual rate of 5% (r = 0.05), compounded monthly (n = 12), left to grow for 10 years (t = 10).
Start with r/n: 0.05 divided by 12 gives the rate applied in any single month — a small fraction of the annual 5%, because only a twelfth of the year has passed. Add 1 to get the growth factor for one month: 1 plus that fraction.
Next comes nt: 12 compounding periods per year multiplied by 10 years gives 120 — the total number of times interest gets calculated and added over the life of the deposit.
The formula then raises that monthly growth factor to the power of 120, and multiplies the result by the principal. Run the whole calculation — €1,000 growing at 5% annually, compounded monthly, over 10 years — and it becomes €1,647.01.
Notice what did the work here. The rate itself never changed; 5% is 5% throughout. What changed the outcome was compounding it monthly rather than annually, and giving it ten years — 120 periods — to stack.
The classic mistakes
Using 5 instead of 0.05 for the rate. The formula wants a decimal. Typing the whole number produces a result so large it’s obviously wrong, but it’s an easy slip when moving from a percentage sign to a calculator.
Mixing months and years in t. t counts years, full stop. If your compounding is monthly, that frequency already lives in n — don’t also convert t into months, or you’ll double-count the time period and badly overstate the exponent.
Forgetting the whole bracket is raised to the power. The exponent nt applies to everything inside the parentheses — (1 + r/n) as a single unit — not to r/n on its own and not to the principal. Raise the wrong part of the expression and the whole result collapses.
What the formula can’t tell you
FV = P(1 + r/n)^(nt) assumes a single lump sum left untouched from start to finish. It has nothing to say about regular contributions — a monthly deposit added on top of the principal needs a second formula, one built to sum the future value of a whole series of payments rather than one starting balance. Most savings and pension calculators handle both calculations behind the scenes, blending the lump-sum formula with the contributions formula so you never have to run them separately. If you’re weighing how compounding frequency affects your own return, or thinking in terms of the Rule of 72 as a rough mental shortcut for how long money takes to double, a calculator that does both jobs at once will get you a realistic answer faster than working the formula by hand.