The Compound Interest Formula
One formula, two pieces: a growth factor for money you already have, and an annuity term for money you keep adding. Work through both once and every compound-growth number — including everything our calculator reports — becomes checkable by hand.
Piece 1: The Lump Sum
| Formula | A = P × (1 + r/n)n·t |
|---|---|
| Pieces | P principal · r annual rate (decimal) · n periods/year · t years |
Worked example: 10,000 at 7% compounded monthly for 10 years. The periodic rate is 0.07 ÷ 12 = 0.005833; the number of periods 12 × 10 = 120. So A = 10,000 × (1.005833)120 = 10,000 × 2.0097 ≈ 20,097. The money doubles in almost exactly ten years — which the Rule of 72 predicts in one division.
Piece 2: Monthly Contributions
| Formula | FV = PMT × ((1 + i)m − 1) ÷ i |
|---|---|
| Pieces | PMT monthly deposit · i monthly rate · m months (end-of-month deposits) |
Worked example: 500 per month under the same conditions: FV = 500 × ((1.005833)120 − 1) ÷ 0.005833 = 500 × 173.08 ≈ 86,542. Total with the lump sum: 20,097 + 86,542 ≈ 106,639 — of which 70,000 is deposits and about 36,639 is interest. Deposits made at the beginning of each month multiply the annuity term by one extra (1 + i), lifting the total to about 107,144.
Nominal vs Effective, and Other Trip Wires
A quoted "7% compounded monthly" is a nominal rate; the effective annual yield is (1 + 0.07/12)12 − 1 ≈ 7.23%. Mixing the two — or rounding intermediate factors — is why manual results drift from calculator results. Keep every quantity in one period system (months, usually), carry full precision to the end, and remember the output is nominal money: inflation quietly shrinks what it buys. For choosing inputs sensibly rather than computing them, see the calculator guide.
Frequently Asked Questions
What is the compound interest formula?
A = P(1 + r/n)^(nt): principal P grows by the periodic rate r/n over n×t periods. With regular monthly
deposits, an annuity term PMT × ((1+i)^m − 1)/i is added on top.
What is the difference between nominal and effective rates?
The nominal rate is the quoted annual figure; the effective rate is what you actually earn after compounding
within the year. 7% compounded monthly is an effective 7.23% per year.
Why does my manual calculation differ from the calculator?
Usually rounding at intermediate steps or mixing periods — an annual rate with monthly deposits needs the
monthly effective rate, not r/12 of the effective annual figure. Keep everything in one period system.