The future value formula is one line of algebra, yet it sits underneath savings accounts, retirement projections, bond math and loan amortization. This page builds it up from the idea of earning interest on interest, explains each symbol, shows the versions you will meet in practice and closes with the mistakes that most often produce wrong answers. The solver above substitutes your own numbers into each step.
The formula, symbol by symbol
- FV — future value: the amount at the end.
- PV — present value: the amount at the start.
- i — interest rate per period, written as a decimal (6% → 0.06).
- n — number of periods the money grows.
- (1 + i)n — the growth factor, also called the future value interest factor (FVIF).
Deriving it from scratch
Start with $1,000 at 6% per period.
- After 1 period: 1,000 + 1,000 × 0.06 = 1,000 × 1.06 = $1,060.00
- After 2 periods: 1,060 × 1.06 = 1,000 × 1.062 = $1,123.60
- After 3 periods: 1,000 × 1.063 = $1,191.02
Each period multiplies the previous balance by the same factor, so after n periods the multiplier is 1.06n. That repeated multiplication — interest earning interest — is what makes compound growth exponential rather than linear. Under simple interest, by contrast, the balance would grow by a flat $60 a period.
Worked example
Using the solver’s default inputs — PV = $1,000, i = 6%, n = 10:
- Growth factor: 1.0610 = 1.790848
- FV = $1,000 × 1.790848 = $1,790.85, of which $790.85 is interest
Add a $100 payment at the start of each period and the future value becomes $1,790.85 + $1,397.16 = $3,188.01, as derived below.
Extending the formula to payments
If you also add a payment PMT at the end of every period, the first payment grows for n − 1 periods, the next for n − 2, and the last for none. Their sum is a geometric series:
The complete equation is therefore:
When payments come at the start of each period (an annuity due), every payment earns one extra period, so the payment term is multiplied by (1 + i). In the example, the annuity factor 13.180795 becomes 13.971643, and $100 × 13.971643 = $1,397.16.
Versions you will meet
| Situation | Formula |
|---|---|
| Single sum, one compounding per period | PV(1 + i)n |
| Annual rate r, compounded m times a year for t years | PV(1 + r/m)mt |
| Continuous compounding | PV × ert |
| Ordinary annuity | PMT × [(1 + i)n − 1] ÷ i |
| Annuity due | PMT × [(1 + i)n − 1] ÷ i × (1 + i) |
| Zero interest | PV + PMT × n |
Common mistakes
- Using a percentage instead of a decimal. (1 + 6)10 is not (1 + 0.06)10.
- Mismatched periods. A 6% annual rate with monthly compounding over 10 years means i = 0.005 and n = 120.
- Forgetting payment timing. Beginning-of-period payments need the extra (1 + i) factor.
- Treating APY as APR. If a rate is already an effective annual yield, compound it once a year.
For full-featured calculations with any compounding frequency, use the future value calculator. The inverse relationship is covered on the present value formula page, and the future value table tabulates (1 + i)n for many rates at once.
This page is for education. Calculations are estimates and are not financial advice.
Frequently asked questions
What is the future value formula?
For a single amount, FV = PV × (1 + i)^n, where PV is the present value, i the interest rate per period as a decimal and n the number of periods. With regular payments added, FV = PV(1 + i)^n + PMT × [(1 + i)^n − 1] ÷ i.
Why is the exponent the number of periods?
Each period multiplies the balance by (1 + i). After one period you have PV(1 + i), after two PV(1 + i)(1 + i), and after n periods the factor has been applied n times — which is (1 + i)^n.
What is the most common mistake with the formula?
Mismatching the rate and the period. If interest compounds monthly, i must be the monthly rate (annual ÷ 12) and n the number of months. Using an annual rate with monthly periods wildly overstates growth.
How does the formula change for continuous compounding?
As compounding becomes infinitely frequent, (1 + r/m)^(m·t) approaches e^(r·t). The continuous version is FV = PV × e^(r·t), with r the annual rate and t in years.
Where does the annuity part of the formula come from?
It is the sum of a geometric series. Payments of PMT grow for n − 1, n − 2, … 0 periods, and adding PMT(1 + i)^k for k = 0 to n − 1 gives PMT × [(1 + i)^n − 1] ÷ i.