Formula 1: lump sum future value
FV = PV x (1 + r)^n
PV is what you have now. r is the rate per compounding period. n is the total number of periods. The expression (1 + r)^n is the growth factor: how many times your money multiplies.
Worked example. $10,000 at 7 percent, compounded annually, for 10 years. r = 0.07, n = 10. FV = 10,000 x 1.07^10 = 10,000 x 1.967151 = $19,671.51. Your money nearly doubles, which lines up with the rule of 72: 72 / 7 = 10.3 years to double.
With compounding frequency. The formula does not change, but n and r do. At 7 percent compounded quarterly for 10 years, r = 0.07 / 4 = 0.0175 per quarter and n = 40 quarters. FV = 10,000 x 1.0175^40 = $20,015.97. More frequent compounding raises the total because each quarter's interest starts earning sooner.
Formula 2: future value of an annuity (monthly deposits)
FV = PMT x (((1 + i)^n - 1) / i)
PMT is the deposit each period, i is the rate per period, and n is the number of deposits. This is the ordinary annuity formula, which assumes deposits land at the end of each period. The bracketed term is the future value interest factor of an annuity: how $1 deposited each period grows.
Worked example. $500 a month at a 7 percent nominal annual rate for 10 years. i = 0.07 / 12 = 0.0058333 per month, n = 120 deposits. FV = 500 x ((1.0058333^120 - 1) / 0.0058333) = 500 x 173.0848 = $86,542.40. Your $60,000 of deposits grew by $26,542.40 of interest.
The most common mistake is using the annual rate as i while counting months for n. If deposits are monthly, the rate must be monthly too. Dividing the nominal annual rate by 12 is the standard convention used by loan and savings calculators.
Formula 3: annuity due (deposits at the start of each period)
FV = PMT x (((1 + i)^n - 1) / i) x (1 + i)
When deposits arrive at the beginning of each period, as with rent-style or first-of-month savings transfers, every payment earns one extra period of interest. That is what the trailing (1 + i) factor does.
Worked example. The same $500 a month at 7 percent nominal for 10 years, but deposited on the first of each month instead of the last. FV = 86,542.40 x 1.0058333 = $87,047.23. The difference is $504.83 over a decade, about one extra monthly deposit. Small timing details add up when compounding gets hold of them.
Putting them together: lump sum plus deposits
Most real plans have both a starting balance and ongoing deposits. The two parts grow independently, so just add them. Start with $10,000 and add $500 a month at 7 percent nominal, monthly compounding, for 10 years:
| Part | Formula | Future value |
|---|---|---|
| Starting $10,000 (lump sum) | 10,000 x 1.0058333^120 | $20,096.61 |
| $500/month (annuity) | 500 x ((1.0058333^120 - 1) / 0.0058333) | $86,542.40 |
| Total | $106,639.01 |
Total deposits were $10,000 + $60,000 = $70,000, so compounding contributed $36,639.01. This combined calculation is exactly what the future value calculator on this site runs.
Quick reference
| Situation | Formula |
|---|---|
| Single deposit | FV = PV x (1 + r)^n |
| Deposits at end of each period | FV = PMT x (((1 + i)^n - 1) / i) |
| Deposits at start of each period | FV = PMT x (((1 + i)^n - 1) / i) x (1 + i) |
| Doubling time estimate | Years = 72 / annual rate |
| Effective annual rate | EAR = (1 + nominal / m)^m - 1 |
On m: it is the number of compounding periods per year. A nominal 7 percent compounded monthly gives an effective annual rate of (1 + 0.07/12)^12 - 1 = 7.229 percent. Use effective rates whenever you compare products with different frequencies.