Basic compound interest formula
The basic formula is A = P × (1 + r)^t. A is the final amount, P the principal, r the effective rate per period and t the number of periods.
Before substituting numbers, choose one time unit. If r is an effective annual rate, t must count years. If r is a monthly effective rate, t must count months. Express percentages as decimals, so 5% becomes 0.05.
The interest earned is A − P when there are no additional cash flows. The formula gives a nominal future amount unless inflation, fees or taxes are modelled separately.
Worked example
$10,000 growing at an effective 5% a year for ten years gives 10,000 × 1.05^10 = $16,288.95 before fees, inflation and taxes.
The calculation can be checked in stages: 1 + 0.05 = 1.05; 1.05^10 = 1.6288946268; and 10,000 × 1.6288946268 = 16,288.9463. Rounding only the final display gives $16,288.95, while rounding every year would introduce avoidable drift.
For a partial year, convert the rate rather than prorating the annual gain. Eighteen months at a 5% effective annual rate uses 10,000 × 1.05^(18/12), because the exponent and the rate share the annual unit.
Nominal rate and multiple compounding periods
For a nominal annual rate j compounded m times per year, A = P × (1 + j/m)^(m×t). The equivalent effective annual rate is (1 + j/m)^m − 1.
At 12% nominal compounded monthly, j = 0.12 and m = 12. The monthly rate is 1%, and the effective annual rate is 1.01^12 − 1 = 12.6825%. If 12% were already an effective annual rate, its equivalent monthly rate would instead be 1.12^(1/12) − 1, about 0.9489%.
The distinction matters whenever two offers or projections quote different rate conventions. Convert both to the same effective period before comparing them; compounding frequency alone does not make one rate larger when the effective annual rates are already equal.
Formula with recurring contributions
For equal end-of-period deposits, FV = PMT × ((1 + r)^n − 1) / r. Beginning-of-period deposits multiply this result by (1 + r). Capliz uses a period-by-period engine for more complex cases.
Using a 5% effective annual rate, the equivalent monthly rate is about 0.4074%. Depositing $200 at the end of every month for 120 months gives 200 × ((1 + 0.004074)^120 − 1) / 0.004074, or approximately $30,872.63. Beginning-of-month deposits give about $30,998.41 because each deposit receives one more month of growth.
When principal and deposits are combined, calculate the future value of the principal and the deposit series separately, then add them. Changing contributions, irregular withdrawals, annual contribution growth and fees are better handled with a period-by-period simulation than with one closed-form expression.
Errors and edge cases
Do not divide an effective annual rate by twelve and treat it as nominal. At a zero rate, final capital equals principal plus contributions. Capliz rounds for display, not after each simulated period.
The annuity expression contains division by r, so r = 0 must be handled by its limit: FV = PMT × n. Rates at or below −100% per period are not valid in this model, and very long horizons or extreme rates can produce mathematically finite but practically uninformative scenarios.
A calculator result can be reproduced only when every convention matches: rate type, compounding period, deposit amount and frequency, deposit timing, duration, fees and inflation treatment. A difference between two tools often reflects one of those assumptions rather than an arithmetic error.
How to check the result
Keep the rate and time unit consistent, distinguish an effective annual rate from a nominal rate, and state when each contribution is made. A manual result should match the Capliz projection when principal, cash flows, timing, rate type and period are identical.