Equal payments at the end of a period
With no starting capital, deposit C at the end of each of n equal periods and let the period rate be i. The first deposit grows for n−1 periods, the next for n−2 and the last does not grow because valuation is immediately after it arrives. Thus F = C × [1 + (1+i) + … + (1+i)ⁿ⁻¹]. For i ≠ 0, F = C × ((1+i)ⁿ − 1) / i. At zero rate use F = C × n, not division by zero.
Move each payment to the beginning
If the same n payments arrive at the beginning of each period, each participates in one additional accrual. The future value is the previous value multiplied by (1+i). The whole sequence moves; no extra payment appears. Both variants contribute the same amount of personal money. With a constant positive rate, earlier payments produce a larger result in this model, but that mathematical fact is not advice to transfer money before required bills are paid.
Numerical example
All conditions are fictional. Deposit 10,000 roubles for 24 periods at a monthly rate of 0.5%, with no fees, deductions or rate change. End-of-month deposits accumulate to 254,319.55 roubles; beginning-of-month deposits to 255,591.15. Personal contributions are 240,000 in both cases. The difference is 1,271.60, not another full contribution. Values are rounded for display; the calculation does not round the internal balance monthly.
A 24-row table verifies the result. For end-of-period deposits, accrue first and then add the deposit; for beginning deposits, add first and then accrue. A common mistake is to count a beginning deposit and then add 24 monthly deposits, which creates a different problem with more personal money.
When the formula stops being convenient
Missed deposits, changing amounts or unequal periods are better handled as separate cash flows. Equal educational months do not establish the contract’s actual interest-start date.
Present value of the same series
For the previous 24 payments of 10,000 at i = 0.005, present value immediately before the first period is C × (1 − (1+i)^(−n)) / i, approximately 225,628.66 roubles. For beginning-of-period payments, multiply by 1+i, giving approximately 226,756.81. At zero rate both equal the total payments. Here annuity means a cash-flow sequence, not a lifetime-income contract.
Payment frequency and interest frequency
If nominal annual rate j compounds m times and payments arrive p times per year, under a consistent equal-period model the payment-period rate is i = (1 + j/m)^(m/p) − 1. Simply dividing j by p describes another model. Real mismatched dates require separate flows. “Monthly” does not mean payment count automatically equals capitalization count; align periods first.
Perpetuity is a separate assumption
For infinitely many end-of-period payments at constant positive i, present value is C/i. Thus 10,000 per period at i=0.005 gives 2,000,000. This is a geometric-series limit, not a lifetime-payment promise or safe pension amount. At i ≤ 0, a positive infinite series has no such finite value; unknown duration, inflation and risk remain.
Real contracts require the date when the rate base changes. Transfer delays, minimum balances, contribution caps and different rates can change the result. Nominal annual percentages cannot always be divided by months; first identify the measure.
Check an actual plan
Record the number of payments, first and last dates, valuation date and accrual order. Reconcile personal contributions independently of income. Add starting capital as a separate term because it was invested longer. Compare schemes only with equal payment counts and valuation dates. Test the zero-interest case: the formula or table must equal the sum of contributions. Automatic-transfer setup date may differ from income receipt, transfer and savings-account posting dates; record planned and actual dates and ensure an early transfer does not leave the required account short.