Future Value of an Annuity: Payment Timing and Formula

Define the payment stream

An annuity in this calculation is a sequence of equal payments separated by equal time intervals. The future value is the accumulated balance at a specified date under an assumed constant rate. It is not automatically the payout of a particular insurance product. Define the payment amount, number of payments, rate per payment period and timing of each payment. The ordinary-annuity formula assumes payments occur at the end of each period, while an annuity due places them at the beginning. One extra period of growth changes the result.

Use the ordinary-annuity formula

For payment M, periodic rate i and n end-of-period payments, future value is FV=M[((1+i)^n-1)/i], provided i is not zero. For i=0, the result is simply Mn. Each payment earns interest for a different number of periods: the last payment has no growth before the endpoint, while the first grows for n-1 periods. The formula is the sum of that geometric sequence. This timing explains why multiplying the total contributions by a lump-sum compound factor overstates the balance.

Verify three annual payments directly

Suppose 1,000 is paid at the end of each year for three years and the assumed annual rate is 5%. At the end of year three, the first payment is worth 1,102.50, the second 1,050 and the third 1,000. Their sum is 3,152.50. The formula gives 1,000[(1.05^3-1)/0.05]=3,152.50. Total contributions are 3,000 and modeled growth is 152.50. This small schedule provides an independent verification of both the formula and the endpoint convention before using a much longer horizon.

Adjust for beginning-of-period payments

For an annuity due with the same rate and number of payments, multiply the ordinary future value by 1+i. In the example, beginning-of-year payments accumulate to 3,152.50×1.05=3,310.125, or 3,310.13 after currency rounding. The extra amount is not caused by an extra payment; all three payments simply earn one additional year's growth. A contribution on the first day of a month and one on the last day are different schedules. State the assumption rather than using the more favorable result without explanation.

Match the rate to the payment frequency

For monthly payments, n counts months and i must be a monthly rate. A nominal annual rate of 12% compounded monthly gives i=0.12/12=0.01. An effective annual rate of 12% instead corresponds to a monthly rate of 1.12^(1/12)-1, approximately 0.9489%. Dividing the effective annual rate by twelve would not preserve the stated annual growth. Likewise, five years of monthly payments means sixty payments, not five. Keep rate convention and payment frequency alongside the calculation to make comparisons meaningful.

Separate existing savings and changing contributions

An initial balance P held for the full n periods contributes P(1+i)^n in addition to the annuity future value. Do not include it again as the first recurring payment unless a real payment is being made then. If contributions increase over time, the level-payment formula no longer describes the full stream. Calculate each dated payment separately or use a suitable growing-annuity model with its own assumptions. Fees, taxes and variable investment returns also need explicit treatment before using the modeled balance as a financial planning estimate.

Read the linked calculator's period convention

The site's Annuity Calculator uses equal annual payments with an annual rate and a number of years. Enter 1,000, 5 and 3 for the worked ordinary-annuity check. It reports future value and total contributions under that convention; it does not automatically turn an annual input into monthly deposits. For a monthly schedule, use a spreadsheet with the matched monthly rate and payment count. Save the assumed rate, timing and horizon with the result. A fixed-rate arithmetic projection is a scenario, not a promise of investment performance or a quote for an annuity contract.

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