Future value of a recurring deposit stream

A balance built from deposits alone

This is the recurring-deposit account in its textbook form: an ordinary annuity, with each deposit arriving at the end of its period and earning only from then on. The last deposit earns nothing at all. What distinguishes it from every lump-sum page is that there is no opening balance to compound — the deposits carry the entire outcome.

Over modest spans the balance is nearly all deposits. Interest needs time between a deposit and the horizon to do anything, and in a schedule most deposits are recent, so the margin above the plain sum of deposits starts thin and only fattens as the count grows. The pack’s vectors show a year of monthly deposits finishing barely above what was paid in — a sobering and accurate picture of the early years of any saving habit.

Time here is counted in deposits, not years, and the cadence is a real input. The annual rate is split across the year by the deposit frequency, and deposits are assumed to land on the same schedule the interest credits. A monthly habit and a quarterly habit are genuinely different schedules — the pack’s vectors carry both — not one figure rescaled.

The closed form this page runs divides by the per-period rate, so it needs a rate that is actually positive. At a rate of nothing the instrument refuses rather than improvising: an interest-free pile of deposits is plain multiplication and does not need an annuity formula. The refusal is the calculator being honest about its own machinery.

When this calculation is used

• Projecting a recurring-deposit account or standing-order savings plan to its final balance.
• Seeing how much of an accumulated balance a payroll deduction would owe to the deposits themselves versus the interest between them.
• Comparing cadences — the same yearly outlay as a monthly habit versus a quarterly one — on the schedule each actually follows.
• Setting expectations for the early years of a savings habit, before compounding has material help to offer.

Worked example

A steady monthly deposit at an everyday annual rate, kept up without a gap for a few years of deposits.

The future value lands above the plain sum of everything deposited, and the margin between the two is the interest — thin relative to the deposits at this span, because most of the schedule is too recent to have earned much. The habit, not the rate, is doing the heavy lifting at this stage.

Re-run the same yearly outlay as a quarterly habit — fewer, larger deposits on a coarser schedule — and the machinery is identical while the answer shifts, because cadence is a real input rather than a display choice; the signed pack’s declared vectors exercise the monthly and the quarterly forms alike. All figures are computed by the verified engine at load — the page holds no numbers of its own.

What each input represents

Assumptions and limits

• Every deposit is identical and lands at the end of its period — the ordinary-annuity convention, with no beginning-of-period option here.
• Deposits and interest crediting share one cadence; a mismatch between the two is outside this closed form.
• The rate is constant across the whole schedule and stated before tax.
• No deposit is missed, no withdrawal is made, and nothing is added outside the schedule.
• The account starts empty: an existing balance belongs to a lump-sum calculation, not this one.

What the guards protect against

• An annual rate at or below zero is refused. The closed form divides by the per-period rate, so a rate of nothing has no formula behind it — the pack declares this refusal as a vector, and the no-interest case is left to plain multiplication where it belongs.
• A deposit count of zero or below is refused: with no deposits there is no schedule to accumulate.
• The deposit amount must be positive and is bounded above, and the cadence is confined to schedules between annual and daily, keeping every input a schedule a bank could offer.

Provenance

Ordinary-annuity future-value closed form

The annual rate is divided by the cadence to give the per-period rate; the deposit is then multiplied by the annuity accumulation factor — the compound factor over the deposit count, less one, over the per-period rate. · The same closed form as given in standard investments and fixed-income texts.

An educational reference on deposit-schedule arithmetic — not financial advice and not a projection of any real account’s terms. The signed pack carries its own citation; the page reports the verification state of the release it mounted rather than asserting one.