How this instrument works
Set money aside every period at a fixed rate and by the target date it has become a single lump sum — that lump sum is what this instrument solves for. A corporate treasurer funding the future retirement of a bond issue, a condo board budgeting toward a roof replacement due in ten years, or a manufacturer reserving cash to replace a machine on a known schedule all run the identical calculation: a level Payment per period, $ compounding at Rate per period, % across a fixed Number of periods, summed into one Future value of the annuity.
Write out the growth factor each deposit ultimately carries — the earliest payment survives n minus 1 compounding periods, the next survives n minus 2, and so on down to the final payment, which survives zero. Sum those factors: S equals (1+r) to the (n−1) plus (1+r) to the (n−2) and onward down to (1+r) plus 1. Multiply that whole sum by (1+r) and subtract the original S from it, and every middle term cancels, leaving (1+r)·S minus S equal to (1+r)ⁿ minus 1. Divide both sides by r and S itself equals ((1+r)ⁿ − 1) ⁄ r — the entire closed form falls out of one subtraction, which is why it only works when every payment and every rate stay identical from the first period to the last.
Nothing in Number of periods requires it to count months: it might tally annual bond-coupon reserve deposits, quarterly contributions approved by a board, or fifty-two weekly payroll deductions, provided Rate per period, % is measured on that identical clock. The result assumes one rate held constant across every period, a deposit made without exception on every single period, and it leaves out fees, taxes, and inflation altogether — it is the plain arithmetic that several narrower calculators on this site each dress up with their own account rules and defaults.
- Enter the level amount set aside at the end of every period into Payment per period, $.
- Set Rate per period, % to what that money actually earns each period — match its length to Number of periods.
- Type how many deposits the plan will make into Number of periods.
- Read Future value of the annuity for the lump sum sitting in the account right after the last deposit lands.
- Raise Number of periods rather than Rate per period, % to see which one moves the total further — periods sit inside the exponent.
Worked example — $500 every period for ten periods
Set Payment per period, $ to 500, Rate per period, % to 6, and Number of periods to 10 — a $500 deposit at the end of each of ten periods, each period crediting 6% on whatever balance is already on deposit. Future value of the annuity reads $6,590.40, the exact balance the instant the tenth deposit posts.
Only $5,000 of that total was ever paid in — ten deposits of $500 — so $1,590.40 came from compounding alone. Extend Number of periods to 20 without touching anything else and the balance does not merely double toward roughly $13,180; it reaches $18,392.80, because ten additional periods compound on a balance that already includes all the growth before it. Double Payment per period, $ back at 10 periods instead, and the result scales in exact proportion, to $13,180.79 — the formula is linear in payment but exponential in period count.
Questions
What is a sinking fund, and why does it use this formula?
A sinking fund is a schedule of level deposits set aside to reach a known sum by a target date — a bond issuer setting cash aside every year to redeem principal at maturity, or a business budgeting for a machine it will replace on a fixed schedule. This formula sizes the fund's ending balance exactly: Payment per period, $ compounded once for every period it has sat on deposit, summed across Number of periods.
Can Number of periods represent something other than months?
Yes — the formula carries no built-in unit. Number of periods might count annual bond-coupon deposits, quarterly board contributions, or weekly payroll deductions; the only requirement is that Rate per period, % is measured on that same interval. A treasurer reserving cash once a year for a bond due in ten years enters 10, not 120, and uses the annual rate directly.
Why does doubling Number of periods more than double the future value?
Because period count sits inside an exponent while payment sits outside one. At 6% per period, $500 deposited for 10 periods reaches $6,590.40; stretching the same deposit to 20 periods does not reach roughly $13,180 — it reaches $18,392.80, since every added period compounds on a balance that already includes all the growth before it. Doubling Payment per period, $ instead, at 10 periods, exactly doubles the result to $13,180.79.
How is this different from compounding a single lump sum?
A single lump sum earns compound interest under future value equals present value times (1 + r) to the n, credited with the full n periods of growth in one shot. This calculator instead sums a whole series of separate deposits, each compounding for a different number of remaining periods — the first earns nearly the full term, the last earns none. Applying the lump-sum formula to a repeated deposit overstates what a real savings plan reaches.
Does the deposit have to land at the end of every period?
This sheet assumes an ordinary annuity — each Payment per period, $ posts at the close of its period, so the very last deposit earns zero interest before Future value of the annuity is read. If deposits instead land at the start of each period, every one of them earns one additional period of compounding, and the true balance equals this result multiplied by (1 + Rate per period, % ÷ 100).
What does this arithmetic leave out?
A single unchanging rate applied to every period, a deposit made without exception on schedule, and no costs deducted anywhere. Real sinking funds sit in accounts whose credited rate moves, real treasurers occasionally skip or delay a contribution, and management fees, taxes on interest earned, and inflation all reduce what the fund is worth by the target date. Treat the output as the deterministic target a level, uninterrupted, cost-free plan would hit — a benchmark, not a forecast.
References
- SEC Investor.gov — Annuities
- Federal Reserve — H.15 selected interest rates
- IRS Topic no. 410 — Pensions and annuities
Read this first: This instrument shows arithmetic, not advice. Real offers add fees, taxes and terms that vary by lender and place — verify the figures against your actual paperwork before deciding anything.