What the required contribution really represents
Every savings plan is one equation with five quantities in it: a starting balance, a contribution, a return, a length of time, and an ending balance. Fix any four and the fifth is determined. Most calculators fix the contribution and report the ending balance. This one does the useful inversion — you fix the ending balance and it reports the contribution.
That inversion is what turns a wish into a budget line. "I want a million dollars" is not actionable; "$820 a month for 30 years at 7%" is, because you can compare it against what is left after your rent. It is also the number a financial planner solves for first, and it is the same calculation an accountant runs when funding a sinking fund to retire a bond issue or replace a roof.
Two components combine to hit the target. Money you already hold compounds on its own for the whole horizon and needs no help from you; that is reported as what today's balance grows to. Whatever gap remains is what the contributions must fill, and the annuity factor converts that gap into a per-period payment. Splitting the answer this way matters because the two components behave very differently: an existing balance benefits from the full run of compounding, while a contribution made in the final year barely compounds at all.
The annuity formula, and why it inverts cleanly
Work forwards first. A balance PV left alone for n periods at periodic rate i becomes PV(1 + i)n. A stream of equal payments PMT made at the end of each of those periods becomes PMT × [((1 + i)n − 1) ÷ i]. The bracketed term is the future-value annuity factor: it is simply the sum of the geometric series 1 + (1+i) + (1+i)² + … + (1+i)n−1, because the first payment compounds for n − 1 periods, the second for n − 2, and the last for none at all.
Add the two and set the sum equal to your target:
FV = PV(1+i)^n + PMT · ((1+i)^n − 1) / i
Because PMT appears linearly and only once, solving for it is a single rearrangement — no iteration, no root-finding. Subtract the compounded starting balance from the target to get the gap, then divide by the annuity factor.
Three details decide whether your answer matches your bank statement. First, the periodic rate is the annual rate divided by the number of contributions per year. That is the market convention for a nominal quoted rate, and it is what every mainstream planning tool uses; it is not the same as compounding the effective annual rate down, which would give a slightly smaller periodic figure. Second, timing: if you pay on the first of the month, every payment earns one extra period of growth, so the required amount falls by a factor of exactly (1 + i). Third, the zero-rate case: at i = 0 the annuity factor is undefined by the formula but has a perfectly good limit of n, so the payment becomes the gap spread evenly. The calculator handles that branch explicitly instead of dividing by zero.
Worked example: a million dollars in 30 years at 7%
You start with nothing, you have 30 years, and you assume a 7% nominal annual return with monthly contributions made at month end.
- Periodic rate. i = 7% ÷ 12 = 0.58333% = 0.00583333.
- Number of contributions. n = 30 × 12 = 360.
- Growth factor. (1.00583333)360 = 8.116498. A dollar paid in at the very start would have multiplied itself eight times over.
- Future value of the starting balance. $0 × 8.116498 = $0, so the contributions must supply the whole target.
- Annuity factor. (8.116498 − 1) ÷ 0.00583333 = 7.116498 ÷ 0.00583333 = 1,219.971. Each $1 per month becomes $1,219.97 by the end.
- Required contribution. $1,000,000 ÷ 1,219.971 = $819.69 per month.
- Total paid in. $819.69 × 360 = $295,088.
- Growth from returns. $1,000,000 − $0 − $295,088 = $704,912, which is 70.5% of the goal.
Now change one thing at a time to feel the sensitivities. Pay on the first of the month instead of the last and the payment falls to $819.69 ÷ 1.00583333 = $814.94 — worth about $4.75 a month for no extra money. Start with $25,000 already invested and that balance alone becomes $25,000 × 8.116498 = $202,912, cutting the gap to $797,088 and the payment to $797,088 ÷ 1,219.971 = $653.37. And cut the assumed return from 7% to 5% and the payment rises to $1,201.55 — a two-point change in an assumption you cannot control moves the budget line by 47%.
How to read the answer without fooling yourself
Read the sensitivity table before you read the headline. The contribution is exact given your inputs, but one of those inputs is a forecast of the next few decades, and the table shows what a one-, two- or three-point miss does to the plan. If the row three points below your assumption is still affordable, you have a plan. If only the optimistic row fits your budget, you have a hope, and the honest fix is a longer horizon or a smaller target rather than a bolder return assumption.
Read the share of the goal from returns second. Over 30 years at a normal equity assumption most of the target is supplied by compounding rather than by you, which is the argument for starting early. Over five years almost all of it comes out of your own pocket, which is the argument for using cash or short bonds rather than equities for a near-term goal — a portfolio that can fall 30% in a year has no business funding a house deposit you need in three.
Read the return assumption as nominal unless you deliberately made it real. If your target is "enough to live on", it is a real target and inflation will erode it, so either raise the target or enter a return net of inflation. The inflation-adjusted return calculator does that conversion properly rather than by simple subtraction. Subtract fund costs too: a 7% gross return in a fund charging 0.75% is a 6.25% assumption here, and the expense ratio drag calculator shows what that difference compounds into.
Finally, if the answer comes back negative, your existing savings alone already overshoot the target on the assumed return. The figure then tells you how much you could withdraw each period and still land on the goal.
Monthly contribution to reach $1,000,000 from a zero balance
| Years to goal | At 5% | At 7% | At 9% |
|---|---|---|---|
| 20 years | $2,433 | $1,920 | $1,497 |
| 25 years | $1,679 | $1,234 | $892 |
| 30 years | $1,202 | $820 | $546 |
| 35 years | $880 | $555 | $340 |
Rounded to the nearest dollar. Read across for the cost of a wrong return assumption; read down for the value of starting five years earlier. Both levers are large across this grid, and only one of them is under your control.
Count the employer match before you count your own money
If part of this goal is funded through a workplace plan, the employer contribution is money you do not have to find. Work out the match first, subtract it from the required contribution here, and budget only the remainder from your own pay. The 401(k) employer match calculator converts a match formula into the dollar figure to subtract. Ignoring it is the most common way people overestimate what a goal will cost them.
Assumptions this calculation makes, and where they break
- A constant return, applied evenly. Real markets deliver the same average through wildly different paths, and the path matters when you are also contributing. Two sequences with identical averages can end tens of thousands apart.
- A level contribution. The payment never rises with your salary or with inflation. If you plan to increase it each year, a step-up schedule reaches the same target with a smaller starting amount.
- No tax and no fees. Enter a return already net of fund costs, and remember that a taxable account loses part of its return to tax each year while a retirement account does not.
- Contributions never stop. The formula assumes every one of the n payments is made. A two-year gap early in a 30-year plan costs far more than a two-year gap at the end.
- The nominal-rate convention. Dividing the annual rate by the number of periods is standard, but it means a 7% quoted rate compounded monthly is an effective 7.229% a year. Do not also add that difference back somewhere else.
- The target is a fixed number of dollars. If what you actually need is a level of spending, inflation makes the dollar target a moving one; solve in real terms instead.
Which of the five variables to solve for
Solving for the contribution is the right move when the date is fixed — a child starts university in a known year, a lease ends, a retirement date is set. When the date is negotiable and the contribution is not, invert the problem the other way and solve for time instead, which is what the investment goal timeline calculator does: given what you can genuinely afford, when do you arrive?
When the target itself is the uncertain quantity — as it is for retirement, where the number depends on your spending and your life expectancy rather than on a purchase price — size the target first with the retirement savings needed calculator, then bring that figure back here to convert it into a monthly amount. Working the other way round produces plans that are precise about the payment and vague about the point.
If you want to run the same fixed monthly plan forwards rather than backwards, the SIP calculator projects the maturity value of a set contribution, and the compound interest calculator handles a lump sum with no contributions at all. Once you start paying money in, the price you get on each purchase is governed by the arithmetic in the dollar-cost averaging calculator.
One structural point worth making. Because the annuity factor grows faster than linearly in n, each extra decade of horizon saves you far fewer dollars than the decade before it. For a $1,000,000 target at 7% from a zero balance, ten years needs $5,778 a month and twenty needs $1,920 — the first extra decade saves $3,858 a month. Thirty years needs $820 and forty needs $381, so the fourth decade saves $439. The proportional cut stays large; the dollar cut collapses to a ninth of what it was. That is why the early years of a plan are the expensive ones to skip, and it is visible in the table above without any exhortation.
Key terms
- Annuity factor
- The future value of $1 paid in every period for n periods at rate i, equal to ((1+i)ⁿ − 1) ÷ i. Dividing your funding gap by it gives the payment.
- Annuity due
- A stream of payments made at the beginning of each period rather than the end. Each payment earns one extra period of growth, so the required amount is smaller by a factor of (1 + i).
- Sinking fund
- A fund built by regular deposits to meet a known future obligation. The payment this calculator returns is what accountants call the sinking-fund payment, and the reciprocal of the annuity factor is the sinking-fund factor.
- Nominal annual rate
- A quoted yearly rate that is divided by the number of compounding periods to get the periodic rate. It is lower than the effective annual rate whenever compounding happens more than once a year.