Solving for the Monthly Payment: The Sinking-Fund Math Behind a Savings Goal

"I need $20,000 in 3 years, I have $2,000 now, and I expect a 5% return — how much a month?" is a solvable equation, not a guess, and the Savings Goal Calculator solves it directly rather than making you iterate on a spreadsheet until the ending balance roughly matches the target.

The sinking-fund formula

The monthly payment comes from the standard sinking-fund annuity formula, solved for the payment: given a monthly rate r and n months, the required monthly contribution is

monthly = (target − initial × (1 + r)^n) × r / ((1 + r)^n − 1)

which is just future-value math run backward — instead of asking "what does this monthly payment grow into," it asks "what monthly payment grows into exactly this target," accounting for the fact that the existing initial balance is already compounding on its own. When the expected return is 0%, the formula collapses to plain division: (target − initial) / months, since there's no compounding to solve around.

Interest accrues before the month's contribution lands

The month-by-month schedule the tool shows computes each month's interest on the balance before adding that month's contribution — meaning a fresh contribution doesn't start earning interest until the following month. This is the more conservative (and more realistic) ordering for most savings accounts, where a deposit made mid-cycle doesn't retroactively earn a full month of interest.

Why inflation adjusts the target, not the payment

Optionally entering an expected annual inflation rate inflates the target forward by (1 + inflation)^(months/12) before solving for the payment — the logic being that a $20,000 goal three years from now, in three-years-from-now dollars, costs more nominal dollars to hit than $20,000 today. This keeps the monthly number honest about what it takes to preserve purchasing power, rather than quietly under-saving because the target stayed a fixed number while prices moved.

Where the freed-up monthly amount comes from

Knowing the number is half the plan — the Subscription & Expense Tracker is often where that monthly contribution actually gets found, and for a goal without a fixed deadline (retirement rather than a target date), the Fire & Retirement Calculator runs the equivalent math the other direction, projecting a timeline instead of solving for a payment.