Retirement drawdown longevity under a growing withdrawal

The race between the return and the withdrawal

The annual return and the annual inflation figure are each brought down to monthly form, and the depletion month falls out of a logarithmic relation between the corpus, the withdrawal and the gap between those two monthly rates. It is a closed-form solution of the same month-by-month walk the full application iterates, matched against that walk in the pack’s own declared vectors.

The month count is the primary answer, taken as a whole number of months by rounding the continuous solution upward — a partial final month still has to be lived through, so the horizon is never understated by truncation. The year figure is the same count expressed in years, and can carry a fraction.

The total withdrawn is not the withdrawal times the months unless inflation is zero. With a growing withdrawal, the later months pay out more than the earlier ones, and the total sums that growth — which is why an inflation assumption changes the lifetime payout even when it barely moves the horizon.

The instrument covers only the finite side of the race. If the return does not exceed the inflation rate, or if the corpus’s real monthly growth already covers the withdrawal, the closed form has nothing truthful to say — the corpus lasts indefinitely on these assumptions — and the calculation declines and points to the application’s iterated projection instead.

When this calculation is used

• Stress-testing a planned withdrawal against a corpus: does it survive a plausible retirement, or fail early.
• Seeing how sharply the horizon shortens when the withdrawal is raised relative to the corpus.
• Measuring what an inflation assumption does to the lifetime amount withdrawn, separately from what it does to the horizon.
• Finding the neighbourhood of the sustainable-withdrawal boundary — the point where the instrument begins refusing because depletion stops happening.

Worked example

A retirement-sized corpus paying a level monthly withdrawal — no inflation growth — with the balance earning a modest fixed return; then the same corpus asked to sustain a much heavier withdrawal.

The first case runs well over a decade before depletion, and because the withdrawal never grows, the total withdrawn is exactly the monthly amount times the month count. The heavier withdrawal collapses the horizon to under a year — the return has no time to matter when the outflow dwarfs it. Add a positive inflation assumption instead, and the total withdrawn rises above the simple product, because the later months pay out more than the first.

All figures are produced by the certified engine at page load, from a release checked against the test vectors declared inside the signed pack — including vectors that pin this closed form to the engine’s own month-by-month walk.

What each input represents

Assumptions and limits

• The return is fixed for the whole horizon: no market sequence, no volatility, no bad-first-decade risk — the single most consequential simplification in retirement arithmetic.
• The withdrawal grows at a constant inflation rate, compounding monthly from its initial value.
• No taxes, fees or contributions touch the corpus during the drawdown.
• The horizon is reported as a whole count of months, rounded upward from the continuous solution.
• This is an educational reference, not financial advice — a real retirement plan needs sequence risk, taxes and personal circumstances that no closed form carries.

What the guards protect against

• The corpus and the withdrawal must each be positive — depletion of nothing, or by nothing, describes no situation.
• Each rate must sit above total loss: a return or inflation figure at or below the floor of the scale is refused as meaningless.
• A return at or below the inflation rate is declined as outside this closed form’s domain — the relation the solution inverts stops holding there.
• When the corpus’s real growth already covers the withdrawal, the calculation refuses with an explanation rather than reporting a horizon: on these assumptions the money outlives any month count, and the honest answer is a referral to the iterated projection, not a large number.

Provenance

Closed-form growing-annuity depletion of a corpus under fixed return and inflation

Monthly return and inflation rates are derived from the annual figures; the depletion month solves a logarithmic relation in the corpus, the withdrawal and the two rates, rounded up to whole months; the total withdrawn sums the growing withdrawal across that horizon, with the level-withdrawal case handled exactly.

Educational reference, not financial advice — a fixed-return depletion horizon, not a retirement plan. The signed pack carries its own citation, including the parity record pinning this closed form to the engine’s iterated walk; the page reports the verification state of the release it mounted rather than asserting one.