The FIRE number
FIRE number = yearly spending ÷ withdrawal rate. At 4% that is 25 × spending: $50,000 ÷ 0.04 = $1,250,000. At 3.5% it is 28.6 ×; at 3%, 33.3 ×. With a pension: (spending − pension) ÷ rate. In the money of a future year: × (1 + inflation)^years.
Growth and the years to FIRE
Each year: balance = balance × (1 + real return) + yearly saving. The saving is added at the end of the year. The year you reach FIRE is the first whole year the balance is at or above the number; if that does not happen within 100 years the calculator says so. Example: $150,000 with $24,000 a year at 5% real reaches $1,250,000 in 21 years.
Coast FIRE
Coast number = FIRE number ÷ (1 + real return)^(years until retirement). From 32 to 60 at 5%: $1,250,000 ÷ 1.05²⁸ = $318,867. If the balance is below it, the calculator keeps adding the yearly saving and reports the first year the balance is at or above that year’s coast number (which rises as the years left fall).
Barista FIRE
Barista number = (spending − part-time earnings after tax) ÷ withdrawal rate. The job must pay = spending − balance × withdrawal rate. Example: ($50,000 − $20,000) ÷ 0.04 = $750,000.
Early retirement and the saving needed
The balance at the chosen age comes from the growth rule above. The yearly saving that lands exactly on the target in n years is (target − balance × (1 + r)^n) × r ÷ ((1 + r)^n − 1), the standard annuity formula; when r is 0 it is (target − balance) ÷ n.
Years to FI from a savings rate
With savings rate s, real return r and withdrawal rate w, income set to 1 and nothing saved yet: spending = 1 − s, target = (1 − s) ÷ w, saving = s a year, and years = ln(1 + target × r ÷ s) ÷ ln(1 + r). At 5% real and 4%: 10% → 51.4 years; 25% → 31.9 years; 50% → 16.6 years; 75% → 7.1 years.
How long the money lasts
A portfolio P paying W a year (rising with inflation) at real return r: if W ≤ P × r the return covers the withdrawal and the money never runs out in this model; otherwise years = ln(W ÷ (W − P × r)) ÷ ln(1 + r); when r is 0, P ÷ W. Example: $1,000,000 paying $50,000 at 3% real lasts about 31 years.
The 4 percent rule
First-year withdrawal = 4% of the portfolio; each later year the previous amount × (1 + inflation). The historical basis: Bengen (1994) found about 4% survived every rolling 30-year period of US returns from 1926 with a 50/50 stock and government bond portfolio; the Trinity study (1998) found 4% inflation-adjusted from a 50/50 stock and corporate bond portfolio succeeded in 95% of 30-year periods from 1926 to 1995, with no taxes or fees.
Real and nominal returns
Real return = (1 + nominal) ÷ (1 + inflation) − 1. 7.5% nominal with 2.5% inflation is 4.88% real. Type real returns everywhere on this site; the inflation box on the “how much do I need” page is used only to show a figure in a future year’s money.
What this leaves out
Taxes on withdrawals and on investment income; fund and platform fees; the order of good and bad years, which can sink a plan whose average return was fine; changes in spending; pensions that start at a later age than retirement; and currencies other than the one you type in. The calculators are arithmetic on your assumptions, not advice, and a plan should be re-run every year with real balances.