Coast FIRE calculator — methodology

The coast number, and why it is solved rather than discounted

coast number = the least starting balance B such that

    project(B, no contributions, your rate, your charge, years to retirement)
      .finalValue  ≥  annual income ÷ withdrawal rate

Found by bisection over that projection, in whole pence: the answer is the smallest penny that reaches the target and one penny less provably does not. Nothing is inverted and no formula is rearranged.

The obvious alternative is target ÷ (1 + rate) ^ years, and it is wrong for the reason this site refuses closed forms everywhere: a platform charge is not linear in the balance and rounding to the penny every month is not linear in anything, so a discounted target agrees with the schedule most of the time and misses it by a few pounds exactly where a reader would check. A number produced by the same loop that produces the table cannot disagree with the table.

Bisection is only trustworthy over a monotone objective, so the objective has to be proved monotone rather than assumed. It is: one period of the loop maps an opening balance through contribution, growth, a capped platform charge and a fixed fee, and each of those stages is weakly increasing in the balance — growth because x + round(x·p) never decreases as x rises for any rate above −100%, the capped charge because it is a pointwise maximum of two weakly increasing functions, the fixed fee because charging only what is there flattens the map at zero without inverting it. Composing them and inducting down the schedule, every balance is weakly increasing in the starting balance, so “finishes at or above the target” is monotone in it. There is no withdrawal in this projection, which matters: the one part of the loop that is not monotone is the latch that stops an income permanently at the first short payment, and with nothing being withdrawn it never arms.

Unreachable is a result, not a clamp. Compounding is the entire mechanism, so where there is none the idea has nothing to work with. At zero growth the coast number is the whole target or more — the platform charge is still being taken, so the pot shrinks and you have to start above the target to finish at it — and far enough below zero no starting pot this page searches grows into the target at all. The page says so, with what the top of the search range would actually reach, rather than returning the search ceiling as though it were an answer.

The year you could stop paying in, which is a scan and not a solve

The coast line is not a fixed number. It falls every year, because there is one year less of compounding left to do the work, while the balance rises. So the year at which contributions become optional is where a falling requirement meets a rising balance, and both sides move.

for each year Y from today to the retirement age:
    B  = the balance the contributing schedule actually reached at the end of Y
    if project(B, no contributions, remaining years).finalValue ≥ target:
        that is the year

A first-crossing scan, for the same reason the projection package’s own term solver scans rather than bisects: the crossing is well defined whatever shape either curve has, and a bisection would have to assume a shape the arithmetic does not promise. The balance is read off the schedule rather than recomputed, so the row a reader can point at in the year-by-year table is the row the answer came from.

The period step, and today’s money

The projection runs in months, and there is one loop. Contributions arrive at the end of each month — the more conservative of the two conventions, fixed here rather than offered as a field and stated on the page; the investment calculator is where the frequency and the timing are choices.

balance ← balance + contribution        at the end of the month
balance ← balance + growth on it        one twelfth of the annual rate, compounded
balance ← balance − platform charge     on the value of the pot

Every figure is a whole number of pence and every row satisfies opening + paid in + growth − charges = closing exactly, so no headline can disagree with the schedule it was read off.

Everything is in today’s money, so the growth rate is a real return. Spending is stated in today’s terms and a withdrawal rate is a real rate, so a nominal growth assumption would put two different sorts of pound into one comparison. On this page that error runs in the flattering direction — a nominal rate makes the coast number look smaller than it is — which is exactly why the field asks for a return after inflation and says so next to the box rather than in a footnote.

Why the calculator asks for two pots

A defined contribution pension cannot be touched until the normal minimum pension age. An ISA can be touched at any age. Coasting is easiest of all to achieve inside a pension, because auto-enrolment and salary sacrifice put most British money where the tax relief is — so a coast number can be entirely genuine and still not let anybody stop working at the age they typed.

That is a distinctively British problem, and it is why this page splits the pot. If the retirement age is below the access age, the calculator runs those years as a drawdown from the accessible side alone and solves for the accessible pot that would pay the income in full for all of them — the same bisection as the coast number, against “the income is never short-paid” instead of a target. That predicate is monotone in the balance too: a larger starting pot never falls short earlier than a smaller one that never falls short at all.

“Income × years” is the shortcut and it is wrong twice: it ignores the growth the bridge earns while it is being spent and the charge taken out of it.

The access age is a field and this page asserts nothing about it. No age of any kind is in the tax engine behind this site, the normal minimum pension age has already been legislated to rise, and some older schemes carry a protected age. A calculator that typed a year in would be publishing an unverified figure wearing the appearance of a statutory one. It is also a live risk to a plan — an age that moves moves the years to be bridged — and no arithmetic here can price that.

The bounds, which are the engine’s legal domain rather than round numbers

• Growth floors at -99%, not −100%. The projection engine rejects an annual rate of exactly −1 — a total loss is not a growth assumption — and the number field’s clamp is inclusive, so a floor of −100 would be an unhandled throw reachable by typing a minus sign.
• The charge is capped at 5.00% a year, three orders of magnitude below the point at which a periodic platform fee would exceed the balance it is charged on and break the monotonicity every solver here depends on.
• The withdrawal rate floors at 0.10%, and an empty box means that rather than zero. Every other field treats an empty box as the neutral value — no growth, no charge, no contribution — because zero is the only value that asserts nothing. A withdrawal rate is the exception: zero is a division by zero and not a rate at all, so the least extreme legal value is the floor.
• The term is capped at 60 years. That is the reader’s constraint rather than the engine’s: a plan longer than a working life is a question nobody asked.

What it does not model

• Any sequence of returns. One rate, every month. This bears on coasting more than on anything else on this site, because coasting is precisely the decision to stop adding the new money that would have absorbed a poor decade. The figure is the best case of a model with no worst case in it.
• Tax, of any kind. Contributions are treated as arriving whole, which flatters a pension — the relief is already inside the figure you enter — and flatters an accessible pot held outside an ISA, whose dividends and gains are taxable. What the eventual income costs depends on the wrapper it comes out of, and the FIRE calculator prices that. Keeping it there is what lets this page be honestly free of statutory figures rather than nearly free of them.
• The ISA subscription limit, which caps how fast the accessible side can be built and is a real constraint on bridging a gap. It is a published figure and it belongs on the pages that carry published figures.
• The state pension — no age, no amount, no forecast.
• Anything about you. No life expectancy, no health, no partner, no property, no redundancy. This is information, not advice: it recommends no rate, no amount, no wrapper, no provider, and it does not recommend stopping your contributions. The default figures are 4.00% and 5% a year after inflation because a page has to open somewhere, not because either is endorsed.