How the maths works

1. The projection loop

This is the heart of it, and it is four steps repeated once a year:

1. Work out this year's withdrawal.
2. Take it out of the balance.
3. Apply this year's investment return to what is left.
4. Carry the result into next year and start again.

Two details in there matter more than they look.

The withdrawal comes out first, then growth applies to the remainder. That is the pessimistic-but-honest convention: it assumes you spend at the start of the year rather than drip-feeding through it. Taking the money out at the end instead would flatter every result on the site by a small amount every year, which compounds into a large amount over thirty.

You cannot withdraw money that isn't there. If the balance is smaller than the planned withdrawal, the calculator takes what remains, marks the portfolio as depleted in that year, and carries on at zero. It never goes negative, and it never quietly borrows to keep the projection looking tidy.

Check it on paper

Start with $1,000,000, withdraw $45,000 a year, return 7%, inflation set to zero. Three years by hand:

Put those numbers into any drawdown calculator on this site and you will get the same figures. If you ever don't, that is a bug worth reporting.

The three spending modes

Different calculators ask you to spend in different ways, and the loop handles three:

• Real (the usual one) — your withdrawal rises with inflation each year, so your purchasing power stays flat. Enter $45,000 with 2.5% inflation and year two draws $46,125.
• Nominal — the same cash amount every year, which means your standard of living quietly falls. Used where a calculator is deliberately showing you that effect.
• Percentage of balance — you take a fixed share of whatever you happen to have, so income moves with markets. This is one of the strategies compared in calculator 02, and it can never run out — it just gets uncomfortably small.

Guardrail strategies sit on top of this as a rule that adjusts the withdrawal before it is taken: cut by a set amount if the drawdown rate has climbed too high, raise it if it has fallen too low.

2. The mortgage arithmetic

The eight mortgage calculators use the standard amortisation formulas — the same ones your lender uses, with no approximation:

• Monthly payment from a balance, a monthly rate and a number of months.
• Balance implied by a given payment over a given term, which is what lets calculator 19 work backwards from "I want the repayment to be £X" to the lump sum required.
• Months remaining for a balance at a given payment — which correctly returns "never" if the payment does not cover the interest.

Check: a $300,000 loan at 5% a year over 20 years gives a monthly payment of $1,979.87. Any mortgage calculator anywhere will agree, because there is only one right answer to that one.

3. Where the simulated market years come from

The calculators that produce a success rate do not assume you earn exactly 7% every year — nobody has ever had that experience. They invent thousands of plausible runs of good and bad years and play your plan through each one.

Why the returns are drawn the way they are

The obvious approach is to draw each year's return from a bell curve around your average. It has one fatal flaw: a bell curve wide enough to include a −40% year also includes a −140% year, and an investment cannot lose more than all of itself.

So the engine draws from a lognormal distribution instead — a bell curve applied to growth factors rather than to percentage changes. The worst possible draw is −100% exactly, never worse, while the upside stays open-ended. This also reproduces something real: markets have more room to surprise on the upside than the downside, which is why long-run charts look the shape they do.

The parameters are converted so that the average and volatility you type in come out the other end as the average and volatility you actually get. That conversion is easy to get wrong and produces a model that quietly returns less than you asked for; it is worth knowing it was done deliberately.

Why you get the same answer twice

The randomness is seeded, which means it is not really random at all — it is a fixed sequence generated from a starting number. Same inputs, same results, every time, on every machine, forever.

This is a deliberate choice with a real cost and a real benefit. The cost is that you are seeing one particular set of imagined futures rather than a fresh draw. The benefit is that the site is honest: a calculator that gave you 84% and then 81% for identical inputs would be impossible to trust, impossible to compare two plans with, and impossible to check.

It is also why the "return sequence" box on calculator 01 exists. Changing that number from 7 to 8 draws an entirely different set of thirty yearly returns. It is there so you can satisfy yourself that the conclusion is not an artefact of one lucky or unlucky draw — change it a few times and watch the headline number move around while the pattern stays put. That pattern is the finding; the specific figure is not.

How many runs

Between 2,000 and 4,000 depending on the page. Beyond a few thousand the answer stops moving in any way that matters, and every extra run costs you time staring at a spinner. Where a page states a number of paths, that is what it ran.

4. How a success rate is built

Run the projection loop a few thousand times, each with a different invented run of market years. Count how many finished with money still in the account. That share is the success rate.

The percentile bands on the charts come from the same set: line up all the balances for a given year, sort them, and read off the value a tenth of the way up, a quarter of the way up, halfway, and so on. That is all a fan chart is — the spread of the same few thousand runs, drawn year by year.

How to read the results covers what a success rate does and does not mean, which matters more than how it is calculated. The one-line version: it is a comparison between two plans, not a probability about your life.

5. The one government rule that is actually implemented

The site contains no tax logic anywhere — with a single exception, and it is worth naming because it is the only place the code encodes a real rule rather than accepting a number you typed.

Forced withdrawals from tax-sheltered accounts. Three of the four countries make you take money out once you reach a certain age, whether you want it or not: required minimum distributions from 73 in the US, RRIF minimums from 71 in Canada, and the minimum pension drawdown from preservation age in Australia. The UK has no equivalent. The relevant tables are transcribed from published sources and re-checked independently — see the tax assumptions audit for the sources and the verification.

How it is modelled matters, and it is a point most calculators get wrong. A forced withdrawal does not destroy the money. It moves it out of the shelter, and you reinvest whatever you did not need to spend. So the real cost is the tax on the surplus, not the withdrawal itself. A model treating the whole forced withdrawal as lost overstates the damage by an enormous margin — sometimes by a factor of ten.

6. What the models leave out

Specific beats general, so here is the actual list.

• Your behaviour. The largest variable in the whole exercise and the one nothing here captures. The simulated version of you never panics, never sells at the bottom, never quietly spends more after a good year, and never adjusts anything even while watching the balance fall for a decade. Real people do all four.
• Bad years clustering together. Each simulated year is drawn independently of the last, so the model does not reproduce the long grinding stretches history actually contains — the 1970s, or Japan after 1990. Real markets have momentum and real volatility comes in clumps. This makes the unlucky end of any fan chart optimistic.
• Fat tails. Genuine crashes are bigger and more frequent than a lognormal distribution implies. Same direction of error as above.
• Tax, in every form. No brackets, no allowances, no means testing. Where a page asks for a tax rate it multiplies by it, and that is the whole of the logic.
• Fees beyond what you enter. Platform charges, fund costs, adviser fees and trading spreads all compound against you, and they are the only genuinely certain number in the exercise.
• Correlation between the things you own. Asset classes are handled at the level of headline return and volatility, not as a matrix of how they move against one another.
• Life. Divorce, redundancy, a parent needing care, an inheritance, a business that does better than expected. None of it is in here, and some of it will happen.

Note that most of these push the same way. Where the models are wrong, they are more often optimistic than pessimistic — which is worth holding in mind when a page tells you a plan works with a comfortable margin.

7. How the charts are drawn

By hand, as plain SVG, from the numbers the calculation produced. There is no charting library and no image service. Nothing on any page of this site is fetched from anyone else's server — no scripts, no fonts, no analytics, no tracking pixels.

That is partly a performance decision and mostly a privacy one: a page that loads nothing from elsewhere cannot leak what you were doing on it. Everything you type stays in your browser, because there is nowhere else for it to go — including the figures carried between calculators, which live in your own browser's storage and are one click away from being erased. More on that on the about page.

8. If you want to check it yourself

Everything is in two files: assets/engine.js for the arithmetic and charts, and assets/locale.js for the country presets and the forced-withdrawal tables. Your browser's "view source" will show you both, in full, unminified and commented. There is no build step and nothing compiled away.

The two paper checks above — the three-year drawdown and the mortgage payment — are the fastest way to confirm the engine is doing what this page says. If you find something that doesn't reconcile, tell me; a calculator that is wrong and confident is worse than no calculator at all.