Four things spend a trader’s capital. Only one sends an invoice.

The capital in question is a trader’s: a pool that is worked, drawn on and topped up, not a pension left alone in the dark. The horizon below is long because that is where these operations become large enough to see, not because the reader is retiring — every mechanism on this page acts on a three-year account exactly as it acts on a thirty-year one, just less visibly.

Take $100,000, a 7.00% nominal return, 3.00% inflation, a flat 25% on gains as they are realised, and thirty years. Every number below follows from those five inputs by arithmetic alone; none of them is measured, and all of them are reproducible with a calculator.

The quoted rate was 7.00%. The rate that survived the price level and the tax collector was 2.18% — 31.2% of the number on the tin. Nothing went wrong in that arithmetic. No crash, no bad year, no mistake. Two ordinary, widely-assumed frictions removed more than two thirds of the compounding rate, and neither of them ever appears as a loss.

And that is the flattering version. The case above assumes the 7.00% arrives smoothly, the same every year. Real returns vary, and a varying series compounds more slowly than a steady one with the same average — so the laboratory in clause 02, which adds volatility, reports medians below every figure in this table. At the settings it opens with, the delivered real rate is nearer 1.5% than 2.2%. The gap between the two is named explicitly in the laboratory’s own ledger, on the line marked dispersion. Neither number is wrong; they answer the same question about two different worlds, and the smooth one is the kinder of them.

Then there is a fourth operation, and it is the one this article exists for. If capital is not merely sitting there — if a trader is adding to it in good years or drawing an income from it — the order in which the returns arrive stops being cosmetic and starts being decisive. The demonstration is in clause 01 and it is not a simulation of anything: it is one fixed list of thirty returns, reordered.

1. K1Terminal capital with no cash flow is order-invariant, exactly. This is not approximately true; it is a property of multiplication.
2. K2With cash flow, terminal capital is starting capital × the product of returns minus each payment compounded by the returns after it. Only the second term depends on order, and it is the whole effect.
3. K3An accumulator wants the bad years first. Someone drawing an income wants them last. Identical mathematics, opposite sign.
4. K4Tax collected annually rather than at disposal costs real money even at an identical rate, because tax paid is capital removed from the compounding base.
5. K5Most tax regimes assess nominal gains, so part of what is taxed is the price level, not a gain.
6. K6A “safe withdrawal rate” is a threshold chosen on a surface. Move the threshold, the assumptions, or the definition of failure, and the number moves with it.

With no money moving in or out, no — not by a cent. Terminal capital is the starting capital multiplied by every annual growth factor in turn, and multiplication does not care about order. It is worth stating flatly, because everything else on this page rests on it:

A thirty-year series can be arranged 30! ways, about 2.65 × 10³² of them, and every single arrangement lands on the same number. Reverse the worst decade of your life to the front and the ending balance is unchanged. This is not an approximation, a tendency or a modelling result: it is a property of multiplication, and it holds for any return series whatsoever, real or simulated, at any horizon.

Add a schedule of payments and the answer inverts completely. With a fixed withdrawal each year, terminal capital splits into two terms:

That is the whole mechanism, and it is worth reading twice. The first term is the capital doing its own compounding, and it is immune to order. The second term is the cost of the payments — and a payment made in year one is multiplied by twenty-nine subsequent growth factors before the horizon arrives, while a payment made in year thirty is multiplied by none. Move the good years around and you change what every payment cost you.

The general form deserves its own statement, because it is not really about withdrawals at all.

Nothing in that statement refers to why the money left. A withdrawal, a tax bill, a management fee, a commission and a spread are the same operation as far as this arithmetic is concerned, and every one of them is correctly priced not at its face value but at its compounded value. It is the reason the $150,000 withdrawn from the bench below costs $268,097.90 of ending capital rather than $150,000. Clause 04 applies the identity to tax and clause 06 to capital that never entered at all; execution friction is the same identity again on a timescale of seconds rather than years.

The effect has an exact size. Swap two adjacent years carrying returns a and b, and terminal capital changes by w × (a − b) × C, where C is the product of the growth factors after them. Over the series below, swapping only the first two years — +23.5% and −1.6% — moves the ending balance by exactly −$5,835.96. Predicted and simulated agree to the cent, because they are the same equation.

Because the size of every swap is w × (a − b) × C and C is always positive, putting the larger return earlier always improves the result. Repeat the argument until no swap helps and you have a proof: with withdrawals, descending order is the best of all possible orderings and ascending order is the worst. No search required.

Now flip the sign. A contribution is a negative withdrawal in the identical equation, so the conclusion flips with it: someone still paying money in ends up with the most capital when the bad years come first. The exact sequence that ruins a person drawing an income is the one that most rewards a person building the account — and a trader is usually the second before becoming the first. It does not follow from this that timing can be arranged; the point is that the same series is not equally good or bad for everybody holding it.

This bites a trader harder than it bites a pensioner. A retirement account is drawn on because the horizon arrived. Trading capital is drawn on because drawing on it is the entire point of holding it. Income taken while the capital is still working is not an edge case here — it is the normal operating condition, and it is exactly the condition under which order stops being cosmetic. A trader taking a monthly draw carries the maximum exposure to the mechanism below and has the least chance of noticing it, because a bad month reads as a bad month rather than as a permanent reduction in the base that everything afterwards has to compound on.

At the settings this page opens with, the arithmetic is stark. The series averages 6.9633% a year and compounds at 5.9427%. Left alone, $100,000 becomes $565,107.36 under every one of those 2.65 × 10³² orderings. Draw $5,000 a year and the as-drawn order finishes with $297,009.45 after paying out $150,000 — while the exact same list read backwards is exhausted in year 20, having paid out only $95,734.42. The reversed ordering delivered less income and ended with nothing.

One number in that paragraph deserves attention on its own. The withdrawals totalled $150,000 in cash, but they cost $268,097.90 of terminal capital — the difference between $565,107.36 and $297,009.45. The extra $118,097.90 is not money anyone spent. It is the compounding those payments would have done had they stayed, and it is the reason a withdrawal is always more expensive than its face value.

The as-drawn ordering is a fortunate one: it sits above the ninetieth percentile of random orderings because its strong years happen to fall early. That is precisely why reversing it is so destructive, and it is precisely why the pair is not the evidence. The permutation distribution is the evidence — a thousand orderings of one list, spreading from nothing to $358,955, with 5.8% of them exhausting the account entirely. Not one of those thousand outcomes has a different average return from any other.

An arithmetic average of annual returns — which is not the rate at which capital grows, and the gap is wider than most people expect. Capital compounds multiplicatively, so what accumulates is not the average of the returns but the average of their logarithms: take logs of the product and the multiplication becomes a sum. The rate that actually compounds is the geometric mean, and it sits strictly below the arithmetic mean whenever the returns vary at all. The frozen series in clause 01 is a worked instance — it averages 6.9633% and compounds at 5.9427%, a gap of 1.02 points produced by nothing but the spread of its own numbers.

Why it is always below, and never above. The logarithm is concave: every chord between two points on it lies underneath the curve. So the average of the logs is at most the log of the average, with equality only in the case where every return is identical. That is Jensen’s inequality, and it is the whole reason a perfectly respectable positive average return can sit above a median outcome that disappoints. Dispersion does not merely spread uncertainty around the average; it moves the result you are actually likely to get, downwards, every time, without ever showing up as a loss.

How large the gap is. Under this page’s stated assumption — annual growth factors drawn independently from a lognormal calibrated so that the simple returns have arithmetic mean m and standard deviation s — the median path’s compounding rate has an exact closed form, with no approximation in it anywhere:

The familiar shorthand for that gap is half the variance: subtract s²/2 from the arithmetic mean. It comes from the second-order term of the logarithm’s expansion, it is an approximation valid for small returns and small volatility, and it is close without being right — at 7% and 15% it predicts a gap of 1.1250 points where the true one is 1.0362. This page uses the exact form and quotes the rule of thumb only to name it. As a check that the model agrees with its own mathematics, the laboratory’s sampled median compounds at 5.9879% against the theoretical 5.9638% — a difference of 0.024 points across 4,000 paths, which is sampling noise and nothing else.

This resolves a discrepancy you may already have spotted. The worked table in clause 00 compounds a smooth 7.00% and arrives at $313,614.83 of purchasing power. The laboratory below, which has volatility in it, reports a median nearer $235,000 and a delivered real rate nearer 1.5% than 2.2%. Neither figure is wrong: the entire difference between them is the 1.0362 points derived above, and the laboratory prints it as a line in its own ledger marked dispersion. The smooth case is the more flattering of the two, which is why this article states its headline finding there rather than here.

A neighbouring question — given a return process like this one, what fraction of capital to commit to it — is the growth-optimal sizing problem, and it belongs to how to size a position. This clause is the other half of the same coin: not how much to risk, but what capital already committed actually delivers over time. Where the number m comes from in the first place is a third question again — a mean estimated from a backtest is an estimate of the past, and how much of a backtest survives contact with the future is its own subject.

The laboratory takes an arithmetic mean and a volatility, and generates annual growth factors from a lognormal distribution calibrated so that the simple returns have exactly that mean and that standard deviation. The choice of lognormal rather than normal is not cosmetic: a normal model of annual returns assigns positive probability to a return below −100%, which is not a possible outcome for capital that cannot go negative. A surprising number of growth calculators contain that defect, and it quietly inflates the left tail into impossibility.

What follows is one laboratory driving every remaining figure on this page. Nothing in it is pre-computed; every number is recalculated from the current control values the moment they change. The shocks are drawn once from a fixed seed and reused, so that moving a control changes the picture because of the control and not because the dice were rolled again.

It divides. It does not subtract. If capital multiplies by (1 + n) and the price level multiplies by (1 + i), then what the capital can buy multiplies by (1 + n) / (1 + i). That is a definition rather than a theory — purchasing power is a ratio of two things that both moved — and it is the exact form of the relation usually quoted in its approximate form as “real = nominal − inflation”.

The approximation is always too generous, and the error compounds. At 7.00% nominal and 3.00% inflation the exact real rate is 3.8835%, not 4.0000%. Over one year the difference is a rounding note. Over thirty years on $100,000 it is $10,724.92 — the naive figure overstates the outcome by 3.42%.

There is a second-order consequence that is not second-order in size. Most tax regimes assess nominal gains. If prices rise 3% and an asset rises 3%, nothing has been gained — and yet a gain has been realised as far as the arithmetic of most tax codes is concerned. Inflation therefore does not merely shrink the result; it manufactures a taxable event out of standing still. The laboratory models this the way the codes generally work: tax is applied to the nominal gain, and the result is deflated afterwards.

The figure below is the argument. Both distributions describe the same 4,000 simulated paths; the only difference between them is division by the price level. On a logarithmic axis that division is a rigid translation — the shape does not change at all, the entire distribution simply slides left by the log of the price level. Inflation is not a risk in that picture. It is a certainty applied to every outcome equally.

At an identical rate on an identical return, collecting annually rather than at disposal costs $54,285.03 of purchasing power over thirty years on $100,000. The rate did not change. The return did not change. Only the collection schedule changed, and it moved the result by more than half the starting capital.

The reason is the same one that governs the withdrawal in clause 01: tax paid is capital removed from the base, and capital removed early is removed from every subsequent year of compounding as well. A tax bill is a withdrawal that happens to be involuntary, and it obeys the identical arithmetic.

Note what this does not say. It does not say deferral is cheaper in tax. It is the opposite: the deferred schedule hands over $165,306.38 against $121,385.04 under the annual one — 36% more tax, in cash — and still finishes $131,764.02 ahead in nominal capital. That is the mechanism at its starkest. The larger bill is paid with money that has already done thirty years of work, while the smaller one was taken in instalments that each stopped compounding the moment they left. The difference between the two schedules is not the tax; it is the compounding on the tax, which is Proposition 2 collecting on an involuntary payment.

Both tax totals above are closed form, not estimates. The annual schedule takes 25% of a 7% gain on a base growing at 5.25%, so the thirty payments sum to 1,750 × (1.0525ⁿ − 1) / 0.0525 with n = 30, which is $121,385.04; the deferred schedule takes 25% of the single $661,225.50 gain. Both are reproducible on a calculator, and the laboratory reports the same quantity for the volatile case in its own readout.

This is the point at which a page like this one has to be careful. Tax treatment differs by jurisdiction, by instrument, by holding period, by the taxpayer’s own circumstances, and it changes. Whether a trading account is assessed as it accrues or only on disposal, whether losses offset gains and over what period, whether there is an allowance, whether income and capital are taxed differently — all of it varies. Nothing here is a statement about what any reader will owe. The rate and the schedule are parameters you set, and the arithmetic simply shows what those parameters imply.

There is no single number, and the reason is not that the question is hard — it is that the question has six arguments. Survival is a surface over withdrawal rate, horizon, expected return, volatility, inflation and tax treatment. Any “safe rate” you have ever been quoted is one point on that surface, selected by somebody who fixed the other five arguments and then chose a threshold for what counts as success.

Two clarifications first, because they are more often conflated than distinguished, and they are not variants of one policy. They are different policies with different failure modes.

• A fixed real withdrawal takes a stated amount each year, increased with the price level. Income is stable; the account can be exhausted. This is the policy every “X% rule” actually describes.
• A percentage of the current balance takes a fraction of whatever is there each year. It can never be exhausted — a fraction of a positive number is always positive — but income is not stable, and after a bad run it falls in exact proportion to the account. The risk has not been removed; it has been moved from the capital onto the income.

The most-quoted number in this territory, the “4% rule”, belongs to the first family, and one sentence is all a trader needs from its history: it came from a 1994 backtest of United States market data, it counts a portfolio finishing its thirtieth year with a single dollar as a success, and its own author has since called it a worst case and revised it to 4.5% tax-free against 4.1% taxable. That the originator split the taxable case from the tax-free one is clause 04’s argument arriving from somebody else’s direction. Provenance, stated where the numbers are: those four figures are taken from encyclopedia summaries of Bengen’s 1994 and 2006 papers (S1, S2), not from the papers themselves — the 1994 original survives online only as a scanned PDF with no extractable text, and was not read. They are reported here as attributed to Bengen, and nothing on this page depends on them.

The trader’s version of this question is not the retirement one. Almost nobody reading this is asking what a portfolio supports for thirty years. The live question is what a working capital base supports while it is still being traded, over the handful of years anyone can actually plan for — and that is the same surface read at its left-hand end, where horizons are short, draws are proportionally much larger, and the answers are far less forgiving than the famous number suggests. Set the horizon to ten years and the draw to 8% and the grid will price it.

The grid below computes the surface for whatever assumptions the laboratory currently holds. The wine line traces the frontier at a threshold you can move yourself. Move it, and watch the “safe rate” move with it. That is the entire argument of this clause, and it is easier to operate than to read.

The real return it was not earning — and that cost is invisible, because it never appears as a loss. $100,000 held as cash for thirty years is still $100,000. At 3% inflation it buys what $41,198.68 buys today. Nothing was lost, on any statement, in any month. The purchasing power simply left, at a rate of 3% a year, without a single debit entry.

This is the register in which capital held idle should be assessed. A cash buffer sitting behind a trading account is not free merely because it is safe, and neither is capital parked between opportunities, nor a reserve held against a drawdown that has not happened. It has a carrying cost equal to the real return forgone on it, and that cost is paid every year whether or not the buffer is ever used. Whether the buffer is worth its carrying cost is a real question with a real answer — it depends on what it prevents — but it cannot be answered at all until the cost is written down as a number rather than treated as zero.

The comparison below puts the same starting capital, over the same horizon, under four treatments, all expressed in today’s money so they can be read against each other. It updates with the laboratory, so the numbers in it are the ones your own assumptions produce rather than the ones this paragraph would prefer to quote. Two features of it are worth naming in advance. The cash bar is the only one that is certain — there are no returns in it to vary — and it is always the shortest. And the distance between the two taxed bars is clause 04’s timing wedge, appearing here as a gap you can see rather than a difference you have to take on trust.

The wine rule marks the only line on that figure that is not an outcome: the starting capital itself, restated in today’s money — the level at which purchasing power has been preserved and no more. It is the hurdle every treatment is really being measured against, and it is the reason a nominal gain can be a real loss without ever looking like one. For the separate question of what it costs to obtain trading capital in the first place, that arithmetic is worked in the real cost, by the numbers; this clause is about capital you already hold.

In at least seven identified places, and the honest thing is to name them rather than let a clean figure imply a precision it does not have. Everything on this page is arithmetic performed on assumptions supplied by the reader. The arithmetic is exact. The assumptions are not the world.

• Returns are independent from year to year. Real return series are not. They cluster: quiet periods follow quiet periods and violent ones follow violent ones, and there is evidence of some reversion over long horizons. Independence tends to make long-horizon survival look worse than historical backtests suggest, which is part of why the grid in clause 05 is more pessimistic at 4% over thirty years than the studies it discusses.
• The lognormal has thin tails. Observed annual returns produce more extreme years than it does. The model will therefore understate the frequency of the very bad runs that sequence risk feeds on.
• One asset, one process. No bonds, no rebalancing, no correlation structure, no change of allocation over a lifetime. The historical studies in clause 05 used mixed portfolios; this does not. What happens when several holdings turn out to move as one is its own mechanism.
• No fees and no costs. No management fee, no spread, no commission, no slippage. Every one of them is a further annual subtraction behaving exactly like the tax in clause 04, and what execution friction actually costs is worked separately.
• Tax is one flat rate on nominal gains. See clause 04 for the full list of what that omits. It is a large list.
• Withdrawals never respond to anything. Each path maintains its real withdrawal through any drawdown, which no person does. This is the single most consequential simplification on the page.
• “Survival” is binary. A path ending with one dollar counts identically to one ending with a million. That is a poor summary of an outcome and it is used here only because it is the criterion the literature it engages with uses.

And a note on precision. The distributions use 4,000 seeded paths and the grid 2,000 per cell. A reported proportion near 50% therefore carries a standard error of roughly 0.8 and 1.1 percentage points respectively. Numbers on this page are quoted to the cent because they are reproducible from the stated seed, not because they are accurate to the cent about anything in the world. Those are different claims and only the first one is being made.

Does the order in which investment returns arrive change the final amount?

Only if money is moving in or out. With no contributions and no withdrawals, terminal capital is the starting capital times the product of all the annual growth factors, and multiplication is commutative, so every possible ordering ends on the same cent. Add a schedule of payments and the ordering becomes one of the largest determinants of the result, because each payment is compounded by the returns that come after it.

Why is the real return not simply the nominal return minus inflation?

Because inflation divides rather than subtracts. If capital multiplies by (1 + n) and prices multiply by (1 + i), purchasing power multiplies by (1 + n) / (1 + i). Subtraction is a first-order approximation and it always flatters. At 7% nominal and 3% inflation the exact real rate is 3.8835%, not 4%, and over thirty years on $100,000 the approximation overstates the result by $10,724.92.

Does it matter whether tax is paid every year or only at the end?

Yes, and by a lot. At a flat 25% on gains, a constant 7% nominal, 3% inflation and thirty years, taxing each year’s gain leaves $191,225.76 of today’s money on a $100,000 start; taxing the whole gain once on disposal leaves $245,510.79. Same rate, same return, $54,285.03 of difference, because tax paid early is capital that is no longer present to compound. Tax rules differ by jurisdiction and by person — these are stated assumptions, not a claim about what anyone owes.

Is there a safe withdrawal rate?

Not as a single number. Survival is a surface over withdrawal rate, horizon, return, volatility, inflation and tax treatment, and any quoted rate is a point selected on it by fixing a success threshold. The familiar 4% figure comes from Bengen’s 1994 backtest of US historical data, refers to a first-year withdrawal indexed to inflation thereafter rather than 4% of the current balance, and counts as success any portfolio not exhausted within thirty years — including one that finishes with a single dollar. Bengen has since called it a worst-case figure and revised it to 4.5% tax-free and 4.1% taxable. Those attributions come from encyclopedia summaries of his 1994 and 2006 papers rather than the papers themselves; the 1994 original survives online only as a scanned PDF with no extractable text and was not read.

Why does a bad year early hurt more than the same bad year late?

Because when you are withdrawing, every payment is compounded by the returns that follow it. A payment made in year one is multiplied by twenty-nine subsequent growth factors before the horizon; a payment made in year thirty is multiplied by none. Formally, swapping two adjacent years carrying returns a and b changes terminal capital by the withdrawal amount times (a − b) times the product of the growth factors after them — which is why putting the larger return earlier always helps.

Does sequence risk affect someone who is still contributing?

Yes, and it runs the other way. A contribution is a negative withdrawal in the same identity, so the sign flips: an accumulator making regular contributions ends with the most capital when the poor returns come first and the strong ones come last, because the contributions are then compounded by the strong years. The multiset of returns that is worst for someone drawing an income is best for someone paying money in.

What does capital held in cash actually cost?

At 3% inflation the price level multiplies by 2.4273 over thirty years, so $100,000 held as cash and earning nothing is worth $41,198.68 of today’s money at the end — a 58.8% loss of purchasing power that never appears as a loss, because the nominal number never moved. The cost of holding capital idle is the real return forgone on it, and it is invisible precisely because it is an opportunity cost rather than a debit.

Most of this page is derived rather than cited: the decomposition, the adjacent-swap identity, the order-invariance result, and every deflation and tax calculation are arithmetic shown in full, and need no authority beyond the working. External sources are used only for genuinely external facts — what a named published study actually said. Full details, with retrieval dates and the exact claim each source supports, are in what-trading-capital-keeps.references.md.

1. S1Trinity study — Wikipedia. Retrieved 27 July 2026. Supports: the 4% figure refers to a first-year withdrawal raised with the consumer price index thereafter, not 4% of the current balance; and that success was defined only as the portfolio not being exhausted within the payout period. Tertiary source, used for its summary of Cooley, Hubbard and Walz (1998).
2. S2William Bengen — Wikipedia. Retrieved 27 July 2026. Supports: Bengen first articulated the 4% rate in the October 1994 Journal of Financial Planning; he later termed it SAFEMAX and revised it to 4.5% tax-free and 4.1% taxable; he has described the figure as a worst-case anchored on a 1968 retiree, and has said the historical average safe rate was closer to 7%. Tertiary source; the primary works are Bengen (1994) and Bengen (2006).
3. S3Fisher equation — Wikipedia. Retrieved 27 July 2026. Supports: the relation between nominal rate, real rate and inflation is commonly stated in the approximate subtractive form, of which the exact multiplicative form used here is the unapproximated version.
4. S4Real and nominal value — Wikipedia. Retrieved 27 July 2026. Supports: a nominal quantity is converted to a real quantity by dividing by the growth factor in a price index — the deflation operation used throughout clause 03.