Sequence of Returns Risk Calculator: Does the Order of Returns Decide Whether Your Money Lasts?

Why the order of returns matters when the average is identical

Withdrawals are what make order matter, and without them it provably does not. With nothing taken out, the ending balance is the starting amount multiplied by every year's return, and multiplication gives the same answer whichever way the years are arranged. Two retirees holding the same returns in opposite orders finish on exactly the same figure, to the last decimal place a computer can represent.

Subtracting a fixed amount every year destroys that identity. A withdrawal taken after a bad year is a larger share of what is left, and the units sold to fund it are gone before any recovery can lift them. The chart at the top of this page is that comparison run on real returns: the window starting in 1969 taken forwards, and the same years taken back to front, with a switch that turns the withdrawals off so the identity can be seen holding.

This is also why an average return is a poor summary of a drawdown. Two plans with the same arithmetic mean, the same volatility and the same worst year can end decades apart purely on when the worst year landed.

How much do the first few years decide?

Repeating the record's worst three years at the start takes the default plan from 8% to 74%. That is the same plan and the same market, changed only by what the opening happened to be.

The third column is what actually happened, and it does not rise with length: those three runs differ in severity as well as in duration, which is why the worst single year outranks the worst five. The last column changes only the length, holding every opening year at -10%, which separates length from severity. Each additional poor year compounds a smaller balance and takes another full withdrawal out of it, so a longer poor run raises the risk rather than averaging it out.

This page conditions on the opening years only. It says nothing about a poor stretch arriving in year twenty, which would need a model that varies the return distribution year by year rather than conditioning on a fixed opening.

What this model does and does not know about order

The probability figures on this page come from a model that treats every year as an independent draw from the same distribution. A model like that has no memory, so it cannot produce sequence risk on its own, because it averages over every ordering rather than favouring any. So the page does not ask it to. The order demonstration is a replay of fixed historical returns with no model involved, and the probability figures are conditional ones, the chance of running out given what the opening years did. Both are exact.

Shuffling the record is the evidence for leaving the model that way. Permuting the 154 observed years into 20,000 random orderings keeps the distribution exactly, same mean, same variance, same worst year, and destroys only the ordering, so it is precisely the independent-years hypothesis with the real returns in it. The actual historical ordering produced a 1.6% failure rate for the 4% rule over rolling 30-year windows; the shuffled orderings average 7%, and 21% of them did as well as or better than history (p = 0.207). If anything, an independent-draws model overstates sequence risk against what actually happened, and the gap is not large enough to conclude anything either way.

What that does not show is markets pulling back towards an average after a bad run. Nothing on this page assumes they do, and the short-lag measurements point the same way: the correlation between one year's real stock return and the next, over 153 year-pairs, is 0.009, which is indistinguishable from zero. That figure is for real stock returns and does not carry across the whole mix slider. Real bond returns measure 0.165 for the inflation reason set out in the assumptions, so a bond-heavy setting rests on a weaker version of this assumption than a stock-heavy one does. A single longer-horizon measurement leans the other way and is recorded as an open question rather than a finding, because any 30-year statistic in this dataset rests on about five non-overlapping windows. The full table is on the methodology page.

What a lump sum could be worth after each horizon

With nothing withdrawn, order stops mattering and only the horizon does. The table below is the distribution of cumulative real total return for a portfolio held 100% in stocks and left alone, over 1871–2024, as a multiple of the amount invested. It is the same repeated convolution the probability engine runs, with the cashflows set to zero.

The band narrows in annualised terms as the horizon lengthens while widening in absolute terms, which is what independent draws produce: the spread of the total grows with the square root of the number of years, so dividing it across more years shrinks it. A narrowing band like that can look like proof that stocks get safer the longer they are held, but the multiple at the bottom of the range keeps falling, and a retiree drawing an income never gets to simply hold on regardless.

Why the top of the range sits above anything that has happened

Because the record is short, and the best of a handful of runs is not the same thing as a 1-in-20 case. The best 30 years in the record compounded at 10.2% a year, turning an amount into 18.6 times itself, starting in 1932. It is the best of about 5 independent runs, because 154 years of history only contains 5 non-overlapping stretches of 30 years.

The largest of 5 draws is expected to land near the 83th percentile, not the 95th. So the right comparison is against the model's 83th percentile, which is 19.1 times the starting amount (10.3% a year) against the 18.6 times history actually delivered. A 1-in-20 case sitting well above the best on record is what the arithmetic requires, rather than a fault in the model.

The bottom of the range works the same way and fits less closely. The worst 30 years in the record, starting 1892, returned 3.2% a year and 2.60 times the starting amount. The model puts its 17th percentile at 3.10 times, so history's worst stretch came out somewhat below where a worst-of-5 is expected, while the best came out almost exactly on it. Neither gap is large enough to read as a verdict on 5 observations, and the two together are the closest thing to a calibration check this evidence allows.

The same arithmetic sets the limit of the exercise. At 50 years the record holds only 3 independent runs, so there is almost nothing to check a 50-year band against. The bands stay computable at any horizon; the evidence that they are right does not stretch that far.

How one year becomes thirty

Convolution does it, and it is computed rather than sampled. Each year is a draw from the fit marked in use in the chart above the assumptions. That chart plots both fits, and only the marked one feeds any figure on this page. Multiplying two independent yearly returns is the same as adding their logarithms, and the distribution of a sum of independent draws is the convolution of their distributions. So the distribution after thirty years is the one-year distribution convolved with itself thirty times. No paths are sampled and no random numbers are drawn anywhere on this page, which is why the same inputs always give the same answer to the last decimal.

With nothing withdrawn, that is the whole story. Each year widens the distribution and nothing else happens to it. Convolution does not care what order its terms come in, so the thirty-year distribution is the same whichever way the years are arranged, which is the same fact the two-retiree chart demonstrates without any model at all.

With withdrawals, each year does two things instead of one. First the same convolution with another year of returns, then the year's spending is subtracted in wealth space and the result re-binned. That subtraction is what breaks the clean convolution, and it is exactly where order enters: the same withdrawal removes a different share of the portfolio depending on what the returns did first. Whatever reaches zero is held there permanently rather than being allowed to go negative, and the weight sitting at zero at the end of the horizon is the chance of running out this page reports.

The paired charts near the top are a picture of exactly that difference. The left is repeated convolution on its own; the right is the same repeated convolution with a withdrawal subtracted every year. The gap between them is what the subtraction costs.

Why the portfolio that lost less can be the one that runs out

Because surviving a bad start depends on what the mix can deliver afterwards, not only on how much it lost. Every mix on this page funds 30 years comfortably at its own long-run rate, so the whole question is what happens after a poor opening, which is this page's subject.

The all-bond row keeps $550,312 through its worst run and the all-stock row keeps only $381,445, and yet the all-bond row is the one likelier to run out, 96% against 74%. What separates them is the last two columns. The bond portfolio needs 5.6% a year from a mix that has compounded at 2.2% and rarely strays far from it, so it almost never gets there. The stock portfolio needs a much higher 9.6% from a mix that has compounded at 7.0%, but its year-to-year spread is wide enough that a real share of outcomes clear it.

One caveat sits on the all-bond row in particular. That row leans on how narrowly a bond portfolio stays near its long-run rate, and real bond returns are the one series where drawing each year independently understates the true spread. A wider spread would cut both ways here, giving a portfolio that needs an unusual return a better chance of one while making more plans fail from a sound start, so treat that figure as approximate rather than as leaning in a known direction.

The relationship is not a straight line between those two ends. Across the mixes above, the chance of running out after a bad opening is lowest at 60% stocks and rises toward both ends, so neither endpoint on its own describes the shape. The worst opening run is also a different historical episode at each mix, which is why the table names the year in every row.

How the mix changes the odds, and why the withdrawal rate changes the answer

Plotted against the stock and bond mix, the chance of running out falls and then rises, and the bottom of that curve moves toward stocks as the withdrawal rate rises. At a 3% rate the lowest point sits at 35% stocks; at 4% it sits at 55%; at 5% it sits at 75%. From 6% upward there is no bottom in the middle, and the curve falls the whole way to the all-stock end.

What the chart plots is one curve per level of bad luck at the start of a retirement. Left to right is how the portfolio is split between stocks and bonds; up and down is the chance the money runs out over the whole 30-year horizon.

The conditioning is what makes it readable, so it is worth stating exactly. For each mix the model takes the 5th-percentile of that mix's own 3 years outcome, which is a single figure covering the opening run as a whole rather than any one year in it, and holds every opening year at the steady yearly return that compounds to it. The rest of the horizon then runs on the full distribution. So a 5th-percentile opening of 3 years is one event of that depth spread evenly across those years, not 3 years of them in a row, which would be a far rarer scenario and a far deeper hole.

The percentile is read from each mix's own distribution rather than from one shared threshold, because a 5th-percentile 3 years is a milder event for a bond portfolio than for a stock one, and holding both to a single number would put two different events on the same axis. The flat opening run is a stand-in as well as a percentile: order inside those years still moves the outcome once withdrawals start, which is the subject of this whole page.

The table further up this page, the one naming a historical year in every row, asks a different question, and neither is a version of the other. That table conditions each mix on the worst opening run that mix actually had, a named historical episode with its years in the order they happened. This chart conditions on a percentile of a fitted distribution. One asks what happened; the other asks what a stated depth of bad luck would imply. Their answers differ for that reason rather than by error.

Two things happen at once between the 4% and 5% rows. The whole curve lifts, from 68% to 95% at the all-bond end, and its bottom slides, from 19% at 55% stocks to 49% at 75% stocks. That is why the withdrawal rate is a control on the chart rather than a fixed assumption. A curve drawn at one rate describes only the plans withdrawing at that rate.

Lengthening the opening run does something different. It lifts the whole curve and leaves the bottom close to where it was. Between an opening run of one year and one of 5 years at a 4% rate, the lowest point travels 5 percentage points, from 50% stocks to 55%, while the chance of running out at that point goes from 13% to 24%. Raising the rate from 3% to 5% moves the same bottom 40 points. The withdrawal rate is what changes the shape of this curve; the length of the opening run changes its height.

The three lines are the same calculation at three depths of bad opening, and they neither cross nor disagree about where the bottom is. At a 4% rate their lowest points sit 5 percentage points apart, one step of the mix axis. That is why the depth is drawn as three fixed lines rather than as a third control. One line that moved would leave the agreement between them invisible.

The all-bond end of every curve is the least reliable part of the chart, for the reason set out in the assumptions below: real bond returns are the one series where drawing each year independently understates the true spread, and that end of the axis rests entirely on it. In 154 years of history the worst single year for an all-bond portfolio is 2022, at -17.0% after inflation; the worst for an all-stock portfolio is 1931, at -38.0%.

What this calculator assumes

A probability invites more trust than a point estimate, so it carries more obligation to say what it rests on. These are all of the model's assumptions, each with its honest limitation attached.

• The probability model treats each year as an independent draw. It has no memory, so it cannot manufacture sequence risk on its own: it averages over every ordering rather than favouring any. That is why the order demonstration on this page is a replay of fixed historical returns rather than a model output, and why the probability figures are conditional ones, the chance of running out given what the opening years actually did. The section on what the model knows about order sets out the measurements.
• The opening years are conditioned on, not predicted. The danger-zone curve holds a chosen return steady across the opening years and then lets the rest of the horizon run on the full distribution. It is a statement about what each opening outcome would imply, not a forecast that any of them is coming, and the curve says nothing about a poor stretch arriving later in a retirement.
• US markets, 1871–2024. The distribution and the replayed windows both come from a single country over a single stretch of history, and the US was the best-performing major stock market of the 20th century largely in hindsight. Of everything listed here this is the likeliest to be flatly wrong about the future, and computing the rest exactly does nothing to fix it.
• Bond-heavy mixes: the spread here is narrower than the real one. Independent draws hold up well for stocks and less well as the mix slider moves toward bonds, and the reason is not the obvious one. Nominal bond returns are close to uncorrelated from one year to the next (0.023). This page works in real terms throughout, and deflating a smooth nominal series by inflation, which is itself persistent (0.332), carries that persistence into the result. Real bond returns measure 0.165, against 0.009 for real stocks. The problem is that inflation is predictable, not that bonds are. A 60/40 real mix measures clean (0.022, with a Ljung-Box p of 0.421), so this is a bond-heavy problem rather than a bonds-at-all one. At high bond weights the true long-run spread is wider than independent draws imply, and a wider spread does not push every figure the same way: it makes more sound plans fail, and it gives a plan that already needs an unusual return a better chance of getting one. Read the bond-heavy end as approximate in both directions rather than as wrong in one.
• Everything is in today’s money. Returns are inflation-adjusted via the CPI series and spending stays constant in real terms, so there is no separate inflation guess to get wrong. Actual historical inflation is already inside the real returns, which is what makes the 1970s window as punishing as it looks.
• One year, one step. A year is a single draw, and there is no within-year sequencing. Someone who retires into a crash in March experiences a path an annual model cannot distinguish from the same year ending flat.
• Annual rebalancing to the target mix. Applying a weighted return to the whole balance quietly assumes the portfolio is rebalanced back to the chosen stock and bond split every year, the same convention the historical simulator uses.
• Withdrawals are fixed in real terms. Spending never flexes: the same inflation-adjusted amount comes out every year however the market behaves. Real retirees tend to trim after a bad year, which is exactly why a fixed-spending figure overstates this particular risk, and a reason to read these numbers as an upper bound rather than a forecast.
• No taxes, fees, or transaction costs. Index returns are used as-is, so the model is optimistic by roughly the total cost drag on a real portfolio.
• Running out means the balance reaches zero within the horizon. Depletion is an absorbing state: once the money is gone it stays gone. The model does not simulate anyone noticing trouble in year eight and adjusting, which is one more reason the raw figures read pessimistic for anyone willing to adapt.
• The distribution is fitted, not the history replayed. The probability figures are computed from a distribution estimated over all 154 years, in a choice of two fits. The smooth normal fit is the starting point, because the observed left tail rests on 4 years out of 154, and a curve drawn through 4 observations in its most extreme region is a weak estimate of that region. The historical fit keeps the tail exactly as it occurred, and switching to it is the sensitivity check against that starting point. The chart above draws both over the same observed years, so a reader can see where they disagree without switching between them. The figures still come from the selected fit alone. Neither is the true distribution. The two-retiree chart is the exception and uses the raw yearly returns themselves.

For the fixed-withdrawal assumption in particular, the historical simulator's dynamic withdrawal strategies show how much adapting spending changes the picture.