Guide · Portfolio arithmetic · Updated 2026-09-19

The same returns in a different order, and one portfolio runs out

Everything below was computed from a real series held offline, and the method is stated so the whole thing can be redone. That matters more than usual on this topic, because it is one where confident numbers circulate freely and almost none of them come with the series they were measured on.

Multiplication commutes. A withdrawal does not.

Start with the case that has no risk of this kind at all. A portfolio of value V with no money going in or out, held across a run of periods with growth factors g, ends at V multiplied by every one of those factors. A product is the same whichever way you multiply it. There is no ordering of the factors that gives a different answer, and no approximation involved in saying so.

Now take a fixed amount W out at the end of each period. The end value is no longer a bare product. It is the starting value times the product of every factor, less the withdrawal times a second quantity: the sum, over every period, of the growth factors that come after that period. Call it S. The first term is order invariant. S is not, and S is the entire risk.

The reason is arithmetic rather than psychological. A withdrawal is a fixed number of units and the portfolio it comes from is not. Taking 11 units after a fall removes a larger share of what remains than taking 11 units after a rise, and the share removed is gone. The recovery, when it arrives, is applied to a permanently smaller base. The damage runs one way only, and no later return undoes it.

Two periods are enough to see it. Growth factors of 1.5 and 0.5 in either order, from 1,000 units. With no withdrawal both orders end at 750 units. With 100 units taken at the end of each period, the rise first gives 600 units and the fall first gives 500. The difference of 100 units is W multiplied by the gap in S, which is 2.5 against 1.5. Every result on the rest of this page is that same sum computed on real data.

The series, and exactly what it is

The data is 3,385 daily index closes for the Nifty 50, from 01 January 2013 to 18 September 2026, held offline in this repository. The index carried three successive names across that window and all three are the same series. Taking the first traded close of each month gives 165 anchor points and therefore 164 consecutive month-to-month growth factors, running from January 2013 to August 2026.

A factor above 1 means the index ended the month higher than it started, and a factor below 1 means lower. 67 of the 164 months are below 1. The weakest is March 2020 at 0.7414 and the strongest is April 2020 at 1.1260, which happen to be consecutive. Compounded end to end the whole series is a factor of 4.042414.

The portfolio convention for everything that follows is deliberately plain. It starts at 1,000 abstract units. Each month the factor is applied and then 11 units are withdrawn. There is no tax, no cost, no rebalancing and no second asset, because adding any of those would introduce assumptions that are not measured here and would blur the one thing being isolated.

That draw is large relative to the portfolio, and that is on purpose. The real data available is under fourteen years, and a small draw over a short window shows almost nothing. A large draw over a short window shows the same structure that a small draw shows over a long one. The level is chosen for visibility and is not a recommended rate. Nothing here suggests what anybody should withdraw.

Forward and reversed, on the same 164 numbers

With the withdrawal switched off, the two runs end at 4,042.41 units each. Not approximately, and not to within rounding: this was computed in exact rational arithmetic from the published closes, so the two products are the same object. Turning the withdrawal on separates them completely.

Forward, the portfolio funds every one of the 164 withdrawals, paying out 1,804 units in total, and still holds 210.23 units at the end. Reversed, it funds 151 withdrawals, paying out 1,661 units, and then cannot fund the next one. It is exhausted in month 152, with 12 months of the window still to go.

The one term that changes

The algebra pins it down with no room for interpretation. The end value is the starting value times the product, less the withdrawal times S. The first term is 4,042.41 units for every ordering that exists. The second term is where the two runs differ.

Forward, S is 348.380, so the withdrawals consume 3,832.18 units of the 4,042.41 the portfolio generates, leaving 210.23. Reversed, S is 379.832, so funding the same 164 withdrawals would require 4,178.16 units, which is 135.74 units more than the portfolio produces. In the real run that shortfall shows up as the value hitting zero in month 152.

The gap between the two values of S is 31.452. Multiplied by the 11 unit draw, that single number is worth 345.97 units, which is more than the forward run finishes with. Order is not a second order effect on this series. It is larger than the entire surviving balance.

The two extreme rows are bounds rather than scenarios. Best months first puts S at 73.829 and ends with 3,230.30 units. Worst months first puts it at 3,182.600 and exhausts the portfolio in month 29. Neither is a path any market would produce. They mark the arithmetic edges of what order alone can be worth.

What order alone is worth, across 50,000 shuffles

The forward and reversed runs are two points. To see the whole distribution, the same 164 factors were shuffled 50,000 times with a fixed seed and the portfolio run through each ordering under identical rules. Every one of those runs contains exactly the same returns, has the same average, the same variability and the same compounded product, and every one of them ends at 4,042.41 units if the withdrawal is switched off.

44.33 per cent of orderings exhausted the portfolio. The median ordering finished with 100.8 units and the best with 1,952.2. That entire spread, from total failure to nearly twice the starting value, is produced by nothing except the sequence. The returns themselves were never varied, not once.

Early declines are the dangerous ones, and the gap is measurable

A shuffle changes everything at once, so it shows that order matters without showing which part of the order matters. The tighter experiment changes one thing. Take the real series, lift out the single weakest month, and put it back at each of the 164 positions in turn, leaving the other 163 months in their original relative order throughout.

Placed at any of the first 61 positions, that one month exhausts the portfolio. Placed at position 62 it survives, barely. Placed last it costs almost nothing and the portfolio finishes with 519.42 units. The identical decline is the difference between ruin and a comfortable finish, depending only on when it lands.

The cause is simple. A decline early is applied to a large balance and then compounded against by every withdrawal that follows, each a larger share of a smaller pool. A decline in the final month is applied once to a balance with almost no withdrawals left to fund. There is no time left for the loss to be multiplied by anything.

The shuffles say the same at scale. Sorting all 50,000 orderings by how their first two years went gives this.

The weakest opening group exhausted 84.5 per cent of the time and the strongest 11.1 per cent, on returns drawn from the identical pool. There is a subtlety worth naming: because the set of returns is fixed, a group with a weak opening must be holding the stronger returns for later, and it still fails more often. The later strength cannot repair what the early weakness did, which is the whole point.

No average can see any of this

This is the part that makes the risk genuinely difficult rather than merely unpleasant. Consider the summary statistics of these 164 factors. The arithmetic mean is 1.009622. The standard deviation is 0.045609. The compounded factor across the window is 4.042414, which is a per month geometric factor of 1.008554. 67 months of 164 are below 1.

Every one of those figures is identical for all 164 factorial orderings. The orderings that exhaust the portfolio and the orderings that finish with 1,952 units share the same mean, the same standard deviation, the same product, the same worst month, the same best month and the same count of falling months. They are the same set of numbers.

So an analysis built on an average return cannot see this risk. Not because it is crude, but because the quantity it rests on does not contain the information. The same holds for anything built on a compound growth figure alone, or on a mean and a variance. Reporting a range around an average does not repair it either, because that is a range of averages. This is not noise around a central case. It is a different quantity.

This is the same class of error as reading a single historical maximum drawdown as though it were a property of a strategy rather than a statistic of one sample path. In both cases a number that looks like a description of the thing is in fact a description of one ordering of it.

What a cash buffer actually does, measured

The standard remedy is a buffer: hold some period of spending in cash so withdrawals need not be funded from a depressed portfolio. The mechanism is sound in principle. Funding a withdrawal from cash while the portfolio is down means the units leave the portfolio later, at a higher value, so fewer units leave in total for the same spending.

So it was measured. A cash sleeve was carved out of the same starting 1,000 units, earning nothing, drawn on in any month the portfolio sat below its running peak and refilled from the portfolio at a new peak. Carving it out of the same total is the fair test: comparing a larger pile of money against a smaller one would measure the extra money rather than the timing.

The sleeve did not work, and the measurement is worth stating plainly rather than softening. On the reversed ordering a twelve month sleeve delayed exhaustion from month 152 to month 161, and on the forward ordering it reduced the finish from 210.23 units to 142.48. Across the shuffles it raised the share of exhausted orderings from 44.33 per cent to 47.06 per cent.

The arithmetic of why is not mysterious. A sleeve changes when a unit leaves the growing pool. It does not change how many units leave, and the total leaving is unchanged at 1,621 units on average. Meanwhile the sleeve sits outside the compounding for the whole window. The timing benefit is real but conditional: it pays only if the fall the sleeve covered is recovered inside the horizon. When it is not, the sleeve has parked capital where it could not grow. On this series, at this draw, the parking cost more than the timing saved.

What did change the outcome was taking less. Cutting the monthly draw by a fifth in any month the portfolio sat below nine tenths of its own running peak turned the reversed ordering from exhaustion in month 152 into a finish of 615.94 units, and lifted the forward ordering from 210.23 to 778.81. Across the shuffles it cut exhaustion from 44.33 per cent to 10.58 per cent, and a three tenths cut brought it to 2.51 per cent.

That is not free and should not be described as one. The average total withdrawn fell from about 1,621 units to about 1,505 under the one fifth cut and about 1,409 under the three tenths cut. The portfolio survived because less came out of it. A cut works harder than a sleeve because it attacks S directly, lowering the withdrawal precisely in the periods where the withdrawal is the largest share of the pool, which are exactly the periods that dominate the sum.

Read together, the two measurements give the buffer a different job from the one usually described. It is not a substitute for reducing the draw. Its value is that it makes the reduction survivable, by supplying spending that does not require selling into a fall in the month somebody discovers they must cut. That is a liquidity and behaviour argument, and a good one. It is not the arithmetic argument, and on this series the arithmetic argument did not hold.

The same arithmetic, mirrored, for somebody still adding

Most writing on this subject stops at the retiree. That leaves out the half of the result that catches people out, because the same term governs the accumulation case with its sign reversed.

Add a fixed contribution C each period instead of withdrawing. The end value is the starting value times the product plus C times S, and it is the identical S. So the ordering that minimises S and therefore maximises the end value for somebody drawing money out is precisely the ordering that minimises the end value for somebody putting money in.

Read the last two columns against each other. The reversed ordering exhausts the withdrawing portfolio in month 152 and leaves the contributing one with 8,221 units, which is more than the forward ordering gives it. Worst months first is catastrophic for the withdrawer, failing in month 29, and is the best of all four for the contributor at 39,051 units. Moving the single worst month of the series to the front gives the contributor the highest end value of any of the 164 positions and the withdrawer the worst.

The consequence is practical. A long decline early in a saving life is favourable, because contributions buy in at lower values and the recovery applies to a larger holding. The identical decline early in a drawing life is the worst thing that can happen. Guidance written for the first case, including the familiar encouragement to welcome a fall while you are still buying, is not merely less useful in the second case. It is inverted. The arithmetic of paying in over time rather than at once and the arithmetic of drawing down are the same equation with opposite signs, and no amount of rewording makes advice from one apply to the other.

What this measurement is, and what it is not

It is one index across one window of under fourteen years, and that window held far more rising months than falling ones. It contains one severe monthly fall and its immediate recovery. A different series or a different window would give different magnitudes, and a more turbulent one would show a wider spread rather than a narrower one, because the spread is driven by the dispersion of the factors being ordered.

What does not depend on the window is the structure. The end value with a fixed cash flow is the order-invariant product term plus or minus the cash flow times an order-dependent sum. That is exact for every series that has ever existed and every one that ever will. The measurement here puts a size on the second term using real returns instead of invented ones, and the size turned out to be larger than the surviving balance.

Nothing on this page is a projection, a recommendation or a plan. No return is stated, no withdrawal rate is suggested, and the units are abstract precisely so that no figure here can be mistaken for an income. Anybody who actually has to fund spending from a portfolio has a specific situation involving tax, other income, health, dependants, horizon and temperament, none of which this arithmetic knows about, and should take advice from somebody qualified who does.

What the arithmetic does give is a better question to ask of any analysis put in front of you. Not what return it assumes, but whether the quantity it rests on can see order at all. If the answer rests on an average, a compound growth figure, or a mean and a variance, then by construction it cannot, and the 44.33 per cent of orderings above are invisible to it. Working out which questions a method is structurally incapable of answering is the part of this that is worth learning, and it generalises well past this one topic. It is the same discipline as choosing the right null to test against.