How sequence of returns risk works

Take five annual returns: minus 20, minus 10, plus 10, plus 20 and plus 30 percent. The simple average is 6 percent. Multiply the growth factors together and a portfolio left completely alone ends at 1.23552 times where it started, a compound annual growth rate of 4.3206 percent.

Now run the same five returns backwards: plus 30, plus 20, plus 10, minus 10, minus 20. The average is still 6 percent. The product is still 1.23552. The compound rate is still 4.3206 percent. Nothing has moved, because the ending value of an untouched portfolio is

$$V_n = V_0 \prod_{t=1}^{n} (1 + r_t)$$

and multiplication does not depend on order.

Two things follow. First, the gap between the 6 percent average and the 4.3206 percent compound rate is not sequence risk at all. It is volatility drag, the ordinary consequence of returns varying around their average, and it is exactly the same in every order. Anyone who points at that gap and calls it sequence of returns risk has named the wrong thing.

Second, order-invariance is why a fund can collapse a period into one annualised return and have it mean something. The figure is built from the fund's own unit prices rather than from anybody's account, so no investor's timing enters it and no reordering of the periods would move it. It describes the fund, not the holder.

There is a condition hiding in the phrase left completely alone, and it is worth saying out loud. Rebalancing is a trade: it sells what rose to buy what fell. A portfolio of several holdings kept at fixed weights therefore has a return series of its own that depends on the order the underlying returns arrived in, with no money entering or leaving anywhere. The order-invariance above belongs to a single stream of returns, which is what one fund reports and what the rest of this page uses.

Sequence of returns risk is what appears the moment that stops being the whole story, and the moment is precisely when money starts moving in or out.

Someone taking a fixed sum out each year is not taking a fixed share of the portfolio. The sum is fixed; the portfolio is not. Divide one by the other and the share moves inversely with the market.

Start with $1,000,000 and take $50,000 at the end of each year. If the first year is minus 20 percent, the pot stands at 80 percent of its opening value when the withdrawal falls due, so $50,000 is 6.25 percent of it. If the first year is plus 30 percent instead, the pot stands at 130 percent and the same $50,000 is 3.85 percent of it. Identical money, and 62.5 percent more of the pot handed over.

Every unit of a fund is worth the same as every other unit, so selling 6.25 percent of the value is selling 6.25 percent of the units. Those units are gone. When the market recovers, it recovers on the units that are left, which is the entire mechanism. A deep drawdown that later reverses is a temporary event for a portfolio nobody is selling from, and a permanent one for the units sold into it. Nothing here promises the reversal. The point is only that the units already sold cannot take part in one if it comes.

This is dollar cost averaging running backwards. A buyer paying in a fixed sum gets more units when prices are low, which is the arithmetic part of its appeal, whatever one makes of the rest of the case for it. A seller taking out a fixed sum gives up more units when prices are low, for the same reason and with the opposite sign.

Across the five year case below, the retiree who met the falls first sold 28.82 percent of the units she started with to pay for five identical withdrawals. The retiree who met the falls last sold 17.25 percent of his. Same fund, same money drawn, and one of them gave up 67 percent more of the holding to draw it.

Both start with $1,000,000 and take $50,000 at the end of every year, 5 percent of the opening balance. Both get the same five returns. Only the order differs. Balances below are a percentage of the opening balance, measured after that year's withdrawal.

A finishes holding 87.95 percent of what she started with and B holds 102.24 percent, a gap of 14.29 percentage points of the opening balance. B's remaining pot is 16.2 percent larger than A's.

The arithmetic behind that gap is exact. With withdrawals of $W$ each period, the ending value is

$$V_n = V_0 \prod_{t=1}^{n} (1 + r_t) - \sum_{t=1}^{n} W \prod_{k=t+1}^{n} (1 + r_k)$$

The first term is the ordinary compound growth of an untouched balance, so it is order-free and equals 123.552 percent for both. Everything that differs sits in the second term: each withdrawal costs its own amount plus every return that came after it. Withdraw ahead of a run of good years and you forfeit those years on that money.

Add up the forfeited growth and A's five withdrawals cost 7.1204 times one year's withdrawal, or 35.602 percent of the opening balance. B's cost 4.2624 times, or 21.312 percent. Subtract each from the common 123.552 percent and you land on 87.95 and 102.24 exactly.

A third retiree earning the same compound rate evenly, 4.3206 percent every single year, ends at $962,963.44, or 96.30 percent. Note which rate that is: the 4.3206 percent compound rate, not the 6 percent average, which no path on this page ever pays anyone. The smoothed answer sits between the two real paths and matches neither of them.

One more thing to hold on to before the next section. Every figure here is nominal. A withdrawal pinned at $50,000 buys less each year, so a retiree holding real spending flat is taking out a rising sum, which the last section returns to.

This is where the concept is most often stated wrongly. It is common to read that sequence risk does not apply while you are still saving. That is true only of a portfolio nobody is adding to. Someone paying in regularly is exposed, and exposed in the opposite direction.

Run the same five returns against a saver paying the same amount in at the end of each year. Under the bad-first order the saver finishes with 7.1204 times one year's payment. Under the good-first order, 4.2624 times. Those are the same two multiples that appeared as the retirees' withdrawal costs, and they match for a reason: the value a payment reaches by the end and the growth a withdrawal forfeits are the same quantity, $\prod_{k=t+1}^{n} (1 + r_k)$, counted once with a plus sign and once with a minus.

That yields a result worth stating plainly. For any set of returns, and as long as the retiree's pot funds every withdrawal to the end, the order that leaves a saver making level contributions best off is exactly the order that leaves a retiree making level withdrawals worst off. Once a pot empties the ending values pile up at zero and the correspondence stops being one for one. Falls early are a discount to a buyer and a permanent sale to a seller. A steady 4.3206 percent would have given the saver 5.4511 times one year's payment, between the two: the smoothed compound rate again lands between the real paths and matches neither.

Be careful what the saver's advantage is not. It applies only to money not yet invested, so it shrinks as the balance grows relative to the payments still to come. A saver with a large pot and few payments left is closer to a retiree's position than to a beginner's, which is the bridge to the next section.

Strip the case back to one moving part. Suppose a portfolio earns 5 percent every year while its owner withdraws 5 percent of the opening balance at the end of every year, the same timing used everywhere on this page. Those cancel exactly, so the balance holds at 100 percent forever. Now replace exactly one of those years with a minus 25 percent year and move it around. The set of returns never changes. Only its position does.

One identical bad year. Arriving last it costs 30 percentage points and the money still lasts. Arriving first it empties the account in year 26 of a 30 year plan. The baseline is deliberately set to break even exactly, which leaves no margin and makes the effect easy to read.

Run the mirror for a saver paying in at the end of each of 30 years and the pattern reverses cleanly. A minus 25 percent year in the first year of saving costs nothing at all, because with the payment landing at the year end there is nothing invested for it to hit. That exact zero is an artefact of the timing convention: move the payments to the start of each year and the first year costs 1.77 percent instead of nothing, while the ranking of every year is unchanged. The same bad year placed last costs 28.14 percent of the balance the saver would otherwise have finished with.

Put the two on one timeline. The most damaging moment for a saver is the last year of paying in. The most damaging moment for a retiree is the first year of drawing down. Those are the same date. That, rather than a rule of thumb, is why the window around the handover carries more sequence exposure than any other stretch of a financial life. What sets the exposure is the size of the balance and the length of the stream it has to fund, not a birthday. Someone whose spending is mostly met from income the portfolio does not have to provide is carrying much less of it, whatever the date on their last payslip.

Nothing removes it. A portfolio funding spending is exposed by construction. What varies is how much, and the idea of a safe withdrawal rate is an attempt to price exactly that: a rate low enough that a bad opening sequence does not empty the account. Any such rate is an output of assumptions about returns, inflation and how long the money must last, never a constant, and the familiar figures come from the history of a small number of markets. The mechanisms below are described so the arithmetic is visible, not recommended, and several of them run on rules that differ by country.

• The withdrawal rate. Units sold in a given year are the withdrawal divided by the unit price, so halving the withdrawal exactly halves the units leaving in every year, good or bad.
• Flexibility. A withdrawal that falls when the market falls sells a smaller share than a fixed one. It is the same lever as the rate, pulled only in the years it matters, and the cost is paid in the thing being flexed: income drops in exactly the years a household may be least able to absorb the drop.
• A cash or short bond buffer. Spending held outside the volatile asset means a fall need not be met by selling that asset. The cost is the return given up on the buffer in every year the fall does not arrive.
• The asset mix. Less variation means less to reorder, and the price is expected return, which is the trade the whole of risk and return is about.
• Income from somewhere else. Every unit of spending covered by a pension, an annuity or a state benefit is spending the portfolio never has to sell for, and it is only as dependable as whoever pays it. In the United States, Social Security works this way for many retirees. What such income costs, whether it keeps pace with prices, and what exists at all are all jurisdictional.
• The handover date. Working longer adds paying-in years and removes drawing-down years, changing both terms at once. It is also the one item here that is often not a choice, and it is priced in years rather than in money.

Inflation belongs here too. A withdrawal that rises with prices is not a fixed sum, and it lifts the share of the fund sold in the later years, which the real return calculator puts numbers on. This page is educational material about how the arithmetic behaves, not advice about any particular portfolio.