1. Sequence risk: the same average return, two different lives

Risk 8: Horizon, contributions and withdrawals

The first six levels of this course share an assumption they never actually wrote down anywhere: that money goes into the portfolio and stays there. Measuring dispersion, sizing a position, surviving a drawdown, understanding leverage, spreading risk across assets and mistrusting value at risk are exercises that work out the same over twenty years as over seventy, provided nobody touches the balance. The moment somebody does, half a dozen conclusions change sign.

This level deals with that, and the right place to start is the arithmetic property that explains everything.

Multiplication is commutative, and that is why the order did not matter

Take two portfolios that deliver exactly the same three annual returns, only in a different order.

• Portfolio A, the one that starts badly: -20%, +10%, +30%.
• Portfolio B, the one that starts well: +30%, +10%, -20%.

Compounding is multiplying factors together. In A, 0.80 x 1.10 x 1.30 = 1.144. In B, 1.30 x 1.10 x 0.80 = 1.144. The same number, because the product of three factors does not depend on the order they are placed in. Starting from 100,000 euros and touching nothing, both end up at 114,400 euros. Not similar: identical to the cent.

And here it is worth pinning down two figures that get confused daily. The arithmetic mean of those returns is (30 + 10 - 20) / 3 = 6.67% a year. The compound return, the one that really describes the outcome, is the cube root of 1.144 minus one, that is, 4.59%. Two points of difference that come from no accounting trick, but from the fact that volatility eats part of the mean, exactly as the level on drawdowns showed. Neither figure depends on the order.

The moment money comes out, the order becomes the dominant factor

Repeat the exercise with a single change: at the end of each year, 10,000 euros are withdrawn to live on.

In portfolio A, the one that starts badly, the first year leaves 100,000 x 0.80 = 80,000 euros, and after the withdrawal 70,000 remain. The second year, 70,000 x 1.10 = 77,000, and after the withdrawal, 67,000. The third, 67,000 x 1.30 = 87,100, and after the last withdrawal 77,100 euros remain.

In portfolio B, the one that starts well, the first year leaves 130,000 euros, and after the withdrawal, 120,000. The second, 132,000, and after the withdrawal, 122,000. The third, 122,000 x 0.80 = 97,600, and after the withdrawal 87,600 euros remain.

Same arithmetic mean, same compound return, same three annual figures, same volatility, same Sharpe ratio, same everything. And 10,500 euros of difference over three years, 13.6% more wealth for the one lucky enough to get its bad year last. This is called sequence risk: the risk that your returns arrive in the wrong order.

Why it happens: every euro withdrawn has a different price

The explanation is no mystery. When you withdraw 10,000 euros at the end of the first year, that money misses out on the growth of the years that remain. In portfolio A, the later years multiply by 1.10 x 1.30 = 1.43; in B, by 1.10 x 0.80 = 0.88. Every euro withdrawn in the first year costs you 1.43 euros of final wealth in A and only 0.88 in B.

Add up the full weight of the three withdrawals. In A, 1.43 + 1.30 + 1 = 3.73; in B, 0.88 + 0.80 + 1 = 2.68. Final wealth is 114,400 minus 10,000 times that coefficient: 77,100 and 87,600 respectively. The difference between the coefficients, 1.05, multiplied by the 10,000 euros of flow, is exactly the 10,500 euros separating the two lives.

The mirror image: whoever is contributing prefers to start badly

Flip the sign of the flow and the symmetry becomes clear. If instead of withdrawing 10,000 euros a year you contribute them, portfolio A ends up at 151,700 euros and B at 141,200. The same difference of 10,500 euros, now in favor of the one that started by collapsing.

It is no coincidence: final wealth is the initial capital compounded, which does not depend on the order, plus the flow multiplied by that coefficient, which does. Flipping the sign of the flow flips the sign of the advantage and leaves its magnitude untouched. Hence the sentence that sums up the whole level: whoever is accumulating wants the bad year to come early and whoever is withdrawing wants it as late as possible.

What this result invalidates

If the order matters, a whole set of familiar figures stops answering the question you actually have:

• A fund's average annual return does not describe what your money earned if you put money in or took money out along the way. They are two different measures: one measures the manager and the other measures your account.
• A backtest with no flows is an exercise about a life nobody lives. Adding withdrawals to the same history can change the ending completely.
• The measures from level 1 and level 6 are blind to order by construction. Volatility, the Sharpe ratio and value at risk give the same number if you shuffle the years.

A real case: retiring in 1966 or in 1982

The canonical example is in the US market. Anyone who retired in the mid-sixties ran into a stock market that went nowhere for sixteen years: the Dow Jones index touched a thousand points in 1966 and did not leave that level behind on any durable basis until the end of 1982. In nominal terms, almost nothing. But over that same stretch US consumer prices multiplied roughly threefold, so that in purchasing power the investor lost around two thirds of what he had. And that person also had to sell units every year in order to eat.

Anyone who retired in 1982 got the opposite: the start of one of the longest bull markets in history, with inflation falling from double digits. Same country, same portfolio, same adviser and same long-run average. The difference between the two retirements is explained by no decision either of them took: it is explained by the date of birth.