4. The horizon changes everything: the same asset, two different risks

Risk 1: Measuring risk

The three previous modules have gradually shifted risk away from the asset and toward the investor. This one completes the move with the variable that shapes it most and that never appears on a product factsheet: how long you can wait and how much money you need along the way.

The same fund, two people, two risks

Marta is 42, pays in a fixed amount every month and will not touch that money for twenty years. Julio is 66, has just retired and will withdraw 5% of the portfolio each year. They both hold the same fund, with the same 16% volatility: identical factsheet, identical chart and identical standard deviation.

For Marta, a 35% fall next year is at worst irrelevant and at best an advantage: her contributions buy more units at a lower price. For Julio, that same fall arrives while he is drawing capital out, and the combined effect is of a different order altogether. The risk is not in the fund but in the relationship between the fund and each person's calendar.

The arithmetic of the horizon: two dispersions pulling in opposite directions

There is a subtlety here that causes more confusion than any other and is settled with two multiplications. Take an asset with an annual volatility of 16% and assume independent periods.

• The uncertainty about the annualized average return shrinks with the square root of the horizon: over 10 years, 16 divided by the square root of 10 is about 5.1 points; over 25 years, 16 divided by 5 is 3.2 points. The longer you wait, the more the annual average settles down.
• The uncertainty about the final capital, by contrast, grows with the square root of the horizon: 16 multiplied by the square root of 10 is about 51 points; over 25 years, 16 times 5 is 80 points. The longer you wait, the wider the fan of possible outcomes.

Both are true at the same time: they describe different things. The classic mistake is to quote the first in order to conclude that time diversifies risk. Paul Samuelson took that apart decades ago and Zvi Bodie closed the argument with an elegant point: if time reduced risk, insuring a floor for your portfolio would get cheaper the longer the term. The opposite happens: a ten-year put option on the index costs more than a one-year one.

"Equities always go up in the long run": the assumptions built into it

The sentence is not false, it is incomplete, and the three missing pieces are the ones doing all the work.

• It assumes your horizon matches the chart's. The charts on display cover a century. Nobody invests for a hundred years. The real horizon, between accumulating and spending, rarely goes beyond thirty.
• It assumes the market you are looking at will carry on existing and that you would have picked it in advance. The American series is the twentieth century's winner; investors in the Russian and Chinese stock markets lost every penny of their capital in 1917 and 1949. Antti Ilmanen and the work of Dimson, Marsh and Staunton keep making the point: extrapolating from the market that did best is survivorship bias in its most expensive form.
• It assumes we are talking about real returns with dividends reinvested, three things that rarely travel together on the same chart.

What long stretches of negative real returns tell us

The evidence on long, disappointing periods is more plentiful than the usual talking points suggest:

• The Nikkei 225 peaked near 38,900 points in December 1989. Thirty years later it was hovering around 23,000, some 40% below, and it did not recover that level until 2024, roughly 34 years on. Reinvested dividends and Japanese deflation improve the real figure, but they do not change the headline.
• The American stock market went through some sixteen years, between 1966 and 1982, of practically zero real return, eaten away by inflation. And from the 2000 peak, the S&P 500 took more than a decade to recover its real level.
• After the 1929 crash, the Dow Jones needed around twenty-five years to get back to its previous nominal level. With dividends and adjusting for the deflation of the thirties the period shortens, but it is still more than a decade.

Sequence risk: when the order changes the result

If no money goes into or out of the portfolio, the order of the returns does not matter: multiplication is commutative. As soon as there are contributions or withdrawals, it matters enormously.

Two retirees start with 100,000 euros and withdraw 10,000 at the end of each year. Both get the same two returns, but in reverse order.

• The first takes the -30% at the start: 70,000, minus 10,000, leaves 60,000. Then the +40%: 84,000, minus 10,000, leaves 74,000.
• The second gets the +40% first: 140,000, minus 10,000, leaves 130,000. Then the -30%: 91,000, minus 10,000, leaves 81,000.

Seven thousand euros of difference, with identical returns, the same mean and the same volatility. With no withdrawals, both would have ended on exactly 98,000. The damage is done by taking capital out of an already reduced base: the euros that leave at the bottom are not there for the recovery.

Case study: buying the Japanese index in December 1989

Picture a 40-year-old Japanese investor who in December 1989 buys his country's index with a thirty-year horizon, convinced that equities always go up in the long run. It is a generous term, longer than almost any retirement plan. In 2019, aged 70 and with the date he needed the money finally arrived, his index was still 40% below the purchase price.

The lesson is not that Japanese equities were a bad investment, nor that the long run is useless: whoever bought that same index in 2012 got an excellent result. It is that the entry price and the exit date are part of the risk, and neither of them appears in the asset's standard deviation.