The tyranny of extreme values

Each box is one year of the S&P 500, from 1928 to 2025. Only price changes are counted, as if companies paid no dividends. Each year sits in the 10-percentage-point column that contains its annual return (for example, from 0% to 10%); negative years are in red. The vertical line is the expected return E[r], the expectation of the annual return r, estimated by the arithmetic mean of the 98 annual returns. The standard deviation, also called volatility, measures the dispersion of annual returns around E[r]: the higher it is, the further returns tend to lie from E[r].

The bars count S&P 500 trading days since 1928, grouped by return (excluding dividends: over a single day, they make almost no difference). The orange curve shows what you would get if returns followed a normal distribution with the same mean and the same standard deviation: the bell curve. From here on, “under the normal distribution” refers to this calculation. The tails of the distribution of returns are the two ends of that distribution: the days very far from the mean.

Between −2% and +2%, where the ordinary days lie, the curve roughly follows the bars: for these days, the normal distribution gives about the right order of magnitude. In the centre, the bars rise above the curve: very calm days are more frequent than the normal distribution predicts. Across all ordinary days, between −2% and +2%, the gap remains small: 93.2% against 90.7%. It is in the tails of the distribution of returns that it goes wrong: the second chart zooms in on the sharp falls, framed in the first chart.

Each interval bounds the return of one trading day, as a percentage. The normal distribution gives how many days should fall outside; the last column counts the days that actually fell outside in the history of the S&P 500.

Under the normal distribution, 95% of days fall within the first interval, 99% within the second, then 99.9%, 99.99% and 99.9999%. At 95%, the normal distribution even predicts slightly more days outside the interval than the historical record shows. The gap then widens: outside the 99.99% interval, the normal distribution predicts 2.5 days; the S&P 500 has 217.

The curve tracks $100 invested in the S&P 500 at the start of the data (30/12/1927), excluding dividends, on a logarithmic scale: the same percentage rise has the same height everywhere. The red dots are the 50 worst days, the green dots the 50 best: these are the extreme days in this step. The timeline at the bottom plots them in two rows. The next chart measures market turbulence day by day.

$100 is invested in the S&P 500 over the selected period, excluding dividends, to compare four investors: always invested, missing the N best days, missing the N worst days, missing both. A day out of the market earns 0%: the money stays in cash. Days out of the market remain in the calculation: the length of the investment does not change.

The result is expressed as a CAGR, the compound annual growth rate: the constant annual rate that takes the starting capital to the final capital.

Under the normal distribution, the number of sharp falls is the same for all three assets: the threshold is set in terms of each asset's own standard deviation.

The annualised return of an investment (its CAGR) is the constant rate that, applied every year, takes $100 to the final capital: $100 shrinking to $62 in 20 years (1929-1948) is equivalent to −2.3% a year for 20 years. It is not the average of the yearly returns, which is higher (0.9% over these 20 years).

For each holding period: the number of investments, the worst and the best (what $100 becomes and the annualised return), then the number of investments that ended below the initial $100. S&P 500 excluding dividends.

Each point on the curve is a $100 investment started at the beginning of the year shown, for the selected holding period; the investments overlap. Green curve: the S&P 500 excluding dividends. Blue curve: 3-month US Treasury bills, rolled over at each maturity, with almost no risk of loss. Below the $100 line, the investment has lost money in nominal dollars, before inflation.

On Yahoo Finance, the Treasury bill series (^IRX) starts in 1960, so this comparison begins in 1961. Dividends are not counted, which puts equities at a disadvantage against Treasury bills.

This last step brings the previous ones together: fat tails in the distribution of returns (step 2), extreme days concentrated in periods of turbulence (step 3), a few days that weigh heavily on the result (step 4), sudden crashes even in gold (step 5) and a horizon that narrows the range of annualised returns without erasing the shocks (step 6). One idea ties them together: the black swan.

1. The average return is not enough. It all depends on the year: from 1928 to 2025, 31 years out of 98 were negative. Investors earn the CAGR (6.3% a year), lower than E[r] (8.1%).
2. The tails of the distribution of returns are fat. Ranked from worst to best, daily returns form a distribution; its tails are its two ends, the days of very sharp falls and very sharp rises. For ordinary days, between −2% and +2%, the normal distribution gives about the right order of magnitude. In the tails of the distribution of returns, extreme days are far more frequent than under the normal distribution: since 1928, there have been 87 one-day falls of at least 5%, compared with 0.29.
3. Extremes come in periods of turbulence. Most extreme days are concentrated in a few periods, such as 1929-1939, 1987, 2008-2009 and 2020. The best days often fall within a few days of the worst (not always: 12 of the 50 best days since 1928 fall more than a month of trading away from each of the 50 worst).
4. A few days weigh heavily. Being out of the market for a few dozen days considerably changes the result over twenty years; nobody can reliably pinpoint those days in advance.
5. Precious metals do not escape extremes. Gold and silver also have their abrupt falls; for diversification, what makes gold useful is its low correlation with equities.
6. The horizon narrows the range of annualised returns. For the S&P 500 from 1928 to 2025, excluding dividends, in 95% of cases they range from −34.8% to +40.0% a year over 1 year and from −0.7% to +12.4% a year over 20 years: extreme years, positive and negative, partly offset each other. 31 of 98 investments end below the initial stake over 1 year, 3 of 79 over 20 years and none over 25 years.
7. Risk is rewarded, not removed. Since 1961, even without counting dividends, equities have usually done better than Treasury bills over 20 years, with no guarantee. Spreading out contributions reduces the effect of the entry point.
8. Markets live in Extremistan. A few rare events account for a large share of gains and losses: nobody can predict when the next black swan will come, hence the value of preparing for it.