How often a fall this bigdraft

Every other page here says what history did. This one asks a different question: if you fit a shape to those returns and ask it about events the record does not contain, what does it say, and how much of the answer is the data rather than the shape? The answer decides how the worst bar on every other page should be read.

The question

A fall of a given size can be described by how often it is expected: a one-in-ten-year fall, a one-in-a-hundred-year fall. Hydrologists have described floods that way for a century, and the phrase "a thousand-year flood" comes from it. The same arithmetic applies to a market, and it carries the same trap. The return period is not measured. It is calculated from an assumed shape, and for events rarer than anything in the record, the answer comes from the shape rather than from the data.

Three shapes, and the record

Three candidates were fitted to this site's own series, each by the values that make the record most likely. A normal distribution, which is what most planning software assumes. A Student t, which is a normal with heavier tails, their heaviness set by one number, its degrees of freedom: the smaller that number, the heavier the tails. And a power law, fitted only to the falls past a threshold, which is the form used for floods and earthquakes. Its own number is a shape: above nought is a power-law tail, and at nought the tail is exponential instead, which is fat but is not a power law.

The monthly series is the 100% growth portfolio this site models, month by month from 1970: all shares, 45% Australian and 55% international, with nothing defensive in it.

Monte Carlo is not a fourth candidate, and the distinction matters. Monte Carlo is a way of computing rather than a distribution: it draws numbers from whatever shape it is given. A Monte Carlo built on a normal is the grey line below, drawn many times over. It is useful on questions about paths, where the order of the years matters, and not on how rare a single fall is.

Months

The dots follow the two fatter shapes. They pass above the normal line at about a 5.3% fall, and the gap widens from there.

Read the first column against the record. A normal fit calls a 13.1% month a once-in-a-thousand-years event. Two months in 56 years were worse than that. It calls a 10.8% month a once-in-a-century event, and four were worse. The worst month in the record, a fall of 24.6%, is one the normal fit puts at one in 273 billion months.

So the thousand-year floods did not become common. They were never thousand-year floods. The ruler was wrong.

What this does not establish is the size of the tail beyond the record. The three lines agree closely where the dots are and separate where they are not, and the further right you read, the more of the line is the assumption and the less of it is the data.

The Student t settles at about five degrees of freedom. It fits far better than the normal, by 57 points of the Akaike criterion, a standard way of comparing two models that charges each one for the parameters it uses; two points is the usual threshold for preferring one to the other, so 57 is decisive. That much is established.

The power law is not established, and this page says so rather than implying otherwise. Fitted to the 68 months that fell more than 3.5%, its shape comes out at +0.12, which would be a power-law tail. But the interval around it runs from −0.06 to +0.45, and it contains nought. A shape of nought is an exponential tail, which is fat but is not a power law. Sixty-eight observations lean towards a power law and cannot establish one. Neither can they rule one out: the interval reaches +0.45, which would be a very heavy tail indeed.

A second measure disagrees with the first, which is worth knowing. The Hill estimator, another way of measuring how heavy a tail is, puts the index at about 2.3 on these months, where the fitted shape implies about 8. Both say the tail is heavy. They do not agree on how heavy, and the record cannot settle it.

Financial years

This is the chart worth spending time on, because it fails. Fifty-six observations, six of them past the threshold the power law is fitted to. To the right of the last dot the three lines separate, and nothing in the data decides between them. The Student t says a once-in-a-thousand-years financial year is a fall of 89.5%, and a little further out it passes a fall of everything, which cannot happen. The third line is not a power law at all: fitted to six falls, its shape comes out NEGATIVE, which describes a tail with a hard ceiling rather than one that keeps going, and the interval around it runs from −0.60 to +1.20, which is to say the six points carry almost no information. Its ceiling near 40% follows from that, and is not a measurement.

Two things follow, and they pull in opposite directions. The tails are fatter than a normal, on every test that can be run. And how much fatter, at the horizons people ask about, is not knowable from 56 years. A tool that answered that question with one number would be reporting its own assumption.

What the record cannot contain

Every line on both charts was fitted to a market that stayed open. The Australian market ran continuously through the twentieth century, which puts it among a small minority. Shanghai closed in 1949. Prague and Leipzig closed in 1945. St Petersburg closed in 1917. A market that closes does not produce a large negative return; it stops producing returns at all, and the holding pays nothing back.

No shape fitted to a surviving market can produce that outcome, and no return period can be attached to it. The horizontal line on the yearly chart is there to mark the boundary rather than to measure anything. What can be said is what the record does not contain, and why the lines above cannot know about it.

Where the overshoot shows, if it shows

A natural guess is that the size of a fall is set by the size of the boom before it, the way a population that outruns what its ground can carry collapses in proportion to the overshoot. Tested on the disaster record itself, with output per person as the measure of the boom, that guess finds nothing: across 147 episodes with enough prior record, how far output had run above its own trend did not predict how far it then fell.

The published work locates the overshoot somewhere else. Schularick and Taylor, on fourteen advanced economies from 1870, found that growth in bank credit relative to output predicts the arrival of a financial crisis where growth in output does not (American Economic Review, 2012). Jordà, Schularick and Taylor then found that the recessions which follow such credit booms are deeper and longer than the rest (Journal of Money, Credit and Banking, 2013). Both findings concern credit, which none of the series on this site measures, so they are cited here and not tested here. What they say for a reader of the charts above is that the state of the system before a fall carries information the return series does not, and that the place to look for it is debt rather than prices.

How this bears on the rest of the site

Every exhibit on the calculators is a backtest: it runs the money through stretches of history that actually happened, and the worst bar on any chart is the worst stretch in the record. Nothing here changes those figures. What it changes is how the worst bar should be read. It is not a floor. It is the worst thing that happened in 56 years, and the shape of the data says the tail continues past it by an amount the same data cannot pin down.

How to reproduce this

Both charts and every figure on this page come from the series this site publishes, and from two scripts in the repository: tools/fit_return_distributions.py for the fits and the tests, and tools/tail_plots.py for the charts. Neither writes anything, and no figure on this page is used by any calculator.