Lesson 18 of 25 · 7 min

Historical replay, and its sample-size problem

Six episodes in a century supports a frequency anywhere between one year in eight and one year in forty-five.

Track 4 lesson 17 said a century contains six or seven genuine equity crises and that this is too few to estimate a distribution. The statement can be made exact, and the exact version is more useful than the qualitative one.

Treat episodes as arrivals at some constant rate. Six in a hundred years gives a central estimate of one every 16.7 years. The standard interval for a count of six runs from 2.20 to 13.06 arrivals per century, which is one episode every 45.4 years at one end and one every 7.66 years at the other. The upper bound is 5.93 times the lower.

That interval already assumes the rate is constant, that episodes are independent, and that the definition of an episode has not drifted over the century. Each of those is questionable, and none of the questioning narrows the interval.

There is a trap in observation counts that makes the sample look larger than it is. A century of monthly data is 1,200 observations, which sounds ample for estimating almost anything. The number of independent tail episodes inside it is still six. Sampling the same crisis more finely produces more rows and no more information about how often crises occur, because the rows are not independent draws — they are pieces of the same event.

A lender validating a haircut over a ten-year window is working with at most one episode. One arrival in ten years carries an interval of 0.03 to 5.57 per decade, which spans a world where this never happens again and a world where it happens every other year.

Key terms

Arrival rate: How often an episode occurs per unit of time. Estimated from a count, and the interval around it is wide whenever the count is small.
Independent episode: A distinct event rather than a further observation of one. The unit that carries tail information, and the unit that a century supplies only six of.
Observation inflation: The appearance of a large sample created by measuring the same episodes more frequently. It raises the row count and not the information content.

Worked example

What six events in a century can and cannot support

Episodes treated as arrivals at a constant rate. Standard interval for a count of six.

Central estimate: One every 16.7 years
Lower bound on the rate: 2.20 per century
Upper bound on the rate: 13.06 per century
Ratio of the bounds: 5.93×
Monthly observations in the same century: 1,200
Ten-year window with one episode: 0.03 to 5.57 per decade

The honest output of a century of data is a range spanning a factor of six, and that range assumes the rate never changed.

Check yourself

An analyst has a century of monthly returns — 1,200 observations — containing six crisis episodes. Why does the large observation count not help in estimating how often crises occur?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.