Probability of Collapse

The “Economix” section of the New York Times online brings up an interesting observation:

“Autumn seems to beget a disproportionate share of American financial crises. But why?”

A quick glance at Wikipedia yielded a page that tallied up the biggest US stock market losses in one day. Starting from the Great Depression and compressing crashes that are within a couple years of one-another (we’ll treat those as part of the same phenomenon and say they began at the start of the earliest member in the series of crashes) the total number of such crashes on the list is 6.

This is over an 80 year period so the frequency of such major crashes is about 0.02 per season (season = 3 months). There have been a total of 80*4 = 320 seasons and of course, 80 fall seasons.

According to the method of counting, these crashes *all* started in the fall months of September/October/November. So the question is, how improbable is this? Assuming that such crashes could in principle be evenly distributed over the different seasons, and assuming that the 6 I’m counting are independent events (questionable assumptions of course), what is the probability that out of 320 seasons, they would all land in the 80 Falls between now and the 1929 crash?

So, take the probability of a big crash occurring in any given season to be

P(crash in a season) ~ 6/320 = 0.02 = 2%.

The number of different ways 1 crash could have occurred on is 320 (possibility of 1 in each season).

The number of different ways 2 crashes could occur is 320*319 / 2 (the first could have occurred in any of 320 seasons, the second has to occur on some other season (we assume–actually in counting up these things I made a different assumption since crashes that were only a couple seasons removed were treated as part of the same overall phenomenon…but anyway…). Divide by two to account for the fact that the order doesn’t matter (we treat the crashes as indistinguishable).

Three crashes: 320*319*318/3!

And so on…

6 crashes: 320*319*318*317*316*315/6! = 1,422,630,723,360 possible ways these outcomes of our interest could happen. Round this to ~ 1.4 x 10^12

Now consider the following *specific* scenario. Imagine 6 crashes happening in a row and then 314 season with no crashes. The probability of such a history is

(0.02)^6 x (1 – 0.02)^314 = (0.02)^6 x (0.98)^314 ~ 1.13 x 10^-13

This will in fact be the probability of any *specific* history involving 6 crashes and 314 crash-free seasons.

Therefore since there are a total of 1.4 x 10^12 ways of getting such histories, the probability of randomly getting any history involving 6 crashes and 314 non-crash seasons is

(1.4 x 10^12) (1.13 x 10^-13) = 0.16 = 16%

BUT we’re not done, we want to know the probability that all of these crashes occurred IN THE FALL given that they occurred at all.

The number of ways a single crash could occur in 80 fall seasons is 80.

The number of ways 6 crashes could occur only in fall seasons is

80*79*78*77*76*75*74/6! = 22,237,014,800 ~ 2.2 x 10^10

So the chance of picking *these* histories out of all of the possible histories of only 6 crashes is

(2.2 x 10^10)(1.13 x 10^-13) ~ 0.003 = 0.3 %

This is the probability of living through a history where 6 crashes occurred AND they occurred in the fall. The probability of living through 6 crashes that occurred in the fall GIVEN that they occurred is given by the formula for conditional probabilities:

P(6 *fall* crashes in 320 seasons GIVEN 6 crashes in 320 seasons)

= P(6 fall crashes) / P(6 crashes over 320 seasons)

~ 0.003 / 0.16 = 0.018 ~ 2% chance.

So given all the assumptions, there was about a 2% chance that given a history with 6 crashes over 80 years, we’ve had all of ours occur in the fall.

Note: you could have gotten close to 2% by taking the probability calculated before of living through a history with 6 crashes and dividing by 4 (you’d get 4%). This basically assumes that the probability of 6 crashes occurring is independent of the probability of a crash occurring in the fall. The assumption isn’t strictly true since once a crash has occurred during some fall season, another crash won’t be considered to have occurred in the same fall season.

If anyone sees an error, please let me know!

Sources:

Wikipedia: Largest Daily Changes in the DJIA

NY Times: Why Do Financial Crises Happen in the Fall?

 

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