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You know you're working with data that are approximately normal if they have a classic bell-curve shape that's fairly symmetrical about the middle.

That's where I was mistaken. I thought a bell curve had to be perfectly symmetrical.

I posted something on the Berkshire board awhile back about there being more stock market crashes than buying panics and Peter L. Bernstein's conclusion, in his book Against the Gods: The Remarkable Story of Risk, that this means "At the extremes, the market is not a random walk."

But it's almost impossible to have enough data to accurately understand what's happening out in the tails. Two sigma events only occur about 1 time out of 19, so to observe 200 of them we need to sample 3,700 independent events. To witness that many 3-sigma events, we need to sample 45,000 independent events. For 4-sigma, we'd need 1.5 million events, and for 5-sigma, we'd need 135 million. There simply aren't enough independent data, on anything, to understand what's happening way out in the tails. That word "independent" is critically important too. Stock market data are notoriously autocorrelated,

The Wall Street Journal had a Stock Market Quarterly Review Section in its April 1st edition. It showed the twenty biggest one day percentage gains and twenty biggest one day percentage losses of all time for the Dow Jones Industrial Average. 1929 had four of the biggest declines but they clearly were not independent of each other. They took place on Oct. 28, Oct. 29, Nov. 6 and Nov. 11. There were two drops back to back in 1933, on July 20 and 21. The last pair was the Oct. 19 and Oct. 26 declines of 1987.

The other twelve big declines are not near each other.

An interesting fact about the twenty biggest gains is that fifteen of them, or 75%, took place in the three year span 1931-1933.


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