The Inevitability of the Unlikely

This was originally posted to the now-defunct Thursday Think Tank Blog by Garbanzo

Do you remember anything from that statistics class you took years ago? If you do, it's probably the "bell curve", aka the "normal distribution". That's a good thing to remember, because the bell curve is a really useful tool for understanding some aspects in the real world. If you were really paying attention, you remember all that stuff about mean and standard deviation, and maybe even skew and central limit theorem.

Attack of the 50 ft woman!

Like me, I'm sure you're concerned about the possibility of a gigantic swimsuit clad supermodel rampaging through our cities. After all, it's happened twice in the movies, once in 1953, and again in 1993. Just how likely is this scenario? Let's look at the height of women, over the age of 20. The statistics are based on the U.S. National Health Statistics Report - October 22, 2008.

As you can see the average height of adult females is around 64 inches, or about 5'4". The laws of statistics tell us that 99.7 percent of women will be between 4'3" and 6'4". Well, what do you suppose the odds are of a woman being 20 feet tall? See if you can find it on the graph - that's about 243 inches. (50 ft didn't fit)

That's right - the chances are infinitesimally small (around 1.0E-399). So, what are the chances of being attacked by a 50 foot woman - zero actually. There are real biological reasons that a human can't grow to be 50 feet tall. She would collapse and suffocate under her own weight. Maybe the 50 foot woman could live in the ocean, eating krill and sticking her nose out of the water to breathe. Water might support her colossal mass like it does for whales, but she could not survive on land. In any case, she wouldn't be much of a menace. 

Like many natural systems systems constrained by biological and physical limits, human height matches the bell curve pretty well

Why Volkswagen was the world's most valuable company: for a few minutes.

Some things don't fit well under a bell curve. There are lots of examples, including executive compensation, attendance of rock concerts, and lottery payouts. What these examples have in common is that a few rare events have an unprecedented impact.

A good example is the stock price of the German auto company Volkswagen. In early 2008, the company stock was trading for around €120 a share. The executives at rival German auto maker Porsche, decided to  take control of Volkswagen. They started by buying all the VW stock they could get their hands on. They also bought stock options to guarantee a reasonable price for shares once the news of the takeover became public. Porsche acquired 43% of all VW stock, and had options for another 32% - controlling a total of 75%. Another 20% was not for sale - leaving only 5% of VW shares on the market.

Stock options are complicated, but allow me to simplify the problem: Options are a type of contract promising to sell stocks in the future, for an agreed price. The seller doesn't have to own any shares when they sell the option - it's just a promise. Some of these sellers are betting that they can buy the stocks in the future more cheaply than they can now. They can make a lot of money if prices go down - or lose a lot if they go up.

Here's how it all went crazy: Porsche called up the people who owed them options, and said  "Gib mir meine Bestände!" ("Give me my stocks, now!" ) The option sellers had to rush out and buy 32% of all VW stock - oops! Only 5% of the stock was for sale. This is called a "short squeeze".

Price history of Volkswagen shares (VOW). Hint: This is not a bell curve.

The buyers were desperate; sellers raised prices - the price shot up to over €1000 a share for a few minutes. Guess who had the most stock to sell? Why Porsche, of course. Porsche sold €1000 shares to the option sellers, so the option sellers could sell it back to them for €120 a share. Not a bad deal for Porsche Automobil Holding SE. (Ironically, some options traders had to sell their Porsche and buy a Volkswagen.)

Life in Mediocristan and Extremistan

The height of women, and the price of VW stock are examples of what economist Nassim Nicholas Taleb  humorously refers to as Mediocristan and Extremistan. These two fictional countries represent the predictable world of things like height, and the insanely unpredictable world of stock prices.

Map of the "Stans"

In Mediocrastan, everything fits under a bell curve. In Mediocrastan rare events are rare - and when they do occur, they can be explained. Yao Defen of China is seven-feet-eight-inches tall. Despite being the tallest woman in the world she belongs in Mediocristan. She she is a mere 90 inches on our graph above (+5 sigma).

Residents of Extremistan include Bill Gates and the price of VW stock on October 23, 2008, and most lottery grand prize winners. Bill Gates is has a net worth of $63 Billion, as compared to the mean U.S. net worth of $556,300*. If Bill Gates's height were proportional to his wealth, he would stand at a height of about 100 miles.

"In Extremistan, rare events are common"

How can this be? Any single rare event is, by definition, rare. But when you take all the possible rare events that can occur, chances of one of them occurring are high. For example, the odds of being struck by lightning are 1 in 280,000. The odds of being killed in an air-crash are 1 in 335,000. Both are pretty long odds. But, when you also add in the possibility of being eaten by a shark, hit by a meteor, bit by a black mamba, eaten by piranhas, run over by a bulldozer, and a thousand other rare events, the odds start adding up.

You probably knew someone who was killed by a rare event - maybe several. The odds of that event were tiny, but given all the possible rare events, and all the people you have known, the chances are pretty high.

Useful information:

• Rare events are common, you just never know which event!
• Options trading is riskier than you may think
• The bell curve does not apply to everything
• Bill Gates is 100 miles tall, financially speaking

Congratulations you're one of those rare people who can read a whole blog post about statistics. Many of the ideas in this post are misrepresentations of The Black Swan: The Impact of the Highly Improbable, by Nassim Nicholas Taleb

*$556,300 is the average U.S. net worth, including Bill Gates and Warren Buffet. Median is $120,300.