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09 August 2012

Uncertainty and It’s Misapplication Part 1




Uncertainty and it’s Misapplication Part 1

In a recent post, Asymmetry of the Stock Market and it’s Implications, we talked about the fact that the stock market (and others) does not statistically follow a Normal Distribution. Although no to my knowledge has described the exact model applicable to financial markets, it is generally agreed that they fit a type of Power Law Distribution.
This asymmetry is not something I, nor any other contemporary, discovered. This fact has been known for some time.

We noted that most forecasting tools are based on a Normal Distribution. Some more sophisticated methods attempt to use an uncertainty factor to account for outliers and their associated risks. There are some things that can’t be known, so practitioners feel they must use a “fudge factor” to account for unlikely events.

This is patently absurd.

As a reminder, here is a typical bell curve;




Let’s suppose that this chart represents the historical price movement of the common stock of Acme Inc. Assuming a normal Distribution, 68.3% of the values will lie within 1 Standard Deviation of the mean. (1 to -1 on the chart);
 

At 4 Sigma, (4 to -4 on the chart),  99.99% of values are covered. That is to say, 99.99 of values will fall within 4 standard deviations of the Mean.


Events or values 6 standard deviations from the mean are considered “impossible”.  The incidence of a 6 Sigma event is 0.0000001973, or approximately 1 in 500 million.
The incidence of a 7 sigma event is 1 in 310 Billion.

If there WERE a 6 Sigma event, our chart would look like this;


The data point at -6 Sigma is what is known as an “outlier”. This is the data that’s typically discarded. At first glance, this would seem reasonable as the chances of this event are about one in 500 million, as we have seen.

Now, what could cause a -6 Sigma event? Maybe Acme’s CFO is indicted for embezzlement. Maybe the headquarters building burns down with all the business records. Maybe.... There are an infinite number of things that could happen. In order to account for this uncertainty, one would have to know everything that could possibly affect Acme Inc, no matter how remote. In other words, one would have to know EVERYTHING. Of course this is impossible. (It’s the same reason we can’t predict the weather more than a few days in advance).

Since there are an infinite number of possible events, there is no way to account for them mathematically. The only certainty is that of you wait long enough, SOMETHING unaccounted for will happen. Attempting to apply a “fudge factor” is ludicrous. If you don’t know what you don’t know, how can you prepare for it? It is unfortunate that the concept of “Unknown Unknowns” is infamously associated with Donald Rumsfeld, firstly because he obviously does not understand it[1], and secondly because it is precisely the reason financial markets are inherently unpredictable, no matter how much technology and science is applied to them.


People are continually losing value in their investments by events that are completely unforeseen; the stock market crash of 1929, Black Monday, Long Term Capitol Management, the Housing Bubble, and on and on. It’s a curious fact that people trick themselves into believing they “should have seen it coming”. “All the signs were there”. Well, all the signs are here now for the next big event. Do you know what it is? No? Neither does anyone else. But it is a certainty, something will happen. This acknowledgement that something will happen is what is missing from practically all investment strategies.

There is a better way.

In part two, we will look at the mis-application of game theory and Gaussian mathematics to the real world and in particular, the financial markets. These are common and widespread mistakes with enormous consequences.






[1] A staffer of Donald Rumsfeld learned the term from its originator, Nassim Taleb.  Had Donald Rumsfeld understood the concept, he would have realized that by invoking it, he was actually making an argument against the invasion of Iraq.