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Decision making through intuition_2/3_Normal distribution


In the previous post we discussed the Power-Law distribution and how the enduro rider ( Graham Jarvis ) waited only the average time required for the situation to resolve, bypassing the stuck rider then. He had the intuition that the length of that situation is likely to be characterized by a heavy tail and it might take long to get out of trouble once the average time is passed. On the contrary, if that event was better described by the Normal distribution, it would be worth waiting a little longer because it would be unlikely to go far beyond the mean value.

A straightforward example of something described pretty well by the Normal distribution is life-expectancy: if the average age is 76 years, after we get to 80 we should not bet on our life lasting other 80 years. Below, a possible distribution describing life-expectancy and its right tail decaying faster than the Power-Law’s:

The Normal ( Gaussian ) distribution is arguably the most frequently used distribution and that is in part due to its properties allowing the approximate representation of many real-life phenomena. However, its popularity hides major pitfalls and they will be the focus of this second post of the series “Decision making through intuition”. The reason of our focus is that the deviations from the ideal case often describe the key and specific dynamics of our systems and they are crucial in order to make effective intuitive decisions.

We want to use an example on financial returns and that is because of the usual higher availability of public data. However, we could apply a similar discussion to other domains: sales, manufacturing, supply-chain, investment management etc...

Below we plot different standardized distributions ( centered on the mean and scaled through the standard deviation ) of the financial returns of a single stock of the S&P 500; they are respectively the distribution of the daily, monthly and quarterly returns of the stock:

The first consideration is that, the distributions change with the time-horizon we consider. Time is the independent variable of our example but, the same consideration may apply to other variables of our possible specific domain. Building different views is important in order to understand how independent variables affect the system.

The second consideration is that, even though they may all seem kind-of Normal distributions adjusted for rel-life deviations, there is much more underneath. The daily returns present long tails; through mathematical tests we could show they act indeed in a Power-Law way beyond five standard deviations. That is an important characteristic: thinking about that distribution as a Normal one may convince us to use it in order to engage in short-term trading. However, that would be very difficult to do considering those Power-Law tails; we would be often right ( around the mean ) and sometimes completely wrong ( deep into the tails ). It is indeed unfortunate that many young people are engaging in so much short-term trading rather than long-term investing (savings).

Quarterly returns present lighter tales and apparently more predictable numbers, however, many deviations will be concentrated in the interval between one and two standard deviations. Long-term returns are indeed often away from predictions but, not by an amount comparable to daily deviations.

The thorough understanding of the different dynamics characterizing the distributions above would be crucial for an investment manager to structure an effective risk-management strategy.

If we want to be effective in managing our processes, we cannot just consider indicators like the average and standard deviation of our numbers ( or data ), we need to study the distributions and their peculiarities. Even tiny details are likely to hide major dynamics driving our systems and they are key in order to make effective intuitive decisions.


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