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Randomness And The Galton Board

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For those who know what the normal/Gaussian distribution looks like but are not 100pct sure what it is

This post is for those who might need a firmer understanding of randomness in a more formal sense before having a better grasp of Monday's post about patterns in US equity and bond markets, as well as the post that I will publish tomorrow about its implications.

Random walks and predictability

Here's the passive versus active debate in a nutshell:

Non-randomness = pattern = predictability = potential for investment outperformance. If markets follow a random walk, which many think they do, then there is no pattern, no predictability, and no scope to outperform.

That said, if there is pattern in financial markets - as there is - and therefore scope for outperformance, this does not mean all active managers outperform. It just means that some can. And indeed some do. Not many. But some. Those who spot the pattern.

Galton boards and the normal distribution

The concept of randomness is encapsulated in the normal distribution - also known as the Gaussian distribution or the bell curve. A great way of understanding how randomness leads to the normal curve/distribution is the Galton board. Its creator, Francis Galton, was, according to Wikipedia, a statistician, sociologist, psychologist, anthropologist, tropical explorer, geographer, inventor, meteorologist, proto-geneticist, psychometrician and a proponent of social Darwinism, eugenics, and scientific racism. He also invented the village fête game, Guess The Weight Of An Ox.

The Galton board is a vertical board with a triangular grid of regularly spaced horizontal bars attached to it - see Chart 1 below. Balls are dropped precisely onto the top spoke - such that probability of bouncing either way is 50pct - then travel downwards, through the grid, bouncing left and right off more bars below - again, with 50pct probability - and, finally, fall into a series of transparent containers at the bottom.

Chart 1 below shows the resulting distribution of the balls across all the containers - the annotations are mine. If the grid was infinite in size and an infinite number of balls were dropped, you would end up with the normal distribution - with a finite grid/number of balls as shown, you get a so-called binomial distribution that approximates to a normal distribution.

(Kurtosis refers to the tailedness of a distribution, the extent to which a probability distribution curve has fat or thin tails. A normal distribution has neither fat nor thin tails, so is said to be mesokurtic. Kurtosis is a measure of pattern. Zero kurtosis - mesokurtosis - means zero pattern i.e. pure randomness).

Chart 1: Galton board and the normal distribution

Momentum pattern

Now, let's add in a pattern, call it momentum. If a ball lands on a left-hand spoke, then instead of the probabilities of then going left or right being 50pct, the probability of going further left is 60pct - and right 40pct. The probabilities on the right side are reversed.

Since balls are more likely to be pushed away from the central vertical line of spokes - the central trend - you will end up with a flatter, wider distribution, as depicted in Chart 2.

It has fat tails - leptokurtosis - which is an indication of pattern. Since the balls are not falling through the grid randomly - 50/50 - but are getting pushed away from the centre, you can predict that more balls are going to end up further from the centre than would be the case with 50/50. We call the pattern momentum because there is momentum behind the balls being pushed outwards. Of course, the probability of being pushed further out is not 100pct, but 60pct. In other words, there is still some randomness - chance - alongside the pattern. Such a series, one that contains both pattern and randomness, is called a Markov Chain.

Chart 2: Galton board with momentum pattern

Mean reversion pattern

Next, we introduce a different pattern, call it mean reversion. Unlike momentum, which nudges balls away from the centre, mean reversion pulls them back towards it - you can see that the probabilities have been switched round.

It should make sense that if balls are being pulled/nudged back to the centre, they are more likely to end up in the more central containers than would be the case with 50/50 or indeed with momentum. So, you end up with a curve that has small tails and a tall, broad body - platykurtosis - as shown in Chart 3.

Chart 3: Galton board with mean reversion pattern

Patterns give you an edge

To conclude, momentum/leptokurtosis and mean reversion/platykurtosis are patterns in statistical series such as financial asset prices or the weather that allow one to predict whether something will move further away from trend or back towards it. In other words, if the equity market in a particular country is trending upwards over the long term as it is prone to do, one can at times predict whether it will move - be pushed - away from this trend - up or down - or will be pulled back towards it.

There is still chance involved so you won't always be right, but identification of these patterns can give you an edge with respect to investment outperformance.

You may be wondering what underlying forces are doing the pushing or pulling mentioned above. Also where long term trends come from. Good questions! I'll cover them in tomorrow's post.

The views expressed in this communication are those of Peter Elston at the time of writing and are subject to change without notice. They do not constitute investment advice and whilst all reasonable efforts have been used to ensure the accuracy of the information contained in this communication, the reliability, completeness or accuracy of the content cannot be guaranteed. This communication provides information for professional use only and should not be relied upon by retail investors as the sole basis for investment.

© Chimp Investor Ltd

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