some Autocorrelation representation by kylemcdonald@flickr (CC)

Kurtosis only?

In an earlier post, I discussed how fat-tails are a reason for Trend Following success (or in technical terms: the excess kurtosis of price distributions).

However, there is something unsatisfying in that explanation: if the kurtosis was the sole source of Trend Following success:

• Random entries should work as well as any other entries
• Strategies such as buying Out-of-The-Money (OTM) options (think Nassim Taleb for example) should exhibit similar performance to Trend Following (with the advantage of being a rather simpler strategy)

Something Extra?

I recently came across this paper (PDF) explaining that Trend Following and OTM options buying are strategies exhibiting similar performance profiles. However, the conclusion of this paper was that Trend Following showed superior performance.

Additionally, there is definitely a measurable edge to Trend Following entries (such as this Donchian breakout e-ratio calculation shows). Random entries would not show such an edge.

So, there must be something extra to the kurtosis story explaining Trend Following success…

Autocorrelation

One hypothesis that I want to investigate further is autocorrelation (also referred to serial correlation).

One of the main principles of Trend Following entries – in the face of conventional wisdom – is:

Buy High and Sell Low

Well, it should really say “Buy High, Sell Higher and Sell Short Low, Buy Back Lower”. The point is that Trend Following entries are made at extremes, in the direction of the extremes.

If market exhibit positive autocorrelation at extremes, it can be derived that following the direction of the extreme moves should provide an edge (positive expectancy). This would explain why Trend Following entries perform better than random entries and why Trend Following is a superior strategy to buying Out-of-The-Money options.

Calculation Project

Now, this sounds all well and fine in theory but does this stack up to verification?

To check this, I am planning to run some calculations on historical prices and see if markets exhibit such autocorrelation at extremes. Another aspect that will be interesting to look into is whether this autocorrelation evolves over time and whether these autocorrelation levels are autocorrelated themselves (ie is there some degree of predictability in the autocorrelation evolution).

Now, please note that I am stepping out of my comfort zone here: my “heavy maths” days are quite far behind me and I know that using statistics can be a minefield (because it is so easy to use it in an incorrect manner). For example, the “standard” correlation calculation (Pearson’s correlation coefficient) only determines linear dependence – although market data is non-linear. Might set myself up for some hardship but as we say in French: “Qui ne risque rien n’a rien” (no pain, no gain).

I am also thinking of getting one or two Econometrics books to give me a headstart on this. But if any of you clever readers have any suggestions or tips on any of the above, please let me know.

Please bear with me and stay tuned.

Another area of statistics which very often comes up in the context of serious trading system development is Robust Statistics. I have not found a good book to leave unread on my bookshelf on Robust Statistics yet so I’ll have to make do with with an unused bookmark on the wikipedia article ;-)

William Eckhardt

Anyway, Bill Eckhardt is a succesful automated trend follower with a strong background in Mathematics (although he apparently never finished his PhD – abandonning his studies for the trading pits). He was also Richard Dennis partner in the Turtle Trading experiment. I re-read his interview in New Market Wizards where he emphasises on a few concepts such as understanding statistics, bet size, and even psychology.

Here are some interesting inssights from the book interview (starting pg 107):

- Statistics are an important element of mathematics for trading:

- “The analysis of commodity markets is prone to pitfalls in statistical inference, and if one uses these tools without having a good foundational understanding, it’s easy to get in trouble.”

- “Classic Statistics work well if you are correct about data distribution assumptions. Because distributions are pathological, we need robust techniques”

- “define what you mean by robust?
A robust statistical estimator is one that is not perturbed much by mistaken assumptions about the nature of the distribution.”

- He mentions the concept of infinite variance in market prices distribution and particularly Mandelbrot and his precursor work on this topic.

- Traders should be more conservative in risk control than might be implied from statistical interpretations using normal distribution

- Less degrees of freedom is better – a degree of freedom is a parameter that produces a different system for every value (e.g. length of Moving Average in crossover systems).

- Chart patterns do not work

- Wished he’d focus more on money management at the start of his trading career (and mentions that in trading, Money Management is the most tractable problem, mathematically speaking)

- Mentions Utility functions and states that all the utility functions used in his risk management model are bounded. He takes the example that if utility functions were unbounded, there would be an amount for which a billionaire would be willing to bet his whole net worth at the flip of a coin.

- Interesting remark about bet size: if you plot system performance against bet size, you obtain a curve in the shape of a rightward-facing cartoon whale, going up in a straight line before dropping dramatically.

Eckhardt's righward-facing whale

- “Trading size is one aspect you dont want to optimize: the optimum comes just before the precipice. You want to be at the left of the optimal point, in the high zone of the straight curve”

- Gives his point of view on psychology and trading.