## Thursday 28 February 2019

### Skew and Trend following

In this post I discuss a well known stylised fact of the investment industry: "Trend following is a positively skewed strategy".

Spoiler alert: yes it is (sort of), but it's much more complicated (and interesting!) than you might think.

## Economists and quant finance 'professionals' often pretend to be scientists (many of them have actual Phds in actual scientific subjects). So, let's pretend to be scientists and actually check to see if the evidence supports our expectations.I'm going to use three types of trend following trading rule: a 2,8 day EWMAC; all the way up to a 64,256 day EWMAC (Exponentially weighted moving average crossover). Finally the results will be calculated over the 40ish futures contracts in my dataset. The whole thing is being done under the auspices of pysystemtrade, and you can find the usual ugly code here.* actually 2,8 is actually a bit expensive to trade, but costs don't affect the calculation of skew since they just shift the distribution of returns to the left a bit.For reasons that will become obvious I'm going to measure skew over different time periods: daily, weekly, monthly, and annual returns.Let's start with the daily returns

 Skew by trading rule, daily returns

## So... WTF?! Negative skew across the board, with significantly negative values for the slower crossovers. Something weird going on here.

Let's check the other time periods out:

 Skew by trading rule, weekly returns
 Skew by trading rule, monthly returns
 Skew by trading rule, annual returns
Interesting. It looks like for bigger time periods the estimate of skew does indeed become positive. We can see this if we plot the median values for each rule, by time period:

 Skew of a trend following rules profits, measured at different time horizons, from left to right: daily, weekly, monthly, annual
The results run from (on the left) daily, to (on the right) annual. Generally, skew gets more positive the slower the time period we use. The exception to this are the very fastest trading rules, which have a 'sweet spot' for skew at the monthly time period.

## Does it make sense that positive skew only appears at certain frequencies of measurement, with a more infrequent measurement required for slower trading strategies? Yes, it does. Think about a fairly slow trend following rule. Maybe it changes it's positions every few months. When it is not changing it's positions, then it's skew of daily returns will be dictated by the skew of the underlying assets.

So if it's trend following say equities (negative skew), then half the time you'd expect to see negative skew of (when it's long), and half the time (when it's short) you'd see positive skew. Overall your skew will be zero (and this result should hold for positive skew assets as well).

However if you start looking at annual returns, you're more likely to see the characteristically positive skew of trend following. The point at which the skew becomes significantly positive will depend on the speed of the trend following rule. With the faster rules we see positive skew with weekly and monthly returns; with the slower rules it isn't until we get to annual returns that the positive skew reveals itself.

(This is not an original finding. See this, written by someone else I used to work with)

But... that doesn't explain one thing. Why is the skew strongly negative at the shorter time frames? It should be zero, or close to it.

The only explanation is that trend following strategies like to be long negatively skewed assets, and short positively skewed assets

This is kind of interesting (well I think it is!). Perhaps the positive returns of trend following (a 'positively skewed' trading strategy) aren't that surprising at all, if it actually loads on to negatively skewed assets. Perhaps trend following is just a way of collecting the negative skew premium.

And... thinking some more... it sort of makes sense. If negative skew assets earn a premium in the market, then on average they will go up more often than they go down. And assets which go up more often than they go down, will tend to exhibit more bullish trends. And assets which exhibit more bullish trends, well they will be bought by trend following strategies.

This is all assuming that negative skew assets are negative before we buy them, and remain so. I will check this in a second.

## What is the conditional relationship between skew and trend following

Let's do the following exercise. We'll find out the median skew, conditional on a trading rule being long or short, for a given trading rule.  I'm going to measure the skew over a period of a month, using daily returns.

First, let's look at the skew of a given instrument in the month after a trading rule has taken it's position. Remember its this skew that matters in determining what the skew of the returns of a trading rule will be (at least for the slower rules, which will 'inherit' the skew of the underlying asset).

 Skew in the month after a trading rule takes it's position; conditioned on the trading rule being short (left hand side) or long (right hand side)

If a trading rule is long, then in the month following the forecast being made the skew is negative. If the rule is short, then the skew is closer to zero, or even positive. The effect is more noticeable for slower rules (faster rules will have changed their position during the following month anyway, perhaps multiple times).
This is a confirmation of our earlier intuition that slower trend following rules are likely to have negative skewed returns, because when they are long the underlying asset is negatively skewed; and when they are short the underlying asset is positively skewed (giving the strategy the opposite: more negatively skewed returns).

## Summary

Trend following rules do indeed have the positive skew you'd expect... but only at the right time horizon. For slower trend following rules you don't see them appear until you are using annual returns. At shorter time horizons they have persistently negative skew.

An asset which is negatively skewed at one time horizon, and positively skewed at another is... weird. Should we want to own it? I guess it depends on our own 'investment horizon'. If you only look at annual returns, you're going to love trend following! If you look at more frequent returns... you'll be less impressed. Given the long drawdowns of trend following strategies, you would be best off looking at your portfolio every 20 years or so :-)

For the slowest trend following rule I use it looks like this occurs because negatively skewed assets have a return premium, which leads to positive drift. So slow trend following rules will have a secular long bias to negatively skewed assets.

For other trend following rules this explanation is wrong. Instead, they tend to short assets whose skew has recently gone negative, and vice versa. It seems likely this is due to sharp selloffs in risky assets creating both negative skew and bearish recent trends. However skew is mean reverting; so the other rules end up being short assets which subsequently have positive skew, and vice versa.

This also means that if you're planning to use negative skew as a trading signal in combination with trend following, it will be a great diversifier! Except for the slowest moving average crossover, the momentum rule will usually do the opposite to a skew trading rule: it will short negative skewed assets, and go long positively skewed assets.

1. the negative skew seems to be related to some form of shorter term reversion, in which case you can try delaying the entry?

1. Sounds complicated and prone to overfitting! I think using three trading rules combined linearly would be simpler: medium speed trend following, short term mean reversion, skew.

2. Great blog post, as always, Rob. I do have question in relation to your final para where you say the skew rule would be a great diversifier as it would do the opposite to a momentum rule. Perhaps I have misunderstood something but doesn’t this mean the returns from a skew rule would be negatively correlated with a momentum rule and therefore they would cancel each other out? I had assumed, partly from reading your previous posts, it is better to have a rule which is uncorrelated rather than -vely correlated?

1. It looks like both medium speed momentum, and loading up on negative skew, both make money; and yet they look like they might be negatively correlated (emphasis - I haven't checked any of this). This is quite rare in finance for the reasons you describe, and bears further investigation (another post). Bear in mind that the negative correlation is only there when conditioned with respect to previous skew; and may not be universally true.

2. Ok, thanks for the clarification, Rob.

3. Rob, your own futures trading system uses various EWMAC variations. Some are using short lookback periods, others are using longer lookback periods. Based on the results of this blog entry, will you make any changes to the EWMAC rules that you use? For example: will you remove certain EWMAC rules, or add certain EWMAC rules?

1. No absolutely not. For two reasons. Firstly, I'm not that bothered about the fact my returns exhibit negative skew at a daily horizon. Secondly, this would be a form on *implicit overfitting* and therefore punishable by death.

2. LOL. "implicit overfitting, punishable by death". That is indeed the most serious showstopper. Thanks for your reply.

4. http://www.sr-sv.com/how-salience-theory-explains-the-mispricing-of-risk/

Interesting timing, some empirical evidence as well to back up the salience/prospect theory you talked about at the start of the article.

5. Thanks for the article.
Wouldn't it serve roughly the same purpose while being less noisy to just add a realized vol signal instead of a skew signal?

1. Perhaps, but since we already scale positions for vol it's less likely that vol will be an important generator of signal information, except perhaps in a non linear (i.e. dangerously overfitted) way.

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