# Correlation Between Return Streams-Where all the Wiggling Matters

What a strange heading for a discussion on correlation. What the heck does ‘wiggling’ have to do with things? Well stick with us in this Primer and you will find out.

In past Primers, we have spoken at length about the power of diversification in balancing risk across a portfolio, and in allowing us to take full advantage of fat tailed environments to accelerate geometric returns. Well, the fundamental principle surrounding these diversification benefits lies in how the return streams of a portfolio move in relation to each other. How they co-vary or correlate as a composite.

If we visually think of return streams as wavelike phenomena whose volatile equity curves range between periods of building drawdowns inter-dispersed with periods of high-water-mark then we could imagine a physics experiment in the bath-tub whereby wave interference (between return streams) amplify or dampen the volatility of the entire portfolio.

Investigating Wave Dynamics in a Portfolio

In the text books you will find that correlation is a statistical measure that expresses the extent to which two variables are linearly related. The Pearson Correlation Coefficient is the popular statistic we use to assess the covariance of two variables (a description of how they more or co-vary together divided by the product of their standard deviations).

So in a nutshell, a single statistic is generated ranging between -1 to 1 that provides a generalised description of how related these movements are. Two variables are said to be perfectly negatively correlated (move in opposition) when the Coefficient is -1, uncorrelated (no related movement) when the Coefficient is 0 and perfectly positively correlated (move in tandem) when the Coefficient is +1.

So if we apply this to a comparison of return streams, two return streams can move in concert together (positively correlated eg. approaching +1), or be diametrically opposed (perfectly anti-correlated) in their relative movements (approaching -1) or be uncorrelated in nature (approaching 0) where the movement is not related.

Now while the degree of correlation is a sign that perhaps two return streams may have some fundamental connection between each other (aka they may be causally related), there is no guarantee that this is the case. Correlation does not imply causation. It may simply be the result of how the two different return streams play out with no causal connection between them. However when you have a subset of strong correlated series to work with, you can be pretty sure that there is some fundamental relationship between the return streams….but never 100% positive.

So let us have a look at this general principle in action. Chart 31 below describes the relationship or the covariance between two return streams, namely the Societe Generale Index (SG Index) and the S&P500TR Index. The SG Index is a composite of trend following programs and the S&P500TR Index is a useful way to describe a buy and hold Program in US Equities. Chart 31 is therefore used as a basis to showcase the very uncorrelated nature between a long only Buy and Hold approach in Equities versus a typical Trend Following Program.

Now the correlation statistic produced is very close to 0. In fact it is slightly negative at -0.09 so this implies that the two return streams are not related. Well this is the information imparted by this single statistics…but look again as the devil once again is in the detail.

Chart 31: Uncorrelated or is it?

When we drill down into the detail by comparing paths of each return stream over the same history, we find that at times the relationship is strongly positively correlated (blue ellipses), strongly negatively correlated (red ellipses) and at times, simply uncorrelated (yellow ellipses). It is the sum total of these discreet elements of time series that produces a low overall correlation statistic between the two return streams, but not a faithful description of how the two return streams actually move together.

It is therefore a blunt instrument and fails to tell us important information that is essential for our trend following pursuits. We need to know when a portfolio can possess risk weakness or when a portfolio can achieve accelerated returns or when a portfolio is simply stagnating  (spinning its wheels).

You see, we trend followers do not use a single correlation statistic to define a directional relationship between a single return stream and alternative return streams. A correlation statistic itself does not include any information about the adverse and beneficial relationships that exist across an entire time series. We therefore assess how each return stream assists or detracts in their inclusion into a portfolio across the time series by mapping the ‘moving relationship’ over the entire history of both equity curves.

By visual mapping we are looking for a number of features. Do equity curves rise or fall under differing market conditions. What we are looking for here is a direct causal relationship between favorable trending conditions and a rising equity curve or unfavorable market conditions and a falling equity curve. We also use a visual mapping process to stress test a portfolio.  Where is there a weakness in the portfolio, and where are the windfalls. A single statistic does not tell us this story. A detailed mapping process of the entire return stream of a portfolio and it’s component return streams does offer us this valuable information.

Correlations are rarely static and change over the course of time, but a single statistic will not convey that information. All the wiggles matter from a trend following perspective in delivering powerful geometric returns.

We have found previously where single statistics do not help our cause as trend followers, such as the simplified use of the positive expectancy equation in Primer 7. When trading out in the fat tails, we need all the details. Our world is a different domain to most alternative trading methods and it is the fine detail that can lead to “Bloom or Gloom”.

In fact, there is a little used term which is frequently swept under the carpet that is even more powerful for a trend follower than correlation. This is a principle called co-integration. This principle is tied to how we design the systems within our trend following portfolios to give us a more static relationship between return streams. as opposed to a moving feast from correlation, upon which we can then generate superior geometric returns. But before we get to co-integration, lets understand correlation a bit better.

## Correlation

Correlation describes best what can be achieved through diversifying into different markets and into different timeframes. There will be certain times when assets are more correlated with each other or not….but there are no guarantees of this persistent relationship. In fact, correlation when explored within non-linear market conditions needs to be very carefully considered.

The adage ‘correlation does not necessarily imply causation’ needs to be kept front of mind with adaptive emerging market conditions. The inter-relationship that exists within the nested systems of financial markets means that at times, causation can rear its ugly head leading to varying correlation relationships. A panic sell in one asset class frequently leads to liquidations in other asset classes to fund these shortfalls and before you know it, correlation goes to 1 across asset classes.

There is no better investment class to observe this feature as the equity markets. During long protracted bull runs, individual stocks tend over time to be less correlated, whereas during market crisis, this relationship evaporates where a sea of red across the entire market is attributed to a very high correlated relationship within the asset class itself, and its impact is also felt across other asset classes.

In terms of diversification at the timeframe level, we all know that trending markets are not simple linear ascents or descents and tend to exhibit a fractal wave nature of persistent overshoots and undershoots in its overall trajectory. The fractal-like nature of markets means that we can find outlier directional moves within any timeframe  and that ‘outliers’ are scale independent. Diversifying into different timeframes therefore allows the divergent trader to benefit from long or short trends arising in any timescale.

Of course, we prefer the higher timeframes (aka the ‘longer look-backs)’ than the shorter timeframes (or the ‘shorter look-backs’) to apply our trend trading process. It is not that there is an absence of outliers in any particular timeframe, but rather it is the influence of noise and mean reversion in the shorter time-scale that significant alters the form of the outlier and makes it very difficult to trade. So this is not a ‘markets don’t trend at shorter time intervals statement’….but rather that ‘trends within shorter time intervals are far more influenced in their general character by noise and mean reversion type of statement’.

## Co-integration

Now as excited as we get about correlation , nothing picks our ears up more than the term co-integration. To obtain cointegration benefits in a portfolio is like the ‘Holy Grail’ for trend followers.

Not many traders talk about co-integration, but they should. There is a degree of confusion expressed by traders in understanding the difference between correlation and cointegration but here is a simple description that seeks to clarify the distinction.

• Correlation – If two stocks are correlated then if stock A has an up day then stock B will have an up day
• Cointegration – If two stocks are cointegrated then it is possible to form a stationary pair from some linear combination of stock A and B.

So here is an example that might allow us to see this difference better which is taken from the following source: gekkoquant.com

“A man leaves a pub to go home with his dog, the man is drunk and goes on a random walk, the dog also goes on a random walk. They approach a busy road and the man puts his dog on a lead, the man and the dog are now co-integrated. They can both go on random walks but the maximum distance they can move away from each other is fixed (ie. length of the lead).

So, in essence the distance/spread between the man and his dog is fixed, also note from the story that the man and dog are still on a random walk, there is nothing to say if their movements are correlated or uncorrelated.

With correlated stocks they will move in the same direction most of the time however the magnitude of the moves is unknown which means they can arbitrarily change and start to mean revert. This is in contrast to co-integration where we say the relationship is “fixed” and that if the relationship deviates from the “fixing” then it will mean revert.”

Now let us discuss ways we can deploy this principle under system diversification.

Think about what is achieved through a perfect hedge. For example, we buy 1 lot of a particular instrument and we sell 1 lot of the same instrument at the same time and hold for a defined period.

What we are achieving here is using a system-design principle to lock in a forced co-integrated relationship between the return streams of each position that never changes over the course of the trade event.

Of course, this example in real application is a ‘fool’s errand’ as the frictional costs associated with taking these trades lead to a negative overall sum game….however it is the design principle that is important to consider here.

So now let’s assume that we have a divergent trend following strategy that is not permanently ‘in the market’, but only enters the market associated with a particular trade signal. This strategy, in isolation, demonstrates a small edge over say 20 years or more to give us confidence in the method, but it has a 30% win rate with a risk/reward of say 1/3 to deliver positive expectancy.

Let us call this strategy a core strategy of a systematic portfolio.

The 30% win rate of the core strategy gives hope to a system designer that there “might” be an alternate ‘divergent’ strategy that could be designed which enters at exactly the same moment as the core strategy but with the opposite directional sign and also possesses a positive expectancy over the same data series.

Remember that a 30% win rate means that somewhere there is 70% of unsuccessful opportunities that might be available to an inverse strategy relationship.  But unlike the theoretical hedged position, you have not created a perfect hedged condition, as the open profit conditions of both systems vary but you have created a co-integrated anchor point at the entry with oscillation about this.

Imagine now how these two return streams co-vary together. There is a stronger causal connection between them and when one system is in drawdown the other will be reaching for the skies.

While the perfect symmetrical relationship is not theoretically possible with the inclusion of the frictional costs of trading, you will be surprised at what less than optimal design-solutions can achieve at the portfolio level.

It is through system design that you can go further in your quest to ‘force co-integration’ benefits into your portfolio with less emphasis applied to the vagaries that simple uncorrelated systems can produce.

Once we can understand this pearl of wisdom, we can now confidently state why we as trend followers insist that we adopt both long and short positions with all return streams that we compile into a portfolio.

It is not because we believe that all markets offer both long and short opportunities equally. We recognise they don’t, particularly with markets such as Equity Indices that have a long bias built into them.

The real reason we are prepared to sacrifice this ‘long bias’ and insist that we trade both long and short is to benefit from the co-integration that exists in a portfolio by this bi-directional stance.

Sufficient numbers of both long and short return streams in a portfolio guarantee a degree of co-integration in the portfolio which is far better that the moving feast that may exist under simple correlated relationships that change over time.

Well that is our argument for why we insist on trading long and short across all markets.

There will be a raised eyebrow or two relating to this statement in our camp, but we all have our different interpretations about trend following philosophy.

Now we are reaching the end of our philosophical musing about trend following and will shortly be putting the philosophy into action in this Primer series where we demonstrate a workflow process used to design and diversified systematic portfolio from scratch, but before we do…I promised a chap on Twitter (looking at you Linus) that I would pinch his term he uses to refer to Trend Following methods. The term called ‘Pain Arbitrage’.

In the next Primer, we will have a look at what is meant by this, and why “pain can be a gain” for trend following folk.

So stay tuned to this series.

The ATS mob

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• […] Correlation Between Return Streams – Where all the Wiggling Matters – Part 11 […]

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• […] Correlation Between Return Streams – Where all the Wiggling Matters – Part 11 […]

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