When Shells are Tails

In the late 15th Century Persian fable of the Scorpion and the Turtle, a turtle meets a scorpion who cannot swim. The scorpion asks to hitch a ride across the river on the turtle’s back. The turtle is accustomed to navigating the ebb and flow of the river and so generously agrees.

Much to the turtle’s dismay, when they are half-way across the river, the scorpion thrusts his stinger towards the turtle’s back. An extreme event which defies all logic – one would assume the scorpion would behave rationally and not risk drowning himself by such a heinous act.

However, much to the scorpion’s surprise, the turtle had his own tail event in store – an impermeable shell to render the attack harmless. No doubt, all of the scorpion’s data would have led him to conclude that his strike would be the end of the matter, but the turtle demonstrated that sometimes a shell can be a tail too! In the world of alternative investment strategies, trend followers (also commonly referred to as CTAs/Managed Futures) seek to navigate the ebb and flow of the markets in much the same way as the turtle. Trends have after all been a pervasive feature of markets for centuries. And just like in the ancient fable, both positive and negative tail events can occur.

The art of managing expectations

Even if tail events may be surprising in that they are very difficult to predict in advance, should we be surprised at the frequency and level of their occurrence in a more general sense? Arguably not. Our research into the performance of large CTAs has found that they can exhibit extreme moves which exceed multiples of the relevant strategy’s volatility target.

This has led us to also consider a more interesting and perhaps less well visited line of enquiry regarding these tail events on a relative basis. Specifically, how does their frequency relative to commonly utilised industry benchmarks compare with that predicted by a normal distribution?

A better understanding here could perhaps help us to reign in our surprise at tail events in either direction, whilst also potentially providing some interesting insights around the limitations of benchmarks and expectations around deviations therefrom.

Thoughtful approaches

Investors take different approaches to determining the merits of adding a particular CTA or group of CTAs to their portfolio. The primary methodology tends to be thoughtful quantitative and qualitative analysis of how a particular strategy return profile and behaviour complements the long-term objective of the relevant investor’s existing and expected portfolio construction. A smoother overall return profile, risk mitigation, uncorrelated returns, diversification and all that good stuff are usually sought through such an approach over a relevant time period.

This is a logical approach to take, and it neatly avoids the nigh impossible task of trying to time when a particular CTA will have a good or a bad year in advance. Trying to time CTA investments is always a risky proposition; it can lead to the unfortunate outcome of hopping from one CTA’s bad year to another CTA’s bad year.

As one of several additional considerations, the use of industry benchmarks has, however, become increasingly popular. There are various benchmarks used, with the most prevalent in the medium-term¹ trend following space tending to be the Société Générale Trend Index (often known as “SG Trend”).² Established in 2000 by Société Générale Prime Services, this benchmark was introduced to measure the performance of some of the largest trend following strategies.

Correlation, coincidence and other utility dilemmas

Where benchmarks are taken into consideration, there are some very important points to bear in mind.

Indices often comprise an evolving set of members. In the case of SG Trend for example, none of the original constituents in 2000 are still members of the index today. If we go back ten years to 2014, only half of the 2024 strategies were components of the index.

This point may seem obvious, but the corollary of it, perhaps less so: the performance of SG Trend historically is not actually relevant to the performance of the index participants today. And so, when we measure a strategy’s performance against the index, to a certain extent we are chasing shadows because even the historic performance of the current index constituents is not in line with the performance of the index!

This signposts us towards another issue worthy of consideration: whilst CTA managers may follow some similar basic principles, they do not (generally speaking) target SG Trend itself as a building block within their portfolio construction approach. As a result, significant divergence from SG Trend at any given time should be expected (both by constituents and non-constituents of the index).

In the absence of an ex-ante relationship between a strategy and any given benchmark, significant levels of return dispersion across CTAs are therefore highly likely to occur. And this can be smoothed or exacerbated depending upon the timeframe over which the ex-post correlation is examined.

Dispersion among CTAs reflects both the variety and complexity of strategies employed. And such dispersion tends to increase during periods of market stress or strong market momentum. While CTAs may share some common exposure to market momentum, their individual approaches combined with market dynamics can lead to significantly different outcomes. There are numerous examples of portfolio design choices which can lead to dispersion, including (without limitation): model speeds, risk allocation weights (by instrument, market, sector and asset class), market universe, risk management (both exogenous and endogenous), amongst many more.

Moreover, self-classification as a “trend follower” does not in reality preclude a manager from allocating a proportion of risk to “non-trend” strategies. A non-trend component (i.e., those broadly defined as orthogonal to pure trend) will likely perform very differently than the trend component of a portfolio. If we compare two hypothetical managers, with one only using pure trend strategies and the other allocating a material proportion of their portfolio to non-trend strategies, we would be unsurprised to find material dispersion between them. An apple and an orange look fundamentally quite similar if they are both rolling down the same hill at high speed, but they will taste quite different if you bite into them.

What’s the frequency, Kurtosis?

Given the above thoughts on dispersion and difference, we thought it would be interesting to undertake a statistical analysis of the relative performance of a group of CTA strategies versus SG Trend and evaluate the potential to experience tail events or large outliers in the distribution of relative performance to see if the above points are borne out.

Since we are primarily concerned with heavy tails, statistical kurtosis is a useful tool. Despite sounding like a mild throat infection, kurtosis is actually considerably more useful as it seeks to quantify the shape of a probability distribution. It is employed in statistical analysis to measure the frequency and magnitude of outcomes residing inside the tails of the relevant distribution and is expressed using the mathematical formula:

where μ₄ is the fourth central moment and σ is the standard deviation. For univariate time series containing Ν samples, kurtosis is given by the following formula:

where Y_i are the outcomes of the distribution, Y_mean is the mean of the outcomes, σ is the standard deviation of the distribution and N is the number of outcomes within the distribution.

These formulae make kurtosis particularly sensitive to the density of extreme values that are found in the tails of the distribution of a dataset. This is useful because it can be used to estimate whether a time series has heavier or lighter tails when compared to a normal distribution.

Types of kurtosis

The kurtosis of a statistical distribution can be classified into three main types (illustrated in Fig 1 below):

Fig. 1: Types of kurtosis, a graphical representation. Source: DG Partners

• Mesokurtic. In a mesokurtic distribution, tail events are expected to occur with the same frequency as in a normal distribution. Measurements of many natural phenomena, such as IQ scores, human height, atmospheric temperatures and the size of snowflakes, fall within this category.

• Leptokurtic. In a leptokurtic distribution, kurtosis exceeds that of a mesokurtic distribution. In other words, extreme outcomes are more frequent than in a normal distribution. Examples include the series of returns of many futures markets.

• Platykurtic. In a platykurtic distribution, kurtosis is less than a mesokurtic distribution. Deterministic signals where surprise measurements are very unlikely to occur are an example of a platykurtic distribution of outcomes.

Research observations

Interestingly, when examining large CTA’s strategies relative to SG Trend, our research suggests that there is a higher frequency of extreme values than one would expect in a normal distribution. In other words, the distribution is leptokurtic. Indeed, we have found that even CTA strategies that demonstrate long-term outperformance relative to SG Trend can have prolonged periods of relative underperformance versus SG Trend.

We have also found that CTA strategies with a high correlation to SG Trend can have leptokurtic distributions of relative returns of comparable size to those CTA strategies that are significantly decorrelated from SG Trend. Or to put it another way, irrespective of their correlation to the benchmark, CTA strategies are likely to experience frequent tail events relative to that same benchmark.

Conclusion

Whilst benchmarks can be an interesting adjunct to an assessment of the appropriateness of a given strategy, there are several potential pitfalls one must be aware of when taking them into account. Given that our observations show leptokurtic distributions of outcomes, we have to expect that tail events will occur with a frequency that is larger than that predicted by a normal distribution.

Understanding this principle enables investors to expect deviations from SG Trend in both a positive and negative direction no matter which trend follower they invest with. And from the manager side, it is good to be honest about this because it enables a constructive dialogue with investors around the fundamentals of a strategy rather than being overly focused upon deviations from indices.

If you are a professional investor interested in seeing the data underlying the above analysis, please do get in touch.

Contact Information

For more information please contact:

Investor.relations@dgpartners.co.uk

Tel: +44-207-408-5200

¹ Defining medium-term trend following is an entirely separate area of discussion, with industry practitioners taking varying approaches. These different methods often result in meaningfully different portfolio outcomes. We will be publishing more on this topic at a later date. ² Bloomberg ticker: “NEIXCTAT”