Trend Following Primer Series – Compounding, Path Dependence and Positive Skew – Part 8
Primer Series Contents
- An Introduction- Part 1
- Care Less about Trend Form and More about the Bias within it- Part 2
- Divergence, Convergence and Noise – Part 3
- Revealing Non-Randomness through the Market Distribution of Returns – Part 4
- Characteristics of Complex Adaptive Markets – Part 5
- The Search for Sustainable Trading Models – Part 6
- The Need for an Enduring Edge – Part 7
- Compounding, Path Dependence and Positive Skew – Part 8
- A Risk Adjusted Approach to Maximise Geometric Returns – Part 9
- Diversification is Never Enough…for Trend Followers – Part 10
- Correlation Between Return Streams – Where all the Wiggling Matters – Part 11
- The Pain Arbitrage of Trend Following – Part 12
- Building a Diversified, Systematic, Trend Following Model – Part 13
- A Systematic Workflow Process Applied to Data Mining – Part 14
- Put Your Helmets On, It’s Time to Go Mining – Part 15
- The Robustness Phase – T’is But a Scratch – Part 16
- There is no Permanence, Only Change – Part 17
- Compiling a Sub Portfolio: A First Glimpse of our Creation – Part 18
- The Court Verdict: A Lesson In Hubris – Part 19
- Conclusion: All Things Come to an End, Even Trends – Part 20
Compounding, Path Dependence and Positive Skew
So far in this Primer series we have steered you towards at least appreciating how we, as Trend Followers, tend to think. We are a strange and sceptical mob who spend a lot of their time just observing the way processes unfold and philosophizing.
Now you may say that philosophy is all well and good but who needs to be a philosopher if we simply want to trade the markets? Well go ahead. Trade what you see or what you think, but don’t come to me crying when you get your ass whipped and then find in your disaster debrief that ‘what you thought’ and ‘how you saw’ was the actual reason you got your ass handed to you on a plate.
A belief system is so very important to your trading endeavours, as it manifests in your behaviour and thus the way you interact with these markets. Financial markets are far too important to leave in the hands of the mere mathematicians and quants. It needs philosophers to tell us a different story and open up new ways to think about trading.
Yes, we are philosophers…and proud of it. You see our whole raison d’etre for our philosophical focus lies in the fact that the Financial Industry to-date, has been woefully inadequate in providing us with the requisite tool kit to apply our trade in the tails of the market distribution and trade the divergent nature of this condition.
A one-size fits all mentality has been applied by industry under the assumption that we all trade the peak of the market distribution of returns and like to predict. Without the correct tools to use, and a litany of ‘convergent baggage’ lying in current statistical science, we just have to be philosophers.
Now while our philosophical viewpoint biases us towards divergent techniques as our preferred method of wealth creation, the real wealth creator responsible for most of the heavy lifting over a successful trading career is not the edge that lies in the technique itself (of trend following) but the path of the return stream that accompanies this trend following edge. It is the path taken which can either assist or detract the compounding effect.
Now many will say that it is the linear ascending equity curve which is the optimal path for compounding to take positive effect over the trade series, but we beg to differ. It is the stepped equity curve that is characteristic of the trend follower that offers greater benefit which includes material favourable outlier impacts embedded in the signature, but never strays too far into adverse drawdowns. The style of equity curve that is delivered by trend following methods at the global portfolio level.
You may say, “well that is not how I understand trend following. The method is inherently volatile, as are all methods with positive skew”, however once again you haven’t understood the ‘complete story’ about path dependence and compounding. Compounding takes time to achieve, but can be accelerated when including the beneficial impact of outliers in the equity curve. Compounded equity is not something delivered in a few hundred trades. It only manifests over thousands of trades, but an accelerated uplift early on it the time series from the benefits received from a few positive outliers, turns a wealth story into a rags to riches tale.
Chart 18: The Volatile but Powerful Risk Signature of Transtrend BV: Enhanced Risk Program
Table 6: Monthly Performance Returns of Transtrend BV: Enhanced Risk Program
Have a look at the equity curve and monthly returns of Transtrend above, a respected trend follower in the industry. Notice how the high returns between 2000 to 2009 have vaulted this Program way above the performance of a buy and hold in the S&P500TR Index. The reason for the accelerated growth profile can be attributed to the number of outliers it caught in its early years, whose benefit via the principle of compounding still are felt today. Notice the volatile nature of its signature particularly between 2015 to 2020 yet it has managed to keep its maximum drawdown low to 16% which has preserved it’s stellar long term performance.
Now, name me a Fund Manager (apart from a handful of ones already listed in Part 6 of this Primer series) that devote their efforts towards convergence as opposed to divergence. Here is a hint, you just won’t find them. The illusory stability of convergence, with its nice linear equity curves at the level of the individual return stream, are examples of approaches that conceal ‘warehoused risk’.
Despite the attractive lure that convergence plays, it is a short term ephemeral signature associated with temporary market stability. A symptom associated with playing around with strategies that have negative skew. The real track record for the convergent player, when viewed over the long term (such as a trading lifetime) is either a story of fantastic blow ups or at best a far bumpier ride than what divergence delivers. Most convergent players only last a few years before they hit the trading graveyard. They never experience the fruits of what compounding can deliver over the long term.
Not to say that we can’t dabble with a bit of convergence if we really know what we are doing, as I know a lot of clever guys who can pull of this stuff off when wisely applied, but without extensive experience in managing the inherent risk of negative skew, the process can compromise the ability to deliver paths that can benefit from the compounding effect.
So clearly, ‘time in the game matters’, and a prerequisite for that condition is to have a philosophy that can maintain an edge over an extensive array of different market conditions, but the edge only needs to be weak. Just sufficient to deliver a suitable path of returns that can then benefit from the heavy lifting of compounding. If we are performance chasers only looking for strategies that ‘currently’ work, or deliver illusory returns offering the false promise of a strong edge for a single market condition, then we are likely to select ‘convergent methods’ or worse still convergent methods that are excessively leveraged.
Once again for our trend following world we need to ‘flip the mindset’, and look for solutions that have stood the test of time, offer a small edge that realistically is present in a ‘mostly’ efficient market, and do not stray too far into the adverse left-hand tail of the distribution of trade returns (not to be confused with the distribution of market returns).
Dangers await in the dreaded zone of the left tail of trade returns. As trend followers, who trade both long or short, our trade distribution results convey our characteristic signature of ‘cutting losses short and letting profits run’. We never let our trade results stray into large losses associated with the left tail where danger resides, but leave ourselves open to the bounty available in the right tail of the distribution of trade results.
Without an edge as a pre-requisite condition in our portfolio development process, the ‘two-edged sword of compounding’ can send our system or portfolio into accelerated oblivion. We discussed what an edge means in our previous Primer.
Most Retail Traders do not consider this little nugget of advice and are only concerned with either the win rate or the positive expectancy of their trading solutions. This is where they typically get left behind by the professionals. They think wealth is tied to profit factor, and forget totally about compounding. Sure, profit factor is nice, as is a system with an edge, but having a return path configured for wealth building takes your wealth to a whole new level.
So, when it comes down to wealth building, it is the principle of compounding that takes the primary role and the compounding effect is intricately connected to path dependence.
We have visited this ‘path dependent’ statement before in this Primer series where we referred to an auto-correlated time series of market data as having path dependence and why we are seeking this serial correlation as trend followers. However now we are referring to the serial correlation that is present in a trade series, as opposed to a market data time series.
Given that the bulk of the heavy lifting in wealth creation can be attributed to compounding, attention then needs to be placed above and beyond the mere edge of the strategy, to now focus on the actual path taken by the returns of a strategy over time which is exhibited by the equity curve of the strategy.
No matter which style of trading strategy you deploy, the method is just the means to achieve an edge, which then allows you to use compounding over time to achieve a far greater end. What we are seeking is a method that offers both a weak edge AND can provide us with the best return over time that provides an optimal path for compounding to work its magic.
So, with both convergent and divergent styles to consider that can offer an edge, which is better suited for compounding? To be able to answer this question, we need to understand what a geometric return is.
A geometric return differs from an arithmetic return by considering the impact of serial correlation in an equity curve and is primarily used for investments that are compounded. It is expressed as:
Serial correlation is a path dependent concept where a time series possess variables that are correlated across time. For example, an equity curve is a representation of the performance of a trading strategy over a particular time horizon. If there is an upward overall bias in the equity curve over time which lead to progressively higher equity, then the equity curve is said to possess positive serial correlation.
Where serial correlation in the equity curve changes direction and becomes negative in direction leading to auto-correlated losses or the serial correlation ceases to exist leaving only an independent series of random returns remaining, an equity curve then either enters drawdown or stagnates.
An equity curve with overall positive serial correlation possesses a bias towards positive equity, with only a few smaller excursions into negative equity. Hopefully, the serial correlation in the trade history is spread throughout the trade history leading to less ulcers for the trader, than those equity curves where serial correlation ‘clusters’ and is haphazardly dispersed through the trade history. Where and when the serial correlation resides, and its direction, are the dominant culprits for the volatility of an equity curve.
Under a serially correlated path, the impact of compounding over that time series at regular intervals is exponentially magnified in accordance with the degree of positive serial correlation present in that series and where it lies (aka its distribution) in the series.
When positive returns that are generated by the strategy are reinvested back into the equity at regular intervals with a positive bias, we hopefully achieve an upwardly rising equity curve. Of course, the trajectory of the equity curve relates to our overall performance. If we have an edge over time (which is a serially correlated positive bias in the series), then the equity curve will rise over time. If however we have negative expectancy, our equity curve will decline in value as we have no profits to reinvest back into the curve.
The intent of profit reinvestment is to progressively magnify the equity which is then periodically compounded at reinvestment intervals. This path dependent feature of investment is critical to wealth building as we want the principles of compounding to do most of the work in wealth creation.
If you consider the compounding process and how it applies at equal intervals over a time series, you could imagine that if an equity curve is rising, then the impact of progressive compounding through reinvesting profits) increases. For example, let’s assume we risk 1% of equity per trade for a particular strategy. As equity builds over time through the reinvestment of profits, if serial correlation persists, then the 1% trade will act on progressively building equity leading to an exponentially rising equity curve over time.
Conversely, if the strategy has a long unfavourable drawdown where much of the time is spend in negative equity with equity declining, there obviously is little profit to reinvest. The 1% trade risk application where any profits are reinvested does not have the ability to generate as much profit from the compounding effect than a favourable equity curve continuously reaching its high watermark.
So, let us have a look at a few different equity curves arising from non-compounded strategies and observe the impact on this series when profits are reinvested in the strategy (aka we turn the series into a compounded series). The variation in Net Wealth between different strategies attributed to compounding will astound you. You will then clearly see how the path of the equity curve over time is critical in the outcome with respect to long term net wealth.
Having this understanding then allows you to compare the equity curves of different trading strategies and identify those that are optimally configured for compounding.
Chart 19: Non-Compounded Equity Curves of Three Separate Systems with an Edge
Chart 19 above displays 3 separate Equity curves derived from 3 separate trading systems that all possess an edge. The non-compounded result of each trading system is very similar where each strategy commences with a single dollar in equity and at the end of 1135 trades all possess similar ending equity balances of approximately $1,144.
To achieve this non-compounded result, all we do is apply an equivalent fixed $ risk per trade over the duration of the time series. All trades are treated equally in terms of their allocated $ risk.
Despite the similar result, each take a different path to achieve this result.
- System 1 is a steady performer over the entire history of trades with no significant drawdown or acceleration in returns.
- System 2 has a slow start with subdued performance up until trade 1,045 (Label C) where it then experiences very strong equity growth.
- System 3 has a very strong start easily outperforming both System 1 and System 2 until trade 1,045 where it starts to enter a steep drawdown.
So now let us have a look at the impact of these 3 different paths when we compound the result. All we are doing is applying a trade risk % to each trade based on the level of equity at the time of placing a trade. This therefore means that we apply a fixed % risk per trade of Equity as opposed to a fixed $ risk per trade which achieves the non-compounded result.
Chart 20: Compounded Equity Curves of the Same Three Separate Systems with an Edge
The compounded result displayed in Chart 20 possesses a totally different performance result to that achieved through the Non-Compounded Result. Clearly, we can see that in terms of overall Net Wealth at the end of the time series, System 1 produces the superior result. This all attributed to the heavy lifting of the compounding affect and the equity path taken by each separate system.
Let us examine why?
The initial trajectory of the System 1 uncompounded path is subdued (Point A), compared to both System 2 and System 3. As a result, when the principles of compounding are applied after each trade to these subdued equity levels, we do not have much growth in the equity curve in the early stages of the trade history (up to Point B). We cannot see this subdued effect early in the time series in the Compounded Chart takes time for compounding to exercise a bias to the time series.
However, when we reach trade 1,045 (Point C) we start to see compounding take hold in the time series with a rapid acceleration in equity. Given the late stage where compounding takes hold, despite the rapid rise in equity from Point C, it does not catch up to the levels of equity achieved by System 2 and System 3.
Unlike System 1, System 3 is a very strong early performer, and you can quickly see how compounding takes effect to lift the equity curve well beyond the alternate strategies. However, we start to see the acceleration diminish from Point B with a rapid deceleration into negative equity by Point C. This under-performance is very detrimental to a compounded equity curve and quickly reduces overall equity of System 3.
System 2 however is a very steady performer and as a result, the compounding affect can accelerate the equity curve along the entire trajectory. Clearly it is therefore the consistent more stable equity curve we are after to deliver superior compounded returns.
However as discussed earlier, an even better path for compounding is one which has significant lifts in the equity curve early on in the time series, yet never suffers any significant adverse volatility that compromises the compounding effect. Having the early-step-ups means that compounding is accelerated and time is provided to turn that acceleration into an exponential growth curve from the get go.
What significantly compromises the compounding effect of an equity curve is a progression into negative equity. Compounding, like leverage, is a two-edged sword. When equity is building, compounding magnifies the $ gains but when the equity curve is entering drawdowns, compounding magnifies the $ losses.
From a close examination of the 3 systems described above you will notice the path taken is critical for the compounding effect to deliver long term wealth. While the compounding bias is slight, it plays a progressively more dominant role over the long term with a larger trade sample size. Given the short-term lifespan of a retail trader, the powerful effects of compounding are rarely experienced. This is why trend following is considered a long-term game of wealth building. Convergent players may achieve fast returns immediately but we are after far superior returns for the long term.
Clearly, if we want to play this trading game and achieve wealth over our lifetimes, then we need to take compounding seriously. The only way we can participate in the fruits of compounding is to be a survivor that can boast a long-term trace record with a persistent edge.
If that edge is compromised by significant extended drawdowns, or if our equity curves are too ‘negatively’ volatile leading to very sharp drops in equity, we will severely compromise our wealth ambition.
While trend following spends a lot of its time in drawdown, the extent and speed of these drawdowns are limited when compared to other methods over the long-term time horizon. This is what makes their return streams ideal for compounding.
Our aim with our Trend Following method is to achieve a portfolio result that embraces positive volatility yet avoids negative volatility. If, at the portfolio level, we can smooth the equity curve to avoid extended periods under-water, and better still retain the characteristic step-up signature, then we can use the miracle of compounding to lift our equity to the heavens.
It is not the volatility of the Equity Curve that bothers a trend follower, as favourable volatility (which results in an equity uplift) is exactly what we want to take advantage of with compounding. What we want to avoid is adverse volatility that detrimentally effects equity levels and diminishes the ability of compounding to magnify the equity of the time series.
Now the long-term sustainability of progressively rising equity curves is significantly hampered by the trading technique you wish to deploy. Some equity curves delivered by convergent methods over the short term, have a deceptive linear ascent while favourable market conditions persist, however this short-term feature is compromised when conditions change and become unfavourable. The linear ascending equity curve is now compromised by significant and fast drawdowns in equity which significantly impede the compounding effect.
This style of equity curve comprising smooth ascents over favourable conditions and then dramatic drawdowns during unfavorable regimes which may also lead to risk of ruin if the favorable condition does not resume is associated with predictive systems such as mean reversion methods.
Fortunately for diversified systematic trend followers, we adopt extensive diversification amongst a myriad of different equity curves that when compiled into a portfolio provides a relatively smooth ascent of portfolio equity including step-ups with far lower and less aggressive drawdowns over the long term.
This feature of diversified trend following systems makes them an ideal candidate for generating exception long term wealth aided by the principles of compounding.
We need to drill down more into how a diversified trend follower manages the volatility present in their portfolio’s equity curve, and to understand this, we need to investigate the concept referred to as Positive Skew.
Now that we understand that is it downside volatility as opposed to upside volatility that we need to be concerned with in delivering the optimal path to wealth building we can throw out the traditional ‘Gaussian’ tools used by industry for risk management such as the Sharpe Ratio which are ambivalent in their treatment of the direction of volatility, and other risk management measures that are not path dependent methods such as the Sortino ratio.
Remember that we operate in the non-linear land of the fat tails where standard statistical measures based on an assumption of ‘Efficient Markets’ no longer applies.
We need to look at risk management tools that explicitly deal with asymmetrical ‘exotic’ market conditions associated with fat tailed behaviour, as opposed to symmetrical risk management tools associated with markets in equilibrium. Our trade performance as trend followers is dictated by our ability to catch outliers and prevent ourselves from wandering into the left tail of the distribution of trade returns. This leads us to the characteristic ‘stepped equity curve’ that is naturally configured to offer the greatest geometric returns through compounding.
One of our biggest weapons we use to allow for an optimal geometric return (or path) is our preference for only trading systems with positive skew.
When discussing positive skew, we are referring to the skewness of the distribution of our trade results. This should not be confused with skewness associated with market data. Given that our divergent methods have a low win rate but a high reward to risk ratio, we find that if we plot our trade results in a histogram, only a handful of successful ‘outlier’ trades are the real drivers of our profitability (as we let profits run).
By far the majority of our trades are either small losers (as we cut losses short) or are breakeven trades or small wins representative of the fact that the vast majority of our trades are just random results. There are only a handful of trades that have been able to catch a trend with endurance backing it.
Chart 21: Distribution of Trade Returns of a Typical Trend Following System with Positive Skew
The result for a plot of our distribution of trade results typically follows the histogram above in Chart 21. If you closely examine the chart, you will see that there are no or very few significant material losses. The vast majority of losses cluster with the range of 0 to -6 R. In this example R is simply a way to express risk$ in equivalent terms. So, a risk of 10R is 5x larger than a risk of 2R. There is only 3 trade events that were worse than -6R attributed to system application errors.
You will also note that the vast majority of winners cluster in the range between 0 to +12R. The total distribution between -13R to +12R leads to a breakeven result for this model. However, it is the few winners that exceed +12R all the way to +45R that lead to overall net wealth of the system.
What you will note from this distribution is that is it constrained in its left tail. This means it is exceedingly difficult to achieve significant material losses that lead to fast and excessive drawdowns.
In statistics, a positively skewed (or right-skewed) distribution is a type of distribution in which most values are clustered around the left tail of the distribution while the right tail of the distribution is longer.
Skewness refers to an asymmetry that deviates from the standard bell curve (or normal distribution) in data (in this case trade results).
Distributions can range between having positive skew to zero skew to negative skew. In essence this can be stated in the following manner.
If your trade distribution has much larger average $wins than average $losses, then the distribution will be positively skewed. If your trade distribution has equivalent $wins to $losses, then the distribution will have zero skew and follow a normal distribution. However, if your trade distribution has smaller average $wins than average $losses then the distribution will have negative skew.
In practical terms a trader with negative skew, provided they have a high win rate, will achieve positive expectancy and the equity curve will grow, however if market conditions change and the Pwin% drops, they become excessively exposed to the left tail of the distribution of trade returns. A sequence of large $losses (which is a symptom of negative skewed strategies) leads to fast drawdowns and potential risk of ruin.
A trader with positive skew however can survive far longer during unfavourable market regimes as the small losses do not build drawdowns to the same material degree as their negative skewed counterparts. Furthermore, the system risk constraints that prevent the trader from entering the left tail make this style of system surprisingly robust and capable of weathering uncertainty.
The speed and nature of drawdown associated with the skewness of the trading system is a major feature that compromises geometric returns.
The upside volatility associated with the large average $wins of a divergent trader contributes to Geometric Returns whereas the large average $losses of a convergent trader can compromise long term wealth ambitions.
Let us have a look at how we can possibly identify positive or negatively skewed trading systems or portfolios by looking at their equity curve.
Have a look at the non-compounded equity curve below in Chart 22. The blue line represents the equity balance from all closed trades and the green line represents the unrealised balance or the ‘mark to market’ value of equity at all points in time. This green line represents the value of all closed trades and the unrealised profits or losses on all currently open trades.
Chart 22: Non-Compounded Equity Curve of a Positively Skewed System
You will notice that there were never any points where the green line wandered significantly below the blue realised balance. This means that over the course of the trade history, there were no points in time where the unrealised risk held by the strategy got out of hand. The strategy simply always cut losses short. There were however times where unrealised equity was higher than the realised balance which is a consequence of leaving some profit on the table. Unfortunately, this is the nemesis of the trend trader. We can never time the end of trends and unfortunately always leave some profit on the table when the trend ends.
Now while we had to leave some profit on the table, we never allowed the strategy to stray towards the left ‘fat tailed’ side of the distribution of trade returns. In practical terms this means that we never exposed ourselves unduly to future risk. We managed risk in the strategy across the entire duration of the equity curve never letting it get out of hand.
The profile of the equity curve however now lends itself to compounding. There are no significant material drawdowns that compromise the compounding effect and there are many upward jumps in the equity curve which accelerate the compounding effect on the series.
Now let us have a look at an equity curve with symptoms of negative skew (Chart 23).
Chart 23: Non-Compounded Equity Curve of a Negatively Skewed System
In this extreme example which applies to a Martingale Technique that is a classic example of an extreme negatively skewed system you will notice that the unrealised equity curve (green) can sit well below the realised balance. We refer to this profile as a strategy that is ‘warehousing risk’. Other common examples include strategies that warehouse risk using averaging down principles.
Convergent strategies such as mean reversion techniques frequently fall into this class given their need to apply tight profit targets and more lenient stop losses (if used at all) to catch the falling knives of mean reversion.
In Chart 23 above, without a regular mark to market (appraisal of unrealised equity) it appears (using the blue line) that the strategy is performing exceedingly well with a steadily rising equity curve and few apparent losses. This strategy however has been closing all trades in profit and holding onto trades in a loss and never realising the risk.
The real story of this strategy is that it possesses extreme negative skew. Many small losses with only a handful of losses that can lead to complete risk of ruin.
Now for the purposes of compounding you can clearly see the result of the fast severe drawdowns and its impact on equity. Such dramatic drawdowns have dire consequences to the compounding effect which is dependent on the path taken by the return stream.
So, what we have learned in this brief introduction to Skew is that the compounding principle favours those systems that display positive skew.
Compounding with Leverage
Now we understand the pivotal role that path dependence plays in delivering wealth through the compounding effect, we can now turn to the impact of when we get too aggressive with our leveraged trend following solutions.
Remember that compounding is a two edged sword. If the impact of compounding over the long term progressively exponentially magnifies the result, we can be left with a death defying ‘pogo stick ride’ with some of the FM’s that play with higher leverage. I refer to these style of highly volatile trend following funds as “Formula One Grand Prix racers with speed wobbles”.
Chart 24: Mulvaney and the pogo stick
Now the Trend Following community love Paul Mulvaney but that is just because we understand his philosophy that goes for the ‘king hits’, however for those that get ulcers when big drawdowns arrive, then they may get concerned. There is no doubt that the Mulvaney Program is one of the top performing Trend Following Programs in the world, but if you want a piece of this extreme action, then do not look at the monthly performance.
Paul clearly knows his stuff given his very long track record and recognises that with high volatility comes very high returns, but this can turn a Trend Following method into one in which encourages investors to time their entry into this style of Program which you want to avoid.
Now as long-term as a Trend Following Program might live, it won’t last forever and there comes a time where we are tempted to just let the reigns go, loosen our risk belts and go for the king hit, like Paul. You see if we put on our racing gloves, we can choose to go-for-broke to demonstrate the real power of these Programs under leverage, however for those investors seeking high Net Wealth in say 50 years time from now, then such a volatile offering may not be the solution as timing entry into the equity curve becomes important.
Other Trend Following Managers deliberately turn down their potential returns by reducing their leverage to cater for a long-term investor seeking a smooth ride where compounding can take the full effect.
So now that we understand that the profile of the equity curve (or more specifically the path of returns) is the major feature that enables compounding to accelerate wealth, we can now explore a central feature that Trend traders focus on to achieve an optimal path of returns. This is what is referred to as a Risk Adjusted approach to maximise geometric returns which we will be exploring in our next Primer episode.
Stay tuned for our next instalment in this Primer Series…..but before I go, I thought I would leave you with a song to pay homage to those trend followers that provide highly leveraged offerings.
Trade well and prosper
The ATS mob