non-stationary riddles - pic by Jaci Berkopec

Non-Stationarity, is defined as a quality of a process in which the statistical parameters (probability distributions) of the process change with time.

One of the consequences is that it might not make much sense to consider and model financial markets as one big distribution over time. And this is where Trading Regimes come in.

The concept behind regimes is that non-stationary time series, such as market data, shift through different “modes” during which market behavior and dynamics are different.

Being able to recognise these trading regimes would a allow a trader to adapt his strategies to better respond to these regime shifts.

Trading Regimes as Filters

Readers interested in the more formal definition of regime switching in econometrics might want to investigate Markov-based systems.

In this post, we will make a looser interpretation of the term: Trading regimes can be thought as providing a context to the trading strategy. This could be any meta-indicator or external variable susceptible to affect or “predict” the performance of a trading strategy. Let’s scratch the surface by making a (non-exhaustive) list of potential ideas.

Directional Trend Filter

OK, we start with a wide departure from Trading Regime in the formal sense of the term.

The directional trend filter however is a basic and popular filter. It is based on the idea that “the trend is your friend”: it considers the trend on a higher timeframe and only allow trades in the direction of the main trend. The trading regimes might be thought of 2 or 3 modes: bullish/uptrend, bearish/downtrend or neutral.

Exceptional Conditions Filter

Another possible filter could be a switch indicating when the market makes an extreme divergence from history. The idea being that a strategy designed and tested on historical norms might degrade during these abnormal periods.

Michael Stokes from MarketSCI discusses his Abnormal Market Filter implementing a smart “shades of grey” concept, with a “dimmer” switch rather than a binary ON/OFF, normal/abnormal switch.

This filter is based on extreme volatilities but I have also seen traders using abnormal correlation filters (when many markets in the portfolio exhibit sudden spurious correlations such as during credit events).

External Factors: Macro Environment

There are many causes to instabilities of financial markets, for example business cycles or monetary policies.

I alluded in a previous post how business cycles could be used in an interesting meta-strategy, trading regime filter:

it would be interesting to measure and test the impact of a macro filter on a Trend Following system (ie something in the vein of: “favour long trades in the system when the macro indicators indicate a period of expansion and short trades during declines”). Maybe exploring a mechanized version of Schumpeter Business Cycles or Kondratiev Waves as a very long-term filter would be a worthy approach.

Similarly, macro indicators such as yield curve, GDP, Monetary supply might provide some input into a Trading Regime model.

Trend Propensity: Fractal Measurements

For a Trend Following system (or a Mean Reversion system) it could be a great advantage to be able to identify those markets that are more likely to trend (or not). The Hurst exponent can be estimated on time series to derive their tendency to revert to the mean or cluster in the same direction.

Each market could be classified as trending or mean-reverting based on the value of its Hurst exponent.

Basic Strategies

This is a concept implicitely used by traders trading the equity curve of their systems. A simple approach is to monitor the equity curve of the system and apply a moving average to it: you can allocate more or less funds based on the current relative position of the equity curve to its moving average. An anti-martingale-type strategy would reduce funds when the system equity curve is below its MA.

This can be taken further by having a collection of “dummy” trading strategies, which performance is monitored to derive the current market regime (identified by which group of strategies is over/under performing). This is a concept explored in the Hack the Market post, linked above.

Price Behaviors

This approach is what triggered this post. One of my current working assumption is how Trend Following represents a trading strategy positioned to benefit from positive kurtosis and positive serial correlation.

One hypothesis I am investigating regarding these characteristics is whether they could be used for Trading Regime identification:

Understanding these market characteristics is a first step towards being able to identify and measure them. This, in turn should be a step to linking Trend Following performance to the state of these market characteristics. Finally, this might be a step towards devising a way for a Trend Following strategy to adapt to these changing market characteristics (this last point makes a very big assumption: market characteristic changes can be predicted with some degree of accuracy).

Hopefully more on that coming soon, with an attempt at calculating serial correlation “in the tails”.

There are of course other price behaviors that might be used for Trading Regime identification (mean, variance, volatility, covariance, cointegration, skew, etc.)

Does the concept make sense?

An important point to consider, for a trader planning on using trading regimes, is the fact that the approach would result in trading a collection of “more specialized” strategies (read: less robust), switching between them as the market switches between its various regimes.

Thinking that such an approach can be more succesful than a “one-size-fits-all”, more robust trading strategy makes a strong assumption about regime persistence, namely that the regime persists longer than the time it takes to identify the shift to the new regime.

In essence, if the regime switch identification lag is too long, you would probably end up applying the wrong specialized trading strategy to the current regime and suffer from under-performance.

I recently found a 30-page paper introducing the ideas and principles of his Leverage Space Model. I thought reading it might be a good way to get back into Vince’s material.

Follows a summary of the paper (PDF download link), which is maths-less (only concepts and principles are discussed). It is a good introduction to Ralph Vince theories.

Vince’s Optimal f

This is what Vince is famous for. It is basically a way to determine trading quantity (aka leverage) using the probability distributions of the trade outcomes.

If f represents the fraction of capital to wager (risk) on each bet (trade), the optimal value is the one which optimises the geometric growth of the bankroll (account balance).

In his previous books, Vince has defined the formula to determine the optimal f. The first part of the paper discusses the optimal f concept and is a good introduction for the non-initiated (showing how over-betting on a game with positive expectancy can and will result in a loss).

Leverage Space Model promises

The optimal f section discusses a single-component approach whereas the Leverage Space Model deals with multiple components portfolio.

It is presented as an improvement on the Modern Portfolio Theory, briefly discussed. This is based on the following advantages:

1. Risk is defined as drawdown (instead of variance in the MPT)
2. The fallacy and danger of correlation is eliminated
3. Valid for any distributional form – Fat-tails are addressed
4. The model is all about Leverage, which is not addressed in the MPT model.

Return aspect

The model starts by building a multi-dimensional terrain, drawing the overall expected return, based on multiple combinations of components in the portfolio and their respective f-values.

In this example the model builds the terrain for 2 simultaneous coin-toss with a payoff of 2:1. The x and y axis represent the respective f-values (leverage) for each of the bets/trades - while the z-axis (vertical) represents the expected return

The maximum portfolio growth is located at the peak of the terrain, resulting from the specific corresponding f-values combination. The terrain construction does not take into account correlation between the instruments – instead, the model uses the joint probability of two scenarios occurring simultaneously, dictated by the price data history.

The Risk Aspect

So far, the model has only looked at returns. To introduce the risk component, you must determine your maximum allowed drawdown. This is a hard and fast rule: no combination should breach that limit.

Using a derivation of the risk of ruin, the model computes the risk of maximum drawdown for each set of f-values (for a specific timeline – as, in the long run, the risk of drawdown tends to 100%). If the risk of drawdown is too high, the specific f-values combination is ignored.

In practice, the initial terrain is truncated: by removing all points breaching the maximum drawdown threshold.

The terrain has been truncated: all areas deemed too risky, from a drawdown perspective, have been removed.

The Algorithm

Vince implements a genetic algorithm to calculate the terrain, by initially calculating the expected return for each set of f-values, and secondly by running the maximum drawdown test on this same set. Once the whole set combination has been run through, the terrain is built (including truncations). The the aim is then to find the optimal set (highest return with lowest f-values).

Comments and Extra Info

The paper is rather short and does not deal with any of the maths behind the models. For this you’d have to get yourself a copy of the Handbook of Portfolio Mathematics which introduces the model in more detail or Vince’s latest book, dedicated to the Leverage Space Model, which has had a not-so-positive review by Max Dama.

The ideas in the paper are an interesting take on position sizing. Vince uses a simple objective/bliss function (CAGR with a binary risk/drawdown filter) to evaluate all possible scenarios of portfolio allocation/leverage. It might be interesting to use the concepts of the model with your own bliss function recipe.

One of Vince’s claim that the MPT does not address leverage sounds a bit simplistic – surely the percentage of Cash as an asset in the portfolio is an implicit measure of leverage. On the other hand, the approach on correlation/joint probability of scenarios sounds interesting and seems to go in the right direction. As Vince says:

Counting on correlation fails you when you need it the most.

Another point that seems missed out is how the model handles non-stationarity of the market. Vince mentions the chronomorphism of market prices distributions (i.e. they change with time) and even draws a betting comparison with blackjack – in which the optimal f curve changes for each card dealt. However there is no mention of how the model takes an adaptive approach to these chronomorphic distributions.

Vince’s homepage contains a link to the java software that implements his model (needs to register/leave email to download) and another one with a spreadsheet example. I have not had time to take a serious look at all those. Please let me know your feedback if you do.

Joshua Ulrich – blogger and reader of this blog (hello there: finally got round to adding you to the blogroll!) – is collaborating with Ralph Vince to port the Leverage Space Model to the R platform. His FOSS Trading blog is definitely worth a read too.

nebulous intricacies

In the last post we looked at the Turtle Trading system and saw that its performance went from outstanding for a long period of time to flat for 20 years. This opens a can of worms:

Does Trend Following work, is it dead, do markets change, does trend following rules need to adapt to these changes?

Let’s look at the different points of view.

The EMH Crowd

The EMH crowd does not believe in anything else than the Random Walk and by definition discards any profit-generating mechanical strategy.

As much as the Efficient Market Hypothesis (EMH) is a cornerstone of most modern financial theory, it has proven to be wrong partly because of some of its assumptions (not all actors in the market are rational, market prices are not fully random and normally distributed, etc.).

One great book that discredits the EMH approach and their descendant theories (CAPM, etc.) is by Mandelbrot: The (mis)behavior of the markets. It lays the arguments against the EMH in an approachable (ie not too much maths) way.

Trend Following is dead/does not work

Curtis Faith declared that “every few years trend following traders experience a period of losses and inevitably some expert will announce the end of trend following.”

Mike Covel also has a similar quote in his Trend Following book: “every 5 years some famous trader blows up and everyone declares trend following to be dead. Then 5 years later some famous trader blows up and everyone declares trend following to be dead, etc. ”

One of the main proponents of the argument against Trend Following is infamous trader Vic Niederhoffer (who “blew up” twice). He has been highly vocal about it, declaring Trend Following as one of the top Stock Market con.
Here is another link from his website to read more about it.

Arguments like this, despite the empirical evidence against it – in the form of Trend Following Wizards success, can be taken as a motivation for healthy skepticism and push you to strengthen your statistical research.

Are markets changing? (and must Trend Following change too?)

The self-professed “Trend Following poster boy” (a.k.a. Michael Covel) authoratively declares that this is a specious argument.

Occasionally, someone trying to promote something or start a debate will argue that trend following rules must always change due to changing market conditions. This is nonsense. It is a specious argument.

This is at best ambiguous. Covel likes to cite Bill Dunn who:

proffered that his basic system rules have not changed since 1974

Now, that is seducing: it seems to sell you the idea that you can develop a system, implement it and trade it for life. However this is not strictly true. As mentioned in this interview:

Dunn annually adjusts the parameters of trading signals and each markets weighting. In February – just as the grains were about to take off – he dumped the entire grain sector. But Dunn has no regrets.

Dunn is also known to have collaborated with Robert Pardo, a strong proponent of Walk-Forward testing (see below: a constant system adjustment).

To clarify: although Trend Following principles will never change, the rules/parameters of a Trend Following system might need to be adjusted to changing market conditions.

How can a Trend Following strategy adapt to the ever-changing markets?

In an Active Trader article, Anthony Garner attempts to discuss:

Do markets change? Is it necessary to undertake continued research and development and adapt a trend-following system to maintain its profitability over the years?

The article is only available for the magazine subscribers, but the result of the equity curve can be found on the Trading Blox forum. By “tuning up” the Turtle system, Garner manages to obtain interesting stats (MAR=2.26, CAGR=35.28%). The main change to the system is the use of a longer-term timeframe.

Practically

I know I would not be happy trading the original Turtle System in the last 20 years and get a 0% return. If you started trading this system “back then”, when and how would you think it is time to switch to a revised system?

Let’s look at options:

1. Walk-Forward

Walk-Forward testing‘s principle is to keep running (in simulation) a “pool” of systems using different rules/parameters. At regular interval, you evaluate what systems are best (performance, robustness, etc.) and trade those until the next re-evaluation. The potential risk with this approach is that you might end up like a dog chasing your tail.

However, this approach would have you switched from the original Turtle system to the new one a while ago.

2. Alternative Walk-Forward

Markets exhibit some degree of inefficiency – and Trend Following is a strategy designed to profit from these inefficiencies. I am still refining my theoritical understanding and explanation of it, but I believe Trend Following’s performance is mostly the result of fat-tailed distributions (distribution kurtosis) and possibly autocorrelation.

If one can associate the evolution of these characteristics to the performance of Trend Following systems it might be possible to adapt the system rules/parameters to the values and evolution of the price distribution characteristics. This is a topic I’d like to investigate further.

3. Mixing Systems

Finally, and this seems to be a strategy adopted by many professionals: mix different systems and different timeframes. Here the rationale is that we cannot predict what systems are going to under/over perform and mixing several ones together will smooth out the equity curve.

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.

Following the article, a reader of the blog (Alex) kindly forwarded me a research paper trying to identify which moments of a distribution (mean/drift, variance, skew, kurtosis) affect the returns of a trend-following strategy. This is an interesting read (10 pages, not too mathematically challenging) which I encourage you to read:

Click to download paper

Summary of the paper

The authors are mostly interested in currencies and in order to free themselves of historical data limitation, they generate artificial price data to simulate different types of price distributions by varying the different underlying moments.

They then apply a standard Triple moving Average Trend Following system to the different time series generated and measure the annualized gross profit for the simulation (over 5,000 trading days) for each type of distribution.

The other parameter measured is the Trading Frequency, from which they derive the auto-correlation characteristic of the underlying data (by applying the logic that a trend following system will trade in and out more frequently in a mean-reverting environment and vice-versa).

Conclusions

By applying some regression analysis to the various results observed, the authors arrive to the equation predicting the return of thee Trend following system:

TMA Result = 38.88Stdev(1 – 6.77TFrq + 0.0392Skew – 0.010Kurtosis + Drift(65.65 + 324,600Drift))

with a standard estimation error of 0.3%

The interpretations are that:

Market volatility (38.88 Stdev) determines the profit (or loss) potential of the trend-following strategy. This relationship is direct, so if market volatility doubles,so does the expected TMA result. Accordingly, it is no longer surprising that trend-following models tend to show the best results across the major currency blocks with high market volatility.

A high Tfreq will have a negative impact on trend model performance.

Skew will enhance performance, while the opposite is true for kurtosis. Drift will increase the value of the equation and thereby contribute positively to the TMA model result.

So it appears that the kurtosis (the source of fat-tails) actually has a negative effect on a trend following model (contrary to that earlier post) and in a relatively large way:

The currency path (auto-correlation/trading frequency) is the most important factor in determining performance (91%). The impact from kurtosis (68%) and drift (56%) is also significant. Skewness is less significant, but still explains 26% of variance on its own. Volatility has no importance at all (0.4%). This might initially come as a surprise, but as illustrated in Equation, it is a multiplication variable and so does not in itself generate trend model profitability (or loss where the path characteristic is unfavorable).

Comments

Mathematical theory is not my strongest suit (despite studying over 10 hrs of Maths per week in my prime!) and I am definitely thinking of getting some refresher training on that. But there are a few points that bother me in that research paper. To the more knowledgeable readers: “Please chime in and tell me where I might be wrong:

Simulated Data
This tends to make me a bit sceptical of the results especially with the authors’ random walk and efficient markets hypothesis (EMH) assumptions. After reading Taleb and especially Mandelbrot (I really enjoyed reading The (mis)behavior of Markets with its debunking of the EMH and alternative explanation of financial markets), I have been converted to the Dark Side of the force: I am not sure we really know how to modelise the data theoritically and this random simulation might be missing specific price data characteristics (granted, the authors empirical verification seems to confirm the assumptions…).

Measure of auto-correlation by proxy
The auto-correlation of the underlying data is actually derived from the trading frequency of the trading system itself. This does not sound right.
Whereas it seems intuitive that the more a trend following strategy trades in and out the least profitable, I am not sure this necessarily and direclty implies auto-correlation in the underlying data.

Only one model tested
Testing the performance of trend following by using one model only might not bee so representative. It would be interesting to see how the results translate to a collection of trend following systems (different styles, different timeframes, additional markets, etc.)

In any case, the finding that kurtosis has a negative effect on trend following returns seems to contradict my earlier post. As per the points above I’ll take the paper’s findings with a pinch of salt – but it might be a useful tool in determining when or not to use trend following strategy (by measuring the characteristics of the price distribution).

Whether trading using these market regimes identification is a valid and robust approach is a different question…

Trend followers position themselves to profit from and capture the “fat tails” exhibited in market returns distribution. In a fat-tail distribution (Power law, Levy or Mandelbrotian distributions), extreme occurrences occur with a probability greater than normal.

Fat-tail vs. normal distribution: notice the thickness of both extremes on the Levy distribution.

If you put in stops and run your profits and trade randomly you make money; and if you put in targets and no stops, and you trade randomly you lose money. So the old saw about cutting losses and running profits has some truth to it.

The basics of trend following is to ride the trend until the end (when it bends) and to protect yourself on the downside by cutting your losses.

This ensures that the location of your trades in the returns distribution will:

• Never venture on the left fat-tail (i.e. no extreme negative return)
• Not be bounded on the right-hand side of the distribution (i.e. allow for extreme positive returns)

As the markets are mostly random, most of the trades will end up in the centre of the distribution curve either side of the horizontal axis – and their return should cancel each other out.

Trend Following’s alpha (the actual strategy return) is generated by extreme movements: By letting trades run on the right-hand side fat-tail and stopping them from “wandering” on the left-hand side one, an overall positive return is generated. This outlines the fact that Trend Following relies on rare extreme returns (outliers) whereas the bulk of trades cancel each other out.

Note that this post simplifies matters to illustrate the fundamental point. Other parameters such as trading costs, etc. obviously need to be considered.

UPDATE: For those readers wanting to investigate this concept a bit further, a later post presents a research paper investigating the effects of the 4 first moments of the price distributions on the return of a Trend Following system. Please read here