I posited in an earlier post that fat tails are one of the main reasons why trend following works. The underlying concept can be summarized as follows: trend following attempts to capture big price moves (a.k.a. trends). Since price distributions are *leptokurtic* (i.e. they exhibit fat-tails) long trends occur at abnormal frequency, providing greater sources of *alpha* for trend followers.

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:

### 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…

Mark// Dec 9, 2009 at 10:28 amThis paper is nonsense because it based on wrong assumptions: random walk and efficient markets hypothesis (EMH). No matter how much analysis you do to simulated data, the conclusion has nothing to do with real trading.

But it is good to have this kind of papers to flow around to discourage more people come into LTTF. That’s why most people don’t make money in the market. They listen to the so-called academic research based on fake data.

Trey// Dec 9, 2009 at 11:30 amKurtosis, per se, doesn’t imply that trend following works. Those outliers moves must be in the same direction of the trend in order to benefit a trend following system, right? If they are against the trend then large moves erode the profitability of TF system.

TF systems need drift in order to profit. Many time series don’t actually have a drift, but many random drifts, similar to streaks in a coin toss.

The regression in that papers appears to suffer from multi-collinearity, which if present renders the results meaningless. Each independent variable is multiplied by stddev. This induces dependency among those variables. The extremely high R2 is evidence of multi-collinearity as well.

There is much to learn from randomly generated data. You’ll see many patterns that you see in the market, which are no more real than the constellation Orion.

Jez// Dec 9, 2009 at 5:43 pm@Mark, I guess you are expressing the same as I felt – just in a “more opinionated way” ;-)

@Trey I personally see kurtosis/fat-tail “at a higher timeframe” the source of trend following, ie if you trade a TF system on EOD data, the fact that monthly price data exhibits a leptokurtic distribution will profit your system, because each outlier monthly return move should generate a trend for your daily system – and the fat-tails mean these moves are more frequent than normal.

Trey// Dec 10, 2009 at 6:49 amScale is irrelevant in this context. My point is that by merely observing a distribution, you can’t deduce that the existence of tails is why trend following supposedly works. It must be in the right direction. The histogram doesn’t tell you this.

One of the well known characteristics of volatility is clustering. This means that there are periods of high volatility and periods of low volatility. Therefore the excess kurtosis that you’re observing in this histogram, both the left and right hand side, tend to occur in the same state, i.e. a state of high volatility. If a market is moving back and forth at very high levels, then this is very bad for a trend following system. It gets whipsawed to death.

Histograms tell you size and frequency but they don’t tell you structure. Structure is what you really want to know. Once you identify a particular structure within a market, you can then build a trading system around it.

Alex// Dec 10, 2009 at 4:07 pm@Trey: I agree with you (have been already commenting in this direction in a previous post of Jez).

Can you specify what you mean by structure?

Jez// Dec 11, 2009 at 4:50 amI think I “got you” now and see what you are saying.

For me, one of the main characeristics of trend following is that it is a strategy that cuts your losses short and let your winners run. From that point of view only (ie ignoring the “trend entry” part of trend following, if you will), a high kurtosis will ensure that large or very large winners occur at a “higher than normal” frequency (ie fat-tail ends of return distributions). Looking at this argument only, random entries with SL but no TP should provide profitable returns.

Note that I have not verified that theory by doing a proper analytical research but this sounds logical to me (and seems confirmed by Trend Followers themselves).

However, this makes a big assumption: that the losses that might result from whipsawing in a trend following system do not outweigh the gains from venturing in the fat-tails. And I believe this is the point you are trying to make with the importance of the direction of volatile moves: if they tend to “move back and forth at very high levels” this will have a detrimental effect to the TF performance system.

Maybe, an additional calculation of the Hurst coefficient/fractal dimension in the time series would help quantify the whipsawing?

Happy to continue the discussion (and hear if you think that does not make sense) as it helps refine my understanding.

Trey// Dec 12, 2009 at 8:45 amAlex – For example, two structures of volatility are clustering and mean-reversion. This leads to two states of volatility, high and low, which could then be incorporated into a strategy, as opposed to a strategy which trades volatility as singular state. Structure isn’t always profitably tradable though.

Jez – Yes. That is correct. Non-directional states tend to outnumber trends which can often lead to ruin.

Stops are a function of the strategy, not the market. Cutting your losses short is a great saying but I think it’s devoid of any real meaning. It’s way too general. Performing a simulation with random entries is a great exercise. Use a stop, run 10,000 simulations of about 300 random entry trades. Perform it on real data and simulated data. With the simulated data, you can alter the drift and volatility.

There is direction and there is volatility. Step one is quantifying them. Step 2 is then determining if there a relationship between them. Correlation is good place to start but it only measures linear dependency. Don’t forget that correlation isn’t causation. Also, if the relationship is non-linear, correlation won’t capture that.

Trading is statistics and time series analysis. I’m amazed how many “trading books” fail to cover these relevant topics.