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).
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).
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:
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…