# Measuring Market Divergence for Systematic Trend Following Screen Out Noise When The Market Cries Wolf

**Motivation**

In his 2012 book, Nicholas Nassim Taleb defined the term antifragile for representing things that benefit from disorder. When this concept is applied to trading, an obvious example can be seen in the relative outperformance of systematic trend following strategies during market crises.

In the CTA and technical analysis domain, divergence is defined as the strength of directional shifts in market prices. The primary objective of trend following is to profit from divergences in prices by capturing outsized swings. On the other hand, the pain point of trend following is getting caught in trendless markets with little or no divergence. One might wonder then – how is it possible to detect regimes of high market divergences for activating trend following strategies?

This week, we examine the Signal to Noise Ratio (StN), which is featured as a method for systematically detecting market divergence in Greyserman and Kaminski’s 2014 book on trend following. After computing this metric for individual commodity futures contracts, we look at how this can be incorporated at the portfolio level to consider aggregate market divergence in the Market Divergence Indicator (MDI). In addition, we also consider the rate of change of StN and MDI as an indicator of whether the market is exhibiting mean reverting or trending tendencies. Using a selection of commodity futures in gold, copper and crude oil, we show that the magnitude of rolling returns from trend following has a strong positive relationship with StN, and that the rate of change of StN and MDI is useful at predicting forward returns from trend following.

**Data and Metric Definitions **

*Data*

Our study uses ten years of daily prices for COMEX Gold, LME Copper, and ICE Brent Crude Oil futures which was obtained from Bloomberg. This comprises of daily closing prices for 2,480 trading days from 6^{th} August 2007 to 4^{th} August 2017.

*Signal to Noise Ratio*

The Signal to Noise Ratio (StN) is the ratio between the trend and individual price changes over a specific period (Greyserman and Kaminski, 2014). For any specific day (time t) in the dataset, StN for a particular lookback period (n) can be calculated with the following formula:

$StN_{t}=\frac{\left | ^{P_{t}-P_{t-n}} \right |}{\sum_{k=0}^{n-1}\left | P_{t-k} – P_{t-k-1} \right |}$

Where $P_{t}$ is the price at time t and n is the lookback window for the signal. For this post, we fix this at 100 days to consider long to medium term trends. In practice, the higher the StN, the higher the quality of the trend.

*Market Divergence Index*

The Market Divergence Index (MDI) is a measure which aggregates StN at a portfolio level for gauge the aggregate level of trend strength in markets. This is computed for any specific day (time t) in the dataset for a particular lookback period (n) with the following formula:

$MDI_{t}=\frac{1}{M}\sum_{i=1}^{M}StN_{t}^{i}(n)$

Where $StN_{t}^{i}(n)$ is the signal to noise ratio for an individual market i, n is the lookback period, and M is the number of included markets. When MDI is higher, this means that there are stronger directional shifts in the market environment.

**Empirical Results**

Given that the performance of trend following strategies depends on the persistence of trends, the higher the StN and MDI, the more profitable the deployment of a trend following program during that period. The following chart plots the time series of 100-day MDI and the rolling 100-day returns of a representative pure trend following system on an equally weighted portfolio of Gold, Copper and Crude Oil futures. Charts of the individual time series for each security is also included at the end. The trend following system is a generic exponential moving average crossover strategy that gives a bullish signal when 50-day EMA is greater than 100-day EMA, and gives a bearish signal otherwise.

Our study obtained similar results to those in Greyserman and Kaminski (2014). Correlation between 100-day MDI and rolling 100-day returns is 0.654. The high correlation figure indicates a direct relationship between market divergence and trend following strategy performance.

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###### Fig 1 – Time Series of 100-day MDI and Rolling 100-day Return from Representative Trend Following System

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According to Greyserman and Kaminski (2014), the distribution of MDI values tends to be positively skewed with fat right tails. In the chart, periods of dramatic market divergence such as the Lehman crisis, the Flash Crash, the Brexit Referendum and the 2016 US Presidential Election coincide with peak MDI values. The histograms below show the distribution of 100-day MDI and 100 day rolling returns from the trend following system over the same period. For our portfolio, MDI has a mean value of 0.104 and standard deviation of 0.094. We see unmistakable positive skewness in the distribution of MDI, which is consistent with the return distribution for the trend following strategy.

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###### Fig 2 – Histograms of 100-day MDI values and 100-day Rolling Returns from Represent Trend Following System

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Greyserman and Kaminski (2014) also stated that the relationship between market divergence and trend following is roughly linear. Linear regression using 100-day rolling returns from the trend following strategy and 100-day MDI showed a very significant positive linear relationship with R-Square of 42.8%. The regression plot is shown below.

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###### Fig 3- Regression Plot of 100-day Rolling Returns from Representative Trend Following System vs. MDI Values with Regression Line in Blue

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Next, we consider average returns from the trend following strategy for each decile of MDI values. To interpret this plot, when MDI is in its 10^{th} quantile (above 90^{th} percentile of values over the period), the average 100-day rolling return from the trend following strategy is 21.6%.

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###### Fig 4 – Average 100-day Rolling Returns from Representative Trend Following System by Decile of MDI

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In addition, the rate of change of StN is also an important metric. This allows us to detect whether the market is increasing or decreasing in divergent behavior, and could be useful metric for identifying changes in regime. Charts below summarize the distribution of 30-day forward returns by decile of MDI rate of change. When MDI is decreasing (as seen in the lower deciles), the market is displays more mean reverting tendencies therefore the trend following system yields negative returns on average. However, when MDI is increasing, the market displays shifts in regime, which translates to positive returns on average. Forward 30-day returns show broadly similar standard deviations across deciles.

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###### Fig 5 – Average 30-day Forward Rolling Returns of Trend Following System by Decile of MDI Rate of Change

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###### Fig 6 – Standard Deviation of 30-day Forward Rolling Returns of Trend Following System by Decile of MDI Rate of Change

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**Conclusion**

Overall, our study finds Signal to Noise ratio and its aggregate measure – the Market Divergence Indicator, to be a reliable regime indicator for deploying trend following strategies at daily intervals. We also believe it could be useful as a filter for finding instruments that fit better with trend following strategies. Going forward, we are interested in studying the behavior of these indicators at intraday timeframes and with larger cross-asset portfolios.

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**References**

- Greyserman, A., Kaminski, K., 2014. Trend Following with Managed Futures: The Search for Crisis Alpha.

**Appendix**

Over the timeframe, correlations between 100-day StN ratios and the rolling 100-day returns of a representative pure trend following strategy for Gold, Copper and Crude Oil were 0.55, 0.66 and 0.65 respectively.

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