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Why Trend Models Diverge

This post is available as a PDF download here.

Summary

In a market note we sent out last weekend, the following graphic was embedded:

What this table intends to capture is the percentage of trend signals that are on for a given model and lookback horizon (i.e. speed) on U.S. equities.  The point we were trying to establish was that despite a very bearish week, trend models remained largely mixed.  For frequent readers of our commentaries, it should come as no surprise that we were attempting to highlight the potential specification risks of selecting just one trend model to implement with (especially when coupled with timing luck!).

But there is a potentially interesting second lesson to learn here which is a bit more academic.  Why does it look like the price-minus (i.e. price-minus-moving-average) models turned off, the time series momentum models partially turned off, and the cross-over (i.e. dual-moving-average-cross) signals largely remained positive?

While this note will be short, it will be somewhat technical.  Therefore, we’ll spoil the ending: these signals are all mathematically linked.

They can all be decomposed into a weighted average of prior log-returns and the primary difference between the signals is the weighting concentration.  The price-minus model front-weights, the time-series model equal weights, and the cross-over model tends to back-weight (largely dependent upon the length of the two moving averages).  Thus, we would expect a price-minus model to react more quickly to large, recent changes.

If you want the gist of the results, just jump to the section The Weight of Prior Evidence, which provides graphical evidence of these weighting schemes.

Before we begin, we want to acknowledge that absolutely nothing in this note is novel.  We are, by in large, simply re-stating work pioneered by Bruder, Dao, Richard, and Roncalli (2011); Marshall, Nguyen and Visaltanachoti (2012); Levine and Pedersen (2015); Beekhuizen and Hallerbach (2015); and Zakamulin (2015).

Decomposing Time-Series Momentum

We will begin by decomposing a time-series momentum value, which we will define as:

We will begin with a simple substitution:

Which implies that:

Simply put, time-series momentum puts equal weight on all the past price changes1 that occur.

Decomposing Dual-Moving-Average-Crossover

We define the dual-moving-average-crossover as:

We assume m is less than n (i.e. the first moving average is “faster” than the second)Then, re-writing:

Here, we can make a cheeky transformation where we add and subtract the current price, Pt:

What we find is that the double-moving-average-crossover value is the difference in two weighted averages of time-series momentum values.

Decomposing Price-Minus-Moving-Average

This decomposition is trivial given the dual-moving-average-crossover.  Simply,

The Weight of Prior Evidence

We have now shown that these decompositions are all mathematically related.  Just as importantly, we have shown that all three methods are simply re-weighting schemes of prior price changes.  To gain a sense of how past returns are weighted to generate a current signal, we can plot normalized weightings for different hypothetical models.

Conclusion

In this brief research note, we demonstrated that common trend-following signals – namely time-series momentum, price-minus-moving-average, and dual-moving-average-crossover – are mathematically linked to one another.  We find that prior price changes are the building blocks of each signal, with the primary differences being how those prior price changes are weighted.

Time-series momentum signals equally-weight prior price changes; price-minus-moving-average models tend to forward-weight prior price changes; and dual-moving-average-crossovers create a hump-like weighting function.  The choice of which model to employ, then, expresses a view as to the relative importance we want to place on recent versus past price changes.

These results align with the trend signal changes seen over the past week during the rapid sell-off in the S&P 500.  Price-minus-moving-average models appeared to turn negative much faster than time-series momentum or dual-moving-average-crossover signals.

By decomposing these models into their most basic and shared form, we again highlight the potential specification risks that can arise from electing to employ just one model.  This is particularly true if an investor selects just one of these models without realizing the implicit choice they have made about the relative importance they would like to place on recent versus past returns.

 


 

  1. Working with prices changes is a bit odd for a number of reasons.  However, we do it here to allow for some transformations later.  One could, in theory, work with log-prices and interpret these derivations as log differences in price, which is perhaps more theoretically sound.

Corey is co-founder and Chief Investment Officer of Newfound Research. Corey holds a Master of Science in Computational Finance from Carnegie Mellon University and a Bachelor of Science in Computer Science, cum laude, from Cornell University. You can connect with Corey on LinkedIn or Twitter.

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