*This post is available as a PDF download here.*

## Summary

- During the week of February 23
^{rd}, the S&P 500 fell more than 10%. - After a prolonged bullish period in equities, this tumultuous decline caused many trend-following signals to turn negative.
- As we would expect, short-term signals across a variety of models turned negative. However, we also saw that price-minus-moving-average models turned negative across a broad horizon of lookbacks where the same was not true for other models.
- In this brief research note, we aim to explain why common trend-following models are actually mathematically linked to one another and differ mainly in how they place weight on recent versus prior price changes.
- We find that price-minus-moving-average models place the greatest weight on the most recent price changes, whereas models like time-series momentum place equal-weight across their lookback horizon.

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:

Simply put, time-series momentum puts equal weight on all the past price changes^{1} 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, *P _{t}*:

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.

- For TSMOM, we can easily see that shorter lookback models apply more weight on less data and therefore are likely to react faster to recent price changes.
- PMAC models apply weight in a linear, declining fashion, with the most weight applied to the most recent price changes. What is interesting is that PMAC(50) puts far more weight on recent prices changes than the TSMOM(50) model does. For equivalent lookback periods, then, we would expect PMAC to react much more quickly. This is precisely why we saw PMAC models turn off in the most recent sell-off when other models did not: they are much more front-weighted.
- DMAC models create a hump-shaped weighting profile, with increasing weight applied up until the length of the shorter lookback period, and then descending weight thereafter. If we wanted to, we could even create a back-weighted model, as we have with the DMAC(150, 200) example. In practice, it is common to see that m is approximately equal to n/4 (e.g. DMAC(50, 200)). Such a model underweights the most recent information relative to slightly less recent information.

## 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.

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