*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.

## Tranching, Trend, and Mean Reversion

By Corey Hoffstein

On April 27, 2020

In Craftsmanship, Momentum, Portfolio Construction, Weekly Commentary

This post is available as a PDF download here.## Summary

In Payoff Diversification (February 10

^{th}, 2020), we explored the idea of combining concave and convex payoff profiles. Specifically, we demonstrated that rebalancing a strategic asset allocation was inherently concave (i.e. mean reversionary) whereas trend-following and momentum was inherently convex. By combining the two approaches together, we could neutralize the implicit payoff profile of our portfolio with respect to performance of the underlying assets.Source: Newfound Research.Payoff Diversification(February 10^{th}, 2020). Source: Kenneth French Data Library; Federal Reserve Bank of St. Louis. Calculations by Newfound Research. Returns are hypothetical and assume the reinvestment of all distributions. Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. The 60/40 portfolio is comprised of a 60% allocation to broad U.S. equities and a 40% allocation to a constant maturity 10-Year U.S. Treasury index. The rebalanced variation is rebalanced at the end of each month whereas the buy-and-hold variation is allowed to drift over the 1-year period. The momentum portfolio is rebalanced monthly and selects the asset with the highest prior 12-month returns whereas the buy-and-hold variation is allowed to drift over the 1-year period.The 10-Year U.S. Treasuries index is a constant maturity index calculated by assuming a 10-year bond is purchased at the beginning of every month and sold at the end of that month to purchase a new bond at par at the beginning of the next month. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results.The intuition behind why rebalancing is inherently mean-reversionary is fairly simple. Consider a simple 50% stock / 50% bond portfolio. Between rebalances, this allocation will drift based upon the relative performance of stocks and bonds. When we rebalance, to right-size our relative allocations we must sell the asset that has out-performed and buy the one that has under-performed. “Sell your winners and buy your losers” certainly sounds mean-reversionary to us.

In fact, one way to think about a rebalance is as the application of a long/short overlay on your portfolio. For example, if the 50/50 portfolio drifted to a 45/55, we could think about rebalancing as holding the 45/55 and overlaying it with a +5/-5 long/short portfolio. This perspective explicitly expresses the “buy the loser, short the winner” strategy. In other words, we’re actively placing a trade that benefits when future returns between the two assets reverts.

While we may not be actively trying to express a view or forecast about future returns when we rebalance, we should consider the performance implications of our choice based upon whether the relative performance of these two assets continues to expand or contract:

Relative Performance ExpandsRelative Performance ContractsRebalance–

+

Do Not Rebalance+

–

Our argument in Payoff Diversification was that by combining strategic rebalancing and momentum / trend following, we could help neutralize this implicit bet.

What we can also see in the table above, though, is that the simple act of

notrebalancing benefits from a continuation of relative returns just as trend/momentum does.Let’s keep that in the back of our minds and switch gears, for a moment, to portfolio tranching. Frequent readers of our research notes will know we have spent considerable time researching the implications of rebalance timing luck. We won’t go into great detail here, but the research can be broadly summarized as, “when you rebalance your portfolio can have meaningful implications for performance.”

Given the discussion above, why that result holds true follows naturally. If two people hold 60/40 portfolios but rebalance them at different times in the year, their results will diverge based upon the relative performance of stocks and bonds between the rebalance periods.

As a trivial example, consider two 60/40 investors who each rebalance once a year. One chooses to rebalance every March and one chooses to rebalance every September. In 2008, the September investor would have re-upped his allocation to equities only to watch them sell-off for the next six months. The March investor, on the other hand, would have rebalanced earlier that year and her equity allocation would have drifted lower as the 2008 crisis wore on.

Even better, she would rebalance in March 2009, re-upping her equity allocation near the market bottom and almost

perfectlytiming the performance mean-reversion that would unfold. The September investor, on the other hand, would be underweight equities due to drift at this point.Below we plot hypothetical drifted equity allocations for these investors over time.

Source: Tiingo. Calculations by Newfound Research.The implications are that rebalancing can imbed large, albeit unintentional, market-timing bets.

In

Rebalance Timing Luck: The Difference between Hired and Firedwe derived that the optimal solution for avoiding the impact of these rebalance decisions is portfolio tranching. This is the same solution proposed by Blitz, van der Grient, and van Vliet (2010).The whole concept of tranching can be summarized with the phrase: “a little but frequently.” In other words, rebalance your portfolio more frequently, but only make small changes. As an example, rather than rebalance once a year, we could rebalance 1/12

^{th}of our portfolio every month. If our portfolio had drifted from a 60/40 to a 55/45, rather than rebalancing all the way back, we would just correct 1/12^{th}of the drift, trading to a 55.42/44.58.^{1}Another way to think about this approach is as a collection of sub-portfolios. For example, if we elected to implement a 12-month tranche, we might think of it as 12 separate sub-portfolios, each of which rebalances every 12 months but does so at the end of a different month (e.g. one rebalances in January, one in February, et cetera).

But why does this approach work? It helps de-emphasize the mean-reversion bet for any given rebalance date. We can see this by constructing the same payoff plots as before for different tranching speeds. The 1-month tranche reflects a full monthly rebalance; a 3-month tranche reflects rebalancing 33.33% of the portfolio; a 6-month tranche reflects rebalancing 16.66% of the portfolio each month; et cetera.

Source: Newfound Research.Payoff Diversification(February 10^{th}, 2020). Source: Kenneth French Data Library; Federal Reserve Bank of St. Louis. Calculations by Newfound Research. Returns are hypothetical and assume the reinvestment of all distributions. Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. The 60/40 portfolio is comprised of a 60% allocation to broad U.S. equities and a 40% allocation to a constant maturity 10-Year U.S. Treasury index. The rebalanced variation is rebalanced partially at the end of each month whereas the buy-and-hold variation is allowed to drift over the 1-year period. The 10-Year U.S. Treasuries index is a constant maturity index calculated by assuming a 10-year bond is purchased at the beginning of every month and sold at the end of that month to purchase a new bond at par at the beginning of the next month. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results.Note how the concave payoff function appears to “unbend” and the 12-month tranche appears similar in shape to payoff of the 90% strategic rebalance / 10% momentum strategy portfolio we plotted in the introduction.

Why might this be the case? Recall that

notrebalancing can be effective so long as there is continuation (i.e. momentum / trend) in the relative performance between stocks and bonds. By allowing our portfolio to drift, our portfolio will naturally tilt itself towards the out-performing asset. Furthermore, drift serves as an interesting amplifier to the momentum signal: the more persistent the relative out-performance, and the larger the relative out-performance in magnitude, the greater the resulting tilt.While tranching naturally helps reduce rebalance timing luck by de-emphasizing each specific rebalance, we can also see that we may be able to naturally embed momentum into our process.

## Conclusion

In portfolio management research, the answer we find is often a reflection of the angle by which a question is asked.

For example, in prior research notes, we have spent considerable time documenting the impact of rebalance timing luck in strategic asset allocation, tactical asset allocation, and factor investing. The simple choice of

when, though often overlooked in analysis, can have a significant impact upon realized results. Therefore, in order to de-emphasize the choice of when, we introduce portfolio tranching.We have also spent a good deal of time discussing the

howaxis of diversification (i.e. process). Not only have we research this topic through the lens of ensemble techniques, but we have also explored it through the payoff profiles generated by each process. We find that by combining diversifying concave and convex profiles – e.g. mean-reversion and momentum – we can potentially create a return profile that is more robust to different outcomes.Herein, we found that tranching the rebalance of a strategic asset allocation may, in fact, allow us to naturally embed momentum without having to explicitly introduce a momentum strategy. What we find, then, is that the two topics may not actually be independent avenues of research about

whenandhow. Rather, they may just different ways of exploring how to diversify the impacts of convexity and concavity in portfolio construction.