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

## Summary

- While rebalancing studies typically focus on the combination of different asset classes, we evaluate a combination of two naïve trend-following strategies.
- As expected, we find that a rebalanced fixed-mix of the two strategies generates a concave payoff profile.
- More interestingly, deriving the
*optimal*blend of the two strategies allows the rebalanced portfolio to out-perform either of the two underlying strategies. - While most rebalancing literature has focused on the benefits of combining asset classes, we believe this literature can be trivially extended to ensembles of strategies.

Two weeks ago, we wrote about the idea of payoff diversification. The notion is fairly trivial, though we find it is often overlooked. Put simply, any and all trading decisions – even something as trivial as rebalancing – create a “payoff profile.” These profiles often fall into two categories: concave strategies that do well in stable environments is maintained and convex strategies that do better in the tails.

For example, we saw that rebalancing a 60/40 stock/bond portfolio earned a premium against a buy-and-hold approach when the spread between stock and bond returns remained narrow. Conversely, when the spread in return between stocks and bonds was wide, rebalancing created a drag on returns. This is a fairly trivial and obvious conclusion, but we believe it is important for investors to understand these impacts and why *payoff *is a meaningful axis of diversification.

In our prior study, we compared two different approaches to investing: strategic rebalancing and momentum investing. In this (very brief) study, we want to demonstrate that these results are also applicable when applied to different variations of the *same *strategy.

Specifically, we will look at two long/short trend following strategies applied to broad U.S. equities. When trend signals are positive, the strategy will be long U.S. equities and short the risk-free rate; when trend signals are negative the strategy will be short U.S. equities and long the risk-free rate. We will use a simple time-series momentum signal. The first model (“21D”) will evaluate trailing 21-day returns and hold for 1 day and the second model (“168D”) will evaluate trailing 168-day returns and holds for 14 days (with 14 overlapping portfolios).^{1} Both strategies implement a full skip day before allocating and assuming implementation at closing prices.

*Source: Kenneth French Data Library. 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. Past performance is not indicative of future results. *

So, what happens if we create a portfolio that holds *both* of these strategies, allocating 50% of our capital to each? Readers of our prior note will likely be able to guess the answer easily: we create a concave payoff profile that depends upon the relative performance between the two strategies. How, specifically, that concave shape manifests will be path dependent, but will also depend upon the rebalance frequency. For example, below we plot the payoff profiles for the 50/50 blend rebalanced weekly and monthly.

*Source: Kenneth French Data Library. 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. Past performance is not indicative of future results. *

If we stop thinking of these as two strategies applied to the same asset and just think of them as two assets, the results are fairly standard and intuitive. What is potentially appealing, however, is that the same literature and research that applies to the potential to create a rebalancing premium between assets can apply to a portfolio of strategies (whether a combination of distinct strategies, such as value and momentum, or an ensemble of the same strategy).

Below, we plot the annualized return of weekly rebalanced portfolios with different fixed-mix allocations to the 21D and 168D strategies. We can see that the curve peaks at approximately 45%, suggesting that a 45% allocation to the 21D strategy and a 55% allocation to the 168D strategy actually maximizes the compound annualized growth rate of the portfolio.

If we follow the process of Dubikovsky and Susinno (2017)^{2} to derive the optimal blend of these two assets – using the benefit of hindsight to measure their annualized returns (7.28% and 7.61% respectively), volatility (17.55% and 17.97% respectively), and correlation (0.1318) – we derive an optimal weight of 45.33%.

Perhaps somewhat surprisingly, even if the correlation between these two strategies was 0.9, the optimal blend would still recommend about 10% to the 21D variation. And, as extreme as it may seem, even if the annualized return of the 21D strategy was just 5.36% – a full 225 basis points below the 168D strategy – the optimal blend would still recommend about 10%. Diversification can create interesting opportunities to harvest return; at least, in expectation.

And, as we would expect, if we have no view as to a difference in return or volatility between the two specifications, we would end up with a recommended allocation of 50% to each.

## Conclusion

While most studies on rebalancing consider the potential benefits of combining assets, we believe that these benefits are trivially extended to strategies. Not just different strategies, however, but even strategies of the same style.

In this brief note, we explore the payoff profile created by combining two naïve long/short trend following strategies applied to broad U.S. equities. Unsurprisingly, rebalancing a simple mixture of the two specifications creates a concave payoff that generally profits when the spread between the two strategies is narrow and loses when the spread is wide.

More interestingly, however, we demonstrate that by rebalancing a fixed-mix of the two strategies, we can generate a return that is greater than either strategy individually. We believe that this potential benefit of ensemble approaches has been mostly overlooked by existing literature and deserves further analysis.

## 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.