Prior research and empirical investment results demonstrate that strategy performance can be highly sensitive to rebalance schedules, an effect called rebalance timing luck (“RTL”). In this paper we extend the empirical analysis to option-based strategies. As a case study, we replicate a popular strategy – the self-financing, three-month put-spread collar – with three implementations that vary only in their rebalance schedule. We find that the annualized tracking error between any two implementations is in excess of 400 basis points. We also decompose the empirically-derived rebalance timing luck for this strategy into its linear and non-linear components. Finally, we provide intuition for the driving causes of rebalance timing luck in option-based strategies.

In past research we have explored the potential benefits of how-based diversification through the lens of pay-off functions.

Specifically, we explored how strategic rebalancing created a concave payoff while momentum / trend-following created a convex payoff. By combining these two approaches, total portfolio payoff became more neutral to the dispersion in return of underlying assets.

We have also spent considerable time exploring when-based diversification through our writing on rebalance timing luck.

To manage rebalance timing luck, we advocate for a tranching methodology that can be best distilled as rebalancing “a little but frequently.”

Herein, we demonstrate that the resulting payoff profile of a tranche-based rebalancing strategy closely resembles that of a portfolio that combines both strategic rebalancing and momentum/trend-following.

While we typically think of tranching as simply a way to de-emphasize the impact of a specific rebalancing date choice, this research suggests that for certain horizons, tranching may also be effective because it naturally introduces momentum/trend-following into the portfolio.

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 Expands

Relative Performance Contracts

Rebalance

–

+

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 not rebalancing 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 perfectly timing 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.

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 not rebalancing 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 how axis 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 when and how. Rather, they may just different ways of exploring how to diversify the impacts of convexity and concavity in portfolio construction.

In past research notes we have explored the impact of rebalance timing luck on strategic and tactical portfolios, even using our own Systematic Value methodology as a case study.

In this note, we generate empirical timing luck estimates for a variety of specifications for simplified value, momentum, low volatility, and quality style portfolios.

Relative results align nicely with intuition: higher concentration and less frequent rebalancing leads to increasing levels of realized timing luck.

For more reasonable specifications – e.g. 100 stock portfolios rebalanced semi-annually – timing luck ranges between 100 and 400 basis points depending upon the style under investigation, suggesting a significant risk of performance dispersion due only to when a portfolio is rebalanced and nothing else.

The large magnitude of timing luck suggests that any conclusions drawn from performance comparisons between smart beta ETFs or against a standard style index may be spurious.

We’ve written about the concept of rebalance timing luck a lot. It’s a cowbell we’ve been beating for over half a decade, with our first article going back to August 7^{th}, 2013.

As a reminder, rebalance timing luck is the performance dispersion that arises from the choice of a particular rebalance date (e.g. semi-annual rebalances that occur in June and December versus March and September).

We’ve empirically explored the impact of rebalance timing luck as it relates to strategic asset allocation, tactical asset allocation, and even used our own Systematic Value strategy as a case study for smart beta. All of our results suggest that it has a highly non-trivial impact upon performance.

This summer we published a paper in the Journal of Index Investing that proposed a simple solution to the timing luck problem: diversification. If, for example, we believe that our momentum portfolio should be rebalanced every quarter – perhaps as an optimal balance of cost and signal freshness – then we proposed splitting our capital across the three portfolios that spanned different three-month rebalance periods (e.g. JAN-APR-JUL-OCT, FEB-MAY-AUG-NOV, MAR-JUN-SEP-DEC). This solution is referred to either as “tranching” or “overlapping portfolios.”

The paper also derived a formula for estimating timing luck ex-ante, with a simplified representation of:

Where L is the timing luck measure, T is turnover rate of the strategy, F is how many times per year the strategy rebalances, and S is the volatility of a long/short portfolio that captures the difference of what a strategy is currently invested in versus what it could be invested in if the portfolio was reconstructed at that point in time.

Without numbers, this equation still informs some general conclusions:

Higher turnover strategies have higher timing luck.

Strategies that rebalance more frequently have lower timing luck.

Strategies with a less constrained universe will have higher timing luck.

Bullet points 1 and 3 may seem similar but capture subtly different effects. This is likely best illustrated with two examples on different extremes. First consider a very high turnover strategy that trades within a universe of highly correlated securities. Now consider a very low turnover strategy that is either 100% long or 100% short U.S. equities. In the first case, the highly correlated nature of the universe means that differences in specific holdings may not matter as much, whereas in the second case the perfect inverse correlation means that small portfolio differences lead to meaningfully different performance.

L, in and of itself, is a bit tricky to interpret, but effectively attempts to capture the potential dispersion in performance between a particular rebalance implementation choice (e.g. JAN-APR-JUL-OCT) versus a timing-luck-neutral benchmark.

After half a decade, you’d would think we’ve spilled enough ink on this subject.

But given that just about every single major index still does not address this issue, and since our passion for the subject clearly verges on fever pitch, here comes some more cowbell.

Equity Style Portfolio Definitions

In this note, we will explore timing luck as it applies to four simplified smart beta portfolios based upon holdings of the S&P 500 from 2000-2019:

Value: Sort on earnings yield.

Momentum: Sort on prior 12-1 month returns.

Low Volatility: Sort on realized 12-month volatility.

Quality: Sort on average rank-score of ROE, accruals ratio, and leverage ratio.

Quality is a bit more complicated only because the quality factor has far less consistency in accepted definition. Therefore, we adopted the signals utilized by the S&P 500 Quality Index.

For each of these equity styles, we construct portfolios that vary across two dimensions:

Number of Holdings: 50, 100, 150, 200, 250, 300, 350, and 400.

Frequency of Rebalance: Quarterly, Semi-Annually, and Annually.

For the different rebalance frequencies, we also generate portfolios that represent each possible rebalance variation of that mix. For example, Momentum portfolios with 50 stocks that rebalance annually have 12 possible variations: a January rebalance, February rebalance, et cetera. Similarly, there are 12 possible variations of Momentum portfolios with 100 stocks that rebalance annually.

By explicitly calculating the rebalance date variations of each Style x Holding x Frequency combination, we can construct an overlapping portfolios solution. To estimate empirical annualized timing luck, we calculate the standard deviation of monthly return dispersion between the different rebalance date variations of the overlapping portfolio solution and annualize the result.

Empirical Timing Luck Results

Before looking at the results plotted below, we would encourage readers to hypothesize as to what they expect to see. Perhaps not in absolute magnitude, but at least in relative magnitude.

For example, based upon our understanding of the variables affecting timing luck, would we expect an annually rebalanced portfolio to have more or less timing luck than a quarterly rebalanced one?

Should a more concentrated portfolio have more or less timing luck than a less concentrated variation?

Which factor has the greatest risk of exhibiting timing luck?

Source: Sharadar. Calculations by Newfound Research.

To create a sense of scale across the styles, below we isolate the results for semi-annual rebalancing for each style and plot it.

Source: Sharadar. Calculations by Newfound Research.

In relative terms, there is no great surprise in these results:

More frequent rebalancing limits the risk of portfolios changing significantly between rebalance dates, thereby decreasing the impact of timing luck.

More concentrated portfolios exhibit larger timing luck.

Faster-moving signals (e.g. momentum) tend to exhibit more timing luck than more stable, slower-moving signals (e.g. low volatility).

What is perhaps the most surprising is the sheer magnitude of timing luck. Consider that the S&P 500 Enhanced Value, Momentum, Low Volatility, and Quality portfolios all hold 100 securities and are rebalanced semi-annually. Our study suggests that timing luck for such approaches may be as large as 2.5%, 4.4%, 1.1%, and 2.0% respectively.

But what does that really mean? Consider the realized performance dispersion of different rebalance date variations of a Momentum portfolio that holds the top 100 securities in equal weight and is rebalanced on a semi-annual basis.

Source: Sharadar. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.

The 4.4% estimate of annualized timing luck is a measure of dispersion between each underlying variation and the overlapping portfolio solution. If we isolate two sub-portfolios and calculate rolling 12-month performance dispersion, we can see that the difference can be far larger, as one might exhibit positive timing luck while the other exhibits negative timing luck. Below we do precisely this for the APR-OCT and MAY-NOV rebalance variations.

Source: Sharadar. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.

In fact, since these variations are identical in every which way except for the date on which they rebalance, a portfolio that is long the APR-OCT variation and short the MAY-NOV variation would explicitly capture the effects of rebalance timing luck. If we assume the rebalance timing luck realized by these two portfolios is independent (which our research suggests it is), then the volatility of this long/short is approximately the rebalance timing luck estimated above scaled by the square-root of two.

Derivation: For variations v_{i} and v_{j} and overlapping-portfolio solution V, then:

Thus, if we are comparing two identically-managed 100-stock momentum portfolios that rebalance semi-annually, our 95% confidence interval for performance dispersion due to timing luck is +/- 12.4% (2 x SQRT(2) x 4.4%).

Even for more diversified, lower turnover portfolios, this remains an issue. Consider a 400-stock low-volatility portfolio that is rebalanced quarterly. Empirical timing luck is still 0.5%, suggesting a 95% confidence interval of 1.4%.

S&P 500 Style Index Examples

One critique of the above analysis is that it is purely hypothetical: the portfolios studied above aren’t really those offered in the market today.

We will take our analysis one step further and replicate (to the best of our ability) the S&P 500 Enhanced Value, Momentum, Low Volatility, and Quality indices. We then created different rebalance schedule variations. Note that the S&P 500 Low Volatility index rebalances quarterly, so there are only three possible rebalance variations to compute.

Source: Sharadar. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.

We see a meaningful dispersion in terminal wealth levels, even for the S&P 500 Low Volatility index, which appears at first glance in the graph to have little impact from timing luck.

Minimum Terminal Wealth

Maximum Terminal Wealth

Enhanced Value

$4.45

$5.45

Momentum

$3.07

$4.99

Low Volatility

$6.16

$6.41

Quality

$4.19

$5.25

We should further note that there does not appear to be one set of rebalance dates that does significantly better than the others. For Value, FEB-AUG looks best while JUN-DEC looks the worst; for Momentum it’s almost precisely the opposite.

Furthermore, we can see that even seemingly closely related rebalances can have significant dispersion: consider MAY-NOV and JUN-DEC for Momentum. Here is a real doozy of a statistic: at one point, the MAY-NOV implementation for Momentum is down -50.3% while the JUN-DEC variation is down just -13.8%.

These differences are even more evident if we plot the annual returns for each strategy’s rebalance variations. Note, in particular, the extreme differences in Value in 2009, Momentum in 2017, and Quality in 2003.

Source: Sharadar. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.

Conclusion

In this study, we have explored the impact of rebalance timing luck on the results of smart beta / equity style portfolios.

We empirically tested this impact by designing a variety of portfolio specifications for four different equity styles (Value, Momentum, Low Volatility, and Quality). The specifications varied by concentration as well as rebalance frequency. We then constructed all possible rebalance variations of each specification to calculate the realized impact of rebalance timing luck over the test period (2000-2019).

In line with our mathematical model, we generally find that those strategies with higher turnover have higher timing luck and those that rebalance more frequently have less timing luck.

The sheer magnitude of timing luck, however, may come as a surprise to many. For reasonably concentrated portfolios (100 stocks) with semi-annual rebalance frequencies (common in many index definitions), annual timing luck ranged from 1-to-4%, which translated to a 95% confidence interval in annual performance dispersion of about +/-1.5% to +/-12.5%.

The sheer magnitude of timing luck calls into question our ability to draw meaningful relative performance conclusions between two strategies.

We then explored more concrete examples, replicating the S&P 500 Enhanced Value, Momentum, Low Volatility, and Quality indices. In line with expectations, we find that Momentum (a high turnover strategy) exhibits significantly higher realized timing luck than a lower turnover strategy rebalanced more frequently (i.e. Low Volatility).

For these four indices, the amount of rebalance timing luck leads to a staggering level of dispersion in realized terminal wealth.

“But Corey,” you say, “this only has to do with systematic factor managers, right?”

Consider that most of the major equity style benchmarks are managed with annual or semi-annual rebalance schedules. Good luck to anyone trying to identify manager skill when your benchmark might be realizing hundreds of basis points of positive or negative performance luck a year.

## 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.