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.
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 10th, 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 10th, 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 1/12th 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 1/12th 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 10th, 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.
During the week of February 23rd, 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.
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.
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.
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.
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.
At Newfound, we adopt a holistic view of diversification that encompasses not only what we invest in, but also how and when we make those investment decisions.
In this three-dimensional perspective, what is correlation-based, how is payoff-based, and when is opportunity-based.
In this piece, we provide an example of what we mean by payoff-based diversification, using a simple strategically rebalanced portfolio and a naïve momentum strategy.
We find that the strategically rebalanced portfolio exhibits a payoff structure that is concave in nature whereas the momentum-based approach exhibits a convex profile.
By combining the two approaches – being careful in how we size positions – we can develop a portfolio that is less sensitive to the co-movement of underlying assets.
At Newfound, we embrace a holistic view of diversification that covers not just what we invest in, but also how and when we make those decisions. What is the diversification most investors are well-versed in and covers traditional, correlation-based diversification between securities, assets, macroeconomic factors, and geographic regions.
We identify when as “opportunity diversification” because it captures the opportunities that are available when we make investment decisions. This often goes overlooked in public markets (which is why we spend so much time writing about rebalance timing luck) but is well acknowledged in private markets where investors often allocate to multiple fund “vintages” to create diversification.
How is generally easy to understand, but sometimes difficult to visualize. We call it “payoff diversification” to acknowledge that when viewed through he appropriate lens, every investment style creates a particular shape. For example, when the return of a call option is plotted against the return of the underlying security, it generates a hockey-stick-like payoff profile.
In this short research note, we are going to demonstrate the payoff profiles of a strategically allocated portfolio and a naïve momentum strategy. We will then show that by combining these two approaches we can create a portfolio that exhibits significantly less sensitivity to the co-movement of underlying assets.
The Payoff Profile of a Strategic Portfolio
Few investors consider a strategically allocated portfolio to be an active strategy. And it isn’t; at least not until we introduce rebalancing. Once we institute a process to systematically returning our drifted weights back to their original fixed mix, we create a strategy and a corresponding payoff profile.
But what does this payoff profile look like? As an example, consider a U.S. 60/40 portfolio comprised of broad U.S. equities and a constant maturity 10-year U.S. Treasury index. If equities out-perform bonds, our equity allocation will increase and our bond allocation will decrease. If equities continue to out-perform bonds, we will benefit relative to our original policy weights. Similarly, if equities under-perform bonds, then our relative equity allocation will decrease. Again, should they continue to underperform, we are well positioned.
However, if we were to rebalance back to our original 60/40 allocation, we would eliminate the opportunity to benefit from the continuation of the relative performance.
On the other hand, consider the case where equities out-perform, our relative allocation to equities increases due to drift, and then equities subsequently under-perform. Now allowing drift has hurt us and we would have been better off rebalancing.
We can visualize this relationship by plotting the return spread between stocks and bonds (x-axis) versus the return spread between a monthly-rebalanced portfolio and a buy-and-hold (drifted) approach (y-axis) over rolling 1-year periods.
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 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.
What we can see is a concave payoff function. When equities significantly out-perform bonds (far right side of the graph), the rebalanced portfolio under-performs the drifted portfolio. Similarly, when bonds significantly out-perform equities (far left side of the graph), the rebalanced portfolio under-performs the drifted portfolio. When the return spread between stocks and bonds is small– a case likely to be more indicative of mean-reversion than positive autocorrelation in the spread – we can see that rebalancing actually generates a positive return versus the drifted portfolio.
Those versed in options will note that this payoff looks incredibly similar to a 1-year strangle sold on the spread between stocks and bonds and struck at 0%. The seller captures the premium when the realized spread remains small but loses money when the spread is more extreme.
The Payoff Profile of Naïve Momentum Following
We can now take the exact same approach to evaluating the payoff profile of a naïve momentum strategy. Each month, the strategy will simply invest in either stocks or bonds based upon whichever had the highest trailing 12-month return
As this approach is explicitly trying to capture auto-correlation in the return spread between stocks and bonds, we would expect to see almost mirror behavior to the payoff profile we saw with strategic rebalancing.
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 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.
While the profile may not be as tidy as before, we can see a convex payoff profile that tends to profit when the return spread is more extreme and lose money when the spread is narrow. Again, those familiar with options will recognize this as similar to the payoff of a 1-year straddle based upon the return spread between stocks and bonds. The buyer pays a premium but captures the spread when it is extreme.
Note, however, the scale of the y-axis. Whereas the payoff profile for the rebalanced portfolio was between -3.0% and +2.0%, the payoff profile for this momentum approach is much larger, ranging between -30.0% and 40.0%.
Creating Payoff Diversification
We have seen that whether we strategically rebalance or adopt a momentum-based approach, both approaches create a payoff profile that is sensitive to the return spread in underlying assets. But what if we do not want to take such a specific payoff bet? One simple answer is diversification.
If we allocate to both the strategically rebalanced portfolio and the naïve momentum portfolio, we will realize both their payoff profiles simultaneously. As their profiles are close mirrors of one another, we may be able to achieve a more neutral outcome.
We have to be careful, however, as to size the allocations appropriate. Recall that the payoff profile of the strategically rebalanced portfolio was approximately 1/10th the size of the naïve momentum strategy. For both profiles to contribute equally, we would want to allocate approximately 90% of our capital to the strategic rebalancing strategy and 10% of our capital to the momentum strategy.
Below we plot the payoff structure of such a mix.
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 mixed portfolio is rebalanced monthly and is a 90% allocation to a rebalanced 60/40 and a 10% allocation to a naïve momentum strategy; 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.
We can see that diversifying how we make decisions results in a payoff structure that is far more neutral to the co-movement of underlying securities in the portfolio. The holy grail, of course, is not just to find strategies whose combination neutralizes sensitivity to the spread in returns, but actually creates a higher likelihood of positive outcomes in all environments.
Conclusion
In this research note, we aimed to provide greater insight into the idea of payoff diversification, the how in our what-how-when diversification framework. To do so, we explored two simple examples: a strategically rebalanced 60/40 allocation and a naïve momentum strategy.
We found that the strategically rebalanced portfolio generates a payoff profile that is convex with respect to the spread in returns between stocks and bonds. In general, the larger the spread, the more likely that rebalancing generates a negative return versus a buy-and-hold approach. Conversely, the smaller the spread, the more likely that rebalancing generates a positive return.
The naïve momentum strategy – which simply bought the asset with the greatest prior 12-month returns – exhibited a convex profile. When the return spread between stocks and bonds was large, the naïve momentum strategy was more likely to out-perform buy-and-hold. Conversely, when the return spread was small, the naïve momentum strategy tended to under-perform.
Importantly, the magnitudes of the payoffs are significantly different, with the naïve momentum strategy generating returns nearly 10x larger than strategic rebalancing in the tails. This difference has important implications for strategy sizing, and we find a portfolio mixture of 90% strategic rebalancing and 10% naïve momentum does a reasonably good job of neutralizing portfolio payoff sensitivity to the spread in stock and bond returns.
Managed futures strategies have historically provided meaningful positive returns during left-tail equity events. Yet as a trading strategy, this outcome is by no means guaranteed.
While trend following is “mechanically convex,” the diverse nature of managed futures programs may actually prevent the strategy from offsetting equity market losses.
We generate a large number of random managed futures strategies by varying the asset classes included. We find that more diverse strategies have, historically, provided a larger offset to negative equity events.
This curious outcome appears to be caused by two effects: (1) diversification allows for increased total notional exposure; and (2) past crises saw coincidental trends across multiple markets simultaneously.
Therefore, for investors trying to offset equity market losses, an allocation to managed futures requires believing that future crises will be marked by a large number of simultaneous trends across multiple assets.
Less diversified strategies – such as just trading S&P 500 futures contracts – appear to work if the volatility target is removed.
Shortly after the 2008 crisis, the appetite for risk management strategies exploded. At the forefront of this trend was managed futures, which had already proven itself in the dot-com fallout. With the Societe Generale Trend Index1 returning 20.9% in 2008, the evidence for CTAs to provide “crisis alpha”2 seemed un-debatable. AUM in these strategies sky-rocketed, growing from $200 billion in 2007 to approximately $325 billion by 2012.
Source: http://managedfuturesinvesting.com
Subsequent performance has, unfortunately, been lack-luster. Since 12/31/2011, the SG Trend Index has returned just 14.2% compared to the S&P 500’s 200.8% total return. While this is an unfair, apples-to-oranges comparison, it does capture the dispersion the strategy has exhibited to the benchmark most investors measure performance against during a bull market.
Furthermore, the allocation to managed futures had to come from somewhere. If investors reduced exposure to equities to introduce managed futures, the spread in performance captures the opportunity cost of that decision. There is hope yet: if the S&P 500 fell 50% over the next year, managed futures would have to return just 32% for their full-period performance (2011-2020) to equalize.
Yet how certain are we that managed futures would necessarily generate a positive return in an S&P 500 left-tail environment? Hurst, Ooi, and Pedersen (2017)3 find that managed futures have generated anything from flat to meaningfully positive results during the top 10 largest drawdowns of a 60/40 portfolio since the late 1800s. This evidence makes a strong empirical case, but we should acknowledge the N=10 nature of the data.
Perhaps we can lean into the mechanicallyconvex nature of trend following. Trend following is a close cousin to the trading strategy that delta-hedges a strangle, generating the pay-off profile of a straddle (long an at-the-money put and call). Even without an anomalous premium generated by autocorrelation in the underlying security, the trading strategy itself should – barring trading frictions – generate a convex payoff.
Yet while mechanical convexity may be true on a contract-by-contract basis, it is entirely possible that the convexity we want to see emerge is diluted by trades across other contracts. Consider the scenario where the S&P 500 enters a prolonged and significant drawdown and our managed futures strategy goes short S&P 500 futures contract. While this trade may generate the hedge we were looking for, it’s possible that it is diluted by trades on other contracts such as wheat, the Japanese Yen, or the German Bund.
When we consider that many investors have portfolios dominated by equity risk (recall that equities have historically contributed 90% of the realized volatility for a 60/40 portfolio), it is possible that too much breadth within a managed futures portfolio could actually prevent it from providing negative beta during left-tail equity events.
Replicating Managed Futures
We begin our study by first replicating a generic trend-following CTA index. We adopt an ensemble approach, which is effectively equivalent to holding a basket of managers who each implement a trend-following strategy with a different model and parameterization.
Specifically, we assume each manager implements using the same 47 contracts that represent a diversified basket of equities, rates, commodities, and currencies.4
We implement with three different models (total return, price-minus-moving-average, and dual-moving-average-cross) and five potential lookback specifications (21, 42, 84, 168, and 336 days) for a total of 15 different implementations.
Each implementation begins by calculating an equal-risk contribution (“risk parity”) portfolio. Weights for each contract are then multiplied by their trend signal (which is simply either +1 or -1).
The weights for all 15 implementations are then averaged together to generate our index weights. Notional exposure of the aggregate weights is then scaled to target a 10% annualized volatility level. We assume that the index is fully collateralized using the S&P U.S. Treasury Bill Index.
Below we plot our index versus the SG Trend Index. The correlation of monthly returns between these two indices is 75% suggesting that our simple implementation does a reasonable job approximating the broad trend-following style of CTAs. We can also see that it captures the salient features of the SG Trend Index, including strong performance from 2001-2003, Q4 2008 and Q1 2009, and the 2014-2015 period. We can also see it closely tracks the shape the SG Trend Index equity curve from 2015 onward in all its meandering glory.
Source: Stevens Analytics. 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. These results do not reflect the returns of any strategy managed by Newfound Research.
Convexity versus Diversification
To explore the impact of diversification in managed futures versus convexity exhibited against the S&P 500, we will create a number of managed futures strategies and vary the number of contracts included. As we are attempting to create a convex payoff against the S&P 500, the S&P 500 futures contract will always be selected.
For example, a 2-contract strategy will always include S&P 500 futures, but the second contract could be 10-year U.S. Treasuries, the Nikkei, the Australian Dollar, Oil, or any of the other 42 futures contracts. Once selected, however, that pair defines the strategy.
For 2-, 4-, 8-, 16-, and 32- contract systems, we generate the performance of 25 randomly selected strategies. We then generate scatter plots with non-overlapping 6-month returns for the S&P 500 on the x-axis and non-overlapping 6-month returns for the managed futures strategies on the y-axis.5 We then fit a 2nd-degree polynomial line to visualize the realized convexity.
(Note that for the single contract case – i.e. just the S&P 500 futures contract – we plot overlapping 6-month returns.)
Source: Stevens Analytics and 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. These results do not reflect the returns of any strategy managed by Newfound Research.
There are two particularly interesting artifacts to note.
First, as the number of contracts goes up, the best-fit model turns from a “smile” to a “smirk,” suggesting that diversification dilutes positive convexity relationships with the S&P 500. This outcome should have been expected, as we generally know how managed futures has done over the 20-year period we’re examining. Namely, managed futures did quite well offsetting losses in 2000-2003 and 2008-2009, but has failed to participate in the 2010s.
Perhaps more interestingly, however, is the increase in left-tail performance of managed futures, climbing from 20% when just trading the S&P 500 futures contract to 150% in the 32-contract case. The subtle reason here is diversification’s impact on total notional exposure.
Consider this trivial example: Asset A and Asset B have constant 10% volatility and are uncorrelated with one another. As they are uncorrelated, any combination of these assets will have a volatility that is less than 10%. Therefore, if we want to achieve 10%, we need to apply leverage. In fact, a 50-50 mix of these assets requires us to apply 1.41x leverage to achieve our volatility target, resulting in 70.7% exposure to each asset.
As a more concrete example, when trading just the S&P 500 futures contract, achieving 10% volatility position in 2008 requires diluting gross notional exposure to just 16%. For the full, 47-contract model, gross notional exposure during 2008 dipped to 90% at its lowest point.
Now consider that trend following tends to transform the underlying distributions of assets to generate positive skewness. Increasing leverage can help push those positive trades even further out in the tails.
But here’s the trade-off: the actual exposure to S&P 500 futures contracts, specifically, still remains much, much higher in the case where we’re trading it alone. In practice, the reason the diversified approach was able to generate increased returns during left-tail equity events – such as 2008 – is due to the fact correlations crashed to extremes (both positive and negative) between global equity indices, rates, commodities, and currencies. This allowed the total notional exposure of directionally similar trades (e.g. short equities, long bonds, and short commodities in 2008) to far exceed the total notional exposure achieved if we were just trading the S&P 500 futures contract alone.
Source: Stevens Analytics. Calculations by Newfound Research.
Our confidence in achieving negative convexity versus equity left-tail events, therefore, is inherently tied to our belief that we will see simultaneously trends across a large number of assets during such environments.
Another interpretation of this data is that because negative trends in the S&P 500 have historically coincided with higher volatility, a strategy that seeks to trade just the S&P 500 futures with constant volatility will lose convexity in those tail events. An alternative choice is to vary the volatility of the system to target the volatility of the S&P 500, whose convexity profile we plot below.
Source: Stevens Analytics and 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. These results do not reflect the returns of any strategy managed by Newfound Research.
This analysis highlights a variety of trade-offs to consider:
What, specifically, are we trying to create convexity against?
Can diversification allow us to increase our notional exposure?
Will diversification be dilutive to our potential convexity?
Perhaps, then, we should consider approaching the problem from another angle: given exposure to managed futures, what would be a better core portfolio to hold? Given that most managed futures portfolios start from a risk parity core, the simplest answer is likely risk parity.
As an example, we construct a 10% target volatility risk parity index using equity, rate, and commodity contracts. Below we plot the convexity profile of our managed futures strategy against this risk parity index and see the traditional “smile” emerge. We also plot the equity curves for the risk parity index, the managed futures index, and a 50/50 blend. Both the risk parity and managed futures indices have a realized volatility of level of 10.8%; the blended combination drops this volatility to just 7.6%, achieving a maximum drawdown of just -10.1%.
Source: Stevens Analytics and 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. These results do not reflect the returns of any strategy managed by Newfound Research.
Conclusion
Managed futures have historically generated significant gains during left-tail equity events. These returns, however, are by no means guaranteed. While trend following is a mechanically convex strategy, the diversified nature of most managed futures programs can potentially dilute equity-crisis-specific returns.
In this research note, we sought to explore this concept by generating a large number of managed futures strategies that varied in the number of contracts traded. We found that increasing the number of contracts had two primary effects: (1) it reduced realized convexity from a “smile” to a “smirk” (i.e. exhibited less up-side participation with equity markets); and (2) meaningfully increased returns during negative equity markets.
The latter is particularly curious but ultimately the byproduct of two facts. First, increasing diversification allows for increased notional exposure in the portfolio to achieve the same target volatility level. Second, during past crises we witnessed a large number of assets trending simultaneously. Therefore, while increasing the number of contracts reduced notional exposure to S&P 500 futures specifically, the total notional exposure to trades generating positive gains during past crisis events was materially higher.
While the first fact is evergreen, the second may not always be the case. Therefore, employing managed futures specifically as a strategy to provide offsetting returns during an equity market crisis requires the belief that a sufficient number of other exposures (i.e. equity indices, rates, commodities, and currencies) will be exhibiting meaningful trends at the same time.
Given its diversified nature, it should come as no surprise that managed futures appear to be a natural complement to a risk parity portfolio.
Investors acutely sensitive to significant equity losses – e.g. those in more traditional strategic allocation portfolios – might therefore consider strategies designed more specifically with such environments in mind. At Newfound, we believe that trend equity strategies are one such solution, as they overlay trend-following techniques directly on equity exposure, seeking to generate the convexity mechanically and not through correlated assets. When overlaid with U.S. Treasury futures – which have historically provided a “flight-to-safety” premium during equity crises – we believe it is a particularly strong solution.
We will begin with a simple substitution:
Which implies that:
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 10th, 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 10th, 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
–
+
+
–
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.
In Rebalance Timing Luck: The Difference between Hired and Fired we 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 1/12th 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 1/12th 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 10th, 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.