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.
A long/flat tactical equity strategy with a portable beta bond overlay – a tactical 90/60 portfolio – has many moving parts that can make attribution and analysis difficult.
By decomposing the strategy into its passive holdings (a 50/50 stock/bond portfolio and U.S. Treasury futures) and active long/short overlays (trend equity, bond carry, bond momentum, and bond value), we can explore the historical performance of each component and diversification benefits across each piece of the strategy.
Using a mean-variance framework, we are also able to construct an efficient frontier of the strategy components and assess the differences between the optimal portfolio and the tactical 90/60.
We find that the tactical 90/60 is relatively close to the optimal portfolio for its volatility level and that its drawdown risk profile is close to that of an unlevered 60/40 portfolio.
By utilizing a modest amount of leverage and pairing it will risk management in both equities and bonds, investors may be able to pursue capital efficiency and maximize portfolio returns while simultaneously managing risk.
Portable beta strategies seek to enhance returns by overlaying an existing portfolio strategy with complementary exposure to diversifying asset classes and strategies. In overlaying exposure on an existing portfolio strategy, portable beta strategies seek to make every invested dollar work harder. This idea can create “capital efficiency” for investors, freeing up dollars in an investor’s portfolio to invest in other asset classes or investment opportunities.
At Newfound, we focus on managing risk. Trend following – or absolute momentum – is a key approach we employ do this, especially in equities. Trend equity strategies are a class of strategies that aim to harvest the long-term benefits of the equity risk premium while managing downside risk through the application of trend following.
We wrote previously how a trend equity strategy can be decomposed into passive and active components in order to isolate different contributors to performance. There is more than one way to do this, but in the most symmetric formulation, a “long/flat” trend equity strategy (one that that either holds equities or cash; i.e. does not short equities) can be thought of as a 100% passive allocation to a 50/50 portfolio of stocks and cash plus a 50% overlay allocation to a long/short trend equity strategy that can move between fully short and fully long equities. This overlay component is portable beta.
We have also written previously about how a portable beta overlay of bonds can be beneficial to trend equity strategies – or even passive equity investments, for that matter. For example, 95% of a portfolio could be invested in a trend equity strategy, and the remaining 5% could be set aside as collateral to initiate a 60% overlay to 10-year U.S. Treasury futures. This approximates a 60/40 portfolio that is leveraged by 50%
Source: Newfound. Allocations are hypothetical and for illustrative purposes only.
Since this bond investment introduces interest rate risk, we have proposed ways to manage risk in this specific sleeve using factors such as value, carry, and momentum. By treating these factors as fully tactical long/short portfolios themselves, if we hold them in equal weight, we can also break down the tactical U.S. Treasury futures overlay into active and passive components, with a 30% passive position in U.S. Treasury futures and 10% in each of the factor-based strategies.
Source: Newfound. Allocations are hypothetical and for illustrative purposes only.
When each overlay is fully invested, the portfolio will hold 95% stocks, 5% cash, and 60% U.S. Treasury futures. When all the overlays are fully short, the strategy will be fully invested in cash with no bond overlay position.
While the strategy has not changed at all with this slicing and dicing, we now have a framework to explore the historical contributions of the active and passive components and the potential diversification benefits that they offer.
Diversification Among Components
For the passive portfolio 50/50 stock/cash, we will use a blend of the Vanguard Total U.S. stock market ETF (VTI) and the iShares Short-term Treasury Bond ETF (SHV) with Kenneth French data for market returns and the risk-free rate prior to ETF inception.
The passive 10-year U.S. Treasury futures is the continuous futures contract with a proxy of the 10-year constant maturity Treasury index minus the cash index used before inception (January 2000). The active long/short bond factors can be found on the U.S. Treasuries section of our quantitative signals dashboard, which is updated frequently.
All data starts at the common inception point in May 1957.
As a technical side note, we must acknowledge that a constant maturity 10-year U.S. Treasury index minus a cash index will not precisely match the returns of 10-year U.S. Treasury futures. The specification of the futures contracts state that the seller of such a contract has the right to deliver any U.S. Treasury bond with maturity between 6.5 and 10 years. In other words, buyers of this contract are implicitly selling an option, knowing that the seller of the contract will likely choose the cheapest bond to deliver upon maturity (referred to as the “cheapest to deliver”). Based upon the specification and current interest rate levels, that current cheapest to deliver bond tends to have a maturity of 6.5 years.
This has a few implications. First, when you buy U.S. Treasury futures, you are selling optionality. Finance 101 will teach you that optionality has value, and therefore you would expect to earn some premium for selling it. Second, the duration profile between our proxy index and 10-year U.S. Treasury futures has meaningfully diverged in the recent decade. Finally, the roll yield harvested by the index and the futures will also diverge, which can have a non-trivial impact upon returns.
Nevertheless, we believe that for the purposes of this study, the proxy index is sufficient for broad, directional attribution and understanding.
Source: Kenneth French Data Library, Federal Reserve Bank of St. Louis, Tiingo, Stevens Futures. Calculations by Newfound Research. Data is from May 1957 to January 2020. 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. 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 50/50 Stock/Cash portfolio is the only long-only holding. While the returns are lower for all the other strategies, we must keep in mind that they are all overlays that can add to the 50/50 portfolio rather than simply de-risk and cannibalize its return.
This is especially true since these overlay strategies have exhibited low correlation to the 50/50 portfolio.
The table below shows the full period correlation of monthly returns for all the portfolio components. The equity and bond sub-correlation matrices are outlined to highlight the internal diversification.
Source: Kenneth French Data Library, Federal Reserve Bank of St. Louis, Tiingo, Stevens Futures. Calculations by Newfound Research. Data is from May 1957 to January 2020. 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. 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.
Not only do all of the overlays have low correlation to the 50/50 portfolio, but they generally exhibit low cross-correlations. Of the overlays, the L/S bond carry and L/S bond momentum strategies have the highest correlation (0.57), and the L/S bond carry and passive bond overlay have the next highest correlation (0.47).
The bond strategies have also exhibited low correlation to the equity strategies. This results in good performance, both absolute and risk-adjusted, relative to a benchmark 60/40 portfolio and a benchmark passive 90/60 portfolio.
Source: Kenneth French Data Library, Federal Reserve Bank of St. Louis, Tiingo, Stevens Futures. Calculations by Newfound Research. Data is from May 1957 to January 2020. 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. 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.
Finding the Optimal Blend
Up to this point, we have only considered the fixed allocations to each of the active and passive strategies outlined at the beginning. But these may not be the optimal holdings.
Using a block-bootstrap method to simulate returns, we can utilize mean-variance optimization to determine the optimal portfolios for given volatility levels.1 This yields a resampled historical realized efficient frontier.
Source: Kenneth French Data Library, Federal Reserve Bank of St. Louis, Tiingo, Stevens Futures. Calculations by Newfound Research. Data is from May 1957 to January 2020. 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. 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.
Plotting the benchmark 60/40, benchmark 90/60, and the tactical 90/60 on this efficient frontier, we see that the tactical 90/60 lies very close to the frontier at about 11.5% volatility. The allocations for the frontier are shown below.
Source: Kenneth French Data Library, Federal Reserve Bank of St. Louis, Tiingo, Stevens Futures. Calculations by Newfound Research. Data is from May 1957 to January 2020. 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. 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.
As expected, the lower volatility portfolios hold more cash and the high volatility portfolios hold more equity. For the 9% volatility level, these two allocations match, leading to the full allocation to a 50/50 stock/cash blend as in the tactical 90/60.
The passive allocation to the Treasury futures peaks at about 60%, while the L/S bond factor allocations are generally between 5% and 20% with more emphasis on Value and typically equal emphasis on Carry and Momentum.
The allocations in the point along the efficient frontier that matches the tactical 90/60 portfolio’s volatility are shown below.
Source: Kenneth French Data Library, Federal Reserve Bank of St. Louis, Tiingo, Stevens Futures. Calculations by Newfound Research. Data is from May 1957 to January 2020. 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. 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.
In this portfolio, we see a higher allocation to passive equities, a smaller position in the tactical equity L/S, and a larger position in passive Treasury futures. However, given the resampled nature of the process, these allocations are not wildly far away from the tactical 90/60.
The differences in the allocations are borne out in the Ulcer Index risk metric, which quantifies the severity and duration of drawdowns.
Source: Kenneth French Data Library, Federal Reserve Bank of St. Louis, Tiingo, Stevens Futures. Calculations by Newfound Research. Data is from May 1957 to January 2020. 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. 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 efficient frontier portfolio has a lower Ulcer Index than that of the tactical 90/60 even though their returns and volatility are similar. However, the Ulcer index of the tactical 90/60 is very close to that of the benchmark 60/40.
These differences are likely due to the larger allocation to the tactical equity long/short which can experience whipsaws (e.g. in October 1987), the lower allocation to passive U.S. equities, and the lower allocation to the Treasury overlay.
In an uncertain future, there can be significant risk in relying too much on the past, but having this framework can be useful for gaining a deeper understanding of which market environments benefit or hurt each component within the portfolio and how they diversify each other when held together.
Conclusion
In this research note, we explored diversification in a long/flat tactical equity strategy with a portable beta bond overlay. By decomposing the strategy into its passive holdings (50/50 stock/bond portfolio and U.S. Treasury futures) and active long/short overlays (trend equity, bond carry, bond momentum, and bond value), we found that each of the overlays has historically exhibited low correlation to the passive portfolios and low cross-correlations to each other. Combining all of these strategies using a tactical 90/60 portfolio has led to strong performance on both an absolute and risk-adjusted basis.
Using these strategy components, we constructed an efficient frontier of portfolios and also found that the “intuitive” tactical 90/60 portfolio that we have used in much of our portable beta research is close to the optimal portfolio for its volatility level. While this does not guarantee that this portfolio will be optimal over any given time period, it does provide evidence for the robustness of the multi-factor risk-managed approach.
Utilizing portable beta strategies can be an effective way for investors to pursue capital efficiency and maximize portfolio returns while simultaneously managing risk. While leverage can introduce risks of its own, relying on diversification and robust risk-management methods (e.g. trend following) can mitigate the risk of large losses.
The fear of using leverage and derivatives may be an uphill battle for investors, and there are a few operational burdens to overcome, but when used appropriately, these tools can make portfolios work harder and lead to more flexibility for allocating to additional opportunities.
If you are interested in learning how Newfound applies the concepts of tactical portable beta to its mandates, please reach out (info@thinknewfound.com).
Over the last several years, we have written several research notes demonstrating the potential benefits of diversifying “specification risk.”
Specification risk occurs when an investment strategy is overly sensitive to the outcome of a single investment process or parameter choice.
Adopting an ensemble approach is akin to creating a virtual fund-of-funds of stylistically similar managers, exhibiting many of the same advantages of traditional multi-manager diversification.
In this piece, we briefly explore whether model specification choices can be timed using momentum within the context of a naïve trend strategy.
We find little evidence that momentum-based parameter specification leads to meaningful or consistent improvements beyond a naively diversified approach.
Over the last several years, we’ve advocated on numerous occasions for a more holistic view of diversification: one that goes beyond just what we invest in, but also considers how those decisions are made and when they are made.
We believe that this style of thinking can be applied “all the way down” our process. For example, how-based diversification would advocate for the inclusion of both value and momentum processes, as well as for different approaches to capturing value and momentum.
Unlike correlation-based what diversification, how-based diversification often does little for traditional portfolio risk metrics. For example, in Is Multi-Manager Diversification Worth It? we demonstrated that within most equity categories, allocating across multiple managers does almost nothing to reduce portfolio volatility. It does, however, have a profound impact on the dispersion of terminal wealth that is achieved, often by avoiding manager-specific tail-risks. In other words, our certainty of achieving a given outcome may be dramatically improved by taking a multi-manager approach.
Ensemble techniques to portfolio construction can be thought of as adopting this same multi-manager approach by creating a set of virtual managers to allocate across.
In late 2018, we wrote two notes that touched upon this: When Simplicity Met Fragility and What Do Portfolios and Teacups Have in Common? In both studies we injected a bit of randomness into asset returns to measure the stability of trend-following strategies. We found that highly simplistic models tended to exhibit significant deviations in results with just slightly modified inputs, suggesting that they are highly fragile. Increasing diversification across what, how, and when axes led to a significant improvement in outcome stability.
As empirical evidence, we studied the real-time results of the popular Dual Momentum GEM strategy in our piece Fragility Case Study: Dual Momentum GEM, finding that slight deviations in model specification lead to significantly different allocation conclusions and therefore meaningfully different performance results. This was particularly pronounced over short horizons.
Tying trend-following to option theory, we then demonstrated how an ensemble of trend following models and specifications could be used to increase outcome certainty in Tightening the Uncertain Payout of Trend-Following.
Yet while more diversification appears to make portfolios more consistent in the outcomes they achieve, empirical evidence also suggests that certain specifications can lead to superior results for prolonged periods of time. For example, slower trend following signals appear to have performed much, much better than fast trend following signals over the last two decades.
One of the benefits of being a quant is that it is easy to create thousands of virtual managers, all of whom may follow the same style (e.g. “trend”) but implement with a different model (e.g. prior total return, price-minus-moving-average, etc) and specification (e.g. 10 month, 200 day, 13 week / 34 week cross, etc). An ancillary benefit is that it is also easy to re-allocate capital among these virtual managers.
Given this ease, and knowing that certain specifications can go through prolonged periods of out-performance, we might ask: can we time specification choices with momentum?
Timing Trend Specification
In this research note, we will explore whether momentum signals can help us time out specification choices as it relates to a simple long/flat U.S. trend equity strategy.
Using data from the Kenneth French library, our strategy will hold broad U.S. equities when the trend signal is positive and shift to the risk-free asset when trends are negative. We will develop 1023 different strategies by employing three different models – prior total return, price-minus-moving-average, and dual-moving-average-cross-over – with lookback choices spanning from 20-to-360 days in length.
After constructing the 1023 different strategies, we will then apply a momentum model that ranks the models based upon prior returns and equally-weights our portfolio across the top 10%. These choices are made daily and implemented with 21 overlapping portfolios to reduce the impact of rebalance timing luck.
It should be noted that because the underlying strategies are only allocating between U.S. equities and a risk-free asset, they can go through prolonged periods where they have identical returns or where more than 10% of models share the highest prior return. In these cases, we select all models that have returns equal-to-or-greater-than the model identified at the 10th percentile.
Before comparing performance results, we think it is worthwhile to take a quick look under the hood to see whether the momentum-based approach is actually creating meaningful tilts in specification selection. Below we plot both aggregate model and lookback weights for the 126-day momentum strategy.
Source: Kenneth French Data Library. Calculations by Newfound Research.
We can see that while the model selection remains largely balanced, with the exception of a few periods, the lookback horizon selection is far more volatile. On average, the strategy preferred intermediate-to-long-term signals (i.e. 181-to-360 day), but we can see intermittent periods where short-term models carried favor.
Did this extra effort generate value, though? Below we plot the ratio of the momentum strategies’ equity curves versus the naïve diversified approach.
We see little consistency in relative performance and four of the five strategies end up flat-to-worse. Only the 252-day momentum strategy out-performs by the end of the testing period and this is only due to a stretch of performance from 1950-1964. In fact, since 1965 the relative performance of the 252-day momentum model has been negative versus the naively diversified approach.
Source: Kenneth French Data Library. 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.
This analysis suggests that naïve, momentum-based specification selection does not appear to have much merit against a diversified approach for our simple trend equity strategy.
The Potential Benefits of Virtual Rebalancing
One potential benefit of an ensemble approach is that rebalancing across virtual managers can generate growth under certain market conditions. Similar to a strategically rebalanced portfolio, we find that when returns across virtual managers are expected to be similar, consistent rebalancing can harvest excess returns above a buy-and-hold approach.
The trade-off, of course, is that when there is autocorrelation in specification performance, rebalancing creates a drag. However, given that the evidence above suggests that relative performance between specifications is not persistent, we might expect that continuously rebalancing across our ensemble of virtual managers may actually allow us to harvest returns above and beyond what might be possible with just selecting an individual manager.
Source: Kenneth French Data Library. 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 explored whether we could time model specification choices in a simple trend equity strategy using momentum signals.
Testing different lookback horizons of 21-through-378 days, we found little evidence of meaningful persistence in the returns of different model specifications. In fact, four of the five momentum models we studied actually under-performed a naïve, diversified. The one model that did out-perform only seemed to do so due to strong performance realized over the 1950-1964 period, actually relatively under-performing ever since.
While this evidence suggests that timing specification with momentum may not be a fruitful approach, it does suggest that the lack of return persistence may benefit diversification for a second reason: rebalancing. Indeed, barring any belief that one specification would necessarily do better than another, consistently re-pooling and distributing resources through rebalancing may actually lead to the growth-optimal solution.1 This potentially implies an even higher hurdle rate for specification-timers to overcome.
Recent research suggests that equity factors exhibit positive autocorrelation, providing fertile ground for the application of trend-following strategies.
In this research note, we ask whether the same techniques can be applied to the active returns of long-only style portfolios.
We construct trend-following strategies on the active returns of popular MSCI style indices, including Value, Size, Momentum, Minimum Volatility, and Quality.
A naïve, equal-weight portfolio of style trend-following strategies generates an information ratio of 0.57.
The interpretation of this result is largely dependent upon an investor’s pre-conceived views of style investing, as the diversified trend-following approach generally under-performs a naïve, equal-weight portfolio of factors except during periods of significant and prolonged factor dislocation.
There have been a number of papers published in the last several years suggesting that positive autocorrelation in factor returns may be exploitable through time-series momentum / trend following. For example,
Ehsani and Linnainmaa (2017; revised 2019) document that “most factors exhibit positive autocorrelation with the average factor earning a monthly return of 2 basis points following a year of losses but 52 basis points following a positive year.”
Renz (2018) demonstrates that “risk premiums are significantly larger (lower) following recent uptrends (downtrends) in the underlying risk factor.”
While this research focuses mostly only long/short equity factors, it suggests that there may be opportunity for long-only style investors to improve their realized results as well. After all, long-only “smart beta” products can be thought of as simply a market-cap benchmark plus a dollar-neutral long/short portfolio of active bets.
Therefore, calculating the returns due to the active bets taken by the style is a rather trivial exercise: we can simply take the monthly returns of the long-only style index and subtract the returns of the long-only market-capitalization-weighted benchmark. The difference in returns will necessarily be due to the active bets.1
Below we plot the cumulative active returns for five popular equity styles: Value (MSCI USA Enhanced Value), Size (MSCI USA SMID), Momentum (MSCI USA Momentum), Minimum Volatility (MSCI USA Minimum Volatility), and Quality (MSCI USA Quality).
The active returns of these indices certainly rhyme with, but do not perfectly replicate, their corresponding long/short factor implementations. For example, while Momentum certainly exhibits strong, negative active returns from 6/2008 to 12/2009, the drawdown is nowhere near as severe as the “crash” that occurred in the pure long/short factor.
This is due to two facts:
The implied short side of the active bets is constrained by how far it can take certain holdings to zero. Therefore, long-only implementations tend to over-allocate towards top-quintile exposures rather than provide a balanced long/short allocation to top- and bottom-quintile exposures.
While the active bets form a long/short portfolio, the notional size of that portfolio is often substantially lower than the academic factor definitions (which, with the exception of betting-against-beta, more mostly assumed to have a notional exposure of 100% per leg). The active bets, on the other hand, have a notional size corresponding to the portfolio’s active share, which frequently hovers between 30-70% for most long-only style portfolios.
The implementation details of the long-only style portfolios and the long/short factor definitions may not perfectly match one another. As we have demonstrated a number of times in past research commentaries, these specification details can often swamp style returns in the short run, leading to meaningful cross-sectional dispersion in same-style performance.
Source: MSCI. Calculations by Newfound Research. Results are hypothetical. Results assume the reinvestment of all distributions. Results are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Past performance is not an indicator of future results. You cannot invest in an index.
Source: MSCI; AQR. Calculations by Newfound Research. Results are hypothetical. Results assume the reinvestment of all distributions. Results are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Past performance is not an indicator of future results. You cannot invest in an index.
Nevertheless, “rhymes but does not replicate” may be sufficient for long-only investors to still benefit from trend-following techniques.
In our test, we will go long the style / short the benchmark (i.e. long active returns) when prior N-month returns are positive and short the style / long the benchmark (i.e. short active returns) when prior N-month returns are negative. Portfolios are formed monthly at the end of each month. Performance results are reported in the table below for 1, 3, 6, 9, and 12-month lookback periods.
Annualized Return
Annualized Volatility
Information Ratio
Maximum Drawdown
Sample Size (Months)
1
Value
1.7%
6.1%
0.28
-15.1%
261
Size
-0.8%
8.2%
-0.10
-44.4%
303
Momentum
-0.2%
7.5%
-0.03
-21.3%
302
Minimum Volatility
-0.1%
5.7%
-0.01
-25.0%
375
Quality
1.3%
3.8%
0.35
-8.9%
302
3
Value
3.3%
6.0%
0.55
-15.5%
261
Size
1.1%
8.2%
0.13
-34.5%
303
Momentum
-0.8%
7.5%
-0.11
-38.0%
302
Minimum Volatility
0.7%
5.7%
0.13
-19.4%
375
Quality
0.9%
3.8%
0.24
-10.1%
302
6
Value
2.9%
6.0%
0.48
-21.0%
261
Size
1.7%
8.2%
0.20
-20.8%
303
Momentum
0.7%
7.5%
0.09
-28.8%
302
Minimum Volatility
0.5%
5.7%
0.09
-27.8%
375
Quality
0.6%
3.9%
0.16
-14.6%
302
9
Value
3.4%
6.0%
0.57
-14.8%
261
Size
2.0%
8.2%
0.24
-27.1%
303
Momentum
1.2%
7.5%
0.16
-23.4%
302
Minimum Volatility
0.9%
5.7%
0.15
-20.8%
375
Quality
0.3%
3.9%
0.07
-14.7%
302
12
Value
3.2%
6.0%
0.54
-11.2%
261
Size
1.8%
8.2%
0.22
-29.9%
303
Momentum
1.9%
7.5%
0.25
-20.0%
302
Minimum Volatility
1.4%
5.7%
0.24
-17.3%
375
Quality
1.3%
3.8%
0.34
-11.0%
302
Source: MSCI. Calculations by Newfound Research. Results are hypothetical. Results assume the reinvestment of all distributions. Results are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Past performance is not an indicator of future results. You cannot invest in an index.
Below we plot the equity curves of the 12-month time-series momentum strategy. We also plot a portfolio that takes a naïve equal-weight position across all five trend-following strategies. The naïve blend has an annualized return of 2.3%, an annualized volatility of 4.0%, and an information ratio of 0.57.
Source: MSCI. Calculations by Newfound Research. Results are hypothetical. Results assume the reinvestment of all distributions. Results are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Past performance is not an indicator of future results. You cannot invest in an index.
This analysis at least appears to provide a glimmer of hope for this idea. Of course, the analysis comes with several caveats:
We assume that investors can simultaneously generate signals and trade at month end, which may not be feasible for most.
We are analyzing index data, which may be different than the realized results of index-tracking ETFs.
We do not factor in trading costs such as impact, slippage, or commissions.
It is also important to point out that the per-style results vary dramatically. For example, trend-following on the size style has been in a material drawdown since 2006. Therefore, attempting to apply time-series momentum onto of a single style to manage style risk may only invite further strategy risk; this approach may be best applied with an ensemble of factors (and, likely, trend signals).
What this commentary has conveniently ignored, however, is that the appropriate benchmark for this approach is not zero. Rather, a more appropriate benchmark would be the long-only active returns of the styles themselves, as our default starting point is simply holding the styles long-only.
The results, when adjusted for our default of buy-and-hold, is much less convincing.
Source: MSCI. Calculations by Newfound Research. Results are hypothetical. Results assume the reinvestment of all distributions. Results are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Past performance is not an indicator of future results. You cannot invest in an index.
What is clear is that the strategy can now only out-perform when the style is under-performing the benchmark. When the portfolio invests in the style, our relative return versus the style is flat.
When a diversified trend-following portfolio is compared against a diversified long-only factor portfolio, we see the general hallmarks of a trend-following approach: value-add during periods of sustained drawdowns with decay thereafter. Trend-following on styles, then, may be more appropriate as a hedge against prolonged style under-performance; but we should expect a cost to that hedge.
Source: MSCI. Calculations by Newfound Research. Results are hypothetical. Results assume the reinvestment of all distributions. Results are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Past performance is not an indicator of future results. You cannot invest in an index.
For some styles, like Minimum Volatility, this appears to have helped relative performance drawdowns in periods like the dot-com bubble without too much subsequent give-up. Size, on the other hand, also benefited during the dot-com era, but subsequently suffered from significant trend-following whipsaw.
Conclusion
Recent research has suggested that equity style premia exhibit positive autocorrelation that can be exploited by trend followers. In this piece, we sought to explore whether this empirical evidence could be exploited by long-only investors by isolating the active returns of long-only style indices.
We found that a naïve 12-month time-series momentum strategy proved moderately effective at generating a timing strategy for switching between factor and benchmark exposure. Per-style results were fairly dramatic, and trend-following added substantial style risk of its own. However, diversification proved effective and an equal-weight portfolio of style trend-following strategies offered an information ratio of 0.57.
However, if we are already style proponents, a more relevant benchmark may be a long-only style portfolio. When our trend-following returns are taken in excess of this benchmark, results deflate dramatically, as the trend-following strategy can now only exploit periods when the style under-performs a market-capitalization-weighted index. Thus, for investors who already implement long-only styles in their portfolio, a trend-following overlay may serve to hedge periods of prolonged style drawdowns but will likely come with whipsaw cost which may drag down realized factor results.
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.