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Measuring Process Diversification in Trend Following

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On July 30, 2018

In Craftsmanship, Portfolio Construction, Risk Management, Weekly Commentary

This post is available as a PDF download here.

Summary­

  • We prefer to think about diversification in a three-dimensional framework: what, how, and when.
  • The “how” axis covers the process with which an investment decision is made.
  • There are a number of models that trend-followers might use to capture a trend. For example, trend-followers might employ a time-series momentum model, a price-minus moving average model, or a double moving average cross-over model.
  • Beyond multiple models, each model can have a variety of parameterizations. For example, a time-series momentum model can just as equally be applied with a 3-month formation period as an 18-month period.
  • In this commentary, we attempt to measure how much diversification opportunity is available by employing multiple models with multiple parameterizations in a simple long/flat trend-following process.

When investors talk about diversification, they typically mean across different investments.  Do not just by a single stock, for example, buy a basket of stocks in order to diversify away the idiosyncratic risk.

We call this “what” diversification (i.e. “what are you buying?”) and believe this is only one of three meaningful axes of diversification for investors.  The other two are “how” (i.e. “how are you making your decision?”) and “when” (i.e. “when are you making your decision?”).  In recent years, we have written a great deal about the “when” axis, and you can find a summary of that research in our commentary Quantifying Timing Luck.

In this commentary, we want to discuss the potential benefits of diversifying across the “how” axis in trend-following strategies.

But what, exactly, do we mean by this?  Consider that there are a number of ways investors can implement trend-following signals.  Some popular methods include:

  • Prior total returns (“time-series momentum”)
  • Price-minus-moving-average (e.g. price falls below the 200-day moving average)
  • Moving-average double cross-over (e.g. the 50-day moving average crosses the 200-day moving average)
  • Moving-average change-in-direction (e.g. the 200-day moving average slope turns positive or negative)

As it turns out, these varying methodologies are actually cousins of one another.  Recent research has established that these models can, more or less, be thought of as different weighting schemes of underlying returns.  For example, a time-series momentum model (with no skip month) derives its signal by averaging daily log returns over the lookback period equally.

With this common base, a number of papers over the last decade have found significant relationships between the varying methods.  For example:

 

Evidence
Bruder, Dao, Richard, and Roncalli (2011)Moving-average-double-crossover is just an alternative weighting scheme for time-series momentum.
Marshall, Nguyen and Visaltanachoti (2014)Time-series momentum is related to moving-average-change-in-direction.
Levine and Pedersen (2015)Time-series-momentum and moving-average cross-overs are highly related; both methods perform similarly on 58 liquid futures contracts.
Beekhuizen and Hallerbach (2015)Mathematically linked moving averages with prior returns.
Zakamulin (2015)Price-minus-moving-average, moving-average-double-cross-over, and moving-average-change-of-direction can all be interpreted as a computation of a weighted moving average of momentum rules.

 

As we have argued in past commentaries, we do not believe any single method is necessarily superior to another.  In fact, it is trivial to evaluate these methods over different asset classes and time-horizons and find an example that proves that a given method provides the best result.

Without a crystal ball, however, and without any economic interpretation why one might be superior to another, the choice is arbitrary.  Yet the choice will ultimately introduce randomness into our results: a factor we like to call “process risk.”  A question we should ask ourselves is, “if we have no reason to believe one is better than another, why would we pick one at all?”

We like to think of it this way: ex-post, we will know whether the return over a given period is positive or negative.  Ex-ante, all we have is a handful of trend-following signals that are forecasting that direction.  If, historically, all of these trend signals have been effective, then there may be no reason to necessarily believe on over another.

Combining them, in many ways, is sort of like trying to triangulate on the truth. We have a number of models that all look at the problem from a slightly different perspective and, therefore, provide a slightly different interpretation.  A (very) loose analogy might be using the collective information from a number of cell towers in effort to pinpoint the geographic location of a cellphone.

We may believe that all of the trend models do a good job of identifying trends over the long run, but most will prove false from time-to-time in the short-run. By using them together, we can potentially increase our overall confidence when the models agree and decrease our confidence when they do not.

With all this in mind, we want to explore the simple question: “how much potential benefit does process diversification bring us?”

The Setup

To answer this question, we first generate a number of long/flat trend following strategies that invest in a broad U.S. equity index or the risk-free rate (both provided by the Kenneth French database and ranging from 1926 to 2018). There are 48 strategy variations in total constructed through a combination of four difference processes – time-series momentum, price-minus-moving-average, and moving-average double cross-over– and 16 different lookback periods (from the approximate equivalent of 3-to-18 months).

We then treat each of the 64 variations as its own unique asset.

To measure process diversification, we are going to use the concept of “independent bets.” The greater the number of independent bets within a portfolio, the greater the internal diversification. Below are a couple examples outlining the basic intuition for a two-asset portfolio:

  • If we have a portfolio holding two totally independent assets with similar volatility levels, a 50% allocation to each would maximize our diversification.Intuitively, we have equally allocated across two unique bets.
  • If we have a portfolio holding two totally independent assets with similar volatility levels, a 90% allocation to one asset and a 10% allocation to another would lead us to a highly concentrated bet.
  • If we have a portfolio holding two highly correlated assets, no matter the allocation split, we have a large, concentrated bet.
  • If we have a portfolio of two assets with disparate volatility levels, we will have a large concentrated bet unless the lower volatility asset comprises the vast majority of the portfolio.

To measure this concept mathematically, we are going to use the fact that the square of the “diversification ratio” of a portfolio is equal to the number of independent bets that portfolio is taking.1

Diversifying Parameterization Risk

Within process diversification, the first variable we can tweak is the formation period of our trend signal.  For example, if we are using a time-series momentum model that simply looks at the sign of the total return over the prior period, the length of that period may have a significant influence in the identification of a trend.  Intuition tells us that shorter formation periods might identify short-term trends as well as react to long-term trend changes more quickly but may be more sensitive to whipsaw risk.

To explore the diversification opportunities available to us simply by varying our formation parameterization, we build equal-weight portfolios comprised of two strategies at a time, where each strategy utilizes the same trend model but a different parameterization.  We then measure the number of independent bets in that combination.

We run this test for each trend following process independently.  As an example, we compare using a shorter lookback period with a longer lookback period in the context of time-series momentum in isolation. We will compare across models in the next section.

In the graphs below, L0 through L15 represent the lookback periods, with L0 being the shortest lookback period and L15 representing the longest lookback period.

As we might suspect, the largest increase in available bets arises from combining shorter formation periods with longer formation periods.  This makes sense, as they represent the two horizons that share the smallest proportion of data and therefore have the least “information leakage.” Consider, for example, a time-series momentum signal that has a 4-monnth lookback and one with an 8-month lookback. At all times, 50% of the information used to derive the latter model is contained within the former model.  While the technical details are subtler, we would generally expect that the more informational overlap, the less diversification is available.

We can see that combining short- and long-term lookbacks, the total number of bets the portfolio is taking from 1.0 to approximately 1.2.

This may not seem like a significant lift, but we should remember Grinold and Kahn’s Fundamental Law of Active Management:

Information Ratio = Information Coefficient x SQRT(Independent Bets)

Assuming the information coefficient stays the same, an increase in the number of independent bets from 1.0 to 1.2 increases our information ratio by approximately 10%.  Such is the power of diversification.

Another interesting way to approach this data is by allowing an optimizer to attempt to maximize the diversification ratio.  In other words, instead of only looking at naïve, equal-weight combinations of two processes at a time, we can build a portfolio from all available lookback variations.

Doing so may provide two interesting insights.

First, we can see how the optimizer might look to combine different variations to maximize diversification.  Will it barbell long and short lookbacks, or is there benefit to including medium lookbacks? Will the different processes have different solutions?  Second, by optimizing over the full history of data, we can find an upper limit threshold to the number of independent bets we might be able to capture if we had a crystal ball.

A few takeaways from the graphs above:

  • Almost all of the processes barbell short and long lookback horizons to maximize diversification.
  • The optimizer finds value, in most cases, in introducing medium-term lookback horizons as well. We can see for Time-Series MOM, the significant weights are placed on L0, L1, L6, L10, and L15.  While not perfectly spaced or equally weighted, this still provides a strong cross-section of available information.  Double MA Cross-Over, on the other hand, finds value in weighting L0, L8, and L15.
  • While the optimizer increases the number of independent bets in all cases versus a naïve, equal-weight approach, the pickup is not incredibly dramatic. At the end of the day, a crystal ball does not find a meaningfully better solution than our intuition may provide.

Diversifying Model Risk

Similar to the process taken in the above section, we will now attempt to quantify the benefits of cross-process diversification.

For each trend model, we will calculate the number of independent bets available by combining it with another trend model but hold the lookback period constant. As an example, we will combine the shortest lookback period of the Time-Series MOM model with the shortest lookback period of the MA Double Cross-Over.

We plot the results below of the number of independent bets available through a naïve, equal-weight combination.

We can see that model combinations can lift the number of independent bets from by 0.05 to 0.1.  Not as significant as the theoretical lift from parameter diversification, but not totally insignificant.

Combining Model and Parameterization Diversification

We can once again employ our crystal ball in an attempt to find an upper limit to the diversification available to trend followers, as well as the process / parameterization combinations that will maximize this opportunity.  Below, we plot the results.

We see a few interesting things of note:

  • The vast majority of models and parameterizations are ignored.
  • Time-Series MOM is heavily favored as a model, receiving nearly 60% of the portfolio weight.
  • We see a spread of weight across short, medium, and long-term weights. Short-term is heavily favored, with Time-Series MOM L0 and Price-Minus MA L0 approaching nearly 45% of model weight.
  • All three models are, ultimately, incorporated, with approximately 10% being allocated to Double MA Cross-Over, 30% to Price-Minus MA, and 60% to Time-Series MOM.

It is worth pointing out that naively allocating equally across all 48 models creates 1.18 independent bets while the full-period crystal ball generated 1.29 bets.

Of course, having a crystal ball is unrealistic.  Below, we look at a rolling window optimization that looks at the prior 5 years of weekly returns to create the most diversified portfolio.  To avoid plotting a graph with 48 different components, we have plot the results two ways: (1) clustered by process and (2) clustered by lookback period.

Using the rolling window, we see similar results as we saw with the crystal ball. First, Time-Series MOM is largely favored, often peaking well over 50% of the portfolio weights.  Second, we see that a barbelling approach is frequently employed, balancing allocations to the shortest lookbacks (L0 and L1) with the longest lookbacks (L14 and L15).  Mid-length lookbacks are not outright ignored, however, and L5 through L11 combined frequently make up 20% of the portfolio.

Finally, we can see that the rolling number of bets is highly variable over time, but optimization frequently creates a meaningful impact over an equal-weight approach.2

Conclusion

In this commentary, we have explored the idea of process diversification.  In the context of a simple long/flat trend-following strategy, we find that combining strategies that employ different trend identification models and different formation periods can lead to an increase in the independent number of bets taken by the portfolio.

As it specifically pertains to trend-following, we see that diversification appears to be maximized by allocating across a number of lookback horizons, with an optimizer putting a particular emphasis on barbelling shorter and longer lookback periods.

We also see that incorporating multiple processes can increase available diversification as well.  Interestingly, the optimizer did not equally diversify across models.  This may be due to the fact that these models are not truly independent from one another than they might seem.  For example, Zakamulin (2015) demonstrated that these models can all be decomposed into a different weighted average of the same general momentum rules.

Finding process diversification, then, might require moving to a process that may not have a common basis.  For example, trend followers might consider channel methods or a change in basis (e.g. constant volume bars instead of constant time bars).

The Importance of Diversification in Trend Following

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On April 30, 2018

In Craftsmanship, Risk & Style Premia, Risk Management, Trend, Weekly Commentary

This post is available as a PDF download here.

Summary­

  • Diversification is a key ingredient to a successful trend following program.
  • While most popular trend following programs take a multi-asset approach (e.g. managed futures programs), we believe that single-asset strategies can play a meaningful role in investor portfolios.
  • We believe that long-term success requires introducing sources of diversification within single-asset portfolios. For example, in our trend equity strategies we employ a sector-based framework.
  • We believe the increased internal diversification allows not only for a higher probability of success, but also increases the degrees of freedom with which we can manage the strategy.
  • Introducing diversification, however, can also introduce tracking error, which can be a source of frustration for benchmark-sensitive investors.

Our friends over at ReSolve Asset Management recently penned a blog post titled Diversification – What Most Novice Investors Miss About Trend Following.  What the team at ReSolve succinctly shows – which we tried to demonstrate in our own piece, Diversifying the What, How, and When of Trend Following– is that diversification is a hugely important component of developing a robust trend following program.

A cornerstone argument of both pieces is that the overwhelming success of a simple trend following approach applied to U.S. equities may be misleading.  The same approach, when applied to a large cross-section of majority international equity indices, shows a large degree of dispersion.

That is not to say that the approach does not work: in fact, it is the robustness across such a large cross-section that gives us confidence that it does. Rather, we see that the relative success seen in applying the approach on U.S. equity markets may be a positive outlier.

ReSolve proposes a diversified, multi-asset trend following approach that is levered to the appropriate target volatility.  In our view, this solution is both theoretically and empirically sound.

That said, here at Newfound we do offer a number of solutions that apply trend following on a single asset class.  Indeed, the approach we are most well-known for (going back to when were founded in August 2008), has been long/flat trend following on U.S. equities.

How do we reconcile the belief that multi-asset trend following likely offers a higher risk-adjusted return, but still offer single-asset trend following strategies?  The answer emerges from our ethos of investing at the intersection of quantitative and behavioral finance.  Specifically, we acknowledge that investors tend to exhibit an aversion to non-transparent strategies that have significant tracking error to their reference benchmarks.

Trend following approaches on single asset classes like U.S. equities (an asset class that tends to dominate the risk profile of most U.S. investors) can therefore potentially offer a more sustainable risk management solution, even if it does so with a lower long-term risk-adjusted return than a multi-asset approach.

Nevertheless, we believe that how a trend following strategy is implemented is critical for long-term success.  This is especially true for approaches that target single asset classes.

Finding Diversification Within Single-Asset Strategies

Underlying Newfound’s trend equity strategies (both our Sector and Factor series) is a sector-based methodology.  The reason for employing this methodology is an effort to maximize internal strategy diversification.  Recalling our three-dimensional framework of diversification – “what” (investments), “how” (process), and “when” (timing) – our goal in using sectors is to increase diversification along the what axis.

As an example, below we plot the correlation between sector-based trend following strategies.  Specifically, we use a simple long/flat 200-day moving average cross-over system.

Correlation matrix of sector-based trend following strategies

Source: Kenneth French Data Library. Calculations by Newfound Research. Trend following strategy is a 200-day simple moving average cross-over approach where the strategy holds the underlying sector long when price is above its 200-day simple moving average and invests in the risk-free asset when price falls below.  Returns are gross of all fees, including transaction fees, taxes, and any management fees.  Returns assume the reinvestment of all distributions.  Past performance is not a guarantee of future results.

While none of the sector strategies offer negative correlation to one another (nor would we expect them to), we can see that many of the cross-correlations are substantially less than one.  In fact, the average pairwise correlation is 0.50.

Average pairwise correlation of sector trend following strategies

Source: Kenneth French Data Library. Calculations by Newfound Research. Trend following strategy is a 200-day simple moving average cross-over approach where the strategy holds the underlying sector long when price is above its 200-day simple moving average and invests in the risk-free asset when price falls below.  Not an actual strategy managed by Newfound. Hypothetical strategy created solely for this commentary and all returns are backtested and hypothetical.  Returns are gross of all fees, including transaction fees, taxes, and any management fees.  Returns assume the reinvestment of all distributions.  Past performance is not a guarantee of future results.

We would expect that we can benefit from this diversification by creating a strategy that trades the underlying sectors, which in aggregate provide us exposure to the entire U.S. equity market, rather than trading a single trend signal on the entire U.S. equity market itself.  Using a simple equal-weight approach among the seconds, we find exactly this.

The increased Sharpe ratio of a diversified trend following strategy

Source: Kenneth French Data Library. Calculations by Newfound Research. Trend following strategy is a 200-day simple moving average cross-over approach where the strategy holds the underlying sector long when price is above its 200-day simple moving average and invests in the risk-free asset when price falls below.  Not an actual strategy managed by Newfound. Hypothetical strategy created solely for this commentary and all returns are backtested and hypothetical.  Returns are gross of all fees, including transaction fees, taxes, and any management fees.  Returns assume the reinvestment of all distributions.  Past performance is not a guarantee of future results.

There are two important things to note.  First is that the simple trend following approach, when applied to broad U.S. equities, offers a Sharpe ratio higher than trend following applied to any of the underlying sectors themselves.  We can choose to believe that this is because there is something special about applying trend following at the aggregate index level, or we can assume that this is simply the result of a single realization of history and that our forward expectations for success should be lower.

We would be more likely to believe the former if we demonstrated the same effect across the globe.  For now, we believe it is prudent to assume the latter.

The most important detail of the chart, however, is that a simple equally-weighted portfolio of the underlying sector strategies not only offered a dramatic increase in the Sharpe ratio compared to the median sector strategy, but also a near 15% boost in Sharpe ratio against that offered by trend following on broad U.S. equities.

Using a sector-based approach also affords us greater flexibility in our portfolio construction.  For example, while a single-signal approach to trend following across broad U.S. equities creates an “all in” or “all out” dynamic, using sectors allows us to either incorporate other signals (e.g. cross-sectional momentum, as popularized in Gary Antonacci’s dual momentum approach) or re-distribute available capital.

For example, below we plot the annualized return versus maximum drawdown for an equal-weight sector strategy that allows for the re-use of capital.  For example, when a trend signal for a sector turns negative, instead of moving the capital to cash, the capital is equally re-allocated across the remaining sectors.  A position limit is then applied, allowing the portfolio to introduce the risk-free asset when a certain number of sectors has turned off.

The trade-off between annualized return and maximum drawdown when capital re-use is allowed

Source: Kenneth French Data Library. Calculations by Newfound Research. Not an actual strategy managed by Newfound.  Hypothetical strategy created solely for this commentary and all returns are backtested and hypothetical.  Returns are gross of all fees, including transaction fees, taxes, and any management fees.  Returns assume the reinvestment of all distributions.  Past performance is not a guarantee of future results.

The annotations on each point in the plot reflect the maximum position size, which can also be interpreted as inversely proportional the number of sectors that have to still be exhibiting a positive trend to remain fully invested.  For example, the point labeled 9.1% does not allow for any re-use of capital, as it requires all 11 sectors to be positive. On the other hand, the point labeled 50% requires just two sectors to exhibit positive trends to remain fully invested.

We can see that the degree to which capital is re-used becomes an axis along which we can trade-off our pursuit of return versus our desire to protect on the downside. Limited re-use decreases both drawdown and annualized return.  We can also see, however, that after a certain amount of capital re-use, the marginal increase in annualized return decreases dramatically while maximum drawdown continues to increase.

Of course, the added internal diversification and the ability to re-use available capital do not come free.  The equal-weight sector framework employed introduces potentially significant tracking error to broad U.S. equities, even without introducing the dynamics of trend following.

Tracking error between U.S. equities and an equal-weight sector portfolio

Source: Kenneth French Data Library. Calculations by Newfound Research. Not an actual strategy managed by Newfound.  Hypothetical strategy created solely for this commentary and all returns are backtested and hypothetical.  Returns are gross of all fees, including transaction fees, taxes, and any management fees.  Returns assume the reinvestment of all distributions.  Past performance is not a guarantee of future results.

We can see that the average long-term tracking error is not insignificant, and at times can be quite extreme.  The dot-com bubble, in particular, stands out as the equal-weight framework would have a significant underweight towards technology.  During the dot-com boom, this would likely represent a significant source of frustration for investors.  Even in less extreme times, annual deviations of plus-or-minus 4% from broad U.S. equities would not be uncommon.

Conclusion

For investors pursuing trend following strategies, diversification is a key ingredient.  Many of the most popular trend following programs – for example, managed futures – take a multi-asset approach.  However, we believe that a single-asset approach can still play a meaningful role for investors who seek to manage specific asset risk or who are looking for a potentially more transparent solution.

Nevertheless, diversification remains a critical consideration for single-asset solutions as well.  In our trend equity strategies here at Newfound, we employ a sector-based framework so as to increase the number of signals that dictate our overall equity exposure.

An ancillary benefit of this process is that the sectors provide us another axis with which to manage our portfolio.  We not only have the means by which to introduce other signals into our allocation process (e.g. overweighting sectors exhibiting favorable value or momentum tilts), but we can also decide how much capital we wish to re-invest when trend signals turn negative.

Unfortunately, these benefits do not come free.  A sector-based framework can also potentially introduce a significant degree of tracking error to standard equity benchmarks.  While we believe that the pros outweigh the cons over the long run, investors should be aware that such an approach can lead to significant relative deviations in performance over the short run.

Diversifying the What, How, and When of Trend Following

By

On April 2, 2018

In Craftsmanship, Risk & Style Premia, Risk Management, Trend, Weekly Commentary

This post is available as a PDF download here.

Summary

  • Naïve and simple long/flat trend following approaches have demonstrated considerable consistency and success in U.S. equities.
  • While there are many benefits to simplicity, an overly simplistic implementation can leave investors naked to unintended risks in the short run.
  • We explore how investors can think about introducing greater diversification across the three axes of what, how, and when in effort to build a more robust tactical solution.

In last week’s commentary – Protect & Participate: Managing Drawdowns with Trend Following – we explored the basics of trend following and how a simple “long/flat” investing approach, when applied to U.S. equities, has historically demonstrated considerable ability to limit extreme drawdowns.

While we always preach the benefits of simplicity, an evaluation of the “long run” can often overshadow many of the short-run risks that can materialize when a model is overly simplistic.  Most strategies look good when plotted over a 100-year period in log-scale and drawn with a fat enough marker.

With trend following in particular, a naïve implementation can introduce uncompensated risk factors that, if left unattended, can lead to performance gremlins.

We should be clear, however, that left unattended, nothing could happen at all.  You could get lucky.  That’s the funny thing about risk: sometimes it does not materialize and correcting for it can actually leave you worse off.

But hope is not a strategy and without a crystal ball at our disposal, we feel that managing uncompensated risks is critical for creating more consistent performance and aligning with investor expectations.

In light of this, the remainder of this commentary will be dedicated to exploring how we can tackle several of the uncompensated risks found in naïve implementations by using the three axes of diversification: what, how, and when. 

The What: Asset Diversification

The first axis of diversification is “what,” which encompasses the question, “what are we allocating across?”

As a tangent, we want to point out that there is a relationship between tactical asset allocation and underlying opportunities to diversify, which we wrote about in a prior commentary Rising Correlations and Tactical Asset Allocation.  The simple take is that when there are more opportunities for diversification, the accuracy hurdle rate that a tactical process has to overcome increases.  While we won’t address that concept explicitly here, we do think it is an important one to keep in mind.

Specifically as it relates to developing a robust trend following strategy, however, what we wish to discuss is “what are we generating signals on?”

A backtest of a naively implemented trend following approach on U.S. equities over the last century has been exceptionally effective.  Perhaps deceivingly so.  Consider the following cumulative excess return results from 12/1969 to present for a 12-1 month time-series momentum strategy.

 

Source: Kenneth French Data Library.  Calculations by Newfound Research.  Past performance is not an indication of future returns.  All performance information is backtested and hypothetical.  Performance is gross of all fees, including manager fees, transaction costs, and taxes.  Performance is net of withholding taxes.  Performance assumes the reinvestment of all dividends.  Benchmark is 50% U.S. equity index / 50% risk-free rate.

While the strategy exhibits a considerable amount of consistency, this need not be the case.

Backtests demonstrate that trend following has worked in a variety of international markets “over the long run,” but the realized performance can be much more volatile than we have seen with U.S. equities.  Below we plot the growth of $1 in standard 12-1 month time-series momentum strategies for a handful of randomly selected international equity markets minus their respective benchmark (50% equity / 50% cash).

Note: Things can get a little whacky when working with international markets.  You ultimately have to consider whose perspective you are investing from.  Here, we assumed a U.S. investor that uses U.S. dollar-denominated foreign equity returns and invests in the U.S. risk-free rate.  Note that this does, by construction, conflate currency trends with underlying trends in the equity indices themselves.  We will concede that whether the appropriate measure of trend should be local-currency based or not is debatable.  In this case, we do not think it affects our overall point.

Source: MSCI.  Calculations by Newfound Research.  Past performance is not an indication of future returns.  All performance information is backtested and hypothetical.  Performance is gross of all fees, including manager fees, transaction costs, and taxes.  Performance is net of withholding taxes.  Performance assumes the reinvestment of all dividends.  Benchmark is 50% respective equity index / 50% U.S. risk-free rate.

The question to ask ourselves, then, is, “Do we believe U.S. equities are special and naive trend following will continue to work exceptionally well, or was U.S. performance an unusual outlier?”

We are rarely inclined to believe that exceptional, outlier performance will continue.  One approach to providing U.S. equity exposure while diversifying our investments is to use the individual sectors that comprise the index itself.  Below we plot the cumulative excess returns of a simple 12-1 time-series momentum strategy applied to a random selection of underlying U.S. equity sectors.

Source: Kenneth French Data Library.  Calculations by Newfound Research.  Past performance is not an indication of future returns.  All performance information is backtested and hypothetical.  Performance is gross of all fees, including manager fees, transaction costs, and taxes.  Performance is net of withholding taxes.  Performance assumes the reinvestment of all dividends.  Benchmark is 50% respective sector index / 50% U.S. risk-free rate.

While we can see that trend following was successful in generating excess returns, we can also see that when it was successful varies depending upon the sector in question.  For example, Energy (blue) and Telecom (Grey) significantly diverge from one another in the late 1950s / early 1960s as well as in the late 1990s / early 2000s.

If we simply equally allocate across sector strategies, we end up with a cumulative excess return graph that is highly reminiscent of the of the results seen in the naïve U.S. equity strategy, but generated with far more internal diversification.

Source: Kenneth French Data Library.  Calculations by Newfound Research.  Past performance is not an indication of future returns.  All performance information is backtested and hypothetical.  Performance is gross of all fees, including manager fees, transaction costs, and taxes.  Performance is net of withholding taxes.  Performance assumes the reinvestment of all dividends.

A potential added benefit of this approach is that we are now afforded the flexibility to vary sector weights depending upon our objective.  We could potentially incorporate other factors (e.g. value or momentum), enforce diversification limits, or even re-invest capital from sectors exhibiting negative trends back into those exhibiting positive trends.

The How: Process Diversification

The second axis of diversification is “how”: the process in which decisions are made.  This axis can be a bit of a rabbit hole: it can start with high-level questions such as, “value or momentum?” and then go deeper with, “which value measure are you using?” and then even more nuanced with questions such as, “cross-market or cross-industry measures?”  Anecdotally, the diversification “bang for your buck” decreases as the questions get more nuanced.

With respect to trend following, the obvious question is, “how are you measuring the trend?”

One Signal to Rule Them All?

There are a number of ways investors can implement trend-following signals.  Some popular methods include:

  • Prior total returns (“time-series momentum”)
  • Price-minus-moving-average (e.g. price falls below the 200 day moving average)
  • Moving-average double cross-over (e.g. the 50 day moving average crosses the 200 day moving average)
  • Moving-average change-in-direction (e.g. the 200 day moving average slope turns positive or negative)

One question we often receive is, “is there one approach that is better than another?”  Research over the last decade, however, actually shows that they are highly related approaches.

Bruder, Dao, Richard, and Roncalli (2011) united moving-average-double-crossover strategies and time-series momentum by showing that cross-overs were really just an alternative weighting scheme for returns in time-series momentum.[1] To quote,

“The weighting of each return … forms a triangle, and the biggest weighting is given at the horizon of the smallest moving average. Therefore, depending on the horizon n2 of the shortest moving average, the indicator can be focused toward the current trend (if n2 is small) or toward past trends (if n2 is as large as n1/2 for instance).”

Marshall, Nguyen and Visaltanachoti (2012) proved that time-series momentum is related to moving-average-change-in-direction.[2] In fact, time-series momentum signals will not occur until the moving average changes direction.  Therefore, signals from a price-minus-moving-average strategy are likely to occur before a change in signal from time-series momentum.

Levine and Pedersen (2015) showed that time-series momentum and moving average cross-overs are highly related.[3] It also found that time-series momentum and moving-average cross-over strategies perform similarly across 58 liquid futures and forward contracts.

Beekhuizen and Hallerbach (2015) also linked moving averages with returns, but further explored trend rules with skip periods and the popular MACD rule.[4] Using the implied link of moving averages and returns, it showed that the MACD is as much trend following as it is mean-reversion.

Zakamulin (2015) explored price-minus-moving-average, moving-average-double-crossover, and moving-average-change-of-direction technical trading rules and found that they can be interpreted as the computation of a weighted moving average of momentum rules with different lookback periods.[5]

These studies are important because they help validate the approach of traditional price-based systems (e.g. moving averages) with the growing body of academic literature on time-series momentum.

The other interpretation, however, is that all of the approaches are simply a different way of trying to tap into the same underlying factor.  The realized difference in their results, then, will likely have to do more with the inefficiencies in capturing that factor and which specific environments a given approach may underperform.  For example, below we plot the maximum return difference over rolling 5-year periods between four different trend following approaches: (1) moving-average change-in-direction (12-month), (2) moving-average double-crossover (3-month / 12-month), (3) price-minus-moving-average (12-month), and (4) time-series momentum (12-1 month).

We can see that during certain periods, the spread between approaches can exceed several hundred basis points.  In fact, the long-term average spread was 348 basis points (“bps”) and the median was 306 bps.  What is perhaps more astounding is that no approach was a consistent winner or loser: relative performance was highly time-varying.  In fact, when ranked 1-to-4 based on prior 5-year realized returns, the average long-term ranks of the strategies were 2.09, 2.67, 2.4, and 2.79 respectively, indicating that no strategy was a clear perpetual winner or loser.

 Source: Kenneth French Data Library.  Calculations by Newfound Research.  Past performance is not an indication of future returns.  All performance information is backtested and hypothetical.  Performance is gross of all fees, including manager fees, transaction costs, and taxes.  Performance assumes the reinvestment of all dividends. 

Without the ability to forecast which model will do best and when, model choice represents an uncompensated risk that we bear as a manager.  Using multiple methods, then, is likely a prudent course of action.

Identifying the Magic Parameter

The academic and empirical evidence for trend following (and, generally, momentum) tends to support a formation (“lookback”) period of 6-to-12 months.  Often we see moving averages used that align with this time horizon as well.

Intuition is that shorter horizons tend to react to market changes more quickly since new information represents a larger proportion of the data used to derive the signal.  For example, in a 6-month momentum measure a new monthly data point represents 16.6% of the data, whereas it only represents 8.3% of a 12-month moving average.

A longer horizon, therefore, is likely to be more “stable” and therefore less susceptible to whipsaw.

Which particular horizon achieves the best performance, then, will likely be highly regime dependent.  To get a sense of this, we ran six time-series momentum strategies, with look-back periods ranging from 6-months to 12-months.  Again, we plot the spread between the best and worst performing strategies over rolling 5-year periods.

 Source: Kenneth French Data Library.  Calculations by Newfound Research.  Past performance is not an indication of future returns.  All performance information is backtested and hypothetical.  Performance is gross of all fees, including manager fees, transaction costs, and taxes.  Performance assumes the reinvestment of all dividends. 

Ignoring the Great Depression for a moment, we can see that 5-year annualized returns between parameterizations frequently deviate by more than 500 bps.  If we dig under the hood, we again see that the optimal parameterization is highly regime dependent.

For example, coming out of the Great Depression, the longer-length strategies seemed to perform best.  From 8/1927 to 12/1934, an 11-1 time-series momentum strategy returned 136% while a 6-1 time-series momentum strategy returned -25%.  Same philosophy; very different performance.

Conversely, from 12/1951 to 12/1971, the 6-1 strategy returned 723% while the 11-1 strategy returned 361%.

Once again, without evidence that we can time our parameter choice, we end up bearing unnecessary parameterization risk, and diversification is a prudent action.

Source: Kenneth French Data Library.  Calculations by Newfound Research.  Past performance is not an indication of future returns.  All performance information is backtested and hypothetical.  Performance is gross of all fees, including manager fees, transaction costs, and taxes.  Performance assumes the reinvestment of all dividends. 

Source: Kenneth French Data Library.  Calculations by Newfound Research.  Past performance is not an indication of future returns.  All performance information is backtested and hypothetical.  Performance is gross of all fees, including manager fees, transaction costs, and taxes.  Performance assumes the reinvestment of all dividends. 

The When: Timing Luck

Long-time readers of our commentary will be familiar with this topic.  For those unfamiliar, we recommend a quick glance over our commentary Quantifying Timing Luck (specifically, the section What is “Timing Luck”?).

The simple description of the problem is that investment strategies can be affected by the investment opportunities they see at the point at which they rebalance.  For example, if we rebalance our tactical strategies at the end of each month, our results will be subject to what our signals say at that point.  We can easily imagine two scenarios where this might work against us:

  1. Our signals identify no change and we remain invested; the market sells off dramatically over the next month.
  2. The market sells off dramatically prior to our rebalance, causing us to move to cash. After we trade, the market rebounds significantly, causing us to miss out on potential gains.

As it turns out, these are not insignificant risks.  Below we plot four identically managed tactical strategies that each rebalance on a different week of the month.  While one of the strategies turned $1 into $4,139 another turned it into $6,797.  That is not an insignificant difference.

Source: Kenneth French Data Library.  Calculations by Newfound Research.  Past performance is not an indication of future returns.  All performance information is backtested and hypothetical.  Performance is gross of all fees, including manager fees, transaction costs, and taxes.  Performance assumes the reinvestment of all dividends. 

Fortunately, the cure for this problem is simple: diversification.  Instead of picking a week to rebalance on, we can allocate to multiple variations of the strategy, each rebalancing at a different point in time.  One variation may rebalance on the 1st week of the month, another on the 2nd week, et cetera.  This technique is called “overlapping portfolios” or “tranching” and we have proven in past commentaries that it can dramatically reduce the impact that timing luck can have on realized results.

Conclusion

Basic, naïve implementations of long/flat trend following exhibit considerable robustness and consistency over the long run when applied to U.S. equities.  The short run, however, is a different story.  While simple implementations can help ensure that we avoid overfitting our models to historical data, it can also leave us exposed to a number of unintended bets and uncompensated risks.

Instead of adding more complexity, we believe that the simple solution to combat these risks is diversification.

Specifically, we explore diversification across three axes.

The first axis is “what” and represents “what we invest across.”  We saw that while trend following worked well on U.S. equities, the approach had less consistency when applied to international indices.  Instead of presuming that the U.S. represents a unique candidate for this type of strategy, we explored a sector-based implementation that may allow for greater internal diversification.

The second axis is “how” and captures “how we implement the strategy.”  There are a variety of approaches practitioners use to measure and identify trends, and each comes with its own pros and cons.  We explore four popular methods and find that none consistently reigns supreme, indicating once again that diversification of process is likely a prudent approach.

Similarly, when it comes to parameterizing these models, we find that a range of lookback periods are successful in the long run, but have varying performance in the short run.  A prudent solution once again, is diversification.

The final axis is “when” and represents “when we rebalance our portfolio.”  Long-time readers recognize this topic as one we frequently write about: timing luck.  We demonstrate that merely shifting what week of the month we rebalance on can have considerable long-term effects.  Again, as an uncompensated risk, we would argue that it is best diversified away.

While a naïve trend following process is easy to implement, we believe that a robust one requires thinking along the many dimensions of risk and asking ourselves which risks are worth bearing (hopefully those that are compensated) and which risks we should seek to hedge or diversify away.

 

 

[1] Bruder, Benjamin and Dao, Tung-Lam and Richard, Jean-Charles and Roncalli, Thierry, Trend Filtering Methods for Momentum Strategies (December 1, 2011). Available at SSRN: http://ssrn.com/abstract=2289097

[2] Marshall, Ben R. and Nguyen, Nhut H. and Visaltanachoti, Nuttawat, Time-Series Momentum versus Moving Average Trading Rules (December 22, 2014). Available at SSRN: http://ssrn.com/abstract=2225551

[3] Levine, Ari and Pedersen, Lasse Heje, Which Trend Is Your Friend? (May 7, 2015). Financial Analysts Journal, vol. 72, no. 3 (May/June 2016). Available at SSRN: https://ssrn.com/abstract=2603731

[4] Beekhuizen, Paul and Hallerbach, Winfried G., Uncovering Trend Rules (May 11, 2015). Available at SSRN: http://ssrn.com/abstract=2604942

[5] Zakamulin, Valeriy, Market Timing with Moving Averages: Anatomy and Performance of Trading Rules (May 13, 2015). Available at SSRN: http://ssrn.com/abstract=2585056

Quantifying Timing Luck

By

On January 22, 2018

In Craftsmanship, Risk Management, Weekly Commentary

This blog post is available as a PDF download here.

Summary­­

  • When two managers implement identical strategies, but merely choose to rebalance on different days, we call variance between their returns “timing luck.”
  • Timing luck can easily be overcome by using a method of overlapping portfolios, but few firms do this in practice.
  • We believe the magnitude of timing luck impact is much larger than most believe, particularly in tactical strategies.
  • We derive a model to estimate the impact of timing luck, using only values that can be easily estimated from portfolios implemented without the overlapping portfolio technique.
  • We find that timing luck looms large in many different types of strategies.

As a pre-emptive warning, this week’s commentary is a math derivation.  We think it is a very relevant derivation – one which we have not seen before – but a derivation nonetheless.  If math is not your thing, this might be one to skip.

If math is your thing: consider this a request for comments.  The derivation here will be rather informal sketch, and we think there are other improvements still lingering.

What is “Timing Luck?”

The basic concept of timing luck is that when we choose to rebalance can have a profound impact on our performance results.  For example, if we rebalance an investment strategy once a month, the choice to rebalance at the end of the month will lead to different performance than had we elected to rebalance mid-month.

We call this performance differential “timing luck,” and we believe it is an overlooked, non-negligible portfolio construction risk.

As an example, consider a simple stock/cash timing model that rebalances monthly, investing in a broad U.S. equity index when its 12-1 month return is positive, and a constant maturity 1-year U.S. Treasury index otherwise.  Depending on which day of the month you choose to rebalance (we will assume 21 variations to represent 21 trading days), your results may be dramatically different.

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

The best performing strategy had an annualized return of 11.1%, while the worst returned just 9.6%.  Compounded over 55 years, and that 150 basis point (“bps”) differential leads to an astounding difference in final wealth.  With a standard deviation between 50-year annualized returns of 0.42%, the 1-year annualized estimate of performance variation due to timing luck is 314bps!

Again, an identical process is employed: the only difference between these results is the choice of what day of the month to rebalance.

That small choice, and the good luck or misfortune it realizes, can easily be the difference between “hired” and “fired.”

Is There a Solution to Timing Luck?

In the past, we have argued that overlapping portfolios can be utilized to minimize the impact of timing luck.  The idea of overlapping portfolios is as follows: given an investment process and a holding period, we can invest across multiple managers that invest utilizing the same process but have offset holding periods.[1]

For example, below each manager has a four time-step holding period, and we utilize four managers to minimize timing luck from a single implementation.

The proof that this approach minimizes timing luck is as follows.

Assume that we have N managers, all following an identical investment process with identical holding period, but whose rebalance points are offset from one another by one period.

Consider that at any point in time, we can define the portfolio of Manager #2 to be the portfolio of Manager #1 plus a dollar-neutral long/short portfolio that captures the differences in holdings between them.  Similarly, Manager #3’s portfolio can be thought of as Manager #2’s portfolio plus a dollar-neutral long/short portfolio.  This continues in a circular manner, where Manager #1’s portfolio can be thought of as Manager #N’s portfolio plus a dollar-neutral long/short.

Given that the managers all follow an identical process, we would expect them to have the same long-term expected return.  Thus, the expected return of the dollar-neutral long/short portfolios is zero.

However, the variance of the dollar-neutral long/short portfolios captures the risk of timing luck.

In allocating capital between the N portfolios, our goal is to minimize timing luck.  Put another way, we want to find the allocation that results in the minimum variance portfolio of the long/short portfolios.  Fortunately, there is a simple, closed form solution for calculating the minimum variance portfolio:

Here, w is our solution (an Nx1 vector of weights), Sigma is the covariance matrix and  is an Nx1 vector of 1s.  To solve this equation, we need the covariance matrix between the long/short portfolios.  Since each portfolio is employing an identical process, we can assume that each of the long/short portfolios should have equal variance.  Without loss of generality, we can assume variances are equal to 1 and replace our covariance matrix, Sigma, with a correlation matrix, C.

The correlations between long/short portfolios will largely depend on the process in question and the amount of overlap between portfolios.  That said, because each manager runs an identical process, we would expect that the long-term correlation between Portfolio #2’s long/short and Portfolio #1’s long/short to be identical to the correlation between Portfolio #3’s long/short and Portfolio #2’s.  Similarly, the correlation between Portfolio #3’s and Portfolio #1’s long/shorts should be the same as the correlation between Portfolio #N’s and Portfolio #2’s.

Following this logic (and remembering the circular nature of the rebalances), we can ignore exact numbers and fill in a correlation matrix using variables:

This correlation matrix has two special properties.  First, being a correlation matrix, it is symmetric.  Second, it is circulant: each row is rotated one element to the right of the preceding row.  A special property of a symmetric circulant matrix is that its inverse – in this case C-1 – is also symmetric circulant.  This property guarantees that C-11 is equal to k1 for some constant k.

Which means we can re-write our minimum variance solution as:

Since the constant  will cancel out, we are left with:

Thus, our optimal solution is an equal-weight allocation to all N portfolios.

Highlighted in gold below, we can see the result of this approach using the same stock/cash example as before.  Specifically, the gold portfolio uses each of the 21 variations as a different sub-portfolio.

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.

While we have a solution for timing luck, a question that lingers is: “how much will timing luck affect my particular strategy?”

The Setup

We assume an active investment strategy with constant portfolio of variance (S2), constant and continuous annualized turnover (T; e.g. 0.5 for 50% annual turnover), and consistent rebalances at discrete frequency (f; e.g. 1/12 for monthly).

We will also assume that the portfolio contains no static components.  This allows us to interpret 100% turnover as meaning that the entire portfolio was turned over, rather than that 50% of the portfolio was turn over twice.

To quantify the magnitude of timing luck, we will calculate the variance of a dollar-neutral, long/short portfolio that is long a discrete implementation (i.e. rebalancing at a fixed interval) of this strategy (D) and short the theoretically optimal infinite overlapping portfolio implementation (M – for “meta”).

As before, the expected return of this long/short is zero, but its variance captures the return differences created by timing luck.

Differences between the Discrete and Continuous Portfolios

The long/short portfolio is defined as (D – M).  However, we would expect the holdings of D to overlap with the holdings of M.  How much overlap will depend on both portfolio turnover and rebalance frequency.

Assume, for a moment, that M does not have infinite overlapping portfolios, but a finite number N, each uniformly spaced across the holding period.

If we assume 100% turnover that is continuous, we would expect that the first overlapping portfolio, implemented at t=1/N, to have (1 – 1/N) percent of its holdings identical to D (i.e. not “turned over”).  On the other hand, the portfolio implemented at t = (N-1)/N will have just 1/N percent of its holdings identical to D.

Thus, we can say that if M contains N discrete overlapping portfolios, we can expect M and D to overlap by:

Which we can reduce,

If we take the limit as N goes to infinity – i.e. we have infinite overlapping portfolios – then we are simply left with:

Thus, the overlap we expect between our discretely implemented portfolio, D, and the portfolio with infinite overlapping portfolios, M, is a simple function of the expected turnover during the holding period.

We can then define our long/short portfolio:

Where Q is the portfolio of holdings in M that are not in D.

We should pause here, for a moment, as this is where our assumption of “no static portfolio elements” becomes relevant.  We defined (1) to be the amount M and D overlap.   Technically, if we allow securities to be sold and then repurchased, (1) represents a lower limit to how much M and D overlap.  As an absurd example, consider a portfolio that creates 100% turnover by buying and selling the same 1% of the portfolio 100 times.  Thus, Q in (6) need not necessarily be unique from D; part of D could be contained in Q.

By assuming that no part of the portfolio is static, we are assuming that over the (very) long run, the average turnover experience over a holding period does not include repurchase of sold securities, and thus (1) is the amount of overlap and D and Q are independent holdings.

This assumption is likely fairer for traditionally active portfolios that focus on security selection, but potentially less realistic for tactical strategies that often sell and re-purchase the same exposure.  More on this later.

Defining,

We can re-write,

Solving for Timing Luck

We can then solve for the variance of the long/short portfolio,

Expanding:

As D and Q both represent viable allocation schemes for the portfolio, we will assume that they share the same long-term portfolio variance, S2.  This assumption may be fair, over the long run, for traditional stock-selection portfolios, but likely less fair for highly tactical portfolios that can meaningfully shift their portfolio risk exposures.

Thus,

Replacing back our definition for a, we are left with:

Or, that the annualized volatility due to timing luck (L) is:

What is Corr(D,Q)?

The least easily interpreted – or calculated – term in our equation is the correlation between our discrete portfolio, D, and the non-overlapping securities found in the infinite overlapping portfolios implementation, Q.

The intuitive interpretation here is that when the securities held in our discrete portfolio are highly correlated to those that are not held but the optimal strategy recommends we hold, then we would expect the difference to have less impact.  On the other hand, if those securities are negatively correlated, then the discrete rebalance choice could lead to significant additional volatility.

Estimating this value, however, may be difficult to do empirically.

One potential answer is to use the intra-portfolio correlation (“IPC”) of an equal-weight portfolio of representative assets or securities.  The intuition here is that we expect each asset to experience, on average, an equivalent amount of turnover due to our assumption that there are no static positions in the portfolio.

Thus, taking the IPC of an equal-weight portfolio of representative securities allows us to express the view that while we do not know which securities will be different at any given point in time, we expect over the long-run that all securities will be “missing” with equal frequency and magnitude, and therefore the IPC is representative of the long-term correlation between D and Q.

Estimating Timing Luck in our Stock/Cash Tactical Strategy

The assumptions required for our estimate of timing luck may work well with traditional security selection portfolios (or, at least, quantitative implementations of factors like value, momentum, defensive etc.), but will it work with tactical portfolios?

Using our prior stock/cash example, let’s estimate the expected magnitude of timing luck.  Using one of the discrete implementations, we estimate that turnover is 67% per year.  Our rebalance frequency is monthly (1/12) and the intra-portfolio correlation between stocks and bonds is assumed to be 0%.  Finally, the long-term volatility of the strategy is about 12.2%.

Using these figures, we estimate:

This is a somewhat disappointing result, as we had calculated prior that the actual timing luck was 314bps.  Our estimate is less than 1/6th of the actual figure!

Part of the problem may be that many of the assumptions we outlined are violated with our example tactical strategy.  We think the bigger problem is that our estimates for these variables, when using a highly tactical strategy, are simply wrong.

In our equation, we assumed that turnover would be continuous.  This is because we are using turnover as a proxy for the decay speed of our alpha signal.

What does this mean?  As an example, value strategies rely on value signals that tend to decay slowly.  When a stock is identified as being a value stock, it tends to stay that way for some time.  Therefore, if you build a portfolio off of these signals, you would expect low turnover.  Momentum signals, on the other hand, tend to decay more quickly.  A stock that is labeled as high momentum this month may no longer be high momentum in three months’ time.  Thus, momentum strategies tend to be high turnover.

This relationship does not necessarily hold for tactical strategies.

In our tactical example, we rebalance monthly because we believe the time-series momentum has a short forecast horizon.  However, with only two assets, the strategy can go years without turnover.  Worse, the same strategy might miss a signal because it is only sampling in a discrete manner and therefore understate true turnover in a continuous framework.

If we were to look at the turnover of a tactical strategy implemented with the same rules but rebalanced daily, we would see a turnover rate over 300%.  This would increase our estimate up to 215bps.  Still well below the realized 314bps, but certainly high enough to raise eyebrows about the impact of timing luck in tactical portfolios not implemented using overlapping portfolios.

We should also remember that timing luck is determined by the difference in holdings between the discrete strategy and the meta strategy.  We had assumed that the portfolios D and Q would have the same volatility, but in a strategy that shifts between stocks and bonds, this most certainly is not the case.  This means that long-run volatility in such a tactical strategy can actually be misleadingly low.

Consider the situation when the tactical strategy goes to cash based upon a short-lived signal; i.e. the meta strategy will not build a significant cash position.  The realized volatility of the strategy will dampen the perceived timing luck, when in reality the volatility difference between the two portfolios is quite large.

In our specific tactical example, we know that when D is stocks, Q is bonds and vice versa.  With this insight, we can re-write equation (10):

Which we can simplify as:

Which is simply just a constant times the variance of a portfolio that is 100% long stocks and -100% short bonds (or vice versa; the variance will be the same).

If we use this equation and the variance of a long/short stock/bond portfolio and our prior estimate of 300% turnover, we get an estimate of timing luck volatility of 191bps.

Note that using this concept, there may be a more generic solution that is possible using some measure of active variance (likely scaled by active share).

Conclusion

In this piece we have demonstrated the potentially massive impact of timing luck, addressed how to solve for it, and derived a model that can be used to estimate the magnitude of timing luck risk in strategies that do not employ an overlapping portfolios technique.

While our derived approach is not perfect – as we saw in its application with our tactical example – we believe it is an important step forward in being able to quantify the potential risk that timing luck creates.

 

 

[1] In reality, we probably wouldn’t hire a different manager to implement the same strategy with different rebalance timing even if we could find such managers. A more feasible solution would be for a single manager to run different sleeves implementing each rebalance iteration.

 

Factor Investing & The Bets You Didn’t Mean to Make

By

On January 16, 2018

In Craftsmanship, Portfolio Construction, Risk & Style Premia, Risk Management, Weekly Commentary

This post is available as a PDF download here.

Summary­­

  • Factor investing seeks to balance specificity with generality: specific enough to have meaning, but general enough to be applied broadly.
  • Diversification is a key tool to managing risk in factor portfolios. Imprecision in the factor definitions means that unintended bets are necessarily introduced.
  • This is especially true as we apply factors across securities that share fewer and fewer common characteristics. Left unmonitored, these unintended bets have the potential to entirely swamp the factor itself.
  • By way of example, we explore a simple value-based country model.
  • While somewhat counter-intuitive, constraints have the potential to lead to more efficient factor exposures.

In quantitative investing, we seek a balance between generality and specificity.  When a model is too specific – designed to have meaning on too few securities or in too few scenarios – we lose our ability to diversify.  When a model is too generic, it loses meaning and forecasting power.

The big quant factors – value, momentum, defensive, carry, and trend – all appear to find this balance: generic enough to be applied broadly, but specific enough to maintain a meaningful signal.

As we argued in our past commentary A Case Against Overweighting International Equity, the imprecision of the factors is a feature, not a bug.  A characteristic like price-to-earnings may never fully capture the specific nuances of each firm, but it can provide a directionally accurate roadmap to relative firm valuations.  We can then leverage diversification to average out the noise.

Without diversification, we are highly subject to the imperfections of the model.  This is why, in the same piece, we argued that making a large regional tilt – e.g. away from U.S. towards foreign developed – may not be prudent: it is a single bet that can take decades to resolve.  If we are to sacrifice diversification in our portfolio, we’ll require a much more accurate model to justify the decision.

Diversification, however, is not just measured by the quantity of bets we take.  If diversification is too naively interpreted, the same imprecision that allows factors to be broadly applied can leave our portfolios subject to the returns of unintended bets.

Value Investing with Countries

If taking a single, large regional tilt is not prudent, perhaps value investing at a country level may better diversify our risks.

One popular way of measuring value is with the Shiller CAPE: a cyclically-smoothed price-to-earnings measure.  In the table below, we list the current CAPE and historical average CAPE for major developed countries.

CAPEMean CAPEEffective Weight
Australia18.517.22.42%
Belgium25.015.40.85%
Canada22.021.43.76%
Denmark36.524.50.73%
France20.921.94.85%
Germany20.620.64.36%
Hong Kong18.218.35.21%
Italy16.822.11.33%
Japan28.943.211.15%
Netherlands23.514.81.45%
Singapore13.922.11.09%
Spain13.418.31.58%
Sweden21.523.01.21%
Switzerland25.921.93.15%
United Kingdom16.515.36.55%
United States30.520.350.30%

Source: StarCapital.de.  Effective weight is market-capitalization weight of each country, normalized to sum to 100%.  Mean CAPE figures use data post-1979 to leverage a common dataset.

While evidence[1] suggests that valuation levels themselves are enough to determine relative valuation among countries, we will first normalize the CAPE ratio by its long-term average to try to account for structural differences in CAPE ratios (e.g. a high growth country may have a higher P/E, a high-risk country may have a lower P/E, et cetera).  Specifically, we will look at the log-difference between the mean CAPE and the current CAPE scores.

Note that we recognize there is plenty to criticize and improve upon here.  Using a normalized valuation metric will mean a country like Japan, which experienced a significant asset bubble, will necessarily look under-valued.  Please do not interpret our use of this model as our advocacy for it: we’re simply using it as an example.

Using this value score, we can compare how over and undervalued each country is relative to each other.  This allows us to focus on the relative cheapness of each investment.  We can then use these relative scores to tilt our market capitalization weights to arrive at a final portfolio.

 

Value ScoreRelative Z-ScoreScaled Z-ScoreScaled Weights
Australia-0.07-0.130.882.31%
Belgium-0.48-1.500.400.37%
Canada-0.030.021.024.15%
Denmark-0.40-1.220.450.36%
France0.050.271.276.65%
Germany0.000.111.115.24%
Hong Kong0.010.131.136.37%
Italy0.271.022.022.92%
Japan0.401.452.4529.59%
Netherlands-0.46-1.430.410.65%
Singapore0.461.652.653.14%
Spain0.311.152.153.68%
Sweden0.070.331.331.75%
Switzerland-0.17-0.450.692.36%
United Kingdom-0.08-0.140.886.22%
United States-0.41-1.250.4524.26%

Source: StarCapital.de.  Calculations by Newfound Research.  “Value Score” is the log-difference between the country’s Mean CAPE and its Current CAPE.  Relative Z-Score is the normalized value score of each country relative to peers.  Scaled Z-Score applies the following function to the Relative Z-Score: (1+x) if x > 0 and 1 / (1+x) if x < 0.  Scaled weights multiply the Scaled Z-Score against the Effective Weights of each country and normalize such that the total weights sum to 100%.

While the Scaled Weights represent a long-only portfolio, what they really capture is the Market Portfolio plus a dollar-neutral long/short factor tilt.

Market Weight+ Long / Short = Scaled Weights
Australia2.42%-0.11%2.31%
Belgium0.85%-0.48%0.37%
Canada3.76%0.39%4.15%
Denmark0.73%-0.37%0.36%
France4.85%1.80%6.65%
Germany4.36%0.88%5.24%
Hong Kong5.21%1.16%6.37%
Italy1.33%1.59%2.92%
Japan11.15%18.44%29.59%
Netherlands1.45%-0.80%0.65%
Singapore1.09%2.05%3.14%
Spain1.58%2.10%3.68%
Sweden1.21%0.54%1.75%
Switzerland3.15%-0.79%2.36%
United Kingdom6.55%-0.33%6.22%
United States50.30%-26.04%24.26%

To understand the characteristics of the tilt we are taking – i.e. the differences we have created from the market portfolio – we need only look at the long/short portfolio.

Unfortunately, this is where our model loses a bit of interpretability.  Since each country is being compared against its own long-term average, looking at the increase or decrease to the aggregate CAPE score is meaningless.  Indeed, it is possible to imagine a scenario whereby this process actually increases the top-level CAPE score of the portfolio, despite taking value tilts (if value, for example, is found in countries that have higher structural CAPE values).  We can, on the other hand, look at the weighted average change to value score: but knowing that we increased our value score by 0.21 has little interpretation.

One way of looking at this data, however, is by trying to translate value scores into return expectations.  For example, Research Affiliates expects CAPE levels to mean-revert to the average level over a 20-year period.[2]  We can use this model to translate our value scores into an annualized return term due to revaluation.  For example, with a current CAPE of 30.5 and a long-term average of 20.3, we would expect a -2.01% annualized drag from revaluation.

By multiplying these return expectations against our long/short portfolio weights, we find that our long/short tilt is expected to create an annualized revaluation premium of +1.05%.

The Unintended Bet

Unfortunately, re-valuation is not the only bet the long/short portfolio is taking.  The CAPE re-valuation is, after all, in local currency terms.  If we look at our long/short portfolio, we can see a very large weight towards Japan.  Not only will we be subject to the local currency returns of Japanese equities, but we will also be subject to fluctuations in the Yen / US Dollar exchange rate.

Therefore, to achieve the re-valuation premium of our long/short portfolio, we will either need to bear the currency risk or hedge it away.

In either case, we can use uncovered interest rate parity to develop an expected return for currency.  The notion behind uncovered interest rate parity is that investors should be indifferent to sovereign interest rates.  In theory, for example, we should expect the same return from investing in a 1-year U.S. Treasury bond that we expect from converting $1 to 1 euro, investing in the 1-year German Bund, and converting back after a year’s time.

Under uncovered interest rate parity, our expectation is that currency change should offset the differential in interest rates.  If a foreign country has a higher interest rate, we should expect that the U.S. dollar should appreciate against the foreign currency.

As a side note, please be aware that this is a highly, highly simplistic model for currency returns.  The historical efficacy of the carry trade clearly demonstrates the weakness of this model.  More complex models will take into account other factors such as relative purchasing power reversion and productivity differentials.

Using this simple model, we can forecast currency returns for each country we are investing in.

FX Rate1-Year RateExpected FX RateCurrency Return
Australia1.2269-0.47%1.2546-2.21%
Belgium1.2269-0.47%1.2546-2.21%
Canada0.80561.17%0.8105-0.60%
Denmark0.1647-0.55%0.1685-2.29%
France1.2269-0.47%1.2546-2.21%
Germany1.2269-0.47%1.2546-2.21%
Hong Kong0.12781.02%0.1288-0.75%
Italy1.2269-0.47%1.2546-2.21%
Japan0.0090-0.13%0.0092-1.88%
Netherlands1.2269-0.47%1.2546-2.21%
Singapore0.75651.35%0.7597-0.42%
Spain1.2269-0.47%1.2546-2.21%
Sweden0.12410.96%0.1251-0.81%
Switzerland1.0338-0.72%1.0598-2.46%
United Kingdom1.37950.43%1.3981-1.33%
United States1.00001.78%1.00000.00%

Source: Investing.com, XE.com.  Euro area yield curve employed for Eurozone countries on the Euro.

Multiplying our long/short weights against the expected currency returns, we find that we have created an expected annualized currency return of -0.45%.

In other words, we should expect that almost 50% of the value premium we intended to generate will be eroded by a currency bet we never intended to make.

One way of dealing with this problem is through portfolio optimization.  Instead of blindly value tilting, we could seek to maximize our value characteristics subject to currency exposure constraints.  With such constraints, what we would likely find is that more tilts would be made within the Eurozone since they share a currency.  Increasing weight to one Eurozone country while simultaneously reducing weight to another can capture their relative value spread while remaining currency neutral.

Of course, currency is not the only unintended bet we might be making.  Blindly tilting with value can lead to time varying betas, sector bets, growth bets, yield bets, and a variety of other factor exposures that we may not actually intend.  The assumption we make by looking at value alone is that these other factors will be independent from value, and that by diversifying both across assets and over time, we can average out their impact.

Left entirely unchecked, however, these unintended bets can lead to unexpected portfolio volatility, and perhaps even ruin.

Conclusion

In past commentaries, we’ve argued that investors should focus on achieving capital efficiency by employing active managers that provide more pure exposure to active views.  It would seem constraints, as we discussed at the end of the last section, might contradict this notion.

Why not simply blend a completely unconstrained, deep value manager with market beta exposure such that the overall deviations are constrained by position limits?

One answer why this might be less efficient is that not all bets are necessarily compensated.  Active risk for the sake of active risk is not the goal: we want to maximize compensated active risk.  As we showed above, a completely unconstrained value manager may introduce a significant amount of unintended tracking error.  While we are forced to bear this risk, we do not expect the manager’s process to actually create benefit from it.

Thus, a more constrained approach may actually provide more efficient exposure.

That is all not to say that unconstrained approaches do not have efficacy: there is plenty of evidence that the blind application of value at the country index level has historically worked.  Rather, the application of value at a global scale might be further enhanced with the management of unintended bets.

 

 

[1] For example, Predicting Stock Market Returns Using the Shiller CAPE (StarCapital Research, January 2016) and Value and Momentum Everywhere (Asness, Moskowitz, and Pedersen, June 2013)

[2] See Research Affiliate’s Equity Methodology for their Asset Allocation tool.

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