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 State of Risk Management
By Justin Sibears
On August 20, 2018
In Portfolio Construction, Risk Management, Weekly Commentary
This post is available as PDF download here.
Summary
I was perusing Twitter the other day and came across this tweet from Jim O’Shaughnessy, legendary investor and author of What Works on Wall Street.
As always. Jim’s wisdom is invaluable. But what does this idea mean for Newfound as a firm? Our first focus is on managing risk. As a result, one of the questions that we MUST know the answer to is how to get more investors comfortable with sticking to a risk management plan through a full market cycle.
Unfortunately, performance chasing seems to us to be just as prevalent in risk management as it is in investing as a whole. The benefits of giving up some upside participation in exchange for downside protection seemed like a no brainer in March of 2009. After 8+ years of strong equity market returns (although it hasn’t always been as smooth of a ride as the market commentators may make you think), the juice may not quite seem worth the squeeze.
While we certainly don’t profess to know the answer to our burning question from above, we do think the first step towards finding one is a thorough understanding on the risk management landscape. In that vein, this week we will update our State of Risk Management presentation from early 2016.
We examine eight strategies that roughly fit into four categories:
The Historical Record
We find that over the period studied (December 1997 to July 2018) six of the eight strategies outperform the S&P 500 on a risk-adjusted basis both when we define risk as volatility and when we define risk as maximum drawdown. The two exceptions are the equity collar strategy and the protective put strategy. Both of these strategies are net long options and therefore are forced to pay the volatility risk premium. This return drag more than offsets the reduction of losses on the downside.
Data Source: Bloomberg, CSI. Calculations by Newfound Research. Past performance does not guarantee future results. Volatility is a statistical measure of the amount of variation around the average returns for a security or strategy. All returns are hypothetical index returns. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses, sales charges, or trading expenses. Index returns include the reinvestment of dividends. No index is meant to measure any strategy that is or ever has been managed by Newfound Research. The Tactical Equity strategy was constructed by Newfound in August 2018 for purposes of this analysis and is therefore entirely backtested and not an investment strategy that is currently managed and offered by Newfound.
Data Source: Bloomberg, CSI. Calculations by Newfound Research. Past performance does not guarantee future results. Drawdown is a statistical measure of the losses experienced by a security or strategy relative to its historical maximum. The maximum drawdown is the largest drawdown over the security or strategy’s history. All returns are hypothetical index returns. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses, sales charges, or trading expenses. Index returns include the reinvestment of dividends. No index is meant to measure any strategy that is or ever has been managed by Newfound Research. The Tactical Equity strategy was constructed by Newfound in August 2018 for purposes of this analysis and is therefore entirely backtested and not an investment strategy that is currently managed and offered by Newfound.
Not Always a Smooth Ride
While it would be nice if this outperformance accrued steadily over time, reality is quite a bit messier. All eight strategies exhibit significant variation in their rolling one-year returns vs. the S&P 500. Interestingly, the two strategies with the widest ranges of historical one-year performance vs. the S&P 500 are also the two strategies that have delivered the most downside protection (as measured by maximum drawdown). Yet another reminder that there is no free lunch in investing. The more aggressively you wish to reduce downside capture, the more short-term tracking error you must endure.
Relative 1-Year Performance vs. S&P 500 (December 1997 to July 2018)
Data Source: Bloomberg, CSI. Calculations by Newfound Research. Past performance does not guarantee future results. All returns are hypothetical index returns. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses, sales charges, or trading expenses. Index returns include the reinvestment of dividends. No index is meant to measure any strategy that is or ever has been managed by Newfound Research. The Tactical Equity strategy was constructed by Newfound in August 2018 for purposes of this analysis and is therefore entirely backtested and not an investment strategy that is currently managed and offered by Newfound.
Thinking of Risk Management as (Uncertain) Portfolio Insurance
When we examine this performance dispersion across different market environments, we find a totally intuitive result: risk management strategies generally underperform the S&P 500 when stocks advance and outperform the S&P 500 when stocks decline. The hit rate for the risk management strategies relative to the S&P 500 is 81.2% in the four years that the S&P 500 was down (2000, 2001, 2002, and 2008) and 19.8% in the seventeen years that the S&P was up.
In this way, risk management strategies are akin to insurance. A premium, in the form of upside capture ratios less than 100%, is paid in exchange for a (hopeful) reduction in downside capture.
With this perspective, it’s totally unsurprising that these strategies have underperformed since the market bottomed during the global market crisis. Seven of the eight strategies (with the long-only defensive equity strategy being the lone exception) underperformed the S&P 500 on an absolute return basis and six of the eight strategies (with defensive equity and the 60/40 stock/bond blend) underperformed on a risk-adjusted basis.
Annual Out/Underperformance Relative to S&P 500 (December 1997 to July 2018)
Data Source: Bloomberg, CSI. Calculations by Newfound Research. Past performance does not guarantee future results. All returns are hypothetical index returns. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses, sales charges, or trading expenses. Index returns include the reinvestment of dividends. No index is meant to measure any strategy that is or ever has been managed by Newfound Research. The Tactical Equity strategy was constructed by Newfound in August 2018 for purposes of this analysis and is therefore entirely backtested and not an investment strategy that is currently managed and offered by Newfound.
Diversifying Your Diversifiers
The good news is that there is significant year-to-year variation in the performance across strategies, as evidenced by the periodic table of returns above, suggesting there are diversification benefits to be harvested by allocating to multiple risk management strategies. The average annual performance differential between the best performing strategy and the worst performing strategy is 20.0%. This spread was less than 10% in only 3 of the 21 years studied.
We see the power of diversifying your diversifiers when we test simple equal-weight blends of the risk management strategies. Both blends have higher Sharpe Ratios than 7 of the 8 individual strategies and higher excess return to drawdown ratios than 6 of the eight individual strategies.
This is a very powerful result, indicating that naïve diversification is nearly as good as being able to pick the best individual strategies with perfect foresight.
Data Source: Bloomberg, CSI. Calculations by Newfound Research. Past performance does not guarantee future results. All returns are hypothetical index returns. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses, sales charges, or trading expenses. Index returns include the reinvestment of dividends. No index is meant to measure any strategy that is or ever has been managed by Newfound Research. The Tactical Equity strategy was constructed by Newfound in August 2018 for purposes of this analysis and is therefore entirely backtested and not an investment strategy that is currently managed and offered by Newfound.
Why Bother with Risk Management in the First Place?
As we’ve written about previously, we believe that for most investors investing “failure” means not meeting one’s financial objectives. In the portfolio management context, failure comes in two flavors. “Slow” failure results from taking too little risk, while “fast” failure results from taking too much risk.
In this book, Red Blooded Risk, Aaron Brown summed up this idea nicely: “Taking less risk than is optimal is not safer; it just locks in a worse outcome. Taking more risk than is optimal also results in a worst outcome, and often leads to complete disaster.”
Risk management is not synonymous with risk reduction. It is about taking the right amount of risk, not too much or too little.
Having a pre-defined risk management plan in place before a crisis can help investors avoid panicked decisions that can turn a bad, but survivable event into catastrophe (e.g. the retiree that sells all of his equity exposure in early 2009 and then stays out of the market for the next five years).
It’s also important to remember that individuals are not institutions. They have a finite investment horizon. Those that are at or near retirement are exposed to sequence risk, the risk of experiencing a bad investment outcome at the wrong time.
We can explore sequence risk using Monte Carlo simulation. We start by assessing the S&P 500 with no risk management overlay and assume a 30-year retirement horizon. The simulation process works as follows:
We plot the distribution of PWRs for the S&P 500 below. While the average PWR is a respectable 5.7%, the range of outcomes is very wide (0.6% to 14.7%). The 95 percent confidence interval around the mean is 2.0% to 10.3%. This is sequence risk. Unfortunately, investors do not have the luxury of experiencing the average, they only see one draw. Get lucky and you may get to fund a better lifestyle than you could have imagined with little to no financial stress. Get unlucky and you may have trouble paying the bills and will be sweating every market move.
Calculations by Newfound Research. Past performance does not guarantee future results. All returns are hypothetical index returns. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses, sales charges, or trading expenses. Index returns include the reinvestment of dividends.
Next, we repeat the simulation, replacing the pure S&P 500 exposure with the equal-weight blend of risk management strategies excluding the equity collar and the protective put. We see quite a different result. The average PWR is similar (6.2% to 5.7%), but the range of outcomes is much smaller (95% confidence interval from 4.4% to 8.1%). At its very core, this is what implementing a risk management plan is all about. Reducing the role of investment luck in financial planning. We give up some of the best outcomes (in the right tail of the S&P 500 distribution) in exchange for reducing the probability of the very worst outcomes (in the left tail).
Calculations by Newfound Research. Past performance does not guarantee future results. All returns are hypothetical index returns. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses, sales charges, or trading expenses. Index returns include the reinvestment of dividends.
Conclusion
There is no holy grail when it comes to risk management. While a number of approaches have historically delivered strong results, each comes with its own pros and cons.
In an uncertain world where we cannot predict exactly what the next crisis will look like, diversifying your diversifiers by combining a number of complementary risk-managed strategies may be a prudent course of action. We believe that this type of balanced approach has the potential to deliver compelling results over a full market cycle while managing the idiosyncratic risk of any one manager or strategy.
Diversification can also help to increase the odds of an investor sticking with their risk management plan as the short-term performance lows won’t be quite as low as they would be with a single strategy (conversely, the highs won’t be as high either).
That being said, having the discipline to stick with a risk management plan also requires being realistic. While it would be great to build a strategy with 100% upside and 0% downside, such an outcome is unrealistic. Risk-managed strategies tend to behave a lot like uncertain insurance for the portfolio. A premium, in the form of upside capture ratios less than 100%, is paid in exchange for a (hopeful) reduction in downside capture. This upside underperformance is a feature, not a bug. Trying too hard to correct it may lead to overfit strategies fail to deliver adequate protection on the downside.