Long/flat trend-following strategies have historically delivered payout profiles similar to those of call options, with positive payouts for larger positive underlying asset returns and slightly negative payouts for near-zero or negative underlying returns.

However, this functional relationship contains a fair amount of uncertainty for any given trend-following model and lookback period.

In portfolio construction, we tend to favor assets that have a combination of high expected returns or diversifying return profiles.

Since broad investor behavior provides a basis for systematic trend-following models to have positive expected returns, taking a multi-model approach to trend-following can be used to reduce the variance around the expected payout profile.

Introduction

Over the past few months, we have written much about model diversification as a tactic for managing specification risk, even with specific case studies. When we consider the three axes of diversification, model diversification pertains to the “how” axis, which focuses on strategies that have the same overarching objective but go about achieving it in different ways.

Long/flat trend-following, especially with equity investments, aims to protect capital on the downside while maintaining participation in positive markets. This leads to a payout profile that looks similar to that of a call option.^{1}

However, while a call option offers a defined payout based on the price of an underlying asset and a specific maturity date, a trend-following strategy does not provide such a guarantee. There is a degree of uncertainty.

The good news is that uncertainty can potentially be diversified given the right combinations of assets or strategies.

In this commentary, we will dive into a number of trend-following strategies to see what has historically led to this benefit and the extent that diversification would reduce the uncertainty around the expected payoff.

Diversification in Trend-Following

The justification for a multi-model approach boils down to a simple diversification argument.

Say you would like to include trend-following in a portfolio as a way to manage risk (e.g. sequence risk for a retiree). There is academic and empirical evidence that trend-following works over a variety of time horizons, generally ranging from 3 to 12 months. And there are many ways to measure trends, such as moving average crossovers, trailing returns, deviations from moving averages, risk adjusted returns, etc.

The basis for deciding ex-ante which variant will be the best over our own investment horizon is tenuous at best. Backtests can show one iteration outperforming over a given time horizon, but most of the differences between strategies are either noise from a statistical point of view or realized over a longer time period than any investor has the lifespan (or mettle) to endure.

However, we expect each one to generate positive returns over a sufficiently long time horizon. Whether this is one year, three years, five years, 10 years, 50 years… we don’t know. What we do know is that out of the multitude the variations of trend-following, we are very likely to pick one that is not the best or even in the top segment of the pack in the short-term.

From a volatility standpoint, when the strategies are fully invested, they will have volatility equal to the underlying asset. Determining exactly when the diversification benefits will come in to play – that is, when some strategies are invested and others are not – is a fool’s errand.

Modern portfolio theory has done a disservice in making correlation seem like an inherent trait of an investment. It is not.

Looking at multiple trend-following strategies that can coincide precisely for stretches of time before behaving completely differently from each other, makes many portfolio construction techniques useless. We do not expect correlation benefits to always be present. These are nonlinear strategies, and fitting them into a linear world does not make sense.

Source: ReSolve Asset Management. Reprinted with permission

From this simple framework, we can break the different performance regimes down as follows:

The Math Behind the Diversification

The expected value of a trend-following strategy can be thought of as a function of the underlying security return:

Where the subscript i is used to indicate that the function is dependent on the specific trend-following strategy.

If we combine multiple trend-following strategies into a portfolio, then the expectation is the average of these functions (assuming an equal weight portfolio per the ReSolve chart above):

What’s left to determine is the functional form of f.

Continuing in the vein of the call option payoff profile, we can use the Black-Scholes equation as the functional form (with the risk-free rate set to 0). This leaves three parameters with which to fit the formula to the data: the volatility (with the time to expiration term lumped in, i.e. sigma * sqrt(T-t)), the strike, and the initial cost of the option.

where d_{1} and d_{2} are defined in the standard fashion and N is the cumulative normal distribution function.

r_{K} is the strike price in the option formula expressed as a percent relative to the current value of the underlying security.

In the following example, we will attempt to provide some meaning to the fitted parameters. However, keep in mind that any mapping is not necessarily one-to-one with the option parameters. The functional form may apply, but the parameters are not ones that were set in stone ex-ante.^{2}

An Example: Trend-Following on the S&P 500

As an example, we will consider a trend-following model on the S&P 500 using monthly time-series momentum with lookback windows ranging from 4 to 16 months. The risk-free rate was used when the trends were negative.

The graph below shows an example of the option price fit to the data using a least-squares regression for the 15-month time series momentum strategy using rolling 3-year returns from 1927 to 2018.

Source: Global Financial Data and Kenneth French Data Library. Calculation by Newfound. Returns are backtested and hypothetical. Returns assume the reinvestment of all distributions. Returns are gross of all fees. None of the strategies shown reflect any portfolio managed by Newfound Research and were constructed solely for demonstration purposes within this commentary. You cannot invest in an index.

The volatility parameter was 9.5%, the strike was 2.3%, and the cost was 1.7%.

What do these parameters mean?

As we said before this can be a bit tricky. Painting in broad strokes:

The volatility parameter describes how “elbowed” payoff profile is. Small values are akin to an option close to expiry where the payoff profile changes abruptly around the strike price. Larger values yield a more gentle change in slope.

The strike represents the point at which the payoff profile changes from participation to protection using trend-following lingo. In the example where the strike is 2.3%, this means that the strategy would be expected to start protecting capital when the S&P 500 return is less than 2.3%. There is some cost associated with this value being high.

The cost is the vertical shift of the payoff profile, but it is not good to think of it as the insurance premium of the trend-following strategy. It is only one piece. To see why this is the case, consider that the fitted volatility may be large and that the option price curve may be significantly above the final payout curve (i.e. if the time-scaled volatility went to zero).

So what is the actual “cost” of the strategy?

With trend-following, since whipsaw is generally the largest potential detractor, we will look at the expected return on the strategy when the S&P 500 is flat, that is, an absence of an average trend. It is possible for the cost to be negative, indicating a positive expected trend-following return when the market was flat.

Looking at the actual fit of the data from a statistical perspective, the largest deviations from the expected value (the residuals from the regression) are seen during large positive returns for the S&P 500, mainly coming out of the Great Depression. This characteristic of individual trend-following models is generally attributable to the delay in getting back into the market after a prolonged, severe drawdown due to the time it takes for a new positive trend to be established.

Source: Global Financial Data and Kenneth French Data Library. Calculation by Newfound. Returns are backtested and hypothetical. Returns assume the reinvestment of all distributions. Returns are gross of all fees. None of the strategies shown reflect any portfolio managed by Newfound Research and were constructed solely for demonstration purposes within this commentary. You cannot invest in an index.

Part of the seemingly large number of outliers is simply due to the fact that these returns exhibit autocorrelation since the periods are rolling, which means that the data points have some overlap. If we filtered the data down into non-overlapping periods, some of these outliers would be removed.

The outliers that remain are a fact of trend-following strategies. While this fact of trend-following cannot be totally removed, some of the outliers may be managed using multiple lookback periods.

The following chart illustrates the expected values for the trend-following strategies over all the lookback periods.

Source: Global Financial Data and Kenneth French Data Library. Calculation by Newfound. Returns are backtested and hypothetical. Returns assume the reinvestment of all distributions. Returns are gross of all fees. None of the strategies shown reflect any portfolio managed by Newfound Research and were constructed solely for demonstration purposes within this commentary. You cannot invest in an index.

The shorter-term lookback windows have the expected value curves that are less horizontal on the left side of the chart (higher volatility parameter).

As we said before the cost of the trend-following strategy can be represented by the strategy’s expected return when the S&P 500 is flat. This can be thought of as the premium for the insurance policy of the trend-following strategies.

Source: Global Financial Data and Kenneth French Data Library. Calculation by Newfound. Returns are backtested and hypothetical. Returns assume the reinvestment of all distributions. Returns are gross of all fees. None of the strategies shown reflect any portfolio managed by Newfound Research and were constructed solely for demonstration purposes within this commentary. You cannot invest in an index.

The blend does not have the lowest cost, but this cost is only one part of the picture. The parameters for the expected value functions do nothing to capture the distribution of the data around – either above or below – these curves.

The diversification benefits are best seen in the distribution of the rolling returns around the expected value functions.

Source: Global Financial Data and Kenneth French Data Library. Calculation by Newfound. Returns are backtested and hypothetical. Returns assume the reinvestment of all distributions. Returns are gross of all fees. None of the strategies shown reflect any portfolio managed by Newfound Research and were constructed solely for demonstration purposes within this commentary. You cannot invest in an index.

Now with a more comprehensive picture of the potential outcomes, a cost difference of even 3% is less than one standard deviation, making the blended strategy much more robust to whipsaw for the potential range of S&P 500 returns.

As a side note, the cost of the short window (4 and 5 month) strategies is relatively high. However, since there are many rolling periods when these models are the best performing of the group, there can still be a benefit to including them. With them in the blend, we still see a reduction in the dispersion around the expected value function.

Expanding the Multitude of Models

To take the example even further down the multi-model path, we can look at the same analysis for varying lookback windows for a price-minus-moving-average model and an exponentially weighted moving average model.

Source: Global Financial Data and Kenneth French Data Library. Calculation by Newfound. Returns are backtested and hypothetical. Returns assume the reinvestment of all distributions. Returns are gross of all fees. None of the strategies shown reflect any portfolio managed by Newfound Research and were constructed solely for demonstration purposes within this commentary. You cannot invest in an index.

Source: Global Financial Data and Kenneth French Data Library. Calculation by Newfound. Returns are backtested and hypothetical. Returns assume the reinvestment of all distributions. Returns are gross of all fees. None of the strategies shown reflect any portfolio managed by Newfound Research and were constructed solely for demonstration purposes within this commentary. You cannot invest in an index.

And finally, we can combine all three trend-following measurement style blends into a final composite blend.

Source: Global Financial Data and Kenneth French Data Library. Calculation by Newfound. Returns are backtested and hypothetical. Returns assume the reinvestment of all distributions. Returns are gross of all fees. None of the strategies shown reflect any portfolio managed by Newfound Research and were constructed solely for demonstration purposes within this commentary. You cannot invest in an index.

As with nearly every study on diversification, the overall blend is not the best by all metrics. In this case, its cost is higher than the EWMA blended model and its dispersion is higher than the TS blended model. But it exhibits the type of middle-of-the-road characteristics that lead to results that are robust to an uncertain future.

Conclusion

Long/flat trend-following strategies have payoff profiles similar to call options, with larger upsides and limited downsides. Unlike call options (and all derivative securities) that pay a deterministic amount based on the underlying securities prices, the payoff of a trend-following strategy is uncertain,

Using historical data, we can calculate the expected payoff profile and the dispersion around it. We find that by blending a variety of trend-following models, both in how they measure trend and the length of the lookback window, we can often reduce the implied cost of the call option and the dispersion of outcomes.

A backtest of an individual trend-following model can look the best over a given time period, but there are many factors that play into whether that performance will be valid going forward. The assets have to behave similarly, potentially both on an absolute and relative basis, and an investor has to hold the investment for a long enough time to weather short-term underperformance.

A multi-model approach can address both of these.

It will reduce the model specification risk that is present ex-ante. It will not pick the best model, but then again, it will not pick the worst.

From an investor perspective, this diversification reduces the spread of outcomes which can lead to an easier product to hold as a long-term investment. Diversification among the models may not always be present (i.e. when style risk dominates and all trend-following strategies do poorly), but when it is, it reduces the chance of taking on uncompensated risks.

Taking on compensated risks is a necessary part of investing, and in the case of trend-following, the style risk is something we desire. Removing as many uncompensated risks as possible leads to more pure forms of this style risk and strategies that are robust to unfavorable specifications.

## Tightening the Uncertain Payout of Trend-Following

By Nathan Faber

On January 28, 2019

In Craftsmanship, Portfolio Construction, Risk Management, Trend, Weekly Commentary

This post is available as a PDF download here.## Summary

IntroductionOver the past few months, we have written much about model diversification as a tactic for managing specification risk, even with specific case studies. When we consider the three axes of diversification, model diversification pertains to the “how” axis, which focuses on strategies that have the same overarching objective but go about achieving it in different ways.

Long/flat trend-following, especially with equity investments, aims to protect capital on the downside while maintaining participation in positive markets. This leads to a payout profile that looks similar to that of a call option.

^{1}However, while a call option offers a defined payout based on the price of an underlying asset and a specific maturity date, a trend-following strategy does not provide such a guarantee. There is a degree of uncertainty.

The good news is that uncertainty can potentially be diversified given the right combinations of assets or strategies.

In this commentary, we will dive into a number of trend-following strategies to see what has historically led to this benefit and the extent that diversification would reduce the uncertainty around the expected payoff.

Diversification in Trend-FollowingThe justification for a multi-model approach boils down to a simple diversification argument.

Say you would like to include trend-following in a portfolio as a way to manage risk (e.g. sequence risk for a retiree). There is academic and empirical evidence that trend-following works over a variety of time horizons, generally ranging from 3 to 12 months. And there are many ways to measure trends, such as moving average crossovers, trailing returns, deviations from moving averages, risk adjusted returns, etc.

The basis for deciding ex-ante which variant will be the best over our own investment horizon is tenuous at best. Backtests can show one iteration outperforming over a given time horizon, but most of the differences between strategies are either noise from a statistical point of view or realized over a longer time period than any investor has the lifespan (or mettle) to endure.

However, we expect each one to generate positive returns over a sufficiently long time horizon. Whether this is one year, three years, five years, 10 years, 50 years… we don’t know. What we do know is that out of the multitude the variations of trend-following, we are very likely to pick one that is

notthe best or even in the top segment of the pack in the short-term.From a volatility standpoint, when the strategies are fully invested, they will have volatility equal to the underlying asset. Determining exactly when the diversification benefits will come in to play – that is, when some strategies are invested and others are not – is a fool’s errand.

Modern portfolio theory has done a disservice in making correlation seem like an inherent trait of an investment. It is not.

Looking at multiple trend-following strategies that can coincide precisely for stretches of time before behaving completely differently from each other, makes many portfolio construction techniques useless. We do not expect correlation benefits to always be present. These are nonlinear strategies, and fitting them into a linear world does not make sense.

If you have pinned up ReSolve Asset Management’s flow chart of portfolio choice above your desk (from Portfolio Optimization: A General Framework for Portfolio Choice), then the decision on this is easy.

Source: ReSolve Asset Management. Reprinted with permissionFrom this simple framework, we can break the different performance regimes down as follows:

The Math Behind the DiversificationThe expected value of a trend-following strategy can be thought of as a function of the underlying security return:

Where the subscript

iis used to indicate that the function is dependent on the specific trend-following strategy.If we combine multiple trend-following strategies into a portfolio, then the expectation is the average of these functions (assuming an equal weight portfolio per the ReSolve chart above):

What’s left to determine is the functional form of

f.Continuing in the vein of the call option payoff profile, we can use the Black-Scholes equation as the functional form (with the risk-free rate set to 0). This leaves three parameters with which to fit the formula to the data: the volatility (with the time to expiration term lumped in, i.e. sigma * sqrt(

T-t)), the strike, and the initial cost of the option.where

dand_{1}dare defined in the standard fashion and_{2}Nis the cumulative normal distribution function.ris the strike price in the option formula expressed as a percent relative to the current value of the underlying security._{K}In the following example, we will attempt to provide some meaning to the fitted parameters. However, keep in mind that any mapping is not necessarily one-to-one with the option parameters. The functional form may apply, but the parameters are not ones that were set in stone ex-ante.

^{2}An Example: Trend-Following on the S&P 500As an example, we will consider a trend-following model on the S&P 500 using monthly time-series momentum with lookback windows ranging from 4 to 16 months. The risk-free rate was used when the trends were negative.

The graph below shows an example of the option price fit to the data using a least-squares regression for the 15-month time series momentum strategy using rolling 3-year returns from 1927 to 2018.

Source: Global Financial Data and Kenneth French Data Library. Calculation by Newfound. Returns are backtested and hypothetical. Returns assume the reinvestment of all distributions. Returns are gross of all fees. None of the strategies shown reflect any portfolio managed by Newfound Research and were constructed solely for demonstration purposes within this commentary. You cannot invest in an index.The volatility parameter was 9.5%, the strike was 2.3%, and the cost was 1.7%.

What do these parameters mean?

As we said before this can be a bit tricky. Painting in broad strokes:

So what is the actual “cost” of the strategy?

With trend-following, since whipsaw is generally the largest potential detractor, we will look at the expected return on the strategy when the S&P 500 is flat, that is, an absence of an average trend. It is possible for the cost to be negative, indicating a positive expected trend-following return when the market was flat.

Looking at the actual fit of the data from a statistical perspective, the largest deviations from the expected value (the residuals from the regression) are seen during large positive returns for the S&P 500, mainly coming out of the Great Depression. This characteristic of individual trend-following models is generally attributable to the delay in getting back into the market after a prolonged, severe drawdown due to the time it takes for a new positive trend to be established.

Source: Global Financial Data and Kenneth French Data Library. Calculation by Newfound. Returns are backtested and hypothetical. Returns assume the reinvestment of all distributions. Returns are gross of all fees. None of the strategies shown reflect any portfolio managed by Newfound Research and were constructed solely for demonstration purposes within this commentary. You cannot invest in an index.Part of the seemingly large number of outliers is simply due to the fact that these returns exhibit autocorrelation since the periods are rolling, which means that the data points have some overlap. If we filtered the data down into non-overlapping periods, some of these outliers would be removed.

The outliers that remain are a fact of trend-following strategies. While this fact of trend-following cannot be totally removed, some of the outliers may be managed using multiple lookback periods.

The following chart illustrates the expected values for the trend-following strategies over all the lookback periods.

Source: Global Financial Data and Kenneth French Data Library. Calculation by Newfound. Returns are backtested and hypothetical. Returns assume the reinvestment of all distributions. Returns are gross of all fees. None of the strategies shown reflect any portfolio managed by Newfound Research and were constructed solely for demonstration purposes within this commentary. You cannot invest in an index.The shorter-term lookback windows have the expected value curves that are less horizontal on the left side of the chart (higher volatility parameter).

As we said before the cost of the trend-following strategy can be represented by the strategy’s expected return when the S&P 500 is flat. This can be thought of as the premium for the insurance policy of the trend-following strategies.

The blend does not have the lowest cost, but this cost is only one part of the picture. The parameters for the expected value functions do nothing to capture the distribution of the data

around– either above or below – these curves.The diversification benefits are best seen in the distribution of the rolling returns around the expected value functions.

Now with a more comprehensive picture of the potential outcomes, a cost difference of even 3% is less than one standard deviation, making the blended strategy much more robust to whipsaw for the potential range of S&P 500 returns.

As a side note, the cost of the short window (4 and 5 month) strategies is relatively high. However, since there are many rolling periods when these models are the best performing of the group, there can still be a benefit to including them. With them in the blend, we still see a reduction in the dispersion around the expected value function.

Expanding the Multitude of ModelsTo take the example even further down the multi-model path, we can look at the same analysis for varying lookback windows for a price-minus-moving-average model and an exponentially weighted moving average model.

And finally, we can combine all three trend-following measurement style blends into a final composite blend.

As with nearly every study on diversification, the overall blend is not the best by all metrics. In this case, its cost is higher than the EWMA blended model and its dispersion is higher than the TS blended model. But it exhibits the type of middle-of-the-road characteristics that lead to results that are robust to an uncertain future.

ConclusionLong/flat trend-following strategies have payoff profiles similar to call options, with larger upsides and limited downsides. Unlike call options (and all derivative securities) that pay a deterministic amount based on the underlying securities prices, the payoff of a trend-following strategy is uncertain,

Using historical data, we can calculate the expected payoff profile and the dispersion around it. We find that by blending a variety of trend-following models, both in how they measure trend and the length of the lookback window, we can often reduce the implied cost of the call option and the dispersion of outcomes.

A backtest of an individual trend-following model can look the best over a given time period, but there are many factors that play into whether that performance will be valid going forward. The assets have to behave similarly, potentially both on an absolute and relative basis, and an investor has to hold the investment for a long enough time to weather short-term underperformance.

A multi-model approach can address both of these.

It will reduce the model specification risk that is present ex-ante. It will not pick the best model, but then again, it will not pick the worst.

From an investor perspective, this diversification reduces the spread of outcomes which can lead to an easier product to hold as a long-term investment. Diversification among the models may not always be present (i.e. when style risk dominates and

alltrend-following strategies do poorly), but when it is, it reduces the chance of taking on uncompensated risks.Taking on compensated risks is a necessary part of investing, and in the case of trend-following, the style risk is something we desire. Removing as many uncompensated risks as possible leads to more pure forms of this style risk and strategies that are robust to unfavorable specifications.