*This post is available as a PDF download here.*

# Summary

- 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.

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 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*are defined in the standard fashion and

_{2}*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.

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 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.

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.

**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.

## Three Applications of Trend Equity

By Corey Hoffstein

On February 25, 2019

In Portfolio Construction, Risk & Style Premia, Trend, Weekly Commentary

This post is available as a PDF download here.## What is Trend Equity?

Trend equity strategies seek to meaningfully participate with equity market growth while side-stepping significant and prolonged drawdowns. These strategies aim to achieve this goal by dynamically adjusting market exposure based upon trend-following signals.

A naïve example of such a strategy would be a portfolio that invests in U.S. equities when the prior 1-year return for U.S. equities is positive and divests entirely into short-term U.S. Treasuries when it is negative.

The TheoryThis category of strategies relies upon the empirical evidence that performance tends to persist in the short-run: positive performance tends to beget further positive performance and negative performance tends to beget further negative performance. The theory behind the evidence is that behavioral biases exhibited by investors lead to the emergence of trends.

In an efficient market, changes in the underlying value of an investment should be met by an immediate, commensurate change in the price of that investment. The empirical evidence of trends suggests that investors may not be entirely efficient at processing new information. Behavioral theory (Figure I) suggests that investors anchor their views on prior beliefs, causing price to underreact to new information. As price continues to drift towards fair value, herding behavior occurs, causing price to overreact and extend beyond fair value. Combined, these effects cause a trend.

Trend equity strategies seek to capture this potential inefficiency by systematically investing in equities when they are exhibiting positively trending characteristics and divesting when they exhibit negative trends. The potential benefit of this approach is that it can try to exploit two sources of return: (1) the expected long-term risk premium associated with equities, and (2) the convex payoff structure typically associated with trend-following strategies.

The Positive Convexity of Trend FollowingAs shown in Figure II, we can see that a hypothetical implementation of this strategy on large-cap U.S. equities has historically exhibited a convex return profile with respect to the underlying U.S. equity index, meaningfully participating in positive return years while reducing exposure to significant loss years.

“Risk Cannot Be Destroyed, Only Transformed.”While the flexibility of trend equity strategies gives them the opportunity to both protect and participate, it also creates the potential for losses due to “whipsaw.” Whipsaws occur when the strategy changes positioning due to what appears to be a change in trend, only to have the market rapidly reverse course. Such a scenario can lead to ”buy high, sell low” and “sell low, buy high” scenarios. These scenarios can be exacerbated by the fact that trend equity strategies may go several years without experiencing whipsaw to only then suddenly experience multiple back-to-back whipsaw events at once.

As Defensive EquityThe most obvious implementation of trend equity strategies is within a defensive equity sleeve. In this approach, an allocation for the strategy is funded by selling strategic equity exposure (see Figure III). Typically combined with other defensive styles (e.g. minimum volatility, quality, et cetera), the goal of a defensive equity sleeve is to provide meaningful upside exposure to equity market growth while reducing downside risk.

This implementation approach has the greatest potential to reduce a policy portfolio’s exposure to downside equity risk and therefore may be most appropriate for investors for whom ”failing fast” is a critical threat. For example, pre-retirees, early retirees, and institutions making consistent withdrawals are highly subject to sequence risk and large drawdowns within their portfolios can create significant impacts on portfolio sustainability.

The drawback of a defensive equity implementation is that vanilla trend equity strategies can, at best, keep up with their underlying index during strong bull markets (see Figure IV). Given the historical evidence that markets tend to be up more frequently than they are down, this can make this approach a frustrating one to stick with for investors. Furthermore, up-capture during bull markets can be volatile on a year-to-year basis, with low up-capture during whipsaw periods and strong up-capture during years with strong trends. Therefore, investors should only allocate in this manner if they plan to do so over a full market cycle.

Implementation within a Defensive Equity sleeve may also be a prudent approach with investors for whom their risk appetite is far below their risk capacity (or even need); i.e. investors who are chronically under-allocated to equity exposure. Implementation of a strategy that has the ability to pro-actively de-risk may allow investors to feel more comfortable with a larger exposure.

Finally, this approach may also be useful for investors seeking to put a significant amount of capital to work at once. While evidence suggests that lump-sum investing (“LSI”) almost always out-performs dollar cost averaging (”DCA”), investors may feel uncomfortable with the significant timing luck from LSI. One potential solution is to utilize trend equity as a middle ground; for example, investors could DCA but hold trend equity rather than cash.

ProsConsAs a Tactical PivotOne creative way of implementing a trend equity strategy is as a tactical pivot within a portfolio. In this implementation, an allocation to trend equity is funded by selling both stocks and bonds, typically in equal amounts (see Figure V). By implementing in this manner, the investor’s portfolio will pivot around the policy benchmark, being more aggressively allocated when trend equity is fully invested, and more defensively allocated when trend equity de-risks.

This approach is often appealing because it offers a highly intuitive allocation sizing policy. The size of the tactical pivot sleeve as well as the mixture of stocks and bonds used to fund the sleeve defines the tactical range around the strategic policy portfolio (see Figure VI).

One benefit of this implementation is that trend equity no longer needs to out-perform an equity benchmark to add value. Rather, so long as the strategy outperforms the mixture of stocks and bonds used to fund the allocation (e.g. a 50/50 mix), the strategy can add value to the holistic portfolio design. For example, assume a trend equity strategy only achieves an 80% upside capture to an equity benchmark during a given year. Implemented as a defensive equity allocation, this up-capture would create a drag on portfolio returns relative to the policy benchmark. If, however, trend equity is implemented as a tactical pivot – funded, for example, from a 50/50 mixture of stocks and bonds – then so long as it outperformed the funding mixture, the portfolio return is improved due to its tilt towards equities.

Implementation as a tactical pivot can also add potential value during environments where stocks and bonds exhibit positive correlations and negative returns (e.g. the 1970s).

One potential drawback of this approach is that the portfolio can be more aggressively allocated than the policy benchmark during periods of sudden and large declines. How great a risk this represents will be dictated both by the size of the tactical pivot as well as the ratio of stocks and bonds in the funding mixture. For example, the potential overweight towards equities is significantly lower using a 70/30 stock/bond funding mix than a 30/70 mixture. A larger allocation to bonds in the funding mixture creates a higher downside hurdle rate for trend equity to add value during a negative equity market environment.

ProsCons## As a Liquid Alternative

Due to its historically convex return profile and potentially high level of tracking error exhibited over short measurement horizons, trend equity may also be a natural fit within a portfolio’s alternative sleeve. Indeed, when analyzed more thoroughly, trend equity shares many common traits with other traditionally alternative strategies.

For example, a vanilla trend equity implementation can be decomposed into two component sources of returns: a strategic portfolio and a long/short trend-following overlay. Trend following can also be directly linked to the dynamic trading strategy required to replicate a long option position.

There are even strong correlations to traditional alternative categories. For example, a significant driver of returns in equity hedge and long/short equity categories is dynamic market beta exposure, particularly during significant market declines (see Figure VII). Trend equity strategies that are implemented with factor-based equity exposures or with the flexibility to make sector and geographic tilts may even more closely approximate these categories.

One potential benefit of this approach is that trend equity can be implemented in a highly liquid, highly transparent, and cost-effective manner when compared against many traditional alternatives. Furthermore, by implementing trend equity within an alternatives sleeve, investors may give it a wider berth in their mental accounting of tracking error, allowing for a more sustainable allocation versus implementation as a defensive equity solution.

A drawback of this implementation, however, is that trend equity will increase a portfolio’s exposure to equity beta. Therefore, more traditional alternatives may offer better correlation- and pay-off-based diversification, especially during sudden and large negative equity shocks. Furthermore, trend equity may lead to overlapping exposures with existing alternative exposures such as equity long/short or managed futures. Investors must therefore carefully consider how trend equity may fit into an already existing alternative sleeve.

ProsCons