Category: Trend Page 4 of 7

G̷̖̱̓́̀litch

On February 19, 2019

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

This post is available as a PDF download here.

Summary

Trend following’s simple, systematic, and transparent approach does not make it any less frustrating to allocate to during periods of rapid market reversals.
With most trend equity strategies exhibiting whipsaws in 2010, 2011, 2015-2016, and early 2018, it is tempting to ask, “is this something we can fix?”
We argue that there are three historically-salient features that make trend following attractive: (1) positive skew, (2) convexity, and (3) a positive premium.
We demonstrate that the convexity exhibited by trend equity strategies is both a function of the strategy itself (i.e. a fast- or slow-paced trend model) as well as the horizon we measure returns over.
We suggest that it may be more consistent to think of convexity as an element than can provide crisis beta, where the nature of the crisis is defined by the speed of the trend following system.
The failure of a long-term trend strategy to de-allocate in Q4 2018 or meaningfully re-allocate in Q1 2019 is not a glitch; it is encoded in the DNA of the strategy itself.

There’s an old saying in Tennessee – I know it’s in Texas, probably in Tennessee – that says, fool me once, shame on – shame on you. Fool me – you can’t get fooled again! — George W. Bush

It feels like we’ve seen this play before. It happened in 2010. Then again in 2011. More recently in 2015-2016. And who can forget early 2018? To quote Yogi Berra, “It’s déjà vu all over again.” We’re starting to think it is a glitch in the matrix.

Markets begin to deteriorate, losses begin to more rapidly accelerate, and then suddenly everything turns on a dime and market’s go on to recover almost all their losses within a few short weeks.

Trend following – like the trend equity mandates we manage here at Newfound – requires trends. If the market completely reverses course and regains almost all of its prior quarter’s losses within a few short weeks, it’s hard to argue that trend following should be successful. Indeed, it is the prototypical environment that we explicitly warn trend following will do quite poorly in.

That does not mean, however, that changing our approach in these environments would be a warranted course of action. We embrace a systematic approach to explicitly avoid contamination via emotion, particularly during these scenarios. Plus, as we like to say, “risk cannot be destroyed, only transformed.” Trying to eliminate the risk of whipsaw not only risks style pollution, but it likely introduces risk in unforeseen scenarios.

So, we have to scratch our heads a bit when clients ask us for an explanation as to our current positioning. After all, trend following is fairly transparent. You can probably pull up a chart, stand a few feet back, squint, and guess with a reasonable degree of accuracy as to how most trend models would be positioned.

When 12-month, 6-month, and 3-month returns for the S&P 500 were all negative at the end of December, it is a safe guess that we’re probably fairly defensively positioned in our domestic trend equity mandates. Despite January’s record-breaking returns, not a whole lot changed. 12-, 6-, and 3-month returns were negative, negative, and just slightly positive, respectively, entering February.

To be anything but defensively positioned would be a complete abandonment of trend following.

It is worth acknowledging that this may all just be Act I. Back when this show was screening in 2011 and 2015-2016, markets posted violent reversals – with the percent of stocks above their 50-day moving average climbing from less than 5% to more than 90% – only to roll over again and retest the lows.

Or this will be February 2018 part deux. We won’t know until well after the fact. And that can be frustrating depending upon your perspective of markets.

If you take a deterministic view, incorrect positioning implies an error in judgement. You should have known to abandon trend following and buy the low on December 24^th. If you take a probabilistic view, then it is possible to be correctly positioned for the higher probability event and still be wrong. The odds were tilted strongly towards continued negative market pressure and a defensive stance was warranted at the time.

We would argue that there is a third model as well: sustainability (or, more morbidly, survivability). It does not matter if you have a 99% chance of success while playing Russian Roulette: play long enough and you’re eventually going to lose. Permanently. Sustainability argues that the low-probability bet may be the one worth taking if the payoff is sufficient enough or it protects us from ruin.

Thus, for investors for whom failing fast is a priority risk, a partially defensive allocation in January and February may be well warranted, even if the intrinsic probabilities have reversed course (which, based on trends, they largely had not).

But sustainability also needs to be a discussion about being able to stick with a strategy. It does not matter if the strategy survives over the long run if the investor does not participate.

That is why we believe transparency and continued education are so critical. If we do not know what we are invested in, we cannot set correct expectations. Without correct expectations, everything feels unexpected. And when everything feels unexpected, we have no way to determine if a strategy is behaving correctly or not.

Which brings us back to trend equity strategies in Q4 2018 and January 2019. Did trend equity behave as expected?

Trend following has empirically exhibited three attractive characteristics:

Positive Skew: The return distribution is asymmetric, with a larger right tail than left tail (i.e. greater frequency of larger, positive returns than large, negative returns).
Convex Payoff Profile: As a function of the underlying asset the trend following strategy is applied on, upside potential tends to be greater than downside risk.
Positive Premium: The strategy has a positive expected excess return.

While the first two features can be achieved by other means (e.g. option strategies), the third feature is downright anomalous, as we discussed in our recent commentary Trend: Convexity & Premium. Positive skew and convexity create and insurance-like payoff profile and therefore together tend to imply a negative premium.

The first two characteristics make trend following a potentially interesting portfolio diversifier. The last element, if it persists, makes it very interesting.

Yet while we may talk about these features as historically intrinsic properties of trend following, the nature of the trend-following strategy will significantly impact the horizon over which these features are observed. What is most important to acknowledge here is that skew and convexity are more akin to beta than they are alpha; they are byproducts of the trading strategy itself. While it can be hard to say things about alpha, we often can say quite a bit more about beta.

For example, a fast trend following system (typically characterized by a short lookback horizon) would be expected to rapidly adapt to changing market dynamics. This allows the system to quickly position itself for emerging trends, but also potentially makes the strategy more susceptible to losses from short-term reversals.

A slow trend following system (characterized by a long lookback period), on the other hand, would be less likely to change positioning due to short-term market noise, but is also therefore likely to adapt to changing trend dynamics more slowly.

Thus, we might suspect that a fast-paced trend system might be able to exhibit convexity over a shorter measurement period, whereas a slow-paced system will not be able to adapt rapidly. On the other hand, a fast trend following system may have less average exposure to the underlying asset over time and may compound trading losses due to whipsaw more frequently.

To get a better sense of these tradeoffs, we will construct prototype trend equity strategies which will invest either in broad U.S. equities or risk-free bonds. The strategies will be re-evaluated on a daily basis and are assumed to be traded at the close of the day following a signal change. Trend signals will be based upon prior total returns; e.g. a 252-day system will have a positive (negative) signal if prior 252-day total returns in U.S. equity markets are positive (negative).

Below we plot the monthly returns of a -short-term trend equity system (21 day)- and a -long-term trend equity system (252 day)- versus U.S. equity returns.

Source: Kenneth French Data Library. Calculations by Newfound Research. Past performance is not a guarantee of future results. All returns are hypothetical and backtested. Returns are gross of all fees. For the avoidance of doubt, neither the Short-Term nor Long-Term Trend Equity strategy reflect any investment strategy offered or managed by Newfound Research and was constructed exclusively for the purposes of this commentary. It is not possible to invest in an index.

We can see that the fast-paced system exhibits convexity over the monthly measurement horizon, while the slower system exhibits a more linear return profile.

As mentioned above, however, the more rapid adaptation in the short-term system might cause more frequent realization of whipsaw due to price reversals and therefore an erosion in long-term convexity. Furthermore, more frequent changes might also reduce long-term participation.

We now plot annual returns versus U.S. equities below.

We can see that while the convexity of the short-term system remains intact, the long-term system exhibits greater upside participation.

To get a better sense of these trade-offs, we will follow Sepp (2018)¹ and use the following model to deconstruct our prototype long/flat trend equity strategies:

By comparing daily, weekly, monthly, quarterly, and annual returns, we can extract the linear and convexity exposure fast- and slow-paced systems have historically exhibited over a given horizon.

Below we plot the regression coefficients (“betas”) for a fast-paced system.

Source: Kenneth French Data Library. Calculations by Newfound Research. Past performance is not a guarantee of future results. All returns are hypothetical and backtested. Returns are gross of all fees. For the avoidance of doubt, the Short-Term Trend Equity strategy does not reflect any investment strategy offered or managed by Newfound Research and was constructed exclusively for the purposes of this commentary. It is not possible to invest in an index.

We can see that the linear exposure remains fairly constant (and in line with decompositions we’ve performed in the past which demonstrate that long/flat trend equity can be thought of as a 50/50 stock/cash strategic portfolio plus a long/short overlay²). The convexity profile, however, is most significant when measured over weekly or monthly horizons.

Long-term trend following systems, on the other hand, exhibit negative or insignificant convexity profiles over these horizons. Even over a quarterly horizon we see insignificant convexity. It is not until we evaluate returns on an annual horizon that a meaningful convexity profile is established.

Source: Kenneth French Data Library. Calculations by Newfound Research. Past performance is not a guarantee of future results. All returns are hypothetical and backtested. Returns are gross of all fees. For the avoidance of doubt, the Long-Term Trend Equity strategy does not reflect any investment strategy offered or managed by Newfound Research and was constructed exclusively for the purposes of this commentary. It is not possible to invest in an index.

These results have very important implications for investors in trend following strategies.

We can see that long-term trend following, for example, is unlikely to be successful as a tail risk hedge for short-term events. Short-term trend following may have a higher probability of success in such a scenario, but only so long as the crisis occurs over a weekly or monthly horizon.³

Short-term trend following, however, appears to exhibit less convexity with annual returns and has lower linear exposure. This implies less upside capture to the underlying asset.

Neither approach is likely to be particularly successful at hedging against daily crises (e.g. a 1987-type event), as the period is meaningfully shorter than the adaptation speed of either of the strategies.

These results are neither feature nor glitch. They are simply the characteristics we select when we choose either a fast or slow trend-following strategy. While trend-following strategies are often pitched as crisis alpha, we believe that skew and convexity components are more akin to crisis beta. And this is a good thing. While alpha is often ephemeral and unpredictable, we can more consistently plan around beta.

Thus, when we look back on Q4 2018 and January 2019, we need to acknowledge that we are evaluating results over a monthly / quarterly horizon. This is fine if we are evaluating the results of fast-paced trend-following strategies, but we certainly should not expect any convexity benefits from slower trend models. Quite simply, it all happened too fast.

Conclusion

When markets rapidly reverse course, trend following can be a frustrating style to allocate to. With trend equity styles exhibiting whipsaws in 2010, 2011, 2015-2016, and early 2018, the most recent bout of volatility may have investors rolling their eyes and thinking, “again?”

“Where’s the crisis alpha?” investors cry. “Where’s the crisis?” managers respond back.

Yet as we demonstrated in our last commentary, two of the three salient features of trend following – namely positive skew and positive convexity – may be byproducts of the trading strategy and not an anomaly. Rather, the historically positive premium that trend following has generated has been the anomaly.

While the potential to harvest alpha is all well and good, we should probably think more in the context of crisis beta than crisis alpha when setting expectations. And that beta will be largely defined by the speed of the trend following strategy.

But it will also be defined by the period we are measuring the crisis over.

For example, we found that fast-paced trend equity strategies exhibit positive convexity when measured over weekly and monthly time horizons, but that the convexity decays when measured over annual horizons.

Strategies that employ longer-term trend models, on the other hand, fail to exhibit positive convexity over shorter time horizons, but exhibit meaningful convexity over longer-horizons. The failure of long-term trend strategies to meaningfully de-allocate in Q4 2018 or rapidly re-allocate in Q1 2019 is not a glitch: it is encoded into the DNA of the strategy.

Put more simply: if we expect long-term trend models to protect against short-term sell-offs, we should prepare to be disappointed. On the other hand, the rapid adaptation of short-term models comes at a cost, which can materialize as lower up-capture over longer horizons.

Thus, when it comes to these types of models, we have to ask ourselves about the risks we are trying to manage and the trade-offs we are willing to make. After all, “risk cannot be destroyed, only transformed.”

Trend: Convexity & Premium

By Corey Hoffstein

On February 11, 2019

In Trend, Weekly Commentary

Available as a PDF download here.

Summary

Trend following is unique among style premia in that it has historically exhibited a convex payoff profile with positive skew.
While the historical premium is anomalous, the convexity makes sense when we use options to replicate trend following strategies.
We explore reasons why frequent rebalancing in trend following strategies is necessary and decompose the return contributions from different portions of the option replication model.
Most of the historical premium associated with trend following comes from the trading impact that is tied to the asset’s Sharpe ratio rather than the convexity.
By separating the impacts of convexity and trading impact, we can gain a deeper understanding of the types of risk exchanges that come with investing using trend following strategies.

Introduction

Unlike many of the other style premia, trend following has historically exhibited a convex payoff profile with positive skew. In less mathematical terms, that means it tends to harvest many small losses (due to reversals) and just a few large gains (when trends take off).

This is unique, as most risk and style premia exhibit the opposite: concave payoffs with negative skew.

To simplify, we can think of concave, negative-skew trades as akin to selling insurance, while convex, positive-skew trades are akin to buying it. With respect to traditional financial literature, the mental model of selling insurance makes quite a bit of sense, as we can think of expected excess returns as being the reward earned for being willing to bear the risk others wish to transfer away.

Trend following, then, is a bit of an anomaly. Not because it exhibits a convex, positive skew profile, but because it does so and has historically exhibited a positive premium. That is not supposed to happen – you don’t expect to profit when you buy insurance – and it leaves many scratching their head asking, “could this be an actual market anomaly?”

Unfortunately, a discussion of why trend following works often conflates the convexity of the strategy with the oddness of the historically positive premium. The latter is, for sure, anomalous. But what we hope to show in this commentary is that the former is just a byproduct of the trading strategy itself and does not require any investor misbehavior.

How will we do this? Whenever we talk about buying or selling insurance, a very natural language to use is that of put and call options. Thus, our goal in this commentary is to approach trend following through the lens of options and demonstrate that simple trend-following strategies can be thought of as naively replicating the pay-off of a straddle.

In doing so, our goal is to differentiate between two key elements of trend following: the convexity of returns it exhibits and the historically positive premium it has generated.

Please note, for all the option geeks out there, that this will be a highly simplified interpretation with lots of hand-waving. For example, we will generally assume that interest rates are zero, dividends do not exist, that price is continuous (i.e. no jumps) and there are no trading frictions. None of this is true, of course, but we do not think it meaningfully takes away from the intuition established.

Pricing is Replication

A foundational principle in financial engineering is the Law of One Price, which states that any two securities with identical future payouts, no matter how the future turns out, must have an identical price. Otherwise, we could construct an arbitrage.

Thus, to price an option, we only need to replicate its payoff. This is, of course, easier said than done, as options have non-linear payoffs. A call option, for example, pays nothing when price is below the strike at maturity and pays the difference between price and the strike otherwise.

Source: Wikipedia

Given access to a risk-free bond and the underlying stock, we cannot easily replicate this payoff with some sort of static portfolio. We can, however, attempt to replicate it using a dynamic trading strategy that adjusts our mixture of the stock and bond over time.

Consider the following example: a stock is priced at $80 and will be worth either $100 or $60 in one year. We have a call option with a strike of $90. This means if the stock ends up at $100, the payoff will be $10 while if the stock ends up at $60, the payoff will be zero.

How can we replicate this?

We need to solve the simultaneous equations:

$100Δ – B(1+r) = 10
$60Δ – B(1+r) = 0

Here, Δ is the number of shares of stock to buy, B is how much to borrow, and r is the risk-free rate. Given r, we can solve the equations for the replicating portfolio. If we assume r=0%, we find that Δ=0.25 and therefore B=$15.

Thus, to replicate the call option, we need to borrow $15 and buy 0.25 shares at $80, for a total cost of $5. Since this portfolio replicates the option payout exactly, this must also be the price of the option!

Of course, these are highly simplified assumptions. But if we collapse the time period down from 1-year to an infinitesimally small unit of time, we can repeat this exercise over and over such that we have a dynamic trading strategy that will replicate the option’s final payoff, and therefore the option’s value over time.

(For a more thorough – but still highly accessible – introduction to this concept, we recommend Emanuel Derman’s The Boy’s Guide to Pricing & Hedging.)

Trend Following is “Long Gamma”

Let us now connect trend following to options.

Consider the following case where the underlying stock price follows a binomial tree and we want to replicate the payoff of a call option with a strike of $80. Again, to simplify things, we will just assume that our risk-free rate is 0%.

Let’s start when price is at $90. At this point, to replicate the final payoff, we have the following two equations:

$100Δ – B = 20
$80Δ – B = 0

Again, solving simultaneously, we find that we need to buy 1 share (at $90) and therefore B=$80. The option, therefore, is equal to $10.

Now let’s consider the bottom case when the stock price is $70,

$80Δ – B = 0
$60Δ – B = 0

In this case, we find that the number of shares is equal to zero and therefore B=$0.

Finally, using these prices for the option in those two states, we can step back to the starting case, where we now know:

$90Δ – B = 10
$70Δ – B = 0

Here we find that the number of shares is equal to 0.5 and B is equal to $35, making the option therefore worth $5.

This highly simplified model tells us that:

The number of shares held in the replicating portfolio informs how sensitive the option price is to movement in the underlying stock. Note that at the initial step the option value was $5 and the number of shares held was 0.5. As price changed by +/- $10, the value of the option changed by +/- $5. The more shares held, the more sensitive the option price is to the stock price change. In option’s parlance, this is known as the option’s “delta.”
The number of shares required by the replicating portfolio also changed based upon changes in the underlying stock price. In option’s parlance, this is known as an option’s “gamma.” This measures how the options delta changes with changes in the stock price.

Note that as the stock price increased, the number of shares required to replicate the option increased. This implies that a call option has positive gamma.

If we repeated this whole exercise but used a put option instead, we would similarly find that a put option has positive gamma: the more price depreciates below the strike, the more shares we need to short to replicate.

Which means that these options can be roughly replicated using a very naïve trend-following strategy.

If we buy a put and a call at the same strike and same maturity, we have constructed a trade known as a “straddle”. We plot an example straddle payoff profile below.

Source: theoptionsguide.com

Since a long straddle is simply the combination of a put and a call, we can replicate its payoff by just replicating both positions independently and summing up our total exposure.

It should come as no surprise that the replication of this straddle is, in essence, a trend following strategy. As the underlying stock price increases, we buy shares, and as underlying stock price decreases, we sell shares.

With this naïve model, we can already see a few interesting trade-offs:

The convexity of trend following may have nothing to do with any sort of market “anomaly,” but rather is a function of the trading strategy employed.
Purchasers of a straddle will realize the payoff minus the up-front cost of the options, which will be a function of implied volatility. The replicating trend-following strategy will realize the same payoff minus the trading costs, which will be a path-dependent function of returns (and, therefore, realized volatility).

For a more nuanced dive into deriving this relationship, we recommend the paper “Tail protection for long investors: Trend convexity at work” by Dao, Nguyen, Deremble, Lempérière, Bouchaud, and Potters (2016).

Straddle-ish

While we have demonstrated that a straddle can replicated with a (continuous) trend-following strategy, it is not the traditional trend-following archetype by any means. Furthermore, trend-following strategies are continuous in nature, while straddles have a defined expiration date.

So, let us consider a more realistic (albeit, still a toy) trend-following implementation. We will go long the S&P 500 when its prior 12-month return is positive and short when it is negative. We will rebalance the strategy at the end of each month. For simplicity, we will assume any available capital is invested in risk-free bonds that return 0%.

How might we translate this strategy into a semi-equivalent straddle replication? One interpretation may be that at the end of each month, we use the price from 12 months ago to set the strike of our straddle. To compute the delta, we will lean on the Black-Scholes equation, where we will assume that the time until expiration is one month, we’ll assume no dividend payment, and we’ll use prior short-term realized volatility as our input for implied volatility.¹ Since the delta will vary between -1 and 1, we will use it as our allocation to the S&P 500, investing remaining capital in risk-free bonds that return 0%.

It is important to note that in the prior section, the simplified trend following strategy replicated the straddle payoff because we were able to delta hedge over infinitesimally small time horizons at zero cost. Here, we are rebalancing monthly, applying a much more static mode of replication.

Below we plot the growth of $1 in each strategy. The correlation in monthly log-returns between the two strategies is 95.8%.

Source: CSI Data. Calculations by Newfound Research. Returns are backtested and hypothetical. Returns assume the reinvestment of all distributions. Returns are gross of all fees except for underlying ETF expense ratios. 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.

We can see that a trend following strategy looks incredibly similar to a strategy that replicates a straddle. Why? Let’s look at the trend response function versus delta.

Note that the x-axis is measured in standard deviations from the strike. This is important. A 10% move in one asset class may be highly significant while a 10% move in another may not. Furthermore, a 10% move in one market environment may be significant while a 10% move in another may not. The delta function captures this as implied volatility is an input to the measure, while the trend signal does not.

We can see that the binary trend signal and the straddle delta are nearly identical when price is greater than 0.6 standard deviations from the strike in either direction. In this model, this can be interpreted as a strong trend where the trend strategy allocation and delta straddle allocation will coincide.

Within that range, however, we can see that the trend signal over-estimates the delta. Therefore, in cases where price continues away from the strike, the binary signal will out-perform and in cases where price reverts back towards the strike, the binary signal will under-perform.

We can use this insight to explore a few questions. For example, why do trend following strategies have to be traded frequently? Let’s consider the case where we only rebalance quarterly. Note what happens to the delta function of the straddle:

We can see that the trend signal and the straddle delta only meet when price is 1.2 standard deviations from strike. The delta is taking into account how long there is until expiration and therefore adjusting itself downward in magnitude in acknowledgement that price might revert back towards the strike. The binary signal does not.

So, is the answer just to rebalance as frequently as possible? After all, as the ratio of the rebalance period to the lookback period goes to zero, the shape of the delta function approaches the binary trend signal. Conversely, as the ratio goes to infinity (i.e. the holding period length far exceeds the lookback), the shape of the delta function approaches y=0 at all points.

The answer in the real world, where prices are not continuous, we are not trading infinitesimally small horizons, and there are trading costs, is “no.”

But if we just statically replicate the remaining time until option expiration, doesn’t that remove the entire long gamma aspect? Have we not lost the convexity created by the trading strategy?

In an attempt to answer this question, we can ask a slightly different one: “how different is the change in delta from rolling into the new straddle versus replicating the original straddle?” We derive the math in the appendix, but under some general assumptions we can say that the deltas will converge when the ratio of the rebalance time-step and the time until option expiration goes to zero.

dt/(T-t) → 0

As this ratio becomes smaller, therefore, the delta change from rolling into the new straddles will approximate the delta change of replicating the prior straddle, and thus conserve the replicating strategy’s natural convexity.

There are two important drivers for this limit here: we want dt to be as small as possible and T-t to be as large as possible.

Unfortunately, this implies that we are delta hedging over an infinitesimally small time period at the beginning of the life of the straddle, a time at which the delta is approximately zero because it is struck at-the-money! On the other hand, if we go towards expiration, T-t goes to zero!

The dilemma at hand, then, is for a simple binary long/short trend strategy to reflect the delta of a straddle, it needs to be close to expiration when the delta function looks more like a step change. However, for the changes in delta from rolling the straddle position to reflect changes in delta from replication, the roll must occur near inception!

Using this information, we can attempt to get a rough approximation of how much a binary trend strategy’s return comes from: (1) the replication of a straddle, (2) excess delta exposure from rolling straddle exposure, and (3) excess delta exposure from using a binary signal.

Specifically, rebalancing our portfolios once a month we will:

Calculate the delta of a 24-month straddle with 12-months left until expiration. This allows the straddle to reflect the same strike price as the rolling and binary trend signals, but with enough life until expiration that dt/(T-t) is small enough that the delta may closely reflect the delta from replication.
Calculate the delta of a 13-month straddle with 1-month until expiration. Calculate the difference in delta exposure from this step to the last and label this the excess delta from taking a rolling approach.
Finally, calculate the binary trend signal and calculate the difference in exposure versus the rolling approach. This will reflect the excess exposure from our binary approximation.

Below we plot the total return from each of these three series.

While we can see that replication has had a meaningful positive return, we can also see that the rolling model has a non-insignificant positive impact on returns. The trend model, despite being a net contributor through 2012, was more-or-less insignificant over the full period.

However, there is a bit of a head-scratcher here worth discussing: why would we expect the replication of a straddle to have a positive return? Positive convexity, sure. But a positive return?

Convexity versus Premium

Bruder and Gaussel (2011)² suggest that any single-asset trading strategy can be broken down into two component pieces: an option profile and trading impact. A simple constant stop-loss level, for example, can be thought of as a perpetual call option payoff with trading costs that alter exposure between long and flat, creating frictions due to non-continuous prices and delays in rebalancing.

For trend-following strategies, they suggest a framework whereby the trend is measured as an exponential moving average of daily returns and exposure is proportional to this measured trend, the asset’s variance, and the investor’s risk tolerance (a suggestion that is theoretically consistent with an optimal Markowitz/Merton strategy).

With these assumptions in hand, they introduce the following proposition:

Using this proposition, we can decompose long/short trend equity strategies into these two component pieces. Using this model, Jusselin, Malongo, Roncalli, Lezmi, Masselin, and Dao (2017)³ demonstrate that theoretical model returns can be decomposed as:

Following this model, we can extract the option payoff (G) and trading impact (g) of a trend following S&P 500 strategy.

What is clear from this analysis is that trend following strategies have two different components driving returns.

The first is the underlying option payoff. In the Bruder and Gaussell (2011) model, the option is similar to a straddle struck on the trend of the underlying asset. We can see in the graph above that this component has a short memory and provides much of the convexity often associated with trend-following returns.

The second component is the trading impact. We can see that this component is low frequency and the driving factor is the realized Sharpe ratio. If we squint our eyes a bit at the return formula, it even resembles the gamma gain minus gamma loss of a traditional straddle delta-hedging strategy.⁴

We can see that the long-term driver of returns in trend-following strategies, then, is not the convexity, but rather the trading impact. Given that trend following has positive gamma, we would expect the trading impact to be positive for returns that exhibit autocorrelation (i.e. trending returns). Interestingly, Jusselin, Malongo, Roncalli, Lezmi, Masselin, and Dao (2017) demonstrate that autocorrelation may not even be a necessary component for positive returns; rather, for this particular trend following model, the trading impact will have positive profit or loss based upon the underlying asset’s Sharpe ratio.

It is worth acknowledging that this analysis is based upon a particular exposure model that is driven by the underlying asset’s realized trend, volatility, and the investor’s risk tolerance. Thus, the delta-based model and binary signal model explored in prior sections will not match as neatly. Nevertheless, it serves as further evidence for trend following’s inherent return convexity with respect to the underlying asset.

Conclusion

Trend following is unique among style premia in that it has historically exhibited a convex payoff profile with positive skew. By replicating example trend following strategies using straddle options, we demonstrated how convexity is inherent to trend following strategies outside of any historical premium.

While the historical premium is anomalous, the convexity makes sense.

By replicating the payoff of a rolling straddle strategy, we saw potential reasons why frequent rebalancing in trend following strategies is necessary and were able to decompose the return contributions of the replication, the rolling model, and a binary trend following approach under more realistic assumptions.

In the simplified model we found that most of the historical premium associated with trend following comes from the trading impact that is tied to the asset’s Sharpe ratio.

A key step in sticking with any trading strategy is an understanding of why it may work in the future. A good historical backtest is nice to see, but there has to be a reason – be it behavioral, economic, or structural – for the backtest to have any reliability.

Using options to isolate the return sources in trend following strategies is a way to separate the impacts of convexity and trading impact while gaining a deeper understanding the types of risk exchanges that come with investing using trend following strategies.

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

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 d₁ and d₂ 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.

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.

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.

Fragility Case Study: Dual Momentum GEM

By Corey Hoffstein

On January 14, 2019

In Craftsmanship, Momentum, Popular, Portfolio Construction, Risk Management, Trend

This post is available as a PDF download here.

Summary

Recent market volatility has caused many tactical models to make sudden and significant changes in their allocation profiles.
Periods such as Q4 2018 highlight model specification risk: the sensitivity of a strategy’s performance to specific implementation decisions.
We explore this idea with a case study, using the popular Dual Momentum GEM strategy and a variety of lookback horizons for portfolio formation.
We demonstrate that the year-to-year performance difference can span hundreds, if not thousands, of basis points between the implementations.
By simply diversifying across multiple implementations, we can dramatically reduce model specification risk and even potentially see improvements in realized metrics such as Sharpe ratio and maximum drawdown.

Introduction

Among do-it-yourself tactical investors, Gary Antonacci’s Dual Momentum is the strategy we tend to see implemented the most. The Dual Momentum approach is simple: by combining both relative momentum and absolute momentum (i.e. trend following), Dual Momentum seeks to rotate into areas of relative strength while preserving the flexibility to shift entirely to safety assets (e.g. short-term U.S. Treasury bills) during periods of pervasive, negative trends.

In our experience, the precise implementation of Dual Momentum tends to vary (with various bells-and-whistles applied) from practitioner to practitioner. The most popular benchmark model, however, is the Global Equities Momentum (“GEM”), with some variation of Dual Momentum Sector Rotation (“DMSR”) a close second.

Recently, we’ve spoken to several members in our extended community who have bemoaned the fact that Dual Momentum kept them mostly aggressively positioned in Q4 2018 and signaled a defensive shift at the beginning of January 2019, at which point the S&P 500 was already in a -14% drawdown (having peaked at over -19% on December 24^th). Several DIYers even decided to override their signal in some capacity, either ignoring it entirely, waiting a few days for “confirmation,” or implementing some sort of “half-and-half” rule where they are taking a partially defensive stance.

Ignoring the fact that a decision to override a systematic model somewhat defeats the whole point of being systematic in the first place, this sort of behavior highlights another very important truth: there is a significant gap of risk that exists between the long-term supporting evidence of an investment style (e.g. momentum and trend) and the precise strategy we attempt to implement with (e.g. Dual Momentum GEM).

At Newfound, we call that gap model specification risk. There is significant evidence supporting both momentum and trend as quantitative styles, but the precise means by which we measure these concepts can lead to dramatically different portfolios and outcomes. When a portfolio’s returns are highly sensitive to its specification – i.e. slight variation in returns or model parameters lead to dramatically different return profiles – we label the strategy as fragile.

In this brief commentary, we will use the Global Equities Momentum (“GEM”) strategy as a case study in fragility.

Global Equities Momentum (“GEM”)

To implement the GEM strategy, an investor merely needs to follow the decision tree below at the end of each month.

From a practitioner stand-point, there are several attractive features about this model. First, it is based upon the long-run evidence of both trend-following and momentum. Second, it is very easy to model and generate signals for. Finally, it is fairly light-weight from an implementation perspective: only twelve potential rebalances a year (and often much less), with the portfolio only holding one ETF at a time.

Despite the evidence that “simple beats complex,” the simplicity of GEM belies its inherent fragility. Below we plot the equity curves for GEM implementations that employ different lookback horizons for measuring trend and momentum, ranging from 6- to 12-months.

Source: CSI Analytics. Calculations by Newfound Research. Returns are backtested and hypothetical. Returns assume the reinvestment of all distributions. Returns are gross of all fees except for underlying ETF expense ratios. 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.

We can see a significant dispersion in potential terminal wealth. That dispersion, however, is not necessarily consistent with the notion that one formation period is inherently better than another. While we would argue, ex-ante, that there should be little performance difference between a 9-month and 10-month lookback – they both, after all, capture the notion of “intermediate-term trends” – the former returned just 43.1% over the period while the latter returned 146.1%.

These total return figures further hide the year-to-year disparity that exists. The 9-month model, for example, was not a consistent loser. Below we plot these results, highlighting both the best (blue) and worst (orange) performing specifications. We see that the yearly spread between these strategies can be hundreds-to-thousands of basis points; consider that in 2010, the strategy formed using a 10-month lookback returned 12.2% while the strategy formed using a 9-month lookback returned -9.31%.

Same thesis. Same strategy. Slightly different specification. Dramatically different outcomes. That single year is likely the difference between hired and fired for most advisors and asset managers.

☞ Explore a diversified approach with the Newfound/ReSolve Robust Equity Momentum Index.

For those bemoaning their 2018 return, note that the 10-month specification would have netted a positive result! That specification turned defensive at the end of October.

Now, some may cry “foul” here. The evidence for trend and momentum is, after all, centuries in length and the efficacy of all these horizons is supported. Surely the noise we see over this ten-year period would average out over the long run, right?

The unfortunate reality is that these performance differences are not expected to mean-revert. The gambler’s fallacy would have us believe that bad luck in one year should be offset by good luck in another and vice versa. Unfortunately, this is not the case. While we would expect, at any given point in time, that each strategy has equal likelihood of experiencing good or bad luck going forward, that luck is expected to occur completely independently from what has happened in the past.

The implication is that performance differences due to model specification are not expected to mean-revert and are therefore expected to be random, but very permanent, return artifacts.¹

The larger problem at hand is that none of us have a hundred years to invest. In reality, most investors have a few decades. And we act with the temperament of having just a few years. Therefore, bad luck can have very permanent and very scarring effects not only upon our psyche, but upon our realized wealth.

But consider what happens if we try to neutralize the role of model specification risk and luck by diversifying across the seven different models equally (rebalanced annually). We see that returns closer in line with the median result, a boost to realized Sharpe ratio, and a reduction in the maximum realized drawdown.

These are impressive results given that all we employed was naïve diversification.

Conclusion

The odd thing about strategy diversification is that it guarantees we will be wrong. Each and every year, we will, by definition, allocate at least part of our capital to the worst performing strategy. The potential edge, however, is in being vaguely wrong rather than precisely wrong. The former is annoying. The latter can be catastrophic.

In this commentary we use the popular Dual Momentum GEM strategy as a case study to demonstrate how model specification choices can lead to performance differences that span hundreds, if not thousands, of basis points a year. Unfortunately, we should not expect these performance differences to mean revert. The realizations of good and bad luck are permanent, and potentially very significant, artifacts within our track records.

By simply diversifying across the different models, however, we can dramatically reduce specification risk and thereby reduce strategy fragility.

To be clear, no amount of diversification will protect you from the risk of the style. As we like to say, “risk cannot be destroyed, only transformed.” In that vein, trend following strategies will always incur some sort of whipsaw risk. The question is whether it is whipsaw related to the style as a whole or to the specific implementation.

For example, in the graphs above we can see that Dual Momentum GEM implemented with a 10-month formation period experienced whipsaw in 2011 when few of the other implementations did. This is more specification whipsaw than style whipsaw. On the other hand, we can see that almost all the specifications exhibited whipsaw in late 2015 and early 2016, an indication of style whipsaw, not specification whipsaw.

Specification risk we can attempt to control for; style risk is just something we have to bear.

At Newfound, evidence such as this informs our own trend-following mandates. We seek to diversify ourselves across the axes of what (“what are we investing in?”), how (“how are we making the decisions?”), and when (“when are we making those decisions?”) in an effort to reduce specification risk and provide the greatest style consistency possible.

A Carry-Trend-Hedge Approach to Duration Timing

By Corey Hoffstein

On October 15, 2018

In Carry, Risk & Style Premia, Trend, Weekly Commentary

This post is available as a PDF download here.

Summary

In this paper we discuss simple rules for timing exposure to 10-year U.S. Treasuries.
We explore signals based upon the slope of the yield curve (“carry”), prior returns (“trend”), and prior equity returns (“hedge”).
We implement long/short implementations of each strategy covering the time period of 1962-2018.
We find that all three methods improve both total and risk-adjusted returns when compared to long-only exposure to excess bond returns.
Naïve combination of both strategies and signals appears to improve realized risk-adjusted returns, promoting the benefits of process diversification.

Introduction

In this strategy brief, we discuss three trading rules for timing exposure to duration. Specifically, we seek to time the excess returns generated from owning 10-year U.S. Treasury bonds over short rates. This piece is meant as a companion to our prior, longer-form explorations Duration Timing with Style Premiaand Timing Bonds with Value, Momentum, and Carry. In contrast, the trading rules herein are simplistic by design in an effort to highlight the efficacy of the signals.

We explore three different signals in this piece:

The slope of the yield curve (“term spread”);
Prior realized excess bond returns; and
Prior realized equity market returns.

In contrast to prior studies, we do not consider traditional value measures, such as real yields, or explicit estimates of the bond risk premium, as they are less easily calculated. Nevertheless, the signals studied herein capture a variety of potential influences upon bond markets, including inflation shocks, economic shocks, policy shocks, marginal utility shocks, and behavioral anomalies.

The strategies based upon our signals are implemented as dollar-neutral long/short portfolios that go long a constant maturity 10-year U.S. Treasury bond index and short a short-term U.S. Treasury index (assumed to be a 1-year index prior to 1982 and a 3-month index thereafter). We compare these strategies to a “long-only” implementation that is long the 10-year U.S. Treasury bond index and short the short-term U.S. Treasury index in order to capture the excess realized return associated with duration.

Implementing our strategies as dollar-neutral long/short portfolios allows them to be interpreted in a variety of useful manners. For example, one obvious interpretation is an overlay implemented on an existing bond portfolio using Treasury futures. However, another interpretation may simply be to guide investors as to whether to extend or contract their duration exposure around a more intermediate-term bond portfolio (e.g. a 5-year duration).

At the end of the piece, we explore the potential diversification benefits achieved by combining these strategies in both an integrated (i.e. signal combination) and composite (i.e. strategy combination) fashion.

Slope of the Yield Curve

In past research on timing duration, we considered explicit measures of the bond risk premium as well as valuation. In Duration Timing with Style Premiawe used a simple signal based upon real yield, which had the problem of being predominately long over the last several decades. In Timing Bonds with Value, Momentum, and Carry we compared a de-trended real yield against recent levels in an attempt to capture more short-term valuation fluctuations.

In both of these prior research pieces, we also explicitly considered the slope of the yield curve as a predictor of future excess bond returns. One complicating factor to carry signals is that rate steepness simultaneously captures both the expectation of rising short rates as well as an embedded risk premium. In particular, evidence suggests that mean-reverting rate expectations dominate steepness when short rates are exceptionally low or high. Anecdotally, this may be due to the fact that the front end of the curve is determined by central bank policy while the back end is determined by inflation expectations.

Thus, despite being a rather blunt measure, steepness may simultaneously be related to business cycles, credit cycles and monetary policy cycles. To quote Ilmanen (2011):

A steep [yield curve] coincides with high unemployment rate (correlation +0.45) and predictsfast economic growth. [Yield curve] countercyclicality may explain its ability to predict near-term bond and stock returns: high required premia near business cycle troughs result in a steep [yield curve], while low required premia near business cycle peaks result in an inverted [yield curve].

Therefore, while estimates of real yield may seek to be explicit measures of value, we may consider carry to be an ancillary measure as well, as a high carry tends to be associated with a high term premium. In Figure 1 we plot the annualized next month excess bond return based upon the quartile (using the prior 10 years of information) that the term spread falls into. We can see a significant monotonic improvement from the 1^stto the 4^thquartiles, indicating that higher levels of carry, relative to the past, are positive indicators of future returns.

Therefore, we construct our carry strategy as follows:

At the end of each month, calculate the term spread between 10- and 1-year U.S. Treasuries.
Calculate the realized percentile of this spread by comparing it against the prior 10-years of daily term spread measures.
If the carry score is in the top two thirds, go long excess bond returns. If the carry score is in the bottom third, go short excess bond returns.
Trade at the close of the 1^sttrading day of the month.

Returns for this strategy are plotted in Figure 2. Our research suggests that the backtested results of this model can be significantly improved through the use of longer holding periods and portfolio tranching. Another potential improvement is to scale exposure linearly to the current percentile. We will leave these implementations as exercises to readers.

Figure 1

Source: Kenneth French Data Library, Federal Reserve of St. Louis. Calculations by Newfound Research. Returns are backtested and hypothetical. Return data relies on hypothetical indices and is exclusive of all fees and expenses. Returns assume the reinvestment of all distributions. The Carry Long/Short strategy does not reflect any strategy offered or managed by Newfound Research and was constructed exclusively for the purposes of this commentary. It is not possible to invest in an index. Past performance does not guarantee future results.

Figure 2

Data from 1972-2018

	Annualized Return	Annualized Volatility	Sharpe Ratio
Long Only	2.1%	7.6%	0.27
CARRY L/S	2.6%	7.7%	0.33

Trend in Bond Returns

Momentum, in both its relative and absolute (i.e. “trend”) forms, has a long history among both practitioners and academics (see our summary piece Two Centuries of Momentum).

The literature covering momentum in bond returns, however, varies in precisely whatprior returns matter. There are three primary categories: (1) change in bond yields (e.g. Ilmanen (1997)), (2) total return of individual bonds (e.g. Kolanovic and Wei (2015) and Brooks and Moskowitz (2017)), and (3) total return of bond indices (or futures) (e.g. Asness, Moskowitz, and Pedersen (2013), Durham (2013), and Hurst, Ooi, Pedersen (2014))

In our view, the approaches have varying trade-offs:

While empirical evidence suggests that nominal interest rates can exhibit secular trends, rate evolution is most frequently modeled as mean-reversionary. Our research suggests that very short-term momentum can be effective, but leads to a significant amount of turnover.
The total return of individual bonds makes sense if we plan on running a cross-sectional bond model (i.e. identifying individual bonds), but is less applicable if we want to implement with a constant maturity index.
The total return of a bond index may capture past returns that are attributable to securities that have been recently removed.

We think it is worth noting that the latter two methods can capture yield curve effects beyond shift, including roll return, steepening and curvature changes. In fact, momentum in general may even be able to capture other effects such as flight-to-safety and liquidity (supply-demand) factors.

In this piece, we elect to measure momentum as an exponentially-weighting average of prior log returns of the total return excess between long and short bond indices. We measure this average at the end of each month and go long duration when it is positive and short duration when it is negative. In Figure 4 we plot the results of this method based upon a variety of lookback periods that approximate 1-, 3-, 6-, and 12-month formation periods.

Figure 3

	MOM 21	MOM 63	MOM 126	MOM 252
MOM 21	1.00	0.87	0.65	0.42
MOM 63	0.87	1.00	0.77	0.53
MOM 126	0.65	0.77	1.00	0.76
MOM 252	0.42	0.53	0.76	1.00

We see varying success in the methods, with only MOM 63 and MOM 256 exhibiting better risk-adjusted return profiles. Despite this long-term success, we can see that MOM 63 remains in a drawdown that began in the early 2000s, highlighting the potential risk of relying too heavily on a specific measure or formation period. In Figure 3 we calculate the correlation between the different momentum strategies. As we found in Measuring Process Diversification in Trend Following, diversification opportunities appear to be available by mixing both short- and long-term formation periods.

With this in mind, we elect for the following momentum implementation:

At the end of each month, calculate both a 21- and 252-day exponentially-weighted moving average of realized daily excess log returns.
When both signals are positive, go long duration; when both signals are negative, go short duration; when signals are mixed, stay flat.
Rebalance at the close of the next trading day.

The backtested results of this strategy are displayed in Figure 5.

As with carry, we find that there are potential craftsmanship improvements that can be made with this strategy. For example, implementing with four tranches, weekly rebalances appears to significantly improve backtested risk-adjusted returns. Furthermore, there may be benefits that can be achieved by incorporating other means of measuring trends as well as other lookback periods (see Diversifying the What, When, and How of Trend Following and Measuring Process Diversification in Trend Following).

Figure 4

Data from 1963-2018

	Annualized Return	Annualized Volatility	Sharpe Ratio
Long Only	1.5%	7.3%	0.21
MOM 21	1.4%	7.5%	0.19
MOM 63	1.8%	7.4%	0.25
MOM 128	1.3%	7.4%	0.18
MOM 252	1.9%	7.4%	0.26

Source: Kenneth French Data Library, Federal Reserve of St. Louis. Calculations by Newfound Research. Returns are backtested and hypothetical. Return data relies on hypothetical indices and is exclusive of all fees and expenses. Returns assume the reinvestment of all distributions. The Momentum strategies do not reflect any strategies offered or managed by Newfound Research and were constructed exclusively for the purposes of this commentary. It is not possible to invest in an index. Past performance does not guarantee future results.

Figure 5

Data from 1963-2018

	Annualized Return	Annualized Volatility	Sharpe Ratio
Long Only	1.5%	7.2%	0.21
MOM L/S	1.7%	6.3%	0.28

Source: Kenneth French Data Library, Federal Reserve of St. Louis. Calculations by Newfound Research. Returns are backtested and hypothetical. Return data relies on hypothetical indices and is exclusive of all fees and expenses. Returns assume the reinvestment of all distributions. The Momentum Long/Short strategy does not reflect any strategy offered or managed by Newfound Research and was constructed exclusively for the purposes of this commentary. It is not possible to invest in an index. Past performance does not guarantee future results.

Safe-Haven Premium

Stocks and bonds generally exhibit a positive correlation over time. One thesis for this long-term relationship is the present value model, which argues that declining yields, and hence increasing bond prices, increase the value of future discounted cash flows and therefore the fair value of equities. Despite this long-term relationship, shocks in economic growth, inflation, and even monetary policy can overwhelm the discount rate thesis and create a regime-varying correlation structure.

For example, empirical evidence suggests that high quality bonds can exhibit a safe haven premium during periods of economic stress. Using real equity prices as a proxy for wealth, Ilmanen (1995) finds that “wealth-dependent relative risk aversion appears to be an important source of bond return predictability.” Specifically, inverse wealth is a significant positive predictor of future excess bond returns at both world and local (U.S., Canada, Japan, Germany, France, and United Kingdom) levels. Ilmanen (2003) finds that, “stock-bond correlations are more likely to be negative when inflation is low, growth is slow, equities are weak, and volatility is high.”

To capitalize on this safe-haven premium, we derive a signal based upon prior equity returns. Specifically, we utilize an exponentially weighted average of prior log returns to estimate the underlying trend of equities. We then compare this estimate to a 10-year rolling window of prior estimates, calculating the current percentile.

In Figure 6 we plot the annualized excess bond return for the month following, assuming signals are generated at the close of each month and trades are placed at the close of the following trading day. We can see several effects. First, next month returns for 1^stquartile equity momentum – i.e. very poor equity returns – tends to be significantly higher than other quartiles. Second, excess bond returns in the month following very strong equity returns tend to be poor. We would posit that these two effects are two sides of the same coin: the safe-haven premium during 1^stquartile periods and an unwind of the premium in 4^thquartile periods. Finally, we can see that 2^ndand 3^rdquartile returns tend to be positive, in line with the generally positive excess bond return over the measured period.

In an effort to isolate the safe-haven premium, we construct the following strategy:

At the end of each month, calculate an equity momentum measure by taking a 63-day exponentially weighted average of prior daily log-returns.
Calculate the realized percentile of this momentum measure by comparing it against the prior 10-years of daily momentum measures.
If the momentum score is in the bottom quartile, go long excess bond returns. If the momentum score is in the top quartile, go short excess bond returns. Otherwise, remain flat.
Trade at the close of the 1^sttrading day of the month.

Returns for this strategy are plotted in Figure 7. As expected based upon the quartile design, the strategy only spends 24% of its time long, 23% of its time short, and the remainder of its time flat. Despite this even split in time, approximately 2/3^rdsof the strategy’s return comes from the periods when the strategy is long.

Figure 6

Source: Kenneth French Data Library, Federal Reserve of St. Louis. Calculations by Newfound Research. Returns are backtested and hypothetical. Return data relies on hypothetical indices and is exclusive of all fees and expenses. Returns assume the reinvestment of all distributions. The Equity Momentum Long/Short strategy does not reflect any strategy offered or managed by Newfound Research and was constructed exclusively for the purposes of this commentary. It is not possible to invest in an index. Past performance does not guarantee future results.

Figure 7

Data from 1962-2018

	Annualized Return	Annualized Volatility	Sharpe Ratio
Long Only	1.5%	7.2%	0.21
Equity Mom L/S	1.9%	5.7%	0.34

Source: Kenneth French Data Library, Federal Reserve of St. Louis. Calculations by Newfound Research. Returns are backtested and hypothetical. Return data relies on hypothetical indices and is exclusive of all fees and expenses. Returns assume the reinvestment of all distributions. The Equity Momentum Long/Short strategy does not reflect any strategy offered or managed by Newfound Research and was constructed exclusively for the purposes of this commentary. It is not possible to invest in an index. Past performance does not guarantee future results.

Combining Signals

Despite trading the same underlying instrument, variation in strategy construction means that we can likely benefit from process diversification in constructing a combined strategy. Figure 8 quantifies the available diversification by measuring full-period correlations among the strategies from joint inception (1972). We can also see that the strategies exhibit low correlation to the Long Only implementation, suggesting that they may introduce diversification benefits to a strategic duration allocation as well.

Figure 8

	LONG ONLY	CARRY L/S	MOM L/S	EQ MOM L/S
LONG ONLY	1.00	0.42	0.33	-0.09
CARRY L/S	0.42	1.00	0.40	-0.09
MOM L/S	0.33	0.40	1.00	-0.13
EQ MOM L/S	-0.10	-0.10	-0.19	1.00

We explore two different implementations of a diversified strategy. In the first, we simply combine the three strategies in equal-weight, rebalancing on a monthly basis. This implementation can be interpreted as three sleeves of a larger portfolio construction. In the second implementation, we combine underlying long/short signals. When the net signal is positive, the strategy goes 100% long duration and when the signal is negative, it goes 100% short. This can be thought of as an integrated approach that takes a majority-rules voting approach. Results for these strategies are plotted in Figure 9. We note the substantial increase in the backtested Sharpe Ratio of these diversified approaches in comparison to their underlying components outlined in prior sections.

It is important to note that despite strong total and risk-adjusted returns, the strategies spend only approximately 54% of their time net-long duration, with 19% of their time spent flat and 27% of their time spent short. While slightly biased long, this breakdown provides evidence that strategies are not simply the beneficiaries of a bull market in duration over the prior several decades.

Figure 9

Data from 1972-2018

	Annualized Return	Annualized Volatility	Sharpe Ratio
Long Only	2.1%	7.6%	0.27
Combined L/S	2.5%	4.3%	0.58
Integrated L/S	3.5%	7.1%	0.49

Source: Kenneth French Data Library, Federal Reserve of St. Louis. Calculations by Newfound Research. Returns are backtested and hypothetical. Return data relies on hypothetical indices and is exclusive of all fees and expenses. Returns assume the reinvestment of all distributions. Neither the Combined Long/Short or Integrated Long/Short strategies reflect any strategy offered or managed by Newfound Research and were constructed exclusively for the purposes of this commentary. It is not possible to invest in an index. Past performance does not guarantee future results.

Conclusion

In this research brief, we continued our exploration of duration timing strategies. We aimed to implement several signals that were simple by construction. Specifically, we evaluated the impact of term spread, prior excess bond returns, and prior equity returns on next month’s excess bond returns. Despite their simplicity, we find that all three signals can potentially offer investors insight for tactical timing decisions.

While we believe that significant craftsmanship improvements can be made in all three strategies, low hanging improvement may simply come from combining the approaches. We find a meaningful improvement in Sharpe Ratio by naively combining these strategies in both a sleeve-based and integrated signal fashion.

Bibliography

Asness, Clifford S. and Moskowitz, Tobias J. and Pedersen, Lasse Heje, Value and Momentum Everywhere (June 1, 2012). Chicago Booth Research Paper No. 12-53; Fama-Miller Working Paper. Available at SSRN: https://ssrn.com/abstract=2174501 or http://dx.doi.org/10.2139/ssrn.2174501

Brooks, Jordan and Moskowitz, Tobias J., Yield Curve Premia (July 1, 2017). Available at SSRN: https://ssrn.com/abstract=2956411 or http://dx.doi.org/10.2139/ssrn.2956411

Durham, J. Benson, Momentum and the Term Structure of Interest Rates (December 1, 2013). FRB of New York Staff Report No. 657. Available at SSRN: https://ssrn.com/abstract=2377379 or http://dx.doi.org/10.2139/ssrn.2377379

Hurst, Brian and Ooi, Yao Hua and Pedersen, Lasse Heje, A Century of Evidence on Trend-Following Investing (June 27, 2017). Available at SSRN: https://ssrn.com/abstract=2993026 or http://dx.doi.org/10.2139/ssrn.2993026

Ilmanen, Antti, Time-Varying Expected Returns in International Bond Markets, Journal of Finance, Vol. 50, No. 2, 1995, pp. 481-506.

Ilmanen, Antti, Forecasting U.S. Bond Returns, Journal of Fixed Income, Vol. 7, No. 1, 1997, pp. 22-37.

Ilmanen, Antti, Stock-Bond Correlations, Journal of Fixed Income, Vol. 13, No. 2, 2003, pp. 55-66.

Ilmanen, Antti. Expected Returns an Investor’s Guide to Harvesting Market Rewards. John Wiley, 2011.

Kolanovic, Marko, and Wei, Zhen, Momentum Strategies Across Asset Classes (April 2015). Available at https://www.cmegroup.com/education/files/jpm-momentum-strategies-2015-04-15-1681565.pdf

Category: Trend Page 4 of 7

G̷̖̱̓́̀litch

Summary­

Conclusion

Trend: Convexity & Premium

Summary­

Introduction

Pricing is Replication

Trend Following is “Long Gamma”

Straddle-ish

Convexity versus Premium

Conclusion

Tightening the Uncertain Payout of Trend-Following

Summary­

Introduction

Diversification in Trend-Following

The Math Behind the Diversification

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

Expanding the Multitude of Models

Conclusion

Fragility Case Study: Dual Momentum GEM

Summary­

Introduction

Global Equities Momentum (“GEM”)

Conclusion

A Carry-Trend-Hedge Approach to Duration Timing

Summary­

Introduction

Slope of the Yield Curve

Trend in Bond Returns

Safe-Haven Premium

Combining Signals

Conclusion

Bibliography

Summary

Summary

Summary

Summary

Summary