Category: Risk & Style Premia Page 2 of 16

Option-Based Trend Following

On June 23, 2020

In Risk Management, Trend, Weekly Commentary

This post is available as a PDF download here.

Summary

The convex payoff profile of trend following strategies naturally lends itself to comparative analysis with option strategies.
To isolate the two extremes of paying for whipsaw – either up front or in arrears – we replicate an option strategy that buys 1-month at-the-money calls and puts based on the trend signal.
We find that while option premiums steadily eat away at the balance of the options portfolio, the avoidance of large whipsaw events gives the strategy a boost at key times over the past 15 years, especially recently.
We examine how this whipsaw cost fits into the historical context of the options strategy and explore some simple ways to shift between the option-based trend following and the standard model.
The extent that whipsaw can be mitigated while still maintaining the potential to earn diversified returns is likely limited, but the optimal blend of trend following and options can be a beneficial guideline for investors to weather both sudden and prolonged drawdowns.

The non-linear payoff of trend following strategies has many similarities to options strategies, and by way of analogy, we can often gain insight into which market environments will favor trend following and why.

In our previous research piece, Straddles and Trend Following, we looked at purchasing straddles – that is, a call option and a put option – with a strike price tied to the anchor price of the trend following model. For example, if the trend following model invested in equities when the return over the past 12 months was positive, for a security that was at $100 12-months ago and is at $120 today, we would purchase a call and a put option with a strike price of $100. In this case, the call would be 20% in-the-money (ITM) and the put would be out-of-the-money (OTM).

In essence, this strategy acted like an insurance policy where the payout was tied to a reversion in the trend signal, and the premium paid when the trend signal was strong was small.

This concept of insurance is an important discussion topic in trend following strategies. The risk we must manage in these types of strategies, either directly through insurance or some other indirect means like diversification, is whipsaw.

In this commentary, we will construct an options strategy that is similar to a trend following strategy. The option strategy will pay a premium up-front to avoid whipsaw. By comparing this strategy to trend following that bears the full risk of whipsaw, we can set a better practical bound for how much investors should expect to pay or earn for bearing this risk.

Methodology and Data

For this analysis, we will use the S&P 500 index for equity returns, the 1-year LIBOR rate as the risk-free rate, and options data on the S&P 500 (SPX options).

To bridge the gap between practice and abstraction, we will utilize a volatility surface calibrated to real option data to price options. We will constrain our SPX options to $5 increments and interpolate total implied variance to get prices for options that were either illiquid or not included in the data set.

For the most part, we will stick to options that expire on the third Friday of each month and will mention when we deviate from that assumption.

The long/short trend equity strategy looks at total returns of equities over 12 months. If this return is positive, the strategy invests in equities for the following month. If the return is negative, the strategy shorts equities for the following month and earns the risk-free rate on the cash. The strategy is rebalanced monthly on the options expiration dates.

For the option-based trend strategy, on each rebalance date, we will purchase a 1-month call if the trend signal is positive or a put if the trend signal is negative. We will purchase all options at-the-money (ATM) and hold them to expiration. The strategy is fully cash-collateralized. Any premium is paid on the options roll date, interest is earned on the remaining account balance, and the option payout is realized on the next roll date.

Why are we now using ATM options when previous research used ITM and OTM options, potentially deeply ITM or OTM?

Here we are looking to isolate the cost of whipsaw in the premium paid for the option while earning a payout that is close to that of the underlying in the event that our trend signal is correct. If we utilized OTM options, then our premium would be lower but we would realize smaller gains if the underlying followed the trend. ITM options would have downside exposure before the protection kicked in.

We are also not using straddles since we do not want to pay extra premium for the chance to profit off a whipsaw. The underlying assumption here is that there is value in the trend following signal. Either strategy is able to capitalize on that (i.e. it’s the control variable); the strategies primarily differ in their treatment of whipsaw costs.

The High Cost of ATM Options

The built-in whipsaw protection in the options does not come cheap. The chart below shows the –L/S trend following strategy–, the –option-based trend strategy–, and the ratio of the two (dotted).Source: DiscountOptionsData.com. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all fees including, but not limited to, management fees, transaction fees, and taxes. Returns assume the reinvestment of all distributions.

During normal market environments and even in prolonged equity-market drawdown periods like 2008, trend following outperformed the option-based strategy. Earning the full return on the underlying equity is generally beneficial.

However, something that is “generally beneficial” can be erased very quickly. In March 2020, the trend following strategy reverted back to the level of the option-based strategy. If you had only looked at cumulative returns over those 15 years, you would not be able to tell much difference between the two.

The following chart highlights these tail effects.

Source: DiscountOptionsData.com. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all fees including, but not limited to, management fees, transaction fees, and taxes. Returns assume the reinvestment of all distributions.

In most months, the option-based strategy forfeits its ~1.5% premium for the ATM option. The 75^th percentile cutoff is 2.2% and the 90^th percentile cutoff is 2.9%. These premiums have occasionally spiked to 6-7%.

While these premiums are not always forfeited without some offsetting gain, they are always paid relative to the trend following strategy.

A 3% whipsaw event in trend should definitely not be a surprise based on the typical up-front cost of the option strategy.

Source: DiscountOptionsData.com. Calculations by Newfound Research.

But What About a 30% Whipsaw?

Now that’s a good question.

Up until March 2020, for the 15 years prior, the largest whipsaws relative to the options strategy were 12-13%. This is the epitome of tail risk, and it can be disheartening to think that now that we have seen 30% underperformance, we should probably expect more at some point in the (hopefully very distant) future.

However, a richer sample set can shed some light on this very poor performance.

Let’s relax our assumption that we roll the options and rebalance the trend strategies on the third Friday of the month and instead allow rebalances and rolls on any day in the month. Since we are dealing with one-month options, this is not beyond implementation since there are typically options listed that expire on Monday, Wednesday, and Friday.

The chart below shows all of these option strategies and how large of an effect that roll / rebalance timing luck can have.

With timing luck in both the options strategies and trend following, there can be large effects when the luck cuts opposite ways.

The worst returns between rebalances of trend following relative to each options strategy highlight how bad the realized path in March 2020 truly was.

In many of the trend following and option strategies pairs, the worst underperformance of trend following over any monthlong period was around 10%.

Returning to the premise that the options strategies are analogous to trend following, we see the same effects of timing luck that we have explored in previous research: effects that make comparing variants of the same strategy or similar strategies more nuanced. Whether an option strategy is used for research, benchmarking, or active investing, the implications of this timing luck should be taken into account.

But even without taking a multi-model approach at this point to the options strategy, can we move toward a deeper understanding of when it may be an effective way to offset some of the risk of whipsaw?

I’d Gladly Pay You Tuesday for a Whipsaw Risk Today

With the two extremes of paying for whipsaw up front with options and being fully exposed to whipsaw through trend following, perhaps there is a way to tailor this whipsaw risk profile. If the risk of whipsaw is elevated but the cost of paying for the insurance is cheap, then the options strategy may be favorable. On the other hand, if option premiums are high, trend following may more efficiently capture the market returns.

The price of the options (or their implied volatilities) is a natural place to start investigating this topic since it encapsulates the premium for whipsaw insurance. The problem is that it may not be a reliable signal if there is no barrier to efficiency in the options market, either behavioral or structural.

Comparing the ATM option implied volatilities with the trend signal (12-month trailing returns), we see a negative correlation, which indicates that the options-based strategy will have a higher hurdle rate of return in strongly downtrending market environments.

Source: DiscountOptionsData.com. Calculations by Newfound Research.

But this is only one piece of the puzzle.

Do these implied volatilities relate to the forward 1-month returns for the S&P 500?

Based on the above scatterplot: not really. However, since we are merely sticking implied volatility in the middle of the trend following signal and the forward return, and we believe that trend following works over the long run, then we must believe there is some relationship between implied volatility and forward returns.

While this monthly trend following signal is directionally correct over the next month 60% of the time, historically, that says nothing about the magnitude of the returns based on the signal.

Without looking too much into the data to avoid overfitting a model, we will set a simple cutoff of 20% implied volatility. If options cost more than that, we will utilize trend following. If they cost less, we will invest in the options strategy.

We will also compare it to a 50/50 blend of the two.

The switching strategy (gray line) worked well until around 2013 when the option prices were cheap, but the risk of whipsaw was not realized. It did make it through 2015, 2016 and 4Q 2018 better than trend following.

When viewed in a broader context of a portfolio, since these are alternative strategies, it does not take a huge allocation to make a difference. These strategies manage equity risk, so we can pair them with an allocation to the S&P 500 (SPY) and see how the aggregate statistics are affected over the period from 2005 to April 2020.

The chart below plots the efficient frontiers of allocations to 100% SPY at the point of convergence on the right of the graph) to 40% SPY on the left of the graph with the remainder allocated to the risk- management strategy.

The Sharpe ratio is maximized at a 35% allocation to the switching strategy, a 25% allocation to the option-based strategy, and 10% for the trend following strategy.

Conclusion

In this research note, we explored the link between trend following and options strategies using 1-month ATM put and call options, depending on the sign of the trend.

The cost of ATM options Is generally 1.5% of the portfolio value, but the fact that this cost can spike upwards of 9% should justify larger whipsaws in trend following strategies. Very large whipsaws, like in March 2020, not only show that the cost can be seemingly unbounded but also that there is significant exposure to timing luck based upon the option roll dates.

Then, we moved on to investigating a simple way to allocate between the two strategies based upon the cost of the options, When the options were cheap, we used that strategy, and when they were expensive, we invested in the trend following strategy. A modest allocation is enough to make a different in the realized efficient frontier.

Deciding to pay the up-front payment of the whipsaw insurance premium, bear the full risk a whipsaw, or land somewhere in between is largely up to investor preferences. It is risky to have a large downside potential, but the added benefit of no premiums can be enough to offset the risk.

An implied volatility threshold was a rather crude signal for assessing the risk of whipsaw and the price of insuring against it. Further research into one or multiple signals and a robust process for aggregating them into an investment decision is needed to make more definitive statements on when trend following is better than options or vice versa. The extent that whipsaw can be mitigated while still maintaining the potential to earn diversified returns is likely limited, but the optimal blend of trend following and options can be a beneficial guideline for investors to weather both sudden and prolonged drawdowns.

Defensive Equity with Machine Learning

By Corey Hoffstein

On May 25, 2020

In Defensive, Risk & Style Premia, Weekly Commentary

This post is available as a PDF download here.

Summary

Defensive equity strategies are comprised of stocks that lose less than the market during bear markets while keeping up with the market during a bull market.
Coarse sorts on metrics such as volatility, beta, value, and momentum lead to diversified portfolios but have mixed results in terms of their defensive characteristics, especially through different crisis periods that may favor one metric over another.
Using non-linear machine learning techniques is a desirable way to identify certain combinations of factors that lead to better defensive equity strategies over multiple periods.
By applying techniques such as random forests and gradient boosting to two sample defensive equity metrics, we find that machine learning does not add significant value over a low volatility sort, given the features included in the model.
While this by no means rules out the benefits of machine learning techniques, it shows how a blanket application of it is not a panacea for investing during crisis periods.

There is no shortage of hypotheses as to what characteristics define a stock that will outperform in a bear market. Some argue that value stocks should perform well, given their relative valuation buffer (the “less far to fall” argument). Some argue for a focus on balance sheet strength while others argue that cash-flow is the ultimate life blood of a company and should be prioritized. There are even arguments for industry preferences based upon economic cyclicality.

Each recession and crisis is unique, however, and therefore the characteristics of stocks that fare best will likely change. For example, the dot-com run-up caused a large number of real-economy businesses to be sorted into the “cheap” bucket of the value factor. These companies also tended to have higher quality earnings and lower beta / volatility than the dot-com stocks.

Common sense would indicate that unconstrained value may be a natural counter-hedge towards large, speculative bubbles, but we need only look towards 2008 – a credit and liquidity event – to see that value is not a panacea for every type of crisis.

It is for this reason that some investors prefer to take their cues from market-informed metrics such as beta, volatility, momentum, or trading volume.

Regardless of approach, there are some philosophical limitations we should consider when it comes to expectations with defensive equity portfolios. First, if we were able to identify an approach that could avoid market losses, then we would expect that strategy to also have negative alpha.¹ If this were not the case, we could construct an arbitrage.

Therefore, in designing a defensive equity portfolio, our aim should be to provide ample downside protection against market losses while minimizing the relative upside participation cost of doing so.

Traditional linear sorts – such as buying the lowest volatility stocks – are coarse by design. They aim to robustly capture a general truth and hedge missed subtleties through diversification. For example, while some stocks deserve to be cheap and some stocks are expensive for good reason, naïve value sorts will do little to distinguish them from those that are unjustifiably cheap or rich.

For a defensive equity portfolio, however, this coarseness may not only reduce effectiveness, but it may also increase the implicit cost. Therefore, in this note we implement non-linear techniques in an effort to more precisely identify combinations of characteristics that may create a more effective defensive equity strategy.

The Strategy Objective

To start, we must begin by defining precisely what we mean by a “defensive equity strategy.” What are the characteristics that would make us label one security as defensive and another as not? Or, potentially better, is there a characteristic that allows us to rank securities on a gradient of defensiveness?

This is not a trivial decision, as our entire exercise will attempt to maximize the probability of correctly identifying securities with this characteristic.

As our goal is to find those securities which provide the most protection during equity market routs but bleed the least during equity market rallies, we chose a metric that scored how closely a stock’s return reflected the payoff of a call option on the S&P 500 over the next 63 trading days (approximately 3 months).

In other words, if the S&P 500 is positive over the next 63 trading days, the score of a security is equal to the squared difference between its return and the S&P 500’s return. If the market’s return is negative, the score of a security is simply its squared return.

To determine whether this metric reflects the type of profile we want, we can create a long/short portfolio. Each month we rank securities by their scores and select the quintile with the lowest scores. Securities are then weighted by their market capitalization. Securities are held for three months and the portfolio is implemented with three tranches. The short leg of the portfolio is the market rather than the highest quintile, as we are explicitly trying to identify defense against the market.

To create a scalable solution, we restrict our investable universe to those in the top 1,000 securities by market capitalization.

We plot the performance below.

Source: Sharadar Fundamentals. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all fees including, but not limited to, management fees, transaction fees, and taxes. Returns assume the reinvestment of all distributions.

We can see that the strategy is relatively flat during bull markets (1998-2000, 2003-2007, 2011-2015, 2016-2018), but rallies during bear markets and sudden market shocks (2000-2003, 2008, 2011, 2015/2016, Q4 2018, and 2020).

Interestingly, despite having no sector constraints and not explicitly targeting tracking error at the portfolio level, the resulting portfolio ends up well diversified across sectors, though it does appear to make significant short-term jumps in sector weights. We can also see an increasing tilt towards Technology over the last 3 years in the portfolio. In recent months, positions in Financials and Industrials have been almost outright eliminated.

Source: Sharadar Fundamentals. Calculations by Newfound Research.

Of course, this metric is explicitly forward looking. We’re using a crystal ball to peer into the future and identify those stocks that track the best on the way up and protect the best on the way down. Our goal, then, is to use a variety of company and security characteristics to accurately forecast this score.

We will include a variety of characteristics and features, including:

Size: Market Capitalization.
Valuation: Book-to-Price, Earnings-to-Price, Free Cash Flow-to-Price, Revenue-to-EV, and EBITDA-to-EV.
Momentum: 12-1 Month Return and 1-Month Return.
Risk: Beta, Volatility, Idiosyncratic Volatility, and Ulcer Index.
Quality: Accruals, ROA, ROE, CFOA, GPOA, Net Margin, Asset Turnover, Leverage, and Payout Ratio.
Growth: Internal Growth Rate, EPS Growth, Revenue Growth.

These 24 features are all cross-sectionally ranked at each point in time. We also include dummy variables for each security to represent sector inclusion as well as whether the company has positive Net Income and whether the company has positive Operating Cash Flow.

Note that we are not including any market regime characteristics, such information about market returns, volatility, interest rates, credit spreads, sentiment, or monetary or fiscal policy. Had we included such features, our resulting model may end up as a factor switching approach, changing which characteristics it selects based upon the market environment. This may be an interesting model in its own right, but our goal herein is simply to design a static, non-linear factor sort.

Random Forests

Our first approach will be to apply a random forest algorithm, which is an ensemble learning method. The approach uses a training data set to build a number of individual decision trees whose results are then re-combined to create the ultimate decision. By training each tree on a subset of data and considering only a subset of features for each node, we can create trees that may individually have high variance, but as an aggregate forest reduce variance without necessarily increasing bias.

As an example, this means that one tree may be built using a mixture of low volatility and quality features, while another may be built using valuation and momentum features. Each tree is able to model a non-linear relationship, but by restricting tree depth and building trees using random subsets of data and features, we can prevent overfitting.

There are a number of hyperparameters that can be set to govern the model fit. For example, we can set the maximum depth of the individual trees as well as the number of trees we want to fit. Fitting hyperparameters is an art unto itself, and rather than go down the rabbit hole of tuning hyperparameters via cross-validation, we did our best to select reasonable hyper parameters. We elected to train the model on 50% of our data (March 1998 to March 2009), with a total of 100 trees each with a maximum depth of 2.

The results of the exercise are plotted below.

Source: Sharadar Fundamentals. Calculations by Newfound Research.

The performance does appear to provide defensive properties both in- and out-of-sample, with meaningful returns generated in 2000-2002, 2008, Q3 and Q4 of 2011, June 2015 through June 2016, and Q4 2008.

We can see that the allocations also express a number of static sector concentrations (e.g. Consumer Defensive) as well as some cyclical changes (e.g. Finances pre- and post-2007).

We can also gain insight into how the portfolio composition changes by looking at the weighted characteristic scores of the long leg of the portfolio over time.

Source: Sharadar Fundamentals. Calculations by Newfound Research.

It is important to remember that characteristics are cross-sectionally ranked across stocks. For some characteristics, higher is often considered better (e.g. a higher earnings-to-price cheaper is considered cheaper), whereas for other factors lower is better (e.g. lower volatility is considered to have less risk).

We can see that some characteristics are static tilts: higher market capitalization, positive operating cash flow, positive net income, and lower risk characteristics. Other characteristics are more dynamic. By 12/2008, the portfolio has tilted heavily towards high momentum stocks. A year later, the portfolio has tilted heavily towards low momentum stocks.

What is somewhat difficult to disentangle is whether these static and dynamic effects are due to the non-linear model we have developed, or whether it’s simply that applying static tilts results in the dynamic tilts. For example, if we only applied a low volatility tilt, is it possible that the momentum tilts would emerge naturally?

Unfortunately, the answer appears to be the latter. If we plot a long/short portfolio that goes long the bottom quintile of stocks ranked on realized 1-year volatility and short the broad market, we see a very familiar equity curve.

It would appear that the random forest model effectively identified the benefits of low volatility securities. And while out-of-sample performance does appear to provide more ample defense during 2011, 2015-2016, and 2018 than the low volatility tilt, it also has significantly greater performance drag.

Gradient Boosting

One potential improvement we might consider is to apply a gradient boosting model. Rather than simply building our decision trees independently in parallel, we can build them sequentially such that each tree is built on a modified version of the original data set (e.g. increasing the weights of those data points that were harder to classify and decreasing the weights on those that were easier).

Rather than just generalize to a low-volatility proxy, gradient boosting may allow our decision tree process to pick up upon greater subtleties and conditional relationships in the data. For comparison purposes, we’ll assume the same maximum tree depth and number of trees as the random forest method.

In initially evaluating the importance of features, it does appear that low volatility remains a critical factor, but other characteristics – such as momentum, free cash flow yield, and payout ratio – are close seconds. This may be a hint that gradient boosting was able to identify more subtle relationships.

Unfortunately, in evaluating the sector characteristics over time, we see a very similar pattern. Though we can notice that sectors like Technology have received a meaningfully higher allocation with this methodology versus the random forest approach.

Source: Sharadar Fundamentals. Calculations by Newfound Research.

If we compare long/short portfolios, we find little meaningful difference to our past results. Our model simply seems to identify a (historically less effective) low volatility model.

Re-Defining Defensiveness

When we set out on this problem, we made a key decision to define a stock’s defensiveness by how closely it is able to replicate the payoff of a call option on the S&P 500. What if we had elected another definition, though? For example, we could define defensive stocks as those that minimize the depth and frequency of drawdowns using a measure like the Ulcer Index.

Below we replicate the above tests but use forward 12-month Ulcer Index as our target score (or, more precisely, a security’s forward 12-month cross-sectional Ulcer Index rank).

We again begin by constructing an index that has perfect foresight, buying a market-capitalization weighted portfolio of securities that rank in the lowest quintile of forward 12-month ulcer index. We see a very different payoff profile than before, with strong performance exhibited in both bull and bear markets.

By focusing on forward 12-month scores rather than 3-month scores, we also see a far steadier sector allocation profile over time. Interestingly, we still see meaningful sector tilts over time, with sectors like Technology, Financials, and Consumer Defensives coming in and out of favor over time.

We again use a gradient boosted random forest model to try to model our target scores. We find that five of the top six most important features are price return related, either measuring return or risk.

Despite the increased emphasis on momentum, the resulting long/short index still echoes a naïve low-volatility sort. This is likely because negative momentum and high volatility have become reasonably correlated proxies for one another in recent years.

While returns appear improved from prior attempts, the out-of-sample performance (March 2009 and onward) is almost identical to that of the low-volatility long/short.

Conclusion

In this research note we sought to apply machine learning techniques to factor portfolio construction. Our goal was to exploit the ability of machine learning models to model non-linear relationships, hoping to come up with a more nuanced definition of a defensive equity portfolio.

In our first test, we defined a security’s defensiveness by how closely it was able to replicate the payoff of a call option on the S&P 500 over rolling 63-day (approximately 3-month) periods. If the market was up, we wanted to pick stocks that closely matched the market’s performance; if the market was down, we wanted to pick stocks that minimized drawdown.

After pre-engineering a set of features to capture both company and stock dynamics, we first turned to a random forest model. We chose this model as the decision tree structure would allow us to model conditional feature dynamics. By focusing on generating a large number of shallow trees we aimed to avoid overfitting while still reducing overall model variance.

Training the model on data from 1999-2009, we found that the results strongly favored companies exhibiting positive operating cash flow, positive earnings, and low realized risk characteristics (e.g. volatility and beta). Unfortunately, the model did not appear to provide any meaningful advantage versus a simple linear sort on volatility.

We then turned to applying gradient boosting to our random forest. This approach builds trees in sequence such that each tree seeks to improve upon the last. We hoped that such an approach would allow the random forest to build more nuance than simply scoring on realized volatility.

Unfortunately, the results remained largely the same.

Finally, we decided to change our definition of defensiveness by focusing on the depth and frequency of drawdowns with the Ulcer Index. Again, after re-applying the gradient boosted random forest model, we found little difference in realized results versus a simple sort on volatility (especially out-of-sample).

One answer for these similar results may be that our objective function is highly correlated to volatility measures. For example, if stocks follow a geometric Brownian motion process, those with higher levels of volatility should have deeper drawdowns. And if the best predictor of future realized volatility is past realized volatility, then it is no huge surprise that the models ultimately fell back towards a naïve volatility sort.

Interestingly, value, quality, and growth characteristics seemed largely ignored. We see two potential reasons for this.

The first possibility is that they were simply subsumed by low volatility with respect to our objective. If this were the case, however, we would see little feature importance placed upon them, but would still expect their weighted average characteristic scores within our portfolios to be higher (or lower). While this is true for select features (e.g. payout ratio), the importance of others appears largely cyclical (e.g. earnings-to-price). In fact, during the fall out of the dot-com bubble, weighted average value scores remained between 40 and 70.

The second reason is that the fundamental drivers behind each market sell-off are different. Factors tied to company metrics (e.g. valuation, quality, or growth), therefore, may be ill-suited to navigate different types of sell offs. For example, value was the natural antithesis to the speculative dot-com bubble. However, during the recent COVID-19 crisis, it has been the already richly priced technology stocks that have fared the best. Factors based upon security characteristics (e.g. volatility, returns, or volume) may be better suited to dynamically adjust to market changes.

While our results were rather lackluster, we should acknowledge that we have really only scratched the surface of machine learning techniques. Furthermore, our results are intrinsically linked to how we’ve defined our problem and the features we engineered. A more thoughtful target score or a different set of features may lead to substantially different results.

Straddles and Trend Following

By Nathan Faber

On May 11, 2020

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

This post is available as a PDF download here.

Summary

The convex payoff profile of trend following strategies naturally lends itself to comparative analysis with option strategies. Unlike options, however, the payout of trend following is not guaranteed.
To compare and contrast the two approaches, we replicate simple trend following strategies with corresponding option straddle strategies.
While trend-following has no explicit up-front cost, it also bears the full brunt of any price reversals. The straddle-based approach, on the other hand, pays an explicit cost to insure against sudden and large reversals.
This transformation of whipsaw risk into an up-front option premium can be costly during strongly trending market environments where the option buyer would have been rewarded more for setting a higher deductible for their implicit insurance policy and paying a lower premium.
From 2005-2020, avoiding this upfront premium was beneficial. The sudden loss of equity markets in March 2020, however, allowed straddle-based approaches to make up for 15-years of relative underperformance in a single month.
Whether an investor wishes to avoid these up-front costs or pay them is ultimately a function of the risks they are willing to bear. As we like to say, “risk cannot be destroyed, only transformed.”

We often repeat the mantra that, “risk cannot be destroyed, only transformed.” While not being able to destroy risk seems like a limitation, the assertion that risk can be transformed is nearly limitless.

With a wide variety of investment options, investors have the ability to mold, shape, skew, and shift their risks to fit their preferences and investing requirements (e.g. cash flows, liquidity, growth, etc.).

The payoff profile of a strategy is a key way in which this transformation of risk manifests, and the profile of trend following is one example that we have written much on historically. The convex payoff of many long/short trend following strategies is evident from the historical payoff diagram.

Source: Newfound Research. Payoff Diversification (February 10^th, 2020). Source: Kenneth French Data Library; Federal Reserve Bank of St. Louis. Calculations by Newfound Research. Returns are hypothetical and assume the reinvestment of all distributions. Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. The 60/40 portfolio is comprised of a 60% allocation to broad U.S. equities and a 40% allocation to a constant maturity 10-Year U.S. Treasury index. The momentum portfolio is rebalanced monthly and selects the asset with the highest prior 12-month returns whereas the buy-and-hold variation is allowed to drift over the 1-year period. The 10-Year U.S. Treasuries index is a constant maturity index calculated by assuming a 10-year bond is purchased at the beginning of every month and sold at the end of that month to purchase a new bond at par at the beginning of the next month. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results.

This characteristic “V” shape in the diagram is reminiscent of an option straddle, where an investor buys a put and call option of the same maturity struck at the same price. This position allows the investor to profit if the price of the underlying security moves significantly in either direction, but they pay for this opportunity in the option premiums.

Source: theoptionsguide.com

The similarity of these payoff profiles is no coincidence. As we demonstrated in Trend – Convexity and Premium (February 11^th, 2019), simple total return trend following signals coarsely approximate the delta of the straddle. For those less familiar with the parlance of options, delta is the sensitivity in the value of the options to changes in the underlying stock. For example, if delta is +1, then the value of the option position will match price changes in the underlying dollar-for-dollar. If delta is -1, then the position will lose $1 for every dollar gained in the underlying and vice versa (i.e. the position is effectively short).

How does this connection arise? Consider a naïve S&P 500 trend strategy that rebalances monthly and uses 12-month total returns as a trend signal, buying when prior returns are positive and shorting when prior returns are negative. The key components of this strategy are today’s S&P 500 level and the level 12 months ago.

Now consider a strategy that buys a 1-month straddle with a strike equal to the level of the S&P 500 12 months ago. When the current level is above the strike, the strategy’s delta will be positive and when the level is below the strike, the delta will be negative. What we can see is that the sensitivity of our options trade to changes in the S&P 500 will match the sign of the trend strategy!

There are two key differences, however. First, our trend strategy was designed to always be 100% long or 100% short, whereas the straddle’s sensitivity can vary between -100% and 100%. Second, the trend strategy cannot change its exposure intramonth whereas the straddle will. In fact, if price starts above the strike price (a positive trend) but ultimately ends below – so far as it is sufficiently far that we can make up for the premium paid for our options – the straddle can still profit!

In this commentary, we will compare and contrast the trend and option-based approaches for a variety of lookback horizons.

Methodology and Data

For this analysis, we will use the S&P 500 index for equity returns, the iShares Short-term U.S. Treasury Bond ETF (ticker: SHV) as the risk-free rate, and monthly options data on the S&P 500 (SPX options).

The long/short trend equity strategy looks at total returns of equities over a given number of months. If this return is positive, the strategy invests in equities for the following month. If the return is negative, the strategy shorts equities for the following month and earns the short-term Treasury rate on the cash. The strategy is rebalanced monthly on the third Friday of each month to coincide with the options expiration dates.

For the (semi-equivalent) straddle replication, at the end of each month we purchase a call option and a put option struck at the level of the S&P 500 at the beginning of the lookback window of the trend following strategy. We can also back out the strike price using the current trend signal value and S&P 500. For example, if the trend signal is 25% and the S&P 500 is trading at $3000, we would set the strike of the options at $2400.

The options account is assumed to be fully cash collateralized. Any premium is paid on the options roll date, interest is earned on the remaining account balance, and the option payout is realized on the next roll date.

To value the options, we employ Black-Scholes pricing on an implied volatility surface derived from available out-of-the money options. Specifically, on a given day we fit a parabola to the implied variances versus log-moneyness (i.e. log(strike/price)) of the options for each time to maturity.

In prior research, we created straddle-derived trend-following models by purchasing S&P 500 exposure in proportion to the delta of the strategy. To calculate delta, we had previously priced the options using 21-day realized volatility as a proxy for implied volatility. This generally leads to over-pricing the options during crisis times and underpricing during more tame market environments, especially for deeper out of the money puts. In this commentary we are actually purchasing the straddles and holding them for one month.

Source: Tiingo and DiscountOptionData.com. Calculations by Newfound Research.

Straddle vs. Trend Following

Below we plot the ratio of the equity curves for the straddle strategies versus their corresponding trend following strategies. When the line is increasing, the straddle strategy is out-performing, and when the line is decreasing the trend strategy is out-performing.

Source: Tiingo and DiscountOptionData.com. Calculations by Newfound Research.

We can see, generally, that trend following out-performed the explicit purchase of options for almost all lookback periods for the majority of the 15-year test period.

It is only with the most recent expiration – March 20, 2020 – that many of the straddle strategies came to out-perform their respective trend strategies. With the straddle strategy, we pay an explicit premium to help insure our position against sudden and large intra-month price reversals. This did not occur very frequently during the 15 year history, but was very valuable protection in March when the trend strategies were largely still long coming off markets hitting all-time-highs in late February.

Shorter-term lookbacks fared particularly well during that month, as the trend following strategy was in a long position on the February 2020 options expiry date, and the straddles set by the short-term lookback window were relatively cheap from a historical perspective.

Source: Tiingo and DiscountOptionData.com. Calculations by Newfound Research.

Note the curious case of the 14-month lookback. Entering March, the S&P 500 was +45% over a 14-month lookback (almost perfectly anchored to December 2018 lows). Therefore, the straddle was struck so deep in the money that it did not create any protection against the market’s sudden and large drawdown.

Prior to March 2020, only the 8- and 15-month lookback window strategies had outperformed their corresponding trend following strategies. In both cases, it was just barely and just recently.

Another interesting point to note is that longer-term straddle strategies (lookbacks greater than 9 months) shared similar movements during many periods while shorter-term lookbacks (3-6 months) showed more dispersion over time.

Overall, many of the straddles exhibit more “crisis alpha” than their trend following counterparts. This is an explicit risk we pay to hedge with the straddle approach and a fact we will discuss in more detail later on.

How Equity Movements Affect Straddles

Before we move into a discussion of how we can frame the straddle strategies, it will be helpful to revisit how straddles are affected by changing equity prices and how this effect changes with different lookback windows for the strategies.

Consider the delta of a straddle versus how far away price is from the strike (normalized by volatility).

Naively, we might consider that the longer our trend lookback window – and therefore the further back in time we set our strike price – the further away from the strike that price has had the opportunity to move. Consider two extremes: a strike set equal to the price of the S&P 500 10 years ago versus one set a day ago. We would expect that today’s price is much closer to that from a day ago than 10 years ago.

Therefore, for a longer lookback horizon we might expect that there is a greater chance that the straddle is currently deeper in the money, leading to a delta closer to +/- 1. In the case of straddles struck at index levels more recently realized, it is more likely that price is close to at the money, leading to deltas closer to 0.

This also means that while the trend following strategy is taking a binary bet, the straddle is able to modulate exposure to equity moves when the trend is less pronounced. For example, if a 12-month trend signal is +1%, the trend model will retain a +1 exposure while the delta of the straddle may be closer to 0.

Source: Tiingo and DiscountOptionData.com. Calculations by Newfound Research.

Additionally, when the delta of a straddle is closer to zero, its gamma is higher. Gamma reflects how quickly the straddle’s sensitivity to changes in the underlying asset – i.e. the delta – will change. The trend strategy has no intra-month gamma, as once the position is set it remains static until the next rebalance.

As we generally expect the straddles struck longer ago to be deeper in the money than those struck more recently, we would also expect them to have lower gamma.

This also serves to nicely connect trend speed with the length of the lookback window. Shorter lookback windows are associated with trend models that change signals more rapidly while longer lookback windows are slower. Given that a total return trend signal can be thought of as the average of daily log returns, we would expect a longer lookback to react more slowly to recent changes than a shorter lookback because the longer lookback is averaging over more data.

But if we think of it through the lens of options – that the shorter lookback is coarsely replicating the delta of a straddle struck more recently – then the ideas of speed and gamma become linked.

Source: Tiingo and DiscountOptionData.com. Calculations by Newfound Research.

The Straddle Strategy as an Insurance Policy

One of the key differences between the trend strategy and the straddle is that the straddle has features that act as insurance against price reversals. As an example, consider a case where the trend strategy has a positive signal. To first replicate the payoff, the straddle strategy buys an in-the-money call option. This is the first form of insurance, as the total amount this position can lose is the premium paid for the option, while the trend strategy can lose significantly more.

The straddle strategy goes one step further, though, and would also buy a put option. So not only does it have a fixed loss on the call if price reverses course, but it can also profit if it reverses sufficiently.

One way to model the straddle strategies, then, is as insurance policies with varying deductibles. There is an up-front premium that is paid, and the strategy does not pay out until the deductible – the distance that the option is struck in the money – is met.

When the deductible is high – that is, when the trend is very strong in either direction – the premium for the insurance policy tends to be low. On the other hand, a strategy that purchases at the money straddles would be equivalent to buying insurance with no deductible.

Source: Tiingo and DiscountOptionData.com. Calculations by Newfound Research.

On average, the 3-month straddle strategy pays annual premiums of about 14% for the benefit of only having to wait for a price reversal of 6% before protection kicks in. Toward the other end of the spectrum, the 12-month strategy has an annual average premium of under 6% with a 16% deductible.

We can also visualize how often each straddle strategy pays higher premiums by looking at the deltas of the straddles over time. When these values deviate significantly from +1 or -1, then the straddle is lowering its insurance deductible in favor of paying more in premium. When the delta is nearly +1 or -1, then the straddle is buying higher deductible insurance that will take a larger whipsaw to payout.

The charts below show the delta over time in the straddle strategies vs. the trend allocation for 3-, 6-, and 12-month lookback windows.

There is significant overlap, especially as trends get longer. The differences in the deltas in the 3-month straddle model highlight its tradeoff between lower deductibles and higher insurance premiums. However, this leads it to be more adaptive at capitalizing on equity moves in the opposite direction that lead to losses in the binary trend-following model.

Source: Tiingo and DiscountOptionData.com. Calculations by Newfound Research.

The chart below shows the annualized performance of the straddle strategies when they underperform trend following (premium) and the annualized performance of the straddle strategies when they outperform trend following (payout). As the lookback window increases, both of these figures generally decline in absolute value.

Source: Tiingo and DiscountOptionData.com. Calculations by Newfound Research.

Even though we saw previously that the 3-month straddle strategy had the highest annual premium, its overall payout when it outperforms trend following is substantial. The longer lookbacks do not provide as much of a buffer due to their higher deductible levels, despite their lower premiums.

When the naïve trend strategy is right, it captures the full price change with no up-front premium. When it is wrong, however, it bears the full brunt of losses.

With the straddle strategy, the cost is paid up front for the benefit to not only protect against price reversals, but even potentially profit from them.

As a brief aside, a simpler options strategy with similar characteristics would be to buy only either a call option or put option depending on the trend signal. This strategy would not profit from a reversion of the trend, but it would cap losses. Comparing it to the straddle strategies highlights the cost and benefit of the added protection.

Source: Tiingo and DiscountOptionData.com. Calculations by Newfound Research.

Buying only puts or calls generally helped both of the strategies shown in the chart. This came in reduced premiums over a time period when trimming premiums whenever possible paid off, especially for the 12-month lookback strategy. However, there are some notable instances where the extra protection of the straddle was very helpful, e.g. August 2011 and late 2014 for the 3-month lookback strategy and March 2020 for both.

Despite the similarities between the options and trend strategies, this difference in when the payment is made – either up-front in the straddle strategy or after-the fact in whipsaw in the trend following strategy – ends up being the key differentiator.

The relative performance of the strategies shows that investors mostly benefitted over the past 15 years by bearing this risk of whipsaw and large, sudden price-reversals. However, as the final moths of data indicates, option strategies can provide benefits that option-like strategies cannot.

Ultimately, the choice between risks is up to investor preferences, and a diversified approach that pairs strategies different convex strategies such as trend following and options is likely most appropriate.

Conclusion

The convex payoff profile of trend following strategies naturally lends itself to comparative analysis with option strategies, which also have a convex payoff profile. In fact, we would argue – as we have many times in the past – that trend following strategies coarsely replicate the delta profile of option straddles.

In this commentary, we sought to make that connection more explicit by building option straddle strategies that correspond to a naïve trend following strategies of varying lookback lengths.

While the trend following approach has no explicit up-front cost, it risks bearing the full brunt of sudden and large price reversals. With the straddle-based approach, an investor explicitly pays an up-front premium to insure against these risks.

When evaluated through the lens of an insurance policy, the straddle strategy dynamically adjusts its associated premium and deductible over time. When trends are strong, for example, premiums paid tend to be lower, but the cost is a higher deductible. Conversely, when trends are flat, the premium is much higher, but the deductible is much lower.

We found that over the 2005-2020 test period, the cost of the option premiums exceeded the cost of whipsaw in the trend strategies in almost all cases. That is, until March 2020, when a significant and sudden market reversal allowed the straddle strategies to make up for 15 years of relative losses in a single month.

As we like to say: risk cannot be destroyed, only transformed. In this case, the trend strategy was willing to bear the risk of large intra-month price reversals to avoid paying any up-front premium. This was a benefit to the trend investor for 15 years. And then it wasn’t.

By constructing straddle strategies, we believe that we can better measure the trade-offs of trend following versus the explicit cost of insurance. While trend following may approximate the profile of a straddle, it sacrifices some of the intra-month insurance qualities to avoid an up-front premium. Whether this risk trade-off is ultimately worth it depends upon the risks an investor is willing to bear.

Tranching, Trend, and Mean Reversion

By Corey Hoffstein

On April 27, 2020

In Craftsmanship, Momentum, Portfolio Construction, Weekly Commentary

This post is available as a PDF download here.

Summary

In past research we have explored the potential benefits of how-based diversification through the lens of pay-off functions.
Specifically, we explored how strategic rebalancing created a concave payoff while momentum / trend-following created a convex payoff. By combining these two approaches, total portfolio payoff became more neutral to the dispersion in return of underlying assets.
We have also spent considerable time exploring when-based diversification through our writing on rebalance timing luck.
To manage rebalance timing luck, we advocate for a tranching methodology that can be best distilled as rebalancing “a little but frequently.”
Herein, we demonstrate that the resulting payoff profile of a tranche-based rebalancing strategy closely resembles that of a portfolio that combines both strategic rebalancing and momentum/trend-following.
While we typically think of tranching as simply a way to de-emphasize the impact of a specific rebalancing date choice, this research suggests that for certain horizons, tranching may also be effective because it naturally introduces momentum/trend-following into the portfolio.

In Payoff Diversification (February 10^th, 2020), we explored the idea of combining concave and convex payoff profiles. Specifically, we demonstrated that rebalancing a strategic asset allocation was inherently concave (i.e. mean reversionary) whereas trend-following and momentum was inherently convex. By combining the two approaches together, we could neutralize the implicit payoff profile of our portfolio with respect to performance of the underlying assets.

Source: Newfound Research. Payoff Diversification (February 10^th, 2020). Source: Kenneth French Data Library; Federal Reserve Bank of St. Louis. Calculations by Newfound Research. Returns are hypothetical and assume the reinvestment of all distributions. Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. The 60/40 portfolio is comprised of a 60% allocation to broad U.S. equities and a 40% allocation to a constant maturity 10-Year U.S. Treasury index. The rebalanced variation is rebalanced at the end of each month whereas the buy-and-hold variation is allowed to drift over the 1-year period. The momentum portfolio is rebalanced monthly and selects the asset with the highest prior 12-month returns whereas the buy-and-hold variation is allowed to drift over the 1-year period. The 10-Year U.S. Treasuries index is a constant maturity index calculated by assuming a 10-year bond is purchased at the beginning of every month and sold at the end of that month to purchase a new bond at par at the beginning of the next month. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results.

The intuition behind why rebalancing is inherently mean-reversionary is fairly simple. Consider a simple 50% stock / 50% bond portfolio. Between rebalances, this allocation will drift based upon the relative performance of stocks and bonds. When we rebalance, to right-size our relative allocations we must sell the asset that has out-performed and buy the one that has under-performed. “Sell your winners and buy your losers” certainly sounds mean-reversionary to us.

In fact, one way to think about a rebalance is as the application of a long/short overlay on your portfolio. For example, if the 50/50 portfolio drifted to a 45/55, we could think about rebalancing as holding the 45/55 and overlaying it with a +5/-5 long/short portfolio. This perspective explicitly expresses the “buy the loser, short the winner” strategy. In other words, we’re actively placing a trade that benefits when future returns between the two assets reverts.

While we may not be actively trying to express a view or forecast about future returns when we rebalance, we should consider the performance implications of our choice based upon whether the relative performance of these two assets continues to expand or contract:

	Relative Performance Expands	Relative Performance Contracts
Rebalance	–	+
Do Not Rebalance	+	–

Our argument in Payoff Diversification was that by combining strategic rebalancing and momentum / trend following, we could help neutralize this implicit bet.

What we can also see in the table above, though, is that the simple act of not rebalancing benefits from a continuation of relative returns just as trend/momentum does.

Let’s keep that in the back of our minds and switch gears, for a moment, to portfolio tranching. Frequent readers of our research notes will know we have spent considerable time researching the implications of rebalance timing luck. We won’t go into great detail here, but the research can be broadly summarized as, “when you rebalance your portfolio can have meaningful implications for performance.”

Given the discussion above, why that result holds true follows naturally. If two people hold 60/40 portfolios but rebalance them at different times in the year, their results will diverge based upon the relative performance of stocks and bonds between the rebalance periods.

As a trivial example, consider two 60/40 investors who each rebalance once a year. One chooses to rebalance every March and one chooses to rebalance every September. In 2008, the September investor would have re-upped his allocation to equities only to watch them sell-off for the next six months. The March investor, on the other hand, would have rebalanced earlier that year and her equity allocation would have drifted lower as the 2008 crisis wore on.

Even better, she would rebalance in March 2009, re-upping her equity allocation near the market bottom and almost perfectly timing the performance mean-reversion that would unfold. The September investor, on the other hand, would be underweight equities due to drift at this point.

Below we plot hypothetical drifted equity allocations for these investors over time.

Source: Tiingo. Calculations by Newfound Research.

The implications are that rebalancing can imbed large, albeit unintentional, market-timing bets.

In Rebalance Timing Luck: The Difference between Hired and Fired we derived that the optimal solution for avoiding the impact of these rebalance decisions is portfolio tranching. This is the same solution proposed by Blitz, van der Grient, and van Vliet (2010).

The whole concept of tranching can be summarized with the phrase: “a little but frequently.” In other words, rebalance your portfolio more frequently, but only make small changes. As an example, rather than rebalance once a year, we could rebalance 1/12^th of our portfolio every month. If our portfolio had drifted from a 60/40 to a 55/45, rather than rebalancing all the way back, we would just correct 1/12^th of the drift, trading to a 55.42/44.58.¹

Another way to think about this approach is as a collection of sub-portfolios. For example, if we elected to implement a 12-month tranche, we might think of it as 12 separate sub-portfolios, each of which rebalances every 12 months but does so at the end of a different month (e.g. one rebalances in January, one in February, et cetera).

But why does this approach work? It helps de-emphasize the mean-reversion bet for any given rebalance date. We can see this by constructing the same payoff plots as before for different tranching speeds. The 1-month tranche reflects a full monthly rebalance; a 3-month tranche reflects rebalancing 33.33% of the portfolio; a 6-month tranche reflects rebalancing 16.66% of the portfolio each month; et cetera.

Source: Newfound Research. Payoff Diversification (February 10^th, 2020). Source: Kenneth French Data Library; Federal Reserve Bank of St. Louis. Calculations by Newfound Research. Returns are hypothetical and assume the reinvestment of all distributions. Returns are gross of all fees, including, but not limited to, management fees, transaction fees, and taxes. The 60/40 portfolio is comprised of a 60% allocation to broad U.S. equities and a 40% allocation to a constant maturity 10-Year U.S. Treasury index. The rebalanced variation is rebalanced partially at the end of each month whereas the buy-and-hold variation is allowed to drift over the 1-year period. The 10-Year U.S. Treasuries index is a constant maturity index calculated by assuming a 10-year bond is purchased at the beginning of every month and sold at the end of that month to purchase a new bond at par at the beginning of the next month. You cannot invest directly in an index and unmanaged index returns do not reflect any fees, expenses or sales charges. Past performance is not indicative of future results.

Note how the concave payoff function appears to “unbend” and the 12-month tranche appears similar in shape to payoff of the 90% strategic rebalance / 10% momentum strategy portfolio we plotted in the introduction.

Why might this be the case? Recall that not rebalancing can be effective so long as there is continuation (i.e. momentum / trend) in the relative performance between stocks and bonds. By allowing our portfolio to drift, our portfolio will naturally tilt itself towards the out-performing asset. Furthermore, drift serves as an interesting amplifier to the momentum signal: the more persistent the relative out-performance, and the larger the relative out-performance in magnitude, the greater the resulting tilt.

While tranching naturally helps reduce rebalance timing luck by de-emphasizing each specific rebalance, we can also see that we may be able to naturally embed momentum into our process.

Conclusion

In portfolio management research, the answer we find is often a reflection of the angle by which a question is asked.

For example, in prior research notes, we have spent considerable time documenting the impact of rebalance timing luck in strategic asset allocation, tactical asset allocation, and factor investing. The simple choice of when, though often overlooked in analysis, can have a significant impact upon realized results. Therefore, in order to de-emphasize the choice of when, we introduce portfolio tranching.

We have also spent a good deal of time discussing the how axis of diversification (i.e. process). Not only have we research this topic through the lens of ensemble techniques, but we have also explored it through the payoff profiles generated by each process. We find that by combining diversifying concave and convex profiles – e.g. mean-reversion and momentum – we can potentially create a return profile that is more robust to different outcomes.

Herein, we found that tranching the rebalance of a strategic asset allocation may, in fact, allow us to naturally embed momentum without having to explicitly introduce a momentum strategy. What we find, then, is that the two topics may not actually be independent avenues of research about when and how. Rather, they may just different ways of exploring how to diversify the impacts of convexity and concavity in portfolio construction.

Why Trend Models Diverge

By Corey Hoffstein

On March 9, 2020

In Risk & Style Premia, Trend, Weekly Commentary

This post is available as a PDF download here.

Summary

During the week of February 23^rd, the S&P 500 fell more than 10%.
After a prolonged bullish period in equities, this tumultuous decline caused many trend-following signals to turn negative.
As we would expect, short-term signals across a variety of models turned negative. However, we also saw that price-minus-moving-average models turned negative across a broad horizon of lookbacks where the same was not true for other models.
In this brief research note, we aim to explain why common trend-following models are actually mathematically linked to one another and differ mainly in how they place weight on recent versus prior price changes.
We find that price-minus-moving-average models place the greatest weight on the most recent price changes, whereas models like time-series momentum place equal-weight across their lookback horizon.

In a market note we sent out last weekend, the following graphic was embedded:

What this table intends to capture is the percentage of trend signals that are on for a given model and lookback horizon (i.e. speed) on U.S. equities. The point we were trying to establish was that despite a very bearish week, trend models remained largely mixed. For frequent readers of our commentaries, it should come as no surprise that we were attempting to highlight the potential specification risks of selecting just one trend model to implement with (especially when coupled with timing luck!).

But there is a potentially interesting second lesson to learn here which is a bit more academic. Why does it look like the price-minus (i.e. price-minus-moving-average) models turned off, the time series momentum models partially turned off, and the cross-over (i.e. dual-moving-average-cross) signals largely remained positive?

While this note will be short, it will be somewhat technical. Therefore, we’ll spoil the ending: these signals are all mathematically linked.

They can all be decomposed into a weighted average of prior log-returns and the primary difference between the signals is the weighting concentration. The price-minus model front-weights, the time-series model equal weights, and the cross-over model tends to back-weight (largely dependent upon the length of the two moving averages). Thus, we would expect a price-minus model to react more quickly to large, recent changes.

If you want the gist of the results, just jump to the section The Weight of Prior Evidence, which provides graphical evidence of these weighting schemes.

Before we begin, we want to acknowledge that absolutely nothing in this note is novel. We are, by in large, simply re-stating work pioneered by Bruder, Dao, Richard, and Roncalli (2011); Marshall, Nguyen and Visaltanachoti (2012); Levine and Pedersen (2015); Beekhuizen and Hallerbach (2015); and Zakamulin (2015).

Decomposing Time-Series Momentum

We will begin by decomposing a time-series momentum value, which we will define as:

We will begin with a simple substitution:

Which implies that:

Simply put, time-series momentum puts equal weight on all the past price changes¹ that occur.

Decomposing Dual-Moving-Average-Crossover

We define the dual-moving-average-crossover as:

We assume m is less than n (i.e. the first moving average is “faster” than the second). Then, re-writing:

Here, we can make a cheeky transformation where we add and subtract the current price, P_t:

What we find is that the double-moving-average-crossover value is the difference in two weighted averages of time-series momentum values.

Decomposing Price-Minus-Moving-Average

This decomposition is trivial given the dual-moving-average-crossover. Simply,

The Weight of Prior Evidence

We have now shown that these decompositions are all mathematically related. Just as importantly, we have shown that all three methods are simply re-weighting schemes of prior price changes. To gain a sense of how past returns are weighted to generate a current signal, we can plot normalized weightings for different hypothetical models.

For TSMOM, we can easily see that shorter lookback models apply more weight on less data and therefore are likely to react faster to recent price changes.
PMAC models apply weight in a linear, declining fashion, with the most weight applied to the most recent price changes. What is interesting is that PMAC(50) puts far more weight on recent prices changes than the TSMOM(50) model does. For equivalent lookback periods, then, we would expect PMAC to react much more quickly. This is precisely why we saw PMAC models turn off in the most recent sell-off when other models did not: they are much more front-weighted.
DMAC models create a hump-shaped weighting profile, with increasing weight applied up until the length of the shorter lookback period, and then descending weight thereafter. If we wanted to, we could even create a back-weighted model, as we have with the DMAC(150, 200) example. In practice, it is common to see that m is approximately equal to n/4 (e.g. DMAC(50, 200)). Such a model underweights the most recent information relative to slightly less recent information.

Conclusion

In this brief research note, we demonstrated that common trend-following signals – namely time-series momentum, price-minus-moving-average, and dual-moving-average-crossover – are mathematically linked to one another. We find that prior price changes are the building blocks of each signal, with the primary differences being how those prior price changes are weighted.

Time-series momentum signals equally-weight prior price changes; price-minus-moving-average models tend to forward-weight prior price changes; and dual-moving-average-crossovers create a hump-like weighting function. The choice of which model to employ, then, expresses a view as to the relative importance we want to place on recent versus past price changes.

These results align with the trend signal changes seen over the past week during the rapid sell-off in the S&P 500. Price-minus-moving-average models appeared to turn negative much faster than time-series momentum or dual-moving-average-crossover signals.

By decomposing these models into their most basic and shared form, we again highlight the potential specification risks that can arise from electing to employ just one model. This is particularly true if an investor selects just one of these models without realizing the implicit choice they have made about the relative importance they would like to place on recent versus past returns.