Category: Momentum Page 3 of 4

Tactical Credit

On June 3, 2019

In Credit, Momentum, Popular, Risk & Style Premia, Weekly Commentary

This post is available as a PDF download here.

Summary

In this commentary we explore tactical credit strategies that switch between high yield bonds and core fixed income exposures.
We find that short-term momentum signals generate statistically significant annualized excess returns.
We use a cross-section of statistically significant strategy parameterizations to generate an ensemble strategy.Consistent with past research, we find that this ensemble approach helps reduce idiosyncratic specification risk and dramatically increases the strategy’s information ratio above the median underlying strategy information ratio.
To gain a better understanding of the strategy, we attempt to determine the source of strategy returns. We find that a significant proportion of returns are generated as price returns occurring during periods when credit spreads are above their median value and are expanding.
Excluding the 2000-2003 and 2008-2009 sub-periods reduces gross-of-cost strategy returns from 2.9% to 1.5%, bringing into question how effective post-of-cost implementation can be if we do not necessarily expect another crisis period to unfold.

There is a certain class of strategies we get asked about quite frequently but have never written much on: tactical credit.

The signals driving these strategies can vary significantly (including momentum, valuation, carry, macro-economic, et cetera) and implementation can range from individual bonds to broad index exposure to credit default swaps. The simplest approach we see, however, are high yield switching strategies. The strategies typically allocate between high yield corporate bonds and core fixed income (or short-to-medium-term U.S. Treasuries) predominately based upon some sort of momentum-driven signal.

It is easy to see why this seemingly naïve approach has been attractive. Implementing a simple rotation between –high-yield corporates– and –core U.S. fixed income– with a 3-month lookback with 1-month hold creates a fairly attractive looking –tactical credit– strategy.

Source: Tiingo. Calculations by Newfound Research. Tactical Credit strategy returns are hypothetical and backtested. Returns gross of all management fees and taxes, but net of underlying fund fees. HY Corporates represents the Vanguard High-Yield Corporate Fund (VWEHX). Core Bonds is represented by the Vanguard Total. Bond Market Index Fund (VBMFX). Returns assume the reinvestment of all distributions.

Visualizing the ratio of the equity curves over time, we see a return profile that is reminiscent of past writings on tactical and trend equity strategies. The tactical credit strategy tends to outperform core bonds during most periods, with the exception of periods of economic stress (e.g. 2000-2002 or 2008). On the other hand, the tactical credit strategy tends to underperform high yield corporates in most environments, but has historically added significant value in those same periods of economic stress.

This is akin to tactical equity strategies, which have historically out-performed the safety asset (e.g. cash) during periods of equity market tailwinds, but under-performed buy-and-hold equity during those periods due to switching costs and whipsaw. As the most aggressive stance the tactical credit strategy can take is a 100% position in high yield corporates, it would be unrealistic for us to expect such a strategy to out-perform in an environment that is conducive to strong high yield performance.

What makes this strategy different than tactical equity, however, is that the vast majority of total return in these asset classes comes from income rather than growth. In fact, since the 1990s, the price return of high yield bonds has annualized at -0.8%. This loss reflects defaults occurring within the portfolio offset by recovery rates.¹

This is potentially problematic for a tactical strategy as it implies a significant potential opportunity cost of switching out of high yield. However, we can also see that the price return is volatile. In years like 2008, the price return was -27%, more than offsetting the 7%+ yield you would have achieved just holding the fund.

Source: Tiingo. Calculations by Newfound Research. Returns gross of all management fees and taxes, but net of underlying fund fees.

Like trend equity, we can think of this tactical credit strategy as being a combination of two portfolios:

A fixed-mix of 50% high yield corporates and 50% core bonds; and
50% exposure to a dollar-neutral long/short portfolio that captures the tactical bet.

For example, when the tactical credit portfolio is 100% in high yield corporates, we can think of this as being a 50/50 strategy portfolio with a 50% overlay that is 100% long high yield corporates and 100% short core bonds, leading to a net exposure that is 100% long high yield corporates.

Thinking in this manner allows us to isolate the active returns of the portfolio actually being generated by the tactical signals and determine value-add beyond a diversified buy-and-hold core. Thus, for the remainder of this commentary we will focus our exploration on the long/short component.

Before we go any further, we do want to address that a naïve comparison between high yield corporates and core fixed income may be plagued by changing composition in the underlying portfolios as well as unintended bets. For example, without specifically duration matching the legs of the portfolio, it is likely that a dollar-neutral long/short portfolio will have residual interest rate exposure and will not represent an isolated credit bet. Thus, naïve total return comparisons will capture both interest rate and credit-driven effects.

This is further complicated by the fact that sensitivity to these factors will change over time due both to the math of fixed income (e.g. interest rate sensitivity changing over time due to higher order effects like convexity) as well as changes in the underlying portfolio composition. If we are not going to specifically measure and hedge out these unintended bets, we will likely want to rely on faster signals such that the bet our portfolio was attempting to capture is no longer reflected by the holdings.

We will begin by first evaluating the stability of our momentum signals. We do this by varying formation period (i.e. lookback) and holding period of our momentum rotation strategy and calculating the corresponding t-statistic of the equity curve’s returns. We plot the t-statistics below and specifically highlight those regions were t-statistics exceed 2, a common threshold for significance.

Source: Tiingo. Calculations by Newfound Research.

It should be noted that data for this study only goes back to 1990, so achieving statistical significance is more difficult as the sample size is significantly reduced. Nevertheless, unlike trend equity which tends to exhibit strong significance across formation periods ranging 6-to-18 months, we see a much more limited region with tactical credit. Only formation periods from 3-to-5 months appear significant, and only with holding periods where the total period (formation plus holding period) is less than 6-months.

Note that our original choice of 63-day (approximately 3 months) formation and 21-day (approximately 1 month) hold falls within this region.

We can also see that very short formation and holding period combinations (e.g. less than one month) also appear significant. This may be due to the design of our test. To achieve the longest history for this study, we employed mutual funds. However, mutual funds holding less liquid underlying securities tend to exhibit positive autocorrelation. While we adjusted realized volatility levels for this autocorrelation effect in an effort to create more realistic t-statistics, it is likely that positive results in this hyper short-term region emerge from this effect.

Finally, we can see another rather robust region representing the same formation period of 3-to-5 months, but a much longer holding length of 10-to-12 months. For the remainder of this commentary, we’ll ignore this region, though it warrants further study.

Assuming formation and holding periods going to a daily granularity, the left-most region represents over 1,800 possible strategy combinations. Without any particular reason for choosing one over another, we will embrace an ensemble approach, calculating the target weights for all possible combinations and averaging them together in a virtual portfolio-of-portfolios configuration.

Below we plot the long/short allocations as well as the equity curve for the ensemble long/short tactical credit strategy.

Note that each leg of the long/short portfolio does not necessarily equal 100% notional. This reflects conflicting signals in the underlying portfolios, causing the ensemble strategy to reduce its gross allocation as a reflection of uncertainty.

As a quick aside, we do want to highlight how the performance of the ensemble compares to the performance of the underlying strategies.

Below we plot the annualized return, annualized volatility, maximum drawdown, and information ratio of all the underlying equity curves of the strategies that make up the ensemble. We also identify the –ensemble approach–. While we can see that the ensemble approach brings the annualized return in-line with the median annualized return, its annualized volatility is in the 14^thpercentile and its maximum drawdown is in the 8^thpercentile.

By maintaining the median annualized return and significantly reducing annualized volatility, the ensemble has an information ratio in the 78^thpercentile. As we’ve demonstrated in prior commentaries, by diversifying idiosyncratic specification risk, the ensemble approach is able to generate an information ratio significantly higher than the median without having to explicitly choose which specification we believe will necessarily outperform.

Given this ensemble implementation, we can now ask, “what is the driving force of strategy returns?” In other words, does the strategy create returns by harvesting price return differences or through carry (yield) differences?

One simple way of evaluating this question is by evaluating the strategy’s sensitivity to changes in credit spreads. Specifically, we can calculate daily changes in the ICE BofAML US High Yield Master II Option-Adjusted Spread and multiply it against the strategy’s exposure to high yield bonds on the prior day.

By accumulating these weighted changes over time, we can determine how much spread change the strategy has captured. We can break this down further by isolating positive and negative change days and trying to figure out whether the strategy has benefited from avoiding spread expansion or from harvesting spread contraction.

In the graph below, we can see that the strategy harvested approximately 35,000 basis points (“bps”) from 12/1996 to present (the period for which credit spread data was available). Point-to-point, credit spreads actually widened by 100bps over the period, indicating that tactical changes were able to harvest significant changes in spreads.

Source: St. Louis Federal Reserve. Calculations by Newfound Research.

We can see that over the full period, the strategy predominately benefited from harvesting contracting spreads, as exposure to expanding spreads had a cumulative net zero impact. This analysis is incredibly regime dependent, however, and we can see that periods like 2000-2003 and 2008 saw a large benefit from short-exposure in high yield during a period when spreads were expanding.

We can even see that in the case of post-2008, switching to long high yield exposure allowed the strategy to benefit from subsequent credit spread declines.

While this analysis provides some indication that the strategy benefits from harvesting credit spread changes, we can dig deeper by taking a regime-dependent view of performance. Specifically, we can look at strategy returns conditional upon whether spreads are above or below their long-term median, as well as whether they expand or contract in a given month.

Source: St. Louis Federal Reserve. Calculations by Newfound Research. Tactical Credit strategy returns are hypothetical and backtested. Returns gross of all management fees and taxes, but net of underlying fund fees. Returns assume the reinvestment of all distributions.

Most of the strategy return appears to occur during times when spreads are above their long-term median. Calculating regime-conditional annualized returns confirms this view.

	Above	Below
Expanding	10.88%	-2.79%
Contracting	1.59%	4.22%

The strategy appears to perform best during periods when credit spreads are expanding above their long-term median level (e.g. crisis periods like 2008). The strategy appears to do its worst when spreads are below their median and begin to expand, likely representing periods when the strategy is generally long high yield but has not had a chance to make a tactical switch.

This all points to the fact that the strategy harvests almost all of its returns in crisis periods. In fact, if we remove 2000-2003 and 2008-2009, we can see that the captured credit spread declines dramatically.

Source: St. Louis Federal Reserve. Calculations by Newfound Research.

Capturing price returns due to changes in credit spreads are not responsible for all of the strategy’s returns, however.

Below we explicitly calculate the yield generated by the long/short strategy over time. As high yield corporates tend to offer higher yields, when the strategy is net long high yield, the strategy’s yield is positive. On the other hand, when the strategy is net short high yield, the strategy’s yield is negative.

This is consistent with our initial view about why these sorts of tactical strategies can be so difficult. During the latter stages of the 2008 crisis, the long/short strategy had a net negative yield of close to -0.5% per month.² Thus, the cost of carrying this tactical position is rather expensive and places a larger burden on the strategy accurately timing price return.

Source: Tiingo. Calculations by Newfound Research. Tactical Credit strategy returns are hypothetical and backtested. Returns gross of all management fees and taxes, but net of underlying fund fees.

From this graph, we believe there are two interesting things worth calling out:

The long-run average yield is positive, representing the strategy’s ability to capture carry differences between high yield and core bonds.
In the post-crisis environments, the strategy generates yields in excess of one standard deviation of the full-period sample, indicating that the strategy may have benefited from allocating to high yield when yields were abnormally large.

To better determine whether capturing changes in credit spreads or carry differences had a larger impact on strategy returns, we can explicitly calculate the –price– and –total return– indices of the ensemble strategy.

Source: St. Louis Federal Reserve. Calculations by Newfound Research. Tactical Credit strategy returns are hypothetical and backtested. Returns gross of all management fees and taxes, but net of underlying fund fees. Total return series assumes the reinvestment of all distributions.

The –price return– and –total return– series return 2.1% and 2.9% annualized respectively, implying that capturing price return effects account for approximately 75% of the strategy’s total return.

This is potentially concerning, because we have seen that the majority of the price return comes from a single regime: when credit spreads are above their long-term median and expanding. As we further saw, simply removing the 2000-2003 and 2008-2009 periods significantly reduced the strategy’s ability to harvest these credit spread changes.

While the strategy may appear to be supported by nearly 30-years of empirical evidence, in reality we have a situation where the vast majority of the strategy’s returns were generated in just two regimes.

If we remove 2000-2003 and 2008-2009 from the return series, however, we can see that the total return of the strategy only falls to 0.7% and. 1.6% annualized for –price return– and –total return– respectively. While this may appear to be a precipitous decline, it indicates that there may be potential to capture both changes in credit spread and net carry differences even in normal market environments so long as implementation costs are kept low enough.

Conclusion

In this commentary, we explored a tactical credit strategy that switched between high yield corporate bonds and core fixed income. We decompose these strategies into a 50% high yield / 50% core fixed income portfolio that is overlaid with 50% exposure to a dollar-neutral long/short strategy that captures the tactical tilts. We focus our exploration on the dollar-neutral long/short portfolio, as it isolates the active bets of the strategy.

Using cross-sectional momentum, we found that short-term signals with formation periods ranging from 3-to-5 months were statistically significant, so long as the holding period was sufficiently short.

We used this information to construct an ensemble strategy made out of more than 1,800 underlying strategy specifications. Consistent with past research, we found that the ensemble closely tracked the median annualized return of the underlying strategies, but had significantly lower volatility and maximum drawdown, leading to a higher information ratio.

We then attempted to deconstruct where the strategy generated its returns from. We found that a significant proportion of total returns were achieved during periods when credit spreads were above their long-term median and expanding. This is consistent with periods of economic volatility such as 2000-2003 and 2008-2009.

The strategy also benefited from harvesting net carry differences between high yield and core fixed income. Explicitly calculating strategy price and total return, we find that this carry component accounts for approximately 25% of strategy returns.

The impact of the 2000-2003 and 2008-2009 periods on strategy returns should not be understated. Removing these time periods reduced strategy returns from 2.9% to 1.6% annualized. Interestingly, however, the proportion of total return explained by net carry only increased from 25% to 50%, potentially indicating that the strategy was still able to harvest some opportunities in changing credit spreads.

For investors evaluating these types of strategies, cost will be an important component. While environments like 2008 may lead to opportunities for significant out-performance, without them the strategy may offer anemic returns. This is especially true when we recall that a long-only implementation only has 50% implicit exposure to the long/short strategy we evaluated in this piece.

Thus, the 2.9% annualized return is really closer to a 1.5% annualized excess return above the 50/50 portfolio. For the ex-crisis periods, the number is closer to 0.8% annualized. When we consider that this analysis was done without explicit consideration for management costs or trading costs and we have yet to apply an appropriate expectation haircut given the fact that this analysis was all backtested, there may not be sufficient juice to squeeze.

That said, we only evaluated a single signal in this piece. Combining momentum with valuation, carry, or even macro-economic signals may lead to significantly better performance. Further, high yield corporates is a space where empirical evidence suggests that security selection can make a large difference. Careful selection of funds may lead to meaningfully better performance than just broad asset class exposure.

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.

Momentum’s Magic Number

By Corey Hoffstein

On July 15, 2018

In Momentum, Risk & Style Premia, Weekly Commentary

This post is available as a PDF download here.

Summary

In HIMCO’s May 2018 Quantitative Insight, they publish a figure that suggests the optimal holding length of a momentum strategy is a function of the formation period.
Specifically, the result suggests that the optimal holding period is one selected such that the formation period plus the holding period is equal to 14-to-18 months: a somewhat “magic” result that makes little intuitive, statistical, or economic sense.
To investigate this result, we construct momentum strategies for country indices as well as industry groups.
We find similar results, with performance peaking when the formation period plus the holding period is equal to 12-to-14 months.
While lacking a specific reason why this effect exists, it suggests that investors looking to leverage shorter-term momentum signals may benefit from longer investment horizons, particularly when costs are considered.

A few weeks ago, we came across a study published by HIMCO on momentum investing¹. Contained within this research note was a particularly intriguing exhibit.

Source: HIMCO Quantitative Insights, May 2018

What this figure demonstrates is that the excess cumulative return for U.S. equity momentum strategies peaks as a function of both formation period and holding period. Specifically, the returns appear to peak when the sum of the formation and holding period is between 14-18 months.

For example, if you were to form a portfolio based upon trailing 6-1 momentum – i.e. ranking on the prior 6-month total returns and skipping the most recent month (labeled in the figure above as “2_6”) – this evidence suggests that you would want to hold such a portfolio for 8-to-12 months (labeled in the figure above as 14-to-18 months since the beginning of the uptrend).

Which is a rather odd conclusion. Firstly, we would intuitively expect that we should employ holding periods that are shorter than our formation periods. The notion here is that we want to use enough data to harvest information that will be stationary over the next, smaller time-step. So, for example, we might use 36 months of returns to create a covariance matrix that we might hold constant for the next month (i.e. a 36-month formation period with a 1-month hold). Given that correlations are non-stable, we would likely find the idea of using 1-month of data to form a correlation matrix we hold for the next 36-months rather ludicrous.

And, yet, here we are in a similar situation, finding that if we use a formation period of 5 months, we should hold our portfolio steady for the next 8-to-10 months. And this is particularly weird in the world of momentum, which we typically expect to be a high turnover strategy. How in the world can having a holding period longer than our formation period make sense when we expect information to quickly decay in value?

Perhaps the oddest thing of all is the fact that all these results center around 14-18 months. It would be one thing if the conclusion was simply, “holding for six months after formation is optimal”; here the conclusion is that the optimal holding period is a function of formation period. Nor is the conclusion something intuitive, like “the holding period should be half the formation period.”

Rather, the result – that the holding period should be 14-to-18 months minus the length of the formation period – makes little intuitive, statistical, or economic sense.

Out-of-Sample Testing with Countries and Sectors

In effort to explore this result further, we wanted to determine whether similar results were found when cross-sectional momentum was applied to country indices and industry groups.

Specifically, we ran three tests.

In the first, we constructed momentum portfolios using developed country index returns (U.S. dollar denominated; net of withholding taxes) from MSCI. The countries included in the test are: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Hong Kong, Ireland, Israel, Italy, Japan, Netherlands, New Zealand, Norway, Portugal, Singapore, Spain, Sweden, Switzerland, the United Kingdom, and the United States of America. The data extends back to 12/1969.

In the second, we constructed momentum portfolios using the 12 industry group data set from the Kenneth French Data Library. The data extends back to 7/1926.

In the third, we constructed momentum portfolios using the 49 industry group data set from the Kenneth French Data Library. The data extends back to 7/1926.

For each data set, we ran the same test:

Vary formation periods from 5-1 to 12-1 months.
Vary holding periods from 1-to-26 months.
Using this data, construct dollar-neutral long/short portfolios that go long, in equal-weight, the top third ranking holdings and go short, in equal-weight, the bottom third.

Note that for holding periods exceeding 1 month, we employed an overlapping portfolio construction process.

Below we plot the results.

Source: MSCI and Kenneth French Data Library. Calculations by Newfound Research. Past performance is not a predictor of future results. All information is backtested and hypothetical and does not reflect the actual strategy managed by Newfound Research. Performance is net of all fees except for underlying ETF expense ratios. Returns assume the reinvestment of all dividends, capital gains, and other earnings.

While the results are not as clear as those published by HIMCO, we still see an intriguing effect: returns peak as a function of both formation and holding period. For the country strategy, formation and holding appear to peak between 12-14 months, indicating that an investor using 5-1 month signals would want to hold for 7 months while an investor using 12-1 signals would only want to hold for 1 month.

For the industry data, the results are less clear. Where the HIMCO and country results exhibited a clear “peak,” the industry results simply seem to “decay slower.” In particular, we can see in the results for the 12-industry group test that almost all strategies peak with a 1-month holding period. However, they all appear to fall off rapidly, and uniformly, after the time where formation plus holding period exceeds 16 months.

While less pronounced, it is worth pointing out that this result is achieved without the consideration of trading costs or taxes. So, while the 5-1 strategy 12-industry group strategy return may peak with a 1-month hold, we can see that it later forms a second peak at a 9-month hold (“14 months since beginning uptrend”). Given that we would expect a nine month hold to exhibit considerably less trading, analysis that includes trading cost estimates may exhibit even greater peakedness in the results.

Does the Effect Persist for Long-Only Portfolios?

In analyzing factors, it is often important to try to determine whether a given result is arising from an effect found in the long leg or the short leg. After all, most investors implement strategies in a long-only capacity. While long-only strategies are, technically, equal to a benchmark plus a dollar-neutral long/short portfolio², the long/short portfolio rarely reflects the true factor definition.

Therefore, we want to evaluate long-only construction to determine whether the same result holds, or whether it is a feature of the short-leg.

We find incredibly similar results. Again, country indices appear to peak between 12-to-14 months after the beginning of the uptrend. Industry group results, while not as strong as country results, still appear to offer fairly flat results until 12-to-14 months after the beginning of the uptrend. Taken together, it appears that this result is sustained for long-only portfolio implementations as well.

Conclusion

Traditionally, momentum is considered a high turnover factor. Relative ranking of recent returns can vary substantially over time and our intuition would lead us to expect that the shorter the horizon we use to measure returns, the shorter the time we expect the relative ranking to persist.

Yet recent research published by HIMCO finds this intuition may not be true. Rather, they find that momentum portfolio performance tends to peak 14-to-18 months after the beginning of the uptrend in measured. In other words, a portfolio formed on prior 5-month returns should hold between 9-to-13 months, while a portfolio formed on the prior 12-months of returns should only hold 2-to-6 months.

This result is rather counter-intuitive, as we would expect that shorter formation periods would require shorter holding periods.

We test this result out-of-sample, constructing momentum portfolios using country indices, 12-industry group indices, and 49-industry group indices. We find a similar result in this data. We then further test whether the result is an artifact found in only long/short implementations whether this information is useful for long-only investors. Indeed, we find very similar results for long-only implementations.

Precisely why this result exists is still up in the air. One argument may be that the trade-off is ultimately centered around win rate versus the size of winners. If relative momentum tends to persist for only for 12-to-18 months total, then using 12-month formation may give us a higher win rate but reduce the size of the winners we pick. Conversely, using a shorter formation period may reduce the number of winners we pick correctly (i.e. lower win rate), but those we pick have further to run. Selecting a formation period and a holding period such that their sum equals approximately 14 months may simply be a hack to find the balance of win rate and win size that maximizes return.

Factor Fimbulwinter

By Corey Hoffstein

On June 11, 2018

In Carry, Defensive, Momentum, Popular, Risk & Style Premia, Trend, Value, Weekly Commentary

This post is available as a PDF download here.

Summary

Value investing continues to experience a trough of sorrow. In particular, the traditional price-to-book factor has failed to establish new highs since December 2006 and sits in a 25% drawdown.
While price-to-book has been the academic measure of choice for 25+ years, many practitioners have begun to question its value (pun intended).
We have also witnessed the turning of the tides against the size premium, with many practitioners no longer considering it to be a valid stand-alone anomaly. This comes 35+ years after being first published.
With this in mind, we explore the evidence that would be required for us to dismiss other, already established anomalies. Using past returns to establish prior beliefs, we simulate out forward environments and use Bayesian inference to adjust our beliefs over time, recording how long it would take for us to finally dismiss a factor.
We find that for most factors, we would have to live through several careers to finally witness enough evidence to dismiss them outright.
Thus, while factors may be established upon a foundation of evidence, their forward use requires a bit of faith.

In Norse mythology, Fimbulvetr (commonly referred to in English as “Fimbulwinter”) is a great and seemingly never-ending winter. It continues for three seasons – long, horribly cold years that stretch on longer than normal – with no intervening summers. It is a time of bitterly cold, sunless days where hope is abandoned and discord reigns.

This winter-to-end-all-winters is eventually punctuated by Ragnarok, a series of events leading up to a great battle that results in the ultimate death of the major gods, destruction of the cosmos, and subsequent rebirth of the world.

Investment mythology is littered with Ragnarok-styled blow-ups and we often assume the failure of a strategy will manifest as sudden catastrophe. In most cases, however, failure may more likely resemble Fimbulwinter: a seemingly never-ending winter in performance with returns blown to-and-fro by the harsh winds of randomness.

Value investors can attest to this. In particular, the disciples of price-to-book have suffered greatly as of late, with “expensive” stocks having outperformed “cheap” stocks for over a decade. The academic interpretation of the factor sits nearly 25% belowits prior high-water mark seen in December 2006.

Expectedly, a large number of articles have been written about the death of the value factor. Some question the factor itself, while others simply argue that price-to-book is a broken implementation.

But are these simply retrospective narratives, driven by a desire to have an explanation for a result that has defied our expectations? Consider: if price-to-book had exhibited positive returns over the last decade, would we be hearing from nearly as large a number of investors explaining why it is no longer a relevant metric?

To be clear, we believe that many of the arguments proposed for why price-to-book is no longer a relevant metric are quite sound. The team at O’Shaughnessy Asset Management, for example, wrote a particularly compelling piece that explores how changes to accounting rules have led book value to become a less relevant metric in recent decades.¹

Nevertheless, we think it is worth taking a step back, considering an alternate course of history, and asking ourselves how it would impact our current thinking. Often, we look back on history as if it were the obvious course. “If only we had better prior information,” we say to ourselves, “we would have predicted the path!”² Rather, we find it more useful to look at the past as just one realized path of many that’s that could have happened, none of which were preordained. Randomness happens.

With this line of thinking, the poor performance of price-to-book can just as easily be explained by a poor roll of the dice as it can be by a fundamental break in applicability. In fact, we see several potential truths based upon performance over the last decade:

This is all normal course performance variance for the factor.
The value factor works, but the price-to-book measure itself is broken.
The price-to-book measure is over-crowded in use, and thus the “troughs of sorrow” will need to be deeper than ever to get weak hands to fold and pass the alpha to those with the fortitude to hold.
The value factor never existed in the first place; it was an unfortunate false positive that saturated the investing literature and broad narrative.

The problem at hand is two-fold: (1) the statistical evidence supporting most factors is considerable and (2) the decade-to-decade variance in factor performance is substantial. Taken together, you run into a situation where a mere decade of underperformance likely cannot undue the previously established significance. Just as frustrating is the opposite scenario. Consider that these two statements are not mutually exclusive: (1) price-to-book is broken, and (2) price-to-book generates positive excess return over the next decade.

In investing, factor return variance is large enough that the proof is not in the eating of the short-term return pudding.

The small-cap premium is an excellent example of the difficulty in discerning, in real time, the integrity of an established factor. The anomaly has failed to establish a meaningful new high since it was originally published in 1981. Only in the last decade – nearly 30 years later – have the tides of the industry finally seemed to turn against it as an established anomaly and potential source of excess return.

Thirty years.

The remaining broadly accepted factors – e.g. value, momentum, carry, defensive, and trend – have all been demonstrated to generate excess risk-adjusted returns across a variety of economic regimes, geographies, and asset classes, creating a great depth of evidence supporting their existence. What evidence, then, would make us abandon faith from the Church of Factors?

To explore this question, we ran a simple experiment for each factor. Our goal was to estimate how long it would take to determine that a factor was no longer statistically significant.

Our assumption is that the salient features of each factor’s return pattern will remain the same (i.e. autocorrelation, conditional heteroskedasticity, skewness, kurtosis, et cetera), but the forward average annualized return will be zero since the factor no longer “works.”

Towards this end, we ran the following experiment:

Take the full history for the factor and calculate prior estimates for mean annualized return and standard error of the mean.
De-mean the time-series.
Randomly select a 12-month chunk of returns from the time series and use the data to perform a Bayesian update to our mean annualized return.
Repeat step 3 until the annualized return is no longer statistically non-zero at a 99% confidence threshold.

For each factor, we ran this test 10,000 times, creating a distribution that tells us how many years into the future we would have to wait until we were certain, from a statistical perspective, that the factor is no longer significant.

Sixty-seven years.

Based upon this experience, sixty-seven years is median number of years we will have to wait until we officially declare price-to-book (“HML,” as it is known in the literature) to be dead.³ At the risk of being morbid, we’re far more likely to die before the industry finally sticks a fork in price-to-book.

We perform this experiment for a number of other factors – including size (“SMB” – “small-minus-big”), quality (“QMJ” – “quality-minus-junk”), low-volatility (“BAB” – “betting-against-beta”), and momentum (“UMD” – “up-minus-down”) – and see much the same result. It will take decades before sufficient evidence mounts to dethrone these factors.

	HML	SMB⁴	QMJ	BAB	UMD
Median Years-until-Failure	67	43	132	284	339

Now, it is worth pointing out that these figures for a factor like momentum (“UMD”) might be a bit skewed due to the design of the test. If we examine the long-run returns, we see a fairly docile return profile punctuated by sudden and significant drawdowns (often called “momentum crashes”).

Since a large proportion of the cumulative losses are contained in these short but pronounced drawdown periods, demeaning the time-series ultimately means that the majority of 12-month periods actually exhibit positive returns. In other words, by selecting random 12-month samples, we actually expect a high frequency of those samples to have a positive return.

For example, using this process, 49.1%, 47.6%, 46.7%, 48.8% of rolling 12-month periods are positive for HML, SMB, QMJ, and BAB factors respectively. For UMD, that number is 54.7%. Furthermore, if you drop the worst 5% of rolling 12-month periods for UMD, the average positive period is 1.4x larger than the average negative period. Taken together, not only are you more likely to select a positive 12-month period, but those positive periods are, on average, 1.4x larger than the negative periods you will pick, except for the rare (<5%) cases.

The process of the test was selected to incorporate the salient features of each factor. However, in the case of momentum, it may lead to somewhat outlandish results.

Conclusion

While an evidence-based investor should be swayed by the weight of the data, the simple fact is that most factors are so well established that the majority of current practitioners will likely go our entire careers without experiencing evidence substantial enough to dismiss any of the anomalies.

Therefore, in many ways, there is a certain faith required to use them going forward. Yes, these are ideas and concepts derived from the data. Yes, we have done our best to test their robustness out-of-sample across time, geographies, and asset classes. Yet we must also admit that there is a non-zero probability, however small it is, that these are false positives: a fact we may not have sufficient evidence to address until several decades hence.

And so a bit of humility is warranted. Factors will not suddenly stand up and declare themselves broken. And those that are broken will still appear to work from time-to-time.

Indeed, the death of a factor will be more Fimulwinter than Ragnarok: not so violent to be the end of days, but enough to cause pain and frustration among investors.

Addendum

We have received a large number of inbound notes about this commentary, which fall upon two primary lines of questions. We want to address these points.

How were the tests impacted by the Bayesian inference process?

The results of the tests within this commentary are rather astounding. We did seek to address some of the potential flaws of the methodology we employed, but by-in-large we feel the overarching conclusion remains on a solid foundation.

While we only presented the results of the Bayesian inference approach in this commentary, as a check we actually tested two other approaches:

A Bayesian inference approach assuming that forward returns would be a random walk with constant variance (based upon historical variance) and zero mean.
Forward returns were simulated using the same bootstrap approach, but the factor was being discovered for the first time and the entire history was being evaluated for its significance.

The two tests were in effort to isolate the effects of the different components of our test.

What we found was that while the reported figures changed, the overall magnitude did not. In other words, the median death-date of HML may not have been 67 years, but the order of magnitude remained much the same: decades.

Stepping back, these results were somewhat a foregone conclusion. We would not expect an effect that has been determined to be statistically significant over a hundred year period to unravel in a few years. Furthermore, we would expect a number of scenarios that continue to bolster the statistical strength just due to randomness alone.

Why are we defending price-to-book?

The point of this commentary was not to defend price-to-book as a measure. Rather, it was to bring up a larger point.

As a community, quantitative investors often leverage statistical significance as a defense for the way we invest.

We think that is a good thing. We should look at the weight of the evidence. We should be data driven. We should try to find ideas that have proven to be robust over decades of time and when applied in different markets or with different asset classes. We should want to find strategies that are robust to small changes in parameterization.

Many quants would argue (including us among them), however, that there also needs to be a why. Why does this factor work? Without the why, we run the risk of glorified data mining. With the why, we can choose for ourselves whether we believe the effect will continue going forward.

Of course, there is nothing that prevents the why from being pure narrative fallacy. Perhaps we have simply weaved a story into a pattern of facts.

With price-to-book, one might argue we have done the exact opposite. The effect, technically, remains statistically significant and yet plenty of ink has been spilled as to why it shouldn’t work in the future.

The question we must answer, then, is, “when does statistically significant apply and when does it not?” How can we use it as a justification in one place and completely ignore it in others?

Furthermore, if we are going to rely on hundreds of years of data to establish significance, how can we determine when something is “broken” if the statistical evidence does not support it?

Price-to-book may very well be broken. But that is not the point of this commentary. The point is simply that the same tools we use to establish and defend factors may prevent us from tearing them down.

Two Centuries of Momentum

By Corey Hoffstein

On March 23, 2018

This post is available as a PDF download here.

A momentum-based investing approach can be confusing to investors who are often told that “chasing performance” is a massive mistake and “timing the market” is impossible.

Yet as a systematized strategy, momentum sits upon nearly a quarter century of positive academic evidence and a century of successful empirical results.

Our firm, Newfound Research, was founded in August 2008 to offer research derived from our volatility-adjusted momentum models. Today, we provide tactically risk-managed investment portfolios using those same models.

Momentum, and particularly time-series momentum, has been in our DNA since day one.

In this Foundational Series piece, we want to explore momentum’s rich history and the academic evidence demonstrating its robustness across asset classes, geographies, and market cycles.

1. What is momentum?

Momentum is a system of investing that buys and sells based upon recent returns. Momentum investors buy outperforming securities and avoid – or sell short – underperforming ones.

The notion is closely tied to physics. In physics, momentum is the product of the mass and velocity of an object. For example, a heavy truck moving at a high speed has large momentum. To stop the truck, we must apply either a large or a prolonged force against it.

Momentum investors apply a similar notion. They assume outperforming securities will continue to outperform in absence of significant headwinds.

2. The Two Faces & Many Names of Momentum

2.1 Relative Momentum

The phenomenon of relative momentum is also called cross-sectional momentum and relative strength.

Relative momentum investors compare securities against each other’s performance. They favor buying outperforming securities and avoiding – or short-selling – underperforming securities.

Long-only relative momentum investors rotate between a subset of holdings within their investable universe. For example, a simple long-only relative strength system example is “best N of.” At rebalance, this system sells its current holdings and buys the top N performing securities of a basket. In doing so, the strategy seeks to align the portfolio with the best performing securities in hopes they continue to outperform.

2.2 Absolute Momentum

Absolute momentum is also referred to as time-series momentum or trend following.

Absolute momentum investors compare a security against its own historical performance. The system buys positive returning securities and avoids, or sells short, negative returning securities.

The primary difference is that relative momentum makes no distinction about return direction. If all securities are losing value, relative momentum will seek to invest in those assets that are going down least. Absolute momentum will seek to avoid negative returning assets.

3. A Brief History of Momentum

3.1 Early Practitioners

Momentum is one of Wall Street’s oldest investment strategies.

In 1838, James Grant published The Great Metroplis, Volume 2. Within, he spoke of David Ricardo, an English political economist who was active in the London markets in the late 1700s and early 1800s. Ricardo amassed a large fortune trading both bonds and stocks.

According to Grant, Ricardo’s success was attributed to three golden rules:

As I have mentioned the name of Mr. Ricardo, I may observe that he amassed his immense fortune by a scrupulous attention to what he called his own three golden rules, the observance of which he used to press on his private friends. These were, “Never refuse an option* when you can get it,”—”Cut short your losses,”—”Let your profits run on.” By cutting short one’s losses, Mr. Ricardo meant that when a member had made a purchase of stock, and prices were falling, he ought to resell immediately. And by letting one’s profits run on he meant, that when a member possessed stock, and prices were raising, he ought not to sell until prices had reached their highest, and were beginning again to fall. These are, indeed, golden rules, and may be applied with advantage to innumerable other transactions than those connected with the Stock Exchange.

The rules “cut short your losses” and “let your profits run on” are foundational philosophies of momentum.

Following in Ricardo’s footsteps are some of Wall Street’s greatest legends who implemented momentum and trend-following techniques.

Charles H. Dow (1851 – 1902) was the founder and first editor of the Wall Street Journal as well as the co-founder of Dow Jones and Company. In his Wall Street Journal column, he published his market trend analysis, which eventually developed into a body of research called Dow theory. Dow theory primarily focuses on the identification of trends as being the key signal for investing.

Jesse Livermore (1877 – 1940) was a stock market speculator in the early 1900s who famously made – and subsequently lost – two massive fortunes during the market panic of 1907 and crash of 1929. He is attributed (by Edwin Lefèvre, in Reminiscences of a Stock Operator) to saying,

[T]he big money was not in the individual fluctuations but in the main movements … sizing up the entire market and its trend.

Livermore claimed that his lack of adherence to his own rules was the main reason he lost his wealth.

In the same era of Livermore, Richard Wyckoff (1873 – 1934) noted that stocks tended to trend together. Thus he focused on entering long positions only when the broad market was trending up. When the market was in decline, he focused on shorting. He also emphasized the placement of stop-losses to help control risk.

He was personally so successful with his techniques, he eventually owned nine and a half acres in the Hamptons.

Starting in the 1930s, George Chestnutt successfully ran the American Investors Fund for nearly 30 years using relative strength techniques. He also published market letters with stock and industry group rankings based on his methods. He wrote,

[I]t is better to buy the leaders and leave the laggards alone. In the market, as in many other phases of life, ‘the strong get stronger, and the weak get weaker.’

In the late 1940s and early 1950s, Richard Donchian developed a rules based technical system that became the foundation for his firm Futures, Inc. Futures, Inc. was one of the first publicly held commodity funds. The investment philosophy was based upon Donchian’s belief that commodity prices moved in long, sweeping bull and bear markets. Using moving averages, Donchian built one of the first systematic trend-following methods, earning him the title of the father of trend-following.

In the late 1950s, Nicholas Darvas (1920 – 1977), trained economist and touring dancer, invented “BOX theory.” He modeled stock prices as a series of boxes. If a stock price remained in a box, he waited. As a stock price broke out of a box to new highs, he bought and placed a tight stop loss. He is quoted as saying,

I keep out in a bear market and leave such exceptional stocks to those who don’t mind risking their money against the market trend.

Also during the 1950s and 1960s was Jack Dreyfus, who Barron’s named the second most significant money manager of the last century. From 1953 to 1964, his Dreyfus Fund returned 604% compared to 346% for the Dow index. Studies performed by William O’Neil showed that Dreyfus tended to buy stocks making new 52-week highs. It wouldn’t be until 2004 that academic studies would confirm this method of investing.

Richard Driehaus took the momentum torch during the 1980s. In his interview in Jack Schwager’s The New Market Wizards, he said he believed that money was made buying high and selling higher.

That means buying stocks that have already had good moves and have high relative strength – that is, stocks in demand by other investors. I would much rather invest in a stock that’s increasing in price and take the risk that it may begin to decline than invest in a stock that’s already in a decline and try to guess when it will turn around.

3.2 Earliest Academic Studies

In 1933, Alfred Cowles III and Herbert Jones released a research paper titled Some A Posteriori Probabilities in Stock Market Action. Within it they specifically focused on “inertia” at the “microscopic” – or stock – level.

They focused on counting the ratio of sequences – times when positive returns were followed by positive returns, or negative returns were followed by negative returns – to reversals – times when positive returns were followed by negative returns, and vice versa.

Their results:

It was found that, for every series with intervals between observations of from 20 minutes up to and including 3 years, the sequences out-numbered the reversals. For example, in the case of the monthly series from 1835 to 1935, a total of 1200 observations, there were 748 sequences and 450 reversals. That is, the probability appeared to be .625 that, if the market had risen in a given month, it would rise in the succeeding month, or, if it had fallen, that it would continue to decline for another month. The standard deviation for such a long series constructed by random penny tossing would be 17.3; therefore the deviation of 149 from the expected value of 599 is in excess of eight times the standard deviation. The probability of obtaining such a result in a penny-tossing series is infinitesimal.

Despite the success of their research on the statistical significance of sequences, the next academic study on momentum was not released for 30 years.

In 1967, Robert Levy published Relative Strength as a Criterion for Investment Selection. Levy found that there was “good correlation between past performance groups and future … performance groups” over 26-week periods. He states:

[…] the [26-week] average ranks and ratios clearly support the concept of continuation of relative strength. The stocks which historically were among the 10 per cent strongest (lowest ranked) appreciated in price by an average of 9.6 per cent over a 26-week future period. On the other hand, the stocks which historically were among the 10 per cent weakest (highest ranked) appreciated in price an average of only 2.9 per cent over a 26-week future period.

Unfortunately, the scope of the study was limited. The period used in the analysis was only from 1960 to 1965. Thus, of the 26-week periods tested, only 8 were independent. In Levy’s words, “the results were extensively intercorrelated; and the use of standard statistical measures becomes suspect.” Therefore, Levy omitted these statistics.

Despite its promise, momentum research went dark for the next 25 years.

4. The Dark Days of Momentum Research

Despite the success of practitioners and promising results of early studies, momentum would go largely ignored by academics until the 1990s.

Exactly why is unknown, but we have a theory: fundamental investing, modern portfolio theory, and the efficient market hypothesis.

4.1 The Rise of Fundamental Investing

In 1934, Benjamin Graham and David Dodd published Security Analysis. Later, in 1949, they published The Intelligent Investor. In these tomes, they outline their methods for successful investing.

For Graham and Dodd, a purchase of stock was a purchase of partial ownership of a business. Therefore, it was important that investors evaluate the financial state of the underlying business they were buying.

They also defined a strong delineation between investing and speculating. To quote,

An investment operation is one which, upon thorough analysis, promises safety of principal and an adequate return. Operations not meeting these requirements are speculative.

Speculative was a pejorative term. Even the title of The Intelligent Investor implied that any investors not performing security analysis were not intelligent.

The intelligent investor began her process by computing a firm’s intrinsic value. In other words, “what is the business truly worth?” This value was either objectively right or wrong based on the investor’s analysis. Whether the market agreed or not was irrelevant.

Once an intrinsic value was determined, Graham and Dodd advocated investors buy with a margin of safety. This meant waiting for the market to offer stock prices at a deep discount to intrinsic value.

These methods of analysis became the foundation of value investing.

To disciples of Graham and Dodd, momentum is speculative nonsense. To quote Warren Buffett in The Superinvestors of Graham-and-Doddsville:

I always find it extraordinary that so many studies are made of price and volume behavior, the stuff of chartists. Can you imagine buying an entire business simply because the price of the business had been marked up substantially last week and the week before?

4.2 Modern Portfolio Theory and the Efficient Market Hypothesis

In his 1952 article “Portfolio Selection,” Harry Markowitz outlined the foundations of Modern Portfolio Theory (MPT). The biggest breakthrough of MPT was that it provided a mathematical formulation for diversification.

While the concept of diversification has existed since pre-Biblical eras, it had never before been quantified. With MPT, practitioners could now derive portfolios that optimally balanced risk and reward. For example, by combining assets together, Markowitz created the efficient frontier: those combinations for which there is the lowest risk for a given level of expected return.

By introducing a risk-free asset, the expected return of any portfolio constructed can be linearly changed by varying the allocation to the risk-free asset. In a graph like the one on the left, this can be visualized by constructing a line that passes through the risk-free asset and the risky portfolio (called a Capital Allocation Line or CAL). The CAL that is tangent to the efficient frontier is called the capital market line (CML). The point of tangency along the efficient frontier is the portfolio with the highest Sharpe ratio (excess expected return divided by volatility).

According to MPT, in which all investors seek to maximize their Sharpe ratio, an investor should only hold a mixture of this portfolio and the risk free asset. Increasing the allocation to the risk-free asset decreases risk while introducing leverage increases risk.

The fact that any investor should only hold one portfolio has a very important implication: given all the assets available in the market, all investors should hold, in equal relative proportion, the same portfolio of global asset classes. Additionally, if all investors are holding the same mix of assets, in market equilibrium, the prices of asset classes – and therefore their expected returns – must adjust such that the allocation ratios of the assets in the tangency portfolio will match the ratio in which risky assets are supplied to the market.

Holding anything but a combination of the tangency portfolio and the risk-free asset is considered sub-optimal.

From this foundation, concepts for the Capital Asset Pricing Model (CAPM) are derived. CAPM was introduced independently by Jack Treynor, William Sharpe, John Lintner, and Jan Mossin from 1961-1966.

CAPM defines a “single-factor model” for pricing securities. The expected return of a security is defined in relation to a risk-free rate, the security’s “systematic” risk (sensitivity to the tangency portfolio), and the expected market return. All other potentially influencing factors are considered to be superfluous.

While its origins trace back to the 1800s, the efficient market hypothesis (EMH) was officially developed by Eugene Fama in his 1962 Ph.D. thesis.

EMH states that stock prices reflect all known and relevant information and always trade at fair value. If stocks could not trade above or below fair value, investors would never be able to buy them at discounts or sell them at premiums. Therefore, “beating the market” on a risk-adjusted basis is impossible.

Technically, MPT and EMH are independent theories. MPT tells us we want to behave optimally, and gives us a framework to do so. EMH tells us that even optimal behavior will not generate any return in excess of returns predicted by asset pricing models like CAPM.

Markowitz, Fama, and Sharpe all went on to win Nobel prizes for their work.

4.3 Growing Skepticism Towards Technical Analysis

Technical analysis is a category of investing methods that use past market data – primarily price and volume – to make forward forecasts.

As a category, technical analysis is quite broad. Some technicians look for defined patterns in price charts. Others look for lines of support or resistance. A variety of indicators may be calculated and used. Some technicians follow specific techniques – like Dow theory or Elliot Wave theory.

Unfortunately, the broad nature of technical analysis makes it difficult to evaluate academically. Methods vary widely and different technical analysts can make contradictory predictions using the same data.

Thus, during the rise of EMH through the 1960s and 1970s, technical analysis was largely dismissed by academics.

Since momentum relies only on past prices, and many practitioners used tools like moving averages to identify trends, it was categorized as a form of technical analysis. As academics dismissed the field, momentum went overlooked.

4.4 But Value Research Went On

Despite CAPM, EMH, and growing skepticism towards technical analysis, academic research for fundamental investing continued. Focus was especially strong on value investing.

For example, in 1977, S. Basu authored a comprehensive study on value investing, titled Investment Performance of Common Stocks in Relation to their Price-Earnings Ratios: A Test of the Efficient Market Hypothesis. Within, Basu finds that the return relationship strictly increases for stocks sorted on their price-earnings ratio. Put more simply, cheap stocks outperform expensive ones.

Unfortunately, in many of these studies, the opposite of value was labeled growth or glamor. This became synonymous with high flying, over-priced stocks. Of course, not value is not the same as growth. And not value is certainly not the same as momentum. It is entirely possible that a stock can be in the middle of a positive trend, yet still be undervalued. Nevertheless, it is easy to see how relatively outperforming and over-priced may be conflated.

It is possible that the success of value research in demonstrating the success of buying cheap stocks dampened the enthusiasm for momentum research.

5. The Return of Momentum

Fortunately, decades of value-based evidence against market efficiency finally piled up.

In February 1993, Eugene Fama and Kenneth French released Common Risk Factors in the Returns on Stocks and Bonds. Fama and French extended the single-factor model of CAPM into a three-factor model. Beyond the “market factor,” factors for “value” and “size” were added, acknowledging these distinct drivers of return.

Momentum was still nowhere to be found.

But a mere month later, Narasimhan Jegadeesh and Sheridan Titman published their seminal work on momentum, titled Returns to Buying Winners and Selling Losers: Implication for Stock Market Efficiency. Within they demonstrated:

Strategies which buy stocks that have performed well in the past and sell stocks that have performed poorly in the past generate significant positive returns over 3- to 12-month holding periods.

The results of the paper could not be explained by systematic risk or delayed reactions to other common factors, echoing the results of Cowles and Jones some 60 years prior.

In 1996, Fama and French authored Multifactor Explanations of Asset Pricing Anomalies. Armed with their new three-factor model, they explored whether recently discovered market phenomena – including Jegadeesh and Titman’s momentum – could be rationally explained away.

While most anomalies disappeared under scrutiny, the momentum results remained robust. In fact, in the paper Fama and French admitted that,

“[momentum is the] main embarrassment of the three-factor model.”

6. The Overwhelming Evidence for Momentum

With its rediscovery and robustness against prevailing rational pricing models, momentum research exploded over the next two decades. It was applied across asset classes, geographies, and time periods. In chronological order:

Asness, Liew, and Stevens (1997) shows that momentum investing is a profitable strategy for country indices.

Carhart (1997) finds that portfolios of mutual funds, constructed by sorting on trailing one-year returns, decrease in monthly excess return nearly monotonically, inline with momentum expectations.

Rouwenhorst (1998) demonstrates that stocks in international equity markets exhibit medium-term return continuations. The study covered stocks from Austria, Belgium, Denmark, France, Germany, Italy, the Netherlands, Norway, Spain, Sweden, Switzerland, and the United Kingdom.

LeBaron (1999) finds that a simple momentum model creates unusually large profits in foreign exchange series.

Moskowitz and Grinblatt (1999) finds evidence for a strong and persistent industry momentum effect.

Rouwenhorst (1999), in a study of 1700 firms across 20 countries, demonstrates that emerging market stocks exhibit momentum.

Liew and Vassalou (2000) shows that momentum returns are significantly positive in foreign developed countries but there is little evidence to explain them by economic developments.

Griffin, Ji, and Martin (2003) demonstrates momentum’s robustness, finding it to be large and statistically reliable in periods of both negative and positive economic growth. The study finds no evidence for macroeconomic or risk-based explanations to momentum returns.

Erb and Harvey (2006) shows evidence of success for momentum investing in commodity futures.

Gorton, Hayashi, and Rouwenhorst (2008) extends momentum research on commodities, confirming its existence in futures but also identifying its existence in spot prices.

Jostova, Niklova Philopov, and Stahel (2012) shows that momentum profits are significant for non-investment grade corporate bonds.

Luu and Yu (2012) identifies that for liquid fixed-income assets, such as government bonds, momentum strategies may provide a good risk-return trade-off and a hedge for credit exposure.

7. Academic Explanations for Momentum

While academia has accepted momentum as a distinct driver of return premia in many asset classes around the world, the root cause is still debated.

So far, the theory for rational markets has failed to account for momentum’s significant and robust returns. It is not correlated with macroeconomic variables and does not seem to reflect exposure to other known risk factors.

But there are several hypotheses that might explain how irrational behavior may lead to momentum.

7.1 The Behavioral Thesis

The most commonly accepted argument for why momentum exists and persists comes from behavioral finance. Behavioral finance is a field that seeks to link psychological theory with economics and finance to explain irrational decisions.

Some of the popular behavioral finance explanations for momentum include:

Herding: Also known as the “bandwagon effect,” herding is the tendency for individuals to mimic the actions of a larger group.

Anchoring Bias: The tendency to rely too heavily on the first piece of information received.

Confirmation Bias: The tendency to ignore information contradictory to prior beliefs.

Disposition Effect: Investors tend to sell winners too early and hold on to losers too long. This occurs because investors like to realize their gains but not their losses, hoping to “make back” what has been lost.

Together, these biases cause investors to either under- or over-react to information, causing pricing inefficiencies and irrational behavior.

7.1.1 Cumulative Advantage & Momentum Beyond Markets

There is strong evidence for momentum being a behavioral and social phenomenon beyond stock markets.

Matthew Salganik, Peter Dodds, and Duncan Watts ran a 14,000 participant, web-based study designed to establish independence of taste and preference in music.

Participants were asked to explore, listen to, and rate music. One group of participants would be able to see how many times a song was downloaded and how other participants rated it; the other group would not be able to see downloads or ratings. The group that could see the number of downloads (“social influence”) was then sub-divided into 8 distinct, random groups where members of each sub-group could only see the download and ratings statistics of their sub-group peers.

The hypothesis of the study was that “good music” should garner the same amount of market share regardless of the existence of social influence: hits should be hits. Secondly, the same hits should be hits across all independent social influence groups.

What the study found was dramatically different. Each social-influence group had its own hit songs, and those songs commanded a much larger market share of downloads than songs did in the socially-independent group.

Introducing social-influence did two things: it made hits bigger and it made hits more unpredictable. The authors called this effect “cumulative advantage.” The consequences are profound. To quote an article in the New York Times by Watts,

It’s a simple result to state, but it has a surprisingly deep consequence. Because the long-run success of a song depends so sensitively on the decisions of a few early-arriving individuals, whose choices are subsequently amplified and eventually locked in by the cumulative-advantage process, and because the particular individuals who play this important role are chosen randomly and may make different decisions from one moment to the next, the resulting unpredictability is inherent to the nature of the market. It cannot be eliminated either by accumulating more information — about people or songs — or by developing fancier prediction algorithms, any more than you can repeatedly roll sixes no matter how carefully you try to throw the die.

7.2 The Limits to Arbitrage Thesis

EMH assumes that any mis-pricing in public markets will be immediately arbitraged away by rational market participants. The limits to arbitrage theory recognizes that there are often restrictions – both regulatory and capital based – that may limit rational traders from fully arbitraging away these price inefficiencies.

In support of this thesis is Chabot, Ghysels, and Jagannathan (2009), which finds that when arbitrage capital is in short supply, momentum cycles last longer.

Similarly, those investors bringing good news to the market may lack the capital to take full advantage of that information. So if there has been good news in the past, there may be good news not yet incorporated into the price.

7.3 The Rational Inattention Thesis

Humans possess a finite capacity to process the large amounts of information they are confronted with. Time is a scarce resource for decision makers.

The rational inattention theory argues that some information may be evaluated less carefully, or even outright ignored. Or, alternatively, it may be optimal for investors to obtain news with limited frequency or limited accuracy. This can cause investors to over- or under-invest and could cause the persistence of trends.

Chen and Yu (2014) found that portfolios constructed from stocks “more likely to grab attention” based on visual patterns induces investor over-reaction. They provide evidence that momentum continuation is induced by visually-based psychological biases.

8. Advances in Cross-Sectional Research

Much like there are many ways to identify value, there are many ways to identify momentum. Recent research has identified methods that may improve upon traditional total return momentum.

52-Week Highs: Hwang and George (2004) shows that nearness to a 52-week high price dominates and improves upon the forecasting power of past returns (i.e. momentum). Perhaps most interestingly, future returns forecast using a 52-week high do not mean-revert in the long run, like traditional momentum.

Liu, Liu, and Ma (2010) tests the 52-week high strategy in 20 international markets and finds that it is profitable in 18 and significant in 10.

Residual Momentum: Using a universe of domestic equities, covering the period of January 1926 to December 2009, Blitz, Huij, and Martens (2009) decomposes stock returns using the Fama-French three-factor model. Returns unexplained by the market, value, and size factors are considered to be residual. The study finds that momentum strategies built from residual returns exhibit risk-adjusted profits that are twice as large as those associated with total return momentum.

Idiosyncratic Momentum: Similar to Blitz, Huij, and Martens, Chaves (2012) uses the CAPM model to correct stocks for market returns and identify idiosyncratic returns. Idiosyncratic momentum is found to work better than momentum in a sample of 21 developed countries. Perhaps most importantly, idiosyncratic momentum is successful in Japan, where most traditional momentum strategies have failed.

9. Using Momentum to Manage Risk

While most research in the late 1990s and early 2000s focused on relative momentum, research after 2008 has been heavily focused on time-series momentum for its risk-mitigating and diversification properties.

Some of the earliest, most popular research was done by Faber (2006), in which a simple price-minus-moving-average approach was used to drive a portfolio of U.S. equities, foreign developed equities, commodities, U.S. REITs, and U.S. government bonds. The resulting portfolio demonstrates “equity-like returns with bond-like volatility.”

Hurst, Ooi, and Pedersen (2010) identifies that trend-following, or time-series momentum, is a significant component of returns for managed futures strategies. In doing so, the research demonstrates the consistency of trend-following approaches in generating returns in both bull and bear markets.

Going beyond managed futures specifically, Moskowitz, Ooi, Hua, and Pedersen (2011) documents significant time-series momentum in equity index, currency, commodity, and bond futures covering 58 liquid instruments over a 25-year period.

Perhaps some of the most conclusive evidence comes from Hurst, Ooi, Pedersen (2012), which explores time-series momentum going back to 1903 and through 2011.

The study constructs a portfolio of an equal-weight combination of 1-month, 3-month, and 12-month time-series momentum strategies for 59 markets across 4 major asset classes, including commodities, equity indices, and currency pairs. The approach is consistently profitable across decades. The research also shows that incorporating a time-series momentum approach into a traditional 60/40 stock/bond portfolio increases returns, reduces volatility, and reduces maximum drawdown.

Finally, Lempérière, Deremble, Seager, Potters, and Bouchard (2014) extends the tests even further, using both futures and spot prices to go back to 1800 for commodity and stock indices. It finds that excess returns driven by trend-following is both significant and stable across time and asset classes.

10. Evidence & Advances in Time-Series Momentum

While the evidence for time-series momentum was significantly advanced by the papers and teams cited above, there were other, more focused contributions throughout the years that helped establish it in more specific asset classes.

Wilcox and Crittenden (2005) demonstrates that buying stocks when they make new 52-week highs and selling after a prescribed stop-loss is broken materially outperforms the S&P 500 even after accounting for trading slippage.

ap Gwilym, Clare, Seaton, and Thomas (2009) explores whether trend-following can be used as an allocation tool for international equity markets. Similar to Faber (2006), it utilizes a 10-month price-minus-moving-average model. Such an approach delivers a similar compound annual growth rate to buy and hold, but with significantly lower volatility, increasing the Sharpe ratio from 0.41 to 0.75.

Szakmary, Shen, and Sharma (2010) explores trend-following strategies on commodity futures markets covering 48 years and 28 markets. After deducting reasonable transaction costs, it finds that both a dual moving-average-double-crossover strategy and a channel strategy yield significant profit over the full sample period.

Antonacci (2012) explores a global tactical asset allocation approach utilizing both relative and absolute momentum techniques in an approach called “dual momentum.” Dual momentum increases annualized return, reduces volatility, and reduces maximum drawdown for equities, high yield & credit bonds, equity & mortgage REITs, and gold & treasury bonds.

Dudler, Gmuer, and Malamud (2015) demonstrates that risk-adjusted time series momentum – returns normalized by volatility – outperforms time series momentum on a universe of 64 liquid futures contracts for almost all combinations of holdings and look-back periods.

Levine and Pedersen (2015) uses smoothed past prices and smoothed current prices in their calculation of time-series momentum to reduce random noise in data that might occur from focusing on a single past or current price.

Clare, Seaton, Smith and Thomas (2014) finds that trend following “is observed to be a very effective strategy over the study period delivering superior risk-adjusted returns across a range of size categories in both developed and emerging markets.”

11. Unifying Momentum & Technical Analysis

Despite their similarities, trend-following moving average rules are often still considered to be technical trading rules versus the quantitative approach of time-series momentum. Perhaps the biggest difference is that the trend-following camp tended to focus on prices while the momentum camp focused on returns.

However, research over the last half-decade actually shows that they are highly related strategies.

Bruder, Dao, Richard, and Roncalli (2011) unites moving-average-double-crossover strategies and time-series momentum by showing that cross-overs were really just an alternative weighting scheme for returns in time-series momentum. To quote,

The weighting of each return … forms a triangle, and the biggest weighting is given at the horizon of the smallest moving average. Therefore, depending on the horizon n2 of the shortest moving average, the indicator can be focused toward the current trend (if n2 is small) or toward past trends (if n2 is as large as n1/2 for instance).

We can see, above, this effect in play. When n2 << n1 (e.g. n2=10, n1=100), returns are heavily back-weighted in the calculation. As n2 approaches half of n1, we can see that returns are most heavily weighted at the middle point.

Marshall, Nguyen and Visaltanachoti (2012) proves that time-series momentum is related to moving-average-change-in-direction. In fact, time-series momentum signals will not occur until the moving average changes direction. Therefore, signals from a price-minus-moving-average strategy are likely to occur before a change in signal from time-series momentum.

Levine and Pedersen (2015) shows that time-series momentum and moving average cross-overs are highly related. It also find that time-series momentum and moving-average cross-over strategies perform similarly across 58 liquid futures and forward contracts.

Beekhuizen and Hallerbach (2015) also links moving averages with returns, but further explores trend rules with skip periods and the popular MACD rule. Using the implied link of moving averages and returns, it shows that the MACD is as much trend following as it is mean-reversion.

Zakamulin (2015) explores price-minus-moving-average, moving-average-double-crossover, and moving-average-change-of-direction technical trading rules and finds that they can be interpreted as the computation of a weighted moving average of momentum rules with different lookback periods.

These studies are important because they help validate the approach of price-based systems. Being mathematically linked, technical approaches like moving averages can now be tied to the same theoretical basis as the growing body of work in time-series momentum.

12. Conclusion

As an investment strategy, momentum has a deep and rich history.

Its foundational principles can be traced back nearly two centuries and the 1900s were filled with its successful practitioners.

But momentum went long misunderstood and ignored by academics.

In 1993, Jegadeesh and Titman published “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency.” Prevailing academic theories were unable to account for cross-sectional momentum in rational pricing models and the premier market anomaly was born.

While momentum’s philosophy of “buy high, sell higher” may seem counterintuitive, prevailing explanations identify its systemized process as taking advantage of the irrational behavior exhibited by investors.

Over the two decades following momentum’s (re)introduction, academics and practitioners identified the phenomenon as being robust in different asset classes and geographies around the globe.

After the financial crisis of 2008, a focus on using time-series momentum emerged as a means to manage risk. Much like cross-sectional momentum, time-series momentum was found to be robust, offering significant risk-management opportunities.

While new studies on momentum are consistently published, the current evidence is clear: momentum is the premier market anomaly.

Antonacci, G., (2012). “Risk Premia Harvesting Through Dual Momentum”, Portfolio Management Associates.

Asness, Clifford S., John M. Liew, and Ross L. Stevens. “Parallels Between the Cross-Sectional Predictability of Stock and Country Returns.” Journal of Portfolio Management, 23, no. 3 (Spring 1997), pp. 79–87.

Asness, Cliff, Tobias Moskowitz, and Lasse H. Pedersen (2013), “Value and Momentum Everywhere,” The Journal of Finance 68(3), 929-985.

Beekhuizen, Paul and Hallerbach, Winfried G., Uncovering Trend Rules (May 11, 2015). Available at SSRN: http://ssrn.com/abstract=2604942

Bhojraj, Sanjeev and Swaminathan, Bhaskaran, Macromomentum: Returns Predictability in Currencies and International Equity Indices (October 2002). Available at SSRN: http://ssrn.com/abstract=356940

Blitz, David and Huij, Joop and Martens, Martin, Residual Momentum (August 1, 2009). Available at SSRN: http://ssrn.com/abstract=2319861

Bruder, Benjamin and Dao, Tung-Lam and Richard, Jean-Charles and Roncalli, Thierry, Trend Filtering Methods for Momentum Strategies (December 1, 2011). Available at SSRN: http://ssrn.com/abstract=2289097

Carhart, Mark M. 1997. “On Persistence in Mutual Fund Performance.” Journal of Finance, vol. 52, no. 1 (March):57–82.

Chaves, Denis. 2012. “Eureka! A Momentum Strategy that Also Works in Japan.” Research Affiliates Working Paper (January 9).

Chen, Li-Wen and Yu, Hsin-Yi, Investor Attention, Visual Price Pattern, and Momentum Investing (August 12, 2014). 27th Australasian Finance and Banking Conference 2014 Paper. Available at SSRN: http://ssrn.com/abstract=2292895

Chui, Andy C.W. and Titman, Sheridan and Wei, K. C. John, Individualism and Momentum Around the World (February 23, 2009). Journal of Finance, Forthcoming. Available at SSRN: http://ssrn.com/abstract=1150076

Clare, Andrew and Seaton, James and Smith, Peter N. and Thomas, Steve, Size Matters: Tail Risk, Momentum and Trend Following in International Equity Portfolios (November 6, 2014). Available at SSRN: http://ssrn.com/abstract=2520075

Dudler, Martin and Gmuer, Bruno and Malamud, Semyon, Risk Adjusted Time Series Momentum (June 22, 2014). Swiss Finance Institute Research Paper No. 14-71. Available at SSRN: http://ssrn.com/abstract=2457647

Erb, Claude, and Campbell Harvey, 2006, The strategic and tactical value of commodity futures, Financial Analysts Journal 62, 69–97.

Faber, Mebane T., A Quantitative Approach to Tactical Asset Allocation (February 1, 2013). The Journal of Wealth Management, Spring 2007 . Available at SSRN: http://ssrn.com/abstract=962461

Gorton, Gary B., Fumio Hayashi, and K. Geert Rouwenhorst, 2008, The fundamentals of commodity futures returns, Working paper, University of Pennsylvania.

Griffin, John M. and Martin, J. Spencer and Ji, Susan, Momentum Investing and Business Cycle Risk: Evidence from Pole to Pole (March 18, 2002). AFA 2003 Washington, DC Meetings; EFA 2002 Berlin Meetings Presented Paper. Available at SSRN: http://ssrn.com/abstract=291225

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Jostova, Gergana and Nikolova, Stanislava and Philipov, Alexander and Stahel, Christof W., Momentum in Corporate Bond Returns (December 12, 2012). Available at SSRN: http://ssrn.com/abstract=1651853

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Category: Momentum Page 3 of 4

Tactical Credit

Summary­

Conclusion

Fragility Case Study: Dual Momentum GEM

Summary­

Introduction

Global Equities Momentum (“GEM”)

Conclusion

Momentum’s Magic Number

Summary­

Out-of-Sample Testing with Countries and Sectors

Does the Effect Persist for Long-Only Portfolios?

Conclusion

Factor Fimbulwinter

Summary­

Conclusion

Addendum

How were the tests impacted by the Bayesian inference process?

Why are we defending price-to-book?

Two Centuries of Momentum

A momentum-based investing approach can be confusing to investors who are often told that “chasing performance” is a massive mistake and “timing the market” is impossible.

1. What is momentum?

2. The Two Faces & Many Names of Momentum

2.1 Relative Momentum

2.2 Absolute Momentum

3. A Brief History of Momentum

3.1 Early Practitioners

3.2 Earliest Academic Studies

4. The Dark Days of Momentum Research

4.1 The Rise of Fundamental Investing

4.2 Modern Portfolio Theory and the Efficient Market Hypothesis

4.3 Growing Skepticism Towards Technical Analysis

4.4 But Value Research Went On

5. The Return of Momentum

6. The Overwhelming Evidence for Momentum

7. Academic Explanations for Momentum

7.1 The Behavioral Thesis

7.1.1 Cumulative Advantage & Momentum Beyond Markets

7.2 The Limits to Arbitrage Thesis

7.3 The Rational Inattention Thesis

8. Advances in Cross-Sectional Research

9. Using Momentum to Manage Risk

10. Evidence & Advances in Time-Series Momentum

11. Unifying Momentum & Technical Analysis

12. Conclusion

Summary

Summary

Summary

Summary