Recently a paper was published by AQR where the authors advocate for an integrated approach to multi-factor portfolios, preferring securities that exhibit strong characteristics across all desired factors instead of a mixed approach, where securities are selected based upon extreme exposure to a single characteristic.

We believe the integrated approach fails to acknowledge the impact of the varying lengths over which different factors mature, ultimately leading to a portfolio more heavily influenced by higher turnover factors.

The Importance of Factor Maturity Cliff Asness, founder of AQR, recently published a paper titled My Factor Philippic. This paper was written in response to the recently popularized article How Can “Smart Beta” Go Horribly Wrong? which was co-authored by Robert Arnott, co-founder of Research Affiliates.

Arnott argues that many popular factors are currently historically overvalued and, furthermore, that the historical excess return offered by some recently popularized factors can be entirely explained by rising valuation trends in the last 30 years. Caveat emptor, warns Arnott: valuations always matter.

Much to our delight (after all, who doesn’t like to see two titans of industry go at it?), Asness disagrees.

One of the primary arguments laid out by Asness is that valuation is a meaningless predictor for factors with high turnover.

The intuition behind this argument is simple: while valuations may be a decent predictor of forward annualized returns for broad markets over the next 5-to-10 years, the approach only works because the basket of securities stays mostly constant. For example, valuations for U.S. equities may be a good predictor because we expect the vast majority of the basket of U.S. equities to stay constant over the next 5-to-10 years.

The same is not true for many factors. For example, let’s consider a high turnover factor like momentum.

Valuations of a momentum basket today are a poor predictor of annualized returns of a momentum strategy over the next 5-to-10 years because the basket of securities held could be 100% different three months from now.

Unless the same securities are held in the basket, valuation headwinds or tailwinds will not necessarily be realized.

For the same reason, valuation is also poor as an explanatory variable of factor returns. Asness argues that Arnott’s warning of valuation being the secret driver of factor returns is unwarranted in high turnover factors.

The paper attempts to conclude the current debate about the best way to build multi-factor portfolios. The first approach is to mix, where a portfolio is built by combining stand-alone factor portfolios. The second approach is to integrate, where a portfolio is built by selecting securities that have simultaneously strong exposure to multiple factors at once.

A figure from the paper does a good job of illustrating the difference. Below, a hypothetical set of stocks is plotted based upon their current valuation and momentum characteristics.

In the top left, a portfolio of deep value stocks is selected. In the top right, the mix approach is demonstrated, where the deepest value and the highest momentum stocks are selected.

In the bottom left, the integrated approach is demonstrated, where the securities simultaneously exhibiting strong valuation and momentum characteristics are selected.

Finally, in the bottom right we can see how these two approaches differ: with yellow securities being those only found in the mix portfolio and blue securities being found only in the integrated portfolio.

It is worth noting that the ETF industry has yet to make up its mind on the right approach.

GlobalX and Goldman Sachs prefer the mix approach in their ETFs (SCIU / GSLC) while JPMorgan and iShares prefer the integrate approach (JPUS / LRGF).

The argument made by those taking the integrated approach is that they are looking for securities with well-rounded exposures rather than those with extreme singular exposures. Integrators argue that this approach helps them avoid holding securities that might cancel each other out. If we look back towards the mix example above (top right), we can see that many securities selected due to strength in one factor are actually quite poor in the other.

Integrators claim that this inefficiency can create a drag in the mix portfolio. Why hold something with strong momentum if it has a very poor valuation score that is only going to offset it?

We find it somewhat ironic that FFPS and Asness both publish for AQR, because we would argue that Asness’s argument points out the fundamental flaw in the theory outlined by integrators. Namely: the horizons over which the premia mature differ.

Therefore, a strong positive loading in a factor like momentum is not necessarily offset by a strong negative loading in a factor like value. Furthermore, by integrating we run the risk of the highest turnover factor actually dominating the integrated selection process.

Data In the rest of this commentary, we will be using industry data from the Kenneth French data library. For momentum scores, we calculate 12 one-month total return and calculate cross-sector z-scores[1]. For valuation scores, we calculate a normalized 5-year dividend yield score and then calculate cross-sector z-scores.[2]

Do Factor Premia Actually Mature at Different Time Periods? In his paper, Asness referenced the turnover of a factor portfolio as an important variable. We prefer to think of high turnover factors as factors whose premium matures more quickly.

For example, if we buy a stock because it has high relative momentum, our expectation is that we will likely hold it for longer than a day, but likely much shorter than a year. Therefore, a strategy built off relative momentum will likely have high turnover because the premium matures quickly.

On the other hand, if we buy a value stock, our expectation is that we will have to hold it for up to several years for valuations to adequately reverse. This means that the value premium takes longer to mature – and the strategy will likely have lower turnover.

We can see this difference in action by looking at how valuation and momentum scores change over time.

We see similar pictures for other industries. Yet, looks can be deceiving and the human brain is excellent at finding patterns where there are none (especially when we want to see those patterns). Can we actually quantify this difference?

One way is to try to build a model that incorporates both the randomness of movement and how fast these scores mean-revert. Fitting our data to this model would tell us about how quickly each premium matures.

One such model is called an Ornstein-Uhlenbeck process (“OU process”). An OU process follows the following stochastic differential equation:

To translate this into English using an example: the change in value z-score from one period to the next can be estimated as a “magnetism” back to fair value plus some randomness. In the equation, theta tells us how strong this magnetism is, mu tells us what fair value is, and sigma tells us how much influence the randomness has.

For our momentum and valuation z-scores, we would expect mu to be near-zero, as over the long-run we would not expect a given sector to exhibit significantly more or less relative momentum or relative cheapness/richness than peer sectors.

Given that we also believe that the momentum premium is realized over a shorter horizon, we would also expect that theta – the strength of the magnetism, also called the speed of mean reversion – will be higher. Since that strength of magnetism is higher, we will also need sigma – the influence of randomness – to be larger to overcome it.

So how to the numbers play out?[3]

For the momentum z-scores:

Theta

Mu

Sigma

NoDur

0.97

0.02

1.00

Durbl

1.00

0.03

1.63

Manuf

1.22

-0.03

0.96

Enrgy

0.98

0.06

1.69

HiTec

1.04

0.03

1.49

Telcm

1.15

-0.07

1.52

Shops

1.22

0.03

1.24

Hlth

0.84

0.11

1.39

Utils

1.48

-0.09

1.61

Other

1.18

-0.09

1.13

Average

1.10

0.00

1.36

For the valuation z-scores:

Theta

Mu

Sigma

NoDur

0.11

-0.20

0.34

Durbl

0.08

0.58

0.49

Manuf

0.13

0.01

0.37

Enrgy

0.07

0.19

0.40

HiTec

0.09

0.23

0.33

Telcm

0.07

0.03

0.38

Shops

0.11

-0.15

0.36

Hlth

0.05

-0.47

0.36

Utils

0.06

-0.35

0.40

Other

0.11

-0.01

0.37

Average

0.08

-0.01

0.38

We can see results that echo our expectations: the speed of mean-reversion is significantly lower for value than momentum. In fact, the average theta is less than 1/10^{th}.

The math behind an OU-process also lets us calculate the half-life of the mean-reversion, allowing us to translate the speed of mean reversion to a more interpretable measure: time.

The half-life for momentum z-scores is 0.27 years, or about 3.28 months. The half-life for valuation z-scores is 3.76 years, or about 45 months. These values more or less line up with our intuition about turnover in momentum versus value portfolios: we expect to hold momentum stocks for a few months but value stocks for a few years.

Another way to analyze this data is by looking at how long the relative ranking of a given industry group stays consistent in its valuation or momentum metric. Based upon our data, we find that valuation ranks stayed constant for an average of approximately 120 trading days, while the average length of time an industry group held a consistent momentum rank was only just over 50 days.

Maturity’s Influence on Integration The scatter plots drawn by FFPS are deceiving because they only show a single point in time. What they fail to show is how the locations of the dots change over time.

With the expectation that momentum scores will change more rapidly than valuation scores, we would expect to see points move more rapidly up and down along the Y-axis than we would see them move left and right along the X-axis. Given this, our hypothesis is that changes in our inclusion score are driven more significantly by changes in our momentum score.

To explore this, we create an integration score, which is simply the sum of the valuation and momentum z-scores. Those industries in the top 30% of integration scores at any time are held by the integrated portfolio.

To distill the overall impact of momentum score changes versus valuation score changes, we need to examine the absolute value of these changes. For example, if the momentum score change was +0.5 and the valuation score change was -0.5, the overall integration score change is 0. Both momentum and value, in this case, contributed equally (or, contributed 50% each), to the overall score change.

So a simple formula for measuring the relative percentage contribution to score change is:

If value and momentum score changes contributed equally, we would expect the average contribution to equal 50%.

The average contribution based upon our analysis is 72.18% (with a standard error of 0.24%). The interquartile range is 59.02% to 91.19% and the median value is 79.47%.

Put simply: momentum score changes are a much more significant contributor to integration score changes than valuation score changes are.

We find that this effect is increased when we examine only periods when an industry is added or deleted from the integrated portfolio. In these periods, the average contribution climbs to 78.46% (with a standard error of 0.69%), with an interquartile range of 70.28% to 94.46% and a median value of 85.57%.

Changes in the momentum score contribute much more significantly than value score changes.

Integration: More Screen than Tilt? The objective of the integrated portfolio approach is to find securities with the best blend of characteristics.

In reality, because one set of characteristics changes much more slowly, certain securities can be sidelined for prolonged periods of time.

Let’s consider a simplified example. Every year, the 10 industry groups are assigned a random, but unique, value score between 1 and 10.

Similarly, every month, the 10 industry groups are assigned a random, but unique, momentum score between 1 and 10.

The integration score for each industry group is calculated as the sum of these two scores. Each month, the top 3 scoring industry groups are held in the integrated portfolio.

What is the probability of an industry group being in the integrated portfolio, in any given month, if it has a value score of 1? What about 2? What about 10? Numerical simulation gives us the following probabilities:

So if these are the probabilities of an industry group being selected in a given month given a certain value score, what is the probability of an industry group not being selected into the integrated portfolio at all during the year it has a given value score?

If an industry group starts the year with a value score of 1, there is 99.1% probability it will never being selected into the integrated portfolio all year.

Conclusion While we believe this topic deserves a significantly deeper dive (one which we plan to perform over the coming months), we believe the cursory analysis highlights a very important point: an integrated approach runs a significant risk of being more heavily influenced by higher turnover factors. While FFPS believe there are first order benefits to the integrated approach, we think the jury is still out and that those first order effects may actually be simply due to an increased exposure to higher turnover factors. Until more a more substantial understanding of the integrated approach is established, we continue to believe that a mixed approach is prudent. After all, if we don’t understand how a portfolio is built and the source of the returns it generates, how can we expect to manage risk?

[1] Z-scoring standardizes, on a relative basis, what would otherwise be arbitrary values.

[2] We use yield versus historical as our measure for valuation as a matter of convenience. However, there are two theoretical arguments justifying this choice. First, the most common measure of value is book-to-market (B/M), which assumes that fair valuation of a company is its book value. Another such model is the dividend discount model. If we assume a constant growth rate of dividends and a constant cost of capital for the company, then book value should be proportional to total dividends, or, equivalently, book-to-market proportional to dividend yield. Similarly, if you assume a constant long-term payout ratio, dividends per share are proportional to earnings per share, which makes yield inversely proportional to price-to-earnings, a popular valuation ratio.

[3] We used maximum likelihood estimation to calculate these figures.

## Multi-Factor: Mix or Integrate?

By Corey Hoffstein

On July 11, 2016

In Portfolio Construction, Risk & Style Premia, Weekly Commentary

This blog post is available as a PDF here.SummaryThe Importance of Factor MaturityCliff Asness, founder of AQR, recently published a paper titled

My Factor Philippic. This paper was written in response to the recently popularized articleHow Can “Smart Beta” Go Horribly Wrong?which was co-authored by Robert Arnott, co-founder of Research Affiliates.Arnott argues that many popular factors are currently historically overvalued and, furthermore, that the historical excess return offered by some recently popularized factors can be entirely explained by rising valuation trends in the last 30 years.

Caveat emptor,warns Arnott: valuationsalwaysmatter.Much to our delight (after all, who doesn’t like to see two titans of industry go at it?), Asness disagrees.

One of the primary arguments laid out by Asness is that valuation is a meaningless predictor for factors with high turnover.

The intuition behind this argument is simple: while valuations may be a decent predictor of forward annualized returns for broad markets over the next 5-to-10 years, the approach only works because the basket of securities stays mostly constant. For example, valuations for U.S. equities may be a good predictor because we expect the vast majority of the basket of U.S. equities to stay constant over the next 5-to-10 years.

The same is

not truefor many factors. For example, let’s consider a high turnover factor like momentum.Valuations of a momentum basket today are a poor predictor of annualized returns of a momentum strategy over the next 5-to-10 years because the basket of securities held could be 100% different three months from now.

Unless the same securities are held in the basket, valuation headwinds or tailwinds will not necessarily be realized.

For the same reason, valuation is also poor as an explanatory variable of factor returns. Asness argues that Arnott’s warning of valuation being the secret driver of factor returns is unwarranted in high turnover factors.

Multi-Factor: Mix or Integrate?On July 2

^{nd}, Fitzgibbons, Friedman, Pomorski, and Serban (FFPS) – again from AQR – published a paper titledLong-Only Style Investing: Don’t Just Mix, Integrate.The paper attempts to conclude the current debate about the best way to build multi-factor portfolios. The first approach is to mix, where a portfolio is built by combining stand-alone factor portfolios. The second approach is to integrate, where a portfolio is built by selecting securities that have simultaneously strong exposure to multiple factors at once.

A figure from the paper does a good job of illustrating the difference. Below, a hypothetical set of stocks is plotted based upon their current valuation and momentum characteristics.

In the top left, a portfolio of deep value stocks is selected. In the top right, the

mixapproach is demonstrated, where the deepest value and the highest momentum stocks are selected.In the bottom left, the integrated approach is demonstrated, where the securities simultaneously exhibiting strong valuation and momentum characteristics are selected.

Finally, in the bottom right we can see how these two approaches differ: with yellow securities being those only found in the mix portfolio and blue securities being found only in the integrated portfolio.

It is worth noting that the ETF industry has yet to make up its mind on the right approach.

GlobalX and Goldman Sachs prefer the mix approach in their ETFs (SCIU / GSLC) while JPMorgan and iShares prefer the integrate approach (JPUS / LRGF).

The argument made by those taking the integrated approach is that they are looking for securities with well-rounded exposures rather than those with extreme singular exposures. Integrators argue that this approach helps them avoid holding securities that might

cancel each other out.If we look back towards the mix example above (top right), we can see that many securities selected due to strength in one factor are actually quite poor in the other.Integrators claim that this inefficiency can create a drag in the mix portfolio. Why hold something with strong momentum if it has a very poor valuation score that is only going to offset it?

We find it somewhat ironic that FFPS and Asness both publish for AQR, because we would argue that Asness’s argument points out the fundamental flaw in the theory outlined by integrators. Namely: the horizons over which the premia mature differ.

Therefore, a strong positive loading in a factor like momentum is not necessarily offset by a strong negative loading in a factor like value. Furthermore, by

integratingwe run the risk of the highest turnover factor actually dominating the integrated selection process.DataIn the rest of this commentary, we will be using industry data from the Kenneth French data library. For momentum scores, we calculate 12 one-month total return and calculate cross-sector z-scores[1]. For valuation scores, we calculate a normalized 5-year dividend yield score and then calculate cross-sector z-scores.[2]

Do Factor Premia Actually Mature at Different Time Periods?In his paper, Asness referenced the turnover of a factor portfolio as an important variable. We prefer to think of high turnover factors as factors whose premium matures more quickly.

For example, if we buy a stock because it has high relative momentum, our expectation is that we will likely hold it for longer than a day, but likely much shorter than a year. Therefore, a strategy built off relative momentum will likely have high turnover because the premium matures quickly.

On the other hand, if we buy a value stock, our expectation is that we will have to hold it for up to several years for valuations to adequately reverse. This means that the value premium takes longer to mature – and the strategy will likely have lower turnover.

We can see this difference in action by looking at how valuation and momentum scores change over time.

We see similar pictures for other industries. Yet, looks can be deceiving and the human brain is excellent at finding patterns where there are none (especially when we

wantto see those patterns). Can we actually quantify this difference?One way is to try to build a model that incorporates both the randomness of movement and how fast these scores mean-revert. Fitting our data to this model would tell us about how quickly each premium matures.

One such model is called an Ornstein-Uhlenbeck process (“OU process”). An OU process follows the following stochastic differential equation:

To translate this into English using an example: the change in value z-score from one period to the next can be estimated as a “magnetism” back to fair value plus some randomness. In the equation, theta tells us how strong this magnetism is, mu tells us what fair value is, and sigma tells us how much influence the randomness has.

For our momentum and valuation z-scores, we would expect mu to be near-zero, as over the long-run we would not expect a given sector to exhibit significantly more or less relative momentum or relative cheapness/richness than peer sectors.

Given that we also believe that the momentum premium is realized over a shorter horizon, we would also expect that theta – the strength of the magnetism, also called the

speed of mean reversion– will be higher. Since that strength of magnetism is higher, we will also need sigma – the influence of randomness – to be larger to overcome it.So how to the numbers play out?[3]

For the momentum z-scores:

ThetaMuSigmaFor the valuation z-scores:

ThetaMuSigmaWe can see results that echo our expectations: the speed of mean-reversion is

significantlylower for value than momentum. In fact, the average theta is less than 1/10^{th}.The math behind an OU-process also lets us calculate the

half-lifeof the mean-reversion, allowing us to translate the speed of mean reversion to a more interpretable measure: time.The half-life for momentum z-scores is 0.27 years, or about 3.28 months. The half-life for valuation z-scores is 3.76 years, or about 45 months. These values more or less line up with our intuition about turnover in momentum versus value portfolios: we expect to hold momentum stocks for a few months but value stocks for a few years.

Another way to analyze this data is by looking at how long the relative ranking of a given industry group stays consistent in its valuation or momentum metric. Based upon our data, we find that valuation ranks stayed constant for an average of approximately 120 trading days, while the average length of time an industry group held a consistent momentum rank was only just over 50 days.

Maturity’s Influence on IntegrationThe scatter plots drawn by FFPS are deceiving because they only show a single point in time. What they fail to show is how the locations of the dots change over time.

With the expectation that momentum scores will change more rapidly than valuation scores, we would expect to see points move more rapidly up and down along the Y-axis than we would see them move left and right along the X-axis.

Given this, our hypothesis is that changes in our inclusion score are driven more significantly by changes in our momentum score.

To explore this, we create an integration score, which is simply the sum of the valuation and momentum z-scores. Those industries in the top 30% of integration scores at any time are held by the integrated portfolio.

To distill the overall impact of momentum score changes versus valuation score changes, we need to examine the absolute value of these changes. For example, if the momentum score change was +0.5 and the valuation score change was -0.5, the overall integration score change is 0. Both momentum and value, in this case, contributed equally (or, contributed 50% each), to the overall score change.

So a simple formula for measuring the relative percentage contribution to score change is:

If value and momentum score changes contributed equally, we would expect the average contribution to equal 50%.

The average contribution based upon our analysis is 72.18% (with a standard error of 0.24%). The interquartile range is 59.02% to 91.19% and the median value is 79.47%.

Put simply: momentum score changes are a much more significant contributor to integration score changes than valuation score changes are.

We find that this effect is increased when we examine only periods when an industry is added or deleted from the integrated portfolio. In these periods, the average contribution climbs to 78.46% (with a standard error of 0.69%), with an interquartile range of 70.28% to 94.46% and a median value of 85.57%.

Changes in the momentum score contribute much more significantly than value score changes.

Integration: More Screen than Tilt?The objective of the integrated portfolio approach is to find securities with the best blend of characteristics.

In reality, because one set of characteristics changes much more slowly, certain securities can be sidelined for prolonged periods of time.

Let’s consider a simplified example. Every year, the 10 industry groups are assigned a random, but unique, value score between 1 and 10.

Similarly, every month, the 10 industry groups are assigned a random, but unique, momentum score between 1 and 10.

The integration score for each industry group is calculated as the sum of these two scores. Each month, the top 3 scoring industry groups are held in the integrated portfolio.

What is the probability of an industry group being in the integrated portfolio, in any given month, if it has a value score of 1? What about 2? What about 10?

Numerical simulation gives us the following probabilities:

So if these are the probabilities of an industry group being selected in a given month given a certain value score, what is the probability of an industry group

notbeing selected into the integrated portfolio at all during the year it has a given value score?If an industry group starts the year with a value score of 1, there is 99.1% probability it will never being selected into the integrated portfolio all year.

ConclusionWhile we believe this topic deserves a significantly deeper dive (one which we plan to perform over the coming months), we believe the cursory analysis highlights a very important point: an integrated approach runs a significant risk of being more heavily influenced by higher turnover factors. While FFPS believe there are first order benefits to the integrated approach, we think the jury is still out and that those first order effects may actually be simply due to an increased exposure to higher turnover factors. Until more a more substantial understanding of the integrated approach is established, we continue to believe that a mixed approach is prudent. After all, if we don’t understand how a portfolio is built and the source of the returns it generates, how can we expect to manage risk?