Factors play an important role for quantitative portfolio construction.

How a factor is defined and how a factor portfolio is constructed play important roles in the results achieved.

Naively constructed portfolios – such as most “academic” factors – can lead to latent style exposures and potentially large unintended bets.

Through numerical techniques, we can seek to develop pure factors that provide targeted exposure to one style while neutralizing exposure to the rest.

In this research note, we implement a regression-based and optimized-based approach to achieving pure factor portfolios and report the results achieved.

Several years ago, we penned a note titled Separating Ingredients and Recipe in Factor Investing (May 21, 2018). In the note we discussed why we believe it is important for investors and allocators to consider not just what ingredients are going into their portfolios – i.e. securities, styles, asset classes, et cetera – but the recipe by which those ingredients are combined. Far too often the ingredients are given all the attention, but mistake salt for sugar and I can guarantee that you’re not going to enjoy your cake, regardless of the quality of the salt.

As an example, the note focused on constructing momentum portfolios. By varying the momentum measure, lookback period, rebalance frequency, portfolio construction, weighting scheme, and sector constraints we constructed over 1,000 momentum strategies. The resulting dispersion between the momentum strategies was more-often-than-not larger than the dispersion between generic value (top 30% price-to-book) and momentum (top 30% by 12-1 prior returns).

Yet having some constant definition for factor portfolios is desirable for a number of reasons, including both alpha signal generation and return attribution.

One potential problem for naïve factor construction – e.g. a simple characteristic rank-sort – is that it can lead to time-varying correlations between factors.

For example, below we plot the correlation between momentum and value, size, growth, and low volatility factors. We can see significant time-varying behavior; for example, in 2018 momentum and low volatility exhibited moderate negative correlation, while in 2019 they exhibited significant positive correlation.

The risk of time-varying correlations is that they can potentially leading to the introduction of unintended bets within single- or multi-factor portfolios or make it more difficult to determine with accuracy a portfolio’s sensitivity to different factors.

More broadly, low and stable correlations are preferable – assuming they can be achieved without meaningfully sacrificing expected returns – because they should allow investors to develop portfolios with lower volatility and higher information ratios.

Naively constructed equity styles can also exhibit time-varying correlations to traditional economic factors (e.g. interest rate risk), risk premia (e.g. market beta) or risk factors (e.g. sector or country exposure).

But equity styles can even exhibit time-varying sensitivities to themselves. For example, below we multiply the weights of naively constructed long/short style portfolios against the characteristic z-scores for the underlying holdings. As the characteristics of the underlying securities change, so does the actual weighted characteristic score of the portfolio. While some signals stay quite steady (e.g. size), others can vary substantially; sometimes value is just more value-y.

Source: Sharadar. Calculations by Newfound Research. Factor portfolios self-financing long/short portfolios that are long the top quintile and short the bottom quintile of securities, equally weighted and rebalanced monthly, ranked based upon their specific characteristics (see below).

In the remainder of this note, we will explore two approaches to constructing “pure” factor portfolios that can be used to generate a factor portfolio that neutralizes exposure to risk factors and other style premia.

Using the S&P 500 as our parent universe, we will construct five different factors defined by the security characteristics below:

Value (VAL): Earnings yield, free cash flow yield, and revenue yield.

All characteristics are first cross-sectionally winsorized at the 5^{th} and 95^{th} percentiles, then cross-sectionally z-scored, and finally averaged (if a style is represented by multiple scores) to create a single score for each security.

Naively constructed style benchmarks are 100% long the top-ranked quintile of securities and 100% short the bottom-ranked quintile, with securities receiving equal weights.

Source: Sharadar. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.

Factor Mimicry with Fama-MacBeth

Our first approach to designing “pure” factor portfolios is inspired by Fama-MacBeth (1973)^{3}. Fama-MacBeth regression is a two-step approach:

Regress each security against proposed risk factors to determine the security’s beta for that risk factor;

Regress all security returns for a fixed time period against the betas to determine the risk premium for each factor.

Similarly, we will assume a factor model where the return for a given security can be defined as:

Where R_{m} is the return of the market and RF_{j} is the return for some risk factor. In this equation, the betas define a security’s sensitivity to a given risk factor. However, instead of using the Fama-MacBeth two-step approach to solve for the factor betas, we can replace the betas with factor characteristic z-scores.

Using these known scores, we can both estimate the factor returns using standard regression^{4} and extract the weights of the factor mimicking portfolios. The upside to this approach is that each factor mimicking portfolios will, by design, have constant unit exposure to its specific factor characteristic and zero exposure to the others.

Here we should note that unless an intercept is added to the regression equation, the factor mimicking portfolios will be beta-neutral but not dollar-neutral. This can have a substantial impact on factors like low volatility (VOL), where we expect our characteristics to be informative about risk-adjusted returns but not absolute returns. We can see the impact of this choice in the factor return graphs plotted below.^{5}

Furthermore, by utilizing factor z-scores, this approach will neutralize characteristic exposure, but not necessarily return exposure. In other words, correlations between factor returns may not be zero. A further underlying assumption of this construction is that an equal-weight portfolio of all securities is style neutral. Given that equal-weight portfolios are generally considered to embed positive size and value tilts, this is an assumption we should be cognizant of.

Source: Sharadar. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.

Attempting to compare these mimic portfolios versus our original naïve construction is difficult as they target a constant unit of factor exposure, varying their total notional exposure to do so. Therefore, to create an apples-to-apples comparison, we adjust both sets of factors to target a constant volatility of 5%.

Source: Sharadar. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.

We can see that neutralizing market beta and other style factors leads to an increase in annualized return for value, size, momentum, and quality factors, leading to a corresponding increase in information ratio. Unfortunately, none of these results are statistically significant at a 5% threshold.

Nevertheless, it may still be informative to take a peek under the hood to see how the weights shook out. Below we plot the average weight by security characteristic percentile (at each rebalance, securities are sorted into percentile score bins and their weights are summed together; weights in each bin are then averaged over time).

Before reviewing the weights, however, it is important to recall that each portfolio is designed to capture a constant unit exposure to a style and therefore total notional exposure will vary over time. To create a fairer comparison across factors, then, we scale the weights such that each leg has constant 100% notional exposure.

As we would generally expect, all the factors are over-weight high scoring securities and underweight low scoring securities. What is interesting to note, however, is that the shapes by which they achieve their exposure are different. Value, for example leans strongly into top decile securities whereas quality leans heavily away (i.e. shorts) the bottom decile. Unlike the other factors which are largely positively sloped in their weights, low volatility exhibits fairly constant positive exposure above the 50^{th} percentile.

What may come as a surprise to many is how diversified the portfolios appear to be across securities. This is because the regression result is equivalent to minimizing the sum of squared weights subject to target exposure constraints.

Source: Sharadar. Calculations by Newfound Research.

While we focused specifically on neutralizing style exposure, this approach can be extended to also neutralize industry / sector exposure (e.g. with dummy variables), region exposure, and even economic factor exposure. Special care must be taken, however, to address potential issues of multi-collinearity.

Pure Quintile Portfolios with Optimization

Liu (2016)^{6} proposes an alternative means for constructing pure factor portfolios using an optimization-based approach. Specifically, long-only quintile portfolios are constructed such that:

They minimize the squared sum of weights;

Their weighted characteristic exposure for the target style is equal to the weighted characteristic exposure of a naïve, equally-weighted, matching quintile portfolio; and

Weighted characteristic exposure for non-targeted styles equals zero.

While the regression-based approach was fast due to its closed-form solution, an optimization-based approach can potentially allow for greater flexibility in objectives and constraints.

Below we replicate the approach proposed in Liu (2016) and then create dollar-neutral long/short factor portfolios by going long the top quintile portfolio and short the bottom quintile portfolio. Portfolios are re-optimized and rebalanced monthly. Unlike the regression-based approach, however, these portfolios do not seek to be beta-neutral.

Source: Sharadar. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.

We can see that the general shapes of the factor equity curves remain largely similar to the naïve implementations. Unlike the results reported in Liu (2016), however, we measure a decline in return among several factors (e.g. value and size). We also find that annualized volatility is meaningfully reduced for all the optimized portfolios; taken together, information ratio differences are statistically indistinguishable from zero.

Source: Sharadar. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.

As with the regression-based approach, we can also look at the average portfolio exposures over time to characteristic ranks. Below we plot these results for both the naïve and optimized Value quintiles. We can see that the top and bottom quintiles lean heavily into top- and bottom-decile securities, while 2^{nd}, 3^{rd}, and 4^{th} quintiles had more diversified security exposure on average. Similar weighting profiles are displayed by the other factors.

Source: Sharadar. Calculations by Newfound Research.

Conclusion

Factors are easy to define in general but difficult to define explicitly. Commonly accepted academic definitions are easy to construct and track, but often at the cost of inconsistent style exposure and the risk of latent, unintended bets. Such impure construction may lead to time-varying correlations between factors, making it more difficult for managers to manage risk as well as disentangle the true source of returns.

In this research note we explored two approaches that attempt to correct for these issues: a regression-based approach and an optimization-based approach. With each approach, we sought to eliminate non-target style exposure, resulting in a pure factor implementation.

Despite a seemingly well-defined objective, we still find that how “purity” is defined can lead to different results. For example, in our regression-based approach we targeted unit style exposure and beta-neutrality, allowing total notional exposure to vary. In our optimization-based approach, we constructed long-only quintiles independently, targeting the same weighted-average characteristic exposure as a naïve, equal-weight factor portfolio. We then built a long/short implementation from the top and bottom quintiles. The results between the regression-based and optimization-based approaches were markedly different.

And, statistically, not any better than the naïve approaches.

This is to say nothing of other potential choices we could make about defining “purity.” For example, what assumptions should we make about industry, sector, or regional exposures?

More broadly, is “purity” even desirable?

In Do Factors Market Time? (June 5, 2017) we demonstrated that beta timing was an unintentional byproduct of naïve value, size, and momentum portfolios and had actually been a meaningful tailwind for value from 1927-1957. Some factors might actually be priced across industries rather than just within them (Vyas and van Baren (2019)^{7}). Is the chameleon-like nature of momentum to rapidly tilt towards whatever style, sector, or theme has been recently outperforming a feature or a bug?

And this is all to say nothing of the actual factor definitions we selected.

While impurity may be a latent risk for factor portfolios, we believe this research suggests that purity is in the eye of the beholder.

## Pursuing Factor Purity

By Corey Hoffstein

On January 6, 2020

In Risk & Style Premia, Weekly Commentary

This post is available as a PDF download here.## Summary

Several years ago, we penned a note titled

Separating Ingredients and Recipe in Factor Investing(May 21, 2018). In the note we discussed why we believe it is important for investors and allocators to consider not just what ingredients are going into their portfolios – i.e. securities, styles, asset classes, et cetera – but the recipe by which those ingredients are combined. Far too often the ingredients are given all the attention, but mistake salt for sugar and I can guarantee that you’re not going to enjoy your cake, regardless of the quality of the salt.As an example, the note focused on constructing momentum portfolios. By varying the momentum measure, lookback period, rebalance frequency, portfolio construction, weighting scheme, and sector constraints we constructed over 1,000 momentum strategies. The resulting dispersion between the momentum strategies was more-often-than-not larger than the dispersion between generic value (top 30% price-to-book) and momentum (top 30% by 12-1 prior returns).

Yet having some constant definition for factor portfolios is desirable for a number of reasons, including both alpha signal generation and return attribution.

One potential problem for naïve factor construction – e.g. a simple characteristic rank-sort – is that it can lead to time-varying correlations between factors.

For example, below we plot the correlation between momentum and value, size, growth, and low volatility factors. We can see significant time-varying behavior; for example, in 2018 momentum and low volatility exhibited moderate negative correlation, while in 2019 they exhibited significant positive correlation.

The risk of time-varying correlations is that they can potentially leading to the introduction of unintended bets within single- or multi-factor portfolios or make it more difficult to determine with accuracy a portfolio’s sensitivity to different factors.

More broadly, low and stable correlations are preferable – assuming they can be achieved without meaningfully sacrificing expected returns – because they should allow investors to develop portfolios with lower volatility and higher information ratios.

Naively constructed equity styles can also exhibit time-varying correlations to traditional economic factors (e.g. interest rate risk), risk premia (e.g. market beta) or risk factors (e.g. sector or country exposure).

But equity styles can even exhibit time-varying sensitivities to

themselves.For example, below we multiply the weights of naively constructed long/short style portfolios against the characteristic z-scores for the underlying holdings. As the characteristics of the underlying securities change, so does the actual weighted characteristic score of the portfolio. While some signals stay quite steady (e.g. size), others can vary substantially; sometimes value is just more value-y.Source: Sharadar. Calculations by Newfound Research. Factor portfolios self-financing long/short portfolios that are long the top quintile and short the bottom quintile of securities, equally weighted and rebalanced monthly, ranked based upon their specific characteristics (see below).In the remainder of this note, we will explore two approaches to constructing “pure” factor portfolios that can be used to generate a factor portfolio that neutralizes exposure to risk factors and other style premia.

Using the S&P 500 as our parent universe, we will construct five different factors defined by the security characteristics below:

^{1}, negative accruals ratio, negative leverage ratio^{2}.All characteristics are first cross-sectionally winsorized at the 5

^{th}and 95^{th}percentiles, then cross-sectionally z-scored, and finally averaged (if a style is represented by multiple scores) to create a single score for each security.Naively constructed style benchmarks are 100% long the top-ranked quintile of securities and 100% short the bottom-ranked quintile, with securities receiving equal weights.

Source: Sharadar. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.Factor Mimicry with Fama-MacBethOur first approach to designing “pure” factor portfolios is inspired by Fama-MacBeth (1973)

^{3}. Fama-MacBeth regression is a two-step approach:Similarly, we will assume a factor model where the return for a given security can be defined as:

Where R

_{m}is the return of the market and RF_{j}is the return for some risk factor. In this equation, the betas define a security’s sensitivity to a given risk factor. However, instead of using the Fama-MacBeth two-step approach to solve for the factor betas, we can replace the betas with factor characteristic z-scores.Using these known scores, we can both estimate the factor returns using standard regression

^{4}and extract the weights of the factor mimicking portfolios. The upside to this approach is that each factor mimicking portfolios will, by design, have constant unit exposure to its specific factor characteristic and zero exposure to the others.Here we should note that unless an intercept is added to the regression equation, the factor mimicking portfolios will be

beta-neutral but notdollar-neutral. This can have a substantial impact on factors like low volatility (VOL), where we expect our characteristics to be informative about risk-adjusted returns but not absolute returns. We can see the impact of this choice in the factor return graphs plotted below.^{5}Furthermore, by utilizing factor z-scores, this approach will neutralize

characteristicexposure, but not necessarily return exposure. In other words, correlations between factor returns may not be zero. A further underlying assumption of this construction is that anequal-weightportfolio of all securities is style neutral. Given that equal-weight portfolios are generally considered to embed positive size and value tilts, this is an assumption we should be cognizant of.Source: Sharadar. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.Attempting to compare these mimic portfolios versus our original naïve construction is difficult as they target a constant unit of factor exposure, varying their total notional exposure to do so. Therefore, to create an apples-to-apples comparison, we adjust both sets of factors to target a constant volatility of 5%.

Source: Sharadar. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.We can see that neutralizing market beta and other style factors leads to an increase in annualized return for value, size, momentum, and quality factors, leading to a corresponding increase in information ratio. Unfortunately, none of these results are statistically significant at a 5% threshold.

Nevertheless, it may still be informative to take a peek under the hood to see how the weights shook out. Below we plot the average weight by security characteristic percentile (at each rebalance, securities are sorted into percentile score bins and their weights are summed together; weights in each bin are then averaged over time).

Before reviewing the weights, however, it is important to recall that each portfolio is designed to capture a constant unit exposure to a style and therefore total notional exposure will vary over time. To create a fairer comparison across factors, then, we scale the weights such that each leg has constant 100% notional exposure.

As we would generally expect, all the factors are over-weight high scoring securities and underweight low scoring securities. What is interesting to note, however, is that the shapes by which they achieve their exposure are different. Value, for example leans strongly into top decile securities whereas quality leans heavily away (i.e. shorts) the bottom decile. Unlike the other factors which are largely positively sloped in their weights, low volatility exhibits fairly constant positive exposure above the 50

^{th}percentile.What may come as a surprise to many is how diversified the portfolios appear to be across securities. This is because the regression result is equivalent to minimizing the sum of squared weights subject to target exposure constraints.

Source: Sharadar. Calculations by Newfound Research.While we focused specifically on neutralizing style exposure, this approach can be extended to also neutralize industry / sector exposure (e.g. with dummy variables), region exposure, and even economic factor exposure. Special care must be taken, however, to address potential issues of multi-collinearity.

Pure Quintile Portfolios with OptimizationLiu (2016)

^{6}proposes an alternative means for constructing pure factor portfolios using an optimization-based approach. Specifically, long-only quintile portfolios are constructed such that:While the regression-based approach was fast due to its closed-form solution, an optimization-based approach can potentially allow for greater flexibility in objectives and constraints.

Below we replicate the approach proposed in Liu (2016) and then create dollar-neutral long/short factor portfolios by going long the top quintile portfolio and short the bottom quintile portfolio. Portfolios are re-optimized and rebalanced monthly. Unlike the regression-based approach, however, these portfolios do not seek to be beta-neutral.

We can see that the general shapes of the factor equity curves remain largely similar to the naïve implementations. Unlike the results reported in Liu (2016), however, we measure a decline in return among several factors (e.g. value and size). We also find that annualized volatility is meaningfully reduced for all the optimized portfolios; taken together, information ratio differences are statistically indistinguishable from zero.

As with the regression-based approach, we can also look at the average portfolio exposures over time to characteristic ranks. Below we plot these results for both the naïve and optimized Value quintiles. We can see that the top and bottom quintiles lean heavily into top- and bottom-decile securities, while 2

^{nd}, 3^{rd}, and 4^{th}quintiles had more diversified security exposure on average. Similar weighting profiles are displayed by the other factors.Source: Sharadar. Calculations by Newfound Research.ConclusionFactors are easy to define in general but difficult to define explicitly. Commonly accepted academic definitions are easy to construct and track, but often at the cost of inconsistent style exposure and the risk of latent, unintended bets. Such impure construction may lead to time-varying correlations between factors, making it more difficult for managers to manage risk as well as disentangle the true source of returns.

In this research note we explored two approaches that attempt to correct for these issues: a regression-based approach and an optimization-based approach. With each approach, we sought to eliminate non-target style exposure, resulting in a pure factor implementation.

Despite a seemingly well-defined objective, we still find that how “purity” is defined can lead to different results. For example, in our regression-based approach we targeted unit style exposure and beta-neutrality, allowing total notional exposure to vary. In our optimization-based approach, we constructed long-only quintiles independently, targeting the same weighted-average characteristic exposure as a naïve, equal-weight factor portfolio. We then built a long/short implementation from the top and bottom quintiles. The results between the regression-based and optimization-based approaches were markedly different.

And, statistically, not any better than the naïve approaches.

This is to say nothing of other potential choices we could make about defining “purity.” For example, what assumptions should we make about industry, sector, or regional exposures?

More broadly, is “purity” even desirable?

In

Do Factors Market Time?(June 5, 2017)we demonstrated that beta timing was an unintentional byproduct of naïve value, size, and momentum portfolios and had actually been a meaningful tailwind for value from 1927-1957. Some factors might actually be priced across industries rather than just within them (Vyas and van Baren (2019)^{7}). Is the chameleon-like nature of momentum to rapidly tilt towards whatever style, sector, or theme has been recently outperforming a feature or a bug?And this is all to say nothing of the actual factor definitions we selected.

While impurity may be a latent risk for factor portfolios, we believe this research suggests that purity is in the eye of the beholder.