This blog post is available as a PDF here.

**Summary**

- The debate rages on over the application of valuation in factor-timing methods. Regardless, diversification remains a prudent recommendation.
- How to diversify multi-factor portfolios, however, remains up for debate.
- The ActiveBeta team at Goldman Sachs finds new evidence that composite diversification approaches can offer a higher information ratio than integrated approaches due to interaction effects at low-to-moderate factor concentration levels.
- At high levels, they find that integrated approaches have higher information ratios due to high levels of idiosyncratic risks in composite approaches.
- We return to old research and explore empirical evidence in FTSE Russell’s tilt-tilt approach to building an integrated multi-factor portfolio to determine if this multi-factor approach does deliver greater factor efficiency than a comparable composite approach.

The debate over factor timing between Cliff Asness and Rob Arnott rages on. This week saw Cliff publish a blog post titled *Factor Timing is Hard* providing an overview of his recently co-authored work *Contrarian Factor Timing is Deceptively Difficult*. Generally in academic research, you find a certain level of hedged decorum: authors rarely insult the quality of work, they just simply refute it with their own evidence.

This time, Cliff pulled no punches.

“In multiple online white papers, Arnott and co-authors present evidence in support of contrarian factor timing based on a plethora of mostly inapplicable, exaggerated, and poorly designed tests that also flout research norms.”

At the risk of degrading this weekly research commentary into a gossip column: Ouch. Burn.

We’ll be providing a much deeper dive into this continued factor-timing debate (as well as our own thoughts) in next week’s commentary.

In the meantime, at least there is one thing we can all agree on – including Cliff and Rob – factor portfolios are better diversified than not.

Except, as an industry, we cannot even agree how to diversify them.

**Diversifying Multi-Factor Portfolios: Composite vs. Integrated**

When it comes to building multi-factor portfolios, there are two camps of thought.

The first camp believes in a two-step approach. First, portfolios are built for each factor. To do this, securities are often given a score for each factor, and when a factor sleeve is built, securities with higher scores receive an overweight position while those with lower scores receive an underweight. After those portfolios are built, they are blended together to create a combined portfolio. As an example, a value / momentum multi-factor portfolio would be built by first constructing value and momentum portfolios, and then blending these two portfolios together. This approach is known as “mixed,” “composite,” or “portfolio blend.”

*Source: Ghayur, Heaney, and Platt (2016)*

The second camp uses a single-step approach. Securities are still given a score for each factor, but those scores are blended into a single aggregate value representing the overall strength of that security. A single portfolio is then built, guided by this blended value, overweighting securities with higher scores and underweighting securities with lower scores. This approach is known as “integrated” or “signal blend.”

*Source: Ghayur, Heaney, and Platt (2016)*

To re-hash the general debate:

- Portfolio blend advocates tend to prefer the simplicity, transparency, and control of the approach. Furthermore, there is a preponderance of evidence supporting single-factor portfolios, but research exploring the potential interaction effects of a signal blend approach is limited and therefore potentially introduces unknown risks.
- Signal blend advocates argue that a portfolio blend approach introduces inefficiencies: that by constructing each sleeve independently, securities with canceling factor scores can be introduced and dilute overall factor exposure. The general argument goes along the line of, “we want the decathlon athlete, not a team full of individual sport athletes.”

Long-time readers of our commentary may, at this point, be groaning; how is this topic not dead yet? After all, we’ve written about it numerous times in the past.

- In
*Beware Bad Multi-Factor Portfolios*we argued that the integrated approach was fundamentally flawed due to the different decay rates of factor alpha (which is equivalent to saying that factor portfolios turnover at different rates). By combining a slow-moving signal with a fast-moving signal, variance in the composite signal becomes dominated by the fast-moving signal.In retrospect, our choice of wording here was probably a bit too concrete. We believe our point still stands that care must be taken in integrated approaches because of relative turnover speed differences in different factors, but it is not an insurmountable hurdle in construction. - In
*Multi-Factor: Mix or Integrate?*we explored an AQR paper that advocated for an integrated approach. We found it ironic that this was published shortly after Cliff Asness had published an article discussing the turnover issues that make applying value-based timing difficult for factors like momentum – an argument similar to our past blog post.In this post, we continued to find evidence that integrated approaches ran the risk of being governed by high turnover factors. - In
*Is That Leverage in my Multi-Factor ETF?*we explored an argument made by FTSE Russell that an integrated approach offered implicit leverage effects, allowing you to use the same dollar to access multiple factors simultaneously.This is probably the best argument we have heard for multi-factor portfolios to date.Unfortunately, empirical evidence suggested that existing integrated multi-factor ETF portfolios did not offer significantly more factor exposure than composite peers.It is worth noting, however, that the data we were able to explore was limited as multi-factor portfolios are largely new. We were not even able to evaluate, for example, a FTSE Russell product despite the fact it was FTSE Russell making the argument.

- Feeling that our empirical test did not necessarily do justice to FTSE Russell’s argument, we wrote
*Capital Efficiency in Multi-Factor Portfolios.*If we were to make an argument for our most underrated article of 2016, this would be it – but that is probably because it was filled with obtuse mathematics.The point of the piece was to try to reconcile FTSE Russell’s argument from a theoretical basis. What we found, under some broad assumptions, was that under all cases, an integrated approach should provide at least as much, and generally much more, factor exposure than a mixed approach due to the implied leverage effect.

So, honestly, how much more can we say on this topic?

**New Evidence of Interaction Effects in Multi-Factor Portfolios**

Well the ActiveBeta Equity Strategies team at Goldman Sachs Asset Management published a paper late last year comparing the two approaches using Russell 1000 securities from January 1979 to June 2016.

Unlike our work, in which we compared composite and integrated portfolios built to match the percentage of stocks selected, Ghayur, Heaney, and Platt (2016) built portfolios to match factor exposure. Whereas we matched an integrated approach that picked the top 25% of securities with a composite approach where each sleeve picked the top 25%, Ghayur, Heaney, and Platt (2016) accounted for expected factor dilution by having the sleeves in the composite approach pick the top 12.5%.

Using this factor-exposure matching approach, their results are surprising. Rather than a definitive answer as to which approach is superior, they find that the portfolio blend approach offers a higher information ratio at lower levels of factor exposure (i.e. lower levels of active risk), while the signal blend approach offers a higher information ratio at higher levels of factor exposure (i.e. higher levels of active risk).

How can this be the case?

The answer comes down to interaction effects.

When a portfolio is built expecting more diluted overall factor exposure – e.g. to have lower tracking error to the index – the percentage overlap between securities in the composite and integrated approaches is higher. However, for more concentrated factor exposure, the overlap is lower.

*Source: Ghayur, Heaney, and Platt (2016)*

Advocates for an integrated approach have historically argued that securities found in Area 3 in the figure above would be a drag on portfolio performance. These are the securities found in a composite approach but not an integrated approach. The argument is that while high in one factor score, these securities are also very low in another, and including them in a portfolio only dilutes overall factor exposure via a canceling effect.

On the other hand, securities in Area 2, found only in the integrated approach, should increase factor exposure because you are getting securities with higher loadings on both factors simultaneously.

As it turns out, evidence suggests this is not the case.

In fact, for lower concentration factor portfolios, Ghayur, Heaney, and Platt (2016) find just the opposite.

*Source: Ghayur, Heaney, and Platt (2016)*

As it turns out, interaction effects give Area 3 *positive active returns* while Area 2 ends up delivering *negative active returns*. To quote,

“The securities held in the portfolio blend and the signal blend can be mapped to the 4×4 quartile matrix (Table 5). The portfolio blend holds securities in the top row (Q4 value) and second-to-last column (Q4 momentum). All buckets provide positive contributions to active return. The mapping is more complicated for the signal blend but is roughly consistent with the diagram in Figure 1 (i.e., holdings will be anything to the right of the diagonal line drawn from the top left to the bottom right of the 4×4 matrix). Examining contributions to active return and risk (not reported), we find that the signal blend suffers from not holding enough of the high value/low momentum (Q4/Q1) stocks and low value/high momentum (Q1/Q4) stocks. The signal blend also incurs significant risk from holding Q3 value/Q3 momentum stocks, which have a negative active return (-0.4%). High momentum/high value (Q4/Q4) stocks earn the highest active return. These stocks offer a greater benefit to the portfolio blend as they are double-weighted.

In terms of active risk contributions, we note that low momentum/high value (Q1/Q4) stocks have a net positive exposure to value, while high momentum/low value (Q4/Q1) stocks have a net positive exposure to momentum. These two groups exhibit a high negative active return correlation and are diversifying (i.e., reduce active risk), while delivering positive active returns. As such, the assertion that avoiding securities with offsetting factor exposures improves portfolio performance is not entirely correct. If factor payoffs depict strong interaction effects, then holding such securities may actually be beneficial, and the portfolio blend benefits from investing in such securities. These contextual relationships are also present to varying degrees in other factor pairings.”

When factor concentration is higher, however, the increased degree of idiosyncratic risk found in Area 1 of the composite approach outweighs the interaction benefits found in Area 3. This effect can be seen in the table below. We see that Shared Securities under Portfolio Blend have an increased Active Return Contribution in comparison to the Signal Blend but also significantly higher Active Risk Contribution. This is due to the fact that Shared Securities represent only 45% of the active weight in the High Factor Exposure example for the Signal Blend approach, but 72% of the weight in the Portfolio Blend. The large portfolio concentration on just a few securities ultimately introduces too much idiosyncratic risk.

*Source: Ghayur, Heaney, and Platt (2016)*

Furthermore, while Area 3 (Securities Held Only in Portfolio Blend) remains a positive contributor to Active Return, it does not have the negative Active Risk contribution as it did in the prior, low factor concentration example.

The broad result that Ghayur, Heaney, and Platt (2016) propose is simple: for low-to-moderate levels of factor exposures, a portfolio blend exhibits higher information ratios and for higher levels of factor exposure, a signal blend approach works better. That being said, we would be remiss if we didn’t point out that these types of conclusions are very dependent on the exact portfolio construction methodology used. There are varying qualities of approaches to building both portfolio blend and signal blend multi-factor portfolios, which brings us back full circle to…

**Re-Addressing FTSE Russell’s Tilt-Tilt Method**

In our initial empirical analysis of FTSE Russell’s leverage argument, we were unable to actually test the theory on FTSE Russell’s multi-factor approach itself due to a lack of data. In our analytical analysis, we used a standard integrated approach of averaging factor scores. FTSE Russell takes the integrated method a step further by introducing a “tilt-tilt” approach, where instead of averaging factor signals to create an integrated signal, they use a multiplicative approach.

This multiplicative approach, however, is not run on normally distributed variables (i.e. factor z-scores) as was the case in our own analysis (and GSAM paper discussed above), but rather on uniformly distributed scores between [0, 1].

This makes things analytically gnarly (e.g. instead of working with normal and chi-squared distributions, we’re working with Irwin-Hall and product of uniform distributions). Fortunately, we can employ a numerical approach to get an idea of what is going on. Below we simulate scores for two factors (assumed to be independent; let’s call them A and B) for 500 stocks and then plot the distribution of resulting integrated and tilt-tilt scoring methods using those scores.

*Source: Newfound Research. Simulation-based methodology.*

What we can see is that while the integrated approach looks somewhat normal (in fact, the Irwin-Hall distribution approaches normal as more uniform distributions are added; e.g. we incorporate more factors), the tilt-tilt distribution is single-tailed.

A standard next step in constructing an index would be to multiply these scores by benchmark weights and then normalize to come up with new, tilted weights. We can get a sense for how weights are scaled by taking each distribution above and dividing it by the distribution average and then plotting scores against each other.

*Source: Newfound Research. Simulation-based methodology.*

The grey dotted line provides guidance as to how the two methods differ. If a point is above the line, it means the integrated approach has a larger tilt; points below the line indicate that the tilt-tilt method has a larger tilt.

What we can see is that for scores below average, tilt-tilt is more aggressive at reducing exposure; similarly for scores above average, tilt-tilt is more aggressive at increasing exposure. In other words, the tilt-tilt approach works to aggressively increase the intensity of factor exposure.

Using index data for FTSE Russell factor indices, we can empirically test whether this approach actually captures the capital efficiency that integrated approaches should benefit from. Specifically, we can compare the FTSE Russell Comprehensive Factor Index (the tilt-tilt integrated multi-factor approach) versus an equal-weight composite of FTSE Russell single-factor indices. The FTSE Russell multi-factor approach includes value, size, momentum, quality, and low-volatility tilts, so our composite portfolio will be an equal-weight portfolio of long-only indices representing these factors.

To test for factor exposure, we regress both portfolios against long/short factors from AQR’s data library. Data covers the period of 9/30/2001 through 1/31/2017.

We find that factor loadings for the tilt-tilt method exceed those for the equal-weight composite.

*Source: FTSE Russell; AQR; calculations by Newfound Research.*

We also find they do an admirable job at capturing a significant share of factor exposure available that would be available in long-only single-factor indices. In other words, if instead of taking a composite approach – which we expect to be diluted – we decide to only purchase a long-only momentum portfolio, how much of that long-only momentum exposure can be re-captured by using this tilt-tilt integrated, multi-factor approach?

We find that for most factors, it is a significant proportion.

*Source: FTSE Russell; AQR; calculations by Newfound Research.*

(Note: The Bet-Against-Beta factor (“BAB”) is removed from this chart because the amount of the factor available in the FTSE Russell Volatility Factor Index was deemed to be insignificant, and so resulting relative proportions exceed 18x).

**Conclusion**

While the jury is still out on factor timing itself, diversifying across factors is broadly considered to be a prudent decision. How to implement that diversification remains in debate.

What makes the diversification concept in multi-factor investing unique, as compared to standard asset class diversification, is that through an integrated approach, implicit leverage can be accessed. The same dollar can be used to introduce multiple factor exposures simultaneously.

While this implicit leverage should lead to portfolios that empirically have more factor exposure, evidence suggests that is not always the case. A new paper by the ActiveBeta team at Goldman Sachs suggests that for low-to-moderate levels of factor exposure, a composite approach may be just as, if not more, effective as an integrated approach. More surprisingly is that this effectiveness comes from beneficial interaction effects exactly in the area of the portfolio that integrated advocates have claimed there to be a drag.

At higher concentration levels of factor exposure, however, the integrated approach is more efficient, as the composite approach appears to introduce too much idiosyncratic risk.

We bring the conversation full circle in this piece by going back to some original research we detailed last fall, testing FTSE Russell’s unique tilt-tilt methodology to integrated mutli-factor investing. In theory, the tilt-tilt method should increase the intensity of factor exposure compared to traditional integrated approaches. While we previously found little empirical evidence supporting the capital efficiency argument for integrated multi-factor ETFs versus composite peers, a test of FTSE Russell index data finds that the tilt-tilt method may provide a significant boost to factor exposure.

## Capital Efficiency in Multi-factor Portfolios

By Corey Hoffstein

On October 17, 2016

In Portfolio Construction, Risk & Style Premia, Weekly Commentary

This blog post is available as a PDF here.Summarycapital efficientthan mixed portfolios in realized return data for several multi-factor ETFs.in theory. We find that for some broad assumptions, they definitively are.This commentary is highly technical, relying on both probability theory and calculus, and requires rendering a significant number of equations. Therefore, it is only available as a PDF download.For those less inclined to read through mathematical proofs, the important takeaway is this: for some broad assumptions, integrated multi-factor portfolios are provably more capital efficient (e.g. more factor exposure for your dollar) than mixed approaches.