*This post is available as a PDF download here.*

# Summary

- The Fama French three-factor model provides a powerful tool for assessing exposures to equity risk premia in investment strategies.
- In this note, we explore alternative specifications of the value (HML) and size (SMB) factors using price-to-earnings, price-to-cash flow, and dividend yield.
- Running factor regressions using these alternate specifications on a suite of value ETFs and Newfound’s Systematic Value strategy, lead to a wide array of results, both numerically and directionally.
- While many investors consider the uncertainty of the parameter estimates from the regression using the three-factor model, most do not consider the uncertainty that comes from the assumption of how you construct the equity factors in the first place.
- Understanding the additional uncertainty is crucial for manager and investors who must consider what risks they are trying to measure and control by using tools like factor regression and make sure their assumptions align with their goals.

In their 1992 paper, *The Cross-Section of Expected Stock Returns*, Eugene Fama and Kenneth French outlined their three-factor model to explain stock returns.

While the Capital Asset Pricing Model (CAPM) only describes asset returns in relation to their exposure to the market’s excess return through the stock’s beta and identifies any return beyond that as alpha, Fama and French’s three-factor model reattributed some of that supposed alpha to exposures to a value factor (High-minus-low or HML) based on returns stratified by price-to-book ratios and a size factor (small-minus-big or SMB) based on returns stratified by market capitalization.

This gave investors a tool to judge investment strategies based on the loadings to these risk factors. A manager with a seemingly high alpha may have simply been investing in value and small-cap stocks historically.

The notion of compensated risk premia has also opened the floodgate of many additional factors from other researchers (such as momentum, quality, low beta, etc.) and even two more factors from Fama and French (investment and profitability).

A richer factor universe opens up a wide realm of possibilities for analysis and attribution. However, setting further developments aside and going back to the original three-factor model, we would be remiss if we didn’t dive a bit further into its specification.

At the highest level, we agree with treating “value” and “size” as risk factors, but there is more than one way to skin a factor.

What is “value”?

Fama and French define it using the price-to-book ratio of a stock. This seems legitimate for a broad swath of stocks, especially those that are very capital intensive – such as energy, manufacturing, and financial firms – but what about industries that have structurally lower book values and may have other potential price drivers? For example, a technology company might have significant intangible intellectual property and some utility companies might employ leverage, which decreases their book value substantially.

To determine value in these sectors, we might utilize ratios that account for sales, dividends, or earnings. But then if we analyzed these strategies using the Fama French three-factor model as it is specified, we might misjudge the loading on the value factor.

“Size” seems more straightforward. Companies with low market capitalizations are small. However, when we consider how the size factor is defined based on the value factor, there might even be some differences in SMB using different value metrics.

In this commentary, we will explore what happens when we alter the definition of value for the value factor (and hence the size factor) and see how this affects factor regressions of a sample of value ETFs along with our Systematic Value strategy.

**HML Factor Definitions**

In the standard version of the Fama French 3-factor model, HML is constructed as a self-financing long/short portfolio using a 2×3 sort on size and value. The investment universe is split in half based on market capitalization and in three parts (30%/40%/30%) based on valuation, in this base case, price-to-book ratio.

Using additional data from the Kenneth French Data Library and the same methodology, we will construct HML factors using sorts based on size and:

- Price-to-earnings ratios
- Price-to-cash flow ratios
- Dividend yields

The common inception date for all the factors is June 1951.

The chart below shows the growth of each of the four value factor portfolios.

*Source: Kenneth French Data Library. 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. *

Over the entire time period – and for many shorter time horizons – the standard HML factor using price-to-book does not even have the most attractive returns. Price-to-earnings and price-to-cash flow often beat it out.

On the other hand, the HML factor formed using dividend yields doesn’t look so hot.

One of the reasons behind this is that the small, low dividend yield companies performed much better than the small companies that were ranked poorly by the other value factors. We can see this effect borne out in the SMB chart for each factor, as the SMB factor for dividend yield performed the best.

(Recall that we mentioned previously how the Fama French way of defining the size factor is dependent on which value metric we use.)

*Source: Kenneth French Data Library. Calculations by Newfound Research. *

Looking at the statistical significance of each factor through its t-statistic, we can see that Price-to-Earnings and Price-to-Cash Flow yielded higher significance for the HML factor than Price-to-Book. And those two along with Dividend Yield all eclipsed the Price-to-Book construction of the SMB factor.

*T-Statistics for HML and SMB Using Various Value Metrics*

| Price-to-Book | Dividend Yield | Price-to-Earnings | Price-to-Cash Flow |

HML | 2.9 | 0.0 | 3.7 | 3.4 |

SMB | 1.0 | 2.4 | 1.6 | 1.9 |

Assuming that we do consider all metrics to be appropriate ways to assess the value of companies, even if possibly under different circumstances, how do different variants of the Fama French three-factor model change for each scenario with regression analysis?

**The Impact on Factor Regressions**

Using a sample of U.S. value ETFs and our Systematic Value strategy, we plot the loadings for the different versions of HML. The regressions are carried out using the trailing three years of monthly data ending on October 2019.

*Source: Tiingo, Kenneth French Data Library. Calculations by Newfound Research. **Past performance is not an indicator of future results. Returns represent live strategy results. Returns for the Newfound Systematic Value strategy are gross of all management fees and taxes, but net of execution fees. Returns for ETFs included in study are gross of any management fees, but net of underlying ETF expense ratios. Returns assume the reinvestment of all distributions.*

For each different specification of HML, the differences in the loading between investments is generally directionally consistent. For instance, DVP has higher loadings than FTA for all forms of HML.

However, sometimes this is not the case.

VLUE looks more attractive than VTV based on price-to-cash flow but not dividend yield. FTA is roughly equivalent to QVAL in terms of loading when price-to-book is used for HML, but it varies wildly when other metrics are used.

The tightest range for the four models for any of the investments is 0.09 (PWV) and the widest is 0.52 (QVAL). When we factor in that these estimates each have their own uncertainty, distinguishing which investment has the better value characteristic is tough. Decisions are commonly made on much smaller differences.

We see similar dispersion in the SMB loadings for the various constructions.

*Source: Tiingo, Kenneth French Data Library. Calculations by Newfound Research. **Past performance is not an indicator of future results. Returns represent live strategy results. Returns for the Newfound Systematic Value strategy are gross of all management fees and taxes, but net of execution fees. Returns for ETFs included in study are gross of any management fees, but net of underlying ETF expense ratios. Returns assume the reinvestment of all distributions.*

Many of these values are not statistically significant from zero, so someone who has a thorough understanding of uncertainty in regression would likely not draw a strict comparison between most of these investments.

However, one implication of this is that if a metric is chosen that does ascribe significant size exposure to one of these investments, an investor may make a decision based on not wanting to bear that risk in what they desire to be a large-cap investment.

**Can We Blend Our Way Out?**

One way we often mitigate model specification risk is by blending a number of models together into one.

By averaging all of our HML and SMB factors, respectively, we arrive at blended factors for the three-factor model.

*Source: Tiingo, Kenneth French Data Library. Calculations by Newfound Research. **Past performance is not an indicator of future results. Returns represent live strategy results. Returns for the Newfound Systematic Value strategy are gross of all management fees and taxes, but net of execution fees. Returns for ETFs included in study are gross of any management fees, but net of underlying ETF expense ratios. Returns assume the reinvestment of all distributions.*

All of the investments now have HML loadings in the top of their range of the individual model loadings, and many (FTA, PWV, RPV, SPVU, VTV, and the Systematic Value strategy) have loadings to the blended HML factor that exceed the loadings for all of the individual models.

The opposite is the case for the blended SMB factor: the loadings are in the low-end of the range of the individual model loadings.

*Source: Tiingo, Kenneth French Data Library. Calculations by Newfound Research. *

So which is the correct method?

That’s a good question.

For some investments, it is situation-specific. If a strategy only uses price-to-earnings as its value metric, then putting it up against a three-factor model using the P/E ratio to construct the factors is appropriate for judging the efficacy of harvesting that factor.

However, if we are concerned more generally about the abstract concept of “value”, then the blended model may be the best way to go.

**Conclusion**

In this study, we have explored the impact of model specification for the value and size factor in the Fama French three-factor model.

We empirically tested this impact by designing a variety of HML and SMB factors based on three additional value metrics (price-to-earnings, price-to-cash flow, and dividend yield). These factors were constructed using the same rules as for the standard method using price-to-book ratios.

Each factor, with the possible exceptions of the dividend yield-based HML, has performance that could make it a legitimate specification for the three-factor model over the time that common data is available.

Running factor regressions using these alternate specifications on a suite of value ETFs and Newfound’s Systematic Value strategy, led to a wide array of results, both numerically and directionally.

While many investors consider the uncertainty of the parameter estimates from the regression using the three-factor model, most do not consider the uncertainty that comes from the assumption of how you construct the equity factors in the first place.

Understanding the additional uncertainty is crucial for decision-making. Managers and investors alike must consider what risks they are trying to measure and control by using tools like factor regression and make sure their assumptions align with their goals.

“Value” is in the eye of the beholder, and blind applications of two different value factors may lead to seeing double conclusions.

## Timing Trend Model Specification with Momentum

By Corey Hoffstein

On December 23, 2019

In Craftsmanship, Risk & Style Premia, Trend, Weekly Commentary

A PDF version of this post is available here.## Summary

Over the last several years, we’ve advocated on numerous occasions for a more holistic view of diversification: one that goes beyond just

whatwe invest in, but also considershowthose decisions are made andwhenthey are made.We believe that this style of thinking can be applied “all the way down” our process. For example, how-based diversification would advocate for the inclusion of both value and momentum processes, as well as for different approaches to capturing value and momentum.

Unlike correlation-based

whatdiversification,how-based diversification often does little for traditional portfolio risk metrics. For example, inIs Multi-Manager Diversification Worth It?we demonstrated that within most equity categories, allocating across multiple managers does almost nothing to reduce portfolio volatility. It does, however, have a profound impact on the dispersion of terminal wealth that is achieved, often by avoiding manager-specific tail-risks. In other words, our certainty of achieving a given outcome may be dramatically improved by taking a multi-manager approach.Ensemble techniques to portfolio construction can be thought of as adopting this same multi-manager approach by creating a set of virtual managers to allocate across.

In late 2018, we wrote two notes that touched upon this:

When Simplicity Met FragilityandWhat Do Portfolios and Teacups Have in Common?In both studies we injected a bit of randomness into asset returns to measure the stability of trend-following strategies. We found that highly simplistic models tended to exhibit significant deviations in results with just slightly modified inputs, suggesting that they are highly fragile. Increasing diversification acrosswhat, how, andwhenaxes led to a significant improvement in outcome stability.As empirical evidence, we studied the real-time results of the popular Dual Momentum GEM strategy in our piece

Fragility Case Study: Dual Momentum GEM, finding that slight deviations in model specification lead to significantly different allocation conclusions and therefore meaningfully different performance results. This was particularly pronounced over short horizons.Tying trend-following to option theory, we then demonstrated how an ensemble of trend following models and specifications could be used to increase outcome certainty in

Tightening the Uncertain Payout of Trend-Following.Yet while more diversification appears to make portfolios more consistent in the outcomes they achieve, empirical evidence also suggests that certain specifications can lead to superior results for prolonged periods of time. For example, slower trend following signals appear to have performed much, much better than fast trend following signals over the last two decades.

One of the benefits of being a quant is that it is easy to create thousands of virtual managers, all of whom may follow the same style (e.g. “trend”) but implement with a different model (e.g. prior total return, price-minus-moving-average, etc) and specification (e.g. 10 month, 200 day, 13 week / 34 week cross, etc). An ancillary benefit is that it is also easy to re-allocate capital among these virtual managers.

Given this ease, and knowing that certain specifications can go through prolonged periods of out-performance, we might ask: can we time specification choices with momentum?

Timing Trend SpecificationIn this research note, we will explore whether momentum signals can help us time out specification choices as it relates to a simple long/flat U.S. trend equity strategy.

Using data from the Kenneth French library, our strategy will hold broad U.S. equities when the trend signal is positive and shift to the risk-free asset when trends are negative. We will develop 1023 different strategies by employing three different models – prior total return, price-minus-moving-average, and dual-moving-average-cross-over – with lookback choices spanning from 20-to-360 days in length.

After constructing the 1023 different strategies, we will then apply a momentum model that ranks the models based upon prior returns and equally-weights our portfolio across the top 10%. These choices are made daily and implemented with 21 overlapping portfolios to reduce the impact of rebalance timing luck.

It should be noted that because the underlying strategies are only allocating between U.S. equities and a risk-free asset, they can go through prolonged periods where they have identical returns or where more than 10% of models share the highest prior return. In these cases, we select all models that have returns equal-to-or-greater-than the model identified at the 10

^{th}percentile.Before comparing performance results, we think it is worthwhile to take a quick look under the hood to see whether the momentum-based approach is actually creating meaningful tilts in specification selection. Below we plot both aggregate model and lookback weights for the 126-day momentum strategy.

Source: Kenneth French Data Library. Calculations by Newfound Research.We can see that while the model selection remains largely balanced, with the exception of a few periods, the lookback horizon selection is far more volatile. On average, the strategy preferred intermediate-to-long-term signals (i.e. 181-to-360 day), but we can see intermittent periods where short-term models carried favor.

Did this extra effort generate value, though? Below we plot the ratio of the momentum strategies’ equity curves versus the naïve diversified approach.

We see little consistency in relative performance and four of the five strategies end up flat-to-worse. Only the 252-day momentum strategy out-performs by the end of the testing period and this is only due to a stretch of performance from 1950-1964. In fact, since 1965 the relative performance of the 252-day momentum model has been negative versus the naively diversified approach.

Source: Kenneth French Data Library. 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.This analysis suggests that naïve, momentum-based specification selection does not appear to have much merit against a diversified approach for our simple trend equity strategy.

The Potential Benefits of Virtual RebalancingOne potential benefit of an ensemble approach is that rebalancing across virtual managers can generate growth under certain market conditions. Similar to a strategically rebalanced portfolio, we find that when returns across virtual managers are expected to be similar, consistent rebalancing can harvest excess returns above a buy-and-hold approach.

The trade-off, of course, is that when there is autocorrelation in specification performance, rebalancing creates a drag. However, given that the evidence above suggests that relative performance between specifications is not persistent, we might expect that continuously rebalancing across our ensemble of virtual managers may actually allow us to harvest returns above and beyond what might be possible with just selecting an individual manager.

Source: Kenneth French Data Library. 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.ConclusionIn this study, we explored whether we could time model specification choices in a simple trend equity strategy using momentum signals.

Testing different lookback horizons of 21-through-378 days, we found little evidence of meaningful persistence in the returns of different model specifications. In fact, four of the five momentum models we studied actually under-performed a naïve, diversified. The one model that did out-perform only seemed to do so due to strong performance realized over the 1950-1964 period, actually relatively under-performing ever since.

While this evidence suggests that timing specification with momentum may not be a fruitful approach, it does suggest that the

lackof return persistence may benefit diversification for a second reason: rebalancing. Indeed, barring any belief that one specification would necessarily do better than another, consistently re-pooling and distributing resources through rebalancing may actually lead to the growth-optimal solution.^{1}This potentially implies an even higher hurdle rate for specification-timers to overcome.