In 2013, the Nobel Prize in economics was shared by Eugene Fama, Lars Peter Hansen, and Robert Shiller "for their empirical analysis of asset prices". The work of Fama and co-author, Ken French has changed the way people think about passive investing and, consequently, interpreting the performance of active investors. I wanted to dig into the data Fama's work is based on, and see what (if any) conclusions I could draw from it.

Ken French graciously posts much of the data the Fama-French (hereafter FF) model was based on to his website. I used the monthly, value-weighted data in the "6 Portfolios Formed on Size and Book-to-Market" file, along with the market and risk-free returns in the monthly "Fama/French Factors" file. The 6 portfolios that are typically used to construct the FF factors are created by annually dividing stocks into two groups based on size and into three groups based on price/book (which they flipped to book/price, and referred to as book-to-market). The size groups are called big and small, and the book-to-market groups are called value (high book-to-market), neutral (medium book-to-market) and growth (low book-to-market). The term "growth" is just a label given to stocks with low book-to-market ratios, without regard to whether or not they were actually growing.

First, let's look at the relative performance of these six portfolios:

The graph shows the performance of each portfolio *relative to the market.* When a line goes up, that portfolio outperformed the market; when a line goes down, that portfolio underperformed. For instance, someone who stayed invested in the Small Neutral portfolio for the entire period would have ended up with about 16 times the assets of someone who stayed invested in the broad market (assuming they started with the same amount), and Small Value did nearly 4 times better than that. Both the Small and Big Growth portfolios underperformed the broader market, as indicated by their values that are below 1. All the portfolios generated positive returns over the 86+ years. (If you graph the portfolios themselves, you mostly get six lines that look a lot like the market itself, which is not that interesting so I didn't include it). During the Depression, Big Growth outperformed, and in the 1999 dotcom bubble, Small Growth spiked and everything that is not growth dropped sharply. The stats, calculated for the entire period and the most recent 22 years, look like this (excess returns are portfolio returns minus the risk-free rate, as measured by the one-month T-bill):

If you are familiar with FF, then you have some idea of what to expect: Value outperforms Growth, and Small outperforms Big. Small Value, in particular, does extremely well. However, there are two features of the graph worth noting: 1) Among Growth stocks, the size premium has worked in the opposite direction for the last 30+ years - Big Growth has outperformed Small Growth. 2) Over about the same timeframe, the difference between Big Value and Big Growth has been close to flat. In English, the value premium in large-cap U.S. stocks has been very small in recent years.

FF next use these six portfolios to construct two more:

High minus Low (HML):

1/2 (Small Value + Big Value) - 1/2 (Small Growth + Big Growth) =

1/2 (Small Value - Small Growth) + 1/2 (Big Value - Big Growth)

Small minus Big (SMB):

1/3 (Small Value + Small Neutral + Small Growth) - 1/3 (Big Value + Big Neutral + Big Growth) =

1/3 (Small Value - Big Value) + 1/3 (Small Neutral - Big Neutral) + 1/3 (Small Growth - Big Growth)

Note the HML and SMB portfolios are "market neutral" in the sense that they are 50% weighted "long" and 50% weighted "short" for a net weight of zero. The top equation is how they are defined on French's website; the bottom one rearranges the terms in a way that makes more sense to me. HML is the average difference in performance of Value and Growth stocks over the different size groupings; and SMB is the average difference in performance of Small and Big stocks over the different value groupings.

At this point, FF take a portfolio's excess returns and regress them onto three variables: the market excess returns, the HML portfolio, and the SMB portfolio. The intercept is alpha. As a sanity check, we can take the six portfolios used in the construction of HML/SMB and see what we get. In addition to all of the data, I chose a sample with January 1992 as a starting point because the FF data from the 1993 paper ended in December 1991, so I thought an out-of-sample test would be a good idea.

The results are, to me, problematic. Our six portfolios were constructed entirely on size and value and so we would expect them to have zero alpha. But four of the six portfolios - all those not in the 'neutral' category - show statistically significant alpha. The Big Value portfolio over the last 22 years, for example, has an alpha of -0.21 percentage points per month, or about 2.5 percentage points a year. If someone were using the 3-factor model to evaluate their performance, and had exactly matched the Big Value portfolio, the FF model would give an alpha of -2.5% per year, even though we know if they exactly matched a passive benchmark, alpha should be zero. And if they had actually outperformed the Big Value benchmark portfolio by 2.5%/year with identical factor exposures, the FF model would give an alpha of zero.

When we look at the performance of the portfolios via which HML/SMB are constructed, it is not hard to see what's going wrong here. FF create HML by averaging the value premium among large and small stocks. The value premium is substantial among small stocks, but almost nonexistent among large stocks, and the average is halfway in between. The Big Value portfolio has a positive HML loading, and the Big Growth portfolio has a negative HML loading. For the model to work, Big Value needs to outperform Big Growth by a considerable amount. It doesn't, and this gets picked up incorrectly as negative alpha for Big Value and positive alpha for Big Growth. It doesn't work to try to use a single factor to capture HML, when the effect is clearly different for large caps vs. small caps. Similarly, SMB is clearly very different for growth stocks than value stocks, but the model only allows for one SMB factor, which turns into a muddied combination of what seem to be multiple forces at work, and this contributes to positive alpha for the Big Growth portfolio, and negative alpha for the Small Growth portfolio.

To be fair to FF, this is MILES ahead of what preceded it, which was the same regression but without any factors for HML or SMB. And when you look at the six portfolios, it's almost impossible to argue that size and value don't matter - they clearly do. But, they interact with each other in a way that the functional form of the 3-factor model does not capture.

The simpler and more intuitive way to incorporate size and value into performance benchmarking is to simply measure a portfolio against a benchmark with similar size and value characteristics. This poses some practical challenges, however. Size and value are continuous, and any attempt to create portfolios based on them requires a decision about how many categories to divide them into. FF chose three for value and two for size. On pg. 9 of 1993's *"**Common risk factors in the returns on stocks and bonds"**, *they have this to say on the topic:

Our decision to sort firms into three groups on BE/ME and only two on ME follows the evidence in Fama and French (1992a) that book-to-market equity has a stronger role in average stock returns than size. The splits are arbitrary, however, and we have not searched over alternatives. The hope is that the tests here and in Fama and French (1992b) are not sensitive to these choices. We see no reason to argue that they are.

Morningstar uses a 3x3 split, which seems to work reasonably well. Another issue is the so-called "style drift"; you can get a snapshot of a portfolio's size/value composition at any given time, but it might be very different at a different time. This can even happen without turnover; Cisco (CSCO) was the classic growth stock twenty years ago, but is probably more of a value stock today. Which benchmark should you use if you held it the entire time? I don't know.

Conclusions

*For passive investors*

First off, for passive investors, simply owning the broad market still seems like a very reasonable approach to me. Small and value stocks are included in the market, so you are still getting some exposure to them. Their history of outperformance certainly seems tempting, but there's some saying about past performance and future results that may apply here. Also, FF seem to argue the higher returns for Small/Value are related to some sort of risk. Possibly these stocks would fare the worst in a serious economic downturn. So, in spite of their past performance, by not overweighting them you may be limiting your exposure to so-called "tail risk".

For those who do want more of that Small Value performance in their portfolio, there are options. The FF portfolios are theoretical and cannot be invested in directly. To see how well some investable options actually track FF Small Value, I ran simple regressions for five passive small value instruments, using the actual fund monthly returns as the dependent variable and the FF SV monthly returns as the independent variable. If you are not too familiar with regression, what we are hoping for in this case is a multiple close to 1; an intercept close to 0; and an R-squared close to 1.

Not surprisingly (Fama's on the board of directors), Dimensional Funds' DFSVX tracks SV the most closely, with an R-squared that's noticeably higher than any of the alternatives. However, both Vanguard alternatives VIOV and VTWV submit perfectly respectable performances over their unfortunately short histories. RZV is a different animal; in part because the S&P "Pure Value" indices weight stocks by their Growth or Value score rather than by market cap. It actually looks like a slightly leveraged version of SV; but the R-squared is much lower, so there may be other things happening there too. JKL's multiple of SV is significantly (four standard errors) lower than 1, so it's not giving investors the same exposure to SV as some of the others. If you want the Cadillac of Small Value, DFSVX is probably the way to go; otherwise, I'd say VIOV's low expense ratio and slightly higher R-squared make it the best of the rest.

*For professional benchmarking*

As a disclaimer, I have no idea if anyone doing serious portfolio measurement actually runs regressions on the FF factors, or the more recently-added UMD (Up minus Down; for momentum) or QMJ (Quality minus Junk; for gross profitability) factors. The above analysis suggests they probably shouldn't but certainly isn't the last word - there may well be a way to make it work. If anyone has, I would be really interested in what/how they do so.

The portfolio-specific approach of trying to benchmark against a portfolio of similar stocks, as Morningstar does, is a reasonable way to proceed, but in addition to the problems discussed earlier, it's in some danger of becoming hopelessly convoluted. FF's most recent paper uses *five* factors (market, SMB, HML, QMJ, and they added CMA, which is Conservative minus Aggressive, for firms that invest a little or a lot), and if you add momentum, that's six. You can also get passive exposure to low-volatility now (which seems to outperform as well), so that's a total of seven. Hypothetically, one could be comparing a fund manager to a portfolio of, say, large-cap, low-profitability, medium-volatility, high book-to-market, moderately-aggressive losers, which seems absurd. But I don't really have any better ideas.

*For personal benchmarking*

I don't have much to say here, other than I don't recommend the 3-factor model, based on the above analysis. If you choose to not benchmark at all, I think that's fine. It may help you focus on your process as opposed to your results, which can only be a good thing. The downside is if you turn out to be terrible at investing, you may be costing yourself money without ever realizing it. Otherwise, you can use the broad market, choose a Morningstar index that you feel matches your style, or pick a combination of passive funds that would be your "next-best" choice if you couldn't buy individual stocks. It's really up to the individual.

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