Fundamental Indexing

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William J. Bernstein

Fundamental Indexing and the Three-Factor Model

A year ago, Rob Arnott, Jason Hsu, and Philip Moore published a piece in Financial Analysts Journal, "Fundamental Indexation," in which they describe a novel way of attacking the well-known tendency of market-cap indexing to overweight large growth stocks.

The usual approach to achieving this end is to "decapitate" traditional market indexes, such as the S&P 500 or Wilshire 5000, by including only those companies meeting predetermined balance-sheet ratios. For example, the Barra Large Value Index includes the bottom half of the S&P 500 sorted by price-to-book ratio (P/B), while the Fama-French Large Value Index includes only those names in the bottom third, more or less, of their large-cap index.

Decapitation, however, has its own disadvantages. First, performing such radical surgery on the equity universe yields grossly under-diversified portfolios. Second, within the universe of remaining stocks, one is still cap weighting, overweighting the overvalued stocks within it. And third, significant transactional costs are incurred when companies "jump the gap" in and out of the universe. Such costs can be mitigated with "hold bands" just outside the universe, but this, in turn, does decrease factor exposure a tad.

Arnott, Hsu, and Moore suggest an innovative fix. Rather than decapitate the cap-weighted universe, they "shape it up" by including the 1,000 largest companies, then weight them according to their "economic footprint," as measured by parameters such as aggregate earnings, sales, book value, dividends, and employees.

Take, for example, book value. If we weight all companies according to their book value, this is equivalent to saying that we begin with cap weighting, then adjust that by the book-to-market ratio (B/M). (Since B = M x [B/M]). B/M is the inverse of P/B.) So if a company has, say, a B/M ratio that is twice the market’s, it will get twice the weighting in a fundamental index as it would in a simple cap-weighted scheme.

While the authors compute returns series for all of the relevant parameters, for the sake of simplicity we’ll stick to the composite index, which combines all of them. The results were impressive: between 1962 and 2004, the annualized return of their composite RAFI (Research Affiliates Fundamental 1000 Index) outpaced the S&P 500 by 197 basis points (bp).

While noting that three-factor regression of their indexes had "exposure to the value factor and, to a lesser extent, the size factor," as well as an "estimated alpha of –0.1 percent" (presumably per year), they softpedal the possibility that a large part of the excess return of their fundamental indexes came from exposure to the two "supplemental" Fama-French factors, while nodding implicitly to it by observing that other value indexes do even worse, with alphas in the –1.5% range.

We believe that insight is gained by closer scrutiny to these regressions, as well as to the flaws in the Three-Factor Model. The fact that certain commercially available indexes of value stocks have alphas lower than those of the fundamental indexes is interesting in and of itself. The critical question is this: Just how much of the excess return of the fundamental indexes is due to factor exposure, and how much return above and beyond this is added by the technique of fundamental indexing?

In order to answer this question, we performed regressions of the RAFI, kindly supplied to us by Robert Arnott and Jason Hsu, as well as the Fama-French Large Value and Large Growth indexes between 1962 and 2004. The results are most instructive.
























In the first place, as Arnott et. al. observed, the alpha of the RAFI is zero. We first note that its size loading is actually slightly negative, consistent with its large-cap bias. Most critically, its value loading is about half that of the Fama-French Large Value Index. Next, as also noted by Arnott et. al., the three-factor return of the Fama-French Large Value Index is –0.12% per month, in line with their estimate of a -1.5% per year alpha for the commercially available indexes. However, the Fama-French Large Growth has a positive alpha that is even larger. There is, of course, nothing particularly "wrong" with the FF LV index, and there is nothing particularly "good" about the FF LG index. Rather, the alphas of these indexes, which are statistically significant, represent a flaw in the model, which does an otherwise excellent, but obviously not perfect, job of predicting an extraordinarily complex system using just three variables. The Three-Factor Model is, as Fama and French have repeatedly emphasized, only a model—it is not reality.

How, then, to reconcile these data? We begin by multiplying the values of the RAFI’s value-factor loading, 0.36, times the return of the value factor for this period, 5.08% per year. This yields a value of 1.83%, perilously close to the 1.97% return margin of the RAFI over the S&P 500.

However, we do need to give the RAFI credit for overcoming the inherently negative alpha of this corner of the Fama-French universe. How much credit? Since the FF LV index has a value factor loading of 0.75, the RAFI 0.36, and the alpha of the FF LV is –0.12% per month, then (0.36/0.75) x 0.12%, or approximately 6 bp per month, or 72 bp per year, needs to be credited back to the RAFI composite. Since the t-stat for the FF LV index is –2.53, that for the RAFI can be estimated at about half of it with the opposite sign—positive, but statistically nonsignificant.

Thus, at a rough approximation, slightly less than two-thirds of the "excess return" of the RAFI over the S&P is due to naïve factor exposure, and slightly more than one-third seems to be inherent to the technique. Unfortunately, this latter effect is not statistically significant, raising the issue of data mining.

In summary, we draw two conclusions from this exercise:

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