Morningstar Advisor - Spring 2008 - 25

I did this by demonstrating that if the variation in market valuation errors exceeds the variation in fair value multiples (used in fundamental index construction), the correlation between fundamental values and fair values will be greater than the correlation between market values and fair values. If that were to hold true, it would show fundamental weighting to be superior to market-cap weighting. Conversely, if the variation in fair value multiples exceeds the variation in market valuation errors, the correlation between market values and fair values would be greater than the correlation between fundamental values and fair values. In that case, market-cap weighting would be clearly superior to fundamental weighting. Ultimately, though, the question of which is greater, the variation in fair value multiples or the variation in market valuation errors, cannot really be answered because it is about unobservable variables. The very fact that the question remains so broadly open, though, illustrates that claims made by proponents of fundamental indexing do not really rise to the level of a theory. Rather, they are demonstrably only a conjecture. The Value Bias between the portfolio’s yield and that of its market-cap-weighted version. I show that this is the ratio of the variance of yield across the stocks in the index to the yield on the market-weighted index. This is not a boundary condition; it is true by construction. Thus, the conclusion that fundamental indexing contains a value bias does not depend on circumstances (that is, on the observed values of variables). It is always true. Best of Both Worlds While fundamental indexing cannot justifiably claim to be a revolutionary new paradigm, it does introduce worthwhile ideas for enhancing capitalization-weighted indexes. Effectively, because both fundamental and market-cap weights contain some information about a stock’s fair value, combining the two weighting schemes can potentially produce results that are better than using either one alone. One indexing methodology that uses that approach is “collared” weighting, which seeks to preserve the benefits of market-cap as well as fundamental weighting while minimizing the disadvantages of each. Sanjay Arya, who is Morningstar’s director of indexes, and I discussed the collared approach in the 2007 Morningstar Index Yearbook and in a Morningstar working paper, “Collared Weighting: A Hybrid Approach to Indexing.” I briefly summarize it here. One major advantage of market-cap weighting is that the portfolio does not require regular rebalancing and, thus, generates less transaction costs compared with a non-cap-weighted weighting scheme. On the other hand, marketcap weighting can lead to volatile levels of diversification because a stock that keeps on rising has an increasingly large portfolio weight. A collared weighting scheme addresses these issues by weighting according to market cap unless a stock’s market-cap weight deviates significantly from its fundamental weight. When market weights get far out of line, collared weighting reins them back in by linking them to fundamental weights. A collared weighting scheme uses fundamental weights to set boundaries on portfolio weights rather than to be the portfolio weights themselves. These boundaries (or collars) are fixed multiples of the fundamental weights, such as a lower bound of half the fundamental weight and an upper bound of twice the fundamental weight. Stocks are held in the index at their market value weights unless such weights fall outside the boundary, in which case they are held at the boundary they violate. The idea is to take advantage of the low transaction costs of market-cap weighting while avoiding concentration in a few large stocks during market run-ups, such as occurred in the late 1990s. On the one hand, collared weighting can help investors avoid exaggerated stakes in overvalued growth stocks. At the same time, collared weighting moderates the value bias (as well as the higher turnover and higher cost) associated with fundamental weighting. Conclusion Historical performance data do give fundamental indexing a sizable edge over capitalizationweighted indexing. But as I mentioned, some critics of fundamental indexing point out that fundamental weighting is simply an alternative way to introduce a value bias into a portfolio. Value-biased portfolios have historically outperformed unbiased portfolios, so it is no surprise that a fundamentally weighted index outperforms a market-cap index of the same stocks in a long-term back test. In my Financial Analysts Journal article, I derive a mathematical formula for the value bias of fundamental weighting. Defining the yield on a stock as the ratio of the fundamental size measure to market value, I define the magnitude of a portfolio’s value bias as the difference Proponents of fundamental indexing, rather than having a theory on which to base their claims, have only a conjecture that market valuation errors are more variable than the variability of fair market multiples. They may have been correct over long historical periods, as the successful back tests of their strategies seem to demonstrate, but they should be much more modest in their claims. In particular, they could argue that it is better to introduce a value bias into a portfolio by using their weighting scheme than by excluding low-yield stocks. In other words, they could promote fundamental indexing as a more disciplined, systematic approach to value investing. But they have not, to date, pursued this more humble line of reasoning. They may have a successful investment strategy, but they have not produced a revolution in investment theory. K Paul Kaplan, Ph.D., CFA, is Morningstar’s vice president of quantitative research and a frequent contributor to the magazine. He last appeared in the Fall 2007 issue with an article titled “Few Mutual Funds Exhibit Serial Correlation.” MorningstarAdvisor.com 25

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Morningstar Advisor - Spring 2008

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