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used as an alternative to market capitalization when calculating a stock’s portfolio weight. Recall that proponents of fundamental indexing assert that fundamental weights can be unbiased estimators of the unobservable fair value weights with “errors” that are statistically independent of market values—the independence assumption. It turns out, however, that the so-called errors are actually restatements of the stocks’ fair value multiples. For example, if the fundamental weights are based on earnings, errors in the fundamental weights are restatements of the fair P/Es of the stocks. This conclusion follows from the very way that such proponents of fundamental indexing define the error for a given stock; namely, as the ratio of the fundamental weight to the fair value weight minus one. (See Jason Hsu, “Cap Weighted Portfolios Are Sub-Opitmal,” Journal of Investment Management, Third Quarter 2006.) Because the fundamental weight is proportional to the fundamental size measure and the fair value weight is proportional to fair value, this “error,” being a restatement of the ratio of the two weights, is therefore also a restatement of the ratio of fair value to the fundamental size variable; in other words, the stock’s fair value multiple. By way of analogy, imagine that we have a collection of gems of various types, qualities, and weights. We find out the market value of each gem, write it on a bag, place the gem in the bag, and seal the bag. Once we have sealed the bags, we cannot tell which gem— or its type or quality—is in which bag. We do have a scale, however, so we weigh each bag and write the weight on the bag as well as each gem’s market value. As with stocks, for each gem, we know its market value and have a fundamental measure of size (weight), but we do not know its fair value or fair value multiple (fair price per ounce). Unless all the gems are of the same type and quality, however, or their market values are completely unrelated to their fair values, there is some relationship between the market values written on the bags and the fair prices per ounce of the gems inside them. Clearly, we would not want to rely on weight alone to assess the value of each bag’s contents. Proponents of fundamental indexing are effectively asking us to rely solely on weight, though, while at the same time assuring us that the market values they are choosing to ignore have no relationship whatsoever to the type or quality of the gems inside the bags. Why Fundamental Indexing Might Work A stock’s fair value multiple, by definition, reflects investors’ assessments of the stock’s risk and future growth prospects. Ideally, such factors should be fully taken into account in portfolio construction. Because one must actually know a stock’s fair value to pinpoint the magnitude of those factors, though, they are effectively unobservable. To engage in portfolio construction, therefore, one must either ignore those factors or take them into account by using proxies. Market-cap weighting inherently takes risk and expected growth into account by using the market values of stocks as proxies for their unobservable fair values. To be sure, if market prices contain noise, then marketcap weights contain errors. Yet fundamental weighting schemes introduce weighting errors of a different type. They throw the baby out MorningstarAdvisor.com 23

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Morningstar Advisor - Spring 2008

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