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 Fundamental indexation has been touted by its proponents as a revolutionary innovation in the field of passive investing.
They claim that weighting stocks by market capitalization inevitably causes a drag on performance, owing to the market's inherent mispricing of stocks. These claims are based on a new paradigm of asset pricing called the "noisy market hypothesis," which argues that stocks' market prices stray from their fair values in a random fashion, and that since they do, market-cap-weighted portfolios—which inherently favor stocks with rising prices—are skewed toward overvalued stocks. Hence, the reasoning goes, a portfolio with weights that are derived independently of market value, such as a fundamentally weighted index, will outperform its market-cap-weighted counterpart.
The fundamental indexers' biggest criticism against traditional, market-cap-weighted indexes stands on rather flimsy ground, though. André Perold of Harvard Business School demonstrated in a paper in Financial Analysts Journal (November/December 2007),1 for example, that even if market prices deviate from fair values, it does not automatically follow that capitalization-weighted indexes load up on overvalued stocks. That is because there is simply no reason to conclude at the outset that a stock that commands a high valuation is overpriced. It is just as likely that the "expensive" stock deserves its premium due to its superior growth prospects, or that a "cheap" stock's fair value could be even lower than its market price. Proponents of fundamental indexation make a case against market-cap weighting by implicitly (and perhaps unknowingly) assuming that a market observer can and does know a stock's fair value, thereby contradicting one of their own assumptions.
The Independence Assumption
As noted, one of the main underpinnings of the fundamental indexation case is the claim that while fundamental weights may not be a perfect expression of stocks' fair values, the errors they include are not correlated with market values. This "independence assumption" is crucial because without it, claims of theoretical superiority of fundamental weighting over market-cap weighting do not hold.
Here's the problem: The independence assumption has absurd implications about the fair values of stocks.
In order for the independence assumption to hold, at least one of the following conditions must be met:
- Valuation ratios, if calculated using fair values, must be the same across stocks; or
- Market values must be completely unrelated to fair values.
Both scenarios are difficult to imagine, to say the least. Companies have different growth and risk characteristics, so even stocks of companies with the same earnings clearly do not deserve the same price multiple. And even after making allowances for pricing errors, market values should clearly have some relationship to fair values. Hence, the independence assumption cannot hold.
Ultimately, this means that fundamental indexation proceeds from logic that is internally inconsistent. In particular, the conclusion that a fundamentally weighted index has a higher expected return than a market-weighted index simply lacks theoretical foundation.
I refer to a valuation ratio calculated using fair value as a "fair value multiple." Mathematically, a fair value multiple is the (unobservable) number M, which we define as a ratio of the (unobservable) fair value of a stock, V, over some observed fundamental measure of company size, such as earnings or book value, F:
V=FM or M=V/F
Put another way, a stock's fair value multiple is the ratio of a stock's "true" fair value relative to a fundamental size measure that is being used as an alternative to market capitalization when calculating a stock's portfolio weight.
Recall that proponents of fundamental indexation assert that fundamental weights can be unbiased estimators of the unobservable "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/E ratios 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-Optimal Portfolios," 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—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.
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