The air is coming out of the argument that fundamental indexing is a revolutionary innovation.

Fundamental indexing has been touted by its proponents as a revolutionary innovation in the field of passive investing. Weighting stocks by market capitalization, they say, inevitably causes a drag on performance, owing to the market's inherent mispricing of stocks. Their claims are based on a 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 because they do, market-cap-weighted portfolios--which inherently favor stocks with rising prices--are skewed toward overvalued stocks. (For our purposes, consider the definition of "fair value" to mean a number that would represent the "true" or intrinsic value of a stock that would perfectly describe its worth.) Hence, the reasoning goes, a portfolio with weightings that are derived independently of market value, such as a fundamental weighted index, will outperform its market-cap-weighted counterpart.

The fundamental indexers' biggest criticism against cap-weighted indexes stands on flimsy ground, though. AndrĂ© Perold of Harvard Business School demonstrated in a paper in Financial Analysts Journal (November/December 2007), for example, that even if market prices deviate from fair values, it does not automatically follow that cap-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 because of its superior growth prospects, or that a "cheap" stock's fair value could be even lower than its market price. Proponents of fundamental indexing make their case against market-cap weighting by implicitly (and perhaps unknowingly) assuming that a market observer 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-indexing case is the claim that while fundamental weights may not be a perfect expression of stocks' fair values, the errors they include--the difference between their fundamentally determined weights and those that would accurately reflect stocks' fair values--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 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 indexing 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 a stock's theoretical 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.