Style indexes under the microscope.

Eugene Fama and Kenneth French's work popularized the value and size effects, the puzzlingly high returns that value and small-cap stocks have earned in almost every market studied. Their research has resonated with indexers, many who overweight value and small-cap stocks in their portfolios. Despite style tilting's popularity, the popular discussion is short on specific numbers. Some tilters point out that small-value stocks as defined by Fama and French have earned 14.2% annualized from 1928 to 2010, 4 percentage points above the U.S. market's return. But that doesn't mean any value index, small-cap or large-cap, will earn a similar value premium--some indexes are more size- or value-laden than others. The style map is a beautiful way to represent value and size tilts, but it doesn't quantify value and size tilts as readily observable numbers. For that, we have to go back to the size and value premiums' academic roots, to the Fama-French model.

**Models and Models**In simple terms, the Fama-French model estimates how sensitive a stock strategy's returns are to three risk factors: the total stock market's returns, value stocks' returns, and small-cap stocks' returns. Value is defined as low price/book; the other factors are self-explanatory. However, since value stocks have both market and size risk mixed into their returns, Fama and French isolate the value-return factor with a strategy that owns a portfolio of low price/book stocks and short-sells a dollar-equivalent portfolio of high price/book stocks, earning the spread between value and growth stocks. The size factor is isolated in a similar manner. An extended version of the Fama-French model, called the Carhart model, adds a momentum factor, which captures the return of a strategy that every month buys the best-performing stocks and short-sells the worst-performing stocks. We'll wield this model from now on.

The Carhart model estimates "betas," numbers that represent a stock strategy's sensitivity to each factor. For example, a value beta of 0.80 means that for each percentage point the value factor rises, the strategy rises by 0.80 percentage points, holding all other factors constant. I've reproduced the annualized returns of the various factors in the table below. The table links betas to actual returns. For example, a strategy with a size beta of 0.50 and a value beta of 1.00 would have earned about 5.17% annualized (0.50*2.50% + 3.92*1.00) from its value and size loadings for the period 1927 to 2011.

Let's look at an actual index. The S&P SmallCap 600 Value had a value beta of 0.61, implying that if it had existed since 1927 and maintained its value loading, it would have earned from its value exposure about 2.39% annualized (0.61*3.92%). The bigger the value loading, the deeper or purer its value tilt.

Source: French Data Library, Author's Calculations. Market represents the U.S. stock market's returns, minus the T-bill rate. Size, value, and momentum factor annualized returns represent strategies that hold small-cap, value, or high-momentum stocks and short-sell large-cap, growth, and low-momentum stocks.

We can't take this model literally. It doesn't say the S&P SmallCap 600 Value will earn 2.39% annualized from the value premium in the future. All it produces are backward-looking estimates. The true value and size premiums will always be obscured to us. But the model gives us another perspective on the relative style purity of various indexes.

**Size, Value, and Momentum**It's worth exploring how the size, value, and momentum factors have behaved over the past century. As the chart below shows, the factors' excess returns can disappear or turn negative for painfully long periods. Witness how the size factor lost money for decades-long periods--from 1983 to 1999 and from 1946 to 1965. Value was more consistent, but even it had decade-long droughts, most recently during the 1990s. The momentum strategy's returns were both greed-inspiring and puzzlingly consistent, but punctuated by horrendous losses during the two great stock market crashes. Besides, its returns were unobtainable for much of the century because of high frictional costs. Value and size tilters need to be committed to their strategies for the long haul and ready to endure years--maybe decades--of underperformance.