Fama and French three-factor model

Background
The Fama-French Three Factor Model is used to explain differences in the returns of diversified equity portfolios. It's a model that compares a portfolio to three distinctive types of risk found in the equity market to assist in categorizing returns. Prior to the three-factor model, the Capital Asset Pricing Model (CAPM) was used as a "single factor" way to explain portfolio returns.

However, several shortcomings of the CAPM model exist when compared to realized returns, and the affect of other risk factors have put this model under criticism. The assumption of a single risk factor limits the usefulness of this model.

In June 1992, Eugene F. Fama and Kenneth R. French published a paper that found that on average, a portfolio’s beta only explains about 70% of its actual returns. For example, if a portfolio was up 10%, about 70% of the return can be explained by the advance of all stocks and the other 30% is due to other factors not related to beta.


 * "Beta," the measure of market exposure of a given stock or portfolio, which was previously thought to be the be-all/end-all measurement of stock risk/return, is of only limited use. Fama/French showed that this parameter did not explain the returns of all equity portfolios, although it is still useful in explaining the return of stock/bond and stock/cash mixes.
 * The return of any stock portfolio can be explained almost entirely by two factors: Market cap ("size") and book/market ratio ("value"). Therefore, a portfolio with a small median market cap and a high book/market ratio will have a higher expected return than a portfolio with a large median market cap and a low book/market ratio.

Note: The terminology by Fama/French is different than common usage. "Book/market ratio" is the inverse of the more familiar "price/book ratio." In other words, a high book/market ratio is the same as a low price/book ratio— a "value". In their paper, high book/market is acronymed "HBM."

In summary, Fama/French viewed both size and value as risk factors, for which one is rewarded with extra return.

The 3-Factor model
To represent the market cap ("size") and book/market ratio ("value") returns, Fama and French modified the original CAPM with two additional risk factors: size risk and value risk.

The original CAPM equation:
 * E(rA) = r(f) + βA(E(rm) - rf)
 * where r(f) is the risk-free rate and
 * E(rm) is the expected excess return of the market portfolio beyond the risk-free rate, often called the equity risk premium.

The Fama and French equation:
 * E(rA) = r(f) + βA(E(rm) - rf) + sASMB + hAHML
 * where SMB is the "Small Minus Big" market capitalization risk factor and
 * HML is the "High Minus Low" value premium risk factor

SMB, Small Minus Big, measures the additional return investors have historically received by investing in stocks of companies with relatively small market capitalization. This additional return is often referred to as the “size premium.”

HML, which is short for High Minus Low, has been constructed to measure the “value premium” provided to investors for investing in companies with high book-to-market values (essentially,the value placed on the company by accountants as a ratio relative to the value the public markets placed on the company, commonly expressed as B/M). (Note terminology usage as mentioned above.)

The key point of the model is that it allows investors to to weight their portfolios so that they have greater or lesser exposure to each of the specific risk factors, and therefore can target more precisely different levels of expected return.

Market risk is a common factor, so it does not appear on the graph. Note that although there are three factors in the model, only two are ever shown.

Categorizing portfolios
One powerful feature of the Three Factor Model is that it provides a way to categorize mutual funds by size and value risks, and therefore predict expected return premiums. This classification provides two main benefits.

Classifying funds into style buckets
Funds (and their fund managers) can be compared by placing them in specific "buckets" based on the style of asset allocation chosen in their portfolios. For this purpose, funds are often plotted on a 3x3 matrix, demonstrating the relative amount of risk represented by different strategies.

The mutual fund rating company Morningstar is the biggest resource for classification. Funds are separated horizontally into three groups through a B/M ranking (value ranking) and vertically based on a ranking of market capitalization (size ranking).



Specifying risk factor helps investor choices
The second advantage of categorizing funds is that investors can easily choose the amount of exposed risk factor when investing in particular funds. This characterization is typically derived by multivariate regression. The historical returns of a specific portfolio are regressed against the historical values of the three factors, generating estimates of the coefficients.



Note how easy it is to see the spectrum of possible strategies with a style graph.

Evaluating fund managers
As shown, the Three-Factor Model allows classification of mutual funds and enables investors to choose exposure to certain risk factors. This model can be extended to measure historical fund manager performance to determine the amount of value added by management.

A new variable, alpha ("α") is added to the equation and the terms rearranged in a form that can be used for regression analysis.


 * E(rA) - r(f) = α + βA(E(rm) - rf) + sASMB + hAHML
 * where α is "effective return" as defined in the CAPM equation

Historical data is utilized in a multivariate regression analysis to determine the value of alpha. A positive alpha indicates that the fund manager is adding to the value of the portfolio versus a result of exposure to the HML or SMB factors. In other words, the three-factor model can help determine the effectiveness of a fund manager.

Articles

 * Small Cap Growth Indexing and the Multifactor Threestep William Bernstein, EF (April 1999)
 * Factor Rotation William Bernstein, EF (Summer 2000)
 * Rolling Your Own: Three Factor Analysis William Bernstein EF (Winter 2001)
 * The Investment Entertainment Pricing Theory William Bernstein EF (Winter 2001)


 * The Cross-Section of Expected Stock Returns: A Tenth Anniversary Reflection William Bernstein, EF (Summer 2002)
 * The Dimensions of Stock Returns Truman A. Clark, DFA (September 2007)
 * Asset Management: Engineering Portfolios for Better Returns Eugene F. Fama Jr. (May 1998)
 * Fama-French Three Factor Model Portfolio Solutions, Rick Ferri

Other references

 * The Bogleheads' Reading List Investing books written by members of the Bogleheads&reg; forum.
 * Source for Graphs (Google Docs Presentation)