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.

Eugene F. Fama and Kenneth R. French found that on average, a portfolio’s beta (single-factor model) 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.

The three-factor model explains up to 95% of returns for a cross-section of equity portfolios of various sizes and styles.

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 model with two additional risk  factors: size risk and value risk.


 * $$r_{it}-r_{ft}=\alpha_i +\beta_{im}(r_{mt}-r_{ft})+\beta_{is}\mathit{SMB}_t+\beta_{ih}\mathit{HML}_t+\epsilon_{it}$$


 * where $$\mathit{SMB}_t$$ is the "Small Minus Big" market capitalization risk factor and $$\mathit{HML}_t$$ is the "High Minus Low" value premium risk factor.

$$\mathit{SMB}$$ (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.”

$$\mathit{HML}$$ (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).

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.

Refer to the Risk Factor Exposure plot below, which represents a universe of opportunities. A portfolio can land anywhere on this plot (the axes values are not restricted) and an expected return can be calculated. The axes represent exposures to the two risk factors. As all equity portfolios take similar market risk (common to both), there is no need for a 3rd axis, &beta; (beta).

The horizontal axis (X-axis) is the "Value" factor, represented as SMB. The left side (more negative) is termed "Growth," the right side (more positive) is "Value."

The vertical axis (Y-axis) is the "Size" factor, represented as HML. The upper side (more positive) is termed "Small Cap," the lower side (more negative) is "Large Cap."

At the plot origin (0, 0) is the "The Market," the baseline reference. The dashed line running through the origin represents equivalent market risk. Values along this line represent the point where the risk factors cancel out. For example, (-0.55, 0.55) represents ((-0.55)*SMB, (+0.55)*HML); which, when added together (and skipping a lot of details) results in the expected return of the market (stated in relative terms, which is 0.00%).

This dashed line is used to define risk relative to the market. The farther up and to the right of the market line you go, the higher the expected return and the higher the risk. Lower and to the left of the line represents less expected return but lower risk, relative to the market. Note that stocks which fit definitions of "Small Cap and Value" represent the highest risk and highest expected return.

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 above, the Three-Factor Model allows classification of mutual funds and enables investors to choose exposure to certain risk factors. This model can also used to measure historical fund manager performance to determine the amount of value added by management.


 * $$r_{it}-r_{ft}=\alpha_i +\beta_{im}(r_{mt}-r_{ft})+\beta_{is}\mathit{SMB}_t+\beta_{ih}\mathit{HML}_t$$


 * where $$\alpha$$, the Y-intercept of the equation, is the Active Return and defined as:


 * $$\alpha$$ = Active Return = (Portfolio Actual Return - Benchmark Actual Return)

In this case, the benchmark is the $$r_{ft}$$, risk-free market return. Historical data is utilized in a multiple regression analysis to determine the value of $$\alpha$$.

Alpha indicates how well the fund manager is capturing the expected returns, given the portfolio's exposure to the $$\mathit{(r_{mt}-r_{ft})}$$, $$\mathit{HML}$$ and $$\mathit{SMB}$$ factors.

If the fund manager captures the factor exposures perfectly, the expected alpha would be zero, minus the expense ratio (ER) of the fund. An alpha greater than this suggests that the fund manager is adding value beyond the underlying factor exposures. In other words, the three-factor model can help determine the effectiveness of a fund manager.

Tracking error is a measure of how closely a portfolio follows the index to which it is benchmarked. See: below for additional details.

Tracking error
Tracking error is measured as the dispersion of a portfolio's returns relative to the returns of its benchmark, and is expressed as the standard deviation of the portfolio's active return (annualized).

A portfolio created to match the benchmark index (e.g. an index fund) that always matches its benchmark's actual return (zero active returns) would have a tracking error of zero.

There are two types of tracking error models: Backward looking tracking error is useful to analyze fund manager performance. This model has little predictive value and may be misleading if used in that fashion.
 * Based on historical performance: Also called backward-looking or ex post tracking error.
 * Predicted future performance: Also called predicted or ex ante tracking error.

A portfolio manager uses a forward-looking estimate of tracking error to accurately reflect the portfolio risk going forward.

This is done by using a multifactor risk model which contains the risks associated with the benchmark index. Statistical analysis of historical returns in the benchmark index are used to obtain the factors and quantify their risk (variances and correlations are involved). The portfolio's current exposure to the various factors are calculated and compared to the benchmark's exposures to the factors. A forward-looking tracking error is then calculated from the differential factor exposures and risks of the factors.

The forward-looking tracking error is useful in risk control and portfolio construction. "What-if" scenarios can be evaluated to optimize the portfolio within the desired level of risk. Although there are no guarantees that the forward-looking tracking error will match the backward-looking historical error over a period of time (for example, one year), the average of forward-looking tracking error estimates obtained at different times during the year will be reasonably close to the backward-looking tracking error estimate obtained at the end of the year.

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
 * Multifactor Investing, Dimensional Fund Advisors, Eugene F. Fama Jr., July 2006 (Eugene F. Fama Jr. is the son of Eugene F. Fama)

Other references

 * Wiki_charts_CAPM_Fama_French_3_factor.ppt (Google Docs) Source file for graphs, MS PowerPoint.