Fama-French three-factor model analysis

 describes aspects of Fama and French three-factor model loading (weighting) factors which determine the  expected return of a portfolio or fund manager performance. These factors are determined by use of a regression analysis. Building a portfolio by determination of loading factors is known as multifactor investing.

Multifactor investing
This article describes the end-to-end process to create and maintain a portfolio. The objective is to match the desired factor loads while optimizing other factors like costs, (negative) alpha, diversification, taxes, etc. The basic steps are:
 * Determine equity / fixed income split - (Asset allocation)
 * Determine Reasonable Targets for Fama-French Factor Tilts
 * Choose Specific Funds for Each Region
 * Choose Global Asset Allocations - Each regional fund must be weighted according to its global allocation
 * Re-adjusting Asset Allocation
 * Maintenance

Portfolio weighting
Factor weightings of a portfolio are the weighted averages of the factor weightings of all the funds in the portfolio. For example, a portfolio consisting of 60% of Fund A, and 40% of Fund B with the following factors:
 * $$Fund_A=60%(1\times(r_{mt}-r_{ft})+0.6\times\mathit{SMB}+0.4\times\mathit{HML})$$
 * $$Fund_B=40%(1\times(r_{mt}-r_{ft})-0.2\times\mathit{SMB}+0.3\times\mathit{HML})$$

Results in portfolio factor weightings of:
 * $$Fund_{A+B}=(60%(1)+40%(1))\times(r_{mt}-r_{ft})+(60%(0.6)+40%(-0.2))\times\mathit{SMB}+(60%(0.4)+40%(0.3))\times\mathit{HML}$$
 * $$Fund_{A+B}=1\times(r_{mt}-r_{ft})+0.28\times\mathit{SMB}+0.36\times\mathit{HML}$$

Regression analysis model
The regression analysis uses the Fama-French three-factor model as follows.

Define the equation:


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

Configuration:
 * Dependent variable ("Y-axis"): $$(r_{it}-r_{ft})$$
 * Independent variables ("X-axis"): $$(r_{mt}-r_{ft})$$, $${SMB}_t$$, $${HML}_t$$

Regression outputs:
 * Y-axis intercept: $$\alpha$$
 * Coefficients (loading factors, the slope of the line): $$\beta_{im}$$(Market), $$\beta_{is}$$ (size), $$\beta_{ih}$$ (value)

Small/Big and High/Low definitions: "The median NYSE size is then used to split NYSE, Amex, and (after 1972) NASDAQ stocks into two groups, small and big (S and B). Most Amex and NASDAQ stocks are smaller than the NYSE median, so the small group contains a disproportionate number of stocks (3,616 out of 4,797 in 1991). Despite its large number of stocks, the small group contains far less than half (about 8% in 1991) of the combined value of the two size groups. We also break NYSE, Amex, and NASDAQ stocks into three book-to-market equity groups based on the breakpoints for the bottom 30% (Low), middle 40% (Medium), and top 30% (High) of the ranked values of BE/ME for NYSE stocks."

Data quality
There are two metrics, R2 and t-values. Use best judgment to determine if the metrics are within acceptable limits. If not, modify input parameters (or assumptions) and repeat the analysis.

Coefficient of determination
The Goodness of fit of a statistical model describes how well it fits a set of observations. In regression, the R2 Coefficient of determination is a statistical measure of how well the regression line approximates the real data points. The lower the R2, the more unexplained movements there are in the returns data, which means greater uncertainty.

An R2 value of 1.0 is a perfect fit. For this analysis, R2 applies to the regression of the complete model. When comparing several portfolios over the same number of samples, the ones with higher R2 are explained more completely by the linear model.

T-statistics
The t-statistic is a ratio of the departure of an estimated parameter from its notional value and its  standard error. For this analysis, the t-statistics apply to each factor.

The confidence levels depend on the number of data points. Refer to the Student's t-distribution Table of selected values on Wikipedia. (Or, do it yourself using TDIST and TINV spreadsheet functions.) For a large number of data points, the t-distribution approaches a normal distribution. A t-value of 1 (or -1 for a negative factor) means the standard error is equal to the magnitude of the value itself.

For example, an HmL of 0.3 with a t-value of 1 means the standard error of that measurement is also 0.3. For 68% of the time (normal distribution assumed), the true value is 0.3 +/-0.3, or between 0.0 and 0.6.

If the HmL result was again 0.3, but the t-value was 3, the standard error is 0.1. For 68% of the time (normal distribution assumed), the true value is 0.3 +/-0.1, or between 0.2 and 0.4.

Expected return
Using the Fama-French three factor model:
 * $$r_{it}-r_{ft}=\alpha_i +\beta_{im}(r_{mt}-r_{ft})+\beta_{is}\mathit{SMB}_t+\beta_{ih}\mathit{HML}_t$$

Move $$r_{ft}$$ to the right side of the equation.


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

where $$r_{it}$$ is the expected return. For example:
 * $$r_{ft}=4.67$$, $$\beta_{im}=0.87$$, $$(r_{mt}-r_{ft})=2.65$$, $$\beta_{is}=0.63$$, $$\mathit{SMB}_t=-8.22$$, $$\beta_{ih}=0.50$$, $$\mathit{HML}_t=-12.04$$, $$\alpha_i=0.05$$


 * $$-4.17%=4.67+(0.87)\times2.65+(0.63)\times(-8.22)+(0.50)\times(-12.04)+0.05$$

Alpha
Alpha is used to evaluate fund manager performance.


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

See: Evaluating fund managers

Software

 * Kenneth R. French - Data Library - the source of the Fama-French factors.

R
RStudio is the recommended tool for performing regression analysis.
 * RStudio, a free and open source integrated development environment (IDE) for R (a free software environment for statistical computing and graphics).
 * Screencast: Fama-French Regression Tutorial Using R, from The Calculating Investor by forum member camontgo.
 * Fama-French Regression example in R, R script by forum member ClosetIndexer
 * Factor Attribution « Systematic Investor
 * Systematic Investor Toolbox, (includes the Three Factor Rolling Regression Viewer by forum member mas)

Spreadsheet

 * Rolling Your Own: Three Factor Analysis William Bernstein EF (Winter 2001) - an excellent tutorial on how to do this in Excel.

Rolling regression viewer

 * mas financial tools, experimental java utility by forum member mas.

Online factor regression analysis tool
Portfolio Visualizer, by forum member pvguy, is an easy-to-use online tool to determine Fama-French factors for one or more assets.

Forum discussions

 * Larry Swedroe - Saint Louis Post-Dispatch 05/06/07, forum post by Larry Swedroe. A tutorial on Fama and French's Three-Factor model, focusing on risk factors as a technique for portfolio diversification.
 * Collective thoughts, forum post by Robert T. The best reference collection of anything you need to know about Fama-French, as well as risk factors, risk exposure and more. Includes both equity and fixed income risk.
 * How to get Fama-French EAFE Factors, with results, tutorial by forum member ClosetIndexer

Other references

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