Fama-French three-factor model analysis

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This article shows how to estimate the Fama and French Three-Factor Model loading (weighting) factors which are typically used to determine the  expected return of a portfolio or fund manager performance. These factors are determined by use of a regression analysis.

Process overview
First, define the Fama-French three-factor model:


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

Then, gather the necessary data and perform a multiple regression analysis to find the output parameters in the following table.

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)

Data quality
There are two metrics, R2 and t-values.

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

Set $$\alpha_i$$ to 0 and move $$r_{ft}$$ to the other side:


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

where $$r_{it}$$ is the expected return.

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

 * 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
 * RStudio, a free and open source integrated development environment (IDE) for R (a free software environment for statistical computing and graphics).
 * Factor Attribution « Systematic Investor
 * Systematic Investor Toolbox, (includes the Three Factor Rolling Regression Viewer by forum member mas)
 * mas financial tools, by forum member mas

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