Fama-French three-factor model analysis

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
The analysis starts by gathering data and performing a multiple regression analysis to find the output parameters in the following table.

Configuration:
 * Dependent variable ("Y-axis"): $$(r_a-r_f)$$
 * Independent variables ("X-axis"): $$(r_m-r_f)$$, $${SMB}$$, $${HML}$$

Regression outputs:
 * $$\alpha$$: A risk-adjusted measure of the so-called active return on an investment.
 * Coefficients (loading factors): $$\beta_a$$ (Market), $$s_a$$ (size), $$h_a$$ (value)

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. As a rough guide:


 * Perfect fit: 1.0
 * Good: > 0.97
 * Acceptable: > 0.95
 * Somewhat usable: > 0.94
 * Rough estimate: >= 0.90
 * Unusable: < 0.90

T-statistics
The t-statistic is a ratio of the departure of an estimated parameter from its notional value and its  standard error.

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, which means 68% of the time the true value is between 0.3-0.3 and 0.3+0.3, ie between 0 and 0.6.

If the HmL result were again 0.3, but the t-value were 3, that would mean the standard error was 0.1, so 68% of the time the true value would be between 0.2 and 0.4. If the HmL factor is approximated (vs. using the Fama-French library), an addtional +/- 0.1 should be added to the uncertainty. (To be reviewed.)

Expected return
From the Fama-French equation:
 * $$r_a=r_f +\beta_a(r_m-r_f)+s_a\cdot\mathit{SMB}+h_a\cdot\mathit{HML}$$

Subtract the average market return from both sides to get the market adjusted expected return (expected return over market):
 * $$(r_a-r_m)=(r_f-r_m) +\beta_a(r_m-r_f)+s_a\cdot\mathit{SMB}+h_a\cdot\mathit{HML}$$

For example (note the sign swap due to $$(r_f-r_m)$$):


 * $$2.69%=(-1.55)+(1.01*5.55)+(0.26*3.73)+(0.31*5.34)$$


 * where: $$\beta_a$$ = 1.01, $$(r_m-r_f)$$ = 5.55, $$s_a$$ = 0.26, $$SMB$$ = 3.73, $$h_a$$ = 0.31, $$HML$$ = 5.34

If the average market return was 5%, the asset's expected return is 7.69%.

Alpha
Alpha is used to evaluate fund manager performance:
 * $$(r_a-r_f)=\alpha+\beta_a(r_m-r_f)+s_a\cdot\mathit{SMB}+h_a\cdot\mathit{HML}$$

Analysis

 * how to download data
 * run script
 * show results, plot
 * get expected return
 * subtract transaction costs(??)
 * new expected return

Software

 * 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