FamaFrench threefactor model analysis
This article is intended for investing enthusiasts who wish to analyze historical equity performance. The analysis is used in an advanced investing technique (tilting) which is strongly not recommended for new investors. Past performance does not predict future performance. 
This article shows how to estimate the Fama and French ThreeFactor Model loading (weighting) factors^{[footnotes 1]} 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.^{[footnotes 2]} Building a portfolio by determination of loading factors is known as multifactor investing.
Contents
Multifactor investing
This article describes the endtoend 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.^{[1]} The basic steps are:
 Determine equity / fixed income split  (Asset Allocation)
 Determine Reasonable Targets for FamaFrench Factor Tilts
 Choose Specific Funds for Each Region
 Choose Global Asset Allocations  Each regional fund must be weighted according to its global allocation^{[2]}
 Readjusting Asset Allocation
 Maintenance
Multifactor investing  a comprehensive tutorial contains numerous referenced examples throughout the article, many of which contain a detailed regression analysis. There is no need to repeat those examples here. The External links section contains an example summary, which include both the Bogleheads and Financial Wisdom Forums. 
Portfolio weighting
Factor weightings of a portfolio are the weighted averages of the factor weightings of all the funds in the portfolio.^{[1]} For example, a portfolio consisting of 60% of Fund A, and 40% of Fund B with the following factors:
 Fund_{A} = 60%(1×(r_{mt}  r_{ft}) + 0.6×SMB+ 0.4×HML)
 Fund_{B} = 40%(1×(r_{mt}  r_{ft})  0.2×SMB+ 0.3×HML)
Results in portfolio factor weightings of:
 Fund_{A+B} = (60%(1)+40%(1))×(r_{mt}  r_{ft}) + (60%(0.6)+40%(0.2))×SMB + (60%(0.4)+40%(0.3))×HML
 Fund_{A+B} = 1×(r_{mt}  r_{ft}) + 0.28×SMB + 0.36×HML
Regression analysis model
The regression analysis uses the FamaFrench threefactor model as follows.
Define the equation:^{[3]}
 R_{it}  R_{ft} = α_{i} + β(R_{mt}  R_{ft})+ β_{is}SMB_{t} + β_{ih}HML_{t} + ε_{it}
Configuration:^{[4]}
 Dependent variable ("Yaxis"): (R_{it}  R_{ft})
 Independent variables ("Xaxis"): (R_{mt}  R_{ft}), SMB_{t}, HML_{t}
Parameter  Description  Regression Input / Output 

(R_{it}  R_{ft})  Excess return: (Asset Return  Risk Free Return), also known as "Risk Adjusted Return."  Inputs: asset return, 30day Tbill return 
α_{i}  Active return: The Yaxis intercept of Excess Return. An investment's return over its benchmark.^{[5]}^{[6]}  Output 
β_{im}  Market loading factor: A measure of the exposure an asset has to market risk (although this beta will have a different value from the beta in a CAPM model as a result of the added factors).  Output 
(R_{mt}  R_{ft})  Market: (Market Return  Risk Free Return)  Input: RmRf data 
β_{is}  Size loading factor: The level of exposure to size risk.  Output 
SMB_{t}  Small Minus Big: The size premium, a factor computed as the average return for the smallest 30% of stocks minus the average return of the largest 30% of stocks in that month.  Input: SMB data 
β_{ih}  Value loading factor: The level of exposure to value risk.  Output 
HML_{t}  High Minus Low: The value premium, a factor computed as the average return for the 50% of stocks with the highest B/M ratio minus the average return of the 50% of stocks with the lowest B/M ratio each month.  Input: HML data 
ε_{it}  A random error, which can be regarded as firmspecific risk.^{[3]}^{[footnotes 3]} This is the part of the return which can't be explained by the factors.^{[7]}  Not applicable.^{[footnotes 4]} 
Regression outputs:^{[4]}
 Yaxis intercept: α
 Coefficients (loading factors, the slope of the line): β_{im} (Market), β_{is} (size), β_{ih} (value)
Data quality
There are two metrics, R^{2} and tvalues. 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 R^{2} Coefficient of determination is a statistical measure of how well the regression line approximates the real data points.^{[8]} The lower the R^{2}, the more unexplained movements there are in the returns data, which means greater uncertainty.
An R^{2} value of 1.0 is a perfect fit. For this analysis, R^{2} applies to the regression of the complete model.^{[footnotes 5]} When comparing several portfolios over the same number of samples, the ones with higher R^{2} are explained more completely by the linear model.
Tstatistics
The tstatistic is a ratio of the departure of an estimated parameter from its notional value and its standard error.^{[9]} For this analysis, the tstatistics apply to each factor.
The confidence levels depend on the number of data points. Refer to the Student's tdistribution Table of selected values on Wikipedia. (Or, do it yourself using TDIST() and TINV() spreadsheet functions.) For a large number of data points, the tdistribution approaches a normal distribution. A tvalue 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 tvalue 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.^{[10]}
If the HmL result was again 0.3, but the tvalue 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.^{[10]}
Applications
Expected return
Using the FamaFrench three factor model:
 R_{it}  R_{ft} = α_{i} + β_{im}(R_{mt}  R_{ft} + β_{is}SMB_{t} + β_{ih}HML_{t}
Move R_{ft} to the right side of the equation.
 R_{it} = R_{ft} + β_{im}(R_{mt}  R_{ft}) + β_{is}SMB_{t} + β_{ih}HML_{t} + α_{i}
where R_{it} is the expected return. For example:^{[11]}
 r_{ft} = 4.67, β_{im} = 0.87, (R_{mt}  R_{ft}) = 2.65, β_{is} = 0.63, SMB_{t} = 8.22, β_{ih} = 0.50, HML_{t} = 12.04, α_{i} = 0.05
 4.17% = 4.67 + (0.87)×2.65 + (0.63)×(8.22) + (0.50)×(12.04) + 0.05
Alpha
Alpha is used to evaluate fund manager performance.
 R_{it}  R_{ft} = α_{i} + β_{im}(R_{mt}  R_{ft})+ β_{is}SMB_{t} + β_{ih}HML_{t}
Software
 Kenneth R. French  Data Library  the source of the FamaFrench 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: FamaFrench Regression Tutorial Using R, from The Calculating Investor by forum member camontgo.
 FamaFrench 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 Bogleheads forum member mas.
Notes
 ↑ A factor is a common characteristic among a group of assets. The FamaFrench factors of size and booktomarket have crosssectional characteristics. Hence, the title of the seminal paper "The CrossSection of Expected Stock Returns" (1992). See: Factors (finance).
 ↑ The concept of regression might sound strange because the term is normally associated with movement backward, whereas in the world of statistics, regression is often used to predict the future. Simply put, regression is a statistical technique that finds a mathematical expression that best describes a set of data. Ref: Perform a regression analysis, from Microsoft.
 ↑ Residual error, uncorrelated with the market return. Also referred to as unsystematic risk, companyspecific risk, companyunique risk, or idiosyncratic risk. Ref: Fabozzi, et al. "Chapter 14.5.1 Decomposition of Total Risk".
 ↑ The residual is the difference between the actual value of the dependent variable for each sample and the estimate of the dependent variable given by the regression equation. Basically, it is the error in the regression estimate of the sample value. The regression is a "least squares" optimization, which means that the intercept and factor loadings are chosen to minimize the squared sum of all the residuals. (From forum member camontgo, via PM.)
 ↑ General guidance on acceptable ranges of R^{2} cannot be recommended. See: What's a good value for Rsquared?, from Duke University.
See also
Bogleheads wiki
 CAPM  Capital Asset Pricing Model
 Expected return
 Fama and French ThreeFactor Model
 Factors (finance)
References
 ↑ ^{1.0} ^{1.1} Multifactor Investing  A comprehensive tutorial, Financial Wisdom Forum, direct link to post.
 ↑ Multifactor Investing  A comprehensive tutorial, direct link to post.
 ↑ ^{3.0} ^{3.1} Frank J. Fabozzi; Edwin H. Neave; Guofu Zhou (eds). "14: Capital Asset Pricing Model". Financial Economics. John Wiley & Sons. © 2011. ISBN 0470596201
 ↑ ^{4.0} ^{4.1} ^{4.2} Rolling Your Own: Three Factor Analysis William Bernstein EF (Winter 2001)
 ↑ ^{5.0} ^{5.1} Womack, Kent L. and Zhang, Ying, Understanding Risk and Return, the CAPM, and the FamaFrench ThreeFactor Model. Tuck Case No. 03111. Available at SSRN: http://ssrn.com/abstract=481881
 ↑ Fabozzi, Frank J., and Harry M. Markowitz (eds). "Chapter 10  Tracking Error and Common Stock Portfolio Management". Equity Valuation and Portfolio Management. John Wiley & Sons. © 2011. ISBN 9780470929919
 ↑ From forum member camontgo, via PM.
 ↑ Goodness of fit, Coefficient of determination, from Wikipedia.
 ↑ tstatistic, standard error, from Wikipedia.
 ↑ ^{10.0} ^{10.1} How to get FamaFrench EAFE Factors, with results, forum discussion, direct link to post.
 ↑ How to get FamaFrench EAFE Factors, with results, Bogleheads Forum, direct link to post.
External links
 Asset Management: Engineering Portfolios for Better Returns Eugene F. Fama Jr. (May 1998)
 Fama–French threefactor model, from Wikipedia
 Capital asset pricing model, from Wikipedia
 Rolling Your Own: Three Factor Analysis William Bernstein EF (Winter 2001)

 The Investment Entertainment Pricing Theory William Bernstein EF (Winter 2001)
Bogleheads forum discussions
 Collective thoughts, forum post by Robert T. The best reference collection of anything you need to know about FamaFrench, as well as risk factors, risk exposure and more. Includes both equity and fixed income risk.
 How to get FamaFrench EAFE Factors, with results, tutorial by forum member ClosetIndexer
Academic papers (external links)  

