Fama-French three-factor model analysis

This article shows how to estimate the Fama and French Three-Factor Model loading (weighting) factors^{[note 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.^{[note 2]} 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.^[1] 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^[2]
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.^[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 Fama-French three-factor model as follows.

Define the equation:^[3]

R_it - R_ft = α_i + β(R_mt - R_ft)+ β_isSMB_t + β_ihHML_t + ε_it

Configuration:^[4]

Dependent variable ("Y-axis"): (R_it - R_ft)
Independent variables ("X-axis"): (R_mt - R_ft), SMB_t, HML_t

**Fama-French Parameters**^[5]^[4]
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, 30-day T-bill return
α_i	Active return: The Y-axis 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: Rm-Rf 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 firm-specific risk.^[3]^{[note 3]} This is the part of the return which can't be explained by the factors.^[7]	Not applicable.^{[note 4]}

Regression outputs:^[4]

Y-axis 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, 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.^[8] 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.^{[note 5]} 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.^[9] 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.^[10]

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.^[10]

Applications

Expected return

Using the Fama-French three factor model:

R_it - R_ft = α_i + β_im(R_mt - R_ft + β_isSMB_t + β_ihHML_t

Move R_ft to the right side of the equation.

R_it = R_ft + β_im(R_mt - R_ft) + β_isSMB_t + β_ihHML_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_isSMB_t + β_ihHML_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 Bogleheads forum member mas.

Notes

^ A factor is a common characteristic among a group of assets. The Fama-French factors of size and book-to-market have cross-sectional characteristics. Hence, the title of the seminal paper "The Cross-Section 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, company-specific risk, company-unique 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 R² cannot be recommended. See: What's a good value for R-squared?, from Duke University.

References

^ ^a ^b Multifactor Investing - A comprehensive tutorial, Financial Wisdom Forum, direct link to post.
^ Multifactor Investing - A comprehensive tutorial, direct link to post.
^ ^a ^b Frank J. Fabozzi; Edwin H. Neave; Guofu Zhou (eds). "14: Capital Asset Pricing Model". Financial Economics. John Wiley & Sons. © 2011. ISBN 0-470596-20-1
^ ^a ^b ^c Rolling Your Own: Three Factor Analysis William Bernstein EF (Winter 2001)
^ ^a ^b Womack, Kent L. and Zhang, Ying, Understanding Risk and Return, the CAPM, and the Fama-French Three-Factor Model. Tuck Case No. 03-111. 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.
^ t-statistic, standard error, from Wikipedia.
^ ^a ^b How to get Fama-French EAFE Factors, with results, forum discussion, direct link to post.
^ How to get Fama-French 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 three-factor 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 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

[1] A factor is a common characteristic among a group of assets. The Fama-French factors of size and book-to-market have cross-sectional characteristics. Hence, the title of the seminal paper "The Cross-Section of Expected Stock Returns" (1992). See: Factors (finance).

[2] 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.

[9] Residual error, uncorrelated with the market return. Also referred to as unsystematic risk, company-specific risk, company-unique risk, or idiosyncratic risk. Ref: Fabozzi, et al. "Chapter 14.5.1 Decomposition of Total Risk".

[11] 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.)

[13] General guidance on acceptable ranges of R² cannot be recommended. See: What's a good value for R-squared?, from Duke University.

[CI-3] Multifactor Investing - A comprehensive tutorial, Financial Wisdom Forum, direct link to post.

[4] Multifactor Investing - A comprehensive tutorial, direct link to post.

[Fabozzi-5] Frank J. Fabozzi; Edwin H. Neave; Guofu Zhou (eds). "14: Capital Asset Pricing Model". Financial Economics. John Wiley & Sons. © 2011. ISBN 0-470596-20-1

[Roll-6] Rolling Your Own: Three Factor Analysis William Bernstein EF (Winter 2001)

[R-n-R-7] Womack, Kent L. and Zhang, Ying, Understanding Risk and Return, the CAPM, and the Fama-French Three-Factor Model. Tuck Case No. 03-111. Available at SSRN: http://ssrn.com/abstract=481881

[Fabozzi10-8] 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

[10] From forum member camontgo, via PM.

[12] Goodness of fit, Coefficient of determination, from Wikipedia.

[14] t-statistic, standard error, from Wikipedia.

[Closet-15] How to get Fama-French EAFE Factors, with results, forum discussion, direct link to post.

[16] How to get Fama-French EAFE Factors, with results, Bogleheads Forum, direct link to post.

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