# Fama-French three-factor model analysis

Fama-French three-factor model analysis describes aspects of Fama and French three-factor model loading (weighting) factors[note 1] which 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]
• 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:

${\displaystyle{\displaystyle Fund_{A}=60\%(1\times(r_{mt}-r_{ft})+0.6\times{% \mathit{SMB}}+0.4\times{\mathit{HML}})}}$
${\displaystyle{\displaystyle Fund_{B}=40\%(1\times(r_{mt}-r_{ft})-0.2\times{% \mathit{SMB}}+0.3\times{\mathit{HML}})}}$

Results in portfolio factor weightings of:

${\displaystyle{\displaystyle Fund_{A+B}=(60\%(1)+40\%(1))\times(r_{mt}-r_{ft})% +(60\%(0.6)+40\%(-0.2))\times{\mathit{SMB}}+(60\%(0.4)+40\%(0.3))\times{% \mathit{HML}}}}$
${\displaystyle{\displaystyle Fund_{A+B}=1\times(r_{mt}-r_{ft})+0.28\times{% \mathit{SMB}}+0.36\times{\mathit{HML}}}}$

## Regression analysis model

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

Define the equation:[3]

${\displaystyle{\displaystyle r_{it}-r_{ft}=\alpha_{i}+\beta_{im}(r_{mt}-r_{ft}% )+\beta_{is}{\mathit{SMB}}_{t}+\beta_{ih}{\mathit{HML}}_{t}+\epsilon_{it}}}$

Configuration:[4]

• Dependent variable ("Y-axis"): ${\displaystyle{\displaystyle(r_{it}-r_{ft})}}$
• Independent variables ("X-axis"): ${\displaystyle{\displaystyle(r_{mt}-r_{ft})}}$, ${\displaystyle{\displaystyle{SMB}_{t}}}$, ${\displaystyle{\displaystyle{HML}_{t}}}$
Fama-French Parameters[5][4]
Parameter Description Regression Input / Output
${\displaystyle{\displaystyle(r_{it}-r_{ft})}}$ Excess return: (Asset Return - Risk Free Return), also known as "Risk Adjusted Return." Inputs: asset return, 30-day T-bill return
${\displaystyle{\displaystyle\alpha_{i}}}$ Active return: The Y-axis intercept of Excess Return. An investment's return over its benchmark.[6][7] Output
${\displaystyle{\displaystyle\beta_{im}}}$ 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
${\displaystyle{\displaystyle(r_{mt}-r_{ft})}}$ Market: (Market Return - Risk Free Return) the excess return on the market, value-weight return of all CRSP firms incorporated in the US and listed on the NYSE, AMEX, or NASDAQ that have a CRSP share code of 10 or 11 at the beginning of month t, good shares and price data at the beginning of t, and good return data for t minus the one-month Treasury bill rate (from Ibbotson Associates). Input: Rm-Rf data
${\displaystyle{\displaystyle\beta_{is}}}$ Size loading factor: The level of exposure to size risk. Output
${\displaystyle{\displaystyle{SMB}_{t}}}$ Small Minus Big: The size premium, is the average return on the three small portfolios minus the average return on the three big portfolios, 1/3 (Small Value + Small Neutral + Small Growth) - 1/3 (Big Value + Big Neutral + Big Growth). Input: SMB data
${\displaystyle{\displaystyle\beta_{ih}}}$ Value loading factor: The level of exposure to value risk. Output
${\displaystyle{\displaystyle{HML}_{t}}}$ High Minus Low: The value premium, is the average return on the two value portfolios minus the average return on the two growth portfolios, 1/2 (Small Value + Big Value) - 1/2 (Small Growth + Big Growth). Input: HML data
${\displaystyle{\displaystyle\epsilon_{it}}}$ 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.[8] Not applicable.[note 4]

Regression outputs:[4]

• Y-axis intercept: ${\displaystyle{\displaystyle\alpha}}$
• Coefficients (loading factors, the slope of the line): ${\displaystyle{\displaystyle\beta_{im}}}$(Market), ${\displaystyle{\displaystyle\beta_{is}}}$ (size), ${\displaystyle{\displaystyle\beta_{ih}}}$ (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.[9] 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.[10] 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.[11]

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.[11]

## Applications

### Expected return

Using the Fama-French three factor model:

${\displaystyle{\displaystyle r_{it}-r_{ft}=\alpha_{i}+\beta_{im}(r_{mt}-r_{ft}% )+\beta_{is}{\mathit{SMB}}_{t}+\beta_{ih}{\mathit{HML}}_{t}}}$

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

${\displaystyle{\displaystyle r_{it}=r_{ft}+\beta_{im}(r_{mt}-r_{ft})+\beta_{is% }{\mathit{SMB}}_{t}+\beta_{ih}{\mathit{HML}}_{t}+\alpha_{i}}}$

where ${\displaystyle{\displaystyle r_{it}}}$ is the expected return. For example:[12]

• ${\displaystyle{\displaystyle r_{ft}=4.67}}$, ${\displaystyle{\displaystyle\beta_{im}=0.87}}$, ${\displaystyle{\displaystyle(r_{mt}-r_{ft})=2.65}}$, ${\displaystyle{\displaystyle\beta_{is}=0.63}}$, ${\displaystyle{\displaystyle{\mathit{SMB}}_{t}=-8.22}}$, ${\displaystyle{\displaystyle\beta_{ih}=0.50}}$, ${\displaystyle{\displaystyle{\mathit{HML}}_{t}=-12.04}}$, ${\displaystyle{\displaystyle\alpha_{i}=0.05}}$
${\displaystyle{\displaystyle-4.17\%=4.67+(0.87)\times 2.65+(0.63)\times(-8.22)% +(0.50)\times(-12.04)+0.05}}$

### Alpha

Alpha is used to evaluate fund manager performance.

${\displaystyle{\displaystyle r_{it}-r_{ft}=\alpha_{i}+\beta_{im}(r_{mt}-r_{ft}% )+\beta_{is}{\mathit{SMB}}_{t}+\beta_{ih}{\mathit{HML}}_{t}}}$

See: Evaluating fund managers

## Software

### R

RStudio is the recommended tool for performing regression analysis.

### 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.

## Notes

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.
3. 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".
4. 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.)
5. General guidance on acceptable ranges of R2 cannot be recommended. See: What's a good value for R-squared?, from Duke University.

## References

1. Multifactor Investing - A comprehensive tutorial, Financial Wisdom Forum, direct link to post.
2. Multifactor Investing - A comprehensive tutorial, direct link to post.
3. 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
4. Rolling Your Own: Three Factor Analysis William Bernstein EF (Winter 2001)
5. 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
6. 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
7. From forum member camontgo, via PM.
8. Goodness of fit, Coefficient of determination, from Wikipedia.
9. t-statistic, standard error, from Wikipedia.
10. How to get Fama-French EAFE Factors, with results, forum discussion, direct link to post.
11. How to get Fama-French EAFE Factors, with results, Bogleheads Forum, direct link to post.