# Fama-French three-factor model analysis

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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. 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
• 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. For example, a portfolio consisting of 60% of Fund A, and 40% of Fund B with the following factors:

$Fund_{A}=60\%(1\times (r_{mt}-r_{ft})+0.6\times {\mathit {SMB}}+0.4\times {\mathit {HML}})$ $Fund_{B}=40\%(1\times (r_{mt}-r_{ft})-0.2\times {\mathit {SMB}}+0.3\times {\mathit {HML}})$ Results in portfolio factor weightings of:

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

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

Define the equation:

$r_{it}-r_{ft}=\alpha _{i}+\beta _{im}(r_{mt}-r_{ft})+\beta _{is}{\mathit {SMB}}_{t}+\beta _{ih}{\mathit {HML}}_{t}+\epsilon _{it}$ Configuration:

• Dependent variable ("Y-axis"): $(r_{it}-r_{ft})$ • Independent variables ("X-axis"): $(r_{mt}-r_{ft})$ , ${SMB}_{t}$ , ${HML}_{t}$ Fama-French Parameters
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
$\alpha _{i}$ Active return: The Y-axis intercept of Excess Return. An investment's return over its benchmark. Output
$\beta _{im}$ 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
$(r_{mt}-r_{ft})$ 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
$\beta _{is}$ Size loading factor: The level of exposure to size risk. Output
${SMB}_{t}$ 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
$\beta _{ih}$ Value loading factor: The level of exposure to value risk. Output
${HML}_{t}$ 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
$\epsilon _{it}$ A random error, which can be regarded as firm-specific risk.[note 3] This is the part of the return which can't be explained by the factors. Not applicable.[note 4]

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. 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. 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. 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.

## Applications

### Expected return

Using the Fama-French three factor model:

$r_{it}-r_{ft}=\alpha _{i}+\beta _{im}(r_{mt}-r_{ft})+\beta _{is}{\mathit {SMB}}_{t}+\beta _{ih}{\mathit {HML}}_{t}$ Move $r_{ft}$ to the right side of the equation.

$r_{it}=r_{ft}+\beta _{im}(r_{mt}-r_{ft})+\beta _{is}{\mathit {SMB}}_{t}+\beta _{ih}{\mathit {HML}}_{t}+\alpha _{i}$ where $r_{it}$ is the expected return. For example:

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

Alpha is used to evaluate fund manager performance.

$r_{it}-r_{ft}=\alpha _{i}+\beta _{im}(r_{mt}-r_{ft})+\beta _{is}{\mathit {SMB}}_{t}+\beta _{ih}{\mathit {HML}}_{t}$ 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.