# How Does the Fama French 3 Factor Model Work?

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The Fama-French Three Factor model is a formula for calculating the rate of return on a given asset. Like many (if not most) such models, it offers an estimated value based on market factors at large. In this case, investors can predict their return on investment based on overall market risk, market size and market value. What follows is a description of how it works. Meanwhile, consider using the services of a financial advisor to make sure you’re taking full advantage of the most appropriate analytical tools.

## What Is the Fama-French Three Factor Model?

The Fama-French Three Factor model is a formula to describe the rate of return on a stock investment. Developed in 1992 by then-University of Chicago professors Eugene Fama and Kenneth French, it is based on the observation that value shares tend to outperform growth shares and small-cap shares tend to outperform large-cap shares. Jumping on those observations the two economists developed their three-factor model as an expansion of the Capital Asset Pricing Model (CAPM). Rather than just gauge market risk as the CAPM does, the Fama-French Three Factor model adds value risk and size risk to the calculation.

The Fama-French Three Factor model calculates an investment’s likely rate of return based on three elements: overall market risk, the degree to which small companies outperform large companies and the degree to which high-value companies outperform low-value companies.

The model uses market capitalization to calculate a company’s size, comparing small-cap firms to large cap-firms. It uses book-to-market to calculate a company’s value, comparing high book-to-market value companies against low book-to-market value companies. Book-to-market is simply the inverse of the price-to-book ratio. This third element is used to distinguish value stocks from growth stocks.

The model is known as a three-factor model, in distinction to the CAPM, which only uses the single factor of market risk to calculate the likely return on investment.

## The Fama-French Three Factor Model Formula

In shorthand this model is expressed as:

• Return = Rf + Ri + SMB + HML

Where:

• Return is the rate of return on your portfolio or investment being measured
• Rf is the risk-free rate, the rate of return given by a zero-risk asset such as a Treasury bond or bill
• Ri is the market risk premium, the rate at which investing in the market at large outperforms investing in a zero-risk asset
• SMB (or Small Minus Big) is the performance of small-cap companies vs. large-cap companies
• HML (or High Minus Low) is the performance of high book-to-market (or “value”) stocks vs. low book-to-market (or “growth”) stocks

Expressed as a complete formula the Fama-French Three Factor model is:

R = Rf + B1(Rm – Rf) + B2(SMB) + B3(HML) + a

In this expanded model we have added a few elements:

• B1, B2, B3 – The market coefficient for each factor of the model
• Rm – Total market return
• a – The investment’s alpha

The market coefficients (B1, B2 and B3) set the Fama-French Three Factor model apart from its predecessor. In the CAPM, the return on investment is calculated using only one element and only one coefficient of risk. In fact, that formula is used as the basis for the Fama-French Three Factor model. The CAPM model is:

R = Rf + B(Rm – Rf)

The Fama-French Three Factor model expands on this concept. Under the CAPM model, the return on your investment is estimated based entirely on overall market risk. The Fama-French Three Factor model estimates an investment’s return based on market risk, market size and investment value.

### Factor 1 – Market Risk

The CAPM makes up the first factor of the Fama-French Three Factor. Its central element, (Rm – Rf), is known as the “market risk premium.” It measures the returns you get by investing in the market (which carries the potential for loss) compared to the returns you would get by investing in a risk-free asset. This difference is your compensation for accepting the market’s risk of loss.

Typically when calculating formulas such as the CAPM and the Fama-French Three Factor model you will use the return on U.S. Treasury bills or bonds as the risk-free rate.

In the CAPM the beta variable, “B1” in the formula above, is calculated based on the volatility of the investment being measured. This represents the risks involved with that investment.

### Factor 2 – Small Minus Big

One of the two key observations of the Fama-French Three Factor model is that small firms tend – over the long term – to outperform large firms when it comes to stock market returns. This element of the model captures that observation.

The SMB factor of the Fama-French Three Factor model measures the degree to which small-cap companies have historically posted excess returns over large-cap companies. It helps to weight the model in favor of small-cap companies, as the Fama-French Three Factor model predicts that investment portfolios with smaller companies will have higher rates of return than portfolios with larger companies.

The SMB beta, “B2” in the formula above, is calculated based on assets in the portfolio being measured compared against small-cap/large-cap returns in the market at large. Generally speaking, a B2 value above zero indicates that return in the investment portfolio being measured is weighted toward small-cap stocks. (For more information on interpreting SMB beta values, see this paper.)

### Factor 3 – High Minus Low

The second key observation in the Fama-French model is that firms with high book-to-market values tend to post stronger returns than those with low book-to-market values. This factor of the model captures that observation.

The HML factor of the Fama-French Three Factor model measures the average return on value portfolios (those with companies that have a high book-to-market value) against the average return on growth portfolios (those with companies that have a low book-to-market value). It helps to weight the model in favor of value stocks, as the Fama-French Three Factor model predicts that investment portfolios with value stocks will have higher rates of return than portfolios with growth stocks.

The HML beta, “B3” in the formula above, is calculated based on assets in the portfolio being measured compared against the value/growth stock returns in the market at large. Generally, a B3 value above zero indicates that the portfolio being measured is weighted toward high book-to-market stocks.

### Calculating SMB and HML

The SMB and the HML are historic market constants. In other words, at any given time you will use the same values for SMB and HML in any given Fama-French Three Factor calculation. At the time of writing, French posted a running calculation of the SMB and HML values on his website here and explained the formula for calculating them here.

### The Alpha

The final variable of the Fama-French Three Factor model, “a,” represents the investment’s risk. This is more formally known as the investment’s alpha. This is a relatively rarely applied variable.

Alpha measures an investment’s ability to beat the market. If a given investment or portfolio manages to post stronger returns than comparable investments in the market at large, this investment has some value that an investor’s market analysis hasn’t captured. The same holds true in reverse for unexpectedly weak returns. This is expressed as the investment’s “alpha.”

In this case, if an investment performs more highly than the Fama-French model suggests, that means it posted returns stronger than an investor would expect based on the investment’s composition as compared to the market’s overall risk, size and value. (Again, the same would hold true in reverse.)

We would express this as the investment’s “alpha,” generally calculated as the percent by which the investment beat expectations. Absent specific reasons to believe that an investment will outperform or underperform the market, the alpha is generally not used in predictive Fama-French Three Factor models.

## Applying the Fama-French Three Factor Model

The Fama-French model is, in essence, a form of modified market constant. When running a Fama-French analysis, you take four constants into account:

1. First, is the risk-free return (Rf). This is how much money you could make by taking effectively zero risk. Any other investments need to use this rate of return as a starting point, otherwise, you would just take your money to the risk-free assets.
2. Second, returns are based on risk (Rm). This is how much the market is beating a risk-free alternative. Third, returns are based on the size as measured by small-cap performance over large-cap performance (SML). Fourth, and finally, returns are based on value as measured by value stock performance over growth stock performance (HML).
3. You then modify these market constants by the composition of the investment which you are analyzing. This is reflected by the beta coefficient which is applied to each constant.
4. When complete, you will have a likely rate of return on your investment as judged by its composition weighted against overall market risk, size and value. This approach has historically proven significantly more accurate than the CAPM model on which it is based, generally approaching predictive reliability of 90% compared with the CAPM’s 70%.

## The Bottom Line

The Fama-French Three Factor model is a formula for calculating the likely return on a stock market investment. It measures this return based on a comparison of the investment to the overall risk in the market, the size of the companies involved and their book-to-market values (the inverse of the price-to-book ratio). Its prediction that investments with smaller and more value-oriented stocks will generally post stronger returns over time has proven highly reliable over time.

## Tips on Investing

• Fama-French Three Factor is complicated, but that doesn’t mean investing has to be intimidating! Financial advisors can help you use a number of models to calculate likely returns. Finding a financial advisor doesn’t have to be hard. With SmartAsset’s matching tool you can find a financial professional near you – in minutes – to make sure that you understand the market inside and out, no matter what kind of stocks you want to buy. If you’re ready then get started now.
• The stock market can be volatile. While it’s important to watch it for patterns, you can take hands-on measures to guard your finances. For example, an asset allocation calculator can help you create and maintain a diversified portfolio that will help buffer your portfolio as the market goes through bullish and bearish phases.