# Fama-French Three-Factor Model

Describes stock performance through market risk and outperformance of small-cap companies and high book-to-market companies

## What is the Fama-French Three-factor Model?

The Fama-French Three-factor Model is an extension of the Capital Asset Pricing Model (CAPM). The Fama-French model aims to describe stock returns through three factors: (1) market risk, (2) the outperformance of small-cap companies relative to large-cap companies, and (3) the outperformance of high book-to-market companies versus low book-to-market companies. The rationale behind the model is that high value and small-cap companies tend to regularly outperform the markets.

The Fama-French three-factor model was developed by University of Chicago professors Eugene Fama and Kenneth French.

In the original model, the factors were specific to four countries: the U.S., Canada, Japan, and the U.K. Subsequently, Fama and French adjusted the factors, and they became available for other regions, including Europe and the Asia-Pacific.

### The Fama-French Three-Factor Model Formula

The mathematical representation of the Fama-French model is:

Where:

= Expected rate of return

r= Risk-free rate

ß = Factor’s coefficient (sensitivity)

(rm – rf= Market risk premium

SMB (Small Minus Big) = Historic excess returns of small-cap companies over large-cap companies

HML (High Minus Low) = Historic excess returns of values stocks (high book-to-price ratio) over growth stocks (low book-to-price ratio)

↋ = Risk

Market risk premium is the difference between the expected return of the market and the risk-free rate. It provides an investor with an excess return as compensation for the additional volatility of returns over and above the risk-free rate in the future.

### #2 SMB (Small Minus Big)

Small Minus Big (SMB) is a size effect based on a market capitalization of a company. SMB measures the historic excess of small-cap companies over big-cap companies. Once SMB is identified, its beta coefficient (β) can be determined via linear regression. Beta coefficient can take positive values, as well as negative ones.

The main rationale behind this factor is that in the long-term, the small-cap companies tend to see higher returns than big-cap companies.

### #3 HML (High Minus Low)

High Minus Low (HML) is a value premium. It represents the spread in returns between companies with a high book-to-market ratio (value companies) and companies with a low book-to-market ratio (growth companies). Equivalent to SMB, once the HML factor is determined, its beta coefficient can be found by linear regression. HML beta coefficient can also take positive or negative values.

The HML factor reveals that the in the long-term, value companies (high book-to-market ratio) enjoy higher returns than growth companies (low book-to-market ratio).

### Importance of the Fama-French Three-factor Model

The Fama-French three-factor model is a great expansion of the capital asset pricing model (CAPM). The model is adjusted for the outperformance tendency. Also, two extra risk factors make this model more flexible relative to CAPM.

According to the model, in the long-term, small companies overperform large companies, and value companies beat growth companies. The studies conducted by Fama and French revealed that the model could explain more than 90% of the diversified portfolios’ returns. Similar to the CAPM, the three-factor model is designed based on the assumption that riskier investments require higher returns.

Nowadays, there are further extensions to the Fama-French three-factor model, including four-factor and five-factor models.

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