Introduction

In this chapter, we delve into the Fama-French three-factor model, a prominent extension of traditional asset pricing theories. Developed by Eugene Fama and Kenneth French, this model enhances our understanding of asset returns by considering additional factors beyond the market. We will explore the components of the model, its rationale, and its practical applications in estimating asset returns.

Fama-French Three-Factor Model Overview

The Fama-French three-factor model extends the Capital Asset Pricing Model (CAPM) by incorporating two additional risk factors: size and value. These factors are grounded in empirical observations of asset returns and provide a more comprehensive framework for explaining variations in returns.

Components of the Model

• Market Risk Premium (Market Factor): Similar to CAPM, the market risk premium is represented by the excess return of the market over the risk-free rate. It captures the systematic risk associated with overall market movements.

• Size Factor (SMB - Small Minus Big): This factor highlights the historical trend of small-cap stocks outperforming large-cap stocks over time. It quantifies the excess return of a portfolio of small-cap stocks over large-cap stocks.

• Value Factor (HML - High Minus Low): The value factor addresses the phenomenon of value stocks outperforming growth stocks. It calculates the excess return of high book-to-market (value) stocks over low book-to-market (growth) stocks.

Formula for Expected Return

The expected return using the Fama-French three-factor model can be calculated as follows:

$$E(R_i)=R_f+\beta_{Mkt}(E(R_{Mkt})-R_f)+\beta_{SMB} E(SMB)+\beta_{HML} E(HML)$$

Where:

• $E(R_i)$ is the expected return of the asset.
• $R_f$ is the risk-free rate.
• $\beta_{Mkt}, \beta_{SMB}$, and $\beta_{HML}$ are the factor sensitivities.
• $E(R_{Mkt})$ is the expected market return.
• $E(S M B)$ is the expected return on the size factor.
• $E(H M L)$ is the expected return on the value factor.

Example: Consider a portfolio with $\beta_{Mkt}=1.2$, $\beta_{SMB}=-0.3$, and $\beta_{HML}=0.5$. If the risk-free rate is 3%, the expected market return is 8%, the expected return on SMB is 5%, and the expected return on HML is 4%, the portfolio’s expected return can be calculated using the formula.

$$E(R_i)= 3.5 + 1.2 \times (8-3.5)+ 0.3 \times 5 + 0.3 \times 4 = 3.5 + 1.8 + 1.5 + 1.2 = 8\%$$