The Capital Asset Pricing Model (CAPM) is a foundational model in finance that explains the relationship between systematic risk and expected return for assets, particularly stocks. However, it has limitations in explaining the variations in stock returns. The Fama-French three-factor and five-factor models expand upon CAPM by introducing additional factors to better capture the nuances of asset returns. Here's a breakdown of how these models expand upon CAPM.

The CAPM is based on the following formula:

(1)  

where

• is the expected return of asset .
• is the risk-free rate.
• is the sensitivity of the asset's returns to the market returns.
• is the expected return of the market portfolio.
• is the market risk premium.

The Fama-French models expand upon CAPM by introducing additional factors to explain asset returns. The Fama-French three-factor model expands on CAPM by adding two more factors:

(2)  

where

• SMB (Small Minus Big): Accounts for the size effect, capturing the excess returns of small-cap stocks over large-cap stocks.
• HML (High Minus Low): Accounts for the value effect, capturing the excess returns of value stocks (high book-to-market ratio) over growth stocks (low book-to-market ratio).

These additional factors address the observation that small-cap stocks and value stocks tend to outperform large-cap and growth stocks, respectively, which CAPM does not fully explain. The Fama-French five-factor model further extends the three-factor model by adding two more factors:

(3)  

where

• RMW (Robust Minus Weak): Accounts for profitability, capturing the excess returns of stocks with robust (high) profitability over those with weak (low) profitability.
• CMA (Conservative Minus Aggressive): Accounts for investment patterns, capturing the excess returns of firms that invest conservatively over those that invest aggressively.

These additional factors are based on empirical evidence showing that more profitable companies and those that invest conservatively tend to yield higher returns.

The expansions provided by the Fama-French models are important for better understanding and predicting stock returns. They allow for a more nuanced risk assessment and portfolio management strategy. These models are extensively used in academic research and practical finance for asset pricing, risk management, and investment decision-making.

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