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What Is the Fama-French Three-Factor Model?

The Fama-French Three-Factor Model is an asset pricing model that expands on the traditional Capital Asset Pricing Model (CAPM) by adding size and value factors to the market risk factor. Developed by Eugene Fama and Kenneth French, this model suggests that, in addition to the overall market's risk premium, smaller companies and those with high book-to-market ratio (value stocks) tend to outperform larger companies and those with low book-to-market ratios (growth stocks). This model falls under the broader financial category of asset pricing models, providing a framework for understanding and explaining the historical expected return of diversified portfolios and individual stocks.

History and Origin

The Fama-French Three-Factor Model emerged from research challenging the explanatory power of the Capital Asset Pricing Model (CAPM). Developed by William F. Sharpe in the 1960s, among others, CAPM posited that a stock's expected return was solely dependent on its sensitivity to the overall market, measured by beta. Sharpe, along with Harry M. Markowitz and Merton H. Miller, received the Nobel Prize in Economic Sciences in 1990 for their foundational work in financial economics, including CAPM, which greatly contributed to portfolio theory.¹²

However, empirical studies in the 1980s and early 1990s began to reveal "anomalies" not fully explained by CAPM, such as the persistent outperformance of small-cap stocks and value stocks. Building on this evidence, Eugene Fama and Kenneth French introduced their influential Fama-French Three-Factor Model in a 1993 paper. They proposed that two additional factors—company size and value (as proxied by the book-to-market ratio)—could better explain the cross-section of stock returns. Their seminal work laid the groundwork for factor investing and continues to be a cornerstone of quantitative finance research. The data and research underlying their model are publicly available through resources like Kenneth French's data library.

Key Takeaways

• The Fama-French Three-Factor Model suggests that stock returns can be explained by three factors: market risk, company size, and value.
• It posits that small-cap stocks and value stocks tend to generate higher returns over the long term compared to large-cap and growth stocks.
• This model extends the traditional Capital Asset Pricing Model (CAPM) by incorporating additional risk factors beyond just market beta.
• It is widely used in academic research and by financial professionals to analyze portfolio performance and estimate the cost of equity.
• The model helps investors understand sources of return that are not captured by exposure to the overall stock market alone.

Formula and Calculation

The Fama-French Three-Factor Model is expressed as a linear regression equation:

Where:

• ( R_{i} ): The expected return of the asset or portfolio (i).
• ( R_{f} ): The risk-free rate of return.
• ( R_{M} ): The expected return of the overall market.
• ( R_{M} - R_{f} ): The excess return of the market, also known as the market risk premium. This factor represents the systematic market risk.
• ( SMB ) (Small Minus Big): The size factor, calculated as the historical excess return of small-cap stocks over large-cap stocks. This accounts for the size premium.
• ( HML ) (High Minus Low): The value factor, calculated as the historical excess return of high book-to-market ratio stocks (value stocks) over low book-to-market ratio stocks (growth stocks). This accounts for the value premium.
• ( \beta_{i,MKT} ), ( \beta_{i,SMB} ), ( \beta_{i,HML} ): The coefficients (betas) for each factor, representing the asset's sensitivity to that specific factor.
• ( \alpha_{i} ): The alpha, or the excess return of the asset not explained by the model's factors.

Interpreting the Fama-French Three-Factor Model

Interpreting the Fama-French Three-Factor Model involves analyzing the sensitivity of an asset or portfolio to each of the three factors: market risk, size, and value.

• Market Beta ((\beta_{i,MKT})): Similar to CAPM, a higher market beta indicates greater sensitivity to overall market movements. A beta of 1 suggests the asset moves in line with the market, while a beta greater than 1 implies higher volatility.
• SMB Beta ((\beta_{i,SMB})): A positive SMB beta indicates that the asset tends to perform better when small-cap stocks outperform large-cap stocks. Investors with a higher exposure to smaller companies would have a positive SMB beta.
• HML Beta ((\beta_{i,HML})): A positive HML beta suggests that the asset tends to perform better when value stocks outperform growth stocks. Portfolios heavily weighted towards companies with high book-to-market ratios would exhibit a positive HML beta.

The alpha ((\alpha_{i})) from the Fama-French Three-Factor Model represents the portion of the asset's return that cannot be explained by its exposure to these three factors. A positive and statistically significant alpha is often seen as an indication of genuine outperformance by an investment manager or unique characteristics of the asset not captured by the model.

Hypothetical Example

Consider an investor evaluating a hypothetical mutual fund, "Growth & Value Fund (GVF)", over a specific period.
Suppose the following values are observed:

• Average monthly return of GVF ((R_{i})) = 1.2%
• Average monthly risk-free rate ((R_{f})) = 0.1%
• Average monthly market return ((R_{M})) = 1.0%
• Average monthly Small Minus Big (SMB) factor return = 0.3%
• Average monthly High Minus Low (HML) factor return = 0.4%

Using regression analysis, the estimated betas for GVF are found to be:

• ( \beta_{i,MKT} ) = 0.95
• ( \beta_{i,SMB} ) = 0.70
• ( \beta_{i,HML} ) = 0.60

Now, we can calculate the expected return of GVF using the Fama-French Three-Factor Model:

Expected Excess Return = ( \beta_{i,MKT}(R_{M} - R_{f}) + \beta_{i,SMB}SMB + \beta_{i,HML}HML )
Expected Excess Return = ( 0.95(1.0% - 0.1%) + 0.70(0.3%) + 0.60(0.4%) )
Expected Excess Return = ( 0.95(0.9%) + 0.21% + 0.24% )
Expected Excess Return = ( 0.855% + 0.21% + 0.24% )
Expected Excess Return = ( 1.305% )

So, the expected return of GVF according to the Fama-French Three-Factor Model is ( 1.305% + 0.1% = 1.405% ).

Since the actual average monthly return of GVF was 1.2%, the alpha ((\alpha_{i})) would be:
( \alpha_{i} = R_{i} - (R_{f} + \text{Expected Excess Return}) )
( \alpha_{i} = 1.2% - (0.1% + 1.305%) )
( \alpha_{i} = 1.2% - 1.405% = -0.205% )

In this hypothetical example, the fund exhibited a negative alpha of 0.205%, meaning its actual returns were slightly lower than what the Fama-French Three-Factor Model would predict, given its exposure to market, size, and value factors. This analysis helps investors understand how much of a fund's return is attributable to systematic factor exposures versus active management or unique stock selection.

Practical Applications

The Fama-French Three-Factor Model has several practical applications across the financial industry:

• Performance Attribution: Investment managers use the model to determine whether their portfolio returns are due to genuine skill (alpha) or simply exposure to common market, size, and value factors. This helps in evaluating the effectiveness of a particular investment strategy.
• Portfolio Construction: Investors can intentionally tilt their portfolios towards small-cap and value stocks if they believe in the persistence of these factor premiums, aligning their asset allocation with specific risk exposures.
• Cost of Equity Estimation: Corporations and financial analysts use the Fama-French Three-Factor Model to estimate the cost of equity for capital budgeting decisions, providing a more refined discount rate than CAPM by accounting for size and value effects.
• ¹¹ Benchmarking: The model allows for the creation of more sophisticated benchmarks that account for specific factor exposures, leading to a more accurate comparison of actively managed funds. For instance, when analyzing a portfolio composed of companies included in the S&P 500, a widely tracked index of U.S. equities, one might consider how the portfolio's factor exposures compare to the broader market and its constituents., Da¹⁰t⁹a for broad market indices like the S&P 500 can be retrieved from sources such as the Federal Reserve Bank of St. Louis.

##⁸ Limitations and Criticisms

Despite its widespread adoption and improved explanatory power over CAPM, the Fama-French Three-Factor Model is not without its limitations and criticisms:

• Omitted Factors: One of the primary criticisms is that the model does not fully capture all sources of return. Notably, it fails to explain the "momentum anomaly," where stocks that have performed well recently continue to outperform, and those that have performed poorly continue to underperform. This led to the development of the Carhart Four-Factor Model, which added a momentum factor.
• ⁷ Factor Definition and Stability: The construction of the SMB and HML factors can be sensitive to the definitions of "small" and "big," and "value" and "growth." The stability of these factor premiums over different time periods and markets has also been debated.
• ⁶ Data Snooping: Some critics argue that the factors identified by Fama and French may be a result of "data snooping," meaning they were discovered through extensive data mining rather than based on underlying economic theory, and thus may not persist in the future.
• Exploratory Power Varies: While the Fama-French Three-Factor Model explains a larger portion of diversified portfolio returns than CAPM (often cited as explaining about 95% of a diversified portfolio's return compared to CAPM's 70%), its explanatory power can vary across different markets and time periods. For⁵ example, some studies suggest its performance might be weaker in certain emerging markets or during periods of financial turmoil.,

T⁴h³e ongoing debate and research in financial economics have led to the proposal of additional factors, such as profitability and investment, further expanding factor models to better explain asset returns.

Fama-French Three-Factor Model vs. Capital Asset Pricing Model (CAPM)

The Fama-French Three-Factor Model builds upon and addresses perceived shortcomings of the Capital Asset Pricing Model (CAPM). Here's a comparison:

While¹