Fama french_three factor_model

What Is the Fama-French Three-Factor Model?

The Fama-French Three-Factor Model is an influential asset pricing model within the broader field of portfolio theory that expands upon earlier frameworks to better explain the expected return of a security or portfolio. Developed by Eugene Fama and Kenneth French, this model posits that in addition to the overall market risk, two other factors—company size and value—influence stock returns. The Fama-French Three-Factor Model seeks to account for the observed empirical tendencies of small-cap companies and value stocks to outperform large-cap companies and growth stocks over time.

History and Origin

The Fama-French Three-Factor Model was developed by Eugene Fama and Kenneth French, professors at the University of Chicago Booth School of Business, and formally introduced in their seminal 1992 paper, "The Cross-Section of Expected Stock Returns." The³⁴ir work built upon the Capital Asset Pricing Model (CAPM), which had long been a cornerstone of asset pricing theory but primarily accounted for only one risk factor: a stock's sensitivity to overall market movements, known as beta.

Fa³³ma and French observed that CAPM did not fully explain the variations in stock returns, particularly the persistent outperformance of small-cap stocks and value stocks. To address these anomalies, they introduced two additional factors. Eugene Fama was recognized for his empirical analysis of asset prices, sharing the Nobel Memorial Prize in Economic Sciences in 2013. Investors and researchers can access historical data for the Fama-French factors through Kenneth French's Data Library.

##³² Key Takeaways

• The Fama-French Three-Factor Model expands on the CAPM by including size and value factors to explain stock returns.
• The three factors are the market risk premium, Small Minus Big (SMB), and High Minus Low (HML).
• ³¹ It suggests that small-cap stocks and value stocks historically tend to outperform the broader market.
• ³⁰ The model can explain a significant portion of the variation in diversified portfolio returns, often cited as over 90%.
• ²⁹ It is widely applied in portfolio management and performance evaluation.

##²⁸ Formula and Calculation

The Fama-French Three-Factor Model is expressed by the following equation:

Where:

• ( E(R_i) ) = Expected return of the security or portfolio ( i )
• ( R_f ) = Risk-free rate (e.g., the yield on short-term government bonds)
• ²⁷ ( R_m ) = Expected return of the overall market portfolio
• ( R_m - R_f ) = Market risk premium (the excess return of the market over the risk-free rate)
• ²⁶ ( SMB ) = "Small Minus Big," a factor that captures the excess return of small-market capitalization stocks over large-cap stocks.
• ( HML ) = "High Minus Low," a factor that captures the excess return of high book-to-market ratio (value) stocks over low book-to-market ratio (growth) stocks.
• ( \beta_1, \beta_2, \beta_3 ) = The respective beta coefficients or factor sensitivities to the market, size, and value factors. The²⁵se are determined through linear regression analysis of historical returns.
• ( \alpha ) = Alpha, the abnormal return not explained by the model's factors (the error term).

Interpreting the Fama-French Three-Factor Model

Interpreting the Fama-French Three-Factor Model involves understanding the significance of each factor's coefficient. A positive beta for the market risk premium ((\beta_1)) indicates that the asset's return is positively correlated with the overall market, similar to traditional CAPM.

Th²⁴e SMB coefficient ((\beta_2)) shows the asset's sensitivity to the size factor. A positive (\beta_2) suggests that the asset's returns tend to be higher when small-cap stocks outperform large-cap stocks. Conversely, a negative (\beta_2) would indicate a tilt towards larger companies. Thi²³s factor acknowledges the "size effect," where smaller companies may exhibit higher potential returns due to higher growth prospects or greater risk.

Th²²e HML coefficient ((\beta_3)) reflects the asset's sensitivity to the value factor. A positive (\beta_3) implies that the asset performs well when value stocks (companies with high book-to-market ratios) outperform growth stocks (companies with low book-to-market ratios). Thi²¹s is known as the "value premium" or "value effect," suggesting that undervalued companies might yield higher returns over time. By ²⁰analyzing these coefficients, investors can gain deeper insights into the underlying drivers of a portfolio's returns beyond just its exposure to the broad market.

##¹⁹ Hypothetical Example

Consider an investment portfolio that aims to generate returns from a diversified selection of U.S. equities. To analyze its performance using the Fama-French Three-Factor Model, historical monthly returns for the portfolio, the risk-free rate, the market risk premium, SMB, and HML factors are gathered.

Let's assume the following hypothetical monthly factor exposures derived from a linear regression of the portfolio's excess returns:

• Beta for Market Risk Premium ((\beta_1)) = 0.95
• Beta for SMB ((\beta_2)) = 0.30
• Beta for HML ((\beta_3)) = 0.15

And assume the following average monthly factor returns over a specific period:

• Average Market Risk Premium ((R_m - R_f)) = 0.75%
• Average SMB Factor Return = 0.20%
• Average HML Factor Return = 0.10%
• Average Risk-Free Rate ((R_f)) = 0.05%

Using the Fama-French Three-Factor Model formula:

Substituting the values:

This calculation suggests that, based on its exposure to the three factors, the portfolio's expected monthly return is approximately 0.8375%. This hypothetical expected return can then be compared to the portfolio's actual historical returns to assess if it generated any alpha (returns not explained by the model's factors).

Practical Applications

The Fama-French Three-Factor Model has various practical applications in finance, especially in the realms of portfolio management and investment analysis. Investors and financial professionals utilize the model to evaluate the performance of investment portfolios and individual securities. By breaking down returns into contributions from market, size, and value factors, the model provides a more nuanced understanding of where returns are originating.

Fo¹⁸r example, a portfolio manager might use the Fama-French Three-Factor Model to determine if a fund's outperformance is due to genuine skill (generating alpha) or simply its inherent exposure to small-cap and value stocks. Thi¹⁷s distinction is critical for performance attribution. The model also aids in constructing factor-tilted portfolios, allowing investors to intentionally allocate capital towards Small Minus Big (SMB) and High Minus Low (HML) factors if they believe these premiums will persist.

Ac¹⁶ademic research consistently examines the application and efficacy of the Fama-French model in various markets. Studies have shown that domestic factor models, including the Fama-French Three-Factor Model, can effectively explain time-series variation in stock returns and provide more accurate pricing compared to simpler models in different countries and regions. Thi¹⁵s makes it a valuable tool for global asset pricing and allocation strategies.

Limitations and Criticisms

Despite its widespread adoption and improved explanatory power over the Capital Asset Pricing Model (CAPM), the Fama-French Three-Factor Model is not without its limitations and criticisms. One primary critique is its empirical nature; while the model effectively describes historical patterns in returns, some argue it lacks a strong theoretical foundation for why these size and value premiums persist.

An¹⁴other limitation is that the model relies on historical data, which may not always be a reliable indicator of future returns, especially during extreme market conditions. The¹³ relationships between the factors themselves can also be time-varying and potentially non-linear, which linear regression analysis might not fully capture.

Fu¹¹, ¹²rthermore, the model assumes that all investors operate with the same information, which is often not the case due to information asymmetry and behavioral biases in real-world markets. Ove¹⁰r time, the empirical performance of the SMB and HML factors has also been debated. For instance, some analysis suggests the size factor performed strongly up to the early 1980s but has largely stagnated since, and the value factor (HML) also saw a significant decline after the 2008 financial crisis. The⁹se observations have led to extensions of the model, such as the Fama-French Five-Factor Model, which adds profitability and investment factors.

##⁸ Fama-French Three-Factor Model vs. Capital Asset Pricing Model (CAPM)

The Fama-French Three-Factor Model and the Capital Asset Pricing Model (CAPM) are both fundamental asset pricing models, but they differ in their scope and explanatory power. CAPM, introduced in the 1960s, is a single-factor model that asserts an asset's expected return is solely determined by its sensitivity to overall market risk (its beta coefficient) and the risk-free rate. It ⁷suggests that the higher the beta, the higher the expected return, as investors require greater compensation for bearing more systematic risk.

The Fama-French Three-Factor Model expands on CAPM by adding two additional factors: Small Minus Big (SMB) and High Minus Low (HML). These factors account for the observed empirical phenomena that small-cap stocks tend to outperform large-cap stocks, and value stocks tend to outperform growth stocks. Res⁶earch has consistently shown that the Fama-French Three-Factor Model provides a more accurate fit to actual stock return data, explaining a greater proportion of the cross-sectional variation in average stock returns compared to CAPM. Whi⁵le CAPM might explain about 70% of return variations in a diversified portfolio, the Fama-French Three-Factor Model can explain over 90%. Thi⁴s enhanced explanatory power makes the Fama-French model a more robust tool for analyzing and attributing investment performance.

FAQs

What are the three factors in the Fama-French model?

The three factors are: the market risk premium (the excess return of the market over the risk-free rate), Small Minus Big (SMB), and High Minus Low (HML). SMB accounts for the tendency of small-cap stocks to outperform, while HML captures the tendency of value stocks to outperform.

Why was the Fama-French Three-Factor Model developed?

It was developed to address the limitations of the Capital Asset Pricing Model (CAPM), which only considered market risk. Eugene Fama and Kenneth French observed that small-cap stocks and value stocks consistently showed higher returns than what CAPM predicted, leading them to incorporate these additional factors to better explain return variations.

How is the Fama-French model used in investment decisions?

Investors and portfolio managers use the Fama-French Three-Factor Model to analyze and attribute portfolio returns. It helps to determine if a portfolio's performance is due to its exposure to systemic market, size, and value factors, or if there's an unexplained portion (alpha) suggesting manager skill. It can also guide portfolio construction by strategically allocating to desired factor exposures, such as tilting towards smaller companies or those categorized as value.

##³# Is the Fama-French Three-Factor Model still relevant today?
Yes, the Fama-French Three-Factor Model remains highly relevant in financial academia and practice. While it has been extended to a five-factor model and further, the core insights about size and value premiums continue to be studied and debated. It serves as a foundational tool for understanding multifactor models and continues to be used in portfolio management and performance evaluation.

##²# What is the efficient market hypothesis and how does it relate to Fama-French?
The efficient market hypothesis posits that asset prices fully reflect all available information, making it impossible to consistently earn abnormal returns. Eugene Fama, a key proponent of this hypothesis, developed the Fama-French Three-Factor Model to explain observed anomalies (like the size and value premiums) within the framework of market efficiency, suggesting these premiums are compensation for taking on specific types of risk, rather than market inefficiencies.¹