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Fama and french three factor model

What Is Fama and French Three Factor Model?

The Fama and French Three Factor Model is an asset pricing model that expands on the traditional Capital Asset Pricing Model (CAPM) by adding two additional factors to explain variations in stock returns: firm size and value. It falls under the broader financial category of portfolio theory, aiming to provide a more comprehensive explanation of expected returns than single-factor models. The Fama and French Three Factor Model suggests that, in addition to market risk, smaller companies and companies with a high book-to-market ratio (often referred to as value stocks) tend to outperform larger companies and those with a low book-to-market ratio (growth stocks). This model posits that these size and value premiums represent distinct sources of systematic risk that are rewarded in the market.

History and Origin

The Fama and French Three Factor Model was developed by Eugene Fama and Kenneth French in response to empirical observations that the Capital Asset Pricing Model (CAPM), while foundational, did not fully capture certain anomalies in stock returns. Specifically, academic research in the 1970s and 1980s noted that small-cap stocks and value stocks tended to exhibit higher returns than predicted by their market beta alone.17

In their seminal 1992 and 1993 papers, most notably "Common Risk Factors in the Returns on Stocks and Bonds," Fama and French proposed augmenting the CAPM with two additional factors: size and value.12, 13, 14, 15, 16 They hypothesized that these factors, like market exposure, captured distinct types of risk for which investors demand compensation. Their work laid the groundwork for factor-based investing and profoundly influenced modern asset pricing research. Kenneth French maintains a data library where researchers and practitioners can access historical factor returns.11

Key Takeaways

  • The Fama and French Three Factor Model extends CAPM by adding factors for company size and value to explain stock returns.
  • It suggests that smaller firms and those with high book-to-market ratio (value stocks) historically tend to yield higher returns.
  • The model includes three factors: the market risk premium (Rm-Rf), the size premium (SMB), and the value premium (HML).
  • It is widely used in academic research, portfolio performance attribution, and factor-based portfolio management.
  • While influential, the Fama and French Three Factor Model has faced criticisms and has been extended with additional factors (e.g., profitability and investment) in later iterations by the same authors.

Formula and Calculation

The Fama and French Three Factor Model describes the expected return of a portfolio or asset as follows:

E(Ri)−Rf=βi(E(Rm)−Rf)+si⋅SMB+hi⋅HML+αiE(R_i) - R_f = \beta_i(E(R_m) - R_f) + s_i \cdot SMB + h_i \cdot HML + \alpha_i

Where:

  • ( E(R_i) ) = Expected return of asset ( i )
  • ( R_f ) = Risk-free rate (e.g., return on a short-term U.S. Treasury bill)
  • ( E(R_m) ) = Expected return of the market portfolio
  • ( E(R_m) - R_f ) = Market risk premium (the excess return of the market over the risk-free rate)
  • ( \beta_i ) = Beta coefficient for asset ( i ) (sensitivity to the market risk premium)
  • ( SMB ) = "Small Minus Big," the size factor. This is the historical excess return of small-cap stocks over large-cap stocks.10
  • ( s_i ) = Size coefficient for asset ( i ) (sensitivity to the size factor)
  • ( HML ) = "High Minus Low," the value factor. This is the historical excess return of high book-to-market (value) stocks over low book-to-market (growth) stocks.9
  • ( h_i ) = Value coefficient for asset ( i ) (sensitivity to the value factor)
  • ( \alpha_i ) = Alpha, the portion of the asset's return unexplained by the model's factors.

The SMB and HML factors are constructed using diversified portfolios of stocks, designed to capture the return differences between small and big companies, and value and growth companies, respectively.8

Interpreting the Fama and French Three Factor Model

Interpreting the Fama and French Three Factor Model involves understanding how each factor contributes to an asset's or portfolio's return. The beta coefficient (( \beta_i )) indicates the asset's sensitivity to overall market movements. A beta greater than 1 suggests higher volatility than the market, while less than 1 indicates lower volatility.

The size coefficient (( s_i )) reveals a portfolio's exposure to the small-cap premium. A positive ( s_i ) suggests the asset tends to perform better when small-cap stocks outperform large-cap stocks, and vice-versa. Similarly, the value coefficient (( h_i )) shows the exposure to the value premium. A positive ( h_i ) indicates the asset performs better when value stocks outperform growth stocks.

The ( \alpha_i ) (alpha) represents the portion of the return that cannot be explained by the three systematic factors. A positive alpha is often interpreted as a manager's skill in generating returns beyond what can be attributed to their exposure to market, size, and value factors, though it can also simply indicate exposure to unmeasured factors or unsystematic risk. The model's utility lies in providing a more nuanced framework for analyzing returns compared to single-factor models, aiding investors in understanding the sources of their portfolio's performance.

Hypothetical Example

Consider an investment fund, "Global Growth Fund," and an analyst is trying to understand its performance using the Fama and French Three Factor Model.
Let's assume the following historical factor returns for a given period:

  • Market Risk Premium (Rm-Rf) = 8%
  • Size Premium (SMB) = 2%
  • Value Premium (HML) = 3%
  • Risk-free rate (Rf) = 1%

Through regression analysis of the fund's historical returns against these factors, the analyst determines the fund's sensitivities:

  • Fund Beta (( \beta_i )) = 1.1
  • Fund Size Coefficient (( s_i )) = -0.5 (indicating a tilt towards larger companies, or negative exposure to the small-cap factor)
  • Fund Value Coefficient (( h_i )) = -0.8 (indicating a tilt towards growth companies, or negative exposure to the value factor)
  • Fund Alpha (( \alpha_i )) = 0.5%

Using the Fama and French Three Factor Model formula:

E(Ri)−Rf=βi(E(Rm)−Rf)+si⋅SMB+hi⋅HML+αiE(R_i) - R_f = \beta_i(E(R_m) - R_f) + s_i \cdot SMB + h_i \cdot HML + \alpha_i

Expected excess return for the fund = ( (1.1 \times 8%) + (-0.5 \times 2%) + (-0.8 \times 3%) + 0.5% )
Expected excess return = ( 8.8% - 1% - 2.4% + 0.5% )
Expected excess return = ( 5.9% )

Therefore, the expected total return for the fund would be ( 5.9% + 1% = 6.9% ). This example shows that while the fund has higher market exposure (beta 1.1), its negative exposures to the size and value factors (consistent with a "growth" fund that invests in larger, less value-oriented companies) reduce its expected return from these dimensions. The positive alpha suggests the fund generated 0.5% more return than expected based purely on its exposure to these three common factors.

Practical Applications

The Fama and French Three Factor Model has numerous practical applications in the financial industry:

  • Performance Attribution: The model is widely used to analyze the performance of investment portfolios and mutual funds. By breaking down returns into contributions from market, size, and value factors, investors can understand if a fund's returns are due to genuine skill (captured by alpha) or simply its inherent exposure to these well-documented risk premiums. This helps differentiate between luck and a manager's consistent ability.7
  • Factor Investing: The model's identification of size and value as persistent return drivers underpins factor-based investment strategies. Investors can construct portfolios that explicitly tilt towards small-cap or value stocks, aiming to capture the premiums associated with these factors, or combine them with other factors identified in later models.6
  • Cost of Equity Estimation: Companies and analysts can use the model to estimate the cost of equity for a firm, which is crucial for valuation and capital budgeting decisions. By determining the company's sensitivities to the market, size, and value factors, a more nuanced expected return can be derived, potentially leading to more accurate valuations.
  • Risk Management and Diversification: Understanding a portfolio's exposure to these factors allows for better risk assessment and management. For example, if a portfolio has a high concentration in small-cap value stocks, the model helps explain the systematic risks it undertakes beyond general market exposure. Academic research and data for these factors are freely available through Kenneth French's data library, allowing for broad application and testing.5

Limitations and Criticisms

Despite its widespread influence, the Fama and French Three Factor Model is not without limitations and criticisms. One significant critique is that it does not fully explain all observed anomalies in stock returns. For instance, the phenomenon of momentum, where stocks that have performed well recently tend to continue to perform well, is not captured by the three factors. This led to the development of the Carhart Four-Factor Model, which adds a momentum factor.4

Furthermore, critics argue that the model's factors (size and value) are empirically derived and not directly linked to a clear theoretical foundation of risk. While Fama and French posited that these factors proxy for underlying economic fundamentals and risks, the precise nature of these risks is debated.3 Some researchers question whether the premiums observed for size and value are truly compensation for systematic risk or simply statistical artifacts of data mining over specific periods.

Additionally, the performance of the size and value factors can vary significantly over different economic cycles and timeframes. For example, the size premium has shown periods of underperformance, and the value premium has also faced challenges in certain market environments.2 This suggests that relying solely on the Fama and French Three Factor Model may not always provide a complete picture of expected returns or portfolio performance. Subsequent research by Fama and French themselves led to the expansion of the model to a five-factor model in 2015, which added profitability and investment factors, acknowledging that more elements might influence stock returns.1

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

The Fama and French Three Factor Model represents a significant advancement over the Capital Asset Pricing Model (CAPM), though both models belong to the family of asset pricing models. The CAPM, introduced by William Sharpe, John Lintner, and Jack Treynor, posits that the expected return of an asset is solely determined by its sensitivity to the overall market's movements, measured by beta. In essence, the CAPM suggests that only systematic market risk is rewarded.

The Fama and French Three Factor Model builds upon the CAPM by recognizing that market beta alone does not fully explain observed variations in stock returns. It introduces two additional factors—size (SMB) and value (HML)—arguing that smaller companies and companies with high book-to-market ratios carry additional systematic risks for which investors are compensated. While CAPM assumes all non-market risk can be eliminated through diversification, the Fama and French Three Factor Model suggests that size and value effects are persistent and distinct from general market exposure. The fundamental difference lies in their scope: CAPM is a single-factor model, whereas the Fama and French Three Factor Model is a multi-factor model that attempts to provide a more nuanced explanation for asset returns by incorporating additional risk premiums.

FAQs

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

The three factors are:

  1. Market Risk Premium (Rm-Rf): The excess return of the broad market over a risk-free rate.
  2. Size (SMB - Small Minus Big): The historical excess return of portfolios of small-capitalization stocks over large-capitalization stocks.
  3. Value (HML - High Minus Low): The historical excess return of portfolios of high book-to-market ratio (value) stocks over low book-to-market ratio (growth) stocks.

Why was the Fama and French Three Factor Model developed?

It was developed to address the limitations of the Capital Asset Pricing Model (CAPM), which struggled to explain observed anomalies where small-cap stocks and value stocks consistently outperformed what their market beta alone would predict. Eugene Fama and Kenneth French proposed that these "size" and "value" characteristics represent additional sources of systematic risk that affect expected returns.

Is the Fama and French Three Factor Model still used today?

Yes, the Fama and French Three Factor Model remains widely used in academic research, portfolio performance attribution, and quantitative investing. While it has been extended by Fama and French themselves (e.g., to a five-factor model) and other researchers (e.g., adding momentum), the original three-factor model is still a foundational tool for understanding and explaining equity returns. It helps investors and analysts identify the sources of returns beyond simple market exposure.

How do I get the data for the Fama and French factors?

Historical data for the Fama and French factors (including Rm-Rf, SMB, and HML) is publicly available. The most common and reliable source is Kenneth French's Data Library, hosted by Dartmouth College. This website provides various datasets, including daily, weekly, monthly, and annual returns for the factors, which can be downloaded for use in financial analysis.

What is alpha in the context of the Fama and French Three Factor Model?

In the Fama and French Three Factor Model, alpha (( \alpha_i )) represents the portion of an asset's or portfolio's return that cannot be explained by its exposure to the three specified factors (market, size, and value). A positive alpha suggests the asset outperformed its expected return given its factor exposures, while a negative alpha indicates underperformance. It is often interpreted as a measure of a fund manager's skill, though it can also arise from exposure to other unmeasured factors or specific, non-systematic events.