Fama-French three-factor model

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Fama-French three-factor model (Fama-French 3-factor model, referred to FF3)

Fama-French Three Factor Model Overview

　　Fama and French 1992 Nian on the US stock market decided to study different equity returns differences in factors found that the stock 's market beta values can not explain different equity returns differences, while the listed company 's market capitalization , book capitalization ratio, price-earnings ratio can be interpreted stock differences in rates of return. Fama and French opinion, the excess income is CAPM in $β does not reflect the risk factor compensation. "$

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Expression Fama-French three-factor model of ^[1]

　　Fama and French 1993, indicating that you can build a three-factor model to explain stock returns. Model believe that a portfolio (including single stock) of excess return by its exposure to three factors to explain these three factors are: the market portfolio $(R m - R f), the market value factors (SMB), Book market ratio factor (HML). This multi-factor equilibrium pricing model can be expressed as:$

　　 $E(R_{it})-R_{ft}=\beta_i [E(R_{mt})-R_{ft}]+s_i^E(SMB_t)+h_i^E(HML_t)$

　　Wherein $R & lt F t represents time t risk-free rate; R & lt m t represents time t market yields; R & lt i t represents assets i yield at time t; E (R & lt m t) - R & lt F t market risk premium, S M B t is the time t market (Size) yields factor mimetic compositions, H M the I t time t book market ratio (book-to-market) yields factor mimetic compositions.$

　　 $beta] I, S I and H I are the coefficients of the three factors, regression model is expressed as follows:$

　　 $R i t - R f t = a i + β i (R m t - R f t) + s i S M B t + h i H M L t + ε i t$

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Assumptions Fama-French Three Factor Model

　　1, theoretical assumptions

　　In discussing the application Fama-French three-factor model, based on " bounded rationality " theoretical assumptions. And draw some basic assumptions on the basis of:

　　(1) there are a lot of investors;

　　(2) all investors to plan their portfolio invested in the same securities holding period;

　　(3) investors invest only within the open financial market assets traded on;

　　(4) there is no securities transaction costs ( commissions and service fees, etc.) and taxes ;

　　(5) For the average investor, securities and rates of return variance and covariance with the same expectations;

　　(6) all investors asked to comment on the economic situation and the securities are the same.

　　2, statistical hypothesis

　　As can be seen from the expression model, FF model belongs to the multiple regression model. The basic assumptions are:

　　(. 1) $(R & lt m - R & lt F), the SMB, the HML and uncorrelated random error term u;$

　　(2) assuming zero mean: $E (\ varepsilon_i) = 0$ ;

　　(3) With the assumed variance, i.e. $\varepsilon$ the variance is a constant: $Var (\ varepsilon_ u) = S ^ 2$ ;

　　(4) assumed that no self-correlation: $cov (\ varepsilon_i, \ varepsilon_j) = 0, i \ ne j$ ;

　　Linear relationship exists between (5) the explanatory variables. I.e., no exact linear relationship between two explanatory variables;

　　(6) assuming random error term $\varepsilon$ with mean zero and variance $\varepsilon_{i}\sim N(0,S^2)$