Choosing Between Fama and French and CAPM Models

How to Compare the Results of CAPM and Fama and

 French Model

We use the following equation for calculation of beta under CAPM:

(Ri-Rf) = α + β(Rm-Rf)

(Ri-Rf) is the actual risk premium on a given stock while B(Rm-Rf) is the expected risk premium as suggested by CAPM. If CAPM correctly predicts the risk premium on a given security, the expected risk premium and actual risk premium will be the same. If so, the value of intercept  α will be zero. For example, if actual risk premium is 15% and expected risk premium is also 15%, then replacing these values in the above equation, will shall get the following result:

(Ri-Rf) = α + β(Rm-Rf)

15% = α + 15%

15% - 15% = α

α = 0 i.e. intercept = 0

In any other case where expected risk premium and actual risk premium differ, the value of intercept will be either positive or negative. In such cases, CAPM will not be truly predicting the risk premium on the stock. Consider another example where actual risk premium is 15% and expected risk premium is 20%, the value of intercept will be:

15% = α + 20%

15% - 20% = α

-5% = i.e intercept = -5%

The above example suggests when CAPM gives expected risk premium above the actual risk premium, the value of intercept will be negative, and vice versa. In both cases where the value of intercept is either positive or negative, CAPM fails to correctly predict the risk premium. In nutshell, we can judge the validity of CAPM by looking at the value of intercept. If the value of intercept is different from zero and is statistically significant, CAPM fails to predict the true risk premium.

How to Check the Validity of Fama and French

Similarly, we can apply the same criterion for checking the validity of Fama and French model. Fama and French model uses the following equation for calculating betas of the three independent variables.

The (Ri-Rf) term denotes the actual risk premium on a given security while the three variables  Collectively represent the expected risk premium. Value of intercept will be zero if actual risk premium and expected risk premium are the same. For example, actual risk premium  (Ri-Rf) = 15%,  and expected risk premium is B(Rm-Rf) = 10%, B(SMB) = 2%, and B(HML)= 3%, then:

15%  = α + 10% + 2% + 3%

15% = α + 15%

15%  - 15% = α

0 = α

When value of the intercept is not zero, it suggests that the Fama and French model does not correctly predict the risk premium on the given security.

The validity of both CAPM and Fama and French models can be judged from the value of intercept. An asset valuation model is considered valid if:

1. Value of intercept is zero or closer to zero OR

2. The intercept is statistically insignificant. The intercept is considered statistically insignificant when its P-value is above 0.1.

Consider the following regression output of Fama and French Model:

The value of intercept is -.007 which shows that Fama and French model predicts 0.7% risk premium above the actual risk premium. However, this value is insignificant as the P-value is 0.78 which is above 0.1. The value of intercept, being insignificant, suggests that Fama and French model is valid. But the two additional variables of Fama and French are insignificant as well. HML and SMB are both insignificant as their P-values are above 0.1. The result shows that SMB and HML have no relationship with the dependent variable (Ri-Rf). This is against the basic foundation of FF model.

How to Check the Validity of CAPM

Our next alternative is to use CAPM and see whether there is any improvement. Applying CAPM to the same data, we get the following regression output:

The intercept has a value of -.007 which suggests that CAPM also gives expected risk premium above the actual risk premium by a value of .7%. But this value is insignificant as the P-value is 0.76 which is above 0.1. The beta is highly significant as the P-value is 0.000001.

In both of the regression outputs of Fama and French and CAPM, the intercept values were insignificant, suggesting that both models correctly predict the risk premium on the given security. However, Fama and French model fails to confirm to its own predictions as both of its main variable showed no relationship with the actual risk premium (Ri-Rf). On the other hand, CAPM seems to be right in its prediction that a security risk premium is dependent upon the risk premium of market portfolio and a security’s beta. This is why CAPM is the preferred model in our case.