In the paper A five-factor asset pricing model from Fama and French (JFE 2015) they say at page 3:

"Treating the parameters in (4) as true values rather than estimates, if the factor exposures $b_i , s_i$ , and $h_i$ capture all variation in expected returns, the intercept $a_i$ is zero for all securities and portfolios $i$."

Why is that? Can someone shed a light at this? Thanks!

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    $\begingroup$ @Nitin I don't think that is true. That is an implication of the fundamental asset pricing equation but not an implication from a linear regression. I think if 100% is captured by covariates then the R2 would be high but the intercept is not necessarily zero. $\endgroup$ – phdstudent Jun 8 '18 at 16:16

Equation (4) from the Fama-French (2015) text is:

$R_{it} – R_{Ft} = a_i + b_i(R_{Mt} – R_{Ft})+s_iSMB_t+h_iHML_t +e_{it}$

If $a_i$ were not zero, then an investor would be able to build portfolios with different levels of non-zero expected returns and yet have 0 exposure to the 3 factors. Hence, the variation in expected returns is not captured entirely by the 3 factors.

Equation (4) splits expected excess returns into the 3 factors from their 1993 paper - market, book value, and size factor. $a_i$ is the average return of the portfolio in excess of the return expected from those three factors for a specific security. The Fama-French 3-factor model is loosely rooted in the Arbitrage Pricing Theory (APT) of Ross (1976). The APT says that an asset’s expected returns are a linear function of the asset’s exposure to a range of factors. If the APT model works, then there will be no security-specific average expected returns, and $a_i$ should be zero for all securities. If it doesn’t work, an investor can build portfolios with positive expected return and zero systematic risk.


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