Why Factor Pricing Models Imply Zero Intercepts
Summary
The document explains the zero-intercept implication in a factor asset-pricing model. It presents the Fama-French regression, where an asset’s excess return is decomposed into exposures to market, size, and value factors, an intercept, and a residual. The intercept represents average excess return not accounted for by the modeled factor exposures.
Under the assumption that the factors capture all variation in expected returns, a nonzero intercept would indicate return unexplained by those factors. The response uses an arbitrage-pricing intuition: portfolios could be formed with zero exposure to the modeled factors but nonzero expected return, contradicting the claim that the factors fully explain expected returns. This is a theoretical implication conditional on the model and its assumptions, not evidence that estimated intercepts in real data must equal zero. The discussion also does not address statistical uncertainty, model misspecification, or practical limits on constructing such portfolios.
Key ideas
- The regression intercept represents average excess return left unexplained by the included factors.
- If the factors fully capture expected-return differences, the model implies zero intercepts.
- A nonzero intercept would suggest return beyond the modeled factor exposures.
- The zero-intercept implication relies on the factor model assumptions and does not guarantee empirical estimates will be zero.
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Full text
# FF 5 factor model Intercept equal 0
# FF 5 factor model Intercept equal 0
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!
## Answer by Tim Wilding (score 1)
https://quant.stackexchange.com/a/40236
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.