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Factor Sensitivity and Risk Premia in Fama–MacBeth Regressions

Article Quant Q&A · Author: Konstantinos
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Summary

The document asks how factor exposures differ from the factor returns themselves in the Fama–French three-factor model. It contrasts a Fama–MacBeth second-stage regression using estimated asset betas with a proposed regression using the factor series directly, and asks how the resulting lambda coefficients should be interpreted.

It presents the modeling question but supplies no answer, derivation, empirical results, or guidance on estimation. The useful subject is the distinction between an asset’s sensitivity to a priced factor and the factor’s realized return, and whether a cross-sectional pricing regression can use one in place of the other. Readers should treat the equations as the setup for a conceptual question, not as evidence that either specification is valid or that the two sets of coefficients have equivalent meanings.

Key ideas

  • Factor sensitivities describe how asset returns respond to factor returns.
  • The question contrasts cross-sectional pricing on estimated betas with a regression on factor returns themselves.
  • The document does not resolve whether the proposed alternative regression is valid.
  • Risk-premium interpretation depends on the variables and regression stage used.

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Factor investingStatistics
Full text
# What is the meaning of factor sensitivity and its risk premium?


# What is the meaning of factor sensitivity and its risk premium?












For the Fama and French three-factor model, $$R_{t}-R_{t,F}=\alpha+\beta_{MKT}\left(r_{t,MKT}-r_{t,f}\right)+\beta_{SMB}R_{t,SMB}+\beta_{HML}R_{t,HML}.$$ I run Fama-MacBeth cross sectional regressions, defining Model A, $$R_{t}-R_{t,F}= \mathrm{intercept} +\lambda_{MKT} \beta_{MKT} + \lambda_{SMB} \beta_{SMB} +\lambda_{HML} \beta_{HML}.$$ Intuitively, what is the difference between the sensitivity of the risk factors and the risk factors themselves? Why can't I run a Fama-MacBeth regressions defining Model B, $$R_{t}-R_{t,F}= \mathrm{intercept} +\lambda_{MKT} {MKT} + \lambda_{SMB} {SMB} +\lambda_{HML} {HML}.$$ If I could run them, though, how would the interpretation of the risk premia ($\lambda$) differ between Models A and B?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.