# 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?

• Lambdas and betas are to be determined by a regression over a universe of stocks and days, in A? So R is a matrix of stock-returns and there is a hidden error term implied. But what more precisely are MKT, SMB and HLM in B? You mean that you have already somehow extracted the factor returns? Then maybe you have already also extracted the lamdas. Model A looks to me as cooking it all from the raw data while Model B is like throwing something ready-made in the micro? Please explain! Commented Aug 16, 2016 at 19:09
• Intuitively a $\beta$ tells you how a security responds each day to movements in a factor, a $\lambda$ tells you over the long run how much you earn by having that Beta. Commented Aug 16, 2016 at 19:18