Carhart four-factor model

From Carhart four-factor model:

Rp − RF = ap + bp1RMRF + bp2SMB + bp3HML + bp4WML + εp

How to interpret the intercept and error term in this model?

And how to transform the above equation to the following equation:

E(Rp) = RF + βp,1RMRF + βp,2SMB + βp,3HML + βp,4WML

Form the second equation, does it suppose to get required return, not expected return?

Thank you!

Rp − RF = ap + bp1RMRF + bp2SMB + bp3HML + bp4WML + εp

  • Rp - RF represents the return premium due to various unsystematic (the 4 factors) + systematic (error term)

  • ap, the intercept, represents the expected return given that there are no surprises in the 4 factors. ap also captures all other factors that are not stated in the model.

And how to transform the above equation to the following equation:

E(Rp) = RF + βp,1RMRF + βp,2SMB + βp,3HML + βp,4WML

Rp - RF equation above represents a variant of the market model equation. The market model is based on historical data and helps you derive a linear equation to plot the line of best fit. The resulting value gives you Historical Req’d Return or Return Premium.

The first equatino is used to derive the Betas so you can use to plug it into the 2nd equation in order to calculate the expected return.

Someone can chime in if i’m wrong.

Thank you.

Carhart model is Fama-French w/price momentum as the 4th factor…

The first equation is excess return version of Carhart model where “ap” is the return in excess of that expected at given level of systematic risk.

If there is equilibrium in the market “ap” becomes zero as there is no excess return. Now replace “ap” with zero in equation 1 and transfer Rf from left of equation 1 to the right. You got equation 2. Now you have expected return at equilibrium level.


I am going through the example Exhibit 3. Sample Carhart Factor Model Attribution but having a hard time understanding how the security selection got calculated.

I got how the FACTOR TILTS RETURN is calculated as being the sumproduct of the factor return and the factor difference between portfolio and benchmark = SUMPROD( (3) , (4) ). However, it says somewhere that “Positive active exposure to the HML factor—the bet on value stocks—contributed 204 bps to the realized active return, about 98% of the 207 bps of total realized active return” and then “the overall active return from security selection is the portion of return not explained by factor sensitivities. In this period, the contribution from selection was slightly negative (–0.05%).” It feels to me that -0.05 is calculated from 207 and at the same time 207 (active return) is calculated as the sum of A. factor tilts return and B. security selection (-0.05)…

can anyone help clarify?