Fixed-Income Return Attribution: The Carhart Four-Factor Model

Time to get specific about multifactor models. After years of practice, you might expect that investment professionals have settled on the best factors to use for explaining asset returns, and they have, to a degree.

Fama and French presented three solid factors, and Carhart added a fourth in 1997. $$\displaystyle R_p - R_F = a_p + b_{p1}RMRF + b_{p2}SMB + b_{p3}HML + b_{p4}WML + \epsilon_p $$ Before even focusing on what the variables are, what difference do you see here between this and a standard arbitrage pricing theory (APT) model?

Exactly. That's the dependent variable here, and there's still an intercept term on the right. So it's not just an algebra trick. This is a reasonable extension to APT, given that it's excess return that compensates risk, not total return.

No. Both models are multifactor models.

No. There's an intercept in both models.

$$ R_p - R_F = a_p + b_{p1}RMRF + b_{p2}SMB + b_{p3}HML + b_{p4}WML + \epsilon_p $$ The Carhart four-factor model explains excess returns as a function of an intercept term, a sensitivity to a market index, or "return of market minus risk-free" (RMRF); a market capitalization factor, "small minus big" (SMB); a book-value-to-price factor, "high minus low" (HML); and a momentum factor, "winners minus losers" (WML). For example, the "small minus big" estimate is the average return on three small-cap portfolios minus the average return on three large-cap portfolios. How will this difference be used here?

You got it!

No. This is the factor.

Then the sensitivity of assets to this factor is estimated. In the CAPM, what would this sensitivity be?

Exactly.

No. It's 0.

These other factors aren't in the CAPM, just the RMRF. Pull out that extra alpha and the other three factors (or set them all to zero), and you're back to the CAPM. So what are factors in the Carhart four-factor model are just anomalies in the CAPM. Finally, in expectation and equilibrium, there shouldn't be any extra alpha at all, so you can expect the following. $$\displaystyle E[R_p] = R_F + b_{p1}RMRF + b_{p2}SMB + b_{p3}HML + b_{p4}WML $$

To summarize: [[summary]]

An intercept is allowed

Multiple factors are being used

The risk premium is being modeled, not return

As a factor

As a sensitivity

0

1

Continue

Continue

Continue