Why SMB and HML Returns Are Already Excess Returns
Summary
The document addresses whether SMB and HML factor returns should have the risk-free rate subtracted when they are used as risk premia in a Fama–French expected-return model. Its explanation focuses on how the factors are constructed: HML is a long position in value stocks combined with a short position in growth stocks, and SMB similarly represents a small-stock portfolio against a large-stock portfolio.
Because each side’s excess return is measured relative to the risk-free rate, subtracting one side from the other cancels that rate. The resulting long-short factor return is therefore already an excess return, which explains why its expected return can serve as the factor risk premium without another risk-free adjustment. This answer provides the conceptual accounting behind the factor returns but does not give a procedure for estimating future premia from historical data or address sampling uncertainty.
Key ideas
- SMB and HML are constructed as returns on long-short portfolios.
- HML represents a long position in value stocks and a short position in growth stocks.
- Subtracting the excess return of one portfolio from another cancels the risk-free rate.
- The resulting SMB or HML factor return is already an excess return and can represent a factor risk premium.
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# How are the risk premiums (SMB, HML) calculated for the Fama-French factor model
# How are the risk premiums (SMB, HML) calculated for the Fama-French factor model
The question is assuming a Fama-French model, how should we calculate the expected return of an asset? To do this according to arbitrage pricing theory requires the risk premiums of the 3 factors, but how is this calculated?
According to Investments by Bodie, Kane, Marcus, the model is
$$ r- r_f = \alpha+ \beta_M (r_M-r_f) + \beta_{HML} r_{HML} + + \beta_{SMB} r_{SMB} +\epsilon $$
They claim $\alpha=0$ for no arbitrage, and the expected return is
$$ E(r)- r_f = \beta_M (E(r_M)-r_f) + \beta_{HML} E(r_{HML}) + \beta_{SMB} E(r_{SMB}). $$ I don't see how this follows. APT says it should be $$ E(r)- r_f = \beta_M RP_M + \beta_{HML} RP_{HML} + \beta_{SMB} RP_{SMB} $$ where RP stands for the risk premium, they call $E(r_M)-r_f$ the market risk premium (I agree). But they call $E(r_{HML})$ the HML risk premium, similarly for SMB. I disagree with this. I think $RP_{HML}=E(r_{HML})-r_f$. This follows from using the "factor portfolio" form of the APT (which is also in the book). While HML doesn't really have a factor portfolio, I still fail to see how you can get the equation stated in the book.
The book justifies this by saying that HML and SMB are already risk premiums, but I don't see why this is true, nor even if it is, how it is relevant given the mathematical fact that this is a factor model to which APT should be applied.
Looking beyond the formulas, how is the risk premium for HML and SMB calculated using real data? If Bodie et al are to be believe, you would just take the sample of the HML factor that is given on the Fama-French website without needing to do any regressions.
Can anyone help?
## Answer by phdstudent (score 1)
https://quant.stackexchange.com/a/75324
Your confusion comes from the fact that HML and SMB are long-short portfolios.
In particular HML is a portfolio that goes long high book to market stocks and shorts low book to market stocks.
Call the return on a portfolio of high book to market stocks (value): $r_H$. Call the return on a portfolio of low book to market stocks (growth): $r_L$.
The excess return on each portfolio is: $r_H - r_f$ and $r_L - r_f$.
Now go long value and short growth to get: $r_{HML} = r_H - r_f - (r_L - r_f)$. So as you can see the risk-free cancels out. So these are already excess returns.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.