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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?

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  • $\begingroup$ @RichardHardy Corrected. $\endgroup$
    – user67149
    Apr 25, 2023 at 3:06

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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.

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  • $\begingroup$ Right! I agree. This raises 2 other questions. 1. It would seem based on this that it is possible to estimate the risk premiums as the mean of the SMB, HML factors. This post seems to disagree (quant.stackexchange.com/questions/37987/…), and isn't it the standard method to do a regression to get the risk premiums, which this seems to get around? 2. Mathematically, if you treat it as an arbitrary factor without knowing where it came from, you'd get $E[r_{HML}]-r_f$, rather $E[r_{HML}]$. $\endgroup$
    – user67149
    Apr 25, 2023 at 3:28
  • $\begingroup$ That's called the fama-macbeth procedure. See this post for more details: quant.stackexchange.com/questions/24343/… $\endgroup$
    – phdstudent
    Apr 25, 2023 at 14:05

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