Fama and French 1997 Cost of Equity

Dear Quantitative Finance Members,

I was wondering if you can clarify me the following issue. I am trying to estimate the cost of equity following "Industry costs of equity" (Fama and French, 1997). I am not sure if I correctly understood the steps that I need to follow. Here they are:

1. Obtain firm returns (from CRSP database) and SMB, HML, Rm-rf from Fama and French website.
2. Run $$R_i -r_f = \beta_0 + \beta_1(R_m-r_f) + \beta_2 SMB + \beta_3 HML$$ over each out of 48 industries, whole sample.
3. Save estimated coefficients of $\beta_0, \beta_1, \beta_2, \beta_3$
4. Estimate fitted Cost of Equity ($CE$) as $$CE = \beta_0 + \beta_1 \times (R_m-r_f) + \beta_2 \times (SMB) + \beta_3 \times (HML)$$

Please, correct me if I am wrong

First of all, the cost of equity is the expected returns on equity of a stock. This means that you could take any estimate of $\mathbb{E}(R_i)$ as cost of equity.

• a first version would be the empirical average ${1\over D} \sum_{d=1}^D R_i(d)$ where $d$ are available days in your database.
• to obtain a more robust version of the cost of equity you can rely on proxies of stable sources of returns.
• if you believe in reliability of FF factors, you can do it this way:

1. estimate the beta to the factors (like you propose in your question; you forgot the epsilon --i.e. residuals--): $$R_i -r_f = \beta_0 + \beta_1(R_m-r_f) + \beta_2 SMB + \beta_3 HML+\epsilon$$

2. plug the empirical averages in place of the daily (or weekly) version of the data you used to estimate your beta (no more epsilon if you used any non-biased version of regression to obtain the betas): $$CE := \mathbb{E}(R_i) = r_f + \beta_0 + \beta_1(\mathbb{E}(R_m)-r_f) + \beta_2 \mathbb{E}(SMB) + \beta_3 \mathbb{E}(HML)$$

• [EDIT] If you want to have one cost of equity by industry / sector, you can simply perform a regression within each sector, thus you will obtain $\mathbb{E}(\beta_i\vert Sector)$ in place of $\beta_i$ (for $i\in\{0,\ldots, 3\}$). As a consequence, you will replace $\mathbb{E}(R_i)$ by $\mathbb{E}(R_i\vert Sector)$. And it is what you want.

• Thank you for the answer. One clarification, should I run main FF eq. within industry-year? Or just within industry? – Alberto Alvarez Jun 24 '18 at 12:27
• just within industry, otherwise your estimate of beta will be very noisy (250 points in a year). Betas should be stable over years. – lehalle Jun 24 '18 at 12:35
• Thanks a lot. The very last question, please. SMB and HML (in the FF website) are constructed taking the whole sample of firm. Do I have to take these values or shall I reconstuct SMB and HML within industry? – Alberto Alvarez Jun 24 '18 at 12:46
• if you do one regression per industry, no real need to rebuild the portfolio, you can take theirs (I will update my answer). By the way: if you like my answer, please vote +1 on it ;{)} – lehalle Jun 24 '18 at 12:53
• Sorry for asking late question...If I use monthly observations of returns, do I need to use monthly observations of FF factors? And if I estimate 5-years trailing Betas do I calculate cost of equity as beta_{t-1}*Market_premim_{t}, am I right? – Alberto Alvarez Sep 21 '18 at 12:41