The Fama-French three-factor model is an asset pricing model developed by Eugene Fama and Kenneth French in 1993 which expands upon the Capital Asset Pricing Model (CAPM). The three factors in the model are the market risk premium, the size factor (SMB, small-minus-big), and the value factor (HML, high-minus-low).
Here’s how you can estimate each of these factors:
1. Market risk premium (MRP):
The market risk premium (MRP) is the excess return of a broad market portfolio over the risk-free rate. It can be estimated as an historical average or using forward-looking methods.
- Historical average: Take a sufficiently long time period and calculate the difference between the average return of the market portfolio (e.g., S&P 500 index) and the risk-free rate (e.g., 3-month T-bill rate).
$$MRP = (\frac{1}{n}) \sum_{i = 1}^{n} (Rm_i - Rf_i)$$
where n is the sample size, Rmi is the market return at time i, and Rfi is the risk-free rate at time i.
- Forward-looking methods: Use models or surveys to forecast expected market returns and risk-free rates, and then calculate the expected market risk premium.
2. Size factor (SMB):
The size factor (SMB) represents the excess return of small-capitalization stocks over big-capitalization stocks.
- First, create two portfolios based on market capitalization: a small-cap portfolio ("Small") and a big-cap portfolio ("Big").
- For each month, calculate the value-weighted average return of both portfolios.
- Compute the difference between the average return of the small-cap and big-cap portfolios.
SMB = RSmall - RBig
3. Value factor (HML):
The value factor (HML) represents the excess return of value stocks (stocks with high book-to-market ratio) over growth stocks (stocks with low book-to-market ratio).
- First, create two portfolios based on book-to-market ratio: a value portfolio ("High") and a growth portfolio ("Low").
- For each month, calculate the value-weighted average return of both portfolios.
- Compute the difference between the average return of the value and growth portfolios.
HML = RHigh - RLow
To build a model for a particular investment, you would use time-series regression analysis to estimate the sensitivity of that investment to the three factors:
Ri − Rf = α + βMRP(Rm − Rf) + βSMBSMB + βHMLHML + ε
where Ri is the return on the investment, Rf is the risk-free rate, α is the asset-specific constant, βMRP is the sensitivity to the market risk premium, βSMB is the sensitivity to the size factor, βHML is the sensitivity to the value factor, and ε is the residual error term.
The estimated coefficients, βMRP, βSMB, and βHML, will provide insight into how the stock’s returns relate to the three factors in the Fama-French model.