Answer to Question #249651 in Macroeconomics for Ricardo J

Question

Can you explain the Fama-French 3 and 5 factor model with math examples?

Answer 1

Solution:

The Fama-French 3-factor model is an asset pricing model that adds size risk and value risk factors to the market risk factors in addition to the market risk factors in the capital asset pricing model. The Fama-French 3-factor model has three components: firm size, book-to-market values, and market excess return. In other words, the three factors are SMB (small minus big), HML (high minus low), and the portfolio's return minus the risk-free rate of return.

The Fama-French 3-factor model seeks to describe stock returns by focusing on three factors: (1) market risk, (2) outperformance of small-cap companies relative to large-cap companies, and (3) outperformance of high book-to-market value companies versus low book-to-market value companies. The model is based on the fact that high-value and small-cap companies consistently outperform the overall market.

 

A math example is as follows:

Expected return = Rf + Ri + SMB + HML

r = r_f + β₁(r_m – r_f) + β₂(SMB) + β₃(HML) + ε

Where:

• r = Expected rate of return
• r_f = Risk-free rate
• ß = Factor’s coefficient (sensitivity)
• (r_m – r_f) = Market risk premium
• SMB (Small Minus Big) = Historic excess returns of small-cap companies over large-cap companies
• HML (High Minus Low) = Historic excess returns of value stocks (high book-to-price ratio) overgrowth stocks (low book-to-price ratio)
• ↋ = Risk

 

The Fama-French 5-factor model is an asset pricing model that adds profitability and investment factors to the three-factor model, in addition to the market risk factors in the capital asset pricing model. The model seeks to improve the Fama-French 3-factor model by adding the two additional factors

 

R_it — R_Ft = a_i + β_i(R_Mt — R_Ft) + s_iSMB_t + h_iHML_t + r_iRMW_t + c_iCMA_t + ε_it

 

Where:

·        R_ft = is the return in month t of one of the portfolios

·        R_Ft = is the riskfree rate

·        Rm - Rf = is the return spread between the capitalization-weighted stock market and cash

·        SMB = is the return spread of small minus large stocks

·        HML = is the return spread of cheap minus expensive stocks

·        RMW = is the return spread of the most profitable firms minus the least profitable

·        CMA = is the return spread of firms that invest conservatively minus firms that invest aggressively.

·        ε = Risk

·        b_i, s_i, h_i, r_i, and c_i, = Factor coefficients.