Fama-French 3-factor model

Motivated by the size and value anomalies, Fama and French argued for a three factor model.

$$R_{i,t} - R_{f,t} = \alpha_i + \beta_i (R_{m,t} - R_{f,t}) + s_i SMB_t + h_i HML_t + \varepsilon_{i,t}$$

• Size factor: SMB (Small Minus Big)
• Value factor: HML (High Minus Low)
• Widely used asset-pricing model for stocks and for evaluation of asset managers

Momentum alphas

Can market risk exposure explain momentum?

$$WML_t = \alpha_{\text{WML}} + \beta_{\text{WML}} (R_{m,t} - R_{f,t}) + \varepsilon_{i,t}$$

Fama-French-Carhart model

The FFC model augments the Fama-French-Carhart model with a momentum factor.

$$r_{i,t} - r_{f,t} = \alpha_i + \beta_i (r_{m,t} - r_{f,t}) + s_i SMB_t + h_i HML_t + m_i WML_t+ \varepsilon_{i,t}$$

• Size factor: SMB (Small Minus Big)
• Value factor: HML (High Minus Low)
• Momentum factor: WML (Winners Minus Losers)

Fama-French 5-factor model

• Industrious researchers have continued to generate firm characteristics that correlate with ex post performance.

• Recently, Fama and French have argued for the following model: $$\begin{align*} R_{i,t} - R_{f,t} = \alpha_i +& \beta_i (R_{m,t} - R_{f,t}) + s_i SMB_t + h_i HML_t \\ &+ r_i RMW_t + c_i CMA_t + \varepsilon_{i,t} \end{align*}$$

• Size factor: SMB (Small Minus Big)

• Value factor: HML (High Minus Low)

• Operating profitability factor: RMW (Robust Minus Weak)

• Investment factor: CMA (Conservative Minus Aggressive)

RMW + CMA cumulative returns

(Data starts in the 1960s due to availability of accounting information.)

E[r] estimates

Using the estimated factor loadings and estimates of the factor risk permia, the factor model’s estimate of expected returns is:

$$E[R_i] = R_f + \hat{\beta}_i \hat{\lambda}_{\text{mkt}} + \hat{s}_i \hat{\lambda}_{\text{smb}} + \hat{h}_i \hat{\lambda}_{\text{hml}} + \hat{r}_i \hat{\lambda}_{\text{rmw}} + \hat{c}_i \hat{\lambda}_{\text{cma}}$$

• Let’s look at an example on the dashboard
• A notebook implementing this approach is on Colab

The procedure

1. Run cross-sectional regressions for 120 months of returns $$R_{it}-R_{ft} = a_t + \sum_{j=1}^N b_{j,t}\cdot \text{characteristic}_{it-1} + e_{it}$$
2. Take average of each characteristic’s time-series of $b_{jt}$s $$\overline{b}_{j}= \frac{1}{120} \sum_{t=1}^{120} b_{j,t}$$
3. Expected return estimate $\tau$ is: $$E[R_{i\tau}] = R_{f\tau} + \overline{a} + \sum_{j=1}^N \overline{b}_{j} \cdot \text{characteristic}_{i\tau-1}$$

• Let’s look at how to implement this