Multifactor Models


Kevin Crotty
BUSI 448: Investments

Where are we?

Last time:

  • The cross-section of expected stock returns
  • Portfolio sorts
  • Cross-sectional regression

Today:

  • Multifactor models
  • Estimating expected returns
  • Characteristic-based models

Multifactor models

Expected returns

  • We are interested in characterizing the risk premium for stocks \[ E[r] = r_f + \text{risk premium} \]
  • Empirically, the CAPM fares poorly in this regard.
  • Today, we will explore some alternatives.

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

SMB and HML

Form 6 portfolios on size (mkt cap) and value (B/M ratio)

Low B/M Medium B/M High B/M
Small Small growth Small value
Large Large growth Large value
  • SMB: \((0.5\cdot SG + 0.5 \cdot SV) - (0.5\cdot LG + 0.5\cdot LV)\)
  • HML: \((0.5\cdot SV + 0.5 \cdot LV) - (0.5\cdot SG + 0.5\cdot LG)\)

SMB + HML cumulative returns

What are the CAPM alphas of HML and SMB?

\[ SMB_t = \alpha_{\text{SMB}} + \beta_{\text{SMB}} (R_{m,t} - R_{f,t}) + \varepsilon_{i,t} \]

\[ HML_t = \alpha_{\text{HML}} + \beta_{\text{HML}} (R_{m,t} - R_{f,t}) + \varepsilon_{i,t} \]

  • Let’s look at today’s first notebook.
  • Recall: non-zero alphas mean that the market portfolio is not mean-variance efficient
    • Investing in a portfolio of the market and a positive alpha portfolio leads to a higher Sharpe ratio.

Momentum

  • Consider sorting stocks based on their returns over the past year
  • Call the top performers “winners”
  • Call the bottom performers “losers”
  • A portfolio that goes long “winners” and short “losers” outperforms
  • This is known as a momentum strategy

Momentum cumulative returns

Momentum alphas

Can market risk exposure explain momentum?

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

What about the size and value factors?

\[ WML_t = \alpha_{\text{WML}} + \beta_{\text{WML}} (R_{m,t} - R_{f,t}) + s_{\text{WML}} SMB_t + h_{\text{WML}} HML_t + \varepsilon_{i,t} \]

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.)

What are the CAPM alphas of HML and SMB?

\[ RMW_t = \alpha_{\text{RMW}} + \beta_{\text{RMW}} (R_{m,t} - R_{f,t}) + \varepsilon_{i,t} \]

\[ CMA_t = \alpha_{\text{CMA}} + \beta_{\text{CMA}} (R_{m,t} - R_{f,t}) + \varepsilon_{i,t} \]

Expected return estimates

Factor models and E[r] estimates

  • For a given stock, we need three ingredients to construct an expected return estimate.
  1. Factor loadings (\(\beta_i, s_i, h_i, r_i, c_i\))
  2. Factor risk premia (\(\lambda_{\text{mkt}}\), \(\lambda_{\text{smb}}\), \(\lambda_{\text{hml}}\), \(\lambda_{\text{rmw}}\), \(\lambda_{\text{cma}}\))
  3. The risk-free rate
  • We have previously discussed the market risk premium.
  • Now we want to estimate the other risk premiums
    • we can use the time-series average return of their respective long-short portfolio

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

Characteristic regressions

Fama-MacBeth cross-sectional approach

  • We could also simply use the cross-sectional relationship between realized returns and lagged characteristics to characterize expected returns.
  • We will use \(N\) characteristics guided by the empirical record so far

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

For next time: Fixed Income: Duration





BUSI 448