# Pre-Built Factor Models

### Pre-Built Factor Models

According to the CAPM Model, we know the \(\mu_i -r_f =\beta_m(\mu_m -r_f)\), where \(\mu_i\) is the expected return of asser i ,\(\mu_m\) is the expected return of market, \(r_f\) is the risk-free rate.By doing so, we know that some other factors can be added to the initial model with the norm form:

\(R_{it}=\alpha_i +\sum_{k=1}^{K}\beta_{ik}f_{ik} +\epsilon_i\)

, where \(R_{it}\) is the excess return on asset i at time t, \(f_{kt}\) is the factor.

Now let us introduce our pre-built factor models.

### Fama-French 3 Factors Model

This model have three factors- market \(\beta\), SMB and HML with introductions below.

**Market \(\beta\)** has the same meaning as CAPM model, which represents the excess return on the whole market.

**SMB (small minus big) ** reflects the risks associated with the size of the company. The calculation method is: based on the last trading day of June each year, the median value of the circulating market value of all stocks in the stock market (that is, the size of the company) is calculated . The stock with value higher than the level is divided into class S, others are divided into class B. For each class, the stocks are divided into three groups based on the book value ratio: the first 30% of the stocks are classified as H, the middle 40% of the stocks are classified as M, and the last 30% of the stocks are classified as L. Hence, we have six groups: S/L, S/M, S/H, B/L, B/M and B/H. Here, we have SMB as

\(\frac{S/L+S/M+S/H}{3}-\frac{B/L+B/M+B/H}{3}\)

**HML(high minus low)** can be defined by method mentioned in SMB with formula:

\(\frac{S/H+B/H}{2}-\frac{S/L+B/H}{2}\)

### Carhart 4 Factors Model

This model have four factors- market \(\beta\), SMB, HML and UMD.

On account of the first three factors have been introduced above, here we mainly introduce UMD.

**UMD **(momentum factors) is recombined for monthly construction. At the end of t-1 month, we calculate the cumulative return(CRET) for the eleven months from t-12 to t-2. According to the size of CRET, the stocks are divided into three groups: the first 30% of the stocks are classified as U, the middle 40% of the stocks are classified as M, and the last 30% of the stocks are classified as D. Then the corresponding t period portfolio yield is calculated. The momentum factor is defined as follows:

\(\frac{B/U+S/U}{2}-\frac{B/D+S/D}{2}\)

### Fama-French 5 Factors Model

This model have five factors- market \(\beta\), SMB, HML, RMW and CMA.

**RMW** represents profit level by dividing stocks into three groups according to profit size-the first 30% of the stocks are classified as R, the middle 40% of the stocks are classified as M, and the last 30% of the stocks are classified as W.

\(RMW =\frac{B/R+S/R}{2}-\frac{B/W+S/W}{2}\)

, where B/R represents the weighted average return of all stocks that belong to both B and R, and others can be similarly defined.

**CMA** is the difference between the returns of firms that invest conservatively and firms that invest aggressively. It divides stocks into three groups according to investment size-the first 30% of the stocks are classified as A, the middle 40% of the stocks are classified as M, and the last 30% of the stocks are classified as C.

\(CMA =\frac{B/C+S/C}{2}-\frac{B/A+S/A}{2}\)

, where B/A represents the weighted average return of all stocks that belong to both B and A, and others can be similarly defined.

### References

https://en.wikipedia.org/wiki/Fama%E2%80%93French_three-factor_model

Fama E, French K.Common risk factors in the returns on stocks and bonds [ J] .Journal of Financial Economics, 1993, 33:3— 56

Fama E, French K.Size and book-to-market factors in earnings and returns [ J] .Journal of Finance , 1995, 50(1):131 —156

徐忠亚, 一种新的定价因子构建方法及在我国的应用[D].南京大学,2017.