Fama and French - Part 1

Can we calculate the FF portfolios?

https://pixabay.com/pt/photos/estoque-negocia%C3%A7%C3%A3o-monitor-neg%C3%B3cios-1863880/

This post is co-authored with Gerson Junior.

The goal

The goal of this post is simple: to learn how to calculate the Fama and French portfolios' returns. Keep in mind that we begin from scratch, so we are not thinking about coding optmization for now.

You can download here the data that we fabricated to learn with. We hope you help us if we make any mistake.

Important: we are using the two-factor 3X2 portfolio, meaning we have two factors (Growth and Size) and six portfolios (sample is split into three groups of Growth against two groups of Size).

First step

To start the process, install and load the following packages.

library(readxl)
library(dplyr)
library(tidyverse)
library(writexl)

Clear your workspace.

# Clean workspace
rm(list = ls())

Then, you have to import our data. Again, this is fabricated by us. You can see we have 10 stocks in two years. We also have each firms' Return, Market-to-book (MtB), and Size. All these values are random.

# import data
data <- read_excel("FF Example.xlsx", sheet="Before")

The first important step is to create deciles for the first sorting step, which is based on MtB. The result is simply a column with the decile the firm is (sorted by MtB). We did that in both years.

# Create deciles by MtB
data <- data  %>% 
        group_by(Year) %>%  
        mutate(MtB_deciles = ntile(MtB, 10))

Then, using these deciles, we can split firms into three groups: Growth (firms until the 3rd decile), Neutral (firms from the 4th decile to the 7th decile), and Value (firms from the 8th decile until the 10th).

# Defining Growth, Neutral, and Value
data$MtB_class = ifelse(data$MtB_deciles<=3 ,"Growth", ifelse(data$MtB_deciles>=8 ,"Value", "Neutral" ))

Next step is a little tricky. Not sure we can explain well in words, perhaps you want to see the data.frame first.

But the idea below is to sort firms within each of the three groups of MtB, and split them into Small or Big. So, for the group Growth, we split firms into Small or Big, then we did the same for the group Neutral, then for the group Value.

# Create deciles by Size within MtB
data <- data  %>% 
        group_by(Year, MtB_class) %>%  
        mutate(Size_deciles = ntile(Size, 2))

# Defining Growth, Neutral and Value within Size
data$Size_class = ifelse(data$Size_deciles<=1 ,"Small","Big")

# Defining Growth, Neutral and Value within Size
data$Size_class = ifelse(data$Size_deciles <= median(data$Size_deciles) ,"Small", "Big")

The product of these first rows is the definition of four new columns. We will use the two new variables were we split the sample.

Defining Portfolios

The row below is to define the 3x2 portfolios.

# Finding portfolios
data$port <- paste0(data$MtB_class,data$Size_class)

Then, we calculate the weights of each firm in each portfolio in each year. Our weight is based on firms' Size.

# generate firm-level time- and portfolio-specific weights
data <- data  %>% group_by(Year, port) %>% mutate (weight = Size/sum(Size))

Final part

Okay, now that we have the six portfolios in each year, we can calculate the return of the SmB (Small minus Big) and HmL (High minus Low) factors.

First, we need to define which stocks we are buying and which we are selling. We use the structure below. For more information about this structure, see French’s site.

$$SmB = \frac{1}{3}(Small Value + Small Neutral + Small Growth) - \frac{1}{3} (Big Value + Big Neutral + Big Growth)$$

$$HML = \frac{1}{2} (Small Value + Big Value) - \frac{1}{2} (Small Growth + Big Growth)$$

Ok, the code below calculates the return of each of the six portfolios, and store then in a new data.frame:

# Six portfolios returns
data$smb_ret <-  data$weight * data$Return

ret <- as.data.frame(tapply(data$smb_ret,
                                list(data$Year, 
                                     data$port), 
                                FUN = sum))

Here, we calculate the return of the SmB portfolio.

# SMB return

ret$smb_buy  <- as.data.frame((ret$GrowthSmall + ret$NeutralSmall + ret$ValueSmall)/1/3)
ret$smb_sell <- as.data.frame((ret$GrowthBig   + ret$NeutralBig   + ret$ValueBig  )/1/3)

ret$smb      <- ret$smb_buy - ret$smb_sell

And here, we calculate the return of the HmL portfolio.

# HML return

ret$hml_buy  <- as.data.frame((ret$ValueSmall + ret$ValueBig)/1/2)
ret$hml_sell <- as.data.frame((ret$GrowthSmall+ ret$GrowthBig)/1/2)

ret$hml      <- ret$hml_buy - ret$hml_sell

You can print portfolios' return in each year:

paste("The returns of the Small minus Big portfolios are, respectively," ,round(ret$smb * 100 ,3),"%")

paste("The returns of the High minus Low portfolios are, respectively," , round(ret$hml * 100 ,3),"%")

The SmB return in year 1 and 2 are, respectively, -4.79% and 6.02%, while the HmL return in year 1 and 2 are, respectively, 7.06% and 1.71%.

You can save the file to learn more about how we did it. This is our result: here .

write_xlsx(data, "FF Example After.xlsx")

What do we learn?

It is not easy to calculate the factors' returns, but it is doable. Our next step is to calculate it using real data from Brazilian stocks.

Thanks for passing by. See ya!

Henrique Castro Martins
Henrique Castro Martins
Assistant Professor of Finance
Gerson Junior
Gerson Junior
PhD Candidate
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