Workshop 5, Financial Econometrics I
Alberto Dorantes, Ph.D.
Sep 8, 2020
Abstract
This is an INDIVIDUAL workshop. In this workshop we will practice the simple regression model, and also we will learn about predicting stock returns based on the Market Regression Model. We will learn about rolling regression to see how beta coefficients can move over time.1 General directions for this Workshop
You will work in RStudio. It is strongly recommended to have the latest version of R and RStudio. Once you are in RStudio, do the following.
Create an R Notebook document (File -> New File -> R Notebook), where you have to write whatever is asked in this workshop. More specifically, you have to:
Replicate all the R Code along with its output.
You have to do whatever is asked in the workshop. It can be: a) Responses to specific questions and/or do an exercise/challenge.
Any QUESTION or any INTERPRETATION you need to do will be written in CAPITAL LETTERS. For ANY QUESTION or INTERPRETATION, you have to RESPOND IN CAPITAL LETTERS right after the question.
- It is STRONGLY RECOMMENDED that you write your OWN NOTES as if this were your personal notebook. Your own workshop/notebook will be very helpful for your further study.
You have to keep saving your .Rmd file, and ONLY SUBMIT the .html version of your .Rmd file. Pay attention in class to know how to generate an html file from your .Rmd.
2 Predicting stock return for the BLACK MONDAY
Last year, on March 9th, 2020 we experienced one of the worst days for most of the financial markets in the world since October 2008. The S&P500 lost close to 8% of its value, while the Mexican IPCyC index lost almost 7%. Experts consider that the main reason for this radical decline of financial markets was due to a lack of agreement between Saudi Arabia and Russia to keep low oil production levels. Another factor is that China, one of the most important crude oil buyers, is radically reducing its demand for oil due to the diffusion of the COVID-19 (Coronavirus disease). Then, Saudi Arabia and Russia started a price war because both want to win the declining but important China oil market. This price war made the oil price going down dramatically in just few hours. From Friday March 6 to Monday March 9, crude oil price went down from about USD 45 per barrel to about USD 32, a decline close to 30%. This caused panic among big mutual funds and investors around the world, so many of these funds started to sell volatile instruments such as commodities and stocks and buy more secure assets such as treasury bills, bonds and gold.
In this workshop you will predict the stock return of Gruma stock on Monday March 9th, given the market return of that day. You have to use a market regression model.
You have to run 2 versions of a Market regression model for the Gruma company: one model using daily data, and another one using monthly data. To run these models we will use historical data from January 2017 to March 6, 2020 (just before the BLACK MONDAY March 9, 2020), and then you have to predict the Gruma stock return for the BLACK MONDAY, March 9, 2020.
Download DAILY stock data of Gruma (GRUMAB.MX) and the IPyC (^MXX) from Yahoo from 2017 to March 11, 2020:
# Remember to load the library you will be using
library(quantmod)
# Obtain daily data
getSymbols(c("GRUMAB.MX", "^MXX"), from="2017-01-01",
to="2020-03-11", periodicity="daily", src="yahoo")## [1] "GRUMAB.MX" "^MXX"# Visualize data
head(GRUMAB.MX, 5)## GRUMAB.MX.Open GRUMAB.MX.High GRUMAB.MX.Low GRUMAB.MX.Close
## 2017-01-02 263.50 264.00 262.01 263.10
## 2017-01-03 263.97 265.00 257.50 258.77
## 2017-01-04 260.06 266.06 257.00 265.74
## 2017-01-05 265.74 275.00 264.46 274.41
## 2017-01-06 266.00 275.00 266.00 274.64
## GRUMAB.MX.Volume GRUMAB.MX.Adjusted
## 2017-01-02 22518 239.4060
## 2017-01-03 420265 235.4660
## 2017-01-04 1120534 241.8083
## 2017-01-05 1175996 250.1682
## 2017-01-06 860582 250.3779head(MXX, 5)## MXX.Open MXX.High MXX.Low MXX.Close MXX.Volume MXX.Adjusted
## 2017-01-02 45642.80 45792.46 45593.26 45695.10 6765800 45695.10
## 2017-01-03 45698.93 46444.72 45690.46 46123.36 164938800 46123.36
## 2017-01-04 46123.36 46587.74 46068.83 46587.74 215385900 46587.74
## 2017-01-05 46586.03 46767.37 46285.88 46719.99 237288300 46719.99
## 2017-01-06 46721.70 46731.90 46027.10 46071.57 201371200 46071.57Visualize the daily stock prices for Gruma:
plot(GRUMAB.MX$GRUMAB.MX.Adjusted)
Now visualize the IPCyC market index:
plot(MXX$MXX.Adjusted)
We can see that Gruma price and the IPCyC index have a negative trend over time, but looking at March 9 it seems that both had important decline.
We can visualize daily returns of Gruma:
r_GRUMA <- diff(log(Ad(GRUMAB.MX)))
plot(r_GRUMA)
We can visualize daily returns of the Market:
r_MXX <- diff(log(Ad(MXX)))
plot(r_MXX)
We can see that the Mexican Market had its worst day at the end of the series, that is March 9th, 2020. Gruma had also a negative return on the same day, but not as dramatic as in the case of the Mexican market index.
Now we can visualize the linear relationship between Gruma returns and the Mexican index return:
plot(as.numeric(r_MXX), as.numeric(r_GRUMA), type = "p",
xlab = "r_MXX", ylab = "r_GRUMA",
main = "Scatter plot between Gruma returns and Market returns")
abline(lm(r_GRUMA~r_MXX), col="red")
Although the scales of both axis have different range values, we can appreciate that the slope of the regression line is not quite big, and it should be less than 1 since the inclination of the line is less than 45 degrees.
Now you have to run the Market Regression Model for Gruma restricting dates up to March 6, 2020 (before the BLACK MONDAY):
# Subset orginal time series of GRUMA returns and assign it to r_GRUMA_bef
r_GRUMA_bef <- r_GRUMA["2017-01-02/2020-03-06"]
# Another way to subset that will give the exact same result
ie <- r_GRUMA[1:799,]
# Do the same for the returns of the market
r_MXX_bef <- r_MXX["2017-01-02/2020-03-06"]
# Use the lm() function to make a linear regression
# bef_bm stands for before black monday
bef_bm <- lm(r_GRUMA_bef ~ as.vector(r_MXX_bef))
# Notice the class of the new object
class(bef_bm)## [1] "lm"# Take a look at the contents of the object
names(bef_bm)## [1] "coefficients" "residuals" "effects" "rank"
## [5] "fitted.values" "assign" "qr" "df.residual"
## [9] "na.action" "xlevels" "call" "terms"
## [13] "model"# Use the function summary() to access the names of bef_bm
summ_bef_bm <- summary(bef_bm)
# We are interested in the coefficients of the model
summ_bef_bm$coefficients## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.000288431 0.0004395498 -0.6561964 5.118874e-01
## as.vector(r_MXX_bef) 0.594998772 0.0500974141 11.8768360 4.566972e-30What you can say about the market risk of GRUMA? EXPLAIN.
What is the regression equation of this model? WRITE it as comments in your R script.
Note that your regression DID NOT include the BLACK MONDAY March 9, 2020. Now check what was the real market daily return on March 9th and March 10th, 2020. You can just show the last 5 days of the data set to see the last daily returns:
tail(r_MXX, 5)## MXX.Adjusted
## 2020-03-04 0.02171823
## 2020-03-05 -0.02472212
## 2020-03-06 -0.02283717
## 2020-03-09 -0.06638094
## 2020-03-10 0.02132713tail(r_GRUMA, 5)## GRUMAB.MX.Adjusted
## 2020-03-04 0.020392005
## 2020-03-05 -0.001206592
## 2020-03-06 -0.025035224
## 2020-03-09 -0.030216764
## 2020-03-10 0.010485221Now using the real data of the market return for the BLACK MONDAY, ESTIMATE what would be the expected return for GRUMA on Monday March 9th, 2020 using the regression equation. How GOOD was your prediction compared with the real Gruma return on that day? DO THE CALCULATION of your PREDICTION in your r file.
## [1] -0.03978325Now, use the predict() function to do the prediction of the GRUMA daily returns. Make sure that your manual estimation of the prediction for the BLACK MONDAY is exactly the same you got using the predict() function. To run the predict() function, just type:
predict(bef_bm, newdata = data.frame(r_MXX_bef = -0.06638094))## 1
## -0.03978501You can also create a data frame that contains the values of the explanatory variable you want to try. In order not to copy and paste the value, you can refer to it by subsetting the original time series of the explanatory variable (in this case, r_MXX) However, the explanatory variable in both lm() and predict() must be the same. Thus, in this case, we must convert the subset of r_MXX (an xts object) to a simple numeric value by using as.numeric().
explanatory_var <- data.frame(r_MXX_bef = as.numeric(r_MXX["2020-03-09"]))
# Then, we can use predict() or predict.lm()
# with the explanatory variable object and get the same result.
predict.lm(bef_bm, newdata = explanatory_var)## 1
## -0.03978501According to the model, GRUMA would have had -3.97% of returns on March 9, 2020 given the real market return of March 9,2020 (-6.63%)
Looking at the real return of GRUMA we displayed above (-3.02%), we can see that the prediction was close to the actual GRUMA return on the Black Monday March 9, 2020 (-3.97% vs -3.02%).
3 Predicting monthly stock return of GRUMA for February and March 2020
This part is OPTIONAL (but recommended to study for exam)
Now download MONTHLY data for GRUMA and the Mexican index from January 2017 to March 2020. Run the Market Regression model using monthly data UP TO JANUARY 2020, so you have to PREDICT (manually and also using predict) the monthly return for GRUMA in February and March, using the real return of the market for February and March. REPORT your results. DID YOU GET similar BETA1 when using monthly data compared to the BETA1 of the daily model? Did you have the same CONCLUSION about the market risk of Gruma using daily and monthly data? EXPLAIN
4 Estimating “moving” betas of the Market Model
Systematic risk of a stock is mainly due to movements in the market. Non-systematic risk of a stock is the idiosyncratic risk of the stock that is NOT related to the market. Beta1 represents the market risk of a stock, or the systematic risk of the stock.
We have to be aware that the beta1 and beta0 regression coefficients of a market model of a stock CAN MOVE over time. This is the reason why we always see or calculate the standard error of the beta0 and beta1 in a regression. Remember that the standard error in the context of a regression is the standard deviation of the corresponding beta.
Then, how the estimated betas of a stock move over time? Is the beta1 of a stock stable or does it radically change over time?
We can use the roll_lm() function of the roll package to automatically run many regression models, one for different rolling time window. For example, to run many regressions with a 24-month rolling window for the Gruma monthly returns starting in 2010, you can do the following in R:
We get the monthly data of Gruma from 2010 to March, 2020:
# Start with a clean environment
rm(list=ls())
# Use getSymbols() to download data
getSymbols(c("GRUMAB.MX", "^MXX"), from="2010-01-01",
to="2020-03-31", periodicity="monthly", src="yahoo")## [1] "GRUMAB.MX" "^MXX"# Calculate returns
r_GRUMA <- diff(log(Ad(GRUMAB.MX)))
r_MXX <- diff(log(Ad(MXX)))Now we run the market regression model using windows of 24-month starting in January 2010. Then, R will run many regression models; one regression model for each 24-month window. If I have 10 years and 3 months of data, then I have about 123 months in my data set, so R will run 100 (123-24+1) regression models since there are 100 24-month windows from Jan 2010 to March 2020.
We need to install the roll R package (install it from RStudio/Packages). We will use the roll_lm() function of thie package, indicating the dependent and independent variables of the regression model (in that order: x, y) as vectors or matrices, as well as size of the windows. We can store the resulting data in a new object:
# Load library
library(roll)
# roll_lm() receives only vectors and matrices, so I coerce the data of returns
rollmktmdl <- roll_lm(as.vector(r_MXX), as.vector(r_GRUMA), width = 24)
# Look at the class of the resulting object
class(rollmktmdl)## [1] "list"# Notice the elements of the list object
names(rollmktmdl)## [1] "coefficients" "r.squared" "std.error"As we are interested in the coefficients, we will create an object that contains them:
rolling_betas <- rollmktmdl$coefficients
# The columns have weird names. Let's change them
colnames(rolling_betas) <- c("b0", "b1")If you use View() to take a look at the coefficients, you will notice that the first 24 rows contain NA values. This is because the first 24 rows were taken in the first window and there was no prior information to compute the regression model for these values.
The roll_lm() function returns the coefficients as vectors. However, we are working with time series, so we have to change the class of the coefficients object (rolling_betas).
rolling_betas <- xts(rolling_betas, order.by = index(r_GRUMA))
# Now rolling_betas is an xts objectI graph the beta1 of Gruma for the 100 24-month windows (windows ending in 2012 to 2020):
plot(rolling_betas$b1, main = "Moving beta1 for Gruma stock", ylab = "Beta 1")
You have to INTERPRET THIS GRAPH (respond in your R script)
I can also see how beta0 moves over time in the 100 24-month windows:
plot(rolling_betas$b0, main = "Moving beta0 for Gruma stock", ylab = "Beta 0")
You have to INTERPRET THIS GRAPH (respond in your R script)
5 Reading
Read again: Basics of Linear Regression Models.
6 Quiz 5 and W5 submission
Go to Canvas and respond Quiz 5 about Basics of Return and Risk. You will be able to try this quiz up to 3 times. Questions in this Quiz are related to concepts of the readings related to this Workshop. The grade of this Workshop will be the following:
- Complete (100%): If you submit an ORIGINAL and COMPLETE HTML file with all the activities, with your notes, and with your OWN RESPONSES to questions
- Incomplete (75%): If you submit an ORIGINAL HTML file with ALL the activities but you did NOT RESPOND to the questions and/or you did not do all activities and respond to some of the questions.
- Very Incomplete (10%-70%): If you complete from 10% to 75% of the workshop or you completed more but parts of your work is a copy-paste from other workshops.
- Not submitted (0%) Remember that you have to submit your .html file through Canvas BEFORE NEXT CLASS.