FINC621 Winter 2018-19 Lab Worksheet 02

Time Series Distributions & Normality (finc621-lab02)

About

This worksheet has three main taks: analyze the time series of returns, assess for normal distribution, and check for exponential behavior of prices time series.

Setup

Remember to always set your working directory to the source file location. Go to ‘Session’, scroll down to ‘Set Working Directory’, and click ‘To Source File Location’. Read carefully the below and follow the instructions to complete the tasks and answer any questions. Submit your work to RPubs as detailed in previous notes.

Note

For clarity, tasks/questions to be completed/answered are highlighted in red color (color visible only in preview mode) and numbered according to their particular placement in the task section. Type your answers outside the red color tags!

Quite often you will need to add your own code chunk. Execute sequentially all code chunks, preview, publish, and submit link on Sakai following the naming convention. Make sure to add comments to your code where appropriate. Use own language!

Any sign of plagiarism, will result in dissmissal of work!

---

Task 1: Plot of Returns & Testing for Normality Distribution

In this task we will look at various type of daily returns calculations for comparison and to test normality.

# Require will load the package only if not installed 
# Dependencies = TRUE makes sure that dependencies are install
if(!require("quantmod",quietly = TRUE))
  install.packages("quantmod",dependencies = TRUE, repos = "https://cloud.r-project.org")

##### 1A) Follow the instructions on p.41 to generate four plots for daily, daily log, weekly, and monthly returns. Select a stock of your choice and a time period long enough (5-10 years) to capture the returns behavior.

library("quantmod")
getSymbols("RPM",src="yahoo",from="2012-11-18",to="2018-11-18")

[1] "RPM"

rpRd = periodReturn(RPM,period="daily")
plot(rpRd, main="RPM Daily Returns")

library("quantmod")
getSymbols("RPM",src="yahoo",from="2012-11-18",to="2018-11-18")

[1] "RPM"

rpRdL = periodReturn(RPM,period="daily",type="log")
plot(rpRdL, main="RPM Daily Log Returns")

library("quantmod")
getSymbols("RPM",src="yahoo",from="2012-11-18",to="2018-11-18")

[1] "RPM"

rpRdw = periodReturn(RPM,period="weekly")
plot(rpRdw, main="RPM Weekly Returns")

library("quantmod")
getSymbols("RPM",src="yahoo",from="2012-11-18",to="2018-11-18")

[1] "RPM"

rpRdM = periodReturn(RPM,period="monthly")
plot(rpRdM, main="RPM Monthly Returns")

##### 1B) For the case of daily log returns only, write down the mathematical formula representing the calculation in the code. Confirm integrity of your mathematical formula by selecting a recent data point from your time series object, substituting the corresponding values in the formula to manually calculate the log return, and comparing both results.

Mathematical formula for daily log return: \[r_{t}=lnP_{t}-lnP_{t-1}\] Where \(r_{t}\) refers to the return value at time “t”, and \(P_{t}\) & \(P_{t-1}\) refer to the price of the stock in two succeeding days. Inserting adjusted close values from the time series object into the formula yields:

.0104012308 = ln(23.40315) - ln(23.16099)

##### 1C) Check the normality of the daily returns using the R function qqnorm() to generate a Q-Q plot. For the function to work properly, you will need to extract first the numeric values from the time series object. Note that a time series object contains both a date and a corresponding value. To extract the numerical value only, on can use the R function as.numeric() on the time series object. Explain what the Y and X axis of the Q-Q plot represent, and share your observation on the normality of the returns distribution.

library("quantmod")
getSymbols("RPM",src="yahoo",from="2012-11-18",to="2018-11-18")

[1] "RPM"

rpRd2 = periodReturn(RPM,period="daily")
rtrn=as.numeric(rpRd2)
qqnorm(rtrn)

In this plot of daily stock returns we see theoretical quantiles, or percentiles, along the X-axis of a standard normal distribution with a mean of zero and standard deviation of one. The daily return data is also divided into quantiles, which are listed in ascending order along the Y-axis.This sample return data is then plotted and one can expect to see a straight line, or nearly straight, if the data represents a normal distribution. In this case the data set is fairly normal, however, it would be described as having fat tails, which indicates higher than normal values at the extremes.

Task 2: Density Distribution

Another way to assess the normality of a distribution, other than a Q-Q plot, is to look at the actual density distribution and compare to a normal distribution.

##### 2A) Follow the example in R Lab 2.7.9/p. 70 to generate the density distribution for your stock of choice. Comment on your results.

library("quantmod")
getSymbols("NFLX",src="yahoo")

[1] "NFLX"

nfRd = periodReturn(NFLX,period="daily",type="log")
#estimate density of daily log ret
dsd=density(nfRd)
#set y limits
yl=c(min(dsd$y),max(dsd$y))
##plot the normal density with mean, stdv of nfRd
plot(dsd,main=NULL,ylim=yl)
a=seq(min(nfRd),max(nfRd),0.001)
points(a,dnorm(a,mean(nfRd),sd(nfRd)), type="l",lty=2)

library("quantmod")
getSymbols("NFLX",src="yahoo")

[1] "NFLX"

nfRd = periodReturn(NFLX,period="weekly",type="log")
#estimate density of weekly log ret
dsd=density(nfRd)
#set y limits
yl=c(min(dsd$y),max(dsd$y))
##plot the normal density with mean, stdv of nfRd
plot(dsd,main=NULL,ylim=yl)
a=seq(min(nfRd),max(nfRd),0.001)
points(a,dnorm(a,mean(nfRd),sd(nfRd)), type="l",lty=2)

library("quantmod")
getSymbols("NFLX",src="yahoo")

[1] "NFLX"

nfRd = periodReturn(NFLX,period="monthly",type="log")
#estimate density of monthly log ret
dsd=density(nfRd)
#set y limits
yl=c(min(dsd$y),max(dsd$y))
##plot the normal density with mean, stdv of nfRd
plot(dsd,main=NULL,ylim=yl)
a=seq(min(nfRd),max(nfRd),0.001)
points(a,dnorm(a,mean(nfRd),sd(nfRd)), type="l",lty=2)

These results are typical as they exhibit “aggregational normality”“, which means the distribution of returns looks more like a normal distribution as we increase the time scale from daily through monthly returns.

Task 3: Exponential Behavior of Prices & Curve Fitting

In general, the price history of a stock, over a sufficiently large time window, tends to follow an exponential curve. Many other economic indicators like GDP, population growth, and inflation also follow exponential growth over a long time. Keep in mind that for investment purposes we care more about returns and not prices.

##### 3A) Follow the example in R Lab 2.7.2/p. 67 or R Labs 2 from book’s website (*) to generate an exponential fit for the Dow Jones Industrial Average DJIA. In case the suggested command in the book does not work, consider using instead the command in the code chunk below to capture the DIJA prices.

library("quantmod")
#Federal Reserve Bank of St Louis
getSymbols("DJIA",src="FRED")

[1] "DJIA"

serie=DJIA["1990/2018"]
price=as.numeric(serie) #extract numeric values of price
time = index(serie) #extract the indices
x=1:length(price)
model=lm(log(price)~x)
expo=exp(model$coef[1]+model$coef[2]*x)
plot(x=time,y=price, main="Dow Jones",type="l")
lines(time,expo,col=2,lwd=2)

##### 3B) Write down the mathematical form representing the exponential function in the code. Substitute for the exact coefficients in the exponential form and clearly label the variables in the function, in particular the time index.

In this code we use the exponential function “exp” in order to replicate the calcuation of price \(P_{t}\) as a function of \(e^{a+bt}\). This R function uses values from the linear model of log prices, captured in the term “model”, as coefficients “a” and “b”, then multiplies the b term by a time value drawn from the index of values for the DJIA.

Mathematical representation: \[P_{t} = e^{(2.25360+0.001066547*t)}\] ##### 3C) Repeat the exercise in 3A) for AAPL Adjusted prices.

library("quantmod")
getSymbols("AAPL",src="yahoo")

[1] "AAPL"

#extract adjusted close prices
AAPLad=AAPL$AAPL.Adjusted
serie=AAPLad["1990/2018"]
price=as.numeric(serie) #extract numeric values of price
time = index(serie) #extract the indices
x=1:length(price)
model=lm(log(price)~x)
expo=exp(model$coef[1]+model$coef[2]*x)
plot(x=time,y=price, main="APPLE",type="l")
lines(time,expo,col=2,lwd=2)