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!

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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")

Attaching package: ‘zoo’

The following objects are masked from ‘package:base’:

    as.Date, as.Date.numeric

Version 0.4-0 included new data defaults. See ?getSymbols.
Learn from a quantmod author: https://www.datacamp.com/courses/importing-and-managing-financial-data-in-r

##### 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.

require(quantmod)
getSymbols("IBM", src="yahoo")  ##data starts from 2009

[1] "IBM"

ibmRd = periodReturn (IBM,period="daily", subset= "2009::")
plot(ibmRd, main="IBM daily returns")

require(quantmod)
getSymbols("IBM", src="yahoo")  ##data starts from 2009

[1] "IBM"

ibmRd = periodReturn (IBM,period="daily", subset= "2009::", type="log")
plot(ibmRd, main="IBM daily log returns")

require(quantmod)
getSymbols("IBM", src="yahoo")  ##data starts from 2009

[1] "IBM"

ibmRd = periodReturn (IBM,period="weekly", subset= "2009::")
plot(ibmRd, main="IBM weekly returns")

require(quantmod)
getSymbols("IBM", src="yahoo")  ##data starts from 2009

[1] "IBM"

ibmRd = periodReturn (IBM,period="monthly", subet= "2009::")
plot(ibmRd, main="IBM 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.

\(r_{t}\) = ln(1 + \(r_{t}\))

According to the generated plots, for 11/01/2018, the daily return for this date = 0.0121285801 and the daily log return for this date = 0.0120556182

so, \(r_{t}\) = ln(1 + 0.0121285801) \(r_{t}\) = 0.0120556182

##### 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.

mean.ibmRd = mean(ibmRd)
var.ibmRd = var(ibmRd)
sd.ibmRd = sd(ibmRd)
require(graphics)
y = as.numeric(ibmRd)
qqnorm(y, main = "Normal Q-Q Plot of IBM Daily Returns", xlab = "Frequency", ylab = "Probability"); qqline(y)

The mean return = 0.000230363878181582 The standard deviation = 0.0128416550217886

The X axis represents the frquency of the data to fall within the standard deviations. 68% of the data falls between -1 and 1 Standard Deviation. 95% of the data falls between -2 and 2 standard deviations, and 99.7% of the data falls between -3 and 3 Standard Deviations.

The Y axis represents the probability that a random data point will fall at a rate of return at different frequency points.

The data for IBM daily returns appears to be normally distributed since 95% of the data (falls within +2 or -2 standard deviations) follows the line of best fit, indicating a normal distribution.

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.

require(quantmod)
nflx = getSymbols("NFLX", src="yahoo")
nfRd= periodReturn(NFLX, period="daily", type="log")
dsd=density(nfRd) #estimate density of daily log return
yl=c(min(dsd$y), max(dsd$y)) #set y limits
plot(dsd, main="NFLX Daily Return", ylim=yl)
a=seq(min(nfRd), max(nfRd), 0.001)
points(a,dnorm(a, mean(nfRd), sd(nfRd)), type="log", lty=2)

plot type 'log' will be truncated to first character

require(quantmod)
nflx = getSymbols("NFLX", src="yahoo")
nfRd= periodReturn(NFLX, period="weekly", type="log")
dsd=density(nfRd) #estimate density of daily log return
yl=c(min(dsd$y), max(dsd$y)) #set y limits
plot(dsd, main="NFLX Weekly Return Distribution", ylim=yl)
a=seq(min(nfRd), max(nfRd), 0.001)
points(a,dnorm(a, mean(nfRd), sd(nfRd)), type="log", lty=2)

plot type 'log' will be truncated to first character

require(quantmod)
nflx = getSymbols("NFLX", src="yahoo")
nfRd= periodReturn(NFLX, period="monthly", type="log")
dsd=density(nfRd) #estimate density of daily log return
yl=c(min(dsd$y), max(dsd$y)) #set y limits
plot(dsd, main="NFLX Monthly Return Distribution", ylim=yl)
a=seq(min(nfRd), max(nfRd), 0.001)
points(a,dnorm(a, mean(nfRd), sd(nfRd)), type="log", lty=2)

plot type 'log' will be truncated to first character

As the return time frame that is observed grows larger in each graph (daily, weekly, monthly), the density decreases.

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.

#Federal Reserve Bank of St Louis
require(quantmod); getSymbols.FRED("DJIA",env=globalenv())

require(quantmod); getSymbols("DJIA", src="FRED")

[1] "DJIA"

serie=DJIA["1978/2001"]
price=as.numeric(serie) #extract numeric values of price
time = index(serie) #extract the indices
x=1:length(price)
model=lm(log(price)~x)

Error in model.frame.default(formula = log(price) ~ x, drop.unused.levels = TRUE) : 
  variable lengths differ (found for 'x')

##### 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.

\(P_{t}\) = \(e^{a + bt}\)

or

ln(\(P_{t}\)) = a + bt

Coefficients were unattained. Failure to debugg for variable lengths differ for “X”

t represents time, so,

1<=t<=2,515

##### 3C) Repeat the exercise in 3A) for AAPL Adjusted prices.

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

[1] "AAPL"

AAPLad=AAPL$AAPL.Adjusted
serie=AAPL["1978/2001"]
price=as.numeric(serie) #extract numeric values of price
time = index(serie) #extract the indices
x=1:length(price)
model=lm(log(price)~x)

Error in model.frame.default(formula = log(price) ~ x, drop.unused.levels = TRUE) : 
  variable lengths differ (found for 'x')

*http://computationalfinance.lsi.upc.edu