Stock Analysis Assignment

Download stock data from http://finance.yahoo.com. In this example, we retrieved a year (from April 10, 2017 to April 6, 2018) of data of Apple Inc. (AAPL), saved as AAPL.csv.

We used the read.csv command to import data and assign them to the object dat. Be sure that the file is in the current directory; if not, use the setwd command (Under Session/Set Work Directory). The head command is for obtaining the first few rows of the data. We show the stock price as a function of time using the plot command. Note that the type is l (el) meaning line not 1 (one).

dat <- read.csv("AAPL.csv")
head(dat)

##         Date   Open   High    Low  Close Adj.Close   Volume
## 1 2017-04-10 143.60 143.88 142.90 143.17  140.9404 18933400
## 2 2017-04-11 142.94 143.35 140.06 141.63  139.4244 30379400
## 3 2017-04-12 141.60 142.15 141.01 141.80  139.5917 20350000
## 4 2017-04-13 141.91 142.38 141.05 141.05  138.8534 17822900
## 5 2017-04-17 141.48 141.88 140.87 141.83  139.6213 16582100
## 6 2017-04-18 141.41 142.04 141.11 141.20  139.0011 14697500

plot(as.Date(dat$Date), dat$Adj.Close, type = "l")

We assign the adjusted close to the vector called close for later convenience. The number of trading days is the length of the vector, denoted by n.

close <- dat$Adj.Close
(n <- length(close))

## [1] 250

In financial news, we often hear relative, or percentage changes. A stock’s rate of return in any time period \(t\) is defined as \[ R_{t} = \frac{S_{t} - S_{t-1}}{S_{t-1}} \] where \(S_{t}\) and \(S_{t-1}\) are the closing prices at time \(t\) and \(t-1\), respectively. Below is the calculation.
A histogram is made for the relative changes.

daily.diff <- close[2:n] - close[1:n-1]
rel.change <- daily.diff/close[1:n-1]
hist(rel.change, breaks = 20)

We can find the five-number summary and produce a box plot as below.

summary(rel.change)

##       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
## -0.0433902 -0.0055524  0.0002952  0.0008015  0.0079969  0.0474718

boxplot(rel.change, horizontal = TRUE)

To find the mean and standard deviation, use the following commands.

(x.bar <- mean(rel.change))

## [1] 0.0008014673

(stdev <- sd(rel.change))

## [1] 0.01321591

The number of trading day is n above

n

## [1] 250

and the number of days when the change is less than the mean is

sum(rel.change < x.bar)

## [1] 130

Express the changes as \(z\)-scores, and overlay the histogram of \(z\) with a standard normal distribution probability density function.

z.score <- (rel.change - mean(rel.change))/sd(rel.change)
hist(z.score, breaks = 20, freq = FALSE)
curve(dnorm(x), col = "blue", add = TRUE)

One financial model assumes that stocks advances with geometric Brownian motion, \[ S_{i+1} = S_{i} + \mu S_{i} \Delta t + \sigma S_{i} \epsilon \sqrt{\Delta t} \] where \(\epsilon\) is a random number drawn from a standardized normal distribution.
Use $168.38 as the initial price (the closing value on April 6, 2018), we simulate a possible outcome for the next year. You should re-run this several times to observe the possible outcomes.

S <- c()
close[n]

## [1] 168.38

S[1] <- close[n]
for (i in 1:n){
  S[i+1] <- S[i] + x.bar*S[i]*1 + stdev*S[i]*rnorm(1)*sqrt(1)
}
plot(S, type = "l")