INTRODUCTION TO R FOR FINANCE Part II

Library

For those who have not installed tidyverse previously: please install and define library i.e. run the below line: install.packages("tidyverse")
library(tidyverse)
Note that installation may take a while (up to 25 minutes on the class computers)

For those who have installed tidyverse previously: please define library i.e. run line (you should do this each time you open new R session):
library(tidyverse)

Data frame

Creating data frame

Create FAANG data frame (Facebook, Amazon, Apple, Netflix, Google) that contains:

Date column (taken from fb data frame)
Close column for each company (closing price)

Get the data files: copy and run the below code to download the separate data frames for each company:

url <- "http://coin.wne.uw.edu.pl/gzakrzewski/data_for_students/"
fb <- read.csv2(paste0(url, "FB.csv"), header = TRUE,  sep = ",", dec = ".") 
amzn <- read.csv2(paste0(url,"AMZN.csv"), header = TRUE,  sep = ",", dec = ".") 
aapl <- read.csv2(paste0(url, "AAPL.csv"), header = TRUE,  sep = ",", dec = ".") 
nflx <- read.csv2(paste0(url,"NFLX.csv"), header = TRUE,  sep = ",", dec = ".") 
goog <- read.csv2(paste0(url,"GOOG.csv"), header = TRUE,  sep = ",", dec = ".")

faang <- data.frame(date = fb$Date[1:10],
                    fb   = fb$Close[1:10],
                    amzn = amzn$Close[1:10],
                    aapl = aapl$Close[1:10],
                    nflx = nflx$Close[1:10],
                    goog = goog$Close[1:10])

Rate of return

Calculating rate of return

Arithmetic rate of return:
\(rr = \frac{price_t-price_{t-1}}{price_{t-1}}\)

Helpful functions:

lag(x) - returns the previous value in a vector
lead(x) - returns the next value in a vector

Example Create vector of returns based on Close vector in goog data frame:

# inspect goog data frame:
head(goog)
# create vector of returns based on Close vector in goog data frame:
goog_rr <- (goog$Close - lag(goog$Close)) / lag(goog$Close)

Example
If you need returns in the data frame, create new vector in faang using goog vector existing in faang. First inspect faang dataframe using head().

head(faang)

##         date    fb   amzn     aapl     nflx     goog
## 1 2014-02-24 70.78 351.78 75.36429 63.85714 602.3374
## 2 2014-02-25 69.85 358.32 74.58000 64.71858 606.0582
## 3 2014-02-26 69.26 359.80 73.90714 64.11285 606.1426
## 4 2014-02-27 68.94 360.13 75.38143 64.60429 605.6657
## 5 2014-02-28 68.46 362.10 75.17714 63.66143 603.8972
## 6 2014-03-03 67.41 359.78 75.39429 63.65572 597.4591

Then create new vector called goog_rr containing rate of return:

faang$goog_rr <- (faang$goog - lag(faang$goog)) / lag(faang$goog)

head(faang)

##         date    fb   amzn     aapl     nflx     goog       goog_rr
## 1 2014-02-24 70.78 351.78 75.36429 63.85714 602.3374            NA
## 2 2014-02-25 69.85 358.32 74.58000 64.71858 606.0582  0.0061772106
## 3 2014-02-26 69.26 359.80 73.90714 64.11285 606.1426  0.0001393794
## 4 2014-02-27 68.94 360.13 75.38143 64.60429 605.6657 -0.0007868263
## 5 2014-02-28 68.46 362.10 75.17714 63.66143 603.8972 -0.0029199160
## 6 2014-03-03 67.41 359.78 75.39429 63.65572 597.4591 -0.0106609383

Excercise 1
calculate logarithmic rate of return for Apple: \(rr = ln(\frac{price_t}{price_{t-1}})\)

aapl_rr <- log(faang$aapl / lag(faang$aapl))

Excercise 2
Create data frame faang_rr containing:

Date as a first vector
rates of returns (arithmetic) as a separate vectors for Facebook, Amazon, Apple and Netflix

faang_rr <- data.frame(date = fb$Date,
                       fb = (fb$Close - lag(fb$Close)) / lag(fb$Close),
                       amzn = (amzn$Close - lag(amzn$Close)) / lag(amzn$Close),
                       aapl = (aapl$Close - lag(aapl$Close)) / lag(aapl$Close),
                       nflx = (nflx$Close - lag(nflx$Close)) / lag(nflx$Close))

Excercise 2a
Add to data frame faang_rr rates of returns (arithmetic) for Google

faang_rr$goog <- (goog$Close - lag(goog$Close)) / lag(goog$Close)

Excercise 3
Create data frame faang_log_rr containings:

Date as a first vector
rates of returns (log) as a separate vectors for each FAANG company

faang_log_rr <- data.frame(date = fb$Date,
                       fb   = log(fb$Close    / lag(fb$Close)),
                       amzn = log(amzn$Close / lag(amzn$Close)),
                       aapl = log(aapl$Close / lag(aapl$Close)),
                       nflx = log(nflx$Close / lag(nflx$Close)),
                       goog = log(goog$Close / lag(goog$Close)))

Other helpful functions

Probabilities (optional)

Normal distribution:
density: dnorm(x, mean = 0, sd = 1, log = FALSE)
distribution: pnorm(q, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)
quantile: qnorm(p, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)
random deviates: rnorm(n, mean = 0, sd = 1)

The normal distribution density: \(f(x) = \frac{1}{\sqrt{2 πσ^2} } * e^{-\frac{(x - μ)^2}{2σ^2}}\)

# X ~ N(0.01, (0.05)^2)
mu_x <- 0.01
sigma_x <- 0.05

# Pr(X < -0.10)
pnorm(-0.10, mean = mu_x, sd = sigma_x)

# Pr(X > 0.10)
1 - pnorm(0.10, mean = mu_x, sd = sigma_x)

# Pr(-0.05 < X < 0.15)
pnorm(0.15, mean = mu_x, sd = sigma_x) - pnorm(-0.05, mean = mu_x, sd = sigma_x)

# Quantiles: 1%, 5%, 10%, 90%, 95% and 99% 
qnorm(c(0.01, 0.05, 0.1, 0.9, 0.95, 0.99), mean = mu_x, sd = sigma_x)

Quantile from empirical distribution: quantile(x, q)

# Create sample of 100 values from Normal distriburion with mean = 0 and sigma = 1
sample <- sort(rnorm(100)) # we sorted the data for better visualisation
# Calculate quantile
quantile(sample, 0.1)

Simulations (optional)

Drawing the distribution:

generate sequence of x, eg. from -5 till 5
use density function to generate the distribution
use plot() to draw the distribution

x <- seq(-5, 5, by = 0.01)
x_norm <- dnorm(x, mean = 0, sd = 1)
plot(x_norm)

x_norm <- dnorm(x, mean = 0, sd = 2)
plot(x_norm)

x_norm <- dnorm(x, mean = 0.05, sd = 0.25)
plot(x_norm)

# Random normal variables mean = 0, sd = 1
y_norm  <- rnorm(1000)
plot(y_norm)

# plot(rnorm(100))

Monte Carlo (optional)

Calculate \(\int_0^1(x)=x^3\)

Monte Carlo solution:

Create a sequence of pairs of random variables coming from uniform dsitribution (x, y): for generating x and y use rununif() function.
check the value of f(x) vs. value of y.

## [1] 0.235

n = 100
x <- runif(n, min = 0, max = 1)
y <- runif(n, min = 0, max = 1)

integral <- sum(y < x ^ 3) / n
# note that condition (y < x ^ 3) returns bolean vector of TRUE i.e. 1 or FALSE i.e. 0 for each elements of vector x and y
integral

Excercise 4
Calculate \(\int_0^{10}(x)=5+4*\frac{x^3}{e^x}*sin(x)\)

n <- 1000
x <- runif(n, 0, 10)
y <- runif(n, 0, 10)

integral <- sum(y < 5 + 4 * x ^ 3/ exp(x) * sin(x)) / (n)
integral

## [1] 0.471

Own functions

Writing simple functions

Calculate PV of CF = 1000 paid at the end of 5th year. Discount rate is 8%.

\(PV = \frac{CF}{(1+i)^t}\)

pv = 1000 / (1 + 0.08) ^ 5
pv

## [1] 680.5832

How to automate calculations in order not to duplicate work: write a function calculationg PV

function_name <- function(par1, par2, ...) {body of the function}

Body part contains actual computations the function should perform.

# the function for single payment, t defined as number of years:
get_PV <- function(cf, i, t) {
  cf / (1 + i) ^ t
}
# cf - cash flow, i - discount rate, t - time.
get_PV(1000, 0.08, 5)

## [1] 680.5832

get_PV(1000, 0.06, 5)

## [1] 747.2582

Excercise 5
Write function calculating present value of a yearly payments. Function should take two arguments:

vector of cash flows cf, eg. cf <- c(12, 12, 12, 12, 12, 112)
discount rate i, eg. i = 0.06
payments are made at end of consecutive years.

Note: length(x) returns the length of vector x (number of elements)

# the vector for sequence of payments:
cf <- c(12, 12, 12, 12, 12, 112)
# the vector of years:
t <- seq(1:length(cf))
t

## [1] 1 2 3 4 5 6

Instead of t we use directly seq(1:length(cf)) in the formula:

get_PV_annual_CF <- function(cf, i) {
  sum(cf / (1 + i) ^ seq(1:length(cf)))
}

get_PV_annual_CF(cf, 0.06)

## [1] 129.5039

Excercise 6
Write function calculating present value of a payments with different frequency of capitalisation (m). Function should take arguments:

vector of cash flows cf, eg. cf <- c(3, 3, 3, 3, 3, 3, 3, 3, 112)
discount rate i, eg. i = 0.06
frequency of capitalisation (m) within a year, eg. m = 4 for quarterly payments, m = 12 for monthly payments.

# the function for sequence of payments:
get_PV_CF <- function(cf, i, m = 1) {
  sum(cf / (1 + i / m) ^ (m * seq(1:length(cf))))
}
cf <- c(3, 3, 3, 3, 3, 3, 3, 3, 112)
get_PV_CF(cf, 0.015, 4)

## [1] 120.3304

Note that you may define default value of the argument, in the get_PV_CF function m is set to 1 by default. It means that if you do not put n argument when executing the function, R will calculate for m=1 (i.e. annual payments). Define function get_PV_CF and try for:

cf <- c(3, 3, 3, 3, 3, 3, 3, 3, 112)
get_PV_CF(cf, 0.015)

## [1] 120.4121

Plotting functions

Simple way with `curve()`

Simple way of ploting a function:

curve(f(x), from = , to = ) draws a curve corresponding to a function over the interval [from, to]

Excercise 7
Plot \(f(x) = 2x ^ 2 - 3x + 4,\) in the interval [-5, 10]

curve(2 * x ^ 2 - 3 * x + 4, -5, 10)

Excercise 8

Polt \(f(x)=(cos(x)+tan(x), -10, 3 * \pi)\) in the interval [-1, 3 \(\pi\) ]

curve(cos(x)+tan(x), -10, 3 * pi)

Example
To plot two functions on one chart use argument add = TRUE in the curve() function.

curve(dnorm(x, mean = 0, sd = 0.5), -5, 5)
curve(dnorm(x, mean = 0, sd = 1), -5, 5, add = TRUE, col = "red")
curve(dnorm(x, mean = 0, sd = 2), -5, 5, add = TRUE, col = "green")

Excercise 9

Plot the PV of CF = 1000 paid at the end of 15th year for discount rates ranging from 0% up to 15%. Add to the plot:

blue line presenting PV of CF = 1000 paid at the end of 8th year for discount rates ranging from 0% up to 15%.
green line presenting PV of CF = 1000 paid at the end of 4th year for discount rates ranging from 0% up to 15%.

curve(get_PV(1000, x, 15), 0, 0.15)
curve(get_PV(1000, x, 8), 0, 0.15, add = TRUE, col = "blue")
curve(get_PV(1000, x, 4), 0, 0.15, add = TRUE, col = "green")

Using `plot()` function

plot() is a basic way to plot objects. To create user defined layout use arguments (for more details see ?plot):

type: visualisation type
col: colour of series
ylab: y label
ylim: y axis scale

head(faang)

##         date    fb   amzn     aapl     nflx     goog       goog_rr
## 1 2014-02-24 70.78 351.78 75.36429 63.85714 602.3374            NA
## 2 2014-02-25 69.85 358.32 74.58000 64.71858 606.0582  0.0061772106
## 3 2014-02-26 69.26 359.80 73.90714 64.11285 606.1426  0.0001393794
## 4 2014-02-27 68.94 360.13 75.38143 64.60429 605.6657 -0.0007868263
## 5 2014-02-28 68.46 362.10 75.17714 63.66143 603.8972 -0.0029199160
## 6 2014-03-03 67.41 359.78 75.39429 63.65572 597.4591 -0.0106609383

plot(x = fb$Close,
     type = "l",
     ylab = "stock price",
     col = "red",
     ylim = c(50, 250))
lines(aapl$Close, type = "l", col = "blue",  lwd = 2)

Excercise 10
Plot rate of return of Apple.
Hint: use existing faang_rr data frame

plot(x = faang_rr$aapl,
     type = "l",
     ylab = "stock price",
     col = "red",
     main = "Apple stock RR")

Solving equations

For solving simple equations we can use uniroot(f, lower = , upper = ) function. The function uniroot searches the interval from lower to upper for a root (i.e., zero) of the function f with respect to its first argument.

Examples:
Find solutions for:

\(2x - 4 = 0\)
\(3\sqrt{log(x)} - 4x = 0\)

f <- function(x) {2 * x - 4}
curve(f, 0, 4)

uniroot(f, lower = -2, upper = 4)$root

## [1] 2

f <- function(x) {3 * sqrt(log(x))  - 4}
curve(f, 1, 10)

uniroot(f, lower = 1, upper = 10)$root

## [1] 5.916694

Excercise 11
Find x that solves the equation:
\(-97+3*x^{-1}+3*x^{-2}+103*x^{-3}=0\)

f <- function(x) {-97+3*x^-1+3*x^-2+103*x^-3}
curve(f, 0.8, 1.8)

uniroot(f, lower = 1, upper = 10)$root

Matrix

Matrix: definition

Matrix as a special case of a two-dimensional array

A matrix is the special case of a two-dimensional array, where the objects are of the same type (note that in data frame the vectors may be of the different data types: numertic, character, …)

How to create matrix:

most frequently used way in our examples: binding consecutive vectors into using cbind function:

vector_1 <- seq(1, 5, by = 1)
vector_2 <- seq(101, 105, by = 1)
vector_3 <- seq(-5, -1, by = 1)

my_matrix <- cbind(vector_1, vector_2, vector_3)
my_matrix

##      vector_1 vector_2 vector_3
## [1,]        1      101       -5
## [2,]        2      102       -4
## [3,]        3      103       -3
## [4,]        4      104       -2
## [5,]        5      105       -1

You can use matrix function creating matrix from single vector (sequence):

vector_4 <- seq(1, 15)
my_matrix_2 <- matrix(data = vector_4, nrow = 5, ncol = 3) # put values by column
my_matrix_2
my_matrix_2 <- matrix(data = vector_4, nrow = 5, ncol = 3, byrow = TRUE) # put values by row
my_matrix_2

Matrix: column names and selection

Column names and selection of values from matrix

Create FAANG matrix (Facebook, Amazon, Apple, Netflix, Google):
Hint: Copy and run the below code to download the data frames (unless you did it previously and have not changed the files):

url <- "http://coin.wne.uw.edu.pl/gzakrzewski/data_for_students/"
fb <- read.csv2(paste0(url, "FB.csv"), header = TRUE,  sep = ",", dec = ".") 
amzn <- read.csv2(paste0(url,"AMZN.csv"), header = TRUE,  sep = ",", dec = ".") 
aapl <- read.csv2(paste0(url, "AAPL.csv"), header = TRUE,  sep = ",", dec = ".") 
nflx <- read.csv2(paste0(url,"NFLX.csv"), header = TRUE,  sep = ",", dec = ".") 
goog <- read.csv2(paste0(url,"GOOG.csv"), header = TRUE,  sep = ",", dec = ".")

Create matrix consosting of first 20 observations of Close price of FAANG companies.

faang <- cbind(fb$Close[1:10],
              amzn$Close[1:10],
              aapl$Close[1:10],
              nflx$Close[1:10],
              goog$Close[1:10])
faang

Name the column of the matrix:

colnames(faang) <- c("fb", "amzn", "aapl", "nflx", "goog")
faang

Selecting:

column: faang[ , 3], faang[ , "aapl"]
row: faang[4, ]
element: intersection of n row and m column faang[n, m]

faang[ , 5]
faang[ , "goog"]
faang[3, 4]

Inspecting the matrix:

head(matrix) - prints begining of matrix
tail(matrix) - pronts last rows of matrix

head(faang)
tail(faang)

Matrix simple operations

Matrix multiplication %*%

Assme there are two matrices (v1 and m1) defined:

v1

##      [,1] [,2] [,3] [,4]
## [1,]    1    2    3    4

m1

##      [,1] [,2] [,3] [,4] [,5]
## [1,]    1    5    9   13   17
## [2,]    2    6   10   14   18
## [3,]    3    7   11   15   19
## [4,]    4    8   12   16   20

Use %*% to multiply the matrices:

v_m <- v1 %*% m1
v_m

##      [,1] [,2] [,3] [,4] [,5]
## [1,]   30   70  110  150  190

Correlation

For correlation calculation one may use cor() function. Example of use for two variables: (we will use faang_rr dataframe created at the begining of the lesson)

# Inspect faang_rr:
head(faang_rr)

##         date           fb          amzn         aapl          nflx
## 1 2014-02-24           NA            NA           NA            NA
## 2 2014-02-25 -0.013139319  0.0185911877 -0.010406600  1.348999e-02
## 3 2014-02-26 -0.008446614  0.0041303331 -0.009021976 -9.359307e-03
## 4 2014-02-27 -0.004620271  0.0009172235  0.019947855  7.665109e-03
## 5 2014-02-28 -0.006962619  0.0054702496 -0.002710124 -1.459433e-02
## 6 2014-03-03 -0.015337351 -0.0064070891  0.002888484 -8.975607e-05
##            goog
## 1            NA
## 2  0.0061772106
## 3  0.0001393794
## 4 -0.0007868263
## 5 -0.0029199160
## 6 -0.0106609383

# Create data frame consisting of two vectors of our interest (Apple and Facebook rates of return):
df <- faang_rr[-1, c(2, 4)] # "-" before the column or row number remove the column / row
# Plot the rates of return against each other to asses correlation visualy:
plot(df)

# Calculate correlations:
cor(df)

##             fb      aapl
## fb   1.0000000 0.4361795
## aapl 0.4361795 1.0000000

Example of use for multi variables:

# Create data frame consisting of all companies (this time keep rates of return for all the FAANG comapnies but remove first row and first column):
df <- faang_rr[-1, -1] # "-" before the column or row number remove the column / row
  
plot(df)

cor(df)

##             fb      amzn      aapl      nflx      goog
## fb   1.0000000 0.5538736 0.4361795 0.4122462 0.6027777
## amzn 0.5538736 1.0000000 0.4557642 0.4701487 0.6460855
## aapl 0.4361795 0.4557642 1.0000000 0.3474320 0.4968930
## nflx 0.4122462 0.4701487 0.3474320 1.0000000 0.4761786
## goog 0.6027777 0.6460855 0.4968930 0.4761786 1.0000000

Excercise 12

Calculate correlations between log rate of returns of FAANG companies.
Create vector containing correlations of Apple with other FAANG companies.

# Create data frame consisting of two vectors of our interest (this time rates of return for all the FAANG comapnies):
df <- faang_log_rr[-1, -1] # "-" before the column or row number remove the column / row
  
# plot(df)

cor(df)
# select rasults for Apple:
cor(df)[ , 3] 
cor(df)[3 , ] 
cor(df)["aapl", ]

INTRODUCTION TO R FOR FINANCE Part II

RS/GZ, contact: gzakrzewski@wne.uw.edu.pl, rslepaczuk@wne.uw.edu.pl

27 Feb 2019

Library

Data frame

Rate of return

Other helpful functions

Probabilities (optional)

Simulations (optional)

Monte Carlo (optional)

Own functions

Plotting functions

Simple way with `curve()`

Using `plot()` function

Solving equations

Matrix

Matrix: definition

Matrix: column names and selection

Matrix simple operations

Correlation

INTRODUCTION TO R FOR FINANCE Part II

RS/GZ, contact: gzakrzewski@wne.uw.edu.pl, rslepaczuk@wne.uw.edu.pl

27 Feb 2019

Library

Data frame

Rate of return

Other helpful functions

Probabilities (optional)

Simulations (optional)

Monte Carlo (optional)

Own functions

Plotting functions

Simple way with curve()

Using plot() function

Solving equations

Matrix

Matrix: definition

Matrix: column names and selection

Matrix simple operations

Correlation

Simple way with `curve()`

Using `plot()` function