Workshop at Statistical Mathods in Finance, 2016

Introduction

This session gives an introduction to the R and walks you through its most essential features.
Screen image of R for Windows looks like:

R is an object-oriented language.
R works fundamentally by the question-and-answer model.
You enter a line with a command and press Enter.
For example: Following code draw 1000 random sample with mean=0 and sd=1. Then draw a histogram of the random sample.

r<-rnorm(n=1000,mean=0,sd=1)
hist(r,col="red")

This is done by entering the code in command prompt.

Arithmetic expression and Standard Calculation

The machine knows that two plus two makes four.
It also knows how to do other standard calculations like:

2+2

## [1] 4

exp(-2)

## [1] 0.1353353

pi

## [1] 3.141593

cos(pi/3)

## [1] 0.5

Assignments

To assign the value 2 to the variable x, you can enter

x<-2
y<-3
x+y

## [1] 5

This symbol <- is known as the assignment operator.

Vector

R handles entire data vector as a single object.
A data vector is simply an array of numbers or characters or factors etc. and a vector variable can be constructed like this:

Nifty.50 <- c(8114.30, 7965.50, 8033.30, 8002.30, 7929.10 )
Nifty.50

## [1] 8114.3 7965.5 8033.3 8002.3 7929.1

The construct c(...) is used to define vectors. It combine values into a vector.
This is not the best way to enter data; nor the preferred way; but short vectors are used for many other purposes, and the c(...) construct is used extensively

Dowload stock price data from Yahoo

library(tseries)
Nifty.50<-get.hist.quote(instrument = "^NSEI"
                        ,start="2016-11-21"
                        ,end="2016-11-25"
                        ,quote="AdjClose"
                        ,provider = "yahoo")
Nifty.50

##            AdjClose
## 2016-11-21   7929.1
## 2016-11-22   8002.3
## 2016-11-23   8033.3
## 2016-11-24   7965.5
## 2016-11-25   8114.3

Compute log-return

log-return is the consecutive differences of prices in log-scale, i.e.,

\[ \begin{eqnarray*} r_t &=& \log(P_t) - \log(P_{t-1})\\ &=& \log\Big(\frac{P_t}{P_{t-1}}\Big) \end{eqnarray*} \] where \(P_t\) is the price of a stock (or value of an index) at time point \(t\).

We can calculate the log-return in R in the following way

ln_rt<-diff(log(Nifty.50))
ln_rt

##                AdjClose
## 2016-11-22  0.009189427
## 2016-11-23  0.003866402
## 2016-11-24 -0.008475662
## 2016-11-25  0.018508197

Compute simple return

Simple return can be calculated as \[ \begin{eqnarray*} R_t &=&\frac{P_t-P_{t-1}}{P_{t-1}}\\ &=&\frac{P_t}{P_{t-1}}-1\\ &=& e^{r_t}-1, \end{eqnarray*} \] wher \(r_t\) is the log-return.
We can calculate the simple return in R in the following way

Rt<-exp(ln_rt)-1
Rt

##                AdjClose
## 2016-11-22  0.009231780
## 2016-11-23  0.003873886
## 2016-11-24 -0.008439845
## 2016-11-25  0.018680535

To present the return regarding percentage you may consider multiply with 100

Rt*100

##              AdjClose
## 2016-11-22  0.9231780
## 2016-11-23  0.3873886
## 2016-11-24 -0.8439845
## 2016-11-25  1.8680535

ln_rt*100

##              AdjClose
## 2016-11-22  0.9189427
## 2016-11-23  0.3866402
## 2016-11-24 -0.8475662
## 2016-11-25  1.8508197

K-period Simple Return

\[ \begin{eqnarray*} R_t(k)&=&\frac{P_t-P_{t-k}}{P_{t-k}}=\frac{P_t}{P_{t-k}}-1\\ 1+R_t(k)&=&\frac{P_t}{P_{t-k}}=\Big(\frac{P_t}{P_{t-1}}\Big)\Big(\frac{P_{t-1}}{P_{t-2}}\Big)...\Big(\frac{P_{t-k+1}}{P_{t-k}}\Big)\\ &=& (1+R_t)(1+R_{t-1})...(1+R_{t-k+1}) \end{eqnarray*} \]

K-period log Return

\[ \begin{eqnarray*} r_t(k)&=&\log\{1+R_t(k)\}\\ &=& \log\{(1+R_t)(1+R_{t-1})...(1+R_{t-k+1})\}\\ &=& \log(1+R_t)+\log(1+R_{t-1})+...+\log(1+R_{t-k+1})\}\\ &=& r_t+r_{t-1}+...+r_{t-k+1} \end{eqnarray*} \]

\(k\)-period log-return is sum of \(k\) single period log-return
But \(k\)-period simple return cannot be expressed as easily as log-return.
log-return has some other advantage as well. We will discuss them later.
Now onwards, we will use log-return for rest of the workshop.

Average Return

mean(ln_rt)

## [1] 0.005772091

Volatility (aka. Standard Deviation)

sd(ln_rt)*100

## [1] 1.126228

Volatility tells us, on average, how much value of an asset can go down or go up.
Since sd(ln_rt)*100 is calculated using daily log-return; so calculated volatility is daily volatility in percentage.
So value of Nifty 50 can go down (or up) by 1.12623% on a given day.

How to compute 30-days Volatility

Suppose \(r_t,r_{t-1},...,r_{t-k+1}\) are \(k\) single period log-return of an asset, where \[ \begin{eqnarray} \mathbb{E}(r_{t-i+1})&=&\mu~~~ \forall i=1,2,...,k\\ \mathbb{V}ar(r_{t-i+1})&=&\sigma^2~~ \forall i\\ \mathbb{C}ov(r_{t-i+1},r_{t-j+1})&=& 0~~~ \forall i\neq j=1,2...,k. \end{eqnarray} \] That is the covariance matrix is \[ \begin{eqnarray} \Sigma=\left(\begin{array}{cccc} \sigma^2 & 0 & ... & 0 \\ 0 & \sigma^2 & ... & 0 \\ \vdots & \vdots & ... & \vdots\\ 0 & 0 & ... & \sigma^2 \end{array}\right)_{k \times k} \end{eqnarray} \]

The \(k\)-period return can be presented in matrix notation as \[ \begin{eqnarray*} r_{t}(k)&=&r_t+r_{t-1}+...+r_{t-k+1}\\ &=&c^T\mathbf{r}, \end{eqnarray*} \] where \(c^T=(1,1,...,1)_{k}\) is an unit vector of order \(k\) and \(\mathbf{r}=\{r_t,r_{t-1},...,r_{t-k+1}\}\). The mean and variance of the \(k\)-period return are \[ \begin{eqnarray*} \mathbb{E}(r_{t}(k))&=& c^T\mathbf{\mu}=k\mu,\\ \mathbb{V}ar(r_{t}(k))&=&c^T\Sigma c =k\sigma^2 \end{eqnarray*} \]

Therefore k-period volatility is \(\sqrt{k}\sigma\).

Monthly volatility of Nifty 50

sqrt(22)*sd(ln_rt)*100

## [1] 5.282477

Yearly volatility of Nifty 50

sqrt(252)*sd(ln_rt)*100

## [1] 17.87831

Why are we using \(k=22\) and \(k=252\) for monthly and yearly volatility of Nifty 50?

email: sourish at cmi .ac. in

Workshop at Statistical Mathods in Finance, 2016

Sourish Das

18-Dec-2016

Requirement

Introduction