Mathematics and Finance Summer School at CMI

Chennai Mathematical Institute

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Requirement

• You should come with your laptop.
• It is assumed that you have a working installation of the R software and of the following R-Packages.
  • tseries
  • copula
  • PortRisk
• You can use the following code in the command prompt or R-console to install the packages: install.packages(c("tseries","copula","PortRisk"))
• Also, we firmly suggest that you should install R-Studio (after installing the base-R).

Introduction

• History of R
• 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")

## 'getSymbols' currently uses auto.assign=TRUE by default, but will
## use auto.assign=FALSE in 0.5-0. You will still be able to use
## 'loadSymbols' to automatically load data. getOption("getSymbols.env")
## and getOption("getSymbols.auto.assign") will still be checked for
## alternate defaults.
## 
## This message is shown once per session and may be disabled by setting 
## options("getSymbols.warning4.0"=FALSE). See ?getSymbols for details.

## 
## WARNING: There have been significant changes to Yahoo Finance data.
## Please see the Warning section of '?getSymbols.yahoo' for details.
## 
## This message is shown once per session and may be disabled by setting
## options("getSymbols.yahoo.warning"=FALSE).

Nifty.50

##            Adjusted
## 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

##                Adjusted
## 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

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

• To present the return in percentage you may consider multiply with 100

Rt*100

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

ln_rt*100

##              Adjusted
## 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 course.

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?

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• email: sourish at cmi .ac. in