Requirement

Introduction

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

Arithmetic expression and Standard Calculation

2+2
## [1] 4
exp(-2)
## [1] 0.1353353
pi
## [1] 3.141593
cos(pi/3)
## [1] 0.5

Assignments

x<-2
y<-3
x+y
## [1] 5

Vector

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

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

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

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

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
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

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

Average Return

mean(ln_rt)
## [1] 0.005772091

Volatility (aka. Standard Deviation)

sd(ln_rt)*100
## [1] 1.126228

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?