Questions:

1. Interpret and compare the VaR95 estimate for each of the three companies with the historical method.

2. Interpret and compare each of the three companies' ES95 estimate with the historical method.

3. Interpret and compare the estimated VAR95 vs ES95 of the portfolio (equally weighted) of the three companies with the historical method.

4. Interpret and compare the VAR95 estimate of the portfolio (equally weighted) of the three companies with the historical method versus the normal distribution method

R code

######4.2. Empirical VaR and ES ####

n <- length(Ra_daily_port_ts) #Number of observations

VaR95 <- quantile(Ra_daily_port_ts,0.95,na.rm=T)

VaR95

VaR99 <- quantile(Ra_daily_port_ts,0.99,na.rm=T)

VaR99

which(Ra_daily_port_ts>=VaR99)

n <- length(Ra_daily_port_ts) #Number of observations

#Useful in back-testing: is the percentage of exceedances "close" to 1-alpha (99%)?

1-length(which(Ra_daily_port_ts>=VaR99))/n

chartSeries(Ra_daily_port_ts,up.col="red",theme = "white")

abline(h=VaR95,col="black")

abline(h=VaR99,col="blue")

ES95 <- mean(Ra_daily_port_ts[Ra_daily_port_ts>=VaR95])

ES95

ES99 <- mean(Ra_daily_port_ts[Ra_daily_port_ts>=VaR99])

ES99

######4.3. VaR and ES assuming Normal Distribution) ####

m=mean(Ra_daily_port_ts,na.rm=TRUE)# empirical mean

s=sd(Ra_daily_port_ts,na.rm=TRUE)# empirical sd

times=index(Ra_daily_port_ts)

n=sum(is.na(Ra_daily_port_ts))+sum(!is.na(Ra_daily_port_ts))

# we generate normal observations with mean m and standard deviation s

x=seq(1,n)

y=rnorm(n, m, s)

plot(times,Ra_daily_port_ts,pch=19,xaxs="i",cex=0.03,col="blue",ylab="Ra_daily_port_ts",xlab="Time",main ='')

segments(x0 =times,x1 =times,y0 =0,y1 =Ra_daily_port_ts,col="blue")

points(times,y,pch=19,cex=0.3,col="red",ylab="X",xlab="n",main ='')

VaR99_Emp <- quantile(Ra_daily_port_ts,0.99,na.rm=T)#empirical

VaR99_Emp

VaR99_Norm <- qnorm(0.99,m,s)# normal

VaR99_Norm

ES99_Emp <- mean(Ra_daily_port_ts[Ra_daily_port_ts>=quantile(Ra_daily_port_ts,0.99,na.rm=T)],na.rm=T)# empirical

ES99_Emp

ES99_Norm <- m+s*(dnorm(qnorm(0.99,0,1),0,1))/0.01# normal

ES99_Norm

######4.4. Backtesting Normal VaR) ####

VaR99_BT_Exp <- round(0.01*n) #Number of observation expected

VaR99_BT_Exp

VaR99_BT_Obs <- length(which(Ra_daily_port_ts>=VaR99_Norm)) #Number of observation observed in the data

VaR99_BT_Obs

#Right-tail Binomial test

1-pbinom(VaR99_BT_Obs-1,n,0.01)

#If p-value<=significance level, then we confirm that we are underestimating VaR under Normality (Normal Distribution)

######4.5. VaR and ES using Package Performance Analytics ####

VaR99_pa <- VaR(Ra_daily_port_ts, p=.99, method="historical", invert = F)

VaR99_ga <- VaR(Ra_daily_port_ts, p=.99, method="gaussian", invert = F)

VaR99_mo <- VaR(Ra_daily_port_ts, p=.99, method="modified", invert = F)

ES99_pa <- ES(Ra_daily_port_ts, p=.99, method="historical", invert = F)

ES99_ga <- ES(Ra_daily_port_ts, p=.99, method="gaussian", invert = F)

VaRES99_qa <- hs(Ra_daily_port_ts, p = 0.99, method = "plain", lambda = 0.98)

VaRES99_fc <- rollcast(Ra_daily_port_ts, p=.99, method="plain",

nout=nout, nwin=nwin)

plot(VaRES99_fc)

trftest(VaRES99_fc)

The companies are: GIS, K and MDLZ

Please provide the modified code that provide this information