load("dataset.RData")

#MEAN RETURN AND RISK OF ASSETS

#select the returns
DJIA_ret <- DJIA[,-c(1,2)]

#compute the mean of each stock
DJIA_mu <- colMeans(DJIA_ret)

#compute the standard deviation of each stock. "2" is used to select the column. "1" would compute the sd of the rows
DJIA_sd <- apply(DJIA_ret, 2, sd)

#we compute the covariance matrix of the returns of the individual stocks
DJIA_cov <- cov(as.matrix(DJIA_ret))

#we plot mean vs. sd on a scatter plot
plot(DJIA_sd, DJIA_mu, xlab = "Standard deviation", ylab = "Mean return", xlim=c(0.04,0.1), pch=19)

#we now create an equally weighted portfolio with the 22 stocks

#as it is equally weighted we just compute the mean return of each stock in each periods to get the returns of this portfolio
EWP_ret <- rowMeans(DJIA_ret)

#we compute the mean and sd
DJIA_mean <- mean(EWP_ret)
DJIA_sd <- sd(EWP_ret)

#we now plot this portfolio on the previous plot
points(DJIA_sd, DJIA_mean, col="red", pch=19)

#COMPUTATION AND PLOTTING OF WEALTH DYNAMICS

#custom function to compute the evolution of wealth
Wealth <- function(Returns){
  wealth <- c(1,Returns)
  wealth <- timeSeries::as.timeSeries(wealth)
  for(i in 2:length(wealth)){
    if (wealth[i-1]>0){
      wealth[i] = wealth[i-1]+wealth[i-1]*wealth[i]
    } else {
      wealth[i] = 0
    }
  }
  wealth <- as.vector(wealth)
  return(wealth)
}

#we use this function to compute the wealth of the equally weighted portfolio

#first we need the dates
Date <- seq(as.Date("2000-01-01"),length=nrow(DJIA)+1,by="months")-1

#we compute the wealth
EWP_wealth <- Wealth(EWP_ret)

plot(EWP_wealth~Date,type="l", lwd=2, col="red", ylab="Wealth")

#MEAN RETURN AND RISK OF A PORTFOLIO WITH GIVEN WEIGHTS

#we now consider a portfolio with five assets: AXP, BA, CAT, CSCO, CVX
library(dplyr)
stocks_ret <- DJIA_ret %>% select(AXP,BA,CAT,CSCO,CVX)

#while it is more convenient to use dplyr, we can also select the columns in other ways using base are
stocks_ret <- DJIA_ret[,1:5]

#we compute the mean return of the stocks of interest
stocks_mu <- colMeans(stocks_ret)

#we compute the covariance matrix of hese stocks
stocks_cov <- cov(stocks_ret)

#we use as set of weight: 0.2 for AXP, 0.1 for BA, 0.3 for CAT, 0.3 for CSCO and 0.1 for CVX
p_w <- c(0.2,0.1,0.3,0.3,0.1)

#using the means as expected returns, we not compute the expected return of the portfolio
p_mu <- t(p_w) %*% stocks_mu

#and we compute the variance of the portfolio
p_var <- t(p_w) %*% stocks_cov %*% p_w

#ESTIMATION WITH ROLLING AND EXPANDING WINDOW

#up to now we used the entire dataset. now we see how to use rolling and expanding windows

WIND <- 120 #window size (initial window size in the case of expanding window)

#rolling window
DJIA_mu_roll <- list()
DJIA_cov_roll <- list()
for (i in 1:(nrow(DJIA_ret)-WIND)) {
  DJIA_mu_roll[[i]] <- colMeans(DJIA_ret[i:(WIND+i-1),])
  DJIA_cov_roll[[i]] <- cov(DJIA_ret[i:(WIND+i-1),])
}

#expanding window
DJIA_mu_exp <- list()
DJIA_cov_exp <- list()
for (i in 1:(nrow(DJIA_ret)-WIND)) {
  DJIA_mu_exp[[i]] <- colMeans(DJIA_ret[1:(WIND+i-1),])
  DJIA_cov_exp[[i]] <- cov(DJIA_ret[1:(WIND+i-1),])
}