load("dataset.RData")
load("SP500.RData")

#COMPUTE THE BETA OF THE STOCKS

#to get the excess market return we use the average excess return of the S&P 500
SP500_ex <- SP500$SP500 - DJIA_ex$Rfree

#now we compute AXP's beta

#first we plot its returns against the returns of the market for illustration purposes
plot(SP500_ex, DJIA_ex$AXP, pch=19, xlab="S&P 500 excess returns", ylab="AXP excess returns", main="AXP")

#we compute AXP's beta (and Jensen's alpha) by regressing its excess returns on the market's excess returns
AXP_reg <- lm(DJIA_ex$AXP ~ SP500_ex)
AXP_alpha <- AXP_reg$coefficients[1]
AXP_beta <- AXP_reg$coefficients[2]

#now we plot the regressed line on the previous plot
abline(AXP_reg, col = "red", lwd=2)

#the line is pretty flat because the beta is low. let's try with BA
plot(SP500_ex, DJIA_ex$BA, pch=19, xlab="S&P 500 excess returns", ylab="BA excess returns", main="BA")
BA_reg <- lm(DJIA_ex$BA ~ SP500_ex)
BA_alpha <- BA_reg$coefficients[1]
BA_beta <- BA_reg$coefficients[2]
abline(BA_reg, col = "red", lwd=2)

#the line is steeper because BA has a higher beta. let's compare the mean return of the two stocks
AXP_mean <- mean(DJIA$AXP)
BA_mean <- mean(DJIA$BA)

AXP_mean
BA_mean

#BA has a higher mean return than AXP, in line with the theory (higher beta should imply higher return)

#now we compute all the betas
betas_stocks <- vector()
DJIA_ex_ret <- DJIA_ex[,-c(1,2)]

for (i in 1:ncol(DJIA_ex_ret)) {
  model <- lm(DJIA_ex_ret[,i] ~ SP500_ex)
  betas_stocks[i] <- model$coefficients[2]
}
names(betas_stocks) <- names(DJIA_ex_ret)

#PLOT THE CAPITAL MARKET LINE AND THE SECURITY MARKET LINE

#for the first plot we need the expected return and SD of the risk-free rate, the stocks and the S&P 500

#we create a vector with the mean risk-free rate and one with the mean market excess return
rfree_mean <- rep(mean(DJIA$Rfree),length(betas_stocks))
SP500_ex_mean <- rep(mean(SP500_ex),length(betas_stocks))

#then we apply the CAPM formula for the expected return
E_ret_stocks <- rfree_mean + betas_stocks*SP500_ex_mean

#we still need the expected return of the risk-free rate and of the SP500
#the first is equal to the mean risk-free rate
E_ret_rfree <- mean(DJIA$Rfree)
names(E_ret_rfree) <- "T-bills"
#the second is equal to the mean return of the SP500
E_ret_SP500 <- mean(SP500$SP500)
names(E_ret_SP500) <- "S&P 500"

#finally we join all the expected returns in one object
E_ret <- c(E_ret_rfree,E_ret_SP500,E_ret_stocks)

#now we compute the SD of the stocks, T-bills and S&P 500
DJIA_sd <- apply(DJIA[,-c(1,2)], 2, sd)
rfree_sd <- sd(DJIA$Rfree)
names(rfree_sd) <- "T-bills"
SP500_sd <- sd(SP500$SP500)
names(SP500_sd) <- "S&P 500"
#and we join them together
sds <- c(rfree_sd,SP500_sd,DJIA_sd)

plot(sds, E_ret, pch=19, xlab="Standard deviation", ylab="Expected return", main="Capital Market Line")
#we color in blue the S&P 500 ("market portfolio") and in green the T-bill ("risk-free")
points(SP500_sd, E_ret_SP500, pch=19, col="blue3")
points(rfree_sd, E_ret_rfree, pch=19, col="green4")
#now we plot the capital market line
abline(coef = c(coef(lm(c(E_ret_rfree, E_ret_SP500) ~ c(rfree_sd, SP500_sd)))), lwd=2, col="orange2")

#to plot the security market line we replace the standard deviation with the betas
#the beta of the risk-free is 0 and the one of the one of the market is 1
betas <- c(0,1,betas_stocks)
names(betas)[1] <- "T-bills"
names(betas)[2] <- "S&P 500"

plot(betas, E_ret, pch=19, xlab="Beta", ylab="Expected return", main="Security Market Line")
#we color in blue the S&P 500 ("market portfolio") and in green the T-bill ("risk-free")
points(1, E_ret_SP500, pch=19, col="blue3")
points(0, E_ret_rfree, pch=19, col="green4")
#now we plot the security market line
abline(coef = c(coef(lm(c(E_ret_rfree, E_ret_SP500) ~ c(0, 1)))), lwd=2, col="orange2")