Portfolio Optimization, Tangency Portfolio, CML with shorts not allowed.

The functions in the previous post have been updated to put in the additional constraints that no portfolio weights can be negative, that is short selling is not allowed.

setwd("C:/Users/admin/Desktop/Data Analytics/QuantitativeFinance/QFLabs")
getwd()

library(ggplot2)
library(quadprog)

GetData20 <- function(CSVfile){
  baseData <- read.csv(CSVfile)
  N0 <- ncol(baseData)
  Close <- baseData[, 3:N0]
  logRet1 <- log(head(Close, -1) / tail(Close, -1))
  return(logRet1)
}

PlotData20 <- function(logRet1){
  Retn <- colMeans(logRet1)
  Risk <- diag(var(logRet1))
  RiskReturn <- as.data.frame(t(rbind(Retn,Risk)))
  plot1 <- ggplot(data = RiskReturn,aes(x = Risk, y = Retn) )
  plot1 <- plot1 + geom_point()
  plot1 <- plot1 + xlab("Risk / Variance") + ylab("Daily Returns") + ggtitle("Risk/Returns")
  plot1
}

rOptPort20 <- function(hRets,pRet, Shorts = TRUE){
  Dmat <- 2*cov(hRets)
  dvec <- rep(0,ncol(hRets))
  if (Shorts){
    ##message("Shorts : TRUE")    ## Shorts are allowed
    Amat <- cbind(rep(1,ncol(hRets)),colMeans(hRets))
    bvec <- c(1,pRet)
  } else {
    ##message("Shorts = FALSE")    ## Shorts are not allowed
    Amat <- cbind(rep(1,ncol(hRets)),colMeans(hRets),diag(1,nrow = ncol(hRets)))
    bvec <- c(1,pRet,rep(0,ncol(hRets)))
  }
  result = tryCatch({
    solve.QP(Dmat = Dmat, dvec = dvec, Amat = Amat, bvec = bvec, meq =2)
  }, warning = function(w){
    message("warning : ",pRet)
    return(NULL)
  }, error = function(w){
    message("infeasible : ", pRet)
    return(NULL)
  }, finally = {
  }
  )
  if (!is.null(result)){
    wP <- result$solution
    varP <- result$value
    
  }
  else{
    wP <- "error"
    varP <- "error"
  }
  retList <- list(wP,varP)
  names(retList) <- c("wP","varP")
  return(retList)
}

EFMinVar20 <- function(hRets, minRet, maxRet,Shorts = FALSE){
  smuP <- seq(minRet,maxRet,length=50)
  svarP <- sapply(smuP,function(x) rOptPort20(hRets,x,Shorts)$varP)
  EffF <- as.data.frame(cbind(smuP,svarP))
  EffF0 <- as.data.frame(EffF[EffF$svarP != "error",])
  EffF0 <- as.data.frame(apply(EffF0, 2, FUN = function(x) as.numeric(as.character(x))))
  minVar <- min(EffF0$svarP)
  L <- EffF0$svarP == minVar
  minRet <- EffF0[L,]$smuP
  minPoint <- as.data.frame(cbind(minRet,minVar))
  minVarwP <- rOptPort20(hRets,minRet,Shorts)$wP
  rList <-list(EffF0,minPoint,minVarwP)
  names(rList) <- c("EFF","minPoint","wP")
  return(rList)
}

EFMinVar20Plot <- function(list1){
  
  ealred  <- "#7D110C"
  
  plot2 <- ggplot(data = list1$EFF,aes(x = svarP, y = smuP) )
  plot2 <- plot2 + geom_point()
  plot2 <- plot2 + geom_point(data = list1$minPoint, aes(x = minVar,y = minRet),color = "red", size=3)
  plot2 <- plot2 + xlab("Variance") + ylab("Returns") + ggtitle("Efficient Frontier - MinVar")
  plot2
}

EFSharpe20 <- function(hRets, minRet, maxRet,RF, Shorts = FALSE){
  smuP <- seq(minRet,maxRet,length=50)
  svarP <- sapply(smuP,function(x) rOptPort20(hRets,x,Shorts)$varP)
  EffF <- as.data.frame(cbind(smuP,svarP))
  EffF0 <- as.data.frame(EffF[EffF$svarP != "error",])
  EffF0 <- as.data.frame(apply(EffF0, 2, FUN = function(x) as.numeric(as.character(x))))
  sharpe <- (EffF0$smuP-RF)/EffF0$svarP
  EFF <- as.data.frame(cbind(EffF0,sharpe,RF))
  L <- EFF$sharpe == max(EFF$sharpe)
  maxSharpe <- EFF[L,]
  wTP <- rOptPort20(hRets,maxSharpe$smuP,Shorts)$wP
  rList <-list(EFF,maxSharpe,wTP)
  names(rList) <- c("EFF","maxSharpe","wTP")
  return(rList)
}

EFSharpe20Plot <- function(list1){
  
  
  plot2 <- ggplot(data = list1$EFF,aes(x = svarP, y = smuP) )
  plot2 <- plot2 + geom_point()
  plot2 <- plot2 + geom_point(data = list1$maxSharpe, aes(x = svarP,y = smuP),colour = "red", pch =24, size=3)
  plot2 <- plot2 + geom_point(data = list1$maxSharpe, aes(x = 0,y = RF),color = "red", pch =24, size=3)
  plot2 <- plot2 + xlab("Variance") + ylab("Returns") + ggtitle("Efficient Frontier - Sharpe")
  plot2 <- plot2 + geom_abline(intercept = (list1$maxSharpe)$RF, slope = (list1$maxSharpe)$sharpe, colour = "red")
  plot2
}

#Testing

logRet <- GetData20("US5.csv")
##logRet <- GetData20("India5.csv")
PlotData20(logRet)

## Shorts = FALSE

z10 <- EFMinVar20(logRet,-0.0005,0.0002, Shorts = FALSE)
z10$wP
cminRet <- (z10$minPoint)$minRet
cminRet
z11 <- rOptPort20(logRet,cminRet,Shorts = FALSE)
z11$wP
EFMinVar20Plot(z10)
cminRet <- (z10$minPoint)$minRet
maxR <- cminRet + max(0.001,cminRet+ 2*abs(cminRet))
z10a <- EFMinVar20(logRet,cminRet, maxR, Shorts = FALSE)
EFMinVar20Plot(z10a)

z12 <- EFSharpe20(logRet,cminRet,maxR,0.0001,Shorts = FALSE)
z12$wTP
sum(z12$wTP)
(z12$maxSharpe)$smuP
(z12$maxSharpe)$svarP
maxSharpeRet <- (z12$maxSharpe)$smuP
EFSharpe20Plot(z12)

After the data is downloaded from the web and stored in  CSV file as explained in an earlier post. define the functions and test the program as follows :

> logRet <- GetData20("US5.csv")
> ##logRet <- GetData20("India5.csv")
> PlotData20(logRet)

Unlike the case were shorts are allowed, in this case, not every desired portfolio return is feasibile when shorts are not allowed. Here we search through a range of returns from -0.0005 to + 0.0002 and detect which return gives us the minimum variance. The means that are infeasible are flagged as sucn and ignored.
> z10 <- EFMinVar20(logRet,-0.0005,0.0002, Shorts = FALSE)
infeasible : -5e-04
.....
infeasible : -0.000242857142857143

Dataframe z10 contains the range of feasible returns and the corresponding variance. The portofolio weights are printed out as follows. The first one though technically negative is very, very small.
> z10$wP
[1] -1.386988e-17  4.295438e-02  1.356919e-01  4.977252e-01  3.236285e-01

we note the return achieved at the point of minimum variance
> cminRet <- (z10$minPoint)$minRet
> cminRet
[1] 0.0001285714

> EFMinVar20Plot(z10)

The efficient frontier is plotted, but this is not very useful because the range of returns that we have taken are "below" the minvariance point.

So in the next few lines, we have defined a range of returns that goes above, or higher than, the return for minimum variance

> cminRet <- (z10$minPoint)$minRet
> maxR <- cminRet + max(0.001,cminRet+ 2*abs(cminRet))
> z10a <- EFMinVar20(logRet,cminRet, maxR, Shorts = FALSE)
infeasible : 0.000761224489795919
.......
infeasible : 0.00112857142857143
> EFMinVar20Plot(z10a)

Next we identify the maximum Sharpe ratio, identify the tangency portfolio and draw the capital market line. Note that the risk free rate is 0.0001

> z12 <- EFSharpe20(logRet,cminRet,maxR,0.0001,Shorts = FALSE)
infeasible : 0.000761224489795919
....
infeasible : 0.00112857142857143
> z12$wTP
[1]  2.515328e-01  0.000000e+00 -1.456853e-17  7.484672e-01 -6.938894e-18
> sum(z12$wTP)
[1] 1
> (z12$maxSharpe)$smuP
[1] 0.0005163265
> (z12$maxSharpe)$svarP
[1] 0.0002621069
> maxSharpeRet <- (z12$maxSharpe)$smuP
> EFSharpe20Plot(z12)

Please note that by setting SHORTS = TRUE, this set of functions should be able to recreate the results given in the previous post.

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These programs can be run on StatAce, the free hosted R environment and get the same result.