Portfolio Optimisation under weight constraints

Question

With a lot of help from contributors to StackOverflow I have managed to put together a function to derive the weights of a 2-asset portfolio which maximises the Sharpe ratio. No short sales are allowed and the sum of weights add to 1. What I would like to do now is to constrain asset A to not being more or less than 10% from a user defined weight. As an example I would like to constrain the weight of asset A to be no less than 54% or more than 66% (i.e 60% +/- 10%). So on the below example I would end up with weights of (0.54,0.66) instead of the unsconstrained (0.243,0.7570) .I assume this can be done by tweaking bVect but I am not so sure how to go about it. Any help would be appreciated.

asset_A <- c(0.034320510,-0.001209628,0.031900161,0.023163947,-0.001872938,-0.010322489,0.006090395,-0.003270854,0.017778990,0.017204915) 

asset_B <- c(0.047103261,0.055175057,0.021019816,0.020602347,0.007281368,-0.006547404,0.019155238,0.005494798,0.025429958,0.014929124)

     require(quadprog)

     HR_solve   <- function(asset_A,asset_B) {
     vol_A    <-  sd(asset_A)
     vol_B   <-  sd(asset_B)
     cor_AB  <-   cor(cbind(asset_A,asset_B),method="pearson")
     ret_A_B      <- as.matrix(c(mean(asset_A),mean(asset_B)))

      vol_AB <- c(vol_A,vol_B)
      covmat <- diag(as.vector(vol_AB))%*%cor_AB%*%diag(as.vector(vol_AB))

      aMat <- cbind(rep(1,nrow(covmat)),diag(1,nrow(covmat)))
      bVec  <- c(1,0,0)
      zeros <- array(0, dim = c(nrow(covmat),1))
      minw <-  solve.QP(covmat, zeros, aMat, bVec, meq = 1 ,factorized = FALSE)$solution
      rp <- as.numeric(t(minw) %*% ret_A_B)
      sp <- sqrt(t(minw) %*% covmat %*% minw)
      wp <- t(matrix(minw))

      sret <- sort(seq(t(minw) %*% ret_A_B,max(ret_A_B),length.out=100))
      aMatt <- cbind(ret_A_B,aMat)

      for (ri in sret[-1]){
      bVect  <- c(ri,bVec)
      result <-  tryCatch({solve.QP(covmat, zeros, aMatt, bVect, meq = 2,factorized = FALSE)},
                            warning = function(w){ return(NULL) } , error = function(w){ return(NULL)}, finally = {} )
     if (!is.null(result)){
     wp <-  rbind(wp,as.vector(result$solution))
     rp <-c(rp,t(as.vector(result$solution) %*% ret_A_B))
     sp <- c(sp,sqrt(t(as.vector(result$solution)) %*% covmat %*% as.vector(result$solution))) }

     }

      HR_weights <- wp[which.max(rp/sp),]
      as.matrix(HR_weights)
    }


HR_solve(asset_A,asset_B)


          [,1]
[1,] 0.2429662
[2,] 0.7570338

Answer 1

Ok I have found a way to do this... if you think there is a more elegant way please let me know...

 require(quadprog)

 HR_solve   <- function(asset_A,asset_B,mean_A,range_A) {
 vol_A    <-  sd(asset_A)
 vol_B   <-  sd(asset_B)
 cor_AB  <-   cor(cbind(asset_A,asset_B),method="pearson")
 ret_A_B      <- as.matrix(c(mean(asset_A),mean(asset_B)))

 vol_AB <- c(vol_A,vol_B)
 covmat <- diag(as.vector(vol_AB))%*%cor_AB%*%diag(as.vector(vol_AB))
 bVec  <-  c(1,0,0)

aMat <- cbind(rep(1,nrow(covmat)),diag(1,nrow(covmat)))

zeros <- array(0, dim = c(nrow(covmat),1))
minw <-  solve.QP(covmat, zeros, aMat, bVec, meq = 1 ,factorized = FALSE)$solution
rp <- as.numeric(t(minw) %*% ret_A_B)
sp <- sqrt(t(minw) %*% covmat %*% minw)
wp <- t(matrix(minw))

sret <- sort(seq(t(minw) %*% ret_A_B,max(ret_A_B),length.out=1000))

min_A <- mean_A * (1-range_A)
max_A <- mean_A * (1+range_A)

aMatt <- cbind(ret_A_B,aMat,-diag(2))
bVec  <- c(1,min_A,0,-max_A,-1)




for (ri in sret[-1]){
bVect  <- c(ri,bVec)


result <-  tryCatch({solve.QP(covmat, zeros, aMatt, bVect, meq = 2,factorized = FALSE)},
                        warning = function(w){ return(NULL) } , error = function(w){ return(NULL)}, finally = {} )
if (!is.null(result)){
wp <-  rbind(wp,as.vector(result$solution))
rp <-c(rp,t(as.vector(result$solution) %*% ret_A_B))
sp <- c(sp,sqrt(t(as.vector(result$solution)) %*% covmat %*% as.vector(result$solution))) }
}

HR_weights <- wp[which.max(rp/sp),]
as.matrix(HR_weights)
 }

Answer 2

I think you should take a look at the link below.

http://economistatlarge.com/portfolio-theory/r-optimized-portfolio/r-code-graph-efficient-frontier

I think you'll learn a lot from that. I'll post the code here, in case the link gets shut down sometime in the future.

# Economist at Large
# Modern Portfolio Theory
# Use solve.QP to solve for efficient frontier
# Last Edited 5/3/13

# This file uses the solve.QP function in the quadprog package to solve for the
# efficient frontier.
# Since the efficient frontier is a parabolic function, we can find the solution
# that minimizes portfolio variance and then vary the risk premium to find
# points along the efficient frontier. Then simply find the portfolio with the
# largest Sharpe ratio (expected return / sd) to identify the most
# efficient portfolio

library(stockPortfolio) # Base package for retrieving returns
library(ggplot2) # Used to graph efficient frontier
library(reshape2) # Used to melt the data
library(quadprog) #Needed for solve.QP

# Create the portfolio using ETFs, incl. hypothetical non-efficient allocation
stocks <- c(
 "VTSMX" = .0,
 "SPY" = .20,
 "EFA" = .10,
 "IWM" = .10,
 "VWO" = .30,
 "LQD" = .20,
 "HYG" = .10)

# Retrieve returns, from earliest start date possible (where all stocks have
# data) through most recent date
returns <- getReturns(names(stocks[-1]), freq="week") #Currently, drop index

#### Efficient Frontier function ####
eff.frontier <- function (returns, short="no", max.allocation=NULL,
 risk.premium.up=.5, risk.increment=.005){
 # return argument should be a m x n matrix with one column per security
 # short argument is whether short-selling is allowed; default is no (short
 # selling prohibited)max.allocation is the maximum % allowed for any one
 # security (reduces concentration) risk.premium.up is the upper limit of the
 # risk premium modeled (see for loop below) and risk.increment is the
 # increment (by) value used in the for loop

 covariance <- cov(returns)
 print(covariance)
 n <- ncol(covariance)

 # Create initial Amat and bvec assuming only equality constraint
 # (short-selling is allowed, no allocation constraints)
 Amat <- matrix (1, nrow=n)
 bvec <- 1
 meq <- 1

 # Then modify the Amat and bvec if short-selling is prohibited
 if(short=="no"){
 Amat <- cbind(1, diag(n))
 bvec <- c(bvec, rep(0, n))
 }

 # And modify Amat and bvec if a max allocation (concentration) is specified
 if(!is.null(max.allocation)){
 if(max.allocation > 1 | max.allocation <0){
 stop("max.allocation must be greater than 0 and less than 1")
 }
 if(max.allocation * n < 1){
 stop("Need to set max.allocation higher; not enough assets to add to 1")
 }
 Amat <- cbind(Amat, -diag(n))
 bvec <- c(bvec, rep(-max.allocation, n))
 }

 # Calculate the number of loops
 loops <- risk.premium.up / risk.increment + 1
 loop <- 1

 # Initialize a matrix to contain allocation and statistics
 # This is not necessary, but speeds up processing and uses less memory
 eff <- matrix(nrow=loops, ncol=n+3)
 # Now I need to give the matrix column names
 colnames(eff) <- c(colnames(returns), "Std.Dev", "Exp.Return", "sharpe")

 # Loop through the quadratic program solver
 for (i in seq(from=0, to=risk.premium.up, by=risk.increment)){
 dvec <- colMeans(returns) * i # This moves the solution along the EF
 sol <- solve.QP(covariance, dvec=dvec, Amat=Amat, bvec=bvec, meq=meq)
 eff[loop,"Std.Dev"] <- sqrt(sum(sol$solution*colSums((covariance*sol$solution))))
 eff[loop,"Exp.Return"] <- as.numeric(sol$solution %*% colMeans(returns))
 eff[loop,"sharpe"] <- eff[loop,"Exp.Return"] / eff[loop,"Std.Dev"]
 eff[loop,1:n] <- sol$solution
 loop <- loop+1
 }

 return(as.data.frame(eff))
}

# Run the eff.frontier function based on no short and 50% alloc. restrictions
eff <- eff.frontier(returns=returns$R, short="no", max.allocation=.50,
 risk.premium.up=1, risk.increment=.001)

# Find the optimal portfolio
eff.optimal.point <- eff[eff$sharpe==max(eff$sharpe),]

# graph efficient frontier
# Start with color scheme
ealred <- "#7D110C"
ealtan <- "#CDC4B6"
eallighttan <- "#F7F6F0"
ealdark <- "#423C30"

ggplot(eff, aes(x=Std.Dev, y=Exp.Return)) + geom_point(alpha=.1, color=ealdark) +
 geom_point(data=eff.optimal.point, aes(x=Std.Dev, y=Exp.Return, label=sharpe),
 color=ealred, size=5) +
 annotate(geom="text", x=eff.optimal.point$Std.Dev,
 y=eff.optimal.point$Exp.Return,
 label=paste("Risk: ",
 round(eff.optimal.point$Std.Dev*100, digits=3),"\nReturn: ",
 round(eff.optimal.point$Exp.Return*100, digits=4),"%\nSharpe: ",
 round(eff.optimal.point$sharpe*100, digits=2), "%", sep=""),
 hjust=0, vjust=1.2) +
 ggtitle("Efficient Frontier\nand Optimal Portfolio") +
 labs(x="Risk (standard deviation of portfolio)", y="Return") +
 theme(panel.background=element_rect(fill=eallighttan),
 text=element_text(color=ealdark),
 plot.title=element_text(size=24, color=ealred))
ggsave("Efficient Frontier.png")

Answer 3

Just change aMat and bVec:

# sset A to be no less than 54% or more than 66%
  aMat <- cbind(rep(1,nrow(covmat)),diag(1,nrow(covmat)),c(1,0),c(-1,0))
  bVec  <- c(1,0,0,.54,-.66)