short-sale constraint with nonpositive-definite matrix in portfolio optimization

Question

I need help about portfolio optimization in R. I have inverted matrix and I want to use it as an input in portfolio optimization. It was non-positive definite before I have handled it. In portfolio selection theory we need inverted matrix I have already have it. The problem for me that to impose a non-negativity constraint (short sale prohibited) on weights. In traditional optimization packages we have to put covariance matrix and it solves. I do not want cope with the covariance matrix by using different technique. I have already inverted matrix. The aim for me to solve portfolio selection codes with putting inverted covariance matrix(precision matrix) under non-negativity constraint (each element of weight vector must be positive) Which method should I use I do not know. If you help me, I would be very happy..

This is my work so far:

I am trying to construct a portfolio weight vector to minimize the variance of the returns.

       w ̂=argmin w'Σ w
s. t.  w'I = 1           #weights sum up to 1                                                 
       w'μ=ρ             #target expected return
       w≥0               #non-negativity(short-sale) constraint

where w is the vector of weights, Σ covariance matrix.

optimization<-function(returns) {
  p <- ncol(x)                    #number of assets
  n <- nrow(x)                    #number of observations
  x <- matrix(data$return,n,assets)
  mean <- colMeans(na.rm=FALSE,x)
  M <- as.integer(10)             #nuber of ports on the eff.front.
  S <- cov(x)                     #covariance matrix
  Rmax<- 0.01                     #max monthly return value
  Dmat   <- solve(S)              #inverse of covariance matrix
  u <- rep(1,p)                   #vector of ones

  These codes are for the Lagrange solutions

  a<- matrix(rep(0,4), nrow=2)    
  a[1,1] <- t(u)%*% Dmat %*%u 
  a[1,2] <- t(mean)%*%Dmat%*%u
  a[2,1] <- a[1,2] 
  a[2,2] <- t(mean)%*%Dmat%*%mean 
  d <- a[1,1]*a[2,2]-a[1,2]*a[1,2] 
  f <- (Dmat%*%(a[2,2]*u-a[1,2]*mean))/d 
  g <- (Dmat%*%(-a[1,2]*u+a[1,1]*mean))/d
  r <- seq(0, Rmax, length=M)
  w <- matrix((rep(0, p*M)), nrow=p)

I tried to find non-negative weights using the codes below:

for(i in 1:M) { w[,i] = f+r[i]*g                    #portfolio weights 
      if (w[,i] <0) {w[,i]=0} else {w[,i]=w[,i]}
    } 

Also, I tried to make a loop using 'while' function in R.

while (w> 0) 
  { for(i in 1:M) { w[,i] = f+r[i]*g }
  print(w)                     
  } 

Unfortunately, I could not get the positive weights. Is there another solution to get positive weights?

Answer 1

Why not just use PortfolioAnalytics, as if your matrix is non positive definite you will have problems using non optimization approaches.

Here is an example taken from my blog: retmat is a matrix of returns

library(PortfolioAnalytics)
moms_portfolio = portfolio.spec(assets=colnames(retmat))
moms_portfolio = add.constraint(portfolio=moms_portfolio,type="full_investment")
moms_portfolio = add.constraint(portfolio=moms_portfolio,type="long_only")
moms_portfolio = add.objective(portfolio=moms_portfolio,name="StdDev",type="risk")

/#Optimize (May want to switch to DEoptim or different optimizer if ROI fails)

optimal_portfolio = optimize.portfolio(retmat,moms_portfolio,optimize_method="ROI")

optimal_portfolio = optimize.portfolio(retmat,moms_portfolio,optimize_method="DEoptim", search_size=5000, trace=TRUE, traceDE=0)

edit 1: Documentation

see here for general optimization with Portfolio analytics: Portfolio Optimization Vignette

Portfolio Optimization With custom moments and objectives