using quadprog for portfolio optimization

Question

I have my input parameters mu (mean vector μ), Q (covariance matrix Q), and tau (risk tolerance τ) and I need to return the vector h (asset weights) that maximizes the following utility function U defined by:

U(h)= −1/2h^T*Q*h + τ*h^T*μ

subject to constraints:

0 ≤ h ≤ 0.1 for all h
and sum of all h is equal to 1: h^T*e = 1

TAU contains numbers from zero to 0.5 in steps of 0.001. How do I define the parameters: Dmat, dvec, Amat and bvec for this problem? I know the finance concepts but not how to program it correctly.

Thank you

This doesn't work as I still have negative weights indicative of short selling :(

frontieropti <- c(NULL)
i <- 1
for (i in 1:nrow(TAU)){
  solQP <- solve.QP(Dmat,TAU[i]*mu, Amat, bvec, meq = 1, factorized = F)
  sol <- c(i,solQP$value)
  frontieropti <- rbind(frontieropti,sol)
  i <- i +1
}
solQP <- solve.QP(Dmat, TAU[1]*mu, Amat, bvec, meq = 1, factorized = F)
solQP

user_7015675 · Answer

Setting up Amat:

na <- 5  ## number of assets

I use only 5 assets and a maximum weight of 40% so that I can show the resulting matrices:

wmin <- 0
wmax <- 0.4
A <- rbind(1,-diag(na), diag(na))
bvec <- c(1, -rep(wmax, na), rep(wmin, na))
cbind(A, bvec)
##                      bvec
##  [1,]  1  1  1  1  1  1.0
##  [2,] -1  0  0  0  0 -0.4
##  [3,]  0 -1  0  0  0 -0.4
##  [4,]  0  0 -1  0  0 -0.4
##  [5,]  0  0  0 -1  0 -0.4
##  [6,]  0  0  0  0 -1 -0.4
##  [7,]  1  0  0  0  0  0.0
##  [8,]  0  1  0  0  0  0.0
##  [9,]  0  0  1  0  0  0.0
## [10,]  0  0  0  1  0  0.0
## [11,]  0  0  0  0  1  0.0

Note that the first row of Amat is for the budget constraint, so you need to set argument meq to 1. Also, solve.QP wants the transpose of Amat, i.e. t(Amat).

So here would be a complete example:

library("quadprog")
library("NMOF")

I start by creating some random data for 30 assets.

na <- 30
R <- randomReturns(na = na, ns = 120, rho = 0.5, sd = 0.03)
mu <- colMeans(R)
V <- cov(R)


wmin <- 0
wmax <- 0.1
A <- rbind(1,-diag(na), diag(na))
b <- c(1, -rep(wmax, na), rep(wmin, na))

TAU <- seq(0, 0.5, by = 0.01) ## choose an appropriate stepsize

It is good practice to initialise data structures before the loop, and not "grow" them. (Even though it does not matter much in this example.)

results <- numeric(length(TAU))
weights <- array(NA, dim = c(na, length(TAU)))
for (i in seq_along(TAU)) {
    solQP <- solve.QP(Dmat = V,
                      dvec = TAU[i]*mu,
                      Amat = t(A),
                      bvec = b, meq = 1)

    ## an equivalent computation    
    ## NMOF::mvPortfolio(mu, V, wmax = 0.1, lambda = c(TAU[i], 0.5))

    results[i] <- solQP$value
    weights[, i] <- solQP$solution
}

Note that, because of round-off error, some results may be negative. So round the results to 8 decimal places, say.

weights <- round(weights, 8)
barplot(weights)

using quadprog for portfolio optimization

1 Answers