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$value)
frontieropti <- rbind(frontieropti,sol)
i <- i +1
}
solQP <- solve.QP(Dmat, TAU[1]*mu, Amat, bvec, meq = 1, factorized = F)
solQP
Setting up Amat
:
na <- 5 ## number of assets
I use only 5 assets and a maximum weight of 40% so that I can show the resulting matrices:
wmin <- 0
wmax <- 0.4
A <- rbind(1,-diag(na), diag(na))
bvec <- c(1, -rep(wmax, na), rep(wmin, na))
cbind(A, bvec)
## bvec
## [1,] 1 1 1 1 1 1.0
## [2,] -1 0 0 0 0 -0.4
## [3,] 0 -1 0 0 0 -0.4
## [4,] 0 0 -1 0 0 -0.4
## [5,] 0 0 0 -1 0 -0.4
## [6,] 0 0 0 0 -1 -0.4
## [7,] 1 0 0 0 0 0.0
## [8,] 0 1 0 0 0 0.0
## [9,] 0 0 1 0 0 0.0
## [10,] 0 0 0 1 0 0.0
## [11,] 0 0 0 0 1 0.0
Note that the first row of Amat
is for the budget constraint, so you need to set argument meq
to 1. Also, solve.QP
wants the transpose of Amat
, ie t(Amat)
.
So here would be a complete example:
library("quadprog")
library("NMOF")
I start by creating some random data for 30 assets.
na <- 30
R <- randomReturns(na = na, ns = 120, rho = 0.5, sd = 0.03)
mu <- colMeans(R)
V <- cov(R)
wmin <- 0
wmax <- 0.1
A <- rbind(1,-diag(na), diag(na))
b <- c(1, -rep(wmax, na), rep(wmin, na))
TAU <- seq(0, 0.5, by = 0.01) ## choose an appropriate stepsize
It is good practice to initialise data structures before the loop, and not "grow" them. (Even though it does not matter much in this example.)
results <- numeric(length(TAU))
weights <- array(NA, dim = c(na, length(TAU)))
for (i in seq_along(TAU)) {
solQP <- solve.QP(Dmat = V,
dvec = TAU[i]*mu,
Amat = t(A),
bvec = b, meq = 1)
## an equivalent computation
## NMOF::mvPortfolio(mu, V, wmax = 0.1, lambda = c(TAU[i], 0.5))
results[i] <- solQP$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)
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