Constrained portfolio optimisation with quadratic solving

Question

I'm trying to grapple with portfolio optimisation in R and I'm struggling to find consensus about how to code constraints etc using solve.QP.

Essentially I have already calculated the covariance matrix of returns of n assets, which I understand to be Dmat, however, I'm unsure on the rest. I would like my constraints to purely be that all weights sum to 1 and they must all be below a certain size (max.allocation) in absolute value, i.e.

|x_i|<max.allocation<=1.

Am I right in thinking, therefore, that my Amat should be:

Amat <- matrix(1,nrow=n)
Amat <- cbind(Amat, -diag(n))

Also, should my bvec be:

bvec <- 1
bvec <- c(bvec, rep(-max.allocation,n))

Am I right to believe that since only one constraint is an equality that meq=1? And finally, my research seems to point to:

dvec <- rep(0,n)

But if this is the case, where do the mean returns of the assets come into the problem?

All I am trying to do is solve for portfolio weights and minimise portfolio standard deviation for a given portfolio return, not plot the efficient frontier.

I've already seen the economistatlarge post and its got me to where I currently am. Apologies as I am new to this solving technique, but any help to simplify and clear up what my parameters need to be would be of great help.

Thanks.

I'm not familiar with this package, so this is tentative, but: I think you need to multiply matrix(1,nrow=n) by max.allocation and prepend a column of [1,rep(-1,n)] in Amat. And if you want to minimize for a portfolio return, I think you need to use the mean returns to express the portfolio return in terms of allocation, and then include that as a constraint. Also, since you're using linear equations, you'll probably have to minimize variance rather than std (which is equivalent, but the set-up will be slightly different). A link to the economistatlarge post might help. — Acccumulation

Erwin Kalvelagen Erwin Kalvelagen · Accepted Answer · 2017-10-07T09:07:31

If you want to add

|x(i)| <= U

then in quadprog you need to split this into:

-x(i) >= -U
 x(i) >= -U

so you end up with 2 additional diagonal structures in Amat.

Constrained portfolio optimisation with quadratic solving

1 Answers