Multivariate portfolio choice via quantiles

We first show how the quantile approach used for univariate optimal portfolio choice can also be useful when dealing with the multivariate case. Specifically, when a related multivariate risk-sharing problem (in the absence of a financial market) can be solved explicitly, then the multivariate optimal portfolio choice is shown to reduce to a one-dimensional problem that can be dealt with using the quantile approach. We then use this finding to develop an efficient algorithm to determine optimal portfolios. We also develop a numerical approach that makes it possible to obtain approximate solutions for general multivariate portfolio selection problems.