Portfolio Selection with a Rank-deficient Covariance Matrix

In this paper, we consider optimal portfolio selection when the covariance matrix of the asset returns is rank-deficient. For this case, the original Markowitz' problem does not have a unique solution. The possible solutions belong to either two subspaces namely the range- or nullspace of the covariance matrix. The former case has been treated elsewhere but not the latter. We derive an analytical unique solution, assuming the solution is in the null space, that is risk-free and has minimum norm. Furthermore, we analyse the iterative method which is called the discrete functional particle method in the rank-deficient case. It is shown that the method is convergent giving a risk-free solution and we derive the initial condition that gives the smallest possible weights in the norm. Finally, simulation results on artificial problems as well as real-world applications verify that the method is both efficient and stable.