On the use of the Moore-Penrose generalized inverse in the portfolio optimization problem

When the number of assets (N) exceeds the number of time periods (T), the sample covariance matrix is singular, and the portfolio optimization problem cannot be solved via traditional mean-variance algebra. In such a case, the Moore-Penrose (MP) generalized inverse becomes handy: In this paper, we critically examine the MP solution of the portfolio optimization problem. Our findings include: i) the MP solution leads to a portfolio of “pseudo-riskfree composite assets”; ii) it is orthogonal to principal components, iii) most importantly, it is poorly diversified. We illustrate our findings using equity market data.