Mean-Variance Portfolio Selection with Either a Singular or Nonsingular Variance-Covariance Matrix

Introduction 1 In derivations of the mean-variance model of portfolio selection, authors from Markowitz [6 and 7] and Tobin [11] to Merton [8] and Black [1] rely on the inverse of the matrix of variances and covariances for the returns on risky securities. Unfortunately, as is shown in this paper, such an inverse does not exist when risk-free combinations can be formed from the risky securities. Accordingly, the general validity of the mean-variance model is challenged by the existence of opportunities for hedging, including those fostered by short sales and the rapidly expanding markets for warrants, options, and futures. Fortunately, the mean-variance model is tractable even when the conventional methods for deriving it fail. Alternative solution procedures presented in this paper are valid with or without riskless securities and with either singular or nonsingular variance-covariance matrices. The important properties of the mean-variance model are shown to extend for the previously omitted cases. In particular, the frontier of mean-variance combinations is always well-defined, is always strictly convex, and (the efficient portion of the frontier) is always positively sloped. In addition, the frontier of mean-variance combinations 2 always can be expressed in terms of a pair of mutual funds which are determined on purely technical grounds.