Correlation Matrices with the Perron-Frobenius Property

The first principal component of stock returns is often identified with the market factor. If this portfolio is to represent the market portfolio, then all its weights must be positive. From the classical Perron-Frobenius theorem, a sufficient condition for the dominant eigenvector to be positive is that all the off diagonal elements are positive. Stock return correlation matrices typically contain negative elements and the frequency of negative elements has varied during the last 20 years. However, it is possible for a correlation matrix with some negative elements to have a positive dominant eigenvector. This paper explores the conditions under which the dominant eigenvector of a correlation matrix has strictly positive weights.