Think of a set R of T historical returns on k assets so that R T × k. There are, at best, k independent sources of variation in this set of returns. If, for example, one of the asset's returns can be expressed as a linear combination of another asset's returns, then there are only k – 1 independent sources of variation. In that case, the k × k covariance matrix would not be invertible since it has less than full rank; its rank is k – 1. Without full rank, we could not solve for the minimum variance portfolio. The reason is that we are asking too much of our data: it can provide us with only k – 1 weights, not k. At the other extreme, if the asset returns are independent, then there are k independent sources of variation (called factors) and the covariance matrix has full rank, whose diagonal elements contain variances of the individual asset returns and zeros otherwise.
The idea of the diagonal covariance matrix is intriguing for two reasons: first, it would suggest that returns are independent and that has important implications for diversification. Second, it simplifies the math; for example, solving the minimum variance portfolio is a matter of ranking the individual variances (or Sharpe ratios), giving highest weight to the lowest variance (highest Sharpe ratio).
Principal components analyzes the covariance properties of a set of returns or factors and finds ...