IEOR 4739 Quick Review of the Principal Components Method

Suppose Q is the covariance matrix for the returns of n assets. Then Q is symmetric (qij = qji for all indices i, j) and positive-semidefinite (v Qv ≥ 0 for any vector v ∈ R – this is denoted Q 0). We want to “explain” Q using a small number of synthetic factors, i.e. factors that are determined from the data itself, as opposed to using economic factors. If r is the number of factors (typically, r will be significantly smaller than n) we get a decomposition of Q of the form