The Theory and Practice of Investment Management: Asset Allocation, Valuation, Portfolio Construction, and Strategies, Second Edition

APPENDIX: PRINCIPAL COMPONENT ANALYSIS IN FINANCE

Principal component analysis (PCA) is a widely used tool in finance. It is useful not only for estimating factor models as explained in this chapter, but also for extracting a few driving variables in general out of many for the covariance matrix of asset returns. Hence, it is important to understand the statistical intuition behind it. To this end, we provide a simple introduction to it in this appendix.

Perhaps the best way to understand the PCA is to go through an example in detail. Suppose there are two risky assets, whose returns are denoted by and , with covariance matrix

That is, we assume that they have the same variances of 2.05 and covariance of 1.95. Our objective is to find a linear combination of the two assets so that it has a large component in the covariance matrix, which will be clear below. For notation brevity, we assume first that the expected returns are zeros; that is,

and will relax this assumption later in this appendix.

Recall from linear algebra that we call any vector (a₁, a₂)′ satisfying

an eigenvector ...