A portfolio can hold many assets and still concentrate most of its risk in a single common movement. To reveal that concentration, a quant needs to turn the asset covariance matrix into uncorrelated factor directions and measure the portfolio's exposure to each. The spectral theorem guarantees that this can be done for every real covariance matrix, even when eigenvalues repeat or some are zero. It is the foundation of principal component analysis (PCA) and of the risk decomposition used in mean–variance portfolio construction.
This lesson states the theorem, derives the portfolio-variance decomposition, and uses a two-asset portfolio to distinguish common-movement risk from spread risk. It also explains what repeated eigenvalues do and do not guarantee about the factors. The full existence proof is in the derivation sub-note.
Learning objectives. By the end of this note you should be able to:
- state the finite-dimensional real spectral theorem in both directions and write its matrix form;
- derive portfolio variance as a sum of factor variances times squared exposures;
- compute and interpret common-movement and spread-risk contributions in a two-asset example;
- distinguish existence of a complete factor basis from uniqueness of individual factors and invertibility of the covariance matrix.
In this lesson
- The risk question before the theorem
- The theorem
- From the theorem to portfolio risk
- A two-asset risk decomposition
- What repeated and zero eigenvalues mean for a risk model
Before you start
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