Covariance Matrices and Linear Combinations

Variances on the diagonal, covariances off it. Symmetric and positive semi-definite.

The one formula that matters

For portfolio weights $\mathbf{w}$:

This quadratic form collapses an entire book into a single risk number, and it is the foundation of portfolio risk management, optimisation and VaR.

Expanded, it says total variance is the sum of individual variances plus every pairwise covariance term. The cross terms are why diversification works and why it can fail: they can reduce total risk substantially, or amplify it when everything is positively correlated.

Positive semi-definiteness

$\mathbf{w}^\top\boldsymbol{\Sigma}\mathbf{w} \geq 0$ for every $\mathbf{w}$, since a variance cannot be negative.

This is not a technicality. An estimated matrix that violates it, which happens easily with missing data or inconsistent estimation windows, implies some portfolio has negative variance. Optimisers exploit such portfolios enthusiastically, producing enormous positions in what the model believes is risk-free. Any risk system must check and repair this.

Estimation is the hard part

With $n$ assets there are $\frac{n(n+1)}{2}$ parameters to estimate. For 500 stocks that is over 125,000 numbers, from a return history that may be a few hundred observations.

The consequence is severe. With fewer observations than assets the sample matrix is singular, and even with more it is dominated by noise. Optimisers then load into whichever spurious low-correlation pair the noise created, which is why naive mean-variance optimisation produces such unstable and extreme portfolios.

The fixes

Factor models impose structure. Rather than estimating every pairwise covariance, model returns as driven by a few common factors:

Now you estimate factor loadings and a small factor covariance matrix instead of thousands of pairs. This is the standard approach and it is why factor models are ubiquitous.

Shrinkage blends the noisy sample estimate toward a simple structured target, trading a little bias for a large variance reduction. The same bias-variance logic that justifies ridge regression.

Longer windows reduce noise but assume stability that markets do not offer, so this trades one error for another.