Linear Algebra for Quantitative Finance › A Complete Example: Minimum Variance Portfolio

Let's bring together the linear algebra concepts to solve a standard portfolio optimization problem: finding the portfolio with minimum variance subject to being fully invested.

The optimization problem is:

$$\min_{\mathbf{w}} \mathbf{w}^T \mathbf{\Sigma} \mathbf{w}$$

subject to:

$$\mathbf{1}^T \mathbf{w} = 1$$

where $\mathbf{1}$ is a vector of ones (enforcing that weights sum to 1, meaning the portfolio is fully invested).

This problem asks which portfolio, among all portfolios with net weights summing to 100%, has the smallest variance. The objective function is the quadratic form we encountered earlier, portfolio variance expressed in matrix notation. The equality constraint fixes net investment but still allows short positions and gross leverage. Nonnegative weights, $w_i\geq0$, prohibit shorts and give gross exposure of one under this net budget. A gross-exposure cap above one limits leverage and shorts but does not prohibit them; a cap of one together with the budget equality excludes both.

The following closed form assumes that $\Sigma$ is symmetric positive definite, so $\Sigma^{-1}$ exists and the minimizer is unique. If $\Sigma$ is singular and merely positive semi-definite, the inverse is undefined and minimizers may be non-unique; use a generalized-inverse, regularized, or constrained formulation instead. Simply replacing the inverse with a pseudoinverse in this normalized expression need not solve the constrained problem. A direction $\mathbf{v}$ with $\Sigma\mathbf{v}=0$ is a nullspace direction. If its coordinates sum to a nonzero value, normalizing by that sum gives budget-one weights with zero variance. Under the positive-definite assumption, Lagrange multipliers give:

$$\mathbf{w}^* = \frac{\mathbf{\Sigma}^{-1} \mathbf{1}}{\mathbf{1}^T \mathbf{\Sigma}^{-1} \mathbf{1}}$$

where:

• $\mathbf{w}^*$: the optimal weight vector that minimizes portfolio variance
• $\mathbf{\Sigma}^{-1}$: the inverse of the covariance matrix
• The numerator $\mathbf{\Sigma}^{-1} \mathbf{1}$ jointly accounts for all variances and covariances; it does not guarantee positive weights or an asset-by-asset ordering
• The denominator $\mathbf{1}^T \mathbf{\Sigma}^{-1} \mathbf{1}$ normalizes these weights to sum to 1

The appearance of $\mathbf{\Sigma}^{-1}$ in the solution formula is typical of quadratic optimization with linear constraints. Intuitively, the inverse covariance matrix reweights assets to account for correlation. An asset that seems low-variance in isolation might contribute more risk than expected if it's highly correlated with other holdings. Under the chosen covariance estimate and net-budget constraint, this adjustment identifies the model's unconstrained minimum-variance allocation; estimation error or practical constraints can change the implementable result.