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Mean-Variance Optimization

Choose weights minimising variance for a target return:

The mean-variance problem
min⁡w  w⊤Σwsubject tow⊤μ=μp,  ∑iwi=1\min_w\; w^\top\Sigma w \quad \text{subject to}\quad w^\top\mu = \mu_p,\; \sum_i w_i = 1

Minimise variance subject to hitting a target return, with the weights summing to one.

A quadratic problem with linear constraints, so it has a closed-form solution. Markowitz's insight, and the origin of modern portfolio theory.

The curve is the solution set of the problem above, one point per target return. Optimising is choosing where on it to stand, which is why the inputs matter more than the algorithm.

What makes it work

The whole framework rests on the asymmetry established earlier: expected return is linear in the weights and ignores correlation, while variance is quadratic and depends on it entirely.

That gap is where diversification lives. You can reduce risk without reducing expected return, and the optimiser finds the best available trade.

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