Portfolio Optimisation: What the Maths Can—and Cannot—Decide

The maths needs a question

Portfolio optimisation finds weights that best meet a specified mathematical objective while satisfying constraints. The objective might concern an estimated risk measure; the constraints define what combinations are allowed. The output answers that formal question. It cannot decide whether the question captures everything a person or institution cares about.

Here the objective is minimum variance, which means minimising the model’s estimated dispersion of portfolio returns. Volatility is the square root of variance. Neither measure describes every kind of risk: liquidity, permanent loss, changing relationships and extreme events can require separate consideration. Calling a portfolio minimum variance is meaningful only relative to its inputs and constraints.

We use two fictional assets, A and B. Weights must be non-negative and sum to one: the model is long-only, fully invested and includes no borrowing or cash alternative. Assume zero correlation between the assets’ returns. Zero correlation removes their covariance term in this calculation; it does not mean the assets are independent or cannot fall together.

A precise answer can depend on uncertain inputs

The model shifts weight away from A because A now has a higher estimated variance. It does not know anything about company quality, future returns or an investor’s circumstances. Expected returns do not enter this particular objective, so the output cannot support a claim that the revised portfolio offers a better return.

The revised minimum volatility is higher than before even though the weights adapt. That is consistent: changing an input changes the problem itself. The new weights minimise the revised variance function among the permitted combinations; they do not restore the old risk estimate or prove that realised risk will match either estimate.

The CVX portfolio mathematics material explains weights, covariance and constrained mean–variance models. Our example uses the narrower minimum-variance objective and a closed-form two-asset calculation. No software solver is needed to follow it. Different objectives or constraints would define different optimisation problems and could produce different weights.

What remains outside the worksheet

Volatility and correlation are estimated rather than fixed properties. Different samples, measurement periods and changing market conditions can alter them. A zero correlation assumption is especially strong: positive co-movement would generally raise variance for a given positive-weight combination compared with this example. The optimisation cannot eliminate uncertainty in the inputs by calculating more decimal places.

Trading costs, taxes, liquidity limits and restrictions on particular holdings are omitted. So are return forecasts and losses outside what variance usefully summarises. A mathematically feasible solution can therefore be impractical or inappropriate outside the model. The educational task is to identify those omissions and examine sensitivity, rather than treat the smallest calculated number as a complete decision.