Markowitz mean-variance optimisation

The idea in plain words

Every extra point of expected return has to be paid for with risk. Mean-variance optimisation states the price you are willing to pay, the risk aversion λ, and then finds the weights that give the most return after subtracting that price. A low λ chases return; a high λ hugs safety.

How it is calculated

Maximise the expected return minus λ/2 times the variance, with the weights adding up to 1 (long only by default, each weight capped by the maximum weight you set). The optimiser solves it by projected gradient steps. Default λ = 3, a common middle choice.

maximise  wᵀμ − (λ ÷ 2) wᵀΣw
subject to  Σwᵢ = 1,  0 ≤ wᵢ ≤ w_max
Unconstrained solution: w = (1 ÷ λ) Σ⁻¹μ

How to use it

• Use it when you have a view on returns you trust and want a portfolio that reflects it.
• Tune λ under Settings › risk aversion: try 1, 3 and 10 to see how the mix moves along the frontier.
• Cap the maximum weight (e.g. 40%) to stop it from putting everything in one ticker.

Drawing conclusions

• Weights that jump a lot when you change the history by a few months mean the inputs, not the markets, are driving them.
• Tickers that get 0% are not bad; they were simply dominated for these estimates.

Watch out for

• Very sensitive to expected returns: a 1-point change in one ticker's return can swing its weight by tens of points.
• It tends to concentrate, so out-of-sample results are often worse than they look (see Walk-forward testing).

A little history

Harry Markowitz, "Portfolio Selection", Journal of Finance, 1952; expanded in his 1959 book. For decades it was more admired than used, because of its sensitivity to the inputs; most later models in this list are answers to that problem.