Portfolio construction
Robust mean-variance optimisation
Mean-variance that assumes the expected returns could be as bad as their statistical uncertainty allows, and optimises for that: fewer extreme bets.
The idea in plain words
Historical average returns have large error bars. Ordinary optimisation treats them as exact and loads up on the tickers whose returns happen to be overestimated. The robust version subtracts a penalty for that uncertainty, so a ticker needs a clearly better return, not just a lucky one, to get a big weight.
How it is calculated
The uncertainty of the mean returns is Ω = Σ ÷ (years of data). κ comes from the chi-square distribution at the confidence you set (95% by default). The objective is the mean-variance one minus κ times the portfolio's uncertainty.
maximise wᵀμ − κ √(wᵀΩw) − (λ ÷ 2) wᵀΣw
Ω = Σ ÷ (T ÷ 252), κ = √χ²_{n}(confidence)How to use it
- When you want mean-variance's logic but not its overconfidence.
- Raise the uncertainty level to make it more cautious, lower it to approach plain mean-variance.
- More years of data shrink Ω and let the expected returns count for more.
Drawing conclusions
- A large difference between this and Markowitz means the expected returns were not statistically distinguishable; trust this one more.
- With little data it drifts towards the minimum-variance portfolio.
Example
Running example (used in every portfolio guide): US500, GER40, gold (XAUUSD) and EURUSD, two years of daily data. Expected returns 8%, 7%, 7% and 1% a year; volatilities 18%, 20%, 15% and 7%. US500 and GER40 move together (correlation about 0.7); gold hardly moves with the indices (about 0.04). With two years of data κ is about 3.1 for four tickers at 95%. The robust portfolio holds 14% US500, 15% gold and 71% EURUSD (2.9% return, 7.0% volatility), while plain Markowitz held 47% US500 and 53% gold: with only two years, the higher returns of the indices and gold are not certain enough to bet on.
The example's inputs are illustrative; the weights and figures are what our optimiser returns for them.
Watch out for
- It can look too timid: robustness costs expected return in-sample.
- It protects against errors in returns, not in correlations.
A little history
Richard Michaud called mean-variance optimisers "estimation-error maximisers" in 1989. Goldfarb and Iyengar (2003) and Ceria and Stubbs (2006) gave the robust formulation with an ellipsoid of uncertainty used here.
Educational content only. No portfolio model knows the future: every one of them works on estimates from the past, and nothing here is investment advice.