Robust mean-variance optimisation

用通俗的话解释

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.

计算方法

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)

如何使用

• 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.

得出结论

• 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.

注意事项

• It can look too timid: robustness costs expected return in-sample.
• It protects against errors in returns, not in correlations.

一点历史

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.