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.
ایده به زبان ساده
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.
روش محاسبه
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)نحوه استفاده
- 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.
نتیجه‌گیری
- 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.
مثال
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.
ورودی‌های مثال نمایشی‌اند؛ وزن‌ها و ارقام همان چیزی است که بهینه‌ساز ما برایشان برمی‌گرداند.
مراقب این‌ها باشید
- It can look too timid: robustness costs expected return in-sample.
- It protects against errors in returns, not in correlations.
کمی تاریخچه
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.
فقط محتوای آموزشی. هیچ مدل سبدی آینده را نمی‌داند: همه با برآوردهایی از گذشته کار می‌کنند و هیچ‌چیز اینجا توصیه سرمایه‌گذاری نیست.