Mean-Variance Optimization and Uncertain Return Estimates
Summary
The discussion examines whether a year or several years of daily returns can provide reliable inputs for a mean-variance portfolio of multiple assets. Its central warning is that sample means are imprecise: the uncertainty around an estimated mean remains substantial even with long histories. Since optimized allocations can change sharply when expected-return inputs move by small amounts, classic mean-variance optimization is especially sensitive to estimation error.
The answer distinguishes expected returns from volatility and correlation estimates. It argues that risk estimates may be more responsive to recent conditions, since volatility and correlations can shift across regimes, while longer samples do not eliminate uncertainty in average returns. It also relates portfolio objectives: with equal expected returns, the minimum-variance portfolio coincides with the maximum-Sharpe portfolio; assuming returns proportional to volatility connects maximum diversification to maximum Sharpe. A maximum-Sharpe portfolio based on trailing returns can therefore resemble a momentum allocation. The response does not directly resolve the question about using excess returns, and these equivalences depend on their stated assumptions.
Key ideas
- Mean-variance allocations can be highly sensitive to small changes in expected-return estimates.
- A long return history does not remove substantial uncertainty in estimated average returns.
- Recent volatility and correlation estimates may better reflect current conditions when regimes shift.
- With equal expected returns, minimum variance and maximum Sharpe lead to the same portfolio.
- A maximum-Sharpe portfolio built from trailing returns can behave like a momentum strategy.
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# Portfolio Optimization and Global Minimum Variance Portfolio (GMV) # Portfolio Optimization and Global Minimum Variance Portfolio (GMV) I have few questions about classic mean-variance-optimization in general. I have a series daily returns of 15 assets and I want to combine these assets in a portfolio. 1) Do you think that 1 year of historical data is enough to estimate reliable estimators for the true mean and variances (I.e. sample mean and sample variance) or should I go for 3 years? 2) Is it possible to estimate the GMV and mean-variance portfolios with excess returns or is that only allowed for the Max. Sharpe Ratio portfolio? Thanks for your help! Thomas ## Answer by demully (score 7) https://quant.stackexchange.com/a/48877 1) To be honest, any horizon is problematic in this respect. Simple sampling statistics 101 will tell you that the standard error around any estimate of true mean returns is the root time * variance. So for eg stocks at 20 vol, that's a +/-40% 1y 95% confidence interval around your sample mean ;-) With 100 years of data, that's still +/-4%! Which is in-line all too many estimates of the equity risk premium... The problem here is as much as methodology as your sample, because the classic problem with the mean-variance optimisation approach from the get-go is that is hugely sensitive to the input assumptions. A couple of percent different on the returns and you get a very different portfolio output suggested. Volatility and correlation regimes also shift over time; but in a sense, that makes it more OK to use shorter-term assumptions for these than for the returns. Because it's more likely their recent behaviour reflects the current paradigm; and these often do stick around for a while. 2) It's easy to calculate the GMV portfolio. It's simply the Max-Sharpe Portfolio if you assume equal returns across all your assets. Then MinVar becomes MaxSharpe! Likewise, assume returns proportional to volatility (ie equal Sharpe across your assets), and Max Diversification becomes MaxSharpe. Seen from the other side, Max Sharpe using last 12m returns is nothing more than a "long Momentum" portfolio.
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