# Mathematical proof of out-of-sample disappointment in portfolio performance being a function of a portfolio's variance

• The minimum-variance portfolio is considered more optimal than the maximum Sharpe ratio (tangency) portfolio on the grounds that its in-sample performance is less likely to disappoint out-of-sample.

Out-of-sample disappointment refers to the difference between performance predicted in-sample and actual performance attained out-of-sample.

• It is also proven by definition that the minimum-variance portfolio's returns (a time series) have lower variance than the variance of the maximum Sharpe portfolio's returns series.

Is the minimum-variance portfolio's out-of-sample performance more consistent with its in-sample performance because it has lower variance than the tangency portfolio? In other words, is the in-and-out-of-sample discrepancy of mean-variance portfolios a function of their defined variance?

If so, how can this be shown mathematically? How can it be proven that the minimum-variance portfolio will always have lower out-of-sample disappointment than the tangency portfolio, similar to how it can be mathematically proved that it has lower variance? Put a different way, how can we show that the minimum-variance portfolio's performance predicted in-sample will always be more consistent with its actual out-of-sample performance than the maximum Sharpe portfolio's in-versus-out-of-sample performance consistency?

• Handwaving here: the metric for 'disappointment' is different in the two cases. The minvar portfolio is built without using any estimate of expected returns, and it's in-sample 'performance' is estimated the same way. The Markowitz portfolio has to estimate the vector of mean returns, and is scored in-sample by the same, adding more noise. Nov 12, 2020 at 18:45
• See also Ao, Li, Zheng, El Karoui, Pav. Tangentially related is Paulsen & Sohl, and Pav. Nov 12, 2020 at 18:48
• those are known. let's go past that. Nov 12, 2020 at 19:22