Portfolio Variance and Out-of-Sample Performance Disappointment

Summary

This document asks whether a minimum-variance portfolio is more likely than a maximum-Sharpe, or tangency, portfolio to deliver out-of-sample performance close to its in-sample estimate. It defines disappointment as the gap between predicted in-sample performance and realized out-of-sample performance. It also notes that the minimum-variance portfolio has lower return variance by construction, then questions whether that fact explains greater consistency across samples.

The text frames a proposed mathematical comparison: can lower portfolio variance guarantee less out-of-sample disappointment, or does that conclusion require additional assumptions? It offers no proof, model, empirical test, or answer. The distinction matters because lower return variance alone does not establish how estimation error, changing conditions, or realized mean returns behave out of sample. The document is therefore a research question about the relationship between portfolio construction and forecast reliability, rather than a demonstrated result.

Key ideas

• The document defines out-of-sample disappointment as the difference between estimated and realized performance.
• It compares minimum-variance and maximum-Sharpe portfolios.
• The minimum-variance portfolio is described as having lower return variance by definition.
• The text asks whether lower variance implies less performance disappointment across samples.
• No mathematical proof or empirical evidence is supplied.

Tags

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


# 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?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.