How Mean-Variance Optimization Amplifies Estimation Errors
Summary
The note explains a quotation describing mean-variance optimization as an estimation-error maximizer. It separates the optimizer’s response to estimated inputs from the sampling uncertainty of those inputs: all else equal, higher estimated returns, negative covariances, and lower variances can make an asset more attractive to the optimizer. The covariance contribution follows from portfolio variance, where negative covariance can offset other risk.
The reply also clarifies that the standard deviation of an individual asset’s returns is not the standard error of its sample mean; the latter depends on sample size. It cautions that the quoted claim about covariance refers to the covariance’s magnitude in the optimization, not the standard error of the covariance estimate. The exchange does not derive sampling-error formulas or assess the broader conditions under which optimization magnifies estimation error, so it is a conceptual clarification rather than a full statistical treatment.
Key ideas
- Mean-variance optimization can favor assets with higher estimated returns, lower variances, and negative covariances, holding other inputs constant.
- Negative covariance can reduce total portfolio variance by offsetting risk from other assets.
- An asset’s return standard deviation differs from the standard error of its estimated mean return.
- The quoted discussion concerns covariance values used by the optimizer, not standard errors of covariance estimates.
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# Mean variance optimisation as error-maximisation: why would negative correlation increase standard error of estimates?
# Mean variance optimisation as error-maximisation: why would negative correlation increase standard error of estimates?
"The unintuitive character of many optimized portfolios can be traced to the fact that MV optimizers are, in a fundamental sense, estimation error maximizers. Risk and return estimates are inevitably subject to estimation error. MV optimization significantly overweights (underweights) those securities that have large (small) estimated returns, negative (positive) correlations and small (large) variances. These securities are, of course, the ones most likely to have large estimation errors” (Michaud, 1989, page 33).
Struggling to interpret this. Say we estimate expected returns with sample mean, covariance matrix with sample covariances. Aren't the standard errors of these statistics highest for the assets with high population variances (which MVO underweights) and high absolute correlation (from my experience in equities these are generally positive)?
## Answer by mark leeds (score 1)
https://quant.stackexchange.com/a/80484
Hi: Michaud says stocks with larger returns, negative covariances and small variances will get over-weighted.
- The larger returns makes sense, right. Ceteris Paribus, the optimizer will prefer the stock with the larger return.
- The negative covariances contribute to making the overall risk smaller ( negative covariances tend to cancel out variance ) which is one of the goals of the MV framework. This is true because Var(X+Y) = Var(X) + Var(Y) + 2 Cov(X,Y).
- Small variances kind of help similarly to negative covariances. Ceteris Paribus, the optimizer will prefer the stock with the smaller variance.
Note that when the return of a stock $i$ is estimated, the sd of it is just $\sigma_{i}$ rather than $\frac{\sigma_{i}}{\sqrt{n}}$. Your formula is for when wants the sd of the average return of stock $i$.
I'm not familiar with your formula for SE(COV(X,Y)) but Michaud is referring to the size of the COV rather than the size of the SE of the COV.
I hope this helps.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.