I hear it said a lot that standard MV optimization "maximizes errors". But I can't find a good explanation for what exactly they mean by this "maximization" of estimation error.

I understand that if you simulate $500$ matrices of returns $T-t$ months into the future from $t$ (now) to $T$ (future), and you do MV optimization on each matrix at $T$ to arrive at $500$ frontiers, then these will differ wildly from the MV optimization at $t$. (Figure 1 here). But what's this saying?


4 Answers 4


I think the original reference of mean-variance portfolios being “error maximizing portfolios” is:

Michaud, R. (1989). “The Markowitz Optimization Enigma: Is Optimization Optimal?” Financial Analysts Journal 45(1), 31–42.

The reason is that even small changes in the estimated means can result in huge changes in the whole portfolio structure.

Have a look at this new piece from Andrew Ang which explains this quite well ("4.1 Sensitivity to Inputs", p. 26-27):

Mean-Variance Investing by Andrew Ang

For a different perspective see this paper from
Mark Kritzman (2006): Are Optimizers Error Maximizers? Hype versus reality?

From the abstract:

Small input errors to mean-variance optimizers often lead to large portfolio misallocations when assets are close substitutes for one another. In fact, when the assets are close substitutes, the return distribution of the presumed optimal portfolio is actually similar to the distribution of the truly optimal portfolio. Contrary to conventional wisdom, therefore, mean-variance optimizers usually turn out to be robust to small input errors when sensitivity is measured properly.

A free version can be found on pages 165-168: Here.

A nice summary of this line of reasoning can be found in
Mark Kritzman (2014): Six Practical Comments About Asset Allocation:

The Myth of Estimation Error:
Cynics often refer to mean-variance optimizers as error maximizers because they believe that small input errors lead to large output errors. This cynicism arises from a misunderstanding of sensitivity to inputs. Consider optimization among assets that have similar expected returns and risk. Errors in the estimates of these values may substantially misstate optimal allocations. Despite these misallocations, however, the return distributions of the correct and incorrect portfolios will likely be quite similar. Therefore, the errors do not matter because the resultant incorrect portfolio is nearly as good as the correct portfolio.
Now consider optimization among assets that have significantly dissimilar expected returns and risk. Errors in these estimates will have little impact on optimal allocations; hence again the return distributions of the correct and incorrect portfolios will not differ much. There may be some cases in which small input errors matter, but in most cases sensitivity to estimation error is more hype than reality [...]

(Unfortunately I haven't found a free version of the paper - if you find one let me know in the comments and I will update the post).

  • $\begingroup$ Do you have any references that you could refer me to that deal with the reasons why re-sampling means that estimation error is less problematic? $\endgroup$
    – user2921
    Sep 17, 2012 at 11:42
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    $\begingroup$ +1. There's alao a 2006 paper by Sebatian Ceria and Robert Stubbs that also illustrates this with an example. They both are at Axioma so you can also find some more research there. $\endgroup$ Sep 17, 2012 at 12:48
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    $\begingroup$ I think of Michaud's resampling as an application of Bayesian techniques. Under Black-Litterman/Entropy Pooling framework, if you take views on every asset and perform unconstrained optimization with equal confidence in the views, then the optimal portfolio is an average of the individual portfolios if you had full confidence in them. You could come up with arbitrary views by sampling from $\mu_{r}\sim N\left(\mu,\frac{\Sigma}{T}\right)$ where $T$ is the number of observations. More observations, less estimation error. $\endgroup$
    – John
    Sep 17, 2012 at 21:14
  • $\begingroup$ @John - Take a look at "Robust Portfolio Construction" by Bernd Scherer who has a takedown of Michaud's resampling technique on theoretical grounds (namely that it is not Bayesian although it smells like bootstrapping): papers.ssrn.com/sol3/papers.cfm?abstract_id=796625 $\endgroup$ Sep 18, 2012 at 19:16
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    $\begingroup$ @vonjd Link to a free version of Kritzman paper: harrykatz.com/Final_Printer_Proof.pdf pages 165-168 $\endgroup$ Dec 22, 2012 at 7:58

One of the most salient empirical examples of "error maximization" is provided by Chopra and Ziemba (1993):

Chopra, Vijay K., and William T. Ziemba. 1993. “The Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice.” Journal of Portfolio Management, vol. 19, no. 2 (Winter):6–11.

The authors compare the performance of mean-variance optimization using (a) historical data and traditional sample estimators against a portfolio formed with (b) perfect information of the future. The authors find after comparing the performance of (a) relative to the clairvoyant portfolio (b),

  1. Using historical returns to estimate the covariance matrix is sufficient.
  2. Using historical returns to estimate the mean return incurs a massive performance shortfall.

Thus, using a shrinkage estimator, or simply setting all returns equal to a constant $\hat{\mu}_i = c$ $\forall i$ (equivalent to the minimum variance portfolio), is a superior alternative.


Let $\mu$ and $\Sigma$ be the expected mean and covariance matrices for a mean-variance optimization. For a standard, unconstrained, utility-based optimization, it can be shown that the optimal weights will equal $$ w=\frac{1}{\lambda}\Sigma^{-1}\mu $$ where $\lambda$ is an arbitrary risk aversion coefficient.

In order to measure the sensitivity of the weights to the expected return, you could calculate $$\frac{\partial w}{\partial\mu}=\frac{1}{\lambda}\Sigma^{-1}$$

As a result of the nature of the inverse of the covariance matrix, this formula suggests that arbitrary changes in $\mu$ tend to lead to large changes in portfolio weights.

  • $\begingroup$ John: Could you pls. expound on "the nature of the inverse of the covariance matrix" - thank you. $\endgroup$
    – vonjd
    Sep 18, 2012 at 11:52
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    $\begingroup$ Well imagine that there is one element in the covariance matrix, so the inverse is one over the variance of the asset. If the standard deviation is 20%, the inverse of the variance is 25. This is why small changes in the means matter. $\endgroup$
    – John
    Sep 18, 2012 at 13:58

For future readers of this question. A concise and clear explanation is actually given by Brandt (2010), who indeed refer to Michaud (1989), who first mentioned the term "error maximization". The following section is from Brandt (2010, p.300):

Michaud (1989) argues that extreme and unstable portfolio weights are inherent to mean–variance optimizers because they tend to assign large positive (negative) weights to securities with large positive (negative) estimation errors in the risk premium and/or large negative (positive) estimation errors in the volatility. Mean–variance optimizers therefore act as statistical “error maximizers”.

The difference you are referring to originates from the estimation error in your volatility matrix. This error is summed over longer horizons, hence the difference will be bigger for longer investment horizons.


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