# Markowitz mean-variance optimization as “error maximization”

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?

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@user2921: You can accept one of the answers if you are satisfied by it :-) – vonjd Jan 28 at 16:18

## 3 Answers

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

EDIT:
For a different perspective see this paper from Mark Kritzman:
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.

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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? – user2921 Sep 17 '12 at 11:42

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+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. – Quant Guy Sep 17 '12 at 12:48

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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. – John Sep 17 '12 at 21:14
– Quant Guy Sep 18 '12 at 19:16

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– Kevin Schmit Dec 22 '12 at 7:58
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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.

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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.

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John: Could you pls. expound on "the nature of the inverse of the covariance matrix" - thank you. – vonjd Sep 18 '12 at 11:52

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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. – John Sep 18 '12 at 13:58
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