Mean-Variance Optimization, Estimation Error, and Portfolio Sensitivity

# Markowitz mean-variance optimization as "error maximization"


# 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?

## Answer by vonjd (score 12)

https://quant.stackexchange.com/a/4133

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 (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.

EDIT 2 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).

## Answer by Bryce (score 5)

https://quant.stackexchange.com/a/4818

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),

- Using historical returns to estimate the covariance matrix is sufficient.

- 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.

## Answer by John (score 3)

https://quant.stackexchange.com/a/4136

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.

## Answer by Pim (score 0)

https://quant.stackexchange.com/a/46435

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.

## Answer by Colm O'C (score 0)

https://quant.stackexchange.com/a/81402

I've worked in the investment management industry for 25 years and like many quants have learned how to make good use of Markowitz optimization. I didn't get to that point by solving some mathematical problem but rather by understanding what the various stakeholders need from a portfolio construction process.

Recently however I wrote a mathematical paper that explains the level of sensitivity one should expect on average. In it I show that under reasonable assumptions reflective of quant practice, a tactical asset allocation optimization taking 1% active risk may typically lead to 100% absolute active holdings (or "active share"). For some, this means that a small active risk leads to unacceptably large active positions. This is precisely what is flagged as sensitivity in practice. Less risk averse investors may not be deterred by the large positions.

This outcome rests heavily on a knowledge of the typical risks and correlations seen among indices, combined with typical investment practice. So this is not something mathematicians or engineers can explain to us quants because they don't know what our data looks like.

Please see my July 2025 paper on SSRN. Although it is mathematical, the next-to-last section explains how to estimate active share in the manner I've illustrated above.

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4915769

## Answer by Colm O'C (score 0)

https://quant.stackexchange.com/a/81404

I didn't complete my previous answer because I wanted to check some reference. Here is a very precise interpretation of the phrase "error maximization" in this context, but I can't provide a reference for it. Using the notation of a previous answer, suppose that $\hat \mu$ is our estimate of $\mu$ so that our estimated optimal portfolio is $$\hat h \equiv {1\over \lambda} \Sigma^{-1}\hat\mu.$$ Then the tracking portfolio between $\hat h$ and the true optimal, call it $h$, is $$\hat h-h \equiv {1\over \lambda} \Sigma^{-1}\hat e,$$ where $e\equiv \hat\mu - \mu.$ But then $\hat h$ is precisely the portfolio that maximizes the error in the expected return of the portfolio, penalized in the usual way for risk. So in this tracking error or active risk sense, Markowitz optimization indeed maximizes errors. It's the worst possible result in a sense. But, given that $\mu$ is unknown, we can't really do better without adding layers of further assumptions and modeling (e.g., Bayesian or robust approaches), not all of which are necessarily valid or beneficial.

The phrase "error maximization" is among the very eloquent expressions of quants' frustration with quant methods. A closely related phrase is Rob Arnott and Jason Hsu's phrase "overweighting overpriced assets and underweighting underpriced asset." This is what indexing is purported to do, but it is mathematically closely related to "error maximization." If stat professors could find such eloquent ways of expressing statistical uncertainty, we would all remember Stat 101 more fondly.