Problems with the stability of the optimal weights

There are two main problems with the Markowitz approach. The first is that, unless positivity constraints are assigned, the Markowitz solution can easily find highly leveraged portfolios (large long positions in a subset of assets financed by large short positions in another subset of assets). Needless to say, given their leveraged nature the returns from these portfolios are extremely sensitive to small changes in the returns of the constituent assets. These leveraged portfolios can therefore be extremely ‘dangerous’.

Positivity constraints are easy to enforce, and fix this problem. However, if the user wants to ‘believe’ in the robustness of the Markowitz approach, it would be nice if better-behaved solutions (at the very least, positive weights) were obtained in an unconstrained manner when the set of investment assets is close to the available investment opportunities (the market portfolio). This is often not the case.

The second, closely related (and practically more vexing) problem is the instability of the Markowitz solution: small changes in inputs can give rise to large changes in the portfolio. Somewhat unkindly, mean-variance optimization has been dubbed an ‘error maximization’ device (Scherer 2002): ‘an algorithm that takes point estimates (of returns and covariances) as inputs and treats them as if they were known with certainty will react to tiny return differences that are well within measurement error’. In the real world, this degree of instability will lead, to begin with, to large transaction costs.