Richard's example is good. The numerical optimizers used in Markowitz optimization are generally prone to weird values, and depend heavily on the exact formulation of the risk metric. Another reason could be that it's overfitting to a single asset in every time period. It is plausible that for sufficiently many assets, you will be able to find one whose risk profile far outstrips the others over a particular look-back window.
A final consideration is that Markowitz optimization is contested these days, with people suggesting it is a not really a profitable technique out of sample, and that it has been arbitraged away in current markets. It is in essence a momentum strategy -- you are betting that assets whose risk profiles were good recently will continue to be good in the future. Depending on timeframe of your strategy and the type of assets you are using, this may or may not be the case. It's important to take all this into consideration when using Markowitz optimization.
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