Do higher moments really matter in portfolio choice?

We present explicit formulas for evaluating the difference between Markowitz weights and those from optimal portfolios, with the same given return, considering either asymmetry or kurtosis. We prove that, whenever the higher moment constraint is not binding, the weights are never the same. If, due to special features of the first and second moments, the difference might be negligible, in quite many cases it will be very significant. An appealing illustration, when the designer wants to incorporate an asset with quite heavy tails, but wants to moderate this effect, further supports the argument.