We evaluate how departure from normality may affect the conditional allocation of wealth. The expected utility function is approximated by a fourth-order Taylor expansion that allows for non-normal returns. Market returns are characterized by a joint model that captures the time dependency and the shape of the distribution. We show that under large departure from normality, the mean-variance criterion can lead to portfolio weights that differ significantly from those obtained using the optimal strategy accounting for non-normality. In addition, the opportunity cost for a risk-adverse investor to use the sub-optimal mean-variance criterion can be very large.