In this paper, we analyze the asset allocation problem under the generalized hyperbolic (GH) distribution of returns and exponential utility. We provide closed-form expressions to compute the optimal portfolio weights; and we introduce two new measures, associated with a more general mean-risk trade-off, that allow us to express the optimal solution as an affine combination of two efficient portfolios: one minimizing risk and the other maximizing mean given a particular level of risk. Also, we prove that optimal portfolio performance is not monotonic in tail behavior since it increases when tails become lighter or heavier with respect to a particular threshold; however, distributions with heavier tails produce more conservative allocations in terms of the weight given to the minimum-risk portfolio increments. Finally, the practical relevance of our paper show that tail behavior greatly affects portfolio construction and performance, and that including non-normality features of short-term asset returns, through a GH distribution, has the potential to significantly improve the investor's certainty equivalent excess return.