We consider a single-location, single-period stock allocation problem (newsvendor-like problem) with n items in which demand rates, holding costs, and backorder costs vary across all products. Inventory levels are replenished at the end of each period instantaneously. We apply robust optimization under an uncertainty set that captures a risk pooling phenomenon across items to this problem. The number of constraints governing the uncertainty set grows linearly in the number of items. A closed form solution is presented for the single and two-item cases. For the general n item problem, we present a 2-approximation algorithm and demonstrate its asymptotic optimality. The experimental results confirm the value of the approximation algorithm and indicate that the average performance is close to optimal.