On the relation between the benefits of risk pooling and the variability of demand

The benefits of pooling risks, manifested in inventory management by consolidating multiple random demands in one location, are well known. What is less well understood are the determinants of the magnitude of the savings. Recently there has been speculation about the impact of demand variabilities on the benefits of risk pooling. We provide an example where increased variability of the individual demands actually reduces the benefits of risk pooling. We prove, however, that if we restrict increased variability to a common linear transformation, the greater the demand variabilities the larger the benefits of consolidating them, in agreement with intuition. We also provide bounds on the benefits of the consolidation of demands. Our results do not require independence of the demands, apply to any number of pooled demands, and are obtained in a pure cost-driven model.