Climatic and other extreme events threaten the globalized economy, which relies on increasingly complex and specialized supply-chain networks. Disasters generate (i) direct economic losses due to reduced production in the locations where they occur, and (ii) to indirect losses from the supply shortages and demand changes that cascade along the supply chains. Firms can use inventories to reduce their risk of shortages. Since firms are interconnected through the supply chain, the level of inventory hold by one firm influences the risk of shortages of the others. Such interdependencies lead to systemic risks in supply chain networks. We introduce a stylized model of complex supply-chain networks in which firms adjust their inventory to maximize profit. We analyze the resulting risks and inventory patterns using evolutionary game theory. We report the following findings. Inventories significantly reduce disruption cascades and indirect losses at the expense of a moderate increase in direct losses. The more fragmented a supply chain is, the less beneficial it is for individual firms to maintain inventories, resulting in higher systemic risks. One way to mitigate such systemic risks is to prescribe inventory sizes to individual firms-a measure that could, for instance, be fostered by insurers. We found that prescribing firm-specific inventory sizes based on their position in the supply chain mitigates systemic risk more effectively than setting the same inventory requirements for all firms.