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Information security managers with fixed budgets must invest in security measures to mitigate increasingly severe threats whilst maintaining the alignment of their systems with their organization's business objectives. The state of the art lacks a systematic methodology to support security investment decision-making. We describe a methodology that integrates methods from multi-attribute utility evaluation and mathematical systems modelling. We illustrate our approach using a collaborative case study with the security managers of a large organization divesting itself of its IT support services. The case study was validated against the experience and observations of the security managers and delivered, according to their judgement, useful results. Specifically, by integrating a mathematical model of system behaviour with an account of the utility of available security investment strategies, the case study has enabled them to understand better the trade-offs between the security performance and the operational consequences of their choices.