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The paper introduces a new notion of vector-valued risk function, a crucial notion in Actuarial and Financial Mathematics. Both deviations and expectation bounded or coherent risk measures are defined and analyzed. The relationships with both scalar and vector risk functions of previous literature are discussed, and it is pointed out that this new approach seems to appropriately integrate several preceding points of view. The framework of the study is the general setting of Banach lattices and Bochner integrable vector-valued random variables. Sub-gradient linked representation theorems and practical examples are provided.