Value at risk: Is a theoretically consistent axiomatic formulation possible?

This note identifies three properties of a risk measure, the acceptance of all of which implies the acceptance of the VaR risk measure; and the rejection of any one of which implies the rejection of the VaR risk measure. First, a risk measure should reflect weak aversion to losses. Second, only sufficiently likely threats matter. Finally, the risk measurement should be unaffected by how promising the upside may look like. These properties, by themselves, constitute a consistent set of axioms that are necessary and sufficient for the acceptance of the VaR risk measure on a given probability space. The axiomatization highlights a peculiar characteristic of VaR: it ignores the upside, while at the same time neglecting the worse of the downside.