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Why is traditional, simple VaR measurement not coherent

Why is traditional, simple VaR measurement not coherent?

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The traditional, simple VaR measure is not coherent because it does not satisfy the sub-additivity condition. Sub-additivity is an evident requirement for a risk measure; or else there would be no risk benefit to adding uncorrelated new trades in a book. When you have two portfolios X and Y so then this benefit can be explained as

ρ(X) + ρ(Y) − ρ(X + Y).

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