coherent risk measure

Before trusting any formula that turns a portfolio into a single 'how risky is this?' number, it is worth asking whether the formula behaves sensibly — does it reward, rather than punish, common-sense actions like spreading your bets? A coherent risk measure is one that has passed four such common-sense tests. The idea, introduced by Artzner, Delbaen, Eber and Heath in the late 1990s, gives us a way to say which risk numbers can be trusted to behave logically and which can quietly mislead.

The four properties, in plain words. First, monotonicity: if portfolio A always loses at least as much as B in every scenario, A's risk number must be at least as big. Second, sub-additivity: the risk of two portfolios combined must not exceed the sum of their separate risks — merging things should never look riskier than keeping them apart, because diversification can only help. Third, positive homogeneity: doubling every position doubles the risk number. Fourth, translation invariance: adding a guaranteed cash buffer of amount k reduces the required capital by exactly k. The single most important of these is sub-additivity, the mathematical statement that diversification reduces risk.

Why actuaries care: the most popular measure, Value at Risk, fails sub-additivity in some cases — you can combine two portfolios and find total VaR larger than the sum, which falsely tells management that diversifying made them worse off, and can even be gamed. Expected shortfall / Tail VaR, by contrast, is coherent. This is a major reason capital frameworks have been shifting toward tail-average measures. A caveat to keep honest: coherence is about logical good behaviour, not accuracy — a coherent measure fed bad assumptions still produces a bad answer; it just produces it consistently.