What are the four axioms of a coherent risk measure, and how does VaR violate subadditivity?

A coherent risk measure satisfies four axioms proposed by Artzner, Delbaen, Eber, and Heath (1999). These axioms ensure the risk measure behaves sensibly for portfolio management and capital allocation.\n\nThe Four Axioms:\n\n1. Monotonicity: If portfolio X always has worse outcomes than Y, then rho(X) >= rho(Y). Intuition: a riskier portfolio should have higher measured risk.\n\n2. Subadditivity: rho(X + Y) <= rho(X) + rho(Y). Intuition: diversification should not increase risk. The risk of a combined portfolio should be at most the sum of individual risks.\n\n3. Positive Homogeneity: rho(lambda x X) = lambda x rho(X) for lambda > 0. Intuition: doubling a position doubles the risk. No economies of scale in risk.\n\n4. Translation Invariance: rho(X + c) = rho(X) - c for constant c. Intuition: adding a risk-free cash amount c reduces the risk measure by exactly c.\n\nVaR Violates Subadditivity — Concrete Example:\n\nConsider two binary credit positions at Greystone Bank:\n\n- Bond A: 4% chance of $10M loss, 96% chance of zero loss\n- Bond B: 4% chance of $10M loss, 96% chance of zero loss\n- Losses are independent\n\nIndividual VaR at 95%:\n- VaR(A) = $0 (the 5th percentile loss is zero, since losses occur only 4% of the time)\n- VaR(B) = $0\n- Sum = $0\n\nCombined portfolio VaR at 95%:\nProbability of zero total loss: 0.96 x 0.96 = 92.16%\nProbability of exactly $10M loss: 2 x 0.04 x 0.96 = 7.68%\nProbability of $20M loss: 0.04 x 0.04 = 0.16%\n\nThe 95th percentile loss falls in the 7.68% region (since 92.16% < 95%), so:\nVaR(A + B) = $10M\n\nSubadditivity violation: VaR(A + B) = $10M > VaR(A) + VaR(B) = $0\n\nDiversification (combining independent risks) appears to INCREASE risk under VaR, which is nonsensical.\n\nExpected Shortfall Passes:\n\nES(A) at 95% = (0.04/0.05) x $10M = $8M\nES(B) at 95% = $8M\nES(A + B) at 95% = (0.0768/0.05) x $10M + (0.0016/0.05) x $20M adjusted = approximately $10.64M\n\nSince $10.64M < $8M + $8M = $16M, subadditivity holds.\n\nExam Tip: The subadditivity violation in VaR typically arises with concentrated, binary-type risks. For normally distributed returns, VaR does satisfy subadditivity. The FRM tests understanding that VaR is not generally coherent.\n\nStudy risk measure theory in our FRM Market Risk course.