What makes a risk measure 'coherent,' and why does Expected Shortfall satisfy the criteria while VaR does not?

A coherent risk measure satisfies four mathematical axioms that ensure it behaves sensibly for portfolio risk management. The concept was formalized by Artzner et al. (1999) and has become central to modern risk measurement theory.

The Four Axioms

1. Monotonicity: If portfolio X always produces losses at least as large as portfolio Y in every scenario, then the risk measure of X should be at least as large as Y's. (Worse outcomes = more risk.)

1. Subadditivity: rho(X + Y) <= rho(X) + rho(Y). The risk of a combined portfolio should not exceed the sum of individual risks. This means diversification never increases measured risk.

1. Positive Homogeneity: rho(lambda X) = lambda rho(X) for lambda > 0. Doubling a position doubles the risk.

1. Translation Invariance: rho(X + c) = rho(X) - c for a cash addition c. Adding risk-free cash reduces risk by exactly that amount.

Why VaR Fails Subadditivity

Consider Westbrook Capital with two options portfolios:

• Portfolio A: Deep out-of-the-money put options on Stock 1. 95% of the time it loses nothing; 5% of the time it loses $10M.
• Portfolio B: Deep out-of-the-money put options on Stock 2 (independent of Stock 1). Same distribution.

95% VaR of A = $0 (the 95th percentile loss is zero)

95% VaR of B = $0

Now combine them into Portfolio A+B. There is a 0.25% chance both lose $10M (total: $20M), a 9.75% chance exactly one loses $10M, and a 90.25% chance neither loses. The 95th percentile loss is now $10M.

VaR(A+B) = $10M > VaR(A) + VaR(B) = $0 + $0 = $0

Diversification increased measured risk under VaR. This is absurd and violates subadditivity.

Why Expected Shortfall Is Coherent

Expected Shortfall (ES) at confidence level alpha is the average loss in the worst (1-alpha) scenarios:

> ES_alpha = E[Loss | Loss > VaR_alpha]

ES satisfies all four axioms, including subadditivity, because averaging across tail scenarios naturally rewards diversification. For the example above:

• ES_95% of A = average loss in worst 5% = $10M * (5%/5%) = $10M
• ES_95% of B = $10M
• ES_95% of A+B = [0.25%$20M + 4.75%$10M] / 5% = [$0.05M + $0.475M] / 5% = $10.5M

$10.5M < $10M + $10M = $20M. Subadditivity holds.

Practical Implications

The Basel Committee replaced VaR with ES in the Fundamental Review of the Trading Book (FRTB) partly because of the subadditivity failure. A bank using VaR might find that its trading desks report low individual VaRs, but the combined firm-wide VaR is paradoxically higher — creating perverse incentives for desks to concentrate rather than diversify risk.

FRM exam tip: Be able to construct a simple numerical example showing VaR's subadditivity violation (like the one above). Know that ES is always >= VaR for the same confidence level, and that ES at 97.5% is roughly comparable to VaR at 99% for many distributions.

Explore our FRM Part II question bank for more risk measures practice.