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VaR: Validation & Alternativesadvanced

Coherent Risk Measures

Artzner, Delbaen, Eber and Heath (1999) asked what properties a sensible risk measure should obey and gave four axioms: monotonicity (a position that always loses more is riskier), subadditivity (the risk of a combined book is no greater than the sum of its parts, so diversification cannot increase risk), positive homogeneity (scaling a position by λ\lambda scales its risk by λ\lambda), and translation invariance (adding cc of risk-free cash lowers the risk number by cc). A measure satisfying all four is coherent. The headline result for risk management: VaR violates subadditivity in general and is therefore not coherent, while expected shortfall is coherent.

Why it matters

The axioms are common-sense demands. Diversification should never be punished, twice the position should carry twice the risk, and holding more safe cash should make you look safer. VaR fails the diversification test, so a risk system built on VaR can tell a desk that splitting a portfolio in two reduces capital even when the combined exposure is more dangerous. Expected shortfall passes all four tests, which is the deep reason regulators shifted toward it.

Formulas

Subadditivity axiom
ρ(X+Y)≤ρ(X)+ρ(Y)\rho(X + Y) \le \rho(X) + \rho(Y)
A coherent measure ρ\rho never assigns a merged book more risk than the sum of the parts. VaR can break this inequality; expected shortfall always satisfies it.
Translation invariance
ρ(X+c)=ρ(X)−c\rho(X + c) = \rho(X) - c
Adding cc units of risk-free cash to a position reduces the required risk capital by exactly cc.

Worked examples

Scenario

A risk manager wants a measure where splitting a portfolio into desks and adding the desk-level capital charges can never understate the firm-wide charge. Which property guarantees this, and does VaR provide it?

Solution

Subadditivity guarantees that desk-level charges sum to at least the firm-wide charge, so aggregation is conservative. Expected shortfall is subadditive and provides this guarantee; VaR is not subadditive in general, so a VaR-based system can let the sum of desk charges fall below the true firm-wide risk, understating capital.

Scenario

In its 2019 Fundamental Review of the Trading Book, the Basel Committee replaced 99% VaR with 97.5% expected shortfall as the market-risk capital measure. Which coherence property drove the switch?

Solution

Subadditivity. Regulators wanted a measure that never penalises diversification and that reflects how deep tail losses run, not just a threshold. VaR can violate subadditivity and is blind beyond the quantile, whereas expected shortfall is coherent and averages the tail. The 97.5% ES level was chosen to sit close to the old 99% VaR under normality while remaining tail-aware, so the change sharpens the risk measure without arbitrarily inflating capital.

Common mistakes

  • ✗VaR is a coherent risk measure. VaR satisfies monotonicity, homogeneity and translation invariance but can fail subadditivity, so it is not coherent in general.
  • ✗Coherence is a purely abstract property with no practical bite. Subadditivity is exactly the diversification principle, so a non-coherent measure can misallocate capital and penalise hedging in real risk systems.
  • ✗Expected shortfall is just a more conservative VaR. ES is structurally different: it averages the tail losses and, unlike VaR, satisfies all four coherence axioms.
  • ✗Translation invariance means risk does not change when you add cash. Adding risk-free cash reduces the risk number one-for-one; it does not leave it unchanged.

Revision bullets

  • •Four axioms: monotonicity, subadditivity, positive homogeneity, translation invariance
  • •Subadditivity formalises the diversification principle
  • •VaR fails subadditivity, so it is not coherent in general
  • •Expected shortfall satisfies all four axioms (coherent)
  • •Coherence is why regulators favour ES over VaR for tail risk

Quick check

Which coherence axiom does VaR fail in general?

The subadditivity axiom ρ(X+Y)≤ρ(X)+ρ(Y)\rho(X+Y) \le \rho(X) + \rho(Y) encodes which financial idea?

Connected topics

VaR DefinitionVaR LimitsExpected Shortfall

Sources

  1. Artzner et al. (1999)
    Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. "Coherent Measures of Risk." Mathematical Finance, 9(3), 203-228, 1999.
    Original paper defining the four axioms and showing VaR fails subadditivity.
  2. Acerbi & Tasche (2002)
    Acerbi, C., & Tasche, D. "On the Coherence of Expected Shortfall." Journal of Banking & Finance, 26(7), 1487-1503, 2002.
    Proves expected shortfall is a coherent risk measure.
  3. BCBS (2019), FRTB
    Basel Committee on Banking Supervision. Minimum Capital Requirements for Market Risk (FRTB). Bank for International Settlements, 2019.
    Replaces 99% VaR with 97.5% expected shortfall, citing coherence and tail-sensitivity.
How to cite this page
Dr. Phil's Quant Lab. (2026). Coherent Risk Measures. Derivatives Atlas. https://phucnguyenvan.com/concept/frm-coherent-measures
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