Properties of Risk Measures

Two portfolios each show VaR of $10 million. Combine them and the VaR jumps to $22 million (diversification increased risk. This is not hypothetical. VaR fails subadditivity, one of four properties required for a "coherent" risk measure. Knowing which property VaR violates) and that TVaR does not, is the core of this lesson.

A risk measure $\rho$ is coherent if it satisfies all four:

$$\rho (X+c)=\rho (X)+c$$

$$\rho (cX)=c\cdot \rho (X),c>0$$

HIGH-FREQUENCY: Subadditivity and VaR's failure of it is the most tested concept here.

$$\rho (X+Y)\le \rho (X)+\rho (Y)$$

$$X\le Y\text{ a.s.}⟹\rho (X)\le \rho (Y)$$

VaR satisfies translation invariance, positive homogeneity, and monotonicity. It fails subadditivity.

KEY: VaR is not a coherent risk measure because it fails subadditivity. TVaR satisfies all four properties and is coherent. This distinction is one of the most frequently tested concepts on FAM.

Common mistakes

• Claiming VaR is coherent. VaR fails subadditivity. On the exam, "VaR is a coherent risk measure" is always false. Trap: selecting True because VaR satisfies the other three properties.
• Forgetting which property VaR violates. VaR satisfies translation invariance, positive homogeneity, and monotonicity. It fails only subadditivity. Trap: selecting "monotonicity" as the failed property.
• Reversing the subadditivity inequality. Subadditivity says $\rho (X+Y)\le \rho (X)+\rho (Y)$. The weak inequality allows equality (comonotonic case). Writing $\ge$ describes super-additivity. Trap: selecting a reversed inequality.

Bottom line

• Coherent = 4 properties: translation invariance, positive homogeneity, subadditivity, monotonicity
• VaR fails subadditivity: not coherent. It still satisfies the other three properties
• TVaR satisfies all four: coherent
• Subadditivity: $\rho (X+Y)\le \rho (X)+\rho (Y)$, so diversification can never increase total risk or capital

Exam shortcut

Memorize the 2-by-4 grid: VaR gets checkmarks on translation invariance, positive homogeneity, and monotonicity, and an X on subadditivity. TVaR gets all checkmarks. If asked for a counterexample, use the binary loss (0 or 100, probability 96/4) (clean, fast, universally applicable. Remember: "VaR has no Sub") it fails subadditivity. "TVaR sees the field beyond the fence", it integrates over the entire tail, capturing diversification effects that VaR misses.

The full lesson (about 1,975 words, 13 min read) adds 3 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2n

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