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State each property that a coherent risk measure must have and describe what each one means.

FRM® Part I · Valuation and Risk Models · Measures of Financial Risk · FRM1.VR.1.e

The idea

A coherent risk measure, ρ, satisfies four properties. Monotonicity: if portfolio X has losses at least as large as Y in every scenario, then ρ(X) is at least ρ(Y). Positive homogeneity: scaling a position by a factor k scales its risk by k, so ρ(kX) = k × ρ(X), which ignores any liquidity effect. Translation invariance: adding k of riskless cash reduces the risk measure by k, so ρ(X + k) = ρ(X) − k, with ρ expressed as capital needed. Subadditivity: ρ(X + Y) ≤ ρ(X) + ρ(Y), so diversification cannot increase risk and merging two books cannot create risk. Expected shortfall is coherent. Standard deviation fails monotonicity, and VaR is not guaranteed to be subadditive. Exam point: recall each name with its meaning, and know which common measures pass or fail.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which condition describes subadditivity of a risk measure rho?

  1. rho(X + Y) <= rho(X) + rho(Y): combining two positions never creates more risk than the sum of the separate risks. - correct
  2. rho(kX) = k x rho(X): doubling a position doubles its risk.Why not: This is positive homogeneity, which concerns scaling a single position rather than combining two.
  3. rho(X + k) = rho(X) - k: adding k of cash reduces risk by k.Why not: This is translation invariance, which concerns adding riskless cash.
  4. If X always loses at least as much as Y, then rho(X) >= rho(Y).Why not: This is monotonicity, which ranks portfolios scenario by scenario.

Subadditivity formalises diversification: the risk of a combined portfolio is at most the sum of the stand-alone risks. The other three properties are monotonicity, positive homogeneity and translation invariance.

A harder one

A portfolio has a 99% expected shortfall of USD 4 million. A manager scales up the risky position by 50% and adds USD 1 million of riskless cash. For a coherent measure, the new risk measure is:

  1. USD 7.0 millionWhy not: Adds the cash instead of subtracting it, getting the sign of translation invariance wrong.
  2. USD 6.0 millionWhy not: Applies positive homogeneity (4 x 1.5) but ignores the effect of the cash.
  3. USD 4.5 millionWhy not: Adds the cash first and then scales (4 - 1) x 1.5, wrongly scaling the riskless cash with the position.
  4. USD 5.0 million - correct

Positive homogeneity: scaling the position by 1.5 gives 4 x 1.5 = 6.0. Translation invariance: adding USD 1 million of riskless cash subtracts 1, giving 5.0. The cash is added to the scaled position, not scaled with it.

The hardest tier

A risk measure rho is coherent. Two positions have rho(X) = USD 10 million and rho(Y) = USD 6 million, and their correlation is unknown. The largest value that rho(2X + 3Y) can take is:

  1. USD 38 million - correct
  2. USD 16 millionWhy not: Adds the stand-alone measures without scaling by the position sizes of 2 and 3.
  3. USD 26.9 millionWhy not: Combines the scaled risks as sqrt(20² + 18²), which assumes zero correlation and is not an upper bound for a general coherent measure.
  4. USD 50 millionWhy not: Applies the larger measure (10) to the total scaling of 2 + 3 = 5, ignoring that Y is the less risky position.

Subadditivity: rho(2X + 3Y) <= rho(2X) + rho(3Y). Positive homogeneity: rho(2X) = 20 and rho(3Y) = 18. The bound is 20 + 18 = 38, reached when the two positions offer no diversification benefit.

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