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IAI Actuarial Core Principles · Economic Modelling · Measures of investment risk

Which statement about coherent risk measures is correct?

Tail Value at Risk is sub-additive and coherent, whereas Value at Risk can fail sub-additivity, so diversification might appear to increase measured risk. This is a key reason expected shortfall is often preferred in practice.

  1. ATail Value at Risk satisfies sub-additivity, whereas Value at Risk can fail itCorrect
  2. BValue at Risk always satisfies sub-additivity for any distributions
  3. CVariance is a coherent risk measure because it is translation invariant
  4. DA coherent measure must rise when a portfolio's holdings are halved
  5. Sub-additivity means the risk of a combined portfolio can exceed the sum of the separate risks

Explanation

Sub-additivity requires risk of a combination to be at most the sum of the individual risks. Tail Value at Risk satisfies it; Value at Risk can violate it for non-elliptical distributions. Variance fails translation invariance and monotonicity.

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