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

Which statement about VaR and Tail VaR as risk measures is correct?

Tail VaR is a coherent risk measure, while VaR can fail subadditivity, so diversification might appear to increase VaR. Tail VaR is also at least as large as VaR and captures the size of tail losses, which VaR ignores.

  1. AVaR is subadditive for all loss distributions, while Tail VaR is not
  2. BTail VaR is coherent, whereas VaR can fail subadditivityCorrect
  3. CNeither measure can fail subadditivity when losses are independent
  4. DTail VaR at a given level is never larger than VaR at the same level
  5. VaR gives information about the size of losses beyond the quantile

Explanation

Tail VaR satisfies monotonicity, translation invariance, positive homogeneity and subadditivity, so it is coherent. VaR can violate subadditivity, for example with skewed or discrete losses. Tail VaR is at least VaR, and VaR ignores loss size beyond the quantile.

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