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

A loss distribution is normal with mean 0 and standard deviation 1. The 95% VaR is 1.645 and the 95% tail value at risk (expected shortfall) equals the mean loss given loss exceeds VaR, which is phi(1.645)/0.05 with phi(1.645)=0.1031. Which statement is correct?

TVaR at 95% is about 2.06, above the VaR of 1.645, because it averages losses in the tail beyond VaR. TVaR is a coherent measure, being subadditive, while VaR in general is not.

  1. ATVaR at 95% is 1.645 because it equals VaR
  2. BTVaR at 95% is about 2.06 and exceeds VaR, and TVaR is coherent while VaR is not in generalCorrect
  3. CTVaR at 95% is about 0.21 and is below VaR
  4. DTVaR at 95% is about 2.06 but, like VaR, it is not subadditive in general
  5. TVaR at 95% is about 1.03 and is half of VaR

Explanation

TVaR = 0.1031/0.05 = 2.062, above VaR of 1.645 as it averages the tail beyond VaR. TVaR is subadditive and coherent, whereas VaR can fail subadditivity. Option 3 wrongly denies subadditivity of TVaR.

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