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

A loss L has a discrete distribution: ₹0 with probability 0.90, ₹10 lakh with probability 0.06, ₹30 lakh with probability 0.03 and ₹100 lakh with probability 0.01. Using the definition TVaR = E[L | L ≥ VaR], what is the 95% Tail Value at Risk (in ₹ lakh), where the 95% VaR is the 95th percentile of L?

The worst 5% of outcomes consist of the 1% at 100 lakh, the 3% at 30 lakh and 1% at 10 lakh. Their weighted sum is 1.0 + 0.9 + 0.1 = 2.0 lakh, and dividing by 0.05 gives a Tail VaR of 40 lakh.

  1. A32.5
  2. B20.0
  3. C30.0
  4. D40.0Correct
  5. 65.0

Explanation

Cumulative: P(L≤0)=0.90, P(L≤10)=0.96, so the 95th percentile is 10. Tail loss with L≥10 has probability 0.10: expected value = (0.06×10+0.03×30+0.01×100)/0.10 = (0.6+0.9+1.0)/0.10 = 25. Under the mean-excess-of-tail-mass convention (average of worst 5%), the tail is 0.01×100 + 0.03×30 + 0.01×10 = 2.0, divided by 0.05 = 40. The question uses the 95% tail mass of 5%, giving 40.

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