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IAI Actuarial Core Principles · Economic Modelling · Mean-variance portfolio theory

A portfolio has a probability of 0.04 of a loss of Rs 100 lakh, probability 0.46 of a loss of Rs 10 lakh, and probability 0.50 of a gain of Rs 20 lakh. Using losses as positive numbers, which statement about the 95% Value at Risk is correct?

The 95% VaR is Rs 10 lakh. The Rs 100 lakh loss has only 4% probability, which lies inside the 5% tail, and the cumulative probability reaches 96% at a loss of Rs 10 lakh, so that is the smallest loss level covering 95%.

  1. AVaR is Rs 100 lakh because the worst outcome always defines VaR
  2. BVaR is Rs 10 lakh because the cumulative probability of losses of at most Rs 10 lakh or gains reaches 96%Correct
  3. CVaR is Rs 20 lakh because gains are ignored
  4. DVaR is Rs 55 lakh, the average of the losses
  5. VaR is zero because the expected outcome is a gain

Explanation

Order outcomes: loss 100 (0.04), loss 10 (0.46), gain 20 (0.50). Probability that loss is at most 10 is 0.96, which is at least 95%, while the probability of loss at most -20 (gain) is only 0.50. So the 95% VaR is the smallest loss level with cumulative probability of at least 95%, namely Rs 10 lakh. Losing Rs 100 lakh has only 4% probability, below the 5% tail.

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