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FRM Part I · FRM Exam Part I · Calculating and Applying VaR

A bank has two independent bonds, each with a 4% chance of defaulting over one year, causing a loss of USD 100 if default occurs and zero otherwise. Defaults are independent. Using the 95% confidence level, what is the VaR of the combined portfolio compared with the sum of the individual VaRs?

Combined VaR is 100, while the individual VaRs sum to zero. Each bond defaults with 4% probability, below the 5% tail, so its VaR is zero. The chance of at least one default is 7.84%, above 5%, so portfolio VaR is 100, violating subadditivity.

  1. ACombined VaR is 200, which exceeds the sum of 0
  2. BCombined VaR is 100, which exceeds the sum of 0Correct
  3. CCombined VaR is 100, which equals the sum of 100
  4. DCombined VaR is 0, which equals the sum of 0

Explanation

Each bond has a 4% loss probability, below the 5% tail, so each individual 95% VaR is 0, sum 0. For the portfolio, P(at least one default) = 1 − 0.96² = 7.84%, which exceeds 5%, so the 95% VaR is 100 (the probability of losing 200 is only 0.16%). Thus the combined VaR of 100 exceeds the sum of 0, showing VaR violates subadditivity.

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