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FRM Part I · FRM Exam Part I · Measures of Financial Risk

Two independent bonds each have a 4% chance of defaulting over one year, with a loss of USD 10 million on default and no loss otherwise. Using a 95% confidence level and the one-year loss distribution, what are the VaR of each bond individually and the VaR of a portfolio holding both bonds?

Each bond has VaR of zero because its default probability of 4% is below the 5% tail. The portfolio has a 7.84% chance of at least one default, so its 95% VaR is USD 10 million, exceeding the sum of zero and illustrating VaR's failure of subadditivity.

  1. AEach bond VaR is USD 0; portfolio VaR is USD 10 millionCorrect
  2. BEach bond VaR is USD 10 million; portfolio VaR is USD 10 million
  3. CEach bond VaR is USD 0; portfolio VaR is USD 0
  4. DEach bond VaR is USD 10 million; portfolio VaR is USD 20 million

Explanation

Each bond has a 4% chance of loss, less than 5%, so the 95% VaR is 0. For the portfolio, P(at least one default) = 1 - 0.96^2 = 7.84%, which exceeds 5%, so the 95% quantile loss is USD 10 million. Portfolio VaR 10 > 0 + 0, so VaR violates subadditivity.

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