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FRM Part II · FRM Exam Part II · Estimating Market Risk Measures: An Introduction and Overview

Two independent bonds each have a 4% chance of a USD 100 million loss and a 96% chance of zero loss. A risk manager computes the 95% VaR of each bond individually and of the combined two-bond portfolio. Which result is correct?

Each bond has a 95% VaR of zero because its 4% default probability is under 5%. The portfolio has a 7.84% chance of at least one default, so its 95% VaR is USD 100 million, exceeding the sum of zero and zero and violating subadditivity.

  1. AEach bond has 95% VaR of 0; the portfolio 95% VaR is USD 100 million, showing a subadditivity violationCorrect
  2. BEach bond has 95% VaR of USD 100 million; the portfolio VaR is USD 200 million
  3. CEach bond has 95% VaR of 0; the portfolio 95% VaR is 0
  4. DEach bond has 95% VaR of USD 100 million; the portfolio VaR is USD 100 million

Explanation

Each bond has a 4% loss probability, below the 5% tail, so individual 95% VaR is 0. For the portfolio, the probability of at least one default is 1 - 0.96^2 = 7.84%, above 5%, so 95% VaR is USD 100 million (a single default). Then 100 > 0 + 0, so subadditivity fails.

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