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

Two independent bonds each default with probability 4% over the horizon, each with a loss of USD 100 if default occurs and zero otherwise. The 95% VaR is defined as the smallest loss level L such that P(loss > L) is at most 5%. What does the comparison of VaR for the individual bonds and for the two-bond portfolio show?

Each bond has a 95% VaR of zero because default probability is 4%, under 5%. The portfolio has a probability of at least one default of 7.84%, so its 95% VaR is 100. That exceeds the sum of individual VaRs of zero, violating subadditivity.

  1. AEach bond has a 95% VaR of 100 and the portfolio has a 95% VaR of 200, showing VaR is subadditive here
  2. BEach bond has a 95% VaR of 0 and the portfolio has a 95% VaR of 100, showing VaR violates subadditivityCorrect
  3. CEach bond has a 95% VaR of 0 and the portfolio has a 95% VaR of 0, showing VaR is additive here
  4. DEach bond has a 95% VaR of 100 and the portfolio has a 95% VaR of 100, showing VaR is subadditive here

Explanation

For one bond, P(loss>0)=4%, which is at most 5%, so VaR is 0. For the portfolio, P(at least one default)=1-0.96^2=7.84%, above 5%. P(loss>100)=0.04^2=0.16%, below 5%, so VaR is 100. Portfolio VaR 100 exceeds the sum of 0+0, violating subadditivity.

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