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

A bank holds two independent loans, each of USD 10 million. Each defaults with probability 4% over the year, causing a total loss of the USD 10 million (no recovery); otherwise the loss is zero. Using a 95% confidence level, which statement about VaR is correct?

Each loan defaults with 4% probability, below the 5% tail, so its VaR is zero. The portfolio has a 7.84% chance of at least one default, so its 95% VaR is USD 10 million. That exceeds the zero sum of individual VaRs, violating subadditivity.

  1. AEach loan has a 95% VaR of zero, but the two-loan portfolio has a 95% VaR of USD 10 million, violating subadditivityCorrect
  2. BEach loan has a 95% VaR of USD 10 million, and the portfolio VaR is USD 20 million
  3. CEach loan and the portfolio have a 95% VaR of zero
  4. DEach loan has a 95% VaR of zero and the portfolio has a 95% VaR of USD 20 million

Explanation

For one loan, P(loss)=4%, below the 5% tail, so 95% VaR is 0. For the portfolio, P(at least one default)=1-0.96^2=1-0.9216=7.84%, above 5%, so the 95% VaR is USD 10 million. P(both default)=0.16%, so USD 20 million is not reached. Sum of individual VaRs is 0 while portfolio VaR is 10 million, violating subadditivity.

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