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

Two independent bonds each have a one-year default probability of 4% and, if default occurs, a loss of USD 100 million; otherwise the loss is zero. Using a 95% confidence level, what are the stand-alone VaR of one bond and the VaR of the portfolio holding both bonds?

Each bond's 95% VaR is zero because the no-default probability of 96% exceeds 95%. For the two-bond portfolio, no-default probability is only 92.16%, so the 95% VaR is USD 100 million. This illustrates VaR violating subadditivity.

  1. AStand-alone VaR USD 0; portfolio VaR USD 100 millionCorrect
  2. BStand-alone VaR USD 100 million; portfolio VaR USD 100 million
  3. CStand-alone VaR USD 0; portfolio VaR USD 200 million
  4. DStand-alone VaR USD 100 million; portfolio VaR USD 200 million

Explanation

For one bond, P(loss=0)=96% which exceeds 95%, so the 95% VaR is 0. For the portfolio, P(no default)=0.96^2=92.16%, and P(at most one default)=1-0.04^2=99.84%. The 95% quantile lies above 92.16% but below 99.84%, so VaR is USD 100 million. Sum of stand-alone VaRs is 0 < 100, showing subadditivity violation.

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