FRM Part I · FRM Exam Part I · Measures of Financial Risk
Two independent bonds each have a 4% chance of default within a year, with a loss of 100 on default and no loss otherwise. Using the 95% confidence level, what are the stand-alone 95% VaR of each bond and the 95% VaR of the combined two-bond portfolio?
Each bond has a 95% VaR of 0 because default probability of 4% is below 5%, but the two-bond portfolio has a 95% VaR of 100 because the chance of no default is only 92.16%. This breaches subadditivity.
- AEach 0; combined 100Correct
- BEach 100; combined 100
- CEach 0; combined 0
- DEach 100; combined 200
Explanation
Each bond has a 96% chance of zero loss, which exceeds 95%, so each 95% VaR is 0. For the portfolio, P(no default) = 0.96^2 = 0.9216, so P(loss of 0) is below 95%, and P(loss of at most 100) = 0.9216 + 2(0.04)(0.96) = 0.9984, so VaR is 100. Combined 100 exceeds the sum of 0, showing subadditivity violation.
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