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

Portfolio A has a one-day 99% VaR of 10 million and Portfolio B has 6 million. The two portfolios are combined and the combined 99% VaR is 17 million. Which statement is most accurate?

The combined VaR of 17 million exceeds the sum of the stand-alone VaRs of 16 million, violating subadditivity. This illustrates that VaR is not a coherent risk measure, whereas expected shortfall always satisfies subadditivity.

  1. AThe result violates subadditivity, which shows VaR is not a coherent risk measureCorrect
  2. BThe result is consistent with a diversification benefit of 1 million
  3. CThe result implies correlation between A and B is negative
  4. DThe result is impossible for any coherent risk measure but is normal for expected shortfall

Explanation

Subadditivity requires combined risk to be no more than the sum of the parts: 17 exceeds 10 + 6 = 16. VaR can violate this for non-elliptical distributions, which is why it is not coherent. Expected shortfall is coherent, so it could not produce such a result; a diversification benefit would require a combined figure below 16.

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