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FRM Part II · FRM Exam Part II · Estimating Market Risk Measures: An Introduction and Overview

A bank has two independent bonds, each with a 4% chance of default within the horizon, and a loss of USD 100 if default occurs (otherwise zero). At the 95% confidence level, VaR is calculated for each bond alone and for a portfolio holding both. Which result is correct?

Each bond has a 95% VaR of zero because default probability of 4% is below the 5% tail, but the portfolio has a 7.84% chance of at least one default, so its VaR is 100. This exceeds the sum of zero, showing VaR violates subadditivity.

  1. AEach bond has VaR of 0, but the portfolio VaR is 100, exceeding the sum of the individual VaRsCorrect
  2. BEach bond has VaR of 100, and the portfolio VaR is 200, equal to the sum
  3. CEach bond has VaR of 0, and the portfolio VaR is 0, so VaR is subadditive
  4. DEach bond has VaR of 100, but the portfolio VaR is 100, below the sum

Explanation

Each bond's loss exceeds zero with probability 4%, below the 5% tail, so its 95% VaR is 0. For the portfolio, P(at least one default) = 1 − 0.96² = 7.84% > 5%, so the 95% VaR is 100. Then 100 > 0 + 0, a violation of subadditivity.

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