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FRM Part II · FRM Exam Part II · VaR and Risk Budgeting in Investment Management

A portfolio manager decomposes the 1-day 95% VaR of a multi-asset portfolio into the contribution of each position using the standard Euler allocation. Which statement about the resulting component VaRs is correct?

Component VaRs sum exactly to total portfolio VaR. VaR is homogeneous of degree one, so Euler's theorem means position size times marginal VaR adds up to the whole. Stand-alone VaRs do not add up this way, and hedging positions can have negative components.

  1. AThey sum exactly to the total portfolio VaRCorrect
  2. BThey sum to the sum of the stand-alone VaRs of the positions
  3. CThey sum to the portfolio VaR only when all correlations equal zero
  4. DThey are always positive for every position in the portfolio

Explanation

Component VaR is the position size times its marginal VaR. Because VaR is homogeneous of degree one, Euler's theorem makes the components add up to total VaR. The stand-alone VaRs sum to more than portfolio VaR when diversification exists, and a hedge position can have a negative component.

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