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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 99% VaR of a two-asset portfolio into component VaRs using the portfolio's marginal VaRs and position sizes. Which statement about the component VaRs is correct?

Component VaRs sum exactly to total portfolio VaR. This follows from Euler's theorem, because VaR is homogeneous of degree one in positions. Stand-alone VaRs only sum to total VaR in the special case of perfect correlation, so they are a different measure.

  1. AThey sum exactly to the total portfolio VaRCorrect
  2. BThey sum to the total portfolio VaR only if all asset correlations equal one
  3. CThey sum to the sum of the stand-alone VaRs of the positions
  4. DThey sum to the VaR of the portfolio after the largest position is removed

Explanation

Component VaR is position size times marginal VaR. Because portfolio VaR is homogeneous of degree one in position sizes, Euler's theorem makes the component VaRs add up exactly to total VaR. The sum of stand-alone VaRs is larger unless correlations are all one, so that option describes a different quantity.

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