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FRM Part II · FRM Exam Part II · Portfolio Risk: Analytical Methods

A portfolio has a 1-day 95% VaR of $4.0 million. The risk manager computes the component VaR of each of its four positions using the standard Euler-based decomposition. Which statement about these component VaRs is correct?

Component VaRs sum exactly to the portfolio VaR. This follows from Euler's theorem because VaR is homogeneous of degree one in position sizes, regardless of correlations. Individual components can be negative for hedges, but the total always reconciles to portfolio VaR.

  1. AThey sum exactly to the portfolio VaR of $4.0 millionCorrect
  2. BThey sum to the weighted average of the standalone VaRs
  3. CThey sum to the portfolio VaR only if all correlations equal zero
  4. DThey are always positive for every position

Explanation

Component VaR is built from the marginal VaR multiplied by the position size, and because VaR is homogeneous of degree one, Euler's theorem makes the components add up to total VaR. It holds for any correlations. Components can be negative for hedging positions, so the last option is wrong.

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