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

A risk manager decomposes portfolio VaR into component VaRs, each equal to the position's weight multiplied by its marginal VaR, and finds that the components add exactly to total portfolio VaR. Which property of the VaR measure makes this exact additive decomposition possible?

Component VaRs sum to total VaR because portfolio VaR is homogeneous of degree one in position sizes. Euler's theorem then says the sum of each weight times its marginal VaR equals total VaR. Subadditivity or coherence alone would not guarantee an exact additive decomposition.

  1. AVaR is homogeneous of degree one in the position sizes, so Euler's theorem appliesCorrect
  2. BVaR is subadditive for all return distributions
  3. CVaR is independent of the correlations among positions
  4. DVaR is a coherent measure under any confidence level

Explanation

Under the parametric approach, scaling every position by a factor scales VaR by the same factor, which is homogeneity of degree one. Euler's theorem then implies that the sum of weight times marginal VaR equals total VaR. Subadditivity is a separate property and does not produce an additive decomposition.

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