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

A portfolio has a total delta-normal VaR of USD 8.0 million. The component VaRs of its three sub-portfolios are USD 4.5 million, USD 2.5 million and USD 1.0 million. Which statement about these figures is correct?

Component VaRs sum exactly to total portfolio VaR. Each is position size times marginal VaR, and because VaR is homogeneous of degree one, Euler's theorem makes the decomposition additive. Here 4.5, 2.5 and 1.0 add to 8.0 million, so the figures reconcile with total VaR.

  1. AThe component VaRs sum to USD 8.0 million, matching total VaR, because component VaR is an additive decompositionCorrect
  2. BThe component VaRs sum to less than total VaR because diversification is excluded
  3. CThe component VaRs sum to more than total VaR because each ignores diversification
  4. DThe component VaRs are not additive, so they cannot be compared with total VaR

Explanation

Component VaR for each position equals weight times marginal VaR, and by Euler's theorem for homogeneous degree-one risk measures these sum exactly to portfolio VaR. Here 4.5+2.5+1.0 = 8.0. Stand-alone VaRs, not component VaRs, would sum to more than total VaR.

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