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

A portfolio manager decomposes total portfolio volatility into each position's contribution, defined as the position weight multiplied by its marginal contribution to risk (MCTR). Which statement about these contributions is correct?

Component contributions, weight times marginal contribution to risk, sum exactly to total portfolio volatility. This follows from Euler's theorem because volatility is homogeneous of degree one in the weights, so the decomposition is additive regardless of correlations.

  1. AThey sum exactly to total portfolio volatility because volatility is homogeneous of degree one in the weightsCorrect
  2. BThey sum to portfolio variance rather than volatility
  3. CThey sum to more than portfolio volatility whenever correlations are below one
  4. DThey sum to zero when the portfolio is fully invested

Explanation

Portfolio volatility is homogeneous of degree one in the weights, so by Euler's theorem the sum of w_i times the marginal contribution equals total volatility. This holds for any correlation structure. The claim that the sum exceeds volatility confuses the contributions with standalone volatilities, which do sum to more than portfolio volatility when correlations are below one.

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