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FRM Part I · FRM Exam Part I · Multivariate Random Variables

For two asset returns X and Y, the standard deviations are σX = 2 and σY = 3. The central moments are E[(X-μX)^2(Y-μY)] = 12 and E[(X-μX)(Y-μY)^2] = -18. What is the coskewness S(X,Y,Y) = E[(X-μX)(Y-μY)^2]/(σX σY^2)?

Coskewness S(X,Y,Y) is the third cross central moment divided by σX times σY squared. That is -18 divided by (2 × 9 = 18), giving -1.0. The other given moment, 12, belongs to S(X,X,Y) and is not used.

  1. A-1.0Correct
  2. B-1.5
  3. C-3.0
  4. D-0.5

Explanation

Coskewness is normalized by σX × σY^2 because X appears once and Y twice: 2 × 9 = 18. So S = -18/18 = -1.0. Using σX^2 σY (12) gives -1.5, which uses the normalizer for S(X,X,Y). Using σX σY (6) gives -3.0, and σX^2 σY^2 (36) gives -0.5. The 12 moment is a distractor for this question.

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