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Relate coskewness and cokurtosis back to the skewness and kurtosis of a single variable.

FRM® Part I · Quantitative Analysis · Sample Moments · FRM1.QA.5.l

The idea

Coskewness and cokurtosis extend skewness and kurtosis to several variables. Coskewness S(X,X,Y) is a standardised third cross-moment: with X twice and Y once, Coskewness = E[(X − μX)² (Y − μY)] ÷ (σX² × σY). Cokurtosis K(X,X,Y,Y) is the fourth analogue: with X twice and Y twice, Cokurtosis = E[(X − μX)² (Y − μY)²] ÷ (σX² × σY²). When all arguments are the same variable they reduce to single-variable measures: S(X,X,X) is the skewness of X and K(X,X,X,X) is its kurtosis. If Y is the negative of X, an odd number of Y factors flips the sign. For n variables there are C(n + k − 1, k) distinct moments of order k, counting the single-variable ones, which matters for portfolio tail risk.

Formulas

Coskewness of two variables

S(X,X,Y) = E[(X − μX)² × (Y − μY)] ÷ (σX² × σY)

X, Y
the two random variables
μX
the mean of X
μY
the mean of Y
σX
the standard deviation of X
σY
the standard deviation of Y

With every argument the same variable it reduces to skewness. If Y is the negative of X, the single Y factor flips the sign.

Number of distinct cross-moments

Count = C(n + k − 1, k)

n
the number of variables
k
the order of the moment, 3 for coskewness and 4 for cokurtosis
C(n + k − 1, k)
the number of combinations of k from n + k − 1

Counts the single-variable skewness or kurtosis measures too. The count grows quickly with the number of assets.

A practice question

From the question bank, with the reason each wrong option is wrong.

How are coskewness and cokurtosis related to the skewness and kurtosis of a single random variable X?

  1. Skewness is the coskewness of X and a constant, and kurtosis the cokurtosis of X and a constant.Why not: Coskewness and cokurtosis relate to X's own moments when all arguments are X, not when one argument is a constant.
  2. Coskewness and cokurtosis are unrelated to single-variable skewness and kurtosis.Why not: They are generalisations: with identical arguments they reduce to the single-variable measures.
  3. Skewness is the coskewness of X and Y when X and Y are uncorrelated.Why not: Uncorrelated variables do not reduce coskewness to skewness; identical arguments are required.
  4. Skewness is the coskewness of X with itself, S(X,X,X), and kurtosis is the cokurtosis of X with itself, K(X,X,X,X). - correct

Coskewness is the standardised third cross-moment of three variables and cokurtosis the fourth cross-moment of four. Setting every argument equal to X gives E[(X-mu)³]/sigma³ and E[(X-mu)^4]/sigma^4, which are skewness and kurtosis.

A harder one

The skewness of X is 0.8. Let Y = -X. What are the coskewness S(X,X,Y) and S(X,Y,Y)?

  1. +0.8 and +0.8Why not: Takes both as skewness of X; the sign of Y's deviation is the opposite of X's, flipping an odd number of factors.
  2. -0.8 and -0.8Why not: Flips the sign in both, but S(X,Y,Y) contains two Y deviations, so the two sign changes cancel.
  3. -0.8 and +0.8 - correct
  4. +0.8 and -0.8Why not: Reverses the signs: S(X,X,Y) contains one Y factor (a sign change); S(X,Y,Y) contains two (no net change).

Because y - y-bar = -(x - x-bar) and sd_Y = sd_X, S(X,X,Y) has one sign flip and equals -skew(X) = -0.8. S(X,Y,Y) has two and equals +skew(X) = +0.8.

The hardest tier

For standardised variables from a bivariate normal distribution with correlation rho, the cokurtosis K(X,X,Y,Y) = E[X² Y²] = 1 + 2 rho². For rho = 0.8, what is K(X,X,Y,Y), and how does it relate to the kurtosis of a single variable as rho approaches 1?

  1. 1.64; it approaches 2.Why not: Uses 1 + rho² instead of 1 + 2 rho²; the limit then misses the single-variable kurtosis of 3.
  2. 2.28; it approaches 1, as for independent variables.Why not: The value is right but at rho = 1, Y = X, so K(X,X,X,X) = 3, not 1 (1 is the independent case).
  3. 3.00; it is 3 for every correlation, as both are normal.Why not: Cokurtosis of normals depends on the correlation; it equals 3 only when rho = 1 (or -1).
  4. 2.28; it approaches 3, the kurtosis of a single normal variable, because Y becomes X. - correct

1 + 2 x 0.8² = 2.28. At rho = 1 the formula gives 3, the kurtosis of a normal variable, since K(X,X,X,X) is just kurtosis; at rho = 0 it gives 1 for independent variables.

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