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Show, with examples, what happens to the covariance and correlation of two random variables when linear transformations are applied to them.

FRM® Part I · Quantitative Analysis · Multivariate Random Variables · FRM1.QA.4.f

The idea

Linear transformations change covariance and correlation in a simple way. For constants a, b, c and d, the covariance of the transformed variables, Cov(aX + b, cY + d), is: Transformed covariance = a × c × Cov(X, Y). Adding a constant has no effect, because the mean shifts with the variable and the deviations stay the same. Multiplying scales the covariance by the product of the multipliers, including their signs. Correlation is unit-free, so it is unchanged when a × c is positive. Its sign flips, with the same magnitude, when a × c is negative. For example, converting two returns from decimals to percent multiplies the covariance by 10,000 and leaves the correlation untouched. The usual mistakes are scaling the correlation by the multipliers, forgetting the sign, or letting constants matter.

Formulas

Covariance after linear transformations

Cov(aX + b, cY + d) = a × c × Cov(X, Y)

a
the multiplier on X
c
the multiplier on Y
b, d
added constants
Cov(X, Y)
the original covariance

Constants drop out and the multipliers keep their signs. Correlation keeps its size and flips sign only when a × c is negative.

A practice question

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

X and Y have covariance Cov(X, Y). What is Cov(X + 5, Y)?

  1. Cov(X, Y) + 5, because the constant shifts every product of deviations.Why not: Adds the constant to the covariance; a shift moves X and its mean together, leaving every deviation unchanged.
  2. 5 x Cov(X, Y), because the constant multiplies the covariance.Why not: Treats an added constant as a multiplier; only multiplicative factors scale covariance.
  3. Cov(X, Y) + Var(Y) x 5, because the constant interacts with Y.Why not: Invents an interaction term; a constant is not random and has zero covariance with anything.
  4. Cov(X, Y); adding a constant does not change covariance. - correct

Covariance is built from deviations from the means. Adding 5 to X raises its mean by 5 too, so (X + 5) - E[X + 5] = X - E[X]. Hence Cov(X + 5, Y) = Cov(X, Y).

A harder one

Cov(X, Y) = 0.006. What is Cov(2X + 5, -3Y + 1)?

  1. -0.036 - correct
  2. 0.036Why not: Drops the negative sign of -3; the covariance is multiplied by 2 x (-3) = -6.
  3. -0.006Why not: Adds the multipliers (2 + (-3) = -1) instead of multiplying them.
  4. 0.216Why not: Squares the multipliers, as in the variance rule; covariance scales with the product ac, not with a² c².

Cov(aX + b, cY + d) = ac Cov(X, Y). With a = 2 and c = -3: -6 x 0.006 = -0.036; the constants 5 and 1 do not matter.

The hardest tier

X and Y have standard deviations 4 and 5 and correlation -0.30. Define U = 3X + 10 and V = -2Y + 7. What are Cov(U, V) and Corr(U, V)?

  1. Cov(U, V) = -36; Corr(U, V) = -0.30.Why not: Uses 3 x 2 x (-6) and ignores the minus sign of the coefficient on Y; the covariance becomes +36 and the correlation flips to +0.30.
  2. Cov(U, V) = 36; Corr(U, V) = -0.30.Why not: Gets the covariance right but forgets that a negative coefficient on V flips the sign of the correlation.
  3. Cov(U, V) = 36; Corr(U, V) = +0.30. - correct
  4. Cov(U, V) = -6; Corr(U, V) = -0.30.Why not: Assumes the linear transformations leave the covariance and correlation unchanged.

Cov(X, Y) = -0.30 x 4 x 5 = -6. Cov(U, V) = 3 x (-2) x (-6) = 36. The signs of the coefficients differ, so the correlation flips from -0.30 to +0.30; the constants have no effect.

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