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Define covariance and correlation, and explain why correlation is not the same thing as dependence.

FRM® Part I · Quantitative Analysis · Measuring Returns, Volatility, and Correlation · FRM1.QA.12.f

The idea

Covariance measures how two variables move together, as the average product of their deviations from their means: Covariance = E[(X − μX) × (Y − μY)]. Its units are the product of the units of X and Y, so its size is hard to interpret. Correlation rescales it to be unit-free and bounded by −1 and +1: Corr = Cov(X, Y) ÷ (σX × σY). Correlation captures only linear association. Variables can be perfectly dependent and still have zero correlation, as when X is symmetric around zero and Y is given by: Y = X². Independence implies zero correlation, but zero correlation implies independence only in special cases such as jointly normal variables. When a question gives the standard deviations and the correlation and asks for the covariance, multiply them all together: Covariance = Corr × σX × σY. Do not read a zero correlation as the absence of any relationship.

Formulas

Covariance from correlation and volatilities

Cov(X, Y) = ρ × σX × σY

ρ
the correlation of X and Y
σX
the standard deviation of X
σY
the standard deviation of Y

Correlation measures only linear association, so a zero covariance does not rule out dependence unless the variables are jointly normal.

A practice question

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

Two assets have return standard deviations of 4% and 5% and a correlation of 0.30. Their covariance is closest to:

  1. 0.0020Why not: Multiplies the standard deviations and leaves out the correlation.
  2. 0.0270Why not: Multiplies the correlation by the sum of the standard deviations instead of the product.
  3. 150.0Why not: Divides the correlation by the product of the standard deviations, the inverse of the correct step.
  4. 0.0006 - correct

Cov = rho x sigma_1 x sigma_2 = 0.30 x 0.04 x 0.05 = 0.0006.

A harder one

Two returns have a covariance of 0.0120, and standard deviations of 15% and 20%. Their correlation is closest to:

  1. 0.08Why not: Divides the covariance by one standard deviation only (0.15).
  2. 0.40 - correct
  3. 0.19Why not: Divides by the sum of the two variances (0.0225 + 0.04) instead of the product of the standard deviations.
  4. 0.034Why not: Divides by the sum of the two standard deviations (0.35).

rho = 0.0120 / (0.15 x 0.20) = 0.0120 / 0.0300 = 0.40.

The hardest tier

X is standard normal. Y equals X multiplied by S, where S is +1 or -1 with equal probability, independent of X. Y is also standard normal, and the correlation of X and Y is zero. Which conclusion is correct?

  1. X and Y are independent, because both are normal and their correlation is zero.Why not: Zero correlation implies independence only if the pair is jointly normal, which is not the case here.
  2. X and Y are uncorrelated but not independent, because |Y| = |X| and the pair is not jointly normal. - correct
  3. X and Y have a correlation of 1 in absolute value, because |Y| = |X|.Why not: The sign of Y is random, so the linear relationship cancels and the correlation is 0.
  4. X and Y are negatively dependent, so their correlation must be below zero.Why not: The sign flips with equal probability, so there is no systematic negative association.

E[XY] = E[S X²] = E[S] E[X²] = 0, so the covariance is 0. But Y is determined by X up to a sign, so they are dependent. Marginal normality does not imply joint normality.

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