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Find the variance when two random variables are combined using weights.

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

The idea

Combine two random variables X and Y with weights a and b, and the variance of aX + bY is: Variance = a² × Var(X) + b² × Var(Y) + 2ab × Cov(X, Y). When you are given a correlation ρ and standard deviations σX and σY instead, use Covariance = ρ × σX × σY. The weights are squared on the variances but appear once, doubled, on the covariance, and their signs matter, so for the difference X − Y, with weights 1 and −1: Spread variance = Var(X) + Var(Y) − 2 × Cov(X, Y). If the variables are uncorrelated, the cross term vanishes and the variances simply add, even when you subtract. The standard deviation is the square root of the variance. A positive correlation lowers the benefit of diversification and, in a long-short position, makes the spread less volatile. Common errors are omitting the 2, not squaring the weights and ignoring the sign of the covariance term.

Formulas

Variance of a weighted combination of two variables

Var(aX + bY) = a² × Var(X) + b² × Var(Y) + 2 × a × b × Cov(X, Y)

a
the weight on X
b
the weight on Y
Var(X)
the variance of X
Var(Y)
the variance of Y
Cov(X, Y)
the covariance, ρ × σX × σY

Weights are squared on the variances but appear once, doubled, on the covariance. A short position gives a negative weight, which flips the sign of the cross term.

A practice question

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

For random variables X and Y and constants a and b, which expression gives Var(aX + bY)?

  1. a² Var(X) + b² Var(Y)Why not: Omits the covariance term, which is correct only when X and Y are uncorrelated.
  2. a Var(X) + b Var(Y) + 2ab Cov(X, Y)Why not: Does not square the weights on the variances; variance scales with the square of a constant.
  3. a² Var(X) + b² Var(Y) + ab Cov(X, Y)Why not: Drops the factor 2 on the cross term; the covariance appears twice in the expansion of (aX + bY)².
  4. a² Var(X) + b² Var(Y) + 2ab Cov(X, Y) - correct

Expanding the variance of aX + bY gives a² Var(X) + b² Var(Y) + 2ab Cov(X, Y). The cross term is what captures diversification: it can reduce or increase the combined variance.

A harder one

A portfolio is 60 percent in asset X (volatility 20 percent) and 40 percent in asset Y (volatility 10 percent). The correlation of the returns is 0.30. What is the portfolio volatility?

  1. 16.00 percentWhy not: Takes the weighted average of the two volatilities, which holds only if the correlation is +1.
  2. 13.74 percent - correct
  3. 12.65 percentWhy not: Leaves out the covariance term 2 w1 w2 rho sigma1 sigma2, i.e. assumes zero correlation.
  4. 1.888 percentWhy not: Reports the portfolio variance (0.01888) as a percentage rather than taking the square root.

Variance = 0.6²(0.04) + 0.4²(0.01) + 2(0.6)(0.4)(0.30)(0.20)(0.10) = 0.0144 + 0.0016 + 0.00288 = 0.01888. Volatility = sqrt(0.01888) = 13.74 percent.

The hardest tier

A trader is long one unit of asset X and short one unit of asset Y, so the P&L per unit is X - Y. The volatilities are 15 percent for X and 12 percent for Y, with correlation 0.80. What is the volatility of the spread X - Y?

  1. 25.63 percentWhy not: Adds the covariance term instead of subtracting it; with weights +1 and -1 the cross term is -2 Cov.
  2. 19.21 percentWhy not: Ignores the correlation entirely, which treats X and Y as uncorrelated.
  3. 9.00 percent - correct
  4. 3.00 percentWhy not: Subtracts the two volatilities, which is exact only when the correlation is +1; here the correlation is 0.80 and the full variance formula is needed.

Var(X - Y) = 0.15² + 0.12² - 2(0.80)(0.15)(0.12) = 0.0225 + 0.0144 - 0.0288 = 0.0081. Volatility = 9.00 percent. A high positive correlation makes the long-short spread much less volatile than either leg.

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