Moments of Sums and Portfolio Variance Explained

Understand Moments of Sums and Portfolio Variance

Key formulas to remember

Mean of a linear combination

E(aX + bY) = a·E(X) + b·E(Y)

Always true. No condition on correlation or independence.

Variance of a scaled variable

Var(aX + b) = a² · Var(X)

The constant b drops out. The multiplier a is squared.

Variance of a sum (general)

Var(X + Y) = Var(X) + Var(Y) + 2·Cov(X, Y)

For a difference, the sign of the covariance term flips: Var(X − Y) = Var(X) + Var(Y) − 2·Cov(X, Y).

Covariance and correlation

Cov(X, Y) = ρ · σX · σY

Correlation ρ = Cov(X, Y) ÷ (σX · σY), between -1 and +1.

Two-asset portfolio variance

σp² = wA²σA² + wB²σB² + 2·wA·wB·ρ·σA·σB

Portfolio volatility σp = √σp². Weights usually sum to 1.

Weighted linear combination

Var(aX + bY) = a²σX² + b²σY² + 2ab·Cov(X, Y)

The same rule with any constants a and b, not only weights.

Variance of a sum of n variables

Var(ΣXi) = Σ Var(Xi) + 2·Σ(i<j) Cov(Xi, Xj)

If all pairwise covariances are zero, variances simply add.

Equal-weight, equal-variance, equal-correlation portfolio

σp² = σ² · [1/n + (1 − 1/n)·ρ]

As n grows large, variance approaches ρ·σ². This is the part that diversification cannot remove.

How to solve Moments of Sums and Portfolio Variance questions

1. 1Write down the combination, for example P = wA·RA + wB·RB, and list the weights or constants.
2. 2List what you are given: variances or volatilities, and either covariance or correlation. Convert volatility to variance by squaring.
3. 3If you have correlation, convert to covariance using Cov = ρ·σA·σB.
4. 4Plug into Var = a²σX² + b²σY² + 2ab·Cov. Watch the sign of b if the position is short or a difference.
5. 5Compute each of the three terms separately and add them.
6. 6Take the square root only if the question asks for volatility or standard deviation.
7. 7Check reasonableness: portfolio volatility must lie between the weighted-average volatility (ρ = 1) and the minimum possible (ρ = -1).
8. 8If a time horizon is involved, scale after finding the variance, for example daily to annual by multiplying variance by the number of periods.

Quickest way: Three-term shortcut with a sanity check

Common mistakes in Moments of Sums and Portfolio Variance

• Adding volatilities instead of variances, or treating σp as the weighted average of volatilities.

  Expected return is a simple weighted average, so students assume risk behaves the same way.

  Fix: Weighted-average volatility is correct only when ρ = +1. Otherwise use the full variance formula and take the square root at the end.

• Forgetting to square the weights.

  Students remember the weights multiply returns and carry them into variance unchanged.

  Fix: Variance scales with the square of the multiplier: Var(aX) = a²Var(X). Only the cross term has the product wA·wB.

• Dropping the factor of 2 on the covariance term.

  The cross term appears twice, as Cov(A,B) and Cov(B,A), but this is easy to forget.

  Fix: Write the formula out every time: 2·wA·wB·ρ·σA·σB.

• Using volatility where variance is needed, or reporting variance when volatility is asked.

  Questions switch between σ and σ² without warning.

  Fix: Underline what is given and what is asked. Square given volatilities; take the root of the final variance if volatility is requested.

• Ignoring the sign for short positions or differences.

  Students assume all weights are positive.

  Fix: Keep the negative sign on the weight. For a long-short pair, the cross term is negative when ρ is positive, so Var(X − Y) = σX² + σY² − 2Cov(X, Y).

• Saying zero correlation means independence, or that diversification removes all risk.

  Both ideas are oversimplified rules of thumb.

  Fix: Zero correlation only rules out a linear relationship. Diversification lowers risk unless ρ = 1, but with positive average correlation a floor of risk remains.

Worked examples