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Work out the variance of a portfolio made up of two assets

CFA® Level I · Portfolio Management · Portfolio Risk and Return: Part I · PM.1.b

The idea

Portfolio variance is the sum of each asset's weighted variance plus twice the product of the weights and the covariance between them. The covariance term is what makes a portfolio different from the sum of its parts: because it can be negative, combining two risky assets can produce a portfolio less risky than either. Expressing covariance as the correlation times the two standard deviations makes the mechanism visible — the lower the correlation, the smaller the third term and the greater the reduction in risk relative to a simple weighted average.

Formulas

Variance of a two-asset portfolio

portfolio variance = w₁² × σ₁² + w₂² × σ₂² + 2 × w₁ × w₂ × ρ × σ₁ × σ₂

w₁, w₂
the weight of each asset
σ₁, σ₂
the standard deviation of each asset's return
ρ
the correlation between the two returns

The last term carries the correlation and is the only one that can be negative. At ρ = 1 the whole expression collapses to the weighted average of the standard deviations and nothing is diversified away; below 1 something is.

A practice question

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

Which term in the two-asset variance formula captures the diversification effect?

  1. The weighted variance of the first assetWhy not: That term describes one asset in isolation and is present however the two move together.
  2. The covariance term between the two assets — correct
  3. The sum of the weightsWhy not: The weights sum to one by construction, which carries no information about how the assets interact.

The two variance terms are always positive. Only the covariance term can be negative, and it is what allows the portfolio to be less risky than a weighted average.

A harder one

Two assets each have a standard deviation of 20% and are held equally weighted with a correlation of zero. What is the portfolio standard deviation?

  1. 20%Why not: The answer if the two moved perfectly together. Zero correlation means the cross term drops out and the risk falls.
  2. 10%Why not: Divides by two rather than by the square root of two. Risk falls with the square root of the number of independent assets.
  3. About 14.1% — correct

Variance is 0.25(0.04) + 0.25(0.04) + 0 = 0.02, and the square root is 0.1414. Zero correlation alone cuts risk by nearly 30% with no loss of expected return.

The hardest tier

Two assets have equal standard deviations and a correlation of exactly -1. What weighting eliminates portfolio risk entirely?

  1. Equal weights of 50% each — correct
  2. All capital in the lower-variance assetWhy not: The standard deviations are equal, so there is no lower-variance asset, and a single holding leaves its own risk intact.
  3. No weighting can eliminate riskWhy not: At a correlation of −1 the two move in exactly opposite directions, so equal weights cancel the movements entirely.

With perfect negative correlation and equal volatilities, equal weights make the two deviations cancel exactly at every outcome. Unequal volatilities would require weights inversely proportional to them.

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