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

CFA® Level I · Portfolio Construction · 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.

How can covariance be expressed using correlation?

  1. As the correlation times the two standard deviations - correct
  2. As the correlation divided by the two variancesWhy not: This divides by the variances and inverts the relationship. Correlation is covariance divided by the two standard deviations, so recovering covariance means multiplying by them.
  3. As the correlation plus the two standard deviationsWhy not: Adding a unitless correlation to two percentages is not a defined operation.

Covariance measures how two assets move together, but its size depends on their units. Correlation rescales it to a number between -1 and +1. Correlation = Covariance / (SD of A x SD of B), where SD is standard deviation, the usual measure of risk. Rearranged: Covariance = Correlation x SD of A x SD of B. Example: 0.5 x 0.20 x 0.10 = 0.01. In portfolio variance, a lower correlation means a smaller covariance term, and that is how diversification reduces risk.

A harder one

Two assets each have a standard deviation of 30%, are equally weighted, and have a correlation of zero. What is the portfolio standard deviation?

  1. 30.00%Why not: The perfect-correlation result. With zero correlation the cross term vanishes and the figure falls to 30 over the square root of two.
  2. 15.00%Why not: Halves the standard deviation. Independent risks combine through the square root, not by division.
  3. 21.21% - correct

With zero correlation, the two assets move independently, so there is no covariance term. Portfolio variance = w1^2 x SD1^2 + w2^2 x SD2^2 + 2 x w1 x w2 x correlation x SD1 x SD2. Here the last term is zero. Variance = 0.5^2 x 30^2 + 0.5^2 x 30^2 = 0.25 x 900 + 0.25 x 900 = 450. Standard deviation = square root of 450 = 21.21%. This is below 30% because the two assets partly cancel each other, but it is not 15%, because risks combine through a square root.

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. A 100% allocation to either assetWhy not: A single asset keeps its full risk. Cancelling risk needs a mix of both.
  3. Weights in proportion to expected returnsWhy not: Expected returns do not decide how risk cancels. The weights must offset the standard deviations.

A correlation of -1 means the assets move in exact opposite directions. If they also swing by the same amount, equal amounts of each cancel perfectly. Formula: Portfolio SD = absolute value of (w1 x SD1 - w2 x SD2). With SD1 = SD2 and w1 = w2 = 50%, this gives 0.5 x SD - 0.5 x SD = 0. Whenever one asset loses, the other gains the same. A single holding keeps all its own risk. If the swings were unequal, the weights would need to be in inverse proportion to them.

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