Questions: Correlation and Covariance Between Assets

Question 1

Two asset pairs have identical expected returns and identical individual variances. Pair A has correlation ρ = +0.9. Pair B has correlation ρ = −0.2. Which pair delivers greater risk reduction when combined in an equal-weight portfolio?

• A. Pair A — the high positive correlation means the assets reinforce each other, creating more stable combined returns
• B. Pair B — the lower (negative) correlation means the assets partially offset each other, reducing portfolio variance more
• C. They are identical — same individual variances means the same portfolio risk regardless of correlation
• D. Pair A — positive correlation means the assets diversify each other by moving in the same direction

Answer: (B)

Portfolio variance = w²σ²_a + w²σ²_b + 2w²·Cov(a,b), and Cov = ρ·σ_a·σ_b. With ρ = −0.2, the covariance term is negative, subtracting from portfolio variance. With ρ = +0.9, it adds substantially. The lower the correlation, the greater the diversification benefit. A common misconception is that 'moving together' is good for a portfolio — synchronized movement means no risk reduction.

Question 2

An investor finds that Stock A and Bond B each have a standard deviation of 20%, and their covariance is 0.02. She computes the correlation as 0.02 / (0.20 × 0.20) = 0.5. Why is this correlation figure more useful for comparing relationships across asset pairs than the raw covariance?

• A. Because covariance can only be negative, while correlation reflects both positive and negative relationships
• B. Because correlation is dimensionless and bounded to [−1, 1], making the strength of relationships directly comparable across asset pairs with different volatilities
• C. Because covariance measures absolute risk while correlation measures relative return
• D. Because correlation adjusts for differences in expected returns between the two assets

Answer: (B)

Covariance's magnitude depends on the units and individual volatilities of both assets. Two high-volatility stocks will have a large covariance in absolute terms even if their co-movement relationship is no tighter than two low-volatility bonds. Dividing by σ_a·σ_b standardizes the measure to [−1, 1], making correlation comparable across all asset pairs regardless of scale.

Question 3

If two assets have a correlation of exactly −1, it is theoretically possible to construct a portfolio with zero variance by choosing the right weights.

• T. True
• F. False

Answer: True

With perfect negative correlation, portfolio variance σ²_p = (wₐσ_a − w_bσ_b)². This equals zero when wₐσ_a = w_bσ_b, i.e., when weights are chosen inversely proportional to the assets' standard deviations. In this portfolio, every gain in one asset is perfectly offset by a loss in the other. ρ = −1 is the theoretical maximum diversification benefit.

Question 4

A portfolio of two assets with zero correlation (ρ = 0) achieves no reduction in portfolio risk compared to holding either asset alone at full weight.

• T. True
• F. False

Answer: False

Zero correlation still provides diversification benefits. With ρ = 0, portfolio variance = w²_a σ²_a + w²_b σ²_b (the covariance term vanishes). For equal weights, σ²_p = (σ²_a + σ²_b)/4, which is typically less than holding either asset alone (σ²_a or σ²_b). No risk reduction only occurs when ρ = +1. Even uncorrelated assets reduce variance when combined.

Question 5

Why does the correlation between two assets — rather than their individual variances — determine how much risk reduction is achieved by combining them in a portfolio?

Model answer: Because portfolio variance includes an interaction term: 2·wₐ·w_b·Cov(a,b) = 2·wₐ·w_b·ρ·σ_a·σ_b. The individual variances set what you'd have if ρ = +1 (no diversification benefit), but correlation determines how much the interaction term reduces that floor. When ρ < 1, the assets partially offset each other's fluctuations, reducing total risk below the weighted average of individual risks.

With ρ = +1, assets move in lockstep and there is no diversification benefit. With ρ = 0, portfolio variance is simply the weighted sum of variances. With ρ < 0, assets actively offset each other, potentially reducing variance dramatically. Individual variances determine how volatile each asset is alone; correlation determines how the combination behaves.