Two assets have equal expected returns and equal standard deviations of 20%. An investor calculates the portfolio variance as (0.5)²(0.04) + (0.5)²(0.04) = 0.02, assuming no interaction between assets. Why is this calculation wrong?
AIt's correct — equal weights produce a simple average of the variances
BIt ignores the covariance term; the true portfolio variance includes 2·w·(1−w)·σ₁·σ₂·ρ, which reduces variance when ρ < 1
CIt uses the wrong weights — the minimum-variance weights are not 50/50
DIt should multiply by 2 to account for holding two assets instead of one
Portfolio variance is NOT the weighted average of individual variances. The correct formula is σₚ² = w²σ₁² + (1−w)²σ₂² + 2w(1−w)σ₁σ₂ρ. The third term — the covariance contribution — reduces total variance whenever ρ < 1. If ρ = 0 and each asset has σ = 20% with equal weights, σₚ² = (0.25)(0.04) + (0.25)(0.04) = 0.02, which is 14.1% volatility — less than either asset individually. Omitting the covariance term ignores the entire engine of diversification.
Question 2 Multiple Choice
As the correlation between two assets decreases from +1 to −1, what happens to the portfolio frontier plotted in mean–standard-deviation space?
AIt shifts rightward, increasing risk at every return level
BIt flattens into a horizontal line, since expected returns don't change
CIt bows further leftward, offering lower portfolio variance at every return level
DIt collapses to a single point representing the risk-free rate
The covariance term 2w(1−w)σ₁σ₂ρ decreases as ρ falls, reducing portfolio variance for any given weight allocation. In the limit ρ = −1, a zero-variance portfolio is achievable. Visually, as correlation decreases, the frontier bows further to the left in mean-SD space — meaning investors can achieve the same expected return at lower risk, or more return at the same risk. This leftward bow is the geometric representation of diversification benefit. At ρ = +1, there's no bow at all — the frontier is a straight line, and holding both assets gives no variance reduction.
Question 3 True / False
A portfolio that lies below the minimum-variance point on the two-asset frontier is dominated: you could achieve the same expected return with less risk.
TTrue
FFalse
Answer: True
The minimum-variance portfolio is the leftmost point of the frontier curve. Portfolios above it (higher expected return for similar or slightly higher risk) are efficient. Portfolios below it are inefficient — for any such portfolio, you could shift weight toward the minimum-variance point and get the same expected return with lower variance, or hold a portfolio above the minimum-variance point with higher return and the same risk. No rational mean-variance investor would choose a dominated portfolio.
Question 4 True / False
When two assets have perfect positive correlation (ρ = 1), combining them in any proportion still reduces portfolio variance below the variance of the higher-variance asset.
TTrue
FFalse
Answer: False
At ρ = 1, the portfolio standard deviation is exactly the weighted average of the individual standard deviations: σₚ = w·σ₁ + (1−w)·σ₂. There is no diversification benefit whatsoever. Portfolio variance lies on a straight line between the two assets' risk-return combinations — it is bounded below by the lower-variance asset's variance (achieved by putting all weight there), but no intermediate combination reduces risk below the lower individual variance. The leftward bow in the frontier disappears entirely at ρ = 1.
Question 5 Short Answer
Why is portfolio variance less than the weighted average of the individual variances when the two assets have correlation less than 1? Identify the key term in the variance formula.
Think about your answer, then reveal below.
Model answer: The portfolio variance formula is σₚ² = w²σ₁² + (1−w)²σ₂² + 2w(1−w)σ₁σ₂ρ. When ρ < 1, the cross-term 2w(1−w)σ₁σ₂ρ is smaller than it would be at ρ = 1, so the total variance is less than the weighted average of σ₁² and σ₂². The lower the correlation, the smaller this term, and the greater the variance reduction — reaching zero variance when ρ = −1.
At ρ = 1, the portfolio variance equals [wσ₁ + (1−w)σ₂]² — exactly the square of the weighted-average standard deviation, meaning no benefit from combining. For ρ < 1, the cross-term shrinks, pulling variance below that level. This is the mathematical basis of diversification: combining assets whose returns are not perfectly correlated (the typical case) produces a portfolio with less variance than a naive weighted-average calculation would suggest.