Questions: Expected Return and Variance of Financial Assets

Question 1

Two assets each have a variance of 0.04. You form an equally weighted portfolio (w₁ = w₂ = 0.5). If the correlation between the assets is 0 (rather than +1), what happens to portfolio variance compared to the weighted average of individual variances?

• A. Portfolio variance equals 0.04 — the same as each individual asset
• B. Portfolio variance is 0.02 — lower than the individual variance
• C. Portfolio variance is 0.04 — equal to the weighted average
• D. Portfolio variance is 0.08 — higher because you hold two risky assets

Answer: (B)

With w₁ = w₂ = 0.5, σ₁² = σ₂² = 0.04, and Cov(r₁,r₂) = ρσ₁σ₂ = 0: σ²_p = (0.5)²(0.04) + (0.5)²(0.04) + 2(0.5)(0.5)(0) = 0.01 + 0.01 = 0.02. The weighted average of individual variances would be 0.04. Zero correlation cuts variance in half — this is diversification at work. The covariance term vanishes entirely, so you get the full benefit of combining two independent risks.

Question 2

Adding a second asset to a portfolio usually reduces portfolio variance, regardless of the correlation between the two assets.

• T. True
• F. False

Answer: False

When the correlation between two assets equals +1, the portfolio variance formula gives σ²_p = (w₁σ₁ + w₂σ₂)², which equals the weighted average of standard deviations squared. There is zero diversification benefit. If you add an asset perfectly correlated with your existing holding, you gain no risk reduction — portfolio risk is simply a weighted blend of the two. Only when correlation is less than +1 does diversification reduce variance below the weighted average.

Question 3

In the two-asset portfolio variance formula σ²_p = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov(r₁,r₂), why is the covariance term the key insight rather than the individual variance terms?

Model answer: The individual variance terms simply scale each asset's own risk by its portfolio weight. The covariance term determines how much those risks offset or amplify each other — it captures whether the assets tend to move together or in opposite directions. Diversification benefit comes entirely from the covariance term: if it is negative, portfolio risk falls below the weighted average of individual risks.

The covariance term 2w₁w₂Cov(r₁,r₂) encodes the interaction between the two assets. When covariance is negative (assets move oppositely), it subtracts from portfolio variance, creating diversification benefit. When covariance is positive (assets move together), it adds to variance, reducing diversification. This is why investors seek assets with low or negative correlation — the benefit of combining them comes entirely from this cross-term, not from the individual variance terms.