Given a two-asset portfolio with weights 0.6 and 0.4, standard deviations 20% and 10%, and correlation 0.5, what is the portfolio standard deviation?

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Study for the Finance and Investment Challenge Test. Approaches include flashcards and multiple-choice questions with hints and explanations. Ready yourself to ace the exam!

Multiple Choice

Given a two-asset portfolio with weights 0.6 and 0.4, standard deviations 20% and 10%, and correlation 0.5, what is the portfolio standard deviation?

Explanation:
The concept being tested is how to combine two assets’ risks into a single portfolio risk using weights, each asset’s standard deviation, and their correlation. The portfolio variance formula is Var(P) = w1^2 σ1^2 + w2^2 σ2^2 + 2 w1 w2 ρ σ1 σ2, and the portfolio standard deviation is the square root of this variance. Plugging in the numbers: w1 = 0.6, σ1 = 0.20; w2 = 0.4, σ2 = 0.10; ρ = 0.5. - w1^2 σ1^2 = (0.6)^2 × (0.20)^2 = 0.36 × 0.04 = 0.0144 - w2^2 σ2^2 = (0.4)^2 × (0.10)^2 = 0.16 × 0.01 = 0.0016 - Covariance term = 2 × 0.6 × 0.4 × 0.5 × 0.20 × 0.10 = 0.0048 Sum of terms = 0.0144 + 0.0016 + 0.0048 = 0.0208. The portfolio standard deviation is sqrt(0.0208) ≈ 0.1442, or about 14.42%. Positive correlation increases the portfolio risk relative to the case with uncorrelated assets (which would give sqrt(0.016) ≈ 12.65%). Here the cross-term raises the risk to roughly 14.42%.

The concept being tested is how to combine two assets’ risks into a single portfolio risk using weights, each asset’s standard deviation, and their correlation. The portfolio variance formula is Var(P) = w1^2 σ1^2 + w2^2 σ2^2 + 2 w1 w2 ρ σ1 σ2, and the portfolio standard deviation is the square root of this variance.

Plugging in the numbers: w1 = 0.6, σ1 = 0.20; w2 = 0.4, σ2 = 0.10; ρ = 0.5.

  • w1^2 σ1^2 = (0.6)^2 × (0.20)^2 = 0.36 × 0.04 = 0.0144

  • w2^2 σ2^2 = (0.4)^2 × (0.10)^2 = 0.16 × 0.01 = 0.0016

  • Covariance term = 2 × 0.6 × 0.4 × 0.5 × 0.20 × 0.10 = 0.0048

Sum of terms = 0.0144 + 0.0016 + 0.0048 = 0.0208. The portfolio standard deviation is sqrt(0.0208) ≈ 0.1442, or about 14.42%.

Positive correlation increases the portfolio risk relative to the case with uncorrelated assets (which would give sqrt(0.016) ≈ 12.65%). Here the cross-term raises the risk to roughly 14.42%.

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