Question

Here are returns and standard deviations for four investments. Standard Deviation Return (%) (%) Treasury bills 6.0 Stock P 9.5 14. Stock Q 16.0 25 Stock R 21.5 27 Calculate the standard deviations of the following portfolios a. 50% in Treasury bills, 50% in stock P. (Enter your answer as a percent rounded to 2 decimal places.) % Standard deviation b. 50% each in Q and R, assuming the shares have: (Do not round intermediate calculations. Enter your answers as a percent rounded to 2 decimal places.) Standard Deviation Perfect positive correlation Perfect negative correlation No correlation %

Answer 1

(a)

Standard deviation for a two-asset portfolio σ_p = (w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov_1,2)^1/2

where σ_p = standard deviation of the portfolio

w₁ = weight of Asset 1

w₂ = weight of Asset 2

σ₁² = variance of Asset 1

σ₂² = variance of Asset 2

Cov_1,2 = covariance of returns between Asset 1 and Asset 2

Cov_1,2 = ρ_1,2 * σ₁ * σ_2, where ρ_1,2 = correlation of returns between Asset 1 and Asset 2

Standard deviation = ((0.50² * 0²) + (0.50² * 0.14²) + (2)(0.50)(0.50)(0)(0)(0.14))^1/2

Standard deviation = 7.00%.

(b)

Standard deviation for a two-asset portfolio σ_p = (w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov_1,2)^1/2

where σ_p = standard deviation of the portfolio

w₁ = weight of Asset 1

w₂ = weight of Asset 2

σ₁² = variance of Asset 1

σ₂² = variance of Asset 2

Cov_1,2 = covariance of returns between Asset 1 and Asset 2

Cov_1,2 = ρ_1,2 * σ₁ * σ_2, where ρ_1,2 = correlation of returns between Asset 1 and Asset 2

With perfect positive correlation

The correlation coefficient ρ_1,2 = 1.

Standard deviation = ((0.50² * 0.25²) + (0.50² * 0.27²) + (2)(0.50)(0.50)(0.25)(0.27)(1))^1/2

Standard deviation = 26.00%.

With perfect negative correlation

The correlation coefficient ρ_1,2 = -1.

Standard deviation = ((0.50² * 0.25²) + (0.50² * 0.27²) + (2)(0.50)(0.50)(0.25)(0.27)(-1))^1/2

Standard deviation = 1.00%.

With zero correlation

The correlation coefficient ρ_1,2 = 0.

Standard deviation = ((0.50² * 0.25²) + (0.50² * 0.27²) + (2)(0.50)(0.50)(0.25)(0.27)(0))^1/2

Standard deviation = 18.40%.