Question

Ebenezer Scrooge has invested 50% of his money in share A and the remainder in share B. He assesses their prospects as follows:

A B

Expected return (%) 18 19

Standard deviation (%) 20 24

Correlation between returns 0.3

a. What are the expected return and standard deviation of returns on his portfolio?

b. How would your answer change if the correlation coefficient were 0 or –0.30?

Answer 1

	A	B
Expected Return (%)	18	19
Standard Deviation (%)	20	24

a) Weight of A in the portfolio = w_A = 0.5, Weight of B in the portfolio = w_B = 0.5

Correlation between A and B = ρ = 0.3

Expected Return of portfolio = E[R_p] = w_A*R_A + w_B*R_B = 0.5*18 + 0.5*19 = 18.5%

Variance of the portfolio is calculated using the formula:

Variance of the portfolio = σ_p² = w_A²*σ_A² + w_B²*σ_B² + 2*ρ*w_A*w_B*σ_A*σ_B

σ_p² = 0.5²*(20%)² + 0.5²*(24%)²​​​​​​​ + 2*0.3*0.5*0.5*20%*24% = 0.01+0.0144+0.0072 = 0.0316

Standard deviation is the square root of the variance

Standard deviation of the portfolio = σ_p = 0.0316^1/2 = 17.7763888346312%

b)

if correlation coefficient = 0

Expected return will not not change but the varaince/standard deviation of the portfolio will change

Expected Return of the portfolio = E[R_P] = 18.5%

Variance of the portfolio = σ_p² = w_A²*σ_A² + w_B²*σ_B²​​​​​​​ + 2*ρ*w_A*w_B*σ_A*σ_B

ρ = 0

σ_p² = 0.5²*(20%)² + 0.5²*(24%)²​​​​​​​ + 2*0*0.5*0.5*20%*24% = 0.01+0.0144 = 0.0244

Standard deviation of the portfolio = σ_p = 0.0244^1/2 = 15.6204993518133%

​​​​​​​if correlation coefficient = -0.3

Expected return will not not change but the varaince/standard deviation of the portfolio will change

Expected Return of the portfolio = E[R_P] = 18.5%

Variance of the portfolio = σ_p² = w_A²*σ_A² + w_B²*σ_B²​​​​​​​ + 2*ρ*w_A*w_B*σ_A*σ_B

ρ = -0.3

σ_p² = 0.5²*(20%)² + 0.5²*(24%)²​​​​​​​ + 2*(-0.3)*0.5*0.5*20%*24% = 0.01+0.0144-0.0072 = 0.0172

Standard deviation of the portfolio = σ_p = 0.0172^1/2 = 13.114877048604%

Following table summarizes the expected return and standard deviation of the portfolio when the correlation coefficient is 0.3, 0 and -0.3:

Correlation Coefficient	Expected Return of portfolio	Standard Deviation of portfolio
0.3	18.50%	17.78%
0	18.50%	15.62%
-0.3	18.50%	13.11%